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Dépôt Institutionnel de l’Université libre de Bruxelles / Université libre de Bruxelles Institutional Repository

Thèse de doctorat/ PhD Thesis Citation APA:

Johnson, G. M. (1979). A numerical method for the iterative solution of inviscid flow problem (Unpublished doctoral dissertation). Université libre de Bruxelles, Faculté des sciences, Bruxelles.

Disponible à / Available at permalink : https://dipot.ulb.ac.be/dspace/bitstream/2013/214100/1/8402132e-e8ae-43d5-ab81-01813e1cb132.txt

(English version below)

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E^JS^-gTSIQUE

Institut von Karman de Dynamique des Fluides

Institut Université

de Aéronautique Libre de Bruxelles

A NUMERICAL METHOD FOR THE ITERATIVE SOLUTION OF INVISCID FLOW PROBLEMS

by

Gary M. JOHNSON

è>HP

<03

^

B.s. United States Air Force Academy M.S. California Institute of Technology Diploma von Karman Institute for Fluid Dynamics

Thèse présentée â la Faculté des Sciences Appliquées de l'Université Libre de Bruxelles en vue de

l'obtention du grade de Docteur en Sciences Appliquées

septembre 1979

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ABSTRACT

The purpose of thls dissertation is to describe a novel numerlcal procedure for the itérative solution of inviscid flou probiens and to demonstrate its utility for the calculation of steady subsonic and transonic flou fields- The nethod is

■ore general than are previously developed itérative methods in that no assunptions concerning the existence of either a veloc- ity potential fonction or a stream fonction are required-

Application of the herein defined surrogate équation technique allous the formulation of stable-i fully-conservative-i type-dependent finite différence équations for use in obtaining numerical solutions to Systems of first-order partial differen- tial equations-i such as the steady-state Euler équations-

Included among the results presented are steadyi two- dimensional solutions both to the Euler équations for subsonici rotational flou and to the small disturbance équations for tran­

sonic flou- For the latter case-i a computational efficiency uell in excess of that obtained by means of the standard perturbation potential approach is indicated-

Possible improvements to and extensions of the method are discussed-

5075^3

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ACKNOWLEDGEMENTS

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Ve ehall not oeaee front exploration And the end of ail our exploring Will be to arrive where we etarted And know the plaoe for the firet time.

- T.S. Eliot

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TABLE OF CONTENTS

ABSTRACT ... i

ACKNOWLEDGEMENTS ... ii

LIST OF TABLES... xi

LIST OF FIGURES... xiii

LIST OF SYMBOLS...xxvii

1. INTRODUCTION ... 1

1.1 Motivating Factors ... 1

1.2 Historlcal Context ... 2

1.3 Review of the Modem Literature... 11

1.3.1 Survey Literature ... 12

1.3.2 Research Literature ... 17

1.3.3 Spécial Considérations ... 34

1.4 Scope of the Présent Study... 37

2. CAUCHY-RIEMANN PROBLEM ... 41

2.1 Potential Formulation ... 41

2.2 Surrogate Equation Formulation ... 42

2.3 Problem Spécification ... 44

2.4 Flnite Différence Solution Procedure ... 47

2.5 Computational Résulta ... 51

2.6 Conclusions... 54

3. EULER PROBLEM... 57

3.1 Equations of Motion . ... 57

3.2 Ratlonale for Application of the Surrogate Equation Technique ... 60

3.3 Second-Order System ... 62

3.4 Boundary Conditions ... 64

3.4.1 Solld Boundarles ... 64

3.4.2 Flow Boundarles... 67

3.5 Problem Spécification ... 68

3.5.1 Physical Aspects ... 68

3.5.2 Coordlnate System ... 69

3.5.3 Transformed Fleld Equations ... 70

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3.6 Solution Procedure ... 71

3.6.1 Flnite Différence Equations ... 72

3.6.2 Boundary Conditions ... 73

3.6.3 Itération Process ... 75

3.7 Computatlonal Résulta ... 79

3.7.1 Convergence Behavior ... 79

3.7.2 Efficiency... 80

3.7.3 Accuracy... 82

3.7.4 Stabllity... 83

3.7.5 Rotatlonal Flow Through Bends ... 83

3.8 Conservation Form... 88

3.9 Toplcs for Further Study... 90

3.9.1 Efficiency Improvement ... 90

3.9.2 Accuracy Enhancement ... 90

3.9.3 Analysis of Stabllity... 92

3.9.4 Alternative Coordlnate Systems ... 92

3.9.5 Implémentation of Conservatlon-Form Algorlthm... 93

3.10 Conclusions... 93

TRANSONIC FLOW... 95

4.1 Motivation... 95

4.2 Small Perturbation Approximation ... 97

4.3 Perturbation Potentlal Equation ... 99

4.3.1 Formulation... 99

4.3.2 Numerical Solution ... 100

4.4 Second-Order System ... 102

4.4.1 Fleld Equations... 102

4.4.2 Boundary Conditions ... 106

4.5 Problem Spécification ... 107

4.6 Discrète Formulation ... 108

4.6.1 Fleld Equations... 109

4.6.2 Boundary Conditions ... 111 4.6.3 Switcbing Approximations ... H5 4.7 Solution Procedure... HO

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4.8 Computatlonal Résulta ... 121

4.8.1 Subcrltlcal Flow... 123

4.8.2 Supercritical Flow... 123

4.8.3 Efficlency... 125

4.8.4 Accuracy... 127

4.8.5 Stablllty... 128

4.9 Application to the Full Euler Equations . . . 129

4.9.1 Equations of Motion... 129

4.9.2 Second-Order System ... 130

4.9.3 Boundary Conditions ... 133

4.9.4 Dlscretlzation and Solution Procedure . 133 4.10 Conclusions... 133

5. EPILOGUE... 135

5.1 Review... 135

5.2 Augmentation... 138

5.3 General Conclusions ... 140

REFERENCES... 143

APPENDICES : A. SPECIFICATION OF THE TERMINOLOGY USED IN EQUATION (10)... 171

B. SPECIFICATION OF THE TERMINOLOGY USED IN EQUATION (11)... 173

C. COMPUTER PROGRAM LISTING FOR POTENTIAL FORMULATION OF CAUCHY-RIEMANN PROBLEM 177 D. COMPUTER PROGRAM LISTING FOR SURROGATE EQUATION FORMULATION OF CAUCHY- RIEMANN PROBLEM... 18 5 E. THE JACOBIAN MATRICES AND SOME OF THEIR PROPERTIES... 199

F. SIMPLIFIED EXPRESSION OF THE EULER EQUATIONS... 203

G. DERIVATION OF THE SURROGATE SECOND- ORDER SYSTEM FOR THE EULER EQUATIONS WRITTEN IN SHEARED COORDINATES .... 205

H. DERIVATION OF THE FINITE DIFFERENCE EQUATIONS FOR THE EULER PROBLEM . . . 207

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J. COMPUTER PROGRÂM LISTING FOR SURROGATE EQUATION VERSION OF TRANSONIC FLOW

PROBLEM... 235 K. COMPUTER PROGRAM LISTING FOR PERTURBATION

POTENTIAL VERSION OF TRANSONIC FLOW

PROBLEM... 249

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LIST OF TABLES :

1 Définition of Computational Meshes for Cauchy-Rlemann Problem 2 Compilation of Computational Résulta for Cauchy-Rlemann

Froblem

3 Nomenclature for Euler Equation Test Cases

4 Définition of Meshes for Transonlc Flow Computations 5 Nomenclature for Transonlc Flow Test Cases

6 Convergence Behavlor of Transonlc Flow Âlgorlthms

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LIST OF FIGURES

1 Three Equivalent Formulations of the Same Boundary Value Froblem

2 Veloclty Vector Diagram 3 Equlpotential Diagram 4 Isovel Diagram

5 Convergence Historiés for Cauchy-Riemann Froblem 6 Normallzed Convergence Historiés for Cauchy-Riemann

Froblem

7 Speed of Solution Algorithme for Cauchy-Riemann Froblem 8 Convergence Historiés for Euler Froblem

9 Normallzed Convergence Historiés for Euler Froblem

10 Effect of Transverse Mesh Point Denslty on Computatlonal Efficlency

11 Effect of Longitudinal Mesh Point Denslty on Computatlonal Efficlency

12 Experimental Accuracy Détermination for Euler Froblem 13 Shercllff Bend - Physlcal Domain

Augmented Incompressible Flov Solution Minimum Entrante Veloclty « 50 M/S

Ratio of Maximum to Minimum Entrante Velocitles • 2.0 14 Shercllff Bend - Veloclty Vectors

