DEDICATION ...................................................... i
NOMENCLATURE ................................................... ii
TABLE OF CONTENTS ............................................... v
LIST OF FIGURES .............................................. viii
1 INTRODUCTION ................................................. 1
2 EARLIER WORK ON TEMPORAL DISCRETIZATION ...................... 4
3 CHOICE OF TEMPORAL DISCRETIZATION METHOD USED IN THIS BOOK ... 9
4 NUMERICAL METHOD ............................................ 12
4.1 Model Equations ........................................ 12
4.2 Staggered Grid Approach ................................ 14
4.2.1 Spatial Discretization .......................... 14
4.2.1a Mass Conservation Equation .............. 16
4.2.1b Momentum Conservation Equations ......... 16
4.2.2 Temporal Discretization ......................... 20
4.2.3 Simultaneous Solution of Mass Conservation and
Momentum Conservation Equations ................. 23
4.2.3a Evaluation of CV Face Velocities ........ 25
4.2.3b Correcting Velocity and Pressure
Fields by enforcing Mass Conservation ... 25
4.2.3c Under-relaxation ........................ 28
4.2.3d Convergence Criteria .................... 29
4.2.3e Algorithm ............................... 30
4.3 Non-staggered Grid Approach (or Co-located Variables
Approach) .............................................. 34
4.3.1 Spatial Discretization .......................... 34
4.3.1a Mass Conservation Equation .............. 34
4.3.1b Momentum Conservation Equations ......... 35
4.3.2 Temporal Discretization ......................... 37
4.3.3 Simultaneous Solution of Mass Conservation and
Momentum Conservation Equations ................. 39
4.3.3a Evaluation of CV Face Velocities ........ 39
4.3.3b Correcting Velocity and Pressure
Fields by enforcing Mass Conservation ... 43
4.3.3c Under-relaxation ........................ 47
4.3.3d Convergence Criteria .................... 48
4.3.3e Algorithm ............................... 49
5 VALIDATION .................................................. 53
5.1 Test Case .............................................. 53
5.2 Grid Dependence Study .................................. 54
5.3 Code Validation Runs ................................... 56
5.3.1 Comparison of the Staggered Grid Code with
FLUENT for Re = 400 ............................. 56
5.3.2 Comparison of the Staggered Grid Code with the
Results of Erturk et al. [6] for Re = 1,000 ..... 56
5.3.3 Comparison of the Non-staggered Grid Code
with the Staggered Grid Code for Re = 400 ....... 57
6 RESULTS AND DISCUSSION ...................................... 58
7 SUMMARY ..................................................... 78
8 RECOMMENDATIONS FOR FUTURE WORK ............................. 79
REFERENCES .................................................. 80
APPENDIX A: RUNGE-KUTTA METHODS ................................ 87
A1 General Form of Runge-Kutta Methods ..................... 88
A2 Explicit Runge-Kutta Methods ............................ 90
A3 Diagonally Implicit Runge-Kutta (DIRK) Methods .......... 90
A4 Singly Diagonally Implicit Runge-Kutta (SDIRK)
Methods ................................................. 91
A5 Explicit first stage, Single diagonal coefficient,
Diagonally Implicit, Runge-Kutta (ESDIRK) Methods ....... 91
A6 Stiffly Accurate Runge-Kutta Methods .................... 92
APPENDIX B: TWO EXAMPLES OF EVALUATION OF DEFERRED-
CORRECTION TERM AND COEFFICIENTS IN THE DISCRETIZED
MOMENTUM EQUATIONS .......................................... 93
Bl Power Law Scheme ........................................ 93
B2 QUICK Scheme ............................................ 96
APPENDIX C: FORMULATION FOR SIMPLE DIRK METHOD USING A
FOUR-STAGE ESDIRK METHOD ................................... 100
CI Staggered Grid Approach
C2 Non-staggered Grid Approach
APPENDIX D: LIST OF REFERENCES AS QUOTED BY BUTCHER [72]
AND/OR BUTCHER AND WANNER [73] ............................. 105
ACKNOWLEDGEMENT ............................................ 106
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