Quasi-Gas Dynamic Equations
The monograph is devoted to modern mathematical models and numerical methods for solving gas- and fluid-dynamic problems based on them. Two interconnected mathematical models generalizing the Navier–Stokes system are presented; they differ from the Navier
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Series Editor K.J. Bathe Massachusetts Institute of Technology, Cambridge, MA, USA
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Tatiana G. Elizarova
Quasi-Gas Dynamic Equations
123
Tatiana G. Elizarova Russian Academy of Sciences Institute of Mathematical Modeling Miusskaya Sq. 4 Moskva Russia 125047 [email protected]
ISSN 1860-482X e-ISSN 1860-4838 ISBN 978-3-642-00291-5 e-ISBN 978-3-642-00292-2 DOI 10.1007/978-3-642-00292-2 Springer Dordrecht Heidelberg London New York Library of Congress Control Number: 2009926099 c Springer-Verlag Berlin Heidelberg 2009 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover design: deblik, Berlin Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)
Preface
The monograph is devoted to modern mathematical models and numerical methods for solving gas- and fluid-dynamic problems based on them. Two interconnected mathematical models generalizing the Navier–Stokes system are presented; they differ from the Navier–Stokes system by additional dissipative terms with a small parameter as a coefficient. The new models are called the quasi-gas-dynamic and quasi-hydrodynamic equations. Based on these equations, effective finite-difference algorithms for calculating viscous nonstationary flows are constructed and examples of numerical computations are presented. The universality, the efficiency, and the exactness of the algorithms constructed are ensured by the fulfillment of integral conservation laws and the theorem on entropy balance for them. The book is a course of lectures and is intended for scientists and engineers who deal with constructing numerical algorithms and performing practical calculations of gas and fluid flows and also for students and postgraduate students who specialize in numerical gas and fluid dynamics. Moscow December 2008
Tatiana Elizarova
v
Contents
1 Construction of Gas-Dynamic Equations by Using Conservation Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Averaging Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.1 Spatial Averages . . . . . . . . . . . . . . . . . . . . . . . .
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