BAIL 2006 Book of Abstracts - Institut für Numerische und ...

BAIL 2006 

International Conference on 

Boundary and Interior Layers 

Göttingen, 

24th - 28th July, 2006 

L. Prandtl 

Book of Abstracts 

Organized by: 

Georg-August Universität Göttingen 

Deutsches Zentrum für Luft- und Raumfahrt (DLR) 

Sponsored by: 

Land Niedersachsen, 

Graduiertenkolleg 1023 

Gesellschaft für Angewandte Mathematik und Mechanik (GAMM) 

German Academic Exchange Service (DAAD) 

Georg-August Universität Göttingen 

Deutsches Zentrum für Luft- und Raumfahrt (DLR)

Greeting 

It gives me great pleasure to welcome all participants to the BAIL 2006 conference in 

Goettingen, Germany. This conference is the latest in a long line of international conferences 

on Boundary and Interior Layers, that have been held in many parts of the world. 

The first three conferences were held in Dublin, Ireland in 1980, 1982 and 1984. These 

were followed by conferences in Novosibirsk, USSR (1986), Shanghai, China (1988), 

Copper Mountain, USA (1992) and Beijing, China (1994). 

After a gap of some years, an international steering committee was formed to advise on 

the creation of a new series of BAIL conferences. This committee made a positive recommendation, 

which has resulted in three further BAIL conferences being held, specifically: 

BAIL 2002 in Perth, Australia, BAIL 2004 in Toulouse, France and this year’s BAIL 2006 

in Goettingen, Germany. 

It is heartening to see how well BAIL 2006 is supported. Indeed, the number of participants 

appears to be growing with each successive conference in the new series. I am 

delighted to thank the members of the International Steering Committee for their good 

advice over the years. It is a particular pleasure also to thank the local organizers and 

their teams for the hard work involved in the organization of each of these conferences. 

Only someone who has organized such a conference is aware of just how much work is 

involved! I believe that the success of the BAIL 2006 organizers, especially in making 

this event attractive to a large number of participants, is an important milestone in the 

development of this series of conferences. On behalf of all participants I thank the organizers 

and, in addition, I wish everyone a productive and enjoyable meeting. 

John J H Miller 

Dublin, Ireland 

July 2006 

BAIL conferences 

• BAIL 2004, Toulouse, France 

• BAIL 2002 , Perth, Australia 

• BAIL VII, Beijing, China (1994) 

• BAIL VI, Colorado, USA (1992) 

• BAIL V, Shanghai, China (1988) 

• BAIL IV, Novosibirsk, USSR (1986) 

• BAIL III, Dublin, Ireland (1984) 

• BAIL II, Dublin, Ireland (1982) 

• BAIL I, Dublin, Ireland (1980) 

3 BAIL 2006

Plenary Session 1 Session 2 Session 3 Session 4 Evening 

Talks 

9:15-10:15 10:45-12:25 14:00 - 14:50 15:10 - 16:25 16:45-18:00 

Sunday 

Welcome 

July 23 

Reception 

(17:00-20:00, 

House 6) 

Monday Room SL P. Huerre Applied Applied Minisymposium Minisymposium Get Together 

Aerodynamics Aerodynamics Stynes, O’Riordan Stynes,O’Riordan Party 

July 24 Room MPI (Room Asymptotic Asymptotic Special 

(Old Town-hall, 

MPI) Methods Methods Flows 

19:00-20:30) 

Public Talk 

Prof. Gersten 

Minisymposium 

Hartmann, Hou- 

Minisymposium 

Hartmann, Hou- 

Flows in Special 

Geometries 

Tuesday Room SL M. Stynes Heat 

Transfer 

Room SL 

(20:00-21:30) 

ston 

Numerical 

Methods 1 

ston 

Numeric. Methods 

for Fluid Flows 

Minisymposium 

Maubach, Tselishcheva 

Minisymposium 

Maubach, 

Tselishcheva 

July 25 Room MPI (Room 

MPI) 

Excursion 

Conference Dinner 

Open 

Discussion 1 

Wall 

functions 2 

Wednesday Room SL W. Wall Numerical 

Methods 2 

July 26 Room MPI (Room Wall 

MPI) functions 1 

Minisymposium 

Shishkin, Hemker 

Open 

Discussion 2 

Minisymposium 

Shishkin, Hemker 

Anisotropic 

Meshes 2 

Thursday Room SL P. Houston Anisotropic 

Meshes 1 Round Tour 

July 27 Room MPI (Room Turbulence (DLR, 

MPI) Modelling Math. Inst.) 

End of Conference 

Friday Room SL Minisymposium: Das,Sengupta 

(9:15-10:30, 11:00-12:15) 

5 BAIL 2006 

July 28 Room MPI

Room School-Lab (SL) Room MPI 

Sunday, 23 July 

17:00 - 20:00 Welcome Reception 

Monday, 24 July 

8:00 - 9:00 Registration 

9:00 - 9:15 Opening 

9:15 - 10:15 Plenary Talk: 

P. Huerre: Dynamics of hot jets: a numerical and theo- 


retical study. 

10:15 - 10:45 Coffee Break 

Session: Applied Aerodynamics 

Session: Asymptotic methods 

P. Svacek: Numerical Approximation of Flow Induced 

M. Hamouda, R. Temam: Boundary layers for the 

Airfoil Vibrations (10:45-11:10) 

Navier-Stokes equations: asymptotic analysis (10:45- 

A. Firooz, M. Gadami: Turbulence Flow for NACA 

11:10) 

4412 in Unbounded Flow and Grow Effect with Diffe- 

N.V. Tarasova: Full asymptotic analysis of the Navier- 

rent Turbulence Models and Two Ground Conditions: 

10:45 - 12:25 

Stokes equations in the problems of gas flows over bodies 

Fixed and Moving Ground Conditions (11:10-11:35) 

with large Reynolds number (11:10-11:35) 

B. Eisfeld: Computation of complex compressible 

N. Neuss: Numerical approximation of boundary layers 

aerodynamic flows with Reynolds stress turbulence 

for rough boundaries (11:35-12:00) 

model (11:35-12:00)




A.-M. Il’in, B.I. Suleimanov: The coefficients of in- 

A. Nastase: Qualitative Analysis of the Navier-Stokes 

10:45 - 12:25 

ner asymptotic expansions for solutions of some singular 

Solutions on Vicinty of their Critical Lines (12:00-12:25) 

boundary value problems (12:00-12:25) 

12:30 - 14:00 Lunch Break 



Z.-H. Yang, Y.-Z. Li, Y. Zhu: Application of Bi- 

C.H. Tai, C.-Y. Chao, J.-C. Leong, Q.S. Hong: 

furcation Method to Computing Numerical Solutions of 

Effects of golf ball dimple configuration on aerodyna- 

Lane-Emden Equation (14:00-14:25) 

mics, trajectory, and acoustics (14:00-14:25) 

14:00 - 14:50 


H. Tian: Uniformly Convergent Numerical Methods 

W.S. Islam, V.R. Raghavan: Numerical Simulation 

for Singularly Perturbed Delay Differential Equations 

of High Sub-critical Reynolds Number Flow Past a Cir- 

(14:25-14:50) 

cular Cylinder (14:25-14:50) 


Session: Special Flows 

Minisyposium: M. Stynes, E. O’Riordan 

B. Rasuo: On Boundary Layer Control in Two- 

H. Wang: A Component-Based Eulerian-Lagrangian 

Dimensional Transonic Wind Tunnels (15:10-15:35) 

Formulation for Compositional Flow in Porous Media 

M. Vasiliev: About unsteady Boundary Layer on a di- 

G.I. Shishkin: A posteriori adapted meshes in the 

approximation of singularly perturbed quasilinear 

15:10 - 16:25 

hedral angle (15:35-16:00) 

K. Mansour: Boundary Layer Solution for Laminar 

parabolic convection-diffusion equations 

Flow through a Loosely Curved Pipe by using Stokes 

W. Layton, I. Stanculescu: Numerical Analysis of 

Expansion (16:00-16:25) 

Approximate Deconvolution Models of Turbulence


16:25 - 16:45 Coffee break 

Minisymposium: M. Stynes, E. O’Riordan 

R.K. Dunne, E. O’Riordan, M.M. Turner: 

A singular perturbation problem arising in the model- 

16:45 - 18:00 

ling of plasma sheats 

19:00 - 21:00 Get Together Party 

Old Town Hall 



Tuesday, 25 July 

Plenary Talk: 

M. Stynes: Convection-diffusion problems, SD- 

9:15 - 10:15 

FEM/SUPG and a priori meshes. 

10:15 - 10:45 Coffee break 

Minisymposium: J.Maubach, I.V.Tselishcheva: 

M. Anthonissen, I. Sedykh, J. Maubach: A con- 

vergence proof of local defect correction for convection- 

Session: Heat Transfer 

M. Hölling, H. Herwig: Computation of turbulent 


diffusion problems 

J. Maubach: On the difference between left and right 

natural convection at vertical walls using new wall 

preconditioning for convection dominated convection- 

functions (10:45-11:10) 

diffusion problems 

O. Shishkina, C. Wagner: Boundary and Interior 

10:45 - 12:25 

A. Hegarty, St. Sikwila, G.I. Shishkin: An adap- 

Layers in Turbulent Thermal Convection (11:10-11:35) 

tive method for the numerical solution of an elliptic 

K. Morinishi: Rarefied Gas Boundary Layer Predicted 

convection diffusion problem 

with Continuum and Kinetic Approaches (11:35-12:00) 

P. Zegeling: An Adaptive Grid Method for the Solar 

Coronal Loop Model 

12:30 - 14:00 Lunch Break


Minisymposium: J.Maubach, I.V.Tselishcheva: 

A.I. Zadorin: Numerical method for the Blasius equa- 

Session: Flows in Special Geometries 

tion on an infinite interval 

D. Kachuma, I. Sobey: Fast waves during transient 

S. Li , L.P. Shishkina, G.I. Shishkin, Parameter- 

flow in an asymmetric channel (14:00-14:25) 

14:00 - 14:50 

uniform method for a singularly perturbed parabolic 

J. Mauss, J. Cousteix: Global Interactive Boundary 

equation modelling the Black-Scholes equation in the 

Layer (GIBL) for a Channel (14:25-14:50) 

presence of interior and boundary layers 


Session: Numer. Methods for Fluid Flows 


P. Knobloch: On methods dimishing spurious oscilla- 

tions in finite element solutions of convection-diffusion 

Minisymposium: R. Hartmann, P. Houston 

equations (15:10-15:35) 

J. Mackenzie, A. Nicola: A Discontinuous Galerkin 

G. Matthies, L. Tobiska: Mass conservation of finite 

Moving Mesh Method for Hamilton-Jacobi Equations 

element methods for coupled flow-transport problems 

R. Schneider, P. Jimack: Anisotropic mesh adaption 

15:10 - 16:25 

(15:35-16:00) 

based on a posteriori estimates and optimisation of node 

M. Olshanskii: An Augmented Lagrangian Based 

positions 

Solver for the low-viscosity incompressible flows (16:00- 

S. Perotto: Layer Capturing via Anisotropic Adaption 

16:25) 

16:25 - 16:45 Coffee break


Session: Numerical Methods 1 

M. Bause: Apects of SUPG/PSPG and GRAD-DIV 

Stabilized Finite Element Approximation of Compres- 

Minisymposium: R. Hartmann, P. Houston 

sible Viscous Flow (16:45-17:10) 

V. Heuveline: On a new refinement strategy for adap- 

F. Nataf, G. Rapin: Application of the Smith Fac- 

tive hp finite element 

torisation to Domain Decomposition Methods for the 

16:45 - 18:00 

R. Hartmann: Discontinuous Galerkin methods for 

Stokes Equations (17:10-17:35) 

compressible flows: higher order accuracy, error estima- 

A. Cangiani, E.H. Georgoulis, M. Jensen: 

tion and adaptivity 

Continuous-Discontinuous Finite Element Methods for 

Convection-Diffusion Problems (17:35-18:00): 


Public Talk: 

Gersten: Vom Kochtopf bis zum Fußballspiel: Epi- 

soden zu der weltweiten Wirkung der Göttinger 

20:00 - 21:30 

Strömungsforscher (in german)


Wednesday, 26 July 

Plenary Talk: 

W. Wall: Variational Multiscale Methods for incom- 

9:15 - 10:15 

pressible flows. 


Session: Numerical Methods 3 

Session: Wall Functions 1 

F. Alizard, J.-Ch. Robinet: Two-dimensional tem- 

T. Knopp: Model-consistent universal wall-function for 

poral modes in nonparallel flows (10:45-11:10) 

RANS turbulence modelling (10:45-11:10) 

Q. Ye: Numerical simulation of turbulent boundary 


Th. Alrutz, T. Knopp: Near wall grid adaption for 

for stagnation-flow in the spray-painting process (11:10- 

wall functions (11:10-11:35) 

11:35) 

Z. Hammouch: Similiarity solutions of a power-law 

10:45 - 12:25 

A.I. Tolstykh, M.V. Lipavskii, E.N. Chigerev: 

non-Newtonian laminar boundary layer flows (11:35- 

Highly accurate 9th-order schemes and their applicati- 

12:00) 

ons to DNS of thin shear layer instability (11:35-12:00) 

B. Scheichl, A. Kluwick: On Turbulent Marginal Se- 

N. Parumasur, J. Banasiak, J.M. Kozakiewicz: 

paration: How the Logarithmic Law of the Wall is Su- 

Numerical and Asymptotic Analysis of Singularly Per- 

perseded by the Half-Power Law (12:00-12:25) 

turbed PDEs of Kinetic Theory (12:00-12:25) 

12:30 - 14:00 Lunch Break


Session: Wall Functions 2 

Open Discussion I 

V.D. Liseykin, Y.V. Likhanova, D.V. Patrakhin, 

I.A. Vaseva: Application of boundary layer-type func- 

How to prevent spurious oscillations in boundary and 

14:00 - 15:00 

tions to comprehensive grid generation codes (14:00- 

interior layers? 

14:25) 

16:00 - 22:00 Excursion + Conference Dinner 



Thursday, 27 July 

Plenary Talk: 

P. Houston: Discontinuous Galerkin Finite Element 

9:15 - 10:15 

Methods for CFD: A Posteriori Error Estimation and 

Adaptivity. 


Session: Anisotropic Meshes 1 

Session: Turbul. Modelling 

H.-G. Roos: A Comparison of Stabilization Methods 

Boguslawski: Sheare Stress Distribution on Sphere 

for Convection-Diffusion-Reaction Problems on Layer- 


Surface at Different Inflow Turbulence (10:45-11:10) 

Adapted Meshes (10:45-11:10) 

H. Lüdecke: Detached Eddy Simulation of Supersonic 

H.-G. Roos, H. Zarin: Discontinuous Galerkin stabi- 

Shear Layer Wake Flows (11:10-11:35) 

lization for convection-diffusion problems (11:10-11:35) 

O. Mierka, D. Kuzmin: On the implementation of 

10:45 - 12:25 

L. Tobiska: Using rectangular Qp elements in the SD- 

turbulence models in incompressible flow solvers based 

FEM for a convection-diffusion problem with a bounda- 

on a finite element discretization (11:35-12:00) 

ry layer (11:35-12:00) 

S.A. Gaponov, G.V. Petrov, B.V. Smorodsky: 

C. Clavero, J.L. Gracia, F. Lisbona: A second order 

Boundary layer interaction with external disturbances 

uniform convergent method for a singularly perturbed 

(12:00-12:25) 

parabolic system of reaction-diffusion type (12:00-12:25) 

12:30 - 14:00 Lunch Break 

14:00 - 14:50 Round Tours (DLR or Mathematical Institute) 

14:50 - 15:10 Coffee Break


Minisymposium: G.I. Shishkin, P. Hemker 

Session: Anisotropic Meshes 2 

G.I. Shishkin: Grid approximation of parabolic equa- 

A.E.P. Veldmann: High-order symmetry-preserving 

tions with nonsmooth initial condition in the presence 

discretization on strongly stretched grids (15:10-15:35) 

of boundary layers of different types 

Th. Apel, G. Matthies: A family of non-conforming 

L.P. Shishkina, G.I. Shishkin: A difference scheme 

finite elements of arbitrary order for the Stokes problem 

of improved accuracy for a quasilinear singularly per- 

15:10 - 16:25 

on anisotropic quadrilateral meshes (15:35-16:00) 

turbed elliptic convection-diffusion equation in the case 

G. Lube: A stabilized finite element method with aniso- 

of the third-kind boundary condition 

tropic mesh refinement for the Oseen equations (16:00- 

D. Branley, A. Hegarty, H. MacMullen and 

16:25) 

G.I. Shishkin: A Schwarz method for a convection- 

diffusion problem with a corner singularity 


Minisymposium: G.I. Shishkin, P. Hemker 

I.V. Tselishcheva, G.I. Shishkin: Domain decom- 

position method for a semilinear singularly perturbed 

elliptic convection-diffusion equation with concentrated 

sources 

Th. Linss, M. Madden: Layer-adapted meshes for 


Open Discussion 2 

Anisotropic mesh generation for advection-dominated 

16:45 - 18:00 

problems and for incompressible flow problems 

time-dependent reaction diffusion 

S. Hemavathi, S.Valarmathi: A parameter-uniform 

numerical method for a system of singularly perturbed 

ordinary differential equations


Friday, 28 July 

Minisymposium: D. Das, T.K. Sengupta 

M.H. Buschmann M. Gad-El-Hak: Turbulent 

Boundary Layers: Reality and Myth 

L. Savic, H. Steinrück: Asymptotic Analysis of the 

9:15 - 10:30 

mixed convection flow past a horizontal plate near the 

trailing edge 



Minisymposium: D. Das, T.K. Sengupta 

T.K. Sengupta, A. Kameswara Rao: Spatio- 

temporal growing waves in boundary-layers by Bron- 

wich contour integral method 

A. Nayak, D. Das: Three-dimensional Temporal 

11:00 - 12:25 

Instability of Unsteady Pipe Flow 

J. Hussong, N. Bleier, V.I.V. Ram: The structure 

of the critical layer of a swirling annular flow in transi- 

tion 

12:25 - 12:40 Closing Session 

12:40 - 14:00 Lunch

Contents 

Greetings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 

Schedule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 

Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 

Plenary Presentations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 

KL. GERSTEN: 

Vom Kochtopf bis zum Fußballspiel (Public Talk) . . . . . . . . . 3 

P. HOUSTON: 

Discontinuous Galerkin Finite Element Methods for CFD: A Posteriori 

Error Estimation and Adaptivity . . . . . . . . . . . . . . 4 

L. LESSHAFFT, P. HUERRE, P. SAGAUT, M. TERRACOL: 

Dynamics of Hot Jets: A Numerical and Theoretical Study . . . . 5 

M. STYNES: 

Convection-diffusion problems, SDFEM/SUPG and a priori meshes 6 

V. GRAVEMEIER, S. LENZ, W.A. WALL: 

Variational Multiscale Methods for incompressible flows . . . . . 8 

Minisymposia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 

MS: N. Kopteva, M. Stynes, E. O’Riordan . . . . . . . . . . . . . . . . . . . . 13 

R.K. DUNNE, E. O’RIORDAN, M.M. TURNER: 

A singular perturbation problem arising in the modelling of plasma 

sheaths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 

G.I. SHISHKIN: 

A posteriori adapted meshes in the approximation of singularly 

perturbed quasilinear parabolic convection-diffusion equations . . 16 

W. LAYTON, I. STANCULESCU: 

Numerical Analysis of Approximate Deconvolution Models of 

Turbulence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 

H. WANG: 

A Component-Based Eulerian-Lagrangian Formulation for Compositional 

Flow in Porous Media . . . . . . . . . . . . . . . . . . 19 

MS: P. Houston, R. Hartmann . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 

R. HARTMANN: 

Discontinuous Galerkin methods for compressible flows: higher 

order accuracy, error estimation and adaptivity . . . . . . . . . . 22 


CONTENTS 

V. HEUVELINE: 

On a new refinement strategy for adaptive hp finite element method 24 

J.A. MACKENZIE, A. NICOLA: 

A Discontinuous Galerkin Moving Mesh Method for Hamilton- 

Jacobi Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 25 

S. PEROTTO: 

Layer Capturing via Anisotropic Mesh Adaption . . . . . . . . . 27 

R. SCHNEIDER, P. JIMACK: 

Anisotropic mesh adaption based on a posteriori estimates and 

optimisation of node positions . . . . . . . . . . . . . . . . . . . 28 

MS: Debopam Das, Tapan Sengupta . . . . . . . . . . . . . . . . . . . . . . . 31 

M.H. BUSCHMANN, M. GAD-EL-HAK: 

Turbulent Boundary Layers: Reality and Myth . . . . . . . . . . 32 

A. NAYAK, D. DAS: 

Three-dimesnional Temporal Instability of Unsteady Pipe Flow . 35 

J. HUSSONG, N. BLEIER, V.V. RAM: 

The structure of the critical layer of a swirling annular flow in 

transition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 

T.K. SENGUPTA, A. KAMESWARA RAO: 

Spatio-temporal growing waves in boundary-layers by Bromwich 

contour integral method . . . . . . . . . . . . . . . . . . . . . . 39 

L. SAVIĆ, H. STEINRÜCK: 

Asymptotic Analysis of the mixed convection flow past a horizontal 

plate near the trailing edge . . . . . . . . . . . . . . . . . 41 

MS: G.I. Shiskin, P. Hemker . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 

D. BRANLEY, A.F. HEGARTY, H. PURTILL, G.I. SHISHKIN: 

A Schwarz method for a convection-diffusion problem with a corner 

singularity . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 

T. LINSS, N. MADDEN: 

Layer-adapted meshes for time-dependent reaction-diffusion . . . 47 

G.I. SHISHKIN: 

Grid Approximation of Parabolic Equations with Nonsmooth Initial 

Condition in the Presence of Boundary Layers of Different 

Types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 

L.P. SHISHKINA, G.I. SHISHKIN: 

A Difference Scheme of Improved Accuracy for a Quasilinear 

Singularly Perturbed Elliptic Convection-Diffusion Equation in 

the Case of the Third-Kind Boundary Condition . . . . . . . . . . 49 

I.V. TSELISHCHEVA, G.I. SHISHKIN: 

Domain Decomposition Method for a Semilinear Singularly Perturbed 

Elliptic Convection-Diffusion Equation with Concentrated 

Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 


CONTENTS 

S. HEMAVATHI, S. VALARMATHI: 

A parameter-uniform numerical method for a system of singularly 

perturbed ordinary differential equations . . . . . . . . . . . . . . 53 

MS: J. Maubach, I, Tselishcheva . . . . . . . . . . . . . . . . . . . . . . . . . 55 

A.F. HEGARTY, S. SIKWILA, G.I. SHISKIN: 

An adaptive method for the numerical solution of an elliptic convection 

diffusion problem . . . . . . . . . . . . . . . . . . . . . 56 

M. ANTHONISSEN, I. SEDYKH, J. MAUBACH: 

A Convergence Proof of Local Defect Correction for Convection- 

Diffusion Problems . . . . . . . . . . . . . . . . . . . . . . . . . 57 

J. MAUBACH: 

On the Difference between Left and Right Preconditioning for 

Convection Dominated Convection-Diffusion Problems . . . . . . 58 

S. LI, L.P. SHISHKINA, G.I. SHISHKIN: 

Parameter-Uniform Method for a Singularly Perturbed Parabolic 

Equation Modelling the Black-Scholes equation in the Presence 

of Interior and Boundary Layers . . . . . . . . . . . . . . . . . . 59 

A.I. ZADORIN: 

Numerical Method for the Blasius Equation on an Infinite Interval 61 

P. ZEGELING: 

An Adaptive Grid Method for the Solar Coronal Loop Model . . . 63 

Contributed Presentations . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 

F. ALIZARD, J.-CH. ROBINET: 

Two-dimensional temporal modes in nonparallel flows . . . . . . 67 

TH. ALRUTZ, T. KNOPP: 

Near-wall grid adaptation for wall functions . . . . . . . . . . . . 69 

TH. APEL, G. MATTHIES: 

A family of non-conforming finite elements of arbitrary order for 

the Stokes problem on anisotropic quadrilateral meshes . . . . . . 71 

M. BAUSE: 

Aspects of SUPG/PSPG and GRAD-DIV Stabilized Finite Element 

Approximation of Compressible Viscous Flow . . . . . . . 72 

L. BOGUSLAWSKI: 

Sheare Stress Distribution on Sphere Surface at Different Inflow 

Turbulence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 

A.CANGIANI, E.H.GEORGOULIS, M. JENSEN: 

Continuous-Discontinuous Finite Element Methods for Convection- 

Diffusion Problems . . . . . . . . . . . . . . . . . . . . . . . . . 75 

C. CLAVERO, J.L. GRACIA, F. LISBONA: 

A second order uniform convergent method for a singularly perturbed 

parabolic system of reaction-diffusion type . . . . . . . . . 77 


CONTENTS 

B. EISFELD: 

Computation of complex compressible aerodynamic flows with a 

Reynolds stress turbulence model . . . . . . . . . . . . . . . . . 79 

A. FIROOZ, M. GADAMI: 

Turbulence Flow for NACA 4412 in Unbounded Flow and Ground 

Effect with Different Turbulence Models and Two Ground Conditions: 

Fixed and Moving Ground Conditions . . . . . . . . . . 81 

S.A. GAPONOV, G.V. PETROV, B.V. SMORODSKY: 

Boundary layer intercation with external disturbances . . . . . . . 85 

Z.HAMMOUCH: 

Similarity solutions of a power-law non-Newtonian laminar boundary 

layer flows . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 

M. HAMOUDA, R. TEMAM: 

Boundary layers for the Navier-Stokes equations : asymptotic 

analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 

M. HÖLLING, H. HERWIG: 

Computation of turbulent natural convection at vertical walls using 

new wall functions . . . . . . . . . . . . . . . . . . . . . . . 91 

A.-M. IL’IN, B.I. SULEIMANOV: 

The coefficients of inner asymptotic expansions for solutions of 

some singular boundary value problems . . . . . . . . . . . . . . 93 

W.S. ISLAM, V.R. RAGHAVAN: 

Numerical Simulation of High Sub-critical Reynolds Number Flow 

Past a Circular Cylinder . . . . . . . . . . . . . . . . . . . . . . 96 

D. KACHUMA, I. SOBEY: 

Fast waves during transient flow in an asymmetric channel . . . . 98 

A. KAUSHIK, K.K. SHARMA: 

A Robust Numerical Approach for Singularly Perturbed Time Delayed 

Parabolic Partial Differential Equations . . . . . . . . . . . 100 

P. KNOBLOCH: 

On methods diminishing spurious oscillations in finite element 

solutions of convection-diffusion equations . . . . . . . . . . . . 102 

T. KNOPP: 

Model-consistent universal wall-functions for RANS turbulence 

modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 

J.-S. LEU, J.-Y. JANG, Y.-C. CHOU: 

Evaporating cooling of liquid film along an inclined plate coverd 

with a porous layer . . . . . . . . . . . . . . . . . . . . . . . . . 106 

V.D. LISEYKIN, Y.V. LIKHANOVA, D.V. PATRAKHIN, I.A. VASEVA: 

Application of boundary layer-type functions to comprehensive 

grid generation codes . . . . . . . . . . . . . . . . . . . . . . . . 108 


CONTENTS 

G. LUBE: 

A stabilized finite element method with anisotropic mesh refinement 

for the Oseen equations . . . . . . . . . . . . . . . . . . . . 109 

H. LÜDEKE: 

Detached Eddy Simulation of Supersonic Shear Layer Wake Flows 111 

K. MANSOUR: 

Boundary Layer Solution For laminar flow through a Loosely 

curved Pipe by Using Stokes Expansion . . . . . . . . . . . . . . 113 

G. MATTHIES, L. TOBISKA: 

Mass conservation of finite element methods for coupled flowtransport 

problems . . . . . . . . . . . . . . . . . . . . . . . . . 115 

J. MAUSS, J. COUSTEIX: 

Global Interactive Boundary Layer (GIBL) for a Channel . . . . . 116 

O. MIERKA, D. KUZMIN: 

On the implementation of turbulence models in incompressible 

flow solvers based on a finite element discretization . . . . . . . . 118 

K. MORINISHI: 

Rarefied Gas Boundary Layer Predicted with Continuum and Kinetic 

Approaches . . . . . . . . . . . . . . . . . . . . . . . . . . 120 

A. NASTASE: 

Qualitative Analysis of the Navier-Stokes Solutions in Vicinity of 

their Critical Lines . . . . . . . . . . . . . . . . . . . . . . . . . 122 

F. NATAF, G. RAPIN: 

Application of the Smith Factorization to Domain Decomposition 

Methods for the Stokes equations . . . . . . . . . . . . . . . . . 124 

N. NEUSS: 

Numerical approximation of boundary layers for rough boundaries 126 

M.A. OLSHANSKII: 

An Augmented Lagrangian based solver for the low-viscosity incompressible 

flows . . . . . . . . . . . . . . . . . . . . . . . . . 128 

N. PARUMASUR, J. BANASIAK, J.M. KOZAKIEWICZ: 

Numerical and Asymptotic Analysis of Singularly Perturbed PDEs 

of Kinetic Theory . . . . . . . . . . . . . . . . . . . . . . . . . . 130 

B. RASUO: 

On Boundary Layer Control in Two-Dimensional Transonic Wind 

Tunnels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 

H.-G. ROOS: 

A Comparison of Stabilization Methods for Convection-Diffusion- 

Reaction Problems on Layer-Adapted Meshes . . . . . . . . . . . 134 

H.-G. ROOS, H. ZARIN: 

Discontinuous Galerkin stabilization for convection-diffusion problems 

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 


B. SCHEICHL, A. KLUWICK: 

On Turbulent Marginal Separation: How the Logarithmic Law of 

the Wall is Superseded by the Half-Power Law . . . . . . . . . . 136 

O. SHISHKINA, C. WAGNER: 

Boundary and Interior Layers in Turbulent Thermal Convection . 138 

M. STYNES, L. TOBISKA: 

Using rectangular Qp elements in the SDFEM for a convectiondiusion 

problem with a boundary layer . . . . . . . . . . . . . . 140 

P. SVÁ ˘CEK: 

Numerical Approximation of Flow Induced Airfoil Vibrations . . 141 

N.V. TARASOVA: 

Full asymptotic analysis of the Navier-Stokes equations in the 

problems of gas flows over bodies with large Reynolds number . . 143 

C.H. TAI, C.-Y. CHAO, J.-C. LEONG, Q.-S. HONG: 

Effects of golf ball dimple configuration on aerodynamics, trajectory, 

and acoustics . . . . . . . . . . . . . . . . . . . . . . . . . 145 

H. TIAN: 

Uniformly Convergent Numerical Methods for Singularly Perturbed 

Delay Differential Equations . . . . . . . . . . . . . . . . 147 

A.I. TOLSTYKH, M.V. LIPAVSKII, E.N. CHIGEREV: 

Highly accurate 9th-order schemes and their applications to DNS 

of thin shear layer instability . . . . . . . . . . . . . . . . . . . . 149 

M. VASILIEV: 

About unsteady Boundary Layer on a dihedral angle . . . . . . . 150 

A.E.P. VELDMANN: 

High-order symmetry-preserving discretization on strongly stretched 

grids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 

Z.-H. YANG, Y.-Z. LI, Y. ZHU: 

Application of Bifurcation Method to Computing Numerical Solutions 

of Lane-Emden Equation . . . . . . . . . . . . . . . . . . 154 

Q. YE: 

Numerical simulation of turbulent boundary for stagnation-flow 

in the spray-painting process . . . . . . . . . . . . . . . . . . . . 156 

Participants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 

Authors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164 


Plenary Presentations

KL. GERSTEN: Vom Kochtopf bis zum Fußballspiel (Public Talk) 

✬ 

✫ 

Öffentlicher Vortrag 

am 25. Juli 2006 – 20:00 Uhr 

im DLR_School_Lab - Bunsenstraße 10 – 37073 Göttingen 

im Rahmen der International Conference Boundary and Interior Layers 2006 (BAIL 2006) 

Vom Kochtopf bis zum Fußballspiel 

Episoden zu den weltweiten Wirkungen der Göttinger Strömungsforschung 

Professor Dipl. Math. Dr.-Ing. Dr.-Ing. E.h. Klaus Gersten 

Ruhr-Universität Bochum 

Im Vortrag werden unter anderem folgende Fragen erörtert: 

� Als Modell für ein Herz mit defekten Herzklappen entwickelte der Arzt Dr. Liebau in den fünfziger 

Jahren eine ventillose Pumpe. Weshalb funktionierte die Pumpe? 

� Wie lässt sich die Bananenflanke beim Fußballspiel physikalisch erklären? 

� Kann die Kraftfahrzeug-Aerodynamik von der Flugzeug-Aerodynamik lernen? 

� Wie kann der Besitzer eines Automobils mit Schrägheck leicht feststellen, ob sein 

Fahrzeug gute aerodynamische Eigenschaften besitzt? 

� Warum sind bestimmte bei der Umströmung von Körpern auftretende Strömungsstrukturen je nach 

Anwendung erwünscht (Überschallflugzeug) oder unerwünscht (Automobil)? 

� Welches für die Göttinger Strömungsforschung charakteristische Konzept liegt allen bisher 

genannten Fragen zugrunde? 

Zur Person: 

Professor Gersten, Jahrgang 1929, studierte Mathematik und Physik an der TH 

Braunschweig. Als Assistent arbeitete er am Institut für Strömungsmechanik der TH 

Braunschweig unter der Leitung von Professor Hermann Schlichting. Nach seiner Promotion 

zum Dr.-Ing. leitete er die Abteilung Theoretische Aerodynamik und wurde anschließend 

Stellvertretender Direktor des Institutes für Aerodynamik der Deutschen Forschungsanstalt 

für Luftfahrt (DFL) Braunschweig. Nach seiner Habilitation für das Fach Strömungsmechanik 

lehrte er dreißig Jahre als Ordentlicher Professor an der Ruhr-Universität Bochum. Während 

dieser Zeit war er als Visiting Professor an der University of Rio de Janeiro, Brazil, der Nagoya 

University, Japan und an der University of Arizona, Tucson, USA tätig. 1992 wurde er zum 

Dr.-Ing. E.h. der Universität Essen ernannt. Er ist Mitglied der Österreichischen Akademie der 

Wissenschaften in Wien. 2003 wurde Professor Gersten der Ludwig-Prandtl-Ring, die 

höchste Auszeichnung im Bereich der Flugwissenschaften, von der Deutschen Gesellschaft 

für Luft- und Raumfahrt Lilienthal-Oberth e.V. (DGLR) verliehen. 

Kontakt: 

DLR, Dr. H. J. Heinemann 

0551- 709 2108, hajo.heinemann@dlr.de 

Universität Göttingen, Prof. Dr. G. Lube 

0551- 39 4503, lube@math.uni-goettingen.de 

Speaker: GERSTEN, KL. 3 BAIL 2006 

✩ 

✪

P. HOUSTON: Discontinuous Galerkin Finite Element Methods for CFD: A Posteriori 

Error Estimation and Adaptivity 

✬ 

✫ 

Discontinuous Galerkin Finite Element Methods for CFD: A Posteriori Error Estimation and 

Adaptivity 

Paul Houston 

School of Mathematical Sciences, University of Nottingham, UK. 

In recent years there has been considerable interest in the mathematical design and 

practical application of nonconforming finite element methods that are based on discontinuous 

piecewise polynomial approximation spaces; such approaches are referred to as 

discontinuous Galerkin (DG) methods. The main advantages of these methods lie in their 

conservation properties, their ability to treat a wide range of problems within the same 

unified framework, and their great flexibility in the mesh-design. Indeed, DG methods can 

easily handle non-matching grids and non-uniform, even anisotropic, polynomial approximation 

degrees, which makes them ideally suited for application within adaptive finite 

element software. 

In this talk we present an overview of some recent developments concerning the a posteriori 

error analysis and adaptive mesh design of h– and hp–version DG finite element 

methods for the numerical approximation of second–order elliptic boundary value problems. 

In particular, we consider the derivation of computable upper and lower bounds 

on the error measured in terms of an appropriate (mesh–dependent) energy norm. The 

proofs of the upper bounds are based on rewriting the method in a non-consistent manner 

using polynomial lifting operators and employing an appropriate decomposition result 

for the underlying discontinuous spaces. Applications to the numerical approximation 

of second–order linear elliptic problems, including Poisson’s equation, Stokes equations, 

nearly–incompressible elasticity, and the time harmonic eddy current problem, as well 

as second–order quasilinear boundary value problems, which typically arise in the modelling 

of non-Newtonian flows, will be considered. Numerical experiments confirming the 

reliability and efficiency of the proposed a posteriori error bounds within an automatic 

mesh refinement algorithm employing both local mesh subdivision and local polynomial 

enrichment will be presented. 

This research has been carried out in collaboration with Dominik Schötzau (University 

of British Columbia), Thomas Wihler (University of Minnesota), and Ilaria Perugia 

(University of Pavia). 

References 

[1] P. Houston, I. Perugia, and D. Schötzau. An posteriori error indicator for discontinuous 

Galerkin discretizations of H(curl)–elliptic partial differential equations. 

Submitted to IMA J. Numer. Anal. 

[2] P. Houston, D. Schötzau, and T. Wihler. Energy norm a posteriori error estimation 

for mixed discontinuous Galerkin approximations of the Stokes problem. J. Sci. 

Comp., 22(1):357–380, 2005. 

[3] P. Houston, D. Schötzau, and T. Wihler. An hp-adaptive mixed discontinuous 

Galerkin FEM for nearly incompressible linear elasticity. Comput. Methods Appl. 

Mech. Engrg., (to appear). 

[4] P. Houston, D. Schötzau, and T. P. Wihler. Energy norm a posteriori error estimation 

of hp-adaptive discontinuous Galerkin methods for elliptic problems. Math. 

Models Methods Appl. Sci., (to appear). 

1 

Speaker: HOUSTON, P. 4 BAIL 2006 

✩ 

✪

L. LESSHAFFT, P. HUERRE, P. SAGAUT, M. TERRACOL: Dynamics of Hot Jets: A 

Numerical and Theoretical Study 

✬ 

✫ 


DYNAMICS OF HOT JETS: A NUMERICAL AND THEORETICAL STUDY 

Lutz Lesshafft 1.2 , Patrick Huerre 1 , Pierre Sagaut 3 and Marie Terracol 2 

1Laboratoire d'Hydrodynamique (LadHyX), CNRS – École Polytechnique, F-91128 Palaiseau, France 

2ONERA, Department of CFD and Aeroacoustics, 29 avenue de la Division Leclerc, F-92322 

Châtillon, France 

3Laboratoire de Modélisation en Mécanique, Université Pierre et Marie Curie, Boîte 162, 4 place 

Jussieu, F-75252 Paris cedex 05, France 

Since the experiments of Monkewitz, Bechert, Barsikow & Lehmann (1990), sufficiently hot circular 

jets have been known to give rise to self-sustained synchronized oscillations induced by a locally 

absolutely unstable region. Numerical simulations (Lesshafft, Huerre, Sagaut & Terracol 2005, 2006) 

have been carried out in order to determine if such synchronized states correspond to a nonlinear 

global mode of the underlying basic flow, as predicted in the context of Ginzburg-Landau amplitude 

evolution equations by Couairon & Chomaz (1997, 1999), Pier, Huerre, Chomaz & Couairon (1998) 

and Pier, Huerre & Chomaz (2001). In the presence of a pocket of absolute instability embedded 

within a convectively unstable jet, global oscillations are generated by a steep nonlinear front located 

at the upstream station of marginal absolute instability. The global frequency is given, within 10% 

accuracy, by the absolute frequency at the front location. For jet flows displaying absolutely unstable 

inlet conditions, global instability is observed to arise if the streamwise extent of the absolutely 

unstable region is sufficiently large: While local absolute instability sets in for ambient-to-jet 

temperature ratios S < 0.453, global modes only appear for S < 0.325. In agreement with theoretical 

predictions, the selected frequency near the onset of global instability coincides with the absolute 

frequency at the inlet, provided that the ratio of jet radius R to shear layer momentum thickness θ is 

sufficiently small (R/ θ ∼ 10, thick shear layers). For thinner shear layers (R/ θ ∼ 25), the numerically 

determined global frequency gradually departs from the inlet absolute frequency. 

References 

Couairon, A. & Chomaz, J.-M. 1997 Absolute and convective instabilities, front velocities and global 

modes in nonlinear systems. Physica D 108, 236-276 

Couairon, A. & Chomaz, J.-M. 1999 Fully nonlinear global modes in slowly varying flows. Phys. 

Fluids 11, 3688-3703. 

Lesshafft, L., Huerre, P., Sagaut, P. & Terracol, M. 2005 Global modes in hot jets, absolute / 

convective instabilities and acoustic feedback. 10 pages 11th AIAA / CEAS Aeroacoustics Conference, 

Monterey, USA, May 23-25, 2005. 

Lesshaft, L., Huerre, P., Sagaut, P. & Terracol, M. 2006 Nonlinear global modes in hot jets. J. Fluid 

Mech., in press. 

Monkewitz, P. A., Bechert, D. W., Barsikow, B & Lehmann, B. 1990 Self-excited oscillations and 

mixing in a heated round jet. J. Fluid Mech. 213, 611-639. 

Speaker: HUERRE, P. 5 BAIL 2006 

✩ 

✪

M. STYNES: Convection-diffusion problems, SDFEM/SUPG and a priori meshes 

✬ 

✫ 

Convection-diffusion problems, SDFEM/SUPG 

and a priori meshes 

Martin Stynes ∗ 

Abstract 

Linear convection-diffusion problems will be briefly described (cf. [15]). The nature and 

structure of their solutions will be examined, including the main features of exponential 

and characteristic/parabolic layers. Next, Shishkin meshes will be described and discussed 

[4, 11, 15]; these piecewise-uniform meshes are suited to the numerical solution of convectiondiffusion 

problems with boundary layers — yet they do not fully resolve these layers. 

Then the Streamline Diffusion Finite Element Method (SDFEM), which is also known 

as the Streamline-Upwinded Petrov-Galerkin method (SUPG), will be introduced. Since 

its inception [6] in 1979, this method has been the subject of a huge number of theoretical 

analyses and numerical investigations that continue to this day; see the references in [12, 13, 

15]. The main body of the talk is a comprehensive survey of the application of the method 

to convection-diffusion problems, including discussions of its strengths and weaknesses, and 

presenting recent theoretical results. In particular the following topics will be addressed: 

References 

• stability and the choice of SDFEM parameter [1, 5, 12] 

• quasioptimality [3, 14] 

• accuracy on general meshes [9, 18] 

• accuracy on Shishkin meshes [5, 10, 16, 17] 

• variants of SDFEM: 

(i) nonconforming spaces [7] 

(ii) nonlinear shock-capturing modifications of SDFEM [2, 8] 

— the references given here are only a subset of those available. 

[1] J. E. Akin and T. E. Tezduyar. Calculation of the advective limit of the SUPG stabilization 

parameter for linear and higher-order elements. Comput. Methods Appl. Mech. Engrg., 

193:1909–1922, 2004. 

[2] E. Burman and A. Ern. Stabilized Galerkin approximation of convection-diffusion-reaction 

equations: discrete maximum principle and convergence. Math. Comp., 74:1637–1652 (electronic), 

2005. 

[3] L. Chen and J. Xu. An optimal streamline diffusion finite element method for a singularly 

perturbed problem. In Z.C Shi, Z. Chen, T. Tang, and D. Yu, editors, Recent Advances in 

Adaptive Computation, volume 383 of Contemporary Mathematics, pages 236–246. American 

Mathematical Society, 2005. 

∗ Mathematics Department, National University of Ireland, Cork, Ireland 

1 

Speaker: STYNES, M. 6 BAIL 2006 

✩ 

✪

M. STYNES: Convection-diffusion problems, SDFEM/SUPG and a priori meshes 

✬ 

✫ 

[4] P.A. Farrell, A.F. Hegarty, J.J. Miller, E. O’Riordan, and G.I. Shishkin. Robust Computational 

Techniques for Boundary Layers. Chapman & Hall/CRC, Boca Raton, 2000. 

[5] S. Franz and T. Linß. Superconvergence analysis of Galerkin FEM and SDFEM for elliptic 

problems with characteristic layers. Technical Report MATH-NM-03-2006, Institut für 

Numerische Mathematik, Technische Universität Dresden, 2006. 

[6] T.J.R. Hughes and A.N. Brooks. A multidimensional upwind scheme with no crosswind 

diffusion. In T.J.R. Hughes, editor, Finite Element Methods for Convection Dominated 

Flows, volume 34 of AMD. ASME, New York, 1979. 

[7] P. Knobloch and L. Tobiska. The P mod 

1 

element: a new nonconforming finite element for 

convection-diffusion problems. SIAM J. Numer. Anal., 41:436–456 (electronic), 2003. 

[8] T. Knopp, G. Lube, and G. Rapin. Stabilized finite element methods with shock capturing 

for advection-diffusion problems. Comput. Methods Appl. Mech. Engrg., 191:2997–3013, 

2002. 

[9] N. Kopteva. How accurate is the streamline-diffusion FEM inside characteristic (boundary 

and interior) layers? Comput. Methods Appl. Mech. Engrg., 193:4875–4889, 2004. 

[10] T. Linß and M. Stynes. Numerical methods on Shishkin meshes for linear convectiondiffusion 

problems. Comput. Methods Appl. Mech. Engrg., 190:3527–3542, 2001. 

[11] J.J.H. Miller, E. O’Riordan, and G.I. Shishkin. Solution of singularly perturbed problems 

with ε-uniform numerical methods–introduction to the theory of linear problems in one and 

two dimensions. World Scientific, Singapore, 1996. 

[12] H.-G. Roos, M. Stynes, and L. Tobiska. Numerical methods for singularly perturbed differential 

equations, volume 24 of Springer Series in Computational Mathematics. Springer- 

Verlag, Berlin, 1996. 

[13] H.-G. Roos, M. Stynes, and L. Tobiska. Numerical methods for singularly perturbed differential 

equations. Springer Series in Computational Mathematics. Springer-Verlag, Berlin, 

Second edition, (to appear). 

[14] G. Sangalli. Quasi optimality of the SUPG method for the one-dimensional advectiondiffusion 

problem. SIAM J. Numer. Anal., 41:1528–1542 (electronic), 2003. 

[15] M. Stynes. Steady-state convection-diffusion problems. Acta Numer., 14:445–508, 2005. 

[16] M. Stynes and L. Tobiska. The SDFEM for a convection-diffusion problem with a boundary 

layer: optimal error analysis and enhancement of accuracy. SIAM J. Numer. Anal., 41:1620– 

1642, 2003. 

[17] M. Stynes and L. Tobiska. Using rectangular Qp elements in the SDFEM for a convectiondiffusion 

problem with a boundary layer. Technical Report 08-2006, Faculty of Mathematics, 

Otto-von-Guericke-Universität, Magdeburg, 2006. 

[18] G. Zhou. How accurate is the streamline diffusion finite element method? Math. Comp., 

66:31–44, 1997. 

Speaker: STYNES, M. 7 BAIL 2006 

2 

✩ 

✪

V. GRAVEMEIER, S. LENZ, W.A. WALL: Variational Multiscale Methods for incompressible 

flows 

✬ 

✫ 

1. Introduction 

Variational Multiscale Methods for incompressible flows 

V. Gravemeier, S. Lenz & W.A. Wall 

Chair for Computational Mechanics 

Technische Universität München 

Boltzmannstr. 15, D-85747 Garching b. München 

http://www.lnm.mw.tum.de 

(vgravem,lenz,wall)@lnm.mw.tum.de 

The numerical simulation of incompressible flows governed by the Navier-Stokes equations requires 

to deal with subgrid phenomena. Particularly in turbulent flows, the scale spectra are 

notably widened and need to be handled adequately to get a reasonable numerical solution. 

Separating the complete scale range into subranges enables a different treatment of any of these 

subranges. In this talk, we will present the general framework of a two- and a three-scale separation 

of the incompressible Navier-Stokes equations based on the variational multiscale method 

as proposed in [Hughes et al. (2000)] and [Collis (2001)]. In a two-scale separation, resolved 

and unresolved scales are distinguished, and in a three-scale separation, large resolved scales, 

small resolved scales, and unresolved scales are differentiated. After having presented the general 

framework, three different approaches to numerical realizations will be addressed (i.e., a 

global, a local, and a new residual-based approach). A comprehensive overview may be found 

in [Gravemeier (2006b)]. 

2. The variational multiscale framework 

A variational form of the incompressible Navier-Stokes equations reads 

BNS (v,q;u,p) = (v,f) Ω ∀(v,q) ∈ Vup (1) 

where Vup denotes the combined form of the weighting function spaces for velocity and pressure 

in the sense that Vup := Vu × Vp. 

In a three-scale separation, which will be focused on in the present abstract, the solution 

functions are separated as 

u = u + u ′ + û, p = p + p ′ + ˆp, (2) 

where (·), (·) ′ , and ˆ (·) denote the large resolved, small resolved, and unresolved scales, respectively. 

The weighting functions are separated accordingly. Due to the linearity of the weighting 

functions, the variational equation (1) may now be decomposed into a system of three variational 

equations reading 

� � ′ ′ 

BNS v,q;u + u + û,p + p + ˆp = (v,f)Ω ∀(v,q) ∈ Vup (3) 

� � � � 

′ ′ ′ ′ ′ 

BNS v ,q ;u + u + û,p + p + ˆp = v ,f ∀ Ω � v ′ ,q ′� ∈ V ′ up (4) 

� � 

′ ′ ˆv, ˆq;u + u + û,p + p + ˆp = (ˆv,f)Ω ∀(ˆv, ˆq) ∈ ˆ Vup (5) 

BNS 

Furthermore, it is assumed that the direct influence of the unresolved scales on the large resolved 

scales is close to zero, relying on a clear separation of the large-scale space and the space of 

unresolved scales. This leads to a simplified equation system. 

1 

Speaker: WALL, W.A. 8 BAIL 2006 

✩ 

✪

V. GRAVEMEIER, S. LENZ, W.A. WALL: Variational Multiscale Methods for incompressible 

flows 

✬ 

✫ 

It is not intended to resolve anything called unresolved a priori. Taking into account the 

effect of the unresolved scales onto the small scales is the only desire. Some alternatives lend 

themselves for this purpose, but the focus in this talk will be on the subgrid viscosity approach 

as a usual and well-established way of taking into account the effect of unresolved scales in 

the traditional LES. In the variational multiscale approach to LES, the subgrid viscosity term 

directly acts only on the small resolved scales. Indirect influence on the large resolved scales, 

however, is ensured due to the coupling of the large- and the small-scale equations. 

3. Practical methods 

At this stage, it should be pointed out that the variational multiscale method is a theoretical 

framework for the separation of scales. Thus, it is essential to develop corresponding practical 

implementations by incorporating the variational multiscale framework into a specific numerical 

method. For such practical methods, it is crucial that the aforementioned separation of the 

different scale groups is actually achieved. Implementations of the three-scale separation have 

already been done within continuous Galerkin finite element methods, discontinuous Galerkin 

finite element methods, finite volume methods, finite difference methods, and spectral methods. 

According to the numerical treatment of the smaller of the resolved scales, a global (see, 

e.g., [Gravemeier (2006a)]) and a local (see, e.g., [Gravemeier et al. (2005)]) approach may be 

distinguished. 

Recently [Calo (2005)], a new residual-based approach has been developed where only resolved 

and unresolved scales are distinguished (i.e., a two-scale separation). This new concept 

approxmiates the unresolved scales by analytical expressions. It may be considered an 

advanced stabilized method which takes into account the nonlinear nature of the Navier-Stokesequations. 

Our results obtained with the residual-based approach are about to be published in 

[Lenz & Wall (2006)]. 

All three approaches (i.e., the global, the local, and the residual-based approach) will be 

presented together with numerical results in this talk. 

References 

[Calo (2005)] Calo, V.M. 2005 Residual-based Multiscale Turbulence Modeling: Finite Volume 

Simulations of Bypass Transition. PhD thesis, Department of Civil and Environmental 

Engineering, Stanford University. 

[Collis (2001)] Collis, S.S. 2001 Monitoring unresolved scales in multiscale turbulence modeling. 

Phys. Fluids 13, 1800-1806. 

[Gravemeier (2006a)] Gravemeier, V. 2006 Scale-separating operators for variational multiscale 

large eddy simulation of turbulent flows. J. Comp. Phys. 212, 400-435. 

[Gravemeier (2006b)] Gravemeier, V. 2006 The variational multiscale method for laminar and 

turbulent flow. Arch. Comp. Meth. Engrg. , in press. 

[Gravemeier et al. (2005)] Gravemeier, V., Wall, W.A. and Ramm, E. 2005 Large eddy 

simulation of turbulent incompressible flows by a three-level finite element method Int. J. 

Numer. Meth. Fluids 48, 1067-1099. 

[Hughes et al. (2000)] Hughes, T. J. R., Mazzei, L. & Jansen, K. E. 2000 Large eddy 

simulation and the variational multiscale method. Comput. Visual. Sci. 3, 47-59. 

[Lenz & Wall (2006)] Lenz, S. and Wall, W.A. 2006 A residual-based variational multiscale 

method and its application to turbulent channel flow. To be submitted to Theoret. Comput. 

Fluid Dyn. 

Speaker: WALL, W.A. 9 BAIL 2006 

2 

✩ 

✪

Minisymposia

Speaker: 

N. Kopteva, M. Stynes, E. O’Riordan 

Robust methods for nonlinear singularly perturbed differential equations 

This minisymposium is concerned with nonlinear singularly perturbed 

differential equations such as semilinear reaction-diffusion problems, 

quasilinear parabolic convection-diffusion equations, flows in porous 

media, and the modelling of catalytic chemical reactions and of turbulence. 

