Recently Tulin and Wu have treated the problem of fully developed cavitation on flat plate and cambered foils.  In these treatments, the length of the cavity is always greater than the chord of the hydrofoil and the cavity is assumed to start at the leading edge of the plate.  The purpose of this note is to extend Tulin's work to account for partial cavitation, i.e., when the cavitation bubble is less than the hydrofoil chord

Acosta, A. J.

English

Caltech Authors - Main

Office of Naval Research  
Department of the Navy 
Contract N6onr-244, Task Order 11 
A NOTE ON PARTIAL CAVITATIOK OF F L A T  PLATE HYDROFOILS 
A. J. Acosta 
Reproduction in  whole or  in  par t  is  permitted for any 
purpose of the United States Government 
Hydrodynamics ~ a b o r a t o r ~  
California Institute of Technology 
Pasadena, California 
Report No. E - 19.9 
October 1955 
Approved by: 
A. Hollander 
Introduction 
i e c e n t l y  ~ u l i k '  and Wu2 have t reated the problem of fully developed 
cavitation on flat  plate and cambered foils, In the se treatments,  the length 
of the cavity is always grea ter  than the chord of the hydrofoil and the cavity i s  
assumed to s t a r t  a t  the leading edge of 'the plate. .The purpose of this  note is  
to extend Tul in ls  work to account for partial  cavitation, i. e ,  , when the cavita- 
tion bubble is l e s s  than the hydrofoil chord. 
Notation 
constants 
cavity length parameter 
chord 
p res su re  coefficient = p - p U 2 
P 2 lift coefficient = lift/c p U 
1 2  cavity number = pm - Pk/Z p U 
cavity length 
p res su re  (pm denotes f ree  s t r e a m  pressu re ,  
p cavity pressure)  k 
f ree  s t r e a m  velocity 
perturbation velocity in x direction 
perturbation velocity in y direction 
perturbation velocity function = u - i v 
physical. plane = x + i y 
angle of attack 
auxiliary plane s 
coordinate of hydrofoil 
Statement of Problem 
The cavity bubble is a surface of constant pressure .  Thus, on the wetted 
portion of the hydrofoil the slope of the foil i s  specified, and on the cavity por-  
tion the magnitude of the velocity i s  specified. The flow is shown schematically 
in Fig.  I .  The assumptions of thin airfoi l  theory will now be made, i. e. , the 
slope, angle of attack, camber and thickness a r e  sma l l  so  that the boundary 
conditions can be written a s  follows: On .the wetted portions 
(i) v = -  dr) U ,  dx  
and on the cavity 
K 
(ii) u = - U . 2 
Equation (i) follows f rom the l inearized Bernoulli equation and the definition 
of cavitation number K. I t  is fur ther  required that the perturbation velocities 
u, v vanish a t  infinity, i. e . ,  
(iii) U, v .--0 a s  2; .--.-a0 , 
and that the velocity be finite a t  the trailing edge. The final res t r ic t ion  is that 
the shape of cavity plus hydrofoil i s  closed. This condition is expressed  
mathematically as 
Method of Solution 
The solution i s  effected by conformal mapping of the complex velocity 
function u - i v  and is the same a s  Tul in ts  procedure.  The hydrofoil and  cavity 
is regarded as a s l i t  in  the physical plane (I. plane) and this s l i t  is mapped onto 
auxiliary planes in such a way a s  to separate the cavity and wetted portions. 
Thus consider the sequence of mappings: 
and 
where 
The relation between the various planes- is shown in  Fig.  2. Equation (1) maps  
the s l i t  in the physical plane onto the r e a l  ax is  in the plane in such a way 
the cavity placed on Re G : < :-- 1 and the upper half < plane is mapped 
onto the ent i re  z plane. The point '4 = i a corresponds to infinity in  the z 
plane. Equation (2) t ransforms the cavity portion of the r e a l  axis in  the 
plane to the upper half of the unit c i rcle  i n  the 4: plane, the wetted portion 
corresponding to r e a l  -1:: < :z 1. The points A and B correspond to  the 
leading and trail ing edges of the hydrofoil and point C designates the end of 
the bubble. In both auxiliary planes the trail ing edge B is a t  infinity. 
