i ’. , I.- 1. . , . i 

A 

DEPARTMENT OF MECHANICAL ENGINEERING AND MECHANICS . -- ~ 

/. e: 

COLLEGE OF ENGINEERING AND TECHNOLOGY 

OLD DOMINION UNIVERSITY i ’ 

NORFOLK, VIRGINIA 23529 

/--- 

r3 3 >./:>dl 

NUKERICAL STUDY OF THREE-DIMENSIONAL SEPARATION 

AND FJBW CONTROL AT A WING/BODY JUNCTION 

BY 

Balakrishnan Lakshmanan, Graduate Research Assistant 

Principal Investigator: Robert L. Ash 

Progress Report 

For the period ended April 30, 1989 

Prepared for 

National Aeronautics and Space Administration 

Langley Research Center 

Hampton, VA 23665 

Under 

Research Contract NAG-1-530 

John B. Anders, Technical Monitor 

HSAD-Viscous Flow Branch 

August 1989 

- - 

(UASA-CR- 18 58 19) bl UMBRICIL STUDI OF I89-26826 TBRBE-DIBEUSIOBIL SEPARATIOI1 AHD PLOP 

COHTBOL AT a UI16/BODY JUYCTIOI Progress 

3eport, period ended 30 -Apr., 1989 (Old Unclas 

DosiPioa IJn1v.C 214 p CSCL OtA 63/02 0224565 

I 
DEPARTMENT OF MECHANICAL ENGINEERING AND MECHANICS 

COLLEGE OF ENGINEERING AND TECHNOLOGY 

OLD DOMINION UNIVERSITY 

NORFOLK, VIRGINIA 23529 

NUMERICAL STUDY OF THREE-DIMENSIONAL SEPARATION 

AND FLOV CONTROL AT A WING/BODY JUNCTION 

BY 

Balakrishnan Lakshmanan, Graduate Research Assistant 

Principal Investigator: Robert L. Ash 

Progress Report 

For the period ended April 30, 1989 

Prepared for 

National Aeronautics and Space Administration 

Langley Research Center 

Hampton, VA 23665 

Under 

Research Contract NAG-1-530 

John B. Anders, Technical Monitor 

HSAD-Viscous Flow Branch 

Submitted by the 

Old Dominion University Research Foundation 

P.O. Box 6369 

Norfolk, Virginia 23508-0369 

August 1989 
NUMERICAL STUDY OF THREE -DIMENSIONAL SEPARATION 

AND FLOW CONTROL AT A WING/BODY JUNCTION 

ABSTRACT 

The problem of three-dimensional separation and flow control at a 

wing/body junction has been investigated numerically using a threedimensional Navier-Stokes code. The numerical code employs an algebraic grid generation technique for generating the grid for unmodified 

fin 

ate 

c grid,generation technique for 

for laminar flow past a blunt f 

junction and an ell ipt 

junction. The results 

filleted 

n/flat p 

junction demonstrate that after grid refinement, the computations agree 

with experiment and reveal a strong dependency of the number of vortices at the junction on Mach number and Reynolds number. The numerical 

results for pressure distribution, particle paths and 1 imiting streamlines for turbulent flow past a swept fin show a decrease in the peak 

pressure and in the extent of the separated flow region compared to the 

laminar case. 

stream1 ine patterns lose much of their vortical character with proper 

filleting. Fillets with a radius of three and one-half times the fin 

leading edge diameter or two times the incoming,boundary layer thickness, significantly weaken the usual "necklace" 

the Mach number and Reynolds number considered in the present study. 

The results for a filleted juncture indicate that the 

interaction vortex for 
This is a progress report for the research project "Studies of Vortex 

Stability and Control." 

November 1, 1988 through April 30, 1989. The work was supported in part by 

the NASA Langley Research Center Grant NAG-1-530. 

monitored by John B. Anders Jr. of the High-speed Aerodynamics Division 

(Viscous Flow Branch). 

The period of performance for the research is 

The research grant was 

ii 
TABLE OF CONTENTS 

Page 

ACKNOWLEDGMENTS...... ...................................... ii 

LIST OF FIGURES ............................................ vi 

LIST OF SYMBOLS.. .......................................... xi 

C h apt er 

1. INTRODUCTION.. ......................................... 1 

2. PHYSICS OF WING/BODY JUNCTURE FLOWS .................... 3 

2.1 Low Speed Flows ................................. 3 

2.1.1 

2.1.2 

Low Speed Unmodified Juncture Flows ...... Low Speed Controlled Juncture Flows.. .... 3 

8 

2.2 High Speed Flows ................................ 1U 

2.2.1 

2.2.2 

High Speed Unmodified Juncture Flows.. ... High Speed Controlled Juncture Flows.. ... 10 

15 

2.3 Turbulence Modeling for NinglBody Juncture 

Flows ......................................... 16 

3. THEORETICAL FORMULATION ................................ 18 

3.1 Governing Equations in Non-Dimensional Form.. ... 

3.3 Thin Layer Approximation.. ...................... 3.2 Turbulence Model ................................ 

3.4 Mesh System. .................................... 3.5 Boundary Conditions.. ........................... 

18 

24 

30 

32 

36 

36 

36 

37 

39 

3.5.1 Inflow Boundary.. ........................ 3.5.2 Outflow Boundary.. ....................... 3.5.3 Sol id Boundary.. ......................... 3.5.4 Symmetry Boundary.. ...................... 

Conditions ...................................... 3.6 Initial Conditions and Blunt Body Starting 

' 39 

42 4. METHOD OF SOLUTION............... ...................... 42 

46 

4.1 Finite Volume Formulation ....................... 4.2 Explicit MacCormack Algorith in................... 

iii 
TABLE OF CONTENTS (continued) 

Page 

4.3 Implicit MacCormack Algorithm ................... 4.4 Operation of Numerical Code ..................... 

4.4.1 General Procedure .......................... 4.4.2 Requirement on Grid ........................ 

5 . PHYSICAL CONDITIONS OF INVESTIGATION ................... 

5.1 Laminar Flow Past a Swept Fin/Flat Plate JunctionValidation with Experiment and Parametric Study ... 

5.1.1 Validation with Experiment ................. 5.1.2 Parametric Study ........................... 

Laminar Flow Past a Blunt Fin-Conditions for 

Additional Results and Parametric Studies ......... 5.2 

5.2.1 Conditions for Additional Results .......... 5.2.2 Parametric Studies ......................... 

5.3 Turbulent Flow Past a Sharp Fin/Flat Plate 

Junction ......................................... 

5.3.1 Flow Past a 16" Sharp Fin .................. 5.3.2 Flow Past a 20" Sharp Fin .................. 

5.4 Turbulent Flow Past a Swept Fin/Flat Plate 

Junction ......................................... 5.5 Turbulent Flow Past a Filleted Fin/Flat Plate 

Junction .......................................... 

5.5.1 Flow Past an Unmodified Junction ........... 5.5.2 Flow Past a Modified Junction .............. 

6 . RESULTS AND DISCUSSION ................................. 6.1 Laminar Flow Past a Swept Fin ..................... 

6.1.1 

6.1.2 

Computation of 0' Swept Fin Flowfield ...... Parametric Effect of Fin Sweep and Boundary 

Layer Thickness ............................ 

6.2 Laminar Flow Past a Blunt Fin ..................... 

6.2.1 

6.2.2 

Comparison of Computations with Experimental 

Data ....................................... Effect of Grid and Flow Parameters ......... 

6.3 Computation of Three Dimensional Sharp Fin Flowfield .............................................. 

iv 

51 

59 

59 

60 

63 

64 

64 

64 

66 

66 

66 

68 

68 

68 

69 

69 

69 

71 

72 

72 

72 

74 

84 

89 

100 

115 
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TABLE OF CONTENTS (concl uded) 

6.3.1 

6.3.2 

Flow Past a 16" Sharp Fin .................. Flow Past a 2U" Sharp Fin. ................. 

6.4 Turbulent Flow Past a Swept Blunt Fin.. ........... 6.5 Turbulent Flow Past a Filleted Fin ................ 

7. CONCLUSIONS AND RECOMMENDATIONS ........................ 

REFERENCES..... ............................................ 

APPENDIX A: MAXIMUM ALLOWABLE TIME STEP FOR EXPLICIT 

ALGORITHM. .................................... 

Page 

115 

125 

140 

159 

188 

190 

195 

V 
2.1 

2.2 

3.1 

3.2 

3.3 

3.4 

3.5 

3.6 

3.7 

3.8 

4.1 

5.1 

5.2 

6.1 

6.2 

6.3 

6.4 

6.5 

LIST OF FIGURES 

F-i gure 

Flow structure at the low speed wing/body junction ... 

Flow structure at the high speed wing/body junction.. 

A hexahedron finite volume v(i,j,k) with coordinate 

systems ............................................. 

Definition of four regions in (J-K) plane ............ 

Definition of Buleev length scale .................... 

Mesh distribution on sharp fin and flat plate .,...... 

Mesh distribution on swept fin and flat plate ........ 

Mesh distribution on filleted fin and flat plate ..... 

Boundary layer profiles on the flat plate at J=J .. 

Region near the nose for setting blunt body starting 

conditions.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 

Distance di k between two adjacent cells ......... 

Geometry of blunt fin/flat plate junction showing 

typical surface locations of comparison with 

experiment ........................................... 

Location of experimental surveys for 20' sharp fin... 

Pressure distribution on the blunt fin leading edge 

and comparison with experiment.. . . . . . . . . . . . . . . . . . . . . . 

Pressure distribution on the flat plate and 

comparison with experiment.. . . . . . . . . . . . . . . . . . . . . . . . . . 

Pressure distribution on the fin leading edge for 

various fin sweeps................................... 

Pressure distribution on the flat plate for various 

fin sweeps.. . , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 

Pressure distribution on the fin leading edge for 

various fin sweeps in a thick laminar boundary layer. 

max 

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12 

21 

25 

27 

33 

34 

35 

38 

40 

47 

65 

70 

73 

75 

76 

78 

80 
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LIST OF FIGURES (Continued) 

F i gure 

6.6 

6.7 

6.8 

6.9 

6.10 

6.11 

6.12 

6.13 

6.14 

6.15 

6.16 

6.17 

6.18 

6.19 

Pressure distribution on the flat plate for various 

fin sweeps in a thick laminar boundary layer ......... 

Mach number contours in the plane of synmetry for various fin sweeps: (a) sweep angle = O', 

(b) sweep angle = 45". .............................. 

Pressure contours in the plane of synmetry for 

various fin sweeps: (a) sweep angle = 0"; (b) sweep 

angle = 45" ......................................... 

Density contours in the plane of synmetry for 

various fin sweeps: (a) sweep angle = 0'; (b) sweep angle =45 .......................................... 

Comparison of pressure distribution along the line 

of symmetry ......................................... 

Pressure distribution on the blunt fin: (a) along 

0 = 0" and 45"; (0) along 41 = YU' and X/D = 2.5 

and 4.5 ............................................. 

Comparison of streamwise velocity profile: 

X/D = -8.0, -7.2, -6.4 and -5.6; (b) along X/D = 

-4.8, -4.0 and -3.2; (c) along X/D = -2.4 and -1.6.. 

Comparison of vertical velocity profile: (a) along 

X/D = -4.8, -4.0 and -3.2; (b) along X/D = -2.4 

and -1.6............................................ 

(a) along 

Limiting streamlines on the flat plate for laminar 

flow past a blunt fin....... ....................... 

Particle paths in the plane of synmetry for various 

grid refinements: (a) 30x50~30 grid; (b) 40x100~30 

grid; (c) 40x100~50 grid ............................ 

Pressure distribution along the line of spmetry for 

various Macn nunbers...................... .......... 

Pressure distribution along the fin stagnation line 

for various Mach nunbers................ ............ 

Streamwise velocity profiles for various Mach nunbers: 

(a) along X/D = -4.0 and -3.2; (b) along X/D = -2.4 

and -1.6 ............................................ 

Particle paths in the plane of synmetry for various 

Mach nunbers: (a) Ita, = 1.2; (b) rcb, = 2.36; 

(c) M, = 3.5 ........................................ 

vi i 

Page 

81 

82 

85 

87 

90 

91 

94 

98 

101 

10 3 

106 

108 

109 

111 
LIST OF FIGURES (Continued) 

F i aure 

6.20 

6.21 

6.22 

6.23 

6.24 

6.25 

6.26 

6.27 

6.28 

6.29 

6.30 

6.31 

6.32 

6.33 

Pressure distribution along the line of synmetry 

for various unit Reynolds nunber .................... 

Pressure distribution along the fin stagnation line 

for various unit Reynolds nunber ................... 

Streamwise velocity profiles for various unit 

Reynolds number: (a) along X/D = -4.0 and -3.2; 

(b) along X/D = -2.4 and -1.6 ...................... 

Particle paths in the plane of sflrnetry fo various 

unit Reynolds numb % r: (a) Re, = 1.2xlf%i'; 

(b) Rea. = 2.5 x 10 m-l; (c) Re, = 5x10 m' .......... 

Pressure distribution on the flat plate along Y/6, = 

5.56 for 16' sharp fin .............................. 

Yaw angle distribution on the flat plate along 

Y/6, = 5.56 for 16' sharp fin ........................ 

Skin friction distribution on the flat plate along 

Pitot pressure distribution at different axial 

locations for 20' sharp fin: (a) station 1; 

(b) station 2; (c) station 3; (d) station 4; 

(e) station 5; (f) station 6; (9) station 7; 

(h) station 8 ....................................... 

Y/d0 = 5.56 for 16' sharp fin ........................ 

Limiting streamline patterns on the plate surface 

for 20' sharp fin ................................... 

Mean streamlines for 20' sharp fin: 

at Z/6, = 0, 0.0048 and 0.52; (b) originating at 

Z/6, = 1.1 ........................................... 

Limiting streamline patterns on the fin surface for 

20' sharp fin ...................................... 

(a) originating 

Velocity magnitude contours on the plate surface for 

20' sharp fin ....................................... 

Comparison of pressure distribution along the fin 

stagnation line for various fin sweeps.............. 

Temperature distribution along the fin stagnation 

line for various fin sweeps....... .................. 

Page 

114 

116 

117 

119 

123 

124 

126 

12 7 

138 

139 

142 

143 

144 

146 

viii 
LIST OF FIWRES (Continued) 

Figure 

6.34 

6.35 

6.36 

6.37 

6.38 

6.39 

6.40 

6.41 

6.42 

6.43 

6.44 

Comparison of pressure distribution on the flat 

plate for various fin sweeps: (a) along Y/D = 0; 

(b) along Y/D = 1.5; (c) along Y/D = 2.5; (d) along 

Y/D = 3.5; (e) along Y/D = 4.5 ..................... 

Temperature distribution along the line of symmetry 

for various fin sweeps ............................... 

Particle paths in the plane of symmetry for various 

fin sweeps: (a) sweep = 0'; (b) enlarged view of the 

flow at the junction for sweep = 0"; (c) sweep = 60". 

Skin friction distribution along the line of symmetry for various sweeps ............................ 

Limiting streamline patterns on the flat plate for 

various fin sweeps: (a) sweep = 0'; (b) sweep = 30"; 

(c) sweep = 60" ...................................... 

Pressure distribution on the fin leading edge for 

various junctions .................................... 

Pressure distribution on the flat plate for various 

junctions ........................................... 

Particle paths in the plane of synmetry for various 

junctions: 

(b) cirular fillet (6/D = 2, R/D = 2); (c) circular 

fillet (6/D = 2, R/D = 3.5); (d) circular fillet 

(6/D = 0.2, R/D := 3.5).: ............................. 

Surface limiting streamlines for various junctions: 

(a) unmodified junction (6/0 = 2); (b) circular 

fillet (6/D = 2, R/D = 2); (c) circular fillet 

(6/D = 2, R/O = 3.5); (d) circular fillet (6/D = 

0.2, R/D = 3.5) ..................................... 

(a) unmodified junction (6/D = 2); 

Instantaneous streamlines for various junctions: (a) unmodified junction (6/D = 2); (b) circular fillet 

(6/D = 2, R/D = 2); (c) circular fillet (6/D = 2, 

R/D = 3.5); (d) circular fillet (6/D = 0.2, R/D = 

3.5); (e) continuous fillet (6/D =2) ................. 

Velocity magnitude contours in the plane of symmetry 

for various junctions: (a) unmodified junction 

(6/D = 2); (b) circular fillet (6/D = 2 

(c) circular fillet (6/D = 2, R/O = 3.5); 

(d) circular fillet (6/D = 0.2, R/D = 3.5) ........... 

R/D = 2); 

ix 

Page 

147 

153 

154 

158 

160 

163 

165 

166 

171 

175 

180 
LIST OF FIGURES (Concluded) 

F i gure 

6.45 Vorticity contours in the plane of synmetry for 

(S/D = 2); (b) circular fillet (6/D = 2, R/D = 2); 

(c) circular fillet (S/D = 2, R/D = 3.5); 

(d) circular fillet (S/D = 0.2, R/D = 3.5) .......... 

various junctions: (a) unmodified junction 

Page 

184 

X 
a 

CF L 

cC P 

c f 

'kl eb 

D 

E 

ei 

F - 

Fkl eb 

Fm ax 

Fouter 

Fwake 

112 

11 s 

qil) 

g(il) 

H 

i 

k 

11 

LIST OF SYMBOLS 

Speed of sound 

Courant Fredrich Levy nunber 

Turbulence model constant 

Skin friction coefficient 

Turbulence model constant 

Van Driest damping factor and d,ameter of the fin 

Total energy per unit mass 

(E = ei + 1/2 (u2 + v2 + w2)) 

Specific internal energy (ei = cvT) 

Flux vector along the local coordinate direction 

Klebanoff intermittency function 

Velocity scale in outer turbulence model 

Outer function in outer turbulence model 

Wake function in outer turbulence model 

. Jacobi an 

Contravariant base vector in the xa direction 

Component of the contravariant base vector - g' along 

the xa direction 

Contravariant unit base vector along the x" direction 

Height of the fin 

Grid volune index in E-direction 

Grid volune index in n-direction 

Turbulence model constant (k = 0.0168); thermal 

conductivity; grid index in c-direction. 