The talks deal with methods that are ”robust” for such problems, 

i.e., that yield accurate approximations of the solution for a broad range 

of values of the singular perturbation parameter. This includes numerical 

methods for which numerical experiments demonstrate robustness 

for a wide range of values of the parameter, even though no theoretical 

proof of convergence exists. 

• Eugene O’Riordan: A singular perturbation problem arising in the modelling of 

plasma sheaths 

• Grigory I. Shishkin: A posteriori adapted meshes in the approximation of singularly 

perturbed quasilinear parabolic convection-diffusion equations 

• Iuliana Stanculescu: Numerical Analysis of Approximate Deconvolution Models 

of Turbulence 

• Hong Wang: An Eulerian-Langrangian Formulation for Compositional Flow in 

Porous Media

R.K. DUNNE, E. O’RIORDAN, M.M. TURNER: A singular perturbation problem 

arising in the modelling of plasma sheaths 

✬ 

✫ 

A singular perturbation problem arising in the 

modelling of plasma sheaths ∗ 

R. K. Dunne † , E. O’Riordan ‡ and M. M. Turner § 

Abstract 

Consider the interaction of a flowing plasma with a planar Langmuir probe [2]. Assume 

that the plasma flows in the plane of the probe surface and that the plasma consists 

solely of positive ions with density n+ and electrons with density ne. Downstream of 

the probe, the ions are moving with a velocity of u = (ui, uv + uF ) and (u0, uF ) is 

the flow velocity of the ions upstream of the probe. We wish to consider the influence 

of the probe on the flow of the ions. Let X be the horizontal distance to the right of 

the probe and Y the distance along the probe (from the tip of the probe). Assuming 

a collision-less plasma and that the ions are cold, the continuity equations for the ion 

density and momentum are [2] 

∂n+ 

∂t + ∇ · (n+u) = 0 

�∂u 

m+n+ 

∂t + (u · ∇)u� = en+E 

where E = (Ex, Ey) = −∇ ˜ φ is the electric field and m+ is the mass of the ions. Our 

interest is in the steady state case and if uF >> uv, we disregard terms involving uv. 

Hence this system is approximated with the system 

∂ 

∂X (n+ui) 

∂n+ 

+ uF = 0 

∂Y 

∂ui ∂ui e 

ui + uF = EX 

∂X ∂Y m+ 

where (if EX >> EY ) then the electric field is determined from solving Poisson’s equation 

− ∂EX 

∂X = ∂2 ˜ φ 

∂X 2 = e(n+ − ne) 

. 

Since me

R.K. DUNNE, E. O’RIORDAN, M.M. TURNER: A singular perturbation problem 

arising in the modelling of plasma sheaths 

✬ 

✫ 

where Te is the electron temperature, k is Boltzmann’s constant, ɛ0 is the permittivity 

of free space and e is the electron charge. 

In a similar fashion to the scaling of the variables used in [3], we introduce the nondimensional 

independent variables x, y and the non-dimensional dependent variables 

n, u and φ, which are defined as follows: 

n = n+ 

n0 

u = ui 

cs 

φ = e ˜ φ 

, x = 

kTe 

X 

L 

y = Y cs 

uF L 

where the ion sound speed cs and the electron Debyre length λD are defined by 

c 2 s = kTe 

, λ 

m+ 

2 D = ɛ0kTe 

. 

n0e2 The length L is a distance sufficiently far from the probe so that the effect of the probe 

on the plasma at this distance is negligible. Introduce the small parameter 

ε = λD 

L . 

After reformulating the problem with the above transformations and formulating 

suitable boundary and initial conditions we propose to examine the related mathematical 

problem : 

Find (u(x, y), n(x, y), φ(x, y)) which satisfy the following system of differential equations 

in the domain (x, y) ∈ (0, 1) × (0, T ] 

∂n ∂(nu) 

+ = 0, 

∂y ∂x 

(x, y) ∈ [0, 1) × (0, T ] 

∂u 

+ u∂u = −∂φ , 

∂y ∂x ∂x 

(x, y) ∈ [0, 1) × (0, T ] 

ε 2 ∂2 φ 

∂x 2 = eφ − n, (x, y) ∈ (0, 1) × (0, T ] 

subject to the following set of boundary and initial conditions 

φ(0, y) = −A, φ(1, y) = 0, y ≥ 0 φ(x, 0) = φ0(x), 0 ≤ x ≤ 1 

n(x, 0) = 1, 0 ≤ x ≤ 1; ny(1, y) = −(nux)(1, y), y ≥ 0 

u(x, 0) = ũ0, 0 ≤ x ≤ 1; uy(1, y) = −φx(1, y), y ≥ 0. 

The parameters ũ0 and A are assumed to be known and the initial condition φ0(x) is 

chosen so that ε 2 φ ′′ 

0(x) = e φ0(x) − 1, φ0(0) = −A, φ0(1) = 0. 

Due to the presence of the singular perturbation parameter ε, layers or sheaths 

appear in the solutions. The system is discretized using simple upwinding and a special 

piecewise-uniform Shishkin-type mesh [1]. Numerical results are presented to display 

the robustness of the numerical algorithm with respect to ε. 

References 

[1] P. A. Farrell, A.F. Hegarty, J. J. H. Miller, E. O’Riordan and G. I. Shishkin, Robust 

Computational Techniques for Boundary Layers, Chapman and Hall/CRC Press, 

Boca Raton, U.S.A., (2000). 

[2] M. A. Lieberman and A. J. Lichtenberg, Principles of plasma discharges and materials 

processing, Wiley and sons, (1994). 

[3] H. Liu and M. Slemrod, KDV Dynamics in the plasma-sheath transition, Appl. 

Math. Lett., 17 (2004) 401-419. 

2 

Speaker: O’RIORDAN, E. 15 BAIL 2006 

✩ 

✪

G.I. SHISHKIN: A posteriori adapted meshes in the approximation of singularly 


✬ 

✫ 

A posteriori adapted meshes in the approximation 

of singularly perturbed quasilinear parabolic 

convection-diffusion equations ∗ 

Grigory I. Shishkin 

A Dirichlet problem on a segment for a quasilinear parabolic 

convection-diffusion equation with a small (perturbation) parameter 

ε multiplying the highest derivative is considered. For this 

problem, a solution of the classical finite difference scheme on a 

uniform mesh converges only under the condition h ≪ ε, where 

h is the step-size of the space mesh; moreover, the order of convergence 

in x is O � ε N −1� , where N + 1 is the number of nodes 

in the uniform mesh with respect to x. 

To improve the accuracy of the approximate solution, we apply 

a posteriori sequential procedure of grid refinement in the 

subdomains that are defined by the gradient of solutions of intermediate 

discrete problems. The correction of the grid solutions 

is performed only on these subdomains, where uniform meshes 

are used. We construct a difference scheme that converges ”almost 

ε-uniformly”, i.e., with an error weakly depending on the 

parameter ε. 

� 

The convergence rate of the constructed scheme is O 

where N1 + 1 and N0 + 1 are the numbers of mesh points with 

respect to x and t, respectively, ν is an arbitrary number from 

(0, 1]. Thus, the scheme on a posteriori adapted meshes converges 

under the condition N −1 ≪ εν , which is essentially weaker 

in comparison with the scheme on uniform meshes. 

ε −ν N −1 

1 

+ N −1/2 

1 

∗ This research was supported in part by the Russian Foundation for Basic Research 

(grant No 04-01-00578, 04–01–89007–NWO a), by the Dutch Research Organisation NWO 

under grant No 047.016.008 and by the Boole Centre for Research in Informatics, National 

University of Ireland, Cork. 

2 

+ N −1 

� 

0 , 

Speaker: SHISHKIN, G.I. 16 BAIL 2006 

✩ 

✪

G.I. SHISHKIN: A posteriori adapted meshes in the approximation of singularly 


✬ 

✫ 

A scheme on a posteriori adapted meshes in the case of a 

linear problem is considered in [1]. 

References 

[1] G.I. Shishkin, A posteriori adapted (to the solution gradient) 

grids in the approximation of singularly perturbed 

convection-diffusion equations, Vychisl. Tekhnol. (Computational 

Technologies), 6 (1), 72-87 (2001) (in Russian). 

Speaker: SHISHKIN, G.I. 17 BAIL 2006 

3 

✩ 

✪

W. LAYTON, I. STANCULESCU: Numerical Analysis of Approximate Deconvolution 

Models of Turbulence 

✬ 

✫ 

Numerical Analysis of Approximate Deconvolution Models of 

Turbulence 

William Layton ∗ and Iuliana Stanculescu † 

Abstract 

If the NSE are averaged with a local, spacial, convolution type filter the resulting 

system is not closed due to the term g ∗ (uu). A deconvolution operator GN is one 

which satisfies: 

u = GN(g ∗ u) + O(δ 2N+2 ) (0.1) 

where δ is the filter width. This yields the closure method: 

g ∗ (uu) = g ∗ (GN(g ∗ u)GN(g ∗ u)) + O(δ 2N+2 ) (0.2) 

We will review several solutions to the ill-possed deconvolution problem, present 

an ”optimal” deconvolution procedure and present numerical analysis and numerical 

experiments with it. 

∗ 

Department of Mathematics, University of Pittsburgh, Pittsburgh, PA,15260, U.S.A.; email: 

wjl@pitt.edu, www.math.pitt.edu/˜wjl 

† 

Department of Mathematics, University of Pittsburgh, Pittsburgh, PA,15260, U.S.A.; email: 

ius1@pitt.edu, www.math.pitt.edu/˜ius1 

Speaker: STANCULESCU, I. 18 BAIL 2006 

1 

✩ 

✪

H. WANG: A Component-Based Eulerian-Lagrangian Formulation for Compositional 

Flow in Porous Media 

✬ 

✫ 

A Component-Based Eulerian-Lagrangian Formulation 

for Compositional Flow in Porous Media 

Hong Wang ∗ 

Introduction Compositional models describe the simultaneous transport of multiple components 

flowing in coexisting phases in porous media [1, 4]. Because each component can transfer 

between different phases, the mass of each phase or a component within a particular phase is 

no longer conserved. Instead, the total mass of each component among all the phases must be 

conserved, leading to strongly coupled systems of transient nonlinear advection-diffusion equations. 

These equations are closely coupled to a set of constraining equations, which are strongly 

nonlinear, implicit functions of phase pressure, temperature, and composition. These equations 

need to be solved in all spatial cells at each iterative step of each time step via thermodynamic 

flash calculation. In industrial applications, upwind methods have commonly been used to stabilize 

the numerical approximations [1, 4]. However, these methods often generate excessive 

numerical dispersion and serious spurious effects due to grid orientation. 

Eulerian-Lagrangian methods symmetrize the transport equations and generate accurate 

numerical solutions even if large time steps and coarse spatial grids are used. They have demonstrated 

excellent performance in the numerical simulations of single-phase flow [3, 5] and immiscible 

two phase flow [2]. However, there exist serious mathematical and numerical difficulties in 

the development of Eulerian-Lagrangian methods for multiphase multicomponent compositional 

flow. We present a component-based Eulerian-Lagrangian formulation for compositional flow, 

which can be used by many Eulerian-Lagrangian methods. 

Numerical experiments We simulate the transport of Methane, Propane, and n-Hexane, 

flowing in coexisting liquid and vapor phases in a horizontal porous medium reservoir Ω = 

(0, 1000) × (0, 1000) ft 2 with a thickness of 1 ft over a time period of 20 years. An injection well 

is located at the upper-right corner of Ω with a volumetric injection rate of Q =15ft 3 /day. A 

production well is located at the lower-left corner with a production rate of Q = −15 ft 3 /day. 

The porosity φ = 0.1. The permeability is 60 md. The effect of capillary pressure is neglected. 

The relative permeability k r,l =(s l ) 2 and k r,v =(1−s l ) 2 . The initial reservoir pressure is 2100 

psia and the reservoir temperature is 350 o K. The composition of the resident fluid is cMethane 

= 0.5, cP ropane =0.2, and cn−Hexane =0.3, which is in liquid phase at the given temperature 

and pressure. The composition of the injected fluid is ¯cMethane = 0.8, ¯cP ropane =0.15, and 

¯cn−Hexane =0.05, which is in vapor phase. 

We use a uniform coarse spatial grid of ∆x =∆y= 25 ft. We use a time step of ∆tel =1 

year for the Eulerian-Lagrangian method, and of ∆tup = 2 days for the upwind method that is 

the largest possible. In Figure 1(a) we present the plots of the overall mole fraction cMethane 

generated by the Eulerian-Lagrangian method at t = 5 and 20 years. In Figure 1(b) we present 

the plots of the normalized molar amount n v Methane = ρv s v c v Methane /(ρ cMethane), where ρ v and 

sv are density and saturation of the vapor phase and ρ is bulk molar density of the fluid mixture. 

Note that nv i represents the fraction of ci in the vapor phase. Thus, nv i = 1 or 0 indicates that 

component i stays in the vapor phase or liquid phase completely. 0

H. WANG: A Component-Based Eulerian-Lagrangian Formulation for Compositional 

Flow in Porous Media 

✬ 

✫ 

Figure 1: The ELLAM ((a)-(b)) and upwind ((c)-(d)) simulation at 5 and 20 years. 

(a) The overall mole fraction of Methane (b) The molar amount of Methane in vapor 

(c) The overall mole fraction of Methane (d) The molar amount of Methane in vapor 

Discussion The numerical results show that the Eulerian-Lagrangian formulation generates 

stable and accurate solutions (overall mole fractions) that have preserved physically reasonable 

moving steep fronts, even if a large time step of ∆tel = 1 year is used. With the (largest 

possible) fine time step ∆tup = 2 days, the upwind method generates qualitatively similar (but 

much more diffusive) overall mole fraction. We observe a similar comparison in terms of the 

normalized molar amounts in the vapor and liquid phases. 

It is instructive to look at the computational efficiency. Although the Eulerian-Lagrangian 

method uses more CPU time than the upwind method in solving the mass balance equations, 

both simulators have to perform the same computations on the pressure system and the thermodynamic 

flash calculations which consume a larger portion of the CPU time per iterative step 

at each time step. These results indicate that the Eulerian-Lagrangian simulator uses less than 

twice the CPU time an upwind simulator uses per time step. Therefore, the Eulerian-Lagrangian 

method generates accurate and stable solutions with steeper fronts using much less CPU time. 

References 

[1] H. Aziz and A. Settari, Petroleum Reservoir Simulation, Applied Science Publishers, 1979. 

[2] M.S. Espedal and R.E. Ewing, Characteristic Petrov-Galerkin sub-domain methods for twophase 

immiscible flow, Comput. Meth. Appl. Mech. Engrg., 64, (1987) 113–135. 

[3] R.E. Ewing, T.F. Russell, and M.F. Wheeler, Simulation of miscible displacement using 

mixed methods and a modified method of characteristics, SPE 12241, (1983), 71–81. 

[4] R. Helmig, Multiphase Flow and Transport Processes in the Subsurface, Springer Verlag, 

Berlin, 1997. 

[5] H. Wang, D. Liang, R.E. Ewing, S.L. Lyons, and G. Qin, An ELLAM-MFEM solution 

technique for compressible fluid flows in porous media with point sources and sinks, J. Comput. 

Phys., 159, (2000) 344–376. 

Speaker: WANG, H. 20 BAIL 2006 

✩ 

✪

Speaker: 

P. Houston, R. Hartmann 

Self-adaptive Methods for PDEs 

Many processes in science and engineering are formulated in terms of 

partial differential equations. Typically, for problems of practical interest, 

the underlying analytical solution exhibits localised phenomena 

such as boundary and interior layers and corner and edge singularities, 

for example, and their numerical approximation presents a challenging 

computational task. Indeed, in order to resolve such localised features, 

in an accurate and efficient manner, it is essential to exploit so-called 

self-adaptive methods. 

Such approaches are typically based on a posteriori error estimates for 

the underlying discretization method in terms of local quantities, such 

as local residuals, computed from the discrete solution. Over the last 

few years, there have been significant developments within this field in 

terms of both rigorous a posteriori error analysis, as well as the subsequent 

design of optimal meshes. In this minisymposium, a number of 

recent developments, such as the design of high-order and hp-adaptive 

finite element methods will be considered, as well as anisotropic mesh 

adaptation and mesh movement. 

• Ralf Hartmann: Discontinuous Galerkin methods for compressible flows: higher 

order accuracy, error estimation and adaptivity 

• Vincent Heuveline: On a new refinement strategy for adaptive hp finite element 

method 

• John Mackenzie: A Discontinuous Galerkin Moving Mesh Method for Hamilton- 

Jacobi Equations 

• Simona Perotto: Layer Capturing via Anisotropic Mesh Adaption 

• Rene Schneider: Anisotropic mesh adaption based on a posteriori estimates and 

optimisation of node positions

R. HARTMANN: Discontinuous Galerkin methods for compressible flows: higher order 

accuracy, error estimation and adaptivity 

✬ 

✫ 

Discontinuous Galerkin methods for compressible flows: 

higher order accuracy, error estimation and adaptivity 

Ralf Hartmann 

Institute of Aerodynamics and Flow Technology, German Aerospace Center (DLR), 

Lilienthalplatz 7, 38108 Braunschweig, Germany 

The Discontinuous Galerkin (DG) method for the compressible Euler equations is extended 

to the symmetric interior penalty (SIP)DG method for the compressible Navier-Stokes equations, 

see Hartmann & Houston [2]. Shock-capturing is used to avoid overshoots near shocks, see Hartmann 

[1]. The nonlinear equations are solved using a fully implicit solver. We demonstrate the 

accuracy of higher order DG discretizations with respect to the approximation of aerodynamical 

force coefficients and to the approximation of viscous boundary layers, see Figure 1 and 

Hartmann & Houston [2]. 

Finally, we demonstrate the use of a posteriori error estimation and goal-oriented (also 

called weighted-residual-based or adjoint-based) adaptive mesh refinement for subsonic flows, 

Hartmann & Houston [3], and for supersonic compressible flows, see Figure 2 and Hartmann [1]. 

References 

[1] R. Hartmann. Adaptive discontinuous Galerkin methods with shock-capturing for the compressible 

Navier-Stokes equations. Int. J. Numer. Meth. Fluids, 2006. To appear. 

[2] R. Hartmann and P. Houston. Symmetric interior penalty DG methods for the compressible 

Navier–Stokes equations I: Method formulation. Int. J. Num. Anal. Model., 3(1):1–20, 2006. 

[3] R. Hartmann and P. Houston. Symmetric interior penalty DG methods for the compressible 

Navier–Stokes equations II: Goal–oriented a posteriori error estimation. Int. J. Num. Anal. 

Model., 3(2):141–162, 2006. 

Speaker: HARTMANN, R. 22 BAIL 2006 

1 

✩ 

✪

R. HARTMANN: Discontinuous Galerkin methods for compressible flows: higher order 

accuracy, error estimation and adaptivity 

✬ 

✫ 

c dp 

0.024 

0.022 

0.02 

0.018 

0.016 

0.014 

0.012 

0.01 

0.008 

0.006 

reference cdp 

DG(3), global refinement 



10000 100000 

number of elements 

|c dp − 0.0222875| 

0.01 

0.001 

0.0001 

(a) (b) 




10000 100000 

number of elements 

Figure 1: Subsonic laminar flow around the NACA0012 airfoil at M = 0.5, α = 0 and Re = 5000: 

Convergence of the pressure induced drag cdp under global refinement for DG(p), p = 1,2,3: (a) 

cdp versus number of elements; (b) Error in cdp (reference cdp − cdp) versus number of elements. 

For more detail cf. Hartmann & Houston [2]. 

8 

4 

0 

-4 

-8 

-4 0 4 8 

8 

4 

0 

-4 

-8 

-4 0 4 8 

(a) (b) 

Figure 2: Supersonic laminar flow around the NACA0012 airfoil at M = 1.2, α = 0 and 

Re = 1000: (a) Residual-based refined mesh of 17670 elements with 282720 degrees of freedom 

and |Jcdp (u) − Jcdp (uh)| = 1.9 · 10 −3 ; (b) Adjoint-based refined mesh for cdp: mesh of 10038 

elements with 160608 degrees of freedom and |Jcdp (u) − Jcdp (uh)| = 1.6 · 10 −4 . For more detail 

cf. Hartmann [1]. 

Speaker: HARTMANN, R. 23 BAIL 2006 

2 

✩ 

✪

V. HEUVELINE: On a new refinement strategy for adaptive hp finite element method 

✬ 

✫ 

On a new refinement strategy for adaptive hp finite 

element method 

V. Heuveline 

University Karlsruhe (TH) 

Institute for Applied Mathematics II 

vincent.heuveline@math.uni-karlsruhe.de 

Abstract 

We consider finite element methods with varying meshsize h as well as varying 

polynomial degree p. Such methods have been proven to show exponentially fast 

convergence in some classes of partial differential equations if an adequate distribution 

of h− and p−refinement is chosen. In order to find hp−refinement strategies 

that show up automaticaly with optimal complexity, it is a first step to establish 

convergent adaptive algorithms. We develop a strategy that automatically construct 

a solution adapted approximation space by combining local h− and p− refinement 

and that can be proven for the 1d case to be convergent with a linear rate. This 

construction is based on an a posteriori error estimate with respect to the error 

in the energy norm. We then extend the proposed approach to the 2d and 3d 

case. Numerical experiments as well as implementation issues are considered in 

that framework. 

Speaker: HEUVELINE, V. 24 BAIL 2006 

1 

✩ 

✪

J.A. MACKENZIE, A. NICOLA: A Discontinuous Galerkin Moving Mesh Method for 

Hamilton-Jacobi Equations 

✬ 

✫ 

A Discontinuous Galerkin Moving Mesh Method for 


1 Introduction 

J.A. Mackenzie and A. Nicola 

Department of Mathematics 

University of Strathclyde 

26 Richmond St, Glasgow 

G1 1XH, U.K. 

jam@maths.strath.ac.uk 

Abstract 

In this talk we consider the adaptive numerical solution of Hamilton-Jacobi (HJ) equations 

φt + H(φx1, . . .,φxd ) = 0, φ(x, 0) = φ0(x), (1) 

where x = (x1, . . . , xd) ∈ IR d , t > 0. HJ equations arise in many practical areas such 

as differential games, mathematical finance, image enhancement and front propagation. It 

is well known that solutions of (1) are Lipschitz continuous but derivatives can become 

discontinuous even if the initial data is smooth. Since generalised solutions are not unique, a 

selection principle is required to pick out the physically relevant solution. For HJ equations 

the most commonly used condition is the vanishing viscosity condition which requires that the 

correct solution should be the vanishing viscosity limit of smooth solutions of corresponding 

viscous problems. The notion of viscosity solutions was introduced by Crandall and Lions 

[5], where the questions of existence, uniqueness and stability of solutions were addressed. 

Crandall and Lions were also the first to study numerical approximations of (1) and 

introduced the important class of monotone methods which were shown to converge to the 

viscosity solution [4]. However, monotonic schemes are well known to be at most first-order 

accurate. 

There is a close relation between HJ equations and hyperbolic conservation laws. With 

this in mind, it not surprising to find that many of the numerical methods used to solve HJ 

equations are motivated by conservative finite difference or finite volume methods for conservation 

laws. Methods that have been proposed include high-order essentially nonoscillatory 

(ENO) schemes, weighted ENO schemes and high resolution central schemes. 

An increasingly popular approach to solve hyperbolic conservation laws is the discontinuous 

Galerkin (DG) finite element method [2], [3]. Recently, Hu and Shu [7] proposed a DG 

method to solve HJ equations by first rewriting (1) as a system of conservation laws 

(wi)t + (H(w))xi = 0, i = 1, . . .,d, w(x, 0) = ∇φ0(x), (2) 

where w = ∇φ. The usual DG formulation would be obtained if w belonged to a space 

of piecewise polynomials. However, we note that wi, i = 1, . . .,d are not independent due 

to the restriction that w = ∇φ. In [7] a least squares procedure was used to enforce this 

condition. More recently it was shown that it is possible to enforce the gradient condition 

using a smaller solution space [12]. Theoretical analysis of the accuracy and stability of the 

method was performed in [11]. 

One of the often cited advantages of DG methods is that since the numerical solution is 

not continuous across inter-element boundaries then, in theory, this makes solution adaptive 

strategies much easier to implement. This has lead to the development of a number of 

adaptive methods based on hp-refinement strategies for hyperbolic conservation laws [1] [6]. 

1 

Speaker: MACKENZIE, J.A. 25 BAIL 2006 

✩ 

✪

J.A. MACKENZIE, A. NICOLA: A Discontinuous Galerkin Moving Mesh Method for 


✬ 

✫ 

An alternative adaptive strategy that has worked well for time dependent problems is 

to use moving meshes. A useful way to construct an adaptive moving mesh is to regard it 

as the image of a uniform mesh covering a computational domain under a time-dependent 

transformation that clusters mesh elements towards areas where improved spatial resolution 

is required. The transformation is often found through a variational formulation where the 

mapping is the minimiser of a functional involving properties of the mesh and the solution 

(see e.g. [10]). To improve the stability and smooth the evolution of the moving mesh 

Russell, Huang and coworkers [9] [8] found it useful to obtain the mapping as the solution of 

parabolic moving mesh PDEs which are modified gradient flow equations for the minimisation 

of a suitable mesh functional. 

For problems where solution discontinuities exist the correct choice of an adaptivity 

criterion, or monitor function, is problematic. It is not unusual to find in the literature that 

many grid adaptation criteria are singular at solution discontinuities. To prevent singularities 

producing degenerate meshes either some form of smoothing procedure is employed or a 

regularised functional is used in the variational formulation. 

The aim of this talk is to consider the use of the DG method of Hu and Shu [7] to solve 

HJ equation using a moving mesh method based on the solution of MMPDEs. The governing 

equation are transformed to include the effect of the movement of the mesh and this is done 

in such a way that the conservation properties of the original equation are not lost. The 

adaptive mesh is driven by a monitor function which is shown to be non-singular in the 

presence of solution discontinuities. To produce an acceptable mesh we smooth the monitor 

function before it is used to drive the adaptive procedure. Numerical examples in one and 

two dimensions are presented to demonstrate the effectiveness of the adaptive procedure.. 

References 

[1] K.S. Bey and J.T. Oden. hp-version discontinuous Galerkin methods for hyperbolic 

conservation laws. Comput. Methods Appl. Mech. Engrg., 133:259–286, 1996. 

[2] B. Cockburn, G.E. Karniadakis, and C.W. Shu. Discontinuous Galerkin methods: Theory, 

Computation and Applications. Springer, 2000. 

[3] B. Cockburn and C.W. Shu. The Runge-Kutta discontinuous Galerkin finite element 

method for conservation laws V: Multidimensional systems. Journal of Computational 

Physics, 141:199–224, 1998. 

[4] M.G. Crandall and P.L. Lions. Two Approximations of Solutions of Hamilton-Jacobi 

Equations. Mathematics of Computation, 43(167):1–19, 1984. 

[5] M.J. Crandall and P.L. Lions. Viscosity Solutions of Hamilton-Jacobi Equations. Transactions 

of the American Mathematical Society, 277(1):1–42, 1983. 

[6] P. Houston, B. Senior, and E. Suli. hp-discontinuous Galerkin finite element methods 

for hyperbolic problems: analysis and adaptivity. Int. J. Numer. Methods in Fluids, 

40:153–169, 2002. 

[7] C. Hu and C.W. Shu. A Discontinuous Galerkin Finite Element Method for Hamilton- 

Jacobi Equations. SIAM Journal on Scientific Computing, 21(2):666–690, 1999. 

[8] W. Huang. Practical aspects of formulation and solution of moving mesh partial differential 

equations. Journal of Computational Physics, 171:753–775, 2001. 

[9] W. Huang and R. D. Russell. Moving mesh strategy based upon a gradient flow equation 

for two-dimensional problems. SIAM Journal on Scientific Computing, 20:998–1015, 

1999. 

[10] P. Knupp and S. Steinberg. Foundations of Grid Generation. CRC Press, Boca Raton, 

1994. 

[11] O. Lepsky, C. Hu, and C. W. Shu. Analysis of the discontinuous Galerkin method for 

Hamilton-Jacobi equations. Appl. Numer. Math., 33:423–434, 2000. 

[12] F. Li and C.W. Shu. Reinterpretation and simplified implementation of a discontinuous 

Galerkin method for Hamilton-Jacobi equations. Applied Mathematics Letters, 18:1204– 

1209, 2005. 

Speaker: MACKENZIE, J.A. 26 BAIL 2006 

✩ 

✪

S. PEROTTO: Layer Capturing via Anisotropic Mesh Adaption 

✬ 

✫ 

Layer Capturing via Anisotropic Mesh Adaption 

Simona Perotto 

MOX - Dipartimento di Matematica “F. Brioschi” 

Politecnico di Milano, Via Bonardi 9 

I-20133 Milano, Italy 

simona.perotto@mate.polimi.it 

The numerical approximation of many problems in Computational Fluid Dynamics (CFD) leads 

often to deal with solutions exhibiting directional features, namely great variations along certain 

directions and less significant changes along the other ones. This is the case of boundary and 

internal layers typical, for instance, of the advection-diffusion and Navier-Stokes equations as 

well as of shocks in the case of Euler equations. In view of an efficient numerical approach to 

this kind of problems, it turns out advisable to resort to a suitably oriented and deformed (i.e., 

anisotropic) computational mesh, matching the local directional features of the solution. For 

a fixed solution accuracy, a considerable reduction of the number of the degrees of freedom is 

shown in the presence of an anisotropic grid, besides a sharper capture of the solution features. 

In this regard, we have introduced a theoretically sound anisotropic framework moving from 

the spectral properties of the standard affine map between the reference and the general mesh 

element [2]. As first step, we have derived suitable anisotropic interpolation error estimates for 

piecewise linear finite elements. Then, in view of an anisotropic mesh adaption driven by an a 

posteriori error estimator, these anisotropic estimates have been merged with the standard dualbased 

approach proposed by R. Becker and R. Rannacher in [1]. Thus the final outcome of our 

analysis consists of an automatic tool able to properly orient and stretch the mesh elements so 

that a goal-functional of the exact solution, representing a quantity of interest, is approximated 

within a user-defined tolerance [3, 4]. 

In this communication the leading ideas of our anisotropic framework are presented together 

with some numerical results associated with the goal-oriented a posteriori analysis. 

References 

[1] R. Becker and R. Rannacher, “A feed-back approach to error control in finite element 

methods: basic analysis and examples”, East-West J. Numer. Math., 4(4), 237–264 (1996). 

[2] L. Formaggia and S. Perotto, “New anisotropic a priori error estimates”, Numer. Math., 

89, 641–667 (2001). 

[3] L. Formaggia and S. Perotto, “Anisotropic error estimates for elliptic problems”, Numer. 

Math., 94, 67–92 (2003). 

[4] L. Formaggia, S. Micheletti and S. Perotto, “Anisotropic mesh adaptation in Computational 

Fluid Dynamics: application to the advection-diffusion-reaction and the Stokes problems.”, 

Appl. Numer. Math., 51 (4), 511–533 (2004). 

1 

Speaker: PEROTTA, S. 27 BAIL 2006 

✩ 

✪

R. SCHNEIDER, P. JIMACK: Anisotropic mesh adaption based on a posteriori estimates 

and optimisation of node positions 

✬ 

✫ 

Anisotropic mesh adaption based on a posteriori estimates and 

optimisation of node positions 

Abstract 

Introduction 

René Schneider ∗ & Peter Jimack † 

Efficient numerical approximation of solution features like boundary or interior layers by means 

of the finite element method requires the use of layer adapted meshes. Anisotropic meshes, like 

for example Shishkin meshes, allow the most efficient approximation of these highly anisotropic 

solution features. However, application of this approach relies on a priori analysis on the thickness, 

position and stretching direction of the layers. If it is impossible to obtain this information 

a priori, as it is often the case for problems with interior layers of unknown position for example, 

automatic mesh adaption based on a posteriori error estimates or error indicators is essential in 

order to obtain efficient numerical approximations. 

Historically the majority of work on automatic mesh adaption used locally uniform refinement, 

splitting each element into smaller elements of similar shape. This procedure is clearly not 

suitable to produce anisotropically refined meshes. The resulting meshes are over-refined in at 

least one spatial direction, rendering the approach far less efficient than that of the anisotropic 

meshes based on a priori analysis. 

Automatic anisotropic mesh adaption is an area of active research, e.g. [2, 1]. Here we present 

a new approach to this problem, based upon using not only an a posteriori error estimate to 

guide the mesh refinement, but its sensitivities with respect to the positions of the nodes in the 

mesh as well. 

Outline of the approach 

The underlying idea is to use techniques from mathematical optimisation to minimise the estimated 

error by moving the positions of the nodes in the mesh appropriately. This basic idea is 

of course not new, but the approach taken to realise it is. 

A key ingredient is the utilisation of the discrete adjoint technique to evaluate the sensitivities 

of an error estimate J = J(uh(s),s) with respect to the node positions s, where uh = uh(s) 

denotes the solution of the discretised PDE, R(uh,s) = 0, which depends upon the node positions 

s. The sensitivities 

� ∂R 

∂uh 

DJ ∂J ∂uh ∂J 

= + 

Ds ∂uh ∂s ∂s 

� T 

∗ Chemnitz University of Technology, Germany 

† School of Computing, University of Leeds, UK 

= ∂J 

∂s 

∂R 

− ΨT , (1) 

∂s 

Ψ = ∂J 

, (2) 

∂uh 

Speaker: SCHNEIDER, R. 28 BAIL 2006 

✩ 

✪

R. SCHNEIDER, P. JIMACK: Anisotropic mesh adaption based on a posteriori estimates 

and optimisation of node positions 

✬ 

✫ 

y 

1 

0.8 

0.6 

0.4 

0.2 

0 

−0.2 

−0.4 

−0.6 

−0.8 

−1 

initial mesh 

−1 −0.5 0 

x 

0.5 1 

y 

1 

0.8 

0.6 

0.4 

0.2 

0 

−0.2 

−0.4 

−0.6 

−0.8 

−1 

final mesh for eps=1.0e−03 

−1 −0.5 0 

x 

0.5 1 

(a) (b) 

Figure 1: Initial (a) and adapted (b) meshes for model problem (3) with a boundary layer around 

the square hole 

are thereby evaluated according to (1), utilising the adjoint solution Ψ which is defined by (2). 

This way, DJ/Ds can be evaluated without computing ∂uh/∂s first, reducing the number of 

equation systems to be solved from O(dim(s)) to just two, independent of dim(s). As the 

number of nodes can easily be larger than one hundred even in extremely coarse meshes (if the 

domain geometry is complicated), this approach is of fundamental importance to make gradient 

based optimisation methods feasible for this type of problem. Fast optimisation algorithms like 

BFGS-type methods can be applied to obtain significant reductions in the error estimate J after 

a few optimisation steps. The aim of this procedure is to provide a mesh with problem adapted 

anisotropic elements, which may then be used as a good basis for further locally uniform adaptive 

mesh refinement. 

Numerical results 

To demonstrate feasibility of the approach it is applied to a number of model problems. For the 

purpose of this abstract we consider a reaction diffusion equation, 

−∆u + 1 

ε2u = 1 

ε2 in Ω := (−1,1) 2 \ (−1 1 

5 , 5 )2 

subject to u = 0 on ΓD := � −1 ⎫ 

⎪⎬ 

� 

1 2 � � 

1 1 2 

5 , 5 \ −5 , (3) 

5 

⎪⎭ 

∂u 

∂n = 0 on ΓN := [−1,1] 2 \ (−1,1) 2 . 

Figure 1 shows an initial mesh for this problem and an adapted one for ε = 10 −3 . Concentration 

of the elements in the boundary layer which forms around the hole at the centre of the domain is 

clearly visible, and significantly increased aspect ratios in the boundary layer may be observed. 

For more detail on the approach and the example we refer to [3]. 

References 

[1] L. Formaggia, S. Micheletti, and S. Perotto. Anisotropic mesh adaptation in computational fluid dynamics: 

Application to the advection-diffusion-reaction and the Stokes problems. Applied Numerical 

Mathematics, 51(4):511–533, December 2004. 

[2] G. Kunert. Toward anisotropic mesh construction and error estimation in the finite element method. 

Numerical Methods for Partial Differential Equations, 18(5):625–648, 2002. 

[3] R. Schneider and P.K. Jimack. Toward anisotropic mesh adaption based upon sensitivity of a posteriori 

estimates. School of Computing Research Report Series 2005.03, University of Leeds, http://www. 

comp.leeds.ac.uk/research/pubs/reports/2005/2005_03.pdf, 2005. 

Speaker: SCHNEIDER, R. 29 BAIL 2006 

✩ 

✪

Speaker: 

Debopam Das, Tapan Sengupta: 

Asymptotic Methods for Laminar and Turbulent boundary Layers 

The minisymposium will have contributions dealing with laminar/turbulent 

flows- their morphology and structures. Specifically, results 

would be presented that discuss the very concept of turbulent 

boundary layers. Also, detailed flow structures arising in mixed convection 

problem involving velocity and thermal boundary/ interior layers 

will be a focus of another contribution. This session will also have 

contributions dealing specifically with temporal and/or spatio-temporal 

growth of waves in boundary/ interior layers. For example, existence 

of spatio-temporal growth of waves in boundary layer is established for 

the first time that is related non-orthogonal modes proposed earlier for 

Couette and channel flows. The focus will be on receptivity and stability 

of equilibrium steady flows as well as unsteady bi-directional pipe 

flows that identifies non-axisymmetric modes. 

• Matthias H. Buschman: Turbulent Boundary Layers: Reality and Myth (Key-Note 

Lecture) 

• Debopam Das: Three-dimensional Temporal Instability of Unsteady Pipe Flow. 

• Venkatesa I. Vasanta Ram: The structure of the critical layer of a swirling annular 

flow in transition. 

• T.K. Sengupta: Spatio-temporal growing waves in boundary-layers by Bromwich 

contour integral method 

• Herbert Steinrueck: Asymptotic Analysis of the mixed convection flow past a horizontal 

plate near the trailing edge.

M.H. BUSCHMANN, M. GAD-EL-HAK: Turbulent Boundary Layers: Reality and 

Myth 

✬ 

International Conference on Boundary and Interior Layers—Computational & Asymptotic Methods 

(BAIL 2006) 

Abstract of keynote lecture for the 

Minisymposium on Asymptotic Methods for Laminar and Turbulent Boundary Layers 

Turbulent Boundary Layers: Reality and Myth 

Matthias H. Buschmann 

Privatdozent, Institut für Strömungsmechanik, Technische Universität Dresden, Dresden, Germany 

Mohamed Gad-el-Hak 

Caudill Professor and Chair of Mechanical Engineering, 

Virginia Commonwealth University, Richmond, VA 23284-3015, USA 

Hundred years after Ludwig Prandtl’s fundamental lecture on boundary layer theory, 1 the meanvelocity 

profile and the shear-stress distribution of the seemingly simplest case of wall-bounded 

flow, the zero-pressure-gradient turbulent boundary layer (ZPG TBL), still appears to be terra 

incognita. Even less is known about confined and semi-confined flows undergoing pressure 

gradients, such as pipe and channel flows and wall-bounded flows approaching pressure-driven 

separation. The problem is of course related to the lack of analytical solutions to the 

instantaneous, nonlinear Navier–Stokes equations that govern the stochastic dependent variables 

of almost all turbulent flows. What little we know about turbulence comes from experiments and 

heuristic modeling, not first-principles solutions. (Direct numerical simulations provide firstprinciples 

integration of the instantaneous Navier–Stokes equations, but are at present limited to 

modest Reynolds numbers and simple geometries.) 

Consider a two-dimensional, isothermal, incompressible, steady flow over a body of length L. 

The fluid may have constant density ρ and constant dynamic viscosity µ. Assuming that the 

characteristic velocity is U, we write the Reynolds number as 

Re= !UL µ (1) 

It was Prandtl’s genius that discovered that, at sufficiently high Reynolds number, a thin shear 

layer exists close to the body. The thickness of this layer δ is much smaller than L. Due to the 

strong streamwise velocity gradient normal to the wall, even a small viscosity such as that for air 

or water can cause considerable viscous shear stress, ! = µ "u "y . Originally Prandtl called this 

layer therefore Reibungsschicht, which literally translates to friction layer. 

Introducing the transformation 

! ! ! ! ! $ 

( x,y 

,u ,v , p ) ! & L x,Re 

% 

" 1 

2 L y,U u,Re " 1 

' 

2 2 

U v,#U p) 

( 

1 Prandtl, L., “Über Flüssigkeitsbewegung bei sehr kleiner Reibung,” Proc. Third Int. Math. Cong., pp. 484–491, 

Heidelberg, Germany, 1904. 

✫ 

M. H. Buschmann & M. Gad-el-Hak, Abstract for BAIL 2006 

Speaker: BUSCHMANN, M.H. 32 BAIL 2006 

(2) 

✩ 

✪


Myth 

✬ 

✫ 

into the Navier-Stokes equations and taking the limit Re → ∞, leads to Prandtl’s boundary layer 

equations 

The boundary conditions are 

uu x + vu y = ! p x + u yy ; 

y = 0 : 

u = v = 0 and 

0 = p y ; 

y ! " : 

The task is now to find physically appropriate solution for eqn. (3). 

ux + v = 0 (3) 

y 

u ! U e x ( ) (4) 

What goals do we have when solving eqn. (3)? One of the main objectives is to find self-similar 

solutions. Boundary layers are self-similar when normalization can be found so that data of 

different physical realizations (e.g., experiments in different wind tunnels, profiles at different 

downstream positions within one experiment) can be collapsed within one single curve. 

Examples are the mean velocity profiles of the Blasius’ laminar zero-pressure-gradient boundary 

layer and the fully-developed turbulent flow in pipes. 

y 

u( x, y ) 

u( x, y ) 

x 1 

3 x 

x 2 

Which physical problems do we face when solving eqn. (3)? We neither know if for a certain 

type of wall-bounded flow a transformation as searched for in the first question exists in general, 

nor do we know what the transformation parameters are. The physically appropriate 

transformation is a non-dimensionalization that is much more than simply changing the 

coordinates. The crucial issue is top choose the scaling based on the physics of the problem. At a 

minimum, the scale basis has to satisfy two criteria. It should consist of characteristic 

parameters and represent problem-intrinsic scales. The foundation of dimensional analysis is the 

Π-theorem formulated by Buckingham. 2 

f ( x 1 ,x 2 ,...xn ) 0 

Here xi denote the n variables of the system having m dimensions and Δi are the non-dimensional 

similarity parameters of the problem. 

2 Buckingham, E., “On physically similar systems: illustrations of the use of dimensional equations,” Phys. Rev., 2. 

Ser., vol. 4, pp. 1119–1126, 1914. 

u( x, y ) 

Π-theorem 

y 

! 

Find proper 

scaling parameters 

Δ and U 

= ( 1 2 n m) 

u( x, y) 

U 

F ! , ! ,... ! " = 0 



✩ 

✪


Myth 

✬ 

Which mathematical techniques do we have to solve eqn. (3)? Basically we have analytical, 

numerical, and asymptotic methods and combination of those. 3 In this paper, we will focus on 

recent advances in analytical and asymptotic approaches. 

During the mid 1990s, a new debate on the aforementioned subject arose. Caused by new 

unconventional approaches questioning one of the cornerstones of modern fluid mechanics—the 

logarithmic law of turbulent boundary layers—several new scalings were developed. In 

conjunction with these theoretical investigations, high-quality experiments in zero-pressuregradient 

turbulent boundary layers and turbulent pipe and channel flows were undertaken. In 

general, the physical picture of wall-bounded flow is now much more complex than was thought 

a decade ago. However, the physical picture seems to be also more controversial than ever 

before. Which of these new approaches will survive and contribute substantially to fluid 

mechanics in the future is still open. 

The present talk will discuss four main schools of thought, which can be summarized as follows: 

(1) Standard Logarithmic Overlap Layer 

Above a certain critical Reynolds number, a pure logarithmic region exists in the 

mean-velocity profile of ZPG TBL and channel and pipe flows. The parameters of 

this logarithmic law are completely Reynolds number invariant. 

(2) Power Law Based On Similarity Assumption 

An inner and an outer power law describe the overlap layer of ZPG TBL and 

channel and pipe flows. Both laws exhibit Reynolds number dependent 

parameters, and never achieve an asymptotic state. 

(3) Asymptotic Invariance Principle 

Full similarity solutions have to be searched separately for the inner and outer 

equations. Because in the limit the outer boundary layer equations are independent 

of Reynolds number, the properly scaled profiles for the outer region must also be 

independent of Reynolds number. 

(4) Higher-Order Asymptotic Matching 

Based on asymptotic matching, a higher-order approach with respect to Reynolds 

number and the wall-normal coordinate can be derived. In the limit of infinite 

Reynolds number, the classical logarithmic law is recovered. At finite Re, 

however, the similarity laws for both the mean and higher-order statistics are 

Reynolds-number dependent. 

The presentation will summarize, analyze and critique the recent theoretical and experimental 

investigations of a variety wall-bounded flows. Based on that analysis, open issues in the field 

will be highlighted. 

3 Gersten, K., and Herwig, H., Strömungsmechanik, Fundamentals and Advances in the Engineering Science, Verlag 

Vieweg, 1992. 

✫ 

Analytical 

methods 

Asymptotic 

methods 

Numerical 

methods 



✩ 

✪

A. NAYAK, D. DAS: Three-dimesnional Temporal Instability of Unsteady Pipe Flow 

✬ 

✫ 

Three-dimensional Temporal Instability of Unsteady Pipe Flow 

By 

Avinash Nayak and Debopam Das 

Department of Aerospace Engineering 

Indian Institute of Technology Kanpur, India 

Introduction: 

In this paper we present temporal three-dimensional linear stability analysis of unsteady 

bi-directional laminar flow in a duct. For a known but arbitrary volume flow rate, analytical 

solution of one-dimensional unsteady flow through a pipe has been obtained by Das & Arakeri 

(1998, 2000). Generalized form of this solution and the solution for unsteady flow through 

annular space between two concentric pipes has been obtained by Nayak (2005). Prior to this 

study, solutions that are available in the literature were for known pressure gradient (Womersley 

1955) and Uchida 1956). Das (1998) and Das & Arakeri (1998) have observed the transitional 

nature of this flow in a pipe and channel where the velocity profiles are inflectional and have 

reverse flow near wall. In their experiments, for flows with average volume flow rate varying like 

a trapezoidal function with time, shows that both axisymmetric and non-axisymmetric mode of 

disturbance grows. In this paper through linear stability analysis of three dimensional disturbances 

some of the observed experimental facts are explained such as growth of helical modes for certain 

cases. 

Linear stability analysis: 

The disturbance stream-function is assumed as 

{ u , v, 

w, 

p} 

= { u( 

r) 

, v( 

r ) , w( 

r ) , p( 

r ) } exp[ 

iα 

( z − ct) 

+ inθ 

] . (1) 

Here, u, v and w are radial, azimuthal and axial perturbation velocity components respectively and 

p is the fluctuating pressure. The disturbance quantities are substituted in perturbed Navier-Stokes 

equations in cylindrical coordinates and the governing equations after simplifications are, 

2 

1 ⎡1+ 

n 

⎤ 2in 

1 ⎡ iv 2 3 3 3 ⎤ 

u′ 

′ + u′ 

− ⎢ + α + iα 

Re( W − c) 

− − 

2 

⎥u 

v 2 2 ⎢u 

+ u′ 

′ 

− u′ 

′ + u′ 

− u 

2 3 4 

r 

⎥ 

⎣ r 

⎦ r α ⎣ r r r r ⎦ 

2 

in ⎡1 

2 3 3 ⎤ ⎡n 

2 

⎤⎧ 

1 ⎡ 1 1 ⎤ in ⎡1 

1 ⎤⎫ 

− 

Re( ) 

2 ⎢ v′ 

′ 

− v′ 

′ + v′ 

− v 

2 3 4 ⎥ + ⎢ + α + iα 

W − c 

2 

⎥⎨ 

+ 

2 ⎢u′ 

′ + u′ 

− u 2 ⎥ 2 ⎢ v′ 

− v 2 ⎥⎬ 

α ⎣r 

r r r ⎦ ⎣ r 

⎦⎩α 

⎣ r r ⎦ α ⎣r 

r ⎦⎭ 

2 ⎡ 2n 

⎤⎧ 

1 ⎡ 1 ⎤ in ⎡1 

⎤⎫ 

i i 

+ ⎢− 

+ iα 

ReW 

′ 

− Re ′ ′ − Re ′ 

= 0 − − − − − −( 

1) 

3 

⎥⎨ 

+ 

2 ⎢u′ 

+ u⎥ 

2 ⎢ v⎥⎬ 

u W uW 

⎣ r 

⎦⎩α 

⎣ r ⎦ α ⎣r 

⎦⎭ 

α α 

2 

1 ⎡1+ 

n 2 

v′′ 

+ v′ 

−⎢ 

+ α + iα 

Re 

2 

r ⎣ r 

2 

in⎡n 

2 

+ ⎢ + α + iαRe 

2 

r ⎣r 

( W −c) 

2 

⎤ 2in 

in ⎡1 

2 1 1 ⎤ n ⎡ 1 1 1 ⎤ 

⎥v 

+ u − 2 2 ⎢ u ′′′ + u′′ 

− u′ 

+ u + 

2 3 4 ⎥ 2 ⎢ v′ 

− v′ 

+ v 

2 3 4 ⎥ 

⎦ r α ⎣r 

r r r ⎦ α ⎣r 

r r ⎦ 

⎤⎧ 

1 ⎡ 1 ⎤ in ⎡1 

⎤⎫ 

n ⎛1 

⎞ 

⎥⎨ 

⎜ ⎟ 

2 ⎢ 

+ 

⎥ 2 ⎢ ⎥⎬ 

⎦⎩α 

⎣ r ⎦ α ⎣r 

⎦⎭ 

α ⎝r 

⎠ 

( W −c) 

u′ 

u + v + Re u W′ 

= 0− 

−− 

−− 

−−( 

2) 

These equations are solved using the finite difference technique and it is written in the form 

of ⎧u 

⎫ 

( [ ] [ ] ) 

. The complex eigenvalues c , are obtained for a particular value of α and 

A − c B = 0 

⎨ ⎬ 

⎩ v ⎭ 

Re, using MATLAB. In these two sets of equations u has the highest 4 th derivative while has 

v 

rd 

the highest 3 derivative. Thus 7 boundary conditions are required to solve these two coupled 

equations. Following are the boundary conditions given for different values of n . 

n 

0 

R=0 

(center-line) 

R=1 

(at wall) 

u = 0 

u = 0 

v = 0 

v = 0 

u ′ = 0 

u ′ = 0 

1 

u + iv = 0 u ′ = 0 v ′ = 0 

−1 

lim ∝ n 

u r 

R=0 

(center-line) 

Speaker: DAS, D. 35 BAIL 2006 

r→0 

✩ 

✪

A. NAYAK, D. DAS: Three-dimesnional Temporal Instability of Unsteady Pipe Flow 

✬ 

✫ 

2 

>2 

R=1 

(at wall) 

R=0 

(center-line) 

R=1 

(at wall) 

R=0 

(center-line) 

R=1 

(at wall) 

u = 0 v = 0 u ′ = 0 

u = 0 v = 0 u ′ + iv′ 

= 0 3 u ′′ + 2iv 

′′ = 0 

u = 0 v = 0 u ′ = 0 

u = 0 v = 0 u ′ = 0 v ′ = 0 

u = 0 v = 0 u ′ = 0 

Table. 1 Boundary conditions for perturbation at wall and the centerline 

For n=0, the two equations decouple to give a fourth order equation in u and a second order 

n−1 

equation in v . For n=1, the seventh condition is u ∝ r when limr → 0 (Batchelor & Gill 

1962). 

4.4 Results and Discussions 

We have considered a case (case2 of Das & Arakeri 1998) in which a perceptible wave was 

observed through flow visualization at time t= t . During this period the piston has stopped after a 

p 

( 2 

trapezoidal motion and the profile considered for analysis is at time 

1 

t t2 

t p t 

4 

− + = ) where, t2 

is 

the time when piston has stopped. The neutral stability curves are shown in figure 1. It is observed 

that critical Reynolds number for n=1 mode is minimum (=398). Figure 2 shows the 

corresponding flow visualization picture of Das & Arakeri (1998). The position of vortex wave at 

0 

90 phases between top and bottom portion of pipe is visible in figure 2. Hence the helical mode 

with n=1 is most unstable in this case. As the neutral curve for n=0 mode is close to n=1 mode for 

some cases n=0 mode might be most unstable mode. Symmetric modes has been observed as most 

unstable in other cases experimentally (Das & Arakeri 1998 case 3 ). The stability analysis will be 

shown in the final paper for these cases, but the neutral curves (figure1) itself indicate the 

possibility of growth of such axisymmetric modes. The calculated wave length is λ 

≈ 3. 

