202
A NUMERICAL METHOD FOR TWO-DIMENSIONAL, CAVITATING, LIFTING FLOWS Daniel Wilson Golden
A NUMERICAL METHODFOR TWO-DIMENSIONAL, CAVITATING,
LIFTING FLOWS
Daniel Wilson Golden
A NUMERICAL METHOD
FOR TWO-DIMENSIONAL, CAVITATING,
LIFTING FLOWS
by
DANIEL WILSON GOLDEN
B.S., California State University, Northridge
1967
SUBMITTED IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE
DEGREE OF OCEAN ENGINEER
AND MASTER OF SCIENCE IN NAVAL ARCHITECTURE
AND MARINE ENGINEERING
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
May, 1975
t&D'JATE SCHOOL\L!FORNlA 93940
A NUMERICAL METHOD
FOR TWO-DIMENSIONAL,
CAVITATING, LIFTING
FLOWS
by
DANIEL WILSON GOLDEN
Submitted to the Department of Ocean Engineering onMay 9» 1975 in partial fulfillment of the requirements forthe degree of Ocean Engineer and degree of Master of Scienceand Marine Engineering.
ABSTRACT
A numerical method for two-dimensional cavitating flowis developed for the flat plate. The linearized boundaryvalue problem is restated as a set of coupled integral equations*The integral equations are approximated numerically. Thenumerical approximation is executed by a Fortran IV computerprogram. The computed results are compared to the analyticsolution. This method should provide insight into developing amethod for three-dimensional cavitating flows and is readilyextendable to cambered profiles.
Thesis Supervisor! Patrick Leehey
Title i Professor of Applied MechanicsProfessor of Naval Architecture
ACKNOWLEDGEMENTS
The guidance and invaluable constructive criticism
provided by Professor Patrick Leehey is gratefully acknowl«
edged. The very useful suggestions of Professor Justin
Kerwin are also gratefully acknowledged.
TABLE OP CONTENTS
Page
ABSTRACT 2
ACKNOWLEDGEMENTS 3
LIST OP FIGURES 5
LIST OF TABLES 6
NOMENCLATURE • 7
CHAPTER I INTRODUCTION 9
CHAPTER II THE LINEARIZED BOUNDARY VALUE PROBLEM 12
CHAPTER III THE INTEGRAL EQUATION FORMULATION 18
CHAPTER IV NUMERICAL APPROXIMATION TO
THE INTEGRAL EQUATIONS, # 2*f
CHAPTER V ANALYTIC SOLUTION 34
CHAPTER VI RESULTS, CONCLUSIONS
AND RECOMMENDATIONS • 39
REFERENCES 66
APPENDIX A COMPUTER PROGRAM 68
APPENDIX B SAMPLE COMPUTER OUTPUT 88
APPENDIX C COMPUTER PROGRAM FOR COMPUTATION OF
CAVITY LENGTH FROM ARBITRARY VALUES OF
ANGLE OF ATTACK AND CAVITATION NUMBER. ,. 9^
Figure No,
1.
2.
3.
4.
5.
6.
7.
8.
9.
10,
11.
12.
13.
14.
15.
16.
LIST OP FIGURES
Page
Coordinate Systems 45
Element Arrangements 46
Variation of Last Source 47
Mapped Planes 48
Vortex Distribution, Jl « o,5 49
Source Distribution, J^ « 0.5 50
Cavity Length - Partial Cavitation 51
Lift Coefficient - Partial Cavitation 52
Cavity Area - Partial Cavitation 53
Vortex Distribution, i= 1.5 55Source Distribution, fl * 1.5 56
Cavity Length - Super Cavitation 57
Lift Coefficient - Super Cavitation 58
Cavity Area - Super Cavitation 59
Numerical Convergence;z-o.s 6l
Computation Time 62
LIST OF TABLES
Table No, Page
1. Values of M and N for 54
figures 5 through 9
2. Values of M and N for
figures 10 through 14 60
3« Computed Results 63
4. Singular Behavior 65
C 1« Results of Cavity
Length Computations 97
NOMENCLATURE
A Cavity Area
c chord length
£C] coefficient matrix
CjL
element of coefficient matrix
Cp pressure coefficient
CL a lift coefficient
f location of vortex control points
f = location of source control points
h = locus of points describing mean camber line
and cavity surface
£ « cavity length
M * number of vortex elements
N number of source elements
P pressure
P*, pressure at infinity
q = velocity
qCJ) source distribution
u,v = perturbation velocity components
U^ inflow velocity
w complex velocity = u-iv
x,y coordinate system
%£ lower boundary of vortex or source element
2, m complex plane « x + iy
8= quantity with a bar is a dimensional quantity
= no bar indicates a nondimensional variable
« variable to be evaluated at x 0*
« variable to be evaluated at x * 0*
variable to be differentiated wrt x
(X angle of attack
y(£\ = vortex distribution
£ * mapped plane S£+ L T(
t)^l s dummy coordinate system for integration
and see Q above
P « fluid density
(5 * cavitation number
- velocity potential
CD » perturbation velocity potential
n = degree of singularity
9
CHAPTER I
INTRODUCTION
The objective of this investigation is to develop a
numerical solution to cavitating flows which can be compared to
a known analytic solution. Therefore, the cavitating two-dimen-
sional flat plate in steady flow has been chosen for this
investigation. Geurst [l,2,3j has given solutions for the
linearized problem for both partial and super cavitation.
