Filtering Techniques for Turbulent Flow Simulation

409 Pages·1990·21 MB·English

by  Alvaro A. Aldama (auth.)

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Lecture Notes in
Engineering
Edited by C. A. Brobbla and S. A. Orszag
56
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A.A. Aldama
Filtering Techniques for
Turbulent Flow Simulation
Springer-Verlag
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Series Editors
C. A. Brebbia . S. A. Orszag
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J. H. Seinfeld . P. Silvester' P. Spanos· W. Wunderlich' S. Yip
Author
Alvaro A. Aldama
Dept. of Civil Engineering
and Operations Research
Princeton University
Princeton, NJ 08544
USA
ISBN-13: 978-3-540-52137-2 e-ISBN-13: 978-3-642-84091-3
001: 10.1007/978-3-642-84091-3
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2161/3020-543210 Printed on acid-free paper.
To ELIZABETH. DAVID AND ABRAHAM
"AND ONE OF THE ELDERS SAITH UNTO
ME, WEEP NOT; BEHOLD, THE LION
OF THE TRIBE OF JUDA, THE ROOT
OF DAVID, HATH PREVAILED TO OPEN
THE BOOK, AND TO LOOSE THE SEVEN
SEALS THEREOF."
REVELATION 5:5
CONTENTS
Chapter 1 INTRODUCTION 1
1.1 Scope of the Study 1
1.2 Report Outline 4
Chapter 2 TURBULENCE MODELING 7
2.1 The Nature of Turbulent Flows 7
2.2 Conventional Turbulence Modeling 12
2.3 Large Eddy Simulation 19
2.4 Summary 41
Chapter 3 TWO-SCALE FILTERING APPROACH THEORY 42
3.1 Introduction 42
3.2 Space-Time Filtering Approach 43
3.3 The Leonard Approximation in the STF
Context 48
3.4 The Approximation of the Cross Terms 53
3.5 The Asymptotic Nature of the Leonard
Approximation 59
3.6 The Asymptotic Nature of the Approximation
of the Cross Terms 65
3.7 Fourier Space Theory 83
3.8 Filtering Effect of Discrete Approximations 88
3.9 Closure Model 95
3.10 Summary 96
Chapter 4 THREE-SCALE FILTERING APPROACH THEORY 98
4.1 Introduction 98
4.2 Three-Scale Approach 100
4.3 Macroscale Dynamics 102
4.4 Mesoscale Dynamics 106
4.5 Relationship between Two- and Three-Scale
Approaches 120
VI
4.6 Closure Model 123
4.7 Summary 124
Chapter 5 DIRECT SIMULATIONS OF PERIODIC BURGERS' FLOW 125
5.1 Introduction 125
5.2 Burgers' Equation as a Mathematical Model
of Turbulence 126
5.3 Shock Development and Energy Equation for
Free Burgers' Flow 129
5.4 Design of Numerical Experiment for Free
and Periodic Burgers' Flow 133
5.5 Computation of Spectral statistics 142
5.6 Filtering of Instantaneous Variables 144
5.7 Generation of Initial Conditions 147
5.8 Numerical Scheme 150
5.9 Numerical Stability Analysis 152
5.10 Results for Free and Periodic Burgers'
Flow 162
5.10.1 statistical Significance and
Spatial Resolution 162
5.10.2 Sensitivity to Time step
Magnitude 169
5.10.3 Time Filtering Effect 169
5.10.4 Velocity Fields 183
5.10.5 Spectral Dynamics and Energy
Evolution 191
5.11 Forced and·· Periodic Burgers' Problem 215
5.12 Generation of Random Forcing 216
5.13 Results for Forced and Periodic Burgers'
Flow 225
5.13.1 statistical significance and
spatial Resolution 226
5.13.2 Time Filtering Effect 235
5.13.3 Velocity Fields 235
VII
5.13.4 Spectral Dynamics and Energy
Evolution 244
5.14 Summary 244
Chapter 6 NUMERICAL EXPERIMENTS WITH TWO-SCALE FILTERING
APPROACHES APPLIED TO PERIODIC BURGERS' FLOW 253
6.1 Introduction 253
6.2 LES Approach for Free Burgers' Flow 254
6.3 STF Approach for Forced Burgers' Flow 257
6.4 Relationship between LES and STF
Approaches 259
6.5 Numerical Scheme 261
6.5.1 Accuracy Requirements 261
6.5.2 Quasi-Implicit Factored Time
Stepping Scheme with
Richardson Extrapolation 262
6.5.3 Quadratic Finite Element
spatial Discretization 269
6.5.4 Green's Function Solution
Algorithm 272
6.5.5 Accuracy Properties 276
6.6 Simulation of Space-Filtered (LS) Free
Burgers' Flow 282
6.6.1 Choice of Grid Size and Time
Step 282
6.6.2 Calibration of Closure Models 284
6.6.3 LS Spectral Dynamics 284
6.7 Simulation of space-Time Filtered (STF)
Forced Burgers' Flow 286
6.7.1 Choice of Grid Size and Time
Step 286
6.7.2 Energy Evolution 293
6.7.3 STF Spectral Dynamics 294
6.7.4 sensitivity Analyses 300
6.8 Summary 300
VIII
Chapter 7 NUMERICAL EXPERIMENTS WITH THREE-SCALE FILTERING
APPROACH APPLIED TO PERIODIC BURGERS' FLOW 306
7.1 Introduction 306
7.2 Space-Filtered Macrosca1e Dynamics 308
7.3 Space-Filtered Mesoscale Dynamics 309
7.4 Numerical Scheme 313
7.5 Simulation of Space-Filtered (Free)
Mesoscale Burgers' Flow 317
7.5.1 Mesoscale Spectral Dynamics 317
7.5.2 Reconstruction of LS Spectral
statistics 317
7.6 Summary 323
Chapter 8 DISCUSSION, CONCLUSION AND RECOMMENDATIONS 329
8.1 Conventional Averaging Procedures Versus
Filtering 329
8.2 Discussion of Results and Summary of
Contributions 341
8.3 Recommendations for Future Research 349
APPENDIX A IMPLICIT FACTORED - QUADRATIC FINITE ELEMENT
FORMULATION OF STF FORCED BURGERS' EQUATION 354
APPENDIX B IMPLICIT FACTORED - QUADRATIC FINITE ELEMENT
FORMULATION OF LES-MESOSCALE EQUATION 370
REFERENCES 380
Chapter 1
INTRODUCTION
1.1 Scope of the Study
The detailed and reasonably accurate computation of large scale
turbulent flows has become increasingly important in geophysical and engi
neering applications in recent years. The definition of water quality
management policies for reservoirs, lakes, estuaries, and coastal waters,
as well as the design of cooling ponds and solar ponds, requires an ade
quate quantitative description of turbulent flows. When the diffusion of
some tracer (be it active, such as temperature or salinity, or passive,
such as dissolved oxygen) is of relevance to a specific application, the
proper determination of the effects of turbulent transport processes has
paramount importance. Thus, for instance, the proper understanding of
lake and reservoir dynamics requires, as a first step, the ability to
simulate turbulent flows. Applications in other areas of geophysical
research, such as meteorology and oceanography are easily identified and
large in number. It should be stressed that, in this context, the analyst
seeks predictive ability to a certain extent. Accordingly, the need for
simulation models that closely resemble the natural processes to be repre
sented has recently become more evident.
Since the late 1960s considerable effort has been devoted to the
development of models for the simulation of complex turbulent flows. This
has resulted in the establishment of two approaches which have been, or