Table Of ContentLecture Notes
in Computational Science 34
and Engineering
Editors
TimothyJ. Barth.Moffett Field. CA
Michael Griebel. Bonn
David E.Keyes.NewYork
Risto M.Nierninen, Espoo
DirkRoose. Leuven
Tamar Schlick. NewYork
Springer-Verlag Berlin Heidelberg GmbH
Volker John
Large Eddy Simulation
of Turbulent
Incompressible Flows
Analytical and Numerical Results
for a Class ofLES Models
t
Springer
Volker John
Institute of Analysis and Numerical Mathematics
Department of Mathematics
Otto-von-Guericke-University Magdeburg
39106 Magdeburg, Germany
e-mail: [email protected]
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Mathematics Subject Classification (2000): 76-02, 76F65, 65M60, 65M55
ISBN 978-3-540-40643-3 ISBN 978-3-642-18682-0 (eBook)
DOI 10.1007/978-3-642-18682-0
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Preface
The numericalsimulation ofturbulent flowsisundoubtedly very important in
applications. Therichness ofscalesinherent inturbulentflowsmakesit impos
sible tosolvethegoverningequations,theNavier-Stokes equations,onpresent
day computers and even on computes in the foreseeable future. Turbulence
models are the tool to modify the Navier-Stokesequationssuch thatequations
arise which can be numericallyapproximated using present-dayhardwareand
software. One kind of modelling turbulence is large eddy simulation (LES)
which aims to compute the large flow structures accurately and models the
interactions of the small flowstructures to the large ones.
ThismonographconsidersaclassofLESmodelswhosederivationismainly
based on mathematical (and not on physical) arguments and in addition the
Smagorinsky model. One main goal ofthis monograph is to present all math
ematical analysis which is known for these models. The second goal is to give
a detailed description of the implementation of these LES models into a fi
nite element code. Since the probably best model within the considered class
of LES models is rather new, it still requires comprehensive numerical tests.
Therefore, the last main topic of this monograph is the presentation of first
numerical studies with this new LES model whichinvestigate, e.g., howgood
a space averaged flowfield is approximated.
The writing of this monograph would have been impossible without the
support offriends and colleagues. Aparticularthank goesto William (Bill) J.
Layton (UniversityofPittsburgh).The participationat thescientificcoopera
tion ofLutzTobiska with Billand hisgroupenabled meto enlargemyfieldsof
research considerably. It was Billwho brought me into contact with LES and
together with whom a number of results presented in this monograph were
obtained. The Deutsche Akademische Austauschdienst (D.A.A.D.) made it
possible for me to pay three longer research visits at the University of Pitts
burgh within the past two years which were essential for the work at this
monograph.I liketo thank Annette and Bill Layton also for their hospitality
during these visits. The computational results were obtained with the code
MooNMD which was developed in our group. My special thanks go to Gunar
VIII Preface
Matthies who often answered questions whicharose in the implementation of
the algorithms. I like to thank Lutz Tobiska who gave useful suggestions for
improving the monograph. For helpful discussions on subjects of this mono
graph,I wouldliketo thank, besides the already mentioned colleagues,Adrian
Dunca, Traian Iliescu and Friedheim Schieweck. Useful suggestions for the
preparationofthe final versionofthis monograph came from MichaelGriebel,
MaxD.Gunzburgerand Tobias Knopp. Iliketo thankalsoWalfredGrambow
for his efforts to provide the computer resources which were necessary to do
the numerical simulations.
Last but not least, I liketo thank my beloved wife Anja for her constant
encouragement and support.Her effortsto solvethe daily problems ofour life
werethe basis offinding sufficienttime to workat this monograph inthe past
two and a half years.