Augmented Incompressible Flow Solution Minimum Entrante Veloclty ■ 50 M/S

Ratio of Maximum to Minimum Entrante Velocitles • 2.0 15 Shercllff Bend - Veloclty Contours

Augmented Incompressible Flow Solution Minimum Entrante Veloclty “ 50 M/S

Ratio of Maximum to Minimum Entrante Velocitles * 2.0 16 Shercllff Bend - Static Pressure Contours

Augmented Incompressible Flow Solution Minimum Entrante Veloclty • 50 M/S

Ratio of Maximum to Minimum Entrante Velocitles •• 2.0 17 Shercllff Bend - Denslty Contours

Augmented Incompressible Flow Solution Minimum Entrance Veloclty “ 50 M/S

Ratio of Maximum to Minimum Entrance Velocitles “ 2.0

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Shercllff Bend - Fhysical Domain Augmented Incompressible Flow Solution Minimum Entrance Veloclty » 50 M/S

Ratio of Maximum to Minimum Entrance Velocltles “ 1.5 Shercllff Bend - Veloclty Vectors

Augmented Incompressible Flow Solution Minimum Entrance Veloclty ■ 50 M/S

Ratio of Maximum to Minimum Entrance Velocltles ” 1.5 Shercllff Bend - Veloclty Contours

Augmented Incompressible Flow Solution Minimum Entrance Veloclty ~ 50 M/S

Ratio of Maximum to Minimum Entrance Velocltles ~ 1.5 Shercllff Bend - Mach Number Contours

Augmented Incompressible Flow Solution Minimum Entrante Veloclty “ 50 M/S

Ratio o£ Maximum to Minimum Entrance Velocltles “ 2.0

Shercllff Bend - Statlc Pressure Contours Augmented Incompressible Flow Solution Minimum Entrance Veloclty - 50 M/S

Ratio of Maximum to Minimum Entrance Velocltles “ 1.5 Shercllff Bend - Denslty Contours

Augmented Incompressible Flow Solution Minimum Entrance Veloclty ■ 50 M/S

Ratio of Maximum to Minimum Entrance Velocltles >1.5 Shercllff Bend - Mach Number Contours

Augmented Incompressible Flow Solutions Minimum Entrance Veloclty • 50 M/S

Ratio of Maximum to Minimum Entrance Velocltles • 1.5 Shercllff Bend - Veloclty Vectors

Surrogate Equation Compressible Flow Solution Minimum Entrance Veloclty ~ 50 M/S

Ratio of Maximum to Minimum Entrance Velocltles • 1.5 Shercllff Bend - Veloclty Contours

Surrogate Equation Compressible Flow Solution Minimum Entrance Veloclty • 50 M/S

Ratio of Maximum to Minimum Entrance Velocltles - 1.5 Shercllff Bend - Statlc Pressure Contours

Surrogate Equation Compressible Flow Solution Minimum Entrance Veloclty ~ 50 M/S

Ratio of Maximum to Minimum Entrance Velocltles “ 1.5 Shercllff Bend - Denslty Contours

Surrogate Equation Compressible Flow Solution Minimum Entrance Veloclty ~ 50 M/S

Ratio of Maximum to Minimum Entrance Velocltles - 1.5

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29 Shercliff Bend - Mach Number Contours

Surrogate Equation Compressible Flow Solution Minimum Entrante Veloclty - 50 M/S

Ratio o£ Maximum to Minimum Entrance Velocltles ■ 1.5 30 Shercllff Bend - Veloclty Vectors

Augmented Incompressible Flow Solution Minimum Entrance Veloclty i* 100 M/S

Ratio of Maximum to Minimum Entrance Velocltles • 1.5 31 Shercllff Bend - Veloclty Contours

Augmented Incompressible Flow Solution Minimum Entrance Veloclty • 100 M/S

Ratio of Maximum to Minimum Entrance Velocltles ■ 1.5 32 Shercllff Bend - Statlc Pressure Contours

Augmented Incompressible Flow Solution Minimum Entrance Veloclty “ 100 M/S

Ratio of Maximum to Minimum Entrance Velocltles • 1.5 33 Shercllff Bend - Density Contours

Augmented Incompressible Flow Solution Minimum Entrance Veloclty “ 100 M/S

Ratio of Maximum to Minimum Entrance Velocltles • 1.5 34 Shercllff Bend - Mach Number Contours

Augmented Incompressible Flow Solution Minimum Entrance Veloclty « 100 M/S*

Ratio of Maximum to Minimum Entrance Velocltles “ 1.5 35 Shercllff Bend - Veloclty Vectors

Surrogate Equation Compressible Flow Solution Minimum Entrance Veloclty • 100 M/S

Ratio of Maximum to Minimum Entrance Velocltles - 1.5 36 Shercllff Bend - Veloclty Contours

Surrogate Equation Compressible Flow Solution Minimum Entrance Veloclty “ 100 M/S

Ratio of Maximum to Minimum Entrance Velocltles • 1.5 37 Shercllff Bend - Statlc Pressure Contours

Surrogate Equation Compressible Flow Solution Minimum Entrance Veloclty • 100 M/S

Ratio of Maximum to Minimum Entrance Velocltles - 1.5 38 Shercllff Bend - Density Contours

Surrogate Equation Compressible Flow Solution Minimum Entrance Veloclty • 100 M/S

Ratio of Maximum to Minimum Entrance Velocltles ” 1.5 39 Shercllff Bend - Mach Number Contours

Surrogate Equation Compressible Flow Solution Minimum Entrance Veloclty ■ 100 M/S

Ratio of Maximum to Minimum Entrance Velocltles - 1.5

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Shercllff Bend - Velocity Contours Augmente! Incompressible Flow Solution Minimum Entrance Velocity " 150 M/S

Ratio of Maximum to Minimum Entrance Veloclties ” 1.5 Shercllff Bend - Statlc Pressure Concours

Augmente! Incompressible Flow Solution Minimum Entrance Velocity • 150 M/S

Ratio of Maximum to Minimum Entrance Veloclties ~ 1.5 Shercllff Bend - Denslty Contours

Augmente! Incompressible Flow Solution Minimum Entrance Velocity • 150 M/S

Ratio of Maximum to Minimum Entrance Veloclties > 1.5 Shercllff Bend - Mach Number Contours

AugmenCed Incompressible Flow Solution Minimum Entrance Velocity « 150 M/S

Ratio of Maximum to Minimum Entrance Veloclties 1.5 Shercllff Bend - Velocity Vectors

Surrogate Equation Compressible Flow Solution Minimum Entrance Velocity « 150 M/S

Ratio of Maximum to Minimum Entrance Veloclties • 1.5 Shercllff Bend - Velocity Concours

Surrogate Equation Compressible Flow Solution Minimum Entrance Velocity • 150 M/S

Ratio of Maximum to Minimum Entrance Veloclties * 1.5 Shercllff Bend - Statlc Pressure Contours

Surrogate Equation Compressible Flow Solution Minimum Entrance Velocity « 150 M/S

Ratio of Maximum to Minimum Entrance Veloclties ~ 1.5 Shercllff Bend - Denslty Contours

Surrogate Equation Compressible Flow Solution Minimum Entrance Velocity ~ 150 M/S

Ratio of Maximum to Minimum Entrance Veloclties ~ 1.5 Shercllff Bend - Mach Number Contours

Surrogate Equation Compressible Flow Solution Minimum Entrance Velocity > 150 M/S

Ratio of Maximum to Minimum Entrance Veloclties - 1.5 Shercllff Bend - Velocity Vectors

AugmenCed Incompressible Flow Solution Minimum Entrance Velocity - 150 M/S

Ratio of Maximum to Minimum Entrance Veloclties >1.5

Shercllff Bend - Velocity Vectors Augmente! Incompressible Flow Solution Minimum Entrance Velocity “ 200 M/S

Ratio of Maximum to Minimum Entrance Veloclties • 1.5

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51 Shercliff Bend - Velocity Contours Augmented Incompressible Flow Solution Minimum Entrante Velocity • 200 M/S