Solution for F l a t  ?late Hydrofoil 
In this case  Condition (i) can be written a s  
where n is the angle of attack. In the g plane, therefore,  u mus t  be con- 
stant on the upper half of the unit c i rc le  and v must  be a constant on the r e a l  
axis.  It is known that a source located a t  the points 5 = f 1 sat isf ies  these 
conditions, and, in  addition, gives a zero velocity a t  infinity. Thus i n  the 
E plane the complex velocity must be of the fo rm 
where A,  B, C, D a r e  r e a l  constants, The introduction of the plane was  
mere ly  to i l lustrate  the type and location of singularit ies required. The velo - 
city function above can be transformed to the C plane where i t  takes the form 
Conditions ( i )  through (iv) can now be applied to determine the constants. 
Application of Conditions (i) and (ii) give 
and 
D =  U a  . 
At the point < = i a, the perturbation"velocities must  vanish, which gives 
two additional relations: 
7F where 0 = -- + tan-' (l/a). Another relation between the constants A and B 2 
i s  provided by the closure requirement of Condition (iv). This  condition can 
be written a s  
d y =  j6 v(x) dx = 0 , 
I 
body body 
body 
where Irn cienotes the imaginary par t  of the integral.  A s  the boundary of the 
foil is t raced  out i n  the countercloclr,wise direction in the z plane, 2: proceeds 
along the r e a l  a x i s  f r o m  positive to negative infir4ty. Thus, the following 
integral must be considered 
where dz/dc has been computed from Eq. ( 1 )  and w ( c )  is to be found from 
q s .  ( 4  (5). The integrand of J contains double poles a t  2= = i a and i s  
such that J can be evaluated by contour integration, i. e . ,  
J = - 2 r r i ( ~ e s .  a t  < =  i a )  . 
The resul t  of these operations i s  
7 ~ a  1/221/2B ( 3 o 7 3-0 J = 3 0  sin - -  i ( c o s y +  sin -) i 
4(a2+ 1)3/47 1 'OS -2- 2 2 1  
J 
s a  - 1/2f 1/2* . 'cos ( 0  q t s in -  0 + i ( c o s  z 8 - 2 (8) 4(a2t  1 p 4  1 
I 
The requirement that 1m J = 0 gives the required additional relation between 
the constants A and B. This  resul t ,  together with Eq. (5), i s  used to 
determine the constants, and so to obtain the relation between the cavity 
length t , cavitation number K and angle of attack a, i. e. ,  
This equation is plotted in Fig. 3. 
Calculation of the Lift 
If the usual limitations of thin airfoil  theory a r e  observed, the lift 
coefficient can be determined from the approximate formula 
body 
Thus 
body 
where R e  denotes the r e a l  par t  of the integral.  F r o m  Sqs .  ( 6 ) ,  (7) i t  is seen 
that 
The lift coefficient become s a f te r  some manipulation 
F o r  8 / c  4 C I the approximate resul t  
i s  obtained. 
Discussion 
F r o m  the plot of ~ / 2 a  vs bubble length i n  Fig.  3 i t  is seen that .U/a 
has  a minimum value when the cavity-chord rat io  (c /c )  is about 0. 74. The 
theory is thus l imited to values of C /c c.0.74, otherwise i t  would be possible 
to have two different cavity lengths for  the same cavitation number and angle 
of attack. Due to the res t r ic t ions  of th? thin airfoi l  assumptions,  the theory 
i s  not likely to be much good for  8 / c  z7 0.5 anyway. Thus the thin a i r fo i l  
theory fails for  C /c 5 1 and to obtain more accurate  information there  a 
more  exact method would have to be used, for  example a hodograph theory. 
Tul ints  r e su l t  (Ref. 1) is plotted for  a = 5.7' in  Fig.  3, and i t  a l so  shows the 
singular behavior near  C / c  = 1. 
The effect of cavitation number on lift coefficient is shown in Fig. 4. F o r  
K large and 8 / c  sma l l  the lift approaches the f la t  plate value. Some experi-  
mental lift measurements  on a sharp-edged Karman-Trefftz profile a r e  also 
plotted on  Fig.  4.3 The agreement  with the flat  plate theory is good for high 
cavitation numbers;  however, the theory is inadequate for  combinations of 
cavitation number and angles of attack that r e su l t  in  comparatively long cavities. 
