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1968

The numerical solution of two-dimensional fluid flow problems The numerical solution of two-dimensional fluid flow problems

E. David Spong

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THE Nt~ERICAL SOLUTION OF

TWO-DIMENSIONAL FLUID FLOW PROBLEMS

BY

EDWARD DAVID SPONG - I Cf 3 8

A

THESIS

submitted to the faculty of

THE UN-IVERSITY OF MISSOURI AT ROLLA

T~o/o Cl ~s-Mx.i ..

in partial fulfillment of the requirements for the

Degree of

MASTER OF SCIENCE IN PROPULSION AND SPACE ENGINEERING

Rolla, Missouri

1968

f) 1/ If Approved by

/(~ Jt ~~(advisor) ---------

ii

ABSTRACT

A numerical solution method is developed for the solu­

tion of two-dimensional, irrotational and compressible

fluid flow problems. The partial differential equation, in

terms of the velocity potential, describing the flow is re­

placed by finite difference equations and the resulting

matrix is solved by Gaussian elimination.

The method is successfully applied to two subsonic

o a 6o flow problems:· a 7.5 wedge and wedge inlet. The

method becomes invalid, as expected, with the appearance of

sonic velocity in the flow field.

An investigation of the definition of the singu­

larities is made. This indicates that the best agreement

~rith the experimental results for the same problem is ob­

tained when the flow directions at the singularities are

assumed to be equal to that of the wedge.

Methods are postulated to remove the restraints asso-

ciated with a limited field size by replacing the boundary

values, after the initial solution, with values extrapolated

from the flow field.

iii

ACKNOvJLEDG EMENT

The author wishes to express his appreciation for the

help and encouragement of Dr. R. H. Howell in the pre­

paration of this thesis.

iv

TABLE OF CONTENTS Page

ABSTRACT •••••••••••••••••••••.••••••.•••••.•••••.••.••• •·•·•. ii

ACKNOvJLEDGEMENT •••••••••••••••••••••••••••••••••••••••• iii

LIST OF FIGURES ••••••••••••••••••••••••••••••••••••••••• v

LIST OF SYMBOLS ............................... ·-· ................ .. vii

I. INTRODUCTION ................................. •·•·•. •. 1

II. NUMERICAL METHODS •.• • • • • • • • • • • • • • • • • • • • • • • • • • •.• •.• •. 4

III. PROBLEM ANALYSIS •••••••••••••••••••••••••••••••••· 8

IV. SOLUTION TECHNIQUE ••••••.•.••••••••••••••••• •·•·•·•.. •. 11

V. DISCUSSION AND RESULTS ••• •·•..................... •. 15

VI. CONCLUSIONS •••••••••••••••••••••.•••••••••••.••• •·• ... 22

VII. RECOMENDATIONS •.•••••••••••••• • •••••••••••••• • • • •.. 23

BIBLIOGRAPHY ••••••••••••••••••••••••••••••••• •·•.. 24

APPENDIX I. DERIVATION OF FINITE DIFFERENCE EQUATIONS • • • • • .• • •.• • • • • • • • • • • • • • • • • • • • 26

APPENDIX II. LISTING OF COMPUTER PROGRAMS •·• •••• •·• 29

APPENDIX III. MODIFIED GAUSSIAN ELIMINATION •••••• 50

VITA ......................................... •· .............. . 75

v

LIST OF FIGURES

Page Figure

1. Details of Flow Field Net ................... •·•·•. 51

2.. Details of Flow Field Net......................... 52

3.

4 •.

5.

6.

8.

9.

10.

Flow Field ........................................... . 0

Details of 7.5 ~edge Test Model •••••••••••••• 0

Pressure Distribution on a 7.5 Wedge at M0 = 0.657. The Effect of Singul-arity Assumptions .............................. .

Mach Number and Pressure Distribution on a 7. 5° Wedge at M0 = 0.657 .................. .

Mach Number and Pressure Distribution on a 7. 5° vJedge at M0 = 0.705 ..................... .

Mach Number and Pressure Distribution on a 7.5• Wedge at M0 = 0.768 ••••••••••••••••••

Mach Number and Pressure Distribution on a 7.5° v..;edge at Mo= 0 • .817. •••••••••••••·····

Mach Number and Pressure Distribution on a 7.5° Wedge at M0 = 0.860 ................. ••·••

0

53

54

55

56

57

58

59

60

11. Pressure Distribution on a 7.5 Wedge ••••••••• 61

12.

13.

14.

15.

16.

0 Pressure Coefficient on a 7.5 Wedge at x/C•0.9.............. •• • • • • • • • • • • • • • • • • ...... 62

Pressure Distribution on a. 7.5• Wedge at M0 = 0.657. The Effect of Boundary Layer..... • • • • • • • .. • • • • • • • •.• • •. •. • • • • • • • •.• • • • • • • • 63

0 Pressure Distribution on a 7.5 Wedge at M0 = 0.657. The Effect of Grid Size •••••••• 64

0 Pressure Distribution on a 7.5 Wedge at Mo = o. 657. The Effect of Singularity Assumptions.................................... 65

0

Velocity Profile along 7.5 Wedge at Mo = 0 • 6 57 • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • ·• ·• • • 66

17. Flow Angle Profile across Field ••••••••••••••• 67

18. 0

Details of 6 Wedge Inlet Model.................. 68

vi

19. Pressure Distribution on the Wedge of a 6o Wedge Inlet at M0 = 0.8 •••••••••••••• •·•... 69

20. Grid Netv.Jork A. 7. r) V.edge with n= 6 m=4. • • • • • • • • • • • • • • • • • • •.• • • •. • • • • • • • • • • • • • • • • • • • 70