Buleev length scale and separation length 

xi 
M 

n 

rl) 

P 

S 

T 

- 

Tw 

t 

U 

U 

V 

W 

LIST C)F SYMBOLS (Continued) 

Length scale in outer turbulence model 

Explicit finite-difference operators in 5, n, and 5 

directions 

Mach number 

Coordinate normal to the surface 

Unit normal at surface 

Static pressure 

Laminar flow Prandtl nunber 

Turbulent flow Prandtl number 

vector of conserved variables 

Heat flux at wal k 

Radius of the fillet at the juncture 

Reynolds number based on diameter (ReD = pU,D/p,) 

Unit Reynolds number based on free stream conditions 

Surf ace normal 

St at i c temper at ur e 

Wall temperature 

time 

Streamwise velocity 

Velocity in x-direction 

Velocity in y-direction 

Velocity in z-direction 

Contravariant velocity component in c-direction 

Contravariant velocity component in n-direction 

Contravariant velocity component in c-direction 

Distance from the flat plate leading edge to fin 

leading edge 

xi i 
LIST OF SYMBOLS (Continued) 

Location of the theoretical inviscid shock at a 

specified spanwise position 

Global Cartesian coordinates ((xll, x2', x3') = 

'snack 

x1 ',x2 ',x3 

kY, d> 

snock x-x xS 

Xl ,x 2 ,x 3 Local body oriented coordinates ( (xl ,x2,x3) = ((,n,<)) 

Greek Symbols 

Y 

6 Boundary layer thickness 

Ratio of specific heats (r = Cp/Cv) 

e Dynamic turbulent eddy viscosity 

e ,E: Inner and outer dynamic turbulent eddy viscosity 

5 Local body oriented coordinate 

io 

n Local body oriented coordinate 

5 Local body oriented coordinate 

A6 Mesh spacing in (-direction 

An Mesh spacing in n-direction 

AC Mesh spacing in c-direction 

Ate, Atn, Atc Time steps used for Le, L, LG operators 

k Von Karman's constant (k=0.4) 

A 

A 

Fin sweep angle 

Second coefficient of viscosity (A = - 2/3 ) 

lJ 

P Density 

Dyn ami c mo 1 ec u 1 ar v i scos i t y 

e Sharp fin leading edge angle 

Peripheral Angle in Buleev length scale and alio angle 

measured in cylindrical coordinates from the fin nose 

Magnitude of vorticity (w = Id x ? I) 

xiii 
LIST OF SYMBOLS (Concluded) 

Subscripts 

0 Evaluated based on upstream conditions 

SeP 

W 

W 

t2 

tw 

Based on separation conditions 

Evaluated at wall 

Evaluated in freestream 

Total conditions behind the bow shock 

Total conditions of the freestream 

xi v 
Chapter 1 

INTRODUCTION 

Three-dimensional separation at intersecting surfaces such as 

wing-fuselage and wing-pylon junctions, and air breathing engine 

inlets is a complex physical phenomenon. The intersecting surface 

flow exhibits complex flow patterns which may significantly affect the 

performance of the aircraft. Major effects include increased pressure 

and heat transfer in the vicinity of the intersections and loss of 

control effectiveness due to flow separation. A variety of simplified 

geometrical configurations have been studied to gain insight into the 

physical aspects of three-dimensional separated flows including (1) 

sharp fin mounted on a flat plate, (2) blunt fin mounted on a flat 

ate, (3) swept compression corner, and (4) semicone affixed to a 

at plate. 

In the present work we concentrate our attention to the flow past 

blunt fin/flat plate junction whch is a typical example of junction 

flow. When a high-speed flow passes over a blunt obstruction mounted 

on a flat plate the detached bow shock formed ahead induces a strong 

adverse pressure gradient to the plate boundary layer. 

disturbance propagates through the subsonic portion of the boundary 

layer and can result in a 3-D separated flow region composed of 

horseshoe vortices near the plate surface and a lambda-type shock 

pattern ahead of the fin. 

stream flow and spiral downstream. 

This 

These vortices entrain a part of the freeThe separation shock emanating 

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from the separated flow region impinges on the fin shock producing a 

shock-shock interaction. Depending on the severity of the incoming 

shock, the shock-shock interaction leads to the formation of 

supersonic jet. This interaction leads to intense heating and high 

pressure locally around the fin leading edge. 

Extensive experimental and computational studies pertinent to the 

physics of the problem have been carried out in the past for flow past 

a wing/body junction and these are reviewed briefly in Chap. 2. 

description of the governing equations and all possible boundary 

conditions employed in the study is given in Chap. 3. 

of the finite volume concept, method of solution and operation of the 

numerical code are given in detail in Chap. 4. The specific physical 

conditions of investigation are described in Chap. 5 and discussion of 

all results are provided in Chap. 6. 

The 

The development 
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Chapter 2 

PHYSICS OF WING/BODY JUNCTURE FLOWS 

In this chapter, the physics of wing/body juncture flows at low 

speed and high speed are discussed. 

studies have been included in the physics of the problem to indicate 

the characteristics of the junction flow. The discussion begins with 

detailed description of low speed unmodified juncture flows for laminar, turbulent and transitional conditions. Next, the description of 

controlled juncture flows are presented. The high speed unmodified 

and modified juncture flows is considered next to clearly highlight 

the difference in the flow structure with reference to low speed. 

Finally the status of turbulence modeling as applied to the computation of junction flow is briefly reviewed. 

Key findings from the previous 

2.1 Low Speed Flows 

2.1.1 Low Speed Unmodified Juncture Flows 

If a protuberance is placed on a flat plate over which there is a 

fluid flow, the protuberance produces an adverse pressure gradient in 

its vicinity. This adverse pressure gradient created upstream of the 

protuberance causes the plate boundary layer to deviate from the 

surface resulting in a three-dimensional separation. The separated 

boundary layer rolls up downstream of the separation line to form a 

system of vortices. These vortices are swept around the base of the 

3 
4 

protuberance and assumes a characteristic shape which has led to its 

name the horseshoe vortex. 

Norman [1]* investigated experimentally the basic features of 

secondary flows and the conditions under which these flows led to 

transition and turbulence. The experimental study defined and described qualitatively the vortex system, the horseshoe vortex system, 

the spiraling-trailing vortex system, the vortex system resulting from 

the instability of the top shear layer, and the interactions of these 

sytems on transition.. 

using smoke visualization technique and observed five types of vortex 

regimes. 

Schwind [Z] studied the low speed junction flow 

Schwind observed that as the velocity is increased the 

number of vortices observed also increases and at higher velocities 

the two clockwise rotating vortices began to oscillate with amplitude 

increasing with velocity. Schwind further points out that although 

the flow passed from,regime 1 to regime 5 types as the velocity is 

increased, the velocity at which the flow changed from one regime to 

another was by no means well defined and different flow regimes could 

exist at one velocity. Baker [3] demonstrated experimentally the 

existence of both steady and unsteady vortex systems. 

behavior of the horseshoe vortex was found to be complex. 

The oscillatory 

As the 

Reynolds nunber is increased the steady horseshoe vortex system Degan 

to oscillate intermittently and randomly at two different frequencies. 

As the Reynolds nunber is increased further, the period of oscillation 

of the vortices became longer, the high frequency oscillation becoming 

more common. At yet higher Reynolds nunber the oscillations became 

irregular and the horseshoe vortex system became turbulent. 

"Numbers in brackets indicate References. 

It was 
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concluded that the frequency of these oscillations is solely determined by the value of Reynolds number and the ratio of diameter of the 

protuberance to the incoming boundary 1 ayer thickness. 

Kaul et al. [4] studied numerically the incompressible junction 

flow using the Navier-Stokes equations. 

generalized coordinates, and solved in time as a hyperbolic system by 

adding a pressure term in the continuity equation. 

marched until steady state was reached. The calculations indicated a 

strong non-uniform pressure loading along the length of the cylinder. 

A new mechanism for the existence of recirculation bubbles behind the 

cylinder end walls with relatively low ratio of cylinder height to the 

incoming boundary 1 ayer thickness was observed. 

The equations were cast in 

The equations were 

The turbulent flow around a winglbody junction is much more 

complex than the laminar counterpart. 

boundary layer interacting with a viscous wake that is pressure driven 

in some regions, Reynold's stress driven in other regions and viscous 

dominated in most other regions. The essential features of the flow 

and the horseshoe root vortex are shown in Fig. 2.1. The wing is 

partially submerged in the oncoming flat plate boundary layer. 

body is sufficiently long that a turbulent boundary layer exists upstream of the wing. 

layer is assumed to be inviscid. 

ing boundary layer wraps around the wing leading edge to form a horseshoe vortex. 

sure gradient created due to the presence of the wing eading to 

three-dimensional separation. 

root vortex, and possibly secondary vortices are estab ished. 

It consists of a turbulent 

The 

The flow around the wing outside the boundary 

The vorticity present in the incomThe oncoming fluid is decelerated by the adverse presIt is in this part that the primary 
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LINE OF INITIAL VORTICITY r 

\ HORSESHOE VORTEX CORE 

ROTATED LINE OF VORTICITY 

Fig. 2.1 Flow structure at the low speed wing/body junction. 
7 

Meanwhile, fluid Off to the side of the wing, passes without 

experiencing much disturbance. Consequently, the fluid elements in 

one part of a vortical line are decelerated relative to fluid in other 

parts. The subsequent wrapping of the vortical line elements around 

the wing leading edge results in the appearance of horseshoe vortex. 

The effect of this skewing of the transverse vorticity line results in 

the generation of streamwise vorticity and a three-dimenstional flow 

velocity. 

velocity, which is directed in the streamwise direction, the flow in 

the root vortex has components in the plane normal to the streamwise 

direction due to the presence of streamwise vorticity. 

vortex experiences viscous diffusion and the effects of counter 

rotation takes place because of the anisotrophy of the Reynold's 

stresses. 

in some sense in equilibrium. 

original state downstream from the wing. Further, there are complex 

Whereas, the flow upstream of the wing has one component 

The horseshoe 

The flow slowly varies in the streamwise direction and is 

The flow does not return to its 

interactions.between the vortices, the separated flow near the 

ing edge and the viscous wake of the wing. The vortices remain 

immersed in the boundary layer from which they originated, and 

not lift up away from the body. 

I 

trai 1- 

they do 

Shabaka and Bradshaw [5, 61 and Oguz [7] made mean flow and 

turbulence measurements in wing-fusel age junctions. The special 

emphasis of the experiment was to investigate the relative importance 

of turbulence terms and examine the spatial extent of the interaction 

flow field. McMahon et al. 

in a wing-fuselage type junction using surface oi l-flow visualization 

technique combined with hot wire anemonerntry. The experimental data 

[8] studied the details of turbulent flow 
8 

collected includes measurement of three mean velocity components and 

six Reynolds stresses at selected streamwise locations. To complete 

the entire picture of the flow field some measurements were taken 

inside the reversed flow region using hot-wire techniques. 

shown that unambiguous measurements of the mean flow direction can be 

made in three-dimensional separated flows. The collected data have 

been used to deduce the flow behavior in and around the juncture. 

Also, the similarities between the mean-flow strain rates and the 

turbulent stresses, and the relative trends among the turbulence 

quantities have been discussed. 

experiments on the behavior of three-dimensional turbulent boundary 

layer in the simplified wing-body junction at low subsonic speed. 

results of the boundary layer behavior in the attached flow region 

have revealed the effects of pressure gradient and curvature of the 

streamline. Briley and McDonald [lo] studied nunerically the wing/ 

body junction flow with swept leading edges using the Navier-Stokes 

equations. A general nonorthogonal coordinate system that resolved 

the viscous sublayer was employed. 

It was 

Hsing and Teng [9] carried out 

The 

The nunerical predictions did not 

agree with measurements well downstream of the leading edge. The 

algebraic turbulence model employed in the study was fairly crude, but 

false diffusion errors, notoriously large in the recirculating region 

may have also contributed. 

2.1.2 Low Speed Controlled Juncture Flows 

The pnysics of low speed controlled juncture flows differ slightly from unmodified juncture flows in that the separation at the wing 

leading edge is eliminated by suitable geometrical modifications to 
9 

Such modifications include fillets, sweep and suction holes the wing. 

in the wing. These simple techniques help in improving the air flow 

around the wing/fuselage junctions and can be easily incorporated in 

existing fleets of aircraft. Preliminary results [ll] indicate that 

there is some reduction in juncture drag at moderate angles of attack 

with the use of leading edge fillets. 

the wing leading edge considerably reduces the extent of separation 

and helps to maintain laminar flow around the juncture. 

The use of sweep and suction at 

Mehta [12] experimentally investigated the effect of wing nose 

shapes on the nature of the generated secondary flow. 

experimental study investigated in detail the correlation of the 

strength, size and position of the horseshoe vortex with the nose 

shape of the wing. 

the horseshoe vortex system in a wing/body junction is strongly 

dependent on the wing nose shape with size and strength of the vortex 

system increasing with the nose bluntness. 

this vortex flow persists right up to the wing trailing edge, with 

very little attenuation, so the only practical way to control it is by 

suitably adjusting the wing leading edge shape. Kubendran and Harvey 

[ll] have made measurements in the wake region of a simulated wingfuselage juncture, with and without leading-edge fillets, to 

understand the effect of leading edge modifications on the flow field 

around the juncture. 

reduction in juncture drag using leading edge fillets at moderate 

The 

The study clearly demonstrated that the nature of 

In most practical cases, 

Preliminary results have indicated some 

angles of attack. 

the surface downstream of the juncture were also noticed when leading 

edge fillets were used. 

Some improvements in the flow characteristics on 

As the fillet size was increased, flow 
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characteristics start deteriorating at some point, and an optimum 

fillet size was needed to achieve an overall improvement in the flow 

field. 

The numerical study of controlled juncture flows using fairings 

at the junction have been studied by Sung and Lin [13] and Kubendran 

et al. [14] using the incompressible Navier-Stokes equations. It has 

been found that the leading edge fairing is quite effective in reducing the leading edge flow separation and non-uniformity of the wave 

velocity profile and the addition of the trailing edge fairing can 

contribute to further improvement. 

2.2 High Speed Flows 

2.2.1 High Speed Unmodified Juncture Flows 

High speed flow past a blunt fin mounted on a flat plate, results 

in a complex, three-dimensional, viscous-inviscid interaction flow 

field. 

the boundary layer to separate from the surface upstream of the fin 

location. 

The ‘detached bow shock formed ahead of the protuberance causes 

This results in a separated flow region composed of horseshoe vortices near the surface and a lambda-type shock pattern in the 

plane of symmetry ahead of the fin. 

wave/turbulent boundary-1 ayer i n t e r a c t i o n are increased pressure loads 

and heat transfer levels on and in the vicinity of the protuberance. 

The existing experimental data suggest that the separated flow is 

very unsteady due to the oscillation of the separation location with 

amp1 itude 1 arger than the undisturbed boundary layer thickness. This 

relatively large movement of the separation location is attributed to 

the pulsating and scavenging action of the flow comprising the 

multiple horseshoe vortices, results in comparatively large 

The major effects of the shock- 
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oscillation of the impinging shock wave. In addition to the pressure 

peaks and heating rates at shock impingement on the fin leading edge, 

there are very large pressures and heating rates on the plate surface 

in the immediate vicinity of the fin root. 

shock wave turbulent boundary 1 ayer interactions have caused 

structural damage on hypersonic wing/body junctions [15]. 

It is believed that such 

Certain aspects of the separated flow ahead of a blunt fin are 

somewhat similar to two-dimensional separated flows ahead of ramps or 

steps. The pressure increase across the fin bow shock wave presents a 

strong adverse pressure gradient to the boundary layer flow on the 

flat plate surface, causing the boundary layer to separate adjacent to 

the plate surface. 

upstream through the subsonic pocket of the boundary layer and leads 

to the separation of the boundary layer on the flat plate. 

sketched in Fig. 2.2, the reversed flow forms a horseshoe vortex in 

which the flow quickly spirals downstream, away from the plane of 

symmetry. Immediately downstream of this vortex, there appears to be 

one or more additional vortices. The reversed flow in the vortices is 

constantly replenished by the separated stream flow, and spirals downstream very rapidly. 

stream into proximity with the surface. 

and large pressure fluctuations have been observed in local regions on 

the surface adjacent to the protuberance. 

visualization techniques suggest that the direction of flow in the 

vortices is predominantly outboard and bears little similarity to twodimensional separated flows in the plane of symmetry. 

ance of the transverse flow masks the nature of the flow along the 

The effects of pressure increase are propagated 

As 

Thus, the vortices bring the high energy air 

Extremely high heating rates 

The existing oil flow 

The predomin- 
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DETAILS OF SUPERSONIC JET r 

I SEPARATED FLOW REOION 

Fig. 2.2 Flow structure at the high speed wing/body junction. 
13 

boundary between two vortices, but there appears a third, small and 

counter rotating horseshoe vortex between the two large vortices. 

flow at the root of the fin is highly three-dimensional and complex. 

There are large pressure, as well as high velocity gradients in this 

region. The compressed separated flow ahead of the fin accelerates to 

supersonic speeds in escaping around the fin to lower pressure 

regions. 

out as it follows the fin root downstream. The fin bow shock causes 

the boundary layer to separate from the surface ahead of the fin, 

resulting in a separated flow region composed of horseshoe vortices 

near the surface, and a lambda-type shock pattern ahead of the fin. 

The shock wave emanating from the separated flow region impinges on 

the fin bow shock, and causes intense heating and high pressure locally around the fin leading edge. 

The 

A strong vortex starts at the root of the fin and spreads 

Young et al. [16], Price and Stallings [17] and Westkaernper [18] 

observed that the extent of turbulent separation is insensitive to 

Mach nunber, Reynolds nunber and boundary layer thickness for a wide 

range of flow conditions, for fin diameters that are larger or comparable to the undisturbed boundary layer thickness. Specifically, for 

Mach numbers from 1.2 to 21, turbulent boundary layers separate 

approximately two diameters ahead of the fin leading edge. 

transitional boundary layer separation ahead of a fin is much more 

extensive than turbulent separation and depends on both Mach number 

and Reynolds number. Limited data available in the literature indicate that the extent of separation ahead of a fin increases with 

increasing Reynolds number based on the separation distance. 

separation lengths exceeding six diameters have been observed by Young 

Laminar or 

Laminar 
14 

et al. [16] for both Mach 3 and Mach 5 flows. 

a strong disturbance to a laminar boundary layer and causes earlier 

transition than for attached, undisturbed boundary layers. In virtually all of the tests carried out in the earlier study [16], transition occurred either ahead of the separation location or in the free 

shear layer bounding the reverse flow region ahead of the fin (transi tional separation). The extent of transitional separation diminished 

toward the turbulent value (a + 20) as the separation location 

approaches the undisturbed boundary 1 ayer transitional location of the 

plate boundary 1 ayer. Mach number, Reynolds number, and boundary 

layer thickness effects on the extent of laminar or transitional 

separation have not been clearly differentiated. 

However, separation is 

SeP 

Sidney [19] investigated experimentally the supersonic junction 

flow problem using flow visualization technique. 

of the experiment was to resolve the fine details of the flow structure. The variation of primary separation distance was discussed as a 

function of Mach nunber, Reynolds nunber and obstacle dimensions. The 

structure of the separated flow upstream of the obstacle was found to 

change with relatively small changes in Reynolds nunber. Experiments 

have been conducted by previous investigators to study the effect of 

transition properties on the diameter of the fin [17, 18, 20-271, 

sweep angle [17, 28, 291, he ght of the protuberance [17, 18, 201, 

Mach nunber [17, 20, 23, 251 and Reynolds nunber [17, 20, 25, 28, 

291. 

confusion over the conditions necessary for the model to be ionsidered 

semi-infinite have caused some difficulty, particularly with respect 

to the vertical scale of the flow field. However most of the 

The major objective 

Attempts have been made to correlate the experimental data, but 
15 

experiments revealed clearly that the dominant parameters controlling 

the centerline flow field are the diameter and sweep angle of the fin. 

