University of Patras - Nemertes
University of Patras - Nemertes
University of Patras - Nemertes
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<strong>University</strong> <strong>of</strong> <strong>Patras</strong><br />
School <strong>of</strong> Engineering<br />
Department <strong>of</strong> Mechanical Engineering and<br />
Aeronautics<br />
A dissertation submitted to the<br />
Department <strong>of</strong> Mechanical Engineering and Aeronautics and the<br />
<strong>University</strong> <strong>of</strong> <strong>Patras</strong> in partial fulfillment <strong>of</strong> the requirements for the degree <strong>of</strong><br />
DOCTOR OF PHILOSOPHY<br />
in<br />
Mechanical Engineering<br />
High-order discontinuous Galerkin<br />
discretization for flows with strong moving<br />
shocks<br />
Author:<br />
Dipl.-Ing<br />
Konstantinos Kontzialis<br />
Supervisor:<br />
Pr<strong>of</strong>.<br />
John A. Ekaterinaris<br />
© Konstantinos Kontzialis. All rights reserved.<br />
<strong>Patras</strong> 2012
1<br />
Thesis committee:<br />
Pr<strong>of</strong>. John A. Ekaterinaris, Department <strong>of</strong> Mechanical Engineering and<br />
Aeronautics, <strong>University</strong> <strong>of</strong> <strong>Patras</strong><br />
Pr<strong>of</strong>. Yannis Kallinderis, Department <strong>of</strong> Mechanical Engineering and<br />
Aeronautics, <strong>University</strong> <strong>of</strong> <strong>Patras</strong><br />
Pr<strong>of</strong>. Kyriakos Giannakoglou, Department <strong>of</strong> Mechanical Engineering,<br />
National Technical <strong>University</strong> <strong>of</strong> Athens<br />
Examining committee:<br />
Pr<strong>of</strong>. John A. Ekaterinaris, Department <strong>of</strong> Mechanical Engineering and<br />
Aeronautics, <strong>University</strong> <strong>of</strong> <strong>Patras</strong><br />
Pr<strong>of</strong>. Yannis Kallinderis, Department <strong>of</strong> Mechanical Engineering and<br />
Aeronautics, <strong>University</strong> <strong>of</strong> <strong>Patras</strong><br />
Pr<strong>of</strong>. Kyriakos Giannakoglou, Department <strong>of</strong> Mechanical Engineering,<br />
National Technical <strong>University</strong> <strong>of</strong> Athens<br />
Pr<strong>of</strong>. Vasilis Kostopoulos, Department <strong>of</strong> Mechanical Engineering and<br />
Aeronautics, <strong>University</strong> <strong>of</strong> <strong>Patras</strong><br />
Pr<strong>of</strong>. Dimitris Saravanos, Department <strong>of</strong> Mechanical Engineering and<br />
Aeronautics, <strong>University</strong> <strong>of</strong> <strong>Patras</strong><br />
Pr<strong>of</strong>. Efstratios Gallopoulos, Department <strong>of</strong> Computer Engineering and<br />
Informatics, <strong>University</strong> <strong>of</strong> <strong>Patras</strong><br />
Pr<strong>of</strong>. John Tsamopoulos, Department <strong>of</strong> Chemical Engineering, <strong>University</strong> <strong>of</strong><br />
<strong>Patras</strong>
Acknowledgments<br />
The present thesis is the culmination <strong>of</strong> years <strong>of</strong> hard work. Many challenges and<br />
difficulties appeared during my research, which undoubtedly were the source <strong>of</strong><br />
inspiration and creation in many aspects <strong>of</strong> my PhD work.<br />
I am deeply thankful to my supervisor Pr<strong>of</strong>essor Yannis Ekaterinaris, a charismatic<br />
person, whose experience proved to be the source <strong>of</strong> guidelines and knowledge<br />
<strong>of</strong> unprecedented value. His patience, encouragement and spirit <strong>of</strong> giving, and most<br />
importantly his calm perseverance, were the key elements for the successful completion<br />
<strong>of</strong> my PhD research.<br />
I would like also to express my gratitude to Pr<strong>of</strong>essor Yannis Kallinderis who<br />
was the first person to initiate me in the wonderful field <strong>of</strong> CFD. His love for excellence<br />
taught me that ideas mature only through continuous and hard work.<br />
I express also my gratitude to the support <strong>of</strong> all the persons whose love and<br />
presence were <strong>of</strong> importance for a deeper understating and continuous effort.<br />
Lastly, I am deeply thankful to my very supportive family for the love and<br />
encouragement to keep on and succeed in my goals.<br />
2
Summary<br />
Supersonic flows over both simple and complex geometries involve features over a<br />
wide spectrum <strong>of</strong> spatial and temporal scales, whose resolution in a numerical solution<br />
is <strong>of</strong> significant importance for accurate predictions in engineering applications.<br />
While CFD has been greatly developed in the last 30 years, the desire and necessity<br />
to perform more complex, high fidelity simulations still remains.<br />
The present thesis has introduced two major innovations regarding the fidelity<br />
<strong>of</strong> numerical solutions <strong>of</strong> the compressible Navier-Stokes equations. The first one is<br />
the development <strong>of</strong> new a priori mesh quality measures for the Finite Volume (FV)<br />
method on mixed-type (quadrilateral/triangular) element meshes. Elementary types<br />
<strong>of</strong> mesh distortion were identified expressing grid distortion in terms <strong>of</strong> stretching,<br />
skewness, shearing and non-alignment <strong>of</strong> the mesh. Through a rigorous truncation<br />
error analysis, novel grid quality measures were derived by emphasizing on the direct<br />
relation between mesh distortion and the quality indicators. They were applied<br />
over several meshes and their ability was observed to identify faithfully irregularlyshaped<br />
small or large distortions in any direction. It was concluded that accuracy<br />
degradation occurs even for small mesh distortions and especially at mixed-type<br />
element mesh interfaces the formal order <strong>of</strong> the FV method is degraded no matter<br />
<strong>of</strong> the mesh geometry and local mesh size.<br />
Therefore, in the present work, the high-order Discontinuous Galerkin (DG)<br />
discretization <strong>of</strong> the compressible flow equations was adopted as a means <strong>of</strong> achieving<br />
and attaining high resolution <strong>of</strong> flow features on irregular mixed-type meshes<br />
for flows with strong moving shocks. During the course <strong>of</strong> the thesis a code was<br />
developed and named HoAc (standing for High Order Accuracy), which can perform<br />
via the domain decomposition method parallel p-adaptive computations for<br />
flows with strong shocks on mixed-type element meshes over arbitrary geometries<br />
at a predefined arbitrary order <strong>of</strong> accuracy. In HoAc in contrast to other DG developments,<br />
all the numerical operations are performed in the computational space,<br />
for all element types. This choice constitutes the key element for the ability to perform<br />
p-adaptive computations along with modal hierarchical basis for the solution<br />
expansion. The time marching <strong>of</strong> the DG discretized Navier-Stokes system is per-<br />
3
formed with the aid <strong>of</strong> explicit Runge-Kutta methods or with a matrix-free implicit<br />
approach.<br />
The second innovation <strong>of</strong> the present thesis, which is also based on the choice<br />
<strong>of</strong> implementing the DG method on the regular computational space, is the development<br />
<strong>of</strong> a new p-adaptive limiting procedure for shock capturing <strong>of</strong> the implemented<br />
DG discretization. The new limiting approach along with positivity preserving limiters<br />
is suitable for computations <strong>of</strong> high speed flows with strong shocks around<br />
complex geometries. The unified approach for p-adaptive limiting on mixed-type<br />
meshes is achieved by applying the limiters on the transformed canonical elements,<br />
and it is fully automated without the need <strong>of</strong> ad hoc specification <strong>of</strong> parameters as<br />
it has been done with standard limiting approaches and in the artificial dissipation<br />
method for shock capturing.<br />
Verification and validation studies have been performed, which prove the correctness<br />
<strong>of</strong> the implemented discretization method in cases where the linear elements<br />
are adequate for the tessellation <strong>of</strong> the computational domain both for subsonic and<br />
supersonic flows. At present HoAc can handle only linear elements since most grid<br />
generators do not provide meshes with curved elements.<br />
Furthermore, p-adaptive computations with the implemented DG method were<br />
performed for a number <strong>of</strong> standard test cases for shock capturing schemes to illustrate<br />
the outstanding performance <strong>of</strong> the proposed p-adaptive limiting approach.<br />
The obtained results are in excellent agreement with analytical solutions and with<br />
experimental data, proving the excellent efficiency <strong>of</strong> the developed shock capturing<br />
method for the DG discretization <strong>of</strong> the equations <strong>of</strong> gas dynamics.<br />
4
To my beloved family, to whom I owe everything.<br />
5
Nomenclature<br />
c k e(t) elemental degrees <strong>of</strong> freedom<br />
∆x e , ∆y e<br />
projections <strong>of</strong> the dual edges in the x, y directions<br />
∆x e,k , ∆y e,k<br />
grid metrics<br />
γ<br />
λ<br />
Λ<br />
Θ e<br />
J<br />
k<br />
adiabatic exponent<br />
thermal conductivity coefficient<br />
diagonal eigenvalue matrix<br />
auxiliary variable for gradient computation<br />
Jacobian matrix<br />
unit vector in an arbitrary direction<br />
L, R left and right eigenvectors<br />
n<br />
U<br />
u<br />
M kj<br />
R<br />
R e<br />
S k<br />
V k<br />
outward normal vector<br />
conservative variable vector<br />
contravariant velocity component<br />
mass matrix<br />
universal gas constant<br />
elemental residual<br />
line integral<br />
volume integral<br />
µ dynamic viscosity <strong>of</strong> the fluid<br />
6
7<br />
ω<br />
skewness angle<br />
e x xx, e x yy, . . . normalized error coefficients<br />
φ<br />
ρ<br />
τ<br />
θ<br />
shearing angle<br />
density<br />
two dimensional viscous stress tensor<br />
mesh rotation angle<br />
Ũ(x, t) solution over the computational domain<br />
Ũ e (x, t) FE interpolant<br />
Ω q st<br />
Ω e<br />
standard element configuration<br />
elemental space<br />
E(x, y) error in gradient computation<br />
b e k (x) elemental basis functions<br />
c<br />
c p<br />
c V<br />
d x , d y<br />
E<br />
e<br />
E x , E y<br />
local speed <strong>of</strong> sound<br />
specific heat at constant pressure<br />
specific heat at constant volume<br />
stretching factors<br />
total energy per unit volume<br />
internal energy per unit mass<br />
error in the computation <strong>of</strong> the x, y derivatives<br />
e x x, e x y, . . . error coefficients<br />
F i<br />
F v<br />
f x , f y<br />
H<br />
h<br />
inviscid flux tensor<br />
viscous flux tensor<br />
grid functions<br />
numerical flux<br />
(specific) enthalpy <strong>of</strong> the gas
8<br />
k<br />
L x , L y<br />
M<br />
Ma<br />
N<br />
p<br />
P r<br />
Boltzmann’s constant<br />
local element lengths<br />
Laplacian approximation<br />
Mach number<br />
order <strong>of</strong> approximation<br />
pressure<br />
Prandtl number<br />
Q, q mesh quality indices<br />
Q 1 , Q 2 , Q m , Q n<br />
number <strong>of</strong> quadrature points<br />
q x , q y<br />
Re<br />
s<br />
T<br />
heat fluxes<br />
Reynolds number<br />
entropy<br />
temperature<br />
u, v Cartesian velocity components<br />
u e<br />
v<br />
field value at the middle <strong>of</strong> each edge <strong>of</strong> the dual contour<br />
test or weighting function<br />
∇ h u, ∇u numerical and analytical value <strong>of</strong> the gradient<br />
S<br />
dual area
Acronyms<br />
TE Truncation Error<br />
FE Finite Element<br />
FD Finite Difference<br />
FV Finite Volume<br />
SD Spectral Difference<br />
SV Spectral Volume<br />
ENO Essentially non-oscillatory<br />
WENO Weighted essentially non-oscillatory<br />
DG Discontinuous Galerkin<br />
RK Runge-Kutta<br />
RKDG Runge-Kutta Discontinuous Galerkin<br />
TVB Total variation bounded<br />
SSP Strong stability preserving<br />
IC Initial conditions<br />
BC Boundary conditions<br />
EC Error coefficients<br />
CV Control Volume<br />
PETSc Portable Extensible Toolkit for Scientific Computing<br />
HoAc High Order Accuracy<br />
9
10<br />
GMRES Generalized Minimum Residual<br />
JFNK Jacobian Free Newton Krylov<br />
LDG Local Discontinuous Galerkin<br />
BO Baumann and Oden<br />
BR1 First Bassi-Rebay scheme<br />
BR2 Second Bassi-Rebay scheme<br />
NIPG Non-symmetric Interior Penalty Method<br />
DOF Degrees <strong>of</strong> freedom<br />
PDEs Partial differential equations<br />
ODEs Ordinary differential equations<br />
MPI Message Passage Interface<br />
GPUs Graphics Processor Units
Contents<br />
Contents 6<br />
1 Introduction 23<br />
1.1 Classical methods in CFD . . . . . . . . . . . . . . . . . . . . . . . . 27<br />
1.2 Why the DG method . . . . . . . . . . . . . . . . . . . . . . . . . . 31<br />
1.3 Thesis outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35<br />
2 Governing equations <strong>of</strong> gas dynamics 37<br />
2.1 Euler equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38<br />
2.1.1 Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38<br />
2.1.2 Polytropic ideal gas . . . . . . . . . . . . . . . . . . . . . . . . 39<br />
2.1.3 Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42<br />
2.1.4 Mathematical character <strong>of</strong> the Euler equations . . . . . . . . . 42<br />
2.2 Navier-Stokes equations . . . . . . . . . . . . . . . . . . . . . . . . . 44<br />
2.2.1 Mathematical character <strong>of</strong> the Navier-Stokes equations . . . . 45<br />
2.3 Non-dimensional form <strong>of</strong> the Navier-Stokes equations . . . . . . . . . 46<br />
2.4 Integral form <strong>of</strong> the compressible flow equations . . . . . . . . . . . . 48<br />
3 Mesh quality measures for the FV method 49<br />
3.1 Gradient computation in the FV method . . . . . . . . . . . . . . . . 52<br />
3.2 The general form <strong>of</strong> the truncation error for the node based FV method 53<br />
3.3 Consistency condition for the node based FV method . . . . . . . . . 56<br />
3.4 Verification <strong>of</strong> the error coefficients analytic expressions . . . . . . . . 59<br />
11
CONTENTS 12<br />
3.5 Elementary Types <strong>of</strong> Mesh Distortion . . . . . . . . . . . . . . . . . . 60<br />
3.6 Stretching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62<br />
3.6.1 Skewness and Shearing . . . . . . . . . . . . . . . . . . . . . . 63<br />
3.6.2 Mesh rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . 64<br />
3.6.3 Mixed-type element mesh interfaces . . . . . . . . . . . . . . . 65<br />
3.7 Direct relation between truncation error and mesh distortion . . . . . 66<br />
3.8 Definition <strong>of</strong> an appropriate index <strong>of</strong> mesh distortion . . . . . . . . . 68<br />
3.8.1 Normalized error coefficients . . . . . . . . . . . . . . . . . . . 68<br />
3.8.2 Mesh quality index . . . . . . . . . . . . . . . . . . . . . . . . 69<br />
3.8.3 Analytic expressions <strong>of</strong> the quality index Q for each type <strong>of</strong><br />
elementary mesh distortion . . . . . . . . . . . . . . . . . . . . 72<br />
3.8.4 Calibration <strong>of</strong> the mesh quality index Q . . . . . . . . . . . . 73<br />
3.9 Application <strong>of</strong> the mesh quality indices . . . . . . . . . . . . . . . . . 74<br />
3.9.1 Channel grids . . . . . . . . . . . . . . . . . . . . . . . . . . . 75<br />
3.9.2 Airfoil grids . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76<br />
3.9.3 Cylinder grids . . . . . . . . . . . . . . . . . . . . . . . . . . . 76<br />
4 Discontinuous Galerkin discretization 87<br />
4.1 Spatial Discontinuous Galerkin discretization . . . . . . . . . . . . . . 88<br />
4.1.1 Discontinuous weak formulation for the Euler equations . . . . 89<br />
4.1.2 The Numerical flux . . . . . . . . . . . . . . . . . . . . . . . . 92<br />
4.1.3 Basis functions and standard elemental configuration . . . . . 95<br />
4.1.4 Elemental operations . . . . . . . . . . . . . . . . . . . . . . . 100<br />
4.1.5 Selection <strong>of</strong> Gauss type numerical integration . . . . . . . . . 102<br />
4.1.6 Mapping between physical and computational space . . . . . . 103<br />
4.2 DG discretization for the Navier-Stokes Equations . . . . . . . . . . . 105<br />
4.2.1 Computation <strong>of</strong> the viscous terms in the DG method . . . . . 105<br />
4.2.2 The LDG method . . . . . . . . . . . . . . . . . . . . . . . . . 107<br />
4.3 The treatment <strong>of</strong> initial and boundary conditions . . . . . . . . . . . 110<br />
4.4 Wall boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . 111
CONTENTS 13<br />
4.5 Far field boundary conditions . . . . . . . . . . . . . . . . . . . . . . 113<br />
4.6 Application <strong>of</strong> Initial Conditions . . . . . . . . . . . . . . . . . . . . . 115<br />
5 Time discretization 117<br />
5.1 Runge-Kutta methods . . . . . . . . . . . . . . . . . . . . . . . . . . 118<br />
5.2 Implicit time marching . . . . . . . . . . . . . . . . . . . . . . . . . . 121<br />
5.3 Newton’s method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122<br />
5.4 Krylov subspace methods . . . . . . . . . . . . . . . . . . . . . . . . 123<br />
5.5 Jacobian-free Newton-Krylov Method . . . . . . . . . . . . . . . . . . 124<br />
5.6 Preconditioning <strong>of</strong> the JFNK method . . . . . . . . . . . . . . . . . . 126<br />
6 Unified limiting 128<br />
6.0.1 Limiting for quadrilateral elements . . . . . . . . . . . . . . . 131<br />
6.0.2 Limiting for triangular elements . . . . . . . . . . . . . . . . . 134<br />
6.1 Positivity preserving limiters for the DG method . . . . . . . . . . . . 136<br />
7 Numerical results 139<br />
7.1 Solution output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140<br />
7.2 Code Verification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141<br />
7.2.1 Convergence study for the Euler equations . . . . . . . . . . . 141<br />
7.2.2 Convergence study for the Navier-Stokes equations . . . . . . 143<br />
7.2.3 Inviscid flow over a cylinder . . . . . . . . . . . . . . . . . . . 144<br />
7.2.4 Standard Sod’s shock tube problem . . . . . . . . . . . . . . . 147<br />
7.2.5 Extreme Sod’s Riemann problem . . . . . . . . . . . . . . . . 149<br />
7.3 Supersonic flow over a cylinder . . . . . . . . . . . . . . . . . . . . . 151<br />
7.4 Double Mach Reflection <strong>of</strong> a strong shock . . . . . . . . . . . . . . . 157<br />
7.5 Inviscid flow at Mach number <strong>of</strong> 3 in a tunnel with a step . . . . . . . 163<br />
7.6 Diffraction <strong>of</strong> a strong shock over a backward facing step . . . . . . . 168<br />
7.7 Shardin’s problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178<br />
7.7.1 Strong Mach 5 shock impingement on a triangle . . . . . . . . 185<br />
7.8 Flat plate boundary layer . . . . . . . . . . . . . . . . . . . . . . . . 189
CONTENTS 14<br />
7.9 Unsteady viscous flow over tandem airfoils . . . . . . . . . . . . . . . 191<br />
8 Conclusions and Future Work 197
List <strong>of</strong> Figures<br />
2.1 Control Volume (CV) Ω. . . . . . . . . . . . . . . . . . . . . . . . . . 48<br />
3.1 Median dual surface for evaluating first-order derivatives at grid point<br />
0 via the FV method. . . . . . . . . . . . . . . . . . . . . . . . . . . . 52<br />
3.2 Mixed-type element channel mesh. . . . . . . . . . . . . . . . . . . . 60<br />
3.3 Distribution <strong>of</strong> the analytic (✷) and the numerical (✸) TE for the<br />
field u(x, y) = xy 2 on the mesh <strong>of</strong> Fig. 3.2 using the median dual. . . 61<br />
3.4 Difference between the analytic and the numerical TE for the field<br />
u(x, y) = xy 2 on the mesh <strong>of</strong> Fig. 3.2 using the centroid dual. . . . . 62<br />
3.5 Stretching for (a) structured and (b) unstructured meshes defined to<br />
facilitate comparison <strong>of</strong> accuracy degradation on them. . . . . . . . . 63<br />
3.6 Skewed mesh, depicting the deviation angle ω from the 180 ◦ angle<br />
between one <strong>of</strong> the two pairs <strong>of</strong> edges sharing point 0 (ω ∈ [0 ◦ , 90 ◦ ]).<br />
The lengths <strong>of</strong> the edges are equal. . . . . . . . . . . . . . . . . . . . 64<br />
3.7 Sheared mesh, depicting displacement <strong>of</strong> the edges causing deviation<br />
from orthogonality expressed by the angle φ (φ ∈ [0 ◦ , 90 ◦ ]). The<br />
lengths <strong>of</strong> the edges are equal. . . . . . . . . . . . . . . . . . . . . . . 64<br />
3.8 Rotated structured mesh depicting non alignment <strong>of</strong> the edges with<br />
the axes <strong>of</strong> the global system (∆x ≠ ∆y and θ ∈ [0 ◦ , 45 ◦ ]). . . . . . . 65<br />
3.9 Common mixed-type element mesh interfaces. . . . . . . . . . . . . . 65<br />
3.10 High aspect ratio quadrilaterals typical <strong>of</strong> a structured mesh in a<br />
boundary layer region. . . . . . . . . . . . . . . . . . . . . . . . . . . 71<br />
3.11 Mesh quality index Q x for the stretched unstructured mesh (◦) and<br />
the stretched structured mesh (□) vs. displacement factor d x (d y = 0). 73<br />
3.12 Mesh quality index Q x and its corresponding normalized error coefficients<br />
vs. the skewness angle ω. . . . . . . . . . . . . . . . . . . . . . 74<br />
15
LIST OF FIGURES 16<br />
3.13 Mesh quality index Q y for the mixed-type element mesh interface <strong>of</strong><br />
Fig. 3.9(a) vs. angle ψ 2 . . . . . . . . . . . . . . . . . . . . . . . . . . 75<br />
3.14 Locally stretched structured channel mesh: (a) mesh geometry, (b)<br />
index q and (c) index Q. . . . . . . . . . . . . . . . . . . . . . . . . . 78<br />
3.15 Locally stretched unstructured channel mesh: (a) mesh geometry, (b)<br />
index q and (c) index Q. . . . . . . . . . . . . . . . . . . . . . . . . . 79<br />
3.16 Mixed-type element channel mesh: (a) mesh geometry, (b) index q<br />
and (c) index Q. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80<br />
3.17 O–type structured mesh around a NACA 0012 airfoil: (a) mesh geometry,<br />
(b) index q and (c) index Q. . . . . . . . . . . . . . . . . . . 81<br />
3.18 Unstructured mesh around a NACA 0012 airfoil: (a) mesh geometry,<br />
(b) index q and (c) index Q. . . . . . . . . . . . . . . . . . . . . . . . 82<br />
3.19 Mixed-type element mesh around a RAE 2822 airfoil: (a) mesh geometry,<br />
(b) index q and (c) index Q. . . . . . . . . . . . . . . . . . . 83<br />
3.20 Structured mesh around a cylinder: (a) mesh geometry, (b) index q<br />
and (c) index Q. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84<br />
3.21 Unstructured mesh around a cylinder: (a) mesh geometry, (b) index<br />
q and (c) index Q. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85<br />
3.22 Mixed-type element mesh around a cylinder: (a) mesh geometry, (b)<br />
index q and (c) index Q. . . . . . . . . . . . . . . . . . . . . . . . . . 86<br />
4.1 Local reconstruction <strong>of</strong> the field at two adjacent elements. . . . . . . 90<br />
4.2 Transformation <strong>of</strong> the physical quadrilateral to the standard quadrilateral<br />
element configuration (η 1 , η 2 ∈ [−1, 1]). . . . . . . . . . . . . . 96<br />
4.3 Transformation <strong>of</strong> the physical domain triangle to the standard triangular<br />
element (ξ 1 , ξ 2 ∈ [−1, 1] with ξ 1 + ξ 2 ≤ 1) and standard<br />
quadrilateral element configuration (η 1 , η 2 ∈ [−1, 1]). . . . . . . . . . 97<br />
4.4 Basis functions over the standard square element: (a) (p, q) = (0, 0),<br />
(b) (p, q) = (1, 0), (c) (p, q) = (2, 0), (d) (p, q) = (0, 1), (e) (p, q) =<br />
(1, 1), (f) (p, q) = (2, 1), (g) (p, q) = (0, 2), (h) (p, q) = (1, 2), (i)<br />
(p, q) = (2, 2). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98<br />
4.5 Basis functions over the standard triangular region: (a) (p, q) = (0, 0),<br />
(b) (p, q) = (1, 0), (c) (p, q) = (2, 0), (d) (p, q) = (0, 1), (e) (p, q) =<br />
(1, 1), (f) (p, q) = (0, 2). . . . . . . . . . . . . . . . . . . . . . . . . . . 99<br />
6.1 Mixed-element mesh patch for estimating parameter M. . . . . . . . 130
LIST OF FIGURES 17<br />
7.1 Computational mesh (blue lines) and visualization mesh (black lines)<br />
for a P 4 expansion <strong>of</strong> the solution over triangular and quadrilateral<br />
elements. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141<br />
7.2 Convergence rate for the DG discretization <strong>of</strong> the Euler equations. . . 142<br />
7.3 Convergence rate for the DG discretization <strong>of</strong> the Navier-Stokes equations<br />
with the LDG method. . . . . . . . . . . . . . . . . . . . . . . . 144<br />
7.4 Mach contours for the inviscid flow over a cylinder using discontinuous<br />
output <strong>of</strong> the solution and the curvature based boundary conditions<br />
in [92]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145<br />
7.5 Comparison between the numerically evaluated pressure coefficient<br />
and the analytical solution for the inviscid flow around a cylinder. . . 146<br />
7.6 Convergence history for the residual in the density field for the inviscid<br />
flow over a cylinder. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147<br />
7.7 Meshes for the two dimensional Sod’s shock tube problem. . . . . . . 148<br />
7.8 Limited elements in time for the Sod’s shock tube problem using the<br />
quadrilateral mesh <strong>of</strong> Fig. 7.7a. . . . . . . . . . . . . . . . . . . . . . 148<br />
7.9 Comparison <strong>of</strong> the exact density and entropy variation at t = 0.25<br />
with the two-dimensional numerical solution for the Sod’s shock tube<br />
problem in the triangular channel mesh <strong>of</strong> Fig. 7.7b. . . . . . . . . . 149<br />
7.10 Density plot for the Sod’s problem with a pressure ratio <strong>of</strong> 100000. . 150<br />
7.11 Pressure plot for the Sod’s problem with a pressure ratio <strong>of</strong> 100000. . 150<br />
7.12 Entropy plot for the Sod’s problem with a pressure ratio <strong>of</strong> 100000. . 150<br />
7.13 Quadrilateral mesh for supersonic flow at Mach 2 around a cylinder. . 151<br />
7.14 Triangular mesh for supersonic flow at Mach 2 around a cylinder. . . 152<br />
7.15 Mixed type mesh for supersonic flow at Mach 2 around a cylinder. . . 152<br />
7.16 Pressure contour lines using 30 equally spaced intervals for flow at<br />
Mach 2 over a cylinder for the numerical solution obtained with a P 3<br />
approximation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153<br />
7.17 Limited elements for the flow at Mach 2 over a cylinder for the numerical<br />
solution obtained with a quadrilateral mesh and a P 3 approximation;<br />
the blue elements are limited the adjacent green elements are<br />
P 1 expansions, the adjacent to green red elements are P 2 , and for the<br />
rest <strong>of</strong> the domain P 3 expansions are employed. . . . . . . . . . . . . 154
LIST OF FIGURES 18<br />
7.18 Limited elements for the flow at Mach 2 over a cylinder for the numerical<br />
solution obtained with a P 2 approximation; the blue elements<br />
are limited the adjacent red elements are P 1 expansions and for the<br />
rest <strong>of</strong> the domain P 2 expansions are employed. . . . . . . . . . . . . 155<br />
7.19 Comparison <strong>of</strong> the pressure distribution along the stagnation line obtained<br />
from P 2 and P 3 approximations <strong>of</strong> the numerical solutions<br />
with different meshes. . . . . . . . . . . . . . . . . . . . . . . . . . . . 156<br />
7.20 Variation <strong>of</strong> the maximum value <strong>of</strong> parameter M for each characteristic<br />
field obtained during convergence to steady state <strong>of</strong> the numerical<br />
solution for the flow at Mach 2 over a cylinder with a quadrilateral<br />
mesh and P 2 solution expansion. . . . . . . . . . . . . . . . . . . . . 157<br />
7.21 Limited elements on the coarse quadrilateral mesh for the double<br />
mach reflection problem at t = 0.2. . . . . . . . . . . . . . . . . . . . 158<br />
7.22 Density contours on the coarse quadrilateral mesh for the double mach<br />
reflection problem at t = 0.2. . . . . . . . . . . . . . . . . . . . . . . . 159<br />
7.23 Density contours for the numerical solution computed on the rectangular<br />
mesh, h = 1/240, and a P 1 approximation for the double mach<br />
reflection problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159<br />
7.24 Limited elements for the numerical solution computed on the rectangular<br />
mesh, h = 1/240, and a P 1 approximation for the double<br />
mach reflection. The blue elements are limited and for the rest <strong>of</strong> the<br />
domain a P 1 approximation <strong>of</strong> the solution is employed. . . . . . . . 160<br />
7.25 Limited elements for the numerical solution computed on the triangular<br />
mesh, h = 1/240, and a P 1 approximation for the double mach<br />
reflection. The blue elements are limited and for the rest <strong>of</strong> the domain<br />
a P 1 approximation <strong>of</strong> the solution is employed. . . . . . . . . . 161<br />
7.26 Density contours for the numerical solution computed on the triangular<br />
mesh, h = 1/240, and a P 1 approximation for the double mach<br />
reflection problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161<br />
7.27 Density contours for the numerical solution computed on the triangular<br />
mesh, h = 1/240, and a P 2 approximation for the double mach<br />
reflection problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162<br />
7.28 Variation <strong>of</strong> the maximum value <strong>of</strong> parameter M for each characteristic<br />
field obtained during the time advancement <strong>of</strong> the numerical<br />
solution for the double mach reflection problem obtained with a uniform<br />
triangular element mesh with a P 2 approximation and h = 1/240.163
LIST OF FIGURES 19<br />
7.29 Limited elements for the numerical solution obtained on a rectangular<br />
mesh and P 1 approximation and h = 1/80 for the flow at Mach<br />
number <strong>of</strong> 3 in a wind tunnel with a forward facing step at t = 2.0. . 164<br />
7.30 Density field for the numerical solution obtained on a rectangular<br />
mesh and P 1 approximation and h = 1/80 for the flow at Mach<br />
number <strong>of</strong> 3 in a wind tunnel with a forward facing step at t = 2.0. . 165<br />
7.31 Limited elements for the numerical solution obtained on a fine triangular<br />
mesh and a P 2 approximation and h = 1/100 for the flow<br />
at M = 3.0 in a wind tunnel with a forward facing step at t = 4.0.<br />
Blue elements: limited solution. Green elements P 1 expansion. Red<br />
elements P 2 expansion. . . . . . . . . . . . . . . . . . . . . . . . . . . 165<br />
7.32 Density field for the numerical solution obtained on a fine triangular<br />
mesh and P 2 approximation and h = 1/100 for the flow at Mach<br />
number <strong>of</strong> 3 in a wind tunnel with a forward facing step at t = 4.0. . 166<br />
7.33 Mixed-type element mesh for the inviscid flow at Mach number <strong>of</strong> 3<br />
in a tunnel with a step. . . . . . . . . . . . . . . . . . . . . . . . . . . 166<br />
7.34 Density field for the numerical solution obtained on a mixed-type<br />
element mesh and a P 1 approximation for the flow at Mach number<br />
<strong>of</strong> 3 in a wind tunnel with a forward facing step at t = 4.0. . . . . . . 167<br />
7.35 Limited elements at t = 0.1 using a mixed-type element mesh for the<br />
inviscid flow at Mach number <strong>of</strong> 3 in a wind tunnel with a forward<br />
facing step. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167<br />
7.36 Variation <strong>of</strong> the parameter M for the TVB limiter applied on the<br />
mixed-type element mesh for the inviscid flow at Mach number <strong>of</strong> 3<br />
in a wind tunnel with a forward facing step. . . . . . . . . . . . . . . 168<br />
7.37 Sample meshes with density contours at t = 0.8 for the diffraction<br />
<strong>of</strong> a strong shock over a backward facing step using a P 2 solution<br />
expansion. The black square shows where the computational domain<br />
was clipped for showing the underlying computational mesh. . . . . . 169<br />
7.38 Limited elements for the diffraction <strong>of</strong> strong shock over a backward<br />
facing step at t = 2.0. Blue elements: limited solution. Green elements<br />
P 1 expansion. Red elements P 2 expansion. . . . . . . . . . . . 170<br />
7.39 Density contours for the diffraction <strong>of</strong> strong shock over a backward<br />
facing step at t = 2.0 with 30 equally spaced intervals from 0.06 to<br />
7.16. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
LIST OF FIGURES 20<br />
7.40 Pressure contours for the diffraction <strong>of</strong> strong shock over a backward<br />
facing step at t = 2.0 with 30 equally spaced intervals from 0.09 to<br />
31.9. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172<br />
7.41 Plot <strong>of</strong> density and pressure along the line defined by points [0, 7] and<br />
[13, 7]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173<br />
7.42 Numerically computed pressure (left) and density (right) gradient<br />
field for the diffraction <strong>of</strong> strong shock over a backward facing step<br />
at t = 2.0, using a rectangular elements mesh. . . . . . . . . . . . . . 173<br />
7.43 Numerically computed pressure (left) and density (right) gradient<br />
field for the diffraction <strong>of</strong> strong shock over a backward facing step<br />
at t = 2.0, using a triangular elements mesh. . . . . . . . . . . . . . . 174<br />
7.44 Variation <strong>of</strong> the second derivative estimate for the diffraction <strong>of</strong> a<br />
strong shock over a backward facing step using a P 2 expansion on a<br />
rectangular elements mesh. . . . . . . . . . . . . . . . . . . . . . . . . 174<br />
7.45 Plot <strong>of</strong> density and pressure along a line defined by the points [2, 5.5]<br />
and [5, 4] at time t = 0.9. . . . . . . . . . . . . . . . . . . . . . . . . . 175<br />
7.46 Vorticity field for the diffraction <strong>of</strong> a Mach 5 shock over a backward<br />
facing step. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177<br />
7.47 Density contours for Schardin’s problem with 30 equal spaced intervals<br />
from 0.6 to 2.4 using a P 2 solution expansion . . . . . . . . . . . 179<br />
7.48 Pressure contours for Schardin’s problem with 30 equal spaced intervals<br />
from 0.4 to 2.14 using a P 2 expansion. . . . . . . . . . . . . . . . 180<br />
7.49 Numerical Schlieren for Schardin’s problem using a P 2 solution expansion.<br />
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181<br />
7.50 Shadowgraph for the Schardin’s problem at t = 128µs. . . . . . . . . 182<br />
7.51 Numerical Schlieren for the Shardin’s problem on the finest mesh and<br />
using P 2 solution expansion. . . . . . . . . . . . . . . . . . . . . . . . 183<br />
7.52 Numerical Schlieren for Schardin’s problem on the refined triangular<br />
mesh with mesh element size in the vortex region equal to h 1 = 0.001. 184<br />
7.53 Numerical Schlieren for Schardin’s problem on the refined triangular<br />
mesh with mesh element size in the vortex region equal to h 2 = 0.0005.185<br />
7.54 Density contours using a P 2 expansion for the flow <strong>of</strong> a strong shock<br />
moving at Mach number <strong>of</strong> 5 around a triangle with 30 equal spaced<br />
intervals from 0.28 to 13.28. Results are depicted at non-dimensional<br />
time. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
LIST OF FIGURES 21<br />
7.55 Limited elements for the flow <strong>of</strong> a strong shock moving at Mach number<br />
<strong>of</strong> 5 around a triangle along with the gradient <strong>of</strong> pressure over 30<br />
equally spaced intervals from 0 to 1000. Results are depicted at nondimensional<br />
time. Blue elements: limited solution. Green elements<br />
P 1 expansion. Red elements P 2 expansion. . . . . . . . . . . . . . . . 188<br />
7.56 Comparison <strong>of</strong> the U-velocity pr<strong>of</strong>ile with the Blasius solution for the<br />
flow over a flat plate. . . . . . . . . . . . . . . . . . . . . . . . . . . . 190<br />
7.57 Comparison <strong>of</strong> the V-velocity pr<strong>of</strong>ile with the Blasius solution for the<br />
flow over a flat plate. . . . . . . . . . . . . . . . . . . . . . . . . . . . 190<br />
7.58 Velocity field vectors and vorticity field for the flat plate boundary<br />
layer solution using a P 3 solution expansion. . . . . . . . . . . . . . . 191<br />
7.59 Visualization mesh for a P 4 computation for the two tandem NACA<br />
0012 airfoils. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192<br />
7.60 Lift and drag coefficients for the forward airfoil. . . . . . . . . . . . . 194<br />
7.61 Lift and drag coefficients for the aft airfoil. . . . . . . . . . . . . . . . 195<br />
7.62 Velocity contours for the flow over NACA0012 tandem airfoils. . . . . 196<br />
7.63 Vorticity contours for the flow over NACA0012 tandem airfoils. . . . 196
List <strong>of</strong> Tables<br />
2.1 Reference quantities used for the non-dimensionalization <strong>of</strong> the Navier-<br />
Stokes equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46<br />
3.1 Expressions for the error coefficients for each type <strong>of</strong> mesh distortion. 66<br />
3.2 Expressions for the error coefficients for the mixed-type element mesh<br />
interfaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67<br />
3.3 Central interface point displacements for improving the accuracy <strong>of</strong><br />
the u y derivative evaluation for the mixed-type element mesh interfaces. 67<br />
3.4 Characteristic local lengths for the mixed-type element mesh interfaces. 70<br />
3.5 Mesh quality index Q expressions for the elementary types <strong>of</strong> mesh<br />
distortion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72<br />
4.1 Basis functions for a third-order approximation over the standard<br />
quadrilateral region. . . . . . . . . . . . . . . . . . . . . . . . . . . . 99<br />
4.2 Basis functions for a third-order approximation over the standard<br />
triangular region. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99<br />
4.3 Types <strong>of</strong> Gauss numerical integration. . . . . . . . . . . . . . . . . . . 102<br />
4.4 DG methods and their interiors fluxes taken for Arnold et. al [6]. . . 108<br />
5.1 Butcher’s table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119<br />
7.1 Results for the sensitivity study regarding the far field boundary distance.<br />
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193<br />
22
Chapter 1<br />
Introduction<br />
The present work focuses on the numerical solution <strong>of</strong> the compressible flow equations<br />
for flows with strong moving shocks and in the a priori evaluation <strong>of</strong> the<br />
geometrical quality <strong>of</strong> the mesh used for the discretization <strong>of</strong> the computational domain.<br />
The high-order Discontinuous Galerkin (DG) method is used for discretizing<br />
the system <strong>of</strong> the compressible Navier-Stokes equations in space along with highorder<br />
explicit and implicit discretization schemes for unsteady problems, in order to<br />
achieve high spatial and temporal resolution <strong>of</strong> the complex flow features appearing<br />
in flows with strong moving shocks.<br />
The numerical solution <strong>of</strong> the equations <strong>of</strong> compressible flow is an extremely<br />
powerful technique for the analysis <strong>of</strong> flows over a wide range <strong>of</strong> speeds around complex<br />
geometries. Until the mid 1980s, two methodologies prevailed the study <strong>of</strong> fluid<br />
motion, namely the experimental and the theoretical methodology. The former uses<br />
and develops techniques for the understating <strong>of</strong> the physical phenomena associated<br />
with a flow field. However, experiments are costly and time consuming and in some<br />
cases, even impossible to perform due to technical difficulties and physical limitations.<br />
An example <strong>of</strong> such cases is the study <strong>of</strong> the flow field over a re-entry vehicle.<br />
The theoretical approach attempts to solve the governing equations <strong>of</strong> fluid flow, or<br />
simplified versions <strong>of</strong> them, using mathematical methods and advanced numerical<br />
algorithms since analytical solutions exist only for very few simple cases <strong>of</strong> limited<br />
23
CHAPTER 1. INTRODUCTION 24<br />
practical interest.<br />
The advent <strong>of</strong> the digital computer and its enormous development over the<br />
last 50 years both in terms <strong>of</strong> speed and memory capabilities, lead to the creation<br />
<strong>of</strong> a third methodology as a branch <strong>of</strong> the theoretical approach for studding fluid<br />
flow, that <strong>of</strong> the numerical simulation, which is known as Computational Fluid Dynamics<br />
(CFD). The numerical solution <strong>of</strong> the governing equations does not have the<br />
limitations <strong>of</strong> the experimental and theoretical methodology. However, it innately<br />
possesses problems originating from the philosophy and practice <strong>of</strong> the numerical<br />
solution.<br />
The philosophy behind a CFD solution to a specific problem is to ”break”<br />
its mathematical continuum representation into a discrete form and numerically<br />
compute a solution. However, this approach possesses an error and the factors that<br />
contribute to this are the following:<br />
ˆ Discretization <strong>of</strong> the system <strong>of</strong> equations <strong>of</strong> fluid flow.<br />
ˆ Discrete representation <strong>of</strong> the geometry <strong>of</strong> the body around which the flow<br />
field is simulated.<br />
ˆ Round-<strong>of</strong>f errors in computer arithmetic and floating point representation <strong>of</strong><br />
numbers.<br />
ˆ Initial conditions and boundary conditions.<br />
ˆ Iterative solution procedures, which are stopped after some specified tolerances<br />
have been met.<br />
ˆ Modeling errors.<br />
Iterative solution procedures are applied when an implicit time discretization<br />