21. Grid Netv.rork B. 7. r) vJedge with n=l6 m=l4. Six Grid Points on Wedge ••••••••••••••. ~ 71

22. Grid Network C. 7.5° ~edge with n:l6 m=l4. Eight Grid Points on Wedge ••••••••••••• 72

23. 0

Grid Network D. 7.5 ~edge with n•l6 m=l4. Undefined Singularities......... • • • • •. • • 73

24. Grid Network E. 6° Wedge Inlet with n•l6 m=l4................................... ... . . ... . 74 25. Augmented Matrix •••••••••••••••••••••••••••••• 50

SYMBOL

A

c

c

Cp

K

M

m

n

p

q

R

r

u

v

v

x,y

0

~

g

f

4> "t

LIST OF SYMBOLS

DESCRIPTION

area

wedge chord

acoustic velocity

coefficient of pressure= ( P-Po )/q0

grid length parameter

Mach number

width of field

length of field

pressure

dynamic pressure= lSpM' /'2

density parameter

radius

velocity component in x-direction

velocity

velocity component in y-direction

coordinates

ratio of specific heats

grid spacing

flow direction

mass density

velocity potential

stream function

V11

UNITS

ft.,.

ft

ft/sec

. lb/irl"

lb/irf

ft

ft/sec

ft/'sec

ft/sec

ft

ft

degrees 3 slug/ft

2 ft /sec

2 ft /sec

SUBSCRIPTS

I

1

in

n

0

t

x,y

FORTRAN SYMBOL

Al

A CAPT

c

CP

C02

CTS

DYL

F~RR

Il

I2

IB

I1·1AX

IPR

KBOUND

)

DESCRIPTION

final

intial

integer assigned to grid point

inlet

length of field

denotes free stream conditions

"1 i V1._

denotes total or stagnation conditions

denotes partial differentiation in the corresponding direction

LIST OF' C01-1PUTER SYMBOLS

ALGEBRAIC SYNBOL OR DEl?INITION

inlet area

inlet capture area

coefficient of matrix

Cp

percent error

boundary definition

print indicator

density derivative indicator

FORTRf~N SYMBOL

LOOP

MO

MFR

M11AX

NMAX

p

PO

PHil

PHI2

PHIX

. PHIY

PTO

R

RO

RRX

RRY

T

-Tl

TO

THETA

V,'Y

vo .

VSQ

y

Yl

ALGEBRAIC STI1BOL OR DEFINITION

iteration count

mass flow ratio

m

n

R~

wedge angle

lip angle

ambient temperature

v

location of wedge leading edge

location of lip leading edge

1x

Ii'ORTRAN SYHBOL

Y2

z.

ALGEBRAIC SYHBOL O:R DE.liNITION

location of shoulder of lip

p

X

I .. INTRODUCTION

The study of compressible fluid flow problems, with

their complex equations and complicated boundary con­

ditions, does not generally lead to analytic solutions.

1

It is evident that in order to obtain a solution some sim­

plifying assumptions must be made. The method of small

perturbations (linearization) (1, chapters 10 and 14)*

provides a relatively simple but approximate solution which

can be applied to both the low subsonic (Mo < 0.6) and to

the supersonic (Mo> 1.5) flow regimes. The method of

characteristics (1, chapter 15) which requires graphical

or numerical calculations, provides an exact solution for

supersonic flow. In the high subsonic and transonic flow

regimes, the solutions that exist are generally for simple

body shapes (2,3,4 ). The complexity of the differential

equations defining the flow in this region indicate that

general analytic solutions will not be found in the near

future.

The relaxation solution technique developed by

R. V. Southwell (5,6) offers complete generality in the

solution of difficult problems involving compressible,

adiabatic and frictionless flow. H. W. Emmons (7) extended

this relaxation technique to mixed flows of both subsonic

and supersonic flow with shock waves. With the advent of

the high-speed, large memory digital computer, the direct

* Denotes references listed in the Bibliography

solution of large order matrices became possible and thus

the actual relaxation of the constraints is no longer

necessary.

2

The solution may be obtained in terms of the velocity

potential or the stream function. The solutions in this

paper ~'ere found in terms of the velocity potential while

Emmons (7) chose the stream function. The velocity po­

tential has the advantage in that three-dimensional pro­

blems may be analyzed but is limited to irrotational flow.

The stream function is restricted to two-dimensions, but

the flo\\' can involve shocks and rotation.

The general solution technique is to obtain a direct

solution of the velocity potential field for an assumed

density field (intially assumed incompressible or some

'guessed' solution if known). A new density field is then

calculated from the velocity potential field.. Using this

density field, a new solution to the velocity potential

field is obtained. The density field is iterated by re­

peated solutions of the velocity potential field until it

converges, whence the final velocity potential solution is

the solution to the given problem.

A general solution technique "'as developed and applied 0

initially to a 7.5 two-dimensional wedge. Experimental

results for this problem are presented by Bryson and

Liepmann (8,9) who also present theoretical results from

(2,3,4). With the experience gained from the wedge study,.

the solution technique was applied to a two-dimensional

ramp inlet. Experimental data for this was obtained from

(10). The solution technique was programmed for the IBM

System 360/50 computer in Fortran IV language. Listings

of the various programs are presented in Appendix II.

3

4

II. NUMERICAL METHODS

R. V. Southv,ell (11) introduced in 1935 an original

and unique approach to physical and engineering cal­

culations. Devised originally for the computation of

stresses in braced frameworks, the notion of 'systematic

relaxation of constraints' (later to be called ''The Re­

laxation Method 11 ) was seen to have much wider applications.

This technique was later formalized (5) for all types of

structural problems, and has since been applied to many

other engineering problems {7,12,13,16,17,19,20).

Emmons (7) in 1944 applied the relaxation method to

compressible, rotational, and inviscid fluid flow problems.