The purpose in considering this problem is to determine
and overcome the numerical difficulties associated with this
problem. Adequately solving the two-dimensional problem is a
preliminary step in developing a numerical solution for the far
more difficult three-dimensional and unsteady flow problems.
Further the numerical method can be applied to foils
of different camber lines. Geurst £2J has obtained a camber
line solution for the case of parabolic camber in partial
cavitation only.
Widnall [k] developed a numerical method for three-
dimensional, unsteady, supercavitating flows. However, Widnall
used assumed cavity lengths and demonstrated that, for long
cavities, lift coefficients were relatively insensitive to
cavity length. Clearly for short super cavities, on the order
of chord length, and partial cavities, a proper statement of
cavity termination is required for the proper solution of
cavitating flows.
10
In this respect this report is a preliminary investiga-
tion into the inclusion of cavity termination in cavitating
flows. The results of this report should be applicable to
three-dimensional and unsteady cavitating flows*
The solution of two-dimensional, cavitating flows pre-
sented herein envolves five distinct steps. These are
i
1
)
statements of the linearized boundary
value problem,
2) solution of the boundary value problem in
the form of coupled integral equations in
source and vortex distributions,
3) a numerical approximation to the integral
equations,
k) execution of the numerical approximation via
FORTRAN computer program,
and
5) validation of the numerical results.
11
Linearizing the boundary value problem and rewriting
the problem in the form of integral equations is not the only
method of solution* Another possible approach is the finite
difference method* Jeppson 5»6| and Mogel and Street [7] have
taken the finite difference method as far as a circular disk
perpendicular to the main stream with an infinite cavity. The
major disadvantage of the finite difference method is the large
programing and computational effort required.
The integral equation method for the linearized problem
represents a far smaller computational effort. It is for this
reason, plus the existence of the analytic solution, that the
integral equation method is used. For an excellent discussion
of the various methods of solution see Birkhoff[8J
•
The following chapters of this report will discuss the
stated steps of the solution.
12
CHAPTER II
THE LINEARIZED BOUNDARY VALUE PROBLEM
Geurst £l,2,3"] has presented a development of the
linearized boundary value problem for both partial and super
cavitation, A similar development is given here, except that
the coordinate system and the nondimensionalization of variables
differ. These differences are strictly a matter of convenience.
Geurst's method of solution relies on a series expansion of the
complex velocity in the neighborhood of infinity. This in
turn requires that the boundary value relations to local slope
on the foil and cavitation number be stated at infinity.
The neighborhood of infinity is not convenient to
computer solutions. Therefore the boundary value relations to
local slope on the foil and cavitation number need to be stated
on the foil and cavity.
The coordinate system and the foil-cavity relation to
the coordinate system are shown in figure 1(a). The inflow
velocity, Q^ , is assumed to be parallel to the x axis and
uniform in the y and z directions. The angle of attack, o(.
is taken to be the nose-tail line for cambered profiles.
First the steady flow form of Bernoulli's equation is
written between infinity and another point in the flowi
tofi+.fo: =^^i%\ ?=/*>/
13
This equation is now rewritten in the form of a nondimensional
pressure coefficient.
On the cavity the pressure coefficient is the cavitation
number, (J"". The velocity potential can be written as the sun of
the potential due to the inflow velocity plus a perturbation
potential
i
(3) (fi-Uo* +
Ik
Where
u x component of the perturbation velocity
v « y component of the perturbation velocity.
It is assumed that the perturbation velocity is very much less
than the inflow velocity, that is a
On the cavity we now havei
using(6)
2W «&=.+**/£ko" ' " "'
°°
Thus equation (2) for the cavity becomes
i
Where
and Uco is now used as a nondimensionalization constant*
15
Equation (8) is the statement of the linearized boundary con-
dition on the cavity surface. The next step is to obtain the
boundary condition on the wetted surface of the foil.
On the foil wetted surface the boundary condition is
flow tangency.
Which becomes
4^- +- S£ +- hio^er orJcr +er~>s = ^or
(10) V - ¥ ^ —
Also, it is assumed that the foil is very thin compared
to the chord length such that the boundary condition in equa-
tion (10) can be applied to the mean camber line.
Further the entire foil cavity system is collapsed to
the x-h plane and the boundary conditions applied on the x-z
plane with all length dimensions nondimensionalized by the chord
length, figure 1(b). Another condition to be satisfied is cavity
termination.
Geurst fcl proved that for the linearized problem, the
re-entrant jet and Riabouchinsky models for cavity termination
reduce to a statement that the cavity closes at its end. This
condition can be written as
(II)
-J 16
i^ xL^r(C«LV«V/ dxJi'foil -o
or
(12) h*-
K
=
The next condition to be satisfied is the condition at
the trailing edge of the foil (x 1), This condition requires
that there be no pressure jump accross the foil at the trailing
edge.
(13) Cp en - c;co =o
In summary the conditions that the solution must
satisfy ares
(CO u+= cr/zo± *-J?
(C2) v- K on wetted surfaceof the foil
(C3) f* - *; -0
(CA) c;co -c;co=o
17
The last condition is that the perturbation potential
satisfies Laplace *s equation
t
The next chapter will give a solution to Laplace's
equation which satisfies the above boundary conditions.