<strong>of</strong> the nonlinear system describing the motion <strong>of</strong> fluids is performed, but they are<br />
terminated after predefined tolerances are met leading to solutions which are close to<br />
the true solution <strong>of</strong> the system. The initial and especially the boundary conditions<br />
have a pr<strong>of</strong>ound impact on the numerical results and their choice and implementation
CHAPTER 1. INTRODUCTION 25<br />
may be the key for the success <strong>of</strong> a numerical solution for a given flow problem. The<br />
limitations imposed by the computer technology in arithmetic operations, requires<br />
special attention on how the mathematical operations are performed in a computer<br />
code. The discrete representation <strong>of</strong> the geometry in a computer model means that<br />
practically one is solving a problem over a polyhedral and not on a curved continuum<br />
geometry as it is in reality. Simplifications made for the mathematical form <strong>of</strong> the<br />
governing equations in order to be practical to solve them for a setting, simply means<br />
that an accurate physical model is not used. Taking all these factors along with the<br />
discretization methodology employed for ”breaking” the continuum approach, it is<br />
evident that their impact on the discretization process has to be accounted for in<br />
order to a priori minimize, as much as possible, their effect on the final solution.<br />
Emphasis on achieving predictive simulations as demonstrated in [94, 109] has<br />
caused research in numerical simulation to focus on the solution algorithms and the<br />
discretization error. The present research work aims at providing tools that increase<br />
the fidelity <strong>of</strong> the discretization <strong>of</strong> the equations <strong>of</strong> gas dynamics. The Finite Volume<br />
(FV) method is currently wide spread in CFD, however its successful application<br />
strongly depends on the quality <strong>of</strong> the mesh used for discretizing the physical space.<br />
For those reasons, a mesh induced error analysis based on a novel approach <strong>of</strong><br />
the present thesis has been conducted in order to equip the CFD community with<br />
better and more accurate metrics for the appropriateness <strong>of</strong> a mesh for a CFD<br />
solution using the FV method. The present work assesses grid quality by computing<br />
appropriate metrics <strong>of</strong> the elements. However, there are two distinguishing aspects<br />
<strong>of</strong> it: (i) the metrics indicating quality are derived directly from related analytic<br />
forms <strong>of</strong> the truncation error and (ii) can give analytic expressions for reducing<br />
the discretization error via re-shaping <strong>of</strong> the elements. The primary issue with TE<br />
analysis is the complexity <strong>of</strong> the related expressions, especially for multi-dimensions<br />
and for general mixed-type element mesh topologies. The present work addresses<br />
this complexity barrier via employment <strong>of</strong> symbolic mathematics s<strong>of</strong>tware [101].<br />
The present work analyzes a generally-distorted mesh (structured, unstructured<br />
and mixed-type) into elementary distortions (stretching, skewness, shearing,<br />
rotation), as well as three common types <strong>of</strong> interfaces, and those were directly related
CHAPTER 1. INTRODUCTION 26<br />
to the TE. The derived analytic expressions are relatively simple and amenable to<br />
future work on improving the mesh. The derived quality indices can be also applied<br />
for assessing the mesh quality for a higher order discretization method as demonstrated<br />
in [99].<br />
Moreover, due to the limits <strong>of</strong> the FV method in terms <strong>of</strong> resolution, especially<br />
for flows with complex features with strong moving shocks and the difficulty in generating<br />
a good structured grid around complex geometries for Finite Difference (FD)<br />
discretizations, a DG code has been developed and named HoAc, which stands<br />
for High Order Accuracy. The developed code uses PETSc [58, 59] for performing<br />
parallel computations using the domain decomposition method. It can handle<br />
mixed-type element meshes employing hierarchical modal basis functions [83] up to<br />
arbitrary order <strong>of</strong> accuracy, for the discretization <strong>of</strong> the compressible Navier-Stokes<br />
equations. Explicit and implicit time schemes are used provided by PETSc and a<br />
novel shock capturing procedure has been developed, implemented and thoroughly<br />
tested. All the computations are performed in the standard element configuration,<br />
thus eliminating the complexity and difficulties arising by direct application <strong>of</strong> the<br />
solution algorithms in the physical space.<br />
The novel shock capturing approach proposed in this work uses the basic TVB<br />
limiter proposed by Cockburn and Shu [82]. However, it overcomes the ambiguity<br />
associated with the a priori specification <strong>of</strong> a parameter that estimates the second<br />
derivative <strong>of</strong> the solution. Application <strong>of</strong> the TVB limiter for the canonical elements<br />
<strong>of</strong> the computational domain, in a dimension per dimension fashion, renders our<br />
approach essentially the same with the original TVB limiter [82] that until now<br />
was used for rectangular elements. For the canonical element <strong>of</strong> the computational<br />
domain, tensor product orthogonal polynomial bases are constructed as described in<br />
detail in [83]. However, in contrast to the original TVB limiter, the second derivative<br />
<strong>of</strong> the numerical solution is estimated for each field, and it is not set to a constant<br />
value, (as suggested in the original paper [82]). Note that in the original approach<br />
[82] <strong>of</strong>ten and for cases with strong shocks and large pressure ratios, the suggested<br />
value must be altered, while in our approach no user intervention is required. Recent<br />
applications for flows with strong discontinuities resulting from high pressure ratios
CHAPTER 1. INTRODUCTION 27<br />
[156] have demonstrated that the constant value <strong>of</strong> the second derivative, needs to<br />
be arbitrarily readjusted for each field in order to avoid divergence. Application <strong>of</strong><br />
the limiter in the transformed domain has the additional advantage that limiting for<br />
rectangular, quadrilateral, and triangular elements is essentially the same, since the<br />
inverse transformation and the collapsed coordinates system introduced in [83] is<br />
used to recover the element in the physical space. The proposed limiting approach<br />
is accurate, effective, and can be applied on distorted mixed-type meshes without<br />
readjusting parameters for a wide range <strong>of</strong> flow problems discussed in the results<br />
section. The extension <strong>of</strong> the proposed limiting approach for three dimensions is<br />
straightforward [90].<br />
In the following sections <strong>of</strong> the introduction chapter, the classical methods used<br />
in CFD are described along with comments on what extent they are affected by the<br />
underlying mesh employed for their application, and their ability to extend them in<br />
orders <strong>of</strong> accuracy higher than two. Furthermore, the selection <strong>of</strong> the discretization<br />
method employed in the current work is advocated and the overview <strong>of</strong> the thesis is<br />
presented.<br />
1.1 Classical methods in CFD<br />
Discretization techniques have been developed since the early 1900s due to the numerous<br />
efforts for developing algorithms for the numerical solution <strong>of</strong> partial differential<br />
equations (PDEs). Currently, the most popular and well established methods<br />
in CFD are the Finite Difference (FD) [97], the Finite Volume (FV) [96] and the<br />
Finite Element (FE) methods [83], and most CFD methods used in practice are at<br />
most second-order accurate in space [145, 148]. A brief description <strong>of</strong> each method<br />
follows.<br />
Finite Difference methods<br />
The most basic discretization technique for a system <strong>of</strong> PDEs is that <strong>of</strong> the FD<br />
method. A structured canonical grid, through the use <strong>of</strong> a generalized coordinate
CHAPTER 1. INTRODUCTION 28<br />
transformation is employed, where the method approximates the analytical form <strong>of</strong><br />
the derivatives appearing in the differential equations with finite differences leading<br />
to a discrete representation <strong>of</strong> the differential form, which can be solved numerically.<br />
There exists a plethora <strong>of</strong> approaches for the discrete estimation <strong>of</strong> the derivatives<br />
and some excellent text books describing these methodologies are those from Hirsch<br />
[63, 64], Anderson [4] and Tannehill et al. [132] to mention a few.<br />
The main advantages <strong>of</strong> the FD methods are that they are easy to program<br />
and efficient in terms <strong>of</strong> computational cost. Due to their efficiency and simplicity<br />
they have been used for computationally intensive problems. It is possible to<br />
construct high-order extensions <strong>of</strong> the FD methods, by adding more grid points in<br />
the computational stencil, since the accuracy <strong>of</strong> the method is determined by the<br />
estimation <strong>of</strong> the derivatives [54, 61, 95, 147]. However, in order to retain high-order<br />
accurate numerical solutions with FD discretizations it is required to have smooth<br />
canonical meshes in the physical domain.<br />
A major drawback, however, <strong>of</strong> the FD method is that it strictly requires structured<br />
grids for its application, thus limiting its use on simple geometrical configurations.<br />
It is possible to apply the method for problems accompanied with complex<br />
geometries. However, this is a difficult task, which practically turns the FD method<br />
inappropriate for computing flows around complex configurations. In principle it is<br />
possible to construct a FD scheme on unstructured grids [98], but this requires a<br />
reconstruction <strong>of</strong> a polynomial function from grid points, which is a complex problem<br />
even in the two dimensional case. Moreover, the high-order versions <strong>of</strong> the FD<br />
method require a regular, smooth grid for stability and accuracy reasons. Consequently,<br />
FD methods are mostly used for fundamental studies on simple geometries.<br />
Usually, in order to ensure stability <strong>of</strong> the scheme the order is reduced near and at<br />
the boundaries, which results in a special grid refinement close to the boundaries so<br />
that to compensate for the loss <strong>of</strong> accuracy, or use <strong>of</strong> one sided differencing schemes<br />
that must be employed in these regions.
CHAPTER 1. INTRODUCTION 29<br />
Finite Volume methods<br />
The FV method is quite popular in the CFD community. The starting point for a<br />
FV discretization <strong>of</strong> a system <strong>of</strong> partial differential equations is the integral form<br />
<strong>of</strong> the system. A tessellation <strong>of</strong> the physical space in elements is performed and in<br />
each element the governing equations are discretized. In contrast to the FD method,<br />
where derivatives are being estimated, fluxes through the element boundaries are<br />
evaluated. There is a plethora <strong>of</strong> choices <strong>of</strong> how these fluxes can be chosen. A very<br />
popular approach for hyperbolic type systems is the upwind method [64, 96] where<br />
the flux estimate is based on the structure <strong>of</strong> the local wave propagation, leading<br />
to robust solution algorithms. Another approach, which however requires an ad hoc<br />
specification <strong>of</strong> parameters for stability reasons and shock capturing, is the artificial<br />
dissipation method [69].<br />
One significant advantage <strong>of</strong> the FV method is its ability to perform well on<br />
structured, unstructured, and mixed-type element meshes in two and three dimensions.<br />
This permits the application <strong>of</strong> the method on problems involving complex<br />
geometries. Formally, the method is first order accurate, but with a reconstruction<br />
procedure [14, 96] <strong>of</strong> the solution at the element boundaries, its accuracy may be<br />
extended to higher order. However, in practice this reconstruction is quite computationally<br />
expensive and difficult to perform, even for moderate orders <strong>of</strong> accuracy<br />
(third-order for instance) [47], and most FV codes on unstructured meshes have<br />
resolution properties belonging to the second-order accurate schemes.<br />
The FV as the FD method requires the construction <strong>of</strong> regular and smooth<br />
meshes. Moreover, due to the difficulty in constructing a higher order FV scheme,<br />
use <strong>of</strong> the second-order accurate FV method for practical flow problems necessitates<br />
the construction <strong>of</strong> fine meshes especially for the solution <strong>of</strong> the Navier-Stokes<br />
equations.
CHAPTER 1. INTRODUCTION 30<br />
Finite Element methods<br />
FE methods use a tessellation <strong>of</strong> the physical space in elements where the weak<br />
form <strong>of</strong> the governing equations is solved. Specifically, the weak form is derived<br />
by multiplying the system <strong>of</strong> the PDEs with a weighting function and integrating<br />
over the elemental space. Then, a polynomial expansion <strong>of</strong> the solution is chosen<br />
and the discrete from <strong>of</strong> the system is derived. The choice <strong>of</strong> the test (also known<br />
as weighting) and the expansion functions dictates which type <strong>of</strong> FE method is<br />
being employed. Versions <strong>of</strong> FE methods are the Galerkin [21, 24, 83], also known<br />
as the Bubnov-Galerkin method, the Petrov-Galerkin [62, 65], also known as the<br />
generalized Galerkin method and the least-squares FE methods [114, 115]. Highorder<br />
versions for FE methods can be constructed by increasing the order <strong>of</strong> the<br />
polynomial used for the solution expansion. Increase <strong>of</strong> the order <strong>of</strong> accuracy in<br />
FE methods comes however at the expense <strong>of</strong> computing cost, algorithmic and<br />
programming complexity.<br />
The Bubnov-Galerkin method uses the same function space both for the test<br />
function and the solution expansion, leading to a symmetric representation <strong>of</strong> the<br />
weak formulation, and it is applicable to any type <strong>of</strong> partial differential equations.<br />
The Petrov-Galerkin version <strong>of</strong> the FE method is used for solving partial differential<br />
equations with odd terms. In this version the formulation method is the<br />
same as the normal Galerkin method. However, the test function and the solution<br />
function approximations are chosen not to be the same, hence they must be approximated<br />
separately and thus, the representation <strong>of</strong> the weak form looses symmetry.<br />
The least squares FE method originates from the idea <strong>of</strong> least squares approximation<br />
developed by Gauss. In this version <strong>of</strong> the FE method the residual <strong>of</strong> the<br />
system obtained from the solution expansion is set as the test function, leading to<br />
a symmetric form <strong>of</strong> the weak representation.<br />
FE methods may be classified into two categories: continuous and discontinuous.<br />
Continuous FE methods employ an expansion set, which is continuous over<br />
the entire domain <strong>of</strong> the physical space being tessellated into non overlapping elements.<br />
On the other hand, discontinuous FE methods perform an approximation <strong>of</strong>
CHAPTER 1. INTRODUCTION 31<br />
the solution <strong>of</strong> the system <strong>of</strong> PDEs at each element separately permitting it to be<br />
discontinuous across inter element boundaries.<br />
Higher-order versions <strong>of</strong> the FE method do not exhibit strong dependence on<br />
the mesh constructed for a field simulation. This is due to the local support <strong>of</strong> the<br />
solution from the local expansion at every element. In the present work, the DG<br />
method which belongs to the version <strong>of</strong> Bubnov-Galerkin FE methods is used and<br />
it will be presented in detail in a following chapter.<br />
1.2 Why the DG method<br />
Accurate prediction <strong>of</strong> aerodynamic and thermal loads on aerospace vehicles is a crucial<br />
stage <strong>of</strong> the design cycle. Wind tunnel investigations and theoretical predictions<br />
are used to estimate these loads. Experimental measurements <strong>of</strong> high speed flows<br />
are costly, difficult to perform, and usually unable to reach realistic flight Reynolds<br />
numbers. On the other hand, highly accurate theoretical predictions with DNS or<br />
LES are limited to simple flow configurations [2] and low Reynolds numbers. Most<br />
<strong>of</strong> these calculations are performed with structured grids that are more efficient<br />
computationally for high resolution simulations. For flows without discontinuities,<br />
high-order compact FD schemes [54, 147] can be used to provide the necessary highorder<br />
spatial accuracy at a reasonable computing cost. On the other hand, flows with<br />
strong discontinuities can be efficiently computed on structured grids using compact<br />
schemes with filters [131, 153] or the FD ENO [73, 130] and weighted ENO (WENO)<br />
[128] discretizations. The extension <strong>of</strong> ENO [1] or WENO [110] methodology to unstructured<br />
meshes is, however, not straightforward and becomes computationally<br />
intensive. Similarly, the high-order FV methods for three dimensional calculations<br />
are quite intensive in memory and computational time, and difficult to implement<br />
[47].<br />
Use <strong>of</strong> standard second-order accurate in space FV and FD methods for the<br />
prediction <strong>of</strong> aerodynamic and thermal loads over more complex configurations, as<br />
mentioned before, is currently widespread. These methods, however, require very
CHAPTER 1. INTRODUCTION 32<br />
high grid densities for the accurate computation <strong>of</strong> the complex flow features such<br />
as those generated from the interaction <strong>of</strong> shock waves with boundary layers even<br />
on relatively simple geometrical configurations. Use <strong>of</strong> unstructured meshes could<br />
reduce computing requirements for accurate computation <strong>of</strong> complex shock interactions,<br />
because they allow selective refinement in critical flow regions. Recently, it<br />
was demonstrated [149], that for these flows only the combination <strong>of</strong> selective grid<br />
refinement and high-order accuracy can achieve the desired level <strong>of</strong> accuracy at a<br />
reduced computing cost for complex shock wave interactions. In addition, Shi et<br />
al. [128] clearly demonstrated that resolution <strong>of</strong> complex flow features, generated<br />
by the impingement <strong>of</strong> a strong shock on a compression ramp, could be equally well<br />
achieved by grid refinement (doubling the resolution along each coordinate direction)<br />
or with the increase <strong>of</strong> the order <strong>of</strong> accuracy. The results <strong>of</strong> Shi et al. [128],<br />
obtained with the WENO scheme, indicate that increase <strong>of</strong> the order <strong>of</strong> accuracy<br />
<strong>of</strong> the spatial discretization scheme by an order <strong>of</strong> two is equivalent to doubling the<br />
resolution <strong>of</strong> the mesh in every direction, as far as the resolution <strong>of</strong> smooth but<br />
complex flow features is concerned. Similar conclusions concerning the enhancement<br />
<strong>of</strong> resolution were drawn for high-order discretizations with the DG method<br />
[53, 137–139] and the closely related Spectral Volume (SV) method [5, 149].<br />
The main advantages <strong>of</strong> the DG [28, 41, 44] method compared to other available<br />
high-order methods is that, as FE type method, it has a compact stencil, it is<br />
applicable to arbitrary unstructured meshes, and it can handle hanging nodes and<br />
curved geometries in a natural manner. Therefore, this method is suitable for grid<br />
adaptation, and appropriate for the computation <strong>of</strong> flows with strong discontinuities.<br />
These significant advantages <strong>of</strong> the DG method over other high-order methods<br />
come unfortunately at the expense <strong>of</strong> implementation complexity, higher computing<br />
cost, and reduction <strong>of</strong> the allowable, by stability limitations, time step for marching<br />
in time. Computation in regions <strong>of</strong> strong shocks and other discontinuities with<br />
the DG method could be stabilized with the application <strong>of</strong> classical total variation<br />
bounded (TVB) limiters [28, 44]. In regions where the TVB limiter is activated the<br />
formal order <strong>of</strong> accuracy <strong>of</strong> the method drops. More recently, alternative approaches<br />
for the computation <strong>of</strong> flows containing discontinuities with high-order accurate DG
CHAPTER 1. INTRODUCTION 33<br />
space discretizations were proposed by applying filters [100, 138], or by adding numerical<br />
dissipation [105, 142]. However, use <strong>of</strong> these approaches increases the overall<br />
computational cost <strong>of</strong> the DG method and does not guarantee that high accuracy<br />
is retained at the shock, where the computation essentially drops to lower order<br />
<strong>of</strong> accuracy, so that preservation <strong>of</strong> monotonicity is ensured. Filters and artificial<br />
dissipation in addition require ad hoc specification <strong>of</strong> tuning parameters.<br />
High resolution <strong>of</strong> strong shocks can also be achieved with the selective increase<br />
<strong>of</strong> grid resolution (h-refinement) and use <strong>of</strong> lower order spatial discretizations<br />
with TVB limiters at the regions <strong>of</strong> discontinuities. This approach appears quite<br />
appropriate for the resolution <strong>of</strong> shocks with DG discretizations and was applied<br />
successfully for hexahedral meshes in [142]. In addition hp-refinement schemes [103]<br />
have appeared in the literature.<br />
The DG method has become popular in recent years due to its high-order <strong>of</strong><br />
accuracy, and the ability to simulate flows around complex geometries with the ease<br />
<strong>of</strong> increasing the order <strong>of</strong> approximation, while keeping a compact stencil. The DG<br />
method has higher computational cost compared with standard second-order FV<br />
methods. However, the compact stencil <strong>of</strong> the DG method provides an essential advantage<br />
for achieving high-order accuracy and parallelization <strong>of</strong> the algorithms, both<br />
with the use <strong>of</strong> domain decomposition through message passage interface (MPI), and<br />
graphics processor units (GPUs) [50, 81, 87]. In this work, parallelization <strong>of</strong> the DG<br />
method through domain decomposition and use <strong>of</strong> MPI was performed.<br />
Accurate predictions <strong>of</strong> skin friction and thermal loads <strong>of</strong> high speed complex<br />
flows in both simple and nontrivial geometries, require high resolution computations.<br />
High-order DG discretizations possess features that make them very attractive for<br />
computation <strong>of</strong> complex flows with strong shocks. A key ingredient that would<br />
make the DG method more suitable for these computations, is the straightforward<br />
application <strong>of</strong> p-adaptive procedures that ensure robust and accurate capturing <strong>of</strong><br />
discontinuities along with high-order <strong>of</strong> accuracy in the smooth parts <strong>of</strong> the flow. In<br />
this spirit, low order expansions and high mesh density could be used in regions <strong>of</strong><br />
discontinuities. On the other hand, resolution <strong>of</strong> smooth but complex flow features,<br />
such as vortices and shear layers that usually evolve in time, could be obtained
CHAPTER 1. INTRODUCTION 34<br />
with relatively uniform mesh density using higher order expansions. Flow features<br />
where p-adaptivity is required could either be detected in the solution as regions<br />
<strong>of</strong> high gradients, such as density and pressure variations in vortical regions, or by<br />
a-posteriori error estimates <strong>of</strong> the FE solution [10, 108, 144]. However, these aspects<br />
<strong>of</strong> the DG method have not been addressed satisfactorily yet, and utilization <strong>of</strong> p-<br />
adaptivity is not currently widespread. Therefore, the ability <strong>of</strong> the DG method<br />
to compute flows with discontinuities in a unified manner for arbitrary mixed-type<br />
meshes requires further development. Moreover, limiting in three dimensions for<br />
arbitrary unstructured meshes has not been demonstrated. The DG method reduces<br />
the solution accuracy near discontinuities. This affects the resolution properties <strong>of</strong><br />
the method and much <strong>of</strong> its potential could be lost when limiting is extended over<br />
large regions. Therefore, narrowing <strong>of</strong> the regions requiring limiting and use <strong>of</strong> p-<br />
adaptive procedures is expected to enhance the potential and broaden utilization <strong>of</strong><br />
the DG method.<br />
The slope limiters for the computation <strong>of</strong> flows with discontinuities used so<br />
far with the DG method have been demonstrated mainly for rectangular elements.<br />
A total variation bounded (TVB) limiting procedure for the DG method was constructed<br />
for solving scalar, one-dimensional, hyperbolic conservation laws by Shu<br />
[129] and Cockburn and Shu [82]. The extension <strong>of</strong> the method to one-dimensional<br />
systems has been applied with satisfactory results [40]. The basic idea behind the<br />
TVB limiter was to use a slope limiter to preserve the monotonicity <strong>of</strong> the solution<br />
averages. The same nonlinear TVB limiting was also applied for two dimensional<br />
problems [27, 32, 41], for second (P 1 polynomial expansion) and third-order (P 2 )<br />
accurate computations with rectangular meshes. One ambiguity <strong>of</strong> this limiting<br />
procedure is that it relies on a priori estimates <strong>of</strong> the numerical solution’s second<br />
derivative. Moreover, the application for triangular meshes [40, 82] is more elaborate<br />
and straightforward application <strong>of</strong> the method for mixed-type meshes has not<br />
been addressed yet.<br />
In an effort to extend the TVB limiter for higher order accurate solutions, Qiu<br />
and Shu [120] applied the same TVB limiter for the one-dimensional Euler equations,<br />
by reconstructing the solution at every element with a Hermite WENO scheme using
CHAPTER 1. INTRODUCTION 35<br />
the average solution at every element. Applying this idea they achieved a third-order<br />
monotone solution. Similar approaches were followed by Zhu and Qiu [157, 158] and<br />
by Luo et al. [100] for multidimensional calculations. These limiting approaches were<br />
demonstrated for meshes with rectangular elements and for triangular meshes using<br />
the approach suggested by Cockburn [40], which is not straight forward to extend<br />
in three dimensions. In addition, the approach <strong>of</strong> Zhu [157, 158] implies increase <strong>of</strong><br />
the stencil width, and one <strong>of</strong> the main advantages <strong>of</strong> the DG method is lost. Biswas<br />
et al. [29] proposed an alternative implementation <strong>of</strong> limiting, where the solution<br />
moments are altered in order to preserve monotonicity. Krivodonova, [91] applied<br />
this concept with promising results, in computations for flows with strong shocks<br />
and with order <strong>of</strong> accuracy higher than two for rectangular elements. On the other<br />
hand Jaffre [68] and Van der Ven [143] added artificial dissipation to stabilize the<br />
DG method.<br />
From the above it is concluded in the present thesis that the DG method as implemented<br />
and enhanced with the newly developed limiting approach is suitable for<br />
the computation <strong>of</strong> high speed complex flow fields around complex geometries. The<br />
present work has made a significant improvement to the application <strong>of</strong> TVB limiter<br />
leading to a p-adaptive shock capturing scheme with outstanding performance.<br />
1.3 Thesis outline<br />
Chapter 2 includes a brief presentation <strong>of</strong> the equations <strong>of</strong> gas dynamics, along with<br />
some basic thermodynamic variables. The mathematical character <strong>of</strong> the system<br />
<strong>of</strong> Euler and Navier-Stokes equations is briefly presented, which dictates the type<br />
<strong>of</strong> boundary conditions to be used in the far field boundaries <strong>of</strong> the computational<br />
domain and the non-dimensional form <strong>of</strong> the equations, which is used in all the<br />
computations in this work is shown. The mesh induced error study is performed<br />
in Chapter 3 and the derived novel mesh quality measures for mixed-type element<br />
meshes are presented. These measures are subsequently used in the construction<br />
<strong>of</strong> the new limiter. Several applications <strong>of</strong> the quality measures are also presented,<br />
which confirm their ability to inspect the quality <strong>of</strong> a mesh for a FV discretiza-
CHAPTER 1. INTRODUCTION 36<br />
tion. The DG discretization <strong>of</strong> the Euler and Navier-Stokes equations is presented<br />
in Chapter 4, where the expansion bases used in the present work are presented in<br />
detail. Also, the numerical operations are described for performing all the computations<br />
in the computational space along with the transformation <strong>of</strong> the elemental<br />
physical configuration to its standard representation in the computational space.<br />
Furthermore, the Local Discontinuous Galerkin (LDG) methodology is presented<br />
for discretizing the system <strong>of</strong> Navier-Stokes equations in the DG framework. Chapter<br />
5 deals with the time marching <strong>of</strong> the DG discretization <strong>of</strong> the compressible<br />
equations <strong>of</strong> gas dynamics and the limiting procedure developed for shock capturing<br />
is presented in Chapter 6. In Chapter 7 numerical results for problems with strong<br />
moving shocks are presented along with verification cases that confirm the correctness<br />
<strong>of</strong> the implemented DG discretization and the limiting procedure in HoAc.<br />
Finally, in Chapter 8 the conclusions <strong>of</strong> the current thesis are presented.
Chapter 2<br />
Governing equations <strong>of</strong> gas<br />
dynamics<br />
The governing equations <strong>of</strong> inviscid and viscous compressible flow, along with some<br />
basic thermodynamic relations, will be presented. They are expressed in a Eulerian<br />
reference frame and they are based on the following conservation laws:<br />
ˆ Conservation <strong>of</strong> Mass<br />
ˆ Conservation <strong>of</strong> Momentum<br />
ˆ Conservation <strong>of</strong> Energy<br />
and mathematically they are derived by using Reynolds transport theorem. Their<br />
mathematical character will be stated, which defines the specification and application<br />
<strong>of</strong> the boundary conditions to be used for a numerical solution. The presented<br />
equations will be given first in differential form followed by their non-dimensionalization<br />
and concluding with their integral form.<br />
37
CHAPTER 2. GOVERNING EQUATIONS OF GAS DYNAMICS 38<br />
2.1 Euler equations<br />
The motion <strong>of</strong> a compressible fluid in two dimensions, without any viscous and<br />
thermal conduction effects, is described by the system <strong>of</strong> Euler equations. The<br />
differential form <strong>of</strong> this system in Cartesian coordinates is:<br />
L(U) = ∂U<br />
∂t + ∇ · F i(U) = 0, (2.1)<br />
where U is the conservative variable vector or state vector:<br />
⎡ ⎤<br />
ρ<br />
U =<br />
ρu<br />
⎢<br />
⎣ρv<br />
⎥<br />
⎦ , (2.2)<br />
E<br />
and F i (U) = [f i<br />
g i ], is the inviscid flux tensor with vector components:<br />
⎡ ⎤ ⎡ ⎤<br />
ρu<br />
ρv<br />
f i =<br />
ρu 2 + p<br />
⎢<br />
⎣ ρuv<br />
⎥<br />
⎦ , g i =<br />
ρuv<br />
⎢<br />
⎣ ρv 2 + p<br />
⎥<br />
⎦ . (2.3)<br />
(E + p)u<br />
(E + p)v<br />
This set <strong>of</strong> four equations is closed with the equation <strong>of</strong> state for a perfect gas.<br />
2.1.1 Energy<br />
The total energy E per unit volume is the sum <strong>of</strong> the internal and kinetic energy <strong>of</strong><br />
the flow:<br />
E = ρe + 1 2 ρ(u2 + v 2 ). (2.4)
CHAPTER 2. GOVERNING EQUATIONS OF GAS DYNAMICS 39<br />
The term 1 2 ρ(u2 + v 2 ) is the kinetic energy, while e is the internal energy per unit<br />
mass . Internal energy includes translational, rotational and vibrational energy and<br />
other forms <strong>of</strong> energy in more complicated situations. In the present work the gas<br />
is assumed to be in local chemical and thermodynamic equilibrium and the internal<br />
energy is a function <strong>of</strong> pressure and density:<br />
e = e(p, ρ). (2.5)<br />
2.1.2 Polytropic ideal gas<br />
For an ideal gas the internal energy is a function <strong>of</strong> temperature only, e = e(T ).<br />
The temperature T is related to pressure p and density ρ by the ideal gas law:<br />
p = RρT, (2.6)<br />
where R is a constant obtained by dividing the universal gas constant R by the<br />
molecular weight <strong>of</strong> the gas. To a good approximation, the internal energy is simply<br />
proportional to the temperature T :<br />
e = c V T, (2.7)<br />
where c V is the specific heat at constant volume . Gases that obey Eqs. (2.6) and<br />
(2.7) are called calorically or thermally perfect. If energy is added to a fixed volume<br />
<strong>of</strong> a thermally perfect gas, then the change in internal energy and temperature is<br />
related via:<br />
de = c V dT. (2.8)<br />
However, if the gas is allowed to expand at a constant pressure, not all <strong>of</strong> the<br />
energy goes into increasing the internal energy. The work done by expanding the<br />
volume 1 by d( 1) is pd( 1 ) and the following relation is obtained:<br />
ρ ρ ρ
CHAPTER 2. GOVERNING EQUATIONS OF GAS DYNAMICS 40<br />
de + pd( 1 ρ ) = c pdT ⇔ d(e + p ρ ) = c pdT, (2.9)<br />
where c p is the specific heat at constant pressure . The quantity:<br />
h = e + p ρ , (2.10)<br />
is called the (specific) enthalpy <strong>of</strong> the gas . For a polytropic gas, c p is constant, so<br />
Eq. (2.9) yields:<br />
h = c p T, (2.11)<br />
and the enthalpy is simply proportional to the temperature. By the ideal gas law<br />
results:<br />
c p − c V = R. (2.12)<br />
The equation <strong>of</strong> state <strong>of</strong> a polytropic gas turns out to depend only on the ratio <strong>of</strong><br />
the specific heats:<br />
and also called the adiabatic exponent .<br />
γ = c p<br />
c V<br />
, (2.13)<br />
The internal energy in a molecule is typically split up between various degrees<br />
<strong>of</strong> freedom (translational, rotational, vibrational, etc.). How many degrees <strong>of</strong> freedom<br />
exist, depends on the nature <strong>of</strong> the gas. The general principle <strong>of</strong> equipartition<br />
<strong>of</strong> energy requires that the average energy in each <strong>of</strong> these is the same leading to<br />
each degree <strong>of</strong> freedom contributing an average energy <strong>of</strong>:<br />
1<br />
2 kT,
CHAPTER 2. GOVERNING EQUATIONS OF GAS DYNAMICS 41<br />
per molecule, where k is Boltzmann’s constant and is equal to 1.3806503·10 −23 m 2 kg/s 2<br />
in SI units . This gives a total contribution <strong>of</strong>:<br />
a<br />
2 kT,<br />
per molecule if there are a degrees <strong>of</strong> freedom. Multiplying this by n, the number<br />
<strong>of</strong> molecules per unit mass (which depends on the gas), gives:<br />
e = a nkT. (2.14)<br />
2<br />
The product nk is precisely the gas constant R, which in comparison with Eq.<br />
(2.7), gives:<br />
c V = a R. (2.15)<br />
2<br />
From Eq. (2.12), results that:<br />
c p =<br />
(<br />
1 + a )<br />
R, (2.16)<br />
2<br />
which leads to the following expression for the adiabatic constant:<br />
γ = c p<br />
c V<br />
= a + 2<br />
a . (2.17)<br />
For air a = 5 and γ = 7 2<br />
= 1.4. Noting that:<br />
T =<br />
p<br />
Rρ , (2.18)<br />
leads to:<br />
e =<br />
p<br />
(γ − 1)ρ . (2.19)
CHAPTER 2. GOVERNING EQUATIONS OF GAS DYNAMICS 42<br />
Using Eq. (2.19) in Eq. (2.4), gives the common form <strong>of</strong> the total energy for a<br />
thermally perfect gas:<br />
E =<br />
p<br />
γ − 1 + 1 2 ρ(u2 + v 2 ). (2.20)<br />
and the pressure is related to total energy and the conservative variables through<br />
the equation <strong>of</strong> state:<br />
)<br />
p = (γ − 1)<br />
(E − u2 + v 2<br />
. (2.21)<br />
2ρ<br />
2.1.3 Entropy<br />
Entropy is a fundamental thermodynamic quantity and indicates the degree to which<br />
its internal energy is available for producing usable work. The specific entropy is<br />
given by:<br />
( ) p<br />
s = c V log + constant. (2.22)<br />
ρ γ<br />
from which follows :<br />
p = κe ( s<br />
c V ) ρ γ , (2.23)<br />
where κ is a constant.<br />
2.1.4 Mathematical character <strong>of</strong> the Euler equations<br />
The system <strong>of</strong> Euler equations is hyperbolic, because the eigenvalues <strong>of</strong> the directional<br />
Jacobian matrix [64]:<br />
K = ∂F i · k, (2.24)<br />
∂U
CHAPTER 2. GOVERNING EQUATIONS OF GAS DYNAMICS 43<br />
are real and distinct for any direction represented by the unit vector k . The diagonal<br />
eigenvalue matrix is:<br />
⎡<br />
⎤ ⎡ ⎤<br />
uk x + vk y u<br />
Λ =<br />
uk x + vk y<br />
⎢<br />
⎣uk x + vk y + c<br />
⎥<br />
⎦ = u<br />
⎢<br />
⎣u + c<br />
⎥<br />
⎦ , (2.25)<br />
uk x + vk y − c u − c<br />
where u, v are the Cartesian velocity components , u is the contravariant velocity<br />
component along the k direction, and c is the local speed <strong>of</strong> sound given by the<br />
following equation:<br />
c =<br />
√ γp<br />
ρ . (2.26)<br />
The right and left eigenvectors <strong>of</strong> the directional Jacobian matrix in Eq. (2.24)<br />
are the following:<br />
⎡<br />
R =<br />
⎢<br />
⎣<br />
⎤<br />
ρ<br />
ρ<br />
k x 0<br />
2c<br />
2c<br />
ρu<br />
− ρkx<br />
2c 2<br />
ρv<br />
− ρky ⎥<br />
2c 2 ⎦<br />
q 1 k x<br />
uρk<br />
2 y − vρk x R 1 R 2<br />
ρu<br />
uk x ρk y + ρkx<br />
2c 2<br />
ρv<br />
vk x −ρk x + ρky<br />
2c 2<br />
, (2.27)<br />
⎡<br />
L =<br />
⎢<br />
⎣<br />
− ukx<br />
ρ<br />
− ukx<br />
ρ<br />
k x − kx<br />
c 2 γ−1<br />
2 q 2<br />
− ukx<br />
ρ<br />
− vky<br />
ρ<br />
+ vky<br />
ρ<br />
− vky<br />
ρ<br />
+ (γ−1)<br />
2ρc q 2<br />
k xρ<br />
uk x(γ−1)<br />
c 2<br />
vk x(γ−1)<br />
c 2<br />
− kx(γ−1)<br />
c 2<br />
k y<br />
ρ<br />
− kx ρ<br />
0<br />
− u(γ−1)<br />
ρc<br />
k y<br />
ρ<br />
− v(γ−1)<br />
ρc<br />
+ (γ−1) q 2ρc 2 − kx − u(γ−1) − ky − v(γ−1)<br />
ρ ρc ρ ρc<br />
γ−1<br />
ρc<br />
γ−1<br />
ρc<br />
⎤<br />
, (2.28)<br />
⎥<br />
⎦<br />
with :
CHAPTER 2. GOVERNING EQUATIONS OF GAS DYNAMICS 44<br />
q 1 =<br />
√<br />
u2 + v 2<br />
,<br />
2<br />
q 2 = 2(u 2 + v 2 ) − q 1 ,<br />
R 1 = qρ<br />
4c + uρk x<br />
+ vρk y<br />
2 2<br />
R 2 = qρ<br />
4c − uρk x<br />
− vρk y<br />
2 2<br />
2.2 Navier-Stokes equations<br />
+ ρc<br />
2γ − 2 ,<br />
+ ρc<br />
2γ − 2 .<br />
The system <strong>of</strong> Navier-Stokes equations describes real compressible flows, which<br />
posses both heat conduction and viscosity effects. The differential form <strong>of</strong> this<br />
system is:<br />
N (U) = ∂U<br />
∂t + ∇ · F i(U) = ∇ · F v (U, ∇U), (2.29)<br />
where U and F i (U) is the conservative state vector and the inviscid flux tensor<br />
respectively, and F v (U) = [f v<br />
vector components :<br />
g v ], is the viscous flux tensor with the following<br />
⎡<br />
⎤ ⎡<br />
⎤<br />
0<br />
0<br />
f v =<br />
τ xx<br />
⎢<br />
⎣ τ<br />
⎥<br />
xy ⎦ , g v =<br />
τ yx<br />
⎢<br />
⎣ τ<br />
⎥<br />
yy ⎦ , (2.30)<br />
uτ xx + vτ xy + q x uτ yx + vτ yy + q y<br />
and τ is the two dimensional viscous stress tensor :<br />
τ =<br />
[<br />
τxx τ xy<br />
τ yx τ yy<br />
]<br />
. (2.31)
CHAPTER 2. GOVERNING EQUATIONS OF GAS DYNAMICS 45<br />
Assuming a Newtonian fluid, for which the Stokes hypothesis is valid, then τ<br />
is symmetric and a linear function <strong>of</strong> the velocity gradients:<br />
τ xx = − 2 µ∇ · u + 2µ∂u<br />
3 ∂x ,<br />
(2.32a)<br />
τ yy = − 2 µ∇ · u + 2µ∂v<br />
3 ∂y ,<br />
where µ is the dynamic viscosity <strong>of</strong> the fluid.<br />
(2.32b)<br />
( ∂u<br />
τ xy = τ yx = µ<br />
∂y + ∂v )<br />
. (2.32c)<br />
∂x<br />
The heat fluxes are modeled according to Fourier’s law:<br />
q x = −λ ∂T<br />
∂x , (2.33)<br />
q y = −λ ∂T<br />
∂y , (2.34)<br />
where the thermal conductivity coefficient λ is related to the dynamic viscosity and<br />
the specific heat at constant pressure c p by the non-dimensional Prandtl number:<br />
P r = µc p<br />
λ . (2.35)<br />
2.2.1 Mathematical character <strong>of</strong> the Navier-Stokes equations<br />
The presence <strong>of</strong> viscosity and heat conduction effects transforms the momentum<br />
and energy equations into second-order partial differential equations. This results<br />
in a hybrid system being parabolic-hyperbolic in time and space, but becoming<br />
elliptic-parabolic in space for steady state problems.
