BAIL 2006 Book of Abstracts - Institut für Numerische und ...
BAIL 2006 Book of Abstracts - Institut für Numerische und ... BAIL 2006 Book of Abstracts - Institut für Numerische und ...
M.H. BUSCHMANN, M. GAD-EL-HAK: Turbulent Boundary Layers: Reality and Myth ✬ Which mathematical techniques do we have to solve eqn. (3)? Basically we have analytical, numerical, and asymptotic methods and combination of those. 3 In this paper, we will focus on recent advances in analytical and asymptotic approaches. During the mid 1990s, a new debate on the aforementioned subject arose. Caused by new unconventional approaches questioning one of the cornerstones of modern fluid mechanics—the logarithmic law of turbulent boundary layers—several new scalings were developed. In conjunction with these theoretical investigations, high-quality experiments in zero-pressuregradient turbulent boundary layers and turbulent pipe and channel flows were undertaken. In general, the physical picture of wall-bounded flow is now much more complex than was thought a decade ago. However, the physical picture seems to be also more controversial than ever before. Which of these new approaches will survive and contribute substantially to fluid mechanics in the future is still open. The present talk will discuss four main schools of thought, which can be summarized as follows: (1) Standard Logarithmic Overlap Layer Above a certain critical Reynolds number, a pure logarithmic region exists in the mean-velocity profile of ZPG TBL and channel and pipe flows. The parameters of this logarithmic law are completely Reynolds number invariant. (2) Power Law Based On Similarity Assumption An inner and an outer power law describe the overlap layer of ZPG TBL and channel and pipe flows. Both laws exhibit Reynolds number dependent parameters, and never achieve an asymptotic state. (3) Asymptotic Invariance Principle Full similarity solutions have to be searched separately for the inner and outer equations. Because in the limit the outer boundary layer equations are independent of Reynolds number, the properly scaled profiles for the outer region must also be independent of Reynolds number. (4) Higher-Order Asymptotic Matching Based on asymptotic matching, a higher-order approach with respect to Reynolds number and the wall-normal coordinate can be derived. In the limit of infinite Reynolds number, the classical logarithmic law is recovered. At finite Re, however, the similarity laws for both the mean and higher-order statistics are Reynolds-number dependent. The presentation will summarize, analyze and critique the recent theoretical and experimental investigations of a variety wall-bounded flows. Based on that analysis, open issues in the field will be highlighted. 3 Gersten, K., and Herwig, H., Strömungsmechanik, Fundamentals and Advances in the Engineering Science, Verlag Vieweg, 1992. ✫ Analytical methods Asymptotic methods Numerical methods M. H. Buschmann & M. Gad-el-Hak, Abstract for BAIL 2006 Speaker: BUSCHMANN, M.H. 34 BAIL 2006 ✩ ✪