Ratio of Maximum to Minimum Entrante Velocities • 1.5 52 Shercliff Bend - Statlc Pressure Contours

Augmented Incompressible Flow Solution Minimum Entrante Velocity • 200 M/S

Ratio of Maximum to Minimum Entrante Velocities - 1.5 53 Shercliff Bend - Denslty Contours

Augmented Incompressible Flow Solution Minimum Entrance Velocity • 200 M/S

Ratio of Maximum to Minimum Entrance Velocities • 1.5 54 Shercliff Bend - Mach Number Contours

Augmented Incompressible Flow Solution Minimum Entrance Velocity « 200 M/S

Ratio of Maximum to Minimum Entrance Velocities >1.5 55 Shercliff Bend - Velocity Vectors

Surrogate Equation Compressible Flow Solution Minimum Entrance Velocity » 200 M/S

Ratio of Maximum to Minimum Entrance Velocities • 1.5 56 Shercliff Bend - Velocity Contours

Surrogate Equation Compressible Flow Solution Minimum Entrance Velocity ~ 200 M/S

Ratio of Maximum to Minimum Entrance Velocities >1.5 57 Shercliff Bend - Statlc Fressure Contours

Surrogate Equation Compressible Flow Solution Minimum Entrance Velocity 200 M/S

Ratio of Maximum to Minimum Entrance Velocities • 1.5 58 Shercliff Bend - Denslty Contours

Surrogate Equation Compressible Flow Solution Minimum Entrance Velocity 200 M/S

Ratio of Maximum to Minimum Entrance Velocities • 1.5 59 Shercliff Bend • Mach Number Contours

Surrogate Equation Compressible Flow Solution Minimum Entrance Velocity • 200 M/S

Ratio of Maximum to Minimum Entrance Velocities • 1.5 60 Shercliff Bend - Velocity Vectors

Augmented Incompressible Flow Solution Minimum Entrance Velocity • 250 M/S

Ratio of Maximum to Minimum Entrance Velocities “ 1.5 61 Shercliff Bend - Velocity Contours*

Augmented Incompressible Flow Solution Minimum Entrance Velocity • 250 M/S

Ratio of Maximum to Minimum Entrance Velocities - 1.5

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Shercllff Bend - Density Contours Augmented Incompressible Plow Solution Minimum Entrance Velocity • 250 M/S

Ratio of Maximum to Minimum Entrance Velocltles • 1.5 Shercllff Bend - Statlc Pressure Contours

Augmented Incompressible Plow Solution Minimum Entrance Velocity • 250 M/S

Ratio of Maximum to Minimum Entrance Velocltles • 1.5

Shercllff Bend - Macb Number Contours Augmented Incompressible Plow Solution Minimum Entrance Velocity • 250 M/S

Ratio of Maximum to Minimum Entrance Velocltles - 1.5 Shercllff Bend - Velocity Vectors

Surrogate Equation Compressible Plow Solution Minimum Entrance Velocity • 250 M/S

Ratio of Maximum to Minimum Entrance Velocltles • 1.5 Shercllff Bend - Velocity Contours

Surrogate Equation Compressible Plow Solution Minimum Entrance Velocity • 250 M/S

Ratio of Maximum to Minimum Entrance Velocltles ~ 1.5 Shercllff Bend - Statlc Pressure Contours

Surrogate Equation Compressible Plow Solution Minimum Entrance Velocity • 250 M/S

Ratio of Maximum to Minimum Entrance Velocltles • 1.5 Shercllff Bend - Density Contours

Surrogate Equation Compressible Plow Solution Minimum Entrance Velocity • 250 M/S

Ratio of Maximum to Minimum Entrance Velocltles - 1.5 Shercllff Bend - Mach Number Contours

Surrogate Equation Compressible Plow Solution Minimum Entrance Velocity ~ 250 M/S

Ratio of Maximum to Minimum Entrance Velocltles • 1.5 Shercllff Bend - Velocity Vectors

Augmented Incompressible Plow Solution Minimum Entrance Velocity “ 275 M/S

Ratio of Maximum to Minimum Entrance Velocltles >1.5 Shercllff Bend - Velocity Contours

Augmented Incompressible Plow Solution Minimum Entrance Velocity “ 275 M/S

Ratio of Maximum to Minimum Entrance Velocltles - 1.5 Shercllff Bend - Statlc Pressure Contours

Augmented Incompressible Plow Solution Minimum Entrance Velocity “ 275 M/S

Ratio of Maximum to Minimum Entrance Velocltles “ 1.5

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73 Shercllff Bend - Denslcy Contours Augmented Incompressible Flow Solution Minimum Entrance Veloclty ” 275 M/S

Ratio of Maximum to Minimum Entrance Velocitles “ 1.5 7A Shercllff Bend - Mach Number Contours

Augmented Incompressible Flow Solution Minimum Entrance Veloclty 275 M/S

Ratio of Maximum to Minimum Entrance Velocitles • 1.5 75 Shercllff Bend - Veloclty Vectors

Surrogate Equation Compressible Flow Solution Minimum Entrance Veloclty “ 275 M/S

Ratio of Maximum to Minimum Entrance Velocitles • 1.5 76 Shercllff Bend - Veloclty Contours

Surrogate Equation Compressible Flow Solution Minimum Entrance Veloclty - 275 M/S

Ratio of Maximum to Minimum Entrance Velocitles •= 1.5 77 Shercllff Bend - Statlc Fressure Contours

Surrogate Equation Compressible Flow Solution Minimum Entrance Veloclty » 275 M/S

Ratio of Maximum to Minimum Entrance Velocitles • 1.5 78 Shercllff Bend - Density Contours

Surrogate Equation Compressible Flow Solution Minimum Entrance Veloclty • 275 M/S

Ratio of Maximum to Minimum Entrance Velocitles • 1.5 79 Shercllff Bend - Mach Number Contours

Surrogate Equation Compressible Flow Solution Minimum Entrance Veloclty ~ 275 M/S

Ratio of Maximum to Minimum Entrance Velocitles • 1.5 80 Shercllff Bend - Physlcal Domain

Augmented Incompressible Flow Solution Minimum Entrance Veloclty • 50 M/S

Ratio of Maximum to Minimum Entrance Velocitles - 1.2 81 Shercllff Bend - Veloclty Vectors

Augmented Incompressible Flow Solution Minimum Entrance Veloclty “ 50 M/S

Ratio of Maximum to Minimum Entrance Velocitles • 1.2 82 Shercllff Bend - Veloclty Contours

Augmented Incompressible Flow Solution Minimum Entrance Veloclty “ 50 M/S

Ratio of Maximum to Minimum Entrance Velocitles • 1.2 83 Shercllff Bend - Statlc Fressure Contours

Augmented Incompressible Flow Solution Minimum Entrance Veloclty “ 50 M/S

Ratio of Maximum to Minimum Entrance Velocitles “ 1.2

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Shercllff Bend - Denslty Contours Augmented Incompressible Flow Solution Minimum Entrance Veloclty ~ 50 M/S

Ratio o£ Maximum to Minimum Entrance Velocltles » 1.2 Shercllff Bend - Mach Mumber Contours

Augmented Incompressible Flov Solution Minimum Entrance Veloclty ~ 50 M/S

Ratio of Maximum to Minimum Entrance Velocltles - 1.2 Shercllff Bend - Veloclty Vectors

Surrogate Equation Compressible Flow Solution Minimum Entrance Veloclty ~ 50 M/S

Ratio of Maximum to Minimum Entrance Velocltles ~ 1.2 Shercllff Bend - Veloclty Contours

Surrogate Equation Compressible Flow Solution Minimum Entrance Veloclty “ 50 M/S

Ratio of Maximum to Minimum Entrance Velocltles > 1.2 Shercllff Bend - Statlc Pressure Contours

Surrogate Equation Compressible Flow Solution Minimum Entrance Veloclty ~ 50 M/S

Ratio of Maximum to Minimum Entrance Velocltles > 1.2 Shercllff Bend - Denslty Contours

Surrogate Equation Compressible Flow Solution Minimum Entrance Veloclty ” 50 M/S

Ratio of Maximum to Minimum Entrance Velocltles “ 1.2 Shercllff Bend - Mach Number Contours

Surrogate Equation Compressible Flow Solution Minimum Entrance Veloclty - 50 M/S

Ratio of Maximum to Minimum Entrance Velocltles 1.2 Shercllff Bend - Physlcal Domain