CHAPTER 2. GOVERNING EQUATIONS OF GAS DYNAMICS 46<br />
2.3 Non-dimensional form <strong>of</strong> the Navier-Stokes<br />
equations<br />
In order to study flows under different conditions around the same geometry and,<br />
moreover, in order to reduce the errors due to the finite precision <strong>of</strong> computers, a<br />
non-dimensionalization <strong>of</strong> the flow variables is performed. In Table 2.1 the reference<br />
quantities used are given, which were taken from [118].<br />
Variable Reference quantity<br />
ρ<br />
ρ ∞<br />
u, v C ∞<br />
E<br />
ρ ∞ C∞<br />
2<br />
p<br />
ρ ∞ C∞<br />
2<br />
x, y L<br />
T<br />
T ∞<br />
µ µ ∞<br />
λ<br />
λ ∞<br />
t<br />
L/C ∞<br />
Table 2.1: Reference quantities used for the non-dimensionalization <strong>of</strong> the Navier-<br />
Stokes equations.<br />
The dimensionless variables, denoted by an over-bar, are the following:<br />
ū, ¯v = u, v , ¯p = p , Ē = E ,<br />
C ∞ ρ ∞ C∞<br />
2 ρ ∞ C∞<br />
2<br />
¯x, ȳ = x, y<br />
L ,<br />
¯t = tC ∞<br />
L ,<br />
¯T =<br />
T<br />
T ∞<br />
, ¯µ = µ<br />
µ ∞<br />
.<br />
Through the non-dimensionalization <strong>of</strong> the equations the following reference numbers<br />
appear:<br />
ˆ Reynolds number Re = ρ∞C∞L<br />
µ ∞<br />
ˆ Prandtl number P r = µ∞cp<br />
λ ∞
CHAPTER 2. GOVERNING EQUATIONS OF GAS DYNAMICS 47<br />
ˆ Mach number Ma = U∞<br />
C ∞<br />
Note that the Reynolds number Re uses c ∞ and therefore Re based on the free<br />
stream velocity (which is the case for experimentally given Reynolds numbers) must<br />
be scaled by the Mach number Ma. The equation <strong>of</strong> state Eq. (2.6) becomes:<br />
¯p =<br />
¯ρ ¯T<br />
γ , (2.36)<br />
and the viscous stresses and heat fluxes are altered, given by:<br />
¯τ xx = − 2 2¯µ ∂ū<br />
¯µ∇ · ū +<br />
3Re Re ∂¯x ,<br />
(2.37a)<br />
¯τ yy = − 2 2¯µ ∂¯v<br />
¯µ∇ · ū +<br />
3Re Re ∂ȳ ,<br />
¯τ xy = ¯τ yx =<br />
¯µ Re<br />
(2.37b)<br />
( ∂ū<br />
∂ȳ + ∂¯v )<br />
, (2.37c)<br />
∂¯x<br />
¯µ ∂<br />
¯q x = −<br />
¯T<br />
ReP r(γ − 1) ∂¯x ,<br />
¯µ ∂<br />
¯q y = −<br />
¯T<br />
ReP r(γ − 1) ∂ȳ .<br />
(2.37d)<br />
(2.37e)<br />
The dynamic viscosity <strong>of</strong> the fluid is related to the temperature <strong>of</strong> the flow<br />
using Sutherland’s law:<br />
where S = 100.55K.<br />
¯µ = µ ( ) 3<br />
T<br />
2 T∞ + S<br />
=<br />
µ ∞ T ∞ T + S , (2.38)
CHAPTER 2. GOVERNING EQUATIONS OF GAS DYNAMICS 48<br />
2.4 Integral form <strong>of</strong> the compressible flow equations<br />
The integral form <strong>of</strong> the Navier-Stokes equations is the basis <strong>of</strong> the FV methods.<br />
Considering an arbitrary fixed control volume Ω as it is shown in Fig. 2.1 and<br />
integrating Eq. (2.29) over it, follows:<br />
n<br />
Ω<br />
∂Ω<br />
Figure 2.1: Control Volume (CV) Ω.<br />
∫ ∫<br />
∫<br />
∂<br />
UdΩ + ∇ · F i dΩ = ∇ · F v dΩ ⇒<br />
∂t Ω<br />
Ω<br />
Ω<br />
∫ ∮<br />
∮<br />
∂<br />
UdΩ + F i · ndS = F v · ndS, (2.39)<br />
∂t Ω<br />
∂Ω<br />
∂Ω<br />
where n is the outward normal vector at the boundary <strong>of</strong> the control volume. Neglecting<br />
the viscous flux tensor in Eq. (2.39), the integral form <strong>of</strong> the Euler equations<br />
is obtained:<br />
∫ ∮<br />
∂<br />
UdΩ + F i · ndS = 0. (2.40)<br />
∂t Ω<br />
∂Ω
Chapter 3<br />
Mesh quality measures for the FV<br />
method<br />
The discretization methods mentioned in the introduction <strong>of</strong> the thesis are all mesh<br />
based methods and their solution quality is affected by the underlying mesh over<br />
which they are applied, especially the FV and the FD method. Thus, field simulations<br />
that incorporate complex physics and geometries poses a formidable challenge<br />
to grid generation [77]. The distribution <strong>of</strong> points and elements can be quite non–<br />
uniform and coarse leading to inaccurate computations. The inaccuracy is <strong>of</strong>ten<br />
assessed after the simulation is performed causing repetitive grid generations and<br />
subsequent field simulations, so it would be desirable to have an assessment <strong>of</strong> the<br />
appropriateness (quality) <strong>of</strong> the mesh before performing the simulation. The presented<br />
study on the mesh quality focuses on the FV method as its use is currently<br />
widespread in CFD.<br />
Grid or mesh quality is affected by two primary factors; the local size <strong>of</strong><br />
the computational elements, and the uniformity <strong>of</strong> the spatial distribution <strong>of</strong> the<br />
points/elements. A primary method for solving the issue <strong>of</strong> coarse mesh resolution<br />
has been adaptive grid refinement [11, 70, 71, 76, 102]. While substantial work<br />
has been performed on this type <strong>of</strong> mesh adaptation, relatively less work has been<br />
devoted to study and improvement <strong>of</strong> the local distribution <strong>of</strong> the points, which<br />
49
CHAPTER 3. MESH QUALITY MEASURES FOR THE FV METHOD 50<br />
basically relates to the shape <strong>of</strong> the elements. Assessment <strong>of</strong> adequacy <strong>of</strong> local resolution<br />
and shape <strong>of</strong> the elements depends on the discretization error. It is imperative<br />
to derive measures <strong>of</strong> this error in the computation.<br />
Two broad categories <strong>of</strong> error indicators are: (i) a priori and (ii) a posteriori<br />
estimation. The two approaches are basically complimentary. A priori error<br />
evaluation can aid in mesh generation, while a posteriori can provide guidance to<br />
mesh adaptation techniques during the simulation. Up to now, works regarding<br />
the a priori grid quality assessment are based on geometric characteristics <strong>of</strong> the<br />
elements such as ratios <strong>of</strong> sizes <strong>of</strong> neighboring elements, as well as on element shape<br />
measures, such as angles and ratios <strong>of</strong> the radii <strong>of</strong> inscribed to prescribed circles<br />
[48, 52, 55, 80, 111]. In the FE method, the quality <strong>of</strong> a mesh is <strong>of</strong>ten given in terms<br />
<strong>of</strong> the element/mesh regularity. This type <strong>of</strong> approach has given measures that can<br />
be computed easily and are quite popular with practical applications.<br />
Truncation error (TE) analysis usually falls under the a posteriori estimation.<br />
The FD discretization method has <strong>of</strong>fered a vehicle for calculating the TE by<br />
performing a Taylor series expansions <strong>of</strong> the solution at the points forming the discretization<br />
stencil [132]. The complexity <strong>of</strong> the expressions has led research works to<br />
focus on simplified model field equations [8, 127, 154]. Nevertheless, those works expressed<br />
the strong dependence <strong>of</strong> the solution and the stability <strong>of</strong> the computations<br />
on the grid and its quality.<br />
Direct relations between TE and mesh distortion parameters, such as departure<br />
from orthogonality and uniformity have been reported in [66, 133]. Significantly less<br />
work exists for unstructured meshes [57] and for the FV method [72, 141]. However,<br />
despite the large number <strong>of</strong> studies on the relation on the TE to the numerical<br />
solution, there have been very few derivations <strong>of</strong> mesh quality measures based on it.<br />
A posteriori error estimation methods include the Richardson extrapolation<br />
[25, 26, 67, 134]. Use is made <strong>of</strong> two or more grids <strong>of</strong> the same domain with the<br />
difference in the yielded solution <strong>of</strong>fering a measure <strong>of</strong> the local error distribution.<br />
Generating and solving on multiple meshes can be quite difficult for practical applications.
CHAPTER 3. MESH QUALITY MEASURES FOR THE FV METHOD 51<br />
Another a posteriori method evaluates the error indirectly via tracking <strong>of</strong> the<br />
numerical solution variation [78]. Local field features, such as boundary layers, shock<br />
waves and vortices are detected using sensors that are based on variations <strong>of</strong> the<br />
computed field parameters, such as the pressure and velocity. The assumption here<br />
is that the discretization error is large where the solution variations are large. The<br />
method has been primarily used for guiding grid adaptation. It is not a method<br />
that can directly yield practical assessment <strong>of</strong> grid quality, taking also into account<br />
that it necessitates expensive computations for large meshes and/or complex fields.<br />
Another school <strong>of</strong> a posteriori error estimation work employs FE discretization<br />
and derives analytic expressions <strong>of</strong> error bounds [3, 106, 116, 117]. Various model<br />
equations have been utilized in order to provide the estimates. Although, the error<br />
bounds do not yield grid quality measures directly, there is potential for such use.<br />
Again, knowledge <strong>of</strong> the solution field is needed.<br />
The goal <strong>of</strong> the mesh induced error study is the derivation <strong>of</strong> a priori quality<br />
measures for mixed-type element meshes for the FV method. The most fundamental<br />
computation is that <strong>of</strong> first-order derivatives. The first derivative is common not<br />
only to fluid flow governing equations, but also to other field equations. The relative<br />
reduced complexity <strong>of</strong> the related mathematics and its commonality led to its choice<br />
in the present work. The FV method will be the method for the mesh induced error<br />
study, as the FV method is currently the most wide spread method in computational<br />
fluid dynamics and moreover, it is the first-order accurate version <strong>of</strong> the DG<br />
discretization. A node-based FV method is chosen as the vehicle for derivation <strong>of</strong><br />
quality measures, since it is known to be sensitive to mesh non-uniformity in terms<br />
<strong>of</strong> its accuracy. The complete expressions for the TE will be presented for meshes<br />
that consist <strong>of</strong> structured, unstructured, or both types <strong>of</strong> elements. The complexity<br />
<strong>of</strong> the analytic expressions lead to study <strong>of</strong> the two–dimensional case.
CHAPTER 3. MESH QUALITY MEASURES FOR THE FV METHOD 52<br />
3.1 Gradient computation in the FV method<br />
A common node based FV evaluation <strong>of</strong> the first-order spatial derivatives employs<br />
the median dual area around a grid point as Fig. 3.1 depicts [15, 63]. This quite<br />
common FV evaluation is chosen to study quality measures. It is expected to reveal<br />
distortions <strong>of</strong> the mesh as it corresponds to a central differencing, and thus, deemed<br />
sufficient for this study.<br />
0 u e<br />
∆x e<br />
∆y e<br />
i-1<br />
e, 1<br />
i<br />
e, 2<br />
i+1<br />
Figure 3.1: Median dual surface for evaluating first-order derivatives at grid point<br />
0 via the FV method.<br />
The evaluation <strong>of</strong> the gradient ∇u is accomplished via the following contour<br />
integration:<br />
∇u ≈ 1 S<br />
∮<br />
∂l<br />
udl ≈ 1 ∑<br />
u e (∆y e î − ∆x e ĵ) ≡ ∇ h u, (3.1)<br />
S<br />
e<br />
where S is the dual area, u e is the field value at the middle <strong>of</strong> each edge <strong>of</strong> the dual<br />
contour , while ∆x e and ∆y e are the projections in the x and y direction <strong>of</strong> the<br />
edges <strong>of</strong> the dual contour joining the centroids <strong>of</strong> each element sharing point 0 with
CHAPTER 3. MESH QUALITY MEASURES FOR THE FV METHOD 53<br />
the middle <strong>of</strong> the grid edges sharing this point . This contour is termed median<br />
dual [15]. Other choices <strong>of</strong> dual surfaces include the surface formed by joining the<br />
centroids <strong>of</strong> the elements sharing point 0 (centroid dual), as well as the surface <strong>of</strong><br />
the union <strong>of</strong> those elements.<br />
Since the goal is deriving a quality measure the specific choice is not important.<br />
A node-based evaluation <strong>of</strong> the derivatives is considered, which will reveal the<br />
inaccuracy due to the distorted elements in a relatively straightforward manner.<br />
3.2 The general form <strong>of</strong> the truncation error for<br />
the node based FV method<br />
The TE in the computation <strong>of</strong> the gradient is defined as:<br />
E(x, y) = ∇ h u(x, y) − ∇u(x, y), (3.2)<br />
with ∇ h u and ∇u being the numerical and the analytical values <strong>of</strong> the gradient,<br />
respectively. The analysis is similar in the x and y directions and employs Taylor<br />
series expansions for u h at the required locations around point 0. The expansions are<br />
substituted into the TE definition <strong>of</strong> Eq. (3.2). The amount <strong>of</strong> operations involved<br />
is very large. This must have been one <strong>of</strong> the main reasons for not seeing complete<br />
TE analysis in previous works. In the present work the hurdle is overcome via use<br />
<strong>of</strong> the symbolic mathematics capability <strong>of</strong> Matlab [101].<br />
The TE terms are grouped according to the following general form given for<br />
the case <strong>of</strong> evaluating the derivative in the x direction (u h x):<br />
E x =e x xu x + e x yu y + e x xyu xy + e x xxu xx + e x yyu yy +<br />
e x xxxu xxx + e x yyyu yyy + e x xyyu xyy + e x xxyu xxy + ...<br />
(3.3)<br />
The terms with the symbol e are functions <strong>of</strong> the metrics <strong>of</strong> the mesh and will be
CHAPTER 3. MESH QUALITY MEASURES FOR THE FV METHOD 54<br />
the link for making the connection <strong>of</strong> the TE with mesh distortion. They will be<br />
termed error coefficients (EC). Their analytic expressions are grouped according to<br />
the order in terms <strong>of</strong> the local mesh size h.<br />
The first two terms are involved in the consistency checks <strong>of</strong> the numerical<br />
approximation:<br />
e x x = 1<br />
2S<br />
e x y = 1<br />
2S<br />
n∑<br />
(∆x e,1 + ∆x e,2 )∆y e − 1,<br />
e=1<br />
n∑<br />
(∆y e,1 + ∆y e,2 )∆y e .<br />
e=1<br />
(3.4a)<br />
(3.4b)<br />
The grid metrics involved in the above expressions are defined as follows:<br />
(∆x e,k ) j (∆y e,k ) l = 1<br />
N e,k<br />
N e,k<br />
∑<br />
(x m | e,k − x 0 ) j (y m | e,k − y 0 ) l ,<br />
m=1<br />
(3.5)<br />
∆y e = y e,2 − y e,1 ,<br />
where x m | e,k , y m | e,k and N e,k are the coordinates and the number <strong>of</strong> nodes participating<br />
in the averaging <strong>of</strong> u at the dual vertex e, k (k = 1, 2). Generally, if a<br />
median dual vertex coincides with the middle point <strong>of</strong> an edge sharing point 0, then<br />
N e,k = 2, and if it coincides with the center <strong>of</strong> a quadrilateral or a triangle then<br />
N e,k = 4, or N e,k = 3, respectively. In Eq. (3.5) it is observed that the metric terms<br />
∆x e,1 , ∆x e,2 , ∆y e,1 and ∆y e,2 are O(h) each.<br />
In the above expressions, n is the number <strong>of</strong> the dual edges, S is the dual area<br />
and it is O(h 2 ), while e, 1 and e, 2 denote the end points <strong>of</strong> each edge <strong>of</strong> the dual<br />
contour, as shown in Fig. 3.1.<br />
The next three terms are first-order:
CHAPTER 3. MESH QUALITY MEASURES FOR THE FV METHOD 55<br />
e x xy = 1<br />
2S<br />
e x xx = 1<br />
2S2!<br />
e x yy = 1<br />
2S2!<br />
n∑<br />
(∆x e,1 ∆y e,1 + ∆x e,2 ∆y e,2 )∆y e ,<br />
e=1<br />
n∑<br />
[(∆x e,1 ) 2 + (∆x e,2 ) 2 ]∆y e ,<br />
e=1<br />
n∑<br />
[(∆y e,1 ) 2 + (∆y e,2 ) 2 ]∆y e .<br />
e=1<br />
(3.6a)<br />
(3.6b)<br />
(3.6c)<br />
For the purpose <strong>of</strong> deriving quality measures, the second-order derivative terms are<br />
retained as well:<br />
e x xxx = 1<br />
2S3!<br />
e x yyy = 1<br />
2S3!<br />
e x xyy = 1<br />
2S2!<br />
e x xxy = 1<br />
2S2!<br />
n∑<br />
[(∆x e,1 ) 3 + (∆x e,2 ) 3 ]∆y e ,<br />
e=1<br />
n∑<br />
[(∆y e,1 ) 3 + (∆y e,2 ) 3 ]∆y e ,<br />
e=1<br />
n∑<br />
[∆x e,1 (∆y e,1 ) 2 + ∆x e,2 (∆y e,2 ) 2 ]∆y e ,<br />
e=1<br />
n∑<br />
[(∆x e,1 ) 2 ∆y e,1 + (∆x e,2 ) 2 ∆y e,2 ]∆y e .<br />
e=1<br />
(3.7a)<br />
(3.7b)<br />
(3.7c)<br />
(3.7d)<br />
When the projections ∆x e and ∆y e are symmetrical relative to point 0, the<br />
computation becomes <strong>of</strong> second-order and the TE expression reduces to the following:
CHAPTER 3. MESH QUALITY MEASURES FOR THE FV METHOD 56<br />
E x = e x xxxu xxx + e x xyyu xyy ,<br />
E y = e y yyyu yyy + e y xxyu xxy .<br />
(3.8)<br />
Meshes that give a TE form given in Eq. (3.8) will be termed ideal meshes. Orthogonal<br />
and equally spaced in each direction quadrilaterals form an ideal mesh. The<br />
equivalent with triangles is an unstructured mesh with equal-length projections <strong>of</strong><br />
their dual edges in the x and y directions. This definition will be used subsequently<br />
to define a quality measure.<br />
Similar expressions for the EC regarding evaluation <strong>of</strong> the derivative in the y<br />
direction hold. The projection ∆y e is replaced with ∆x e and a minus sign is placed<br />
in front <strong>of</strong> the sums. The EC e y x and e y y are given below:<br />
e y x = − 1<br />
2S<br />
e y y = − 1<br />
2S<br />
n∑<br />
(∆x e,1 + ∆x e,2 )∆x e ,<br />
e=1<br />
n∑<br />
(∆y e,1 + ∆y e,2 )∆x e − 1.<br />
e=1<br />
(3.9a)<br />
(3.9b)<br />
3.3 Consistency condition for the node based FV<br />
method<br />
A consistent discretization <strong>of</strong> the gradient requires the TE to vanish as the mesh<br />
size approaches zero. For zero mesh size all the EC become equal to zero, except<br />
for e x x, e x y in Eq. (3.4) and e y x, e y y in Eq. (3.9), and the TE in Eq. (3.2) becomes:
CHAPTER 3. MESH QUALITY MEASURES FOR THE FV METHOD 57<br />
E x = e x xu x + e x yu y ,<br />
E y = e y yu y + e y xu x .<br />
(3.10)<br />
It is noted that the EC e x x, e y x, e x y, e y y remain unchanged for fixed–shape mesh elements<br />
even if the local mesh size is changing. For a consistent discretization, it is required<br />
that:<br />
e x x = e x y = e y x = e y y = 0. (3.11)<br />
Following Eq. (3.11) the expressions <strong>of</strong> the EC e x x in Eq. (3.4) and e y y in Eq.<br />
(3.9) show that in order for them to vanish on a general mesh, the area evaluation<br />
<strong>of</strong> the following two expressions must yield the same result.<br />
S x = 1 n∑<br />
(∆x e,1 + ∆x e,2 )∆y e ,<br />
2<br />
S y = − 1 2<br />
e=1<br />
n∑<br />
(∆y e,1 + ∆y e,2 )∆x e .<br />
e=1<br />
(3.12)<br />
Using the median dual surface, from Eq. (3.12) it results that the two forms<br />
for evaluating the dual area produce the same result. Specifically, their difference is<br />
the following:<br />
∑<br />
[x e,2 y e,2 − x e,1 y e,1 − 2x 0 (y e,2 − y e,1 )] = 0, (3.13)<br />
e<br />
and so e x x = e y y = 0 on a mixed-type element mesh. Carrying out the algebra in the<br />
expression <strong>of</strong> the EC e x y in Eq. (3.4) using the median dual results:
CHAPTER 3. MESH QUALITY MEASURES FOR THE FV METHOD 58<br />
e x y = 1<br />
2S<br />
Analogously, the EC e y x in Eq.<br />
∑<br />
[ye,2 2 − ye,1 2 − 2y 0 (y e,2 − y e,1 )] = 0. (3.14)<br />
e<br />
(3.9) results to be always zero on a mixed-type<br />
element mesh. So, the median dual surface leads to a consistent approximation <strong>of</strong><br />
∇u.<br />
Using the centroid dual, from Eq.<br />
(3.12) it results that the two forms for<br />
evaluating the dual area do not produce the same result on structured meshes.<br />
Their difference considering a structured mesh is:<br />
1<br />
8<br />
∑<br />
[x i (y i+1 − y i−1 ) + y i (x i+1 − x i−1 )] ≠ 0, (3.15)<br />
i<br />
and so e x x ≠ 0, e y y ≠ 0 on a structured mesh. Carrying out the algebra in the<br />
expression <strong>of</strong> the EC e x y in Eq. (3.4) using the centroid dual it gives:<br />
e x y = 1<br />
8S<br />
∑<br />
[y i (y i+1 − y i−1 )] ≠ 0. (3.16)<br />
i<br />
Analogously, the EC e y x in Eq. (3.9) is not always zero on structured meshes. So,<br />
the consistency <strong>of</strong> the centroid dual depends on the geometry <strong>of</strong> a structured mesh<br />
in the approximation <strong>of</strong> ∇u.<br />
Using the centroid dual, from Eq.<br />
(3.12) it results that the two forms for<br />
evaluating the dual area produce the same result on unstructured meshes.<br />
difference <strong>of</strong> the two forms in Eq. (3.12) considering an unstructured mesh is:<br />
1<br />
6<br />
The<br />
∑<br />
[x i (y i+1 − y i−1 ) + x i (y i+1 − y i−1 )] = 0, (3.17)<br />
i<br />
and so e x x = e y y = 0 on unstructured meshes. Carrying out the algebra in the<br />
expression <strong>of</strong> the EC e x y in Eq. (3.4) using the centroid dual it gives:<br />
e x y = 1<br />
6S<br />
∑<br />
[y i (y i+1 − y i−1 )] = 0. (3.18)<br />
i
CHAPTER 3. MESH QUALITY MEASURES FOR THE FV METHOD 59<br />
Analogously, the EC e y x in Eq. (3.9) is always zero on unstructured meshes. So, the<br />
centroid dual gives a consistent approximation <strong>of</strong> ∇u on unstructured meshes.<br />
At mixed-type element mesh interfaces it is difficult to check analytically the<br />
consistency <strong>of</strong> the approximation <strong>of</strong> ∇u using the centroid dual. Despite this fact,<br />
it is anticipated that the computation will be inconsistent due to the inconsistency<br />
<strong>of</strong> the centroid dual expressions on structured meshes.<br />
3.4 Verification <strong>of</strong> the error coefficients analytic<br />
expressions<br />
It is <strong>of</strong> importance to check the complex expression <strong>of</strong> the TE using analytic field<br />
functions u(x, y). The TE is computed by substituting the analytic values <strong>of</strong> the<br />
spatial derivatives and the values <strong>of</strong> the EC in Eq. (3.3). This will be termed<br />
the analytic TE. The TE is also computed directly by subtracting the analytic<br />
expression for ∇u from the FV expression ∇ h u (Eq. (3.2)). This second approach<br />
will be termed numerical TE.<br />
Let us consider a field function u(x, y), which resembles a boundary layer type<br />
<strong>of</strong> flow field with a linear variation in the streamwise direction:<br />
u(x, y) = xy 2 . (3.19)<br />
The differences between the analytic and the numerical TE values are calculated for<br />
the mixed-type element mesh <strong>of</strong> Fig. 3.2 using both the median and the centroid<br />
dual. A feature <strong>of</strong> this mesh is a local lateral displacement in the vicinity <strong>of</strong> an<br />
interface point.<br />
For the median dual the distribution <strong>of</strong> the analytic and the numerical TE is shown<br />
in Fig. 3.3. It is observed that the derived expressions are coincident within plotting<br />
accuracy.<br />
Check <strong>of</strong> consistency “experimentally” via successive reductions <strong>of</strong> the size <strong>of</strong>
CHAPTER 3. MESH QUALITY MEASURES FOR THE FV METHOD 60<br />
Figure 3.2: Mixed-type element channel mesh.<br />
the mesh elements may be impractical for large meshes. An indirect check <strong>of</strong> the<br />
consistency is proposed here by computing the difference between the analytic and<br />
numerical TE, without including in the analytic expression the error coefficients<br />
“responsible” for consistency (e x x and e x y). When including these terms, which are<br />
independent <strong>of</strong> the local mesh size, the difference will be zero within machine accuracy.<br />
If it is not, as the case is for the centroid dual discretization (see Fig. 3.4), a<br />
local inconsistency is revealed. The “spikes” in Fig. 3.4 correspond to points in the<br />
vicinity <strong>of</strong> the locally distorted mesh interface. The same inconsistency was revealed<br />
for the case <strong>of</strong> a distorted structured mesh.<br />
Similar results were observed using other field functions, as well. This provides<br />
assurance that the derived complex expressions are correct. A further check <strong>of</strong> their<br />
validity, even though indirect, will be provided when checking the subsequently<br />
derived quality measures.<br />
3.5 Elementary Types <strong>of</strong> Mesh Distortion<br />
The general mesh depicted in Fig. 3.1 exhibits skewness, stretching, <strong>of</strong>fset <strong>of</strong> point<br />
0 from the center <strong>of</strong> the control surface, as well as interfaces between structured
CHAPTER 3. MESH QUALITY MEASURES FOR THE FV METHOD 61<br />
0.01<br />
0.005<br />
Error<br />
0<br />
-0.005<br />
-0.01<br />
0 20 40 60 80 100 120 140 160 180 200 220<br />
Node<br />
Figure 3.3: Distribution <strong>of</strong> the analytic (✷) and the numerical (✸) TE for the field<br />
u(x, y) = xy 2 on the mesh <strong>of</strong> Fig. 3.2 using the median dual.<br />
and unstructured elements. It is important in the context <strong>of</strong> the present work to<br />
“isolate” each type <strong>of</strong> mesh “defect” and relate it directly to the TE. This is very<br />
important for future work on improving the mesh during or immediately after its<br />
generation. Distortion types are easier to define on a quadrilateral mesh, compared<br />
to a triangular for which the <strong>of</strong>fset <strong>of</strong> point 0 from the dual centroid is a reasonable<br />
measure. Regarding the interfaces between quadrilateral and triangular elements,<br />
three distinct types were identified and studied. It should be noted that skewness,<br />
stretching and center point <strong>of</strong>fset are distortions that are not independent from each<br />
other. Application <strong>of</strong> one <strong>of</strong> them to an ideal mesh causes appearance <strong>of</strong> the rest.<br />
Nevertheless, this categorization <strong>of</strong>fers a clear way to express the TE as a direct<br />
function <strong>of</strong> them and to come up with ways to reduce those distortions. It should<br />
be noted that the present work considers interior point configurations. Boundary<br />
“central” points (node 0 in Fig. 3.1) are not examined as the application <strong>of</strong> boundary<br />
conditions <strong>of</strong>ten “discards” the local discretization.
CHAPTER 3. MESH QUALITY MEASURES FOR THE FV METHOD 62<br />
0<br />
Difference in Error<br />
-0.001<br />
-0.002<br />
0 20 40 60 80 100 120 140 160 180 200 220<br />
Node<br />
Figure 3.4: Difference between the analytic and the numerical TE for the field<br />
u(x, y) = xy 2 on the mesh <strong>of</strong> Fig. 3.2 using the centroid dual.<br />
3.6 Stretching<br />
Stretching is a common type <strong>of</strong> deviation from a uniformly–spaced mesh and it is<br />
readily defined for structured grids. In order to compare quadrilaterals and triangles<br />
in terms <strong>of</strong> sensitivity to stretching, the stretching type <strong>of</strong> mesh distortion is defined<br />
in a similar way as Fig. 3.5 depicts. In the quadrilateral case, the stretching is<br />
considered as the displacement <strong>of</strong> the edges sharing the center point 0 quantified<br />
by the stretching factors d x and d y . These edges are displaced in a structured<br />
way retaining their orientation, while stretching in the triangle case is defined via<br />
displacing the center node according to Fig. 3.5(b).
CHAPTER 3. MESH QUALITY MEASURES FOR THE FV METHOD 63<br />
dx∆x<br />
•<br />
∆y<br />
0<br />
dy∆y<br />
∆y<br />
∆x<br />
(a)<br />
∆x<br />
d x h<br />
•<br />
d y h<br />
h<br />
0<br />
(b)<br />
Figure 3.5: Stretching for (a) structured and (b) unstructured meshes defined to<br />
facilitate comparison <strong>of</strong> accuracy degradation on them.<br />
3.6.1 Skewness and Shearing<br />
The four edges sharing the center point 0 in a quadrilateral grid usually meet forming<br />
angles that deviate from 180 ◦ as Fig. 3.6 illustrates. This type <strong>of</strong> distortion is<br />
typically called skewness. The angle ω quantifies the degree <strong>of</strong> skewness and ranges<br />
from 0 ◦ to 90 ◦ . Shearing is quite different from skewness. It preserves the 180 ◦<br />
angle between the edges and deviates the mesh from being orthogonal (Fig. 3.7).<br />
The angle φ quantifies this deviation and ranges from 0 ◦ to 90 ◦ . It will be seen that<br />
shearing and skewness affect the accuracy very differently.
CHAPTER 3. MESH QUALITY MEASURES FOR THE FV METHOD 64<br />
0 ω<br />
∆x<br />
Figure 3.6: Skewed mesh, depicting the deviation angle ω from the 180 ◦ angle<br />
between one <strong>of</strong> the two pairs <strong>of</strong> edges sharing point 0 (ω ∈ [0 ◦ , 90 ◦ ]). The lengths <strong>of</strong><br />
the edges are equal.<br />
3.6.2 Mesh rotation<br />
Non–alignment <strong>of</strong> a structured mesh to a specific direction dictated by the form<br />
<strong>of</strong> the field is another type <strong>of</strong> distortion examined. The quadrilateral elements<br />
are rotated with respect to the x–axis <strong>of</strong> the coordinate system by an angle θ as<br />
illustrated in Fig. 3.8. This angle varies from 0 ◦ to 45 ◦ .<br />
0<br />
φ<br />
∆x<br />
Figure 3.7: Sheared mesh, depicting displacement <strong>of</strong> the edges causing deviation<br />
from orthogonality expressed by the angle φ (φ ∈ [0 ◦ , 90 ◦ ]). The lengths <strong>of</strong> the<br />
edges are equal.