18
CHAPTER III
THE INTEGRAL EQUATION FORMULATION
This chapter presents the development of the integral
equation formulation of the problem. This development proceeds
from the solution for the velocity field induced by a two-
dimensional distribution of vorticies and sources in space*
Then it can be shown that this velocity field will satisfy the
boundary conditions* This results in a set of coupled integral
equations*
First consider the flow to be in the complex (?) plane,
Where
i
and the complex velocity is given byi
urc?) - u — iv
Then the appropriate general form of the solution is \ 9 ] i
19or
(15) Wfi)^/ t*~£V J§Jo
where
,
20When equations (16) and (17) are added together and
the real part is taken, the result ist
t % 2TCX) r -L.I Jill J -
or, with the cavity boundary condition,
(IB) J?
Again the integral in equation (19) is a Cauchy principal
value integral* For a flat plate at an angle of attack
(20) h~(X)=:-
21
and this is the case taken throughout the rest of this report*
The source distribution q(x) represents the slope difference
between the cavity surface and the mean camber line of the
foil at x. While the vortex distribution Y (x) represents the
difference in the x component of perturbation velocity between
the upper and lower surface at x.
(ao %(x) = xr'+tx) - v~(z)
(nz) V(y) - -u+(x) + u~(x)
These statements follow directly from equations (16) and (17)
•
When equations (2), (7) and (22) are combined the jump in
pressure on the foil is$
(!3) C*CV -c;(x) = -ZY(x)and the coefficient isi
(n) CL - -2.1 YwJz'o
The combination of condition (C 3) and equation (27) gives, for
the closure condition,
as) I $cz)Jx = oJo'o
It is clear from equation (23) that condition (C b) ist
M X(i) - o
22
Equations (18), (19). (20), (25) and (26) constitute a complete
problem statement for the steady flow cavitation of a two-
dimensional flat plate, only, it is convenient to rearrange
equations (18) and (19) into a form that contains the ratio
of angle of attack to cavitation number.
CM i = - ^ + ^"f%^~ Jfx-S7
«*> 0**r-i*t. + £t^* dsand
(*9)
Equations (27) • (28) and (29) . subject to Y(j) = o , are
sufficient to obtain cavity length, the source distribution
and the vortex distribution as functions of the ratio cx/
23
The next section will develop the numerical approxima-
tion to equations (27), (28) and (29) and the method of solution.
24
CHAPTER IV
NUMERICAL APPROXIMATION TO THE INTEGRAL EQUATIONS
The method used in the numerical approximation is to
approximate the integral equations (27) * (28) and (29) by a
finite set of linear algebraic equations with N unknown source
densities, M unknown vortex densities and the ratio oc/cr- .
The numerical integration procedure requires that cavity length
be the independent variable and -^- be computed.
This method requires an assumption about the functional
form of the source and vortex distributions. It is assumed that
the vortex and source distributions are constant over a small
element of the foil or cavity, figure 2. The last vortex
element is assumed to be linear with a value of zero at the
trailing edge to satisfy equation (26). The amplitude of each
Tfl , ^i is unknown.
In each vortex element a control point is placed at
which the flow tangency boundary condition is satisfied. In
each source element a control point is placed at which the
cavity boundary condition is satisfied. The vortex control points
are at 2j and the source control points are at ^s \ • The
relations for the control point locations are
i
1 ) for the vortex control
(30) -Kj = Xq + (x,jtl -&j)f j o
25
and
2) for the sources
(so *,j = ŷ + ( ^y>/ ~-*ej)fs °>V ^ J *%- = * ft" n>V
/
26
The source equations are t
Xj - g d$ 3 oj-O \-or j > M
and the closure condition
(35) ) I Jt-dx - O
W**
Equations (32) through (35) are the numerical approximation to
the integral equations* When the indicated integration is per-
formed equations (32) , (33) • (3*0 and (35) become
(36) o-- z?r£ - *%r + %rM-£r) +fi £M#%t)+
(3T) o=anf -7Tp+ *$r{.l+Ct-t)JU,(-£f)] +AH
-*-L£J"(£3bL) , ^ = o for *>N
27
hi
(38) If - - *|. + |L x.^ +2.^^jfc -= fo r j > A/
and
P?)
/v
*=/
«— (*&+' ~ xm) - O
Equations (36) through (39) are linear algebraic equations in
the unknowns -^ , -^ and -^- • It is convenient to rewrite
these equations in matrix form*
m
1 7a
f,<
Iff
- H
1
-
C1
1 — r3C
1
1
\ 1 %a*
if
o_c
C)
28The coefficient matrix, [C] , is formed from the coeffi-
cients of £ , ii , 1 , 1 and J£ in equations (36)through (39).
The first submatrix, (I), is formed as follows by the
coefficients of %- in equations (36) and (37).
F-
W) cjj = 4" (r=r) 5 J *
f«) C„„ = /+ Ci-DJUifjif)
(13) £)(. - ^( x^-x^)3
i*M*»Jj**
M C^-T^^(^r> +"' 3 1*«
The second submatrix, (II), is given by the coefficient
of Ij/v- in equations (36) and (37)
i
(45) CJ)jt^ = -Tf, l£ j& N
(46) CJl ~ O 5 /-j^M^ M+I£l±M+h
29
Submatrix (III) is the coefficient of
30
Submatrix (V) contains the coefficients of X inequation (38)
i
and
X, — ^-JLi(51) Cm+JjMi-l - *^\ Xj-XjH+i) -i
f,
(52) Cm+^M+J - £"'l-fs )
A1 +j i= M+L
The last submatrix (VI), contains the closure condition,
equation (39)
»
C53) CmWj j - Xjgiti ~ *jli j M + i * L £r M+N
otherwise.