Dolling and Bogdonoff [30] studied experimentally blunt fin induced 

shock wave/turbulent boundary layer interaction. 

of the experiment was to investigate the spanwise development of the 

disturbed flow field and to determine its dependance on the model 

geometry and incoming flow conditions. The second objective of the 

experiment was to determine the vertical extent of the interaction on 

the fin. The results show that, on the test surface near the fin and 

on the fin itself, the leading edge diameter plays a dominant role in 

determining the interactions scale and characteristics. 

The first objective 

Hung and Buning [31] studied the blunt fin induced shock wave/- 

turbulent boundary layer interaction problem using the 30 compressible 

Navier-Stokes equations. 

boundary layer thickness by an order of magnitude showed that the size 

of the horseshoe vortex and the spatial extent of the interaction are 

dominated by inviscid flow and only weakly dependant on the Reynolds 

number. Hung [32] also noted that by changing the blunt fin to a flat 

faced fin the extent of separation increased drastically, and the main 

horseshoe vortex bifurcated into two vortices rotating in the same 

direction. 

It was noted that varying the incoming 

2.2.2 High Speed Control>led Juncture Flows 

Very little work is done in high speed controlled juncture flows. 

Price and Stallings [17] and Stollery [33] studied the effect of fin 

sweep in turbulent boundary layers and noted that the separation of 

the flow ahead of the fin can be reduced drastically by leading edge 
16 

sweep. Gaitonde and Knight [34] studied numerically the effect of 

suction to control the sharp fin/flat plate junction flow. 

tation indicated that the particular bleed schedule of five percent 

employed in the study was incapable of controlling or significantly 

The compumodifying the undesirable separation and the large vortical structure. 

It may be possible to significantly influence the flow field by minor 

modifications to the configuration such as raising the fin a slight 

distance from the flat plate could provide interesting results. 

2.3 Turbulence Modeling for Wing/.Body Juncture Flows 

Complex flows of the type found in wing/body junction are markedly influenced by three-dimensionality, pressure gradients, and viscous 

and turbulence effects. An important effect of calculating the flow 

described above is turbulence modeling. The simplest model used for 

the calculation of turbulent shear flows 

models [35]. These models have been emp 

ing two-dimensional flows. However, the 

are algebraic eddy viscosity 

oyed primarily for calculatformulation of eddy viscosity 

models contain a large amount of empiricism that is generally not 

valid in complex three-dimensional flows. Three other types of turbulence models have been developed in the literature that are applicable 

to junction flow, such as two-equation models [36], algebraic Reynolds 

stress models [37], and the full Reynolds stress transport equations 

[38]. Of the models available, only the algebraic eddy viscosity 

model and the two equation models have been employed in the computation of wing/body junction flow [39]. 

Reynolds stress transport equations to three-dimensional flows involve 

the solution of six additional partial differential equations for the 

The application of the full 
17 

stresses and greatly increase the computational effort required. The 

algebraic Reynolds stress models have been developed as an attractive 

alternative to the full Reynolds stress equations. However, this 

model has not been refined sufficiently to make them applicable to 

three-dimensional flow fields. In particular, more work needs to be 

done before these models can adequately compute the near wa 1 

t urbul ence. 

The literature review indicates that most of the exper 

carried out in the junction flow lack detailed measurements 

ments 

in the 

separated flow region and do not sufficiently portray the structure of 

the vortical flow. The same level of criticism can be levelled 

against the previous computational studies in that they fail to 

examine the effect of grid refinement and turbulence model in the flow 

field simulation. 

flow have been studied experimentally and computationally in the past, 

quantitative information about their effect on the vortex strength is 

lacking in most of the earlier studies. 

A1 though techniques for controlling the junction 

The primary objective of the present study is to obtain further 

understanding of the structure of resulting three-dimensional separation and to compute the junction flow with sufficient resolution. 

The specific objective of the study is to investigate numerically the 

effect of fillets and sweep in controlling the juncture horseshoe 

vortex and to provide quantitative information on the amount of weakening of the vortex. 
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Chapter 3 

PHYSICAL MODEL AND THEORETICAL FORMULATION 

In this chapter, a brief description of the governing partial 

differential equations representing the flow, along with the boundary 

and initial conditions employed in the present study are presented. 

The theoretical formulation of the problem starts with the nondimensional form of the compressible three-dimensional Navier-Stokes equations cast in the generalized body-oriented coordinate system. Brief 

discussions on the turbulence model, mesh system, and thin-layer 

approximation are also provided. 

3.1 Governing Equations i n Non-Dimensi onal Form 

In the absence of body forces and source terms, the governing 

equations for the conservation of mass, momentum, and energy in a 

stationary control volune v, surrounded by a control surface S, can 

be expressed in integral form as [40) 

=o u+b) .nds - 

at 

where 

1 

2 

E=ei+-(u +V +w) 

18 

(3.1) 
19 

b =O 

-0 

-1 -1 

b = - y M (Re Pr) p grad ei + pu + M Re =e m -m 

The nondimensionalization of the governing equations is carried out by 

normalizing the Cartesian velocity components (u,v,w) by sound speed 

am 9 the density p by pm, the specific internal energy ei and the 

total energy E by a,, and p by p,am. The viscosity coefficients X and p are normalized with respect to the molecular 

viscosity p,. The constant y, is the ratio of specific heats, Re 

is the Reynolds number based on freestream velocity, and Pr is the 

Prandtl nunber. The density P, static pressure p and the internal 

energy e are related through the following equations of state: 

2 2 

i 

ei = T/y(y-1) (3.3) 
$=T 

20 

(3.4) 

where T is normalized by the freestream static temperature. 

Equation (3.1) is a simple expression for the basic conservation laws 

of fluid flow and is valid for a global control volune and also for 

each locally discretized control volune (i, j, k) shown in Fig. 3.1. 

For the sake of notational convenience, the superscript without a 

prime is used for the transformed coordinate (x' ,x2 ,x3) interchangeable with the local coordinates (E, rl, 5) and the superscript with a 

prime is used for the global coordinates (x,y,z). A similar notation 

convention is used for all other tensor quantities. For example, 

i' I I I 

u = (u' ,3 ,u3 ) = (u,v,w). 

The partial differential equation corresponding to Eq. (3.1) 

written in tensor convention is expressed as follows: 

or 

i 

In eq. (3.5) (g1I2) is the Jacobian and 9 is the contravariant 

base vector in the xi direction. Here, the Einstein sunmation convention is used for repeated indices. Since Eq. (3.5) is a 
21 

11 

Fig. 3.1 A hexahedron finite volume v(i,j,k) with coordinate systems 
22 

homogeneous equation in 4, it can be written as 

Equation (3.6) provides the basis for the implicit procedure. 

Eq. (3.5) explicitly, one has 

Writing 

continuity 

(3.7a) 

i' momentum for x direction 
23 

energy 

- yMoD ' aei MA m 1/2 1 

--- - - div gu.(g) 3 

(i') Re ( 9) l/*ga. 

Re Pr axi 

a. a. where g is the component of the contravariant base vector p 

along the x direction and gil is the contravariant unit base 

(i') i' 

vector. 

The above system of equations is assuned to be valid for turbulent flows if the molecular transport coefficients are replaced by 

' = lJa. + IJt (3.8a) 

(3.8b) Pr Pr Pr 

a,t 

where ut represents the turbulent eddy viscosity and 

lent Prandtl number. The molecular dynamic viscosity is evaluated 

using the Sutherland's law of viscosity. 

Prt the turbu- 
A turbulence mode 

3.2 Turbulence Model 

for the three-dimensiona configuration of 

Fig. 2.2 is complicated and is still under developnent. As z + m 

the fin surface influences the turbulence; on the otherhand as 

the turbulence is dominated by the flat plate surface. 

tion between the fin and flat plate, the effect of turbulence is truly 

three-dimensional. The two-1 ayer turbulent eddy viscosity model of 

Baldwin and Lomax [35] is extended to generalized coordinates to 

determine the turbulent transport coefficients. This model is 

particularly advantageous for certain shock-induced separated flows 

z + 0 

In the juncwhich the length scales are not 

for the 

on of the 

real istic 

that contain a multiple region in 

clearly defined. However, the pr 

amplification of turbulent intens 

flow through the bow shock wave. 

turbulence models for the present 

sent work does not account 

ty after a sudden compress 

The implementation of more 

work awaits further study. 

24 

In each (J-K) plane, the computational domain is like a corner 

formed by two perpendicular walls as sketched in Fig. 3.2. For the 

sake of convenience coordinates (y,z) are used to define each (J-K) 

plane. The two-1 ayer algebraic turbulence eddy viscosity model is 

applied in each region in the following manner. 

Inner Region 

For the inner region, the eddy viscosity model is expressed as 
*. 

N 

25 

Location of FmaX 

I 

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Y.J 

Fig. 3.2 Definition of four regions in (J-K) plane 
26 

where k = 0.4 is Von Karman's constant, II is Buleev's length scale 

[41-431, D is the Van Driest damping factor and w is the absolute 

value of the vorticity (W = I? x 5 I), i.e., 

2 2 2 

IWI d($ - E) +(: - !!) +(E - $) (3.10) 

The general definition of the Buleev length scale is 

(3.11) 

where s defines the distance from the field point (x,y,z) to a point 

on the boundary, and 0 

3.3. For a two-dimensional external corner flow, the Buleev length 

scale a is the distance normal to the surface. For an open corner 

formed by two orthogonal planes, the Buleev length scale is given by 

is the pheripheral angle as shown in Fig. 

a= 2Y (3.12) 

112 

y + 2 + (y2 + z2) 

This formula is designed to account for the size of turbulence eddies 

or the turbulence mixing length near the corner under the influence of 

both wall s. 
27 

Fig. 3.3 Definition of Buleev length scale 
28 

(a) = ' Fkleb 

Outer Reaion 

1 i- 5.5 

In the outer region the outer eddy viscosity is given by 

outer = PK 'cp Fwake Fkleb (3.13) E: 

where K is the Clauser constant, C is an additional constant. 

The outer wake function, Fwake, is defined by 

CP 

F IC wkamaxuz/F max. 

%ax max. = the smal ler of Fwake (3.144 

The quantity Fmax 

profile and amax is the maximum value of E where the maximum value 

Of Fouter 

is the maximum value of F(y) that occurs in a 

occurs. The relation for Fouter is 

Fouter =ewD (3.14b) 

The wake function is a product of a length scale 

velocity scale Fmax representative of the energy-containing eddies 

in the outer portion of the ooundary layer. In effect, the distribution of vorticity is used to determine length scales so that the 

need to find the ooundary layer edge is eliminated. The Klebanoff 

intermittency factor is given by: 

emax and the 

(3.15) 
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where Ckleb is a constant. The quantity Ue is the difference 

between maximum and minimum total velocity in the profile. 

(3.16) min - ( &J2 + v2 + w2 

max ue = (iU2 + v2 + w2 

The second term in Ue is taken to be zero (except in wakes). The 

numerical value of the constants appearing in the foregoing relation 

are given as follows: 

K = 0.0168, Ccp = 1.6, 'kleb = 0.3 . 

The inner and outer eddy viscosity laws are combined to form the 

composite turbulence eddy viscosity model in the following manner. 

and "outer First, profiles of 

ordinate line orthogonal to the boundary. 

the boundary where cinner exceeds touter is denoted the "cross-over 

point." The turbulent viscosity E is equal to einner for all the 

points in the region between the boundary and the crossover point, and 

is equal to E 

crossover point. 

are determined on each coThe first point nearest to 

for all the points above and including the outer 

In the vicinity of the corner, the (J-K) plane is divided into 

four regions I, 11, I11 and IV as shown in Fig. 3.2. 

11, D is evaluated at z = 0. The search for Fmax and the corresponding 11 proceeds from the wall, either from y = 0 for 

region IV, or from z=O for region I. The values of Fmdx in region 

I1 and I11 are treated as constants equal to the value of Fmax at M 

and N, respectively. 

In regions I and 

max 
30 

3.3 Thin Layer Approximation 

Prandtl's classical boundary layer equations are obtained from 

the full Navier-Stokes equations by performing an order of magnitude 

analysis of the terms and neglecting terms of the order of the inverse 

power of the Reynolds nunber. This procedure leads to (1) neglecting 

the streamwise diffusion compared to normal diffusion; (2) the 

replacement of a normal momentun equation with the assunption of zero 

pressure gradient across the boundary layer thickness. The concept of 

thin-layer approximation to the full Navier-Stokes equation is similar 

to the boundary layer approximation with the only exception of retaining all three momentun equations. This procedure makes no assunption 

about the pressure variation across the boundary layer thickness. The 

thin-1 ayer approximation is particul arly advantageous for computing 

certain high Reynolds number flows and separated turbulent flows 

because it removes the troublesome singularities encountered at the 

separation points and permits straight forward computation of separated flows. As a result of the thin-layer approximation, the governing equations become simple and less complicated than those based on 

approximations that attempt to use streanwise gradients para1 le1 to 

the body surface. The considerable simplification in nunerical algorithm and physical concept motivate the straight forward implementation of the approximation into computer codes for high Reynolds number 

flows. 

In the present research work this concept is extended in a 

three coordinate directions for a general coordinate system. A 

viscous terms containing the cross derivatives a2/ax ax when 

are neglected while retaining the normal derivatives a2/ax ax 

ij 

ij 

1 

1 the 

i#j 

when 
31 

i = j in each direction. For example, along each coordinate 

direction the X div - u in Eq. (3.7b) will be expressed as 

neglected ret ai ned neglected 

------- I 

aul ax3 

ax3 ax1 

a$ ax3 

ax3 ax* 

-0 1 

I 

- 

I I 

The above expression can be expressed in 

direction as follows: 

a tensor form for each X - 

It should be noted, that not all the retained terms are larger 

than the neglected terms. Retention of all the second order normal 

derivative terms make the approximation convenient and consistent such 

that all dominant terms are retained for a general coordinate system 

except at the juncture between two walls. Near the juncture between 
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two walls the neglected cross derivatives can be of the same order of 

magnitude as the normal derivatives. 

carries with it a low momentun fluid and hence neglecting the cross 

derivatives will not alter significantly the basic physics of the flow 

field. 

layer), the viscous terms can be completely neglected in any particular direction, and the present algorithm will reduce back to a thinlayer approximation for the presence of viscous shear layers in two or 

only one direction. 

But the flow at the juncture 

When not needed (because of the absence of a thin shear 

3.4 Mesh System 

For simple configurations like swept and sharp fins on a flat 

plate, an algebraic grid generation technique is used to generate the 

mesh. 

distribution. 

This technique offers an easy control of the grid spacing and 

A modified elliptic grid generation technique [44] with 

direct control over the grid spacing is used to generate the mesh at a 

filleted finjflat plate junction. 

(-) 

continuous fillets in the plane of synmetry. 

The equation of the super ellipse 

Xn 

a b 

+ ( I)" = 1, is used to generate the grid for circular and 

The three-dimensional 

grid for a sharp fin (Fig. 3.4) is obtained by simply stacking 

horizontal grid in the direction normal to the flat plate surface. 

The three-dimensional grid structure for a swept and filleted fin 

(Fig. 3.5 and 3.6) is obtained by simply rotating the grid in the 

plane of symnetry about the center of the fin leading edge. 

is clustered near the fin and plate surfaces using an exponential grid 

The grid 

stretching formula to provide adequate resolution of the viscous 
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- 

Fig. 3.4 Mesh distribution on sharp fin and flat plate 
FLAT PLATE 

Fig. 3.5 Mesh distribution on swept fin and flat plate 

34 

'RY 
35 

8 

FLAT PLATE 

Fig. 3.6 Mesh distribution on filleted fin and flat plate 
36 

effects. Since the flow is assuned to be steady and synmetrical only 

one-half of the flow is computed. 

3.5 Boundary Conditions 

The physical boundary conditions to be imposed are problem 

dependent. 

blunt fin/flat plate geometries. In this case, a supersonic laninar 

boundary layer develops on the flat plate. 

leading edge generates a bow shock wave which interacts with the 

boundary layer on the flat plate. 

At present, the 30 Navier-Stokes code has been applied to 

The deflection of the fin 

The boundary conditions applied may be categorized as follows: 

3.5.1 Inflow Boundary 

The incoming flow is supersonic (except in the subsonic portion 

of the boundary layer) and the inflow boundary is essentially perpendicular to the oncoming flow. 

point from outside towards the computational domain and the appropriate boundary condition is to specify u, v, w, C, and ei. 

blunt fin, the upstream boundary layer profile is obtained from a twodimensional compressible boundary 1 ayer code [45]. 

Consequently all flow characteristics 

For the 3-D 

3.5.2 Outflow Boundary 

The outgoing flow is assuned to be supersonic (except in the 

subsonic portion of the boundary layer), and the outflow boundary is 

essentially perpendicular to the outgoing flow. In this case all the 

flow characteristics point from inside to outside of the computational 

domain, and thus the outflow variables must be determined from the 
37 

interior flow solutions by zeroth order extrapolation. 

assumed to be infinite in length and height so that zero-gradient 

The fin is 

boundary conditions are appropriate at the outer boundaries in the 

corresponding directions. The outer boundary of J = Jmax is 

situated far enough to avoid any influence on the interaction region. 

Theoretically one can prescribe a boundary layer profile on the 

isolated flat plate at each x-location along the outer boundary as 

shown in Fig. 3.7. These profiles are obtained from solving the 

compressible boundary layer equations for flow over a flat plate. 

3.5.3 Solid Boundary 

At a solid wall boundary the following boundary conditions are 

enforced 

-=o aP 

an 

(3.18) 

aT - T = Tw or kan - qw 

where = (u, v, w) is the Cartesian velocity vector, p is the 

static pressure, Tw is the specified wall temperature, qw is the 

specified wall heat flux and k is the coefficient of thermal conductivity. The boundary condition for the static pressure is an 

approximation to the exact boundary condition derivable from the 

normal component of the momentun equations, and has been applied for a 

variety of flows involving shock wave boundary layer interaction. 
8 

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38 

I 

-xo- x*=(x-xoI/o 

Fig. 3.7 Boundary layer profiles on the flat plate at J=JmaX 
39 

3.5.4 Symmetry Boundary 

The fin is at zero angle of attack, and the following boundary 

conditions apply on the plane of symetry. 

V.n - = 0 

3.6 Initia 

an 

-= aP 0 

an 

(3.19) 

Conditions and Blunt Body Starting Condi ,ions 

The simplest way to start a computational solution is by assuning 

that all initial conditions are the same as the incoming flow 

solutions. This may result in over compression near the stagnation 

region followed by an over expansion around the shoulder. This may 

cause the nmerical scheme to become unstable, thereby resulting in 

negative density. 

time step during start-up by gradually enforcing the no slip wall 

boundary condition. 

shock as a discontinuity and started with an assuned pressure distribution, in addition to the soft start technique. 

research work, a simple formula is employed to avoid over expansion 

around the shoulder [40]. 

region, the normal velocity component is set to zero on the surface of 

To avoid this, Pulliam and Stegar [46] used a small 

Rizk, Chaussee, and McRae [47] treated the bow 

In the present 

As depicted in Fig. 3.8 near the n'ose 
40 

C I 

-X L 

Fig. 3.8 Region near the nose for setting blunt body starting 

conditions 
41 

the body at C, and then increased linearly to the incoming flow 

velocity at a predetermined location, say D. 

at a point between C and D is 

The formula for velocity 

r 1 

where LCD is the fixed length between two points C and D and 11 is 

the variable length from the surface. The density and energy are set 

equal to the incoming flow conditions. This simple device may be 

interpreted as a defect in normal momentun and hence kinetic energy in 

the small region near the blunt nose. 

nonphysical flow field developnent and allows large time steps during 

the start up phase. Though different possible boundary conditions 

were discussed the exact boundary conditions used in the present study 

This device alleviates the 

are the sane as given in the previous nunerical study of supersonic 

junction flow [40]. 
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Chapter 4 

METHOD OF SOLUTION 

In this section, the finite volume formulation is developed for 

solving the compressible Navier-Stokes equations at high Reynolds 

number. 