This brilliant paper (7) is the fundamental for all nu­

merical solutions to fluid flow problems. His general

technique is to initially obtain a solution to the given

problem in terms of the stream function but assuming an

irrotational and incompressible flow field. The Cauchy

Riemann equations are then used to calculate the velocity

potential •. The co-ordinates ~,.~for the incompressible

flow field are then used as the co-ordinate system for the '

compressible case thus avoiding the difficulties of com­

plicated boundaries, since they become straight lines in

the transformed plane.

The theoretical results obtained by Emmons on a NACA

0012 airfoil {13) were compared with experimental results

{12) sho¥ring very good agreement for low Mach numbers but

at higher Mach numbers the theory predicts higher negative

5

pressures than actually existed. Amick ( 12) suggested~ that

this poor agreement may be due to neglecting the boundary

layer or to limitations of the relaxation method.

Emmons' method has been sucessfully applied to various

problems (7,13,14), but as he points out (13), "Although

the relaxation method appears to be adequate to solve the

very involved differential equations and boundary con­

ditions describing the flo\<r of a compressible fluid, the

calculations are too involved to permit the investigation

of a very wide range of interesting cases without the use

of high speed calculating machines". The increase in avail­

ability of digital computers in recent years has meant that

relaxation solutions are no longer obtained by "hand"

methods, and although some of Emmons' techniques are not

suited to machine use, his basic approach is still sound.

There are many references (15,16,17,18,19,20) to

numerical solution techniques but invariably the subject

matter is concerned with the actual mathematical technique

for solving the set of linear difference equations rather

than the technique of setting up the difference equations~

Most flow problems have boundary surfaces with abrupt

changes in slope which cause the major difficulty in

setting up the difference equations because they appear as

singularities (discontinuites) to the numerical solution.

Fox (17) notes, "A favored but somewhat inelegant method of

coping with such a discontinuity is effectively to ignore

it, and to mitigate its affect by using a small interval

in a region near the point of discontinuity". Fox also

discusses some methods that have been used to cope with

singularities occuring in the solution of Laplace's

Equation. He concludes by stating, "This and other work

6

on singularities bas concentrated on special problems which

arise in scientific contexts, and no general theory seems

to be availableu.

The best solution technique for a given set of linear ~

equations depends on the form of the equations to be

solved. Polacbak (18) and Poole (20) discuss various

methods in relation to particular equations, together with

the optimum technique for various problems in terms of the

speed of convergence to a given solution. The two basic

methods of solution are termed direct and indirect.

Gaussian elimination (21) is typical of the direct type

whereby a solution is obtained directly and is accurate to

the extent-of the round-off and significance errors. There

are numerous indirect methods but probably.the best known

is the Gauss-Siedel method (21). In this method, each

unknown is made the subject of the equation defined at its

particular grid point. The solution is initially guessed

and then improved by sequential substitution. Even if the

process converges only an approximate solution can be ob-

tained by this method and the accuracy of the solution is

determined by the numbe~ of iterations. The solution of

the equations for this paper was obtained with Gaussian

elimination modified to conserve inversion time and storage

space. For problems with a larger number of unknowns (in

excess of 300) and a similarly sparse matrix, an iterative

method would be preferable provided .that convergence could

be established.

7

8

III. PROBLEM ANALYSIS

The analysis is limited to two-dimensional flow al­

though three-dimensional solutions are possible using the

velocity potential. Further studies will develop a method

for the solution of three-dimensional flow problems from

the method presented below.

The differential equation, in terms of the velocity

potential for two-dimensional,. irrotational, inviscid,

adiabatic, frictionless, compressible, steady flow (1),

takes the form:

<?~x. + <\>~~ + ~ rs~t~ -+ <P~ ~~I~ - o (1)

The density ~ is related to velocity potential ~ by: I

Sl'?t:. : [l-tO'-l)( <?~ + <P:)/zc~)-z> .. , (2)

The flow field t,o be analy~ed is covered by an orthogonal

grid and each intersection (grid point) is assigned a num­

ber. The values of cb, ~ ,V and 9 at each point are assumed

to apply to the area surrounding the point.(Figure la).

Equation (1) is expressed in finite difference form for the

following boundary conditions. The derivation of these

equations is given in Appendix I.

1. Central grid point.

a) Variable grid spacing.

<\>i.J2./[f>iT' l&,., t5,_,))4-~""} + ~i-\{2./Lb,_, (bii't ..- &, .. ,)- Rl(, ~ +

q>,,.W\ i_ 2./ [ ~i....,. l &,.~ +O,.ft)]+ 12~1-t d>i-~ [ z. /[ &i-~ ( b•~~ + ~i .. ~)-~ 1-r ~i i -2./( hi,. •. b;.,)-Z/(&, .. "'.bi .. ft)~ =o (3}

where

t<x. = t ~ '+l -~,_,)I [ ~' ( bi-' + &i ... \)1.]

R~ = { ~i+~- 'i$1-Y\)/[ ~\ ( Oi*i'\ 4- ~i-~t]

b) Constant grid spacing. <&=1)

q,,+, [• +R:1C.] + d>i-•[l-l<x.] + d>i+.,[l+~~J +

<t>i - ~ U - R~] + (\>·, [- 4J ~ 0

R1 = t ~'+' - ~-,_,)I l 4 ~;)

R~ = L qi+W\ - ~'._"' )/ ( 4s>i)

2. Solid boundary. Figure l(b).

a) e> 0

<Pi+t [K,] + lbi""'[K1./&~-~-K,] + q,i~i'\[.1 fbi+n1 +

(4)

( 5)

( 6)

<Pi~-~[u-~1.)/bi ... ,1 +~i L-'/&,~,-'/oi+~1 = o (7>

K, = (cos1.e t '/&i-1 +R1)- S\~ec.ose[,/Oitv,-k?~j]/(&i-\-4-bi)

K'z. = ~i4-v. If>''"'~

Q,. = [S>\- ~;-,.Kl.- ~'4-~-1() -'Kz"J]/(~i bi-•)

Q~ = (~i+n-~i)/ts>i.bi+il\)

b) 9= 0

cPi+• [( l/Oi-1 +Rx.)/{&;_, tb·,) + ¢;_, [l&;j&·:, -£ .. )/(bi-1-+ bi) J +

<Pt+~l' /t~i..-~ 1 + ct>; [.-' lb~-1 -'/&~.!-\~ 1 :: 0

R-x. = lS'\ - ~ ,_\) I ( ~\.bi-1)

R~ = <..~~+\" -· s-i )/ ( ~\-Di+~ .. )

(8)

(9)

(10)

10

3.. Field boundary.

a) Constant velocity potential boundary. Fi~~re 2(a).

