Numerical Simulation of Viscous Shock Layer Flows PDF
Preview Numerical Simulation of Viscous Shock Layer Flows
NUMERICAL SIMULATION OF VISCOUS SHOCK LAYER FLOWS FLUID MECHANICS AND ITS APPLICATIONS Volume 33 Series Editor: R. MOREAU MADYIAM Ecole Nationale Superieure d' Hydraulique de Grenoble Borte Postale 95 38402 Saint Martin d' Heres Cedex, France Aims and Scope of the Series The purpose of this series is to focus on subjects in which fluid mechanics plays a fundamental role. As well as the more traditional applications of aeronautics, hydraulics, heat and mass transfer etc., books will be published dealing with topics which are currently in a state of rapid development, such as turbulence, suspensions and multiphase fluids, super and hypersonic flows and numerical modelling techniques. It is a widely held view that it is the interdisciplinary subjects that will receive intense scientific attention, bringing them to the forefront of technological advance ment. Fluids have the ability to transport matter and its properties as well as transmit force, therefore fluid mechanics is a subject that is particulary open to cross fertilisation with other sciences and disciplines of engineering. The subject of fluid mechanics will be highly relevant in domains such as chemical, metallurgical, biological and ecological engineering. This series is particularly open to such new multidisciplinary domains. The median level of presentation is the first year graduate student. Some texts are monographs defining the current state of a field; others are accessible to final year undergraduates; but essentially the emphasis is on readability and clarity. For a list ofr elated mechanics titles, see final pages. Numerical Simulation of Viscous Shock Layer Flows by YURI P. GOLOV ACHOV A.F.Ioffe Physico-Technical Institute, Russian Academy of Sciences, St. Petersburg, Russia SPRINGER-SCIENCE+BUSINESS MEDIA. B.V. A C.I.P. Catalogue record for this book is available from the Library of Congress. ISBN 978-90-481-4594-2 ISBN 978-94-015-8490-6 (eBook) DOI 10.1007/978-94-015-8490-6 Printed on acid-free paper AII Rights Reserved © 1995 Springer Science+Business Media Dordrecht Originally published by Kluwer Academic Publishers in 1995 Softcover reprint ofthe hardcover Ist edition 1995 No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, inc1uding photocopying, recording or by any information storage and retrieval system, without written permission from the copyright owner. Contents Preface vii Basic notations and scales ix 1.1 Latin letters IX 1.2 Greek letters X 1.3 Subscripts Xl 1.4 Superscripts Xl 1.5 Scales xi 1 Simulation of supersonic flow around a body using the Navier-Stokes equations 1 1.1 Formulation of the problem 1 1.2 Finite-difference methods 11 1.3 Simulation of rarefied flows 30 2 Viscous shock layer models and computational methods 41 2.1 Reduced Navier-Stokes equations 41 2.2 Parabolized equations 50 2.3 Multisweep methods 67 2.4 Thin viscous shock layer model 78 2.5 Self-similar solutions 87 3 Applications to plane and axisymmetric flows 99 3.1 Flow around a blunted body in the aerodynamic wake 99 3.2 Flow from a supersonic source past a spherical bluntness 111 3.3 Viscous shocked flows 121 3.4 Flows with mass injection 130 3.5 Unsteady flows 141 4 Simulation of three-dimensional flows 154 4.1 Notation of the governing equations 154 4.2 Methods for solving the non-stationary equations 157 4.3 Methods for solving the parabolized equations 165 4.4 Thin viscous shock layer model 177 VI CONTENTS 4.5 Examples of calculations 183 4.6 Self-similar solutions 209 4.7 Degenerate three-dimensional flows 227 4.8 Approximate methods for prediction of three-dimensional flow characteristics 239 5 Physical and chemical effects 248 5.1 Mathematical description of high-temperature flows 248 5.2 Non-equilibrium chemically reacting flow 259 5.3 Non-equilibrium weakly ionized flow 301 5.4 Radiating flow 312 References 329 Index 341 Preface The book is concerned with mathematical modelling of supersonic and hyper sonic flows about bodies. Permanent interest in this topic is stimulated, first of all, by aviation and aerospace engineering. The designing of aircraft and space vehicles requires a more precise prediction of the aerodynamic and heat transfer characteristics. Together with broadening of the flight condition range, this makes it necessary to take into account a number of gas dynamic and physical effects caused by rarefaction, viscous-inviscid interaction, separation, various physical and chemical processes induced by gas heating in the intensive bow shock wave. The flow field around a body moving at supersonic speed can be divided into three parts, namely, shock layer, near wake including base flow, and far wake. The shock layer flow is bounded by the bow shock wave and the front and lat eral parts of the body surface. A conventional approach to calculation of shock layer flows consists in a successive solution of the inviscid gas and boundary layer equations. When the afore-mentioned effects become important, implementation of these models meets difficulties or even becomes impossible. In this case, one has to use a more general approach based on the viscous shock layer concept. In this approach, which is the topic of the book, the whole of the flow ahead of the body is calculated using viscous gas dynamics equations, which allows one to avoid matching procedures, to take into account an interaction between the flow regions and to investigate complex stationary and non-stationary flows including inner shocks, separation and recirculation zones. Excepting the simplest problems, the viscous shock layer models are implement ed with numerical methods. It is worth noting here that the role of numerical simulation in solving gas dynamic problems is rapidly increasing due to impressive progress in computer capacity and efficiency of numerical algorithms. At relatively low expenses, numerical simulation provides comprehensive data on the flow un der study and allows one to investigate a wide range of flight conditions, including those which can not be reproduced in laboratory experiments. The first viscous shock layer calculations were carried out for hypersonic stagna tion region flows ahead of blunt bodies in the early