Table Of Content'~~~1!f
Numerical Computation
of
INTERNAL AND EXTERNAL
FLOWS
Volume 2: Computational Methods for lnviscid and
Viscous Flows
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WILEY SERIES IN
NUMERICAL METHODS IN ENGINEERING
Consulting Editors
R. H. Gallagher, Worcester Polytechnic Institute.
Worcester.M assachusetts.U SA
and ';1:, !.~,
O. C. Zienkiewicz, Departmento f Civil Engineering. ' , J.,.. "
UniversityC ollegeo f Swansea ,
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Rock Mechanics in EngineeringP ractice ..
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Edited by K. G. Stagg and O. C. Zienkiewicz ,
Optimum Structural Design: Theory and Applications
Edited by R. H. Gallagher and O. C. Zienkiewicz
Finite Elements in Fluids
Vol. I Viscous Flow and Hydrodynamics
Vol. 2 Mathematical Foundations, Aerodynamics and Lubrication
Edited by R. H. Gallagher. J. T. Oden, C. Taylor, and O. C. Zienkiewicz
Finite Elements in Geomechanics
Edited by G. Gudehu.~
Numerical Methods in Offshore Engineering
Edited by O. C. Zienkiewicz. R. W. Lewis, and K. G. Stagg
Finite Elements in Fluids
Vol. 3
Edited by R. H. Gallagher. O. C. Zienkiewicz. J. T. Oden. M. Morandi Cecchi.
and C. Taylor
Energy Methods in Finite Element Analysis
Edited by R. Glowinksi. E. Rodin. and O. C. Zienkiewicz
Finite Elements in Electrical and Magnetic Field Problems
Edited by M. V. K. Chari and P. Silvester
Numerical Methods in Heat Transfer
Vol. I
Edited by R. W. Lewis. K. Morgan. and O. C. Zienkiewicz
Finite Elements in Biomechanics
Edited by R. H. Gallagher, B. R. Simon, P. C. Johnson. and J. F. Gross
Soil Mechanics- Transient and Cyclic Loads
Edited by G. N. Pande and O. C. Zienkiewicz
Finite Elements in Fluids
Vol. 4
Edited by R. H. Gallagher. D. Norrie. J. T. Oden. a, dO. C. Zienkiewicz
Foundations of Structural Optimization: A Unified Approach
Edited by A. J. Morris
Numerical Methods in Heat Transfer
Vol. II
Edited by R. W. Lewis. K. Morgan, and B. A. Schrefler
Numerical Methods in Coupled Systems
Edited by R. W. Lewis. E. Hinton, and P. Bettess
New Directions in Optimum Structural Design
Edited by E. Atrek. R. H. Gallagher. K. M. Rag.~dell. and O. C. Zienkiewicz
Mechanics of Engineering Materials
Edited by C. S. Desai and R. H. Gallagher
Numerical Analysis of Forming Processes
Edited by J. F. T. Pittman. O. C. Zienkiewicz, R. D. Wood. and J. M. Alexander
Finite Elements in Fluids
Vol. 5
Edited by R. H. Gallagher. J. T, aden, o. C. Zienkiewicz.
T Kawai. and M. Kawahara
Mechanics of Geomaterials: Rocks, Concretes, Soils
Edited by Z. P. Baiant
Finite Elements in Fluids
Vol. 6
Edited by R. H. Gallagher. G. F. Carey, J. T. aden, and O. C. Zienkiewicz
Numerical Methods in Heat Transfer
Vol. III
Edited by R. W. Lewis and K. Morgan
Accuracy Estimates and Adaptive Refinements in Finite Element Computations
Edited by I. Babuska. J. Gago, E. R. de Arantes e Oliveira, and O. C. Zienkiewicz
The Finite Element Method in the Deformation and Consolidation
of Porous Media
R. W. Lewi.~ and B. A. Schrefler
Numerical Methods for Transient and Coupled Problems
Edited by R. W. Lewis. E. Hinton, P. Bettess, and B. A. Schrefler
Microcomputers in Engineering Applications
Edited by B. A. Schrefler and R. W. Lewis
Numerical Computation of Internal and External Flows
Vol. 1: Fundamentals of Numerical Discretization
C. Hirsch
Finite Elements in Fluids
Vol. 7
Edited by R. H. Gallagher, R. Glowinski. P. M. Gresho.