Surrogate Equation Compressible Flow Solution Minimum Entrance Veloclty “ 50 M/S

Ratio of Maximum to Minimum Entrance Velocltles - 1.2 Lengthened Domain

Shercllff Bend - Veloclty Vectors

Surrogate Equation Compressible Flow Solution Minimum Entrance Veloclty « 50 M/S

Ratio of Maximum to Minimum Entrance Velocltles 1.2 Lengthened Domain

Shercllff Bend - Veloclty Contours

Surrogate Equation Compressible Flow Solution Minimum Entrance Veloclty ” 50 M/S

Ratio of Maximum to Minimum Entrance Velocltles » 1.2 Lengthened Domain

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94 Shercllff Bend - Statlc Fressure Contours Surrogate Equation Compressible Flow Solution Minimum Entrance Velocity ■ 50 M/S

Ratio o£ Maximum to Minimum Entrance Velocities • 1.2 Lengthened Domain

95 Shercliff Bend - Density Contours

Surrogate Equation Compressible Flow Solution Minimum Entrance Velocity ■ 50 M/S

Ratio o£ Maximum to Minimum Entrance Velocities ~ 1.2 Lengthened Domain

96 Shercll££ Bend - Mach Mumber Contours

Surrogate Equation Compressible Flow Solution Minimum Entrance Velocity ~ 50 M/S

Ratio o£ Maximum to Minimum Entrance Velocities • 1.2 Lengthened Domain

97 Shercll££ Bend - Velocity Vectors

Surrogate Equation Compressible Flow Solution Minimum Entrance Velocity “ 50 M/S

Ratio o£ Maximum to Minimum Entrance Velocities ~ 1.2 Truncated View o£ Lengthened Domain

98 Shercll££ Bend - Velocity Contours

Surrogate Equation Compressible Flow Solution Minimum Entrance Velocity « 50 M/S

Ratio o£ Maximum to Minimum Entrance Velocities - 1.2 Truncated View o£ Lengthened Domain

99 Shercli££ Bend - Statlc Pressure Contours Surrogate Equation Compressible Flow Solution Minimum Entrance Velocity » 50 M/S

Ratio o£ Maximum to Minimum Entrance Velocities - 1.2 Truncated View o£ Lengthened Domain

100 Shercli££ Bend - Density Contours

Surrogate Equation Compressible Flow Solution Minimum Entrance Velocity • 50 M/S

Ratio o£ Maximum to Minimum Entrance Velocities - 1.2 Truncated View o£ Lengthened Domain

101 Shercll££ Bend - Mach Number Contours

Surrogate Equation Compressible Flow Solution Minimum Entrance Velocity ~ 50 M/S

Ratio o£ Maximum to Minimum Entrance Velocities - 1.2 Truncated View o£ Lengthened Domain

102 Circulât Arc Bend - Physlcal Domain

Surrogate Equation Compressible Flow Solution Minimum Entrance Velocity “ 50 M/S

Ratio o£ Maximum to Minimum Entrance Velocities “ 1.0 Thlrty Degree Bend

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103 Clrcular Arc Bend - Veloclty Vectors

Surrogate Equation Compressible Flow Solution Minimum Entrance Veloclty • 50 M/S

Ratio of Maximum to Minimum Entrance Velocltles “ 1.0 Thlrty Degree Bend

104 Clrcular Arc Bend - Veloclty Contours

Surrogate Equation Compressible Flow Solution Minimum Entrance Veloclty • 50 M/S

Ratio of Maximum to Minimum Entrance Velocltles “ 1.0 Thlrty Degree Bend

105 Clrcular Arc Bend - Statlc Pressure Contours Surrogate Equation Compressible Flow Solution Minimum Engrance Veloclty • 50 M/S

Ratio of Maximum to Minimum Entrance Velocltles ~ 1.0 Thlrty Degree Bend

106 Clrcular Arc Bend - Denslty Contours

Surrogate Equation Compressible Flow Solution Minimum Entrance Veloclty • 50 M/S

Ratio of Maximum to Minimum Entrance Velocltles ~ 1.0 Thlrty Degree Bend

107 Clrcular Arc Bend - Mach Number Contours Surrogate Equation Compressible Flow Solution Minimum Entrance Veloclty • 50 M/S

Ratio of Maximum to Minimum Entrance Velocltles » 1.0 Thlrty Degree Bend

108 Clrcular Arc Bend - Physlcal Domain

Surrogate Equation Compressible Flow Solution Minimum Entrance Veloclty ~ 50 M/S

Ratio of Maximum to Minimum Entrance Velocltles • 1.0 Thlrty Degree Bend

Reflned Computatlonal Mesh

109 Clrcular Arc Bend - Veloclty Vectors

Surrogate Equation Compressible Flow Solution Minimum Entrance Veloclty » 50 M/S

Ratio of Maximum to Minimum Entrance Velocltles • 1.0 Thlrty Degree Bend

Reflned Computatlonal Mesh

110 Clrcular Arc Bend - Veloclty Contours

Surrogate Equation Compressible Flow Solution Minimum Entrance Veloclty 50 M/S

Ratio of Maximum to Minimum Entrance Velocltles • 1.0 Thlrty Degree Bend

Reflned Computatlonal Mesh

111 Clrcular Arc Bend - Statlc Pressure Concours Surrogate Equation Compressible Flow Solution Minimum Entrance Veloclty • 50 M/S

Ratio of Maximum to Minimum Entrance Velocltles • 1.0 Thlrty Degree Bend

Reflned Computatlonal Mesh

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112 Circular Arc Bend - Denslty Contours

Surrogate Equation Compressible Flow Solution Minimum Entrante Velocity « 50 M/S

Ratio of Maximum to Minimum Entrance Velocltles “ 1.0 Thirty Degree Bend

Refined Computatlonal Mesh

113 Circular Arc Bend - Mach Mumber Contours Surrogate Equation Compressible Flow Solution Minimum Entrance Velocity ~ 50 M/S

Ratio o£ Maximum to Minimum Entrance Velocltles “ 1.0 Thirty Degree Bend

Refined Computatlonal Mesh 114 Transonlc Flow Problem

Computatlonal Domain 115 Wall Boundary Condition

Computatlonal Molécules

116 Surface Pressure Coefficient Distribution Surrogate Equation Formulation

Subcritlcal Flow Case Extra Coarse Grid

117 Surface Pressure Coefficient Distribution Surrogate Equation Formulation

Subcritlcal Flow Case Coarse Grid

118 Surface Pressure Coefficient Distribution Surrogate Equation Formulation

Subcritlcal Flow Case Medium Grid

119 Surface Pressure Coefficient Distribution Surrogate Equation Formulation

Subcritlcal Flow Case Fine Grid

120 Surface Fressure Coefficient Distribution Perturbation Fotentlal Formulation

Subcritlcal Flow Case Extra Coarse Grid

121 Surface Pressure Coefficient Distribution Perturbation Fotentlal Formulation

Subcritlcal Flow Case Coarse Grid

122 Surface Pressure Coefficient Distribution Perturbation Fotentlal Formulation

Subcritlcal Flow Case Medium Grid

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123

124

125

126

127

128

129

130

131

132

133

134

Surface Pressure Coefficient Distribution Perturbation Potential Formulation

Subcritical Flow Case Fine Grid

Comparison of Surface Pressure Coefficient Distributions Subcritical Flow Case

Fine Grid

Surface Pressure Coefficient Distribution Surrogate Equation Formulation

Supercrltical Flow Case Extra Coarse Grid

Surface Fressure Coefficient Distribution Surrogate Equation Formulation

Supercrltical Flow Case Coarse Grid

Surface Fressure Coefficient Distribution Surrogate Equation Formulation

Supercrltical Flow Case Medium Grid

Surface Pressure Coefficient Distribution Surrogate Equation Formulation

Supercrltical Flow Case Fine Grid

Surface Pressure Coefficient Distribution Perturbation Potential Formulation

Supercrltical Flow Case Extra Coarse Grid

Surface Fressure Coefficient Distribution Perturbation Potential Formulation

Supercrltical Flow Case Coarse Grid

Surface Fressure Coefficient Distribution Perturbation Potential Formulation

Supercrltical Flow Case Medium Grid

Surface Pressure Coefficient Distribution Perturbation Potential Formulation

Supercrltical Flow Case Fine Grid

Comparison of Surface Pressure Coefficient Distributions Supercrltical Flow Case

Fine Grid

Convergence Behavlor of Transonic Flow Algorithme Subcritical Flow Case

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135 Convergence Behavior o£ Transonlc Flow Âlgorlthms Supercrltical Flov Case