To solve equation (40) for the unknown vortex and source
distributions and the ratio
31On the surface this procedure appears to be straight-
forward and should produce a solution without difficulty.
However, this is not the case*
Difficulties in obtaining a meaningful numerical solution
lie in the selection of control point locations within each
source or vortex element* Control point locations are critical
to approximating the singular behavior of the analytic solution*
It was found that vortex control points should be at 90 percent
of the element length and source control points at the element
midpoint* However, special care must be taken with a few of the
source control points.
The critical control points are the first and last
source control points* A systematic variation of the control
point position determined that the last source control point
should be placed at 10 percent of the element length* As an
example of the importance of this control point, the variation
of the computed ratio of ^ with control point location isshown in figure 3 for a cavity 0*5 times chord length*
The selected control point position is at 10 percent
of the element length* Variation of the first source control
point indicated the placement should be at 90 percent of the
element length* Since the source distribution is singular at
the cavity leading edge and termination, these control points
should be placed away from the singularities*
32Variation of other control points (second and next to
last source, last vortex, first source past trailing edge for
super cavity) showed the best placement to be the generalized
locations. Thus the fractions f and fs arei
f 0.90 for all vortex elements,
f8 0.90 for first source control point,
fs 0.10 for last source control point,
and
fs * 0.50 for all other source control points.
The computer program that executes the solution to the
described numerical procedure is listed in Appendix A. This
program is composed of a main program, which performs Input/
Output operations and logic control, and three subroutines.
The first subroutine, CPGEN, determines the vortex and
source element sizes and the locations of the control points.
The required input information for this subroutine is the cavity
length, the number of vortex elements and the number of source
elements.
The second subroutine, MATRIX, computes the coefficient
matrix, Lc], and the boundary condition vector. The required
input information for MATRIX is the output of CPGEN and the
number of vortex and source elements.
The last subroutine, RMINV, inverts the coefficient
matrix. This subroutine is a standard MIT matrix inversion
routine [ill • Tne inverted matrix is then multiplied by the
boundary condition vector, in the main program, to obtain
the solution vector for £ , lL and -A- •A special note on the program notation is made here*
The vector XU(I) is the upper boundary of a vortex or source
element. This vector is given byt
XU(I) =r XL(I + 1)
through an assignment statement in CPGEN.
33
CHAPTER V J
ANALYTIC SOLUTION
The method of gauging the numerical results is the
analytic solution for the cavitating flow of two-dimensional
flat plate. The first half of this chapter will discuss Geurst's
Solution for partial cavitation £1.2] • Tne second half will
discuss Geurst's solution for super cavitation £ 3J •
Geurst solves the partial cavitation problem by a
conformal mapping of the ^ plane to the t plane as follows*
First the 2 plane is given byt
—/ corresponds -t-o le&.
where
,
35
„/tt:^T
X g
—
- coSz S
anda
JB_-cr
With equation (57) is substituted into equations (21)
and (22) over the cavity and wetted surface of the foil, the
vortex and source distributions can be obtained. Figures 5
and 6 show the vortex and source distributions for cavity equal
to one-half chord length* These figures show the leading edge
and cavity termination singularities. One of the difficulties
for the numerical procedure is to reasonably approximate these
four singularities.
Other quantities of interest are the ratio ^L , the
lift coefficient and the cavity area. These are given ast
and
^J^- = 4r {(i + sinf)(-/ + 3sinS) cost 5in$ +
+ -L cotan&(l+Sin&)(l+3sihS-Zsin 2 &- £s//)*£)l
36The equation for cavity area given in£l] is in error
and was corrected to equation (60) in [2 J • Plots of equations
(58) through (60) are shown in figures 7» 8 and 9«
To obtain a solution to the super cavitating problem*
Geurst performs a similar mapping (figure 4(b) )defined byi
W C -ffThe solution for the complex velocity is then#
where
b = ~o — I ~ ^°^an ^
SL
' 2 r- cos xS
rr O. L
and
^ = f^H\tc^ScosC^--j:)-t si* (%-£)]
37The vortex and source distributions are obtained using
the same procedure as with the partial cavitation case. Figures
10 and 11 depict these distributions* In the super cavitation
case only a leading edge singularity appears in the vortex
distribution. Again the source distribution shows the leading
edge and cavity termination singularities.
The quantities -^= , lift coefficient and cavity area
are t
c^ ^^jc^-s
(61)
and
v Are* _ CO s &
38
Figures 12, 13 and lk depict equations (63) through (65).
As with partial cavitation when the cavity approaches chord
length the lift coefficient and cavity area become singular.
In chapter VI the computed results are compared to the
analytic solution.
39CHAPTER VI
RESULTS, CONCLUSIONS AND RECOMMENDATIONS
The primary result of this investigation is a numerical
method for two-dimensional cavitating flows which gives computed
values close to the analytic solution. The computed results
have been plotted in figures 3 and 5 through 16. In figures 5
through 1^, the analytic results of Geurst are also shown*
Figures 5 and 6 show the computed vortex and source
distributions for a cavity of one-half chord length. Each com-
puted point is plotted at the control point location* except
for the last vortex element which is plotted at x^M • It is to
be remembered that each computed point represents a constant
value from x^ to XjlH • These two figures show that the com-
puted distributions fit the analytic solution reasonably well*
However, the largest magnitude elements are not shown on the
figures (see the sample computer output in Appendix B for their
values). These elements do not appear to approximate the
proper singular behavior.