MacCormack's [48] explicit-implicit scheme in time split fashion to 

reduce the three-dimensional equations to three sets of locally onedimensional equations. 

The governing equations are integrated numerically using the 

4.1 Finite Volune Formulation 

A finite volume cell, indexed by (i,j,k), is defined by eight 

corner points connected with straight lines, as shown in Fig. 3.1. 

Each edge of the finite volme cell is identified by a coordinate 

3 location (it ,x2 ,xk) interchangeable with ,< ), where j jk 

x1 i = si = (i-1) AE, i = 1, IL 

(4.1) 

and the grid spacing along the local coordinate direction is given by 

42 
 AX^ = Arl = l/(JL-1) 

43 

(4.2) 

AX3 = AC = l/(KL-1) 

The enclosed surface of each finite volune cell consists of a family 

of three coordinate surfaces that delineates the hexahedronal mesh 

cell. The above formula yields a unit cube for the transformed computational domain. The choice of a cube of unit dimension in the computational domain is arbitrary, and the other investigators have chosen 

the lengths of each side to yield Ax1 = Ax2 = AX3 = 1 so that the 

volume of each computational cell is given by dv = g Ax1 Ax2 Ax3. 

It should be understood in Eq. (3.5) that g is the volume of the 

1/2 1 

cell dv (i,j,k), the term g g represents the contravariant 

surface normal for the surface x'= constant and - u . g 

sponding contravariant velocity evaluated at the bounding surfaces. 

1/2 

112 

- 

a. is the corre 

112 1 

Therefore, in Eqs. (3.7) (9) g is the surface area for Xa. = - 11 

constant projected on the Cartesian coordinate of x' = constant. 

1/2 112 a. 

In the finite difference approach, (g ) and g g are 

defined at each grid point location, and are typically evaluated by a 

This leads to 

- 

two-point central difference in all three directions. 

an inconsistancy in the volune and surface normal calcu 

that the geometry conservation law is not satisfied and 

difference scheme cannot recapture the free stream. Th 

tency does not occur in the present approach. 

ations such 

hence the 

s inconsisNote that any open-surface element for a given boundary enclosing 

the surface has a unique, effective surface vector - S that is independent of the shape of the surface. This is because, applying the 
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divergence theorem to a constant vector, the integral of the outwardoriented surface normal over a closed surface vanishes as given below 

Is 0 fi ds = V (0 d v = 0 for + = constant (4.3) 

For example, the surface vector S 5876 

the choice of which the partitioning surface diagonal is used to 

in Fig. 3.1 is independent of 

na 

the 

define cell volunes. Indeed whether all the four vert 

plane or not, the effective surface vector is equal to 

cross-product of its diagonal segments, i .e., 

ces lie 

one-ha1 f 

(4.4) 

Given eight corner points of each finite volune cell, a simple 

way to calculate the volune is to divide each face into two planar 

triangles, 

each face since the diagonals of four nonplanar points do not intersect. 

laps, neighboring cells must have the sane surface partitioning. In 

the present study a simple formula developed by Kordulla and Vinokur 

[49] is employed to calculate the volune of each cell as 

The volune is then dependent on which diagonal is used on 

In order for neighboring cells to be contiguous, without over- 
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With the volune and surface evaluated in integral form, Eq. (3.5) 

can be discretized as 

where gi ,j,k 

center of the cell and (q -- u + b). - S is the corresponding flux evaluated at the surface of each finite volune cell. In the present study, 

the discretized finite volume formulation, Eq. (4.6) is used for 

computation. 

Eq. (3.5) will be used. 

evaluated are based on Eqs. (3.7a) -(3.7c) with corresponding interpretations of (9) as the volune at the centroid and (9) CJ as 

the surface normal evaluated at the boundary of each finite volune 

cell. 

is a vector of the dependent variable located at the 

In later discussions, for the purpose of explanation, 

However, details of terms which have to be 

1 /2 112 

It is interesting to see that the finite volume formulation does 

not require a global coordinate transformation. 

data needed concerning the mesh are the three Cartesian coordinates of 

each of the eight vertices of every cell in the mesh system. 

Moreover, the surface and volune of each cell are well defined and 

consistently evaluated, in contrast to a finite difference approach 

where the transformation coefficient and the Jacobian are evaluated 

from surrounding points. 

In fact, the only 

1 /2 

In finite volune formulation, only the cell volune (9) and 

1/2 

the surface normal (g )CJ' are needed for inviscid flow coinputations. To evaluate the viscous terms, the distance between two 
46 

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adjacent mesh cells is needed. In the present approach, instead of 

calculating this distance from the centroid, the vector between two 

adjacent cells (i,j,k) and (i+l,j,k) (see Fig. 4.1) is set in the same 

direction as the surface normal to the surface 

by 

Sb, and is evaluated 

When shape variations are large, a more appropriate formula should be 

used for evaluating the metrics as given by the following formula. 

This implies that in a viscous dominated region, for instance near the 

wall, the grid lines would be required to be nearly orthogonal for an 

accurate evaluation of viscous terms. 

4.2 Explicit MacCormack Algorithm 

The MacCormack's explicit algorith is implemented using a synmetric sequence of time split operators. 

generally known as the method of alternating directions, has been most 

popularly used to split the complex operators into a sequence of 

simpler ones. This concept is applied here to reduce the set of 

three-dimensional equations, Eq. (3.5), into three sets of one-dimensional equations. 

The concept of splitting, 

Equation (3.5) can be split into three local ly one-dimensional 

(LOD) operators as follows: 
47 

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c Fig 4.1 Distance d between two adjacent cells -i ,j,k 
48 

iA 

LX: g,t + = 0 (no sunmation i n i ) i = 1, 2, or 3 (4.9) 

Here, each of the split operators contains the spatial derivatives of 

flux in only one direction. 

scheme, the sumnation convention will not be used in this section. 

Since, this concept is limited to the LOD 

iA Let At be the time step, n the time level, and - F the flux 

i 

across the cell surface x = constant. The LOD predictor-corrector scheme in a generalized coordinate system can be expressed as 

follows: 

Predictor: 

explicit 

update 

Corrector: 

expl ici t 

update 

A f" = - At( A- :E) 

AX 

(4. loa) 

(4. lob) 

Here, the overbars in the corrector sweeps indicate that the quantities are determined with updated predictor values. The A+ and A 
49 

indicate forward and backward two-point differences. 

operators are all one-sided by themselves, the resulting algorithm is 

second order accurate in space and time. 

i 

A1 though the 

Each of the Lx operators discussed above consists of a 

predictor-corrector set, and requires no block or scalar tridiagonal 

inversion. Because of the noncommutativity of Lx operators, the 

combined nunerical scheme is accurate only to first order. 

the second-order accuracy, a symnetric sequence of LOD operators, 

developed by Strang [50] is used. 

- q is updated in time according to the following relation: 

i 

To retain 

Specifically, the solution vector 

A 

(4.11) 

By this procedure the solution is advanced two time steps from t to 

t + 2At. 

1. Temporal Accuracy 

In order to retain a second order accurate scheme in time for the 

operator sequence of Eq. (4.11), the total time applied to each 

operator must be the same, i.e., 

At =Atrl =At =At (4.12) 5 5 

2. Numerical Stability 

Because of the complexity of the full Navier-Stokes equation, it 

is not possible to obtain a closed-form stability expression for the 

MacCormack scheme applied to these equations. 

set of equations, can be subjected to von Newnann type stability 

However, the linearized 
I' 50 

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8 

analysis. In order to maintain nunerical stability, the time for each 

operator must satisfy the following stability condition: 

At 6 At 

CFL s 

(4.13) 

where the maximum allowable time steps At and At , 

are given in Appendix A. 

~FL 

The explicit MacCormack's algorithm of has several advantages. 

First, the method is second-order accurate in time and space. Second, 

it is relatively easy to code the explicit algorith. 

explicit algorithm is easily vectorizable. Fourth, the algorithm is 

robust, in the sense that its overall stability is relatively insensitive to the initial conditions employed. In practical terms, this 

implies that the initial condition need not be close to the final 

solution. In all the computations to date on three-dimensional 

problems, the flow initial conditions are obtained by simply propagating the prescribed boundary layer at the outer boundary of J = 

Third, the 

throughout the mesh. Jm ax 

Computations involving the compressible Navier-Stokes equations 

sometimes "blow up" because of large gradients, for example when the 

flow field contains strong shock waves. 

practical to refine the mesh in these regions, particularly if they 

are far removed from the region of interest. For such situations a 

fourth order damping term is added to the scheme to suppress the 

In many instances, it is not 
51 

nunerical oscillations. This smoothing term has a very small 

magnitude except in regions of pressure oscillations where the truncation error is already degrading the computed solution, and should be 

nearly zero as steady state is approached. 

4.3 Implicit MacCormack Algorithm 

For computations involving full Navier-Stokes equations, grid 

lines have to be clustered in the direction normal to the wall. In 

such situations, the explicit CFL stability limit becomes very 

restrictive and a large number of iterations are required to reach a 

steady state solution. 

shows that there are no restrictions on the time of integration and 

that it is unconditionally stable. 

Bi-diagonal Scheme and Locally One-Dimensional Time Splitting 

The stability analysis of the implicit schemes 

The bi-diagonal impl icit scheme in generalized coordinates at 

time level n is introduced into explicit predictor and corrector 

steps as follows: 

Predictor: 

expl icit 

update 

A An+l 

P"+l = - 

+ 

(4.14a) 
Corrector: 

52 

The matrices I '2 - I are'rnatrices with positive eigenvalues. 

where 

0 

The si are the similarity transformation matrices which diagonalize 

the Eulerian Jacobian pI in Eq. (3.6) by setting p = x = 0. The 

i diagonal terms of the matrix I 01 

sentative eigenvalues defined by 

iA 

contains three distinct repre- 
i 2 pyM- gii 

-- 

RePr pAxi At 

U 

ii i 

where gii = g '3 . The eigenvalues of the diagonal matrix I 1 I 

for i = 1, 2, 3 can be expressed as follows: 

for i=l x1 = A3 = Ab = 1 1 1 1 

1 

u -a lA2= lAu+a, and A5 = 'A 

2 

u-a and A5 = 2A 

u+ a 

3 

3~4 = 3~ u+ a and A5 = 'A u-a 

A 

Each transformation matrix of Mi for i = (1, 2, 3) is split into 

three matrices which can be expressed as follows: 
where 

0 

P-l 

0 

0 

-0 U 

0 

0 

P -l 

0 

-6 V 

0 

0 

0 

P -1 

-0 w 

a = 0.5(u2+v2+~2) and 0 = (y-1) 

for i = 1 

0 

0 

0 

0 

0 

0 

0 

0 

1 

0 

-e1(3') 0 el(ll) 0 I 

0 0 0 

54 
55 

- 

0 

0 

1 

0 

1 - 

for i = 2 

=o M2C 

- 

1 

0 

0 

0 

b 

0 0 

1 0 

0 pa 

M2,. = 

0 0 

0 -pa 

P 

1 

0 

0 

0 

0 

0 

0 

0 

1 

0 

0 0 0 0 

e2(2') -e2(1') 0 0 

0 0 0 1 

I 
56 

for i = 3 

0 0 

1 0 

0 1 

0 0 

0 0 

0 0 

e3(3') 0 

0 -a2 

0 0 

0 0 

pa 1 

-D a 1 

0 0 

0 0 0 1 

112 

-1 / (see) is the component of the normal L 

e(il) - (ill Here, 

contravariant vector in the i' direction. The M matrices transform 

6$ from conservative to nonconservative variables; the matrices 4 

Mit account for the orientation of the cell faces, xi = constant, 

and matrices MiC then transform from nonconservative to characteristic forms. It is interesting to see that the matrices M and MiC 

are the sane in the Cartesian coordinate system and non-orthogonal 

curvilinear coordinate system; the matrices MiT account for the 

generalized coordinate transformation. 

for i = 1, 2, 3 are expressed as follows: 

The characteristic variables 
57 

for i = 1 

for i = 2 

for i = 3 

Now it is clear that 

1/ 2 

6p + pa6i . 93/(!333) 

6p - pa6i . g3/(!333) 

1/ 2 I A jp Mi transforms the inviscid portion of ' j 

into characteristic form 6% jc only when i = j. This implies that, 

in contrast to the unsplit formulation, LOO time splitting makes 
a 

1 

IR 

;I 

58 

variables consistent with the characteristic variables by means of the 

similarity transformation employed in the implicit steps [Eqs. (4.14a) 

and (4.14b)l. 

Two points should be noted here. First, it is an insurmountable 

task to develop the similarity transformation for each entire viscous 

Jacobian 4. To keep the right hand side of Eqs. (4.14a) and 

(4.14b) as simple and straightforward as possible, we follow here the 

suggestion given by MacCormack to consider only the Eulerian Jacobian 

of ig - with the addition of a small nunber of dominant viscous terms to 

the Eulerian eigenvalues. Second, the impl icit procedure is skipped 

whenever the explicit stability conditions are satisfied locally, and 

the algorithm requires virtually no more computation time than an 

explicit method. 

iA 

Each spl it operator described above consists of an expl icitimpl icit predictor-corrector set, is second-order accurate in time and 

space, is unconditionally stable, and requires no block or scalar 

tridiagonal inversion. Because of the noncomutativity of split 

operators, the simple combined nunerical scheme, Lxl Lx2 Lx3 is only 

first order accurate. To maintain, the second order accuracy of the 

method, a synmetric sequence of the LOD operators, developed by Strang 

[50] are employed. Here, the solution is advanced from time t to t 

+ 2At as follows 

(4.15) 
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59 

Each locally one-dimensional split operator is applied line by line in 

each sweep direction and, hence, only needs one level plus three 

single arrays of dependent variables. 

MacCorrnack scheme requires, in general, two time levels of variable 

plus a level of right-hand-side residuals for a total of three levels 

of variables stored. 

Without this splitting, the 

4.4 Operation of Nunerical Code 

4.4.1 General Procedure 

The computation of three-dimensional shock 

interactions using the numerical code is accomp 

as follows: 

wave boundary layer 

ished in three phases 

1. Mesh Generation and Initial Flow Condi,ion 

Depending on the investigation of the flow past a 

specific configuration, the user generates the grid using a 

separate grid generation code. Once the coordinates of the 

mesh points are known, the volune and surface normal for each 

finite volune cell can be easily computed. The information 

concerning the storage architecture for the physical coordinates is provided with the nunerical code. 

condition, including the upstrem (inflow) profiles, and the 

profiles at 

user. 

ation the upstrem profile may be a uniform flow, or a 

boundary 1 ayer flow generated by solving the two-dimensional 

compressible boundary layer equations. The information conThe initial 

J= J, on the flat plate are provided by the 

Depending upon the particular geometry under consider- 
60 

cerning the storage for the flow field and surface normals is 

provided in the nunerical code. 

2. Computation of the Three-Dimensional Flow Field Using the 

Numeric a1 Code 

The nunerical code is used to integrate the governing 

equation in time until a steady state solution is obtained. 

The user of the code must ensure that reasonable criteria for 

the achievement of steady flow have been met. 

3. Graphic Display of Flow Field 

The user must develop his own graphics code to post 

process the output generated by the three-dimensional NavierStokes code. 

graphics code with the graphics library on the host 

computer. 

In this regard, the user may interface his 

4.4.2 Requirements on Grid 

The body-oriented generalized coordinate system must provide 

adequate resolution to unravel the basic physical aspects of the flow 

field. A complete set of mesh c r i t e r i a f o r judging the acceptability 

of the grid system does not exist at present, except for the process 

of mesh refinement which is generally not feasible for three-dimensional Navier-Stokes simulations. The determination of the acceptability of a given grid system, therefore, must be based on experience 

and previous nuner ical computations. 

There are, however, a nunber of necessary criteria which may be 

identified as follows: 

1, Resolution of Boundary Layers in the Direction Normal to the 

Boundary. 
61 

The flow field in the direction normal to the boundary 

is generally characterized by the steep flow gradients. This 

is based on previous experience that a minimum of fifteen 

(15) grid points be placed in the boundary layer depending on 

the amount of resolution needed to resolve steep flow gradients. For a turbulent flow, the y of the first mesh point 

should be located less than three wall units from the 

+ 

boundary for accurate resolution of the viscous sublayer. 

The distance of the first mesh point from the wall in the 

2. Resolution of Boundary Layers in the Cross flow Direction. 

In the nunerical simulation of the three-dimensional 

flow field, a typical maximum grid spacing in the cross flow 

direction is provided. 

where 6 is the boundary layer thickness to accurately 

resolve the spatial features of the flow field. This 

criteria, however, has not been extensively tested and more 

Stringent requirements may need to be imposed depending upon 

the flow configuration. 

Resolution of the Boundary Layers in the Streanwise Direction 

This is kept between 0.26 to 0.86, 

3. 

The streamwise grid spacing depends strongly on the 

character of the flow field. In the vicinity of interaction 

region, the streamwise grid spacing is determined by the size 

of the interaction region. For nominally, two-dimensional 

shock-boundary layer interaction the size is a function of 

pressure rise, Reynolds nunber, and may be a fraction of the 
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boundary 1 ayer thickness. For a three-dimensional shockboundary layer interaction, the size of the interaction 

region is 1 arger than a corresponding two-dimensional interaction with the same pressure rise. 

4. Resolution of the Inviscid Flow 

The criteria for grid spacing in the inviscid region 

depends upon the particular flow configuration, and must be 

determined on a case-by-case basis. For exmple, the existence of a bow shock upstream of the blunt leading edge at 

high Mach nunbers may impose a stringent local constraint on 

the inviscid grid spacing. 
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Chapter 5 

PHYSICAL CONDITIONS OF INVESTIGATION 

For understanding the flow past a wing/body junction, three 

specific simp ified geometries were selected. These include: (1) 

blunt swept f n/flat plate junction, (2) sharp fin/flat plate 

junction, and (3) filleted fin/flat plate junction. The special cases 

of blunt and sharp fins are used to obtain specific results for which 

extensive measurements on velocity and pressure distribution, skin 

friction, and surface yaw angles are readily available in the literature. 

junction tne various parameters control 1 ing the interaction are D 

(diameter of the fin), A (fin sweep angle), 6 (incoming boundary layer 

thickness), M, (Mach number), and Re, (unit Reynolds number). The 

parameters controlling the filleted fin-induced interactions are R 

(f i 1 let radius) and 6 (incoming boundary 1 ayer thickness). Si nce 

three dimensional computations are expensive, only limited parametric 

studies were carried out for selected values of sweep angle, fillet 

radius, Mach number, Reynolds number, and Mesh resolution. The range 

of values in each set of parameters is carefully selected to portray 

its dramatic influence in controlling the intersection flow fields. 