136 Order of Accuracy Estimation Surrogate Equation Formulation Subcrltlcal Flov Case

137 Order of Accuracy Estimation

Perturbation Potentlal Formulation Subcrltlcal Flov Case

138 Order of Accuracy Estimation Surrogate Equation Formulation Supercrltical Flov Case

139 Order of Accuracy Estimation Perturbation Potentlal Formulation Supercritical Flov Case

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matrix élément of block-trIdiagonal matrix Jacobian matrix

scalar element of tridiagonal matrix, dépendent variable

matrix element of block-tridiagonal matrix Jacobian matrix

Bcalar element of tridiagonal matrix

matrix element of block-tridiagonal matrix Jacobian matrix,

constant

scalar element of tridiagonal matrix, speed of Sound

vector element of block-vector, computational domain

scalar element of vector, distance between channel walls total energy per unit volume, block-vector of Inhomogeneous terms spécifie internai energy

block-vector of unknowns vector of unknowns

components of vector f (n-1,2) vector of unknowns

transformation matrix vector of unknowns,

function in sheared coordlnate System Identlty matrix

mesh index in x-directlon mesh index in y-direction, aesh index in n-directlon constant

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constant

length o£ computatlonal domain In z-dlrection length o£ computatlonal domain In y-direction trldlagonal matrix,

maximum value o£ 1, Mach number,

matrix,

block-trIdlagonal matrix Maximum value o£ j, matrix

block-trIdlagonal matrix, matrix

statlc pressure

boundary value £unctlon (n~l...4) vector o£ Inhomogeneous terms,

block-vector o£ Inhomogeneous terms mesh ratio

block-vector o£ Inhomogeneous terms, slmllarlty trans£ormatlon matrix trans£ormatlon matrix

unl£orm stream veloclty

veloclty component In x-dlrectlon,

perturbation veloclty component In x-dlrectlon veloclty component In y-direction,

perturbation veloclty component In y-direction vector o£ conservative variables

spatial coordlnate spatial coordlnate spatial coordlnate

nonllnear term In transonlc small disturbance équation

ratio o£ specl£lc beats Indlcates £lnlte dl££erence

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6

3 H X P

* U

m/s

n T

*

-1

1 j

Indlcates finlte différence

Indicates partial différentiation aheared coordinate

eigenvalue fluid density vector of unknovns scalar unknown, potential function,

perturbation potential function Btream function

accélération parameter approximate equallty meters per second

Superacripts

itération index transpose

indlcates unrelaxed value, Carteslan coordinate Indicates crltical value, spécial value

Inverse

indlcates différentiation wlth respect to x degrees of arc

Subscripts

mesh index In x-dlrectlon mesh index In y-direction

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1. INTRODUCTION

1.1 Motlvatlng Factors

The présent study Is concerned wlth the numerical solution of steady Invlscid flow problems. Many Important

physlcal situations encountered In modem engineering and applled science may be accurately modeled wlthin the constralnts of

steady Invlscid flow theory. Timely substantiation of thls daim is provlded by the generally good results presently being ob- tained from the use of such mathematlcal models for the design and analysis of transonlc alrfolls.

Most Work, both past and présent, has, however, dealt wlth the subset of invlscid flows whlch are Irrotational and, hence, for whlch a velocity potentiel fonction exlsts. Whlle many flows of interest may be successfully modeled wlthin thls addltlonal restriction, many others, constltutlng in ail probabil- Ity a larger class, cannot. Certalnly ail those flows of practl- cal Interest where slgnlflcant gradients of entropy or total

enthalpy may occur requlre a more general model than one based on potentlal flow theory. Cases In thls latter category are virtu- ally certain to occur in many Internai flows, in particular in those through modem turbomachinery, and may also occur in external flows, especlally when the flow over a number of Interactlng

components Is consldered.

Of course, the hyperbollc partial differentlal équations describlng supersonlc steady invlscid flow problems may be solved, for both potentiel and more general flow situations, by means of exlstlng mathematlcal and numerical techniques. Consequently, such flows are not the object of the présent study. Rather, it is the subsonlc and transonlc flow problems, descrlbed respectively by elllptlc and mixed elllptlc-hyperbolic équations, wlth whlch the research descrlbed herein deals.

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At thls Juncture, the Issue of computatlonal efflclency makes its Importance felt. Thls is due to the fact that, at the ezpense of computation tlmes whlch may be qulte long, solutions to the subsonlc and transonlc Invlscld flow problems mentloned above may be obtalned by means of a tlme-accurate computation of a temporal asymptote to the unsteady équations of motion. In contradlstinctlon to thls approach, the présent research descrlbes a method for the direct solution of the steady équations. By proceedlng In such a fashlon, and hence avoldlng the resolution of the translent physlcal States between the Initial State and the deslred final State whlch are obtalned when the unsteady équa­

tions are solved, a means Is provlded for the more efficient solu­

tion of steady Invlscld subsonlc and transonlc flow problems.

As wlll be descrlbed subsequently In thls report, the method Is based on the création of a higher-order System whlch serves as a surrogate for the flrst-order partial dlfferentlal équations of Invlscld flow theory. The stimulus for such an approach was provlded by Prof. J.J. Smolderen and Dr. H.J. Wirz, to whom the author la deeply Indebted.

Before proceedlng to a revlew of the relevant modem llterature, the problem under study, havlng been in existence for a rather long tlme, Is consldered In its historlcal context.

1.2 Historlcal Context

Insofar as the continuum hypothesls Is valld, the motion of an Invlscld fluld la descrlbed by the conservation équations for mass, momentum and energy, collectlvely referred to as the Euler équations. These équations, based on physlcal prlnclples developed by Daniel Bernoulli and D'Alembert and formulated mathematlcally by the prollflc Swlss mathematician and algorlst, Leonard Euler (Ref. 1), hâve been the subject of much study and many solutions. However, from the eighteenth

century untll modem tlmes, rather than address these équations In thelr full generallty, most efforts at solution hâve been dlrected at varlous restrlcted forma, as wlll be discussed sub­

sequently .

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While Che unsteady Euler équations are of hyperbolic type at ail flow velocitles, their temporally Invariant, or steady, version changes type from elliptic to hyperbolic as the flow velocity exceeds the local velocity of Sound. However, prior to the invention of the steam turbine, late in the nlne- teenth century, and the advent of hlgh-speed propellers, in the early twentieth century, the actuel physlcal occurence of flows with velocltles greater than sonie vas llmlted malnly to the field of ballistlcs. In fact, supersonic flow was so poorly un- derstood that the possiblllty of acceleratlng the efflux of a simple nozzle to greater than sonie velocity was verfled as late as 1889, with the introduction of the De Laval nozzle. Confusion concernlng such flows perslsted among theoretlclans and experl- mentallsts allke, at least until Prandtl (Ref. 2) began contrlb- utlng to their illumination. Hence, most work conducted prior to the présent century concentrated on elther purely subsonlc steady flow problème, or on the unsteady wave propagation prob­

lème of acoustlcs.

Potentiel Flow

In partlcular, solutions to subsonlc steady flow prob­

lème were most often sought under the addltional restriction that the fluid motion be Irrotational. Slnce the irrotatlonal motion of a fluid withln a slmply connected région Is characterIzed by the existence of a slngle-valued velocity potentiel fonction, thls assumptlon allowed a considérable simplification from the general case. In fact, where appropriate, the further assumptlon of incompressible flow allowed the réduction of the governlng équation to the simple Laplace équation for the velocity poten­

tiel. Hence, in this case, the full power of harmonie functlon theory could be brought to bear on certain fluid mechanlcs prob­

lème. A sampling of the résulté of potentiel flow theory, in­

dicative of the trends prévalent from ^the mid-elghteenth through the early twentieth century, may be found in the exposltory works of Lamb (Ref. 3) and Mllne-Thomson (Ref. 4).