Near the singularities the distributions of sources and
vorticies can be approximated by l/xn . For the leading edge
n has the value one-quarter and for cavity termination n
has the value one-half tl]. The computed values of n are
given in Table ^ for three values of M and N, From Table 4 it
is apparent that the behavior of the two closest elements does
not match closely the analytic solution's singular behavior.
40Figures 7i 8 and 9 compare the computed values of d./v~ %
c
This conclusion points out one of the problems with
the numerical method used herein. The method reported on requires
that the vortex and source elements in the combined cavity-
foil region be of equal size. An independent variation of the
source and vortex element sizes cannot be performed. Thus it is
not possible to determine whether decreasing the size of the
vortex elements* the source elements, or both* results in the
improved convergence. It is recommended that an investigation
of effects of independent variations of the vortex and source
elements be done. This can be achieved through a moderate re-
vision of the computer program*
Also, since the singularities do not appear to be well
matched, a further investigation of the location of control
points should be considered. It seems desirable to have an
analytic basis for the control point locations*
Prom the general form of the computed vortex and source
distributions, this investigator is convinced that assuming
piecewise linear distributions will produce an immediate improve-
ment in the computational results. It is, therefore, recommended
that this numerical method be modified to incorporate piece-
wise linear vortex and source distributions*
The results for the super cavitation case are very
similar to the partial cavitation case. The convergence to the
analytic solution becomes worse as the cavity length approach
chord length. The same conclusions and recommendations made
for the partial cavity apply to the super cavity*
41
Another purpose of this investigation has been to
develop a method of allowing arbitrary values of angle of attack
and camber as inputs and then determine the cavity length. An'
itterative procedure to accomplish this has been developed.
The method and the computer program are in Appendix C with the
computed results. The method and program are a preliminary effort
provided to prove only that such a method is possible.
In summary these are five recommendations for future
investigation.
1
)
A program be written that allows the vortex
and source element sizes to be determined
independently.
2) An analytic and further numerical investigation
of control point locations.
3) Piecewise linear distributions of sources
and vorticies should be investigated.
4) That the method herein not be applied to the
three-dimensional case without further
investigation.
42
43
5) The program should be modified to include
cambered profiles* In this case the ratio
of */cr cannot be computed as an unknown.
Either the angle of attack or cavitation
number must be known along with the camber
line. It is not clear that the control
point locations will be the same as with
the flat plate.
6) Further development of the method of
computing cavity length from arbitrary
values of oc and cr •
The conclusions drawn from this investigation are t
1) the method gives computed values close to
the analytic solution,
2) the distribution of sources and vorticies
is probably the best test of computed results,
3) the method shows poor convergence for
cavity lengths near chord length,
44
4) an iterative procedure to determine cavity
length from an input of angle of attack
and cavitation number is possible
•
*5
Partial
Cavity
cavit
4o
MDimensional
Super
Cavity
avity
y$ PartialCavity
&*Lv=h
At 'v**h„
SuperCavity
U^Vz M*=
x
Assumed VortexDistribution
V
47
T 1 r i 1 1 1 r
.14
Cf .13
,12
Jl -
stable <"
source
distribution
^oscillating
source
distribution
± x
M=20N=IO
J .2 3 4 .5 .6 .7 .8 3's
figure 3Variation of Last
Source
Control Point for 9.=0.S
lower foil
surface
"? t-M
CO, i)
cavity
upper foilsurface
48
1°)
Partial Cavitation
Cbpj
i7l +
cavitu foilX'u~ ^ v \
COjO
W
HS
cavitu
1C-bjO)
Super Cavitation
fiqure 4Mapped Planes
t 1 r t 1 1 1 1 r
©
t—^~r
*9
Q\ dn
o«Q S
v© 5
_l L.
o ^ ^ ^ ^
f\j ^
M* tI
10
ft
V.
"M
v0
I 1
00 0>
50
JL
— Geursto computed
M=20N-IO
N-17
J ,
51
4
.9
.8
75
.7
.6
.5
.4 -
— Gears to Computed
[for values of M and Nsee Table /]
.3
.2
J
i r
1_J I I L.os a/a .i
figure 7 Cavity Length- Partial Cavitation
52
2TTQC
I lI
i i I l I i
—Geursto Computed
—[for values of Mand N r —
see Table /]/
2.0 — / -
—
p "
1.5* -
o /-
^^o mm
1.0 -
1 i 1 1 I l 1 1 1
.5
figure 8Lift Coefficient -Partial Cavitation
1.0
53
.5 1.0
figure 9 Cavity Arecft- Partial Cavitation
54
Table 1
Values of M and N for figures 5 through 9
Cavity length
Si
No. of Vortexelements
M
No. of Sourceelements
N
0.10.20.30.40.50.60.70.750.80.90.95
3030303054252530303030
1515151534202025252525
n" 1 1 1 1 1 1 1 1
Jl
o
o
- \ ° —
- OII
\ °—
5
o
—Geurst
Computed
V °—
__
o^w m
1 1 1 1 1 1 1 1 i
55
- On
-CD
-N.