22 

δ 

matches with the wavelength observed in experiment, the value of which is approximately 3.0. 

alpha (α) 

8 

7 

6 

5 

4 

3 

2 

Neutral Stability Curve for n 0, 1 & 2 

n=0 

n=1 

n=2 

1 

200 300 400 500 600 700 800 900 1000 1100 1200 

Reynolds number (Re) 

Fig.1 Neutral Stability Curve for different n Fig. 2 Flow viz. picture of Das & Arakeri (case2) 

REFERENCES 

1. Batchelor, G. K. & Gill, A. E. 1962. J. Fluid Mech. 14, 529. 

2. Das, D. 1998 Ph.D Thesis, Department of Mechanical Engg, Indian Institute of Science, Bangalore. 

3. Das, D. & Arakeri, J. H. 1998. J. Fluid Mech. 374, 251-283. 

4. Das, D. & Arakeri, J. H. 2000 Unsteady Jl. Applied Mech. 67, 274-281. 

5. Nayak, A. 2005 MTech report Dept of Aerospace Engg, Indian Institute of Technology, Kanpur. 

6. Uchida, S. 1956 Z. Angrew. Math. Phys. 7, 403-422. 

7. Weinbaum, S. & Parker, K. 1975. J. Fluid Mech. 69, 729-752. 

Speaker: DAS, D. 36 BAIL 2006 

✩ 

✪

J. HUSSONG, N. BLEIER, V.V. RAM: The structure of the critical layer of a swirling 

annular flow in transition 

✬ 

✫ 

The structure of the critical layer of a swirling annular 

flow in transition 

J. Hussong, N. Bleier and V. Vasanta Ram 

Ruhr-Universität Bochum, D-44780 Bochum, Germany 

Subject of our paper falls under the heading of transition of the swirling flow 

in an annulus. Two transition mechanisms with some fundamental differences, 

which we call the Taylor and the Tollmien-Schlichting mechanisms, are generally 

in competition with each other in this flow. The salient difference referred to 

is the presence of a critical layer in the Tollmien-Schlichting mechanism. In 

contrast, such a critical layer is absent in the Taylor mechanism. The focus of 

attention in our work is the modification swirl causes to the structure of this 

critical layer in annular flow. 

The characteristic feature of the critical layer is that the propagation velocity 

of the disturbance wave is the same as the local velocity in the basic flow. 

Mathematically, this results in the equations for the propagation of small amplitude 

disturbances exhibiting a turning point, i.e. the coefficient of a crucial 

term in the governing differential equations crossing a zero value (see eg. [5], 

[6], [2], [3], [7]). For this reason viscous effects gain importance in the critical 

layer which may be regarded as an internal layer in the flow. The viscous effects 

have to be taken properly into account in the critical layer in order to arrive at 

the stability characteristics of the flow of interest undergoing transition. 

Our basic flow is the fully developed flow with swirl in the uniform annular 

gap between concentric circular cylinders. In the flow in this geometrical configuration, 

swirl comes into existence when the axial pressure gradient driving 

the flow acts in conjunction with a rotation of a cylinder about its own axis. 

We restrict our attention for the present to the case when the outer cylinder is 

set in rotation at an angular velocity Ωa and the inner cylinder is pulled axially 

at a translational velocity Vwi. Both of these flow boundary conditions tend to 

raise the critical Reynolds number of the flow. The geometrical and flow pa- 

rameters in our problem are, in a self-explanatory notation: the transverse curvature 

parameter ɛR = Ra−Ri 

Uref x (Ra−Ri) 

, the Reynolds number Re = Ra+Ri 2 ν with 

Uref x = (Ra−Ri)2 � � 

� dPG 

Uref �, 

ϕ 

8µ dx the swirl parameter Sa = Uref x with Uref ϕ = ΩaRa and 

the translational wall velocity parameter Twi = Vwi 

Uref x . 

We have approached the transition problem in this flow configuration by 

conducting a modal analysis of the dynamics of the propagation of disturbances 

in the basic flow in question, starting from small amplitude disturbances for 

which the governing equations may be linearised. The dispersion relationship 

for this linearized problem may formally be written as 

F(ɛR, Re, Sa, Twi, λx, nϕ, ω) = 0, 

Speaker: RAM, V.V. 37 BAIL 2006 

✩ 

✪

J. HUSSONG, N. BLEIER, V.V. RAM: The structure of the critical layer of a swirling 

annular flow in transition 

✬ 

✫ 

where ω is the frequency, λx the disturbance wave number in the axial direction 

and nϕ the corresponding quantity in azimuthal direction which has to be an 

integer. nϕ = 0 represents an axisymmetric disturbance whereas nϕ �= 0 corresponds 

to helical disturbances. We have solved the linearised equations for 

the disturbance numerically by the spectral collocation method using Matlab 

over a range of parameters ɛR, Sa and Twi to determine the dependence of the 

critical Reynolds number on the parameters Sa and Twi from which the wave 

velocity and the location of the critical layer at this Reynolds number have been 

obtained. A sample of the computational results is presented in Fig. 1. 

c = omega / lambda x 

0.3 

0.29 

0.28 

0.27 

0.26 

0 0.1 0.2 

S 

a 

0.3 0.4 

c = omega / lambda x 

0.3 

0.28 

0.26 

0.24 

0.22 

0 0.1 0.2 

T 

wi 

0.3 0.4 

Figure 1: wave velocity for Twi = 0.1 (left) and Sa = 0.1 (right) 

An asymptotic analysis of the critical layer, conducted along the same lines 

as for the corresponding case in planar flow (see eg. [1], [4]), brings out the following 

points as the outstanding features of the critical layer under the influence 

of swirl: 

• Swirl exerts no influence on the scale of thickness of the critical layer 

through its axisymmetric disturbance mode, nϕ = 0 

• For the nonaxisymmetric modes, nϕ �= 0, the thickness of the critical 

layer retains the same scale as in the case of classical planar flow , viz. 

1 − O(λxRe) 3 , as long as the product of the transverse curvature and swirl 

1 − parameters, ɛRSa, remains small to within O(Re 3 ) 

Results obtained from this asymptotic analysis of the critical layer are set 

against the computational results outlined above. 

References 

[1] Drazin, Reid, Hydrodynamic stability, Cambridge Univ. Pr., 1982. 

[2] Hinch, E. J., Perturbation Methods, Cambridge Univ. Pr., 1992. 

[3] Holmes, M. H., Introduction to Perturbation Methods, 20 in the Series Texts in Applied 

mathematics, Springer, 1995. 

[4] Maslowe, S. A., Critical layers in shear flows. In Annual Review of Fluid Mechanics, 18, 

1986, 405-432. 

[5] Nayfeh, A. H., Perturbation Methods, John Wiley, 1973. 

[6] O Malley Jr., R. E., Introduction to Singular Perturbations, 14 in the Series Applied Mathematics 

and Mechanics, Academic Press, 1974. 

[7] Verhulst, F., Methods and Applications of Singular Perturbations, 50 in the Series Texts 

in Applied mathematics, Springer, 2000. 

2 

Speaker: RAM, V.V. 38 BAIL 2006 

✩ 

✪

T.K. SENGUPTA, A. KAMESWARA RAO: Spatio-temporal growing waves in 

boundary-layers by Bromwich contour integral method 

✬ 

✫ 

��Ô�ÖØÑ�ÒØÓ��ÖÓ×Ô��Ò��Ò��Ö�Ò�ÁÁÌÃ�ÒÔÙÖÍÈ ËÔ�Ø�ÓØ�ÑÔÓÖ�Ð�ÖÓÛ�Ò�Û�Ú�×�Ò�ÓÙÒ��ÖÝÐ�Ý�Ö×�Ý�ÖÓÑÛ�� ÌÃË�Ò�ÙÔØ��Ò��Ã�Ñ�×Û�Ö�Ê�Ó ÓÒØÓÙÖ�ÒØ��Ö�ÐÑ�Ø�Ó� 

ÔÐ�Ý×Ô�Ø��Ð�ÖÓÛØ��Ú�ÒÛ��ÒØ��×�×ÔÓ×��×�×Ô��Ø�Ñ��Ô�Ò��ÒØÔÖÓ�Ð�Ñ ÁØ�×Û�ÐÐ�ÒÓÛÒØ��ØÙÒ×Ø��Ð�Þ�ÖÓÔÖ�××ÙÖ��Ö��ÒØ�ÓÙÒ��ÖÝÐ�Ý�Ö��× ÁÒ�� 

�℄�ÓÖ×ÓÑ��ÒØ�ÖÒ�ÐÓÛ×�Ø��×Ó�Ø�Ò��ÒÓÒ��ØÙÖ��Ø��ØÛ��Ð�Ø��ÓÛ ��×ÔÐ�Ý×Ð�Ò��Ö×Ø��Ð�ØÝ�ÝÒÓÖÑ�ÐÑÓ��Ò�ÐÝ×�×ØÖ�Ò×�Ø�ÓÒÓÙÖ×�Ù�ØÓ×Ù Ô�ÖÔÓ×�Ø�ÓÒÓ�ÒÓÒÒÓÖÑ�Ð��Ý�Ò�ÑÓ��×Ø��Ø�Ò��×ÔÐ�ÝÚ�ÖÝ��ØÖ�Ò×��ÒØ �Ò�Ö�Ý�ÖÓÛØ��℄ÁÒØ��ÔÖ�×�ÒØÛÓÖ�Û�×�ÓÛ�Ý�ÖÓÑÛ��ÓÒØÓÙÖ�ÒØ��Ö�Ð Ñ�Ø�Ó�Ó�ÓÑÔÐ�Ø�×Ô�Ø�ÓØ�ÑÔÓÖ�Ð�ÔÔÖÓ��℄Ø��Ø×Ù�ØÖ�Ò×��ÒØ�ÖÓÛØ� Ó��Ò�Ö�Ý�Ð×ÓØ��×ÔÐ��ÓÖ�ØÔÐ�Ø��ÓÙÒ��ÖÝÐ�Ý�ÖÛ��Ò�ÐÐØ��ÒÓÖÑ�Ð ÑÓ��×��×ÔÐ�Ý×Ô�Ø��Ð×Ø��Ð�ØÝ Ø��×ØÙÖ��Ò�×ØÖ��Ñ�ÙÒØ�ÓÒ�Ý �Ø��Ý��ÖÑÓÒ�×ÓÙÖ��ØØ��Û�ÐÐ�Ø��ÖÙÐ�Ö�Ö�ÕÙ�ÒÝ¬Á�Û�Ö�ÔÖ�×�ÒØ Ï��ÒÚ�×Ø��Ø�Ø��Ö�×ÔÓÒ×�Ó��Þ�ÖÓÔÖ�××ÙÖ��Ö��ÒØ�ÓÙÒ��ÖÝÐ�Ý�Ö�Ü 

Û��Ö��Ö�Ò��Ø�Ø��ÖÓÑÛ��ÓÒØÓÙÖ×�ÓÐÐÓÛ��Ò�Ú�ÐÙ�Ø�Ò�Ø��ÓÚ��Ò Ø��Ö�Ð�ÒØ��ÓÑÔÐ�Ü«�Ò�¬ÔÐ�Ò�¬�ÔÔ��Ö�Ò�Ú��Ø��Û�ÐÐ�ÓÙÒ��ÖÝ �Ü�Ý�Ø��Ö�«�¬�Ý��«Ü ¬Ø�«�¬ 

Ì��ÔÓ×��ÔÖÓ�Ð�Ñ�××ÓÐÚ��ÝÐ�Ò��Ö�Þ�Ò�Ø��Æ�Ú��ÖËØÓ��×�ÕÙ�Ø�ÓÒ�ÒØ�� ℄�Ò��×ÓÒ�Ó�Ø��Ð��ÒØÛ�Ý×Ó�ØÖ��Ø�Ò��ÙÐÐ×Ô�Ø�ÓØ�ÑÔÓÖ�ÐÔÖÓ�Ð�Ñ× ×Ô�ØÖ�ÐÔÐ�Ò�×�Ò��ÜÔÖ�××�Ò��Ø�×Ø��ÇÖÖËÓÑÑ�Ö��Ð��ÕÙ�Ø�ÓÒ��Ú�Ò�Ý ÓÒ��Ø�ÓÒ�℄Ì��Ñ�Ø��Ñ�Ø��Ð��×�×Ó��ÖÓÑÛ��ÓÒØÓÙÖ�ÒØ��Ö�Ð�×��Ú�Ò�Ò 

×Ô�Ø��ÐÒÓÖÑ�ÐÑÓ��×��Ú��Ò�Ú�ÐÙ�Ø��Ý�Ö��×��Ö�Ø��Ò�ÕÙ��ÑÔÐÓÝ�Ò� Ñ�ÒØØ��Ò�××À�Ö��Ð�×�Ù×�ÓÙÒ��ÖÝÐ�Ý�Ö�×ÓÒ×��Ö��Û�Ó×��Ü�×Ø�Ò� Û��Ö�ÍÝ�×Ø��Ñ��ÒÓÛ�Ò�Ø��Ê�ÝÒÓÐ�×ÒÙÑ��Ö�×��×��ÓÒ��×ÔÐ�� Í ¬�«� «� Í��«Ê��Ú «� «�� 

�ÒÌ��Ð� Ø��Ò�ÙØÖ�ÐÙÖÚ��Ò�Ø��ÓØ��Ö×�ØÓÖÖ�×ÔÓÒ��Ò�ØÓ¬��ÒØ�¬��×� ��ÑÔ��ÓÒ�×�ØÓÖÖ�×ÔÓÒ��Ò�ØÓ¬��ÒØ�¬��×�Ò� �ÒÌ��Ð� ÓÑÔÓÙÒ�Ñ�ØÖ�ÜÑ�Ø�Ó��℄�ÓÖÊ�� ÓÖØ��×�Ô�Ö�Ñ�Ø�ÖÓÑ��Ò�Ø�ÓÒ×�Ü�×Ø�Ò�×Ô�Ø��ÐÑÓ��×�Ö��ÐÐ �Ò�¬��Ò��Ò��Ú�Ò 

��Õ�Ò�� ××ÓÐÚ��ÐÓÒ�Ø��ÖÓÑÛ��ÓÒØÓÙÖ×�� ¬Ö��¬�� ×Ù�Ø��Ø��Ù×�Ð�ØÝÔÖ�Ò�ÔÐ��×ÒÓØ �«Ö��«�� Ö��ÐÓÛ 

Ö�ÓÒ×ØÖÙØØ��×ØÙÖ��Ò�¬�Ð��Ò�Õ ÑÓ��×�××Ø��Ð�Ø��ÓÒØÓÙÖ�ÒØ��Ö�ÐÖ�×ÙÐØ×�Ò¬�ÙÖ�×�ÓÛ��ÐÓ�Ð×ÓÐÙØ�ÓÒ� Ú�ÓÐ�Ø��Ò�Û�Ú�×ØÖ�Ú�Ð�ÒØ��ÓÖÖ�Ø��Ö�Ø�ÓÒÇ�Ø��Ò��×�Ö�Ù×��ØÓ �Ò¬�ÙÖ��ÓÖØ��×�Ó�¬��Ï��Ð�Ø��Ø��Ð�×�ÓÛ×�ÐÐØ�Ö��×Ô�Ø��Ð Ø�Ñ�Ø��Ý�Ò�Û�Ú�ÓÖÖ�×ÔÓÒ�×ØÓÑÓ��Ò�Ø��Ô��ØÓÖÖ�×ÔÓÒ�×ØÓ ��Ý�Ò�Û�Ú��Ò��Ø�ÑÔÓÖ�ÐÐÝ�ÖÓÛ�Ò�Û�Ú�Ô��Ø��Ø�Ö×ÙÆ��ÒØ �Ò�Ø��Ú�ÐÓ�ØÝ¬�Ð��Ö��×ÔÐÓØØ�� 

Speaker: SENGUPTA, T.K. 39 BAIL 2006 

✩ 

✪

T.K. SENGUPTA, A. KAMESWARA RAO: Spatio-temporal growing waves in 

boundary-layers by Bromwich contour integral method 

✬ 

✫ 

ÑÓ��Ó�Ø��Ð� �ØØ��Ø��ØÐ��ÖÐÝ��Ò�Ø��×Ø��ÑÓ��×��Ò��Ò�ÓØ��Ö×Ø��Ø�Ö�×� Ö�Ø�ÁÒ¬�ÙÖ��ÒÚ�Ö×�Ä�ÔÐ��ØÖ�Ò×�ÓÖÑÓ�Ø��Ø��××�ÓÛÒ�ÓÖØ��×��Ò�Ð Ó�Ø��×��Ò�Ð �Ù�ØÓ�ÒØ�Ö�Ø�ÓÒ×Ó�ÑÓ��×Ø��Ø×�ÓÛÙÔ�×Ø��×Ô�Ø�ÓØ�ÑÔÓÖ�ÐÔÖÓÔ��Ø�ÓÒ ÅÓ�� ×Ò�Ú�ÖÚ�×��Ð��Ù�ØÓ�Ø×�ÜØÖ�Ñ�ÐÝÐ�Ö��Ý 

�Ò�Ø��Ö�ÒØ�Ö�Ø�ÓÒ×�ÑÔ��×�Þ�Ò�Ø��Ò��ØÓÔ�Ö�ÓÖÑ×Ô�Ø�ÓØ�ÑÔÓÖ�Ð�Ò�ÐÝ×�× �Ò×Ø��Ó�Ø�ÑÔÓÖ�ÐÓÖ×Ô�Ø��ÐØ��ÓÖ��× Ê��Ö�Ò�×� ÓÑÔ�Ö�×ÓÒ�ÓÖØ��×ÝÑÔØÓØ�Ô�ÖØÓ�Ø��×ÓÐÙØ�ÓÒÓÑÔÖ�×�Ò�Ó��«�Ö�ÒØÑÓ��× ÁÒØ��¬Ò�ÐÔ�Ô�ÖÛ�Û�ÐÐÔÖ�×�ÒØ�Ò�ÐÝ×�×�ÓÖØ��ÐÓ�Ð×ÓÐÙØ�ÓÒ�Ò��Ø��Ð�� 

�℄ÄÆÌÖ��Ø��Ò�Ø�ÐË��Ò�� ℄ÌÃË�Ò�ÙÔØ��Ø�ÐÈ�Ý×�×�ÐÙ��×� Ä�ÔÐ��ÁÒØ��Ö�Ð��Ñ�Ö��ÍÒ�ÚÈÖ�××�� ℄�Î�Ò��ÖÈÓÐÀ�Ö�ÑÑ�ÖÇÔ�Ö�Ø�ÓÒ�Ð��ÐÙÐÙ×��×��ÓÒÌÛÓ×�� 

0.002 

u′ 

0 

Table 1: 

β0 αr αi 

1) 0.0621 413 0.0696 594 

0.05 

2) 0.1607 670 0.0015 206 

3) 0.1894 256 0.3226 357 

0.15 4) 0.2728 701 0.1675 585 

5) 0.3940 036 0.0104 936 

t = 110.0 

-0.002 

0 100 200 

x 

300 400 

0.002 

0 

-0.002 

5 4 

0 100 200 

x 

300 400 

u′ 

t = 411.6 

0.002 

0 

-0.002 

5 4 

0 100 200 

x 

300 400 

u′ 

0.002 

t = 637.7 

0 

-0.002 

5 4 

0 100 200 

x 

300 400 

Figure 1: Streamwise disturbance velocity plotted as a function of x at different 

t for β0 = 0.15,Re = 1000 at y = 0.278δ ∗ 

φ′ 

0.6 

0.4 

0.2 

u′ 

3 4 5 

t = 801.1 

0 

0 0.1 0.2 

α 

0.3 0.4 0.5 

Figure 2: FFT of streamwise disturbance velocity plotted as a function of α 

Speaker: SENGUPTA, T.K. 40 BAIL 2006 

t = 801.1 

✩ 

✪

L. SAVIĆ, H. STEINRÜCK: Asymptotic Analysis of the mixed convection flow past a 

horizontal plate near the trailing edge 

✬ 

✫ 

Asymptotic Analysis of the mixed convection flow past a horizontal plate 

near the trailing edge 


Lj. Savić and H. Steinrück 

Institute of Fluid Mechanics and Heat Transfer 

Vienna University of Technology 

Resselgasse 3, 1040 Vienna, Austria 

herbert.steinrueck@tuwien.ac.at 

The flow near the trailing edge of a horizontal heated plate which is aligned under a small angle 

of attack φ to the oncoming parallel flow with velocity U∞ in the limit of large Reynolds Re 

and large Grashof Gr = gβ∆TL 3 /ν 2 number will be investigated (see figure 1). As usual β 

and ν denote the isobaric expansion coefficient and the kinematic viscosity, respectively. The 

difference between the plate temperature and the temperature of the oncoming fluid is ∆T and 

L is the length of the plate. A measure for the influence of the buoyancy onto the boundary 

layer flow along a horizontal plate is the buoyancy parameter K = GrRe−5/2 PSfrag replacements 

as defined in [4]). 

U∞ = 1, θ∞ = 0 

φ 

g 

−1 θp = 1 

0 

y 

Re −1/2 

K 

x 

Figure 1: Mixed convection flow past a horizontal plate. 

The starting point of the analysis are the Navier Stokes equations for an incompressible fluid 

using Bousinesq’s approximation to take buoyancy forces into account and the energy equation. 

Additionally to the above mentioned dimensionless parameters the Prandtl number Pr, which 

is assumed to be of order one, and the angle of attack φ enter the problem. 

To analyze the flow behavior near the trailing edge in the frame work of interacting boundary 

layers ([1, 2] it turns out that the buoyancy parameter K and the angle of attack φ have to be of 

the order Re −1/4 . Thus we define the reduced buoyancy parameter κ = K Re 1/4 and the reduced 

inclination parameter λ = φK √ Re. We note that the magnitude of φ is not only dictated by 

the trailing edge analysis, but it is also a consequence of the analysis of the far field (see [5]). As 

we will see the inclination parameter λ will play no role in the trailing edge analysis. Only for 

positive values of λ an outer potential flow field exists [5]. We remark that in case of symmetric 

flow conditions (upper side of the plate heated, lower side cooled) the interaction mechanism 

would allow K to be larger, namely of order Re −1/8 . However, the scope of the present paper 

will be limited to the flow near the trailing edge. 

2. Asymptotic structure of the solution 

For the analysis of the flow field near the trailing edge the velocities, the pressure and the 

temperature are decomposed into a symmetric and anti-symmetric part. To leading order for 

the symmetric part the classical triple deck problem [1, 2] is obtained. For the anti-symmetric 

part (difference of the flow quantities above and below the plate) a new interaction mechanism 

is obtained. The difference of the displacement thicknesses between the upper and lower side 

Speaker: STEINRÜCK, H. 41 BAIL 2006 

1 

✩ 

✪

L. SAVIĆ, H. STEINRÜCK: Asymptotic Analysis of the mixed convection flow past a 

horizontal plate near the trailing edge 

✬ 

✫ 

of the plate ∆A interacts not only via the potential flow in the upper deck but also via the 

hydrostatic pressure in the main deck with the pressure ∆p in the lower deck. The interaction 

law for the pressure difference between the upper and lower side can be written in the form 

∆A ′ (x (3) ) + √ 3ash(x (3) � 

) x (3)� 1/3 

= 

� � 0 1 ∆ˆp(ξ) + 2as|ξ| 

− 

π −∞ 

1/3 

x (3) dξ − 

− ξ 

1 

� ∞ As(ξ) − ash(ξ)|ξ| 

π −∞ 

1/3 

x (3) � 

dξ 

− ξ 

Where As(x (3) ) is the displacement thickness obtained for the classical trailing edge problem 

[3, 1] and x (3) is the coordinate in the lower, main and upper deck parallel to the plate. Here as 

is a constant and h(.) the heayside function. It turns out that the difference pressure ∆p has a 

discontinuity at the trailing edge. Thus new sub-layers (in the main and lower deck) to resolve 

this discontinuity are introduced. 

In the following table we give an overview of the most important layers needed for the 

asymptotic analysis. We introduce the following notation according to the stretching factor as 

a power of the Reynolds number: x (α) = Re α/8 x and y (β) = Re β/8 y. 

α β flow region 

0 0 potential flow region 

0 4 boundary layer and wake 

3 3 upper deck 

3 4 main deck 

3 5 lower deck 

4 4 main deck - trailing edge 

5 5 lower deck - trailing edge 

Table 1: Scales of the different flow regions 

The new results of the analysis will be the local behavior of the difference pressure at the 

trailing edge. For both of the newly introduced sub-layers elliptic equations for the local pressure 

variation can be derived and will be solved numerically. Thus on triple deck scales there will be 

a flow around the trailing edge. Thus the perturbation of the classical triple deck problem by 

small buoyancy effects behaves quite differently than by small angles of attack [3]. 

References 

[1] K. Stewartson, On the flow near the trailing edge of flat plate, Mathematika, 16, 106-121 

(1969). 

[2] A.F. Messiter, Boundary layer flow near the trailing edge of flat plate, SIAM J. Appl. Math., 

18, 241-257, (1970). 

[3] R. Chow and R. E. Melnik, Numerical Solutions of Triple-Deck Equations for Laminar 

Trailing Edge Stall, Grumman Research Dept., Report RE-526J (1976). 

[4] W. Schneider and M. G. Wasel, Breakdown of the boundary-layer approximation for mixed 

convection flow above a horizontal plate, Int. J. Heat Transfer, 28, 2307-2313 (1985). 

[5] Lj. Savić and H. Steinrück, Mixed convection flow past a horizontal plate, Theoretical and 

Applied Mechanics, 32, 1-19 (2005). 

Speaker: STEINRÜCK, H. 42 BAIL 2006 

2 

✩ 

✪

G.I. Shiskin, P. Hemker 

Robust Methods for Problems with Layer Phenomena and Additional Singularities 

Speaker: 

The minisymposium will be concerned with singularly perturbed 

multiscale problems having additional singularities. A complicated 

geometry or unboundedness of the domain and/or the lack of sufficient 

smoothness (or compatibility) of the problem data may result in 

singular solutions that have their own specific scales, besides boundary/interior 

layers. We intend to examine techniques for constructing 

numerical methods that converge parameter-uniformly (in the maximum 

norm). 

The following research aspects will be also considered: (i) As a rule, 

such parameter-uniformly convergent numerical methods have too low 

order of uniform convergence, which restricts their applicability in 

practice. With this respect, methods how to increase the accuracy of 

parameter-uniformly convergent numerical methods will be considered. 

(ii) When standard numerical methods, for example, domain decomposition 

methods are used to find solutions of parameter-uniformly convergent 

discrete approximations, the decomposition errors of the discrete 

solutions and the number of iterations required to solve the discrete 

problem depend on the perturbation parameter and grow when it tends 

to zero. We will consider decomposition methods preserving the property 

of parameter-uniform convergence. Domain decomposition and 

local defect correction techniques allows us to reduce the construction 

of robust numerical methods for multiscale problems to locally robust 

methods for monoscale problems on the specific subdomains. Other aspects 

and applications will be also under consideration. Problems for 

partial differential equations with different types of boundary and interior 

layers will be considered. To construct special numerical methods, 

fitted meshes, which are a priori and a posteriori condensing in the layer 

regions, are used. 

• Deirdre Branley: A Schwarz method for a convection-diffusion problem with a 

corner singularity 

• Thorsten Linss: Layer-adapted meshes for time-dependent reaction-diffusion 

• Grigory I. Shishkin: Grid Approximation of Parabolic Equations with Nonsmooth

Initial Condition in the Presence of Boundary Layers of Different Types 

• Lidia P. Shishkina: A Difference Scheme of Improved Accuracy for a Quasilinear 

Singularly Perturbed Elliptic Convection-Diffusion Equation in the Case of the 

Third-Kind Boundary Condition 

• Irina V. Tselishcheva: Domain Decomposition Method for a Semilinear Singularly 

Perturbed Elliptic Convection-Diffusion Equation with Concentrated Sources 

• S. Valarmathi: A parameter-uniform numerical method for a system of singularly 

perturbed ordinary differential equations

D. BRANLEY, A.F. HEGARTY, H. PURTILL, G.I. SHISHKIN: A Schwarz method 

for a convection-diffusion problem with a corner singularity 

✬ 

✫ 

A Schwarz method for a convection-diffusion problem with a corner 

singularity. ∗ 

Deirdre Branley 1 , Alan F. Hegarty 1 , Helen Purtill 1 and Grigory I. Shishkin 2 . 

1 Department of Mathematics and Statistics, University of Limerick, Plassey, Limerick, Ireland. 

deirdre.branley@ul.ie, alan.hegarty@ul.ie, helen.purtill@ul.ie, 

2 Institute of Mathematics and Mechanics, Ural Branch of Russian Academy of Sciences 

16 S. Kovalevskaya Street, Ekaterinburg 620219, Russia 

shishkin@imm.uran.ru 

We are concerned with two dimensional steady state convection-diffusion problems with 

singular outflow boundary conditions. It is well known that, where the boundary conditions 

are sufficiently smooth and compatible, such problems can be solved with uniform accuracy 

with respect to the small parameter ε using a standard finite difference operator on special 

piece-wise uniform meshes [1, 2]. Where the outflow boundary data are only weakly regular and 

compatible, parameter-uniform solutions may also be obtained by this method [2]. However, 

orders of convergence are relatively small and pointwise errors relatively large in this case. 

Numerical methods for singularly perturbed problems comprising domain decomposition and 

Schwarz iterative technique have been examined by a number of authors, for example in [1], [3], 

[4] and [5]. In particular, MacMullen et al. [5] constructed a parameter-uniform Schwarz method 

for singularly perturbed linear convection-diffusion problems in two dimensions with sufficiently 

smooth and compatible boundary data. We examine experimentally the performance of such 

methods extended to the class of singularly perturbed convection-diffusion problems with more 

general boundary conditions described below. 

We consider problems of the form 

Lu ≡ ε∆uε + a(x, y).∇u = f 

in a domain Ω, the unit square, with Dirichlet boundary conditions, where all components of a 

are strictly positive. Such problems exhibit regular layers along the outflow boundaries, as well 

as a corner boundary layer at the outflow boundary corner. We deal with outflow boundary 

conditions, where the first derivatives are not compatible at the outflow boundary corner. We 

implement domain decomposition methods to isolate the neighbourhood of the singularity, along 

with a Schwarz iterative technique, with the aim of developing a Schwarz method to produce 

parameter-uniformly accurate solutions on the whole domain in the oresence of such a singularity. 

References 

[1] J.J.H. Miller, E.O’Riordan, G.I. Shishkin, Fitted Numerical Methods for Singular Perturbation 

Problems, World Scientific, Singapore, 1996. 

[2] P.A. Farrell, A.F. Hegarty, J.J.H. Miller, E. O’Riordan, G.I. Shishkin, Robust Computational 

Techniques for Boundary Layers, Chapman & Hall, CRC, Boca Raton, FL, 2000. 

[3] J.J.H. Miller, E. O’Riordan, G.I. Shishkin, S. Wang, A parameter-uniform Schwarz method 

for a singularly-perturbed reaction-diffusion problem with an interior layer, Appl. Num. 

Math., 35 (2000), 323-337. 

∗ This research was supported in part by the Irish Research Council for Science, Engineering and Technology 

and by the Russian Foundation for Basic Research under grant No. 04-01-00578. 

1 

Speaker: BRANLEY, D. 45 BAIL 2006 

✩ 

✪

D. BRANLEY, A.F. HEGARTY, H. PURTILL, G.I. SHISHKIN: A Schwarz method 

for a convection-diffusion problem with a corner singularity 

✬ 

✫ 

[4] H. MacMullen, J.J.H. Miller, E. O’Riordan, G.I. Shishkin, A second-order parameter uniform 

overlapping Schwarz method for reaction-diffusion problems with boundary layers, J. 

Comput. & Appl. Math., 130 (2001), 231-244. 

[5] H. MacMullen, E. O’Riordan, G.I. Shishkin, The convergence of classical Schwarz methods 

applied to convection-diffusion problems with regular boundary layers, Appl. Num. Math., 

43 (2002), 297-313. 

Speaker: BRANLEY, D. 46 BAIL 2006 

2 

✩ 

✪

T. LINSS, N. MADDEN: Layer-adapted meshes for time-dependent reaction-diffusion 

✬ 

✫ 

Layer-adapted meshes for time-dependent reaction-diffusion 

Torsten Linß 1 , Niall Madden 2 

1 Institut für Numerische Mathematik, Technische Universität Dresden 

e-mail: torsten.linss@tu-dresden.de 

2 Department of Mathematics, National University of Ireland Galway 

e-mail: niall.madden@math.nuigalway.ie 

ABSTRACT 

We consider singularly perturbed reaction-diffusion problems of the type 

ut + Lεu = f in (0, 1) × (0, T], 

where Lεv := −ε 2 vxx + rv, subject to boundary conditions 

and initial condition 

u(0, t) = γ0(t), u(1, t) = γ1(t) in (0, T] 

u(·, 0) = u 0 

in (0, 1) 

where 0 < ε ≪ 1, r(x) > ρ 2 > 0 for x ∈ [0, 1]. The nature of the differential equation changes when 

ε → 0 giving rise to boundary layers that require special attention in the design of numerical methods, 

in particular local refinement of the meshes used. 

We study an inverse monotone difference scheme on arbitrary meshes. An maximum-norm error bound 

is derived that allows easy classification of various layer-adapted meshes proposed in the literature. 

For example, this general result implies 

�u − U� ∞ ≤ C 

� 

τ + N −2 ln 2 N for Shishkin meshes, 

τ + N −2 for Bakhvalov meshes, 

where τ is the maximal time-step size and N the number of mesh points used in the space discretization. 

Speaker: LINSS, T. 47 BAIL 2006 

✩ 

✪

G.I. SHISHKIN: Grid Approximation of Parabolic Equations with Nonsmooth Initial 

Condition in the Presence of Boundary Layers of Different Types 

✬ 

✫ 

Grid Approximation of Parabolic Equations with Nonsmooth Initial 

Condition in the Presence of Boundary Layers of Different Types ∗ 

Grigory I. Shishkin 

Institute of Mathematics and Mechanics, 

Ural Branch of Russian Academy of Sciences, 

Yekaterinburg 620219, Russia 


A Dirichlet problem for a singularly perturbed parabolic equation with the perturbation 

vector-parameter ε, ε = (ε1, ε2), is considered on a semiaxis. The highest derivative of the 

equation and also the first derivative with respect to x contain respectively the parameters 

ε1 and ε2, which take arbitrary values in the half-open interval (0, 1] and the segment [−1, 1]. 

Depending on the parameter ε2, the type of the equation may be reaction-diffusion or convectiondiffusion. 

The first order derivative of the initial function has a discontinuity of the first kind at 

the point x0. 

For small values of the parameter ε1, a boundary layer appears in a neighbourhood of the 

lateral part of the domain boundary. Depending on the ratio between the parameters ε1 and 

ε2, these layers may be regular, parabolic or hyperbolic (characteristic scales of these boundary 

layers also depend on the ratio between ε1 and ε2). 

In a neighbourhood of the set S γ , that is, the characteristic of the reduced equation outgoing 

from the point (x0, 0), the parabolic transient layer arises. 

Using the method of piecewise uniform meshes condensing in a neighbourhood of the layer, 

we construct a special difference scheme that converges ε-uniformly. 

Numerical methods for problems with different types of boundary layers for elliptic convectiondiffusion 

equations in the case of sufficiently smooth boundary data are studied, e.g., in [1]. 

References 

[1] G.I. Shishkin, “Grid approximation of a singularly perturbed elliptic equations with convective 

terms in the presence of various boundary layers”, Comput. Maths. Math. Phys., 

45 (1), 104–119 (2005). 

∗ This research was supported in part by the Russian Foundation for Basic Research (grants No. 04-01-00578, 

04–01–89007–NWO a), by the Dutch Research Organisation NWO under grant No. 047.016.008 and by the Boole 

Centre for Research in Informatics, National University of Ireland, Cork. 

Speaker: SHISHKIN G.I. 48 BAIL 2006 

1 

✩ 

✪

L.P. SHISHKINA, G.I. SHISHKIN: A Difference Scheme of Improved Accuracy for a 

Quasilinear Singularly Perturbed Elliptic Convection-Diffusion Equation in the Case of 

the Third-Kind Boundary Condition 

✬ 

✫ 

A Difference Scheme of Improved Accuracy for a Quasilinear 

Singularly Perturbed Elliptic Convection-Diffusion Equation 

in the Case of the Third Kind Boundary Condition ∗ 

Lidia P. Shishkina and Grigory I. Shishkin 

Institute of Mathematics and Mechanics, 



Lida@convex.ru and shishkin@imm.uran.ru 

A boundary value problem for a quasilinear singularly perturbed elliptic convection-diffusion 

equation on a strip is considered. The third kind boundary condition admitting both Dirichlet 

and Neumann conditions is given on the domain boundary. For small values of the perturbation 

parameter ε, a boundary layer arises in a neighbourhood of the outflow part of the boundary. For 

such a problem, the base (nonlinear) difference scheme constructed by classical approximations 

of the problem on piecewise uniform meshes condensing in the layer converges ε-uniformly with 

an order of accuracy not higher than 1. 

Our aim is for this boundary value problem to construct grid approximations that converge 

ε-uniformly with an order of convergence close to two. 

Using the�Richardson technique, we construct a (nonlinear) scheme that converges ε-uniformly 

at the rate O N −2 

1 ln 2 N1+N −2 

� 

2 , where N1+1 is the number of nodes in the mesh with respect 

to x1 and N2+1 is the number of mesh points on a unit interval along the x2-axis. Based on the 

nonlinear Richardson scheme, a linearized iterative scheme is constructed where the nonlinear 

term is computed using the unknown function taken at the previous iteration. The solution of 

this iterative scheme converges to the solution of the nonlinear Richardson scheme at the rate of 

a geometry progression ε-uniformly with respect to the number of iterations. Thus, the number 

of iterations required for solving the problem (as well as the accuracy of the resulting solution) 

is independent of the parameter ε. 

The use of lower and upper solutions of the linearized iterative Richardson scheme as a 

stopping criterion allows us during the computational process to define a current iteration under 

which the same ε-uniform accuracy of the solution is achieved as for the nonlinear Richardson 

scheme. To construct the improved scheme, the technique developed in [1]–[4] for a Dirichlet 

problem is applied. 

References 

[1] G.I. Shishkin, “The method of increasing the accuracy of solutions of difference schemes 

for parabolic equations with a small parameter at the highest derivative”, USSR Comput. 

Maths. Math. Phys., 24 (6), 150–157 (1984). 

[2] G.I. Shishkin, ”Finite-difference approximations of singularly perturbed elliptic equations”, 

Comp. Math. Math. Phys., 38 (12), 1909–1921 (1998). 

[3] G.I. Shishkin and L.P. Shishkina, “A high-order Richardson method for a quasilinear singularly 

perturbed elliptic reaction-diffusion equation”, Differential Equations, 41 (7), 1030– 

1039 (2005). 


04–01–89007–NWO a) and by the Dutch Research Organisation NWO under grant No. 047.016.008. 

Speaker: SHISHKINA, L.P. 49 BAIL 2006 

1 

✩ 

✪

L.P. SHISHKINA, G.I. SHISHKIN: A Difference Scheme of Improved Accuracy for a 

Quasilinear Singularly Perturbed Elliptic Convection-Diffusion Equation in the Case of 

the Third-Kind Boundary Condition 

✬ 

✫ 

[4] L. Shishkina and G. Shishkin, “The discrete Richardson method for semilinear parabolic singularly 

perturbed convection-diffusion equations”, in: Proceedings of the 10th International 

Conference “Mathematical Modelling and Analysis” 2005 and 2nd International Conference 

“Computational Methods in Applied Mathematics”, R. Čiegis ed., Vilnius, “Technika”, 

2005, pp. 259–264. 

2 


✩ 

✪

I.V. TSELISHCHEVA, G.I. SHISHKIN: Domain Decomposition Method for a Semilinear 

Singularly Perturbed Elliptic Convection-Diffusion Equation with Concentrated 

Sources 

✬ 

✫ 

Domain Decomposition Method for a Semilinear Singularly Perturbed 

Elliptic Convection-Diffusion Equation with Concentrated Sources ∗ 

Irina V. Tselishcheva and Grigory I. Shishkin 

Institute of Mathematics and Mechanics, Ural Branch of Russian Academy of Sciences 


tsi@imm.uran.ru, shishkin@imm.uran.ru 

For singularly perturbed boundary value problems in a composed domain (in particular, 

with concentrated sources) whose solution has several singularities such as boundary and interior 

layers, it is of keen interest to construct a parameter-uniform numerical method based on 

a domain decomposition technique so that each subdomain in the decomposition contains no 

more than a single singularity. Because of the thin layers, standard numerical methods applied 

to problems of this type yield unsatisfactorily large errors for small values of the singular perturbation 

parameter ε. The fact that the partial differential equation is nonlinear complicates 

the solution process. 

We develop monotone linearized schemes based on an overlapping Schwarz method for a 

semilinear singularly perturbed elliptic convection-diffusion equation on a compound strip in the 

presence of concentrated sources acting inside the domain. We first study a special (base) scheme 

comprising a standard finite difference operator on a piecewise-uniform fitted mesh and an 

overlapping domain decomposition scheme constructed on the basis of the former that converge εuniformly 

at the rates O � N −1 � � −1 

and O N 

1 lnN1 + N −1 

2 

1 ln N1 + N −1 

2 + qt� , respectively. Here 

N1 +1 and N2 +1 are the number of mesh points in x1 and the minimal number of mesh points 

in x2 on a unit interval, q < 1 is the common ratio of a geometric progression, independent of ε, 

t is the iteration count. For these nonlinear schemes we construct monotone linearized schemes 

of the same ε-uniform accuracy, in which the nonlinear term is computed from the unknown 

function taken at the previous iteration. 

The linearized schemes are monotone, which admits to construct their upper and lower 

solutions. We apply the technique of upper and lower solutions to find aposteriori the (optimal) 

number of iterations T in the linearized scheme for which the accuracy of its solution is the same 

(up to a constant factor) as that for the base scheme, where T = O (ln(min[N1, N2])) (see also 

[1] for a reaction-diffusion problem). Thus, the number of required iterations is independent of 

ε. With respect to total computational costs, the iterative method is close to a solution method 

for linear problems, since the number of iterations is only weakly depending on the number of 

mesh points used. The linearized iterative schemes inherit the ε-uniform rate of convergence of 

the nonlinear schemes. 

The decomposition schemes can be computed sequentially and in parallel (so that the suproblems 

on the overlapping subdomains are solved independently of each other). 

Note that schemes of the overlapping domain decomposition method were considered earlier 

by the authors in [2] for linear problems and in [3, 4] for nonlinear problems. 

References 

[1] G.I. Shishkin and L.P. Shishkina, “A high-order Richardson method for a quasilinear singularly 

perturbed elliptic reaction-diffusion equation”, Differential Equations, 41 (7), 1030– 

1039 (2005). 


04–01–89007–NWO a) and by the Dutch Research Organisation NWO grant No. 047.016.008. 

Speaker: TSELISHCHEVA, I.V. 51 BAIL 2006 

1 

✩ 

✪

I.V. TSELISHCHEVA, G.I. SHISHKIN: Domain Decomposition Method for a Semilinear 

Singularly Perturbed Elliptic Convection-Diffusion Equation with Concentrated 

Sources 

✬ 

✫ 

[2] G.I. Shishkin and I.V. Tselishcheva, “Parallel methods for solving singularly perturbed 

boundary value problems for elliptic equations”, (in Russian) Mat. Model., 8 (3), 111–127 

(1996). 

[3] I.V. Tselishcheva and G.I. Shishkin, “Monotone domain decomposition schemes for a singularly 

perturbed semilinear elliptic reaction-diffusion equation with Robin boundary conditions”, 

in: Proceedings of the 10th International Conference “Mathematical Modelling 

and Analysis” 2005 and 2nd International Conference “Computational Methods in Applied 

Mathematics”, R. Čiegis ed., Vilnius, “Technika”, 2005, pp. 251–258. 

[4] I.V. Tselishcheva and G.I. Shishkin, “Monotone domain decomposition schemes for a 

quasilinear singularly perturbed elliptic convection-diffusion equation with a concentrated 

source”, in: Grid Methods for Boundary Value Problems and Applications, Proceedings of 

the VI-th All-Russian Workshop, Kazan, Kazan State University, 2005, pp. 251-255 (in 

Russian). 

Speaker: TSELISHCHEVA, I.V. 52 BAIL 2006 

2 

✩ 

✪

S. HEMAVATHI, S. VALARMATHI: A parameter-uniform numerical method for a 

system of singularly perturbed ordinary differential equations 

✬ 

✫ 

¨�©�¨�¤��¤�§ ��§ ¨�¤��¢��§ ¨�©��£�� ¤�¢ ��©�¨�§ 

¡£¢¥¤¦¢¥§ 

¡�¨�¤�©¦��¤��¨�� ¤��£¢¥¤�� ¨�¤¦¨��©¦��¢��¨��£¢�©�� 

��£��¢�¤�� 

�� 

��¦�� 

�� 

��¦�� 

�� 

��¥�� 

�� 

��¥��£��¥�� 

�� 

�� 

�� 

�� ¦�� 

�� 

�� 

�� 

�� 

�� 

��¥�� 

�� 

�� 

� 

� 

�� 

�� 

��£�� 

��£�� 

�� 

�� 

�� 

�� 

� 

��£��£�� 

�� 

�� 

� �� 

� �� £�� 

�� 

� 

�� 

�� 

�� 

�� 

��¦� � �� ¡ �� £¢�� 

�� 

��¥¤��£¢�� 

�� §¦��©¨�� 

�� 

�� 

��£��§�� 

��£��§�� 

� 

Speaker: VALARMATHI, S. 53 BAIL 2006 

�� 

�� 

✩ 

✪

S. HEMAVATHI, S. VALARMATHI: A parameter-uniform numerical method for a 

system of singularly perturbed ordinary differential equations 

✬ 

�� 

��¥�� £�� 

�� 

��¦�� ¦�� 

��£¢�� 

¡�� 

� £©¨ � £ � ¢¤£¦¥§£ 

� � � � � ¨�� ¦��¦�� 

�� 

�� 

� 

�� ¥�� 

�� 

� �� 

��¥�� 

�� ¦��¨�� 

�� 

�� 

��§�� 

��¥�� 

� �� ¨�� ©��§��§�� 

�� 

��§�� §�� 

� � �� 

��©��©�� 

�� ¡�� ¦�� 

�� 

�� 

�� 

��§�� 

�� 

�� §��§�� §�� 

�� §�� 

�� 

�� ¨�� 

�� 

�� 

�� ¥�� 

��¤�� 

�� 

�� 

�� 

� 

�� 

�� 

�� 

��§¦�� 

�� 

✫ 

��¦�� ¢§¦�� 

��§¦�� 

� 

Speaker: VALARMATHI, S. 54 BAIL 2006 

✩ 

✪

J. Maubach, I, Tselishcheva 

Robust Numerical Methods for Problems with Layer Phenomena and Applications 

Speaker: 

Numerical modeling of many processes and physical phenomena leads 

to boundary value problems for PDEs having non-smooth solutions with 

singularities of thin layer type. Among them are convection-dominated 

convection-diffusion problems, Navier-Stokes equations and boundarylayer 

equations at high Reynolds number, the drift-diffusion equations 

of semiconductor device simulation, flow problems with lift, drag, transition 

and interface phenomena, phenomena in plasma fluid dynamics, 

mathematical models for the spreading of pollutants, combustion, shock 

hydrodynamics or transport in porous media and other related problems. 

The solutions of these problems contain thin boundary and interior 

layers, shocks, discontinuities, shear layers, or current sheets, etc. 

The singular behaviour of the solution in such local structures generally 

gives rise to difficulties in the numerical solution of the problem in 

question by traditional methods on uniform meshes and requires the use 

of highly accurate discretization methods and adaptive grid refinement 

techniques. The problem of resolving layers, which is of great practical 

importance, is still not solved satisfactorily for a wide class of problems 

with layer phenomena and applications, which the minisymposium is 

concerned with. 

• Alan Hegarty: An adaptive method for the numerical solution of an elliptic convection 

diffusion problem 

• Joseph Maubach: A Convergence Proof of Local Defect Correction for Convection- 

Diffusion Problems 

• Joseph Maubach: On the Difference between Left and Right Preconditioning for 

Convection Dominated Convection-Diffusion Problems 

• Lidia P. Shishkina: Parameter-Uniform Method for a Singularly Perturbed Parabolic 

Equation Modelling the Black-Scholes equation in the Presence of Interior and 

Boundary Layers 

• Alexander I. Zadorin: Numerical Method for the Blasius Equation on an Infinite 

Interval 

• Paul Zegeling: An Adaptive Grid Method for the Solar Coronal Loop Model

A.F. HEGARTY, S. SIKWILA, G.I. SHISKIN: An adaptive method for the numerical 

solution of an elliptic convection diffusion problem 

✬ 

✫ 

An adaptive method for the numerical solution of an elliptic convection 

diffusion problem ∗ 

Alan F. Hegarty 1 , Stephen Sikwila 1 and Grigory I. Shishkin 2 . 

1 Department of Mathematics and Statistics, University of Limerick, Plassey, Limerick, Ireland. 

alan.hegarty@ul.ie, stephen.sikwila@ul.ie, 

2 Institute of Mathematics and Mechanics, Ural Branch of Russian Academy of Sciences 



A two dimensional elliptic linear convection diffusion problem: 

Pε : 

� 

ε∆uε + a.∇uε + d(x, y)uε = f(x, y), 

uε = g, 

(x, y) ∈ Ω; 

(x, y) ∈ Γ, 

is considered, where the diffusion parameter 0 < ε ≤ 1 is typically small. The use of special 

piecewise uniform meshes appropriately condensed in the boundary layer regions together 

with montone finite difference operators is well known to result in numerical methods which 

are uniformly convergent with respect to ε (see e.g., [1]). Such methods depend on a priori 

knowledge of the location of any boundary layers. Since such information may not be available 

for more complicated problems, it is of interest to examine whether robust numerical results can 

be obtained for the model problem (1), without using the a priori information. 

An adaptive technique is presented, based on piecewise uniform meshes, where the location 

of the transition point between the coarse and fine meshes is computed iteratively. This is an 

extension of the one dimensional method presented in [2] and [3]. 

Numerical experiments are presented which indicate that the computational solutions obtained 

are uniformly in ε convergent and also that the number of iterations required does not 

increase significantly for small ε. 

References 

[1] P.A. Farrell, A.F. Hegarty, J.J.H. Miller, E. O’Riordan, G.I. Shishkin, Robust Computational 

Techniques for Boundary Layers, Chapman & Hall, CRC, Boca Raton, FL, 2000. 

[2] S. Sikwila, A.F. Hegarty and G.I. Shishkin, Novel robust adaptive techniques for the numerical 

solution of convection diffusion problem, Proceedings of BAIL 2004 Conference, 

Toulouse, July 2004, ONERA, 2004. 

[3] S. Sikwila, Novel robust layer resolving adaptive mesh methods for convection diffusion 

problems in one and two dimesnions, Ph.D. thesis, University of Limerick, 2005. 

∗ This work was supported by the Russian Foundation for Basic Research under grant No. 04-01-00578. 

Speaker: HEGARTY, A.F. 56 BAIL 2006 

1 

(1) 

✩ 

✪

M. ANTHONISSEN, I. SEDYKH, J. MAUBACH: A Convergence Proof of Local 

Defect Correction for Convection-Diffusion Problems 

✬ 

✫ 

A Convergence Proof of Local Defect Correction 

for Convection–Diffusion Problems 

M. Anthonissen 1 , I. Sedykh 2 and J. Maubach 1 

1 Department of Mathematics and Computer Science, Eindhoven University of Technology, 

Eindhoven, The Netherlands 

m.j.h.anthonissen@tue.nl, j.m.l.maubach@tue.nl 

2 Department of Mechanics and Mathematics, Moscow State University, Moscow, Russia 

Examples of partial differential equations with solutions that are rapidly varying functions 

of the spatial or temporal coordinates appear e.g. in combustion, shock hydrodynamics or 

transport in porous media. For boundary value problems with solutions that have one or a few 

small regions with high activity, a fine grid is needed in regions with high activity, whereas a 

coarser grid would suffice in the rest of the domain. Rather than using a truly nonuniform grid, 

we study a method called Local Defect Correction (LDC) that is based on local uniform grid 

refinement. 

Several properties hold for the LDC fixed point iteration for convection-diffusion equations in 

two dimensions. We study the convergence behavior of the LDC method as an iterative process 

and derive an upper bound for the norm of the iteration matrix for the linear two-dimensional 

convection-diffusion equation with constant coefficients on the unit square. The research results 

can be extended to the case of non-constant coefficients. 

Speaker: MAUBACH, J. 57 BAIL 2006 

1 

✩ 

✪

J. MAUBACH: On the Difference between Left and Right Preconditioning for Convection 

Dominated Convection-Diffusion Problems 

✬ 

✫ 

On the Difference between Left and Right Preconditioning for Convection 

Dominated Convection–Diffusion Problems 

Joseph Maubach 

Department of Mathematics and Computer Science, Eindhoven University of Technology, 

Eindhoven, The Netherlands 

j.m.l.maubach@tue.nl 

For convection dominated convection-diffusion problems in several space dimensions discretized 

with finite differences on a locally fine tensor–grid, an approximate inverse operator 

Gn is calculated from the linear operator An. The spectral condition numbers κ2(GnAn) 

and κ2(AnGn) estimated numerically are different: κ2(GnAn) . = O(log h −1 ) and κ2(AnGn) . = 

O(h −1 ). This is corroborated by a proof for the one-dimensional case for a modified approximate 

inverse operator Gn. 

Speaker: MAUBACH, J. 58 BAIL 2006 

1 

✩ 

✪

S. LI, L.P. SHISHKINA, G.I. SHISHKIN: Parameter-Uniform Method for a Singularly 

Perturbed Parabolic Equation Modelling the Black-Scholes equation in the Presence 

of Interior and Boundary Layers 

✬ 

✫ 

Parameter–Uniform Method for a Singularly Perturbed Parabolic Equation 

Modelling the Black–Scholes equation in the Presence of Interior and 

Boundary Layers ∗ 

Shuiying Li 1 , Lidia P. Shishkina 2 and Grigory I. Shishkin 2 

1 Department of Computational Science, 

National University of Singapore, 

Singapore 117543 

scip1075@nus.edu.sg 

2 Institute of Mathematics and Mechanics, 



Lida@convex.ru and shishkin@imm.uran.ru 

Solutions of regular parabolic equations with nonsmooth initial data are of limited smoothness 

themselves (in a neighbourhood of the nonsmoothness of the data). When the equation 

is singularly perturbed, its solution has singularities such as initial, boundary and/or interior 

layers with own scales. These problems belong to the class of singularly perturbed multiscale 

problems. The multiscale behaviour of solutions complicates the construction and study of 

efficient patameter-uniform numerical methods. 