Colbitz, Volker John
July 2003
Contents
1 Introduction. .............................................. 1
1.1 Short Remarks on the Nature and Importance of Turbulent
Flows. ... .. .. ... .... 1
1.2 Remarks on the Direct Numerical Simulation (DNS) and the
k - e Model 1
1.3 Large Eddy Simulation (LES) ...... .. ............. ........ 3
1.4 Contents of this Monograph. .. ................... ... ..... 5
2 Mathematical Tools and Basic Notations 11
2.1 Function Spaces 11
2.2 Some Tools from Analysis and Functional Analysis 14
2.3 Convolution and Fourier Transform. ....................... 17
2.4 Notations for Matrix-Vector Operations. ................... 18
3 The Space Averaged Navier-Stokes Equations and the
Commutation Error 21
3.1 The Incompressible Navier-Stokes Equations 22
3.2 The Space Averaged Navier-Stokes Equations in the Case
a =]Rd 23
3.3 The Space Averaged Navier-Stokes Equations in a Bounded
Domain 25
3.4 The Gaussian Filter 29
3.5 Error Estimate of the Commutation Error Term in the
LP(]Rd) Norm 31
3.6 Error Estimate of the Commutation Error Term in the
H-l (Jl) Norm .......................................... 41
3.7 Error Estimate for a Weak Form of the Commutation Error 43
4 LES Models Which are Based on Approximations in
Wave Number Space 47
4.1 Eddy Viscosity Models 48
X Contents
4.1.1 The Smagorinsky Model " 48
4.1.2 The Dynamic Subgrid Scale Model 49
4.2 Modelling of the Large Scale and Cross Terms 51
4.2.1 The Taylor LES ModeL 52
4.2.2 The Second Order Rational LES Model 54
4.2.3 The Fourth Order Rational LES Model 57
4.3 Models for the Subgrid Scale Term 59
4.3.1 The Second Order Fourier Transform Approach 59
4.3.2 The Fourth Order Rational LES Model 60
4.3.3 The Smagorinsky Model. ........................... 61
4.3.4 Models Proposed by Iliescu and Layton 61
5 The Variational Formulation ofthe LES Models ........... 63
5.1 The Weak Formulation of the Equations. ................... 64
5.2 Boundary Conditions for the LES Models 65
5.2.1 Dirichlet Boundary Condition. ...................... 66
5.2.2 Outflow or Do-Nothing Boundary Condition 67
5.2.3 Free Slip Boundary Condition....................... 67
5.2.4 Slip With Linear Friction and No Penetration
Boundary Condition ............................... 67
5.2.5 Slip With Linear Friction and Penetration With
Resistance Boundary Condition ..................... 69
5.2.6 Periodic Boundary Condition .. ..................... 69
5.3 Function Spaces for the LES Models ....................... 70
6 Existence and Uniqueness ofSolutions ofthe LES Models. 73
6.1 The Smagorinsky Model. ................................. 74
6.1.1 A priori error estimates 74
6.1.2 The Galerkin Method .............................. 78
6.2 The Taylor LES Model. .................................. 92
6.3 The Rational LES Model ................................. 96
7 Discretisation ofthe LES Models .......................... 99
7.1 Discretisation in Time by the Crank-Nicolson or the
Fractional-Step O-Scheme 100
7.2 The Variational Formulation and the Linearisation of the
Time-Discrete Problem 102
7.3 The Discretisation in Space 106
7.4 Inf-Sup Stable Pairs of Finite Element Spaces 108
7.5 The Upwind Stabilisation for Lowest Order Non-Conforming
Finite Elements 114
7.6 The Implementation of the Slip With Friction and
Penetration With Resistance Boundary Condition 117
7.7 The Discretisation of the Auxiliary Problem in the Rational
LES Model 119
Contents XI
7.8 The Computation of the Convolution in the Rational LES
Model 120
7.9 The Evaluation ofIntegrals, Numerical Quadrature 122
8 Error Analysis of Finite Element Discretisations of the
LES Models 125
8.1 The Smagorinsky Model. , 126
8.1.1 The Variational Formulation and Stability Estimates 129
8.1.2 Goal of the Error Analysis and Outline ofthe Proof 136
8.1.3 The Error Equation 137
3 =a
8.1.4 The Case 'Vw E L3(0,T jL (n)) and ao(8) 139
8.1.5 The Case 'Vw E L3(0,r ,L3(n)) and ao(8) >o 142
8.1.6 The Case 'Vw E L2(0,r, LOO (n)) and ao(8) ~ a 149
8.1.7 Failures of the Present Analysis in Other Interesting
Cases 152
8.1.8 A Numerical Example 154
8.2 The Taylor LES Model 158
9 The Solution ofthe Linear Systems 163
9.1 The Fixed Point Iteration for the Solution ofLinear Systems ..163
9.2 Flexible GMRES (FGMRES) With Restart 165
9.3 The Coupled Multigrid Method 168
9.3.1 The Transfer Between the Levelsof the Multigrid
Hierarchy 169
9.3.2 The Vanka Smoothers 177
9.3.3 The Standard Multigrid Method and the Multiple
Discretisation Multilevel Method 182
9.3.4 Schematic Overview and Parameters 183
9.4 The Solution of the Auxiliary Problem in the Rational LES
Model 185
10 A Numerical Study of a Necessary Condition for the
Acceptability ofLES Models 189
10.1 The Flow Through a Channel 189
10.2 The Failure of the Taylor LES Model " 191
10.3 The Rational LES Model 194
10.3.1 Computations With the Smagorinsky Subgrid Scale
Model 194
10.3.2 Computations With the Iliescu-Layton Subgrid Scale
Model 196
10.3.3 Computations Without Model for the Subgrid Scale
Term 197
10.4 Summary 197