A. NAYAK, D. DAS: Three-dimesnional Temporal Instability of Unsteady Pipe Flow ✬ ✫ Three-dimensional Temporal Instability of Unsteady Pipe Flow By Avinash Nayak and Debopam Das Department of Aerospace Engineering Indian Institute of Technology Kanpur, India Introduction: In this paper we present temporal three-dimensional linear stability analysis of unsteady bi-directional laminar flow in a duct. For a known but arbitrary volume flow rate, analytical solution of one-dimensional unsteady flow through a pipe has been obtained by Das & Arakeri (1998, 2000). Generalized form of this solution and the solution for unsteady flow through annular space between two concentric pipes has been obtained by Nayak (2005). Prior to this study, solutions that are available in the literature were for known pressure gradient (Womersley 1955) and Uchida 1956). Das (1998) and Das & Arakeri (1998) have observed the transitional nature of this flow in a pipe and channel where the velocity profiles are inflectional and have reverse flow near wall. In their experiments, for flows with average volume flow rate varying like a trapezoidal function with time, shows that both axisymmetric and non-axisymmetric mode of disturbance grows. In this paper through linear stability analysis of three dimensional disturbances some of the observed experimental facts are explained such as growth of helical modes for certain cases. Linear stability analysis: The disturbance stream-function is assumed as { u , v, w, p} = { u( r) , v( r ) , w( r ) , p( r ) } exp[ iα ( z − ct) + inθ ] . (1) Here, u, v and w are radial, azimuthal and axial perturbation velocity components respectively and p is the fluctuating pressure. The disturbance quantities are substituted in perturbed Navier-Stokes equations in cylindrical coordinates and the governing equations after simplifications are, 2 1 ⎡1+ n ⎤ 2in 1 ⎡ iv 2 3 3 3 ⎤ u′ ′ + u′ − ⎢ + α + iα Re( W − c) − − 2 ⎥u v 2 2 ⎢u + u′ ′ − u′ ′ + u′ − u 2 3 4 r ⎥ ⎣ r ⎦ r α ⎣ r r r r ⎦ 2 in ⎡1 2 3 3 ⎤ ⎡n 2 ⎤⎧ 1 ⎡ 1 1 ⎤ in ⎡1 1 ⎤⎫ − Re( ) 2 ⎢ v′ ′ − v′ ′ + v′ − v 2 3 4 ⎥ + ⎢ + α + iα W − c 2 ⎥⎨ + 2 ⎢u′ ′ + u′ − u 2 ⎥ 2 ⎢ v′ − v 2 ⎥⎬ α ⎣r r r r ⎦ ⎣ r ⎦⎩α ⎣ r r ⎦ α ⎣r r ⎦⎭ 2 ⎡ 2n ⎤⎧ 1 ⎡ 1 ⎤ in ⎡1 ⎤⎫ i i + ⎢− + iα ReW ′ − Re ′ ′ − Re ′ = 0 − − − − − −( 1) 3 ⎥⎨ + 2 ⎢u′ + u⎥ 2 ⎢ v⎥⎬ u W uW ⎣ r ⎦⎩α ⎣ r ⎦ α ⎣r ⎦⎭ α α 2 1 ⎡1+ n 2 v′′ + v′ −⎢ + α + iα Re 2 r ⎣ r 2 in⎡n 2 + ⎢ + α + iαRe 2 r ⎣r ( W −c) 2 ⎤ 2in in ⎡1 2 1 1 ⎤ n ⎡ 1 1 1 ⎤ ⎥v + u − 2 2 ⎢ u ′′′ + u′′ − u′ + u + 2 3 4 ⎥ 2 ⎢ v′ − v′ + v 2 3 4 ⎥ ⎦ r α ⎣r r r r ⎦ α ⎣r r r ⎦ ⎤⎧ 1 ⎡ 1 ⎤ in ⎡1 ⎤⎫ n ⎛1 ⎞ ⎥⎨ ⎜ ⎟ 2 ⎢ + ⎥ 2 ⎢ ⎥⎬ ⎦⎩α ⎣ r ⎦ α ⎣r ⎦⎭ α ⎝r ⎠ ( W −c) u′ u + v + Re u W′ = 0− −− −− −−( 2) These equations are solved using the finite difference technique and it is written in the form of ⎧u ⎫ ( [ ] [ ] ) . The complex eigenvalues c , are obtained for a particular value of α and A − c B = 0 ⎨ ⎬ ⎩ v ⎭ Re, using MATLAB. In these two sets of equations u has the highest 4 th derivative while has v rd the highest 3 derivative. Thus 7 boundary conditions are required to solve these two coupled equations. Following are the boundary conditions given for different values of n . n 0 R=0 (center-line) R=1 (at wall) u = 0 u = 0 v = 0 v = 0 u ′ = 0 u ′ = 0 1 u + iv = 0 u ′ = 0 v ′ = 0 −1 lim ∝ n u r R=0 (center-line) Speaker: DAS, D. 35 BAIL 2006 r→0 ✩ ✪
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- Page 17 and 18: Contents Greetings . . . . . . . .