A singularly perturbed parabolic convection-diffusion equation with the second derivative 

multiplied by a singular perturbation parameter ε, which ranges in (0, 1], arises when we model 

the Black-Scholes equation upon a European call option (see, e.g., [1]). There are a few singularities 

in the problem, i.e., a single discontinuity of the first derivative of the initial condition 

at a point x0, the unbounded growth of the initial condition at infinity and the unbounded 

domain in space. The solution of this initial value problem grows exponentially without bound 

as x → ∞. Thus, we deal with a singularly perturbed multiscale problem that has different 

types of singularities. 

In this presentation we consider a boundary value problem (in a bounded domain) for this 

singularly perturbed parabolic convection-diffusion equation with an additional singularity generated 

by the discontinuity of the first derivative of the initial function. For such a problem, 

singularities such as a boundary and an interior layer with own specific scales arise. For small 

values of the parameter ε, the interior layer due to nonsmooth initial data appears in a neighbourhood 

of the characteristic of the reduced equation passing through the point (x0, 0), and 

the regular boundary layer appears in a neighbourhood of the outflow boundary through which 

the convective flux leaves the domain. 

By using the singularity splitting method and piecewise uniform meshes condensing in the 

boundary layer, we construct a special difference scheme that allows us to approximate εuniformly 

the solution of the problem under consideration, as well as the first derivative of 

the solution. The corresponding theoretical and numerical results are provided in the paper. 

The description of the technique applied in this study can be found in [2]. Preliminary results 

for a similar problem with no boundary layer were given in [3]. 

∗ This research was supported in part by the Russian Foundation for Basic Research (grant No 04-01-00578, 

04–01–89007–NWO a) and by the Dutch Research Organization NWO under grant No 047.016.008. 

1 


✩ 

✪

S. LI, L.P. SHISHKINA, G.I. SHISHKIN: Parameter-Uniform Method for a Singularly 

Perturbed Parabolic Equation Modelling the Black-Scholes equation in the Presence 

of Interior and Boundary Layers 

✬ 

✫ 

References 

[1] J.J.H. Miller, G.I. Shishkin, “Robust numerical methods for the singularly perturbed Black- 

Scholes equation”, in: Proceedings of the Conference on Applied Mathematics and Scientific 

Computing, Springer, Dordrecht, 2005, pp. 95–105. 

[2] G.I. Shishkin, “Grid approximation of parabolic convection-diffusion equations with piecewise 

smooth initial conditions”, Doklady Akad. Nauk, 405 (1), 1–4 (2005); transl. in Doklady 

Mathematics, 72 (3) (2005). 

[3] Li Shuiying, D.B. Creamer, G.I. Shishkin, “Discrete approximations of a singularly perturbed 

Black-Scholes equation with nonsmooth initial data”, in: An International Conference 

on Boundary and Interior Layers — Computational and Asymptotic Methods BAIL 

2004, ONERA, Toulouse, 5th-9th July, 2004, pp. 441-446. 


2 

✩ 

✪

A.I. ZADORIN: Numerical Method for the Blasius Equation on an Infinite Interval 

✬ 

✫ 

Numerical Method for the Blasius Equation on an Infinite Interval ∗ 

A.I. Zadorin 

Omsk Branch of Sobolev Institute of Mathematics, 

Siberian Branch of Russian Academy of Sciences, Omsk, Russia 

zadorin@iitam.omsk.net.ru 

The Blasius equation on a semi-infinite interval is considered. This problem can be considered 

as a model problem for constructing a numerical method for problems in unbounded domains. 

The Blasius problem have been investigated in many papers. For example, G.I. Shishkin [1] 

studied the asymptotic behavior of differential and difference solutions to get a difference scheme 

with a finite number of mesh points for a sufficiently long interval. We apply the method of 

separation of the set of solutions that satisfy the limit boundary condition at infinity to transform 

the problem to a problem on a finite interval (see, e.g., [2, 3]). Difficulties are concerned with 

the nonlinearity of the differential equation and with an unbounded coefficient multiplying the 

second derivative. We propose to make a transformation of the independent variable to avoid 

the last problem. 

So, consider the Blasius problem 

u ′′′ (x) + u(x)u ′′ (x) = 0, 

u(0) = 0, u ′ (0) = 0, lim 

x→∞ u′ (x) = 1. 

Let u(x) = v(x) + x, w(x) = v ′ (x). Then problem (1) take the form 

v ′ (x) = w(x), v(0) = 0, 

w ′′ (x) + [v(x) + x]w ′ (x) = 0, w(0) = −1, lim w(x) = 0. 

x→∞ 

To transfer the limit condition at infinity to a finite point, consider the linear problem 

εu ′′ (x) + [a(x) + x]u ′ (x) = f(x), 

u(0) = A, lim u(x) = 0. 

x→∞ 

Suppose that there is a unique solution of (3), ε > 0, ∃ lim a(x), lim f(x) = 0. To avoid the 

x→∞ x→∞ 

difficulty with unboundedness of the coefficient multiplying the first derivative, we use the new 

variable t = x2 /2. Then problem (3) can be written in the form 

where 

εu ′′ (t) + b(t)u ′ (t) = F(t), u(0) = A, lim 

t→∞ u(t) = 0, (4) 

b(t) = a(√ 2t) 

√ 2t + 1 + 1 

2t , F(t) = f(√ 2t)/(2t). 

By the next equation we can restrict the set of solutions of the differential equation (4) satisfying 

the limit condition at infinity: 

εu ′ (t) + g(t)u(t) = β(t), (5) 

∗ Supported by the Russian Foundation for Basic Research under grant No. 04-01-00578. 

Speaker: ZADORIN, A.I. 61 BAIL 2006 

1 

(1) 

(2) 

(3) 

✩ 

✪

A.I. ZADORIN: Numerical Method for the Blasius Equation on an Infinite Interval 

✬ 

✫ 

where g(t), β(t) are solutions of auxiliary singular Cauchy problems and can be found as series 

in (2t) −0.5 with a given accuracy. We can use equation (5) as the exact boundary condition for 

a finite interval. Then we return to the variable x and get the exact restriction of problem (3) 

to a finite interval as follows: 

εu ′′ (x) + [a(x) + x]u ′ (x) = f(x), 

u(0) = A, 

ε 

L u′ (L) + g(L)u(L) = β(L). 

Return to problem (2). Consider the iterative method 

v ′ n(x) = wn−1(x), vn(0) = 0, 

w ′′ 

n(x) + [vn(x) + x]w ′ n(x) = 0, wn(0) = −1, lim 

x→∞ wn(x) = 0. 

It is proved that method (7) has the property of convergence. At each iteration, we can transform 

problem (7) to a problem for a finite interval, as it was done for a linear problem. One can use 

a difference scheme to solve the problem obtained on a finite interval. Theoretical results are 

confirmed by results of numerical experiments. 

References 

[1] G.I. Shishkin, “Grid approximation of the solution to the Blasius Equation and of its Derivatives”, 

Computational Mathematics and Mathematical Physics, 41 (1), 37–54 (2001). 

[2] A.A. Abramov, N.B. Konyukhova, “Transfer of admissible boundary conditions from a singular 

point of linear ordinary differential equations”, Sov. J. Numer.Anal. Math. Modelling, 

1, (4), 245–265 (1986). 

[3] A.I. Zadorin, “The transfer of the boundary condition from the infinity for the numerical 

solution of second order equations with a small parameter” (in Russian), Siberian Journal 

of Numerical Mathematics, 2 (1), 21–36 (1999). 

Speaker: ZADORIN, A.I. 62 BAIL 2006 

2 

(6) 

(7) 

✩ 

✪

P. ZEGELING: An Adaptive Grid Method for the Solar Coronal Loop Model 

✬ 

✫ 

An Adaptive Grid Method for the Solar Coronal Loop Model 

Paul Zegeling 

Department of Mathematics, Faculty of Science, Utrecht University, Utrecht, The Netherlands 

zegeling@math.uu.nl 

Many interesting phenomena in plasma fluid dynamics can be described within the framework 

of magnetohydrodynamics. Numerical studies in plasma flows frequently involve simulations 

with highly varying spatial and temporal scales. As a consequence, numerical methods on 

uniform grids are inefficient to be used, since (too) many grid points are needed to resolve the 

spatial structures, such as boundary and internal layers, shocks, discontinuities, shear layers, or 

current sheets. For the efficient study of these phenomena, adaptive grid methods are needed 

which automatically track and spatially resolve one or more of these structures. An interesting 

application within this framework can be found in and around our sun. There are several cases 

for which we can expect steep boundary and internal layers (see figure 1). 

The first one deals with the expulsion of the magnetic flux by eddies in a solar magnetic 

field model. This model addresses the role of the magnetic field in a convecting plasma and the 

distortion of the field by cellular convection patterns for various (small) values of the resistivity 

(magnetic diffusion coefficient). This situation is of importance around convection cells just 

below the solar photosphere. Steep boundary and internal layers are formed when the magnetic 

induction reaches a steady-state configuration. 

In this presentation we discuss another phenomenon that takes place a little bit farther from 

the solar interior, viz., in the solar corona. It is known that the temperature gradually decreases 

from the center of the sun down to values of around 10 4 degrees Kelvin at the foot of the corona 

(see figure 2). From that point, however, it surprisingly increases dramatically again up to several 

millions of degrees Kelvin forming a non-trivial transition zone (boundary layer) between the 

photosphere and the chromosphere. Moreover, because of the extreme temperatures, the solar 

corona is highly structured with closed magnetic structures which are generally known as coronal 

loops. It can be derived that the temperature T and pressure distribution P in the loop as a 

function of a mass-coordinate z satisfy the following PDE model: 

5 P 

2 T 

∂T ∂P P ∂ 3 

− = ɛ (T 2P 

∂t ∂t T ∂z ∂T 

∂z ) + EH − P 2 χ(T), (1) 

where P(z,t) = P0(t) − µz, EH is a heating function, χ(T) the radiative loss function and 

ɛ a small parameter representing the thermal conductivity in the loop. Near the base of the 

loop there are two adjacent boundary layers where the temperature increases very quickly when 

moving upward in the loop; in these thin layers the pressure in nearly constant. We will examine 

the nature of this special boundary layer via the theory of significant degenerations and also in 

terms of a dynamical system of the steady-state of PDE (1) in which a non-trivial saddle-center 

connection occurs. A complicating factor is the fact that we also need to take into account the 

so-called loop-condition: 

� 1 T 

L = 2MR 

0 P 

dz = constant, (2) 

with gasconstant R, total mass in the loop M and (half) looplength L. 

To support and confirm the theory, we have applied an adaptive grid technique, based on 

an equidistribution principle with additional smoothing properties, to numerically simulate the 

forming of the thin boundary layer. 

Speaker: ZEGELING, P. 63 BAIL 2006 

1 

✩ 

✪

P. ZEGELING: An Adaptive Grid Method for the Solar Coronal Loop Model 

✬ 

✫ 

Convection cells 

Coronal loops 

Figure 1: Convection cells and coronal loops in, and on top of, the sun in the form of magnetic 

field lines. 

Figure 2: The temperature distribution and the steep transition zone in the outer layers of the 

sun. 

References 

[1] P.A. Davidson, An introduction to magneto-hydrodynamics, Cambridge Univ. Press (2001). 

[2] J.W. Pakkert, P.C.H. Martens and F. Verhulst, The thermal stability of coronal loops by 

nonlinear diffusion asymptotics, Astron. Astrophys., 179, 285-293 (1987). 

[3] F. Verhulst and P.A. Zegeling, An asymptotic-numerical approach to the coronal loop problem, 

Math. Meth. in the Appl. Sciences, 13, 431-439 (1990). 

[4] N.O. Weiss, The expulsion of magnetic flux by eddies, Proc. of Roy. Soc. A, 293, 310-328 

(1966). 

[5] P.A. Zegeling, On resistive MHD models with adaptive moving meshes, J. of Scientific 

Computing, 24, 263-284 (2005). 

[6] P.A. Zegeling and R. Keppens, Adaptive method of lines for magnetohydrodynamic PDE 

models, Chapter 4, 117-137, in Adaptive method of lines, Chapman & Hall/CRC Press 

(2001). 

Speaker: ZEGELING, P. 64 BAIL 2006 

2 

✩ 

✪

Contributed Presentations

F. ALIZARD, J.-CH. ROBINET: Two-dimensional temporal modes in nonparallel 

flows 

✬ 

✫ 

Two-dimensional temporal modes in nonparallel flows 

F. Alizard and J.-Ch. Robinet 

SINUMEF Laboratory, ENSAM-PARIS 151, Boulevard de l’Hôpital, 75013 PARIS, FRANCE 

Frederic.Alizard@paris.ensam.fr, 


To describe the evolution of a two-dimensional wavepacket in flow such as growing boundary 

layer, the classical stability approach is based on the assumption of locally parallel or weakly non 

parallel base flow. These approach was used by Gaster (1982) to characterize the spatio-temporal 

dynamic of a perturbation in a boundary layer. However this approach could failed if a wave 

length of any perturbation is larger than a characteristic length of the spatial inhomogeneity of 

the base flow. Consequently a more general eigenvalue problem was developed by some authors 

as Lin & Malik (1995), Theofilis et al. (2000) and Erhenstein & Gallaire (2005) where two spatial 

directions are inhomogeneous. In this abstract we will focus on two-dimensional modes for a 

convectively unstable attached boundary layer. After the description of the numerical method, 

preliminary results on temporal linear stability modes depending on two space directions are 

computed for a boundary layer flow along a flat plat. 

2. Generalities and Basic Flow 

At the conference we will be interested exclusively in computations of convective instabilities 

in nonparallel flows. Two types of flows will be studied: a flat plate boundary-layer without 

pressure gradient, shown as a preliminary result and a separated boundary-layer. The twodimensional 

Navier-Stokes equations for incompressible fluids in the stream function-vorticity 

formulation are considered: 

∂ω 

∂ω 1 

+ u∂ω + v = 

∂t ∂x ∂y Re 

� 

∂2ω ∂x2 + ∂2ω ∂y2 � 

and ∆ψ = ω, (1) 

where ω and ψ are the vorticity and the stream function respectively. System (1) is closed 

by classical boundary conditions on ψ and ω (Briley (1971)). A second order finite differences 

scheme was used for the vorticity transport equation as well as the poisson equation of stream 

function. An A.D.I algorithm has been employed to solve the transport equation and the poisson 

equation. Preliminary results, shown on part 4, are realized on an attached boundary layer at 

Re=610, with a grid (450x200). 

3. Linearized Perturbed Flow and Numerical Procedure 

The proposed stability analysis is based on the classical perturbations technique where the 

instantaneous flow (q) is the superposition of the basic flow (Q), data of this problem, and 

unknown fluctuations (ˆq): q(x, y, t) =Q(x, y)+ˆq(x, y)exp(−iΩt), where Ω is the circular global 

frequency of the fluctuation. The two-dimensional generalized eigenvalue problem is obtained 

by the linearized Navier Stokes equations: 

� � 

∆2d 

div(û) = 0 and − U.grad û − gradU.û − gradˆp + iΩû =0, (2) 

Re 

1 

Speaker: ALIZARD, F. 67 BAIL 2006 

✩ 

✪

F. ALIZARD, J.-CH. ROBINET: Two-dimensional temporal modes in nonparallel 

flows 

✬ 

✫ 

Figure 1: Discrete global spectrum from a convectively 

instable boundary layer. The real 

part of the eigenvalue is represented on the Xaxis 

and the imaginary part on the Y-axis. 

Figure 2: Real part of the horizontal velocity 

eigenfunction of a discrete eigenvalue from the 

spectrum ( Fig. 1). 

with U(x, y) =[U, V ] t and û(x, y) =[û, ˆv] t . The boundary conditions on velocity disturbances 

and constraints on pressure (T. N. Phillips & G. W. Roberts (1993), C. Canuto et al. (1988)) 

complete the eigenvalue value problem. The partial differential stability equations (2) are discretized 

using an algorithm based on the collocation method based on Chebyshev Gauss-Lobatto 

grid. The algebraic eigenvalue problem (A − ΩB) X = 0 is solved by the QZ algorithm. 

4. Two-Dimensional Temporal Modes, results and perspectives 

The figure 1 represents a global spectrum of the boundary layer. The stable discrete eigenvalues 

appear in concordance with the fact a boundary layer is globally stable. These discrete values 

represent spatio-temporal convective modes as it can been shown fig. 2 (structure similar to a 

spatial exponential growth). At the conference, this analysis will be also applied on a separated 

incompressible boundary layer which is not absolutely unstable but only convectively unstable. 

References 

[1] M. Gaster , “The development of a two-dimensional wavepacket in a growing boundary 

layer ”, Proc. R. Soc. London, 3, 317–332 (1982). 

[2] Lin R.S. and Malik M.R , “On the stability of attachment-line boundary layers. ”, J. Fluid 

Mech., 3,239–255 (1995). 

[3] Theofilis V. Hein S and Dallmann U , “On the origins of unsteadiness and three dimensionality 

in a laminar separation bubble ”, Proc. R. Soc. London, 3,3229–3246 (2000). 

[4] U. Ehrenstein and F. Gallaire , “On two dimensional temporal modes in spatially evolving 

open flows: the flat-plate boundary layer ”, J. Fluid Mech., (2005). 

[5] Briley WR., “A numerical study of laminar separation bubbles using the Navier-Stokes 

equations.”, J. Fluid Mech., 3,713–736 (1971). 

[6] Timothy N. Phillips and Gareth W.Roberts, “The Treatment of Spurious Pressure Modes in 

Spectral Incompressible Flow Calculations.”, Journal of Computational Physics, 3,150–164 

(1993). 

[7] C. Canuto and M. Y. Hussaini and A. Quarteroni and T. A. Zang, “Spectral Methods in 

Fluids Dynamics”, Springer, (1988). 

Speaker: ALIZARD, F. 68 BAIL 2006 

2 

✩ 

✪

TH. ALRUTZ, T. KNOPP: Near-wall grid adaptation for wall functions 

✬ 

✫ 

Near-wall grid adaptation for wall-functions 

Th. Alrutz, T. Knopp 

Institute of Aerodynamics and Flow Technology 

DLR (German Aerospace Center) 

Bunsenstr. 10, 37073 Göttingen, Germany 

Thomas.Alrutz@dlr.de 

A near-wall grid adaptation method for RANS turbulence modelling using wall-functions is proposed. 

Universal wall functions allow a considerable acceleration of the flow solver, but their predictions 

may deviate from the results with near-wall integration in flow situations which are beyond the underlying 

modelling assumptions of wall-functions. In aerodynamic flows these include (i) stagnation points 

(flow impingement) and subsequent transition from laminar to turbulent flow, (ii) large surface curvature 

in conjunction with strong pressure gradients (as can be observed around leading edge of an airfoil), (iii) 

strong adverse pressure gradients leading to separation, and (iv) regions of separated flow. Boundary 

layer theory shows that the deviation from the universal wall-law becomes more significant as the walldistance 

of the first node above the wall is increased. But close agreement with the universal wall-law 

can be achived if the first node is close enough to the wall. 

To reduce the modelling error of universal wall-functions, a near-wall grid-adaptation technique is pro- 

Ýposed. �Ý 

Regions of strong surface curvature, large pressure gradients and recirculating flow are detected 

by a flow and geometry based sensor. Then in critical regions, the nodes of the prismatic nearwall 

grid are shifted towards the wall so that a user-specified target value forÝ is ensured, where 

Ù��with wall-distance of the first nodeÝ , friction velocityÙ�and viscosity�. This 

approach is applied to a transonic airfoil flow with shock induced separation [5] and to a subsonic highlift 

airfoil close to stall [6]. Thereby, the predictions around the leading edge (suction peak) and of the 

separation point and the recirculation region can be improved significantly. 

y + 

100 

80 

60 

40 

20 

y + (1) = 20 

y + (1) = 40 

y + (1) = 60 

0 

0 0.25 0.5 0.75 1 

x/c 

y + 

60 

40 

20 

y + (1) = 20 

y + (1) = 40 

y + (1) = 60 

0 

0 0.25 0.5 0.75 1 

x/c 

Figure 1: RAE2822 case 10 [5]: Distribution ofÝ for Menter SST model [4] without adaptation 

(left) and withÝ-adaptation of the structured (prismatic) near-wall grid (right). 

References 

[1] T. Knopp, T. Alrutz and D. Schwamborn, “A grid and flow adaptive wall-function method for RANS 

turbulence modelling”, Journal of Computational Physics (submitted). 

Speaker: ALRUTZ, TH 69 BAIL 2006 

1 

✩ 

✪

TH. ALRUTZ, T. KNOPP: Near-wall grid adaptation for wall functions 

✬ 

✫ 

0.006 exp. 

low-Re 

y + (1) = 1 

y + (1) = 20 

0.004 

c f 

0.002 

0 

y + (1) = 40 

y + (1) = 60 

c f = 0 

0 0.25 0.5 0.75 1 

x/c 

0.006 exp. 

low-Re 

y + (1) = 1 

y + (1) = 20 

0.004 

c f 

0.002 

0 

y + (1) = 40 

y + (1) = 60 

c f = 0 

0 0.25 0.5 0.75 1 

x/c 

Figure 2: RAE2822 case 10 [5]: Prediction for skin friction coefficient�for Spalart-Allmaras-Edwards 

model [3] without adaptation (left) and withÝ -adaptation of the prismatic near-wall grid (right). 

c p 

-4.2 

-3.8 

-3.4 

-3 

exp. 

low-Re 

y + (1) = 12 

y + (1) = 24 

y + (1) = 40 

y + (1) = 80 

0 0.05 0.1 0.15 

x/c 

Ý ÆÝ Ü×�Ô� Ü×�Ô� 

SA-Edwards Menter SST 

low-Re 33 0.771 0.866 

1 33 0.770 0.866 

4 28 0.755 0.863 

7 26 0.759 0.868 

12 24 0.761 0.861 

24 21 0.787 0.861 (0.864) 

50 19 (0.771) 0.881 (0.873) 

70 17 (0.788) 0.903 (0.867) 

Figure 3: A-airfoil [6]: Detail of pressure coefficientÔfor SST model [4] on adapted grid (left). Right: 

Prediction of the separation point without adaptation and withÝ -adaptation (values in brackets). 

[2] Th. Alrutz, “Hybrid grid adaptation in TAU”, In: MEGAFLOW - Numerical flow simulation for 

aircraft design, Notes on Numerical Fluid Mechanics and Multidisciplinary Design. N. Kroll and 

J.K. Fassbender, Eds., (2005). 

[3] J.R. Edwards and S. Chandra, “Comparison of eddy viscosity-transport turbulence models for threedimensional, 

shock separated flowfields”, AIAA Journal, 34, 756–763 (1996). 

[4] F.R. Menter, “Zonal two equation�/�turbulence models for aerodynamic flows”, AIAA Paper 

1993-2906, (1993). 

[5] P. H. Cook, M. A. McDonald and M.C.P. Firmin, “Aerofoil RAE 2822 - Pressure distributions and 

boundary layer and wake measurements”, AGARD Advisory Report AR-138, A6.1-A6.77 (1979). 

[6] Ch. Gleyzes, “Opération décrochage - Résultats de la 2ème campagne d’essais à F2 – Mesures de 

pression et vélocimétrie laser”, RT-DERAT 55/5004 DN, ONERA, (1989). 

Speaker: ALRUTZ, TH 70 BAIL 2006 

2 

✩ 

✪

TH. APEL, G. MATTHIES: A family of non-conforming finite elements of arbitrary 

order for the Stokes problem on anisotropic quadrilateral meshes 

✬ 

✫ 

A family of non-conforming finite elements of arbitrary order for the Stokes 

problem on anisotropic quadrilateral meshes 

Thomas Apel 1 and Gunar Matthies 2 

1 Institut für Mathematik und Bauinformatik, Universität der Bundeswehr München 

2 Fakultät für Mathematik, Ruhr-Universität Bochum, 

The solution of the Stokes problem in polygonal or polyhedral domains shows in general a 

singular behaviour near corners and edges of the domain. Both edge singularities and layers 

are anisotropic phenomena since the solution changes only slightly in one direction while the 

derivatives in the perpendicular direction(s) are large. These anisotropic behaviour can be well 

approximated on anisotropic triangulations. 

Let Ω ⊂ R 2 be a bounded polygonal domain. We consider the Stokes problem 

−△u + ∇p = f in Ω, 

div u = 0 in Ω, 

u = 0 on ∂Ω, 

where u and p are the velocity and the pressure, respectively, while f is a given force. 

We solve this problem by using finite element methods on triangulations with special properties. 

Let the domain Ω be partitioned by an admissible triangulation which consists of shape 

regular and isotropic macro-cells. Each macro-cell is further refined by applying admissible 

patches which are adapted to boundary layers and corner singularities, respectively. For a 

detailed description of such meshes, we refer to [1]. 

We are interested in solving the Stokes problem by non-conforming finite element methods 

of higher order. To this end, we use the families with finite element pairs of arbitrary order 

which were given recently in [2]. Each pair consists of a non-conforming space of order r for 

approximating the velocity and a discontinuous, piecewise polynomial pressure approximation 

of order r − 1. 

For the stability of finite element methods for solving the Stokes problem and its relatives, it 

is necessary that the discrete spaces for the velocity and the pressure fulfil an inf-sup condition. 

In order to get error estimates with constants which are independent of the aspect ratio of the 

underlying mesh, it is important that on the one hand the inf-sup constant is independent of 

the aspect ratio and that on the other hand the approximation error and the consistency error 

can be bounded by expressions whose constants don’t depend on the aspect ratio. 

Considering the families given in [2], we will show that we obtain for one family optimal 

error estimates on anisotropic meshes, i.e., the constant are independent of the aspect ratio. 

The other families give only error estimates with constant which depend on the aspect ratio, 

i.e., the constants blow up with increasing aspect ratio. 

We will show by means of numerical results how the inf-sup constant behaves on special 

families of triangulation with increasing aspect ratio. 

References 

[1] Th. Apel, S. Nicaise, The inf-sup condition for low order elements on anisotropic meshes, 

CALCOLO, 41, 89–113 (2004). 

[2] G. Matthies, Inf-sup stable nonconforming finite elements of higher order on quadrilaterals 

and hexahedra. Bericht Nr. 373, Fakultät für Mathematik, Ruhr-Universität Bochum 

(2005). 

Speaker: MATTHIES, G. 71 BAIL 2006 

1 

✩ 

✪

M. BAUSE: Aspects of SUPG/PSPG and GRAD-DIV Stabilized Finite Element 

Approximation of Compressible Viscous Flow 

✬ 

✫ 

Aspects of SUPG/PSPG and GRAD-DIV Stabilized 

Finite Element Approximation of Compressible Viscous Flow 

Markus Bause 

Institut für Angewandte Mathematik, Universität Erlangen-Nürnberg 

Martensstr. 3, 91058 Erlangen 

bause@am.uni-erlangen.de 

In this contribution various aspects of a theoretical analysis and numerical study of 

SUPG/PSPG and grad-div stabilized finite element approximations (cf. [3]) of steady 

and unsteady compressible isothermal viscous flow are addressed. After temporal discretization 

of the time variable by means of the implicit Euler method, the equations of 

compressible viscous flow are solved by an iteration between a generalized Oseen problem 

for the velocity and a hyperbolic transport equation for the perturbation from the mean 

density (cf. [1]). Such a splitting-type approach seems attractive for computations because 

it offers a reduction to simpler problems for which highly refined numerical methods either 

are known or can be built from existing discretization techniques for similar equations, 

and because of the guidance that can be drawn from an existence theory based on it. 

In the case of steady motions of a compressible viscous gas, decribed by the equations 

∇ · (ρv) = 0 , ρv · ∇v − µ∆v − (λ + µ)∇∇ · v + ∇p = ρf , 

p = kρ , v|∂Ω = 0 , 

� 

Ω ρ dx = M 

the iteration for solving this system reads as: Let ρ = 1 + σ. Put v0 = 0, σ0 = 0. For 

given vn, σn compute vn+1, σn+1 by 

(i) solving the generalized Oseen-system 

∇ · vn+1 = −∇ · (σnvn) , 

(1 + σn)vn · ∇vn+1 + 1 

2 ∇ · ((1 + σn)vn)vn+1 − µ∆vn+1 + ∇πn+1 = (1 + σn)f , 

vn+1|∂Ω = 0, 

� 

Ω πn+1 dx = 0 

(ii) and, then, solving the hyperbolic transport equation 

kσn+1 + (λ + 2µ)∇ · (σn+1vn+1) = πn+1 − µ∇ · vn+1 . 

For the approximation of the separated Oseen problem, SUPG/PSPG and grad-div stabilized 

higher order finite techniques based on LBB-stable elements are used and analyzed. 

The transport equation is discretized by SUPG stabilized finite element methods. Error 

estimates are provided. Numerical results for realistic steady and unsteady benchmark 

problems (driven cavity flow, flow over a backward facing step and DFG-benchmark) are 

given for a large scale of Reynolds numbers. 

References 

[1] M. Bause, J. G. Heywood, A. Novotny and M. Padula, On some approximation schemes 

for steady compressible viscous flow, J. Math. Fluid Mech., 5 (2003), pp. 201–230. 

[2] M. Bause, Stabilized finite element approximation of compressible viscous flow, to appear, 2006. 

[3] T. Gelhard et al., Stabilized finite element schemes with LBB-stable elements for incompressible 

flow, J. Comput. Appl. Math. 177 (2005), pp. 243–267. 

Speaker: BAUSE, M. 72 BAIL 2006 

✩ 

✪

L. BOGUSLAWSKI: Sheare Stress Distribution on Sphere Surface at Different Inflow 

Turbulence 

✬ 

✫ 


SHEARE STRESS DISTRIBUTION ON SPHERE SURFACE AT DIFFERNT INFLOW 

TURBULENCE 

L. Bogusławski 

Poznań University of Technology, Chair of Thermal Engineering, 60 965 Poznań, Poland; 

e-mail: leon.boguslawski@put.poznan.pl 

Momentum and heat transfer processes on surfaces are sensitive on intensity of turbulence of flow 

above surface. Descriptions of share stress or heat transfer distributions usually assume certain level of 

intensity of turbulence of free flow which overflows surface. When structure of flow is formed as 

developed flow for typical channels one can assume that turbulence level and structure of turbulent 

flow is repeated. In such case detailed knowledge of flow turbulence is not necessary because 

Reynolds number indicated average flow similarity and similarity of turbulence by the way. 

Unfortunately for most technical applications level of turbulence and its structure can vary in wide 

borders. More over this level is difficult to prediction based on channel geometry especially when any 

promoters of turbulence occur. Experimental data indicate that increase of turbulence intensity cause 

increasing heat and momentum transfer coefficients even when average flow velocity does not change. 

To estimate influence of turbulence on local distribution of shear stress a sphere was chosen as the 

simplest, repeatable geometry. Flow was generated by a free round jet. Level of turbulence, in the jet 

axis, change from about 0.5% near the nozzle outlet till about 20% far away from the nozzle. 

Changing of the average flow velocity at the nozzle outlet it is possible to keep constant value of 

velocity at different distances from the nozzle outlet. Turbulent fluctuations of a flow velocity were 

measured by means of a constant temperature anemometer. The local shear stress distribution on 

sphere surface was measured by used a surface sensor connected to the constant temperature 

anemometer as well. 

For low level of turbulence the shear stress distribution was similar to literature data. Increase of 

turbulence cause increase of a local value of shear stress. The local share stress distributions and its 

turbulent fluctuations for two chosen turbulence level are presented in figure 1 as an example. The 

Reynolds number of average flow was the some for both cases. Increasing of inflow turbulence level 

cause increasing of local, average shear stress distributions and equalizing distribution of ‘rms’ of 

turbulent fluctuations on rather high level. 

o 

3.0 

2.5 

2.0 

1.5 

1.0 

0.5 

d = 0.03 m, Re = 3 10 4 , Tu = 2.5 % 

0.0 

0.0 

0 30 60 90 120 150 180 

φ 

3.5 

3.0 

2.5 

2.0 

1.5 

1.0 

0.5 

' rms / ' orms 

0.0 

0.0 

0 30 60 90 120 150 180 

Figure 1. Distribution of the local average shear stress and its turbulent fluctuations on the sphere 

surface at different inflow turbulence level. 

o 

3.0 

2.5 

2.0 

1.5 

1.0 

0.5 

d = 0.03 m, Re = 3 10 4 , Tu = 12.3 % 

Speaker: BOGUSLAWSKI, L. 73 BAIL 2006 

φ 

3.5 

3.0 

2.5 

2.0 

1.5 

1.0 

0.5 

' ms / ' orms 

✩ 

✪

L. BOGUSLAWSKI: Sheare Stress Distribution on Sphere Surface at Different Inflow 

Turbulence 

✬ 

✫ 

External turbulence of inflow jet increasing flow turbulence near sphere surface where flow 

accelerate. This cause increase of local shear stress and its fluctuations. Increasing of shear stress 

transfer fluctuations indicated that turbulence of external flow intensify momentum processes near the 

wall of sphere. In flow deceleration zone of flow >90 o for both presented in figure 1 turbulence levels 

the time average value and its fluctuations rise in this some way. Both runs near parallel. For 

acceleration zone of flow near sphere ( ~60 o ) shear stress reach maximum value. In this some region 

of flow the flow acceleration reduced turbulent fluctuations of momentum transfer generated by 

external flow turbulence. 

Distributions for different level of turbulence without average velocity change will be presented and 

discussed. For such conditions only influence of turbulence are indicated. 

Analysis of the power spectrum of turbulent fluctuations of shear stress on sphere surfaces for 

different flow intensity at chosen locations will be presented and will be compared with the power 

spectrum of inflow turbulence. 

References 

[1] S.Whitaker. Forced convection heat transfer correlation for flow in pipes, past flat plates, single 

cylinders, single spheres and flow in packed beds and tube bundles, J. of AICHE, vol. 18, 361- 

371, 1972. 

[2] L. Bogusławski. Losses of heat from sphere surface at different inflow conditions (in polish), 

XVI Thermodinamics Conference, Kołobrzeg , vol. 1, 135-140, 1996. 

[3] L. Bogusławski. Measurements technique of stagnation point heat transfer and its 

fluctuations by means of a constant temperature sensor, Proceedings of Turbulent Heat 

Transfer Conference, 1-6, San Diego, May 1996. 

[4] H. Giedt. Trans. ASME, 71, 375, 1949 

Speaker: BOGUSLAWSKI, L. 74 BAIL 2006 

✩ 

✪

A.CANGIANI, E.H.GEORGOULIS, M. JENSEN: Continuous-Discontinuous Finite 

Element Methods for Convection-Diffusion Problems 

✬ 

✫ 

Continuous-Discontinuous Finite Element Methods for 

Convection-Diffusion Problems 

Abstract 

Andrea Cangiani, Emmanuil H. Georgoulis and Max Jensen 

February 14, 2006 

Standard (conforming) finite element approximations of convection-dominated convection-diffusion 

problems often exhibit poor stability properties that manifest themselves as non-physical oscillations 

polluting the numerical solution. Various techniques have been proposed for the stabilisation of finite 

element methods (FEMs) for convection-diffusion problems, see for example, Morton [12] and Roos, 

Stynes and Tobiska [14] for a complete survey. Common techniques are Petrov-Galerkin methods, 

like the streamline upwind Petrov-Galerkin (SUPG) method introduced by Hughes and Brooks [8], 

exponential fitting [13], ad hoc meshing, like graded meshes [15] and Shishkin type meshes [11], and 

adaptive mesh refinement (see, e.g., [5] and [6]). More recently, the residual-free bubble method of 

Brezzi et al [2], [1], [4] and the variational multiscale method of Hughes and co-authors [9],[10]. 

During the last decade, families of discontinuous Galerkin finite element methods (DGFEMs) 

have been proposed for the numerical solution of convection-diffusion problems, due to many attractive 

properties they exhibit. In particular, DGFEMs admit good stability properties, they offer 

flexibility in the mesh design (irregular meshes are admissible) and in the imposition of boundary 

conditions (Dirichlet boundary conditions are weakly imposed), and they are increasingly popular 

in the context of hp-adaptive algorithms. 

The above mentioned attractive features of DGFEMs come at the price of the higher number 

of degrees of freedom used. For instance, for piece-wise linear approximations in d-dimensions, 

DGFEMs require 2 d times more degrees of freedom than conforming FEMs and their stabilised 

variants. The relative difference in the number of degrees of freedom reduces when considering 

higher order local polynomial degree in the approximations; nevertheless, the conforming FEMs 

always require less degrees of freedom than their DGFEM counterparts. 

The issue of the number of degrees of freedom required by DGFEMs has recently been addressed 

by T.J.R. Hughes and co-workers in [7], where the Multiscale Discontinuous Galerkin (MDG) finite 

element method framework is introduced. MDG uses local, element-wise problems to develop a 

transformation between the degrees of freedom of the discontinuous space and a related, smaller, 

continuous space. The transformation enables a direct construction of the global matrix problem 

in terms of the degrees of freedom of the continuous space. It is proved that the method has the 

accuracy and stability of the original DGFEM. The drawback of this approach is the computational 

overhead of the solution of the local element-wise problems. 

We propose a numerical scheme for linear convection-diffusion problems which couples continuous 

and discontinuous Galerkin finite elements (CG-DG) in a different manner. Depending on the 

local variation of the solution, the scheme locally uses either the computationally less expensive 

continuous finite elements or the computationally more costly but also more stable discontinuous 

Galerkin discretisation. Given that good stability properties are required near features of almost 

(d − 1)-dimensional character (such as boundary and/or interior layers) discontinuous Galerkin 

discretisation is only used in the neighborhoods of such features, whereas standard conforming FEM 

is used away from the layers. Thus, the increased overhead from the use of DGFEMs is balanced by 

their minimal use, limited to the neighborhoods of layers. 

This work introduces the continuous-discontinuous Galerkin (CG-DG) finite element method, 

and presents the first results in the analysis of this approach. In particular, we derive an a-priori 

error analysis of the CG-DG blending technique, along with numerical experiments that evaluate 

the accuracy and efficiency of the CG-DG approach in practice. 

Speaker: GEORGOULIS, E.H. 75 BAIL 2006 

1 

✩ 

✪

A.CANGIANI, E.H.GEORGOULIS, M. JENSEN: Continuous-Discontinuous Finite 

Element Methods for Convection-Diffusion Problems 

✬ 

✫ 

References 

[1] Brezzi, F., Marini, D., and Süli, E. Residual-free bubbles for advection-diffusion problems: 

the general error analysis. Numer. Math. 85, 1 (2000), 31–47. 

[2] Brezzi, F., and Russo, A. Choosing bubbles for advection-diffusion problems. Math. Models 

Methods Appl. Sci. 4, 4 (1994), 571–587. 

[3] Brooks, A. N., and Hughes, T. J. R. Streamline upwind/Petrov-Galerkin formulations 

for convection dominated flows with particular emphasis on the incompressible Navier-Stokes 

equations. Comput. Methods Appl. Mech. Engrg. 32, 1-3 (1982), 199–259. FENOMECH ’81, 

Part I (Stuttgart, 1981). 

[4] Cangiani, A., and Süli, E. Enhanced RFB method. Numer. Math. 101(2) (2005), 273–308. 

[5] Eriksson, K., Estep, D., Hansbo, P., and Johnson, C. Introduction to adaptive methods 

for differential equations. In Acta numerica, 1995, Acta Numer. Cambridge Univ. Press, 

Cambridge, 1995, pp. 105–158. 

[6] Giles, M. B., and Süli, E. Adjoint methods for PDEs: a posteriori error analysis and 

postprocessing by duality. In Acta numerica, 2002, vol. 11 of Acta Numer. 2002, pp. 145–236. 

[7] Hughes, Thomas J. R.and Scovazzi, G., Bochev, P. B., and Buffa, A. A multiscale 

discontinuous Galerkin method with the computational structure of a continuous galerkin 

method. ICES Report 05-16 (2005). 

[8] Hughes, T. J. R., and Brooks, A. A multidimensional upwind scheme with no crosswind 

diffusion. In Finite element methods for convection dominated flows (Papers, Winter Ann. 

Meeting Amer. Soc. Mech. Engrs., New York, 1979), vol. 34 of AMD. Amer. Soc. Mech. Engrs. 

(ASME), New York, 1979, pp. 19–35. 

[9] Hughes, T. J. R., Feijóo, G. R., Mazzei, L., and Quincy, J.-B. The variational multiscale 

method—a paradigm for computational mechanics. Comput. Methods Appl. Mech. Engrg. 

166, 1-2 (1998), 3–24. 

[10] Hughes, T. J. R., and Stewart, J. R. A space-time formulation for multiscale phenomena. 

J. Comput. Appl. Math. 74, 1-2 (1996), 217–229. 

[11] Madden, N., and Stynes, M. Efficient generation of shishkin meshes in solving convectiondiffusion 

problems. Preprint of the Department of Mathematics, University College, Cork, 

Ireland no. 1995-2 (1995). 

[12] Morton, K. W. Numerical solution of convection-diffusion problems, vol. 12 of Applied Mathematics 

and Mathematical Computation. Chapman & Hall, London, 1996. 

[13] O’Riordan, E., and Stynes, M. A globally uniformly convergent finite element method for 

a singularly perturbed elliptic problem in two dimensions. Math. Comp. 57, 195 (1991), 47–62. 

[14] Roos, H.-G., Stynes, M., and Tobiska, L. Numerical Methods for Singularly Perturbed 

Differential Equations. Springer-Verlag, Berlin, 1996. Convection-diffusion and flow problems. 

[15] Schwab, C., and Suri, M. The p and hp versions of the finite element method for problems 

with boundary layers. Math. Comp. 65, 216 (1996), 1403–1429. 

2 

Speaker: GEORGOULIS, E.H. 76 BAIL 2006 

✩ 

✪

C. CLAVERO, J.L. GRACIA, F. LISBONA: A second order uniform convergent 

method for a singularly perturbed parabolic system of reaction-diffusion type 

✬ 

✫ 

A second order uniform convergent method for a singularly perturbed 

parabolic system of reaction–diffusion type ∗ 

C. Clavero, J.L. Gracia and F. Lisbona 

Department of Applied Mathematics 

University of Zaragoza. Spain 

clavero@unizar.es, jlgracia@unizar.es, lisbona@unizar.es 

Abstract 

In this work we are interested in solving singularly perturbed parabolic boundary value problems 

given by 

⎧ 

⎪⎨ L�ε�u ≡ 

⎪⎩ 

∂�u 

∂t + Lx,�ε�u = � f, (x, t) ∈ Q = Ω × (0, T ] = (0, 1) × (0, T ], 

�u(0, t) = �g0(t), �u(1, t) = �g1(t), ∀t ∈ [0, T ], 

(1) 

�u(x, 0) = �0, ∀x ∈ Ω, 

where 

Lx,�ε ≡ 

� 

−ε1 ∂2 

∂x 2 

−ε2 ∂2 

∂x 2 

� 

� � 

a11(x) a12(x) 

+ A, A = 

, 

a21(x) a22(x) 

with 0 < ε1 ≤ ε2 ≤ 1. The components of the functions �g0(t), �g1(t), the right hand side 

�f(x, t) = (f1(x, t), f2(x, t)) T and the matrix A are supposed smooth enough. The positivity 

condition 

ai1 + ai2 ≥ αi > 0, aii > 0, i = 1, 2, aij ≤ 0 if i �= j, (2) 

is satisfied by matrix A. Otherwise, we consider the transformation �v(x, t) = �u(x, t)e −α0t with 

α0 > 0 sufficiently large so that (2) holds. 

This problem is a simple model of the classical linear double–diffusion model for saturated 

flow in fractures porous media (Barenblatt system) developed in [1]. The first equation describes 

the flow in the matrix and the second one the flow in the fracture system. The coupling terms 

are given by the matrix 

� � 

1 −1 

A = 

, 

−1 1 

which describes the interchange of fluid between the two systems. The permeabilities ε1 and ε2 

in these equations could be very small and also they can have different magnitudes. 

It is well known that the exact solution of these problems has a multiscale character, i.e., 

there are narrow regions (the boundary layer regions) where the solution has strong gradients 

and in the rest of the domain the solution varies smoothly. Therefore, it is necessary to dispose 

of efficient numerical methods (uniformly convergent methods) to approximate the solution 

independently of the values of the diffusion parameters ε1 and ε2. 

Recently some papers study uniform convergent numerical methods to solve singularly perturbed 

elliptic linear systems on a special piecewise uniform Shishkin mesh. In the analysis of 

these type of problems the three following different cases can be distinguished depending on the 

relation between the singular perturbation parameters ε1 and ε2: 

[i.] ε1 = ε, ε2 = 1, [ii.] ε1 = ε2 = ε [iii.] ε1, ε2 arbitrary. 

∗ This research will be partially supported by the project MEC/FEDER MTM 2004-01905 and the Diputación 

General de Aragón. 

1 

Speaker: CLAVERO, C. 77 BAIL 2006 

✩ 

✪

C. CLAVERO, J.L. GRACIA, F. LISBONA: A second order uniform convergent 

method for a singularly perturbed parabolic system of reaction-diffusion type 

✬ 

✫ 

In [7] and [6] it was developed a first order uniformly convergent method in cases [ii.] and [iii.] 

respectively. In [8], [4] and [5] a second order uniformly convergent scheme was obtained for 

cases [i.], [ii.] and [iii.] respectively. 

In [3] a decomposition of the exact solution of problem (1), into its regular and singular 

components, was given. From this decomposition it follows which is the asymptotic behaviour 

of each one of these components with respect to the singular perturbation parameters ε1 and ε2. 

Moreover, in that work a first order in time and second order in space (except by a logarithmic 

factor) uniformly convergent method was developed, using the classical Euler and central differences 

discretizations respectively. In order to increase the order of uniform convergence of this 

numerical scheme, here we replace the Euler scheme by the Crank-Nicolson method in the time 

discretization. This method has been used in the framework of singularly perturbed problem; for 

instance, in [2] to solve 1D evolutionary problems of convection–diffusion type. Some numerical 

experiments will be showed, which illustrate in practice the improvement in the uniform order 

of convergence of the new scheme. 

Keywords: Singular perturbation, reaction-diffusion problems, uniform convergence, coupled 

system, Shishkin mesh. 

AMS classification: 65N12, 65N30, 65N06 

References 

[1] G.I. Barenblatt, I.P. Zheltov and I.N. Kochina, “Basic concepts in the theory of seepage of 

homogeneous liquids in fissured rocks”, J. Appl. Math. and Mech., 24, 1286–1303 (1960). 

[2] C. Clavero, J.L. Gracia and J.C. Jorge, “Second order numerical methods for one– 

dimensional parabolic singularly perturbed problems with regular layers”, Numerical Methods 

for Partial Differential Equations, 21 149–169 (2005). 

[3] J.L. Gracia and F. Lisbona “A uniformly convergent scheme for a system of reaction– 

diffusion equations”, submitted. 

[4] T. Linß and N. Madden, “An improved error estimate for a numerical method for a system 

of coupled singularly perturbed reaction–diffusion equations”, Comp. Meth. Appl. Math. 3 

417–423 (2003). 

[5] T. Linß and N. Madden, “Accurate solution of a system of coupled singularly perturbed 

reaction-diffusion equations”, Computing, 73, 121–133 (2004). 

[6] N. Madden and M. Stynes, “A uniformly convergent numerical method for a coupled system 

of two singularly perturbed linear reaction-diffusion problems“, IMA J. Numer. Anal., 23, 

627–644 (2003). 

[7] S. Matthews, J.J.H. Miller, E. O’Riordan, G.I. Shishkin, A parameter robust numerical 

method for a system of singularly perturbed ordinary differential equations, in: Analytical 

and Numerical Methods for Convection–Dominated and Singularly Perturbed Problems 

(J.J.H. Miller, G.I. Shishkin and L. Vulkov, eds.), Nova Science Publishers, New York, 2000, 

219–224. 

[8] S. Matthews, E. O’Riordan and G.I. Shishkin, “A nunerical method for a system of singularly 

perturbed reaction–diffusion equations“, J. Comput. Appl. Math., 145, 151–166 

(2002). 

2 

Speaker: CLAVERO, C. 78 BAIL 2006 

✩ 

✪

B. EISFELD: Computation of complex compressible aerodynamic flows with a 

Reynolds stress turbulence model 

✬ 

✫ 

�� 

�� 

�� 

�� 

� 

�� 

�� 

�� 

� 

�� 

�� 

�� 

�� 

�� 

�� 

�� 

�� 

�� 

�� 

� 

�� 

�� 

�� 

�� 

�� 

�� 

�� 

�� 

�� 

�� 

�� 

�� 

� 

�� 

�� 

�� 

�� 

�� 

� 

� 

�� 

� 

�� 

�� 

�� 

�� 

�� 

�� 

�� 

Speaker: EISFELD, B. 79 BAIL 2006 

✩ 

✪

B. EISFELD: Computation of complex compressible aerodynamic flows with a 

Reynolds stress turbulence model 

✬ 

✫ 

�� 

�� 

�� 

�� 

Speaker: EISFELD, B. 80 BAIL 2006 

✩ 

✪

A. FIROOZ, M. GADAMI: Turbulence Flow for NACA 4412 in Unbounded Flow and 

Ground Effect with Different Turbulence Models and Two Ground Conditions: Fixed 

and Moving Ground Conditions 

✬ 

✫ 

Turbulence Flow for NACA 4412 in Unbounded Flow and Ground Effect with 

Different Turbulence Models and Two Ground Conditions: Fixed and 

Moving Ground Conditions 

Abdolhamid Firooz, Academic board 

(ahfirooz@hamoon.usb.ac.ir) 

Mojgan Gadami, B.S. student 

(mozhgan_gadami@yahoo.com) 

Mechanical Engineering Department, University of Sistan & Baloochestan, Zahedan, Iran. 

Abstarct 

In this paper, the turbulence fluid flow around a 

two dimensional 4412 airfoil on different angles of 

attack near and far from the ground with the RANS 

(Reynolds averaged Navier-stokes) equations is 

calculated. Realizable K−ε turbulence model with 

Enhanced wall treatment and Spalart-Allmaras model 

are used (Re=2 × 10 6 

). Equations are approximated 

by finite volumes method, and they are solved by 

segregated method. The second order upwind method 

is used for the convection term, also for pressure 

interpolation the PRESTO method is used, and the 

relation between pressure and velocity with SIMPLEC 

algorithm is calculated. 

The computational domain extended 3C upstream 

of the leading edge of the airfoil 5C, downstream from 

the trailing edge, and 4C above the pressure surface. 

Distance from below the airfoil was defined with H/C 

where C is chord, and H is ground distance to the 

trailing edge. 

Velocity inlet boundary condition was applied 

upstream with speed of (U ∞ =29.215) and outflow 

boundary condition was applied downstream. The 

pressure and suction side of the airfoil and above and 

below's boundaries of domain were defined 

independently with no slip wall boundary condition. 

Moving wall with speed of (U ∞ =29.215) for above, 

and fixed or moving wall for below the airfoil was 

used. 

An unstructured mesh arrangement with 

quadrilateral elements was adopted to map the flow 

domain in ground effect and unbounded flow. A 

considerably fine C-type mesh was applied to achieve 

sufficient resolution of the airfoil surface and boundary 

layer region. Particular attention was directed to an 

offset 'inner region' encompassing the airfoil, and also 

C-type mesh was applied on near the airfoil at above 

and bottom, which it’s domain depends on the H/C in 

ground effects condition. Continuing downstream from 

leading edge and continuing far from above the airfoil 

H-type mesh was applied. 

Distance from the wall-adjacent cells must be 

determined by considering the range over which the 

log-law is valid. The distance is usually measured in 

the wall unit, y + (= ρu τ y / µ ). By increasing the 

grid numbers and changing the type of arranging mesh, 

adapting, around the airfoil a proper y + value, is 

obtained, and with this value solution results have good 

agreement with experimental data. Fig (1), Fig (2). 

The aerodynamic characteristic of an airfoil in 

ground proximity is known to be much different from 

that of unbounded flow. The condition of the wind 

tunnel bottom, I. e., moving or fixed relative to the 

airfoil would influence the performance of the airfoil in 

ground effect. The presence of boundary layer when air 

is flowing over bottom of the wind tunnel would be 

different from the real situation for a flying WIG. 

Proper velocity in moving ground boundary 

condition is considered with moving ground, and 

boundary layer is considered with fixed ground, and in 

the moving ground the boundary layer's effect is 

omitted, and so is the proper velocity in the fix ground. 

A grid independence analysis was conducted using 

some meshes of varying cell number. Each mesh was 

processed using the Realisable K-ε turbulence model 

with Enhanced wall treatment and Spalart-Allmaras 

model. Application of each mesh generally produces 

accurate predictions of the lift coefficients with 

comparison to the experimental data. It can be seen 

that the error associated with the predicated lift 

coefficient decreases with mesh refinement. 

In order to validate the present numerical data the 

computational results for NACA 4412 in unbounded 

flow is compared with published numerical results and 

experimental data. Fig (3). 

Fig (4) shows Cp variation on surface of the airfoil 

at four relative ground high computed (α =8). As the 

Speaker: FIROOZ, A. 81 BAIL 2006 

✩ 

✪




✬ 

✫ 

wing approaches the ground, the pressure on the 

pressure side of wing gradually increases due to slowdown 

of flow in region, while the pressure on the 

suction side of wing gradually decreases resulting in 

Lift increase that is regarded as the advantage of the 

WIG vehicle. 

The velocity fields around this section in ground 

effect with H/C=.2 for two different ground conditions 

at α =8 are shown in Fig (5) and (6). The different in 

the velocity field near the wing surface due to the 

bottom condition differences can not be clearly seen in 

Fig (5) and (6), but a boundary layer developed on the 

fix ground can be clearly seen in Fig (6), on the other 

hand, for the moving ground with oncoming 

undisturbed velocity as seen in Fig (6), the velocity 

decreases with increasing high. 