From equations (5) and (6)

t <t>'"''~· R~1 + q,,o\>t'\ L' -T R~1 + ~\-~J,\- Q~1 + <P,, G 4-1 :. ¢>1. t1 + l<J(.1 ( 11)

Rx. =- l ~ '""' - ~ o) I t 4 ~, )

b) Constant velocity boundary. Fig~re 2(b).

From equations (5) and (6)

4>,+, L' +R~ 1 +- tbi-, [l-Rx.1 + ~ i-"[1-e~1 + ~i (-41 =

-(~1 + 'io.-x.} L'-+ e~J

Rx. =· l~W-t- ~i-1 )/ ( 4~~)

R':S = l~o- ~i-VI)/(4~i)

(12)

(13)

(14)

ll

IV. SOLUTION TECHNIQUE

The solution of a fluid flow problem is obtained in

two steps: various assumptions must be made to ensure a

valid solution and the appropriate equations from Section

III must be chosen for the particular problem to be solved.

A description of the assumptions and method developed for 0

the solution of the subsonic flow field around a 7.5 wedge

is given below. This method has also been applied success-0

fully to a 6 wedge inlet.

A. ASStJMPTIONS

Apart from the basic assumption of replacing a dif­

ferential equation with a difference equation, the assump­

tions that affect the final solution all relate to the

field boundaries and problem body surfaces.

1. Field Boundaries

By assuming that the fluid outside of the problem

field is at free stream conditions, additional restraints

are being placed upon the problem unless the field is in­

finite in both directions. The effects of these extra

restraints are discussed in Section V where methods are pro-

posed to remove them •.

2. Body Surfaces

where body surfaces change direction gradually, the

assumption that the velocity vector is tangential to the

surface (Appendix I) enables the difference equations to

be defined. Where singularities occur, the equation at the

singular point can no longer be defined since the flow

12

direction is not unique. Methods have been devised (17) to

avoid the singularity by retaining a few terms in the ana­

lytic series solution at the point. This, however, only

applies to the Laplace equation in terms of r and 9 where

a solution can be obtained by separation of variables •.

Other methods must be devised for variable density problems.

One common method is to ignore the discontinuity and add

extra grid points before and after it. Another, which is

investigated in this paper, is to assume that the flow di­

rection is the mean of the initial and final boundary di­

rections. The latter method has the advantage in that it

does approximate the real problem since the ever present

boundary layer will tend to round off the "edges". Solu­

tions to problems, for which experimental data is avail­

able, have been obtained using various assumptions for the

singular points. These solutions are presented and dis­

cussed in Section V.

3. Density Field

While the initial density field assumption does not

affect the final solution, it does affect the speed of con­

vergence. A reasonable initial assumption is an incompress­

ible flow field; however if a better "guess" is availabler

it should be used. When solutions to a set of similar pro­

blems are being obtained, such as a series of Mach numbers

closely spaced for a given boundary configuration, most of

the iteration time for the second and suqsequent problems

can be avoided by using the previous density fi~ld solution

as the first assumption for the current problem.

B., NETHOD

13

1. The given flow field is covered with an orthogonal net.

The choice of grid size and number of grid points is dis­

cussed in Section V.

2. Grid points are numbered consecutively across the rows

starting at the bottom left hand corner. Arrays are al­

located for flow direction, density ratio, velocity, and

velocity potential. Each element in the density array is

initially set equal to the free stream value (or an ap­

proximate solution, if known).

3. To keep the solution general, the matrix for the finite

difference equations is set up for the flow field in Figure

3, using the appropriate equations from Section III. The

assumed boundary conditions are undisturbed flow at one

grid length outside of the field and a solid boundary at

the lower edge. (Note: if the matrix was inverted at this

point, the only possible solution would be a parallel flow

field at free stream velocity. This is used as a check on

the solution technique) •.

4.. Using the appropriate equations of Section III, the

matrix is then modified by changing the coefficients that

apply to the particular boundary conditions of the problem

to be solved.

5. The matrix is inverted to obtain the velocity potential

field.. If the density field was initially assumed constant,

the first solution is the incompressible flow field solution~

14

using equation (2). Each element in the density field is

compared v..ri th its previous value and the maximum change is

used to determine if a sufficiently accurate solution has

been obtained. If not, the problem is iterated from step

(3) through step (6) until either the accuracy test is

achieved or a prescribed number of iterations are exceeded.

7. With the final solution of<?, all other flow parameters

can be calculated.

C. RESULTS

The validity of a solution obtained by finite dif­

ference methods can usually be justified if experimental

data is available in a similar flow regime. 'When a so­

lution is required for a problem with no available ex­

perimental data,~ a more fundamental approach must be made •.

The solution will be valid providing that the finite

difference equations converge to the corresponding dif­

ferential equations and if no assumptions (irrotation etc.)

are violated. Forsyth and Wasow (23) have investigated and

demonstrated convergence for several types of difference

equations.

The accuracy of the solution depends on the size of

the grid used. Lee (15) has investigated the effect of

grid size and presents methods in which extrapolation is

used to consolidate results from several grid sizes. The

effect of grid size is also investigated and discussed in

Section V of this paper •.

