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Scholars' Mine Scholars' Mine
Masters Theses Student Theses and Dissertations
1968
The numerical solution of two-dimensional fluid flow problems The numerical solution of two-dimensional fluid flow problems
E. David Spong
Follow this and additional works at: https://scholarsmine.mst.edu/masters_theses
Part of the Propulsion and Power Commons
Department: Department:
Recommended Citation Recommended Citation Spong, E. David, "The numerical solution of two-dimensional fluid flow problems" (1968). Masters Theses. 6814. https://scholarsmine.mst.edu/masters_theses/6814
This thesis is brought to you by Scholars' Mine, a service of the Missouri S&T Library and Learning Resources. This work is protected by U. S. Copyright Law. Unauthorized use including reproduction for redistribution requires the permission of the copyright holder. For more information, please contact [email protected].
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
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____ s. oo~ 4
Jr1(1)=1 I P, ( T L + 1 ) = I ,_, A X
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105
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131
11? 133
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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\ •••• xxx •••. • x~ .......................... . ~. \ • •• • • XXX. • • • .)(\ ••• • •••• • • • • • • ••• • • • •••••• • •
\
r=·xx~:·,~·_: ... ~·-~ ·~. ~- ...... x
~ -- ---- - -- - : -
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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VITA
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 •.