Anderson, John Murray (1992) Numerical simulation of imperfect gas flows. PhD thesis http ...

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Anderson, John Murray (1992) Numerical simulation of imperfect gas
flows.

PhD thesis

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NUMERICAL SIMULATION OF IMPERFECT GAS FLOWS
by
John Murray Anderson, B. Eng.
Thesis submitted to the Faculty of Engineering,
the University of Glasgow, for the degree of Doctor of Philosophy.
August 1992
© 1. Murray Anderson, 1992
ACKNOWLEDGEMENTS
DECLARATION 11
ABSTRACT 11l
NOMENCLATURE v
ClwpterOne
INTRODUCTION 1
1.1 The Evolution of Hypersonic Flight. 1
1.2 Definition of Hypersonic Real Gas Flows. 5
1.2-a Hypersonic Flows. 5
1.2-b Real Gas Flows. 6
1.3 The Need for Hypersonic Flow Simulations. 7
1.4 Some General Numerical Aspects of Flow Simulations. 9
1.5 The Scope of This Study. 10
1.6 The Structure of the Dissertation. 11
Chapter Two
PHYSICAL GAS DYNAMICS 12
2.1 The Internal Structure of Gases. 12
2.2 The Conservation Laws for Mass, Momentum and Energy. 13
2.2-a Viscous Equations. 14
2.2-b Inviscid Equations. 15
2.3 The Conservation Laws for Component Species. 16
2.4 The Law of Mass Action. 17
2.5 Equations of State. 18
Chapter Three
OOMPUTATIONAL GAS DYNAMICS 20
3.1 The Mathematical Character of the Flow Equations. 20
3.1-a Linearization of the Equations of Motion. 20
3.1-b Variable Transformations and the Compatibility Relations. 21
3.1-c A Comment on the Characteristic Variables. 24
3.2 Discretization of the Equations of Motion. 26
3.2-a Temporal Discretization. 26
3.2-b Spatial Discretization. 27
3.3 Shock Waves. 28
Chapter Four
EQUILIBRIUM AIR MODELS 29
4.1 Curve Fitting Techniques and Data Base Models. 29
4.1-a Grabau Type Transition Functions. 30
4.1-b Sixteen Coefficient Curve Fits. 32
4.1-c Twenty Four Coefficient Curve Fits. 33
4.2 Physically Based Techniques. 34
4.3 The Chemical Components. 35
4.4 Thermodynamic Propenies of the Component Species. 36
4.4-a Direct Calculation of Species Properties from Partition
Functions. 36
4.4-b Polynomial Curve Fits for the Species Propenies. 38
4.5 Equilibrium Species Concentrations. 40
4.6 Low Temperature Considerations. 41
4.6-a Very low temperature: In Kpl < -100, T < 530 K. 42
4.6-b Low temperature: In Kp3 < -100, T < 1 000 K. 42
4.6-c High temperature: T> 1 000 K 43
4.7 Solution of the Equilibrium Equations. 43
4.8 Thermodynamic Propenies of the Mixture. 46
4.9 Equilibrium and Frozen Speeds of Sound. 48
4.10 Validation. 55
Chapter Five
IMPLEMENTA TION OF EQUILIBRIUM AIR MODELS IN THE
CFD ENVIRONMENT 57
5.1 Inversion of the State Equations. 57
5.2 Calculation of the X and Derivatives. 61
J(
5.3 The Quasi-One-Dimensional Nozzle Problem. 63
5.4 Discretization of the Equations of Motion. 64
5.4-a Numerical Discretization. 65
5.4-b Artificial Dissipation. 66
5.5 Boundary Conditions. 67
5.5-a Subsonic Characteristic Boundary Conditions. 68
5.5-b Supersonic Characteristic Boundary Conditions. 70
5.5-c Evaluation of Boundary Treatments - Supersonic Exhaust. 71
5.5-d Evaluation of Boundary Treatments -Subsonic Exhaust. 74
5.6 Hypersonic Expansion Nozzle. 76
Chapter Six
NONEQUILIBRIUM THERMOCHEMICAL MODELLING 79
6.1 Translational and Rotational Relaxation in Chemical Equilibrium. 80
6.2 Curve Fits for the Transport Coefficients. 81
6.3 Transport Coefficients of the Component Species. 82
6.3-a Curve Fitting Methods. 83
6.3-b Theoretical Techniques for Viscosity. 84
6.3-c Theoretical Techniques for Thermal Conductivity. 85
6.4 Transport Coefficients of the Gas Mixture. 86
6.4-a Estimation of Mixture Viscosity. 86
6.4-b Estimation of Mixture Thermal Conductivity. 87
6.5 Recommendations on the Calculation of the Transport Coefficients. 88
6.6 Chemical Nonequilibrium. 89
6.6-a Additional Chemical Reactions. 89
6.6-b Modelling the Production Terms. 90
6.7 Solution Techniques for the Chemical Nonequilibrium Equations. 92
Chapter Seven
CONCLUSIONS AND FUTURE RESEARCH 93
7.1 Conclusions. 93
7.1-a Curve Fitting Techniques for Equilibrium Mixture Properties. 93
7.1-b Physically Based Techniques for the Equilibrium Mixture
Properties. 94
7.1-c Transport Coefficients. 94
7.1-d Recommendations. 95
7.2 Future Research. 95
7.2-a Physical Aspects. 95
7.2-b Numerical Aspects. 96
REFERENCES 98
APPENDIX ONE 103
APPENDIX TWO 106
APPENDIX THREE 107
APPENDIX FOUR 127
APPENDIX FIVE 129
APPENDIX SIX 131
APPENDIX SEVEN 132
FIGURES 133
ACKNOWLEDGEMENTS
The author would like to thank Professor Bryan E. Richards, Mechan Professor of
Aerospace Engineering, for supervising this project.
I would like to express my sincere gratitude to Professor Roderick A. MCD.
Galbraith, head of Aerospace Engineering, for his support and encouragement in the
later stages of this work.
My thanks also extend to the academic and research staff of the Aerospace
Engineering Department, many of whom have contributed either through discussion or
by technical assistance to this work. I would particularly like to single out Dr Qin
Ning and Dr Zhi Jian Wang for propounding much invaluable information regarding