sixties. Later on, this approach was substantially developed. A number of viscous shock layer models have been suggested which differ from each other in details of the flow description, range of validity and mathematical properties. Appropriate numerical methods have been elaborated as well. In spite of a wide use of the viscous shock layer approach, it has not been expounded in a systematic way. Dealing with a wider range of prob lems, available books on supersonic and computational gas dynamics include only Vll V III PREFACE fragments of this approach. The author attempts to make up for this deficiency. The book presents a description of the viscous shock layer models and appropri ate numerical methods. The basic ideas of mathematical modelling are elucidated with a number of calculation examples which demonstrate also some interesting and important features of supersonic and hypersonic flows about bodies. Our at tention is focused mainly on the flows about blunt bodies, since even if such nose tip is not made especially beforehand, it is usually generated in a flight due to aerodynamic heating. It should be stressed here that the author did not intend to give a comprehensive analysis of all relevant gas dynamic phenomena. The main purpose was to demonstrate application of the viscous shock layer approach. The book consists of five chapters. The first of them presents a formulation of the problem with the use of full Navier-Stokes equations. Modelling of turbulent flows with Reynolds equations and applicability of the continuous medium ap proach to rarefied flow regimes are also discussed. The chapter contains a review of finite-difference methods for solving full Navier-Stokes equations, description of the implicit unidirectional method suggested by the author and discussion of some grid generation techniques. Chapter 2 presents viscous shock layer models using the reduced Navier-Stokes equations. The appropriate numerical methods are considered including the time-marching, global relaxation and space-marching procedures. The self-similar solutions are discussed as well. To clarify basic ideas and their numerical implementation, they are expounded through the first two chapters with reference to a two-dimensional problem. Chapter 3 presents calcu lation results for some axisymmetric and plane flows (blunt-body problem in non uniform flow fields, flows with mass injection, unsteady flows). Chapter 4 extends the above viscous shock layer models and numerical methods to three-dimensional flows. Some kinds of degenerate three-dimensional flows and approximate methods for prediction of aerodynamic characteristics are also discussed. Chapter 5 deals with physical and chemical processes which are important at hypersonic flight speed. The book is intended mainly for researchers working in the field of super- and hypersonic computational gas dynamics. It can be of use for post- and undergrad uates of the proper specialities. The reader is supposed to be acquainted with the fundamentals of the finite-difference numerical methods, gas dynamics and physics of high-temperature gases. The book is based, to a great extent, on the results obtained by the author and his colleagues at the loffe Physico-Technical Institute. I thank them for useful collaboration and I thank other researchers both in Russia and abroad for kind permission to reproduce some of their results. I am much obliged to Professor G A Tirskii for perusal the manuscript and helpful remarks. Many thanks are also due to A M Kuzmin and T V Serova for preparation of the camera ready copy of the manuscript. Yu P Golovachov 1 May 1995 Basic notations and scales 1.1 LATIN LETTERS a, b, c, d parameters of the oncoming stream non-uniformity Cj skin friction coefficient CH heat transfer coefficient (Stanton number) C wall pressure coefficient p C drag coefficient x speed of sound; speed of light C mass fraction of the i-th species Ci C; mass fraction of the j-th element Cp specific heat capacity at constant pressure Cv specific heat capacity at constant volume D bow shock velocity with regard to the body surface D diffusion coefficient Da Damkohler number E total specific energy; energy of the electronic state e specific internal energy G mass injection rate determinant of the metric tensor; degeneracy of the electronic state 9 g .. gij covariant and contravariant components of the metric tensor 'J' H total specific enthalpy; Lame coefficient; altitude; average body surface curvature h specific enthalpy; Planck constant I identity matrix I radiation intensity Ji mass diffusion flux of the i-th species J~ mass diffusion flux of the j-th element J J Jacobian of the coordinate transformation Kp Planck mean absorption coefficient KR Rosseland mean absorption coefficient Kn Knudsen number Kpi equilibrium constant of the i-th reaction k wall-to-stagnation temperature ratio; Boltzmann constant kj, kr forward and reverse reaction rate constants ix x BASIC NOTATIONS AND SCALES kw effective wall catalycity coefficient I mean free path of the particles; distance between the bodies M Mach number molar mass Avogadro number electron number density distance from the body surface stress tensor Prandtl number pressure q heat flux qi general curvilinear coordinates R body nose radius; universal gas constant R* specific gas constant Re Reynolds number Se Schmidt number Sh Strouhal number s distance along the body contour T viscous stress tensor T temperature t time u, v, W gas velocity components V gas velocity vector Wi mass production rate of the i-th species xC>! Cartesian coordinates 1.2 GREEK LETTERS 0:' angle of attack f3 angle of side-slip; sweep angle 'Y gas specific heat ratio; intermittency coefficient 8 boundary layer thickness; hypersonic flow parameter 8~ Kronecker function J c bow shock distance from the body surface; body surface emissivity ( normalized Dorodnitsin's variable 'TJ Dorodnitsin's variable Be cone half-angle K. body surface curvature; absorption coefficient A heat conductivity; radiation wavelength J.L viscosity v unit normal on the bow shock surface stoichiometric coefficient; radiation frequency e/I distance from the. body surface normalized by the shock layer thickness; Dorodnitsin's variable
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