Meanwhile the difference in the Lift simulated by 

the fixed and moving bottom conditions is negligible 

but the Drag force simulated by the moving bottom is 

to some extent larger than that of the fixed one. Also it 

is concluded that on different angles of attack Lift 

force of the airfoil increases as it approaches the 

ground, and the Drag force decreases. 

Figure (2): adapted C-grid for unbounded flow (α =5) Figure (1): C-grid for bounded flow ( α =5) 

Figure (3): CL vs. angel of attack 

(Experimental data ( Abbott, I.H., and von Doenhoff, A.E., 

Theory of Wing Sections, Dover, New York, 1959.) 


✩ 

✪




✬ 

✫ 

Figure (4): Surface pressure distributions for NACA 4412 at the different ground clearance 

(α =8, R=2× 10 6 ) 

Figure (5): Velocity vector field for NACA 4412 in ground effect 

(α=8, Re=2×10 6 , H/C=.2, moving ground) 

Figure (6): Velocity vector field for NACA 4412 in ground effect 

6 

(α =8, Re = 2 × 10 , H/C=.2, fix ground ) 


✩ 

✪




✬ 

✫ 

EFERENCES 

Chawal M. D., Edwards L. C. and Franke M, 

E 1990 Wind Tunnel Investigation of Wing 

–in Ground Effect, Journal of Aircraft, Vol 

27 ,No.4, pp.289-293 

Chorin A. J., 1967 A Numerical Method for 

Solving Incompressible Viscous Flow 

Problems, Journal Computational Physics 

Vol.2 No.2, pp,14-23 

Chun H. H., Chang C. H. and Paik K. J.,1999, 

Longitudinal Stability of a Wing in Ground 

Effect Craft, J. of The Society of Naval 

Architects of Korea, Vol.36,No.3,pp.60-70 

RINA, London, U., total of 19 papers include 

machines 2000, Saint Petersburg State Marine 

Technical University , Russia 

Sowdon A., 1995 A Simple Method to remove the 

Boundary Layer on a Ground Plate, Papers of 

Ship Research Institute Japan , Vol.32 Vo.2, pp. 

53-78 

Stinton D. 1998 The Anatomy of the Aeroplan , 

2nd Edition published Blacewell Science, pp. 

86-92 

Staufenbiel, D, 1996 Comment on Aerodynamic 

Characteristics of a Two Dimensional Airfoil 

with Ground Effect, Journal of Aircraft 

Aerodynamic of Wing-In-Ground Effect Vehicles 

Rheinisch-West-falische Technische 

Hochschule, project Rept. 78/1, Aachen, 

Germany, 1978 

Steinbach, D 1996 Comment on Aerodynamic 

Characteristics of a Two-Dimensional Airfoil 

with Ground Effect , Journal of Aircraft, 

Vol.34, No.3, pp.455-456 

Thomas J. L., Paulson J,w. and Margason R, J., 

1979 Powered Low-Aspect-ratio Wing in 

Ground Effect (WIG) Aerodynamic 

Characteristics, NASA TM78793 

Turner T. R., 1966 Endless-Belt Technique for 

Ground Simulation, NASA SP-116 

Speziale, C.G., Abid, R. & Anderson, E.C. 1992, 

.Critical Evaluation of Two-Equation Models 

for Near-Wall Turbulence., AIAA J., Vol. 30 

No. 2, pp. 324-331. 

Wilcox, D.C. 1988, .Reassessment of the scaledetermining 

equation for advanced turbulence 

models., AIAA J., Vol. 26 No. 11, pp. 1299- 

1310. 

Pajayakrit, P. & Kind, R.J. 2000, .Assessment and 

Modification of Two-Equation Turbulence 

Models., AIAA J., Vol. 38 No. 6, pp. 955-963. 

Gorski, J. & Nguyen, P. 1991, .Navier-Stokes 

Analysis of Turbulent Boundary Layer Wake 

for Two-Dimensional Lifting 

Bodies.,Eighteenth Symposium on Naval 

Hydrodynamics, Ann Abor, USA, pp. 633-643. 

Gersten, K. & Schlichting, H. 2000, .Boundary 

Layer Theory., 8th Ed., Springer-Verlag, 

Berlin, Germany. 

Abbott, I.H., and von Doenhoff, A.E., Theory of 

Wing Sections, Dover, New York, 1959. 

Akimoto, H. and Kubo, S., 1998 Characteristics 

Study of Twp-dimensional in Surface Effect by 

CFD Simulation, Journal of the Society of 

Naval Architects of Japan, Vol. 184,pp.a7-54 

Bagley J. A., 1961 The Pressure Distribution 

on Two-Dimensional Wings Near Ground, 

Ministry of Aviation Aeronautical 

Research Council R&M, NO,3238,40 

pages 

Chang R, H 2000 Numerical Simulation of 

Turbulent Flow around Two-Dimensional 

Wings In Ground Effect Wing Different 

Ground Boundary Condition, MSc Thesis, 

Dept of Naval Architecture & ocean 

Engineering Pusan National University, 

Korea 


✩ 

✪

S.A. GAPONOV, G.V. PETROV, B.V. SMORODSKY: Boundary layer intercation with 

external disturbances 

✬ 

✫ 

BOUNDARY LAYER INTERACTION WITH EXTERNAL DISTURBANCES 

S.A.Gaponov, G.V.Petrov, B.V.Smorodsky 

Institute of Theoretical and Applied Mechanics SB RAS, Novosibirsk 630090, 

Russia; E-mail: gaponov@itam.nsc.ru, Fax:+7(3832)342268 

Generation of initial unstable eigen disturbances inside the boundary layer (BL) by external 

perturbations is an actual problem nowadays in the investigation of laminar-turbulent transition. 

Morkovin [1] was the first who has formulated so called BL receptivity problem. Up to now a lot of 

experimental and theoretical works studying subsonic BL has been performed. The detailed review of 

these efforts can be found in [2,3]. The knowledge about the supersonic BL is much shorter at the 

present moment. The existing papers are devoted mainly to the investigation of external acoustic field 

interaction with a supersonic BL on a smooth flat plate [4-6]. However an oncoming supersonic flow 

always comprises not only acoustic but also vortical and thermal (entropy) perturbations [7]. This 

paper presents results concerning irrotational external disturbances with zero damping in the direction 

of the main flow. Present paper is devoted to the investigation of disturbance excitation in sub- and 

supersonic BL by external vortical and thermal waves. Interaction of the acoustic waves with the BL 

on the non-smooth surface has also been investigated and is reported here.Studies were conducted 

both in an approaching of parallel basic flow and taken in consideration dependence of the main flow 

in boundary layer on longitudinal coordinate. In the second case were used of an equation of stability 

offered in [8]. 

For the acoustic field we distinguish two different kinds of interaction with the BL. In the first case 

the incidence angle of the acoustic wave onto the BL is finite. In the second case the incidence angle is 

zero. If the incidence angle is equal to zero, then in the result of the interaction the normal to the-wall 

disturbance velocity at the BL outer edge is non-zero and it is an additional source of the streamwise 

acoustic field. Such interaction leads to a very fast amplification of initial wave. For nonzero incidence 

angles we have performed computations of the reflection coefficient. It is important to note here that 

in some cases such reflectance is greater than unity. The amplitude of mass flux perturbation inside the 

BL excited by an external acoustic wave is practically always an order of magnitude larger than the 

initial wave amplitude in the free stream. Comparison of theoretical results with the experiment [6] 

leads authors to necessary to consider the problem of diffraction of acoustic waves on a plate leading 

edge. Using Fourier transformation and the equations of gas dynamics, one can show, that mass flux 

fluctuations along the plate could be completely determined by means of the jump of normal-to-thewall 

disturbance velocity at the plate leading edge. The normal velocity disturbance upstream of the 

plate leading edge ( x < 0 ) is determined by the distribution of the mass flux. The obtained formulas 

give the relations of the mass flux in the region x > 0 with its distribution at x < 0 . It was shown that 

intensity of the mass flux fluctuation is reduced with distance from the leading edge and is dependent 

of the orientation of the incident wave. 

Tollmien-Schlichting (TS) wave excitation in the supersonic BL by a pair of acoustic waves has 

been also investigated theoretically. The receptivity coefficient and the range of wave parameters 

where TS-wave generation takes place have been computed. It was obtained, that at enough large 

values of a Reynold's number, in narrow range of the acoustic wave numbers, the excited disturbances 

practically do not differ from an own increasing disturbance. Outside of this range the TS wave is not 

excited. The parameters of excited disturbances hardly differ from TS wave parameters near a neutral 

point, the coincidence is observed only at essential move away from neutral area downstream 

Beside in the paper investigation of the sound interaction with BL on the flat plate with roughness 

is consider also. The problem of the disturbance generation by small amplitude spatially periodic wall 

roughness has been formulated and solved in the linear approximation. The parameters of sound 

irradiated by the model with roughness in the supersonic flow have been computed. The peculiarities 

of the disturbances in the case when the angle between the wave fronts and the mean flow is close to 

the Mach angle have been investigated specially. It has been shown that the amplitude of the 

disturbance streamline inside the BL is greatly reduced in the near-wall sub-layer. At M = 1 mass 

flux perturbations in the main part of the BL are close to zero also. Considerable reduction of the 

Speaker: GAPONOV, S.A. 85 BAIL 2006 

✩ 

✪

S.A. GAPONOV, G.V. PETROV, B.V. SMORODSKY: Boundary layer intercation with 

external disturbances 

✬ 

✫ 

M where M = M cos χ and 

disturbance amplitude takes place for oblique roughness at = 1 

χ is the angle between the roughness wave front and the plate leading edge. At the BL outer edge the 

disturbance amplitude is almost zero because it is already damped inside the BL. The problem of 

nonlinear interactions of the disturbances induced by sound and roughness was investigated. It has 

been found that the parameter, which defines the ability to the instability wave excitation (receptivity 

coefficient) is strongly dependent of the acoustics incidence angle and the roughness wave front 

orientation. Computations show that even at high subsonic external flow velocities the receptivity is 

strongly dependent also on the direction of sound propagation. At M > 1 the instability could be 

excited by disturbances with the wavelength much smaller than TS wave wavelength. Maximal 

receptivity corresponds to the case when the angle between the roughness wave front and the mean 

flow is close to the Mach angle. 

The paper presents results of the detailed investigation of the interaction of supersonic BL with 

external vortical and thermal perturbations. The boundary conditions for the disturbance at the BL 

outer edge are under special consideration. It has been found that inside the BL the streamwise 

velocity and mass flux perturbations can exceed much the amplitude of the external vorticity waves. 

The disturbance excitation rate is reducing with increasing Mach number. The influence of the 

external thermal wave onto the flow inside the BL is considerably The results of the computations of 

the intensity and the spatial dimensions of the streamwise streaky structures in the BL are in 

agreement with measurements [9]. 

This work has been performed under the financial support of RFBR and the Russian Federation 

President Council (projects 05-01-00079-a, NSh-9642003.1). 

References 

1. Morkovin M.V. Critical evaluation of transition from laminar to turbulent shear layer with 

emphasis on hypersonically trevelling bodies. Tech. Rep. AFFDL № 68-149, 1969. 

2. Kachanov Yu.S. Physical mechanisms of laminar-boundary-layer transition. Annu.Rev.Fluid 

Mech., Vol.26, 1994, p.411. 

3. Boyko A.V., Greek G.R., Dovgal A.V., Kozlov V.V. Turbulence origin in near-wall flows. 

Novosibirsk: Science, 1999 (in Russian). 

4. Gaponov S.A. On the interaction of a supersonic boundary layer with acoustic disturbances. 

Thermophysics and Aeromechanics, Vol.2, №3, 1995, pp.181-188. 

5. Fedorov A.V., Hohlov A.P. Excitation of unstable modes in a supersonic boundary layer by 

acoustic waves. Fluid Dynamics, №4, 1991, pp.67-71 (in Russian). 

6. Semionov N.V., Kosinov A.D., Maslov A.A. Experimental investigation of supersonic boundary 

layer receptivity. Transitional boundary layers in aeronautics, edited by R.A.W.M.Henkes and 

J.L. van Ingen. North-Holland, Amsterdam, 1996, pp.413-420. 

7. Kovasznay L.S.G. Turbulence in supersonic flow. J.Aeron.Sci., Vol.20, №10, 1953, pp.657-674. 

8. Petrov G.V. New parabolized system of equations of stability of a compressible boundary 

layer. J. Appl.Mech. Techn. Phys. Vol.41, №1, 2000, p. 55-61. 

9. Westin K.J.A., BakchinovA.A., Kozlov V.V., Alfredsson P.H. Experiments on localized 

disturbances in a flat plate boundary layer. Pt 1: The receptivity and evolution of a localized free 

stream disturbances. Europ. J. Mech., B. Fluids, Vol.17, №6, 1998, pp.823-846. 

Speaker: GAPONOV, S.A. 86 BAIL 2006 

✩ 

✪

Z.HAMMOUCH: Similarity solutions of a power-law non-Newtonian laminar boundary 

layer flows 

✬ 

✫ 

Similarity solutions of a power-law non-Newtonian laminar boundary 

layer flows 

Z. Hammouch 

LAMFA, CNRS UMR 6140, Université de Picardie Jules Verne, 

Faculté de Mathématiques et d’Informatique, 33, rue Saint-Leu 80039 Amiens, France 

Abstract 

A steady–state laminar boundary layer flow, governed by the Ostwald-de Waele power–law model of a non–Newtonian 

fluid past a semi-infinite plate is considered. The Blasius method is used to find similarity solutions. Under appropriate 

assumptions, partial differential equations are transformed into an autonomous third–order nonlinear degenerate ordinary 

differential equation with boundary conditions. We establish the existence of an infinite family of unbounded global 

solutions. The asymptotic behavior is also discussed. Some properties of those solutions depend on the power–law index. 

Keywords: Boundary–layer; Power–law fluid; Degenerate differential equation; Global existence; Asymptotic behavior; Similarity 

solutions. 

MSC: 34B15; 34B40; 76D10; 76M55 . 

PACS: 47.15, 47.27 Te . 

Speaker: HAMMOUCH, Z. 87 BAIL 2006 

1 

✩ 

✪

M. HAMOUDA, R. TEMAM: Boundary layers for the Navier-Stokes equations : 

asymptotic analysis 

✬ 

✫ 

Boundary layers for the Navier-Stokes equations : asymptotic 

analysis. 

M. Hamouda ♯ and R. Temam ∗♯ 

∗ Laboratoire d’Analyse Numérique, Université de Paris–Sud, Orsay, France. 

♯ The Institute for Scientific Computing and Applied Mathematics, 

Indiana University, Bloomington, IN, USA. 

Abstract 

In this talk, we consider the asymptotic analysis of the solutions of the Navier- 

Stokes problem, when the viscosity goes to zero; we consider the flow in a channel of 

R 3 , in the non-characteristic boundary case. More precisely, a complete asymptotic 

expansion, at all orders, is given in the linear case. For the full nonlinear Navier-Stokes 

solution, we give a convergence theorem up to order 1, thus improving and simplifying 

the results of [TW]. 

MSC : 76D05, 76D10, 35C20. 

We consider the Navier-Stokes equations in a channel Ω∞ = R 2 × (0, h) with a permeable 

boundary, making the boundaries z = 0, h, non-characteristic. More precisely we 

have 

⎧ 

⎪⎨ 

⎪⎩ 

which is equivalent to 

⎧ 

⎪⎨ 

⎪⎩ 

∂u ε 

∂t − ε∆uε + (u ε .∇) u ε + ∇p ε = f, in Ω∞, 

div u ε = 0, in Ω∞, 

u ε = (0, 0, −U), on Γ∞, 

u ε is periodic in the x and y directions with periods L1, L2, 

u ε | t=0 = u0, 

∂v ε 

∂t − ε∆vε − UD3v ε + (v ε .∇) v ε + ∇p ε = f, in Ω∞, 

div v ε = 0, in Ω∞, 

v ε = 0, on Γ∞, 

v ε is periodic in the x and y directions with periods L1, L2, 

v ε | t=0 = v0. 

Here Γ∞ = ∂Ω∞ = R 2 × {0, h} and we introduce also Ω and Γ : 

Ω = (0, L1) × (0, L2) × (0, h), Γ = (0, L1) × (0, L2) × {0, h}. 

Speaker: HAMOUDA, M. 88 BAIL 2006 

1 

(0.1) 

(0.2) 

✩ 

✪



✬ 

✫ 

We assume that f and u0 are given functions as regular as necessary in the channel 

Ω∞, and that U is a given constant; at the price of long technicalities, we can also consider 

the case where U is nonconstant everywhere. 

Theorem 0.1 For each N ≥ 1, there exists C > 0 and for all k ∈ [0, N] an explicit given 

function θ k,ε such that : 

�v ε N� 

− ε k (v k + θ k,ε �L∞ (0,T ;L2 (Ω)) ≤ C ε N+1 , (0.3) 

�v ε − 

k=0 

N� 

k=0 

ε k (v k + θ k,ε � L 2 (0,T ;H 1 (Ω)) ≤ C ε N+1/2 , (0.4) 

where L 2 (Ω) = (L 2 (Ω)) 3 , H 1 (Ω) = (H 1 (Ω)) 3 , and C denotes a constant which depends 

on the data (and N) but not on ε. Here v ε denotes the solution of the linearized problem 

of (0.2) and v k the ones of the limit problems (ε = 0) at all orders. 

The second result concerns the solution of the full nonlinear problem (0.2) : 

Theorem 0.2 For v ε solution of the Navier-Stokes problem (0.2), there exist a time T∗ > 

0, correctors function θ 0,ε and θ 1,ε explicitly given, and a constant κ > 0 depending on the 

data but not on ε, such that : 

�v ε − (v 0 + θ 0,ε ) − ε(v 1 + θ 1,ε )� L ∞ (0,T∗; L 2 (Ω)) ≤ κ ε 2 , (0.5) 

�v ε − (v 0 + θ 0,ε ) − ε(v 1 + θ 1,ε )� L 2 (0,T∗; H 1 (Ω)) ≤ κ ε 3/2 . (0.6) 

We recall here the limit problem which is the Euler problem and its solution v0 satisfies : 

⎧ 

∂v 

⎪⎨ 

⎪⎩ 

0 

∂t − UD3v 0 + (v 0 .∇) v 0 + ∇p 0 = f, in Ω∞, 

div v 0 = 0, in Ω∞, 

v 0 3 = 0, on Γ0, 

v 0 = 0, on Γh, 

v 0 (0.7) 

is periodic in the x and y, directions with periods L1, L2. 

It is obvious that we can not expect a convergence result between v ε and v 0 (in H 1 (Ω) for 

example) and more precisely we are leading with a boundary layer problem close to some 

parts of the boundary Γ. This is the main object of our work. 

At the following order v 1 is solution of 

⎧ 

⎪⎨ 

⎪⎩ 

∂v 1 

∂t − UD3v 1 + (v 1 .∇) v 0 + (v 0 .∇) v 1 + ∇p 1 

= − ∂ϕ0 

∂t + UD3ϕ 0 − (v 0 .∇) ϕ 0 − (ϕ 0 .∇) v 0 + ∆v 0 , in Ω∞, 

div v 1 = 0, in Ω∞, 

v 1 3 = 0, on Γ0, 

v 1 = 0, on Γh, 

v 1 is periodic in the x and y, directions with periods L1, L2. 

The function ϕ 0 is a known function at this level. 

2 


(0.8) 

✩ 

✪



✬ 

✫ 

References 

[F] K.O. Friedrichs, The mathematical strucure of the boundary layer problem in Fluid 

Dynamics (R. von Mises and K.O. Friedrichs, eds), Brown Univ., Providence, RI 

(reprinted by Springer-Verlag, New York, 1971). pp. 171-174. 

[Li2] J. L. Lions, Problèmes aux limites dans les équations aux dérivées partielles. Les 

Presses de l’Université de Montréal, Montreal, Que., 1965. Reedited in [?]. 

[O] R. E. O’Malley, Singular perturbation analysis for ordinary differential equations. 

Communications of the Mathematical Institute, Rijksuniversiteit Utrecht, 5. Rijksuniversiteit 

Utrecht, Mathematical Institute, Utrecht, 1977. 

[TW] R. Temam and X. Wang, Boundary layers associated with incompressible Navier- 

Stokes equations: the noncharacteristic boundary case. J. Differential Equations 179 

(2002), no. 2, 647–686. 

[VL] M.I. Vishik and L.A. Lyusternik, Regular degeneration and boundary layer for linear 

differential equations with small parameter, Uspekki Mat. Nauk, 12 (1957), 3-122. 

3 


✩ 

✪

M. HÖLLING, H. HERWIG: Computation of turbulent natural convection at vertical 

walls using new wall functions 

✬ 

✫ 

Computation of turbulent natural convection at vertical walls using new wall 

functions 

M. Hölling, H. Herwig 

Technical Thermodynamics 

Hamburg University of Technology 

Denickestraße 15, 21073 Hamburg, Germany 

m.hoelling@tu-harburg.de 

h.herwig@tu-harburg.de 

Turbulent natural convection at vertical walls has been under investigation in the last decades 

but is still not sufficiently understood. The viscous sublayer, i.e. the region very close to the wall, 

can be described properly, see e.g. Tsuji and Nagano [1], since there the turbulent fluctuations 

are damped by the wall and the governing equations can be solved directly. But in the more 

interesting region in which turbulence dominates, the flow cannot be described properly. George 

and Capp [2] offer analytical temperature and velocity profiles which have become a kind of 

standard for natural convection. But, it was shown by Versteegh and Nieuwstadt [3] and by 

Henkes and Hoogendoorn [4] that at least the velocity profile is erroneous. We employ a different 

approach presented in Hölling and Herwig [5] to describe the turbulence affected region of the 

flow field. 

The starting point for the analysis is a channel with a hot and a cold wall of infinite extent 

as used by Versteegh and Nieuwstadt [3] for their DNS study. The governing equations then 

reduce to: 

0 = ∂ 

� 

∂y 

0 = ∂ 

∂y 

∂y − v′ T ′ 

� 

a ∂T 

� 

ν ∂u 

∂y − u′ v ′ 

� 

+ gβ � � 

T − T0 

It is found that the temperature field consists of a viscosity influenced wall layer and a fully 

turbulent outer layer. The temperature profile is obtained by matching of gradients between 

these layers that reveals a logarithmic profile. It is in good agreement with DNS as well as 

experimental temperature profiles from various studies. 

For the velocity profile a different approach is chosen; the profile is not obtained by matching 

of gradients. Instead the momentum equation (2) is rewritten in such a way that the temperature 

profile and the Reynolds stresses are expressed as a function of the wall distance. The Reynolds 

stresses are modelled using the eddy viscosity approach. A constant turbulent Prandtl number 

is assumed as can be concluded from DNS data. Then the eddy viscosity is directly linked to the 

turbulent thermal diffusivity and therefore is a linear function of wall distance. Once all terms 

are expressed as a function of wall distance the momentum equation can be integrated and a 

velocity profile emerges. This profile is in good agreement with DNS and experimental data. 

Straightforward numerical solutions without adequate near wall treatment, like with FLU- 

ENT 6.2, to reproduce the DNS data of Versteegh and Nieuwstadt [3] for Ra = 5.0 · 10 6 show 

that even with fine grids also in the viscous sublayer only poor agreement can be achieved. 

Thus, we conclude that it would be diserable to have a new approach and improved near wall 

treatment. 

Therefore, we apply the new universal profiles as wall functions for CFD calculations. They 

are implemented in the two dimensional CAFFA code of Ferziger and Peric [6] that uses the kω-turbulence 

model and the Boussinesq-approximation. The standard k-ω model is used inspite 

Speaker: HÖLLING, M. 91 BAIL 2006 

1 

(1) 

(2) 

✩ 

✪

M. HÖLLING, H. HERWIG: Computation of turbulent natural convection at vertical 

walls using new wall functions 

✬ 

✫ 

T in K 

380 

370 

360 

350 

340 

330 

8×200 

4×200 

DNS 

320 

0 0.05 

y in m 

0.1 

u in m/s 

0.25 

0.2 

0.15 

0.1 

0.05 

0 

−0.05 

−0.1 

−0.15 

−0.2 

8×200 

4×200 

DNS 

−0.25 

0 0.05 

y in m 

0.1 

Figure 1: Temperature and velocity profile of Versteegh and Nieuwstadt [3] for Ra = 5.0 · 10 6 

compared to results of the modified CAFFA code using two grid sizes. 

that it is inadequate for natural convection. Instead of trying to find another modification of 

an existing turbulence model, the deficiences could be compensated by modifying the boundary 

conditions for k and ω. 

The results are again compared to the DNS data of Versteegh and Nieuwstadt [3] and good 

agreement is found. Figure 1 shows the temperature and velocity profile for Ra = 5.0 · 10 6 

together with the results of the modified CAFFA code. Two different grid sizes were used and 

despite the very few control volumes (8 and 4 CVs) across the channel width the profiles are 

matched very well. Also the wall gradients, i.e. the Nusselt number and the shear stress, are 

correct within 4 %. 

References 

[1] T. Tsuji and Y. Nagano, “Characteristics of a turbulent natural convection boundary layer 

along a vertical flat plate”, Int. J. Heat Mass Transfer 31, 1723–1734 (1989). 

[2] W.K. George and S.P. Capp, “A theory for natural convection turbulent boundary layers 

next to heated vertical surfaces”, Int. J. Heat Mass Transfer 22, 813–826 (1979). 

[3] T.A.M. Versteegh and F.T.M. Nieuwstadt, “A direct numerical simulation of natural convection 

between two infinite vertical differentially heated walls: scaling laws and wall functions”, 

Int. J. Heat Mass Transfer 42, 3673–3693 (1999). 

[4] R.A.W.M. Henkes and C.J. Hoogendoorn, “Numerical determination of wall functions for 

the turbulent natural convection boundary layer”, Int. J. Heat Mass Transfer 33, 1087– 

1097 (1990). 

[5] M. Hölling and H. Herwig, “Asymptotic analysis of the near wall region of turbulent natural 

convection flows”, J. Fluid Mech. 541, 383–397 (2005). 

[6] J.H. Ferziger and M. Peric, “Computational methods for fluid dynamics”, 2nd ed., Springer, 

Berlin (1999). 

Speaker: HÖLLING, M. 92 BAIL 2006 

2 

✩ 

✪

A.-M. IL’IN, B.I. SULEIMANOV: The coefficients of inner asymptotic expansions for 

solutions of some singular boundary value problems 

✬ 

✫ 

�� 

�� 

�� 

�� 

ux + u 3 − tu − x = 0, (1) 

uxx = u 3 − tu − x (2) 

�� 

�� 

�� 

�� 

ε 2 (Ux1x1 + Ux2x2) + εbUx1 + f(x1, x2, U) = 0, 

�� f(x1, x2, U) = 0 �� Uj(x1, x2). �� 

�� U0(0, 0) = 0 �� f(x1, x2, U) = 

x1 + x2U − U 3 + · · · . �� x2 < 0 �� 

�� x2 > 0. 

�� b �= 0 �� 

z = x1 

, 

ε3/5 x2 

t = , 

ε2/5 U 

w = . 

ε1/5 �� bwz = w3 − tw − z 

�� 

�� 

�� b = 0 �� 

z = x1 x2 U 

, t = , v = 

ε3/4 ε1/2 ε1/4 �� wzz = w3 − tw − z �� 

�� 

�� 

�� 

�� H(x, t) 

H 3 − tH − x = 0 (3) 

�� x 2 + t 2 → ∞. �� 

�� t �� 

�� t �� 

�� u(x, t) − H(x, t) → 0 �� x → ∞. 

u(x, t) = as x 1/3 (1 + 

∞� 

cj(t)x −j/3 ), x → ∞. 

j=2 

�� |t| < T �� 

(x, t). 

�� t → −∞ �� u(x, t) �� 

�� 

u(x, t) = |t| 1/2 

� 

f(s) + 

� 

∞� 

j=1 

vj(s) 

t4j � 

, 

Speaker: IL’IN, A.M. 93 BAIL 2006 

✩ 

✪



✬ 

✫ 

�� 

s = x|t| −3/2 , f 3 (s) + f(s) − s = 0. 

�� t → ∞ �� u(x, t) �� 

�� 

�� 

u(x, t) = t 1/2 

� ∞� 

g(s) + t −4j ∞� 

vj(s) + t −2k � 

wk(ξ) 

j=1 

s = x|t| −3/2 , g(s) 3 − g(s) − s = 0, w0(ξ) = −1 + tanh ξ 

√ 2 , ξ = st 2 

�� t > T > 0. �� 

�� 

�� 

�� 

�� 

�� H(x, t) �� 

�� t � 0� 

�� H(x, t) �� 

k=0 

t > 0, x < xc(t) = 2 

√ 27 t 3/2 

�� 

t > 0, x > xc(t). 

�� H(x, t) �� 

�� x = xc(t). 

�� t → −∞ �� 

x = xc(t) �� u(x, t) = 

v(s, t)|t| 1/2 �� s = x |t| −3/2 , 

∞� 

v(s, t) = |t| 

k=0 

−5k/2 v − 

k (s). 

�� s = 2 √ �� 

27 

�� 

z = (s − s∗)t 5/3 , r = ( 1 

√ 3 + v(s, t))t 5/6 . 

�� 

(z − z0) = ηt −5/6 , w = r(z)t −5/6 = (v(s) + 1/ √ 3) 

�� z0 �� Ai(−3 1/6 z)� 

�� 

1. u(x, t) = as t 1/2 

∞� 

k=0 

2. u(x, t) = as t 1/2 [− 1 

√ 3 + t −5/6 

� 

|t| −5k/2 v + 

k (s), 

� 

∞� 

r0(z) + t −5j/6 � 

rj(z) ] 


j=1 

✩ 

✪



✬ 

✫ 

�� 

3. u(x, t) = as t 1/2 [− 1 

√ 3 + 

∞� 

t −5k/6 wk(ln t, η)]. 

�� 

�� 

�� 

�� 

�� 


� 

k=0 

✩ 

✪

W.S. ISLAM, V.R. RAGHAVAN: Numerical Simulation of High Sub-critical Reynolds 

Number Flow Past a Circular Cylinder 

✬ 

✫ 

Numerical Simulation of High Sub-critical Reynolds Number Flow 

Past a Circular Cylinder 

Wan Saiful Islam* and Vijay R. Raghavan** 

Faculty of Mechanical Engineering 

Kolej Universiti Teknologi Tun Hussein Onn 

86400 Parit Raja, Malaysia 

*wsaiful@kuittho.edu.my 

**vijay@kuittho.edu.my 

ABSTRACT 

Few areas in fluid mechanics have received more attention than that of flow past a bluff body. In 

particular, flow across a circular cylinder in unconfined and confined flow is a classical problem, 

and has been studied experimentally, visually and numerically [1-6]. Although the geometry is 

apparently simple, this problem has not yielded to closed form analytical solution except at very 

low Reynolds numbers because of the complexity associated with adverse pressure gradients, 

separation, eddy shedding, recirculation and reattachment. There are a very large number of 

attempts reported in the literature using a variety of numerical approaches viz., finite difference, 

finite element and finite volume methods. However, there is room for improvement in the 

agreement with experiments that has been obtained hitherto [5,6]. A numerical solution that gives 

good agreement is also likely to be useful for benchmarking existing codes and new ones that 

may be written. The purpose of the present study is to find a satisfactory solution in the entire 

range of sub-critical Reynolds numbers for flow over a circular cylinder. 

In earlier attempts to establish benchmark solutions and to obtain agreement with published data 

over a range of Reynolds numbers, various turbulence models including large eddy simulation 

(LES) had been considered and both 2-D and 3-D had been tried [4-6]. The results were obtained 

in the form of general appearance of the wake flow, examination of the velocity magnitudes in 

the near-field and far-field, eddy frequencies, Strouhal numbers and detailed local distributions of 

pressure. However, most of these authors have carried out their work in the more convenient 

Speaker: ISLAM, W.S. 96 BAIL 2006 

✩ 

✪

W.S. ISLAM, V.R. RAGHAVAN: Numerical Simulation of High Sub-critical Reynolds 

Number Flow Past a Circular Cylinder 

✬ 

✫ 

Reynolds number range of 40 to 1000 [2,3]. As one goes to higher Re ranges, results are seen to 

deviate progressively more widely from experimental results. 

In the present study simulations are performed for unsteady, two-dimensional (2-D) flow past a 

circular cylinder in a confined duct with appropriate blockage ratios. At Reynolds numbers as 

high as 100,000 the numerical solutions obtained agree remarkably well with experiments, not 

only in the global sense in the form of CD, but also locally in terms of pressure distribution. The 

paper describes how the agreement was obtained and these results might serve as a benchmark for 

validating CFD codes. 

REFERENCES 

1. Achenbach, E., “Distribution of Local Pressure and Skin Friction Around a Circular 

Cylinder in Cross-Flow up to Re = 5×10 6 ”, Journal of Fluid Mechanics, 34, 4, 625- 

639, 1968. 

2. Son, J.S. and Hanratty, T.J., “Numerical Solution for the Flow around a Cylinder at 

Reynolds Numbers of 40, 200 and 500”, Journal of Fluid Mechanics, 35, 2, 369-386, 

1969. 

3. Fornberg, B., “Steady Viscous Flow Past a Circular Cylinder up to Reynolds 

Number 600”, Journal of Computational Physics, 61, 297-320, 1985. 

4. Chou, M.-H. and Huang, W., “Numerical Study of High-Reynolds-Number Flow 

Past a Bluff Object”, International Journal for Numerical Methods in Fluids, 23, 711- 

732, 1996. 

5. Selvam, R.P., “Finite Element Modeling of Flow Around a Circular Cylinder using 

LES”, Journal of Wind Engineering and Industrial Aerodynamics, 67&68, 129-139, 

1997. 

6. Breuer, M., “Numerical and Modeling Influences on Large Eddy Simulations for the 

Flow Past a Circular Cylinder”, International Journal of Heat and Fluid Flow, 19, 

512-521, 1998 

Speaker: ISLAM, W.S. 97 BAIL 2006 

✩ 

✪

D. KACHUMA, I. SOBEY: Fast waves during transient flow in an asymmetric channel 

✬ 

✫ 

Abstract 

Fast waves during transient flow in an asymmetric channel 

Dick Kachuma & Ian Sobey 

Oxford University Computing Laboratory 

Wolfson Building, Parks Road, 

Oxford UK OX1 3QD 

dick.kachuma@comlab.ox.ac.uk 

ian.sobey@comlab.ox.ac.uk 

The use of stepped channels together with unsteady laminar flow provides a powerful mixing mechanism 

that is particularly applicable to processes where the fluid contains delicate elements, for example, applications 

involving mass transfer in blood or in a cell culture. In such channel flows there are parameter 

regimes where the flow is described by the two-dimensional unsteady Navier–Stokes equations. Sobey 

(1985) showed both experimentally and numerically that a standing wave of separated regions developed 

behind a channel step during oscillatory flow and called the resulting flow a vortex wave. Included in his 

experimental observations were vortex waves of extreme longitudinal extent and he conjectured that the 

wave formed was, under the correct parameter conditions, virtually undamped in the streamwise direction. 

We have undertaken calculations in a slightly different parameter region and find that a sequence of 

two events occurs: one is the formation of a vortex wave of finite extent (typically 3-5 vortices alternating 

on the two walls behind the step), the second is a subsequent rapidly propagating wave of regular but 

slightly smaller vortices. The speed of propagation of this second wave is such that its resolution would 

have been beyond that of the apparatus used in Sobey (1985). In describing these waves we shall refer to 

the vortex wave as a V-wave and the second wave as a KH-wave. An example of the waves is illustrated 

in Figure 1 where contours of a flow are shown. 

V-wave KH-wave 

Figure 1: Instantaneous streamlines 

In order to understand the genesis of KH-waves we have undertaken a study of starting flows in 

which the fluid is accelerated from rest to either steady channel flux or a flux with a small oscillatory 

component. We have tested two hypotheses: one that the KH-wave results from the evolution of an 

inviscid rotational core flow that is described by an evolutionary linearised Kortweg-de Vries (KdV) 

equation. That this might be a plausible hypothesis comes from results by Tutty and Pedley (1994) 

which show the evolution of waves in solutions of an evolutionary KdV equation that models such a core 

flow. The second hypothesis is that the KH-wave results from an Orr-Sommerfeld type instability of 

nearly parallel but non-Poiseuille like flow. This has lead us to study stability of a base flow 

u0(y) = (1 − σy)(1 − y 2 ), −1 ≤ y ≤ 1, 

where σ = 0 is Poiseuille flow and σ > 1 indicates reverse flow near one wall. 

The development of KH-waves has been considered in channels with smoothly varying and sharp 

corners and using different numerical methods, as well as with different numerical resolution. The paper 

is divided into four sections. 

Speaker: KACHUMA, D. 98 BAIL 2006 

1 

✩ 

✪

D. KACHUMA, I. SOBEY: Fast waves during transient flow in an asymmetric channel 

✬ 

✫ 

(1) We describe the numerical solution of the unsteady Navier–Stokes equations and illustrate the 

development of a KH-wave. 

(2) We integrate velocities in time to obtain particle paths and use these to help interpret the development 

of the flows. 

(3) We describe solutions of an evolutionary linearised KdV equation and their interpretation. 

(4) We consider solutions of an Orr-Sommerfeld equation as the extent of a reverse flow region is 

varied and investigate the consequences of instability in the base flow on subsequent flow development. 

The results we have indicate that it is unlikely that the KH-wave is the result of evolution of an 

inviscid rotational core flow and instead, that it is more likely the result of a linear instability mechanism 

described by an Orr–Sommerfeld equation but with growth rates that are orders of magnitude greater 

than those for disturbances to symmetric Poiseuille flow and with instability occuring at relatively low 

Reynolds number. The complexity of unsteady flows calculated from the full Navier–Stokes equations 

is remarkable and it is likely that flows in other parameter regimes are dominated by entirely different 

mechanisms. 

References 

Sobey I.J. (1985). Observation of waves during oscillatory channel flow. J. Fluid Mech., 151:395–426. 

Tutty O.R. and Pedley T.J. (1994). Unsteady flow in a nonuniform channel: A model for wave generation. 

Phys. Fluids, 6:199–208. 

Speaker: KACHUMA, D. 99 BAIL 2006 

2 

✩ 

✪

A. KAUSHIK, K.K. SHARMA: A Robust Numerical Approach for Singularly Perturbed 

Time Delayed Parabolic Partial Differential Equations 

✬ 

✫ 

A Robust Numerical Approach for Singularly 

Perturbed Time Delayed Parabolic Partial 

Differential Equations 

Aditya Kaushik † Kapil K. Sharma ‡ 

† Department of Mathematics, Kurukshetra University, Kurukshetra-136 119, India 

‡ MAB, University Bourdauex 1, 351 Cours de Libration F-33405, Cedex Talence, France 

[ † a-kaushik@lycos.com ‡ sharmal@math.u-bordeaux1.fr ] 

Summary 

In this paper a numerical study for a class of initial boundary value problems for 

singularly perturbed unsteady parabolic partial differential equation with variable coefficients 

having time delayed reaction term, is initiated on a rectangular domain. Such 

problems arise in diverse area of science and engineering that take into account not just 

the present state of physical systems but also its past history. These models are described 

by certain class of functional differential equations often called delay differential equation 

and from mathematical perspective they are singularly perturbed. In most application in 

the life sciences, a delay is introduced when there are some hidden variables and processes 

which are not well understood but are known to cause a time lag. The delay differential 

is versatile in mathematical modeling of processes in various application field, where they 

provide the best and sometimes the only realistic simulation of observed phenomena. The 

singularly perturbed differential difference equation with delay arises in general in the 

modeling of various real life phenomena’s, for instance, in studying heat or mass transfer 

process in composite materials with small heat conduction or diffusion, in drift diffusion 

model of semiconductor devices, in fluid flow problems, physiological kinetics, in various 

branches of biosciences and population dynamics, control theory, chemical kinetics, etc. 

Some modelers ignore the ”lag” effect and use an differential equations model as a substitute 

for a delay differential equation model. Kuang [1], comments under the heading 

”Small delays can have large effects” on the dangers that researchers risk if they ignore 

lags which they think are small. There are inherent qualitative differences between delay 

differential equations and finite systems of differential equations that make such a strategy 

risky. 

The concept of singular perturbation is not new, indeed, it has been a formidable tool in 

the solution of some important applied mathematical problems. A singular perturbation 

is a modification of partial differential equation by adding a multiple ǫ times a higher 

order term. In accordance with the informal principle that the behavior of solutions is 

governed primarily by the highest order terms, a solution u ǫ of the perturbed problem will 

often behave analytically quite differently from a solution of the original equation, and 

exhibits boundary or the transition layers in the outflow boundary reasons when the perturbation 

parameter specifying the problem tends to zero [2]. Due to the presence of the 

perturbation parameter and in particular time delay, the primary mathematical methods 

fails to provide the desired results. Thus, the quest for some new numerical techniques 

Speaker: KAUSHIK, A. 100 BAIL 2006 

1 

✩ 

✪

A. KAUSHIK, K.K. SHARMA: A Robust Numerical Approach for Singularly Perturbed 

Time Delayed Parabolic Partial Differential Equations 

✬ 

✫ 

to handle the difficulties occurring due to presence of these two parameters and development 

of a robust numerical method to solve such type of problem, has found special 

relevance. To sort out both the difficulties, a numerical method consisting of standard 

finite difference operator on a uniform mesh is constructed. The first step in this direction 

consists of discretizing the time variable with the backward Euler’s method with constant 

time step. This produces a set of stationary singularly perturbed semidiscrete problem 

which is further discretized in space using standard finite difference operator on a uniform 

mesh. An extensive amount of analysis is carried out in order to establish the convergence 

and stability of the method proposed. A set of numerical experiment is carried out in 

support of the predicted theory and to show the effect of delay on the boundary layer 

behavior of the solution for the initial boundary value problem considered which validates 

computationally the theoretical results. 

[1] Y. Kuang, Delay differntial equations with applications in population dynamics, Academic 

Press Inc.,(1993). 

[2] H. G. Roos, M. Stynes, and L. Tobiska, Numerical Methods for Singularly Perturbed 

Di.erential Equa- tions. Convection-Di.usion and Flow Problems, Springer-Verlag, Berlin, 

1996. 

Speaker: KAUSHIK, A. 101 BAIL 2006 

2 

✩ 

✪

P. KNOBLOCH: On methods diminishing spurious oscillations in finite element 

solutions of convection-diffusion equations 

✬ 

✫ 

On methods diminishing spurious oscillations 

in finite element solutions of convection–diffusion equations 

Petr Knobloch 

Charles University, Faculty of Mathematics and Physics, Department of Numerical Mathematics, 

Sokolovská 83, 186 75 Praha 8, Czech Republic 

e-mail: knobloch@karlin.mff.cuni.cz 

We discuss the application of the finite element method to the numerical solution of the scalar 

convection–diffusion equation 

−ε∆u + b · ∇u = f in Ω, u = ub on Γ D , ε ∂u 

∂n = g on ΓN . (1) 

Here Ω is a bounded two–dimensional domain with a polygonal boundary ∂Ω, Γ D and Γ N are disjoint 

and relatively open subsets of ∂Ω satisfying meas1(Γ D ) > 0 and Γ D ∪ Γ N = ∂Ω, n is the outward unit 

normal vector to ∂Ω, f is a given outer source of the unknown scalar quantity u, ε > 0 is the constant 

diffusivity, b is the flow velocity, and ub, g are given functions. 

Despite the apparent simplicity of problem (1), its numerical solution is by no means easy if convection 

is strongly dominant (i.e., if ε ≪ |b|). In this case, the solution of (1) typically possesses interior and 

boundary layers whose widths are usually significantly smaller than the mesh size and hence the layers 

cannot be resolved properly. In particular, it is well known that the classical Galerkin finite element 

discretization of (1) is inappropriate in the convection–dominated regime since the discrete solution is 

typically globally polluted by spurious oscillations. Although, during the last three decades, an extensive 

research has been devoted to the development of methods which diminish spurious oscillations in 

the discrete solutions of (1), the numerical solution of (1) is still a challenge when convection strongly 

dominates diffusion. The broad interest in solving problem (1) is caused not only by its actual physical 

meaning but also by the fact that it represents a simple model problem for convection–diffusion effects 

which appear in many more complicated problems arising in applications. 

Initially, stabilizations of the Galerkin discretization of (1) imitated upwind finite difference techniques. 

However, like in the finite difference method, the upwind finite element discretizations remove 

the unwanted oscillations but the accuracy attained is often poor since too much numerical diffusion 

is introduced. According to our experiences, one of the most successful upwinding techniques is the 

improved Mizukami–Hughes method, see Knobloch [9]. It is a nonlinear Petrov–Galerkin method for 

conforming linear triangular finite elements P1 which satisfies the discrete maximum principle on weakly 

acute meshes. In contrast with many other upwinding methods for P1 elements satisfying the discrete 

maximum principle, the improved Mizukami–Hughes method adds much less numerical diffusion and 

provides rather accurate solutions, cf. Knobloch [10]. 

One of the most efficient procedures is the streamline upwind/Petrov–Galerkin (SUPG) method developed 

by Brooks and Hughes [2] which is a higher–order method possessing good stability properties 

achieved by adding artificial diffusion in the streamline direction. Unfortunately, the SUPG method 

does not preclude spurious oscillations localized in narrow regions along sharp layers. Although these 

oscillations are usually small in magnitude, they are not permissible in many applications (e.g., chemically 

reacting flows, free–convection computations, two–equations turbulence models, compressible 

flow problems with shocks). Therefore, various terms introducing artificial crosswind diffusion in the 

neighborhood of layers have been proposed to be added to the SUPG formulation in order to obtain a 

method which is monotone or which at least reduces the local oscillations (cf. e.g. [3, 4] and the references 

there). This procedure is often referred to as discontinuity capturing (or shock capturing). These 

discontinuity–capturing methods can be divided into three groups according to whether the additional 

artificial diffusion is isotropic, or orthogonal to streamlines, or based on edge stabilizations. We have 

demonstrated for P1 finite elements in [7] that the edge stabilization methods add too much artificial 

diffusion. Among the remaining two groups of methods, the best methods for P1 finite elements seem 

Speaker: KNOBLOCH, P. 102 BAIL 2006 

1 

✩ 

✪

P. KNOBLOCH: On methods diminishing spurious oscillations in finite element 

solutions of convection-diffusion equations 

✬ 

✫ 

to be the methods of do Carmo and Galeão [5], of Almeida and Silva [1] and slight modifications of the 

methods of Codina [6] and of Burman and Ern [3], see also John and Knobloch [8]. 

Our aim is to continue the discussion of properties of the various methods diminishing spurious 

oscillations in finite element solutions of (1) mentioned above. We would like to concentrate on three 

aspects: 

• properties of methods of Mizukami–Hughes type satisfying the discrete maximum principle where 

we mention the improved Mizukami–Hughes method of [9] for P1 finite elements and will discuss 

its generalization to other types of finite elements; 

• comparison of the above–mentioned discontinuity–capturing methods for higher order finite elements; 

• choice of stabilization parameters where we will present a novel approach to the choice of the 

SUPG stabilization parameter in outflow boundary layers. 

The above described discussion will be performed for the two–dimensional case and the results obtained 

in three–dimensions will be mentioned only briefly. 

References 

[1] R.C. Almeida, R.S. Silva, A stable Petrov–Galerkin method for convection–dominated problems, 

Comput. Methods Appl. Mech. Eng. 140 (1997) 291–304. 

[2] A.N. Brooks, T.J.R. Hughes, Streamline upwind/Petrov–Galerkin formulations for convection 

dominated flows with particular emphasis on the incompressible Navier–Stokes equations, Comput. 

Methods Appl. Mech. Eng. 32 (1982) 199–259. 

[3] E. Burman, A. Ern, Nonlinear diffusion and discrete maximum principle for stabilized Galerkin 

approximations of the convection–diffusion–reaction equation, Comput. Methods Appl. Mech. 

Eng. 191 (2002) 3833–3855. 

[4] E.G.D. do Carmo, G.B. Alvarez, A new stabilized finite element formulation for scalar convection– 

diffusion problems: The streamline and approximate upwind/Petrov–Galerkin method, Comput. 

Methods Appl. Mech. Eng. 192 (2003) 3379–3396. 

[5] E.G.D. do Carmo, A.C. Galeão, Feedback Petrov–Galerkin methods for convection–dominated 

problems, Comput. Methods Appl. Mech. Eng. 88 (1991) 1–16. 

[6] R. Codina, A discontinuity–capturing crosswind–dissipation for the finite element solution of the 

convection–diffusion equation, Comput. Methods Appl. Mech. Eng. 110 (1993) 325–342. 

[7] V. John, P. Knobloch, A comparison of spurious oscillations at layers diminishing (SOLD) methods 

for convection–diffusion equations: Part I, Preprint Nr. 156, FR 6.1 – Mathematik, Universität des 

Saarlandes, Saarbrücken, 2005. 

[8] V. John, P. Knobloch, On discontinuity–capturing methods for convection–diffusion equations, submitted 

to the Proceedings of the conference ENUMATH 2005, Santiago de Compostela, July 18–22, 

2005. 

[9] P. Knobloch, Improvements of the Mizukami–Hughes method for convection–diffusion equations, 

Preprint No. MATH–knm–2005/6, Faculty of Mathematics and Physics, Charles University, 

Prague, 2005. 

[10] Knobloch, P.: Numerical solution of convection–diffusion equations using upwinding techniques 

satisfying the discrete maximum principle. Submitted to the Proceedings of the Czech–Japanese 

Seminar in Applied Mathematics 2005, Kuju Training Center, Oita, Japan, September 15–18, 2005. 

Speaker: KNOBLOCH, P. 103 BAIL 2006 

2 

✩ 

✪

T. KNOPP: Model-consistent universal wall-functions for RANS turbulence modelling 

✬ 

✫ 

Model-consistent universal wall-functions for RANS turbulence modelling 

T. Knopp 

Institute of Aerodynamics and Flow Technology 

DLR (German Aerospace Center) 

Bunsenstr. 10, 37073 Göttingen, Germany 

Tobias.Knopp@dlr.de 

Ý 

A model-consistent universal wall-function method for RANS turbulence modelling [1] is presented, 

which gives solutions almost independent of the spacing of the first grid node above the wall (denoted by 

in plus-units) and allows a considerable solver acceleration and reduction of memory consumptions. 

Model-consistent wall-functions are turbulence model specific, ensuring almost grid-independent 

predictions for boundary layer flows at zero pressure gradient, e.g. the flow by Wieghardt [5]. 

0.0035 

C f 

0.0025 

1 2 3 4 

x 

Exp. 

low Re 

y + (1) = 1 

y + (1) = 4 

y + (1) = 9 

y + (1) = 17 

y + (1) = 23 

y + (1) = 40 

0.0035 

C f 

0.0025 

1 2 3 4 

x 

Exp. 

low Re (fine grid) 

y + (1) = 1 

y + (1) = 4 

y + (1) = 9 

y + (1) = 17 

y + (1) = 23 

y + (1) = 40 

Figure 1: Skin friction �for equilibrium layer [5]. Almost grid-independent prediction for Spalart- 

Allmaras (SA) model with Edwards modification [3] (left) and for Menter baseline�-�model [4] (right). 

The underlying approximations, i.e., (i) one-dimensional boundary-layer flow and (ii) near-wall equilibrium 

stress balance by neglecting the streamwise pressure gradient, are assessed by investigation of a flat 

plate turbulent boundary flow with adverse pressure gradient and separation devised by [2]. 

p + 

0.1 

0.05 

p + 3 

= v / (rho ut ) * dp / dx 

p + = 0 

station 1 at x1 = 1.3 




station 5 at x = 2.4 5 




station 9 at x = 2.76 

9 

station 10 at x = 2.8 

10 

0 

0 1 2 

x 

y + 

10 0 

10 1 

10 2 

Ù �Ù�Ù�vs.Ý Figure 2: Pressure gradient parameterÔ ��Ù��Ô��Ü(left) and corresponding velocity profiles 

�ÝÙ��for the adverse pressure gradient flow by Kalitzin et al. [2]. 

u + 

25 

20 

15 

10 

5 

RANS station 4 






RANS ZPG bou. layer 

The method is then applied successfully to aerodynamic flows with separation including a transonic flow 

Speaker: KNOPP, T. 104 BAIL 2006 

1 

✩ 

✪

T. KNOPP: Model-consistent universal wall-functions for RANS turbulence modelling 

✬ 

✫ 

c p 

-1 

-0.5 

0 

0.5 

exp. 

low-Re 

y + (1) = 1 

y + (1) = 20 

y + (1) = 40 

y + (1) = 60 

1 

0 0.25 0.5 

x/c 

0.75 1 

0.006 

0.004 

c f 

0.002 

0 

exp. 

low-Re 

y + (1) = 1 

y + (1) = 20 

y + (1) = 40 

y + (1) = 60 

c f = 0 

0 0.25 0.5 0.75 1 

x/c 

Figure 3: Distribution ofÔ(left) and�(right) for SA-Edwards model [3] for RAE2822 case 10 [6]. 

0.006 

0.004 

c f 

0.002 

0 

0.25 0.5 0.75 

x/c 

exp. 

low-Re 

y + (1) = 1 

y + (1) = 4 

y + (1) = 7 

y + (1) = 10 

c f = 0 

0.006 

0.004 

c f 

0.002 

0 

0.25 0.5 0.75 

x/c 

exp. 

low-Re 

y + (1) = 1 

y + (1) = 20 

y + (1) = 40 

y + (1) = 80 

c f = 0 

Figure 4: Prediction for�on grids with varying near-wall spacing for SST model [4] for A-airfoil [7]. 

with shock induced separation [6] and a subsonic highlift airfoil close to stall [7]. 