- Page 19 and 20: CONTENTS S. HEMAVATHI, S. VALARMATH
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- Page 38 and 39: G.I. SHISHKIN: A posteriori adapted
- Page 40 and 41: W. LAYTON, I. STANCULESCU: Numerica
- Page 42 and 43: H. WANG: A Component-Based Eulerian
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- Page 48 and 49: J.A. MACKENZIE, A. NICOLA: A Discon
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- Page 53 and 54: Speaker: Debopam Das, Tapan Sengupt
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M.H. BUSCHMANN, M. GAD-EL-HAK: Turbulent Bo<strong>und</strong>ary Layers: Reality and<br />
Myth<br />
✬<br />
Which mathematical techniques do we have to solve eqn. (3)? Basically we have analytical,<br />
numerical, and asymptotic methods and combination <strong>of</strong> those. 3 In this paper, we will focus on<br />
recent advances in analytical and asymptotic approaches.<br />
During the mid 1990s, a new debate on the aforementioned subject arose. Caused by new<br />
unconventional approaches questioning one <strong>of</strong> the cornerstones <strong>of</strong> modern fluid mechanics—the<br />
logarithmic law <strong>of</strong> turbulent bo<strong>und</strong>ary layers—several new scalings were developed. In<br />
conjunction with these theoretical investigations, high-quality experiments in zero-pressuregradient<br />
turbulent bo<strong>und</strong>ary layers and turbulent pipe and channel flows were <strong>und</strong>ertaken. In<br />
general, the physical picture <strong>of</strong> wall-bo<strong>und</strong>ed flow is now much more complex than was thought<br />
a decade ago. However, the physical picture seems to be also more controversial than ever<br />
before. Which <strong>of</strong> these new approaches will survive and contribute substantially to fluid<br />
mechanics in the future is still open.<br />
The present talk will discuss four main schools <strong>of</strong> thought, which can be summarized as follows:<br />
(1) Standard Logarithmic Overlap Layer<br />
Above a certain critical Reynolds number, a pure logarithmic region exists in the<br />
mean-velocity pr<strong>of</strong>ile <strong>of</strong> ZPG TBL and channel and pipe flows. The parameters <strong>of</strong><br />
this logarithmic law are completely Reynolds number invariant.<br />
(2) Power Law Based On Similarity Assumption<br />
An inner and an outer power law describe the overlap layer <strong>of</strong> ZPG TBL and<br />
channel and pipe flows. Both laws exhibit Reynolds number dependent<br />
parameters, and never achieve an asymptotic state.<br />
(3) Asymptotic Invariance Principle<br />
Full similarity solutions have to be searched separately for the inner and outer<br />
equations. Because in the limit the outer bo<strong>und</strong>ary layer equations are independent<br />
<strong>of</strong> Reynolds number, the properly scaled pr<strong>of</strong>iles for the outer region must also be<br />
independent <strong>of</strong> Reynolds number.<br />
(4) Higher-Order Asymptotic Matching<br />
Based on asymptotic matching, a higher-order approach with respect to Reynolds<br />
number and the wall-normal coordinate can be derived. In the limit <strong>of</strong> infinite<br />
Reynolds number, the classical logarithmic law is recovered. At finite Re,<br />
however, the similarity laws for both the mean and higher-order statistics are<br />
Reynolds-number dependent.<br />
The presentation will summarize, analyze and critique the recent theoretical and experimental<br />
investigations <strong>of</strong> a variety wall-bo<strong>und</strong>ed flows. Based on that analysis, open issues in the field<br />
will be highlighted.<br />
3 Gersten, K., and Herwig, H., Strömungsmechanik, F<strong>und</strong>amentals and Advances in the Engineering Science, Verlag<br />
Vieweg, 1992.<br />
✫<br />
Analytical<br />
methods<br />
Asymptotic<br />
methods<br />
Numerical<br />
methods<br />
M. H. Buschmann & M. Gad-el-Hak, Abstract for <strong>BAIL</strong> <strong>2006</strong><br />
Speaker: BUSCHMANN, M.H. 34 <strong>BAIL</strong> <strong>2006</strong><br />
✩<br />
✪
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