That is, i t  does not predict  the lift maximum found experimentally a t  the lower 
cavitation parameters .  However, i t  is possible that r e a l  fluid phenomena a r e  
partially responsible for this result .  
These r e su l t s  a r e ,  s t r ic t ly  speaking, applicable only to f la t  plate 
hydrofoils with cavitation starting a t  the sha rp  leading edge. The effect of 
cavitation on prof,ile shapes with thickness i s ,  no doubt, much m o r e  compli- 
cated. 
References 
1. Tulin, M, ?. , "Supercavitating Flow P a s t  Fo i l s  and St ru ts t t ,  
paper no. 16, rJymposium on Cavitation in Hydrodynamics, 
1955, National Yhysical Laboratory, Teddington, Zngland, 
2. Wu, T. Yao- tsu, "A F r e e  Streamline Theory for Two-Dimensional 
Fully Cavitated Hydrofoilst1, California Institute of Technology, 
Hydrodynamics Laboratory Report No. 21-17, July, 1955, 
3,  Balhan, J, , I1ivIetingen Aan Jn ige  Bij  Scheepsschroenen 
Gebruikelijke l ro f i e l en  In vlokke Strorning idit En Zonder 
Cavitatie", Ned. Schcepsbonrokundig Proefstation Te 
Wageningen, 2ublicatie No, 99, 195 1. 
i y  l Z P L A N E  
CAVITY  7 1 
Fig. 1 - Hydrofoil with par t ia l  
cavitation. 
-lJ HYDROFO l L 
i. 
I Y l  
Z P L A N E  
CAVITY  7 
t -c/2 
[ P L A N E  
/ 
/ --+- 
0 A C B 
2 - c  
\\c/2 d 
Fig. 2 - Representation of hydrofoil in 
the z- and auxiliary planes. 
0.5 1 .O 
CAVITY-  CHORD R A T I O  - l / c  
= X 
Fig. 3 - Cavity length in  t e r m s  of cavitation number 
and angle of a t tack for  a flat  plate hydrofoil. 
77 ---.
Fig. 4 - Ratio of cavitating to noncavitating lift coefficient 
v s  cavitation number for a flat plate hydrofoil. 
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A Note on Partial Cavitation of Flat Plate Hydrofoils

Office of Naval Research  
Department of the Navy 
Contract N6onr-244, Task Order 11 
A NOTE ON PARTIAL CAVITATIOK OF F L A T  PLATE HYDROFOILS 
A. J. Acosta 
Reproduction in  whole or  in  par t  is  permitted for any 
purpose of the United States Government 
Hydrodynamics ~ a b o r a t o r ~  
California Institute of Technology 
Pasadena, California 
Report No. E - 19.9 
October 1955 
Approved by: 
A. Hollander 
Introduction 
i e c e n t l y  ~ u l i k '  and Wu2 have t reated the problem of fully developed 
cavitation on flat  plate and cambered foils, In the se treatments,  the length 
of the cavity is always grea ter  than the chord of the hydrofoil and the cavity i s  
assumed to s t a r t  a t  the leading edge of 'the plate. .The purpose of this  note is  
to extend Tul in ls  work to account for partial  cavitation, i. e ,  , when the cavita- 
tion bubble is l e s s  than the hydrofoil chord. 