References 

[1] T. Knopp, T. Alrutz and D. Schwamborn, “A grid and flow adaptive wall-function method for RANS 

turbulence modelling”, Journal of Computational Physics (submitted). 

[2] G. Kalitzin, G. Medic, G. Iaccarino and P. Durbin, “Near-wall behaviour of RANS turbulence models 

and implications for wall functions”, Journal of Computational Physics, 204, 265-291 (2005). 

[3] J.R. Edwards and S. Chandra, “Comparison of eddy viscosity-transport turbulence models for threedimensional, 

shock separated flowfields”, AIAA Journal, 34, 756–763 (1996). 

[4] F.R. Menter, “Zonal two equation�/�turbulence models for aerodynamic flows”, AIAA Paper 

1993-2906 (1993). 

[5] D. E. Coles and E. A. Hirst (Eds.), Computation of Turbulent Boundary Layers - 1968 AFOSR-IFP- 

Stanford Conference, Stanford (1969). 

[6] P. H. Cook, M. A. McDonald and M.C.P. Firmin, “Aerofoil RAE 2822 - Pressure distributions and 

boundary layer and wake measurements”, AGARD Advisory Report AR-138, A6.1-A6.77 (1979). 

[7] Ch. Gleyzes, “Opération décrochage - Résultats de la 2ème campagne d’essais à F2 – Mesures de 

pression et vélocimétrie laser”, RT-DERAT 55/5004 DN, ONERA (1989). 

Speaker: KNOPP, T. 105 BAIL 2006 

2 

✩ 

✪

J.-S. LEU, J.-Y. JANG, Y.-C. CHOU: Evaporating cooling of liquid film along an 

inclined plate coverd with a porous layer 

✬ 

✫ 

Evaporating cooling of liquid film along an inclined plate covered with a 

porous layer 

ABSTRACT 

Jin-Sheng Leu 

Department of Mechanical Engineering 

Air Force Institute of Technology 

Kaohsiung, Taiwan 82042 

Jiin-Yuh Jang* and Yin Chou 

Department of Mechanical Engineering, 

National Cheng-Kung University 

Tainan, Taiwan 70101 

The purpose of this work is to evaluate the heat and mass enhancement of liquid film 

evaporation by covering a porous layer on the plate (as shown in Fig. 1). There is an extensive 

literature for liquid film evaporating flow based on simplified 1-D and 2-D mathematical models. 

Wassel and Mills [1] illustrated a 1-D design methodology for a counter-current falling film 

evaporative cooler. Yan and Soong [2] presented their numerical solution for convective heat and 

mass transfer along an inclined heated plate with film evaporation with more rigorous treatments of 

the equations governing the liquid film and liquid-gas interface. The complete two-dimensional 

boundary layer model for the evaporating liquid and gas flows along an inclined plate was studied 

recently by Mezaache and Daguenet [3]. Their parametric study focused on the effects of inlet 

conditions such as gas velocity, liquid mass flow rate and inclined angles and their interaction with 

both isothermal and heated walls. Zhao [4] studied the coupled heat and mass transfer in a stagnation 

point flow of air through a heated porous bed with thin liquid film evaporation. 

Until now, there seems to be no related theoretical analysis to evaluate the feasibility of 

utilizing porous materials for the heat transfer enhancement of falling liquid evaporation. This has 

motivated the present investigation. The present study analyzes the liquid film evaporation flow along 

a vertical isothermal plate covered with a thin liquid-saturated porous layer. Liquid and gas streams 

are approached by two coupled laminar boundary layers. The non-Darcian inertia and boundary effects 

are included to describe the hydraulic characteristic of the liquid-saturated porous medium. Then, the 

governing equations (tabulated in Table1) are discretized to a fully implicit difference representation, 

in which the upwind scheme is used to model the axial convective terms, while second-order central 

difference schemes are employed for the transverse convection and diffusion terms. Newton 

linearization procedure is used to linearize the nonlinear terms of governing equation.The numerical 

solution is obtained by utilizing a fully implicit finite difference method and examined in detail for the 

effects of porosity ε, porous layer thickness δ, ambient relative humidity φ and Lewis number Le on 

the average heat and mass transfer performance. 

The numerical results conclude that the latent heat flux is the dominant mode for the 

present study. The cases for lower ε and δ would produce higher interfacial temperature and mass 

concentration, and thus enhance the �heat and mass transfer performances across the film interface. The 

influence of ε on the Nusselt number (Nu) and Sherwood number(Sh) is gradually more significant 

as δ is increased. An applicable range of porous layer thickness δ=0.001~0.005 is suggested for the 

practical application (as shown in Fig. 2). For the effect of ambient relative humidity φ, it is observed 

that a lower φ leads to a higher Nu but lower Sh. However, the influence on Nu and Sh appears to be 

less significant than those of ε and δ. In addition, as the Lewis number is increased(Le>1), a larger 

heat transfer rate is achieved. 

* Professor, author to whom correspondence should be addressed 

Tel: 886-6-2088573 Fax: 886-6-2342232 

1 

E-mail: jangjim@mail.ncku.edu.tw 

Speaker: JANG, J.-Y. 106 BAIL 2006 

✩ 

✪

J.-S. LEU, J.-Y. JANG, Y.-C. CHOU: Evaporating cooling of liquid film along an 

inclined plate coverd with a porous layer 

✬ 

✫ 

REFERRENCE 

[1]. Wassel, A. T., and Mills, A. F., "Design Methodology for a Counter-current Falling Film 

Evaporative Condenser". ASME J. of Heat Transfer, Vol. 109, pp.784-787, 1987. 

[2] Yan, W. M., and Soong, C. Y., “Convection heat and mass transfer along an inclined heated plate 

with film evaporation”, Int. J. Heat Mass Transfer, Vol.38, pp.1261-1269, 1995. 

[3] Mezaache, E., and Daguenet, M., “Effects of Inlet Conditions on Film Evaporation along an 

Inclined Plate”, Solar Energy, Vol. 78, pp.535-542, 2005. 

[4]. Zhao, T. S., “Coupled Heat and Mass Transfer of a Stagnation Point Flow in a Heated Porous Bed 

with Liquid Film Evaporation”, Int. J. Heat Mass Transfer, Vol.42, pp.861-872, 1999. 

Table 1: The governing equations for liquid film 

and gas steam regions 

Liquid 

film 

region 

Gas 

stream 

region 

2 

µ l ∂ u l µ l ρlC 

2 

0 = ρlg 

cos ϕ + − u 

2 l − u l 

ε ∂y 

K K 

2 

∂Tl 

∂ Tl 

u l = αe 

∂x 

2 

∂y 

∂u 

g ∂v 

g 

+ = 0 

∂x 

∂y 

2 

∂u 

g ∂u 

g ∂ u g 

u g + vg 

= ν g 

∂x 

∂y 

2 

∂y 

2 

∂Tg 

∂Tg 

∂ Tg 

ug 

+ vg 

= αg 

∂x 

∂y 

2 

∂y 

2 

∂ω 

∂ω 

∂ ω 

ug 

+ vg 

= D 

∂x 

∂y 

2 

∂y 

where the subscript “l”, “g” represents the 

variables of the liquid and gas stream, 

respectively. ω, ρ ,ν ,α and D are the mass 

concentration, density, kinematic viscosity, 

thermal diffusivity and mass diffusivity of the 

gas. ε and K is the porosity and permeability of 

the porous medium, C is the flow inertia 

parameter. 

Nu 

600 

500 

400 

300 

200 

δ =0.001 

0.002 

0.005 

0.01 

0.02 

100 

0.2 0.4 0.6 

ε 

0.8 1 

2 

liquid 

film 

X = 0 

X 

Tw 

air 

Y 

Tl 

Ti a T 

porous layer 

Figure 1 Schematic diagram of the physical system 

Sh 

35 

30 

25 

20 

15 

10 

= 0.001 

0.002 

0.005 

0.01 

0.02 

5 

0.2 0.4 0.6 

ε 

0.8 1 

(a) (b) 

Figure 2 The coupled effects of porosity ε and thickness δ on the average (a) Nusselt and 

(b)Sherwood numbers with Reg =50000 and φ=70%. 

Speaker: JANG, J.-Y. 107 BAIL 2006 

δ 

✩ 

✪

V.D. LISEYKIN, Y.V. LIKHANOVA, D.V. PATRAKHIN, I.A. VASEVA: Application 

of boundary layer-type functions to comprehensive grid generation codes 

✬ 

✫ 

Liseykin V.D., Likhanova Yu.V, Patrakhin D.V., Vaseva I.A. 

Application of boundary layer-type functions to comprehensive 

grid generation codes 

The paper presents recent results related to the development 

of algorithms and codes for generating both structured and unstructured 

grids with the use of operator Beltrami [1]. Control of 

grid properties is realized by monitor metrics formulated with the 

use of boundary layer-type functions. Formulas for the metrics 

providing generation of grids adapting to vector fields, gradients, 

and/or values of physical quantities are presented. Applications 

of adaptive grids to fluid dynamics and plasma related problems 

are demonstrated. 

The work over the paper is supported by CRDF (RU-M1- 

2579) and RFBR (06-01-00205) grants. 

[1] V.D. Liseikin ”A Computaional Differential Geometry Approach to 

Grid Generation”, 2004, Berlin, Springer. 

Speaker: LISEYKIN, V.D. 108 BAIL 2006 

1 

✩ 

✪

G. LUBE: A stabilized finite element method with anisotropic mesh refinement for the 

Oseen equations 

✬ 

✫ 

Int. Conference on Boundary and Interior Layers 


G. Lube, G. Rapin (Eds) 

c○ University of Göttingen, Germany, 2006 

A stabilized finite element method 

with anisotropic mesh refinement for the Oseen problem 

G. Lube 

Institut für Numerische und Angewandte Mathematik, 

Georg-August-Universität Göttingen, 

Lotzestrasse 16-18, D-37083 Göttingen, Germany 

lube@math.uni-goettingen.de 

Nonstationary incompressible flow problems can be split into auxiliary problems of Oseen 

type. We start with a quasi-optimal error estimate of Cea-type for conforming stabilized Galerkin 

methods of SUPG/PSPG-type with equal-order interpolation of velocity/pressure on general 

meshes. Then we present some new results for this class of methods on hybrid meshes with 

structured anisotropic mesh refinement in boundary layers. The layer meshes are supposed to 

be of tensor-product type. In particular, we prove a modified inf-sup condition with a constant 

independent of the viscosity and of critical parameters of the mesh. Full proofs of the main 

results can be found in [1]. 

Some numerical tests for a laminar and a turbulent channel flow problem confirm the theoretical 

results. In Fig. 1, we show some results for the three-dimensional channel flow at Reτ = 395 

using an improved statististical turbulence model based on the unsteady Reynolds averaged 

Navier-Stokes model (URANS), the v 2 − f model by P. Durbin, see also [3]. The results are in 

good agreement with the DNS data in [2] and partly better than those reported in [3]. 

Achnowledgement: This paper is based on joint work with Th. Apel (Neubiberg), T. 

Knopp (DLR Göttingen) and R. Gritzki (TU Dresden). 

References 

[1] Apel, Th., Knopp, T., Lube, G.: Stabilized finite element methods with anisotropic mesh 

refinement for the Oseen problem, submitted to Appl. Num. Math. 2006 

[2] Kim, J., Moin, P., Moser, R.: Turbulence statistics in fully developed channel flow at low 

Reynolds number, J. Fluid Mech. 177 (1987) 133–166. 

[3] Laurence, L.R., Uribe, J.C., Utyuzhnikov, S.V.: A robust formulation of the v2 − f model, 

J. Flow, Turbulence and Combustion 23 (2004) 169–185. 

Speaker: LUBE, G. 109 BAIL 2006 

1 

✩ 

✪

G. LUBE: A stabilized finite element method with anisotropic mesh refinement for the 

Oseen equations 

✬ 

✫ 

30 ParallelNS 

x Laurence 

25 

DNS 

U + 

20 

15 

10 

5 

0 

10 -1 

x x 

x 

x 

x 

x 

x 

x 

x 

x 

x 

x 

x 

x 

x 

x 

x 

x 

x 

x 

x 

x 

x 

x x 

x x 

y + 

10 0 

10 1 

10 2 

0.35 ParallelNS 

x Laurence 

0.3 

DNS 

0.25 

0.2 

ε + 

0.15 

0.1 

0.05 

x 

x 

x 

x xx xxx x x x x xx x x x 

x 

x 

x 

x 

x 

x 

y + 

x 

x 

x 

10 3 

x x x x 

0 

0 20 40 60 80 100 

k + 

v 2+ 

6 ParallelNS 

x Laurence 

5 

DNS 

4 

3 

2 

1 

x 

x 

x 

x 

x 

xx x 

x 

x 

x 

x 

x 

x 

x 

x 

x 

x 

x 

x 

x 

x 

y + 

x 

x 

x 

x 

x 

x 

x x x 

0 x 

0 100 200 300 400 

2 ParallelNS 

x Laurence 

DNS 

1.5 

1 

0.5 

x 

x x x x 

x x 

x x 

x 

x 

x 

x 

x 

x x 

x 

x x 

x 

x x x x x 

x 

x 

x 

x 

x xx 

0 

0 100 200 300 400 

Figure 1: Plot of u + ,k + ,ǫ + ,(v 2 ) + vs. y + := yuτ 

ν compared to DNS-data in [2] and results in [3] 

Speaker: LUBE, G. 110 BAIL 2006 

2 

y + 

✩ 

✪

H. LÜDEKE: Detached Eddy Simulation of Supersonic Shear Layer Wake Flows 

✬ 

✫ 

Detached Eddy Simulation of Supersonic Shear Layer Wake Flows 

Heinrich Lüdeke 

∗ DLR Institute of Aerodynamics 

and Flow Technology 

Lilienthalplatz 7, D-38108 Braunschweig 

e-mail: heinrich.luedeke@dlr.de 

Tel. +49 (531) 295-3315 

Key words: detached-eddy simulation, axisymmetric base flow, compressible wake, turbulent 

separation 

One challenge of numerical investigations of unsteady super- and hypersonic flow fields is the 

study of the turbulent wake at complex vehicle configurations. A recent promising approach is the 

technique of detached-eddy simulation (DES) proposed by Spalart et.al. Detached-eddy simulation 

is a hybrid approach for the modelling of turbulent flow fields at complex geometries. The idea 

is to combine the best features of both, the Reynolds-averaged Navier-Stokes (RANS) and the 

large eddy simulation (LES) approach to predict massively separated unsteady flow fields at high 

Reynolds-numbers especially in the wake of Re-entry vehicles during descent. In this study the 

Spalart Allmaras one equation turbulence model is used as an accurate and efficient base for DES. 

The intention of the current work is an investigation of the shear flow in the wake of a blunt 

cylinder at M =2.4 at high Reynolds numbers. This is used as a basic configuration for re-entry 

vehicles with a blunt base. The typical time averaged flow field for this configuration is shown 

in fig. 1 with pressure contours and streamlines. The large turning angle behind the base causes 

separation and a region of reverse flow as visible in the wake. The point at the axis of symmetrie 

where the streamwise velocity is zero is considered to be the shear layer reattachment point. In 

this region the flow is forced to turn along the axis of symmetry causing a reattachment shock 

to be formed. The detailed experimental data base, provided by Herrin and Dutton [1], is used 

for comparison of numerically predicted and measured data. The practical applicability of the 

approach is demonstrated considering as example an axisymmetric re-entry capsule. 

All simulations were carried out by the hybrid structured-unstructured DLR Tau code which 

is extensively validated for sub- trans- and hypersonic cases [2]. For the axisymmetric cylinderconfiguration 

structured and unstructured grids have been generated at different resolution of the 

turbulent wake. They are designed with a similar resolution used by Forsythe and Sqires in [3]. 

Especially the near wake of the cylinder is refined extensively. The results have shown good grid 

convergence for the cylinder cases for all averaged quantities, namely the velocity components and 

the pressure, as well as for the diagonal elements of the Reynolds stress tensor and the resolved 

turbulent kinetic energy in the free shear layer. 

In the wake the turbulent kinetic energy and turbulent intensities in radial and streamwise 

direction, computed by extracting the standard deviation of the velocitys over the time, was successffully 

compared with the measurements. Furthermore the pressure level along the base was 

compared with the experimental data of Herrin and Dutton [1] 

Finally simulations of the SARA capsule, a facility conceived to be a recovery orbital platform 

to perform orbital flights, are carried out to give a practical demonstratin of DES modelling for 

hypersonic flight [4]. To ensure turbulent wake flow the trajectory point at M =5.1, Re =20· 106 was chosen. The influence of unsteady loads, generated by the turbulent wake flow was investigated 

for this vehicle. A preliminary snapshot of the vorticity contours in the wake is shown in fig. 2. 

1 

Speaker: LÜDEKE, H. 111 BAIL 2006 

✩ 

✪

H. LÜDEKE: Detached Eddy Simulation of Supersonic Shear Layer Wake Flows 

✬ 

✫ 

References 

[1] J.L. Herrin, J.C. Dutton: Supersonic Base flow Experiments in the Near Wake of aa Cylindrical 

Afterbody. AIAA Journal, Vol 32, No. 1, January 1994. 

[2] A. Mack, V. Hannemann: Validation of the unstructured DLR-TAU-Code for Hypersonic 

Flows, AIAA 2002-3111, 2002. 

[3] J.R. Forsythe, K.A. Hoffmann, K.D. Squires: Detached-Eddy Simulation with compressibility 

Corrections Applied to a Supersonic Axisymmetric Base Flow. AIAA 02-0586, 2002. 

[4] Alessandro La Neve, Flávio de Azevedo Corrêa Junior: SARA Experiment Module Project: 

Using Skills to Enlarge Experiences. International conference on Engineering Education, 

Manchester, U.K., August 18-21, 2002 

Expansion Waves 

Separation Bubble 

Reattachment Shock 

cp: -0.18 -0.14 -0.10 -0.07 -0.03 0.01 0.05 

Figure 1: Flow topology and Cp contours of 

the axisymmetric cylinder wake at M∞ =2.4, 

Re =1.4 · 10 6 . 

vorticity: 1600 2106 2773 3650 4804 6325 8326 10960 14427 18991 25000 

Figure 2: Instantanious vorticity contours in the 

SARA wake at M∞ =5.1, Re =20· 10 6 . 

Speaker: LÜDEKE, H. 112 BAIL 2006 

2 

✩ 

✪

K. MANSOUR: Boundary Layer Solution For laminar flow through a Loosely curved 

Pipe by Using Stokes Expansion 

✬ 

✫ 

Boundary Layer Solution For laminar flow through a 

Loosely curved Pipe by Using Stokes Expansion 

By Kamyar Mansour 

Department of Aerospace Engineering and New Technologies Research Center 

Amir Kabir University of Technology 

Tehran, Iran, 15875-4413 

mansour@aut.ac.ir 

And 

Flow Research and Engineering 

P.O. Box # 20543 Palo Alto, CA, 94309 

Keywords: Curved pipe, high Dean Number 

We consider fully developed steady laminar flow through a toroidal pipe of small curvature ratios 

.The solution is expanded up to 40 terms by computer in powers of Dean Number. The major 

conclusion of this investigation is that the friction ratio in a loosely coiled pipe grows 

asymptotically as the 1/4 power of the similarity parameter and not as the 1/2 power as previously 

deduced from boundary-layer analysis. This work confirmed the results obtained by [1]. The goal 

of this analysis is to provide as complete a description as possible of the flow. The analysis 

yields a solution for all values of Reynolds number from zero to infinity in a continuous fashion 

The paradox concerns the discrepancy between the solution obtained using the extended Stokes 

series method [1] and that obtained using boundary layer techniques [2],[3],[4],[5],[6]and 

experimental work of [7],[8],[9],[10]and numerical work of [11]for the ratio of the friction factors 

in coiled tubes to that in straight one in steady, fully developed laminar flow. . In the ensuing 

debates several papers of[12],[13].[14],[15],[16].[17],[18].[19],[20],[21],[22],[23] various 

explanations and new evidence have been given; however, the paradox , the resolution of which 

is important still remain as a open problem 

The author in [20] raise the possibility of cause of this paradox is the use of only 24 terms to 

estimate the asymptotic limit obtained by [1]. In this paper we extend the Stokes series from 24 

terms used by [1] to 40 terms. We confirm the major result of the friction ratio in a loosely coiled 

pipe grows asymptotically as the 1/4 power of the similarity parameter and not as the 1/2 power 

and that confirm the result of [1]. 

What is particularly important in problems of this type is the presence of analyticity. Not every 

stokes expansion, for examples that of the flow past sphere as described by the full Navier-Stokes 

equation are analytic in Reynolds number. In this case, method of matched asymptotic expansions 

is required and can be automated 

An alternative explanation of the departure of the experiments from our curve might be that for 

more tightly coiled pipes the steady laminar flow is succeeded not by turbulent flow, but by an 

intermediate regime of unsteady laminar motion, with higher friction. Taylor [25] expressed the 

possibility that there may exist an intermediate flow regime between the laminar and turbulent 

ranges. He observed a transition from a steady laminar flow to a laminar vibrating flow as the 

speed increased. The onset of turbulence accrued only at a significantly higher speed. 

The effect of diameter ratios a/L on the relation between the friction factor and the Dean 

number has been investigated numerically by [16] and is found to be negligibly small. They 

concluded the friction factor ratio is relatively insensitive to the diameter ratios at a given Dean 

number. However in 1988 [18] for corresponding curved pipe problem tried to make experiment 

to match the series extension, would have to satisfy that is the flow be fully developed at high 

Dean number namely bigger than 500 and laminar as well as a/L less than .03. But his result adds 

another mystery to this paradox and they obtained data, which lay closer to the present method. In 

Speaker: MANSOUR, K. 113 BAIL 2006 

✩ 

✪

K. MANSOUR: Boundary Layer Solution For laminar flow through a Loosely curved 

Pipe by Using Stokes Expansion 

✬ 

✫ 

this paper we have shown that what proposed by [20] as the cause of the discrepancy lies in the 

use of the only first 12 terms for corresponding curved problem is not correct. In this work we 

increase number of terms from 12 in [1] to 20 and still the major conclusion of the discrepancy 

between the solution obtained using the extended stoke series method and that obtained using 

other method persist as it was the case for the rotating pipe[26]. 

REFERENCES 

[1] Van Dyke, M. 1978 Extended stokes series: laminar flow through a loosely coiled pipe. J.Fluid 

Mech.,86, 129-145 

[2] Adler,M.”Flow in curved tubes” Z.Angew.Math.1934 Mech.,14,257 

[3]Barua,S.N. 1963 On secondary flow in stationary curved pipes. Quart. J. Mech. Appl. Maths. 26. 61-77 

[4] Mori, Y. & Nakayama, W. 1965 study on forced convective heat transfer in curved pipes (1 st report, 

laminar region). Intl. J. of Heat Mass Transfer 8, 67-82 

[5] Ito, H., 1969 Laminar Flow in curved pipes Z. agnew. Math. Mech. 11. 653-663 

[6] Smith, F. T. 1976 Steady motion within a curved pipe. Proc. Roy. Soc. A 347, 345-370 

[7] White, C. M. 1929 Streamline flow through curved pipes. Proc. Roy. Soc. A 123, 845-663. 

[8] Hasson,D. 1955 “Streamline flow resistence in coils. Research 8(1) supplement p.S1 

[9] Trefethen, L. 1957b Flow in rotating radial ducts: report R55GL 350 on laminar flow in rotating, 

heated horizontal, and bent tubes, extended into transition and turbulent regions. Gen. Elec. Co. 

Rep. 55GL350-A. 

[10] Ito, H.,1959 Friction factors for turbulent flow in curved pipes.Trans.ASMEJ.BasicEngng.81,123-134 

[11] Collins, W. M. &Dennis, S. C. R. 1975 The steady motion of a viscous fluid in a curved tube. 

Quart. J. Mech. Appl. Maths. 28, 133-156. 

[12] Dennis, S.C.R. 1980 “Calculation of the steady flow through a curved tube using a new finite 

difference method. J.Fluid. Mech.,99, 449-467 

[13] Dennis, S.C.R. &Ng,M.C. 1982 “Dual solution for steady laminar flow through a curved tube.Q.J. 

Mech. Appl. Maths 35,305-324 

[14] Nandakumar, K. & Masliyah, J. H. 1982 Bifurcation in steady laminar flow through curved tubes.J. 

Fluid Mech. 119,475-490 

[15] Mansour, K. 1985 Laminar flow through a slowly rotating straight pipe. J. Fluid Mech. 150, 1-24. 

[16] Soh,W.Y.&Berger,S.A. 1987 “fully developed flow in a curved pipe of arbitrary 

curvature ratio.Intl J. Numer.Mech. Fluids 7,733 

[17] Winters,K.H. 1987 A bifurcation study of laminar flow in a curved 

tube of rectangular cross-section. J. Fluid Mech. 180, 343-369 

[18] Ramshankar, R. & Sreenvasan, K. R. 1988 A paradox concerning the extended Stokes series solution 

for the pressure drop in coiled pipes. Phys. Fluids, 1339-1347. 

[19] Van Dyke, M. 1989 some paradoxes in viscous flow theory. “Some unanswered question in fluid 

mechanics “ (ed. L. M. Trefethan & R. L. Panton).Appl. Mech. Rev. 43, 153-170 

[20] S. Jayanti & G.F. Hewitt. "On the paradox concerning friction factor ratio in laminar flow in coils", 

Proc. R. Soc. Lond. A (1991) 432, 291-299 

[21] Mansour, K. 1993 Using Stokes Expansion For Natural Convection inside a two-dimensional cavity. 

Fluid Dynamics Research, 1-33 

[22 ] Dennis,S.C.R. and Riley,N. 1993 ”On the fully developed flow in a curved pipe 

at large Dean number” Proc. Roy. Soc. A.434, 473-478 

[23] Ishigaki, H. "Analogy between laminar flows in curved pipes and orthogonal rotating pipes". J. Fluid 

Mech. (1994), vol.268, pp. 133-145 

[24] Dean, W.R. 1927 Note on the motion of fluid in a curved pipe. Phil.Mag. (7) 4,208-223 

[25] Taylor, G.I. 1929 The criterion for turbulence in curved piped. Proc. R. Soc. Lond. A 124, 243-249. 

[26] Mansour, K. 2002 Boundary layer solution using Stokes expansion for Laminar flow through a slow- 

rotating pipe, Proceedings of BAIL 2002 ,An international conference on boundary and interior layers 

computational & Asymptotic methods, Perth, Western Australia. 

Speaker: MANSOUR, K. 114 BAIL 2006 

✩ 

✪

G. MATTHIES, L. TOBISKA: Mass conservation of finite element methods for coupled 

flow-transport problems 

✬ 

✫ 

Mass conservation of finite element methods for coupled flow-transport 

problems 

Gunar Matthies 1 and Lutz Tobiska 2 

1 Fakultät für Mathematik, Ruhr-Universität Bochum 

2 Institut für Analysis und Numerik, Otto-von-Guericke-Universität Magdeburg 

We consider a coupled flow-transport problem in a bounded domain Ω ⊂ R d , d = 2,3. The 

system is described by the instationary, incompressible Navier–Stokes equations 

and the time-dependent transport equation 

ut − ν△u + (u · ∇)u + ∇p = f in Ω × (0,T], 

div u = 0 in Ω × (0,T], 

u = ub on ∂Ω × (0,T], 

u(0) = u 0 

in Ω, 

ct − ε△c + u · ∇c = g in Ω × (0,T], 

(cu − ε∇c) · n = cI u · n on Γ− × (0,T], 

ε∇c · n = 0 on Γ+ × (0,T], 

c(0) = c 0 

in Ω. 

Here, u and p denote the velocity and the pressure of the fluid, respectively, ν and ε are small 

positive numbers, c is the concentration, cI the concentration at the inflow boundary Γ− := 

{x ∈ ∂Ω : u · n < 0}, and Γ+ := ∂Ω \ Γ−. We assume that ub is the restriction of a divergence 

free function onto the boundary ∂Ω. Note that the velocity u from the Navier–Stokes equations 

enters the transport equation as a convection field. 

Due to incompressibility constraint, the weak solution c of (2) satisfies the global mass 

conservation property 

� � � � 

d 

cdx + cIu · ndγ + cu · ndγ = g dx. (3) 

dt Ω Γ− 

Γ+ 

Ω 

It is well-known [1], that the finite element solutions satisfy in general the incompressibility 

constraint and the global mass conservation property (3) only approximately. 

Different discretisation methods [2, 3, 4] for the instationary, incompressible Navier–Stokes 

equations and stabilised schemes for the transport problem like SDFEM will be studied numerically. 

References 

[1] C. Dawson, S. Sun, M. F. Wheeler, Compatible algorithms for coupled flow and transport. 

Comput. Methods Appl. Mech. Engrg., 193, 2565–2580 (2004). 

[2] G. Matthies, L. Tobiska, The inf-sup condition for the mapped Qk/P disc 

k−1 element in arbitrary 

space dimensions. Computing, 69, 119–139 (2002). 

[3] G. Matthies, L. Tobiska, Inf-sup stable non-conforming finite elements of arbitrary order 

on triangles. Numer. Math., 102, 293–309 (2005). 

[4] G. Matthies, Inf-sup stable nonconforming finite elements of higher order on quadrilaterals 

and hexahedra. Bericht Nr. 373, Fakultät für Mathematik, Ruhr-Universität Bochum 

(2005). 

Speaker: MATTHIES, G. 115 BAIL 2006 

1 

(1) 

(2) 

✩ 

✪

J. MAUSS, J. COUSTEIX: Global Interactive Boundary Layer (GIBL) for a Channel 

✬ 

✫ 

Global Interactive Boundary Layer (GIBL) for a Channel 

J. Mauss ♦ and J. Cousteix ♣ 

♦ IMFT and UPS, 118 route de Narbonne, 31062 Toulouse Cedex - France 

Phone: 05 61 55 67 94 - mauss@cict.fr 

♣ ONERA and SUPAERO, 2 av. É. Belin, 31055 Toulouse Cedex - France 

Phone: 05 62 25 25 80 - Jean.Cousteix@onecert.fr 

We consider a laminar, steady, two-dimensional flow of an incompressible Newtonian fluid 

in a channel at high Reynolds number. When the walls are slightly deformed, adverse pressure 

gradients are generated and separation can occur. The analysis of the flow structure has been 

done essentially by Smith [4]. Later, a systematic asymptotic analysis has been performed by 

Saintlos and Mauss [3]. With the Successive Complementary Expansion Method, SCEM, we 

assume a uniformly valid approximation (UVA) based on generalised expansions. This method, 

developed by Cousteix and Mauss [1], has been used by Dechaume et al. [2]. 

Navier-Stokes dimensionless equations can be written 

div −→ V =0, (grad −→ V ) · −→ V = − grad Π+ 1 

Re △ −→ V . (1) 

The basic plane Poiseuille flow is 

v (x) = u0 = y − y 2 , v (y) =0, Π=Π0 = − 2x 

Re + p0 . (2) 

The flow is perturbed, for instance, by indentations of the lower and upper walls such as 

yl = εF (x, ε) , yu =1− εG(x, ε) , (3) 

where ε is a small parameter. If we seek a solution in the form 

v (x) = u0(y)+εu(x, y, ε) , v (y) = εv(x, y, ε) , Π − p0 = − 2x 

the Navier-Stokes equations become 

+ εp(x, y, ε) , 

Re 

(4) 

∂u ∂v 

+ = 0 , (5a) 

∂x ∂y 

� 

ε u ∂u 

� 

� 

∂u ∂u du0 ∂p 1 ∂2u + v + u0 + v = − + 

∂x ∂y ∂x dy ∂x Re ∂x2 + ∂2u ∂y2 � 

, (5b) 

� 

ε u ∂v 

� 

� 

∂v ∂v 

1 ∂2v + v + u0 = −∂p + 

∂x ∂y ∂x ∂y Re ∂x2 + ∂2v ∂y2 � 

. (5c) 

It is clear that, for high Reynolds numbers, the reduced equations are of first order leading 

to a singular perturbation. In the core flow, we are looking for approximations coming from 

asymptotic generalised expansions such as 

u = u1(x, y, ε)+··· , v = v1(x, y, ε)+··· , p = p1(x, y, ε)+··· . (6) 

Formally, neglecting terms of order O(ε, 1/Re) , for the core flow, we obtain 

∂u1 

∂x 

+ ∂v1 

∂y 

∂u1 du0 

u0 + v1 

∂x dy 

∂v1 

u0 

∂x 

1 

= 0 , (7a) 

= −∂p1 

∂x 

= −∂p1 

∂y 

, (7b) 

. (7c) 

Speaker: MAUSS, J. 116 BAIL 2006 

✩ 

✪

J. MAUSS, J. COUSTEIX: Global Interactive Boundary Layer (GIBL) for a Channel 

✬ 

✫ 

It is very interesting to observe the singular behaviour of the solution of (7a–7c) as we 

approach the boundaries. For instance, when y → 0, we have 

u1 = −2p10 ln y + c10 + ··· , 

where p10 and c10 are functions of x and ε. 

The generalised asymptotic expansions for the velocity are given by 

v (x) = u0(y)+εu(x, y, ε)+··· , v (y) = εv(x, y, ε)+··· . (8) 

Using the SCEM, the problem consists of solving the continuity equation together with the 

momentum equation 

∂u ∂v 

+ = 0 , (9a) 

∂x ∂y 

� 

ε u ∂u 

� 

∂u ∂u du0 

+ v + u0 + v = −∂p1 

∂x ∂y ∂x dy ∂x + ε3 ∂2u . (9b) 

∂y2 But, now, we have to solve simultaneously (9a–9b) and the core equations (7a–7c). The same 

form as Prandtl’s equations is recovered if we let 

∂Π 

U = u0 + εu , V = εv , 

∂x = −2ε3 + ε ∂p1 

, (10) 

∂x 

leading to 

U ∂U ∂U 

+ V = −∂Π 

∂x ∂y ∂x + ε3 ∂2U . (11) 

∂y2 The boundary conditions are now U = V = 0 on the walls. As four conditions must be satisfied, 

it is clear that the pressure gradient must be adjusted in order to ensure the global mass flow 

conservation in the channel. Calculations have been performed with a simplified model coming 

from the triple deck theory. An example is given in Fig. 1 which gives the evolution of the skinfriction 

coefficient along the lower and upper walls of a channel whose lower wall is deformed. 

References 

Cf 

upper 

wall 

0,0 

2 Re 

lower 

wall 

1.2 

1.0 

0.8 

0.6 

0.4 

0.2 

R =10 3 

x/L 

-5 -4 -3 -2 -1 0 1 2 3 4 5 

Figure 1: Application of GIBL in a channel whose lower wall is deformed 

[1] J. Cousteix and J. Mauss. Approximations of the Navier-Stokes equations for high Reynolds 

number flows past a solid wall. Jour. Comp. and Appl. Math., 166(1):101–122, 2004. 

[2] A. Dechaume, J. Cousteix, and J. Mauss. An interactive boundary layer model compared to 

the triple deck theory. Eur. J. of Mechanics B/Fluids, 24:439–447, 2005. 

[3] S. Saintlos and J. Mauss. Asymptotic modelling for separating boundary layers in a channel. 

Int. J. Engng. Sci., 34(2):201–211, 1996. 

[4] F.T. Smith. On the high Reynolds number theory of laminar flows. IMA J. Appl. Math., 

28(3):207–281, 1982. 

Speaker: MAUSS, J. 117 BAIL 2006 

2 

✩ 

✪

O. MIERKA, D. KUZMIN: On the implementation of turbulence models in incompressible 

flow solvers based on a finite element discretization 

✬ 

✫ 

On the implementation of turbulence models in incompressible 



O. Mierka and D. Kuzmin 

Institute of Applied Mathematics (LS III), University of Dortmund 

Vogelpothsweg 87, D-44227, Dortmund, Germany 

omierka@math.uni-dortmund.de 

kuzmin@math.uni-dortmund.de 

The numerical implementation of turbulence models involves many algorithmic components 

all of which may have a decisive influence on the quality of simulation results. In particular, 

a positivity-preserving discretization of the troublesome convective terms and nonlinear 

sources/sinks is an important prerequisite for the robustness of the numerical algorithm. This 

paper presents a detailed numerical study of several eddy viscosity models implemented in the 

open-source software package FeatFlow (http://www.featflow.de) using algebraic flux correction 

to enforce the positivity constraint. The underlying finite element discretization and 

iterative solution techniques are presented and relevant algorithmic details are revealed. 

2. Algebraic flux correction schemes 

The design of high-resolution finite element schemes for numerical simulation of turbulent incompressible 

flows on the basis of eddy viscosity models is addressed. A robust positivity-preserving 

algorithm is developed building on the algebraic flux correction paradigm for scalar transport 

problems [1]. It is explained how to get rid of nonphysical oscillations in the vicinity of steep 

gradients and to remove excessive artificial diffusion in regions where the solution is sufficiently 

smooth. To this end, the discrete operators resulting from a standard Galerkin discretization 

of convective terms are modified so as to enforce the desired matrix properies without violating 

mass conservation. 

3. Implementation of eddy viscosity models 

The developed algebraic flux correction techniques are applied to high Reynolds number flows 

that can be described by the incompressible Navier-Stokes equations coupled with an eddy 

viscosity model of turbulence. A global Multilevel Pressure Schur Complement (discrete projection) 

method is employed to enforce the incompressibility constraint at the discrete level [2]. 

The turbulent eddy viscosity is introduced in several different ways using 

• the RANS approach as represented by various modifications of the k − ε model; 

• Large Eddy Simulation (LES) with explicit and implicit subgrid scale modelling. 

In particular, the advantages of Monotonically Integrated LES algorithms as compared to explicit 

subgrid scale models of Smagorinsky type are explored. The implementation of the standard k−ε 

model is based on a block-iterative algorithm featuring a positivity-preserving representation of 

sink terms [2]. The main highlight of the present paper is a unified solution strategy for strongly 

coupled PDE systems which result from 2D/3D finite element discretizations of RANS and 

LES models on unstructured meshes. Special emphasis is laid on the near wall treatment and 

Speaker: MIERKA, O. 118 BAIL 2006 

1 

✩ 

✪

O. MIERKA, D. KUZMIN: On the implementation of turbulence models in incompressible 


✬ 

✫ 

implementation of initial/boundary conditions. A set of representative benchmark problems is 

employed to evaluate the performance of the turbulence models under consideration as applied 

to incompressible flows at high Reynolds numbers. 

4. Numerical examples 

The first example deals with a 3D simulation of the turbulent incompressible flow past a backward 

facing step (Re = 44,000) using the standard k −ε model with logarithmic wall functions. 

The numerical solutions displayed in Figure 1 are in a good agreement with those published in 

the literature [3]. 

Figure 1: Backward facing step simulation results (Re = 44,000). Contour lines of a),b) - 

turbulent kinetic energy; c),d) - turbulent eddy viscosity. a),c) - reference solution [3]. 

A preliminary code validation for Chien’s low-Reynolds number modification was performed 

for a simple channel flow at Re = 13,750 and compared with Kim’s [4] DNS data in Figure 2. 

Numerical experiments using both versions of the k −ε model as well as Large Eddy Simulation 

with explicit and implicit subgrid scale modelling are currently under way. The results of an 

in-depth comparative study will be reported at the Conference. 

Figure 2: Channel flow simulation results (Re = 13,750) compared with reference data [4]. 

References 

[1] D. Kuzmin and M. Möller, Algebraic flux correction I. Scalar conservation laws. In: D. 

Kuzmin, R. Löhner and S. Turek (eds.) Flux-Corrected Transport: Principles, Algorithms, 

and Applications. Springer, 2005, 155-206. 

[2] S. Turek and D. Kuzmin, Algebraic flux correction III. Incompressible flow problems. In: D. 

Kuzmin, R. Löhner and S. Turek (eds.) Flux-Corrected Transport: Principles, Algorithms, 

and Applications. Springer, 2005, 251-296. 

[3] F. Ilinca, J.-F. Hétu and D. Pelletier, A Unified Finite Element Algorithm for Two-Equation 

Models of Turbulence. Comp. & Fluids 27-3 (1998) 291–310. 

[4] J. Kim, P. Moin and R. D. Moser, Turbulence statistics in fully developed channel flow at 

low Reynolds number. J. Fluid Mech. 177 (1987) 133–166. 

2 

Speaker: MIERKA, O. 119 BAIL 2006 

✩ 

✪

K. MORINISHI: Rarefied Gas Boundary Layer Predicted with Continuum and Kinetic 

Approaches 

✬ 

✫ 

Ê�Ö�¬�� × �ÓÙÒ��ÖÝ Ä�Ý�Ö 

ÈÖ�� Ø�� Û�Ø� �ÓÒØ�ÒÙÙÑ �Ò� Ã�Ò�Ø� �ÔÔÖÓ� ��× 

ÃÓ�� ÅÓÖ�Ò�×�� 

��Ô�ÖØÑ�ÒØ Ó� Å� ��Ò� �Ð �Ò� ËÝ×Ø�Ñ �Ò��Ò��Ö�Ò� 

ÃÝÓØÓ ÁÒ×Ø�ØÙØ� Ó� Ì� �ÒÓÐÓ�Ý 

Å�Ø×Ù��×�� Ë��ÝÓ �Ù ÃÝÓØÓ � � �� Â�Ô�Ò 

Ì�Ä� � �� 

� Ñ��Ð� ÑÓÖ�Ò�×� ��Ø � �Ô 

ÆÙÑ�Ö� �Ð ×�ÑÙÐ�Ø�ÓÒ Ó� ÐÓÛ ×Ô��×ÓÛ× ��ÓÙØ Ñ� ÖÓ ��Ú� �× �× ÓÒ� Ó� Ø�� Ö� �ÒØ Ò�Û 

�ÖÓÒØ��Ö× Ó� ÓÑÔÙØ�Ø�ÓÒ�Ð Ù�� ÝÒ�Ñ� × �� ÁØ Ñ�Ý ÔÖÓÚ��× �××�ÒØ��Ð ÙÒ��Ö×Ø�Ò��Ò� ��ÓÙØ 

Ø�� Ù�� Ú�ÓÖ �ÖÓÙÒ� Ñ� ÖÓ �Ð� ØÖÓ Ñ� ��Ò� �Ð ×Ý×Ø�Ñ× Å�ÅË Ë�Ò � Ø�� ÓÛ× ��ÓÙØ Ø�� 

Ñ� ÖÓ ��Ú� �× �Ù×Ø Ö�Ò�� ÖÓÑ ×Ð�Ô ØÓ ØÖ�Ò×�Ø�ÓÒ�Ð ÓÛ Ö��Ñ�× ÒÙÑ�Ö� �Ð Ñ�Ø�Ó�× ��×�� ÓÒ Ø�� 

�ÓÐØÞÑ�ÒÒ �ÕÙ�Ø�ÓÒ ÓÖ Ø�� Æ�Ú��Ö ËØÓ��× �ÕÙ�Ø�ÓÒ× ×Ù�� Ø ØÓ Ú�ÐÓ �ØÝ ×Ð�Ô �Ò� Ø�ÑÔ�Ö�ØÙÖ� 

�ÙÑÔ �ÓÙÒ��ÖÝ ÓÒ��Ø�ÓÒ× ÑÙ×Ø �� Ù×�� Ì��× Ô�Ô�Ö ��× Ö��× Ø�� ÓÑÔ�Ö�×ÓÒ Ó� Ú�ÐÓ �ØÝ 

�Ò� Ø�ÑÔ�Ö�ØÙÖ� ��×ØÖ��ÙØ�ÓÒ �Ò Ø�� Ö�Ö�¬�� × �ÓÙÒ��ÖÝ Ð�Ý�Ö ÔÖ�� Ø�� Û�Ø� ×Ù �ÒÙÑ�Ö� �Ð 

Ñ�Ø�Ó�× 

Ì�� Ö� Ø ×�ÑÙÐ�Ø�ÓÒ ÅÓÒØ� ��ÖÐÓ �ËÅ� Ñ�Ø�Ó� � ℄ Û�� × Û��ÐÝ ��Ò Ù×�� ÓÖ 

×�ÑÙÐ�Ø�Ò� �� ×Ô�� Ö�Ö�¬�� ÓÛ× �� ÓÑ�× � ÔÓÓÖ ×�ÑÙÐ�Ø�ÓÒ ØÓÓÐ �ÓÖ Ø�� ÐÓÛ ×Ô��ÓÛ× 

��ÓÙØ Ñ� ÖÓ ��Ú� �× �� Ù×� �Ù�� ×�ÑÔÐ� ×�Þ� �× Ö�ÕÙ�Ö�� ØÓ Ö��Ù � Ø�� ×Ø�Ø�×Ø� �Ð × �ØØ�Ö Ó� 

Ø�� ËÅ� Ñ�Ø�Ó� ØÓ � Ð�Ú�Ð Ó� Ø�� ×Ñ�ÐÐ ��Ò��× Ó� ÓÛ ÕÙ�ÒØ�Ø��× �Ò Ø�� ÐÓÛ ×Ô��ÓÛ× � ℄ 

Ì�� Ñ�Ø�Ó� ��×�� ÓÒ � ��Ò�Ø� ÑÓ��Ð �ÓÐØÞÑ�ÒÒ �ÕÙ�Ø�ÓÒ � ℄ �× �Ö�� ÖÓÑ Ø�� ×Ø�Ø�×Ø� �Ð 

× �ØØ�Ö �Ò� �× ��¬Ò�Ø�ÐÝ ×ÙÔ�Ö�ÓÖ ØÓ Ø�� ËÅ� Ñ�Ø�Ó� �ÓÖ Ø�� ÐÓÛ ×Ô�� ÓÛ× Ì��×� ��Ò�Ø� 

Ñ�Ø�Ó�× �Ò �� ÔÔÐ�� ØÓ ×Ð�Ô �Ò� ØÖ�Ò×�Ø�ÓÒ�Ð Ö��Ñ� ÓÛ× 

Ì�� Æ�Ú��Ö ËØÓ��× Ñ�Ø�Ó�× Ñ�Ý ÔÖ�� Ø ÓÛ× �Ò Ø�� ×Ð�Ô Ö��Ñ� �� Ø�� Ú�ÐÓ �ØÝ ×Ð�Ô �Ò� 

Ø�ÑÔ�Ö�ØÙÖ� �ÙÑÔ �ÓÙÒ��ÖÝ ÓÒ��Ø�ÓÒ× �Ö� �ÒØÖÓ�Ù �� Ì�� ÓÒ��Ø�ÓÒ× Ñ�Ý �� ÛÖ�ØØ�Ò �×� 

Ù × ÙÛ � �Ú ÃÒ 

�Ú � 

� Ô ÌÛ Ì × Ì Û � � Ø 

� Ø 

 

 

Ù 

Ò � 

ÃÒ 

Ö � ÃÒ � 

� 

ÈÖ � Ô ÌÛ Û��Ö� Ò �Ò� × ��ÒÓØ� Ø�� ÒÓÖÑ�Ð �Ò� Ø�Ò��ÒØ��Ð ��Ö� Ø�ÓÒ× ØÓ Ø�� ×ÙÖ�� Ö�×Ô� Ø�Ú�ÐÝ ÃÒ �× Ø�� 

ÃÒÙ�×�Ò ÒÙÑ��Ö �Ò� �× Ø�� Ö�Ø�Ó Ó� ×Ô� �¬ ��Ø× Ì�� ÑÓÑ�ÒØÙÑ � ÓÑÑÓ��Ø�ÓÒ Ó�Æ ��ÒØ 

� Ú �Ò� Ø�� Ò�Ö�Ý � ÓÑÑÓ��Ø�ÓÒ Ó�Æ ��ÒØ � Ø �Ö� ×�Ø ØÓ ÙÒ�ØÝ �Ò �ÐÐ Ø�� ÓÑÔÙØ�Ø�ÓÒ ÔÖ�×�ÒØ�� 

��Ö� 

�ÐØ�ÓÙ�� ÔÔÐ� ��Ð�ØÝ Ó� Ø�� Æ�Ú��Ö ËØÓ��× Ñ�Ø�Ó�× ØÓ Ø�� ÓÛ× �Ò Ø�� ØÖ�Ò×�Ø�ÓÒ�Ð Ö��Ñ� 

�× ÙÒ �ÖØ��Ò Ø�� Æ�Ú��Ö ËØÓ��× Ñ�Ø�Ó�× �Ö� ÓÑÔÙØ�Ø�Ò�ÐÐÝ ×�Ú�Ö�Ð ÓÖ��Ö× Ó� Ñ��Ò�ØÙ�� ÑÓÖ� 

�Ò�ÜÔ�Ò×�Ú� Ø��Ò Ø�� Ò�Ø� Ñ�Ø�Ó�× Ì�Ù× �Ø �× ��ÖÐÝ Ñ��Ò�Ò��ÙÐ ØÓ ÓÑÔ�Ö� Ø�� ÔÖ�� Ø�ÓÒ Ó� 

Ø�� Æ�Ú��Ö ËØÓ��× Ñ�Ø�Ó�× Û�Ø� Ø��Ø Ó� Ø�� Ò�Ø� Ñ�Ø�Ó�× 

Ë�Ú�Ö�Ð ÒÙÑ�Ö� �Ð �ÜÔ�Ö�Ñ�ÒØ× �Ö� ÔÖ�Ð�Ñ�Ò�ÖÝ �ÖÖ�� ÓÙØ �ÓÖ ×Ù�×ÓÒ� �Ò� ×ÙÔ�Ö×ÓÒ� ÓÛ× 

�Ò Ø�� ×Ð�Ô �Ò� ØÖ�Ò×�Ø�ÓÒ�Ð Ö��Ñ�× �ÓÖ �Ò �Ü�ÑÔÐ� �� ×�ÓÛ× Ø�� ÓÑÔ�Ö�×ÓÒ Ó� Ø�� 

Ú�ÐÓ �ØÝ ��×ØÖ��ÙØ�ÓÒ Ò��Ö � ÐÓÛ�Ö Û�ÐÐ Ó� � ÔÐ�Ò� �ÓÙ�ØØ� ÓÛ Ì�� ÃÒÙ�×�Ò ÒÙÑ��Ö ��×�� ÓÒ 

Ø�� ÒÒ�Ð ��Ø �× � �Ò� Ø�� Å� �ÒÙÑ��Ö ��×�� ÓÒ Ø�� Ú�ÐÓ �ØÝ Ó� Ø�� ÙÔÔ�Ö Û�ÐÐ �× � 

Ì�� Æ�Ú��Ö ËØÓ��× ÔÖ�� Ø�ÓÒ Û�ÐÐ Ó�Ò �� Û�Ø� Ø�� Ò�Ø� ÔÖ�� Ø�ÓÒ �Ò Ø�� ÙÐ� ÓÛ Ö��ÓÒ 

Û��Ð� Ø�� ÔÖ�� Ø�ÓÒ ��Ú��Ø�× �ÖÓÑ Ø�� Ò�Ø� ÔÖ�� Ø�ÓÒ �Ò Ø�� ÃÒÙ�×�Ò Ð�Ý�Ö 

Ì�� ×� ÓÒ� �Ü�ÑÔÐ� �× � ×ÙÔ�Ö×ÓÒ� ÓÛ ÓÚ�Ö � �Ö ÙÐ�Ö ÝÐ�Ò��Ö �Ø � �Ö�� ×ØÖ��Ñ Å� �ÒÙÑ 

��Ö Ó� � �Ò� � ÃÒÙ�×�Ò ÒÙÑ��ÖÓ� � ��ÙÖ�× �Ò� ×�ÓÛ Ø�� ÓÑÔ�Ö�×ÓÒ Ó� Ø�Ò��ÒØ��Ð 

Ú�ÐÓ �ØÝ ��×ØÖ��ÙØ�ÓÒ �ÐÓÒ� Ø�� ÒÓÖÑ�Ð Ð�Ò�× �ÖÓÑ Ø�� ÝÐ�Ò��Ö ×ÙÖ�� Ø � � �� Ò�� 

Ö�×Ô� Ø�Ú�ÐÝ Ì��Æ�Ú��Ö ËØÓ��× ÔÖ�� Ø�ÓÒ �× �Ò �ÓÓ� ��Ö��Ñ�ÒØ Û�Ø� Ø�� Ò�Ø� ÔÖ�� Ø�ÓÒ× �Ø 

Speaker: MORINISHI, K. 120 BAIL 2006 

Ì 

Ò 

�Ì Û 

Ì 

× 

✩ 

✪

K. MORINISHI: Rarefied Gas Boundary Layer Predicted with Continuum and Kinetic 

Approaches 

✬ 

y 

__ 

H 

0.20 

0.15 

0.10 

0.05 

Mw=0.2 , Kn=0.1 

Kinetic 

DSMC 

NS-Slip 

0.00 

0.00 0.05 0.10 

u / Uw 

0.15 0.20 

��ÙÖ� � �ÓÑÔ�Ö�×ÓÒ Ó� Ø�� Ú�ÐÓ �ØÝ ��×ØÖ� 

�ÙØ�ÓÒ �ÓÖ � ÔÐ�Ò� �ÓÙ�ØØ� ÓÛ 

y 

__ 

D 

1.00 

0.75 

0.50 

0.25 

q = 90 

Kinetic 

DSMC 

NS-Slip 

0.00 

0.0 0.2 0.4 

u / U 

0.6 0.8 

��ÙÖ� � �ÓÑÔ�Ö�×ÓÒ Ó� Ø�Ò��ÒØ��Ð Ú�ÐÓ �ØÝ 

��×ØÖ��ÙØ�ÓÒ �Ø � � �� 

y 

__ 

D 

1.00 

0.75 

0.50 

0.25 

q = 45 

Kinetic 

DSMC 

NS-Slip 

0.00 

0.0 0.2 0.4 

u / U 

0.6 0.8 

��ÙÖ� � �ÓÑÔ�Ö�×ÓÒ Ó� Ø�Ò��ÒØ��Ð Ú�ÐÓ �ØÝ 

��×ØÖ��ÙØ�ÓÒ �Ø � � �� 

Kinetic 

NS-Slip 

2.0 

2.0 

��ÙÖ� �� ÓÑÔ�Ö�×ÓÒ Ó� Ø�ÑÔ�Ö�ØÙÖ� ÓÒ 

ØÓÙÖ× 

� � �� Û��Ð� ×ÓÑ� ��× Ö�Ô�Ò Ý �× Ó�×�ÖÚ�� Ø � � �� ÙÖ� � ÔÐÓØØ�� Ø�� Ø�ÑÔ�Ö�ØÙÖ� 

ÓÒØÓÙÖ× �ÖÓÙÒ� Ø�� Ö ÙÐ�Ö ÝÐ�Ò��Ö Ì�� Ö�×ÙÐØ× Ó� Ø�� Æ�Ú��Ö ËØÓ��× �ÕÙ�Ø�ÓÒ× Û�Ø� Ø�� Ú� 

ÐÓ �ØÝ ×Ð�Ô �Ò� Ø�ÑÔ�Ö�ØÙÖ� �ÙÑÔ �ÓÙÒ��ÖÝ ÓÒ��Ø�ÓÒ× Û�ÐÐ ��Ö�� Û�Ø� Ø�Ó×� Ó� Ø�� Ò�Ø� ÑÓ��Ð 

�ÓÐØÞÑ�ÒÒ �ÕÙ�Ø�ÓÒ �Ò Ø�� ÖÓÒØ Ö��ÓÒ �Ò ÐÙ��Ò� Ø�� ÔÓ×�Ø�ÓÒ Ó� �ÓÛ ×�Ó � Û��Ð� ��× Ö�Ô�Ò Ý 

�× Ó�×�ÖÚ�� Ò Ø�� Û�� Ì�� Ò×�ØÝ �Ò Ø�� Û�� ×ÐÓÛ�Ö Ø��Ò Ø�� Ö�� ×ØÖ��Ñ ��Ò×�ØÝ Ì��× 

Ð�Ö�� Ö�Ö�� Ø�ÓÒ �«� Ø Ð��× Ø�� Æ�Ú��Ö ËØÓ��× �ÕÙ�Ø�ÓÒ× ØÓ ��Ð ØÓ ÔÖ�� Ø Ø�� Û�� ÓÛ ¬�Ð� 

�ÙÐÐ ��× Ö�ÔØ�ÓÒ Ó� Ø�� ÒÙÑ�Ö� �Ð Ñ�Ø�Ó�× �Ò� �ÙÖØ��Ö ÒÙÑ�Ö� �Ð Ö�×ÙÐØ× ��ÓÙØ ×Ù�×ÓÒ� �Ò� 

×ÙÔ�Ö×ÓÒ� ÓÛ× �Ò ×Ð�Ô �Ò� ØÖ�Ò×�Ø�ÓÒ�Ð Ö��Ñ�× Û�ÐÐ �� ÔÖ�×�ÒØ�� Ò Ø�� ¬Ò�Ð Ô�Ô�Ö 

✫ 

Ê��Ö�Ò �× 

� ℄ ��Ö� � � ÅÓÐ� ÙÐ�Ö ��× �ÝÒ�Ñ� × ÇÜ�ÓÖ� ÍÒ�Ú�Ö×�ØÝ ÈÖ�×× �� 

� ℄ ��Ò Â �ÓÝ� Á � �Ò� �� È �ÓÑÔÙØ�Ø�ÓÒ Ó� Ê�Ö�¬�� × �ÐÓÛ× �ÖÓÙÒ� � Æ�� 

��Ö�Ó�Ð �Á�� ÂÓÙÖÒ�Ð � � � � � � 

� ℄ ÅÓÖ�Ò�×�� Ã �Ò� Ç�Ù �� À � �ÓÑÔÙØ�Ø�ÓÒ�Ð Å�Ø�Ó� �Ò� ÁØ× �ÔÔÐ� �Ø�ÓÒ ØÓ �Ò�ÐÝ×�× 

Ó� Ê�Ö�¬�� × �ÐÓÛ× ÈÖÓ �Ø� ÁÒØ�ÖÒ�Ø�ÓÒ�Ð ËÝÑÔÓ×�ÙÑ ÓÒ Ê�Ö�¬�� × �ÝÒ�Ñ� × 

ÍÒ�Ú�Ö×�ØÝ Ó�ÌÓ�ÝÓ ÈÖ�×× �� 

Speaker: MORINISHI, K. 121 BAIL 2006 

1.6 

1.6 

2.0 

1.2 

1.2 

1.6 

1.6 

1.2 

1.2 

✩ 

✪

A. NASTASE: Qualitative Analysis of the Navier-Stokes Solutions in Vicinity of their 

Critical Lines 

✬ 

✫ 

QUALITATIVE ANALYSIS OF NAVIER-STOKES SOLUTIONS 

IN VICINITY OF THEIR CRITICAL LINES 

by Adriana Nastase 

Aerodynamik des Fluges, RWTH-Aachen, Germany 

Fax: 0049/241/809-2173, E-Mail: nastase@lafaero.rwth-aachen.de 

Own developed, zonal, spectral solutions for the compressible stationary 

Navier-Stokes layer (NSL) over flattened flying configurations (FCs), are here 

proposed. A new spectral coordinate η was introduced in the NSL : 

η = x3 − Z(x1, x2) 

(0 ≤ η ≤ 1) (1) 

δ(x1, x2) 

and the dimensionless axial, lateral and vertical velocities uδ, vδ and wδ, the 

density function R = lnρ and the absolute temperature T on the upper NSL 

(which is here only considered) are expressed in the following spectral forms, 

as in [1], [2], namely : 

N� 

ui η i 

N� 

vi η i 

N� 

wi η i 

, 

uδ = ue 

i=1 

, vδ = ve 

i=1 

N� 

R = Rw + (Re − Rw) 

i=1 

ri η i 

, wδ = we 

i=1 

N� 

, T = Tw + (Te − Tw) 

i=1 

ti η i 

. (2a-e) 

Hereby ue, ve, we, Re and Te are the edge values, which can be obtained 

from the outer potential flow, Rw and Tw are the given values of R and T on 

the wall of the FC and ui, vi, wi, ri and ti are the unknown spectral coefficients 

of the velocity’s components, which are used to fulfill the partial differential 

equations (PDEs) of the NSL. The first and the second derivatives of the velocity’s 

components uδ, vδ, wδ are linear functions versus the spectral coefficients 

ui, vi, wi of the velocity’s components. 