15

V. DISCUSSION AND RESULTS

The usefulness of the numerical solution method de-

veloped in Sections III and IV was investigated by applying

it to two subsonic flow problems. The solution method is

limited to problems in subsonic flow since the velocity

potential is the solution variable and can only be defined

for irrotational and hence shock free flow. With the ex-0

perience gained from the 7.5 wedge problem, the method was

a 60

then applied to wedge inlet. 0

A. T\t;O-DIMEl':SIOl' AL 7. 5 \'~:EDGE (Figure 4)

The wedge problem was chosen because it is applicable

to many real systems (aircraft wings, induction systems,

etc.) and also due to the availability of analytical (4)

and experimental data. (8,9). Bryson (8) discusses the ex-

perimental proceedure used for his tests and demonstrates

that a t\oro-dimens ional flo"' field \oras closely approximated.

He also notes that models were chosen such, " ••• that viscous

influences would not materially affect the flow over them".

1. Singularities

The major difficulty in solving this problem is caused

by the singularities at the leading edge and shoulder of

the wedge. A set of solutions, at constant Mach number,

was obtained for a range of assumed flow angles at these

points. The results are presented in Figure 5 and com­

pared with Bryson's experimental data. These results show

that of the eight grid points on the surface of the wedge,

two (x/C = 0.429. and 1.0) are unaffected by the flow angles,

16

tv.ro others (x/C = 0 and 0. 572) are slightly affected and the

remainder are very sensitive. It can be seen that no as-

sumption will give the correct solution at all chord sta-

tions.

The boundary layer will not affect the results at the

leading edge of the vedge but at the shoulder, as Shapiro

(22) notes, the boundary layer will have a rounding effect.

This explains why the best results are obtained with angles 0 0

of 7.5 and 5.0 at x/C= 0 and 1.0 respectively. Since a

prediction of the effective flow angle at the shoulder is

not possible, it was considered that optimum results were

obtained when the flow angles at the singularities were

assumed equal to that of the wedge.

2. General results 0

Solutions for the 7.5 wedge at several Mach numbers

are shown in Figures 6 through 10 where they are compared

with the experimental results of Bryson (8,9). These re­

sults show reasonable agreement with the experimental data;

the main disagreement being at the beginning and end of the

wedge, as expected •. When a composite plot is made of the

pressure distribution (Figure 11), the theoretical pressure

at the shoulder of the wedge decreases with increasing Mach

number \<.7hereas the experimental data shows the opposite

trend. Although the theoretical curves appear to cross at

the same chord location in Figure 11, it is believed that

this "rould not occur if a more accurate plot could be made.

The incorrect trend is further emphasized in Figure 12

17

v,rhich compares both Bryson's (8,9) and Cole's (4) data. It

is seen that Cole's theoretical data disagrees v,rith Bryson's

experimental data for Hach numbers below 0.768 due to the

rounding effect of the boundary layer as discussed above.

However, above this Mach nu~ber ~he agreement is excellent.

This rounding effect also explains the disagreement between

the theoretical results of this paper and Bryson's data for

Mach numbers below 0.768.

Above M0 = 0.768 however, the disagreement is primarly

due to the appearance of sonic velocity on the wedge.

Bryson (8) discusses at length the mechanism which must ter­

minate a supersonic zone in a subsonic flow field. For

nonviscous solutions he concludes that this must be a shock

wave.. This clearly· violates the assumption of irrotational

flow and thus solutions above M0 = 0.768 will be invalid to

some degree; the error increasing with free stream Mach

number. Cole (4) remarks, " ••• that the drag as computed

from the entropy changes of the shocks ••• should agree with

the pressure drag on the front portion of the wedge". The

effects of neglecting the shock waves in the theoretical

analysis of this paper are thus clearly seen in the low pres­

sure distribution for Mo> 0.768 relative to the experiment­

al data;_ resulting in a low pressure drag •.

Another possible reason for the discrepancy between

the theoretical and experimental results could be due to

the boundary layer on the wedge which was not considered

in the theoretical analysis. The displacement thickness of

18

the boundary layer v7as approximated by varying the vredge 0 0

surface slope from 7.5 at the beginning to 8 at the

shoulder. The results of this are shown in ?igure 13 where

it can be seen that the boundary layer effect is insig­

nificant.

In the theoretical analysis the partial derivatives of

density v..rere obtained from the tv!O point formula (linear).

To assess the loss in accuracy due to this method, a so­

lution was obtained using the three point formula (para­

bola) for the density slopes, but the change in the final

solution v.1as so small it could not be shown when plotted •.

3. ?ield size - Number of grid points

Undoubtedly part of the disagreement betv..reen the theo­

retical and experimental results lies with the number of

grid points in the field. The solution with the largest

number of grid points undertaken so far was obtained with

network C (Figure 22), which has 224 unkno,•.'ns. This re­

quires a matrix of approximately 7400 entries and V7hile a

larger size is possible, the inversion time with Gaussian

elimination would be considerable.. Solutions at one Mach

number were obtained for grid netv.Jorks A, B, c, and D.

The compnrision between networks A and B (or C) shows the.

effect of a large increase in total grid points. The two

solutions agree in the central portion of the v..redge, but at

the ends, the best solution is obtained using a large num­

ber of grid points (B or C). It is also seen from Figure

14 that grid network c, which has two more points on the

19

surface of the wedge than B, gives an improved solution for

the end points. A solution was obtained with the discon­

tinuities undefined (grid netvrork D); this solution is

shown in ?igure 15 Y-'here it is compared with the solution

using grid netvmrk c. There appears to be no advantage in

this technique.