The flow field simulation of the selected cases can be easily 

attempted on existing super computers without requiring excessive 

computational resources. 

In a typical three-dimensional flow past a fin/flat plate 

63 
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For the specific 

junction, comparative 

following experimenta 

problem 

results 

cond i t 

5.1 Laminar Flow Past a Swept Fin/Fl at P1 ate Junction-- 

Validation with Experiment and Parmetric Study 

5.1.1 Validation with Experiment 

of flow past a blunt fin/flat plate 

were obtained corresponding to the 

ons [16]: 

2 M m = 2.95, po -= 22261 N/m , To -= 298 K, 

6 = 0.225 cm, D = 1.875 cm, H = 7.5 cm, A = 0" 

64 

Figure 5.1 shows a simplified blunt fin/flat plate junction and 

various locations along which the computed results are compared with 

the experimental measurements. The experimental investigation 

inc udes data on pressure distribution along the fin stagnation line 

and selected spanwise locations. To supplement the measurements, oil 

flo.. visualization technique is also used to reveal the global 

structure of the flow on the fin and flat plate surface. 

5.1.2 Parametric Study 

The parametric studies were conducted to investigate the effect 

of fin sweep and incoming boundary layer thickness. 

for which specific results were obtained are as follows: 

The conditions 

Effect of Fin Sweep 

2 M, = 2.95, po -= 22261 N/m , To -= 298 K 
65 

Fig 5.1 Geometry of blunt fin/flat plate junction showing typical 

surface locations of comparison wi th-experiment 
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66 

6 = 0.225 cm, D = 1.875 cm, H = 7.5 cm, A = 0' to 45' 

Effect of Incoming Boundary Layer Thickness 

2 M OD = 2.95, po m= 2226 N/m , Tom= 298 K 

6 = 0.675 cm, D = 1.875 cm, H = 7.5 cm, A = 0' and 45' only 

5.2 Laminar Flow Past a Blunt Fin - Conditions for Additional 

Results and Parametric Studies 

5.2.1 Conditions for Additional Results 

The following specific conditions were selected to obtain additional results and for parametric studies: 

2 M rn = 2.36, po QD = 24000 N/m , To -= 294 K 

6 = 0.186 cm, D = 0.635 cm, H = 5 cm 

This specific case was selected to validate the code with the experimental measurements of velocity distribution in the separated region. 

The experimental measurements were obtained by oil flow visualization, 

Schlieren observations, and laser anemometry and the results are 

available in [52]. 

5.2.2 Parametric Studies 

The paranetric studies were conducted by using the conditions of 

Sec. 5.2.1 and changing the various parameters. 
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67 

Effect of Grid Refinement 

To investigate the effect of grid refinement, the conditions of 

Sec. 5.2.1 were used and results were obtained for the following three 

grids: 

30 x 50 x 30, 40 x 100 x 30, 40 x 100 x 50 

Effect of Mach Number 

The freestrean stagnation temperature was changed corresponding 

to each Mach number while keeping the other conditions constant as 

given in Sec. 5.2.1. The results were obtained for the following 

cases : 

= 286 K To 

To 

M = 1.2, m 

M = 2.36, = 294K m 

M = 3.5, To 307 K m 

Effect of Reynolds Number 

The unit Reynolds nunber of the freestrean is altered by changing 

the tunnel stagnation pressure while keeping the other freestrean 

conditions constant as given under Sec. 5.2.1. 

obtained for the following cases: 

The results were 

= 1.25 x lo6/, 2 

Po - = 12,000 N/m , Rea. 
68 

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!I 

= 24,000 N/m 2 , = 2.5 x 106/m 

= 48,000 N/m 2 , = 5.0 x 1h-n 

Po - 

Po 

I' 

5.3 Turbulent Flow Past a Sharp Fin/Plate Plate Junction 

5.3.1 Flow Past a 16' Sharp Fin 

For the case of a 16' sharp fin induced shock wave/boundary layer 

interaction, comparative results were obtained corresponding to the 

. following experimental conditions [53]: 

M = 4.0, Po = 1180822 N/m 2 , To 336 K W 

6 = 0.46 cm, H = 3.7 cm, e = 16' 

The experimental study gathered data on wall pressure, surface yaw 

angle, and skin friction distribution at a specified spanwise location 

throughout the interaction region. The experimental study employed 

oil flow visualization to understand the flow structure, Preston tube 

for measuring skin friction, and cobra probes for flow field yaw 

angles. 

5.3.2 Flow Past a 20' Sharp Fin 

The strong interaction generated by a 20' sharp fin on a flat 

plate is investigated corresponding to the following conditions [54]: 
I 

I 

8 

1 

8 

I 

I 

I 

I 

8 

l 

I 

8 

I 

1 

I 

8 

1 

I 

69 

M = 2.93, Po 688320 N/m 2 , To OD= 260 K OD 

6 = 1.27 cm, H = 10 cm, e = 20' 

Figure 5.2 shows the axial location of experimental surveys where the 

computed pressure distribution is compared with experiment as a 

function of the Z- coordinate normal to the flat plate. 

ments were carried out using high Reynolds nunber supersonic tunnel to 

measure the pressure and yaw angle as a function of the coordinate 

normal to the fl at pl ate. 

The experi5.4 Turbulent Flow Past a Swept Fin/Flat Plate Junction 

For the case of swept fin induced shock wave/turbulent boundary 

layer interaction, comparative results were obtained corresponding to 

the following experimental conditions [17): 

2 

6 = 5 cm, D = 5 cm, H = 35 cm, A = 0' to 60' 

M OD = 3.71, po OD= 203055 N/m , To 339 K 

5.5 Turbulent Flow Past a Filleted Fin 

5.5.1 Flow Past an Unmodified Junction 

For the case of low Reynolds nunber flow past a blunt fidflat 

plate junction, results were obtained corresponding to the following 

experimental conditions [33]: 

2 = 27000 N/m , To 312 K Po M m = 2.5, 
LINE 

70 

FIN SHOCK 7 

OF COALESCENCE 

Fig 5.2 Location of experimental surveys for 20" sharp fin 
71 

6 = 5 cm, 0 = 2.5 cm, H = 20 cm 

5.5.2 Flow Past a Modified Junction 

For the modified junction, the effects of fillet radius and incoming boundary layer thickness were investigated. 

Effect of Fillet Radius 

Two circular fillets with a radius (R = 5 cm, 9 cm) and a conjunction. The incoming flow 

n the blunt f n/flat plate junction case. 

ti nuous f i 1 1 et were used 

conditions were kept as 

to modify the 

Effect of Incoming Boundary Layer Thickness 

Keeping the fillet radius (R = 9 cm) constant, the incoming 

boundary layer thickness was decreased by an order of magnitude from 5 

cm to 0.5 cm to investigate its effect in controlling the juncture 

vortex. 
Chapter 6 

RESULTS AND DISCUSSION 

The entire results obtained for the physical conditions described 

in Chap. 5 are presented and discussed in this chapter. 

physical conditions where experimental results were available, the 

For the 

nunerical results were obtained to validate the nunerical scheme and 

the computer code developed in this study. Extensive results were 

obtained for other physical conditions to study the behavior of threedimensional separation for various high speed juncture flows. The 

effect of different parameters in controlling the intersection flow 

field were also investigated. 

6.1 laminar Flow Past a Swept Fin 

6.1.1 Computation of 0' Swept Fin Flow Field 

The incoming flow conditions for this case are given in Sec. 

5.1.1. 

Figure 6.1 shows the comparison of pressure distribution along the fin 

The computations were carried out using a 40 x 40 x 40 mesh. 

leading edge. The leading edge pressure is normalized by the pitot 

tube pressure behind the bow shock wave. The results show that the 

computed pressure distribution agrees we1 1 with the experimental 

values. The existence of a peak pressure on the fin leading edge is 

noticed around Z/D = 1.5. The minimum pressure is seen to occur 

around Z/D = 0.25. In the downward direction from this point, the 

pressure is seen to increase slowly towards the plate surface. 

computed results, therefore, appear to demonstrate the physically 

The 

realistic trend. 

72 
I 

I 

I 

3.0 

Z/D 2.0- 

73 

-0 Experiment (Young, 

- et al, 1968) 

-Present 

- 

p,= 22261 N/m2, 1-r 108K 

MI= 2.95. b= 0.225cm 

- 

- 

I 

a 

- 

Solution at 1.000 

- 

I I 

11 

I 

1 

1 

I 

I 

1 

I 

c 

I 

8 

I 

I 

I 

1.0 

0.0 

Converged solution kf0.1 0.8 1.2 1.6 

P'Pt2 

Fig 6.1 Pressure distribution on the blunt fin leading edge and 

comparison with experiment 
74 

As pointed out by Hung and Buning i311, the existence of peak 

pressure on the fin is due to multicompression of the separat on 

shock. 

to extreme expansion created by the horseshoe vortex leading to a 

The decrease of pressure from the peak to the minimum is due 

reversed supersonic zone on the fin. The pressure increase in the 

direction towards the plate surface is due to recompression from 

supersonic to subsonic flow. This creates an adverse pressure 

gradient in the direction toward the plate, resulting in a supersonic 

zone on the fin. 

The computed pressure distribution on the flat plate along the 

line of symmetry is compared with the experimental measurement for an 

unswept fin in Fig. 6.2. The results show good overall agreement 

except near the juncture point. The main features of the flow such as 

upstream influence, pressure plateau, and the pressure rises across 

the fin bow shock and plate separation shock are all well simulated by 

the nunerical code. The existence of low pressure following the separation pressure rise is due to the reversed high speed flow leading to 

a reversed supersonic zone above the plate surface. 

6.1.2 Parametric Effect of Fin Sweep and Boundary Layer Thickness 

The pressure distribution on the fin leading edge is illustrated 

in Fig. 6.3 for various fin sweep angles corresponding to the conditions given in Sec. 5.1.2. 

fin case using an identical mesh. It is seen that as the fin'sweep 

increases, the location and magnitude of the peak and minimum pressures decrease drastically, 

magnitude of the heating rate and peak pressure on the fin leading 

The results are compared with the blunt 

As pointed out by Edney [55], the 
75 

12.0 

8.0 

P 

pa 

- 

4.0 

0.0 

- 

0 Experiment (Young. et at, 1968) 

p,= 22261 N/m2, T-= lO8K 

M,= 2.95. b = 0.225cm 

- - Present 

- 1 

- Converged solution 

Solution at 1,000 - 

- 

0 

I I I I I 

0. 

I I I 

Fig. 6.2 Pressure distribution on the flat plate and comparison 

with experiment 
76 

Z/D 

3.0 

2.0 

1.0 

0.0 

lOIK 

0.4 0.8 1.2 1.6 

P/Pt2 

Fig 6.3 Pressure distribution on the fin leading edge for various 

fin sweeps 
77 

edge depend essentially on the geometrical parameters such as the fin I sweep. 

thinner and may strike the fin leading edge far downstream of the 

As the fin sweep angle increases the supersonic jet becomes 

impingement point where, because of diffusion its effect on the heat 

transfer and peak pressure may be considerably lower than for the 

unswept case. 

above a certain critical value, the resulting shear layer may miss the 

fin leading edge entirely, and hence, high pressure and heat transfer 

do not occur. 

Edney further observed that as the fin sweep increases 

Comparison of the pressure distribution on the flat plate is 

shown in Fig. 6.4 for various fin sweep angles. 

the fin sweep angle increases, the separation point on the plate 

surface moves downstream. Also, the peak pressure at the corner 

It is noted that as 

decreases drastically with increasing sweep angle. A1 though no 

comparison of pressure distribution has been made with experimental 

measurements for the swept fin case, the computed results agree qualitatively in trend with the Price and Stallings [17] data for a turbu1 ent boundary 1 ayer. 

To further demonstrate the effect of fin sweep on the interaction 

flow field, the code was run for a thick incoming laminar boundary 

layer for conditions given in Sec. 5.1.2. 

ness, in the previous case, was very small. 

grid in this region created some instability in the nunerical simulation. 

effect of the fin sweep on the interaction flow field, the code was 

run for 0' and 45' fin sweeps. 

using a 40 x 40 x 40 mesh. 

lhe boundary layer thickHence, packing a fine 

To remedy this situation and also to observe the detailed 

The computations were carried out 
Sweep 0’ 

12.9 ’ 

30’ 

4 5’ 

- 1 --- 

--e-- 

-.- 

p, = 22261 N/m2, T, = 108K 

M, = 2.95. b = 0.225cm 

1 

i 

I I I I I I I I 

0.0 0.0‘ -8.0 -6.0 -4.0 -2.0 

X/D 

Fig. 6.4 Pressure distribution on the flat plate for various 

fin sweeps 

78 
1 ead 

stra 

79 

Figure 6.5 shows the computed pressure distribution on the fin 

ng edge for 0' and 45' swept fins. The results clearly demone the dranatic influence of the fin sweep on the pressure distribution. It is seen that as the fin sweep increases, the peak pressure 

and its location along the fin leading edge decreases dramatically. 

The location and magnitude of minimm pressure on the fin leading edge 

also shows a drastic variation in the 45' sweep case as compared to 

the 0' sweep case. 

Figure 6.6 shows the comparison of pressure distribution on the 

flat plate ahead of the fin leading edge for 0' and 45' swept fins. 

It is seen that as the fin sweep angle increases the point of separation of the incoming flow moves closer to the fin leading edge. 

as the sweep increases, no definite first peak in pressure is 

observed. The final pressure jump across the fin leading edge shock 

also shows a drastic reduction in the pressure for the 45' swept fin 

as compared to the 0' swept fin. 

through a severe adverse environment, since the fin bow shock is 

aligned normal to the incoming flow for the 0' sweep fin. 

of the 45' swept fin, the incoming flow goes through the oblique bow 

shock formed ahead of the fin and suffers less severe change as 

compared to the 0' swept fin. 

level of the peak pressure of the flow ahead of the 45" swept fin. 

Also, 

This is because the flow goes 

In the case 

This is the reason for reduction in the 

Figure 6.7a and 6.7b show the Mach nunber contours in the plane 

of symnetry for 0' and 45' swept fins. The separation shock, fin bow 

shock, and the two reversed supersonic zones (in the case of 0' sweep) 

have been clearly identified. As the sweep angle increases, the Mach 

nunber contours show clearly the decrease in the extent of the 
4.0 

3.0 

Z/D 2.0 

1.0 

0.0 

I 

I 

I 

I 

\ 

= 2226 N/m2. TI =108K 

\ 

\ 

\ 

80 

'. TI =108K 

= 2.95, b = .675cm 

- sweep 0' 

---- sweep 45' - 

I I I I I I 

0.4 0.8 1.2 1.6 

P/Pt 2 

Fig. 6.5 Pressure distribution on the fin leading edge for various 

fin sweeps in a thick laminar boundary layer 
8 

I 

8.0 

P/p, L*o0.0 

81 

- 

- ---- sweep 45' 

- sweep 0' 

p, f 2226 N/m2, 1, = 108K 

M,= 2.95. b= .675cm 

- 

I I I I I f I 1 

Fig. 6.6 Pressure distribution on the flat plate for various fin 

sweeps in a thick laminar boundary layer 
8 

I 

1 

6 

1 

3.2 

2.1 

Z/D 

1.6 

0.8 

82 

6RIG:MAL PACE IS 

OF POOR QUALITY 

BOW SHOCK 

, SEPARATION SHOCK 7 

\ 

\\ SUPERSONIC ZONES 

0.0 c . L". 

-3.6 -2.4 -1.2 0.0 -6.0 -4.8 

xi0 

Fig. 6.7 Mach number contours in the plane of symmetry for 

various fin sweeps: (a) sweep angle = 0" 
4.0 

3.2 

2.4 

ZID 

1.6 

83 

ORIGINAL PAG€ IS 

OF POOR QUALtTY 

I 

BOW SHOCK - 

I 

I 

rt 

I SWEPT FIN LEADING EDGE 

11 

r 

SUPERSONIC ZONE 

0.8 

0 

-6.0 -&.8 -3.6 -2.4 -1.2 0 

XI0 

Fig. 6.7(continued) (b) sweep angle = 45" 
t 84 

separated region. It is noted that the two reversed supersonic zones 

(one on the fin and one on the plate) are eliminated in the 45' swept 

fin case. A large variation in the Mach nunber is noted behind the 

fin bow shock for the 0' sweep fin as compared to the 45' swept fin. 

The normalized static pressure contours in the plane of synmetry 

are shown in Figs. 6.8a and 6.8b for 0' and 45' swept fins. The 

regions of high pressure at the fin leading edge and low pressure at 

the corner are clearly seen in the contour plots. As compared to the 

unswept fin case, the pressure behind the shock for the 45' swept fin 

does not show large differences in the peak pressure. 

ments demonstrate clearly the ameliorating effect of the fin sweep 

because the leading edge becomes more and more like a slender ellipse 

in the freestrean direction with the increase in the sweep angle. 

This means with a swept blunt fin on a flat plate, there is a reduction in the upstream influence of the bow shock, and the effect of 

bluntness associated with the fin is compressed into a small interaction region. 

the interaction and hence the pressure levels throughout the whole 

flow field. 

metry are shown in Figs. 6.9a ind 6.9b. 

These two arguIn fact, the sweep clearly reduces the magnitudes of 

The corresponding density contours in the plane of syn6.2 Laminar Flow Past a Blunt Fin 

In this section, nunerical results are compared with available 

experimental data. Also, the results of grid refinement and other 

parametric studies are presented and discussed. 
I 85 

'I 

4.0 7- I I 1 I 

BOW SHOCK3.2 - ! 

2.L 

I 

I 

210 I 

1.6 - 

.:" 

0.3 

/' /' I 0.8 I- /' ,,,I' , 

XI0 

Fig. 6.8 Pressure contours in the plane of symmetry for various fin 

sweeps: (a) sweep angle = 0" 
86 

I 

I 

8 4.0 I I I I I 

i. 

3.1 i 

0.8 1 

BOW SHOCK 

SWEPT FIN LEADING EDGE 

I 

I ~ 

I 

Ii 

0 I 325 [:f . :5 , 

-6.0 -4.8 -3.6 -2.4 

XID 

Fig. 6.8(continued) (b) sweep angle = 45" 

-1.2 0 

ORIGINAL PAGE IS 

OF POOR QUALITY 
8 

8 87 

oWlGl;4Af P2,GE IS 

OF POOR QUALITY 

1.0 ' I I E 1 

- I 

/- ;

f 

I 3.2 - - 

,' .. 