The impulse equations of the NSL, which are PDEs of second order, are 

now considered. If the spectral forms of the velocity’s components uδ, vδ and 

wδ are used, the impulse equations are reduced to three equivalent quadratical 

algebraic equations (QAEs) with slightly variable coefficients, versus the spectral 

coefficients ui, vi and wi, as in [1], [2]. In these equations the free terms 

are proportional to the gradients of the pressure p in the direction of the axis 

of coordinates O xi and, therefore, these terms have greater variations than 

the other coefficients of these equations. The influence of the variation of free 

terms and of one of coefficients of the linear term of the QAE over the existence 

of real values of the spectral coefficients and the performing of the qualitative 

analysis of the asymptotical behaviours of the three-dimensional PDEs of the 

compressible NSLs in the vicinity of their singular points and lines, are treated 

here by using the qualitative analysis of the equivalent QAEs. 

The visualizations of asymptotical behaviours of these equivalent QAEs are 

made in a here introduced ”Euclidian M-orthogonal space of the NSL’s free 

spectral coefficients”, which are here treated as variables. 

Further the assumption is made that all QAEs of the same type (elliptical 

Speaker: NASTASE, A. 122 BAIL 2006 

✩ 

✪

A. NASTASE: Qualitative Analysis of the Navier-Stokes Solutions in Vicinity of their 

Critical Lines 

✬ 

✫ 

or hyperbolical), of the same size (same space dimension M), with the same 

number of positive eigenvalues, for which the free terms a are varied from 

−∞ to +∞ and all the other coefficients are maintained constant, have, in the 

vicinity of their singular points, qualitatively, similar asymptotical behaviours. 

The visualizations of the asymptotical behaviours of elliptical and hyperbolical 

QAEs with variable coefficients, are made versus their principal coordinates. 

Their critical points and lines are obtained by cancelling of their great 

determinants ∆i. 

The visualization of elliptical QAEs with variable free coefficients is represented, 

in two-dimensional cuts, in form of coaxial ellipses, which collapse, if 

the free term a is equal to the critical value ac , which is located in the common 

center C of the coaxial ellipses. If a < ac , the visualization of the elliptical 

QAEs in two-dimensional cuts are coaxial ellipses, which all approach the critical 

point a = ac , when the free term increases. If a = ac , the elliptical QAEs 

collapse in this critical point (black point). If a > ac , the elliptical QAEs have 

no more real solutions and the spectral velocity’s components are partially or 

totally imaginary for each value of M. 

The visualization of the hyperbolical QAEs in the vicinity of their critical 

point (b = bc) is totally different. If their free coefficients b < bc , the 

two-dimensional cuts are coaxial hyperbolas with two branches (for M = 2), 

which approach their common asymptotes, if b increases. Up M ≥ 3 break 

and rebreak of hyperhyperboloids occur. The two-dimensional cuts in hyperboloids 

(M = 3) or hyperhyperboloids (M > 3) are coaxial hyperbolas 

with two sheets or coaxial ellipses, because the hyperboloids and hyperhyperboloids 

can be with one or two sheets. If b = bc , the two-dimensional cuts in 

the hyperhyperboloids degenerate in their asymptotical lines. If b > bc , the 

two-dimensional cuts are coaxial hyperbolas with two sheets, which are jumping 

in the opposite double angles of their asymptotical lines or coaxial ellipses, 

which are approaching their critical point, located in their common center. 

The asymptotical behaviours of the elliptical and hyperbolical QAEs with 

variable values of the coefficients of their free terms and of one of their linear 

terms in the vicinity of their critical parabolas are also very different. 

The elliptical QAEs (for M = 2) are visualized in form of coaxial ellipses, 

which collapse in each point of their critical parabola and, inside this parabola, 

there are no more real solutions. A black parabolic hole occurs. 

The hyperbolic QAEs (for M = 2) degenerate in their asymptotes, in each 

point of their critical parabola. By crossing of this parabola, the hyperbolas 

approach their asymptotes in one of their double angles and, after jumping, 

they are going away from these asymptotes. 

The collapse points and lines of the elliptical QAE are useful for the determination 

of the position of detachment lines and for the beginning of transition. 

The saddle points and lines of the hyperbolical QAE are useful for the determination 

of the position of characteristic and shock surfaces and for the 

bifurcation. 

REFERENCES 

1. NASTASE, A., New Zonal, Spectral Solutions for the Navier-Stokes Partial Differential 

Equations, Proc. of the International Conference on Boundary and 

Interior Layers, ONERA Toulouse, 2004, France. 

2. NASTASE, A., Zonal, Spectral Solutions for Navier-Stokes Layer and Applications, 

4th European Congress on Computational Methods in Applied Sciences 

and Engineering (ECCOMAS), Jyvskyl, 2004, Finland. 

Speaker: NASTASE, A. 123 BAIL 2006 

✩ 

✪

F. NATAF, G. RAPIN: Application of the Smith Factorization to Domain Decomposition 

Methods for the Stokes equations 

✬ 

✫ 

Application of the Smith Factorization to Domain 

Decomposition Methods for the Stokes Equations 

FRÉDÉRIC NATAF, GERD RAPIN 

Laboratoire J. L. Lions, Université Pierre et Marie Curie, 75252 Paris Cedex 05, 

France; nataf@ann.jussieu.fr 

Math. Dep., NAM, University of Göttingen, D-37083, Germany; 

grapin@math.uni-goettingen.de 

In this talk the Smith factorization is used systematically to derive a new domain decomposition 

method for the Stokes problem. In two dimensions the key idea is the 

transformation of the Stokes problem into a scalar bi-harmonic problem. We show, 

how a proposed domain decomposition method for the bi-harmonic problem leads to 

a domain decomposition method for the Stokes equations which inherits the convergence 

behavior of the scalar problem. Thus, it is sufficient to study the convergence of 

the scalar algorithm. The same procedure can also be applied to the three dimensional 

Stokes problem. 

As transmission conditions for the resulting domain decomposition method of the 

Stokes problem we obtain natural boundary conditions. Therefore it can be implemented 

easily. 

A Fourier analysis and some numerical experiments show very fast convergence of 

the proposed algorithm. Our algorithm shows a more robust behavior than Neumann- 

Neumann or FETI type methods. 

The last decade has shown, that Neumann-Neumann type algorithms, FETI, and 

BDDC methods are very efficient domain decomposition methods. Most of the early 

theoretical and numerical work has been carried out for scalar symmetric positive definite 

second order problems, see for example [8, 5]. Then, the method was extended 

to different other problems, like the advection-diffusion equations [1], plate and shell 

problems [12] or the Stokes equations [10, 11]. 

In the literature one can also find other preconditioners for the Schur complement 

of the Stokes equations (cf. [11, 2]). A more complete list of domain decomposition 

methods for the Stokes equations can be found in [10, 13]. Also FETI [6] and BDDC 

methods [7] have been applied to the Stokes problem with success. 

Our work is motivated by the fact that in some sense the domain decomposition 

methods for Stokes are less optimal than the domain decompoition methods for scalar 

problems. Indeed, in the case of two subdomains consisting of the two half planes it 

is well known, that the Neumann-Neumann preconditioner is an exact preconditioner 

for the Schur complement equation for scalar equations like the Laplace problem (cf. 

[8]). A preconditioner is called exact, if the preconditioned operator simplifies to the 

identity. Unfortunately, this does not hold in the vector case. It is shown in [9] that the 

standard Neumann-Neumann preconditioner for the Stokes equations does not possess 

this property. 

Our aim in this talk is the construction of a method, which preserves this property. 

Thus, one can expect a very fast convergence for such an algorithm. And indeed, the 

numerical results clearly support our approach. In this talk the ideas of [4] are extended 

Speaker: RAPIN, G. 124 BAIL 2006 

1 

✩ 

✪

F. NATAF, G. RAPIN: Application of the Smith Factorization to Domain Decomposition 

Methods for the Stokes equations 

✬ 

✫ 

in several directions. We also give some hints how this approach can be applied to the 

Oseen equations. For an application to the compressible Euler equations see [3]. 

References 

[1] Y. Achdou, P. Le Tallec, F. Nataf, and M. Vidrascu. A domain decomposition 

preconditioner for an advection-diffusion problem. Comput. Methods Appl. Mech. 

Engrg, 184:145–170, 2000. 

[2] M. Ainsworth and S. Sherwin. Domain decomposition preconditioners for p and 

hp finite element approximations of Stokes equations. Comput. Methods Appl. 

Mech. Engrg., 175:243–266, 1999. 

[3] V. Dolean and F. Nataf. A New Domain Decomposition Method for the Compressible 

Euler Equations. M 2 AN, 2005. accepted. 

[4] V. Dolean, F. Nataf, and G. Rapin. New constructions of domain decomposition 

methods for systems of PDEs. C.R. Acad. Sci. Paris, Ser I, 340:693–696, 2005. 

[5] Ch. Farhat and F.-X. Roux. A Method of Finite Element Tearing and Interconnecting 

and its Parallel Solution Algorithm. Internat. J. Numer. Methods Engrg., 

32:1205–1227, 1991. 

[6] J. Li. A Dual-Primal FETI method for incompressible Stokes equations. Numer. 

Math., 102:257–275, 2005. 

[7] J. Li and O. Widlund. BDDC algorithms for incompressible Stokes equations, 

2006. submitted. 

[8] J. Mandel. Balancing domain decomposition. Comm. on Applied Numerical 

Methods, 9:233–241, 1992. 

[9] F. Nataf and G. Rapin. Construction of a New Domain Decomposition Method 

for the Stokes Equations, 2005. Submitted to the Proceedings of DD16. 

[10] L.F. Pavarino and O.B. Widlund. Balancing neumann-neumann methods for incompressible 

stokes equations. Comm. Pure Appl. Math., 55:302–335, 2002. 

[11] P . Le Tallec and A. Patra. Non-overlapping domain decomposition methods for 

adaptive hp approximations of the Stokes problem with discontinuous pressure 

fields. Comput. Methods Appl. Mech. Engrg., 145:361–379, 1997. 

[12] P. Le Tallec, J. Mandel, and M. Vidrascu. A Neumann-Neumann Domain Decomposition 

Algorithm for Solving Plate and Shell Problems. SIAM J. Numer. 

Anal., 35:836–867, 1998. 

[13] A. Toselli and O. Widlund. Domain Decomposition Methods - Algorithms and 

Theory. Springer, Berlin-Heidelberg, 2005. 

Speaker: RAPIN, G. 125 BAIL 2006 

2 

✩ 

✪

N. NEUSS: Numerical approximation of boundary layers for rough boundaries 

✬ 

✫ 

Numerical approximation of boundary layers 

for rough boundaries 

N. Neuss 

April 5, 2006 

In physical problems, interesting phenomena often occur at boundaries or 

interfaces between different media. Often these phenomena are complicated 

due to the nature of the process or due to the intricate geometry of the 

interface. Therefore, they are usually described by effective boundary or 

interface laws. 

In this talk we will discuss some instructive cases in a quasi-periodic 

setting, where the constants in those effective conditions can be calculated 

from the microscopic setting. 

First, we consider the Poisson problem 

−∆u ε = f, x ∈ Ω ε 

u ε = 0, x ∈ ∂Ω ε 

where Ω ε is a domain with a boundary ∂Ω ε featuring a micro-structure of size 

ε. A good approximation to u ε is given by the solution u eff to the problem 

−∆u eff = f, x ∈ Ω 

u eff = c(x) ∂ueff 

∂n 

, x ∈ ∂Ω 

where Ω is an approximating domain with smooth boundary ∂Ω and the 

function c : ∂Ω → R can be computed solving auxiliary problems taking into 

account the fine-scale structure of ∂Ω ε , see [?]. 

Similarly, solving the Navier-Stokes equation in such a domain Ω ε can be 

replaced by solving it on Ω with a Navier boundary condition. Related to this 

is the Beavers-Joseph problem, where one can derive an effective interface 

law for a rough interface between free flow and porous medium flow, see [?]. 

Speaker: NEUSS, N. 126 BAIL 2006 

1 

✩ 

✪

N. NEUSS: Numerical approximation of boundary layers for rough boundaries 

✬ 

✫ 

Finally, the same idea can be used to obtain effective boundary conditions 

for numerical approximations Ωh of arbitrary domains Ω. This allows us to 

obtain good O(h 2 ) approximation errors although the domain Ωh only needs 

to approximate Ω up to order O(h). 

References 

[1] W. Jäger, A. Mikelić, N. Neuss: Asymptotic Analysis of the Laminar 

Viscous Flow Over a Porous Bed. SIAM J. Sci. Comp. 22,6, pp. 2006- 

2028 (2001). 

[2] N. Neuss, M. Neuss-Radu, A. Mikelić: Effective Laws for the Poisson 

Equation on Domains with Curved Oscillating Boundaries. To appear in 

Applicable Analysis. 

Speaker: NEUSS, N. 127 BAIL 2006 

2 

✩ 

✪

M.A. OLSHANSKII: An Augmented Lagrangian based solver for the low-viscosity 

incompressible flows 

✬ 

✫ 

An Augmented Lagrangian based solver for the low-viscosity 


Maxim A. Olshanskii ∗ 

We describe an effective solver for the finite element discretization of the Oseen problem 

−ν∆u + (w · ∇) u + ∇p = f in Ω (1) 

div u = 0 in Ω (2) 

u = g on ∂Ω (3) 

with a known, divergence-free vector function w. Discretization of (1)–(3) using LBB-stable 

finite elements is based on the following stabilized finite element formulation 

L(uh, ph; vh, qh) + � 

στ (−ν∆uh + w·∇uh + ∇ph − f, w·∇vh)τ 

with 

τ∈Th 

= (f, vh) ∀ vh ∈ Vh, qh ∈ Qh (4) 

L(u, p; v, q) = ν(∇u, ∇v) + ((w · ∇) u, v) − (p, div v) + (q, div u) 

and a suitable choice of the stabilization parameters στ . With certain assumptions finding FE 

solution results in solving the linear system of the form 

� � � � � � 

A BT u f 

= . (5) 

B O p 0 

Linear systems of the form (5) are often referred to as generalized saddle point systems. In 

recent years, a great deal of effort has been invested in solving systems of this form. Most of 

the work has been aimed at developing effective preconditioning techniques; see [1, 3] for an 

extensive survey. In spite of these efforts, there is still considerable interest in preconditioning 

techniques that are truly robust, i.e., techniques which result in convergence rates that are 

largely independent of problem parameters such as mesh size and viscosity. In this paper we 

describe a promising approach based on an augmented Lagrangian formulation [4]: 

� A + γB T W −1 B B T 

B O 

� � u 

p 

with a positive-definite matrix W and parameter γ > 0. The system (6) has precisely the same 

solution as the original one (5). Rather than treating (5), a block-triangular preconditioner is 

constructed for solving (6) with a Krylov subspace iterative method. 

The success of this method crucially depends on the availability of a robust multigrid solver 

for the (1,1) block (submatrix) in (6); we develop such a method by building on previous work 

by Schöberl [5], together with appropriate smoothers for convection-dominated flows. This 

multigrid iteration will be used to define a block preconditioner for the outer iteration on the 

∗ Department of Mechanics and Mathematics, Moscow State M. V. Lomonosov University, Moscow 119899, 

Russia (Maxim.Olshanskii@mtu-net.ru). The work was supported in part by the Russian Foundation for Basic 

Research and the Netherlands Organization for Scientific Research grants NWO-RFBR 047.016.008 

Speaker: OLSHANSKII, M.A. 128 BAIL 2006 

1 

� 

= 

� f 

0 

� 

(6) 

✩ 

✪

M.A. OLSHANSKII: An Augmented Lagrangian based solver for the low-viscosity 


✬ 

✫ 

coupled saddle point system. We will show that this approach is especially appropriate for 

discretizations based on discontinuous pressure approximations, but can be used to construct 

preconditioners for other discretizations using continuous pressures. As an example of finite 

element (FE) method with discontinuous pressures we will use the isoP2-P0 pair. In this case, 

our numerical experiments demonstrate a robust behavior of the solver with respect to h and 

ν for some typical wind vector functions w in (1). Further, the isoP2-P1 finite element pair 

is used for the continuous pressure-based approximation. For this case numerical experiments 

show an h-independent convergence rates with mild dependence on ν, when the viscosity becomes 

very small. We note that this approach does not require a sophisticated preconditioner for the 

pressure Schur complement of (5) or (6). 

Some spectral estimates for the preconditioned problem are shown; basic components for 

building appropriate multigrid method are discussed. The role of parameter γ in (6) is addressed. 

We also present a comparison with one of the best available preconditioning techniques and 

coupled multigrid methods (Vanka multigrid), showing that our method is quite competitive in 

terms of convergence rates, robustness, and efficiency. Finally we discuss that using SUPG type 

stabilization in finite element formulation (4) is vital not only for capturing effects of unresolved 

subscales in solution to (1)–(3), thus improving accuracy of the discrete solution, but also for 

developing robust and effective iterative methods. 

This presentation is based on the collaborative research with M.Benzi [2]. 

References 

[1] M. Benzi, G. H. Golub and J. Liesen, Numerical solution of saddle point problems, Acta 

Numerica, 14 (2005), pp. 1–137. 

[2] M. Benzi, M.A. Olshanskii, An augmented lagrangian-based approach to the Oseen problem, 

(submitted), available at www.mathcs.emory.edu/~molshan 

[3] H. C. Elman, D. J. Silvester and A. J. Wathen, Finite Elements and Fast Iterative Solvers: 

with Applications in Incompressible Fluid Dynamics, Numerical Mathematics and Scientific 

Computation, Oxford University Press, Oxford, UK, 2005. 

[4] M. Fortin and R. Glowinski, Augmented Lagrangian Methods: Applications to the Numerical 

Solution of Boundary-Value Problems, Studies in Mathematics and its Applications, 

Vol. 15, North-Holland, Amsterdam/New York/Oxford, 1983. 

[5] J. Schöberl, Multigrid methods for a parameter dependent problem in primal variables, 

Numer. Math., 84 (1999), pp. 97–119. 

2 

Speaker: OLSHANSKII, M.A. 129 BAIL 2006 

✩ 

✪

N. PARUMASUR, J. BANASIAK, J.M. KOZAKIEWICZ: Numerical and Asymptotic 

Analysis of Singularly Perturbed PDEs of Kinetic Theory 

✬ 

✫ 

Numerical and Asymptotic Analysis of Singularly Perturbed PDEs of 

Kinetic Theory 

N. Parumasur, J. Banasiak and J.M. Kozakiewicz 

University of KwaZulu-Natal, Durban, 4041, South Africa parumasurn1@ukzn.ac.za 

We consider the numerical solution of various equations occurring in kinetic theory. The 

numerical algorithm is applied in conjuction with a modified asymptotic procedure. Various 

mathematical derivations and numerical algorithms are provided for the following singularly 

perturbed models of kinetic theory 

∂tu + Au + Su + 1 

Cu = 0 (1) 

ε 

where u is the particle distribution, ∂t is the time derivative and the operators A , S , and C 

describe attenuation, streaming and collisions of particles, respectively. We begin by outlining 

the features of a modified asymptotic method [2, 1] which is quite useful when solving (1). 

Let P be a bounded operator in the Banach space X having zero as its simple isolated 

eigenvalue and the corresponding eigenspace V . Then X can be expressed as a direct sum 

X = V ⊕ W, where both V and W are invariant subspaces of the operator C and C is one-to-one 

from W onto itself. Let P be the spectral projection associated with the eigenvalue λ = 0 so 

that 

V = PX, W = QX, 

where Q = I − P is the complementary projection. We use a projection method [2] to write (1) 

as a system of evolution equations in subspaces V and W . Applying the projections P and Q 

on both sides of (1), successively, we obtain 

with the initial conditions 

∂tv = P(A + S)Pv + P(A + S)Qw 

ε∂tw = εQ(A + S)Qw + εQ(S + A)Pv + QCQw, (2) 

v(0) = o v, w(0) = o w, 

where o v = P o u, o w = Q o u. Taking into account that the projected operators PSP, PAQ and 

QAP vanish for most types of linear equations we obtain the following form of (2) 

∂tv = PAPv + PSQw 

ε∂tw = εQSPv + εQSQw + εQAQw + QCQw (3) 

v(0) = o v, w(0) = o w, 

Next we apply the modified asymptotic approach to (3). We represent the solution of (3) as a 

sum of the bulk and the initial layer parts: 

v(t) = ¯v(t) + ˜v(τ), w(t) = ¯w(t) + ˜w(τ), (4) 

where the variable τ in the initial layer part is given by τ = t/ε. The bulk solution will be 

considered as a function of ρ of order zero and the function ¯w (N) will be assumed to be of the 

form 

¯w (N) N� 

(t) = ε n Wnρ(t), (5) 

Speaker: PARUMASUR, N. 130 BAIL 2006 

n=0 

1 

✩ 

✪

N. PARUMASUR, J. BANASIAK, J.M. KOZAKIEWICZ: Numerical and Asymptotic 

Analysis of Singularly Perturbed PDEs of Kinetic Theory 

✬ 

✫ 

where the superscript N indicates the order of the approximation and W are time-independent 

bounded linear operators from V to W . Substituting this expansion into the first equation in 

(3) yields 

N� 

∂tρ = PAPρ + ε n PSQ(Wnρ). (6) 

n=0 

Expressing the time derivative ∂tρ in (6) in powers of ε and comparing terms of the same power 

in ε yields at first order 

W0 = 0, W1 = −(QCQ) −1 QSP. (7) 

The operator W1 can be evaluated since QCQ is invertible on the subspace W. Using (7) in (6) 

gives the equation 

∂tρ = PAPρ − εPSQ(QCQ) −1 QSPρ. (8) 

A similar procedure yields the initial layer terms 

and the initial condition for (8) 

˜v0(τ) ≡ 0, ˜v1(τ) = PSQ(QCQ) −1 e τQCQ o w, 

¯v(0) = o v −εPSQ(QCQ) −1 o w . (9) 

We apply the procedure to a wide range of problems of kinetic theory. 

References 

[1] J. Banasiak, J. Kozakiewicz, N. Parumasur, Diffusion Approximation of Linear Kinetic 

Equations with Non-equilibrium Data – Computational Experiments, Transport Theory 

Statist. Phys. (accepted). 

[2] J. R. Mika, New asymptotic expansion algorithm for singularly perturbed evolution equations, 

Math. Methods Appl. Sci. 3 (1981) 172-188. 

2 

Speaker: PARUMASUR, N. 131 BAIL 2006 

✩ 

✪

B. RASUO: On Boundary Layer Control in Two-Dimensional Transonic Wind Tunnels 

✬ 

✫ 

Speaker: RASUO, B. 132 BAIL 2006 

✩ 

✪

B. RASUO: On Boundary Layer Control in Two-Dimensional Transonic Wind Tunnels 

✬ 

✫ 

Speaker: RASUO, B. 133 BAIL 2006 

✩ 

✪

H.-G. ROOS: A Comparison of Stabilization Methods for Convection-Diffusion- 

Reaction Problems on Layer-Adapted Meshes 

✬ 

✫ 

A Comparison of Stabilization Methods for Convection-Diffusion-Reaction 

Problems on Layer-Adapted Meshes 

Hans-G. Roos, TU Dresden 

The use of layer adapted meshes allows to prove robust convergence results of Galerkin finite element 

methods for convection-diffusion-reaction problems. However, the robust solution of the generated 

discrete problems which are in general nonsymmetric is a nontrivial task. 

The situation improves if one uses some stabilization. But the question is: which of all the existing 

stabilization techniques is optimal? 

Speaker: ROOS, H.-G. 134 BAIL 2006 

✩ 

✪

H.-G. ROOS, H. ZARIN: Discontinuous Galerkin stabilization for convectiondiffusion 

problems 

✬ 

✫ 

Discontinuous Galerkin stabilization for convection–diffusion 

problems 

Hans–Görg Roos 

Institut für Numerische Mathematik, Technische Universität Dresden, 

Germany 

Helena Zarin 

Department of Mathematics and Informatics, University of Novi Sad, 

Serbia and Montenegro 

Abstract. A convection–diffusion problem with Dirichlet boundary conditions 

posed on a unit square is considered. The problem is discretized using a combination 

of standard Galerkin FEM and h–version of the nonsymmetric discontinuous 

Galerkin FEM with interior penalties on a layer–adapted mesh. With 

specially chosen penalty parameters for edges from the coarse part of the mesh, 

we prove uniform convergence (in the perturbation parameter) in an associated 

norm. Numerical tests support our theoretical results. 

References 

[1] Roos, H.–G., Zarin, H., The discontinuous Galerkin method for singularly 

perturbed problems. Numerical Mathematics and Advanced Applications 

(eds. M. Feistauer et al.), Proceedings of the ENUMATH Conference 2003 

(Prague, August 18-22, 2003), Springer Verlag, 2004, pp. 736–745 

[2] Zarin, H., Roos, H.–G., Interior penalty discontinuous approximations of 

convection–diffusion problems with parabolic layers. Numerische Mathematik, 

100(4), 2005, pp. 735–759 

1 

Speaker: ZARIN, H. 135 BAIL 2006 

✩ 

✪

B. SCHEICHL, A. KLUWICK: On Turbulent Marginal Separation: How the Logarithmic 

Law of the Wall is Superseded by the Half-Power Law 

✬ 

✫ 

Abstract submitted to the Local Organization Committee of the BAIL 2006 Conference 1 

On Turbulent Marginal Separation: How the Logarithmic Law of the Wall is 

Superseded by the Half-Power Law ∗ 

B. Scheichl and A. Kluwick 

Institute of Fluid Mechanics and Heat Transfer 

Vienna University of Technology 

Resselgasse 3/E322, A-1040 Vienna, Austria 

bernhard.scheichl@tuwien.ac.at 

1. The Asymptotic Theory of Marginally Separating Turbulent Boundary Layers 

A novel rational theory of the incompressible nominally steady and two-dimensional turbulent 

boundary layer (TBL) along a smooth and impermeable surface and exposed to an adverse 

pressure gradient, which is impressed by the prescribed external potential free-stream flow, has 

been developed recently by the authors. This asymptotic flow description exploits the Reynoldsaveraged 

Navier–Stokes equations by taking the limit Re→∞ where Re denotes a Reynolds 

number formed by using a global length and velocity scale characteristic for the external bulk 

flow. In the following all quantities are non-dimensional with that global reference scales. 

The so-called classical theory, see for instance [1], is capable of describing a strictly attached 

TBL only as it employs the assumption of an asymptotically small streamwise velocity defect 

with respect to the external flow in the fully turbulent main part of the TBL. The new theory, 

however, is an extension of the classical approach insofar as it is essentially based only on the 

hypothesis that the turbulent time-mean motion is governed locally by a single velocity scale. As 

an important consequence, by taking a streamwise velocity deficit of O(1), which is a necessary 

characteristic of flows that may even undergo marginal separation, the boundary layer thickness 

ismeasuredbyasmallparameterdenotedby αwhichisseentobeindependentof Re as Re→∞. 

It then can be shown that in the primary limit α→0, Re −1 = 0 the TBL is represented by a 

two-tiered wake-type flow; remarkably, it thus closely resembles a turbulent free shear layer. The 

description of the outer main layer is addressed in [2]. There it is demonstrated analytically and 

numerically by adopting a local viscous/inviscid interaction strategy that in the primary limit 

considered marginal separation is associated with the occurrence of closed reverse-flow regions 

where the surface slip velocity Us, which is a quantity of O(1) in general, assumes negative 

values along a streamwise distance of O(α 3/5 ). Here we add that the overall slip velocity at the 

base of the whole wake region comprising both layers is written as us = Us + O(α 3/4 ), where 

the perturbations reflect the effect of the inner wake having a thickness of O(α 3/2 ). 

2. The Near-Wall Flow Regime for Finite Values of Re 

It is the main purpose of our contribution to rigorously elucidate how high but finite values of 

Re affect the near-wall flow regime in order to provide a rational basis for the calculation of the 

wall shear stress, in particular immediately up- and downstream of separation and reattachment. 

We note that the investigation of the latter flow situation is not only a challenge to settle an, 

for the time being, unsolved problem in turbulent boundary layer theory, which has attracted 

many researchers in the past, but also of basic engineering relevance, as none of the presently 

adopted turbulence models are applicable when the wall shear stress changes sign. 

In Figure 1 the resulting four-layer TBL structure is depicted where the case of the oncoming 

firmly attached flow chosen. We start the analysis by focusing on the viscous wall layer adjacent 

∗ This research is granted by the Austrian Science Fund (FWF) under project no. P 16555-N12. 

Speaker: SCHEICHL, B. 136 BAIL 2006 

✩ 

✪

B. SCHEICHL, A. KLUWICK: On Turbulent Marginal Separation: How the Logarithmic 

Law of the Wall is Superseded by the Half-Power Law 

✬ 

PSfrag replacements 

✫ 

Abstract submitted to the Local Organization Committee of the BAIL 2006 Conference 2 

y 

OW 

y∼ δ = O(α) 

y 

u∼us 

IW α u∼Us 

IL VWL 

3/2 u = O(√y) u/uτ ∼ κ−1 lny 

u 

Figure 1: Asymptotic splitting and streamwise velocity 

component u of the initially attached turbulent boundary 

layer with thickness δ, which evolves along the surface 

given by y = 0. The notations OW, IW, IL, and 

VWL mark the outer and the inner wake, the intermediate 

and the viscous wall layer, respectively. The 

asymptotic relationships are explained in the text. 

to the surface where convective terms are negligibly small and Reynolds stresses have the same 

magnitude as the molecular shear stress. Let y and τw denote, respectively, the coordinate 

perpendicular to the surface and the wall shear stress. As far as low-order results and the 

attached case τw > 0 are concerned, the asymptotic description of the wall layer turns out to be 

fully analogous to that in the classical theory. That is, sufficiently far from the positions where 

τw vanishes the streamwise velocity component u is expanded in the usual manner according to 

u/uτ ∼ u + 0 (y+ )+··· , sgn(τw)uτ =|τw| 1/2 = (|∂u/∂y|y=0/Re) 1/2 , y + = y|uτ|Re = O(1). (1) 

Most important, the hypothesis stated above is seen to require the celebrated logarithmic 

match with the fully turbulent flow regime on top of the viscous wall layer, 

u + 0 ∼ A± −1 lny + + B±, A± > 0, y + →∞. (2) 

Here the subscripts + and−denote the cases τw > 0 and τw < 0, respectively. Therefore, A+ 

equals the v. Kármán constant, and the values of B± shall refer to a perfectly smooth surface. 

Note that for separated flows (τw < 0) an asymptotic behavior akin to (2) was already proposed 

in [3] on semi-empirical grounds. Moreover, the velocity components u in the wall layer and the 

intermediate layer of thickness τw, see Figure 1, match provided that the skin friction law 

uτ/us∼ A±ɛ+O(ɛ 2 ), ɛ = 1/ln|u 3 τRe|→0, Re→∞, (3) 

holds. This relationship between uτ and the aforementioned slip velocity us represents a generalization 

of the well-known classical result which, in principle, is included in (3). 

Let η = y/u 2 τ = O(1) be the coordinate characteristic for the intermediate layer. The match 

of u with the inner wake reveals a half-power law, u/uτ = O( √ η) for η→∞. Such a behavior 

is believed to hold also on top of the viscous wall layer of a separating TBL. On the other 

hand, the viscous wall and the intermediate layer collapse and, in turn, (3) ceases to be valid 

when us/uτ = O(1), that is, for us = O(Re −1/3 ). As a highlight of the asymptotic analysis, it is 

pointed out how that merge of the near-wall layers generates a new strongly viscosity-affected 

flow regime near the locations where τw = 0. There (1) is replaced by the appropriate expansion 

u/up∼ u × 0 (p× ,y × )+··· , p × = (up/uτ) 3 , up = (px/Re) 1/3 , y × = yupRe = O(1). (4) 

Herein the local value px = O(1) of the leading-order streamwise pressure gradient enters the 

velocity scale up. Finally, then the generalized logarithmic law (2) is gradually transformed into 

References 

u × 0 ∼ C(p× ) � y × + D(p × ), C(p × ) > 0, D(p × )∼us/up = O(1), y × →∞. (5) 

[1] G.L. Mellor, “The Large Reynolds Number, Asymptotic Theory of Turbulent Boundary Layers” J. Engn 

Sci., 10, 851–873 (1972). 

[2] B. Scheichl and A. Kluwick, “Turbulent Marginal Separation and the Turbulent Goldstein Problem”, AIAA 

paper 2005-4936 (2005), also AIAA J. (submitted in extended form). 

[3] R.L.Simpson, “AModelfortheBackflowMeanVelocityProfile”, TechnicalNote, AIAA J., 21(1), 142–143 

(1983). 

Speaker: SCHEICHL, B. 137 BAIL 2006 

✩ 

✪

O. SHISHKINA, C. WAGNER: Boundary and Interior Layers in Turbulent Thermal 

Convection 

✬ 

✫ 

Boundary and Interior Layers in Turbulent Thermal Convection ∗ 


O. Shishkina & C. Wagner 

DLR - Institute for Aerodynamics and Flow Technology, 

Bunsenstrasse 10, 37073 Göttingen, Germany 

Olga.Shishkina@dlr.de, Claus.Wagner@dlr.de 

Numerous scientifical problems and industrial applications require solutions of the Rayleigh- 

Bénard problem, i.e. turbulent convection of fluids heated from below and cooled from above, 

with the Rayleigh number (Ra) from 10 5 up to 10 20 . For a review on this classical problem and 

for the references to earlier literature we refer to the paper Ahlers, Grossmann and Lohse [1]. 

Since the diffusion coefficient in the Navier-Stokes equation, which is inversely proportional to 

the square root of Ra, is very small, the solution - both the temperature and the velocity fields - 

have very thin boundary layers near the horizontal walls. For moderate Rayleigh numbers (from 

10 5 up to 10 8 ) interior layers, i.e. thermal plumes, are also observed. The boundary layers, the 

thermal plumes and the turbulent background are indicated, respectively, by high, moderate and 

small values of the temperature gradient norm and, hence, by large, moderate and small values 

of the thermal dissipation rate. By means of direct numerical simulations (DNS) we invesigate 

boundary and interior layers which take place in turbulent Rayleigh-Bénard convection. 

2. Governing equations and the numerical method 

The governing dimensionless equations for the Rayleigh-Bénard problem in Boussinesq approximation 

can be written in cylindrical coordinates (z, r, ϕ) as follows 

ut + u · ∇u + ∇p = Γ −3/2 Ra −1/2 P r 1/2 ∇ 2 u + T z, (1) 

Tt + u · ∇T = Γ −3/2 Ra −1/2 P r −1/2 ∇ 2 T, (2) 

∇ · u = 0, (3) 

where u is the velocity vector, T the temperature, ut and Tt their time derivatives, p the 

pressure. Here Ra = αgH 3 ∆T/(κν) denotes the Rayleigh number, P r = ν/κ the Prandtl 

number, Γ = D/H the aspect ratio with H the height and D the diameter of the cylindrical 

container. Further, α is the thermal expansion coefficient, g the gravitational acceleration, ∆T 

the temperature difference between the bottom and the top plates, ν the kinematic viscosity and 

κ the thermal diffusivity. The dimensionless temperature varies between +0.5 at the bottom 

plate to −0.5 at the top plate. An adiabatic lateral wall is prescribed by ∂T/∂r = 0. Finally, on 

the solid walls the velocity field vanishes according to the impermeability and no-slip conditions. 

To investigate turbulent Rayleigh-Bénard convection in cylindrical containers of the aspect 

ratios Γ = 10 and Γ = 5 for Ra from 10 5 to 10 7 and P r = 0.7 we conducted DNS. The simulations 

were performed with the fourth order accurate finite volume method developed for solving (1) 

– (3) in cylindrical coordinates on staggered non-equidistant grids. For the numerical method 

used in the DNS we refer to [2]. 

∗ The work is supported by Deutsche Forschungsgemeinschaft (DFG) under the contract WA 1510-1 

Speaker: SHISHKINA, O. 138 BAIL 2006 

1 

✩ 

✪

O. SHISHKINA, C. WAGNER: Boundary and Interior Layers in Turbulent Thermal 

Convection 

✬ 

✫ 

Ra = 10 5 Ra = 10 6 Ra = 10 7 

Figure 1: Snapshots of the temperature, −0.37 ≤ T ≤ 0.37, for Γ = 10 and z = H/(2Nu). 

Colour scale spreads from blue (negative values) through white (zero) to red (positive values). 

3. Mesh generation 

In [3] it was proven that the ratio of the area averaged (over the top or the bottom plates) to 

the volume averaged thermal dissipation rate ɛθ = Γ −3/2 Ra −1/2 P r −1/2 (∇T ) 2 is greater than 

or equal to the Nusselt number (Nu = Γ1/2Ra1/2P r1/2 〈uzT 〉 t,S − Γ−1 � � 

∂T 

∂z ≥ 1) for all Ra, 

t,S 

P r and Γ. It means that the largest values of ɛθ take place in the boundary layers near the 

horizontal walls. To resolve these boundary layers special meshes are required. 

The grid equidistribution ansatz (see for example [4]) enables to detect the boundary layers 

and construct appropriate meshes for their resolution. In our solution-adapted mesh generation 

algorithm we used grid equidistribution of the arc-length of the mean temperature profiles 

computed on equidistant meshes. The algorithm produces meshes that lead to principally more 

accurate solutions in comparison with the equidistant meshes. 

4. Numerical experiments 

Using the solution-adapted meshes with 110, 192 and 512 nodes in z-, r- and ϕ-directions, 

respectively, we conducted the DNS of three-dimensional Rayleigh–Bénard convection in wide 

cylindrical containers of the aspect ratios Γ = 10 and Γ = 5 and for moderate Rayleigh numbers 

from 10 5 to 10 7 . The snapshots of the temperature field on the borders between the lower 

thermal boundary layers and the bulk are presented in Fig. 1 as they were obtained in the DNS 

for Γ = 10. The coherent flow patterns reveal the horizontal extension of hot and cold plumes. 

Analysing spatial distribution of ɛθ for different Ra we conclude that the role of the thermal 

plumes in thermal convection decreases with growing Ra. Some other physical results obtained 

in the DNS of turbulent Rayleigh–Bénard convection in wide containers are discussed in [3]. 

References 

[1] G. Ahlers, S. Grossmann & D. Lohse, “Hochpräzision im Kochtopf: Neues zur turbulenten 

Wärmekonvektion”, Physik Journal, 1, 31–37 (2001). 

[2] O. Shishkina & C. Wagner, “A fourth order accurate finite volume scheme for numerical 

simulations of turbulent Rayleigh-Bénard convection”, C. R. Mecanique, 333, 17–28 (2005). 

[3] O. Shishkina & C. Wagner, “Analysis of thermal dissipation rates in turbulent Rayleigh- 

Bènard convection”, J. Fluid Mech., 546, 51–60 (2006). 

[4] N. Kopteva & M. Stynes, “A robust adaptive method for a quasilinear one-dimensional 

convection-diffusion problem”, SIAM J. Numer. Anal., 39, 1446–1467 (2001). 

Speaker: SHISHKINA, O. 139 BAIL 2006 

2 

✩ 

✪

M. STYNES, L. TOBISKA: Using rectangular Qp elements in the SDFEM for a 

convection-diusion problem with a boundary layer 

✬ 

✫ 

Using rectangular Qp elements in the SDFEM for a convection-diffusion 

problem with a boundary layer 

Martin Stynes 1 and Lutz Tobiska 2 

1 National University of Ireland, Cork, Ireland. 

m.stynes@ucc.ie 

2 Otto-von-Guericke Universität Magdeburg, Postfach 4120, D-39016 Magdeburg, Germany 

tobiska@mathematik.uni-magdeburg.de 

We consider the convection-diffusion problem 

−ε∆u + b · ∇u + cu = f on Ω = (0,1) 2 , u = 0 on ∂Ω, 

where the parameter ε lies in the interval (0,1], the function b(x,y) = (b1(x,y),b2(x,y)) with 

b1(x,y) > β1 > 0 and b2(x,y) > β2 > 0, c(x,y) ≥ 0 on ¯ Ω, and c(x,y) − (div b(x,y))/2 ≥ c0 > 0 

on ¯ Ω. We assume that the functions b, c, and f are sufficiently smooth. Layer-adapted meshes 

are usually used for solving the singularly perturbed boundary value problem since its solution 

typically has boundary layers at the sides x = 1 and y = 1 of Ω. 

The convergence properties of the streamline diffusion finite element method (SDFEM; the 

method is also known as SUPG) on a rectangular Shishkin mesh are analyzed. The trial functions 

in the SDFEM are piecewise polynomials that lie in the space Qp, i.e., are tensor products 

of polynomials of degree p in one variable, where p > 1. In [1, 2], for sufficiently small ε 

(ε ≤ N −1/2 ln 2 N), the error bound 

�u I − u N �SD ≤ C N −2 ln 2 N 

has been proven in the case p = 1, where u I is the nodal interpolant of the solution u, u N is 

the SDFEM solution, and � · �SD is the streamline-diffusion norm. This error bound is based on 

anisotropic interpolation estimates, superconvergence of the piecewise linear nodal interpolation, 

and a detailed study of the behaviour of the solution in the different parts of the domain. 

The main objective is to extent the error analysis to the case p > 1 by using a nonstandard 

interpolation. A detailed study of the approximation and superconvergence properties leads to 

the estimate 

�u I − u N �SD ≤ C N −(p+1/2) , p > 1. 

Comparisons are made between this result and the corresponding result for the case p = 1, which 

turns out to be exceptional [3]. Moreover, we discuss the possibilities to derive similar estimates 

for �u − Pu N �SD and �u − u N �d,SD, where Pu N denote a suitable postprocessed numerical 

solution and � · �d,SD a discrete version of the streamline-diffusion norm, respectively. 

References 

[1] M. Stynes and L. Tobiska, “Analysis of the streamine-diffusion finite element method for 

a convection-diffusion problem with exponential layers”, East-West J. Numer. Math., 9, 

59–76 (2001). 

[2] M. Stynes and L. Tobiska, “The SDFEM for a convection-diffusion problem with a boundary 

layer: Optimal error analysis and enhancement of accurary”, SIAM J. Numer. Anal., 41, 

1620–1642 (2003). 

[3] L. Tobiska, “Analysis of a new stabilized higher order finite element method for advectiondiffusion 

equations”, Tech. Rep. 05-36, Department of Mathematics, Otto-von-Guericke 

University (2005). 

Speaker: TOBISKA, L. 140 BAIL 2006 

1 

✩ 

✪

P. SVÁ ˘CEK: Numerical Approximation of Flow Induced Airfoil Vibrations 

✬ 

✫ 

Numerical Approximation of Flow 

Induced Airfoil Vibrations 

Petr Sváček 

Department of Technical Mathematics, Karlovo náměstí 13, 

121 35 Praha 2, CTU, Faculty of Mechanical Engineering. 

The strong interaction between the aerodynamic field and the aero-elastic field 

plays an important role in the design of aerospace vehicles. The fluid-structure 

interaction is usually a complex nonlinear problem. The aero-elastic stability 

of aerospace vehicles has a great impact on their design. In this paper we are 

interested in the interaction of two dimensional incompressible viscous laminar 

flow and a solid airfoil. The airfoil can rotate and oscillate in vertical direction. 

The numerical simulation of such a problem is very challenging topic - it 

consists of discretization and stabilization of the Navier-Stokes equations for a 

high Reynolds number. Also the nonlinear discrete problem has to be treated 

carefully. Moreover, the computational domain is time dependent and one has 

to recompute it for each time step together with the used grid. In order to take 

the grid motion into account the Arbitrary Lagrangian-Eulerian formulation of 

the Navier-Stokes equation is used. 

The most difficult part of the fluid-structure interaction is a fluid flow simulation. 

We consider viscous laminar incompressible two dimensional flow which is fully 

time dependent 

D A u 

Dt − ν△u + ((u − wg) · ∇) u + ∇p = 0, in Ωt, 

∇ · u = 0, in Ωt 

equipped with appropriate boundary conditions. 

The above problem is discretized by the finite element method(FEM). Nevertheless, 

the Galerkin FEM leads to unphysical solutions if the grid is not fine 

enough in regions of strong gradients (e.g. boundary layer). In order to obtain 

physically admissible correct solutions it is necessary to apply suitable mesh 

refinement combined with a stabilization technique giving stable and accurate 

schemes. In our paper we present a special version of the GLS stabilization 

method for Navier-Stokes equations, see [1], [2]. 

Further, the computation of the force acting on the airfoil requires correct evaluation 

of boundary integral of the stress tensor. A straightforward evaluation 

of the stress tensor integral may lead to inaccurate results. This obstacle is 

avoided with the aid of a weak formulation of the force acting on the profile. 

Speaker: SVÁ ˘CEK, P. 141 BAIL 2006 

1 

✩ 

✪

P. SVÁ ˘CEK: Numerical Approximation of Flow Induced Airfoil Vibrations 

✬ 

✫ 

The fluid forces acting on the profile causes its motion or deformation, which 

depends on the airfoil properties. We simulate the situation, when the airfoil is 

a solid body with two degrees of freedom. Its motion is obtained as the solution 

of two ordinary differential equations coupled with the Navier-Stokes system. 

We discuss the GLS stabilization of the FEM, evaluation of forces acting on 

the profile, time discretization and the solution of the discrete problem. The 

obtained results are compared with available data. 

References 

[1] Sváček, P., Feistauer, M. Application of a stabilized FEM to problems 

of aeroelasticity. In: Feistauer, M., Dolejˇsí, V., K., N. (Eds.),Numerical 

Mathematics and Advanced Applications, ENUMATH2003. Springer, Heidelberg, 

pp. 796–805, 2004 

[2] Sváček, P. , Feistauer, M., Horáček, J., Numerical simulation of flow 

induced airfoil vibrations with large amplitudes. Journal of Fluids and 

Structures(submitted), 2004. 

Speaker: SVÁ ˘CEK, P. 142 BAIL 2006 

✩ 

✪

N.V. TARASOVA: Full asymptotic analysis of the Navier-Stokes equations in the 

problems of gas flows over bodies with large Reynolds number 

✬ 

✫ 

Full asymptotic analysis of the Navier-Stokes equations 

in the problems of gas flows over bodies 

with large Reynolds number 

N.V.Tarasova 

Baltic State Technical University 

1, 1st Krasnoarmeiskaya ul., 190005 St Petersburg, Russia 

E-mail: natali_t3@professor.ru, tsrknv@bstu.spb.su 

Investigation of the gas flows over bodies with large Reynolds number in most cases can be 

significantly simplified when the flow area is divided into two parts: the external inviscid one 

and the narrow area near the body surface well-known as viscous boundary layer. As this takes 

place the Navier-Stokes equations describing such flows are splitted into the Eulier equations for 

the external flow area and the Prandtl boundary layer equations. Such splitting of the problem 

is obtained on the base of the matched asymptotic expansion method proposed by Van Dike for 

hypersonic flow [1]. All gas parameters are written in the form of power series in the standard 

small parameter ε = 1/ √ Re∞ (Re∞ is the Reynolds number in the free stream flow) and the 

system of the Navier-Stokes equations reduces to the sequences of partial differential equation 

systems in the external flow area and internal flow area. As a result, in a first approximation we 

derive the Euler equations (in external flow area) and the Prandtl boundary layer equations. 