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Near the turn of the century, the introduction o£ the hodograph method (Refs. S, 6) provided another means of obtaining particular solutions to planar inviscid flow problems. This method is based on the observation that, given a System of two first-order partial differential équations with dépendent vari­

ables U and V and indépendant variables x and y , if the System is homogeneous and has coefficients whlch are functlons of U and v alone, then, in any région where the Jacoblan :

is not zéro, the System may be transformed Into an équivalant llnear System by Interchanging the rôles of dépendant and in­

dépendant variables. Hence, provided the Jacoblan does not vanlsh, the hodograph method may be used to linearize the équa­

tions descrlbing the inviscid, Irrotational, Isentroplc motion of elther an unsteady one-dlemnsional or a steady two-dimenslonal flow. Thorough discussions of the hodograph transformation hâve been provided by both Courant and Friedrlchs (Ref. 7) and Llght- hill (Refs. 8, 9).

Desplte Its limitations, many useful results hâve been obtalned by means of the hodograph method. Of particular Interest is its use, Inltlated by Trlcomi (Ref. 10), in the study of tran- sonlc, or mlxed, flows. In fact, early transonlc flow solutions with application to planar nozzle design were produced in this

fashlon by both Llghthlll (Ref. 11) and Cherry (Ref. 12). Fur- thermore, a hodograph design method for transonlc shock-free airfoils has been developed by Boerstoel and Hulzlng (Ref. 13) as an extension of earller work done by Nleuwland (Ref. 14).

For addltional information on recent applications of the hodo­

graph method, one may consult the proceedlngs of the Symposia

« Transsonica (Refs. 15, 16) and the llterature to be cited sub- sequently.

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Contemporaneously with the appearence of the hodograph method, Massau (Ref. 17) introduced what vas to become known as the method of characterlstlcs. Thls élégant method, popularlzed by Prandtl and Busemann (Ref. 18) for the case of Invlscld super- Bonlc flow In two dimensions, provlded a very general technique for obtalnlng solutions to hyperbollc partial dlfferentlal équa­

tions. In partlcular, by application of the method of character­

lstlcs to a System of hyperbollc partial dlfferentlal équations In two Indépendant variables, familles of so-called characterls- tlc curves may be found. As, when wrltten along these curves, the original équations contaln no derlvatlves normal to them, the solution process Is slmpllfled. The method has also been generallzed to hlgher dimensions. Detalled discussions of both the theory and method of characterlstlcs may be found In Refs.

7, 19, and 20. Meyer (Ref. 21) provides an Interestlng revlew of applications of the method of characterlstlcs to compressible flow problème In two Indépendant variables and also dlscusses the treatment of flows contalnlng statlonary shock waves wlth Bupersonlc flow downstream. An Interestlng extension of the method to analytlc quasl-llnear partial dlfferentlal équations In two variables and of mlxed type may be found In the work of Svenson (Ref. 22).

Slnce Its Inceptlon, the method of characterlstlcs has proven to be of great utlllty In flndlng solutions to invlscld flow problems of both an unsteady and steady supersonlc nature.

In fact, the generality and power of the method are such that automated versions, created for use with modem digital computers are in widespread and contlnuing use today.

Finite-Dlfferençe_Methods

Startlng in the early twentleth century, a new trend began In the solution of fluld flow problems. Motlvated, perhaps by the dlfflculty of obtalnlng exact theoretical solutions to

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flow problems of practlcal Importance, researchers began to make use of numerlcal analysis and In partlcular the flnlte-dlfference method to obtaln approzlmate solutions. In thls way the elegance of the functlonal relatlonshlps provlded by exact solutions was sacrlflced to the pover and broad range of appllcablllty of the approzlmate numerlcal methods. The ploneerlng work of Richardson

(Ref. 23) In developlng an Itérative flnlte-dlfference solution procedure for elllptlc équations provides an example of one of

the earllest applications of modem numerlcal analysis to the solution of problems Involvlng partial dlfferentlal équations.

In 1928, wlth the appearance of the, now classlc, paper of Courant, Frledrlchs and Lewy (Ref. 24), a large contribution was made to the theoretlcal foundatlon of fInlte-dlfference methods. It Is Interestlng to note, however, that although thls paper has had a profound Influence on flnlte-dlfference solution procedures, Its authors were prlmarlly Interested In utlllzlng flnlte-dlfference formulations as an artifice In the proof of existence and unlqueness theorems for partial dlfferentlal équa­

tions. Discussions of the slgnlflcance of thls paper wlth regard to flnlte-dlfference solution procedures for hyperbollc, para- bollc and elllptlc Systems may be found In the works of Lax

(Ref. 25), Wldlund (Ref. 26), and Farter (Ref. 27), respectlvely.

Prlor to the Introduction of the electronlc computer, the main emphasls In numerlcal work was placed on the solution of elllptlc équations and, In partlcular, on Systems of second or hlgher order. Ferhaps the beat known of the early methods for the numerlcal solution of elllptlc partial dlfferentlal équa­

tions by hand calculation Is the relaxation method developed by Southwell. First Introduced to facllltate stress calculations In frameworks (Ref. 29), the method was later extended to fluld flow problems (Ref. 30) and popularlzed by Southwell's wartlme papers (Ref. 31, 32). In the late fourtles, Emmons (Refs. 33-35) applled relaxation techniques to solve fInlte-dlfference équations representlng transonlc flow problems. In thls manner, he obtalned

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the flrst solutions for mlzed flows vlth embedded shock waves.

It is Interestlng to note that whlle Emnion's method représenta a very advanced State of the art of hand computation, when pro- grammed for automatlc computation on digital computers, It suffers from convergence dlfflcultles (Ref. 36).

The wartlme advent of electronlc computers made the numerlcal solution of tlme-llke problems more practlcal. One can also easlly Imagine that the need for detalled descriptions of such translent phenomena as explosions and heat transfer stlmulated Interest In the further development of methods for the numerlcal solution of hyperbollc and parabollc partial dlf- ferentlal équations, perhaps at the expense of methods for the elllptlc équations descrlblng steady-state phenomena.

An Instrumental rôle In applylng the new computing machines to the problems of mathematlcal physlcs and, ln,partlc- ular, to fluld dynamlcs was played by von Neumann, who In a paper coauthored wlth Goldstlne (Ref. 37) noted that

Our présent analytlcal methods seem unsultable for the solution of the Impor­

tant problems arlslng In connection wlth non-llnear partial dlfferentlal équations and, In fact, wlth vlrtually ail types of non-llnear problems In pure mathematlcs.

The truth of thls statement Is partlcu- larly strlklng In the fleld of fluld dynamlcs. Only the most elementary prob­

lems hâve been solved analytlcally In thls fleld. Furthermore, It seems that In al- most ail cases where llmlted successes were obtalned wlth analytlcal methods,

these were purely fortultous, and not due to any Intrlnslc sultablllty of the method to the milieu. . . .

The advance of analysis Is, at thls moment, stagnant along the entlre front of non-llnear problems. That thls phenomenon Is not of a translent nature but that we are up agalnst an Important conceptual dlfflculty Is clear from the fact that, although the main mathematlcal dlfflcul- tles In fluld dynamlcs hâve been known slnce the tlme of Rlemann and of Reynolds,

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and alchough as brilllant a mathematlcal physiclst as Rayleigh has spent the major part of his W-fe's effort In combattlng them, yet no décisive progress has been made agalnst them ...

von Neumann helped to formulate many of the baslc prlnclples of the fleld whlch has now corne to be known as com- putatlonal fluld dynamlcs. In partlcular, he contrlbuted to the development of numerlcal methods for the solution of partial dlfferentlal équations of hyperbollc and parabollc type and he adapted and generallzed the method of stablllty analysis Intro- duced by Courant, Frledrlchs and Lewy (Ref. 24) and whlch Is now known as the von Neumann method. Due to the fact that much of von Neumann's work dld not appear In the open llterature In a

timely fashlon, the flrst detalled description of the von Neumann method of stablllty analysis Is to be found In the work of O'Brlan, Hyman and Kaplan (Ref. 38).

von Neumann (Ref. 39) and von Neumann and Rlchtmyer (Ref. 40) proposed and developed what Is now known as the shock capturlng method for the treatment of Invlscid flows contalning shock dlscontinultles. By Introduclng dlsslpatlve mechanlsms Into the Invlscid équations of motion, they replaced the surface of dlscontlnulty by a thln transition layer In whlch quantltles change rapldly, but not dlscontlnuously. Slnce the équations of motion may be applled In the transition layer, no boundary con­

ditions are requlred there. Thls method, together wlth the more recently developed and compétitive shock flttlng method (Refs.