CHAPTER 3. MESH QUALITY MEASURES FOR THE FV METHOD 65<br />
∆y<br />
y<br />
∆y<br />
0<br />
θ<br />
x<br />
∆x<br />
∆x<br />
Figure 3.8: Rotated structured mesh depicting non alignment <strong>of</strong> the edges with the<br />
axes <strong>of</strong> the global system (∆x ≠ ∆y and θ ∈ [0 ◦ , 45 ◦ ]).<br />
3.6.3 Mixed-type element mesh interfaces<br />
Change in the topology <strong>of</strong> the elements creates special interfaces which require<br />
examination in terms <strong>of</strong> the local accuracy <strong>of</strong> the discretization [75]. The elements<br />
forming the interface (quadrilaterals and triangles) are considered to have the same<br />
size so that the effect <strong>of</strong> the changing topology is isolated and studied. Three cases<br />
<strong>of</strong> such interfaces are identified in two dimensions and are illustrated in Fig. 3.9.<br />
ψ 2 ψ 1<br />
h<br />
h<br />
h<br />
0<br />
0<br />
0<br />
h<br />
h<br />
h<br />
h<br />
(a)<br />
h<br />
(b)<br />
h<br />
(c)<br />
Figure 3.9: Common mixed-type element mesh interfaces.
CHAPTER 3. MESH QUALITY MEASURES FOR THE FV METHOD 66<br />
3.7 Direct relation between truncation error and<br />
mesh distortion<br />
The general expressions <strong>of</strong> the error coefficients (EC) given in Eqs. (3.6) and (3.7)<br />
can yield simpler expressions directly related to each type <strong>of</strong> mesh distortion presented<br />
here. This is very important to subsequent work on improving the grid.<br />
Table 3.1 gives the expressions <strong>of</strong> the EC for a mesh exhibiting only one type<br />
<strong>of</strong> distortion for each case (a row <strong>of</strong> the table). The related parameters d x , d y , ω<br />
are present in the expressions. For clarity <strong>of</strong> the table, the terms related to the<br />
higher order EC are not given, and it is just noted if they are zero or not. It is<br />
observed that stretching and skewness reduce the formal second order <strong>of</strong> the FV<br />
discretization to first. The error is proportional to the stretching parameters d x and<br />
d y . In the case <strong>of</strong> skewness, the relationship with ω is not linear. Finally, for the<br />
case <strong>of</strong> shearing and non–alignment <strong>of</strong> a structured mesh, no reduction <strong>of</strong> the order<br />
<strong>of</strong> accuracy occurs. However, the structure <strong>of</strong> the TE changes (terms e x xxx, e x yyy, e x xyy<br />
and e x xxy) but the discretization remains second order.<br />
Mesh e x xy e x xx e x yy e x xxx e x yyy e x xyy e x xxy<br />
Fig. 3.5(a)<br />
1<br />
2 d y∆y d x ∆x 0 ≠ 0 0 ≠ 0 ≠ 0<br />
Fig. 3.5(b) d y h d x h 0 ≠ 0 0 ≠ 0 ≠ 0<br />
Fig. 3.6<br />
3 sin (2ω)∆x<br />
8 cos (ω)+8<br />
[cos (ω)−1]∆x<br />
2<br />
[1−cos (ω)]∆x<br />
4<br />
≠ 0 ≠ 0 ≠ 0 ≠ 0<br />
Fig. 3.7 0 0 0 ≠ 0 0 ≠ 0 ≠ 0<br />
Fig. 3.8 0 0 0 ≠ 0 ≠ 0 ≠ 0 ≠ 0<br />
Table 3.1: Expressions for the error coefficients for each type <strong>of</strong> mesh distortion.<br />
The EC corresponding to evaluation <strong>of</strong> the derivative u y <strong>of</strong> the three types <strong>of</strong>
CHAPTER 3. MESH QUALITY MEASURES FOR THE FV METHOD 67<br />
mixed-type element mesh interfaces are presented in Table 3.2. For the first type<br />
<strong>of</strong> interface, it is considered that ψ 1 = ψ 2 . It is observed that mesh interfaces yield<br />
first order accuracy for the evaluation <strong>of</strong> the derivative u y .<br />
Mesh e y xy e y xx e y yy e y xxx e y yyy e y xyy e y xxy<br />
Fig. 3.9(a) 0 − h 48<br />
0 ≠ 0 ≠ 0 ≠ 0 ≠ 0<br />
Fig. 3.9(b) 0 0 − h 10<br />
0 ≠ 0 ≠ 0 ≠ 0<br />
Fig. 3.9(c) − h 6<br />
0 0 0 ≠ 0 ≠ 0 ≠ 0<br />
Table 3.2: Expressions for the error coefficients for the mixed-type element mesh<br />
interfaces.<br />
However, it is possible to relocate node 0, so that the order <strong>of</strong> accuracy <strong>of</strong> the u y<br />
evaluation is improved, but only for the mesh interfaces <strong>of</strong> Figs. 3.9(b),(c). In Table<br />
3.3 the displacements <strong>of</strong> point 0 for improving the accuracy <strong>of</strong> the u y evaluation are<br />
given. It should be noted that the results <strong>of</strong> Table 3.3 concern interfaces with equal<br />
“heights” (h) <strong>of</strong> the elements.<br />
Mesh ∆x ∆y<br />
Fig. 3.9(a) non existent non existent<br />
Fig. 3.9(b) 0 2(−10 + 3 √ 11)h<br />
Fig. 3.9(c) − h 6<br />
0<br />
Table 3.3: Central interface point displacements for improving the accuracy <strong>of</strong> the<br />
u y derivative evaluation for the mixed-type element mesh interfaces.
CHAPTER 3. MESH QUALITY MEASURES FOR THE FV METHOD 68<br />
3.8 Definition <strong>of</strong> an appropriate index <strong>of</strong> mesh<br />
distortion<br />
The focus <strong>of</strong> the present work is on a priori evaluation <strong>of</strong> a grid with regards to<br />
the shape and topology <strong>of</strong> its elements. The other aspect <strong>of</strong> grid quality, namely<br />
the local resolution is not addressed here. Therefore, the defined measure (index) <strong>of</strong><br />
mesh distortion should be independent <strong>of</strong> the local size <strong>of</strong> the mesh. Other properties<br />
that this index should have include:<br />
1. simple mathematical form,<br />
2. direct relation with the TE, as well as the mesh distortion parameters,<br />
3. ability to capture distortions in any direction, and<br />
4. ability to detect relatively small distortions in the presence <strong>of</strong> larger ones.<br />
3.8.1 Normalized error coefficients<br />
The EC <strong>of</strong> Eqs. (3.6) and (3.7) are divided by the appropriate power <strong>of</strong> a characteristic<br />
local length scale L. Care must be taken in order to account for directionally-sized<br />
(high aspect ratio) local mesh elements. This implies appropriate use <strong>of</strong> two length<br />
scales (L x ,L y ) expressing the local size in the x and y direction, respectively.<br />
The “normalized” EC <strong>of</strong> Eqs. (3.6) and (3.7) are defined as :<br />
e x xx = ex xx<br />
L x<br />
,<br />
e x yy = ex yy<br />
L y<br />
,<br />
e x xy = ex xy<br />
L y<br />
, (3.20)<br />
e x xxx = ex xxx<br />
L 2 x<br />
, e x yyy = ex yyy<br />
L 2 y<br />
, e x xxy = ex xxy<br />
L 2 y<br />
, e x xyy = ex xyy<br />
. (3.21)<br />
L 2 y<br />
The denumerators in the above definitions have been chosen based on the mesh<br />
metrics appearing in the EC expressions and the order <strong>of</strong> each EC.
CHAPTER 3. MESH QUALITY MEASURES FOR THE FV METHOD 69<br />
Careful consideration is needed to define the characteristic local length. It is<br />
incorrect to define a single length for every direction, as then, on meshes with cells<br />
<strong>of</strong> high aspect ratio, the normalized EC would either be overestimated or underestimated.<br />
The present work uses an approach that resembles the FV evaluation <strong>of</strong><br />
the derivative. It defines the following grid functions :<br />
f x = ∆x e,k |∆x e,k |, f y = ∆y e,k |∆y e,k |, k = 1, 2. (3.22)<br />
Then, the lengths L x ,L y are computed in a similar manner to Eq. (3.1) with u being<br />
replaced by f x for the x–component and by f y for the y–component:<br />
L x = 1 ∑<br />
f x,e ∆y e ,<br />
S<br />
e<br />
L y = − 1 ∑<br />
f y,e ∆x e (3.23)<br />
S<br />
e<br />
It should be noted that for a uniform structured mesh with element sizes ∆x and<br />
∆y, the characteristic lengths are:<br />
L x = ∆x and L y = ∆y.<br />
For the case <strong>of</strong> a uniform (honeycomb shape) triangular mesh <strong>of</strong> edge size h, the<br />
local lengths are:<br />
L x = 35h<br />
48<br />
and<br />
L y = 7√ 3h<br />
16 .<br />
Also, for the examined mixed-type element mesh interfaces the characteristic local<br />
lengths are given in Table 3.4.<br />
3.8.2 Mesh quality index<br />
The normalized EC are grouped to yield a single number that characterizes the<br />
quality <strong>of</strong> the grid. Two such groupings are being presented and evaluated. The<br />
first is expressed via use <strong>of</strong> the first-order error coefficients:
CHAPTER 3. MESH QUALITY MEASURES FOR THE FV METHOD 70<br />
Mesh L x L y<br />
Mesh interface <strong>of</strong> Fig. 3.9(a)<br />
11<br />
12 h h<br />
Mesh interface <strong>of</strong> Fig. 3.9(b) h h<br />
Mesh interface <strong>of</strong> Fig. 3.9(c) h h<br />
Table 3.4: Characteristic local lengths for the mixed-type element mesh interfaces.<br />
Q x = |e x xx| + |e x yy| + |e x xy|,<br />
(3.24a)<br />
Q y = |e y xx| + |e y yy| + |e y xy|,<br />
(3.24b)<br />
where the first expression regards the u x derivative evaluation, and the second concerns<br />
the u y computation. On meshes which yield second-order accuracy in the<br />
evaluation <strong>of</strong> the derivatives u x and u y , both Q x and Q y are equal to zero.<br />
The second index that is presented expresses the deviation <strong>of</strong> a mesh from<br />
being ideal via the ratio <strong>of</strong> a form <strong>of</strong> the EC appearing on a general distorted mesh<br />
to a form <strong>of</strong> those coefficients for an ideal mesh. Specifically:<br />
q x = |ex xx| + |e x yy| + |e x xy| + |e x xxx| + |e x yyy| + |e x xyy| + |e x xxy|<br />
, (3.25a)<br />
|e x xxx| + |e x xyy|<br />
q y = |ey xx| + |e y yy| + |e y xy| + |e y xxx| + |e y yyy| + |e y xyy| + |e y xxy|<br />
. (3.25b)<br />
|e y xxx| + |e y xyy|<br />
Index q expresses the deviation <strong>of</strong> the mesh locally from being ideal. It should be
CHAPTER 3. MESH QUALITY MEASURES FOR THE FV METHOD 71<br />
noted that q x and q y are equal to one for an ideal mesh, and have values greater<br />
than one for a general distorted grid.<br />
It is interesting to examine the values <strong>of</strong> the indices for a structured mesh with<br />
high aspect ratio elements that is typical <strong>of</strong> boundary layer regions and illustrated<br />
in Fig. 3.10.<br />
0<br />
2∆y<br />
2∆x<br />
Figure 3.10: High aspect ratio quadrilaterals typical <strong>of</strong> a structured mesh in a<br />
boundary layer region.<br />
The relevant computation here is that <strong>of</strong> the derivative u y . The relevant EC<br />
in Eq. (3.8) for this case are:<br />
e y xxy = (∆x)2<br />
8<br />
, e y yyy = (∆y)2 ,<br />
6<br />
which yields the mesh quality index Q y to be zero and the q y to be one, which are<br />
the best possible quality index values.<br />
The experiment is repeated employing the triangular mesh created by subdividing<br />
the quadrilaterals <strong>of</strong> Fig. 3.10 along their diagonals. The corresponding error<br />
coefficients are:<br />
e y yyy = (∆y)2<br />
6<br />
, e y xyy = − ∆x∆y<br />
6<br />
, e y xxy = (∆x)2 .<br />
6<br />
It is observed that the order <strong>of</strong> accuracy remains second (Q y = 0). However, more<br />
non–zero EC appear and the value <strong>of</strong> q y is not that <strong>of</strong> the ideal mesh, but is greater<br />
than one.
CHAPTER 3. MESH QUALITY MEASURES FOR THE FV METHOD 72<br />
3.8.3 Analytic expressions <strong>of</strong> the quality index Q for each<br />
type <strong>of</strong> elementary mesh distortion<br />
The index Q has a simpler mathematical form and thus amenable to a direct relation<br />
to each type <strong>of</strong> mesh distortion. These expressions are given in Table 3.5. The<br />
equations for the stretched unstructured and the skewed structured mesh cases are<br />
given in Appendix A.<br />
Mesh<br />
Q<br />
Stretched structured [Fig. 3.5(a)]<br />
Stretched unstructured [Fig. 3.5(b)]<br />
Skewed [Fig. 3.6]<br />
Q x = | dx | + |<br />
1+d 2 x<br />
Q x (Eq. A-1)<br />
Q x (Eq. A-2)<br />
d y<br />
2+2d 2 y<br />
|<br />
Sheared [Fig. 3.7] Q x , Q y = 0<br />
Non–aligned structured [Fig. 3.8] Q x , Q y = 0<br />
Mesh interface [Fig. 3.9(a)] Q y = 1<br />
44<br />
Mesh interface [Fig. 3.9(b)] Q y = 144<br />
533<br />
Mesh interface [Fig. 3.9(c)] Q y = 1 6<br />
Table 3.5:<br />
distortion.<br />
Mesh quality index Q expressions for the elementary types <strong>of</strong> mesh<br />
It is observed that, the sheared and rotated (non–aligned) structured meshes do not<br />
exhibit reduction <strong>of</strong> accuracy and thus the indices are equal to zero. The unstructured<br />
mesh exhibits higher sensitivity in mesh distortion as it is depicted graphically<br />
in Fig. 3.11.<br />
The variation <strong>of</strong> the index with respect to the skewness angle is given in the Appendix<br />
A, and it is shown graphically in Fig. 3.12. A peak in the degradation <strong>of</strong>
CHAPTER 3. MESH QUALITY MEASURES FOR THE FV METHOD 73<br />
0.5<br />
0.4<br />
0.3<br />
Q x<br />
0.2<br />
0.1<br />
Unstructured<br />
Structured<br />
0<br />
0 0.1 0.2 0.3 0.4 0.5<br />
d x<br />
Figure 3.11: Mesh quality index Q x for the stretched unstructured mesh (◦) and the<br />
stretched structured mesh (□) vs. displacement factor d x (d y = 0).<br />
accuracy is observed at about 80 ◦ . It is observed that the normalized error coefficient<br />
e x xy reduces as the skewness angle approaches 90 ◦ , which leads to the reduction <strong>of</strong><br />
Q x in the same region in the graph.<br />
Finally, the influence <strong>of</strong> the mesh interface angle ψ 2 on degradation <strong>of</strong> accuracy<br />
is shown in the graph <strong>of</strong> Fig. 3.13. This concerns the mixed-type element<br />
mesh interface shown in Fig. 3.9(a). The curve is symmetric around the value <strong>of</strong><br />
tan −1 (2) ≈ 63 ◦ for which Q y is minimum.<br />
3.8.4 Calibration <strong>of</strong> the mesh quality index Q<br />
Any practical use <strong>of</strong> the grid quality index to judge a mesh requires knowledge <strong>of</strong> the<br />
range <strong>of</strong> permissible values <strong>of</strong> Q. A calibration is needed which is based on upper<br />
values <strong>of</strong> stretching d x , d y and skewness ω that are considered acceptable for typical<br />
field solvers. The proposed calibration has a conservative character as the upper<br />
bound for index Q is chosen as max(Q x , Q y ). An upper value <strong>of</strong> 20% stretching<br />
1<br />
means a value <strong>of</strong> d x and d y <strong>of</strong> , which when substituted in the expression for Q<br />
11
CHAPTER 3. MESH QUALITY MEASURES FOR THE FV METHOD 74<br />
0.8<br />
Q x<br />
Normalized EC and Q x<br />
0.6<br />
0.4<br />
0.2<br />
e x xy<br />
e x 2x<br />
e x 2y<br />
0<br />
0 20 40 60 80 100<br />
ω<br />
Figure 3.12: Mesh quality index Q x and its corresponding normalized error coefficients<br />
vs. the skewness angle ω.<br />
yields a value <strong>of</strong> approximately 0.13. Similarly for a maximum skewness angle ω <strong>of</strong><br />
20 ◦ , a value <strong>of</strong> Q ≈ 0.17 is found.<br />
Considering the case <strong>of</strong> a stretched unstructured mesh and applying the same<br />
stretching (center node displacement factor) <strong>of</strong> 20% a value <strong>of</strong> Q ≈ 0.24 is found.<br />
Finally, for the mixed-type element mesh <strong>of</strong> Fig. 3.9(a), setting ψ 2 = tan −1 (2) ± 20 ◦<br />
with ψ 1 = 2 · tan −1 (2) − ψ 2 , an upper bound <strong>of</strong> Q ≈ 0.16 is derived. Therefore, an<br />
upper bound value for Q <strong>of</strong> around 0.20 is a reasonable choice for all types <strong>of</strong> mesh<br />
distortion.<br />
3.9 Application <strong>of</strong> the mesh quality indices<br />
The quality index Q that is based on the first-order error terms appears to be quite<br />
simpler to use compared to the index q that uses both the first and the second<br />
order terms. However, a decision to adopt Q cannot be made until both <strong>of</strong> them<br />
are implemented with general distorted meshes. This section employs three types<br />
<strong>of</strong> grids; structured, unstructured, as well as hybrid. The hybrid grids consist <strong>of</strong><br />
separate quadrilateral and triangular layers, which is quite typical in applications.
CHAPTER 3. MESH QUALITY MEASURES FOR THE FV METHOD 75<br />
0.25<br />
0.2<br />
0.15<br />
0.1<br />
0.05<br />
Q y<br />
ψ 2<br />
0<br />
30 40 50 60 70 80 90 100<br />
Figure 3.13: Mesh quality index Q y for the mixed-type element mesh interface <strong>of</strong><br />
Fig. 3.9(a) vs. angle ψ 2 .<br />
The geometries involved are channel, airfoil and cylinder. It is important that the<br />
employed grids exhibit (i) both small and large magnitude distortions, as well as (ii)<br />
local and more global ones. The contours <strong>of</strong> the functions Q ≡ max(Q x , Q y ) and<br />
q ≡ max(q x , q y ) are plotted for all cases.<br />
In order to compare the effectiveness—fidelity <strong>of</strong> the indices for the different<br />
cases, the same contour levels are plotted, that is the minimum, maximum and<br />
number <strong>of</strong> levels are the same for all grids for each index. Further, both indices<br />
(Q and q) are shown with the same number <strong>of</strong> contour levels, which is essential for<br />
comparing the two on the same mesh.<br />
3.9.1 Channel grids<br />
The structured channel grid shown in Fig. 3.14(a) is uniform everywhere except<br />
in the middle where a point is displaced. Both indices capture this distortion with<br />
index Q–contours being more focused.<br />
The same displacement is applied to an almost uniform triangular mesh shown
CHAPTER 3. MESH QUALITY MEASURES FOR THE FV METHOD 76<br />
in Fig. 3.15(a). Similarly here, both indices capture the primary distortion along<br />
with smaller ones with q capturing a bit broader area than Q. This is due to the<br />
second order EC included in the definition <strong>of</strong> index q.<br />
The third mesh is hybrid with a straight line transitioning from the quadrilaterals<br />
to the triangles shown in Fig. 3.16(a). The index Q is more focused on the<br />
interface region than index q does.<br />
3.9.2 Airfoil grids<br />
An O–type structured mesh for the NACA 0012 airfoil is employed next. There is<br />
a sudden expansion <strong>of</strong> the quadrilaterals size in the region close to the surface as<br />
shown in Fig. 3.17. The same observation is made here, namely Q captures the<br />
areas <strong>of</strong> mesh stretching and skewness (leading and trailing edge regions) stronger<br />
compared to index q.<br />
The unstructured mesh around the NACA 0012 shown in Fig. 3.18 is a good<br />
quality mesh with small distortions everywhere. This case was devised to check the<br />
indices’ sensitivity to relatively small (“background”) distortions. It is observed that<br />
both indices show sensitivity to these non–uniformities with q being more sensitive<br />
to them.<br />
Finally, a hybrid mesh around the RAE 2822 airfoil is employed (Fig. 3.19).<br />
This is a viscous mesh with a very good preservation <strong>of</strong> the order <strong>of</strong> accuracy in<br />
the boundary layer region, except at the mesh interfaces, as indicated by index<br />
Q. However, index q shows a large deviation from the ideal computation for the<br />
boundary layer region due to the non alignment <strong>of</strong> the structured mesh in that area<br />
with the axes <strong>of</strong> the system <strong>of</strong> reference.<br />
3.9.3 Cylinder grids<br />
The cylinder mesh has mild distortions away from the body surface. It is an interesting<br />
case to check the performance <strong>of</strong> the quality indices at the farfield. Fig. 3.20
CHAPTER 3. MESH QUALITY MEASURES FOR THE FV METHOD 77<br />
illustrates the structured mesh which exhibits skewness along lines ”diagonally” <strong>of</strong>f<br />
the surface. The grid is <strong>of</strong> good quality close to the cylinder and this is indicated by<br />
both indices. It is observed that the four “skewness lines” are captured more clearly<br />
by index Q.<br />
The next case involves a triangular grid exhibiting a large stretching area close<br />
to the surface, as well as a mild mesh distortion <strong>of</strong> irregular shape away from the<br />
cylinder (Fig. 3.21). The two indices do recognize both types <strong>of</strong> distortion, and<br />
their contours “follow” the mild distortion pretty closely.<br />
Finally, the hybrid grid <strong>of</strong> Fig. 3.22 exhibits irregularity <strong>of</strong> the interface between<br />
the quadrilaterals and the triangles, as well as quite strong distortions in the<br />
farfield. It is observed that both indices recognize both types <strong>of</strong> strong distortions;<br />
the circular–shaped interface and the “random” ones in the triangles zone.
CHAPTER 3. MESH QUALITY MEASURES FOR THE FV METHOD 78<br />
(a)<br />
(b)<br />
(c)<br />
Figure 3.14: Locally stretched structured channel mesh: (a) mesh geometry, (b)<br />
index q and (c) index Q.
CHAPTER 3. MESH QUALITY MEASURES FOR THE FV METHOD 79<br />
(a)<br />
(b)<br />
(c)<br />
Figure 3.15: Locally stretched unstructured channel mesh: (a) mesh geometry, (b)<br />
index q and (c) index Q.
CHAPTER 3. MESH QUALITY MEASURES FOR THE FV METHOD 80<br />
(a)<br />
(b)<br />
(c)<br />
Figure 3.16: Mixed-type element channel mesh: (a) mesh geometry, (b) index q and<br />
(c) index Q.
CHAPTER 3. MESH QUALITY MEASURES FOR THE FV METHOD 81<br />
(a)<br />
(b)<br />
(c)<br />
Figure 3.17: O–type structured mesh around a NACA 0012 airfoil: (a) mesh geometry,<br />
(b) index q and (c) index Q.
CHAPTER 3. MESH QUALITY MEASURES FOR THE FV METHOD 82<br />
(a)<br />
(b)<br />
(c)<br />
Figure 3.18: Unstructured mesh around a NACA 0012 airfoil: (a) mesh geometry,<br />
(b) index q and (c) index Q.
CHAPTER 3. MESH QUALITY MEASURES FOR THE FV METHOD 83<br />
(a)<br />
(b)<br />
(c)<br />
Figure 3.19: Mixed-type element mesh around a RAE 2822 airfoil: (a) mesh geometry,<br />
(b) index q and (c) index Q.
CHAPTER 3. MESH QUALITY MEASURES FOR THE FV METHOD 84<br />
(a)<br />
(b)<br />
(c)<br />
Figure 3.20: Structured mesh around a cylinder: (a) mesh geometry, (b) index q<br />
and (c) index Q.
CHAPTER 3. MESH QUALITY MEASURES FOR THE FV METHOD 85<br />
(a)<br />
(b)<br />
(c)<br />
Figure 3.21: Unstructured mesh around a cylinder: (a) mesh geometry, (b) index q<br />
and (c) index Q.
CHAPTER 3. MESH QUALITY MEASURES FOR THE FV METHOD 86<br />
(a)<br />
(b)<br />
(c)<br />
Figure 3.22: Mixed-type element mesh around a cylinder: (a) mesh geometry, (b)<br />
index q and (c) index Q.
Chapter 4<br />
Discontinuous Galerkin<br />
discretization<br />
The equations governing compressible flow (Eqs. (2.1) and (2.29)) are discretized<br />
with the DG method. The DG method is a FE method which was first introduced<br />
in 1973 by Reed and Hill [122] and further developed to the so called Runge–Kutta<br />
DG (RKDG) method by Cockburn ans Shu in a series <strong>of</strong> papers [28, 40–44]. It uses<br />
discontinuous piecewise polynomials as basis functions and relies on the choice <strong>of</strong><br />
the numerical fluxes, which handle effectively the interactions across the element<br />
interfaces to achieve stable and highly accurate numerical solutions. In the last few<br />
years, the DG method attracted significant interest in the computational mechanics<br />
community, because it <strong>of</strong>fers certain advantages compared to the FV, FD methods<br />
as well as to the classical continuous FE method.<br />
A remarkable advantage <strong>of</strong> this method is that the local support <strong>of</strong> the numerical<br />
solution at the element level, makes it particularly suitable for flow simulations<br />
with high-order <strong>of</strong> accuracy. Moreover, mesh distortion does not affect the quality<br />
<strong>of</strong> the solution considerably as the order <strong>of</strong> the scheme increases [139, 150, 151] to<br />
very high-orders. Furthermore, hp-adaptation is easy to implement permitting high<br />
resolution <strong>of</strong> the flow field at selected flow regions.<br />
Inspired from the DG method other methods with similar features such as<br />
87
CHAPTER 4. DISCONTINUOUS GALERKIN DISCRETIZATION 88<br />
the spectral difference (SD) [98] and the SV [5, 149] methods have been recently<br />
developed. In this work the DG method is used because it is based on the FE<br />
discretization framework and it is straightforward to extend in three dimensions.<br />
The locality <strong>of</strong> the method makes it well suited for parallel implementation. Due<br />
to the large amount <strong>of</strong> computations required for the evaluation <strong>of</strong> the terms appearing<br />
in the semi-discrete form <strong>of</strong> the equations, it is imperative to implement<br />
the algorithms <strong>of</strong> the DG method for parallel computation. The DG method can<br />
easily be implemented on mixed-type element meshes, with the use <strong>of</strong> edge based<br />
data structures.<br />
By way <strong>of</strong> summary the DG method has several potential advantages compared<br />
to other methods including:<br />
ˆ Spectral accuracy on arbitrary meshes.<br />
ˆ Local h-refinement with the use <strong>of</strong> hanging nodes.<br />
ˆ Local p or hp-refinement.<br />
ˆ Boundary conditions are imposed weakly through the numerical flux.<br />
ˆ Local conservation laws allow for different fidelity models on neighboring elements.<br />
ˆ Highly parallelizable.<br />
4.1 Spatial Discontinuous Galerkin discretization<br />
As a FE method the DG discretization relies upon certain definitions, which are<br />
the building blocks <strong>of</strong> all the FE methods. These definitions are provided here in<br />
accordance to Ciarlet [37].<br />
Definition 1 (FE) A finite element is defined as a triplet {Ω e , P e , Σ e }, where :<br />
ˆ Ω e is a subset <strong>of</strong> R d .
CHAPTER 4. DISCONTINUOUS GALERKIN DISCRETIZATION 89<br />
ˆ P e is a vector space <strong>of</strong> functions p e : Ω → R m for some positive integer m.<br />
ˆ Σ e is a set <strong>of</strong> linearly independent linear forms U i , 1 ≤ i ≤ N defined over<br />
the space P e . The set Σ e is P -unisolvent, that is given any real scalar a there<br />
exists a unique function p e ∈ P e , such that U i (p e ) = a.<br />
Definition 2 (FE Mesh) The set T h = {Ω 1 , Ω 2 , . . . , Ω n } over a domain Ω with a<br />
piece wise polynomial boundary defines a tessellation <strong>of</strong> Ω:<br />
Ω = ⋃ e<br />
Ω e .<br />
Each element Ω e has a polynomial order N e .<br />
Definition 3 (FE interpolant) Given a unisolvent finite element {Ω e , P e , Σ e }, let<br />
B = {b 1 , b 2 , . . . , b n } be the set <strong>of</strong> basis that span the space P e . Let v ∈ V where<br />
P e ⊂ V , be a function for which the linear forms U i (p e ) are defined. The local FE<br />
interpolant is defined as:<br />
n∑<br />
U e = c e i b i .<br />
i=1<br />
From the linearity <strong>of</strong> the forms b i , it follows the linearity <strong>of</strong> the interpolation operator<br />
U e .<br />
4.1.1 Discontinuous weak formulation for the Euler equations<br />
A FE mesh T h over an arbitrary domain Ω is considered, where the physical field<br />
is approximated by a function Ũ(x, t), constructed by a locally continuous approximation<br />
(FE interpolant) Ũe(x, t) over each Ω e . For a DG spatial discretization<br />
Ũ(x, t) is considered discontinuous at the inter element boundaries as depicted in<br />
Fig. 4.1. Symbolically, it may be written as:
CHAPTER 4. DISCONTINUOUS GALERKIN DISCRETIZATION 90<br />
Ũ(x, t) = ∑ e<br />
Ũ e (x, t).<br />
Considering a single element Ω e , the approximation Ũe(x, t) substituted in the system<br />
<strong>of</strong> Euler equations (2.1), results in an approximation error, or residual, <strong>of</strong> the<br />
field over the elemental region, equal to:<br />
R e (Ũe(x, t)) = L(Ũe(x, t)), (4.1)<br />
leading to the residual over domain Ω:<br />
R(Ũ(x, t)) = ∑ e<br />
R e (Ũe(x, t)). (4.2)<br />
Ũ h 1<br />
Ũ h 2<br />
Ω 1<br />
Ω 2<br />
Figure 4.1: Local reconstruction <strong>of</strong> the field at two adjacent elements.<br />
As a FE method, the DG discretization, minimizes the approximation error over<br />
the computational domain in the weighted residual sense. That is, forming the<br />
Legendre inner product <strong>of</strong> R(Ũ(x, t)) with an arbitrary test or weighting function<br />
v, and setting it equal to zero:
CHAPTER 4. DISCONTINUOUS GALERKIN DISCRETIZATION 91<br />
∫<br />
Ω<br />
R(Ũ(x, t))vdΩ ≡ (R(Ũ(x, t)), v) = 0,<br />
leads to:<br />
∫<br />
Ω<br />
[<br />
v ∂Ũ<br />
∂t + v∇ · F (Ũ) ]<br />
dΩ = 0 ⇒<br />
∫<br />
Ω<br />
v ∂Ũ<br />
∫<br />
∂t<br />
∫Ω<br />
dΩ − ∇v · F (Ũ)dΩ + ∇ · vF (Ũ)dΩ = 0.<br />
Ω<br />
Using Gauss’ theorem, the last equation results in:<br />
∫<br />
Ω<br />
v ∂Ũ<br />
∮<br />
∂t<br />
∫Ω<br />
dΩ − ∇v · F (Ũ)dΩ + vF (Ũ) · n∂l = 0, (4.3)<br />
∂Ω<br />
where n is the outward normal vector at the domain boundary. Alternatively, Eq.<br />
(4.3) may be written as:<br />
∑<br />
e<br />
[ ∫ ∫<br />
∮<br />
]<br />
v ∂Ũe<br />
Ω e<br />
∂t dΩ e − ∇v · F (Ũe)dΩ e + vF (Ũe) · n e ∂l = 0,<br />
Ω e ∂Ω e<br />
with n e denoting the outward normal vector at the element boundaries. However,<br />
the discontinuous nature <strong>of</strong> the solution Ũ, permits to write the residual for every<br />
element as a separate equation:<br />
Eq.<br />
equations.<br />
∫<br />
Ω e<br />
v ∂Ũe<br />
∂t dΩ e −<br />
∫<br />
∮<br />
∇v · F (Ũe)dΩ e +<br />
Ω e<br />
vF (Ũe) · n e ∂l = 0.<br />
∂Ω e<br />
(4.4)<br />
(4.4) represents the discontinuous weak formulation for the system <strong>of</strong> Euler<br />
The local approximation <strong>of</strong> the field is constructed by a linear combination <strong>of</strong><br />
elemental basis functions b e k (x), according to the previous definitions:
CHAPTER 4. DISCONTINUOUS GALERKIN DISCRETIZATION 92<br />
Ũ e (x, t) = ∑ k<br />
c k e(t)b e k(x), (4.5)<br />
where the coefficient vector c k e(t) denotes the degrees <strong>of</strong> freedom (DOF) <strong>of</strong> the<br />
numerical solution to be advanced in time at every element. Substituting Ũe(x, t)<br />
in Eq. (4.4), yields:<br />
∫<br />
Ω e<br />
vb e k<br />
∂ĉ k ∫<br />
∮<br />
e<br />
∂t dΩ e − ∇v · F (Ũe)dΩ e + vF (Ũe) · n e ∂l = 0, (4.6)<br />
Ω e ∂Ω e<br />
The test function v and the elemental expansion function b e k (x) belong to the same<br />
FE space:<br />
X = {f ∈ L 2 (Ω) : f e ∈ Pe m (Ω e ) ∀Ω e ∈ T h }, (4.7)<br />
where Pe<br />
m (Ω e ) denotes the local space <strong>of</strong> polynomial functions <strong>of</strong> degree at most m<br />
and L 2 (Ω) represents the space <strong>of</strong> functions, which are squared Lebesgue integrable<br />
over Ω e . Then a system <strong>of</strong> 4DOF × 4DOF equations results for the system <strong>of</strong> Euler<br />
equations in two dimensions.<br />
4.1.2 The Numerical flux<br />
An important component for a DG discretization <strong>of</strong> the system <strong>of</strong> Euler equations<br />
is the evaluation <strong>of</strong> the numerical flux. Generally, a numerical flux that satisfies the<br />
following conditions is suitable for a solution algorithm for conservation laws based<br />
on the DG discretization:<br />
ˆ Consistent.<br />
ˆ Conservative.
CHAPTER 4. DISCONTINUOUS GALERKIN DISCRETIZATION 93<br />
The condition <strong>of</strong> consistency requires that the numerical flux H should approximate<br />
the line integral in Eq. (4.21). Typically a requirement <strong>of</strong> Lipschitz continuity is<br />
imposed, so that if a constant D exists the following inequality is satisfied:<br />
)<br />
∣<br />
∣H<br />
(Ũ+ e , Ũ− e<br />
(∣ )<br />
∣∣<br />
− F (U) ∣ ≤ Dmax Ũ + ∣<br />
− U∣ , ∣Ũ− − U∣<br />
. (4.8)<br />
The requirement for conservation in the formulation <strong>of</strong> the numerical flux is mandatory<br />
in order to obtain physically correct solutions for flows with discontinuities.<br />
The numerical fluxes may be chosen based on the exact or approximate Riemann<br />
solvers, developed for the FD and FV methodologies [135]. Qiu et al. in [119]<br />
examined several numerical fluxes starting from the exact Godunov flux up to more<br />
sophisticated fluxes from approximate Riemann solvers, such as the Roe [124] and<br />
the HLLC [136] flux and it was demonstrated that the local Lax-Friedrichs (LLF)<br />
flux is the most efficient in terms <strong>of</strong> CPU time, and that at orders <strong>of</strong> approximation<br />
higher than 3, the behavior <strong>of</strong> the resulting scheme with the LLF flux being almost<br />
the same with other less diffusive, but more computationally expensive numerical<br />
fluxes. This must have been the main reason that most RKDG works in the literature<br />
to be formulated with the LLF flux, as it has been done in the present work.<br />
The LLF flux is:<br />
)<br />
H<br />
(Ũ+ e , Ũ− e = 1 2<br />
[ ] + (fi + f − i )n x + (g + i + g − 1<br />
i )n y −<br />
2 k ( )<br />
U + e − U − e , (4.9)<br />
where the superscripts (+) and (-) denote the exterior and interior elements respectively,<br />
sharing the same edge and k is the spectral radius (maximum eigenvalue) <strong>of</strong><br />
the flux Jacobian given in Eq. (2.24) along the direction n normal to the edge and<br />
is equal to:<br />
k = u · n + c,
CHAPTER 4. DISCONTINUOUS GALERKIN DISCRETIZATION 94<br />
and it is obtained from the flux Jacobian evaluated at the Roe averaged state:<br />
ρu ˜ =<br />
√<br />
ρ+ u + + √ ρ − u −<br />
√<br />
ρ+ + √ ρ − , (4.10)<br />
ρv ˜ =<br />
√<br />
ρ+ v + + √ ρ − v −<br />
√<br />
ρ+ + √ ρ − , (4.11)<br />
˜h =<br />
√<br />
ρ+ h + + √ ρ − h −<br />
√<br />
ρ+ + √ ρ − , (4.12)<br />
˜c = (γ − 1)<br />
√<br />
˜h − 1 2 ( ρu2 ˜ + ρv ˜<br />
2 ), (4.13)<br />
with h = E + p.<br />
The Roe flux [124] has also been implemented. Roe’s flux is:<br />
)<br />
H<br />
(Ũ+ e , Ũ− e = 1 2<br />
[ ] + (fi + f − i )n x + (g + i + g − 1<br />
i )n y −<br />
2 LΛR ( )<br />
U + e − U − e , (4.14)<br />
where L, R are the left and right eigenvectors (Eqs. (2.28) and (2.27)) and Λ is the<br />
matrix <strong>of</strong> eigenvalues given in Eq. (2.25) computed at the Roe average state. The<br />
Harten and Hyman entropy fix is used to avoid the expansion shock through a sonic<br />
point, alternating the eigenvalues in matrix Λ according to:<br />
The quantity ɛ is determined as follows:<br />
λ i = λ2 i + ɛ 2<br />
. (4.15)<br />
2ɛ<br />
ɛ = max ( 0.01, λ i − λ + i , λ− i − λ i<br />
)<br />
. (4.16)<br />
where λ i is the eigenvalue computed using the Roe averaged variables and λ + i , λ− i<br />
are the eigenvalues computed using the elements’ traces. The index i refers to the i<br />
characteristic field <strong>of</strong> the system.