Notation 
constants 
cavity length parameter 
chord 
p res su re  coefficient = p - p U 2 
P 2 lift coefficient = lift/c p U 
1 2  
cavity number = pm - Pk/Z p U 
cavity length 
p res su re  (pm denotes f ree  s t r e a m  pressu re ,  
p cavity pressure)  k 
f ree  s t r e a m  velocity 
perturbation velocity in x direction 
perturbation velocity in y direction 
perturbation velocity function = u - i v 
physical. plane = x + i y 
angle of attack 
auxiliary plane s 
coordinate of hydrofoil 
Statement of Problem 
The cavity bubble is a surface of constant pressure .  Thus, on the wetted 
portion of the hydrofoil the slope of the foil i s  specified, and on the cavity por-  
tion the magnitude of the velocity i s  specified. The flow is shown schematically 
in Fig.  I .  The assumptions of thin airfoi l  theory will now be made, i. e. , the 
slope, angle of attack, camber and thickness a r e  sma l l  so  that the boundary 
conditions can be written a s  follows: On .the wetted portions 
(i) v = -  dr) U ,  dx  
and on the cavity 
K (ii) u = - U . 2 
Equation (i) follows f rom the l inearized Bernoulli equation and the definition 
of cavitation number K. I t  is fur ther  required that the perturbation velocities 
u, v vanish a t  infinity, i. e . ,  
(iii) U, v .--0 a s  2; .--.-a0 , 
and that the velocity be finite a t  the trailing edge. The final res t r ic t ion  is that 
the shape of cavity plus hydrofoil i s  closed. This condition is expressed  
mathematically as 
Method of Solution 
The solution i s  effected by conformal mapping of the complex velocity 
function u - i v  and is the same a s  Tul in ts  procedure.  The hydrofoil and  cavity 
is regarded as a s l i t  in  the physical plane (I. plane) and this s l i t  is mapped onto 
auxiliary planes in such a way a s  to separate the cavity and wetted portions. 
Thus consider the sequence of mappings: 
and 
where 
The relation between the various planes- is shown in  Fig.  2. Equation (1) maps  
the s l i t  in the physical plane onto the r e a l  ax is  in the plane in such a way 
the cavity placed on Re G : < :-- 1 and the upper half < plane is mapped 
onto the ent i re  z plane. The point '4 = i a corresponds to infinity in  the z 
plane. Equation (2) t ransforms the cavity portion of the r e a l  axis in  the 
plane to the upper half of the unit c i rcle  i n  the 4: plane, the wetted portion 
corresponding to r e a l  -1:: < :z 1. The points A and B correspond to  the 
leading and trail ing edges of the hydrofoil and point C designates the end of 
the bubble. In both auxiliary planes the trail ing edge B is a t  infinity. 
Solution for F l a t  ?late Hydrofoil 
In this case  Condition (i) can be written a s  
where n is the angle of attack. In the g plane, therefore,  u mus t  be con- 
stant on the upper half of the unit c i rc le  and v must  be a constant on the r e a l  
axis.  It is known that a source located a t  the points 5 = f 1 sat isf ies  these 
conditions, and, in  addition, gives a zero velocity a t  infinity. Thus i n  the 
E plane the complex velocity must be of the fo rm 
where A,  B, C, D a r e  r e a l  constants, The introduction of the plane was  
mere ly  to i l lustrate  the type and location of singularit ies required. The velo - 
city function above can be transformed to the C plane where i t  takes the form 
Conditions ( i )  through (iv) can now be applied to determine the constants. 
Application of Conditions (i) and (ii) give 
and 
D =  U a  . 
At the point < = i a, the perturbation"velocities must  vanish, which gives 
two additional relations: 
7F 
where 0 = -- + tan-' (l/a). Another relation between the constants A and B 2 
i s  provided by the closure requirement of Condition (iv). This  condition can 
be written a s  
d y =  j6 v(x) dx = 0 , 
I 
body body 
body 
where Irn cienotes the imaginary par t  of the integral.  A s  the boundary of the 
foil is t raced  out i n  the countercloclr,wise direction in the z plane, 2: proceeds 
along the r e a l  a x i s  f r o m  positive to negative infir4ty. Thus, the following 
integral must be considered 
where dz/dc has been computed from Eq. ( 1 )  and w ( c )  is to be found from 
q s .  ( 4  (5). The integrand of J contains double poles a t  2= = i a and i s  
such that J can be evaluated by contour integration, i. e . ,  
J = - 2 r r i ( ~ e s .  a t  < =  i a )  . 
The resul t  of these operations i s  
7 ~ a  1/221/2B ( 3 o 7 3-0 J = 3 0  sin - -  i ( c o s y +  sin -) i 
4(a2+ 1)3/47 1 'OS -2- 2 2 1  
J 
s a  - 1/2f 1/2* 
. 'cos ( 0  q t s in -  0 + i ( c o s  z 8 - 2 (8) 4(a2t  1 p 4  1 I 
The requirement that 1m J = 0 gives the required additional relation between 
the constants A and B. This  resul t ,  together with Eq. (5), i s  used to 
determine the constants, and so to obtain the relation between the cavity 
length t , cavitation number K and angle of attack a, i. e. ,  
This equation is plotted in Fig. 3. 