J. T. aden, and O. C. Zienkiewicz
Mathematical Modeling of Creep and Shrinkage of Concrete
Edited by Z. P. Baiant
Numerical Computation of Internal and External Flows
Vol. 2: Computational Methods for lnviscid and Viscous Flows
C. Hirsch
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Numerical Computation
of
INTERNAL AND EXTERNAL
FLOJiJiS
Volume 2: Computational Methods for
Inviscid and Viscous Flows
CharlesH irsch
Departmento f Fluid Mechanics,
Vrije Universiteit Brussel,
BrusselsB, elgium
A Wiley-lnterscience Publication
JOHN WILEY & SONS
Chichester. New York. Brisbane. Toronto. Singapore
Copyright C 1990 by John Wiley & Sons Ltd.
Baffins Lane, Chichester, West Sussex POl9 IUD. England
Reprinted November 1991
Reprinted August 1992
Reprinted August 1994
7 /1
All rights reserved.
3
No part of this book may be reproducedb y any means. _
or transmitted. or translated into a machine language ~ 7
without the written permission of the publisher. II
-S'7'?
Other Wiley Editoritll Offices l P '( ~
John Wiley & Sons. Inc.. 605 Third Avenue.
New York. NY 10158-001U2.S A v. <-
Jacaranda Wiley Ltd. G.P.O. Box 859. Brisbane.
Queensland 400 I. Australia
John Wiley & Sons (Canada) Ltd. 22 Worcester Road.
Rexdale. Ontario M9W ILl. Canada
John Wiley & Sons (SEA) Pte Ltd. 37 Jalan Pemimpin #05-04.
Block B. Union Industrial Building. Singapore 2057
Librtlry o!CoIIxre.f.f CtlttlloXillK-ill-P"blictltioil Dtlttl:
Hirsch. Ch.
Numerical computation of internal and external flows.
(Wiley series in numerical methods in engineering)
"A Wiley-interscience publication:'
Contents: v. I. Fundamentals of numerical discretiza-
tion-v. 2. Computational methods for inviscid and
viscous flows.
I. Fluid dynamics-Mathematical models. I. Title.
II. Title: Numerical computation of internal and external
flows. III. Series.
TA357.H574 1988 620.1'064 87-23116
ISBN 0471917621 (v. I)
ISBN 0471923516 (v. 2)
ISBN 0471924520 (pbk.:v. 2)
Briti.fh Librtlry CtlttllOK"illK ill P"blictltion Dtlttl:
Hirsch. Ch. (Charles)
Numerical computation of internal and external flows.
Vol. 2. Computational method~ for invi~cid and vi~cou~ tlow~
I. Fluids. Flow. Simulations. Numerical methods
I. Title
532'.051 '0724
ISBN 0471923516 (Pbk: 0471924520)
Typeset by Thomson Press (India) Ltd
Printed and bound in Great Britain by
Redwood Books, Trowbridge, Wiltshire
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CONTENTS
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PREFACE xv
NOMENCLATURE xix
PART V: THE NUMERICAL COMPUTATION OF POTENTIAL
FLOWS 1
Chapter1 3 The Mathematical Formulations of the Potential Flow Model 4
13.1C onservative Form of the Potential Equation 4
13.2 The Non-conservative Form of the Isentropic Potential Flow Model 6
13.2.1 Small-perturbation potential equation 7
13.3 The Mathematical Properties of the Potential Equation 9
13.3.1 Unsteady potential flow 9
13.3.2 Steady potential flow 9
13.4 Boundary Conditions 14
13.4.1 Solid wall boundary condition 14
13.4.2 Far field conditions 15
13.4.3C ascadea nd channel flows 17
13.4.4 Circulation and Kutta condition 18
13.5 Integral or Weak Formulation of the Potential Model 18
13.5.1 Bateman variational principle 19
13.5.2A nalysis of some properties of the variational integral 20
Chapter1 4 The Discretization of the SubsonicP otential Equation 26
14.1 Finite Difference Formulation 27
14.1.1N umerical estimation of the density 29
14.1.2C urvilinear mesh 31
14.1.3 Consistencyo f the discretization of metric coefficients 34
14.1.4 Boundary conditions-curved solid wall 36
14.2 Finite Volume Formulation 38
14.2.1 Jamesona nd Caughey'sf inite volume method 39
14.3 Finite Element Formulation 42
14.3.1T he finite element-Galerkin method 43
14.3.2 Least squareso r optimal control approach 47
14.4 Iteration Schemef or the Density 47
IX