CHAPTER 4. DISCONTINUOUS GALERKIN DISCRETIZATION 95<br />
4.1.3 Basis functions and standard elemental configuration<br />
The choice <strong>of</strong> the basis functions affects the behavior and computational efficiency <strong>of</strong><br />
the DG discretization [83, 150]. Three types <strong>of</strong> basis functions may be constructed,<br />
while satisfying the unisolvency <strong>of</strong> the FE interpolation: modal, nodal and mixed<br />
modal-nodal.<br />
ˆ Modal bases are a set <strong>of</strong> approximating polynomials which have the property<br />
that by increasing the order <strong>of</strong> approximation, all the previous orders are still<br />
included in the approximation set.<br />
For example, the basis defined by the<br />
following set: {1, x, y, xy, x 2 , y 2 } is a modal hierarchical base.<br />
ˆ Nodal bases are a set <strong>of</strong> approximating polynomials, which have the property<br />
<strong>of</strong> being equal to unity at specific points and zero at other points in the approximation<br />
space. Lagrange interpolation polynomials is an example <strong>of</strong> nodal<br />
bases.<br />
ˆ Mixed modal-nodal bases are a combination <strong>of</strong> the above polynomial sets.<br />
When considering <strong>of</strong> applying a p-type refinement to the numerical solution, it<br />
is advantageous to use modal hierarchical expansion bases. Such bases may be<br />
constructed using the orthogonal set <strong>of</strong> Jacobi polynomials [83]. This approach was<br />
followed in the present work. The basis functions are defined over the standard<br />
square elemental configuration, which permits their systematic construction and<br />
implementation <strong>of</strong> any numerical operation involving them.<br />
Application <strong>of</strong> a Gauss type numerical integration, requires the definition <strong>of</strong> a<br />
standard elemental region spanning the domain [−1, 1] × [−1, 1]. In Fig. 4.2, the<br />
transformation <strong>of</strong> the physical quadrilateral to its standard elemental configuration<br />
Ω q st is depicted, and in Fig. 4.3, the transformation <strong>of</strong> the physical triangle to<br />
its standard elemental configuration Ωst t is shown. In this figures the Ω q st region,<br />
where the basis functions are to be constructed and all the numerical operations are<br />
performed is also shown.
CHAPTER 4. DISCONTINUOUS GALERKIN DISCRETIZATION 96<br />
n 2<br />
n 1<br />
C<br />
x = x(n 1 , n 2 )<br />
(-1,1) (1,1)<br />
D<br />
C<br />
D<br />
B<br />
(0,0)<br />
y<br />
q<br />
A<br />
y = y(n 1 , n 2 )<br />
A<br />
(-1,-1)<br />
B<br />
(1,-1)<br />
O<br />
x<br />
p<br />
Ω q e<br />
Ω q st<br />
Figure 4.2: Transformation <strong>of</strong> the physical quadrilateral to the standard quadrilateral<br />
element configuration (η 1 , η 2 ∈ [−1, 1]).<br />
For quadrilateral elements, the basis is constructed using the tensorial product<br />
<strong>of</strong> Legendre polynomials ˆψ p (n), which are subset <strong>of</strong> the Jacobi polynomials.<br />
according to a specific indexing, over Ω q st [83]:<br />
b k (η 1 , η 2 ) = ˆψ p (η 1 ) ˆψ q (η 2 ), −1 ≤ η 1 , η 2 ≤ 1, (4.17)<br />
with:<br />
k = p + q(N + 1), 0 ≤ p, q ≤ N.<br />
If a local approximation <strong>of</strong> order N is applied, then the number <strong>of</strong> bases in X is<br />
equal to (N + 1) 2 .<br />
For triangular elements, the basis is defined over Ωst t and constructed using<br />
the tensorial product <strong>of</strong> Jacobi polynomials over the region Ω q st (Fig. 4.3) [49]:<br />
b k (ξ 1 , ξ 2 ) = Pp 0,0 (η 1 )( 1 − η 2<br />
) p Pq 2p+1,0 (η 2 ), (4.18)<br />
2<br />
with the use <strong>of</strong> collapsed coordinates [83]:
CHAPTER 4. DISCONTINUOUS GALERKIN DISCRETIZATION 97<br />
(-1,1)<br />
C<br />
ξ 2<br />
(0,0)<br />
ξ 1<br />
A<br />
B<br />
(-1,-1)<br />
(1,-1)<br />
x = x(ξ 1 , ξ 2 )<br />
Ω t st<br />
n 1 = 2 1+ξ 1<br />
1−ξ 2<br />
− 1<br />
y = y(ξ 1 , ξ 2 )<br />
n 2 = ξ 2<br />
C<br />
B<br />
x = x(n 1 , n 2 )<br />
(-1,1) (1,1)<br />
D<br />
C<br />
y<br />
A<br />
Ω t e<br />
y = y(n 1 , n 2 )<br />
q<br />
A<br />
(-1,-1)<br />
(0,0)<br />
n 2<br />
n 1<br />
B<br />
(1,-1)<br />
O<br />
x<br />
p<br />
Ω q st<br />
Figure 4.3: Transformation <strong>of</strong> the physical domain triangle to the standard triangular<br />
element (ξ 1 , ξ 2 ∈ [−1, 1] with ξ 1 + ξ 2 ≤ 1) and standard quadrilateral element<br />
configuration (η 1 , η 2 ∈ [−1, 1]).<br />
ξ 1 = (1 + η 1)(1 − η 2 )<br />
2<br />
and according to the following indexing [93]:<br />
− 1, ξ 2 = η 2 ,<br />
k = p + (N + 1)q − q q − 1 , 0 ≤ p, q ≤ N with p + q ≤ N.<br />
2<br />
For a triangular element the number <strong>of</strong> bases for a local approximation <strong>of</strong> order N,<br />
is equal to (N+1)(N+2)<br />
2<br />
.<br />
In Tables 4.1 and 4.2 the analytical form <strong>of</strong> the basis functions, over the standard<br />
quadrilateral and triangular region, for a third order expansion are given, and
CHAPTER 4. DISCONTINUOUS GALERKIN DISCRETIZATION 98<br />
plotted in Figs. 4.4 and 4.5, respectively.<br />
(a) (b) (c)<br />
(d) (e) (f)<br />
(g) (h) (i)<br />
Figure 4.4: Basis functions over the standard square element: (a) (p, q) = (0, 0),<br />
(b) (p, q) = (1, 0), (c) (p, q) = (2, 0), (d) (p, q) = (0, 1), (e) (p, q) = (1, 1), (f)<br />
(p, q) = (2, 1), (g) (p, q) = (0, 2), (h) (p, q) = (1, 2), (i) (p, q) = (2, 2).
CHAPTER 4. DISCONTINUOUS GALERKIN DISCRETIZATION 99<br />
p, q b k p, q b k p, q b k<br />
(0, 0) 1 (1, 0) η 1 (2, 0)<br />
3n 2 1 −1<br />
2<br />
(0, 1) η 2 (1, 1) η 1 η 2 (2, 1)<br />
3n 2 1 +1<br />
2<br />
η 2<br />
(0, 2)<br />
3n 2 2 −1<br />
2<br />
(1, 2) η 1<br />
3n 2 2 +1<br />
2<br />
(2, 2)<br />
(3n 2 1 +1) (3n 2 2 +1)<br />
2 2<br />
Table 4.1: Basis functions for a third-order approximation over the standard quadrilateral<br />
region.<br />
p, q b k p, q b k p, q b k<br />
1−η<br />
(0, 0) 1 (1, 0) η 2<br />
3<br />
1 (2, 0) (η 2 2 1 − 1) 2 + 3η 1 − 2( 1−η 2<br />
) 2<br />
2<br />
(0, 1)<br />
3η 2 +1<br />
1−η<br />
(1, 1) η 2 5η 2 +3<br />
5<br />
2 1 (0, 2) (η 2 2 2 2 − 1) 2 + 6η 2 − 3<br />
Table 4.2: Basis functions for a third-order approximation over the standard triangular<br />
region.<br />
(a) (b) (c)<br />
(d) (e) (f)<br />
Figure 4.5: Basis functions over the standard triangular region: (a) (p, q) = (0, 0),<br />
(b) (p, q) = (1, 0), (c) (p, q) = (2, 0), (d) (p, q) = (0, 1), (e) (p, q) = (1, 1), (f)<br />
(p, q) = (0, 2).
CHAPTER 4. DISCONTINUOUS GALERKIN DISCRETIZATION 100<br />
4.1.4 Elemental operations<br />
In order to handle arbitrary geometries, it is necessary to numerically evaluate all<br />
the integrals appearing in Eq. (4.6). These are the mass matrix M kj , the volume<br />
integral V k and the surface (line) integral S k :<br />
M kj =<br />
∫<br />
b e kb e jdΩ e ,<br />
Ω e<br />
(4.19)<br />
∫<br />
V k = ∇b e k · F (Ũe)dΩ e ,<br />
Ω e<br />
(4.20)<br />
S k =<br />
∮<br />
b e kF (Ũe) · n e ∂l,<br />
∂Ω e<br />
(4.21)<br />
where in place <strong>of</strong> the test function v the elemental basis function b e j has been substituted.<br />
In the present work, Gaussian quadrature rules are used for numerical<br />
integration.<br />
For quadrilateral elements, the integrals given in Eqs. (4.19) and (4.20) are<br />
then computed by the weighted summations (η = η 1,i , η 2,m ):<br />
∫ 1 ∫ 1<br />
M kj = b k b j |J|dη 1 dη 2 =<br />
−1 −1<br />
Q<br />
∑ 1 Q<br />
∑ 2<br />
w i w m b k (η)b j (η)|J(η)|,<br />
i=1 m=1<br />
(4.22)<br />
∫ 1 ∫ 1<br />
V k = J −1 · ∇b k · F (Ũ e )|J|dη 1 dη 2 =<br />
−1 −1<br />
Q<br />
∑ 1 Q<br />
∑ 2<br />
w i w m J −1 (n) · ∇b k (η) · F (Ũ e (η)|J(η)|,<br />
i=1 m=1<br />
(4.23)
CHAPTER 4. DISCONTINUOUS GALERKIN DISCRETIZATION 101<br />
and for triangular elements, the same integrals are computed by the following<br />
weighted summations:<br />
∫ 1<br />
M kj =<br />
∫ −ξ2<br />
−1 −1<br />
∫ 1 ∫ 1<br />
−1 −1<br />
Q<br />
∑ 1 Q<br />
∑ 2<br />
i=1 m=1<br />
b k b j |J|dξ 1 dξ 2 =<br />
b k b j |J|( 1 − η 2<br />
)dη 1 dη 2 =<br />
2<br />
( 1 − η2,m<br />
w i w m b k (η)b j (η)|J(η)|<br />
2<br />
)<br />
,<br />
(4.24)<br />
∫ 1<br />
V k =<br />
∫ −ξ2<br />
−1 −1<br />
∫ 1 ∫ 1<br />
−1 −1<br />
Q<br />
∑ 1 Q 2<br />
i=1 m=1<br />
J −1 · ∇b k · F (Ũ e )|J|dξ 1 dξ 2 =<br />
J −1 · ∇b k · F (Ũ e )|J|( 1 − η 2<br />
)dη 1 dη 2<br />
2<br />
∑<br />
( 1 −<br />
w i w m J −1 η2,m<br />
(n) · ∇b k (η) · F (Ũ e (η))|J(n)|<br />
2<br />
)<br />
,<br />
(4.25)<br />
where |J| is the Jacobian <strong>of</strong> the transformation, J −1 is the inverse <strong>of</strong> the Jacobian<br />
matrix <strong>of</strong> the transformation from the physical to the computational space, w i , w m<br />
are the quadrature weights and Q 1 and Q 2 is the number <strong>of</strong> quadrature points<br />
along the η 1 and η 2 direction respectively. Matrix M is called the mass matrix and<br />
possesses a diagonal structure only for straight sided triangles and for straight sided<br />
quadrilaterals with parallel edges.<br />
The line integral in Eq. (4.21) is computed by the following weighted summation,<br />
both for triangles and quadrilaterals:
CHAPTER 4. DISCONTINUOUS GALERKIN DISCRETIZATION 102<br />
S k =<br />
Q m<br />
∑<br />
m=1<br />
∫ 1<br />
−1<br />
b k F (Ũe) · n e |l e |dn =<br />
w m b k H(Ũ+ e , Ũ− e ) · n c e|l e |,<br />
(4.26)<br />
where Q m is the number <strong>of</strong> quadrature points, |l e | is the edge length, H the numerical<br />
flux at the element boundary, Ũ+ e and Ũ− e are the element traces on an edge shared<br />
by two elements and n c e is the normalized outward vector <strong>of</strong> the local element edge<br />
at the computational space.<br />
Clearly, in our implementation all numerical computations <strong>of</strong> the DG method<br />
are carried out in the canonical computational domain <strong>of</strong> the square element. This<br />
choice gives a significant advantage for use <strong>of</strong> mixed-type element meshes and the<br />
novel implementation <strong>of</strong> the limiters for arbitrary elements that will be shown later.<br />
4.1.5 Selection <strong>of</strong> Gauss type numerical integration<br />
Gauss numerical integration is classified according to the way the end points <strong>of</strong> the<br />
domain <strong>of</strong> integration are handled [83]. The classification is given in Table 4.3, with<br />
the corresponding order <strong>of</strong> accuracy using Q n quadrature points.<br />
Type End points Order <strong>of</strong> accuracy<br />
Gauss-Legendre (GL) no end points 2Q n − 1<br />
Gauss-Radau (GR) only one end point 2Q n − 2<br />
Gauss-Lobatto-Legendre (GLL) both end points 2Q n − 3<br />
Table 4.3: Types <strong>of</strong> Gauss numerical integration.
CHAPTER 4. DISCONTINUOUS GALERKIN DISCRETIZATION 103<br />
The GL and GLL type have been implemented for both the surface and the<br />
line integrations in the present work. Moreover, in Eqs. (4.22), (4.24) and (4.26),<br />
it is seen that the product <strong>of</strong> two polynomials <strong>of</strong> order N is integrated. Thus, the<br />
same number <strong>of</strong> quadrature points is used for volume and surface integrations and<br />
equal to:<br />
Q 1 = Q 2 = Q m = N + 1, (4.27)<br />
for a GL integration, and:<br />
Q 1 = Q 2 = Q m = N + 2, (4.28)<br />
for a GLL integration.<br />
4.1.6 Mapping between physical and computational space<br />
The computation <strong>of</strong> the integrals in Eqs. (4.22) to (4.25) requires the evaluation <strong>of</strong><br />
the Jacobian matrix <strong>of</strong> the transformation at every quadrature point. This necessitates<br />
the construction <strong>of</strong> a mapping from the physical to the computational space.<br />
Usually, according to the polynomial degree N B <strong>of</strong> the physical space representation,<br />
curved elements are classified in sub-, iso- and super-parametric elements, by<br />
comparing the order N B with that <strong>of</strong> the order N <strong>of</strong> the discretization scheme. In<br />
detail:<br />
N B < N : sub-parametric element<br />
N B = N : iso-parametric element<br />
N B > N : super-parametric element<br />
For straight sided quadrilateral elements, the mapping from the physical configuration<br />
<strong>of</strong> the element Ω e to the standard elemental region (see Fig. 4.2), is given<br />
by the following relations:
CHAPTER 4. DISCONTINUOUS GALERKIN DISCRETIZATION 104<br />
(1 − η 1 ) (1 − η 2 )<br />
x =x A<br />
2 2<br />
(1 − η 1 ) (1 + η 2 )<br />
+ x D<br />
2 2<br />
(1 + η 1 ) (1 − η 2 )<br />
+ x B<br />
2 2<br />
+ x C<br />
(1 + η 1 )<br />
2<br />
(1 + η 2 )<br />
,<br />
2<br />
(4.29)<br />
(1 − η 1 ) (1 − η 2 )<br />
y =y A<br />
2 2<br />
(1 − η 1 ) (1 + η 2 )<br />
+ y D<br />
2 2<br />
(1 + η 1 ) (1 − η 2 )<br />
+ y B<br />
2 2<br />
+ y C<br />
(1 + η 1 )<br />
2<br />
(1 + η 2 )<br />
,<br />
2<br />
(4.30)<br />
and for straight sided triangular elements (see Fig. 4.3) by:<br />
(1 − η 1 ) (1 − η 2 )<br />
x = x A<br />
2 2<br />
(1 + η 1 ) (1 − η 2 )<br />
+ x B<br />
2 2<br />
1 + η 2<br />
+ x C , (4.31)<br />
2<br />
(1 − η 1 ) (1 − η 2 )<br />
y = y A<br />
2 2<br />
(1 + η 1 ) (1 − η 2 )<br />
+ y B<br />
2 2<br />
The Jacobian matrix <strong>of</strong> the transformation is defined as:<br />
1 + η 2<br />
+ y C . (4.32)<br />
2<br />
⎡<br />
⎢<br />
J = ⎣<br />
∂x<br />
∂η 1<br />
∂y<br />
∂η 2<br />
⎤<br />
∂x<br />
∂η 2<br />
⎥<br />
∂y<br />
∂η 2<br />
⎦ , (4.33)<br />
with its inverse equal to:<br />
J −1 = 1<br />
|J|<br />
⎡<br />
⎢<br />
⎣<br />
∂y<br />
∂η 2<br />
− ∂y<br />
∂η 1<br />
− ∂x<br />
∂η 2<br />
∂x<br />
∂η 1<br />
⎤<br />
⎥<br />
⎦ . (4.34)<br />
In the present work, in order to reduce the amount <strong>of</strong> operations required for<br />
the evaluation <strong>of</strong> the surface integrals, the product <strong>of</strong> the elements <strong>of</strong> the inverse
CHAPTER 4. DISCONTINUOUS GALERKIN DISCRETIZATION 105<br />
Jacobian <strong>of</strong> the transformation with the gradient <strong>of</strong> the weighting functions and<br />
the quadrature weights at each quadrature point, is computed and stored in a preprocessing<br />
step.<br />
4.2 DG discretization for the Navier-Stokes Equations<br />
Following the same procedure for the derivation <strong>of</strong> the weak form for the system<br />
<strong>of</strong> the Euler equations, Eq. (4.6), in the DG discretization, the weak form for the<br />
system <strong>of</strong> Navier-Stokes equations is the following:<br />
∫<br />
vb e ∂ĉ k ∫<br />
∮<br />
e<br />
k<br />
Ω e<br />
∂t dΩ e − ∇v · F (Ũe)dΩ e + vF (Ũe) · n e ∂l =<br />
Ω e ∂Ω<br />
∫<br />
∮<br />
e<br />
− ∇v · F v (Ũe, ∇Ũe)dΩ e + vF v (Ũe, ∇Ũe) · n e ∂l.<br />
Ω e ∂Ω e<br />
(4.35)<br />
From the previous system it is observed that the inclusion <strong>of</strong> the viscous and thermal<br />
conduction effects has lead to a system including the solution gradient. This<br />
complicates the solution <strong>of</strong> the Navier-Stokes system in the DG framework, since,<br />
the solution gradient has to be computed and appropriate numerical fluxes for the<br />
viscous part must be chosen to provide a stable and accurate discretization scheme.<br />
4.2.1 Computation <strong>of</strong> the viscous terms in the DG method<br />
The computation <strong>of</strong> the viscous terms in the DG method requires a special attention<br />
as has been demonstrated in [155]. Specifically, considering the one dimensional heat<br />
equation:<br />
u t − u xx = 0, (4.36)
CHAPTER 4. DISCONTINUOUS GALERKIN DISCRETIZATION 106<br />
with periodic boundary conditions and following the same procedure for the derivation<br />
<strong>of</strong> the weak form in the DG method as for the purely hyperbolic system <strong>of</strong> the<br />
Euler equations, yields:<br />
∫<br />
E<br />
v du h<br />
dt<br />
∮∂E<br />
dE + v du h<br />
dx<br />
∫E<br />
de − dv<br />
dx<br />
du h<br />
dE = 0. (4.37)<br />
dx<br />
where u h denotes the solution expansion <strong>of</strong> u. It is obvious that someone now has<br />
to deal with the derivative du h<br />
in the computation <strong>of</strong> the integrals appearing in<br />
dx<br />
Eq. (4.37). Due to the absence <strong>of</strong> an upwind mechanism in the heat equation, the<br />
natural choice is to define the numerical flux for u x as follows:<br />
(<br />
û x = 1 (duh ) +<br />
+<br />
2 dx<br />
( ) ) − duh<br />
. (4.38)<br />
dx<br />
If a P 0 expansion <strong>of</strong> the solution was chosen then this would lead to the analytical<br />
value <strong>of</strong> u h x to be equal to zero and hence the discretization would be innately<br />
inconsistent. Furthermore, in [155] it was demonstrated that this holds for orders<br />
<strong>of</strong> approximation greater than 0.<br />
Several schemes for the discretization <strong>of</strong> the viscous terms have been proposed<br />
in the literature, such as the schemes developed by Bassi and Rebay [17, 19, 20],<br />
the Local DG method (LDG) <strong>of</strong> Cockburn and Shu [43], which is a generalization<br />
<strong>of</strong> the ideas <strong>of</strong> Bassi and Rebay, and the method developed by Baumann and Oden<br />
(BO) [22, 23] and [107]. In the BO DG method, extra terms were added in the<br />
numerical flux that accounted for the jump in the solution between the elements,<br />
but for a P 0 solution expansion the scheme is inconsistent. Moreover, for a P N order<br />
solution expansion, the order <strong>of</strong> accuracy <strong>of</strong> the BO scheme is N for even N, that<br />
is sub-optimal, and N + 1 for odd N. Furthermore, in [155] it was demonstrated<br />
that the BO method has larger errors than the LDG method. For those reasons,<br />
in this work the LDG method is used for discretizing the system <strong>of</strong> Navier-Stokes<br />
equations in a consistent, stable and accurate way.
CHAPTER 4. DISCONTINUOUS GALERKIN DISCRETIZATION 107<br />
4.2.2 The LDG method<br />
The basic idea <strong>of</strong> the LDG method is to perform an order reduction <strong>of</strong> the initial<br />
system into a degenerate first order system and then discretize it within the DG<br />
framework to compute the solution gradient. For doing so, the following auxiliary<br />
variable Θ e at each element is introduced:<br />
⎡ ⎤<br />
∇ρ e<br />
Θ e = ∇Ũe =<br />
∇u e<br />
⎢<br />
⎣∇v ⎥<br />
e ⎦ , (4.39)<br />
∇T e<br />
which in essence is the solution gradient at each element. Multiplying the above<br />
equation by the same weighting function v appearing in Eq. (4.35) and integrating<br />
over the elemental space Ω e , obtain:<br />
∫<br />
∫<br />
vΘ e dΩ e −<br />
Ω e<br />
Ω e<br />
v<br />
⎡ ⎤<br />
∇ρ e<br />
∇u e<br />
⎢<br />
⎣∇v ⎥<br />
e ⎦ dΩ e = 0,<br />
∇T e<br />
which after application <strong>of</strong> Gauss’ theorem, leads to:<br />
⎡ ⎤<br />
ρ e<br />
∫<br />
∫<br />
∮<br />
vΘ e dΩ e + ∇v<br />
u e<br />
⎢<br />
Ω e Ω e ⎣v ⎥<br />
e ⎦ dΩ e −<br />
T e<br />
∂Ω e<br />
v<br />
⎡ ⎤<br />
ρ e<br />
u e<br />
⎢<br />
⎣v ⎥<br />
e ⎦ n e∂l = 0. (4.40)<br />
T e<br />
The evaluation <strong>of</strong> the line integral in Eq. (4.40) along with the numerical viscous<br />
flux computation has a pivotal role in the accuracy and stability <strong>of</strong> the resulting DG<br />
scheme.<br />
Arnold et. al in [6] performed an analysis <strong>of</strong> the resulting schemes depending<br />
on the choice <strong>of</strong> the line integral evaluation. Furthermore, in [6] it was proven
CHAPTER 4. DISCONTINUOUS GALERKIN DISCRETIZATION 108<br />
that two main requirements have to be met by the numerical fluxes in order to<br />
obtain a stable scheme, which furthermore achieves an optimal rate <strong>of</strong> convergence.<br />
These requirements are that the fluxes have to be consistent and conservative. The<br />
examined formulations are given in Table 4.4, where the following notation <strong>of</strong> jump<br />
[] and averaging {} operators for scalar and vector fields is used.<br />
No. Method û ˆΘ<br />
1 Bassi-Rebay (BR1) {u h } {Θ}<br />
2 Brezzi et. al 1 {u h } {Θ} − a r [u h ]<br />
3 LDG {u h } − β · [u h ] {Θ} + β[Θ] − a j [u h ]<br />
4 IP {u h } {∇ h u h } − a j [u h ]<br />
5 Bassi et al. (BR2) {u h } {∇ h u h } − a r [u h ]<br />
6 Baumann-Oden {u h } + n κ · [u h ] {∇ h u h }<br />
7 NIPG {u h } + n κ · [u h ] {∇ h u h } − a j [u h ]<br />
8 Babuska-Zlamal (u h | K )∂K −a j [u h ]<br />
9 Brezzi et. al 2 (u h | K )∂K −a j [u h ]<br />
Table 4.4: DG methods and their interiors fluxes taken for Arnold et. al [6].<br />
ˆ for a scalar variable s:<br />
[s] = s + n + + s − n − = n(s + − s − ),<br />
{s} = 1 2<br />
(<br />
s + + s −) ,
CHAPTER 4. DISCONTINUOUS GALERKIN DISCRETIZATION 109<br />
ˆ for a vector variable g:<br />
[g] = g + · n + + g − · n − = n(g + − g − ),<br />
{g} = 1 2<br />
(<br />
g + + g −) .<br />
The functional operators a r and a j in Table 4.4 are penalty terms which their appearance<br />
in the gradient variable Θ is <strong>of</strong> significant importance for obtaining a stable<br />
scheme. The methods which do not contain any <strong>of</strong> these terms are only weakly stable<br />
and it is obvious that the current chosen LDG method may be or may not be<br />
weakly stable. All the methods with penalty terms may be interpreted as interior<br />
penalty (IP) methods. The penalty method generally imposes a penalty term in<br />
the formulation in order to enforce the solution and must automatically satisfy the<br />
boundary conditions. This is achieved by the presence <strong>of</strong> the penalty term which<br />
adds more artificial dissipation to the jump at the boundary so that to smear out<br />
the discontinuity that appears there.<br />
In the present work, the utilization <strong>of</strong> the LDG method permits, as it easily<br />
verified from Table 4.4, the application <strong>of</strong> the BR1 scheme which in the literature<br />
is known as the first Bassi-Rebay scheme. However, Cockburn and Shu in [43]<br />
demonstrated some deficiencies <strong>of</strong> the BR1 scheme:<br />
ˆ Convergence order <strong>of</strong> N only for odd weighting function.<br />
ˆ Spread stencil.<br />
Both problems may be remedied by using an upwind like fashion on the choice <strong>of</strong><br />
fluxes in the gradient computation and in the viscous flux evaluation. In the present<br />
work this is achieved by the following use <strong>of</strong> the parameters β = ±0.5 and a j = 0.
CHAPTER 4. DISCONTINUOUS GALERKIN DISCRETIZATION 110<br />
Recently in the literature appeared the so called hybridizable DG (HDG) [38,<br />
39, 104] methods, which aim at alleviating the high computational cost <strong>of</strong> the DG<br />
discretization for the viscous terms. In HDG methods the final system is produced<br />
in terms <strong>of</strong> the DOF <strong>of</strong> the approximate traces <strong>of</strong> the field variables, that is the<br />
solution approximation at the inter element boundaries. Since the approximate<br />
traces are defined at the edges/faces only and are single valued, the HDG methods<br />
include significantly fewer coupled unknowns than other DG methods for higher<br />
order terms.<br />
The HDG methods are devised using a hybridization procedure <strong>of</strong> the DG<br />
methodology to discretize a system <strong>of</strong> equations. This procedure starts by introducing<br />
first the approximate traces and then define a total numerical flux in terms<br />
<strong>of</strong> them. A continuity requirement is weakly imposed for the normal component <strong>of</strong><br />
the numerical flux and this leads to a large nonlinear system <strong>of</strong> equations for the<br />
approximate variables (including their spatial gradient) and the numerical fluxes.<br />
After linearization <strong>of</strong> the final system the local variables and their spatial gradient<br />
may be condensed in an element by element fashion and obtain a reduced global<br />
system involving only the approximate traces as the unknowns.<br />
4.3 The treatment <strong>of</strong> initial and boundary conditions<br />
According to the flow conditions that have to be met at specific regions <strong>of</strong> the<br />
computational domain, appropriate boundary conditions (BC) have to be applied<br />
to the system <strong>of</strong> the governing equations. In the present work, the implementation<br />
<strong>of</strong> Neumann and Dirichlet BC is performed in a weak manner. This follows from the<br />
flux formulation <strong>of</strong> the present DG discretization leading to a natural weak inclusion<br />
<strong>of</strong> the BC through the numerical fluxes. It is noted that the weak enforcement <strong>of</strong><br />
Dirichlet BC leads to solutions that still ”feel” the influence <strong>of</strong> the boundary through<br />
the numerical fluxes, but in a manner that is consistent with the accuracy <strong>of</strong> the<br />
interior solution, thus, leading to improved solutions away from the boundaries [9,
CHAPTER 4. DISCONTINUOUS GALERKIN DISCRETIZATION 111<br />
45]. Also, the application <strong>of</strong> Neumann BC comes naturally in the DG discretization<br />
again through the flux formulation <strong>of</strong> the scheme.<br />
The technique for weak enforcement <strong>of</strong> BC requires the use <strong>of</strong> ghost elements,<br />
where a solution state U g is determined such that the appropriate boundary values<br />
at every quadrature point over a boundary edge are set. Then, the underlying<br />
Riemann problem is solved according to the main discretization scheme.<br />
It is noted, however, that a polyline representation <strong>of</strong> the wall geometry is<br />
not appropriate for the DG discretization [16, 18, 92, 93], as the DG method is<br />
highly sensitive to the accuracy <strong>of</strong> the boundary representation. This necessitates<br />
either use <strong>of</strong> curved elements or refinement <strong>of</strong> the mesh at the regions where the<br />
wall curvature is high. In the present work, the last approach is followed as only<br />
straight sided elements are used.<br />
The application <strong>of</strong> the initial conditions (IC) in the DG method may be performed<br />
in two ways that will be explained in the current chapter. These are the<br />
collocation and the discrete Galerkin projection. Both may be applied to solution<br />
expansions <strong>of</strong> nodal or modal hierarchical bases, but the first is more suitable for<br />
solution expansions with nodal bases which approximate the solution at specific<br />
points in the computational element, while the second is more suitable for modal<br />
hierarchical bases.<br />
4.4 Wall boundary conditions<br />
For a flow simulation over an arbitrary geometry, the BCs that have to be satisfied<br />
at the boundary segments defining the geometry depend on the physical modeling<br />
being performed for the flow field. Specifically, for inviscid flow numerical solutions<br />
it is required the flow to be tangent at the geometry, leading to the employment <strong>of</strong><br />
the so called slip-wall BC, which is a Dirichlet type condition and simply expresses<br />
that the flow cannot penetrate through the segments defining the geometry. For the<br />
DG discretization this means that at the quadrature points on the edges defining the<br />
wall geometry the velocity field must be enforced to be tangent. This is expressed
CHAPTER 4. DISCONTINUOUS GALERKIN DISCRETIZATION 112<br />
by the following Dirichlet BC:<br />
v · n = 0, (4.41)<br />
The weak enforcement <strong>of</strong> the BC condition <strong>of</strong> tangency in the velocity field is implemented<br />
at every quadrature point at the wall edges by the solution <strong>of</strong> a Riemann<br />
problem. Specifically, a ghost state is created at the opposite side <strong>of</strong> the computational<br />
domain at a wall edge, where the density, pressure and the magnitude <strong>of</strong> the<br />
velocity field are extrapolated from the interior solution. This corresponds to the<br />
following Neumann BC for the pressure at the wall:<br />
∂p<br />
∂n = 0.<br />
Then the velocity components at the ghost state are determined so that the averaged<br />
velocity field at the quadrature points to be tangent to the edge. Analytically, the<br />
ghost state is the following:<br />
ρ g = ρ,<br />
u g = u t cos (a) + u n sin (a) ,<br />
v g = −u n cos (a) + u t sin (a) ,<br />
(4.42)<br />
P g = P,<br />
where u t , u n are the velocity components tangent and normal to the edge and a is<br />
the angle formed by the wall segment with the x coordinate axis.<br />
For viscous flow computations, the flow must adhere to the geometry resulting<br />
to the application <strong>of</strong> another Dirichlet condition, namely the no-slip wall condition.<br />
The weak enforcement <strong>of</strong> the no-slip wall condition is performed by constructing the
CHAPTER 4. DISCONTINUOUS GALERKIN DISCRETIZATION 113<br />
following ghost state:<br />
ρ g = ρ,<br />
u g = −u t cos (a) + u n sin (a) ,<br />
v g = −u n cos (a) − u t sin (a) ,<br />
(4.43)<br />
P g = P,<br />
and the underlying Riemann problem is solved. Furthermore, for viscous computations<br />
additional considerations have to be taken regarding the heat flux in the<br />
flow field at the boundaries <strong>of</strong> the computational domain. This coordinates the use<br />
either adiabatic, isothermal etc. conditions that are enforced to the Navier-Stokes<br />
system through Neumann or Dirichlet BC. For instance, an adiabatic wall requires<br />
that there is no heat flux normal to the wall and this is expressed as follows:<br />
dT<br />
dn<br />
= 0, (4.44)<br />
and implemented by requiring at the quadrature points on the wall edges the temperature<br />
gradient normal to the wall to be zero. For isothermal walls or for flows<br />
with specific temperature distribution on the wall boundary, the temperature field<br />
must be set at every quadrature point to the prescribed values.<br />
4.5 Far field boundary conditions<br />
At the regions <strong>of</strong> the computational domain where the inlet and outlet <strong>of</strong> the flow<br />
is discretized, BC depending on the local characteristic structure <strong>of</strong> the flow field<br />
must be applied [64, 113, 140]. In the current DG discretization, their application,<br />
also follows that <strong>of</strong> the weak enforcement, and they are based on the assumptions<br />
<strong>of</strong> a locally one dimensional isentropic flow. Also, the application <strong>of</strong> the far field BC
CHAPTER 4. DISCONTINUOUS GALERKIN DISCRETIZATION 114<br />
is performed on the primitive variables [ρ, u, v, p] T .<br />
Considering the system <strong>of</strong> Euler equations and their eigenvalues given in Eq.<br />
(2.25), it is observed that for a supersonic flow all the characteristic waves are<br />
traveling in one direction. This dictates the conditions at the inlet to be imposed<br />
by the conditions <strong>of</strong> the free stream flow and at the outlet all the conditions to be<br />
extrapolated from the interior <strong>of</strong> the domain. However, for the case <strong>of</strong> subsonic flow,<br />
the inlet and outlet regions are crossed by waves coming both from the interior and<br />
the exterior <strong>of</strong> the domain. This situation necessitates modeling <strong>of</strong> the waves crossing<br />
the far field boundaries from the exterior, in order to determine the appropriate BC<br />
at subsonic inlets and outlets. The Riemann invariants [64] are used for determining<br />
the far field BC according to the previously made assumptions:<br />
J + = u + + 2<br />
γ − 1 c+ ,<br />
J − = u − − 2<br />
γ − 1 c− ,<br />
(4.45)<br />
where the + and - superscript refer to the appropriate side <strong>of</strong> the inlet or outlet<br />
region. The primitive, and thus the conservative variables, are then calculated according<br />
to the following algorithm:<br />
1: u b = J + +J −<br />
2<br />
2: c b = g−1<br />
4<br />
(J + − J − )<br />
3: p b = ρc2<br />
γ<br />
4: S b = p b<br />
ρ γ<br />
( ) 1<br />
c 3 γ<br />
5: ρ b = b<br />
γS
CHAPTER 4. DISCONTINUOUS GALERKIN DISCRETIZATION 115<br />
4.6 Application <strong>of</strong> Initial Conditions<br />
The use <strong>of</strong> a modal bases set requires determination <strong>of</strong> the initial values <strong>of</strong> the<br />
DOF according to the IC over the computational domain. Two methods may be<br />
used for the initialization <strong>of</strong> the DOF: collocation projection and discrete Galerkin<br />
projection.<br />
Considering a two dimensional function Ũ(x, t) which does not lie within the<br />
polynomial space <strong>of</strong> the expansion basis, then there will be an error R e (Ũ) in the<br />
approximation <strong>of</strong> the function over an arbitrary element where a solution expansion<br />
Ũ e (x, t) is employed:<br />
R e (Ũ) = Ũ(x, t) − Ũe(x, t). (4.46)<br />
Following the method <strong>of</strong> weighted residuals the Legendre inner product <strong>of</strong> Eq.<br />
(4.46) with an arbitrary function v is formed:<br />
∫<br />
Ω<br />
∫<br />
(Ũ(x, t) − Ũ e (x, t))<br />
vdΩ = R e (U)vdΩ, (4.47)<br />
Ω<br />
and setting the right hand side equal to zero, the following system is obtained for<br />
the determination <strong>of</strong> the expansion coefficients:<br />
∫<br />
Ω<br />
If the function v in Eq.<br />
∫<br />
Ũ e (x, t)vdΩ =<br />
Ω<br />
Ũ(x, t)vdΩ. (4.48)<br />
(4.47) is set equal to the Dirac delta function at<br />
the quadrature points, it is then implied that ∫ R(u)dΩ = 0 and the expansion<br />
Ω<br />
coefficients are obtained by:<br />
Û k e = B −1 · U k e, (4.49)<br />
where the matrix B i,j = b i (η j ). This approach corresponds to the collocation projection.<br />
If the function v is set equal to the basis function b i then the discrete
CHAPTER 4. DISCONTINUOUS GALERKIN DISCRETIZATION 116<br />
Galerkin projection is obtained with B i,j = ∫ b Ω i(η)b j (η)dΩ. In the present work<br />
the Galerkin projection has been used for applying the IC due to the use <strong>of</strong> a modal<br />
bases set.