4. Field size - ~edge size

The v.redge thickness to field v!idth ratio is 0.132,

0.051, and 0.071 for grid networks A, B, and C respectively,

while for the test model the ratio was 0.006. The numerical

solutions obtained for this paper thus have additional re­

straints due to the limited size of the field. These re­

straints ~rould tend to increase the velocity tov-rards the

rear of the wedge since, in essence, the solution is that

of a "channel" problem. Figures 16 and 17 show the cal­

culated velocity profile along the ramp and the calculated

flow direction across the field. It is interesting to note

from Figure 16 that if the assumed flo\•l angle at the sin­

gularities is increased, both the low and high velocity

points are displaced outwards from the wedge. The discon­

tinuities in velocity and flow direction shown on Figure 16

and 17 are caused by the relative size of the wedge to the

field.

The obvious way to remove these restraints is to in­

crease the field size relative to the wedge. By extra­

polating the results from Figure 17, the required field

size is estimated to be 10 times the width of network c.

20

This number of grid points is obviously impractical and thus

some other method must be devised.. One such method of in­

creasing the field size without the proportional increase

in grid points is to vary the grid size within the field.

This can obviously be done in the usual way but it does

tend to lead to matrices which have a very wide band with

a necessary increase in storage area (Appendix III). It

is postulated that another technique would be to replace

the boundary conditions at the edge of the field with values

extrapolated from within the field after the initial so­

lution has been obtained; or more crudely, to replace the

boundary conditions with the corresponding values at the

edge of the field and then to re-~olve the problem. No

justification can be given for these techniques, but in

further research beyond this paper it is intended to see if

improved results are obtained using them. Another pos­

sibility is to initially solve the problem for a field with

a reasonable number of grid points and then to make this

field one grid square (or rectangle) in a much larger field.

This larger field is then solved to obtain the necessary

boundary values for the smaller field. The smaller field

is thcL re-solved with the new boundary values.

B. T\\·0-DIMEl\!SIONAL \';EDGl~ INLET (Figure 18)

The tvJo-dimensional wedge inlet is an extension of the

wedge problem previously investigated. The experimental

data available (10) is limited to one Mach number and one

wedge angle. Solutions have been obtained with grid net-

21

·work E for three of the mass flov1 ratios tested and are pre­

sented v1ith the theoretical results in Figure 19. The flow

angles at the leading edge and shoulder of the wedge are

assumed to be equal to that of the wedge (the assumption • 0

v?hlch gave the best results for the 7. 5 wedge). As with 0

the 7.5 wedge problem, excellent agreement is obtained

over the central portion of the wedge. Experimental data

is also available for other (lower) mass flow ratios but

reasonable theoretical solutions have not be~btained for

these. The density field does not converge in the high

velocity region on the outside of the lip surface probably

because of the existence of shock waves in the flow. To

obtain solutions for these mass flow ratios will require a

method with the stream function as the solution variable.

22

VI. CONCLUSIONS

A. The numerical solution method developed in Sections III

and IV is successful in the solution of subsonic fluid flow

problems.. When sonic velocity is exceeded in the flow

field, the method is no longer valid.

B.. The singularity assumptions considered for the '\>Tedge

affect the solution on the wedge surface, but do not affect

the values at the singular points,. themselves, to any de­

gree.

C.. The most accurate results were obtained by assuming

that the flow direction at the singular points is the same

as that of the wedge.

D. The accuracy of the solution does not depend on the

singularities being defined with grid points.

E •. The inclusion of the boundary layer displacement thick­

ness does not affect the solution to the problem considered.

F. For the density derivatives evaluated at the surface

of the v.redge, a t'\>Jo-point (linear) formula is sufficiently

accurate •.

G.. Gaussian elimination is limited to problems with less

than 300 grid points.

23

VII. RECOMMENDATIONS

A. The definition of the singularities should be developed

further.

B. Other problems should be investigated including those

in three dimensions.

C. The method of solving the difference equations should

be changed to an indirect (iterative) technique so that

problems with a large number of grid points may be

solved.

D. A similar method should be developed with the stream

fur..ction as the solution variable. This vTill enable

two-dimensional rotational flow fields to be inves­

tigated.

1.

2.

3.

4.

5.

6.

7·

8 •.

10.

11.

24

BIBLIOGR1~?hY

Shapiro,. A~ h.~ The Dy~amics and Thermodynamics of Comnressible ii'luid .:"lovr~ Vol 1, Ronald Press Com­pany, 1953.

Guderley '· G., and Yoshihara, H.:· "The ?low over a Wedge Profile at Much l~u:m.ber One." Jour. Aero •. Sci., vol. 17, no. 11, Nov. 1950, pp. 723-736 •.

Tsien, H. s., and Baron, J.: 11Airfoils in Slightly Supersonic Flov7, 11 Jour. Ae:-o. Sci., vol. 16, no. 1, Jan. 1949, pp. 55-61.

Cole, J. D.: "Drag of a :?ini te vJedge at High Subsonic Speeds. 11 Jour. Hath. and ?hys., vol. 30, no. 2, July 1951, pp. 79-93.

Southviell, R. V.: Relaxation Net hods In. Engineering Scienceo Oxford University Press, 1940.

Christopherson, D. G., and South'\vell, R. V. :· "Relaxation Methods applied to Engineering Problems. III-Problems involving T'\vo Independent Variables." Proc. Roy. soc., vol. 168, no. 934, 1938, pp. 317-350.

Emmons, H. W. :· The Numerical Solution of Compressible b,luid Flovl Pro bJ.errs. NACA TN 932, May 1944.

Bryso~, A. E.:· An Exnerimental Investigation of Tran­sonlc ~low nast Two-Dimensional ~edge and Circular­Arc Sections using a Mach-Zehnder Interferometer, NACA TN 2560, Nov. 1951.

Liepmann,. H •. w •. , and Bryson, A •. E •. Jr.:· "Transonic Flow past v~·edge Sections •. " Jour. Aero. Sci., vol. 17, no. 12, Dec. 1950, pp. 745-755. ·

Subsonic-Transonic Dra~ of Su.ersonic Inlets. Pratt and Whitney Aircraft Company, TDM 1973, 19 6.