, ,/' BOW SHOCK- 2.L - ,/' 

SUPERSONIC ZONES 

r 

-6.0 -4.8 -3.6 -2.4 -1.2 0 

XI0 

Fig. 6.9 Density contours in the plane of symmetry for various fin 

sweeps: (a) sweep angle = 0" 
4.0 

3.2 - 

2.4 

ZID 

1.6- 

0.8 

88 

ORIGINAL PAGE IS 

OF POOR QUALITY 

0 

-6.0 -4.8 -3.6 -2.4 -1.2 0 

XI0 

Fig. 6.9(continued) (b) sweep angle = 45" 
I 

8 

8 

I' 

It 

I 

I 

8 

89 

6.2.1 Comparison of Computations with Experimental Data 

The base computations for this case have been carried out using a 

30 x 50 x 30 mesh. The tunnel flow conditions are given in Sec. 

5.2.1. 

and comparison with the experiment are shown in Fig. 6.10. It is 

apparent that in the experiment the pressure begins to rise at about 

ten diameters upstream of the fin leading edge. 

occurs, in the experiment, further downstream (X/D = - 6.8) between 

the regions of initial pressure rise and pressure plateau. 

sure reaches a valley downstream of the plateau region and then 

increases across the detached bow snock. The secondary separation in 

the reversed flow occurs between the pressure valley and the pressure 

plateau. The computed results using the base grid (30 x 50 x 30) are 

in reasonable agreement with the experiment in the region upstream of 

the pressure plateau. As observed in the experiment, the location of 

separation and initial pressure rise are closely predicted by the 

computations. 

mental results is not very good downstrean of the plateau. Further, 

the computed results fail to predict the length of the plateau region 

acc ur ate1 y . 

The computed pressure distribution along the axis of symmetry 

The separation 

The presHowever, the agreement between the computed and experiThe computed pressure distribution along the fin stagnation line 

(4 = 0'), and along various 9 = constant and X/D = constant lines on 

the fin surface, are shown in Figs. 6.11a and 6.11b. For 4 = 0' and 

45', the fin leading edge pressure is referenced to the total pressure, pt2, behind the normal shock. Otherwise, it is referenced to 

the freestrean static pressure p,.It is clear from the computed 

results that the peak pressure on the fin surface decreases 
90 

1.2 

1.1 

1.0 

0.9 

0.8 

GRID 

30x50~30 

--- 40x100~50 

1.1 0 EXPT CaZCAN AND HOLT, 19841 

1.3 

p, = 24000 N/ms, 1, = 283K 

M, = 2.36, b* .la6 em 

- 

- 

--. 

- 

- 

0 0 0°, 

\j 

0.7 

-12 

1 

-10 -8 -6 -4 -2 0 

X/D 

Fig. 6.10 Comparison of pressure distribution along the line of 

symmetry 
I 

8 

I 

I 

1 

8 

I 

1 

I 

8 

I 

1 

I 

I 

I 

8 

I 

8 

e 91 

GRID 

- --- 30x50~30 

40x100~30 COMPUTED 

40x100~50 1 -.- 

--- 

p, = 24000 N/m2, T, = 283K 

.2 .c .6 .8 1.0 1.2 

p'pt 2 

Fig. 6.11 Pressure distribution on the blunt fin: (a) along 4 = 0" 

and 45" 
92 

GRlO 

---- 30x50~30 

--- 40x100x50 

40x100~30 COMPUTED 1 -- 

@=go' X/D=2.5 XID=L.S 

6 

4 

2 

0 

I .5 1.0 1.5 2.0 2.5 3.0 

1.0 1.5 2.0 2.5 

1.0 1.5 2.0 

II - 

111 u 

T, = 283K 

cm 

Pt 2 

Fig. 6.ll(continued) (b) along 0 = 90" and X/D = 2.5 and 4.5 
I 

8 

93 

substantially as the flow expands around the blunt fin. 

pressure on the fin surface at (I = 0' is about 1.1 and at (I = 45' it 

is approximately 0.6. 

drops at (I = 45" but not as much as the peak pressure. 

drops below the value of the total pressure pt2 behind the normal 

shock. The computed results along the fin side face (Fig. 6.11b) show 

that the pressure drops below the value of freestream static pressure. 

As the flow continues to move downstream the extremely low pressure on 

the fin surface vanishes downstream of the shoulder. 

The peak 

The magnitude of the minimum pressure also 

The pressure 

The computed and the experimentally measured streamwise velocity 

profiles along the line of synmetry are shown in Fig. 6.12. The 

streamwise velocity component is nondimensionalized with reference to 

the freestream velocity Ue; Ue is nominally 560 m/sec. 

lines along the vertical axis show the displaced origin to avoid overcrowding of the figures. The streamwise velocity profiles along X/D = 

-8.0 and -7.2 (Fig. 6.12a) indicate preseparation behavior. The 

velocity profile at X/D = -6.4 indicates that the separation occurs in 

close vicinity of X/D = -6.4. 

occurs around X/D = -6.8. However, it was difficult to make repeatable measurements at that location due to the inherent unsteadiness of 

the flow. Also, due to the small scale of the reversed flow region at - 

X/D = -6.4, it was found difficult to detect the reversed flow region 

in the vicinity of the primary separation line. The velocity profile 

at X/D = -5.6 clearly shows the reversed flow region. 

profiles at X/D = -4.8, -4.0, and -3.2 are shown in Fig. 6.12b. These 

results show that the magnitude and height of the reversed flow region 

increases with increasing X/D. 

The dashed 

In the experiment, the separation 

The velocity 

The streamwise velocity profiles 
94 

GRID 

40x100~30 

40x100~50 

X/D EXPT (OZCAN AND HOLT, 1984) 

-8.0 30x50~30 

-7.2 A 

-6.4 

6 

Z(mm) 5 

4 

3 

2 

1 

IO - 

Fig. 6.12 Compari son of streamwi se velocity prof i 1 e: 

X/D = -8.0, -7.2, -6.4 and -5.6 

(a) a1 ong 
9 

8 

7 

6 

5 

4 

3 

2 

1 

Z(mm1 

95 

- X/D EXPT (OZCAN AND HOLT, 19841 GRID 

-4.8 ---- 30x50~30 

. -4.0 A COMPUTED -.- 40x100~30 

-3.2 {--e 40x100~50 

. p, = 24000 Nlm', T, = 283K 

M, = 2.36, b= 186 cm 

------- 

I I IO 

-.75-.50 0.25 0 .25 .50 .75 1.0 

u/u, 

Fig. 6.12(continued) (b) along X/D = -4.8, -4.0 and -3.2 
96 

at X/D = -2.4 and -1.6 are given in Fig. 

measurements show that the maximum magnitude of the reverse velocities 

at these two locations are approximately equal. At these locations, 

computed results deviate significantly from measurement. The height 

of the reversed flow region decreases with increasing X/D. 

measured streamwise velocity profiles at X/D = -3.2, -2.4, and -1.6 

display significant kinks between the outer edge of the reversed flow 

region and the edge of the boundary layer. 

attributed to the inherent unsteadiness present in the flow. 

computed velocity profiles do not display such behavior because the 

flow is assumed to be steady. 

6.12~. The experimental 

The 

This phenomena is 

The 

The vertical velocity profiles at X/D = -4.8, -4.0, -3.2, -2.4, 

and -1.6 are shown in Figs. 6.13a and 6.13b. 

occurs around the outer edge of the boundary layer. 

the velocity then starts decreasing in the z-direction towards the 

freestrean. 

the reversed flow region Fig. 6.12a,b,c increases with increasing 

X/D. 

The maximum velocity 

The magnitude of 

The maximum velocity toward the plate and the height of 

The postulated mean streanline pattern shown in Fig. 2.2 requires 

positive streamwise velocities close to the flat plate slightly upstrean of secondary and tertiary separation lines. 

observed in the junction demonstrates that across the boundary layer, 

in the vicinity of secondary and tertiary separation lines, the vertical velocities remain positive. Moreover, the vortical pattern also 

requires that the vertical velocity profile change its sign more than 

once during a streamwise traverse along various z = constant lines. 

However, the present vertical velocity profiles do not show such 

The flow structure 
I 

I 

1 

I 

I 

I I x/D Exp.(Otcan and Holt. 19841 GRID 

-2.c . 30X50X30 

-1.6 A LOXlOOX30 

p,=2cOOO Nlm’. T,=283K 

M,=2.36. bs.186 cm 

A 

A 

A 

3 

2 

1 

IO 

-.75 -.SO -.25 0 .25 .SO .75 1.0 

UIU, 

Fig. 6.12(continued) (c) along X/D = -2.4 and -1.6 

c 

3 

2 

1 

0 II 
98 

0 Exp.(Orcan and Holt, 19841 

---- p,.=2&000 Nlm’. T.=283K 

COMPUTED M,=2.36, b=.186 cm 

12.5 1 --- COXlOOX50 

XIDs-3.2 

10.0 

7.5 

5.0 

2.5 

0 

I -2.5 0 2.5 5.0 7.5 10.0 12.5 15.0 17.5 

~ 

-5.0 -2.5 0 2.5 5.0 7.5 10.0 

Ill L I I I I 1 

-5.0-2.5 0 2.5 5.0 7.5 

Fig. 6.13 Comparison of vertical velocity profile: (a) along 

X/D = -4.8, -4.0 and -3.2 
99 

I 

0 Exp.(Ozcan and Holt, 198k) 

---- 30X50X30 

-.- 

--- Lox1oox5o 

10 - 

- ZImml 

10.0 15.0 20.0 I -5.0 0 5 .O 

II 

-5.0 

- 

0 5.0 

Fig, 6.13(continued) (b) along X/D = -2.4 and -1.6 
100 

behavior. 

are time dependent, but the time averaged surface locations where the 

vortices are shed remain approximately constant. Thus, the stream1 ine 

pattern given in the plane of symnetry can be considered as one of the 

possible streanlines which succeed each other in time. 

Peake [56] have shown such a postulated sequence of the cyclic vortical structures in low speed laninar flow past a circular cylinder. 

The unsteady nature of high speed laminar flow can be similar to low 

speed laninar flow. HOwever, further study is required to understand 

the unsteady nature of this type of flow field. The limiting stream1 ine patterns on the plate surface shown in Fig. 6.14 depicts the 

separation and reattachment lines on the flat plate. The primary 

separation, corresponding to the most upstream separation location of 

the plate boundary layer, occurs roughly at about X/D = -6.2. 

reattachment line of the flow is located very close to the fin. 

It can be hypothesized that the locations of vortex centers 

Tobak and 

The 

6.2.2 Effect of Grid and Flow Parameters 

In this subsection, the results of the parmetric study of grid 

resolution and variation in Mach and unit Reynolds nunbers are presented. 

freestrean conditions given in Sec. 5.2.2. 

For various cases, the flow field is investigated for the 

The effects of grid refinement on the wall pressure distribution 

along the line of synmetry are shown in Fig. 6.10. These results 

demonstrate that computations using three different grids yield identical results except for the variation in the magnitude of the pressure valley. The computed pressure distribution along Q = 0" and 45" 

are shown in Fig. 6.11a. Further plots of pressure distribution along 
10 1 

r REATTACHMENT 

15 

10 

Y 

0 

- 

5 

c 

0 

-15 -10 -5 0 

x10 

Fig. 6.14 Limiting streamlines on the flat plate for laminar flow 

past a blunt fin 
I 

lE 1 

I 

E 

a 

102 

+ = 90' and downstrean of the fin shoulder (X/D = 2.4 and 4.5) are 

shown in Fig. 6.11b. The pressure distribution along + = 0' is most 

sensitive to the grid refinement. 

The sensitivity of the grid on streanwise velocity profiles at 

various locations upstream of the fin leading edge are shown in Fig. 

6.12. The most upstream separation location of the flat plate 

boundary layer is insensitive to the grid refinement. The computed 

results generally appear to move towards the experimental data with 

refined mesh. However, at X/D = -2.4 and -1.6 (Fig. 6.12c), the 

nunerical prediction using fine mesh (40 x 100 x 50) still display 

large discrepancies with the experiment. 

files and comparison with the experimental data for various streamwise 

locations are shown in Fig. 6.13. 

solution at X/D = -4.0 and -3.2 show some discrepancy with the experiment for z-locations less than 2.5 mm. 

grid are able to improve the prediction with the experiment at these 

locations. ' 

The vertical velocity proThe coarse grid (30 x 50 x 30) 

Computations using the fine 

The nunerically simulated particle paths in the plane of spmetry 

for various grids studied are shown in Fig. 6.15. 

the presence of two clockwise rotating primary vortices and a third 

counter clockwise rotating vortex at the corner. These results demonstrate that the structure of the flow is relatively insensitive to the 

grid employed . 

The solutions show 

The grid refinement study discussed in the previous paragraphs 

established that a 40 x 100 x 30 mesh is sufficient to predict the 

experimental measurements. 

distribution in the plane of symnetry is shown in Fig. 6.16 using a 40 

The effect of Mach nunber on the pressure 
103 

PRIMARY VORTICES- CORNER VORTEX 2 

Fig. 6.15 Particle paths in the plane of symmetry for various 

grid refinements: (a) 30x50~30 grid 
Fig. 6,15(continued) (b) 40x100~30 grid 

104 
II 

I 

il 

IE 

105 

PRIMARY 

Fig. 6.15( continued) 

VORTICES 

( C) 40x100~50 grid 
1.8 

1.4 

P - 1.2 

p, 

1.0 

.8 

COMPUTED M, 

1.2 

2.L 

3.5 

--A 

--- -- 

Re, = 2.5 x 10 / m , T, = 2 8 3 K 

106 

.6 

- 

1 I I I I I 1 

X/D 

12 -10 -8 -6 --I -2 0 

Fig. 6.16 Pressure distribution along the line of symmetry for 

various Mach numbers 
107 

x 100 x 30 mesh. 

decrease the length of the most upstream separation distance. Also 

The effect of increasing the Mach nunber is to 

the magnitude of the pressure plateau fol lowing separation increases 

with an increase in the Mach number. The pressure distribution along 

the fin stagnation line shown in Fig. 6.17 indicate that as the Mach 

number increases the z-location corresponding to the peak pressure 

decreases. However, the level of peak pressure remains constant for 

r4 = 2.36 and 3.5. The minimun pressure on the fin also shows a 

drastic reduction with increase in the Mach nunber. Also, the variation in pressure from a peak to a minimun increases with increasing 

Mach number. 

QD 

The streanwise velocity profiles for various Mach nunbers are 

shown in Figs. 6.18a and 6.18b at selected X/D locations. 

of increasing the Mach nunber is to increase the maximum reverse 

velocity towards the flat plate. 

plane of synmetry) at various Mach nunbers are shown in Fig. 6.19. 

The results ‘of Figs. 6.19b and 6.1% demonstrate that the flow at 

higher Mach nunbers (MOD = 2.36 and 3.5) display no abrupt change in 

the vortical structure. 6.19a), only a 

single vortex rotating in the clockwise direction can be seen as 

opposed to mu1 ti-vortex patterns observed at higher Mach nunbers. 

Similar trends in flow patterns were observed by Baker [3] in low 

speed laninar flow where the nunber of vortices in the junction 

increases with freestream velocity. 

The effect 

The simulated particle paths (in the 

However, at MQD = 1.2 (Fig. 

The effect of unit Reynolds nunber on the pressure distribution 

along the line of symnetry is shown in Fig. 6.20. 

increasing the Reynolds nunber is to move the location of the first 

The effect of 
10 

8 

6 

Z 

0 

- 

4 

2 

0 

108 

COMPUTED M, 

1.2 

2.L 

3.5 

--- 

--- -- 

. Rq=2.5x1O6/rn, T,=283K 

I I I ‘ii 

.2 .4 .6 .8 1.0 1.2 

p’pt 2 

Fig. 6.17 Pressure distribution along the fin stagnation line for 

various Mach numbers 
109 

-6 

-4 

-2 

-0 11 

COMPUTED M, 

1.2 

2.4 

3.5 

--- 

XID=-3.2 , --- -- 

xIo=--L.o /I 

IO 2L -1.0 -.5 0 .5 1.0 

UIU, 

Fig. 6.18 Streamwise velocity profiles for various Mach numbers: 

(a) along X/D = -4.0 and -3.2 
COMPUTED M, 

--- 1.2 

- 2.L 

--- -- 3.5 

1 lo 

10 

8 

6 

zlmml 

L 

2 

c I 

i I\ 

i ;I ReI = 2.5 x 10 I m, T, =28 3 K 

0 .5 1.0 -1.0 -.s 

UIU, 

Fig. 6.18(continued) (b) along X/D = -2.4 and -1.6 
i 

PRIMARY VORTEX 

Fig. 6.19 Particle paths in the plane of symmetry for various 

Mach numbers: (a) MOD = 1.2 

111 
112 

Fig. 6.19(continued) (b) M, = 2.36 
Fig. 6.19(continued) (c) M, = 3.5 

113 

~ --_ -------- 
114 

COMPUTED R.1 

1.2x106 

----e 5x106 

2.5~10~ 

1.4 

M,= 2.36. T-= 283K 

.6 I I I I 1 I 

-12 -10 -8 -6 -4 -2 0 

XI0 

Fig. 6.20 Pressure distribution along the line of symmetry for 

various unit Reynolds number 
8 

II 

t 

t 

i 

I 

I 

I 

8 

I 

t 

8 

8 

I 

I 

t 

t 

8 

e 

115 

peak in pressure slightly inboard toward the fin leading edge. 

rnagni tude of the pressure plateau remains approximately constant. The 

pressure distribution along the fin stagnation line (Fig. 6.21) show 

that as the Reynolds number is 

to peak pressure moves slightly upward. 

Z/D = 0.5 also shows a substant a1 variation to the increase in the 

unit Reynolds nunber. 

The 

ncreased, the z-location corresponding 

The minimum pressure around 

The streanwise velocity profile at various X/D locations shown in 

Fig. 6.22 demonstrates that the maximun reverse velocity toward the 

plate increases with an increase in the Reynolds nunber. Also, the 

height of the reversed flow region above the flat plate decreases with 

an increase in the Reynolds nunber. The computed particle paths, in 

the plane of symnetry (Fig. 6.23), show the effect of Reynolds nunber 

on the flow structure. 

numbers (Fig. 6.23b and 6.23~) show the presence of three vortices, in 

contrast to the situation at the lowest Reynolds nunber (Fig. 6.23a) 

which depicts only two vortices. 

The flow structures at two higher Reynolds 

6.3 Computation of Three-Dimensional Sharp Fin Flow Field 

6.3.1 Flow Past a 16' Sharp Fin 

The flow past a 16' sharp fin is simulated using the conditions 

given in Sec. 5.3.1. 

x 45 mesh. 

interaction region is set at a distance of less than 5 wall units from 

the fin and flat plate surface. The present computations are compared 

with the previous numerical work of Horstman [56] and the experimental 

measurements of Peake [53] for identical flow conditions. 