Chapter 5<br />
Time discretization<br />
For time advancement <strong>of</strong> the compressible flow equations either the method <strong>of</strong> lines<br />
or the space-time DG method could be used. For space-time DG discretizations<br />
the solution is not only discretized in space but in time as well. This leads to<br />
a discretization scheme where the DOFs are scalar but the basis functions <strong>of</strong> the<br />
solution expansion depend both in space and time. For more details in the spacetime<br />
DG discretization refer to [86, 143].<br />
For the method <strong>of</strong> lines, the space discretization is performed first and the<br />
time is considered continuous. Then, a set <strong>of</strong> ODEs <strong>of</strong> size equal to the number <strong>of</strong><br />
DOFs in Eq. (4.6) for the Euler equations and in Eqs. (4.35) and (4.40) for the<br />
Navier-Stokes equations in the current implementation is obtained, which can be<br />
advanced in time either with explicit or implicit methods for ODEs.<br />
Use <strong>of</strong> explicit time integration schemes has been widespread in DG discretizations.<br />
However, explicit schemes suffer from severe stability limitations. The allowable<br />
time step due to CFL-stability limits is very small and few cases may be solved<br />
at a reasonable computing time. On the other hand, the application <strong>of</strong> implicit time<br />
integration does not have limitations imposed by CFL-stability limits, at least for<br />
linear conservation laws, and thus, can allow larger time steps, which are <strong>of</strong>ten determined<br />
only by accuracy considerations. Moreover, implicit methods are suitable<br />
for stiff problems.<br />
117
CHAPTER 5. TIME DISCRETIZATION 118<br />
5.1 Runge-Kutta methods<br />
Runge-Kutta (RK) methods belong to a class <strong>of</strong> multistage single step methods.<br />
They apply numerical integration rules over a specified set <strong>of</strong> stages, where progressively<br />
more accurate approximation <strong>of</strong> the solution’s slope is performed. They are<br />
classified as explicit and implicit, with the solution at every stage being well defined<br />
for small enough step size. Although, their implementation is expensive in terms<br />
<strong>of</strong> storage and processing, they are widely applied in computational mechanics, due<br />
to their exceptional properties in terms <strong>of</strong> stability accompanied with high-order<br />
accuracy.<br />
The generic form <strong>of</strong> an initial value problem for a scalar equation is the following:<br />
dy<br />
=f (t, y (t)) , t ∈ [a, b] ,<br />
dt<br />
y (a) = y 0 ,<br />
(5.1)<br />
and the numerical solution sought is denoted as y : [a, b] → R. Next, defining the<br />
following constants: q ∈ N, τ i ∈ R, a ij ∈ R, b i ∈ R with i, j = 1, . . . , q, and usually<br />
0 ≤ τ i ≤ 1 for and arbitrary function ψ (s) : R → R the following integrals:<br />
∫ ti<br />
0<br />
ψ (s) ds,<br />
∫ 1<br />
0<br />
ψ (s) ds,<br />
are approximated by the following summations, respectively:<br />
q∑<br />
a ij ψ (s) ,<br />
j=1<br />
q∑<br />
b i ψ (τ i ) , i = 1, . . . , q.<br />
j=1<br />
The constants a ij , τ i and b i define q + 1 quadrature rules, where τ i are the<br />
quadrature points and b i and a ij are the quadrature weights for the approximations<br />
<strong>of</strong> the integrals <strong>of</strong> ψ over the interval [0, 1] and [a, b], respectively. Each set <strong>of</strong> those<br />
constants define a RK method and it is customary to depict them in a tabular form<br />
known as Butcher’s table (Table 5.1). It is noted that the index i refers generally
CHAPTER 5. TIME DISCRETIZATION 119<br />
to the stages <strong>of</strong> the RK method.<br />
a 11 a 12 . . . a 1q τ 1<br />
a 21 a 22 . . . a 2q τ 2<br />
. . . .<br />
a q1 a q2 . . . a qq τ q<br />
b 1 b 2 . . . b q<br />
Table 5.1: Butcher’s table<br />
Discretizing the interval [a, b] into N equally spaced elements <strong>of</strong> size h = b−a<br />
N ,<br />
the nodal coordinates are equal to t n = a + nh, n = 0, . . . , N. Let y n be the solution<br />
given by a RK method as t n . Furthermore, as it has been mentioned that a RK<br />
method during each stage computes progressively a better approximation <strong>of</strong> the<br />
solution’s slope, the following constant is defined:<br />
t n,i = t n + τ 1 h, i = 1, . . . , q, (5.2)<br />
which simply expresses the parameterization <strong>of</strong> the sub interval [t n , t n+1 ] to the RK<br />
stages.<br />
Integrating Eq. (5.1) over [t n , t n+1 ], follows:<br />
y ( t n,i) − y (t n ) =<br />
∫ t n +τ i h<br />
which after some algebra may be written as:<br />
y ( t n,i) = y (t n ) +<br />
∫ τi<br />
0<br />
t n<br />
f (t, y (t)) dt,<br />
f (t + sh, y (t n + sh)) ds. (5.3)<br />
Applying the quadrature formulae with constants a ij and τ j , the approximation<br />
y n,i <strong>of</strong> y (t n,i ) results to be equal to:<br />
q∑<br />
y n,i = y n + h a ij f ( t n,j , y n,j) , i = 1, . . . , q. (5.4)<br />
j=1
CHAPTER 5. TIME DISCRETIZATION 120<br />
Integrating now Eq. (5.1) from t n to t n+1 , by applying the variable transformation<br />
t = t n + hs, and approximating the final integral over the interval [0, 1]<br />
with the quadrature rule defined by the points τ i and the weights b i , the following<br />
approximation y n+1 <strong>of</strong> y (t n+1 ) follows:<br />
q∑<br />
y n+1 = y n + h b i f ( t n,i , y n,i) . (5.5)<br />
i=1<br />
As a result, the RK method described by Butcher’s table produces a sequence<br />
<strong>of</strong> approximations given by the following relations:<br />
y 0 = y n = y 0 ,<br />
y n,i = y n + h ∑ qa ij f ( t n,j , y n,j) , 1 ≤ i ≤ q,<br />
j=1<br />
q∑<br />
y n+1 = y n + h b i f ( t n,i , y n,i) .<br />
i=1<br />
(5.6)<br />
In the present implementation <strong>of</strong> the DG discretization <strong>of</strong> the system <strong>of</strong> the<br />
compressible flow equations, explicit time advancement <strong>of</strong> the solution is performed<br />
with high-order strong stability preserving (SSP) Runge-Kutta methods, that preserve<br />
the monotonicity <strong>of</strong> the spatial discretization in any norm or semi-norm coupled<br />
with first-order forward Euler time stepping [85]. That is, if the monotonicity<br />
requirement:<br />
||u n+1 || ≤ ||u n ||, (5.7)<br />
is met for a step size <strong>of</strong> ∆t F E with a first-order forward Euler time stepping, then a<br />
high-order Runge-Kutta method is SSP if there exists a number a ∈ (0, 1] for which<br />
the step size ∆t under ∆t ≤ a∆t F E satisfies the monotonicity requirement.<br />
Extensive use <strong>of</strong> the explicit third order accurate SSP Rung-Kutta method
CHAPTER 5. TIME DISCRETIZATION 121<br />
proposed by Shu and Osher [130] is made:<br />
u (1) = u n + ∆tR(u n ),<br />
u (2) = 3 4 un + 1 4 u(1) + 1 4 ∆tR(u(1) ),<br />
(5.8)<br />
u n+1 = 1 3 un + 2 3 u(2) + 2 3 ∆tR(u(2) ),<br />
where R denotes the residual <strong>of</strong> the system after the DG discretization. Furthermore,<br />
SSP explicit Runge-Kutta methods provided by the PETSc library package are<br />
employed [58, 59].<br />
5.2 Implicit time marching<br />
By grouping together the time dependent and spatial contributions resulting from<br />
the DG discretization <strong>of</strong> either the Euler or the Navier-Stokes equations given in<br />
Eqs. (4.6) and (4.35) respectively, the following system <strong>of</strong> ODEs results:<br />
M · dU dt<br />
+ R(U) = 0, (5.9)<br />
where M is the mass matrix given by Eqs. (4.22) and (4.24) and R is the residual<br />
including all the spatially discretized terms.<br />
Considering for simplicity the backward implicit Euler time discretization <strong>of</strong><br />
Eq. (5.9), obtain:<br />
M · U n+1 − U n<br />
δt<br />
+ R(U n+1 ) = 0. (5.10)<br />
This is a system <strong>of</strong> nonlinear equations <strong>of</strong> the form:<br />
F (U) = 0, (5.11)
CHAPTER 5. TIME DISCRETIZATION 122<br />
and must be solved numerically using a Newton like method that is described next.<br />
5.3 Newton’s method<br />
The Newton iteration for Eq. (5.11) is derived from a multidimensional Taylor series<br />
expansion about a point U k :<br />
F (U k+1 ) = F (U k ) + F ′ (U k ) · (U k+1 − U k ) + higher order terms. (5.12)<br />
Neglecting higher order and setting the right hand side equal to zero, yields Newton’s<br />
method, which basically is an iterative procedure over a sequence <strong>of</strong> linear systems:<br />
J (U k ) · ∆U k = −F (U k ) , U k+1 = U k + ∆U k , k = 0, 1, . . . , (5.13)<br />
given an initial guess for the solution U 0 , J ≡ F ′<br />
= ∂F<br />
∂u<br />
being the approximate<br />
Jacobian <strong>of</strong> the system, U is the state vector to be found and k is the nonlinear<br />
iteration index. The Newton iteration is terminated based on the drop in the norm<br />
<strong>of</strong> the nonlinear function F under a specified tolerance or a sufficiently very small<br />
Newton correction ∆U.<br />
The sequence <strong>of</strong> linear systems is almost always solved<br />
using an iterative method belonging to the class <strong>of</strong> Krylov subspace methods.<br />
Forming each element <strong>of</strong> the Jacobian matrix requires the analytic or discrete<br />
derivatives <strong>of</strong> the system with respect to the state vector U. However, the analytic<br />
determination <strong>of</strong> the derivatives in order to form the Jacobian matrix is error-prone<br />
and even impossible, as the derivatives <strong>of</strong> an nondifferentiable function may need to<br />
be determined. Therefore in this work the Jacobian free method [36, 88] employing<br />
discrete evaluation <strong>of</strong> the Jacobian is used.
CHAPTER 5. TIME DISCRETIZATION 123<br />
5.4 Krylov subspace methods<br />
Krylov subspace methods are approaches for solving large linear systems and first<br />
appeared in the 1950s and became very popular afterward since Reid [123] introduced<br />
them as iterative methods for the solution <strong>of</strong> large linear systems. They can<br />
be considered as projection or generalized projection methods for solving a linear<br />
system A · x = b using the Krylov subspace K j defined as:<br />
K j = span ( r 0 , A · r 0 , A 2 · r 0 , A 3 · r 0 , . . . , A j−1 · r 0<br />
)<br />
,<br />
where r 0 = b − A · x 0 is the system’s residual. These methods require only matrixvector<br />
products during the iteration and this is the key element that favors use <strong>of</strong><br />
Newton’s method in a Jacobian free context as it will be shown.<br />
A wide spectrum <strong>of</strong> iterative methods belongs to the class <strong>of</strong> Krylov methods<br />
[13, 84, 125]. Two primary points <strong>of</strong> classification appear regarding the solution <strong>of</strong><br />
symmetric and non-symmetric systems and the procedure followed for the construction<br />
<strong>of</strong> the bases <strong>of</strong> the Krylov subspace. The bases may be constructed using the<br />
long-recurrence Arnoldi orthogonalization procedure, which generates orthonormal<br />
bases <strong>of</strong> the Krylov subspace. Specifically, the Arnoldi iteration [7] uses the stabilized<br />
Gram-Schmidt process to produce a sequence <strong>of</strong> orthonormal vectors q 1 , q 2 , q 3 , . . .,<br />
known as the Arnoldi vectors such that the vectors q 1 , q 2 , q 3 , . . . , q j span the Krylov<br />
subspace K j . The algorithm for producing the Arnoldi vectors is the following:<br />
1: Start with an arbitrary vector q 1 with norm 1.<br />
2: Repeat for k = 2, 3, . . .<br />
3: q k ← Aq k−1<br />
4: for j to k − 1 do<br />
5: h j,k−1 ← q ∗ j q k<br />
6: q k ← −h j,k−1 q j<br />
7: end for<br />
8: h k,k−1 ← ||q k ||
CHAPTER 5. TIME DISCRETIZATION 124<br />
9: q k ← 1<br />
h k,k−1<br />
q k<br />
The widely used Generalized Minimal RESidual method (GMRES) [126] is an<br />
Arnoldi-based method. In GMRES, the Arnoldi basis vectors form the trial subspace<br />
out <strong>of</strong> which the solution is constructed. One matrix-vector product per iteration<br />
is required to create a new trial vector, and the iterations are terminated based on<br />
a by-product estimate <strong>of</strong> the residual that does not require explicit computation <strong>of</strong><br />
the constructed solution, which is a beneficial feature <strong>of</strong> the algorithm. The residual<br />
minimization property <strong>of</strong> the method is in the Euclidean norm, but requires the<br />
storage <strong>of</strong> all previous Arnoldi basis vectors, leading unfortunately to high requirements<br />
in storage. This necessitates the use <strong>of</strong> efficient preconditioning <strong>of</strong> the linear<br />
system.<br />
5.5 Jacobian-free Newton-Krylov Method<br />
The Jacobian-free Newton-Krylov (JFNK) method is a synergistic combination <strong>of</strong><br />
the Newton method for solving a nonlinear system <strong>of</strong> equations and Krylov subspace<br />
methods for the solution <strong>of</strong> the linear system leading to the Newton correction.<br />
The specific advantage <strong>of</strong> the JFNK method is that the Jacobian <strong>of</strong> the system<br />
is not formed and stored, but only the Jacobian-vector product during the Krylov<br />
iteration is used, which is approximated numerically. This means that the gains<br />
both in processing time and memory requirements are extremely large. However,<br />
the application <strong>of</strong> the Krylov iteration necessitates adequate preconditiong <strong>of</strong> the<br />
linear system to be solved for the successful application <strong>of</strong> the JFNK method.<br />
The origins <strong>of</strong> the JFNK method can be found in efforts motivated by the<br />
solution <strong>of</strong> ODEs and PDEs [30, 31, 33, 56]. The primary goal in all cases has been<br />
the ability to perform a Newton iteration without the need to form the Jacobian.<br />
This has helped the application <strong>of</strong> higher-order implicit integration methods for<br />
ODEs as it is performed in the present work.<br />
In the JFNK approach a Krylov method is used to solve the linear system<br />
resulting from the linearization <strong>of</strong> the initial nonlinear system. An initial residual,
CHAPTER 5. TIME DISCRETIZATION 125<br />
r 0 is defined, given an initial guess for, ∆u 0 , for the Newton correction:<br />
r 0 = −F (u) − J · ∆u 0 (5.14)<br />
Using the GMRES method for the solution <strong>of</strong> the linear system for determining<br />
the Newton correction and denoting by j the GMRES iteration index, the iteration<br />
for the solution <strong>of</strong> the linear system minimizes ‖J · ∆U j + F (U) ‖ 2 within a subspace<br />
<strong>of</strong> small dimensions compared to the total number <strong>of</strong> unknowns. The Newton<br />
correction ∆U j is drawn from the Krylov subspace spanned by the following vectors:<br />
{<br />
r0 , J · r 0 , J 2 · r 0 , . . . , J j−1 · r 0<br />
}<br />
,<br />
and can be written as:<br />
j−1<br />
∑<br />
∆U j = ∆U 0 + β i J i · r 0 , (5.15)<br />
where the scalars β i minimize the residual. Initially the Newton correction ∆U 0 is<br />
set to zero, which asymptotically is a reasonable guess as the approach corrections<br />
should converge to zero at late Newton iterations. Moreover, in practice ∆U j is<br />
determined as a linear combination <strong>of</strong> the orthonormal Arnoldi vectors produced by<br />
GMRES.<br />
Close examination <strong>of</strong> Eq. (5.15) reveals that during the GMRES iterations the<br />
action <strong>of</strong> the Jacobian is taking place in the form <strong>of</strong> matrix-vector products, which<br />
may be approximated as follows [31, 33]:<br />
i=0<br />
J · v ≈<br />
F (U + ɛv) − F (U)<br />
, (5.16)<br />
ɛ<br />
where ɛ is a small perturbation. Eq. (5.16) is simply a first-order Taylor series<br />
expansion approximation to the Jacobian times a vector. For instance, considering<br />
a nonlinear system <strong>of</strong> two equations with two variables, the Jacobian <strong>of</strong> the system<br />
is the following:
CHAPTER 5. TIME DISCRETIZATION 126<br />
J =<br />
⎡<br />
⎢<br />
⎣<br />
∂F 1<br />
∂u 1<br />
∂F 1<br />
∂u 2<br />
∂F 2<br />
∂u 1<br />
∂F 2<br />
∂u 2<br />
JFNK does not require the formation <strong>of</strong> this matrix, instead a vector is formed<br />
that approximates the multiplication <strong>of</strong> this matrix by a vector. Specifically, approximating<br />
F (U + ɛv) − F (U) with a first-order Taylor series expansion about U,<br />
gives:<br />
⎤<br />
⎥<br />
⎦ .<br />
which simplifies to:<br />
F (U + ɛv) − F (U)<br />
ɛ<br />
≈<br />
⎡<br />
⎢<br />
⎣<br />
F 1 (u 1 ,u 2 )+ɛv 1<br />
∂F 1<br />
∂u 1<br />
+ɛv 2<br />
∂F 1<br />
∂u 2<br />
−F 1 (u 1 ,u 2 )<br />
ɛ<br />
F 2 (u 1 ,u 2 )+ɛv 1<br />
∂F 2<br />
∂u 1<br />
+ɛv 2<br />
∂F 2<br />
∂u 2<br />
−F 2 (u 1 ,u 2 )<br />
ɛ<br />
⎤<br />
⎥<br />
⎦ ,<br />
⎡<br />
⎢<br />
⎣<br />
v 1<br />
∂F 1<br />
∂u 1<br />
+ v 2<br />
∂F 1<br />
∂u 2<br />
v 1<br />
∂F 2<br />
∂u 1<br />
+ v 2<br />
∂F 2<br />
∂u 2<br />
⎤<br />
⎥<br />
⎦ = J · v.<br />
It is apparent that the error in the approximation is proportional to ɛ and its evaluation<br />
follows the approach proposed in [126] and [74].<br />
5.6 Preconditioning <strong>of</strong> the JFNK method<br />
Preconditioning in the JFNK method [36, 46, 121, 146] has the purpose <strong>of</strong> reducing<br />
the number <strong>of</strong> GMRES iterations by efficiently clustering close to unity the eigenvalues<br />
<strong>of</strong> the iteration matrix during the solution <strong>of</strong> the linear system. The JFNK<br />
approach has as its main goal the avoidance <strong>of</strong> forming the Jacobian matrix <strong>of</strong> the<br />
system and an effective preconditioner for the linear solution part <strong>of</strong> the method<br />
is necessary. In the present work the preconditioner used is formed by numerically<br />
computing the elements <strong>of</strong> the Jacobian matrix corresponding only to perturbations<br />
<strong>of</strong> the elemental DOFs, using coloring techniques provided by PETSc [58, 59] and
CHAPTER 5. TIME DISCRETIZATION 127<br />
applying an ILU(p) (for p = 0, . . . , 4) factorization procedure for computing the preconditiong<br />
matrix. However, it the experience gained in this work it was observed<br />
that this procedure did not work efficiently especially for large time step sizes.<br />
Unlike the matrix-based GMRES algorithm the Jacobian-free approach enjoys<br />
more flexibility as it allows to approximate the preconditioner matrix without affecting<br />
the quadratic convergence <strong>of</strong> the Newton algorithm. Preconditioning deals<br />
only with the solution <strong>of</strong> the linear system at every Newton iteration and in the<br />
present work advantage <strong>of</strong> this feature was taken by freezing the evaluation <strong>of</strong> the<br />
preconditioner for a number <strong>of</strong> Newton iterations (Jacobian lagging). Significant<br />
efficiency improvements have been reached by the lagging procedure because the<br />
matrix evaluation, for a DG code, is a rather computational expensive task [46].
Chapter 6<br />
Unified limiting<br />
The construction <strong>of</strong> the basis functions and the implementation <strong>of</strong> the DG method<br />
in the computational domain for the standard square element, allows straightforward<br />
numerical treatment <strong>of</strong> mixed-type meshes in a unified way. The expansion<br />
coefficients are defined in the computational space and are unique for every element.<br />
A Taylor series expansion [91] reveals that in the absence <strong>of</strong> discontinuities the solution<br />
coefficients are estimates <strong>of</strong> the solution derivatives. Thus, limiting <strong>of</strong> the<br />
solution coefficients amounts to limiting the solution derivatives.<br />
A significant innovation <strong>of</strong> this work is that a unified procedure was introduced<br />
that allows to preserve monotonicity <strong>of</strong> the solution in a TVB in the mean (TVBM)<br />
sense. A slope limiting procedure, which is transparent to the element type was<br />
developed and explained in this chapter. Specifically, the limiting operations are<br />
performed for every element over the standard square element configuration, where<br />
it is imposed that the slope <strong>of</strong> the solution at the element edges does not exceed the<br />
variation <strong>of</strong> the mean solution across them.<br />
Every element is checked for limiting, separately in each direction, by using<br />
essentially the same TVB limiter proposed by Cockburn and Shu [41, 129]:<br />
m(a 1 , a 2 , . . . , a n ) =<br />
{<br />
a 1<br />
m(a 1 , a 2 , . . . , a n )<br />
if |a 1 | ≤ ML 2 x,y,<br />
otherwise,<br />
(6.1)<br />
128
CHAPTER 6. UNIFIED LIMITING 129<br />
The significant modification <strong>of</strong> the present work is, however, that the parameter<br />
M ≥ 0 for each field is not set to a fixed value, but it is estimated as is shown<br />
bellow. Furthermore, L x,y in Eq. (6.1), is not the rectangular element length (∆x) 2<br />
in the physical space as in the original limiter, but represents the characteristic local<br />
lengths <strong>of</strong> the element in the physical space. The function m(a 1 , a 2 , · · · , a n ), is the<br />
usual minmod function:<br />
⎧<br />
⎨<br />
m(a 1 , a 2 , . . . , a n ) =<br />
⎩<br />
s min<br />
1≤j≤n |a j| if sgn(a 1 ) = sgn(a 2 ) = . . . = sgn(a n ) = s,<br />
0 otherwise.<br />
(6.2)<br />
According to the analysis <strong>of</strong> Cockburn and Shu [40, 129] for scalar conservation<br />
law problems the parameter, M, is an estimate <strong>of</strong> the second derivative <strong>of</strong> the<br />
solution. It is furthermore required that for systems <strong>of</strong> conservation laws to perform<br />
limiting <strong>of</strong> the solution coefficients (solution moments), in the characteristic space.<br />
Evaluation <strong>of</strong> the parameter M is performed for each characteristic field in<br />
the regular computational domain. Limiting for each characteristic variable is performed<br />
separately along each direction. The directional Jacobian, F(u) · n on the<br />
element edges, the corresponding eigenvalues, and eigenvectors are evaluated at the<br />
Roe’s averaged state from the mean interior u − and the exterior u + states in order<br />
to transform the conservative space state variables to the characteristic space<br />
variables. An element is flagged for limiting, if the limiter in Eq. (6.1) returns an argument<br />
other than the first, essentially when the slope is smaller than an appropriate<br />
measure <strong>of</strong> the second derivative value.<br />
For p-adaptive computations <strong>of</strong> arbitrary order <strong>of</strong> approximation P p , p ≥ 1,<br />
only the P 1 expansion <strong>of</strong> the solution is used for elements flagged for limiting in order<br />
to preserve monotonicity. Then, if the solution at an element has been limited, the<br />
approximation order at that element is set to P = 1.<br />
For elements neighboring<br />
limited elements, a gradually higher order <strong>of</strong> approximation could be used until<br />
progressively the order <strong>of</strong> approximation increases to the preset order <strong>of</strong> accuracy<br />
P p .<br />
It is emphasized that for a multistage time marching method and for calcu-
CHAPTER 6. UNIFIED LIMITING 130<br />
lations with a preset order <strong>of</strong> approximation P p , if an element, e, with order <strong>of</strong><br />
approximation p e > 1 is flagged for limiting during a stage, then the solution coefficients<br />
and the residual <strong>of</strong> the discretization corresponding to the higher order part<br />
<strong>of</strong> the approximation are set to zero. The rationale behind this is to keep the order<br />
<strong>of</strong> the approximation strictly equal to P 1 for all limited elements.<br />
For the effective use <strong>of</strong> the TVB limiter, it is crucial to have appropriate<br />
estimates <strong>of</strong> the solution’s second derivative for each field in the characteristic space,<br />
e.g. the parameters M wj , where w i , j = 1, 2, 3, 4 denotes the characteristic fields.<br />
The estimates for the second derivative are given bellow. Furthermore, because<br />
limiting is performed in the computational domain, an estimate <strong>of</strong> the local mesh<br />
size, L x,y , required. In this work L x,y was estimated as suggested by Eq. (3.23).<br />
The value <strong>of</strong> the parameter M (the estimate <strong>of</strong> second derivative <strong>of</strong> the solution<br />
in Eq. (6.1)) is evaluated at every element k, and for each field in the characteristic<br />
space, using central differences <strong>of</strong> the mean solution. Considering the mixed-type<br />
mesh patch given in Fig. 6.1, the parameter M is computed as follows:<br />
Figure 6.1: Mixed-element mesh patch for estimating parameter M.
CHAPTER 6. UNIFIED LIMITING 131<br />
∑<br />
M =<br />
(u e − u k )<br />
∣<br />
∣ , (6.3)<br />
e<br />
where index e counts the neighboring elements <strong>of</strong> k and, which actually is an approximation<br />
<strong>of</strong> the sum <strong>of</strong> the second derivatives <strong>of</strong> the solution. Specifically, performing<br />
a Taylor series expansion [79] <strong>of</strong> the sum given in Eq. (6.3) relative to elements k,<br />
results in:<br />
M = e x u x + e y u y + e xy u xy + e xx u xx + e yy u yy + . . . , (6.4)<br />
where, e x , e y , e xy , . . . , etc are coefficients induced by the local mesh geometry. For a<br />
single type mesh e x = e y = e xy = 0, except for at mesh interfaces. It is observed,<br />
that:<br />
M ≈ u xx + u yy .<br />
In practical computations however, in order to account for the mesh distortion,<br />
truncation error and floating points arithmetic, the following slightly modified value<br />
for the parameter M is used:<br />
ˆM = M +<br />
b<br />
L 2 x,y<br />
, (6.5)<br />
where b is a constant. All results were obtained using the same value b = 0.1.<br />
Therefore in essence in Eq. (6.1), the first condition is replaced by |a 1 | ≤ ML 2 x,y+0.1,<br />
where M is the estimate <strong>of</strong> Eq. (6.3), and L x,y the element size estimate.<br />
6.0.1 Limiting for quadrilateral elements<br />
For a P 1 approximation over a quadrilateral element, the solution expansion is:<br />
u q h = cq 0b q 0 + c q 1b q 1 + c q 2b q 2 + c q 3b q 3. (6.6)<br />
For the standard square element, the bases functions b q j , j = 1, . . . , 4 in Eq. (6.6)<br />
relative to the local Cartesian system (η 1 , η 2 ) <strong>of</strong> the canonical computational domain
CHAPTER 6. UNIFIED LIMITING 132<br />
are:<br />
b q 0 = 1,<br />
b q 1 = η 1 ,<br />
b q 2 = η 2 ,<br />
b q 3 = η 1 η 2 .<br />
Modification <strong>of</strong> the c q j coefficients in Eq. (6.6) is carried out by the TVB limiter as<br />
shown below. It is observed that the base b q 3 is not linear (see Fig. 4.4). Therefore,<br />
the solution coefficient c q 3 could be set to zero if a quadrilateral element is flagged for<br />
limiting. Otherwise, the coefficient c q 3 could be modified with the same procedure<br />
used for the modification <strong>of</strong> the coefficients c q 1 c q 2 considering additional quadrature<br />
points. In this work the c q 3 was set to zero. The solution coefficient c q 1 corresponds to<br />
the u x solution derivative therefore the coefficient c q 1 is limited along the η 1 direction.<br />
For doing so, the P 1 part <strong>of</strong> the solution at the midpoints <strong>of</strong> the edges BC and AD<br />
(see Fig. 4.2) in Ω st is used. The midpoints <strong>of</strong> the edges are chosen for limiting the<br />
solution coefficients, for the following reasons:<br />
ˆ Any ambiguity and closure problems are completely avoided.<br />
ˆ It is computationally efficient.<br />
According to Eq.<br />
solution is equal to:<br />
(6.6), at the midpoints <strong>of</strong> edges BC and AD in Ω st the<br />
U BC = u q (1, 0) − c q 0 = c q 1,<br />
(6.7a)<br />
U AD = u q (−1, 0) − c q 0 = −c q 1,<br />
(6.7b)<br />
and the value for the coefficient c q 1 is expressed in terms <strong>of</strong> the midpoint values as:<br />
c q 1 = U BC − U AD<br />
. (6.8)<br />
2
CHAPTER 6. UNIFIED LIMITING 133<br />
Limiting <strong>of</strong> the solution along the η 1 direction is performed by limiting the midpoint<br />
values U BC and U AD , using Eq. (6.1), as follows:<br />
Ũ BC = m(U BC , c q 0,(i+1,j) − cq 0,(i,j) , cq 0,(i,j) − cq 0,(i−1,j) ),<br />
(6.9a)<br />
Ũ AD = −m(−U AD , c q 0,(i+1,j) − cq 0,(i,j) , cq 0,(i,j) − cq 0,(i−1,j) ),<br />
(6.9b)<br />
where c q 0,(i,j)<br />
denotes the mean solution for the element (i, j) in the computational<br />
domain Ω st . The new limited value <strong>of</strong> c q 1 is computed by substituting in Eq. (6.8)<br />
the limited values from Eqs. (6.9).<br />
Along the η 2 direction the solution coefficient c q 2 is limited using a similar<br />
procedure as for the c q 1 coefficient. Specifically, at the midpoints <strong>of</strong> the edges AB<br />
and CD the solution expansion is equal to:<br />
U AB = u q (0, −1) − c q 0 = −c q 2,<br />
(6.10a)<br />
U CD = u q (0, 1) − c q 0 = c q 2,<br />
(6.10b)<br />
and the value <strong>of</strong> coefficient c q 2 is expressed in terms <strong>of</strong> the midpoint values as:<br />
c q 2 = U CD − U AB<br />
. (6.11)<br />
2<br />
Limiting <strong>of</strong> the values U AB and U CD using Eq. (6.1) is performed as follows:<br />
Ũ AB = −m(−U AB , c q 0,(i,j) − cq 0,(i,j−1) , cq 0,(i,j+1) − cq 0,(i,j) ),<br />
(6.12a)<br />
Ũ CD = m(U CD , c q 0,(i,j) − cq 0,(i,j−1) , cq 0,(i,j+1) − cq 0,(i,j) ).<br />
(6.12b)
CHAPTER 6. UNIFIED LIMITING 134<br />
The new limited value for c q 2 is computed by substituting in Eq. (6.11) the limited<br />
values from Eqs. (6.12).<br />
6.0.2 Limiting for triangular elements<br />
For a P 1 approximation over a triangular element, the solution expansion:<br />
u t h = c t 0b t 0 + c t 1b t 1 + c t 2b t 2, (6.13)<br />
where the basis functions relative to the local Cartesian system (η 1 , η 2 ) <strong>of</strong> Ω st (see<br />
Fig. 4.3 ) are:<br />
b t 0 = 1,<br />
b t 1 − η 2<br />
1 = η 1 ,<br />
2<br />
b t 2 = 3η 2 + 1<br />
.<br />
2<br />
For triangular elements, the coefficients c t 1 and c t 2 correspond to u x and u y ,<br />
respectively. The midpoints <strong>of</strong> the edges are chosen for limiting the solution, and a<br />
similar procedure to that applied for the case <strong>of</strong> quadrilaterals is followed. Specifically,<br />
along the η 1 direction, coefficient c t 1 is limited, by limiting the P 1 part <strong>of</strong><br />
the solution. Then, from the expansion given in Eq. (6.13), the solution at the<br />
midpoints <strong>of</strong> the edges AD and BC is equal to:<br />
U BC = u t (1, 0) − c t 0 = ct 1<br />
2 + ct 2<br />
2 , (6.14a)<br />
U AD = u t (−1, 0) − c t 0 = − ct 1<br />
2 + ct 2<br />
2 , (6.14b)<br />
and coefficient c t 1 is expressed in terms <strong>of</strong> the midpoint values as:
CHAPTER 6. UNIFIED LIMITING 135<br />
c t 1 = U BC − U AD . (6.15)<br />
Limiting <strong>of</strong> the values U AD and U BC is performed using Eq. (6.1) as follows:<br />
Ũ AD = m(U AD , c t 0,(i,j) − c t 0,(i,j−1), c t 0,(i,j+1) − c t 0,(i,j)),<br />
(6.16a)<br />
Ũ BC = m(U BC , c t 0,(i,j) − c t 0,(i,j−1), c t 0,(i,j+1) − c t 0,(i,j)),<br />
(6.16b)<br />
and the new value for c t 1 is computed by substituting in Eq. (6.15) the limited values<br />
from Eqs. (6.16).<br />
Along the η 2 direction the coefficient c t 2 is limited, but the P 1 part <strong>of</strong> the<br />
solution across the edge AB is limited only with the mean solution variation across<br />
that edge. The final value <strong>of</strong> the c t 2 coefficient is:<br />
c t 2 = m(U AB,(i,j) , c t 0,(i,j) − c t 0,(i,j−1), c t 0,(i,j) − c t 0,(i,j−1)). (6.17)<br />
Clearly, the approach described for quadrilateral and triangular elements could<br />
be extended for hexahedral and tetrahedral elements. Again, in three dimensions,<br />
use <strong>of</strong> collapsed coordinates must be employed and limiting will be performed in a<br />
dimension per dimension fashion for the standard cubical element.<br />
Remark 1. As has been noted the parameter M is computed in the characteristic<br />
space. For doing so, first the difference <strong>of</strong> the mean solution is formed at<br />
every element edge and then transformed to the characteristic space.<br />
Remark 2. If the solution at an element is limited then the order <strong>of</strong> accuracy<br />
does not drop to P 0 , since a modified slope is preserved for the approximation.
CHAPTER 6. UNIFIED LIMITING 136<br />
6.1 Positivity preserving limiters for the DG method<br />
The assumption <strong>of</strong> a thermally perfect gas for the equations <strong>of</strong> gas dynamics lead to<br />
the relation between the total energy, pressure and kinetic energy in a flow field given<br />
by Eq. (2.21). It is easily verified that the pressure p is a concave function <strong>of</strong> the state<br />
variables, e.g. defining two states W 1 = [ρ 1 , u 1 , v 1 , E 1 ] T and W 2 = [ρ 2 , u 2 , v 2 , E 2 ] T<br />
it is implied from Jensen’s inequality for 0 ≤ s ≤ 1, that:<br />
p (sW 1 + (1 − s) W 2 ) sp (W 1 ) + (1 − s) p (W 2 ) if ρ 1 , ρ 2 0, (6.18)<br />
and the set <strong>of</strong> admissible states is defined by:<br />
G = {W | ρ, p > 0}, (6.19)<br />
which is convex and this is the key observation for the positivity preserving limiters<br />
for the density and pressure fields developed by Zhang and Shu in [156] where the<br />
solutions coefficients are limited in such a way, so that:<br />
ˆ For smooth solutions accuracy is preserved.<br />
ˆ The scheme remains conservative.<br />
ˆ Limiting for positivity <strong>of</strong> density and pressure is performed locally at every<br />
element.<br />
As in [156] the TVB limiter is applied before the positivity limiters, which will<br />
be described hereafter, but taking into consideration that for the overall success <strong>of</strong> a<br />
positivity preserving scheme, a numerical flux that is preserving the positivity <strong>of</strong> the<br />
variables must be employed, such as the Lax-Friedrichs or the local-Lax-Friedrichs<br />
(Eq. (4.9)). Furthermore, in [112] the positivity <strong>of</strong> an Euler explicit scheme was<br />
analyzed and a sufficient condition was derived for the mean solution <strong>of</strong> the state<br />
variables to be in the set G:
CHAPTER 6. UNIFIED LIMITING 137<br />
∆t<br />
∆x || (|u| + c) || ∞ ≤ aα, (6.20)<br />
where a ∈ (0, 1]. SSP high order Runge-Kutta methods then are suitable for the<br />
positiveness <strong>of</strong> high order time accurate discretizations.<br />
The first step is to limit the coefficients for the density field. This is accomplished by<br />
first computing the minimum value <strong>of</strong> the density ρ min looping over the quadrature<br />
points on the edges. Then the following parameter is defined:<br />
( c<br />
ρ )<br />
0 − ε<br />
θ 1 = min<br />
c ρ , 1 , (6.21)<br />
0 − ρ min<br />
where ε = min (1.0 −13 , c ρ 0, p, ) and p is the mean value <strong>of</strong> the computed pressure.<br />
Then, the final coefficients for the density expansion are modified as follows:<br />
˜c ρ i = θ 1c ρ i , (6.22)<br />
where the index i counts all the bases except for the one corresponding to the mean<br />
solution.<br />
The second step is to limit the pressure. This requires the alteration <strong>of</strong> all the<br />
solution coefficients for every conservative variable. This is accomplished as follows.<br />
Define:<br />
s = (1 − t)W + tq, (6.23)<br />
where W is the mean solution and q is the conservative variables state vector with<br />
the limited density solution. First, calculate:<br />
t = the solution <strong>of</strong> p(s) = ε, if p(q) < ε, (6.24)<br />
then, modify the solution coefficients for each field except for the ones corresponding
CHAPTER 6. UNIFIED LIMITING 138<br />
to the mean solution as:<br />
˜c i = θ 2 c i , (6.25)<br />
where θ 2 is the limiting function given by the following equation:<br />
θ 2 = min (1, t). (6.26)<br />
This limiter was applied only for quadrilateral elements by Zhang and Shu<br />
[156]. In this work, applications for unstructured triangular meshes are demonstrated<br />
with excellent results.