Traditionally the model of incompressible gas flow is used both in the external flow area 

and inside the boundary layer if the Mach number in the free stream M∞ is less than 0.3 and 

the characteristic temperature drop is small enough, otherwise the model for compressible gas 

flow is used. At the same time in practice in many problems of gas flows over bodies, such as 

calculation of induced convection in heat exchanges, we have hyposonic gas flows in which the 

gas temperature across the boundary layer on the body surface can be changed significantly that 

leads to considerable changes of the gas density. As a result, the gas flow outside the boundary 

layer can be considered as incompressible but inside the layer as essentially compressible one. 

In the described case the researchers usually use the common model of the compressible gas 

both inside the boundary layer and in the external inviscid flow area. However, such approach 

implies the necessity to solve the compressible Euler equations in the external area that leads to 

significant computational difficulties if the Mach number in the free stream flow is small enough 

(in the case of hyposonic flows). 

The main aim of this paper is to carry out the strict asymptotic analysis based on the method 

of matched asymptotic expansions of the complete system of the Navier-Stokes equations in all 

possible situations: 

1) flows with the Mach number greater than 0.3 and the large temperature drop across the 

boundary layer, 

2) flows with the Mach number greater than 0.3 and the small temperature drop across the 


3) flows with the Mach number less than 0.3 and the large temperature drop across the 


4) flows with the Mach number less than 0.3 and the small temperature drop across the 


and as a result to suggest the setting of the problem for the Euler equations and the boundary 

layer equations including the procedure of sewing together the appropriate gas parameters. 

Cases 1) and 4) are seen to give us the well-known models for the compressible and incompressible 

gas flow in both areas respectively. Because of this, they have not been the major subject 

of investigation in the present study. At the same time cases 2) and 3) give us some intermediate 

situations which are of great interest. This problem is not trivial for example for case 3) because 

Speaker: TARASOVA, N.V. 143 BAIL 2006 

1 

✩ 

✪

N.V. TARASOVA: Full asymptotic analysis of the Navier-Stokes equations in the 

problems of gas flows over bodies with large Reynolds number 

✬ 

✫ 

the application of the Euler equations for incompressible flow outside the boundary layer and the 

classical equations of the compressible boundary layer near the body surface makes impossible 

the agreement between both equation systems. 

To consider the proposed problems from common positions another special dimensionless 

variables for gas parameters in the Navier-Stokes equations are introduced. These new variables 

are varied inside the studied flow area usually from 0 to 1 (in other words, they can vary to the 

value of the order of unit). The main idea to use such special dimensionless variables is that in 

this case all governing parameters that can be both small and not small are indicated explicitly 

as the coefficients in the equation system. Among these parameters are M 2 ∞ and ∆T/T0 (∆T is 

the temperature drop and T0 is the characteristic value of the temperature in the area). 

It should be noticed that in the hyposonic flows the gravity force can affect significantly the 

gas flow (in studies of free and induced convection). In term of asymptotic analysis it means 

that it is necessary to take account of one more parameter, for example 1/Fr (Fr is the Frud 

number) that can be both small and not small. 

The full asymptotic analysis is carried out for all mentioned situations on the base of 

comparison of all parameters mentioned above (M 2 ∞, ∆T/T0, 1/Fr) with the standard small 

parameter ε in order of magnitude. As a result, the model for gas flow in both areas is formulated 

and an attempt to construct the procedure of the agreement of the solutions in both areas is 

made. It should be stressed that the equation systems derived under the assumptions 1) - 4) 

differ from each other. 

This model constructed for the cases 2) and 3) possibly can occupy the intermediate place 

between two classical approaches when a gas is considered as incompressible or compressible one 

over the whole flow field. 

The equations describing hyposonic flows (M → 0) with the arbitrary values of the Reynolds 

number (Re), ∆T/T0 and 1/Fr were derived in [2] from the Navier-Stokes equations on the base 

of the asymptotic analysis with the Mach number (M) as a small parameter. In this work an 

attempt to compare the model constructed for cases 2) and 3) with the results obtained in [2] 

when considered flow with the large Reynolds number was made. 

References 

1. M. Van Dyke. 1962 In: Hypersonic Flow Research (Ed. F.R.Riddell). Academic Press. 

2. A.I. Zhmakin, Yu.N. Makarov. 1985 Numerical modelling of hyposonic flows of viscous gas, 

Dokl. AN SSSR, Mekh. Zh. i Gaza 280 (4), 827–830. [in Russian] 

Speaker: TARASOVA, N.V. 144 BAIL 2006 

2 

✩ 

✪

C.H. TAI, C.-Y. CHAO, J.-C. LEONG, Q.-S. HONG: Effects of golf ball dimple 

configuration on aerodynamics, trajectory, and acoustics 

✬ 

✫ 

International Conference: BAIL 2006 


- Computational & Asymptotic Methods - 

Effects of golf ball dimple configuration on aerodynamics, trajectory, and acoustics 

*Chang-Hsien Tai + Chih-Yeh Chao ++ Jik-Chang Leong + Qing-Shan Hong + 

*Corresponding author 

Department of Vehicle engineering, National Pingtung University of Science and Technology + 

Department of Mechanical engineering, National Pingtung University of Science and Technology ++ 

1, Hseuh Fu Road, Neipu Hsiang, Pingtung Taiwan, R.O.C. 

Fax: 886-8-7740398 E-mail: chtai@mail.npust.edu.tw 

Abstract 

In many reports about golf ball, including the 

history of its development, have introduced the 

standards on golf ball specification. However, 

there is not a single well-documented solid 

requirement found for the design of golf ball 

surface. Not only have a lot of reports discussed 

the material and structure of a golf ball, but also 

most of the golf ball manufacturers improve their 

products by modifying the number of layers 

beneath the golf ball surface and their materials. 

Even so, there are relatively very few papers 

focused on the influence of different concave 

surface configurations on the aerodynamic 

characteristics of the golf ball. Furthermore, the 

noise a golf ball generates in a tournament is very 

likely to affect the emotion and hence the 

performance of the golf ball player. For these 

reasons, this study investigates the performance of 

a golf ball based on the CFD method with the 

validation using a wind tunnel. 

In 1938, Goldstein [1] had proposed an 

important parameter – the spin ratio. In 

corporation with different Reynolds numbers, this 

parameter makes the study of life and drag effects 

feasible for whirling smooth bodies. In his book, 

Jorgensen [2] especially emphasized that the main 

objective of concaved surfaces on a golf ball is to 

generate small scale turbulence. When flying, this 

turbulence postpones air separation, reduces the 

low pressure region trailing the golf ball, and 

eventually lowers the air drag. Warring [3] used 

numerical approach to perform a series of studies 

related to golf ball using Excel spreadsheets. The 

goal of his paper was to provide guidance for golf 

ball players and manufacturers so that their golf 

ball was capable of flying for a longer distance. In 

the study of acoustics, Singer, et al. [4] calculated 

the noise level from a source using a hybrid grid 

system with the help of Lighthill’s acoustics 

analytic approach. On the other hand, Montavon, 

et al. [5] combined CFD method and 

Computational Aeroacoustics Approach (CAA) to 

simulate noise generation from a cylinder. Using 

CFX-5 with LES (Large Eddy Simulation) as their 

turbulence model and Ffowcs-Williams Hawkings 

formulation, they had successfully shown that their 

predicted sound levels agreed very well with 

theoretical ones for Reynolds numbers about 1.4× 

10 5 . 

Figure 1 shows the flow field around a typical 

golf ball (Case 1). In Case 2, additional dimples 

are added onto the golf ball considered in Case 1. 

The orientation of these additional dimples is 

depicted in Figure 3. It is found, based on Figure 2, 

that the flow field associated to Case 2 is no longer 

symmetrical because of the presence of the 

additional dimples. Figure 3 demonstrates the 

distribution of lift and drag coefficients of Cases 1 

and 2. Clearly, the addition of small dimples 

increases the drag. This implies that the golf ball in 

Case 2 suffers more serious drag effect at low 

trajectory speeds. The lift the golf ball in Case 2 

experiences at moderate Reynolds numbers 

increases so greatly that it becomes greater than 

that for Case 1. The life force in overall is 

therefore greater for Case 2 than Case 1. Although 

the drag imposed on the golf ball is always smaller 

for Case 1 than for Case 2, the drag in Case 1 is 

only about 38.5% less than that in Case 2. 

However, the lift in Case 2 is 103% greater than 

that in Case 1. This somewhat indicates the lift 

effect is 2.68 times of the drag effect. The overall 

performance of the golf ball for Case 2 is much 

greater than that for Case 1. Therefore, the former 

golf ball is capable of traveling further, as shown 

in Figure 4. 

Speaker: HONG, Q.-S. 145 BAIL 2006 

✩ 

✪

C.H. TAI, C.-Y. CHAO, J.-C. LEONG, Q.-S. HONG: Effects of golf ball dimple 

configuration on aerodynamics, trajectory, and acoustics 

✬ 

✫ 

This thesis used structured and non-structured 

grids to come out with the most appropriate grid 

systems for the current golf balls simulations. Then, 

numerical simulations were carried out to estimate 

the aerodynamics parameters and noise levels for 

various kinds of golf balls having different dimple 

configurations. With the obtained aerodynamics 

parameters, the flying distance and trajectory for a 

golf ball were determined and visualized. The 

results showed that structured grids produced more 

accurate results. In terms of dimple layout, the lift 

coefficient of the golf ball increased if small 

dimples were added between the original large 

dimples. When launched at small angles, golf balls 

with deep dimples were found to have greater lift 

effect than drag effect. Therefore, the golf balls 

would fly further until a critical depth of 0.25 mm. 

As far as noise generation was concerned, deep 

dimples produced lower noise levels. 

Keywords: golf ball, CFD, dimple, flying 

trajectory, noise 

Fig.1 The velocity vector contour on rotation of 

Case1(Re=1×10 5 ) 

Fig.2 The velocity vector contour of 

Case2(Re=1×10 5 ) 

International Conference: BAIL 2006 


- Computational & Asymptotic Methods - 

Drag coefficient 

Fig.3 Life, Drag coefficient of Case1 and Case2 

Height (m) 

0.7 

0.6 

0.5 

0.4 

0.3 

0.2 

0.1 

10 3 0 

250 

200 

150 

100 

50 

0 

Case1-Drag coefficient 

Case2-Drag coefficient 

Case1-Lift coefficient 

Case2-Lift coefficient 

10 4 

Reynolds number 

Case1 = 215.2m 

Case2 = 262.1m 

50 100 150 200 250 

Distance (m) 

Fig.4 Flying trajectory of Case1 and Case2 

Reference 

[1] Goldstein, S., “Modern Developments in Fluid 

Dynamics,” Vols. I and II. Oxford: Clarendon 

Press, 1938. 

[2] Jorgensen, T. P., “The Physics of Golf, 2nd 

edition,” New York: Springer-Verlag, pp. 71-72, 

1999. 

[3] Warring, K. E., “The Aerodynamics of Golf Ball 

Flight,” St. Mary’s College of Maryland, pp. 1-37, 

2003. 

[4] Singer, B. A., Lockard, D. P. and Lilley, G. M., 

“Hybrid Acoustic Predictions,” Computers and 

Mathematics with Application 46, pp. 647-669, 

2003. 

[5] Montavon, C., Jones, I. P., Szepessy, S., 

Henriksson, R., el-Hachemi, Z., Dequand, S., 

Piccirillo, M., Tournour, M. and Tremblay, F., 

“Noise propagation from a cylinder in a cross 

flow: comparison of SPL from measurements and 

from a CAA method based on a generalized 

acoustic analogy,” IMA Conference on 

Computational Aeroacoustics, pp. 1-14, 2002. 

Speaker: HONG, Q.-S. 146 BAIL 2006 

10 5 

0.16 

0.14 

0.12 

0.1 

0.08 

0.06 

0.04 

0.02 

Lift coefficient 

✩ 

✪

H. TIAN: Uniformly Convergent Numerical Methods for Singularly Perturbed Delay 


✬ 

Uniformly Convergent Numerical Methods for Singularly Perturbed Delay 

Differential Equations ∗ 

1 Introduction 

Hongjiong Tian 

Department of Mathematics, Shanghai Normal University, 

100 Guilin Road, Shanghai 200234, P. R. China 

hjtian@shnu.edu.cn 

Singularly perturbed problems form a special class of problems containing a small parameter 

which may tend to zero. Singularly perturbed delay differential equations (DDEs) has arose in 

many fields, such as in the study of an “optically bistable device” [1] and in a variety of models 

for physiological processes or diseases [2]. Such problems include a subclass of what we frequently 

thought of as “stiff” equations. We will concentrate on uniformly convergent numerical methods 

for linear and nonlinear singularly perturbed delay differential equations with a fixed lag. 

2 Uniformly convergent schemes 

Linear problems 

Consider linear singularly perturbed delay differential equations 

Ly(t) ≡ εy ′ (t) + a(t)y(t) = b(t)y(t − 1) + g(t), 0 ≤ t ≤ T, 

y(t) = φ(t), −1 ≤ t ≤ 0, 

where ε > 0 is a small parameter, a(t) ≥ α > 0, b(t), φ(t) and g(t) are smooth functions. We 

concentrate on the following difference schemes 

and 

L h yi ≡ εσi(ρ)D+yi + a(ti)yi+1 = b(ti)yi−m + g(ti), i ≥ 0, 

y−j = φ(t−j), j = 0, 1, 2, . . . , m, 

L h yi ≡ εσi(ρ)D+yi + a(ti+1)yi+1 = b(ti+1)yi+1−m + g(ti+1), i ≥ 0, 

y−j = φ(t−j), j = 0, 1, 2, . . . , m, 

where the step length h satisfies the constraint 1 = mh with a positive integer m, ti = ih, ρ = h 

ε 

and 

σi(ρ) = 

� 

ρa(0) 

1−exp(−ρa(0)) 

ρa(j) 

1−exp(−ρa(j)) 

Nonlinear problems 

Consider nonlinear problems of the form 

exp(−ρa(0)), i = 0, 1, · · · , m − 1, 

exp(−ρa(j)), i = jm, jm + 1, · · · , (j + 1)m − 1, j = 1, 2, · · · . 

εy ′ (t) = f(t, y(t), y(t − 1)), 0 ≤ t ≤ T, 

y(t) = φ(t), −1 ≤ t ≤ 0, 

∗ The work of this author is supported in part by E-Institutes of Shanghai Municipal Education Commission 

(No. E03004), Shanghai Municipal Science and Technology Commission (No.03QA14036), Science and Technology 

Foundation of Shanghai Higher Education (No.03DZ21), and The Special Funds for Major Specialties of Shanghai 

Education Committee. 

✫ 

1 

Speaker: TIAN, H. 147 BAIL 2006 

(1) 

(2) 

(3) 

(4) 

(5) 

✩ 

✪

H. TIAN: Uniformly Convergent Numerical Methods for Singularly Perturbed Delay 


✬ 

✫ 

where we assume that −fy(t, y, z) ≥ α > 0 for all t ≥ 0 and all real y, z. We now linearize (5) 

and introduce the Newton sequence {ys(t)} ∞ 

s=0 for the initial guess y0(t) satisfying the initial 

condition y0(t) = φ(t), t ∈ [−1, 0]. This is done by defining ys+1(t), for all s ≥ 0, to be the 

solution of the linear problem 

�Lsys+1(t) ≡ ε dys+1(t) 

dt − fy(t, ys(t), ys(t − 1))ys+1(t) 

−fz(t, ys(t), ys(t − 1))ys+1(t − 1) 

= f(t, ys(t), ys(t − 1)) − fy(t, ys(t), ys(t − 1))ys(t) 

−fz(t, ys(t), ys(t − 1))ys(t − 1), t > 0, 

ys+1(t) = φ(t), −1 ≤ t ≤ 0. 

We may show that not only the convergence of this sequence is quadratic, but also its proportionality 

constant is independent of s and ε. If the initial guess y0(t) is sufficiently close to y(t), 

converges to y(t). 

then the Newton sequence {ys(t)} ∞ 

s=0 

3 Numerical experiments 

Consider 

We take initial guess as 

y0(t) = 

� 

εy ′ (t) = −y(t) + y 2 (t − 1), t ≥ 0, 

y(t) = 2, t ∈ [−1, 0]. 

t 

− 4 − 2e ε , t ∈ [0, 1), 

16 − (8 + 16 t−1 

t−1 

t−1 

−2 

ε )e− ε − 4e ε , t ∈ [1, 2). 

The true solution and the numerical solution using the optimal scheme after two iterations are 

plotted in Figure 1. This figure indicates that the uniformly convergent scheme works well also 

for nonlinear problem. 

References 

y 

18 

16 

14 

12 

10 

8 

6 

4 

Numerical and analytic solutions:ε y’(t)=−y(t)+y 2 (t−1),ε=10 −2 

Numerical solution 

True solution 

2 

0 0.2 0.4 0.6 0.8 1 

t 

1.2 1.4 1.6 1.8 2 

Figure 1: Comparison between numerical and analytical solutions. 

[1] M. W. Derstine, H. M. Gibbs, F. A. Hopf, D. L. Kaplan, Bifurcation gap in a hybrid optical 

system, Phys. Rev., A, 26(1982)3720–3722. 

[2] M. C. Mackey, L. Glass, Oscillation and chaos in physiological control systems, Science, 

197(1977)287–289. 

Speaker: TIAN, H. 148 BAIL 2006 

2 

(6) 

✩ 

✪

A.I. TOLSTYKH, M.V. LIPAVSKII, E.N. CHIGEREV: Highly accurate 9th-order 

schemes and their applications to DNS of thin shear layer instability 

✬ 

✫ 

Highly accurate 9th-order schemes and their applications to 

DNS of thin shear layer instability 

A.I.Tolstykh , M.V.Lipavskii, E.N.Chigerev 

Computing Center of Russian Academy of Sciences, Vavilova str.40, 119991 Moscow GSP-1, 

Russia e-mail tol@ccas.ru 

ABSTRACT 

The principle of constructing arbitrary-order approximations and schemes is outlined. Its 

essence is forming linear combinations of basis operators from certain types of one-parametric 

operators families by fixing distinct values of the parameter. The details of the procedure first 

proposed in [1] can be found in [2],[3] where the linear combinations were referred to as multioperators. 

The multioperators were designed for parallel machines providing approximation 

orders which are linear functions of numbers of processors involved in calculations. 

In the present talk, extremely accurate ninth-order multioperators-based schemes for fluid 

dynamics equations are presented, the basis operators being Compact Upwind Differencing 

(CUD) ones from [4]. The schemes preserve upwinding and conservation properties of CUD 

schemes; they are characterized by very small phase & amplitude errors for physically relevant 

wave numbers supported by grids and damping spurious oscillations. They are capable to 

resolve properly small scale phenomena using reasonable meshes and allow to perform highaccuracy 

unsteady calculations for large time intervals. Their properties make them very useful, 

in particular, for thin layers, DNS and LES computations. 

Illustrative examples followed by direct numerical simulations of thin incompressible 2D shear 

layers instability are presented. The Navier-Stokes calculations were carried out for various high 

Reynolds number flows with complete resolution of turbulent scales as well as for zero molecular 

viscosity. In the latter case, ninth-order dissipative mechanism was responsible for generating 

small-scale vorticity. The results obtained for large time intervals show the full history of the 

flow development with rolling-up,pairing,generation and decaying of turbulence.The resulting 

energy and enstrophy spectra are discussed. 

References 

[1] A. I. Tolstykh, Multioperator high-order compact upwind methods for CFD parallel calculations, 

in Parallel Computational Fluid Dynamics, Elsevier, Amsterdam, 1998, pp. 383-390. 

[2] A. I. Tolstykh, On ultioperators principle for constructing arbitrary-order difference 

schemes, Applied Numerical Mathematics 46 (2003),pp.411-423 

[3] A. I. Tolstykh, Centered prescribed-order approximations with structured grids and resulting 

finite-volume schemes, Applied Numerical Mathematics, 49(2004),pp.431-440. 

[4] A. I. Tolstykh, High accuracy non-centered compact difference schemes for fluid dynamics 

applications, World Scientific, Singapore, 1994. 

Speaker: TOLSTYKH, A.I. 149 BAIL 2006 

✩ 

✪

ABOUTUNSTEADYBOUNDARYLAYERONADIHEDRALANGLE 

beingatrest.Letdenotethelinearangleof?.Itisassumed,thattheanglemovesin bythesuddenmotionofthedihedralangle?withtheconstantvelocityUintheuid KeldyshInstituteofAppliedMathematics,Miusskajasq.4,Moscow125047,Russia Theunsteadyowoftheviscousincompressibleuidisconsidered.Thisowiscaused vasiliev@keldysh.ru M.M.Vasiliev 

investigatedrstbyStokes[2].Steadyboundarylayerontherightdihedralanglewas consideredbyLoytsjansky[3]. theangle?.Since?isinniteinthedirectionofaxis0zandtheowiscausedonlyby thedirectionoftheedge(0z)andonlyonevelocitycomponentofuidwinthisdirection isdierentfromzero.Suchowsarecalledbylayered[1]. 

coordinatez.Accordingtomadeassumptions,themotionequationsreducetofollowing thewallmotion,weshallassumethatallhydrodynamicfunctionsareindependentfrom Letusintroducethecylindricalcoordinates(r;#;z).Weconnectthiscoordinateswith Theunsteadylayeredowcausedbythesuddenmotionofaninniteatplateis 

wheretisthetime-coordinate,s=r=p,{kinematicviscouscoecient. intersectionthewingandthefuselageofanaircraftatenoughdistancesfromtheleading andthetrailingedgesofthewing. oneequation: 

Inthisworkthepowergeometrymethods[4]areusedforobtainingofself-similar Theconsideredowsimulatesroughlyaboundarylayerintheneighborhoodofthe @w @t? @2w @s2+1s@w @s+1s2@2w @#2!=0: (1) 

solutionsofboundary-valueproblems.Thesemethodshavesimplealgorithms.They wereappliedsuccesifullebothtolinearandnonlinearproblemsinworks[5]{[9]and 

wasobtainedintheform wherew=U(1;2).Solutionoftheequation(2)bycorrespondingboundaryconditions others. Fortheself-similarvariables1=s=pt;2=#theequation(1)is 

where==andfunction1isdeterminedasaresultofthesolutionoftheboundaryvalueproblemfortheequation 12@2 @12+11+1212@1+@2 w=U1(1)cos(#); @22=0: (3) (2) 

TheanalyticalsolutionoftheconsideredprobleminCartesiancoordinates(x;y;z)was obtainedonlyincase==2.Thissolutionis where 12100+11+121210?21=0; w=Uerferf; (5) (4) 

=x 2pt;=y 2pt;erf'=2 p' Z0e?l2dl: 

M. VASILIEV: About unsteady Boundary Layer on a dihedral angle 

✬ 

✫ 

Speaker: VASIELIEV, M. 150 BAIL 2006 

✩ 

✪

!+1. Trans.Cambr.Phil..IX,8.1851. Itischeckedthatasymptoticsofthesolutions(3)and(5)coinsidebothbyr!0and 

Moscow,1998(Russian).=ElsevierScience,Amsterdam,2000. [1]Schlichting,H.,Grenzschicht-Theorie.VerlagG.Braun.Karlsruhe.1951. [2]Stokes,G.G.,Ontheeectofinternalfrictionofuidonthemotionofpendumlums. [3]Loytsjansky,L.G.,LaminarBoundaryLayer.Fizmatgiz,Moscow,1962(Russian). [4]Bruno,A.D.,Powergeometryinalgebraicanddierentialequations.Fizmatlit, References 

bypowergeometry.Proc.oftheInternationalConferenceonBoundaryandInterior Layers(BAIL2002),Perth,WesternAustralia,2002,pp.251-256. oftheInternationalConferenceonBoundaryandInteriorLayers(BAIL2002),Perth, WesternAustralia,2002,pp.251-256. [5]Bruno,A.D.,Algorithmicanalysisofsingularperturbationsandboundarylayers [6]Vasiliev,M.M.,Asymptoticsofsomeviscous,heatconductinggasows//Proc. 

Toulouse,France,2004,9pp. 2003,pp.93-101. //Proc.oftheInternationalConferenceonBoundaryandInteriorLayers(BAIL2004), equationsbymethodsofpowergeometry//ProcofISAAC-2001,WorldScientic,Singapore, [8]Bruno,A.D.,Shadrina,T.V.,Theaxiallysymmetricboundarylayeronaneedle [9]Vasiliev,M.M.,Ontheself-similarsolutionofsomemagnetohydrodynamicproblems [7]Vasiliev,M.M.,Abouttheobtainingself-similarsolutionsoftheNavier-Stokes 

Toulouse,France,2004,6pp. //Proc.oftheInternationalConferenceonBoundaryandInteriorLayers(BAIL2004), 

M. VASILIEV: About unsteady Boundary Layer on a dihedral angle 

✬ 

✫ 

Speaker: VASIELIEV, M. 151 BAIL 2006 

✩ 

✪

A.E.P. VELDMANN: High-order symmetry-preserving discretization on strongly 

stretched grids 

✬ 

✫ 

High-order symmetry-preserving discretization on strongly stretched grids 

Arthur E.P. Veldman 

Institute of Mathematics and Computing Science, University of Groningen 

P.O. Box 800, 9700 AV, Groningen, The Netherlands 

Many physical phenomena feature thin boundary layers, in which the solution varies much more 

rapidly than elsewhere in the domain of interest. A ‘natural’ approach is to adapt a computational grid 

to the variations of the solution. In this way one obtains grids with a large diversity in mesh size. Also 

the size of adjacent mesh cells can be quite different, i.e. the grid shows strong stretching. 

‘Traditional’ discretization methods (based on Lagrangian interpolation) focus on minimizing local 

truncation error, but experience has shown that these methods prefer low grid stretching rates (e.g. [2] 

and the refernces therein). The problems arise because this approach does not take into account the 

properties of the discrete system matrices that arise after discretization. An alternative approach is to 

develop discretization schemes with the properties of the system matrix in mind - a general name for 

this philosophy is mimetic discretization. Here certain properties of the analytic operator are mimiced 

in their discrete counterpart. 

At RuG we have chosen to retain the symmetry properties of the operator, in our application a 

combination of convection and diffusion. In particular, we discretize convection such that the discrete 

version remains skew-symmetric. An almost immediate consequence is that the system matrix remains 

diffusively stable (hence never can become singular) on any grid. In formula: Let the system under study 

be given by 

dφ 

+ Lφ = 0, 

dt 

then the evolution of the kinetic energy � φ �H= φ∗Hφ, where H represents the local mesh size, is given 

by 

d 

dt � φ �= −φ∗ ((HL) ∗ + HL)φ. 

With skew-symmetric convection, the symmetric part (HL) ∗ + (HL) of the system matrix comes only 

from diffusion, and the above assertion follows. 

Another consequence is that the convective discretization does not produce unphysical numerical diffusion, 

which unavoidably will interfere with the physical diffusion. This strategy has been applied e.g. 

in direct numerical simulation of turbulent flow, where the small-scale balance between convection and 

diffusion is quite delicate [3]. 

In the presentation we would like to illustrate the performance of various symmetry-preserving finitevolume 

methods: second- and fourth-order central schemes (that we apply in DNS), but also higher-order 

upwind schemes. It is less known that on contracting and expanding grids the traditional upwind method 

produces negative diffusion [4]; a symmetry-preserving upwind version (a suitable combination of skewsymmetric 

odd derivatives and symmetric even derivatives) will repair this. 

For instance, the fourth-order symmetry-preserving discretization of a first-order derivative ∂φ/∂x 

becomes 

∂φ 

∂x ≈ −φi+2 + 8φi+1 − 8φi−1 + φi−2 

. 

−xi+2 + 8xi+1 − 8xi−1 + xi−2 

After muliplication with the denominator (as is usual in a finite-volume setting), the convective contribution 

to the coefficient matrix is clearly seen to become skew symmetric. 

As a main example we consider a one-dimensional convection-diffusion equation on the unit interval at 

a diffusion coefficient k = 0.001. Our ‘favorite’ grid is the Shishkin grid, consisting of piecewise-uniform 

grid regions with an abrupt (very large) change in mesh size (e.g. [1]). Figure 1 shows a comparison between 

the individual solutions of second- and fourth-order traditional methods and symmetry-preserving 

methods on an abrupt Shishkin grid and on a (smoothly stretched) exponential grid. Both grids contain 

Speaker: VELDMANN, A.E.P. 152 BAIL 2006 

1 

✩ 

✪

A.E.P. VELDMANN: High-order symmetry-preserving discretization on strongly 

stretched grids 

✬ 

✫ 

28 grid points, with half of the grid points located in [1 − 10k, 1]). On both grids the ‘traditional’ 

fourth-order method suffers from an almost singular system matrix. 

1.2 

1 

0.8 

0.6 

0.4 

0.2 

0 

error=3.5e−02 

2nd Lagrange 

−0.2 

0 0.2 0.4 0.6 0.8 1 

50 

0 

−50 

−100 

error=7.4e+01 

4th Lag (+ exact bc) 

−150 

0 0.2 0.4 0.6 0.8 1 

1.2 

1 

0.8 

0.6 

0.4 

0.2 

0 


2nd symm−pres 

−0.2 

0 0.2 0.4 0.6 0.8 1 

1.2 

1 

0.8 

0.6 

0.4 

0.2 

0 

4th symm−pres (+ 2nd bc) 


−0.2 

0 0.2 0.4 0.6 0.8 1 

1.4 

1.2 

1 

0.8 

0.6 

0.4 

0.2 


2nd Lagrange 

0 

0 0.2 0.4 0.6 0.8 1 

200 

0 

−200 

−400 

−600 

error=4.6e+02 

4th Lag (+ exact bc) 

−800 

0 0.2 0.4 0.6 0.8 1 

1.2 

1 

0.8 

0.6 

0.4 

0.2 

0 


2nd symm−pres 

−0.2 

0 0.2 0.4 0.6 0.8 1 

1.2 

1 

0.8 

0.6 

0.4 

0.2 

0 

4th symm−pres (+ 2nd bc) 


−0.2 

0 0.2 0.4 0.6 0.8 1 

Figure 1: Discrete solutions for k = 0.001 at an abrupt grid (left) and an exponential grid (right). The ’traditional’ 

discretization, especially the fourth-order version, has problems with the stretching of the grid. 

Figure 2 shows a grid-refinement study of the second- and fourth-order traditional and symmetrypreserving 

methods on both typs of grids (abrupt and exponential). The much more regular and forgiving 

character of the symmetry-preserving discretization is evident. 

global error 

10 0 

10 −5 

Abrupt: d=10k 

10 −2 

mean mesh size 

10 −1 

10 0 

10 −5 

Exponential: d=10k 

2L 

2SP 

4L 

4SP 

10 −2 

mean mesh size 

Figure 2: The global error as a function of the mean mesh size. Half of the grid points is located in the boundary 

layer of thickness d. Four methods are shown: 2L (second-order Lagrangian), 2S (second-order symmetrypreserving), 

4L (fourth-order Lagrangian with exact boundary conditions) and 4S (fourth-order symmetry-preserving 

with second-order boundary treatment). 

References 

[1] P.A. Farrell, J.J.H. Miller, E. O’Riordan and G. I. Shishkin: A uniformly convergent finite difference 

scheme for a singularly perturbed semilinear equation. SIAM J. Num. Anal. 33 (1996) 1135–1149. 

[2] A.E.P. Veldman and K. Rinzema: Playing with nonuniform grids. J. Eng. Math. 26 (1992) 119–130. 

[3] R.W.C.P. Verstappen and A.E.P. Veldman: Symmetry-preserving discretisation of turbulent flow. J. 

Comput. Phys. 187 (2003) 343–368. 

[4] G. Golub, D. Silvester and A. Wathen: Diagonal dominance and positive definiteness of upwind approximations 

for advection diffusion problems. In: D.F. Griffiths and G.A. Watson (eds.) Numerical 

Analysis: A R. Mitchell 75th Birthday Volume, World Scientific, Singapore (1996) 125–132. 

Speaker: VELDMANN, A.E.P. 153 BAIL 2006 

2 

10 −1 

✩ 

✪

Z.-H. YANG, Y.-Z. LI, Y. ZHU: Application of Bifurcation Method to Computing 

Numerical Solutions of Lane-Emden Equation 

✬ 

✫ 

Application of Bifurcation Method to Computing Numerical 

Solutions of Lane-Emden Equation ∗ 

Zhong-hua Yang, Ye-zhong Li, Ying Zhu 

Department of Mathematics,Shanghai Normal University 

Shanghai,200234,China 

Abstract 

In this paper, the Lane-Emden equations of index p 

� ∆u + u p = 0, (x, y) ∈ Ω 

u|∂Ω = 0, (x, y) ∈ ∂Ω 

where Ω is a bounded open domain in R2 , p > 0, are concerned. Equation (0.1) describes the 

behavior of the density of a gas sphere in hydrostatic equilibrium in appropriate units. The 

index p, which is called the polytropic index in astrophysics, is larger than 1 

2 . It means that no 

polytropic stellar system can be homogeneous in galactic dynamics([3],[7]).Using the Liapunov- 

Schmidt method and symmetry-breaking bifurcation theory, we compute and visualize multiple 

solutions of Lane-Emden equation on a bounded domain of R2 with a homogeneous Dirichlet 

boundary condition, which plays an important role in stellar structure and evolution theory. 

The domains we consider here include the unit square and the square cut by small square. 

The critical point theory was applied to prove the existence and multiplicity of solutions 

under various assumptions([2], [8]). But what distribution and structure they have and how to 

compute them have attracted the attention of many mathematicians, physicists and engineers. 

Due to the multiplicity, degeneracy and instability of the critical points with higher Morse index, 

the computation of multiple solutions encounters essential difficulties and is truly challenging. 

Since 90’s of last century, numerical works to compute numerical solutions of (0.1) appeared 

in the literature. The mountain-pass algorithm, the scaling iterative algorithm ,the monotone 

iteration, the direct iteration algorithm and the research extension method([4],[5],[6]) are used 

to compute the solutions of (0.1). But in these algorithms “ good guess of solution” of (0.1), 

which is a difficult task, is needed. Therefore only few solutions of (0.1) are computed yet. 

In this paper, we try to use the bifurcation method to overcome this difficulty. Our main 

idea is to embed (0.1) into nonlinear elliptic BVP with parameter λ of the form 

� 

F (u, λ) = ∆u + λu + up = 0, (x, y) ∈ Ω, 

(0.2) 

u|∂Ω = 0, (x, y) ∈ ∂Ω. 

According to the bifurcation theory (0.2) has nontrivial solution branches which bifurcate from 

its bifurcation points, so we can compute the solutions of (0.1) by using continuation method([1]) 

along these nontrivial solution branches of (0.2) until λ = 0. Many new solutions of (0.1) with 

different symmetry are computed by the bifurcation method. 

∗ Supported by Shanghai Development Foundation for Science and Technology (No.03QA14036), Science 

Foundation of Shanghai Municipal Education Commission(05DZ07), Shanghai Leading Academic Discipline 

Project(T0401). 

Speaker: YANG, Z.-H. 154 BAIL 2006 

1 

(0.1) 

✩ 

✪

Z.-H. YANG, Y.-Z. LI, Y. ZHU: Application of Bifurcation Method to Computing 

Numerical Solutions of Lane-Emden Equation 

✬ 

✫ 

On the other hand, Lyapunov-Schmidt reduction is used to get the bifurcation equation of 

(0.2), from which we can easily obtain the initial guesses for computing directly the different 

solutions of (0.1) with different symmetry. Therefore the bifurcation method also offers a more 

effective way to computing the multiple solutions of Lane-Emden equation and other nonlinear 

elliptic boundary value problem. 

References 

[1] E.L.,Allogower, K.Georg, Numerical continuation Methods, An Introduction, Springer Series 

in Computational Mathematics(Springer, Berlin), 1990. 

[2] A.Ambrosetti, P.H. Rabinowitz, Dual variational mehtods in critical point theory and application, 

J.Funct. Anal., Vol.14(1973), 327-381. 

[3] S.Chandrasekhar, An Introduction to the stellar structure, Dover Publication, Inc., NY, 

1967. 

[4] Chuanmiao Chen, Ziqing Xie, Structure of multiple solutions for nonlinear differential equations. 

Science in China Ser.A Mathematics Vol.47 Supp.(2004), 172-180. 

[5] Goong Chen,Jianxin Zhou, Wei-ming Ni, Algorithms and visualization for solutions of nonlinear 

elliptic equations.International Journal of Bifurcation and Chaos,Vol.10,No.7(2000) 

1565-1612. 

[6] Y. Li, J.X. Zhou, A minimax method for finding multiple critial points and its applications 

to semilinear PDEs, SIAM J.Sci. Comput., Vol.23(2002), 840-865. 

[7] R. Kippenhaln, A. Weigert, Stellar Structure and Evolution, Springer-Verlag, New York, 

Berlin, 1990. 

[8] P.H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential 

Equations, CBMS Regional Conf. Series in Math.65, Amer.Math.Soc., Providence, 

1986. 

Speaker: YANG, Z.-H. 155 BAIL 2006 

2 

✩ 

✪

Q. YE: Numerical simulation of turbulent boundary for stagnation-flow in the spraypainting 

process 

✬ 

✫ 

Numerical simulation of turbulent boundary for stagnation-flow 

in the spray-painting process 

Q. Ye 

Institut für Industrielle Fertigung und Fabrikbetrieb Universität Stuttgart 

Nobelstr. 12, 70569 Stuttgart, Germany 

(Abstract for submission to the BAIL 2006) 

In order to optimise the painting process, which amounts to a high percentage of fixed and 

flexible costs in automotive production, numerical simulations of spray painting for the 

automotive industry, especially using high-speed rotary bell and electrostatically supported 

methods, have been performed [1]. Previous numerical studies were mainly concerned with 

the calculation of the two-phase turbulent flow of the spray jet, the modelling of the 

electrostatic field including space charge and the prediction of the film thickness distribution 

on the coated work piece. Less attention was paid to the near-wall turbulent flow in the 

painting process, which is, however, quite important for particle deposition. 

The current numerical investigation is aimed at the turbulent boundary of the stagnationairflow. 

A CFD code (Fluent) based on Reynolds-averaged Navier-Stokes equations (RANS) 

has to be used because of the complicated turbulent flow in the spray-painting process (Fig. 1). 

A two-dimensional turbulent channel flow first is calculated using different turbulent models 

with near-wall functions. The simulated results are compared with the DNS data [2] (Fig.2), 

in order to obtain a suitable model for the investigation of the real near-wall flow. The 

complicated three-dimensional turbulent flow is then calculated using a real geometry of the 

atomizer, high-speed rotary bell, and simple target geometry, e.g., a flat plate. The influences 

of the near-wall mesh resolution and the near-wall functions on the velocity distributions (Fig. 

3) and the turbulent magnitudes close to the target are analysed. The computational results 

provide useful information for the further study of the particle deposition in spray-painting 

processes. 

up 

down 

Figure 1: Contours of the air velocity magnitude (m/s) 

Speaker: YE, Q. 156 BAIL 2006 

✩ 

✪

Q. YE: Numerical simulation of turbulent boundary for stagnation-flow in the spraypainting 

process 

✬ 

✫ 

normalized k 

velocity(m/s) 

k-yplus 

5 

4.5 

4 

3.5 

3 

2.5 

2 

1.5 

1 

0.5 

0 

1.00E-03 1.00E-02 1.00E-01 1.00E+00 

yplus 

1.00E+01 1.00E+02 1.00E+03 

3.5 

3 

2.5 

2 

1.5 

1 

0.5 

velocity magnitude 

0 

-0.5 -0.3 -0.1 0.1 0.3 0.5 

z(m) 

standard-wall 

DNS 

enhanced-wall 

Reynolds-stress 

k-omega 

Figure 2: Comparison of non-dimensional turbulent kinetic energy in the fully 

developed turbulent channel flow between different turbulent models 

enhanced -w all 

non-equilibrium 

Figure 3: Comparison of velocity magnitudes along the painting target at a distance of 

0.1 mm to the wall for two different near-wall functions 

[1] Q. Ye, J. Domnick, A. Scheibe, K. Pulli: Numerical Simulation of the Electrostatic Spraypainting 

Process in the Automotive Industry. High-Performance Computing in Science 

and Engineering’04, Springer-Verlag Berlin, Heidelberg, , 2004, pp. 261-275. 

[2] J. Kim, P. Moin, R. Moser: Turbulence statistics in fully developed channel flow at low 

Reynolds number. J. Fluid Mech. 177, pp.133-166. 1987. 

Speaker: YE, Q. 157 BAIL 2006 

✩ 

✪

Participants


Mr. Alizard Frédéric SINUMEF ENSAM Paris Paris FRANCE 

Mr. Alrutz Thomas DLR - Institute of Aerodynamics and Flow Technology Goettingen Germany 

Mr. Apel Thomas Universität der Bundeswehr München Institut f. Mathematik und BauinformatikNeubiberg Germany 

Mr. Bause Markus Universitaet Erlangen-Nuernberg Institut fuer Angewandte Mathematik Erlangen Germany 

Mr Boguslawski Poznan University of Technology Poland 

Mr. Buschmann Matthias H. ILK Dresden / TU Dresden Luft-und Kältetechnik /Strömungsmechanik Dresden Germany 

Mr. Clavero Carmelo UNIVERSIDAD DE ZARAGOZA CENTRO POLITÉCNICO SUPERIOR ZARAGOZA SPAIN 

Mr. Das Debopam Departmnet of Aerospace Engg Indian Institute of Technology Kanpur Kanpur India 

Mr Dillmann Andreas DLR - Institute of Aerodynamics and Flow Technology Goettingen Germany 

Mr. Donkor Jackson CAPSTONE MINISTRIES TRANS-AFRICAN COLLEGE ACCRA GHANA 

Mr. Eisfeld Bernhard DLR Braunschweig Inst. f. Aerodynamik u. Strömungstechnik Braunschweig Germany 

Mr. Firooz Abdolhamid University of Sistan & Baloochestan Mechanical Engineering Department Zahedan Iran 

Mr. Fynn Richard LUGANSK STATE MEDICAL UNIVERSITY LUGANSK UKRAINE 

Mr. Gaponov Serguei INSTITUTE OF THEORETICAL AND APPLIED MECHANICS Novosibirsk Russia 

Mr Georgoulis Emanuil University of Leicester United Kingdom 

Mr Gersten Klaus Ruhr-Universitaet Bochum Bochum Germany 

Mr. Gracia Jose Luis UNIVERSIDAD DE ZARAGOZA ESCUELA POLITÉCNICA DE TERUEL TERUEL SPAIN 

Ms. Hammouch zakia LAMFA UMR CNRS 6140 Faculté de Mathématiques et d'Info, UPJV Amiens France 

Mr. Hamouda Makram Faculty of Science of Bizerte Jarzouna Tunisia 

Mr. Hartmann Ralf German Aerospace Center (DLR) Inst. of Aerodynamics & Flow Technology Braunschweig Germany 

Mr. Hegarty Alan Dept of Mathematics and Statistics University of Limerick Limerick Ireland 

Mr Heinemann Hans-J. DLR - Institute of Aerodynamics and Flow Technology Goettingen Germany 

Mr. Hemker Pieter W CWI CWI Amsterdam Netherlands 

Mr. Herwig Heinz Hamburg University of Technology Thermofluiddynamics Hamburg Germany 

Mr. Heuveline Vincent Universität Karlsruhe (TH) Institut für Angewandte Mathematik II Karlsruhe Germany 

Mr. Hölling Marc Hamburg Univesity of Technology Thermofluiddynamics Hamburg Germany 

Mr. Houston Paul University of Nottingham School of Mathematical Sciences Nottingham England 

Mr. Huerre Patrick Ecole Polytechnique Laboratoire d'Hydrodynamique (LadHyX) Palaiseau France 

Mr Il'in A.M. State University Chelyabinsk Russia 

Mr Jang Jiin-Yuh Department of Mechanical Engineering Air Force Inst. of Technology Kaohsiung Taiwan 

Mr. Kachuma Dick Computing Laboratory Oxford University Oxford United Kingdom 

Mr. Knobloch Petr Charles University Faculty of Mathematics and Physics Praha 8 Czech Republic 

Mr. Knopp Tobias DLR (German Aerospace Center) Aerodynamics and Flow Technology Goettingen Germany 

Mr. Kroll Norbert DLR Inst. f. Aerodynamik u. Strömungstechnik Braunschweig Germany 

Mr. Linss Torsten Technische Universität Dresden Institut für Numerische Mathematik Dresden Germany 

Mr. Liseykin Vladimir Institute of Computational Technologies Novosibirsk Russia 

Mr Leong Jik-Chang National Pingtung University Pingtung Taiwan 

Mr. Lube Gert University of Goettingen Institut for Numerical and Applied Mathe Goettingen Germany 

Mr. Luedeke Heinrich DLR Aerodynamics and Flow Technology Braunschweig germany 

Mr. Mackenzie John Department of Mathematics University of strathclyde Glasgow U.K 

Mr. Mansour kamyar Department of Aerospace Engineering Amirkabir University of technology Tehran, 15875-4413 Iran 


Mrs.Maragatha Meenakshi Ponnusamy P. Maragatha Meenakshi Government of Tamilnadu, India FIP Teacher Fellow, St.Josephs college Tiruchirappalli, Tamil Nadu, India 

Mr. Matthies Gunar Ruhr-Universität Bochum Fakultät für Mathematik Bochum Deutschland 

Mr. maubach joseph technical university of Eindhoven faculty of mathematics and computer scie Eindhoven netherlands 

Mr Mauss J. IMFT Toulouse France 

Mr. Mierka Otto Fachbereich Mathematik, LSIII Universität Dortmund Dortmund Germany 

Mr. Miller John Royal College of Surgeons in Ireland INCA Dublin 2 Ireland 

Mr. Morinishi Koji Dept. Mechanical & System Engineering Kyoto Institue of Technology Kyoto Japan 

Mrs Nastase Adriana RWTH Aachen Aaachen Germany 

Mr. Neuss Nicolas University Kiel Scientific computing Kiel Germany 

Mr. Olshanskii Maxim Moscow M.V.Lomonosov State University Dept. Mech. & Math. Moscow Russia 

Mr. O'Riordan Eugene Dublin City University Dublin Ireland 

Mr. Parumasur Nabendra University KwaZulu-Natal Durban South Africa 

Mrs Perotto Simona Politecnico di Milano Milano Italy 

Mr. Rapin Gerd University Goettingen NAM Goettingen Germany 

Mr. Rasuo Bosko University of Belgrade/Faculty of Mechan Aeronautical Department Belgrade, 35 Serbia + Montenegro 

Mr. Roos Hans-G. TU Dresden Dresden Germany 

Mr Saiful-Islam Wan Salim KUiTTHO Batu Pahat Malaysia 

Mr. Scheichl Bernhard F. Vienna University of Technology Inst. of Fluid Mechanics & Heat Transfer Vienna Austria 

Mr. Schneider Rene TU-Chemnitz, Fakultät für Mathematik Chemnitz Germany 

Mr. Sengupta Tapan Dept. of Aerospace Engineering Indian Institute of Technology Kanpur KANPUR India 

Mr. Shishkin Grigory Russian Academy of Sciences, Ural Branch Institute of Mathematics and Mechanics Ekaterinburg Russia 

Mrs.Shishkina Lidia Russian Academy of Sciences, Ural Branch Institute of Mathematics and Mechanics Ekaterinburg Russia 

MrsShishkina Olga DLR Aerodynamics and Flow Technology Goettingen Germany 

Ms. Stanculescu Iuliana University of Pittsburgh Pittsburgh USA 

Mr. Steinrück Herbert Vienna University of Technology Fluid Mechanics and Heat Transfer Vienna Austria 

Ms. Stephens Meghan Maths Dept National University of Ireland, Galway Galway Ireland 

Mr. Stynes Martin Department of Mathematics National University of Ireland Cork Ireland 

Mr. Svacek Petr CTU in Prague, Faculty of Mech. Eng. Department of Technical Mathematics Praha 2 Czech Republic 

Mrs.Tarasova Natalia Baltic State Technical University St Petersburg Russia 

Mr. Tian Hongjiong Department of Mathematics Shanghai Normal University Shanghai P. R. China 

Mr. Tobiska Lutz Otto-von-Guericke University Department of Mathematics Magdeburg Germany 

Mr Tolstykh A.I. Comp. Center of Russian Acad. of Science Moscow Russia 

Mrs.Tselishcheva Irina Russian Academy of Sciences, Ural Branch Institute of Mathematics and Mechanics Ekaterinburg Russia 

Mrs.Valarmathi Sigamani Church of South India Bishop Heber College Tiruchirappalli India 

Mr. Vasanta Ram Venkatesa Iyengar Ruhr University Bochum Institut fuer Thermo- und Fluiddynamik Bochum Germany 

Mr Vasliiev M.M. Keldysh Inst. of Appl. Math. Moscow Russia 

Mr. Veldman Arthur University of Groningen Mathematics and Computing Science Groningen The Netherlands 

Mr. Wall Wolfgang A. TU München Chair for Computational Mechanics Garching Germany 

Mr. Wang Hong Department of Mathematics University of South Carolina Columbia USA 

Mr. Yang Zhong-hua Shanghai Normal Univ. Shanghai P.R.China 

Mrs.Ye Qiaoyan Fraunhofer Institut IPA Stuttgart Germany 

Mr. Zadorin Alexander Russian Academy of Sciences Institute of Mathematics SB RAS Omsk Russia 

Mrs.Zarin Helena Faculty of Science and Mathematics Dept. of Mathematics and Informatics Novi Sad Serbia 


Mr. Zegeling Paul Dep. of Math. Utrecht Universitz Utrecht Netherlands 

Mr. Georgoulis Emanuil University of Leicester U.K. 

Mr. Boguslawski Leon Poznan University of Technology Poland 


Authors 

Alizard, F., 67 

Alrutz, Th, 69 

Anthonissen, M., 57 

Apel, Th., 71 

Banasiak, J., 130 

Bause, M., 72 

Bleier, N., 37 

Boguslawski, L., 73 

Branley, D., 44 

Buschmann, M.H., 32 

Cangiani, A., 75 

Chao, C.-Y., 145 

Chigerev, E.N., 149 

Chou, Y.-C., 106 

Clavero, C., 77 

Cousteix, J., 113 

Das, D., 35 

Dunne, R.K., 14 

Eisfeld, B., 79 

Firooz, A., 81 

Gad-el-Hak, M., 32 

Gadami, M., 81 

Gaponov, S.A., 85 

Georgoulis, E.H., 75 

Gersten, Kl., 3 

Gracia, J.L., 77 

Gravemeier, V., 8 

Hölling, M., 91 

Hammouch, Z., 87 

Hamouda, M., 88 

Hartmann, R., 22 

Hegarty, A.F., 44, 56 

Hemavathi, S., 53 

Herwig, H., 91 

Heuveline, V., 24 

Hong, Q.-S., 145 

Houston, P., 4 

Huerre, P., 5 

Hussong, J., 37 

Il’in, A.M., 93 

Islam, W.S., 96 

Jang, J.-Y., 106 

Jensen, M., 75 

Jimack, P., 28 

Kachuma, D., 98 

Kameswara Rao, A., 39 

Kaushik, A., 100 

Kluwick, A., 136 

Knobloch, P., 102 

Knopp, T., 69, 104 

Kozakiewicz, J.M., 130 

Kuzmin, D., 118 

Lüdeke, H., 111 

Lenz, S., 8 

Lesshafft, L., 5 

Leu, J.-S., 106 

Li, S., 59 

Li, Y.-Z., 154 

Likhanova, Y.V., 108 

Linß, T., 47 

Lipavskii, M.V., 149 

Lisbona, F., 77 

Liseykin, V.D., 108 

Lube, G., 109

Mackenzie, J.A., 25 

Madden, N., 47 

Mansour, K., 113 

Matthies, G., 71, 115 

Maubach, J., 57, 58 

Mauss, J., 116 

Mierka, O., 118 

Morinishi, K., 120 

Nastase, A., 122 

Nataf, F., 124 

Nayak, A., 35 

Neuss, N., 126 

Nicola, A., 25 

O’Riordan, E., 14 

Olshanskii, M.A., 128 

Parumasur, N., 130 

Patrakhin, D.V., 108 

Perotta, S., 27 

Petrov, G.V., 85 

Purtill, H., 44 

Raghavan, V.R., 96 

Ram, V.V., 37 

Rapin, G., 124 

Rasuo, B., 132 

Robinet, J.-Ch., 67 

Roos, H.-G., 134, 135 

Sagaut, P., 5 

Savić, L., 41 

Scheichl, B., 136 

Schneider, R., 28 

Sedykh, I., 57 

Sengupta, T.K., 39 

Sharma, K.K., 100 

Shishkin, G.I., 16, 44, 48, 49, 51, 56, 59 

Shishkina, L.P., 49, 59 

Shishkina, O., 138 

Sikwila, S., 56 

Smorodsky, B.V., 85 

Sobey, I., 98 

Stanculescu, I., 18 

Steinrück, H., 41 

Stynes, M., 6 

Stynes, M., 140 

Svá˘cek, P., 141 

Tai, C.H., 145 

Tarasova, N.V., 143 

Temam, R., 88 

Terracol, M., 5 

Tian, H., 147 

Tobiska, L., 117, 140 

Tolstykh, A.I., 149 

Tselishcheva, I.V., 51 

Turner, M.M., 14 

Valarmathi, S., 53 

Vaseva, I.A., 108 

Vasieliev, M., 150 

Veldmann, A.E.P., 152 

Wagner, C., 138 

Wall, W.A., 8 

Wang, H., 19 

Yang, Z.-H., 154 

Ye, Q., 156 

Zadorin, A.I., 61 

Zarin, H., 135 

Zegeling, P., 63 

Zhu, Y., 154

BAIL 2006 Book of Abstracts - Institut für Numerische und ...

Create successful ePaper yourself

Delete template?

Save as template?