IT
5rn
-r\,
3
56
t—
r
T—I—i—I—I—i—i—
r
£ o. o o
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O CDC5 C> c> C>
1 1 l1
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9.
-
a-
7.-
6.-
6Of 5>
4.-
A
— Geursto Computed
[for values of
M and N seeTable 2]
57
.4 j_ .6X
figure 12
Cavity Lengthy Super Cavitation
58
1.6
1.4
1.2
CL 1.0
2TT0C
.2
Geust
Computed
Ifor values of
M and Nsee Table 2]
.2 .6 A
figure 13
Lift Coefficient -Super Cavitation
1.6
1.4
1.2
1.0
infer*
.6
.4
.2
"i r
Geurst
59
o Computedfor values ofM and Nsee Table 2]
.8 10A , .6h
figure 14
Cavity Area-Super Cavitation
60
Table 2
Value 8 of M and N for figures 10 through 14-
No. of Vortex No. of SourceCavity length elements elements
SiM N
1.05 20 251.10 20 251.50 15 301.60 15 81.80 152.00 15 302.50 15 303.00 15 303*50 15 304.00 15 305.00 15 30
61
6
Q ' 1 1 1
JO oM-29 N d-
.098\° *#=/.59 -
.096 -
.094 -
.092 — -
.09 -
1 1i i^
20 40 60 SOnumber of equations
figure 15
Numerical Convergence, 2=0.5
PSVtteoreticar 00*58 )
62
300-
,.200cpu time
[/O^sec]
100
80
60
40
20
20 40 60 80number of equations
figure 16
Computation time
Table 3
Computed Results
63
cavity No, of No, of cx c^ Area. e lapsedlength Vortex source
a~ Ztfcx al?/0<time
elementsM
elementsN (tw^O
*.1 30 15 .0380 1.039 4.66x10"^
3.45xl0~240
•1~ 45 5 • 0485 1.002 57.2*
1115 .0569 1.066 .0129 41
.2- 10 .0627 1.035 .0119 61•3* 30 15 .0718 1.0913 .0237 42.i* 30 15 .0844 1.12 .0368 40• 5 10 5 .107 1.09 .0412 2.5 20 10 .0984 1.14 .0495 15• 5
4015 .0953 1.15 .0520 42
.5 20 .0936 1.16 .0533 94• 5 62 31 .0916 1.18 .0549 324.5 30 25 .0796 1.33 .0590 72•5» 27 17 .0913 1.19 .0536 38.5 54 34 .0891 1.20 .0558 283.5 25 20 .0819 1.30 .0576 42.6 30 25 .0882 1.39 .0785 70• 6» 30 15 .1047 1.20 .0695 *3.6* 25 20 .0909 1.36 .0765 40.7 30 25 .0952 1.48 .101 70•7* 30 15 .112 1.26 .0898 42•7 • 25 20 .0982 1.44 .0984
40
•75.8* 38 13
•°2Z?.0996
1.5371.61
.114
.1296873
• 8_ 25 20 .103 1.56 .125 43.9 30 25 .0981 1.88 .167 73• 9 * 25 20 .102 1.81 .161 41.95 30 25 .0906 2.22 .199 71.95 25 20 .0947 2.12 .191 40
Points plotted in the figures.
64
Table 3 continued
cavity No. of No. of (X cL elapsedlength Vortex
elementsM
sourceelements
N
cr ZL?f
65
Table 4
Singular Behavior
M N n*leadingedge
Vortex
n*cavity
terminationVortex
n*leadingedgesource
n*cavity
terminationsource
20
27
60
10
17
30
.093
.099
-.106
3ia
27.5
-60.5
-.79^
-.716
-.604
36
35.5
57.7
Based on only the two elements closest tothe singularity (e.g. for the leading edgevortex distribution the elements are thefirst to vortex densities) and J? * 0.5
•
66
REFRENCES
J. A. Geurst, Linearized Theory for Partially CavitatedHydrofoils, International Shipbuilding Progress, Vol. 6,No. 60, Aug. 1959.
J. A. Geurst, Linearized Theory of Two- Dimensional CavityPlows, Ph.D Thesis, Technical University, Delft,The Netherlands, May 1961.
J. A. Geurst, Linearized Theory for Fully CavitatedHydrofoils, International Shipbuilding Progress,Vol. 7 1 No. 65, Jan. i960.
S.E. Widnall, Unsteady Loads on Hydrofoils IncludingFree Surface Effects and Cavitation, MIT Fluid DynamicsLaboratory Report No. 64-2, June 1964.
R.W. Jeppson, Techniques for Solving Free-Streamline,Cavity, Jet and Seepage Problems by Finite Differences,Department of Civil Engineering, Stanford University,Report No. 68, Sept. 1968.
R.W. Jeppson, Finite Difference Solutions to FreeJet and Confined Cavity Flows Past Disks withPreliminary Analysis of the Results, Utah WaterResearch Laboratory, Utah State University,No. PRWG - 76-1, Nov. 1969.
T.R. Mogel and R.L. Street, A Numerical Method forSteady-State Cavity Flows, Department of CivilEngineering, Stanford University Technical ReportNo. 155. Feb. 1972.
G. Birkhoff, Mathraatical Analysis of Cavitation,Proceedings of the IUTAM Symposium, Leningrade,June 1971.
I.H. Abbot and A.E. Von Doenhoff, Theory of WingSections, Chs 3 +4 • Dover Publications, Inc.,New York, 1959.