Chapter 7<br />
Numerical results<br />
In the present work a CFD solver based on the previously described DG discretization<br />
was developed and named HoAc as an abbreviation for High Order ACcuracy.<br />
The unified limiting procedure has been implemented in HoAc . Many cases <strong>of</strong> high<br />
speed flows with strong shocks have been analyzed and are presented in this chapter<br />
to demonstrate the improvements obtained with the proposed limiting approach.<br />
The results confirm the outstanding shock capturing and positivity preserving capabilities<br />
<strong>of</strong> the novel limiting approach. Moreover, applications for supersonic inviscid<br />
and subsonic viscous cases will be presented.<br />
In the computations with the new limiting approach, the value M in Eq. (6.1)<br />
was computed at each time step for each characteristic field. The variation <strong>of</strong> the<br />
computed maximum value <strong>of</strong> the Laplacian for each field in the characteristic space:<br />
W 1 ≡ δw 1 = δρ−δp/c 2 , W 2 ≡ δw 2 = −k x δu−k y δv, W 3 ≡ δw 3 = k x δu+k y δv+δp/ρc,<br />
and W 4 ≡ δw 4 = −k x δu − k y δv + δp/ρc, where k = [k x , k y ] is an arbitrary direction<br />
was recorded and will be presented in many cases.<br />
First, verification tests <strong>of</strong> the spatial DG discretization implemented in the<br />
standard space Ω st in HoAc will be presented followed by verification and validation<br />
tests for the limiting procedure, which are performed by comparing the analytical<br />
solution for the standard Sod’s shock tube problem and again for the Sod’s problem<br />
with an extreme pressure ratio. The supersonic flow over a cylinder is considered<br />
139
CHAPTER 7. NUMERICAL RESULTS 140<br />
next employing a P 2 and a P 3 expansion <strong>of</strong> the solution, followed by standard test<br />
cases involving flows with strong shocks proposed in [152]. These standard test cases<br />
for shock capturing schemes include the double mach reflection <strong>of</strong> a Mach 10 strong<br />
shock and a supersonic flow at Mach number <strong>of</strong> 3 in a wind tunnel with a forward<br />
facing step. The Shardin’s problem [60] is solved next, which also confirms the very<br />
good resolution <strong>of</strong> the developed limiting procedure, followed by a test case <strong>of</strong> an<br />
impinging shock at Mach Mach number <strong>of</strong> 5 around the same geometry. Finally, the<br />
diffraction <strong>of</strong> a strong shock over a backward facing step at Mach, Mach number <strong>of</strong><br />
5 is considered for quadrilateral, rectangular and triangular meshes using P 2 and P 3<br />
expansions <strong>of</strong> the solution. Furthermore, it is noted that in all test cases the LLF<br />
flux has been applied and the GLL quadrature points are used for the computation<br />
<strong>of</strong> the integrals. Also, for the application <strong>of</strong> the IC Galerkin projection is employed.<br />
7.1 Solution output<br />
Most present CFD solvers store the solution at the element center or at the nodes in<br />
the mesh. However, a DG method computes the solution in a more general fashion,<br />
as for modal expansions especially the solution coefficients do not have a specific<br />
physical point <strong>of</strong> relevance.<br />
As the solution is discontinuous across the element edges two approaches have<br />
been followed in the present work for outputting and post processing <strong>of</strong> the computed<br />
results: (i) averaging <strong>of</strong> the solution at the nodes <strong>of</strong> the computational mesh<br />
and (ii) a discontinuous output <strong>of</strong> the solution, which is based on outputting the solution<br />
at points over the computational element, whose coordinates are determined<br />
by the inverse transformation <strong>of</strong> equally spaced points in the standard element configuration.<br />
Both approaches give results that are very similar on fine meshes, and for<br />
that, in test cases where fine meshes have been used averaged print <strong>of</strong> the solution<br />
is used.<br />
The discontinuous output <strong>of</strong> the solution leads to a visualization mesh that<br />
is finer than the computational mesh. In Fig. 7.1 a portion <strong>of</strong> a mixed-type ele-
CHAPTER 7. NUMERICAL RESULTS 141<br />
ment computational mesh is shown together with the visualization mesh for a P 4<br />
expansion <strong>of</strong> the solution over every computational element.<br />
Figure 7.1: Computational mesh (blue lines) and visualization mesh (black lines)<br />
for a P 4 expansion <strong>of</strong> the solution over triangular and quadrilateral elements.<br />
In Appendix B the pseudo code for constructing the visualization mesh over a<br />
quadrilateral and a triangular element is provided.<br />
7.2 Code Verification<br />
In this section the verification <strong>of</strong> the implemented DG discretization for the Euler<br />
and the Navier-Stokes equations will be presented, which confirm the correctness <strong>of</strong><br />
the solution algorithms employed in the HoAc solver.<br />
7.2.1 Convergence study for the Euler equations<br />
For verifying the correctness <strong>of</strong> the spatial DG discretization <strong>of</strong> the Euler equations,<br />
a stationary isentropic vortex is considered described by the following IC in the flow<br />
variables:
CHAPTER 7. NUMERICAL RESULTS 142<br />
ρ =<br />
[1 −<br />
] 1<br />
γ−1<br />
(γ − 1) β2<br />
e (1−r2 ) , (7.1)<br />
8γπ<br />
(δu, δv) = β “<br />
2π e 1−r<br />
2 ”<br />
2<br />
[− (y − y 0 ) , (x − x 0 )] , (7.2)<br />
p = ργ<br />
γ . (7.3)<br />
As the above IC satisfy exactly the Euler equations the residual <strong>of</strong> the system in<br />
every field must drop accordingly to the order <strong>of</strong> discretization using a refined mesh.<br />
In Fig. 7.2 the convergence rates for the residual <strong>of</strong> the system corresponding<br />
to the density field is given starting from a P 1 and going up to a P 4 expansion <strong>of</strong> the<br />
solution using the LLF flux. It is observed that the design order <strong>of</strong> the discretization<br />
is achieved.<br />
1<br />
0.01<br />
2<br />
L 2<br />
Error<br />
0.0001<br />
3<br />
P1<br />
P2<br />
P3<br />
P4<br />
1e-06<br />
4<br />
5<br />
1e-08<br />
0.1 1<br />
∆x,∆y<br />
Figure 7.2: Convergence rate for the DG discretization <strong>of</strong> the Euler equations.
CHAPTER 7. NUMERICAL RESULTS 143<br />
7.2.2 Convergence study for the Navier-Stokes equations<br />
The correctness <strong>of</strong> the results for viscous computations is verified by considering the<br />
test case corresponding to the Poiseuille flow, which simply refers to a flow between<br />
two parallel plates under a pressure gradient. An analytical solution <strong>of</strong> this problem<br />
exists under the assumptions <strong>of</strong> incompressible flow and constant viscosity and in<br />
conservative variables is the following:<br />
⎡<br />
U exact =<br />
⎢<br />
⎣<br />
ρ 0<br />
γ−1<br />
⎤<br />
ρ<br />
y (y − b)<br />
0<br />
. (7.4)<br />
(<br />
p0 + dp x) ⎥<br />
+ ρ dx 2 u2 ⎦<br />
1 dp<br />
2µ 0 dx<br />
As HoAc solves the compressible Navier-Stokes equations, the introduction <strong>of</strong> a<br />
source term S to the right hand side <strong>of</strong> the system given in Eq. (2.29) is necessary<br />
in order to retrieve numerical results from the compressible Navier-Stokes equations<br />
corresponding to the incompressible solution given in Eq. (7.4). The source term S<br />
is equal to:<br />
⎡<br />
S =<br />
⎢<br />
⎣<br />
1<br />
2µ<br />
( dp<br />
dx<br />
) 2<br />
[<br />
1<br />
k−1<br />
0<br />
0<br />
0<br />
y (y − b) −<br />
(2y−b)2<br />
2<br />
⎤<br />
. (7.5)<br />
⎥<br />
] ⎦<br />
.<br />
In Fig. 7.3 the convergence rates for the DG discretization <strong>of</strong> the compressible<br />
Navier-Stokes equations using the LDG method are given for a P 1 up to P 4<br />
expansion <strong>of</strong> the solution, with respect to the element length. It is observed that<br />
the design order <strong>of</strong> the method is achieved.
CHAPTER 7. NUMERICAL RESULTS 144<br />
0.1<br />
0.01<br />
L 2<br />
Error<br />
0.001<br />
0.0001<br />
1e-05<br />
1e-06<br />
1e-07<br />
2<br />
3<br />
4<br />
5<br />
P1<br />
P2<br />
P3<br />
P4<br />
1e-08<br />
0.1 1<br />
∆x,∆y<br />
Figure 7.3: Convergence rate for the DG discretization <strong>of</strong> the Navier-Stokes equations<br />
with the LDG method.<br />
7.2.3 Inviscid flow over a cylinder<br />
The potential <strong>of</strong> the discontinuous output <strong>of</strong> the solution is depicted for the computation<br />
<strong>of</strong> the flow over a cylinder at a low Mach number (M ∞ = 0.14). Solutions are<br />
computed with a fixed mixed-type linear element mesh with quadrilateral elements<br />
in the vicinity <strong>of</strong> the cylinder and starting from a P 1 and going up to a P 3 expansion<br />
<strong>of</strong> the solution. It is noted that the curvature based boundary conditions proposed<br />
in [92] have been applied, due to the use <strong>of</strong> straight sided elements, otherwise an<br />
unsteady wake is formed and a steady state solution is not reached.
CHAPTER 7. NUMERICAL RESULTS 145<br />
(a) Computational mesh<br />
(b) P 1 expansion<br />
(c) P 2 expansion<br />
(d) P 3 expansion<br />
Figure 7.4: Mach contours for the inviscid flow over a cylinder using discontinuous<br />
output <strong>of</strong> the solution and the curvature based boundary conditions in [92].<br />
It is observed in Fig. 7.4 that as the order <strong>of</strong> the solution expansion increases so does<br />
the quality <strong>of</strong> the numerical results. Furthermore, in Fig. 7.5 the computed pressure<br />
coefficient at every quadrature point along the edges defining the cylinder geometry<br />
is compared with the analytical solution for the pressure distribution around the<br />
cylinder: C p = 1 − 3 sin 2 (θ). It is clear that as the order <strong>of</strong> the approximation<br />
increases, the numerically evaluated pressure distribution approaches that given by<br />
the analytical solution.
CHAPTER 7. NUMERICAL RESULTS 146<br />
Figure 7.5: Comparison between the numerically evaluated pressure coefficient and<br />
the analytical solution for the inviscid flow around a cylinder.<br />
This test case was solved using an implicit first order backward Euler scheme,<br />
with a step size <strong>of</strong> dt = 1.0 using an unpreconditioned matrix-free approach. The<br />
large number <strong>of</strong> iterations observed in the convergence history plot in Fig. 7.6 for<br />
the density field is attributed to the placement <strong>of</strong> the far field boundary at a distance<br />
<strong>of</strong> 20 unit lengths from the cylinder center.
CHAPTER 7. NUMERICAL RESULTS 147<br />
Figure 7.6: Convergence history for the residual in the density field for the inviscid<br />
flow over a cylinder.<br />
7.2.4 Standard Sod’s shock tube problem<br />
The standard Sod’s shock tube problem with IC [ρ, u, v, p] L = [1, 0, 0, 1] and [ρ, u, v, p] R =<br />
[0.125, 0, 0, 1], which is an exact solution <strong>of</strong> the 1D Euler equations is solved in two<br />
dimensional triangular and quadrilateral meshes, which are shown in Fig. 7.7. This<br />
problem is the first validation test for the developed limiting procedure.
CHAPTER 7. NUMERICAL RESULTS 148<br />
(a) Quadrilateral mesh<br />
(b) Triangular mesh<br />
Figure 7.7: Meshes for the two dimensional Sod’s shock tube problem.<br />
The evolution <strong>of</strong> the limiting procedure for a P 1 numerical solution is depicted<br />
in Fig. 7.8 for the computations performed on the quadrilateral mesh.<br />
0.25<br />
0.2<br />
Time<br />
0.15<br />
0.1<br />
0.05<br />
0<br />
-1 -0.5 0 0.5 1<br />
Length<br />
Figure 7.8: Limited elements in time for the Sod’s shock tube problem using the<br />
quadrilateral mesh <strong>of</strong> Fig. 7.7a.<br />
It is observed that the limiting procedure follows the discontinuities appearing in<br />
this Riemann problem. The contact discontinuity is smeared as time advances and<br />
limiting is performed at the start and end <strong>of</strong> the contact. Similarly, at the start<br />
and end <strong>of</strong> the rarefaction wave the elements are limited. For the smooth parts <strong>of</strong><br />
the flow limiting has been applied in few elements. The final density and entropy<br />
pr<strong>of</strong>iles at time t = 0.25 are shown in Fig. 7.9 and compared with the exact result
CHAPTER 7. NUMERICAL RESULTS 149<br />
for the computations performed on the triangular channel mesh. The numerical<br />
solution was recorded along the center line <strong>of</strong> the channel mesh shown in Fig. 7.7b.<br />
The P 1 computed density distribution <strong>of</strong> Fig. 7.9a agrees quite well with the exact<br />
result. The comparison <strong>of</strong> the most sensitive entropy distribution given in Fig. 7.9b<br />
shows that only small oscillations exist at the location <strong>of</strong> the contact and that the<br />
shock is essentially captured within two elements.<br />
2<br />
1<br />
Exact Solution<br />
Numerical Solution P1<br />
1.8<br />
Numerical Solution P1<br />
Exact Solution<br />
1.6<br />
Density<br />
0.5<br />
Entropy<br />
1.4<br />
1.2<br />
1<br />
0<br />
-1 -0.5 0 0.5 1<br />
Length<br />
(a) Density<br />
-1 -0.5 0 0.5 1<br />
Length<br />
(b) Entropy<br />
Figure 7.9: Comparison <strong>of</strong> the exact density and entropy variation at t = 0.25 with<br />
the two-dimensional numerical solution for the Sod’s shock tube problem in the<br />
triangular channel mesh <strong>of</strong> Fig. 7.7b.<br />
7.2.5 Extreme Sod’s Riemann problem<br />
In order to demonstrate the proper implementation <strong>of</strong> the positivity preserving<br />
limiters a one-dimensional Riemann problem with an exact solution for an extreme<br />
pressure ratio was considered. This problem was solved with a two dimensional<br />
rectangular mesh and it is noted the TVB limiter is applied before the application<br />
<strong>of</strong> the positivity limiters. For this problem, as the solution was advanced in time<br />
it was verified that the positivity preserving limiters were activated. The Sod’s<br />
shock tube problem is considered with the following IC [ρ, r, v, p] l = [1.0, 0, 0, 100],<br />
[ρ, u, v, p] r = [0.125, 0, 0, 0.001], and a specific heat ratio γ = 1.4. For this shock<br />
tube problem an extreme pressure ratio p l /p r = 10 5 exists. In Figs. 7.10, 7.11 and<br />
7.12 the exact and the numerical solution computed with a P 1 expansion and 800
CHAPTER 7. NUMERICAL RESULTS 150<br />
equally sized elements are compared. The computed density, pressure, and entropy<br />
are in good agreement with the exact result.<br />
1<br />
Numeical Solution<br />
Exact Solution<br />
Density<br />
0.5<br />
0<br />
-1 -0.5 0 0.5 1<br />
Length<br />
Figure 7.10: Density plot for the Sod’s problem with a pressure ratio <strong>of</strong> 100000.<br />
100<br />
Numerical Solution<br />
Exact Solution<br />
Pressure<br />
50<br />
0<br />
-1 -0.5 0 0.5 1<br />
Length<br />
Figure 7.11: Pressure plot for the Sod’s problem with a pressure ratio <strong>of</strong> 100000.<br />
100<br />
Entropy<br />
50<br />
Numerical Solution<br />
Exact Solution<br />
0<br />
-1 -0.5 0 0.5 1<br />
Length<br />
Figure 7.12: Entropy plot for the Sod’s problem with a pressure ratio <strong>of</strong> 100000.
CHAPTER 7. NUMERICAL RESULTS 151<br />
For this problem an explicit third-order accurate SSP Runge-Kutta method<br />
was employed with 25 stages provided by the PETSc library package. The time<br />
step used in the computations was equal to dt = 2.0 · 10 −7 and the final time <strong>of</strong> the<br />
simulation was equal to t = 5.0 · 10 −2 . This very low step size was needed as the<br />
maximum Mach number developed in the flow approached the value <strong>of</strong> 12.<br />
7.3 Supersonic flow over a cylinder<br />
The inviscid, supersonic flow at Mach number <strong>of</strong> 2 over a cylinder is considered<br />
with a P 2 and P 3 expansion <strong>of</strong> the solution in order to demonstrate the developed<br />
p-adaptive limiting procedure for arbitrary shaped elements, which is noted again<br />
that it is being performed in the characteristic variable space and over the standard<br />
square element <strong>of</strong> the computational space. Three types <strong>of</strong> meshes quadrilateral,<br />
triangular and a mixed-type element mesh were used for the computations which<br />
are shown in Figs. 7.13, 7.14 and 7.15. An impulsive start <strong>of</strong> the numerical solution<br />
is performed with the following IC: [ρ, u, v, p] = [1.0, 2.0, 0.0, 0.714285714]<br />
Figure 7.13: Quadrilateral mesh for supersonic flow at Mach 2 around a cylinder.
CHAPTER 7. NUMERICAL RESULTS 152<br />
Figure 7.14: Triangular mesh for supersonic flow at Mach 2 around a cylinder.<br />
Figure 7.15: Mixed type mesh for supersonic flow at Mach 2 around a cylinder.<br />
The computed pressure field, which is the same for the numerical solutions obtained<br />
with different meshes, is shown in Fig. 7.16 for a P 3 solution expansion over the<br />
quadrilateral mesh.
CHAPTER 7. NUMERICAL RESULTS 153<br />
Figure 7.16: Pressure contour lines using 30 equally spaced intervals for flow at Mach<br />
2 over a cylinder for the numerical solution obtained with a P 3 approximation.<br />
It is emphasized that once an element is flagged for limiting based on the value<br />
<strong>of</strong> the estimate <strong>of</strong> the Laplacian (second derivative) M, this element is marked as<br />
limited. For a P 2 global expansion the elements next to the limited elements are<br />
P 1 and for the rest <strong>of</strong> the domain are P 2 . For computations with preset higher<br />
order P n expansions and for elements neighboring the P 2 elements the expansion<br />
will be raised to P 3 and progressively to P n . This is demonstrated in Fig. 7.17 for<br />
a computation with P 3 global approximation for a quadrilateral mesh.
CHAPTER 7. NUMERICAL RESULTS 154<br />
Figure 7.17: Limited elements for the flow at Mach 2 over a cylinder for the numerical<br />
solution obtained with a quadrilateral mesh and a P 3 approximation; the blue<br />
elements are limited the adjacent green elements are P 1 expansions, the adjacent<br />
to green red elements are P 2 , and for the rest <strong>of</strong> the domain P 3 expansions are<br />
employed.<br />
The limited elements for a P 2 computation with the triangular and quadrilateral<br />
are shown in Fig. 7.18.
CHAPTER 7. NUMERICAL RESULTS 155<br />
(a) Mixed-type mesh (b) Quadrilateral mesh (c) Triangular mesh<br />
Figure 7.18: Limited elements for the flow at Mach 2 over a cylinder for the numerical<br />
solution obtained with a P 2 approximation; the blue elements are limited<br />
the adjacent red elements are P 1 expansions and for the rest <strong>of</strong> the domain P 2<br />
expansions are employed.<br />
It can be seen that limiting has been applied only for elements neighboring the<br />
shock. For a P 2 global expansion the elements next to the limited elements and for<br />
the rest <strong>of</strong> the domain are P 2 and only the limited elements are P 1 . It is noted<br />
that on the mixed-type element mesh the limited elements are not visible as they<br />
are concentrated in the region where the mesh is very much refined.<br />
A comparison <strong>of</strong> the computed pressure distribution along the stagnation line<br />
obtained with different meshes and polynomial expansions is shown in Fig. 7.19.<br />
The agreement <strong>of</strong> the computations is very good. The computation on the refined<br />
mixed-type element mesh has the sharpest capturing <strong>of</strong> the shock.
CHAPTER 7. NUMERICAL RESULTS 156<br />
Figure 7.19: Comparison <strong>of</strong> the pressure distribution along the stagnation line obtained<br />
from P 2 and P 3 approximations <strong>of</strong> the numerical solutions with different<br />
meshes.<br />
The variation <strong>of</strong> the Laplacian estimate, in essence the parameter M in the<br />
TVB limiter in the current test case, for each field in the characteristic space W i , is<br />
shown in Fig. 7.20 for the P 2 computation using quadrilateral elements.
CHAPTER 7. NUMERICAL RESULTS 157<br />
Figure 7.20: Variation <strong>of</strong> the maximum value <strong>of</strong> parameter M for each characteristic<br />
field obtained during convergence to steady state <strong>of</strong> the numerical solution for the<br />
flow at Mach 2 over a cylinder with a quadrilateral mesh and P 2 solution expansion.<br />
It must be emphasized that until now numerical solutions with the DG method<br />
and the TVB limiters were presented only for rectangular elements and for triangular<br />
elements following the elaborate procedure <strong>of</strong> Cockburn [82], but from the current<br />
test case it is observed that the developed limiting approach works quite well for<br />
elements <strong>of</strong> arbitrary shape, mixed-type element meshes and p-adaptive solutions.<br />
Furthermore, a crisp resolution <strong>of</strong> the shocks is obtained since M < 3 (in contrast to<br />
the suggested value <strong>of</strong> M = 50 by [82]) and fewer elements are limited as depicted<br />
in Fig. 7.18.<br />
7.4 Double Mach Reflection <strong>of</strong> a strong shock<br />
Next, the complex shock pattern generated by the reflection <strong>of</strong> a moving shock at<br />
Mach number <strong>of</strong> 10 inclined at an angle <strong>of</strong> θ = 60 ◦ with respect to a horizontal wall<br />
[152] is solved numerically in order to investigate the potential <strong>of</strong> the p-adaptive
CHAPTER 7. NUMERICAL RESULTS 158<br />
limiting procedure for a flow with a strong moving shock. This is a standard test<br />
problem for shock capturing schemes. It originated by experimental and numerical<br />
studies <strong>of</strong> reflections <strong>of</strong> planar shock waves from wedges. The computational domain<br />
is a rectangle with a width <strong>of</strong> four units and a height <strong>of</strong> one unit. The shock is initially<br />
set up to be inclined at an angle <strong>of</strong> 60 ◦ to the x-axis, and has a Mach number <strong>of</strong><br />
10. Therefore, the IC are: [ρ, u, v, p] = [8.0, 8.25 cos (30 ◦ ) , −8.25 cos (30 ◦ ) , 116.5] for<br />
x < 1 6 + y √<br />
3<br />
and [ρ, u, v, p] = [1.4, 0, 0, 1.0] for x 1 6 + y √<br />
3<br />
. At x = 0 the boundary<br />
conditions are specified to the values corresponding to the post-shock state, and at<br />
x = 4 extrapolation <strong>of</strong> the computed variables is performed. The lower y boundary<br />
for x 1 is a reflecting wall and for x < 1 the fluid values are set by the initial<br />
6 6<br />
post-shock conditions. The upper y boundary is constructed to follow the flow <strong>of</strong> the<br />
diagonal shock such that there is no interaction between the shock and this boundary.<br />
Given the IC here, the intersection <strong>of</strong> the diagonal shock and the upper boundary at<br />
time t occurs at x s (t) = 1 + 1+20t √<br />
6 3<br />
. For x x s (t), the values at the upper y boundary<br />
are set by the initial pre-shock conditions. For x < x s (t), the state variable values<br />
are set by the initial post-shock conditions. In [82, 91, 119] this problem was solved<br />
using a DG discretization <strong>of</strong> the Euler equations on rectangular meshes, but in the<br />
present work is solved using quadrilateral, rectangular and triangular meshes. Both<br />
P 1 and P 2 approximations were used for the numerical solution that was run to a<br />
final time <strong>of</strong> t = 0.2 using the third order explicit SSP Runge-Kutta developed in<br />
[130]. For demonstration, a solution obtained in a relatively coarse quadrilateral<br />
mesh is shown first. The limited elements are shown in Fig. 7.21.<br />
Figure 7.21: Limited elements on the coarse quadrilateral mesh for the double mach<br />
reflection problem at t = 0.2.
CHAPTER 7. NUMERICAL RESULTS 159<br />
It is seen that even for a coarse mesh limiting is applied in a relatively narrow region<br />
around the discontinuities. The final density at t = 0.2 obtained from the numerical<br />
solution with the coarse mesh is shown in Fig. 7.22.<br />
Figure 7.22: Density contours on the coarse quadrilateral mesh for the double mach<br />
reflection problem at t = 0.2.<br />
The numerical solution obtained on a mesh with rectangular elements <strong>of</strong> size<br />
h = 1/240, with a P 1 expansion <strong>of</strong> the solution, for the density field is shown in Fig.<br />
7.23<br />
Figure 7.23: Density contours for the numerical solution computed on the rectangular<br />
mesh, h = 1/240, and a P 1 approximation for the double mach reflection<br />
problem.<br />
and the elements where limiting <strong>of</strong> the solution was performed are shown for time<br />
t = 0.1 and t = 0.2 in Fig. 7.24,
CHAPTER 7. NUMERICAL RESULTS 160<br />
(a) Limited elements at t = 0.1<br />
(b) Limited elements at t = 0.2<br />
Figure 7.24: Limited elements for the numerical solution computed on the rectangular<br />
mesh, h = 1/240, and a P 1 approximation for the double mach reflection. The<br />
blue elements are limited and for the rest <strong>of</strong> the domain a P 1 approximation <strong>of</strong> the<br />
solution is employed.<br />
where the domain decomposition employed for parallel computation is highlighted.<br />
Clearly, seamless limiting is achieved for parallel processing. The computed density<br />
<strong>of</strong> Fig. 7.23 agrees well with other numerical solutions in the literature [82, 91, 119]<br />
obtained with high resolution shock capturing schemes <strong>of</strong> equivalent order.<br />
A numerical solution for an isotropic mesh with triangular elements whose<br />
edge length is equal to h = 1/240 was also obtained for a P 1 solution expansion.<br />
The limited elements at time t = 0.1 and t = 0.2, which are marked in Fig. 7.25,<br />
show that limiting is performed at narrow zones around the discontinuities. The<br />
numerical solution for the density field computed on the triangular mesh is shown<br />
in Fig. 7.26.
CHAPTER 7. NUMERICAL RESULTS 161<br />
(a) Limited elements at t = 0.1<br />
(b) Limited elements at t = 0.2<br />
Figure 7.25: Limited elements for the numerical solution computed on the triangular<br />
mesh, h = 1/240, and a P 1 approximation for the double mach reflection. The blue<br />
elements are limited and for the rest <strong>of</strong> the domain a P 1 approximation <strong>of</strong> the<br />
solution is employed.<br />
Figure 7.26: Density contours for the numerical solution computed on the triangular<br />
mesh, h = 1/240, and a P 1 approximation for the double mach reflection problem.<br />
This test case was also run on the same triangular mesh using a P 2 expansion<br />
<strong>of</strong> the solution. The density field at t = 0.2 is shown in Fig. 7.27 which agrees very<br />
well with the P 1 solution obtained with the rectangular mesh and with the results<br />
given in [119, 152].
CHAPTER 7. NUMERICAL RESULTS 162<br />
Figure 7.27: Density contours for the numerical solution computed on the triangular<br />
mesh, h = 1/240, and a P 2 approximation for the double mach reflection problem.<br />
The variation <strong>of</strong> the parameter M in the application <strong>of</strong> the TVB limiter for<br />
the P 2 solution on the isotropic triangular mesh is depicted in Fig. 7.28, and it is<br />
observed a wide variation <strong>of</strong> its values for each characteristic field.
CHAPTER 7. NUMERICAL RESULTS 163<br />
Figure 7.28: Variation <strong>of</strong> the maximum value <strong>of</strong> parameter M for each characteristic<br />
field obtained during the time advancement <strong>of</strong> the numerical solution for the double<br />
mach reflection problem obtained with a uniform triangular element mesh with a<br />
P 2 approximation and h = 1/240.<br />
7.5 Inviscid flow at Mach number <strong>of</strong> 3 in a tunnel<br />
with a step<br />
The inviscid supersonic flow at Mach number <strong>of</strong> 3 in a tunnel with a step has been<br />
computed with rectangular, triangular and mixed-type element meshes in order to<br />
demonstrate the flexibility and potential <strong>of</strong> the proposed limiting approach. This<br />
is another standard test case for shock capturing schemes and was introduced by<br />
Emery in [51]. The wind tunnel is one unit length wide and three unit lengths
CHAPTER 7. NUMERICAL RESULTS 164<br />
long. The step is 0.2 unit lengths high and is located 0.6 length units from the<br />
left-hand end <strong>of</strong> the tunnel. The corner <strong>of</strong> the step is the center <strong>of</strong> a rarefaction fan<br />
and hence is a singular point <strong>of</strong> the flow field which is seriously affected by large<br />
numerical errors generated just in the neighborhood <strong>of</strong> this singular point. This<br />
may be alleviated in two ways: (i) by using a fine mesh in the region close to the<br />
corner or (ii) by applying special boundary conditions near the corner <strong>of</strong> the step as<br />
demonstrated by Woodward and Collela in [152]. Specifically, the density is reset<br />
so that the entropy has the same value in the zone just to the left and below the<br />
corner <strong>of</strong> the step and the magnitudes <strong>of</strong> the velocities are reset also, but not their<br />
direction, so that the sum <strong>of</strong> enthalpy and kinetic energy per mass has the same<br />
value as in the same zone used to set the entropy. In the present work the first<br />
method is followed.<br />
The numerical solution with the following IC [ρ, u, v, p] = [1.4, 3, 0, 1.0] for the<br />
tunnel with a step was obtained with a P 1 and P 2 expansions, because for computations<br />
with higher order expansions severe time step limitations are encountered<br />
due to the very small elements that had to be used at the corner in order to reduce<br />
the artificial entropy layers generated by the corner singularity.<br />
The first computation was performed with a P 1 expansion <strong>of</strong> the solution for<br />
a mesh with rectangular elements with the edge length equal to h = 1/80. In Figs.<br />
7.29 and 7.30 the limited elements and the numerical solution for the density field<br />
are shown at t = 2.0, respectively.<br />
Figure 7.29: Limited elements for the numerical solution obtained on a rectangular<br />
mesh and P 1 approximation and h = 1/80 for the flow at Mach number <strong>of</strong> 3 in a<br />
wind tunnel with a forward facing step at t = 2.0.
CHAPTER 7. NUMERICAL RESULTS 165<br />
Figure 7.30: Density field for the numerical solution obtained on a rectangular mesh<br />
and P 1 approximation and h = 1/80 for the flow at Mach number <strong>of</strong> 3 in a wind<br />
tunnel with a forward facing step at t = 2.0.<br />
The limited elements <strong>of</strong> Fig. 7.29 closely match the captured discontinuities <strong>of</strong> Fig.<br />
7.30. At later times <strong>of</strong> the simulation the artificial entropy layer generated by the<br />
singularity at the corner creates a Mach stem. In an effort to alleviate the error by<br />
the singularity at the corner, a numerical solution on a isotropic fine triangular mesh<br />
with an average edge length <strong>of</strong> h = 1/100 and for a P 2 expansion <strong>of</strong> the solution<br />
was performed. The limited elements for this numerical solution along with the<br />
computed density field at t = 4 are shown in Figs. 7.31 and 7.32:<br />
Figure 7.31: Limited elements for the numerical solution obtained on a fine triangular<br />
mesh and a P 2 approximation and h = 1/100 for the flow at M = 3.0 in a<br />
wind tunnel with a forward facing step at t = 4.0. Blue elements: limited solution.<br />
Green elements P 1 expansion. Red elements P 2 expansion.
CHAPTER 7. NUMERICAL RESULTS 166<br />
Figure 7.32: Density field for the numerical solution obtained on a fine triangular<br />
mesh and P 2 approximation and h = 1/100 for the flow at Mach number <strong>of</strong> 3 in a<br />
wind tunnel with a forward facing step at t = 4.0.<br />
It is observed that the Mach stem still appears in the flow field. This lead to a third<br />
effort for a numerical solution on a mixed-type element mesh refined at the corner<br />
with triangular elements as shown in Fig. 7.33.<br />
Figure 7.33: Mixed-type element mesh for the inviscid flow at Mach number <strong>of</strong> 3 in<br />
a tunnel with a step.<br />
The average edge length size in the quadrilateral region <strong>of</strong> the mesh is h = 1/40 and<br />
in the triangular region the edge length is equal to h = 3.0 · 10 −3 . The computed<br />
density field at t = 4 is shown in Fig. 7.34. It is observed that the Mach stem has<br />
diminished considerably.
CHAPTER 7. NUMERICAL RESULTS 167<br />
Figure 7.34: Density field for the numerical solution obtained on a mixed-type<br />
element mesh and a P 1 approximation for the flow at Mach number <strong>of</strong> 3 in a wind<br />
tunnel with a forward facing step at t = 4.0.<br />
Furthermore, in Fig. 7.35 the limited elements on the mixed-type element<br />
mesh computation are shown at t = 0.1, where the TVB limiter has smoothly ”fireup”<br />
at the narrow region <strong>of</strong> the developing shock wave in the part <strong>of</strong> the mesh<br />
covered by quadrilateral and triangular elements.<br />
Figure 7.35: Limited elements at t = 0.1 using a mixed-type element mesh for the<br />
inviscid flow at Mach number <strong>of</strong> 3 in a wind tunnel with a forward facing step.<br />
The variation <strong>of</strong> the parameter M versus the number <strong>of</strong> time steps is shown<br />
in Fig. 7.36 where it is observed that there is a wide variation <strong>of</strong> its value for each
CHAPTER 7. NUMERICAL RESULTS 168<br />
characteristic field and differs considerably from the constant value <strong>of</strong> M = 50 as<br />
suggested in [82].<br />
Figure 7.36: Variation <strong>of</strong> the parameter M for the TVB limiter applied on the mixedtype<br />
element mesh for the inviscid flow at Mach number <strong>of</strong> 3 in a wind tunnel with<br />
a forward facing step.<br />
7.6 Diffraction <strong>of</strong> a strong shock over a backward<br />
facing step<br />
The effectiveness <strong>of</strong> the positivity preserving limiters combined with the TVB limiter<br />
for shock capturing is tested for the diffraction <strong>of</strong> a Mach number <strong>of</strong> 5.09 right<br />
moving shock over a backward facing step. This problem is solved on meshes with<br />
rectangular, quadrilateral and triangular elements. The IC for this problem is a<br />
right moving shock wave. The element size for the solutions was h = 0.025 for the<br />
rectangular and triangular meshes and <strong>of</strong> comparable size for the region close to the
CHAPTER 7. NUMERICAL RESULTS 169<br />
corner for the mesh with quadrilateral elements. Sample meshes along with contour<br />
lines for the density field at t = 0.8 <strong>of</strong> the diffraction <strong>of</strong> the shock wave are given in<br />
Fig. 7.37.<br />
(a) Quadrilateral elements mesh<br />
(b) Rectangular elements mesh<br />
(c) Triangular elements mesh<br />
Figure 7.37: Sample meshes with density contours at t = 0.8 for the diffraction<br />
<strong>of</strong> a strong shock over a backward facing step using a P 2 solution expansion. The<br />
black square shows where the computational domain was clipped for showing the<br />
underlying computational mesh.
CHAPTER 7. NUMERICAL RESULTS 170<br />
A P 2 expansion <strong>of</strong> the solution has been used in all cases. For demonstration,<br />
a solution with P 3 expansion was also computed. The elements for which limiting<br />
was applied at time t = 2.0 for the computations performed with different meshes<br />
are shown in Fig. 7.38.<br />
(a) Quadrilateral elements<br />
(b) Rectangular elements<br />
(c) Triangular elements<br />
Figure 7.38: Limited elements for the diffraction <strong>of</strong> strong shock over a backward<br />
facing step at t = 2.0. Blue elements: limited solution. Green elements P 1 expansion.<br />
Red elements P 2 expansion.<br />
Comparing Fig. 7.38 with the computed density at the same time in Fig. 7.39:
CHAPTER 7. NUMERICAL RESULTS 171<br />
(a) Quadrilateral elements<br />
(b) Rectangular elements<br />
(c) Triangular elements<br />
Figure 7.39: Density contours for the diffraction <strong>of</strong> strong shock over a backward<br />
facing step at t = 2.0 with 30 equally spaced intervals from 0.06 to 7.16.<br />
it is seen that limited elements were obtained at the shock locations. Comparing<br />
Fig. 7.39 with the pressure distribution in Fig. 7.40:
CHAPTER 7. NUMERICAL RESULTS 172<br />
(a) Quadrilateral elements<br />
(b) Rectangular elements<br />
(c) Triangular elements<br />
Figure 7.40: Pressure contours for the diffraction <strong>of</strong> strong shock over a backward<br />
facing step at t = 2.0 with 30 equally spaced intervals from 0.09 to 31.9.<br />
the location <strong>of</strong> the contact discontinuity can be identified. It appears no limiting was<br />
performed in the regions <strong>of</strong> the contact discontinuities. The line plot <strong>of</strong> Fig. 7.41 for<br />
the density and pressure along a line starting from the left side <strong>of</strong> the computational<br />
domain, ending at its right side and being above the step, also demonstrates the<br />
existence and quite good capturing <strong>of</strong> the contact discontinuity, which is further<br />
justified by the numerically computed gradient <strong>of</strong> pressure and density in Figs. 7.42<br />
and 7.43.