41, 42), where the surface of dlscontlnulty is retained, continue to be useful tools, partlcularly for the analysis of Invlscid transonlc flow.

Another development of considérable Importance to the future course of computatlonal fluld dynamlcs was the populari- zation of the use In numerlcal methods of the conservation form of the dlfferentlal équations by Lax (Ref. 43) and Lax and

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In the late fiftles and early slxtles, Morawetz made a considérable contribution to the mathematlcal foundatlons of inviscld transonlc flow theory by shovlng that continuons exter- nal transonlc flows past two-dlmensional profiles do not, In general, exlst and that If such a flow can be found, It Is Iso- lated In the sense that there are no neighbourlng solutions of a continuons nature (Refs. 45-48).

Thls development notwlthstandlng, the Interest in tran- sonic aerodynamlcs contlnued, spurred on by both mllitary and commercial requlrements. The so-called transonlc controversy whlch ensued In the wake of Morawetz’ work was settled on the basls of experimental evldence generated malnly In the United Klngdom (Refs. 49, 50) and the Netherlands (Ref. 51). By the end of the slxtles a consensus had been reached that It was In- deed possible to design "shock-free" alrfoll sections whlch would hâve only negllglble wave drag in a usefully large nelghbourhood of the design conditions.

In 1970, two Works of considérable importance to the computation of Inviscld transonlc flow appeared. Magnus and Toshlhara (Ref. 52), applying an idea of unknown orlgln whlch had prevlously been popularlzed by Crocco (Ref. 53) as a possible means for the numerlcal solution of vlscous flow problème, In­

troduced a procedure for obtainlng a steady Inviscld flow solu­

tion as the temporal llmit of an unsteady flow wlth steady bound- ary conditions. Slnce only the unsteady Euler équations needed to be solved, the dlfficultles of deallng wlth partial dlfferen- tial équations of mlxed elllptic-hyperbollc type were completely avoIded.

Hendroff (Ref. 44). The use of thls form of the équations, Introduced earller by Courant and Frledrlchs (Ref. 7), is béné­

ficiai in may circumstances, includlng the treatment of tran- aonlc flows by means of shock capturing methods.

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Subséquent experlence wlth thls sort of tlme-accurate solution procedure for steady invlscld flow problems bas shown it to be rellable but costly. This helghtens the Importance of the contemporaneous work of Murman and Cole (Ref. 54). Uslng the potentlal formulation of the steady transonlc small distur­

bance équations, they exhlblted a method of qulte general appll- cablllty for Iteratlvely solvlng partial dlfferentlal équations of mlxed type. The ablllty to treat mlxed elllptlc-hyperbollc équations numerlcally provlded the basls for a large body of research almed at reduclng the computatlonal tlme requlred to obtaln solutions to steady Invlscld transonlc flow problems.

Thls contlnulng Interest In the rapld solution of mlxed équations has also caused a resurgence of Interest In the solution of elllp- tlc équations, at least Insofar as they are applicable to fluld flow problems of current Interest.

From thls brlef revlew of some of the mllestones In development of solution procedures for Invlscld flow problems, several points become apparent. As noted by Goldstlne and von Neumann (Ref. 37), analytlcal methods hâve made relatlvely lit progress wlth the dlfflcult and non-llnear problems posed by f mechanlcs. Perhaps the most general analytlcal tool thus far developed Is the method of characterIstlcs, yet Its Implementa tlon normally requîtes the extensive use of approxlmate numerl calculations .

the

t le luld

cal

The Introduction of digital computers has enabled great progress to be made, In partlcular, as regards the solution of partial dlfferentlal équations of évolution type, such as the unsteady compressible Euler équations. By Imposlng steady bound- ary conditions, steady-state solutions may also be obtalned as temporal llmlts to the unsteady équations, albelt as the resuit of a posslbly lengthy and hence costly process.

The numerlcal solution équations, such as the potentlal tlon,: . by means of relaxation or

of second order steady-state or perturbation potentlal equa-

other Itérative techniques Is

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also now fairly commonplace. However, not ail Invlscid flows are potentlal. In £act, It Is more llkely that the majority of In- vlscld flows of pracClcal Interest are rotatlonal. For such flows, one Is In general faced wlth tbe task of obtainlng a solution to tbe complété first-order Euler System, ratber tban one of tbe bls- tnrlcally utlllzed approximations. In cases wbere tbe Euler équa­

tions are of a byperbollc nature, tbls présents no partlcular prob- lem. However, in cases such as steady subsonlc or transonic flow, whlcb contaln réglons wbere the Euler équations are of elliptlc

type, relatlvely little progress bas been made. Xhose pertinent developments whlcb bave occurred slnce the initiation of tbls study and, in partlcular, non-tlme-accurate solutions of the unsteady équations of motion wlll be mentioned in the final chapter of tbls report.

As the purpose of the présent work is to contrlbute to the search for a remedy to the deficlency of itérative solution pro cedures for the steady Euler équations, we wlll now proceed to a more detailed revlew of the recent llterature relevant to tbls task

1.3 Revlew of the Modem Llterature

A general introduction to the mathematlcal aspects of subsonlc and transonic gas dynamlcs may be found in the work of Bers (Ref. 55). The fundamentals of the numerlcal treatment of such flows are dlscussed in Refs. 56-58. Further details important to the numerlcal solution of subsonlc and transonic flow problems may be found in the texts of Richtmyer and Morton (Ref. 59), For- sythe and Hasow (Ref. 60), Isaacson and Relier (Ref. 61) and Varga (Ref. 62). Refs. 55-57, 60 and 62 also contaln extensive and use- ful bibliographie material. Price (Ref. 63) has provided a review of the English language llterature on numerlcal analysis and re- lated toplcs.

The developments in the llterature dlscussed here may, perhaps, be seen in clearer perspective when vlewed against the totallty of research presently belng conducted in computatlonal fluld dynamlcs and, in partlcular, computatlonal aerodynamlcs. To thls end, one may consult the recent survey of American actlv- Ities in the field of computatlonal aerodynamlcs, wrltten by Gessow and Morris (Ref. 64).

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1.3.1 §J}£Z®Z_ti£®EiîîîE£

General surveys of the developments in compressible flow research may be found in the papers o£ Farren (Ref. 65) and Lighcbill (Ref. 9). The former provides, in particular, much information on the practical aspects of transonic aerodynamics, while the latter emphasizes the theory of compressible fluid flow. Both of these réf­

érencés are now somewhat dated in that they appeared before the use of computational methods became widespread in fluid dynamics. However, references to be cited subsequently will relieve this deficiency.

Transonlç_Flow

It appears that most recent survey articles deal with transonic rather than strictly subsonic flow. This is not sur- prising since the main area of application for computational meth­

ods in aerodynamics has been transonic flow technology. Here the hlgh cost of expérimentation first tipped the économie balance in favor of the juclclous introduction of flow simulation.

Sprelter (Ref. 66) has provided a summary of the baslc con­

cepts and principal résulta avallable prior to 1959 concerning the transonic flow past thln wlngs and slender bodles. An évaluation is made thereln of varions methods for the approxlmate solution of the nonllnear small disturbance équations and a crltlcal évaluation of experimental résulta is also presented. Much Interestlng Information on early developments in the study of transonic flow is contalned in Holder's Reynolds-Prandtl lecture (Ref. 50). In particular, Hôlder dlscusses the Importance of varions compresslbllity corrections and slmllarlty laws in the early work on compressible and transonic flow. Information on more recent applications of analytlc methods in aircraft aerodynamics may be found in Ref. 67.

The fundamentals of the hodograph transformation, as mentloned prevlously, hâve been thoroughly dlscussed by Courant and Frledrlchs (Ref. 7) and by Lighthlll (Refs. 8, 9). Germain

(Ref. 68) has revlewed some of the mathematlcal problème asso- clated with the application of the hodograph method to the study

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of transonic flow. Surveys of the application of hodograph theory to the design of shock-free transonic airfoils hâve been made recently by Boerstoel (Refs. 69 , 70). The analysis, or direct, problem for transonic airfoils in the hodograph plane has been discussed by Nocilla, Geymonat and Gabutti (Ref. 71).