Southv.re11, R. V. : "Stress-Ca1cula t ion in Frame'\o.'orks by the Method of 'Systematic Relaxation of Con­straints t •. " Proc •. Roy,, Soc., (A) 151 (1935), pp. 56-9 5.

12 •. Amick, J. 1.: Comparison of the Experimental Pressure Distribution on an NACA 0012 Profile at High Speeds ,,rith that Calculated bv the Relaxation Method. NACA TN 2174, August 1950.

13. Emrr.ons, H. vi .. : Flm,r of a Comnressible Fluid past a Sy~~etrical Airfoil in a Wind Tunnel and in Free Air, NACA TN 174-6, 191t8.

14 •.

16.

17.

18.

19.

20 •.

21.

22.

23.

25

Emmons, H .. Vi.: The Theoreti_cC'll li'low of a Frictionless, Adinbatic l)erfect Gas insj.de of a Two-Dimensional Hvper:1olic T.foz?:J.e. IJACA TN 1003, 19 •

Lee, J,. A.: Numerical Analysis for Computers. Reinhold Pub. Corp., 1956,

Bickley,~. G., Michaelson, s., and Osborne, M. R.: "Finite-Difference Methods for the Numerical so­lution of Boundary-Value Problems, 11 Proc •. Roy •. Soc., A 262, 1961.

Fox, 1.: Numerical Solution of Ordinarv and Partial Differential Eauations. Pergamon Press, 1962.

Seeger, R. J., and Temple, G.: Research Frontiers in Fluid Dynamics. Interscience Publishers, 1965.

Hamza, V, and Richley, E. A.: ITumerical Solution of ~vo-Dimensional Poisson Eauation: Theory and Ap­Plication to Electrostatic-Ion-Engine Analysis, NASA TN D-1323, October 1962.

Poole, V.'. G.: ~~umerical Experiments v.'ith Several Iterative Methods for Solving Partial Difference Bauationso The University of Texas Computation Center, AROD Report 3772.16, August 1965.

Conte, S.D.: Elementary Numerical Analysis. McGraw­Hill Book Company, 1965.

Shapiro, A. H.: The Dvnamics and Thermodvnamics of Comnressible Ij'luid Flovi. Vol II, Ronald Press Com­pany, 1953 •.

Forsythe,. G. E., and vJasm .. r, W, R. :· Finite-Difference Nethods for Partial Differential Eauations, John Wiley, N.Y., 1960.

26

APPENDIX I

DERIVATION OF FINITE DIFFhRbNCb I:QUATIONS

Standard finite difference methods are used throughout the

analysis (21).

A. Equations {3) and (4)

Consider Figure l(a).

4>z. :. l~,+\ -~,_,)I l oi1:1 + o\·-,)

~u.: 2. { ( ~'+'- ~, )/ bi~' - L<l>i- cp,_\ )} bi-,1 / ( &i+• + ~;.,) ·

~~ : t~i~~- ~i-W\) /l~ i-t"-\- bi-~)

<t>'J!f ~ 2.{l1bi~~- ~' )/&,~ ... -l~i- £t>,·-~ )\lbj_V\) I (bi.rk+ Oi-~J

'S':J/~ = l~'"'"'- ~~·-v.)/(~i ( ~i~~+ oi-~)]

~x./~ = l~\-t\ -~,·-,)/ \..~·, l~, .... + ~i-.)]

Now substituting into equation (1) and \tTith %.

R.~::. l~\-4-1- ~i-t)/ (.lSi+• + Oi-\) ~,]

we obtain equation (3)

(4)

c\>;., t 2./ [ l>i+~ ( b;., + b,_, )J + g,_] + d>;_, tz./ c &;_, (S; ., + bi-•)- R. ... 1 +

ljli.~ t z./ (. hi•~ l b;,~ -11\i-~)} t R~] + $; _., { 2.1\:. b i-~ l h;..,. + S; -~) -% ~ +

<P• l- 2./ t Oi-+1- o; ... ,) - z..f ( b·,·h\. bi--~) ~ =- o < 3)

Consider Figure l(b).

The flo~~.r direction at i is defined as g , thus

V· I ;:: ' ~\ l9r-o.d. ) i .:::. lv.~)i

- \Vl cose LV = vi. T .:::. = ~l(.

v = Vi .j = \ y \ s,.:. e = ¢>~

Therefore,

l~x\ =(<Pi+'- G>i-1). CosB/[( bi4-Di_,)/cosG]

(¢\j)i = lcPi""'- ¢i-•J· si~e /l( Si+bi-1 )/ cose]

27

To obtain the second order derivatives of<}>and also the

first derivative of'S a value of4> and ~ /~t is interpolated

between grid points i-n-1 and i-1 (termed * conditions).

~ .q, = <P ,_, . ~·,.V\ I b i-t)\-1 + ~ i+Vl-1 ( I - 0 i+~ J Oi-t~-1)

~* = 5?\-,. b\+V\ I bi+V\-\ + ~i·H\-1 ( \- b,+~ I Oi+"·i)

hence

~~ = 2.{ l¢i+' -~i-,)/l&i-\ to·,) co~-z.e -t<i>i -Q* )/&·,_,~fbi-a

d)~~ = 2. { l ~\+~- <?i) I bi+~ - ( <l>i 1-\ - ~,·_,) /(o I ~,-t bi) SlV\9 cose}/ bi+~ ~-x./~ = l~i- ~ -v) I(Di-•· ~i)

~':i}C?::: (s>; ...... -~·. )/lbl+"" s>;)

On substitution into equation (1), equations (7) and

(8) are obtained.