The computations were performed using a 40 x 45 

The y' location of the first mesh point ahead of the 
Z 

D 

- 

10 

COMPUTED Rei 

--- 1.2x106 

2.5~10~ 

----- 5x106 

M,=2.36. 1,=283K !I 

.2 .b .6 1.0 1.2 

P 

Pt2 

- 

Fig. 6.21 Pressure distribution along the fin stagnation line for 

various unit Reynolds number 
117 

2- 

1 

UIU, 

6 

L 

2 

0 " 

Fig. 6.22 Streamwise velocity profiles for various unit Reynolds 

number: (a) along X/D = -4.0 and -3.2 
8 

8 

1 

I 

1 

I 

1 

8 

8 

I 

I 

8 

I 

I 

8 

I 

1 

8 

e 

-6 

-4 

-2 

-0 11 

10 

8 

6 

t(mml 

L 

2 

IO 

COMPUTED Rei 

XIDt-1.6 

1.2x106 

2.5~10~ 

5x106 

-_- 

----- 

M,=2.36, T,=283K 

-1.0 -.5 0 .5 1.0 

UI u, 

Fig. 6.22(continued) (b) along X/D = -2.4 and -1.6 
8 

I 

1 

I 

1 

I 

I 

8 

I 

8 

8 

8 

8 

I 

8 

I 

8 

1 

I 

- 

119 

- 

.I____.--- 

.---- 

PRIMARY VORTEX 1 CORNER VORTEX 

Fig. 6.23 Particle paths in the plane of symmetry for various unit 

Reynolds number: (a) Re, = 1.2x106m-1 
8 

1 

8 

1 

Fig. 

120 

6.23(continued) (b) ReR = 2.5x106m-1 
I 

8 

8 

8 

1 

I 

8 

I 

8 

1 

8 

8 

I 

I 

I 

8 

8 

1 

B 

Fig. 

121 

---__ 1 

PRIMARY 

6.23( continued) 

VORTICES CORNER 

(c) Re, = 5~10~rn-l 

VORTEX J 
I 

I 

8 

I 

8 

I 

1 

8 

I 

1 

8 

1 

8 

I 

8 

8 

I 

1 

I 

122 

The comparison of present results with previous nunerical and 

experimental investigations are given in Figs. 6.24-6.26. Figure 6.24 

shows the wall pressure distribution on the flat plate. 

pressure is nondimensionalired with respect to the freestream static 

pressure. The abscissa ( x-xshock ) is normalized with respect to the 

incoming boundary layer thickness 6 upstrem of the interaction and 

The wall 

0 

is the axial location of the inviscid shock wave at a specified shock X 

spanwi se station. 

The pressure distribution curve predicts a gradual pressure rise 

due to separation of the incoming boundary layer. 

reaches a plateau followed by a final compression as the flow goes 

past the fin leading edge oblique shock wave. All of the features 

observed in the experiment are closely predicted by the present nunerical simulation. It is interesting to see that the present calculation agrees very closely with the experiment as compared to the 

previous numerical Simulation. A typical inviscid solution is also 

shown in Fig; 6.24. It is clear that the present calculations do not 

give a crisp representation of the shock structure due to the nature 

of the scheme employed in the present study. 

The pressure then 

A comparison of the computed surface yaw angles, tan-' (v/u), 

with the experimental data is shown in Fig. 6.25. The maximum 

computed surface angle (from the present computation) is about 67.5" 

whereas previous computations have predicted an angle of about 88'. 

The maximum measured angle from the experiment is about 48'. Peake 

[53] observed some nonuniformities in the experiment due to upstream 

nozzle effects. 

section to improve the tunnel flow. Because of these conditions, the 

A vortex generator was used upstream of the test 
EXPERIMENT (Poako. 19761 i-INVISCID 

123 

SOLUTION 

I 

1 ---- COMPUTATION (Horrtman.1979~ I 

- 

I ‘I I 

PRESENT 

p,=7777 Nlm’. T,=80 K 

/ 

/ 

2.0 - 

- 

I * I/’ I I 

L 1.0 - 

-8 -4 0 -12 

Fig. 6.24 Pressure distribution on the flat plate along Y/S, = 5.56 

for 16” sharp fin 
124 

8 

I 

8 

1 

1 

1 

I 

I 

i 

120 

105 

90 

75 

CY.dog 60 

L5 

30 

15 

0 

0 EXPERIMENT (Poako, 1976) 

_--- C 0 M PUTATIO N 1 H or s t m an , 1 9 7 9 1 

- PRESENT 

p,=7777 N/m2, T,=80 K 

M,=L, bo =O.C6cm 

I-\ 

I\ .- 

I /I 

/ 

/ 

/ 

/ 

/ I I I I I I 

-10 -8 -6 -4 -2 0 2 I 

Fig. 6.25 Yaw angle distribution on the flat plate along Y/s, = 5.56 

for 16" sharp fin 
I 

I 

I 

8 

I 

I 

1 

I 

I 

1 

I 

I 

1 

8 

I 

I 

8 

1 

8 

125 

quality of the M = 4 experimental data may not be satisfactory. The 

symbols with arrows represent the uncertainty in the experimental 

data. 

mean experimental data is about 40%. The computed skin friction 

results from the present and previous nunerical simulations are compared with the experimental data (Fig. 6.26). The experimental 

results are seen to fall between the two sets of computed results. 

QD 

The maximum difference between the computed results and the 

6.3.2 Flow Past a 20' Sharp Fin 

For the case of 20' sharp fin induced shock wavelboundary layer 

interaction, comparative results were obtained corresponding to the 

conditions given in Sec. 5.3.2. 

50 mesh. 

of less than 5 wall units from the fin and plate surface. The present 

computational results are compared with the previous computational 

study of Knight et al. [57] and the experimental work of Shapey [54] 

for identical flow conditions. 

The computations employed a 30 x 50 x 

The y location of the first mesh point is set at a distance 

The computed pitot pressure profile at station 1 and 2 (Fig. 5.2) 

located upstrean of the interaction region are given in Figs. 6.27a 

and 6.27b. The computed and experimental pitot profiles at stations 3 

through 8 (Fig. 5.2) are displayed in Figs. 6.27~-6.27h. The 

abscissa in all figures is the pitot pressure p normalized by the 

upstream freestrean pitot pressure p 

distance measured from the flat plate surface normalized by the upstrean boundary layer thickness 60, 

general 60 is not the appropriate vertical scaling factor for this 

complex interaction flow field. 

P 

The ordinate represents the P' 

It should be noted that in 

As suggested by the experimental data 
I 

I 

I 

1 

I 

I 

1 

I 

I 

I 

I 

I 

I 

I 

1 

I 

I 

1 

8 

126 

0 EXPERIMENT (Peak., 1976) 

C 0 M P UT AT1 0 N I H or s t man ,19 7 9 1 ---- 

3*0 [ - PRESENT 

2.0 

0 

/ p,=7777 Nlm'. T,=80K / / 

/ 

- M,rL. bo=0.L6cm / 

/ 

/ 

/ 

/ 

/ / 0 / 

0 

0 0 

0 

i I I 

-10 -8 -6 -4 -2 

I I I 

0 2 L 

(x -x s h ock)lbo 

Fig. 6.26 Skin friction distribution on the flat plate along Y/s, = 

5.56 for 16" sharp fin 
c 

3 

2 

b’o2 

1 

127 

Fig. 6.27 Pitot pressure distribution at different axial locations 

for 20” sharp fin: (a) station 1 
Fig. 

- 

q=688320 Nlm’. T,=260 K 

M,= 2.9 3, b0=1.26 cm - 

- 

128 

6.27( continued) (b) station 2 
fi 

I 

P 

T 

8 

1 

8 

E 

E 

c 

8 

f 

E 

c 

8 

e 

i 

1 

Q 

129 

of McClure and Dolling [58] the appropriate vertical scaling factor is 

given by Z Re,1/3/60 where 60 is the experimentally measured boundary 

0 

layer thickness immediately upstream of the shock (without the fin 

effects) at the specified spanwise location. 

the appropriate scaling factor was motivated by the desire to clearly 

The selection of do as 

demonstrate the vertical extent of the interaction relative to the 

height of the incoming boundary layer at the location upstream of the 

fin interaction. Similarly, the profiles are shown at selected 

is the shock and 'shock streamwise locations xs/So, where xs = x-x 

location of the theoretical inviscid shock at a specified spanwise 

location. 

Bogdonoff [59], Dolling and Bodgonoff [60], and Lu and Settles [61] 

indicate that the appropriate scaling is given by Xs = Xs ResO /60. 

It should be noted that the experiments by Settles and 

I 1/3 

Figure 6.27~ shows the comparison of computed pitot pressure 

distribution with the experiment at station 3 which is coincident with 

the line of coalescence. 

study are also included. 

results and the previous experimental and nunerical results are quite 

good. Figure 6.27d shows the comparison of the results at station 4, 

located approximately one-third of the distance between the line of 

coalescence and the fin oblique shock wave at the specified spanwise 

location. The computed and experimental profiles display a modest 

"overshoot" outside the boundary 1 dyer, associated with the compression system ahead of the shock wave. 

The results from the previous nunerical 

In general the agreement between the present 

The comparison of computed pressure profiles with experiment along station 5, located at approximately two-thirds of the distance between the line of coalescence and 
'F 3 

130 

0 EXPERIMENT (Shapoy. 19871 

---- COMPUTED (Knight, ot al. 19871 

- PRESENT 

q.688320 N/m*, T,*260K 

M,=2.93. bo =l.26cm i 

.5 1.0 1.5 2.0 

PP'PP, 

Fig. 6.27(continued) (c) station 3 
131 

4- 

EXPERIMENT IShapey, 1987) 

COMPUTED (Knight, et a1.1987) 

688320 Nlm', T,=260K 

3- PRESENT 

f2- M-12.93, bo z1.26 cm 

60 

1- 

0 I I 1 

1.5 2.0 .5 1.0 

Fig. 6.27(continued) (d) station 4 
132 

the inviscid shock wave at the specified spanwise position is given in 

Fig. 6.27e. The experimental profiles display a slightly S-shaped 

behavior near the wall, which is less apparent in the computed 

profiles at this location. The overshoot in the pitot pressure is 

more pronounced at this location. In both locations, the computed 

profiles of pitot pressure p are in reasonable agreement with the 

experimental data, and accurately predict the observed overshoot in 

P 

. However, the pressure profiles are smeared near the location of pP 

Z/6 = 1.5 due to the shock capturing nature of the algorithm. 

The c a1 c ul ated and experiment a1 pi tot pressure prof i 1 es at 

station 6 are shown in Fig. 6.27f. Due to the close proximity of this 

station to the fin leading edge shock wave, uncertainties in pitot 

pressure measurements were noticed in the experimental data outside 

the boundary 1 ayer edge. 

procedure are needed to resolve this issue. But within the boundary 

layer, reasonable agreement is obtained between the experimental and 

the nunerical results. 

apparent, with reasonable agreement between the experimental and 

nunerical results. The computed pitot pressure profiles at stations 7 

and 8 are compared with experimental data in Figs. 6.279 and 6.27h. 

The computed results are in reasonable agreement with the experimental 

data with the discrepancy appearing in the inviscid portion due to the 

shock capturing nature of the algorithn and the proximity of station 7 

to the shock (Xs = 0.6 60). 

Further improvements in the experimental 

The S-shaped nature of the profile is again 

On the basis of the success of validation of the present 

numerical results with the available data, further results were 

obtained to study the details of the instantaneous and limiting 
133 

c3- 

I 

I 

I- PRESENT 

I I p,=688320 Nlm’, T,=260K 

WI 7 M,=2.93. b0=l.26cm 

.\ 

Fig. 6.27(continued) (e) station 5 
134 

0 EXPERIMENT (Shapey, 19871 

- --- COMPUTED (Knight. et 01,1987) 

- PRESENT 

M,=2.93. bo =1.26cm 

3- 

1- 

1 I 

.5 1.0 1.5 2.0 

Fig. 6.27(continued) (f) station 6 
135 

4- 

3- 

g2- 

b0 

1- 

0 EXPERIMENT (Shapoy, 1987J 

- --- COMPUTED (Knight, ot a1,1987J 

- PRESENT 

p,=688320 Nlm*. T,=260K 

M,r2.93. bo=1.26cm 

I 

.5 1.0 1.5 2.0 0' 

PP'PP, 

Fig. 6.27(continued) (9) station 7 
136 

COMPUTED (Knight, et 01.1987) 

PRESENT 

v. I p-=688320 Nlm’. T,=260K 

M,=2.93. bO=1.26cm 32- 

b0 

1- 

0 J 

.5 1.0 1.5 2.0 

PP’PP.. 

Fig. 6.27(continued) (h) station 8 
137 

streanline patterns. The instantaneous and limiting streamlines 

clearly depict the complex nature of the interaction flow. 

6.28 shows the computed limiting streanline patterns obtained using 

the Baldwin-Lomax turbulence model, The line of coalescence and line 

of divergence are clearly observed. These specific nunerical ly simulated flow structures are in general agreement with the experimental 

results using a kerosene 1 anpbl ack visualization technique. 

In particular, the experimental line of coalescence at Y/6,= 10 (the 

approximate spanwise extent of the experimental kerosene 1 ampbl ack 

visualization) is xs = -5.7 6,, whereas the computed lines of coalescence using the Baldwin-Lomax model occur at Xs = -6.45 60. 

computations performed by Knight et al. [57] using the Baldwin-Lomax 

and Jones-Launder models predict the lines of coalescence at Xs = -4.5 

6o and -3.5 60, respectively. 

Figure 

The 

A series of calculated mean streamlines are displayed in Fig. 

6.29. 

In Fig. 6.29a, the particle traces of 18 streanlines are shown. Six 

streamlines originate from the surface, upstrean of the interaction, 

at equal spanwise increments and serve to define the line of coalescence. Six additional stream1 ines are released imnediately above the 

previous six at a height of 0.0048 6,. 

clearly cross the line of separation and continue to roll downstream 

in the clockwise direction (as viewed from the fin leading edge). 

Another set of streanlines is released at a height of 0.52 6, from tk 

plate surface. 

rotational motion of the fluid particles. The streanlines clearly 

display a vortical structure. 

The vertical scale of the figure has been enlarged for clarity. 

These latter streanlines 

These particles clearly display a counter clockwise 

It is interesting to see that the 
138 

20 - 

15 - 

0 -- 

5- 

0 -- 

REATTACHMENT, 

I I I L I I I I 1 1 

-5 0 5 10 15 20 25 

Fig. 6.28 Limiting streamline patterns on the plate surface for 

20" sharp fin 
139 

Fig. 6.29 Mean streamlines for 200 sharp fin: 

Z/s, = 0, 0.0048 and 0.52 

(a) originating at 
140 

particles originating from the higher z-locations are swept beneath 

the particles originating near the surface. Another set of additional 

streamlines originating upstrean at a height of Z/6 = 1.1 are shown in 

Fig. 

these particles move approximately parallel to the surface. A1 so, the 

stream1 ines in this case exhibit considerable rotation within the 

boundary layer as compared to the case of the 16' sharp fin, due to 

the strong nature of the interaction. 

limiting streanlines on the fin surface. 

tence of separation on the fin surface. 

computed velocity magnitude contours on the flat plate. 

flow features such as separation shock, fin shock, and reattachment 

shock are clearly evident. 

6.29b. Unlike the particles originating at lower values of z, 

Figure 6.30 shows the computed 

This again depicts the exisFigure 6.31 shows the 

The simulated 

6.4 Turbulent Flow Past a Swept Blunt Fin 

For the case of the swept fin induced shock wave/boundary layer 

interaction, comparative results were obtained for the specific conditions of Sec. 5.4. The computations were carried out using a 40 x 

45 x 45 mesh. The pressure distribution along the fin stagnation line 

are compared in Fig. 6.32 for various fin sweep angles. The pressure 

is nondimensionalized by the total pitot pressure behind the bow 

shock, pt2. The agreement between the nunerical results and the 

experimental data is seen to be good except for the zero sweep blunt 

fin where the level of pressure is underpredicted in the boundary 

1 ayer and overpredicted in the inviscid region. The experimental 

measurements [17] indicated that 
I 

141 

FLAT PLATE \ \ 

\ 

\ 

Fig. 6.29(continued) (b) originating at Z/s, = 1.1 
142 

FLAT PLATE / 

Fig. 6.30 Limiting streamline patterns on the fin surface for 20" 

sharp fin 
143 

/? 

/ 4 

SEPARATION SHOCK 

RE AT TACH M EN T 

Fig. 6.31 Velocity magnitude contours on the plate surface for 20" 

sharp fin 
8 

t 

I 

8 

# 

t 

1 

t 

8 

I 

1 

1 

8 

8 

8 

8 

1 

4 

a 

1.6 

1.2 

Z 

0 

- 

.8 

.L 

0 

144 

I SWEEP COYPUTED EXP.(Pdco and 

i 

~~-1983 Nlm'. Tm=90K 

.L .a 1.2 1.6 2.0 

pip t 2 

Fig. 6.32 Comparison of pressure distribution along the fin 

stagnation l i n e f o r various fin sweeps 

9laIWy.. lU7) 
145 

"an irregular pressure distribution does in 

fact occur and the location of peak pressure 

within the boundary layer is a function of sweep 

as well as Mach number, Lack of data in the 

region near the wall prevents identifying a peak 

for any of the other models with this diameter. 

more complete experimental investigation would 

have to be performed to define, for a certainty, 

the flow model responsible for this irregular 

A 

pressure distribution." 

A close exanination of the figure reveals that as the fin sweep from 

the angle increases, the magnitude of the peak pressure decreases 

considerably from 1.2 to 0.2. The variat on in pressure from the 

position of peak to minimun is substantia ly reduced by increasing fin 

sweeps. 

surface from the position of minimun is not seen for fin sweep angles 

greater than 30'. 

stagnation line (Fig, 6.33) also shows some improvements due to the 

sweep. 

The increase in pressure in the direction toward the plate 

The computed temperature distribution along the fin 

The comparison of pressure distribution on the flat plate along 

the line of symnetry (Y/O = 0) and off the line of symnetry (Y/D = 1.5 

to 4.5) are illustrated in Fig. 6.34. Again the agreement is seen to 

be very good. 

features such as upstrean influence, pressure rise due to flow separation from the flat plate, pressure plateau and double peaks in pressure off the line of synmetry. 

reveal a low pressure region between the pressure plateau and the 

The code is able to simulate all the pertinent flow 

Further exanination of the results 
146 

SWEEP 0' 

8'75 [ 1 30' 

5*25i 3.50 

Z 

0 

- 

p,=1983 Nlm'. T,=SOK 

M,=3.71, b=Scm 

I I I I 

.a .9 1.0 1.1 1.2 

Fig. 6.33 Temperature distribution along the fin stagnation 1 ine 

for various fin sweeps 
4 

8 

8 

8 

8 

8 

8 

I 

I 

I 

8 

I 

1 

I 

8 

I 

8 

8 

8 

p,=1983 Nlm’, T,=90K 

M,=3.71. b=5cm 

1 1 1 1 1 1 1 1 JO 

-4 -3 -2 -1 0 

x IO 

Fig. 6.34 Comparison of pressure distribution on the flat plate 

for various fin sweeps: (a) along Y/D = 0 

147 
148 

final compression along the line of synmetry (Fig. 6.34a). The low 

pressure observed is due to the existence of the reversed high speed 

flow region. However, for sweep angles greater than 30' the low pressure region is not seen. The disappearance of the low pressure trough 

region for high fin sweep angles demonstrates the weakening of the 

main horseshoe vortex with the sweep. Further examination of the 

results i 11 ustrates that the sweep considerably decreases the upstrem 

influence. The appearance of double peaks in pressure (Figs. 6.34b6.34e) on the flat plate is due to the primary horseshoe vortex moving 

downstrean in a helical fashion. The low pressure region between the 

two pressure peaks is caused by a high velocity under the core of the 

vortex [31]. It is seen that the sweep decreases the spanwise extent 

of the interaction considerably. 

double pressure peaks for higher sweeps. Some of the trends exhibited 

in Fig. 6.34 will become apparent after discussions of the flow structure. The computed surface temperature distribution along the line of 

spmetry is shown in Fig. 6.35. The improvement in the surface temperature is clearly apparent. 