67
10. N.I. Muskelishvili, Singular Integral Equations,Ch. II, Noordhoff, Groningen, 1953
•
11. Memorandum AP-67 revision 2, Massachusetts Instituteof Technology Information Processing Center,Nov, 22, 197*
•
68
APPENDIX A
COMPUTER PROGRAM
69
Appendix A includes the flow charts for the computer
program and the computer program listing. The flow charts
are provided to give a broad overview of the program flow.
The numbers between blocks in the flow charts indicate
approximate statement numbers in the program listing,
A flow chart for RMINV is not provided since it is a standard
matrix inversion routine.
70
FLOW CHQRT) MAIN PROGRAM
Contpohe
Any c0nhr0(
Computecoefficient
ComputeinverseOf- Coffificiffttna.tr\tc
C&IAXUV)
®
71
MfllA/j CONTINUED
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compose
and */?-
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compose
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72
F/.C?W CHART) CPGEhJ
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t for
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foil region
Ho
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73
FLOW CHART^ MATRIX
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APPENDIX B
SAMPLE COMPUTER OUTPUT
89
Appendix B contains the sample computer output* This
output contains the results of a convergence test for
cavity lengths equal to £ chord length and 1.5 times chord
length. Note that for a partial cavity source control
points beyond cavity termination are listed in the output.
These control points are superfluous and not used in the
program. The same is true of control points for the
vortex elements beyond chord length in super cavitation.
The listing of source and vortex densities has the
same order as the listing of control points.
NO. CF VORTICIES= 10NO. OF SOURCES= 5CAVITY LENGTH= 0.50000
90
*** DETERMIMANT=-. 1 3 1 7091 E+09***
LIFT COEFICIENT/ (2*PI*ALFA) = 0. 1092649E + 1
AREA OF CAVITY/ (2*PI*ALFA) =C. U1 17717E-01
TOTAL SODRCE STRENGTH/SIGMA = C . 2272427E-06
ANGLE OF ATTACK/SIGMA =0. 1067560E+00
CAVITY LENGTH =0. 500O000E+00
ELAPSED TIME= 2 1/100 SEC.
91NO. OF VORTICIES= 20NO. CF SOURCES = 1CCAVITY LENGTH= 0.50000
XL(I) X(I) XO(I) XS(I)0.0 0.28125E-01 0.31250E-01 0.28125E-010. 31250E-01 0.59375E-01 0.62500E-01 0.46875E-010.62500E-Q1 0. 11875E+00 C.12500E+00 0.93750E-010.12500E+00 0.18125E+00 0.18750E+00 0.15625E+000. 18750E+00 0.24375E+00 C.25000E+00 0.21875E*00C.25000E+00 0.30625E+00 0.31250E*00 0.28125E+000. 31250E+00 0.36875E+00 C.37500E+00 0.34375E*000.37500E+00 0.43125E+00 0.43750E+00 0.40625E+000.43750E+00 0.46562E+00 0.U6875E+00 0.45313E+00O.U6875E*00 0.49687E+00 C.50000E+00 0.47187E+000.50000E+00 0.52812E+00 0.53125E+00 0.51563E+000.53125E+00 0.55937E+00 C.56250E+00 0.54688E+000.56250E+00 0.61875E+00 0.62500E+00 0.59375E+000.62500E+00 0.68125E+00 0.68750E+00 0.65625E+000. 68750E+00 0.74375E+00 C.75000E+00 0.71875E+000.75000E+00 0.80625E+00 0.81250E+00 0.78125E+000.81250E+00 0.86875E+00 0.87500E+00 0.84375E+000.87500E+00 0.93125E+00 0.93750E+00 0.90625E+000.93750E+00 0.96562E+00 0.96875E+00 0.95313E*000.96875E+00 0.99687E+00 C.10000E+01 0.98438E+00
*** DETERMINANT^. 3532951E+ 1 7***
VORTEX DENSITIES/SIGMA:•0.93763E+00C6856CE+000.64077E+0C•0.U59U6E-020.63426E-01
-0.874U4E+00-0.66212E+00-0.55656E+00-0.59571E-01-0.51625E-01
SOURCE DENSITIES/SIGMA:0.80515E+00 0.53662E+000.109U9E+00 0.76209E-02-0.56007E+00 -G.13084E+01
LIFT COEFTCIENT/(2*PI*ALFA)
AREA CF CAVITY/ (2*PI*ALFA)
TOTAL SOORCE STRENGTH/SIGMA
ANGLE OF ATTACK/SIGHA
CAVITY LENGTH
ELAPSED TIME= 15 1/100 SEC.