CHAPTER 7. NUMERICAL RESULTS 173<br />
30<br />
25<br />
Density<br />
Pressure<br />
Density, Pressure<br />
20<br />
15<br />
10<br />
5<br />
0<br />
0 2 4 6 8 10 12<br />
Length<br />
Figure 7.41: Plot <strong>of</strong> density and pressure along the line defined by points [0, 7] and<br />
[13, 7].<br />
Figure 7.42: Numerically computed pressure (left) and density (right) gradient field<br />
for the diffraction <strong>of</strong> strong shock over a backward facing step at t = 2.0, using a<br />
rectangular elements mesh.
CHAPTER 7. NUMERICAL RESULTS 174<br />
Figure 7.43: Numerically computed pressure (left) and density (right) gradient field<br />
for the diffraction <strong>of</strong> strong shock over a backward facing step at t = 2.0, using a<br />
triangular elements mesh.<br />
The variation <strong>of</strong> the maximum value <strong>of</strong> the parameter M is shown in Fig.<br />
7.44 versus the iteration number for the simulation on the mesh with rectangular<br />
elements and a P 2 expansion.<br />
30<br />
W 1<br />
25<br />
W 3<br />
W 3<br />
W 4<br />
20<br />
M 15<br />
10<br />
5<br />
0<br />
0 5000 10000<br />
Iteration<br />
Figure 7.44: Variation <strong>of</strong> the second derivative estimate for the diffraction <strong>of</strong> a strong<br />
shock over a backward facing step using a P 2 expansion on a rectangular elements<br />
mesh.
CHAPTER 7. NUMERICAL RESULTS 175<br />
It is observed that there is a wide variation <strong>of</strong> the maximum value <strong>of</strong> the parameter<br />
M for each field as the solution advances in time. In contrast to [156] where fixed a<br />
high value <strong>of</strong> the parameter M was prescribed (M = 100), with the present limiting<br />
approach these values were evaluated during the time advancement <strong>of</strong> the solution<br />
and limiting was performed only where the variation <strong>of</strong> the computed solution was<br />
very large. These regions are most <strong>of</strong> the time shock waves.<br />
Additional computations on the rectangular mesh were performed with P 1 and<br />
P 3 expansions <strong>of</strong> the solution, in order to examine the effect on the results <strong>of</strong> the<br />
combined application <strong>of</strong> the shock capturing and positivity preserving limiters. In<br />
Fig. 7.45 a plot over a line defined by the points [2, 5.5] and [5, 4] at time t = 0.9 <strong>of</strong><br />
density and pressure is shown.<br />
14<br />
12<br />
Pressure P2<br />
Density P1<br />
Denstiy P2<br />
Density P3<br />
Density, Pressure<br />
10<br />
8<br />
6<br />
4<br />
2<br />
0<br />
0.5 1 1.5 2 2.5<br />
Length<br />
Figure 7.45: Plot <strong>of</strong> density and pressure along a line defined by the points [2, 5.5]<br />
and [5, 4] at time t = 0.9.<br />
Due to the fine mesh and the non-limiting <strong>of</strong> the solution at the slip line as it is<br />
evident for the P 2 computation from Fig. 7.38, the resolution <strong>of</strong> the computations<br />
is the same. However, the location <strong>of</strong> the shock has been captured in a different<br />
position for the P 1 computation in comparison with the P 2 and P 3 computations,<br />
which actually for the last two is the same.<br />
The large-scale wiggles at time t = 2.0 <strong>of</strong> Fig. 7.39(a) are due to the irregular<br />
and coarse mesh since the quadrilaterals expand away from the corner. The enlarged
CHAPTER 7. NUMERICAL RESULTS 176<br />
plots <strong>of</strong> Fig. 7.37 also show that some wiggles could be introduced from irregular<br />
meshes (see Fig. 7.39(a)). For the finer and regular rectangular and triangular<br />
meshes <strong>of</strong> Figs. 7.39(b),(c) the wiggles are not caused by the mesh or the TVB<br />
limiter. Note that in Fig. 7.37 the slope limiter and the positivity preserving<br />
limiter have been applied already many times until t = 0.9, but no wiggles in<br />
the computed solution appear. The wiggles at the end <strong>of</strong> the simulation at time<br />
t = 2.0 are caused by small scale disturbances that are generated in the region<br />
between the shock and the contact discontinuity, and they are not well resolved.<br />
In the absence <strong>of</strong> physical viscosity, these disturbances cannot diffuse. Depending<br />
on the numerical viscosity introduced by the numerical scheme, and the resolution<br />
<strong>of</strong> the mesh, some disturbances may dissipate and the rest appear as wiggles in<br />
the computed density and pressure plots. The computed vorticity field at time<br />
t = 2.0, shown bellow in Fig. 7.46, for the simulations performed on the rectangular<br />
and triangular meshes, demonstrate that for the rectangular element mesh these<br />
disturbances are concentrated in the region 1.5 < x < 9 marked by a red line<br />
in Fig. 7.46(a). For the computation with the triangular mesh, which provides<br />
higher resolution per unit area, the onset <strong>of</strong> a second contact discontinuity after<br />
the Mach stem is evident. Further away, this contact could not be resolved in the<br />
computations. For the part <strong>of</strong> the domain were the contact remains unresolved (the<br />
region 5.5 < x < 11 also marked by a red line in Fig. 7.46(b)) wiggles are found in<br />
the computed solution.
CHAPTER 7. NUMERICAL RESULTS 177<br />
(a) Rectangular elements<br />
(b) Triangular elements<br />
Figure 7.46: Vorticity field for the diffraction <strong>of</strong> a Mach 5 shock over a backward<br />
facing step.
CHAPTER 7. NUMERICAL RESULTS 178<br />
intervals<br />
7.7 Shardin’s problem<br />
The effectiveness <strong>of</strong> the novel p-adaptive limiting procedure is further examined for<br />
the standard Schardin’s problem [60] in order to validate the use <strong>of</strong> the limiting<br />
procedure for shock capturing. For this problem a right moving shock at a Mach<br />
number <strong>of</strong> 1.34 impinges on a triangle. An unsteady flow is produced accompanied<br />
by the generation <strong>of</strong> vortices and complex wave patterns resulting from the diffraction<br />
<strong>of</strong> the shock at the triangle’s corners and multiple interactions between the<br />
reflected shock waves and the generated vortices. The complete problem setup and<br />
the experimental results are described in detail in [34].<br />
For this problem two types <strong>of</strong> meshes have been used with different orders <strong>of</strong><br />
expansion for the solution. In order to reduce the problem size only half <strong>of</strong> the<br />
triangle was considered and symmetry conditions were imposed. An unstructured<br />
triangular mesh with a P 1 expansion and approximately 200000 elements and a<br />
quadrilateral mesh with a P 2 expansion and approximately 285000 elements. Both<br />
meshes have an average mesh length <strong>of</strong> h = 0.002. The reason for employing a P 1<br />
expansion on the triangular mesh is that triangles provide almost double resolution<br />
over the same area in the computational domain compared to quadrilateral elements.<br />
The computed density and pressure fields at time t = 128µs are given in<br />
Figs. 7.47 and 7.48. Indeed the computed density field by both triangular and<br />
quadrilateral mesh numerical solutions with P 1 and P 2 expansions respectively are<br />
<strong>of</strong> comparable resolution.
CHAPTER 7. NUMERICAL RESULTS 179<br />
(a) P 1 expansion. Triangular elements<br />
(b) P 2 expansion. Quadrilateral elements<br />
Figure 7.47: Density contours for Schardin’s problem with 30 equal spaced intervals<br />
from 0.6 to 2.4 using a P 2 solution expansion
CHAPTER 7. NUMERICAL RESULTS 180<br />
(a) P 1 expansion. Triangular elements<br />
(b) P 2 expansion. Quadrilateral elements<br />
Figure 7.48: Pressure contours for Schardin’s problem with 30 equal spaced intervals<br />
from 0.4 to 2.14 using a P 2 expansion.<br />
It is observed that both expansion sets and mesh types reveal the same structure <strong>of</strong><br />
the flow field behind the triangle, with the exception, that the results produced by<br />
the P 2 expansion capture each discontinuity in a narrower region. Furthermore, in<br />
Fig. 7.49 where the numerical Schlieren for both polynomial expansions is shown:
CHAPTER 7. NUMERICAL RESULTS 181<br />
(a) P 1 expansion. Triangular elements<br />
(b) P 2 expansion. Quadrilateral elements<br />
Figure 7.49: Numerical Schlieren for Schardin’s problem using a P 2 solution expansion.<br />
it can be observed that the complex wave pattern produced by the interaction <strong>of</strong><br />
the vortices and the impinging and reflected shock waves is basically the same,<br />
but that <strong>of</strong> the P 2 expansion has resolved better the vortices as also the diffracted<br />
waves and the slip lines. However, in the P 2 computation wiggles appear at the slip
CHAPTER 7. NUMERICAL RESULTS 182<br />
lines locations, but not for the P 1 computation. This is due to the fact that the<br />
combination <strong>of</strong> a fine mesh and higher order expansion lead to smaller numerical<br />
viscosity. This observation is consistent with the trends found in high resolution<br />
inviscid flow computations carried out with increasing order <strong>of</strong> accuracy and mesh<br />
resolution using DG [82] and WENO discretizations [12]. The numerical results<br />
presented are in very good agreement with the experimental results given in [34],<br />
where the shadowgraph at t = 128µs shown in Fig. 7.50:<br />
Figure 7.50: Shadowgraph for the Schardin’s problem at t = 128µs.<br />
compares very well with the numerical results depicted previously in Fig. 7.49,<br />
and the numerical results obtained by a high fidelity numerical simulation for this<br />
problem performed by Chaudhuri et al. [35] obtained using a WENO scheme on a 8.4<br />
million points mesh. However, the P 2 results in Fig. 7.49 which were obtained with<br />
the 285000 element mesh required approximately 20 CPU hours on 50 processors<br />
in contrast to the results <strong>of</strong> [35] that required approximately 500 CPU hours using<br />
20-40 processors.<br />
In order to demonstrate that the accuracy is preserved in smooth regions <strong>of</strong> the<br />
flow field for the Schardin’s problem with the present shock capturing approach, a
CHAPTER 7. NUMERICAL RESULTS 183<br />
grid and p-refinement study was performed on the triangular element mesh. Specifically,<br />
at the region behind the triangle where the vortex is formed, a refinement <strong>of</strong><br />
the mesh was performed. The highly refined mesh and the computed Schlieren with<br />
P 2 expansion is shown in Fig. 7.51. Clearly, the shear layer is captured with 4 cells,<br />
while the shocks are captured with two cells.<br />
Figure 7.51: Numerical Schlieren for the Shardin’s problem on the finest mesh and<br />
using P 2 solution expansion.<br />
The first mesh refinement performed using an element size equal to h 1 = 0.001<br />
in the refined mesh region, while for the rest <strong>of</strong> the domain the size <strong>of</strong> the elements<br />
was h = 0.002 as before. Numerical solutions were obtained using P 1 and P 2<br />
solution expansions. The numerical Schlieren at time t = 128µs is shown in Fig.<br />
7.52, for both expansions, and it is apparent that a comparison between Figs. 7.52<br />
and 7.49(a) reveals that the solution on the refined mesh has resolved better the<br />
vortex formation behind the triangle. Furthermore, a solution was obtained with a<br />
more refined mesh having an element size <strong>of</strong> h 2 = 0.0005 in the region behind the<br />
triangle. As in the previous mesh refinement study, two numerical solutions were<br />
obtained using P 1 and P 2 solution expansion. The numerical Schlieren at a time<br />
t = 128µs is shown in Fig. 7.53 where it is observed that both solution expansions<br />
have resolved even finer and more complex flow structures.
CHAPTER 7. NUMERICAL RESULTS 184<br />
(a) P 1 expansion.<br />
(b) P 2 expansion.<br />
Figure 7.52: Numerical Schlieren for Schardin’s problem on the refined triangular<br />
mesh with mesh element size in the vortex region equal to h 1 = 0.001.
CHAPTER 7. NUMERICAL RESULTS 185<br />
(a) P 1 expansion.<br />
(b) P 2 expansion.<br />
Figure 7.53: Numerical Schlieren for Schardin’s problem on the refined triangular<br />
mesh with mesh element size in the vortex region equal to h 2 = 0.0005.<br />
7.7.1 Strong Mach 5 shock impingement on a triangle<br />
In this test case the same geometry employed in the computations for the standard<br />
Schardin’s problem is used, but the IC correspond to a right moving shock at a Mach
CHAPTER 7. NUMERICAL RESULTS 186<br />
number <strong>of</strong> 5.09. The triangular mesh has been used but with a P 2 expansion <strong>of</strong> the<br />
solution as higher gradients in the unsteady flow behind the triangle were expected.<br />
Furthermore, the positivity limiters proposed in [156] were employed, otherwise the<br />
computations were not able to advance in time long enough due to the appearance<br />
<strong>of</strong> negative density and pressure.<br />
The computed density field is shown in Fig. 7.54. In Fig. 7.55, the elements<br />
where the solution was limited along with the computed pressure gradient gray scale<br />
contours are shown for three characteristic non dimensional times. The limited<br />
elements are marked with blue color, for elements neighboring limited elements P 1<br />
approximation is used and these elements are marked with green color. For the rest<br />
<strong>of</strong> the domain, marked with red color, P 2 approximation is used. It is observed that<br />
limiting has been performed mostly at the elements where the impinging shocks<br />
(detected by pressure gradients) as well as through elements, which the reflected<br />
shocks are crossing. Moreover, in Fig. 7.55 the numerical Schlieren is shown where<br />
it is evident that limiting has not been performed over the elements where the<br />
contact discontinuities are formed.
CHAPTER 7. NUMERICAL RESULTS 187<br />
(a) t = 0.06<br />
(b) t = 0.08<br />
(c) t = 0.1<br />
Figure 7.54: Density contours using a P 2 expansion for the flow <strong>of</strong> a strong shock<br />
moving at Mach number <strong>of</strong> 5 around a triangle with 30 equal spaced intervals from<br />
0.28 to 13.28. Results are depicted at non-dimensional time.
CHAPTER 7. NUMERICAL RESULTS 188<br />
(a) t = 0.06<br />
(b) t = 0.08<br />
(c) t = 0.1<br />
Figure 7.55: Limited elements for the flow <strong>of</strong> a strong shock moving at Mach number<br />
<strong>of</strong> 5 around a triangle along with the gradient <strong>of</strong> pressure over 30 equally spaced intervals<br />
from 0 to 1000. Results are depicted at non-dimensional time. Blue elements:<br />
limited solution. Green elements P 1 expansion. Red elements P 2 expansion.
CHAPTER 7. NUMERICAL RESULTS 189<br />
7.8 Flat plate boundary layer<br />
The viscous flow over a flat plate with no pressure gradient at M = 0.2 and Re = 10 4<br />
is considered to further validate the viscous calculation with HoAc by comparing the<br />
velocity pr<strong>of</strong>iles for a P 1 , P 2 and a P 3 solution expansion. The free stream conditions<br />
are: [ρ, u, v, p] = [1, 0.2, 0, 0.714285714] with γ = 1 and the computational domain<br />
4<br />
has dimensions <strong>of</strong> [−2, 2] × [0, 10] with half <strong>of</strong> the lower horizontal line representing<br />
a symmetry boundary and the other half an adiabatic wall. At x = −2 the flow<br />
velocities and pressure are specified, while the density is extrapolated from the<br />
interior. At x = 2 and y = 10 all the computed variables are extrapolated except<br />
for the pressure which is specified at its mean stream value. A discontinuous output<br />
<strong>of</strong> the solution is employed as a relatively coarse mesh is used for the computations.<br />
Specifically, quadrilateral elements are used with a height <strong>of</strong> h = 10 −3 for the first<br />
layer <strong>of</strong> elements and a stretching factor <strong>of</strong> 1.2 in the direction normal to the wall.<br />
In Figs. 7.56 and 7.57 the computed velocity pr<strong>of</strong>iles are shown with comparison<br />
to the Blasius solution. It is observed that the P 1 computation hardly agrees with<br />
the Blasius solution especially for the V-velocity pr<strong>of</strong>ile. But, the P 2 computation<br />
is close to analytical solution and the P 3 computation achieves the best resolution<br />
for the boundary layer.
CHAPTER 7. NUMERICAL RESULTS 190<br />
Figure 7.56: Comparison <strong>of</strong> the U-velocity pr<strong>of</strong>ile with the Blasius solution for the<br />
flow over a flat plate.<br />
Figure 7.57: Comparison <strong>of</strong> the V-velocity pr<strong>of</strong>ile with the Blasius solution for the<br />
flow over a flat plate.
CHAPTER 7. NUMERICAL RESULTS 191<br />
The velocity field vectors along with the vorticity field for the P 3 computation<br />
is depicted in Fig. 7.58, where it is observed that the boundary layer thickness<br />
grows along the plate and the velocity pr<strong>of</strong>iles are nicely resolved by the numerical<br />
solution.<br />
Figure 7.58: Velocity field vectors and vorticity field for the flat plate boundary<br />
layer solution using a P 3 solution expansion.<br />
7.9 Unsteady viscous flow over tandem airfoils<br />
This problem was a test case that was solved and presented in the first International<br />
Workshop on High Order CFD Methods [89]. The geometry <strong>of</strong> the problem consists<br />
<strong>of</strong> two tandem NACA 0012 airfoils with the forward airfoil at 10 ◦ angle <strong>of</strong> attack.<br />
The free stream flow conditions correspond to a Mach number M = 0.2 and a<br />
Reynolds number Re = 10 4 . The IC applied to this test case correspond to a C 1<br />
initial solution for the velocity field:<br />
⎧<br />
√ γp∞<br />
⎨<br />
V = (u, v) = M ∞ (1, 0) (<br />
ρ ∞ ⎩ sin<br />
)<br />
πd<br />
2δ 1<br />
1, if d > δ 1 ,<br />
, if d ≤ δ 1 ,<br />
(7.6)<br />
with δ 1 = 0.05 and d being the distance to the closest wall with the density and<br />
pressure initialed to their free stream values. A mixed-type linear element mesh<br />
was generated around the geometry <strong>of</strong> the airfoils with quadrilateral elements in the<br />
region close to the airfoils in order to achieve better resolution for the boundary layer.
CHAPTER 7. NUMERICAL RESULTS 192<br />
In Fig. 7.59 the visualization mesh for a P 4 solution expansion over quadrilateral<br />
and triangular elements is shown.<br />
Figure 7.59: Visualization mesh for a P 4 computation for the two tandem NACA<br />
0012 airfoils.<br />
For this specific problem a sensitivity study regarding the far field boundary<br />
distance was carried out first in order to resolve the lift and drag coefficients within<br />
0.01 counts using a P 2 solution expansion over each element. The far field boundary<br />
was formed by a circle enclosing the airfoils. Three simulations were performed in<br />
order to examine the effect <strong>of</strong> the far field distance from the airfoils. These corresponded<br />
to the following radii <strong>of</strong> the circles: R = 25, R = 50, R = 100 times<br />
the chord length <strong>of</strong> the airfoil, which was assumed equal to unity. The sensitivity<br />
study was performed using an explicit third-order SSP Runge-Kutta scheme with 16<br />
stages up to t = 100, using a time step size <strong>of</strong> dt = 1.0e − 3, corresponding to nondimensional<br />
units. The cost <strong>of</strong> the computations is expressed in work units defined<br />
by comparing the execution time T 2 , excluding the initialization, post-processing<br />
data preparation time and file I/O time, to TauBench time T 1 , which is an unstructured<br />
grid benchmark. The respective kernels are derived from Tau - a Navier-Stokes<br />
solver, which has been developed at DLR in Germany.
CHAPTER 7. NUMERICAL RESULTS 193<br />
Work units = T 2<br />
T 1<br />
.<br />
In Table 7.1 the work units, number <strong>of</strong> elements, number <strong>of</strong> DOF and the L 2 error<br />
for the whole time <strong>of</strong> the simulation for the aerodynamic coefficients for both airfoils<br />
are given. All the simulations were performed using 8 cores. From the results given<br />
in Table 7.1, the distance <strong>of</strong> the far field for the time accurate simulations was chosen<br />
to be R = 50.<br />
R Elem Work Units DOF C l forward C d forward C l aft C d aft<br />
25 2708 1163.49 146496 – – – –<br />
50 2788 1312.3 150336 5.4880e-7 1.0240e-7 2.1443e-6 1.6780e-7<br />
100 2914 1649.78 156384 6.8725e-8 1.0516e-8 1.2237e-7 3.0579e-8<br />
Table 7.1: Results for the sensitivity study regarding the far field boundary distance.<br />
Furthermore, a p-refinement study was performed using a P 2 , P 3 and P 4<br />
solution expansion for a time accurate computation using an explicit third order<br />
SSP Runge–Kutta method with 16 stages, provided by PETSc, and a small step<br />
size equal to dt = 5.0e − 4 for a simulation time equal to t = 100. The time history<br />
for the lift and drag coefficients for the forward and the aft airfoil are given in Figs.<br />
7.60 and 7.61, respectively.
CHAPTER 7. NUMERICAL RESULTS 194<br />
(a) Lift coefficient t = 0 − 100 (b) Lift coefficient t = 40 − 100<br />
(c) Drag coefficient t = 0 − 100 (d) Drag coefficient t = 40 − 100<br />
Figure 7.60: Lift and drag coefficients for the forward airfoil.
CHAPTER 7. NUMERICAL RESULTS 195<br />
(a) Lift coefficient t = 0 − 100 (b) Lift coefficient t = 80 − 100<br />
(c) Drag coefficient t = 0 − 100 (d) Drag coefficient t = 80 − 100<br />
Figure 7.61: Lift and drag coefficients for the aft airfoil.<br />
It is observed that a time periodic response appears to develop for both airfoils after<br />
time t = 80. In Figs. 7.62 and 7.63 the velocity and vorticity contours at t = 80 are<br />
shown for a P 2 and a P 4 solution expansion using a discontinuous output, where<br />
the difference between the resolved flow fields is apparent.
CHAPTER 7. NUMERICAL RESULTS 196<br />
(a) P 2 Solution<br />
(b) P 4 Solution<br />
Figure 7.62: Velocity contours for the flow over NACA0012 tandem airfoils.<br />
(a) P 2 Solution<br />
(b) P 4 Solution<br />
Figure 7.63: Vorticity contours for the flow over NACA0012 tandem airfoils.
Chapter 8<br />
Conclusions and Future Work<br />
The present work developed and evaluated a priori indicators <strong>of</strong> mesh distortion.<br />
This was accomplished by deriving the complete truncation error expression for the<br />
FV discretization <strong>of</strong> the field gradient in two dimensions. The complexity <strong>of</strong> the<br />
mathematical operations involved was overcome via use <strong>of</strong> symbolic mathematics<br />
s<strong>of</strong>tware.<br />
The presented work analyzed a generally-distorted mesh (structured, unstructured<br />
and mixed-type element) into elementary distortions (stretching, skewness,<br />
shearing, rotation), as well as three common types <strong>of</strong> interfaces, and those were directly<br />
related to the TE. The derived analytic expressions are relatively simple and<br />
amenable to future work on improving the mesh. This is especially true for index<br />
Q for which analytic expressions were presented directly relating it to the degree <strong>of</strong><br />
stretching and skewness, as well as to mixed-type element mesh interfaces.<br />
The two indices (Q and q) “followed” the grid distortion quite faithfully with<br />
Q being more focused on these local areas. Index q marks broader areas <strong>of</strong> the mesh<br />
due to the second order error terms included in its definition. It also picks-up small<br />
(background) distortions which may not be <strong>of</strong> interest. Given the fact that Q is<br />
quite simpler to calculate, it is the recommended quality index. A disadvantage <strong>of</strong><br />
Q is that it cannot indicate the quality <strong>of</strong> distorted meshes that still yield second<br />
order accurate computation <strong>of</strong> the gradient. Shearing and rotation do change the<br />
197
CHAPTER 8. CONCLUSIONS AND FUTURE WORK 198<br />
TE (i.e. only index q “picks them” up) but they do not reduce the order <strong>of</strong> accuracy.<br />
However, the goal is to fix meshes that degrade the accuracy to a lower order.<br />
The following observations were also made for both indices: (i) irregularlyshaped<br />
small distortions were captured faithfully, (ii) small distortions were detected<br />
even though much larger ones existed in the mesh, and (iii) distortions were captured<br />
in any direction.<br />
Employment <strong>of</strong> the centroid dual discretization must be avoided on general<br />
structured and mixed-type element meshes due to its inconsistency in the evaluation<br />
<strong>of</strong> the field gradient.<br />
Future work involves two directions: (i) working the analytical expressions for<br />
the three-dimensional case for mixed-type element (structured/unstructured) grids,<br />
and (ii) using the analytic expressions <strong>of</strong> the present work for a priori improvement<br />
<strong>of</strong> the mesh.<br />
In the present work a CFD solver was developed and named HoAc (an abbreviation<br />
for High order Accuracy) based on the DG discretization. The code was<br />
verified and its ability to achieve the design order <strong>of</strong> the DG discretization for up<br />
to fifth order accurate expansion <strong>of</strong> the solution for the Euler and Navier-Stokes<br />
equations was demonstrated. Furthermore, in the examined cases the ability <strong>of</strong> the<br />
HoAc solver to provide very accurate solutions on problems with complex flow<br />
features was demonstrated.<br />
The HoAc solver has the ability to perform parallel computations via the<br />
domain decomposition method on super computers and massively parallel systems<br />
using the MPI programming paradigm. It can handle arbitrary mixed-type element<br />
meshes over arbitrary geometries in two dimensions and solve unsteady problems<br />
for the Euler and Navier-Stokes equations with the aid <strong>of</strong> high order explicit and<br />
implicit time marching schemes. This was mainly achieved by the fact that all the<br />
operations for the DG discretization in HoAc are being performed in the standard<br />
square element configuration where the expansion bases are defined and constructed.<br />
Also, the capability to produce a high order discontinuous output <strong>of</strong> the solution has<br />
been implemented, which appeared to be fairly adequate for coarse meshes when a
CHAPTER 8. CONCLUSIONS AND FUTURE WORK 199<br />
high order approximation <strong>of</strong> the solution is employed.<br />
Furthermore, in the present work the shock capturing capability <strong>of</strong> the DG<br />
discretization was addressed and improved with a new limiting approach which<br />
is suitable for mixed-type meshes. A shock capturing p-adaptive DG scheme was<br />
developed and implemented in HoAc. In the new limiting approach, limiting is performed<br />
with a slight but essential modification <strong>of</strong> a TVB limiter extensively tested<br />
for DG discretizations. Limiting is carried out in the standard computational space<br />
where all types <strong>of</strong> elements transform to standard squares. Application <strong>of</strong> the new<br />
limiting approach for standard test cases for shock capturing schemes demonstrated<br />
that the new approach is robust, accurate, and it does not require adjustment <strong>of</strong><br />
parameters. Moreover, it provides capabilities for p-adaptation because the solution<br />
accuracy could be increased away from regions <strong>of</strong> discontinuities that are clearly<br />
marked since accurate estimates <strong>of</strong> the second derivative are obtained. In addition,<br />
the developed limiting algorithm was combined with new positivity preserving limiters<br />
especially developed for the DG method and test cases involving high speed<br />
flows with strong shocks were examined with exceptional results. Specifically, these<br />
cases were solved using unstructured triangular meshes and up to the time <strong>of</strong> the<br />
present work nowhere in the literature the positivity preserving limiters appeared<br />
to be applied on triangular meshes.<br />
Future work in the DG discretization and implementation <strong>of</strong> the current thesis<br />
will involve: (i) the extension <strong>of</strong> the developed limiting approach in three dimensions,<br />
both for shock capturing and positivity preservation for density and pressure, (ii)<br />
the ability <strong>of</strong> the HoAc solver to work on meshes with curved elements, (iii) the<br />
extension <strong>of</strong> the HoAc solver to handle three dimensional flows around complex<br />
geometries using mixed-type element meshes with curved elements, (iv) the ability<br />
to perform high order DG simulations for turbulent flows and (v) the extension<br />
<strong>of</strong> the current implementation <strong>of</strong> HoAc for its use with Graphics Processor Units<br />
(GPUs).
APPENDIX A<br />
Mesh quality index Q x for the stretched unstructured mesh <strong>of</strong> Fig. 3.5(b):<br />
Q x = 96<br />
7<br />
dx<br />
∣|1 + 2dx| (1 + 2dx) + 4(1 + dx 2 ) + |2dx − 1| (1 − 2dx) ∣ +<br />
∣ 32√ ∣∣∣<br />
dy<br />
3<br />
7 (2dy − √ 3)[|2dy − √ 3| + | √ 3 + 2dy|] + 2dy<br />
∣ .<br />
(A-1)<br />
Mesh quality index Q x for the skewed mesh <strong>of</strong> Fig. 3.6:<br />
Q x = 1 − cos2 (ω)<br />
2[cos 2 (ω) + 1] + cos (ω) − 1<br />
cos 2 (ω) − cos (ω) − 4 −<br />
3 sin (2ω)<br />
2[cos (ω) + 1][cos 2 (ω) − cos (ω) − 4] .<br />
(A-2)<br />
200
APPENDIX B<br />
Discontinuous Output for Quadrilateral Elements<br />
For an approximation order N, a quadrilateral element is subdivided into (N + 1) 2<br />
quadrilateral sub-elements. This element division is based on a listing Q q <strong>of</strong> equally<br />
spaced points according to Eq. (B-1):<br />
Q q = p + q(N + 2), 0 ≤ p, q ≤ N + 1, (B-1)<br />
over the standard square element configuration. Each point then defines a node<br />
whose coordinates in the physical space are computed by Eq. (4.29). Then the<br />
connectivity <strong>of</strong> the quadrilateral sub-elements is defined according to a structured<br />
mesh. Finally, the solution is projected according to Eq. (4.5) and printed at every<br />
new node. This leads to a print out <strong>of</strong> the solution that is discontinuous across the<br />
inter-element boundaries.<br />
Discontinuous Output for Triangular elements<br />
As for a quadrilateral element a triangular element is subdivided into (N + 1) 2<br />
triangular sub-elements whose nodes are based on a listing Q t <strong>of</strong> equally spaced<br />
points over the standard triangular element configuration given in the following<br />
equation:<br />
201
CHAPTER 8. CONCLUSIONS AND FUTURE WORK 202<br />
Q t = p + (N + 2)q −<br />
q(q − 1)<br />
, 0 ≤ p, q ≤ N with p + q ≤ N. (B-2)<br />
2<br />
The physical coordinates <strong>of</strong> the nodes are computed from Eq. (4.31) and the following<br />
pseudo code is provided for the connectivity list <strong>of</strong> the triangular sub-elements:<br />
m=0;<br />
for (i=0; i< # elements; i++) { // loop elements<br />
for (q = 0; q < N+1; q++)<br />
for (p = 0; p < N+1; p++)<br />
if (p + q < N + 1) { // if in tria<br />
print(Q_t(p, q) + m, Q_t(p+1, q, sys) + m, Q_t(p,q + 1, sys) + m);<br />
if (p + q + 2 < N + 2)<br />
print(Q_t(p + 1, q) + m, Q_t(p + 1, q + 1) + m, Q_t(p, q + 1) + m);<br />
} // if in tria<br />
m += (N + 2) * (N + 3) / 2;<br />
} // loop elments<br />
As for the quadrilateral elements, the solution is projected according to Eq. (4.5)<br />
and printed at every new node resulting in a discontinuous solution print.
APPENDIX C<br />
Convergence analysis<br />
The discretization error is defined as:<br />
Du = u n − u = O (h p ) ,<br />
(C-1)<br />
where p is the order <strong>of</strong> the discretization and u n and u is the numerical solution<br />
obtained with a discretization size h and the exact solution respectively.<br />
The exact solution is necessary in order to compute the order <strong>of</strong> the discretization,<br />
but this is limited to cases that can be solved analytically. The formal order p may<br />
be determined by comparing two discretization errors with differenet sizes h 1 and<br />
h 2 , as follows:<br />
D 1<br />
which after applying logarithms yields:<br />
= hp 1<br />
, (C-2)<br />
D 2<br />
h p 2<br />
and p may be calculated as:<br />
ln D 1<br />
= ln hp 1<br />
, (C-3)<br />
D 2<br />
h p 2<br />
p = − ln D 1 − ln D 2<br />
ln h 1 − ln h 2<br />
.<br />
(C-4)<br />
203
CHAPTER 8. CONCLUSIONS AND FUTURE WORK 204<br />
In Eq. (C-4) scalar error values are needed, which are introduced as integral<br />
values <strong>of</strong> a norm ||∆u|| p <strong>of</strong> the solution error over each element:<br />
||∆u|| p = 1 ∑<br />
(∫<br />
N e<br />
n<br />
Ω e<br />
[Du] p dΩ e<br />
) 1/p<br />
, 1 ≤ n ≤ N e . (C-5)<br />
where N e is the total number <strong>of</strong> elements used in the mesh.
APPENDIX D<br />
HoAc run time options<br />
HoAc relies on PETSc for parallel computations and time marching <strong>of</strong> the solution.<br />
So, any run time option referring to PETSc is readily applied to HoAc as well. A<br />
sample input file for HoAc is given below:<br />
*CONTROL_EQUATION<br />
/*[0,1]=[Euler,NS]*/<br />
1<br />
*END<br />
*CONTROL_FLOWCOND<br />
/*gamma*/ /*P_inf*/ /*U_inf*/ /*V_inf*/ /*rho_inf*/ /*C_inf*/<br />
1.400000e+0 7.142857e-01 0.2 0.000000e+00 1.000000e+00<br />
1.000000e+00<br />
*END<br />
*CONTROL_SOLUTION<br />
/*n_out*/ /*PN*/ /*[0,1]=[start,restart]*/ /*[0,1]=[Roe,LLF]*/ /* # parts*/<br />
5000 1 0 1 16<br />
*END<br />
*CONTROL_TIME<br />
/*End time */ /*Problem type */ /* Output type<br />
[0,1]=[continuous,discontinuous]*/<br />
1.0 0 1<br />
*END<br />
205
CHAPTER 8. CONCLUSIONS AND FUTURE WORK 206<br />
*CONTROL_BOUNDARY_WALL<br />
/*[0,1]=[Straight,CBC]*/<br />
0<br />
*END<br />
*CONTROL_NAVIER_STOKES<br />
/*Re*/ /*Pr*/ /*T_inf*/ /*LDG coefficient*/<br />
100.000000 7.200000e-01 1.0 0.5<br />
*END<br />
*CONTROL_BOUNDARY_IN_SPEC<br />
/*rho_spec_in*/ /*u_spec_in*/ /*v_spec_in*/ /*p_spec_in*/<br />
0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00<br />
*END<br />
*CONTROL_BOUNDARY_OUT_SPEC<br />
/*rho_spec_out*/ /*u_spec_out*/ /*v_spec_out*/ /*p_spec_out*/<br />
0.000000e+00 0.000000e+00 0.000000e+00 0.000000e+00<br />
*END<br />
where form the comments provided in, the user is easy to check what a value means.<br />
Furthermore, in order to give more flexibility to the user additional run time options<br />
are provided which handle the DG discretization in HoAc . These are given in the<br />
following sample options file along with other PETSc commands for an implicit or<br />
explicit run, where it is noted that the comments should not exist when a simulation<br />
is to be performed.<br />
-llf_flux %% for Local Lax Friedrichs<br />
-roe_flux %% for Roe flux<br />
-explicit %% for an expicit run<br />
-explicit_type 0 %% For using the SSP RK develop by Shu and Osher<br />
-explicit_type 1 %% For using the SSP RK provided by PETSc<br />
-gl %% for Gauss-Legendre quadrature<br />
-gll %% for Gauss-Lobatto-Legendre quadrature<br />
-ts_type ssp
CHAPTER 8. CONCLUSIONS AND FUTURE WORK 207<br />
-ts_ssp_type rks3<br />
-ts_ssp_nstages 9<br />
-ts_dt 1.0e-4<br />
-dt 1.0e-4 %% if explicit_type is 0<br />
-order 1 %% Order <strong>of</strong> approximation<br />
-parts 16 %% Number <strong>of</strong> partitions<br />
-end_time 2000.0 %% End time <strong>of</strong> simulation<br />
-n_out 50000 %% Output frequency<br />
-shock_lim %% Limiter for shock capturing<br />
-pos_lim %% Limiters for positivity preservation<br />
-restart %% Switch for restarting a solution<br />
-snes_monitor<br />
-snes_atol 1.0e-8<br />
-snes_rtol 1.0e-10<br />
-snes_mf<br />
-ksp_type fgmres<br />
-pc_type none<br />
-mat_mffd_compute_normu<br />
-ksp_max_it 30<br />
-ksp_gmres_restart 200<br />
HoAc :<br />
At the command line the user then types the following command for running<br />
mpiexec -n {# parts} ./hoac {input file} -options_file {file_name}
Bibliography<br />
[1] R. Abgrall. On Essentially Non-oscillatory Schemes on Unstructured Meshes:<br />
Analysis and Implementation. Journal <strong>of</strong> Computational Physics, 114(1):45–<br />
58, 1994.<br />
[2] N. Adams. Direct simulation <strong>of</strong> the turbulent boundary layer along a compression<br />
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