This is due to diminishing of the 

The computed particle paths in the plane of spmetry are shown in 

Fig. 6.36 for blunt and 60' swept fins. 

flow separation on the flat plate ahead of the fin, leading to the 

formation of the primary horseshoe vortex (Fig. 6.36a). A small 

secondary vortex is also seen near the juncture formed by the fin and 

flat plate (Figs. 6.36a and 6.36b). 

counterclockwise direction as compared with the primary horseshoe 

vortex. 

ahead of the blunt fin with its core located about 0.2 diameters above 

The results clearly show the 

The secondary vortex rotates in a 

The primary vortex elongates to a length of two diameters 
SWEEP COMPUTED EXP.(Prlco and StaIUngo, lW7) 3.5 r 

3.0 

p,=1983 Nlm'. T,=90K 

M,=3.71. b=5cm 

2.5 

2.0 

1.5 

1.0 

t ' 0.5 1 1 1 1 1 1 1 1 1 1 1 J 

-1.5 0.0 1.5 3.0 1.5 6.0 7.5 

Fig. 6.34(continued) (b) along Y/D = 1.5 

149 
I 

I 

8 

I 

8 

8 

I 

I 

8 

8 

I 

I 

I 

1 

8 

1 

8 

1 

8 

- 

I I I I I I I I I I I I 

150 

3.5 r 

SWEEP COMPUTED EXP.(Prlco mnd Strlllngs. 1067) 

p,=1983 Nlm’, T,=90K t 

M,=3.7 1. b=5cm 2.5 1 

t 

! 2.0 

p, 

1.5 

1.0 

L m 

XI0 

Fig. 6.34(continued) (c) along Y/D = 2.5 
I 

I 

8 

I 

111 

8 

I 

8 

I 

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151 

3.5 

2.5 

1.5 

1.0 

p,=1983 Nlm'. T,=90K 

M,=3.71. b=5cm 

t 

0.5 I I I I I I I I I I I I 1 

-1.5 0.0 1.5 3.0 L.5 6.0 7.5 

XCD 

Fig. 6.34(continued) (d) along Y/D = 3.5 
- 

Fig. 6.34(continued) (e) along Y/D = 4.5 

152 
153 

- SWEEP 0' 

p,=1983 Nlm'. T,=90K 

M,=3.71. b-5cm 

-L.O -3.0 -2.0 -1.0 0 

Fig. 6.35 Temperature distribution along the line of symmetry for 

various fin sweeps 
I I I I I I I I 1 

-8 -6 -4 -2 0 

X/D 

Fig. 6.36 Particle paths in the plane of symmetry for various 

fin sweeps: (a) sweep = 0" 

154 

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STAGNATION LINE 
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Fig. 

155 

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STAGNATION LINE - 

PRIMARY VORTEX SECONDARY VORTEX 

LLINE OF SYMMETRY 

6.36(continued) (b) enlarged view of the flow at the junction for 

sweep = 0" 
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the flat plate. As the fin sweep is increased to 60', the extent of 

the separation on the flat plate is reduced considerably (Fig. 6.36~ 

and Fig, 6.37). This leads to the weakening of the horseshoe vortex 

(Fig. 6.36~). 

The computed skin friction distribution along the line of synmetry is illustrated in Fig. 6.37 for various fin sweeps. It is seen 

that with an increase in the fin sweep, the upstrem separation point 

(where Cf value changes sign) moves closer to the fin leading edge. 

Also, for fin sweep angles greater than 0', no definite reattachment 

point (where the Cf value starts increasing from a negative value) is 

observed. 

The horseshoe vortex transports fresh high momentun fluid continuously into the separated flow region (Fig. 6.36a). 

momentum fluid accelerates and moves away from the plane of synmetry, 

it leads to the formation of two reversed supersonic zones [31, 62, 

631, one on the fin and the other on the flat plate. The decrease in 

pressure from the peak to a rninimun observed in Fig. 6.32 is not only 

due to a lower stagnation pressure in the incoming boundary layer 

profile, but also due to the extreme expansion caused by the horseshoe 

vortex. This expansion leads to the formation of a reversed supersonic zone on the fin. From the computed particle paths in the plane 

of synmetry for the blunt fin (Fig. 6.36a), it is seen that the 

streamlines of low momentun fluid never impinge directly on the fin 

surface. 

As this high 

They spiral into the horseshoe vortex and continue to roll 

up downstrean in the form of a necklace vortex. The increase in pressure in the direction toward the plate surface observed in Fig. 6.32 

is due to recompression of the flow from supersonic to subsonic speed. 
157 

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-6 --L -2 0 

XlO 

STAGNATION LINE 

Fig. 6.36(continued) (c) sweep = 60" 
Fig. 

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158 

45. 

60. 

6.37 Skin friction distribution along the 

for various sweeps 

Jo.04 

1 ine of symmetry 
159 

It is this adverse pressure gradient that leads to the formation of 

the secondary vortex at the corner (Fig. 6.36b). 

angle increases, the variation in pressure is reduced substantially. 

This weakens the horseshoe vortex which leads to the disappearance of 

the supersonic zones [62, 631. 

As the fin sweep 

The computed limiting streamline patterns are shown in Fig. 6.38 

for blunt, 30' and 60' swept fins. The separation and reattachment 

lines are also marked in the figure. The foot trace left by the 

horseshoe vortex as the flow goes around the fin is clearly observed 

in the area bounded by the separation and reattachment line. It is 

apparent that the spanwise influence exerted by the horseshoe vortex 

on the juncture flow is reduced considerably for high sweep angles. 

In particular at a sweep angle of A = 60', the flow beyond the spanwise location, Y/D = 3.0 (Fig, 6.38~) is unaffected by the juncture 

effect. 

the disturbed flow persists up to Y/D = 6.0 (Fig. 6.38a). 

However, in the case of the blunt fin (A = O'), the extent of 

6.5 Turbulent Flow Past a Filleted Fin 

For the case of the filleted fin induced shock wave/boundary 

1 ayer interaction, comparative results were obtained for the specific 

conditions given in Sec. 5.5. 

a 40 x 45 x 45 mesh. The pressure distribution along the fin stagnation line is compared in Fig. 6.39 for unmodified (blunt fin) and 

modified (filleted) junctions. The fin surface pressure is again 

nondimensionalized by the total pitot pressure behind the bow shock 

(p, = 0.12 pt2). 

point of impingement of the stagnation streamline on the fin leading 

The computations were carried out using 

The location of the peak pressure corresponds to the 
160 

'0 

R EAT TACH M ENT 1 

-8 

-6 

--L 

-2 

SEPARATION \ / 

I I I I I tlN 

Fig. 6.38 

-8 -6 -4 -2 0 

X/D 

Limiting streamline patterns on the flat plate for various 

fin sweeps: (a) sweep = 0" 
161 

7 RE ATTACH M EN T 

SEPARATION 7 

8 

6 

2 

0 

1 I I I 1 

-8 -6 -4 -2 0 

Fig. 6.38(continued) (b) sweep = 30" 
162 

-8 

-6 

Y 

-c 0 

-2 

-0 

REATTACHM EN1 

S EPA R All 0 N 

I I I I I 

-8 -6 -4 -2 0 

X/D 

Fig. 6.38(continued) (c) sweep = 60" 
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UNMODlFl ED JUNCTION 

CIRCULAR FILLET 

(b/D = R / D = 2 1 

CIRCULAR FILLET 

(b/ 0 = 2, R / 0 = 3.5 I 

(b/D=2) 

-0- CIRCULAR FILLET 

w WD=Q 2. R/D=3.5) I 

/- 

0.0 I I I I I I I I 1 

.25 .so .75 1.00 1.25 

Fig. 6.39 Pressure distribution on the fin leading edge for various 

junctions 
164 

edge. 

leading edge moves upward for the filleted junction. 

the flow approaching the junction is deflected upward instead of being 

rolled into the horsehose vortex. For the case of the small boundarylayer thickness (6/D = 0.2), the results illustrate a considerable 

increase in fin leading edge pressure distribution as compared to the 

unmodified junction. Similar trend in results were observed in blunt 

fin/flat plate studies [31] in which decreasing the incoming boundary 

layer thickness by a factor of five showed a considerable increase in 

the pressure distribution on the fin leading edge. 

It is seen that the location of the peak pressure on the fin 

This is because 

The pressure distribution along the line of synmetry are compared 

in Fig. 6.40 for various types of junctions. 

pressure drop, following the pressure plateau, decreases as the fillet 

size increases. This indicates a reduction in speed of the reversed 

flow region and overall acceleration of the juncture flow. 

It is seen that the 

The computed particle paths in the plane of symnetry are shown in 

Fig. 6.41 for unmodified and modified junctions. The influence of 

filleting on the flow structure is dramatic as compared to unmodified 

junction. 

deforms and the flow streamlines within the boundary layer lose much 

of their vortical character with proper filleting. In another study 

[32], in which the leading edge shape was changed from a blunt to a 

square fin, the results showed a considerable increase in the separation length. This resulted in a bifurcation of the main horseshoe 

vortex into two vortices rotating in the same direction. 

present case, the main horseshoe vortex does not bifurcate even though 

the bluntness of the leading edge is changed by filleting. 

The particle paths demonstrate that the horseshoe vortex 

For the 

The 
- UNMODIFIED JUNCTION 

(bl0=2) 

CIRCULAR FILLET 

IblO = R / 0 = 2 I 

--- CIRCULAR FILLET 

(bID=2, R1013.5) 

(b/O=Q 2. RI0=3.51 

----- 

-0- CIRCULAR FILLET 

165 

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/ 

i p,= 1580 N/m2, TI= 138K 

P 2.50 - 

p. 

-8.0 -6.0 -4.0 -2.0 0 

Fig. 6.40 Pressure distribution on the flat plate for various 

junctions 
166 

tI 46 

2 

0 

-8 -6 --L 

X/D 

-2 0 

Fig. 6.41 Particle paths in the plane of symmetry for various , junctions: (a) unmodified junction (s/D = 2) 
167 

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-8 -6 -4 -2 0 

8 

6 

2 

2 

0 

XID 

Fig. 6,4l(continued) (b) circular fillet (s/D = 2, R/D = 2) 

1 

8 

1 
-8 -6 -4 -2 0 

XI0 

Fig. 6.4l(continued) (c) circular fillet (s/D = 2, R/D = 3.5) 

168 

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I I I I I I I I I 

-a -6 -4 -2 0 

Fig. 6.4l(continued) (d) circular fillet (s/D = 0.2, R/D = 3.5) 

169 

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surface limiting streamlines are shown in Fig. 6.42 for unmodified and 

modified junctions. It is apparent that the separation on the fin 

surface is prevented completely in the case of the modified junction 

as compared to the unmodified junction. 

stremlines are shown in Fig. 6.43 for unmodified and modified 

junctions employing circular and continuous fillets. These results 

show the influence of fillets on the helical vortex structure. The 

computed velocity magnitude contours (Fig. 6.44) clearly show that 

with filleting the supersonic zones observed in the case of the blunt 

fin/flat plate junction weaken as the fillet size is increased at the 

juncture. The vorticity contours (Fig. 6.45) demonstrate that the 

magnitude of vorticity in the vortex is reduced by a factor of three 

in the case of the modified junction as compared to the unmodified 

j unction. 

The computed instantaneous 
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LU 

FLAT PLATE 

Fig. 6.42 Surface limiting streamlines for various junctions: 

(a) unmodified junction (S/D = 2) 

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/ 

FLAT PLATE 

Fig. 6.42(continued) (b) circular fillet (s/D = 2, R/D = 2) 
173 

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1 

I 

Fig. 6.42(continued) (c) circular fillet (s/D = 2, R/D = 3.5) 

P 

8 
174 

FLAT PLATE 

Fig. 6.42(continued) (d) circular fillet (s/D = 0.2, R/D = 3.5) 
17 5 

CTURE HORSESHOE VORTEX 

Fig. 6.43 Instantaneous streamlines for various junctions: 

(a) unmodified junction (s/D = 2) 
Fig. 6.43(continued) 

176 

’\ FLAT PLATE / 

(b) circular fillet (s/D = 2, R/D = 2) 
177 

Fig. 6.43(continued) (c) circular fillet (s/D = 2, R/D = 3.5) 
178 

FLAT PLATE 

Fig. 6.43(continued) (d) circular fillet (s/D = 0.2, R/D = 3.5) 
179 

Fig. 6.43( 

\ JUNCTURE HOR 

FLAT PLATE 

continued) (e) continuous fillet (s/D = 2) 

c - 3 
180 

- 

- 

- 

- 

I SI 

-8 

-6 

Z 

--c 

-2 

-0 

ERSONIC ZONES 

-8 -6 -4 -2 0 

xi0 

Fig. 6.44 Velocity magnitude ccmtours in the plane of symmetry 

for various junctions: (a) unmodified junction (s/D = 2) 
I BOW SHOCK--I 

STAGNATION LII 

I 

I SEPARATION SHOCK1 I 1 I 1 

-6 -4 

x/u 

I I I I 

-2 0 

-8 

181 

2 

0 

Fig. 6.44(continued) (b) circular fillet (s/D = 2, R/D = 2) 
182 

I I I I I I I 1 I 

-4 -2 0 

X/D 

-8 -6 

Fig. 6.44(continued) (c) circular fillet (s/D = 2, R/D = 3.5) 
183 

2 i 0 

I I I I I I I 1 1 

-6 -4 -2 0 -8 

X/D 

Fig. 6.44(continued) (d) circular fillet (6/D = 0.2, R/D = 3.5) 
184 

7 STAGNATION LINE 

I I I 1 I I I I I 

-8 -6 -4 -2 0 

X/D 

Fig. 6.45 Vorticity contours in the plane of symmetry for various 

functions: (a) unmodified junction (s/O = 2) 

-8 

- 

-6 

- 

Z 

-c - D 

- 

-2 
' -6 I 

z 

STAGNATION LINE LE 

-8 -6 --L -2 0 

XI0 

Fig. 6.45(continued) (b) circular fillet (S/D = 2, R/D = 2) 
1 

I 

8 

STAGNATION LINE 1 

-8 -6 --L -2 0 

XI0 

Fia. 6.45(continued) (c) circular fillet (6/D = 2, R/D = 3.5) 

186 
STAGNATION LINE 1 

-8 -6 --L -2 0 

x10 

Fig. 6.45(continued) (d) circular fillet (s/D = 0.2, R/D = 3.5) 

187 
Chapter 7 

CONCLUSIONS AND RECOMMENDATIONS 

The supersonic flow past a wing/body junction is simulated 

numerically using a three-dimensional Navier-Stokes code. Results 

have been obtained for simplified geometries such as blunt fin/flat 

plate junction, sharp fin/flat plate junction, and filleted junction. 

Extensive parametric studies of grid resolution, Mach and Reynolds 

number variation have been conducted to understand their effects in 

modifying the interaction flow field. The influence of fillets and 

sweep on the computed flow field have been investigated extensively 

for a wide range of geometrical and flow conditions. 

computational results show that better agreement with experimental 

results can be obtained by refining the mesh. 

the fin sweep reduces considerably the spanwise extent of the interaction flow field. Some improvements in surface temperature are also 

observed due to sweep. The computed particle paths and limiting 

streamlines for swept junctions demonstrate clearly the dramatic 

influence of fin sweep in weakening the juncture horseshoe vortex. 

The results for sharp fin/flat plate junction show that the structure 

of three-dimensional separation is markedly different from that exhibited by the blunt fin/flat plate junction. 

filleted junctures clearly show that the flow streamlines lose much of 

their vortical character with proper filleting. 

strated that fillets with a radius of three and one-half times the fin 

In general, the 

It has been noted that 

The numerical study of 

It has been demon188 
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189 

leading edge diameter are required to weaken the usual necklace vortex 

interaction for the Mach number and Reynolds number considered in the 

present study. 

Based upon the present study, the following areas of research may 

be identified for further studies: 

1. The sensitivity of grid refinement on the computed flow 

structure should be investigated using a much finer grid than 

used in the present study to define the minimum nunber of 

grid points required for an accurate resolution of flow 

features. 

2. The simulated structure of the shock is smeared due to the 

nature of the algorithm applied in the present study. 

nunerical technique, such as a high resolution TVD scheme, 

can be incorporated in the code for providing a crisp representation of the shock. 

The flow structure should be examined for physical conditions 

when there are four or more vortices present in the flow 

field. 

The effect of changes in the turbulence model, such as a 

nonlinear k- model, can be examined to investigate its 

effect on the flow field simulation. 

Since the flow is unsteady, one should investigate the oscillatory behavior of the horseshoe vortex. 

It would be desirable to examine the chemistry and real gas 

effects using appropriate mathematical models. 

A 

3. 

4. 

5. 

6. 
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47, 

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17993-03, October 1987. 
APPENOIX A 

MAXIMUM ALLOWABLE TIME STEP FOR EXPLICIT ALGORITHM 

An approximate linearized stability analysis [51] yields the 

following: 

= CFL .min At l,J,k 5 At 

~CFL 

= CFL .min i,j,k Atrl At 

CFL 

At = CFL .min At CFL 1,J,k 5 

min where i,j,k denotes the minimum value evaluated over all grid 

volumes which are updated by the explicit scheme, and 

- 

195 
I 

I 

I 

8 

1 

8 

I 

I 

I 

8 

1 

1 

8 

8 

I 

8 

I 

1 

8 

and 

u = (u E, + v cy + w Ep Jt,2 +E Y + 5 2 E 

196 

04.4) 

(A.5) 

u 5 = (u 5, + v cy + w 5p 4- 

and 

The constant CFL is typically chosen to be 0.9. 
I 

I 

I 

8 

I 

I 

1 

8 

I 

8 

1 

1 

I 

1 

D 

8 

t 

8 

8 

AUTOBIOGRAPHICAL STATEMENT 

The author was born on August 27, 1957 in Madras, India. He 

received his Bachelor’s degree in Mechanical Engineering from Bhopal 

University, India in August 1979. He came to the United States of 

America in August 1980 to pursue higher education. 

Master’s degree in Mechanical Engineering from Old Dominion University 

in May 1983. 

Mechanical Engineering at the Old Dominion University. During his 

graduate studies, the author was involved in research work pertaining 

to applied computational fluid dynamics sponsored by the Naval Air 

Development Center and the NASA Langley Research Center. 

He graduated with a 

In August 1983, he enrolled in the Doctoral program in 

197