In 1962, Hall and Sutton (Ref. 72) provided a survey of methods for the calculation of transonic flow in ducts and nozzles. This paper is noteworthy for Its thorough study of early approxlmate solution procedures, including those based on trun- cated sériés expansions. It also contalns a substantial blbllo- graphy. What may be consldered as a sequel to this work vas publlshed in 1976 by Brown and Hamilton (Ref. 73). Thelr work includes a survey of modem computatlonal procedures and a quite extensive bibliography. Further bibliographie material and sur­

vey Information on transonic nozzle flow may be found in the doctoral dissertations of Fannlng (Ref. 74) and Klopfer (Ref. 75).

The latter reference contalns a survey of Soviet literature on this toplc.

As for other internai flow problems, physlcal information on the development and structure of transonic flow in cascades may be found in the work of Dvorak (Ref. 76). Contributions concern- Ing the latest computatlonal and experimental results are con- tained in Refs. 77, 78. Of particular interest Is the review of flnite différence methods for the computation of transonic potentlal flow in turbomachlnery, presented by Murman (Ref. 79).

In 1969, Cole, who had earlier helped to bring attention to the unsuitablllty of llnearized theory for the study of tran- Bonlc flow (Ref. 80), discussed the progress made in the previous twenty years from the point of view of transonic small disturbance theory and formulated the boundary value problem for the steady flow past a thln alrfoil (Ref. 81). Several years later, Murman

(Ref. 36) surveyed the tremendous progress whlch had taken place slnce Cole's revlew. In particular, he reported on the use of relaxation methods for the computation of steady transonic flow.

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the ploneering work In thls area havlng been accompllshed in the Intérim by Murman and Cole (Ref. 54). Also discussed, were cime dépendent computacional techniques and various approximace methods.

Yoshihara (Refs. 82, 83 ) has surveyed computacional methods for Cwo- and three-dimensional Cransonic flows with shocks.

Here comparisons are made among the unsCeady procedure of Magnus and Yoshihara (Ref. 52) and the sCeady relaxation procedures of Mur­

man and Cole (Ref. 54), for the small disturbance équation,and of Garabedian and Korn (Ref. 84), Steger and Lomax (Ref. 85 ) and Jameson (Ref. 86),for the full potentlal équation. The abillty of the various methods to résolve embedded shocks Is stressed.

A thorough review of of the various relaxation metho two- and three-dimensional flow (Ref. 87). A perusal of this r great momentum whlch had been g val slnce the fundamenCal work

the s tatus of the appllca tion ds to the computation of steady s w as glven in 1974 by Ba i ley evi ev serv es to emphaslze the aln ed over the four year inter- of Mu rman and Cole.

Hall and Firmln (Ref. 88) revlewed the deve of a relaxation solution for steady three-dimensional flow uslng a transonic small perturbation approxlmatl also discussed the introduction of rotated différence by Albone (Ref. 89) and Jameson (Ref. 90). In 1976, presenced an updated survey of computacional methods Ing sCeady transonic flow, includlng a discussion of solution procedures belng developed as alternatives t successive line over-relaxaclon technique Introduced in transonic flow computations by Murman and Cole.

lopment transonic on. They

operators Hall (Ref. 91) for comput- che fasc

O the for use

Computacional research carrled out at the Courant Instltute and concerned with both the design and analysis prob- lems for supercritlcal alrfolls has been recently reviewed by Garabedian (Refs. 92, 93). The design aspects of this work hâve also been discussed by Holt (Ref. 94).

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The 1976 von Karman Instltute Lecture Sériés produced three detailed revlew papers on varions aspects of transonic flow computations. Jameson (Ref. 95) discussed solution procedures for the small disturbance équation and the exact potential équa­

tion in considérable mathematical detail. He also presented in­

formation on the growing field of fast solution procedures.

Ballhaus (Ref. 96), in a complementary présentation, discussed, among other topics, the analysis of inviscid unsteady transonic flows, the finite volume method, and airfoil design by numerical optimization. Topics concerned with three-dimensional transonic flows were discussed by Schmidt (Ref. 97).

A thoughtful survey of transonic aerodynamics and a wealth of bibliographie material hâve also recently been provided by Wu and Moulden (Ref. 98).

Review articles concerning the application of intégral équation methods to transonic flow computations hâve been written by Zierep (Ref. 99), Holt (Refs. 94,100), N^rstrud (Ref.101) and Nixon and Hancock (Ref.102).

Nieuwland and Spee (Ref. 103) hâve written a teview of the analytical, experimental and computational work concerned with the design and analysis of transonic airfoils. Their présentation is quite thorough and enlightening, contains an extensive bib- liography, and is to be highly recommended.

Other useful surveys of varions aspects of transonic flow research may be found in Refs. 104 through 108.

In 1971, Newman and Allison (Ref.109) produced an anno- tated bibliography consisting of approximately 700 entries relating to Bteady inviscid external transonic flows.

Further sources of information on transonic flow in- clude Ref. 110, the proceedings of the Symposia Transsonica

(Refs. 15, 16) and the well-known texts of Guderley (Ref. lll) and Ferrari and Trlcoml (Ref. 112).

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Subsonlc Flow

The revlew llCerature concerned solely wlth steady

subsonlc Invlscld flow Is somewhac less plentlful and less diverse.

Thls may be a conséquence of the remarkable extenc to whlch en- glneers and designers were able to utlllze simple llnear théories and résulta drawn from Incompressible flow theory, modlfled by certain correction factors expressing the Influence of compressl- blllty. In fact, It appears that the problem of transonlc flow, wlth Its Intrlnslc nonllnearlty, presented the flrst grave

Impedlment for thls approach.

In the course of the subséquent remarks, one wlll no­

tice that most of the exlstlng methods for solvlng subsonlc In- vlscld flow problems are llmlted In appllcabl11ty to potentiel flows. One justification for thls State of affaire Is that, If one chooses to neglect boundary layer effects, an Important class of external aerodynamlc flows may be reasonably approxlmated by Irrotatlonal flows. Expedlency has no doubt also contrlbuted to the dominance of the potentiel flow assumptlon.

An excellent general survey of the non-computational methods of subsonlc Invlscld flow theory may be found In the text wrltten by Pal (Ref. 113). A thorough revlew of llnearlzed theory for homoentroplc, Irrotatlonal, Invlscld fluld motion has been conducted by Ward (Ref. 114).

The hodograph method, dlscussed prevlously In the context of transonlc flow, has also been used In purely subsonlc Invlscld, Irrotatlonal flow. An early discussion of Its use In such a case may be found In a paper by Tslen (Ref. 115).

Hlrschel (Ref. 116) has echnlques for the numerlcal solu roblems. These ar e, In general Inlte-element and panel methods.

ecelve llttle atte ntlon In thls

glven an Introduction to severa tlon of subsonlc Invlscld flow terms, the fInlte-dlfference,

Flnlte-element methods wlll study. Shen (Ref. 117) has.

1

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however, recently publlshed an interesting revlew article con- cernlng the application of flnlte-element methods to fluld mechanlcs problems.

Loeve (Refs. 118, 119) bas recently provided two survey articles dealing, respectively, with direct calculation methods for subsonic potentlal flow around swept wlngs and a general revlew of the use of both panel and fInlte-difference methods for the calculation of the subsonic flow over wlng-body combi­

nations. A further revlew of calculation procedures for wlng- body aerodynamlc Interactions has been publlshed by Ashley and Rodden (Ref. 120).

A qulte detalled discussion of the application of the panel method to the calculation of both Incompressible and com­

pressible potentlal flow over wlngs and wlng-body combinations may be found In the work of Kraus (Ref. 121 ). The use of another slngularity method, the vortex lattlce method, has recently been reviewed by Lucchl (Ref. 122).

The use of intégral methods In the calculation of sub­

sonic invlscid flow has been dlscussed by both Holt and Masson (Ref. 123) and Melnlk and Ives (Ref. 124).

It appears that the main body of effort proceeded from treatlng subsonic potentlal flow in simple geometries to treatlng subsonic and transonic potentlal flows in falrly complex geomet­

ries. The study of the more general class of rotatlonal, sub­

sonic invlscid flows has not been vigorously pursued.

Havlng cited a hopefully représentative and useful, but certalnly not exhaustive, sélection from the survey llterature, we now proceed to a revlew and brlef discussion of the prlmary

llterature pertinent to the présent research.

1.3.2 Research Llterature

As mentloned prevlously, the problems of transonic flow

Figure

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