Calculation of the Lift 
If the usual limitations of thin airfoil  theory a r e  observed, the lift 
coefficient can be determined from the approximate formula 
body 
Thus 
body 
where R e  denotes the r e a l  par t  of the integral.  F r o m  Sqs .  ( 6 ) ,  (7) i t  is seen 
that 
The lift coefficient become s a f te r  some manipulation 
F o r  8 / c  4 C I the approximate resul t  
i s  obtained. 
Discussion 
F r o m  the plot of ~ / 2 a  vs bubble length i n  Fig.  3 i t  is seen that .U/a 
has  a minimum value when the cavity-chord rat io  (c /c )  is about 0. 74. The 
theory is thus l imited to values of C /c c.0.74, otherwise i t  would be possible 
to have two different cavity lengths for  the same cavitation number and angle 
of attack. Due to the res t r ic t ions  of th? thin airfoi l  assumptions,  the theory 
i s  not likely to be much good for  8 / c  z7 0.5 anyway. Thus the thin a i r fo i l  
theory fails for  C /c 5 1 and to obtain more accurate  information there  a 
more  exact method would have to be used, for  example a hodograph theory. 
Tul ints  r e su l t  (Ref. 1) is plotted for  a = 5.7' in  Fig.  3, and i t  a l so  shows the 
singular behavior near  C / c  = 1. 
The effect of cavitation number on lift coefficient is shown in Fig. 4. F o r  
K large and 8 / c  sma l l  the lift approaches the f la t  plate value. Some experi-  
mental lift measurements  on a sharp-edged Karman-Trefftz profile a r e  also 
plotted on  Fig.  4.3 The agreement  with the flat  plate theory is good for high 
cavitation numbers;  however, the theory is inadequate for  combinations of 
cavitation number and angles of attack that r e su l t  in  comparatively long cavities. 
That is, i t  does not predict  the lift maximum found experimentally a t  the lower 
cavitation parameters .  However, i t  is possible that r e a l  fluid phenomena a r e  
partially responsible for this result .  
These r e su l t s  a r e ,  s t r ic t ly  speaking, applicable only to f la t  plate 
hydrofoils with cavitation starting a t  the sha rp  leading edge. The effect of 
cavitation on prof,ile shapes with thickness i s ,  no doubt, much m o r e  compli- 
cated. 
References 
1. Tulin, M, ?. , "Supercavitating Flow P a s t  Fo i l s  and St ru ts t t ,  
paper no. 16, rJymposium on Cavitation in Hydrodynamics, 
1955, National Yhysical Laboratory, Teddington, Zngland, 
2. Wu, T. Yao- tsu, "A F r e e  Streamline Theory for Two-Dimensional 
Fully Cavitated Hydrofoilst1, California Institute of Technology, 
Hydrodynamics Laboratory Report No. 21-17, July, 1955, 
3,  Balhan, J, , I1ivIetingen Aan Jn ige  Bij  Scheepsschroenen 
Gebruikelijke l ro f i e l en  In vlokke Strorning idit En Zonder 
Cavitatie", Ned. Schcepsbonrokundig Proefstation Te 
Wageningen, 2ublicatie No, 99, 195 1. 
i y  l Z P L A N E  
CAVITY  7 1 
Fig. 1 - Hydrofoil with par t ia l  
cavitation. 
-lJ HYDROFO l L 
i. 
I Y l  
Z P L A N E  
CAVITY  7 
t -c/2 
[ P L A N E  
/ / 
--+- 
0 A C B 
2 - c  
\\c/2 d 
Fig. 2 - Representation of hydrofoil in 
the z- and auxiliary planes. 
0.5 1 .O 
CAVITY-  CHORD R A T I O  - l / c  
= X 
Fig. 3 - Cavity length in  t e r m s  of cavitation number 
and angle of a t tack for  a flat  plate hydrofoil. 
77 ---.
Fig. 4 - Ratio of cavitating to noncavitating lift coefficient 
v s  cavitation number for a flat plate hydrofoil. 
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Supercavitating Flow P a s t Fo i l s and St ru ts t t , paper no. 16, rJymposium on Cavitation in Hydrodynamics,

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A Note on Partial Cavitation of Flat Plate Hydrofoils

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