numerical aspects of this work. I would also like to acknowledge Miss Margaret
Simpson and Mrs Emily Garman, secretaries to the Aerospace Engineering
Department, who have provided invaluable secretarial assistance, and the advisory
staff of the Computing Service Department for their assistance with computational
matters.
This work has been supported financially by the Science and Engineering Research
Council and British Aerospace pIc under a CASE award, and latterly by the Science
and Engineering Research Council and the Ministry of Defence under grant number
GRIF 90035.
I am indebted to the many friends I have in Glasgow, and especially to Dr Fiona J.
Watson and Miss Susannah Miles for proof-reading this document.
Finally, I would like to thank my family for their patience and understanding
during the periods when progress in this work was slow. I would particularly like to
express my appreciation of the support provided by my parents throughout the
duration of this project.
-1-
DECLARA TION
The research contained in this thesis was carried out independently by the author
during the period from October 1987 to August 1992. No portion of this work has
been submitted in support of any other degree or qualification at this or any other
university or institute of learning.
:r.~~
1. Murray Anderson
August 1992.
-11-
ABSTRACT
This dissertation reports research into the thermochemical modelling of imperfect air
with applications to computational fluid dynamics (CFD). The work is broadly
separated into two topics; physical modelling of imperfect gases and numerical aspects
of simulating the flow of such gases.
Various levels of physical modelling are considered. Primarily, state equation
models for high temperature air in vibrational and chemical equilibrium are examined.
The most popular techniques currently used for modelling the thermodynamic state of
such gases are based on either look up tables or curve fits. Available curve fit data are
therefore examined and used in the validation of more physically based methods.
A six species, three reaction, ionization free air state model is developed based on
the solution of the laws of mass action to compute the flow chemistry. The calorically
imperfect behaviour of the component species is modelled using both available curve
fit data and statistical mechanics expressions, and it is concluded that species property
curve fits are more appropriate for applications within the CFD environment.
However, existing models based on this approach are limited by their inability to
compute the derivative information required for the formation of the flux Jacobian and
for the calculation of sonic speed. These limitations are overcome by developing
innovative expressions for the required thermodynamic derivatives. Novel equations
describing the frozen and equilibrium speeds of sound in a chemically reacting
imperfect gas are also developed.
In order to apply any state model within the numerical solution of the flow
equations, it is necessary to identify the correct dependent and independent variables.
The nature of any model based on the solution of the laws of mass action requires that
the equilibrium temperature is used as an independent variable. Techniques for
inverting the state equations to give temperature as a dependent variable are therefore
investigated, and a novel algorithm is developed based on a Newton-Raphson iteration
for this inversion.
Many modern algorithms for the solution of hyperbolic and hyperbolic-parabolic
systems of equations rely on mathematical properties associated with calorically perfect
state equations. Algorithms developed on the basis of perfect gas behaviour must
therefore be modified to account for thermal imperfections in high temperature air.
The principal modifications identified here are the restructuring of the flux Jacobian to
account for temperature variations of the ratio of specific heats, the nonhomogeneous
nature of the flux vector with respect to the conserved variables and the lack of a
closed form for the characteristic variables.
-lll-
In order to illustrate the application of equilibrium gas models within the CFD
environment, a solution of the quasi-one-dimensional Euler equations is presented for
supersonic and hypersonic nozzle flows with and without a shock wave present.
Particular attention is given to the characteristic treatment of boundary conditions, as
this is an area in which perfect and equilibrium gas models require distinct treatments.
The scheme used is a trapezoidal time/central space differenced one with added
artificial dissipation.
In addition to modelling the thermodynamic state equations for reacting gases,
some progress towards the modelling of translational nonequilibrium is described.
Methods for evaluating the transport coefficients of air in chemical equilibrium are
addressed. Chemically relaxing inviscid flows are also examined and techniques are
proposed for the solution of such flow problems.
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