0.79342E+00C.63783E+00C.65990E+000.71848E-01C. 37786 E-01
C.33202E+00C.89198E-01
0.1137383E+01
C4946671E-01
0.1750886E-06
=C.9841657E-01
=C.5000000E+00
0.7247UE+000.65261E+000.11008E+000.70911E-01•0.38749E-01
0.20736E+00•0.30393E+00
92NC. OF VORTICIES= 10NO. OF SOURCES^ 20CAVITY LENGTH= 1.50000
XL (I) X(T) XO(T) XS(I)0.0 0.56250E-01 0.62500E-01 0.56250E-010.62500E-01 0. 11875E+00 0. 12500E+00 0.93750E-01C.12500E+00 0.23750E+00 0.25000E+00 0.18750E+000.25000E+00 0.36250E+00 C.37500H-00 0.31250E+000.37500E+00 0.48750E+00 0.50000E+00 0.43750E+000.50000E+00 0.61250E+00 C.62500E+0O 0.56250E+000.62500E+00 0.73750E+00 C.75000E4-00 0.68750E+00C.7500CE+00 0.86250E+00 0.87500E400 0.81250E*000.87500E+00 0.93125E+00 C.93750E+00 0.90625E+000.93750E+00 0.99375E+00 0.10000F+01 0.96875E+000.13000E+01 0.10281E+01 0.10313E+01 0.10156E+010. 10313E+01 0. 10594E+01 C. 106251*01 0.10469E+010.10625E+01 0.11187E+01 0.11250E+01 0.10938E+010. 11250E+01 0. 11812E + 01 0. 11875 E+01 0.11563E+010. 11875E+01 0.12437E+01 0. 12500E+01 0.12188E+01C.12500E+01 0.13062E+01 0.13125E+01 0. 12813E+010. 13125E+01 0. 13687S+01 C.13750E+01 0.13438E+010.13750E+01 0.143122+01 0. 14375E+01 0. 14063E+010.14375E+01 0.14656E+01 0.14688E+01 0. 14531E+010. 14688E+01 0. 14969E+01 C.15000E+01 0. 14719E+01
*** DETERMINANTS. 3851246E+13***
VORTEX DENSITIES/SIGHA:-0. 11499F«-01 -0.10381E+01-0.62812E+00 -0.54372E+00-0.26088E+C0 -C.25544E+00
SOURCE DENSITIES/SIGMA:0.13267E+01 0.96160E+000.50964E+00 C.43215E+000.25084E+00 0.23408E+00-C.16451E+00 -0.37803E+00-0.98532E+00 -0.15239E+01
LIFT C0EFICIENT/(2*PI*ALFA)
AREA OF CAVITY/ (2*PI*ALFA)
TOTAL SCARCE S1RENGTH/SIGM
A
ANGLE OF ATTACK/SIGMA
CAVITY LENGTH
ELAPSED TIME= 15 1/100 SEC.
C.87268F+00C.46015E+00.
0.73057E+00C.36176E+C00.19550E-040.49625E4-00G.29826E+01
=0.5062250E*00
=C.2296074E*00
-.1788139E-06
=0.38 439 80E-t-00
=0. 1503000E+01
0.72748E+000,37014E*00
0.60112E+000.28510E+000.33281E-010.75565E*000.56046E+01
NC. OF VORTICIES= 15NO. OP SOURCES= 30CAVITY LENGTH= 1.50000
*** DETERMINANTS. 7520241E* 19***
LIFT COEFICIENT/ (2*PI*ALFA) =0. 5145274E+00
AREA OF CAVITY/ (2*PI*ALFA) =C . 2330343 E+00
TOTAL SOURCE STFENGTH/SIGM A =G. 14305 1 1 E-05
ANGLE OF ATTACK/SIGMA =0. 3788325E+00
CAVITY LFNGTH =0. 1 500000E+01
ELAPSED TIME= 39 1/100 SEC.
93
9*
APPENDIX C
COMPUTER PROGRAM FOR COMPUTATION
OP
CAVITY LENGTH FROM ARBITRARY VALUES
OF
ANGLE OF ATTACK AND CAVITATION NUMBER
95
The iterative method for computing cavity length from
arbitrary values of angle of attack and cavitation number is
based on assuming an initial cavity length ( JL 0.5 ) then
computing the next cavity length ast
(-c) 4,^[^-/In equation (1 C) the value of (-gr)
tis the value computed
from £L • This works well for moderate length partial cavities.
For super cavities the method used is to set upper
and lower boundaries on the cavity length. Then the next
assumed value of cavity length isi
The lower and upper boundaries on cavity length are
determined by comparing (-£-)t to {^.) input. For (JjL) input
greater than £jlj. the actual cavity length is greater than Jti
and so long as £L is greater than/4#-/(the lower boundary for the
cavity length is then jii • For(#) t>/)^^ less than (S^.
Jti becomes the new upper boundary. Thus the boundaries on
cavity length will converge to a solution.
The following program listing uses this procedure. This
program also uses the subroutines CPGEN, MATRIX and RMINY.
96
The iterative procedure for the super cavitating case
was developed to overcome apparent nonedivergence of the method
used in partial cavitation* However, the resulting nonconver-
gence was based on theoretical cavity lengths only slightly
greater than chord length (e.g. Jl tac 1*0*0 • In this region
of cavity length
Table C 1
Re suit8 of Cavity Length Computations
97
input computed
cr
computedcavitylength
No. ofiterations
Geurstcavitylength
Percentdifference
.025
.05
.075
.10
.15
.20
.25
.333
.501.0
.0249
.0498
.0745
.998
.1498
.199
.249
.335
.498
.9998
• 0444.145.311.730
1.061.121.191.341.8754.39
433
I96648
.045
.155
.3751.041.0991.161.251.452.05.0
-1.3-6.5-17.1-29.8-3.5-3.4-4.8-7.6-6.25-12.0
98
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Thesis 163^18G5414 Golden
A numerical methodfor two-dimensional,cavi tati ng, 1 i ftingf 1 ows
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thesG5414
A numerical method for two-dimensional,
3 2768 002 13068 4DUDLEY KNOX LIBRARY