Qi +I [K,] + ~i-1 l K'2. I &t., - ~~1 + Q>i +f\ L' I b ~ ... ~] +

<Pi ... )\-\ u~- Kz.)/ bi-,J + ~\ [.\-1 ~-, -l/ &i+VI J ~ o (7)

K, : l cos~e L 11 Oi-l+ l2:t.} - siY\eeose i 11 bi+V\- R~~1/loi-d ~i)

\{'2. = Oi +VI I 0 i+V\- I

R-x..- (9i - ~i-\· 1<''2.- ~;+~-I (I- K'-)1/ ( ~i. bi-•)

28

(8)

} .. PPEl•.'DIX II

LISTING OF' CONPUT:&."ft PROGRAMS

A. T~·:O-DIHB:FSIQ!\;AL kEDGE

29

r"·

.C

;~ .......... ;-.;Y

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----· S • OO't 0 S.004l

S.004? S • () 0 1t)

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_ .S. 0052 S.005~ s. 0 0 .) 1t s.0055

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___ S • 0 0 bO . __ 5.0061 S. OOf., "­$.0061

.. S • 0 0 (:It S.OOA5 S.006A S.0067

__ $.0063 S.OOA0 S.00 7 0 S.007l S.007? S.0071 S.0074 S.0075

__ _s. 0076 S.0077 S.0078 S.007CJ S.OOR8 s.00°l s.ooqz S.00P~

____ s. oo~ 4

Jr1(1)=1 I P, ( T L + 1 ) = I ,_, A X

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C S [ T U n S T /1 N D A!=: 0 ~H'.,_ T r{I X f- 0 n P l~ P. ~ L L t= L rt. 0\,J c

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O'J l 0 I = 1 , "-:C C(T}=O.t) ( 0 I•)T J N!J:: . _ .... ····--- ----· .. _ .. --·---· _ _ ...... _ _ __ .... _ . () Q 4 () I = 1 , I t-~ t. X OYL ( T l=l.O P(J)=ro THl:T'\ {I )=0.0 CONTI "Jt IF DJ 10~ l=JXMNPl,I~AX F= I- I r.HI. Xf"N Pl = I + N t·': ~ X .. T ft-' P = P H I 1 + V () t,d.: P{ IN)=Tl=MP V( IN>=VO P { T ~'J ) =P 0 THfTf>.( l\1)=0.0 crH!T T t--!' 1 r. Cl\tt !JFLTA 1'1=0 . DO 1 4 0 I ::-:- l , T ~) ~X P( T )=O.rJ IPl=I+l 1'~1=1-l l P N= I+"''·~ /l X I~~: f\.'= T -r-.; ~1 f\ X K = T tll ::< ~:?] +N flAX n 1

... K P t-.' = K + ~ ~ ',1 /\ X

!<' '•1 k= K - f\.: '1/\ X K.Pl=K+l !<'M.l=K-) F P- 1 = 4 • >:' Q ( J } C(K)=-4.0 JF( I-NPI\Y) 131,13) ,] 1? I.(KPN)=?.O

_. GQ TG 1 3:; ..... -··· ________________ ·-··- ___________ . I F ( I - T M J\ X r.• ~ J ) 1 -~ L •. , l ·:~ 4 , 1 ~ 3 R R Y = ( P 0- ~~ ( I r.~ r-J l ) I r- R. T F-= l- J ~·!'...X~~ N ( { f< ~.~ ~ ) :::: 1 0 ()- D D y P ( I l = - P ( T P N ) ~' ( 1 • C + f? R Y ) GO TO 11':> R R Y= ( R ( T P "'! l - r~ ( T ~1 N ) ) IF P. J ({KPN)=t.O+PRY

s.oo~5 S.OQ!jr) s. oo~n s.ooqn s.oo.qq S.OOQO s.ooqt

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S.0013 FNO

50

MODI :<'IBD Gi' .. USSIAN ELIHil:ATIOL

The augmented matrix for the solution of partial

difference equations is of the band type and takes the form

shown in ?igure 25.

r.:~ ) .

\

I XX"" •••• ::.{_\ .................................. x \ XXX ••••• X~ • • ••••• • • • •• • • • •••••••••• • • • • • •.'.

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\\

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\

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l••••••••••••••e••••••••••••••·~·····xxx •• ~ \ ...................•.......... ·~ ..•. . xxx .

1x ,

••••••••••••••••••••••••••••••• ~ ••••• xxx;x • • • • • • • • • • • •• • • • • • • • • • • • • • • • • • • • ~;<.. • • • • XXX

"--n-1 nxm

Figure 25. Augmented Matrix.

nxm

In Figure 25 the dots represent zeros and n,m are the

length and width of the problem field respectively. The

total matrix thus requires a storage area of (m~n+l)x(mxn),

but a large amount of this is wasted since it is zeros.

HO'\<Iever, by defining a one;...dir.aensional matrix as indicated

by the slanted dotted lines a considerable saving in stor­

age area can be made. The storage area is thus (mxn)x(2n~l)

if the solution vector is used as the right hand side of

the equations.

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The author, Ed,..rard David Spong, VJas born on September

23, 1938 in Farnborough, Hampshire, England. He received

his primary education at ti'arnborough Grammer School and

then attended the Royal Aircraft Establishment Technical

Colleg: as an under-graduate apprentice. He received an

Upper Second Class Honours Degree in Aeronautical Engineer­

ing as an external student from London University in 1961.

After graduation, he VJorked as an aerodynamicist at the

English Electric Guided Weapons Division at Luton, Bedford­

shire. In 1963, he joined the Wright Aeronautical Division

of Curtiss Wright Engines in Wood-Ridge, New Jersey. He is

presently employed at McDonnell Douglas Corporation in

St. Louis, Missouri, which he joined in 1964 as an engine.er

in the Propulsion Department. He enrolled at the Univer­

sity of Missouri at Rolla as an extension student in the

Spring of 1965 •.