ON METHODS OF CALCULATING FORM OF SLENDER AXISYMMETRIC CAVITIES^

Abstract

Devoted to theoretical study of axial-symmetric cavitation flows in linearizing statement. Main attention is given to problem on determining form of axial-symmetric cavities in infinite steady flow. Investigation was undertaken by the limiting transfer in equations of boundary problem at which cavitator and cavity surfaces are considered as surfaces of thin body. Orders of smallness of terms of boundary problem equation are estimated. Problem was simplified and it solution reduced to solution of integro-differential equation for form of cavity. A simple model is developed of flow near surface of thin axial-symmetric bodies and cavities. Simple ultimate solutions to the problem were found and also more exact solutions in form of asymptotic serieses. Literat. 12, Fig. 3. The present paper is devoted to theoretical study on axial-symmetric cavitation flows in linearizing statement. Investigation was performed by passage to the limit in equations of a boundary problem at which surfaces of cavitator and cavity are considered as surfaces of a thin body. Main attention is given to determination of thin cavities form in infinite steady flow. Investigation on the one hand is directed to obtain ultimate dependences which are convenient for cavitations and to make them as much illustrative as it possible using simple models. On the other hand considerable attention was attached to determination of more exact asymptotic solutions. On formulation of the problem as problem of perturbations. Surfaces of cavitator and cavity in flow which are considered as surfaces of thin bodies consist of two relatively independent parts. It enables to introduce two parameters of thinness-separately for cavitator  and cavity  (for disk    and for infinitely thin body 0  ). Depending on ratio   problem on determination of thin cavities form turns out problem of regular [2] at  

Full text

^Reprinted with translation from: J. Hydromechanics, No.32, 1975, pp.47-54, Publ. house NAUKOVA DUMKA, Kiev, Ukraine
ON METHODS OF CALCULATING FORM OF
SLENDER AXISYMMETRIC CAVITIES^
G.V. LOGVINOVICH, V.V. SEREBRYAKOV
Institute of Hydromechanics NAS of Ukraine
Received 15.05.1974
Devoted to theoretical study of axial-symmetric cavitation flows in linearizing statement. Main attention is given to problem on deter-
mining form of axial-symmetric cavities in infinite steady flow. Investigation was undertaken by the limiting transfer in equations of
boundary problem at which cavitator and cavity surfaces are considered as surfaces of thin body. Orders of smallness of terms of bound-
ary problem equation are estimated. Problem was simplified and it solution reduced to solution of integro-differential equation for form
of cavity. A simple model is developed of flow near surface of thin axial-symmetric bodies and cavities. Simple ultimate solutions to
the problem were found and also more exact solutions in form of asymptotic serieses. Literat. 12, Fig. 3.
The present paper is devoted to theoretical study on axial-
symmetric cavitation flows in linearizing statement. Inves-
tigation was performed by passage to the limit in equations
of a boundary problem at which surfaces of cavitator and
cavity are considered as surfaces of a thin body. Main at-
tention is given to determination of thin cavities form in
infinite steady flow. Investigation on the one hand is di-
rected to obtain ultimate dependences which are convenient
for cavitations and to make them as much illustrative as it
possible using simple models. On the other hand consider-
able attention was attached to determination of more exact
asymptotic solutions.
On formulation of the problem as problem of perturba-
tions. Surfaces of cavitator and cavity in flow which are
considered as surfaces of thin bodies consist of two rela-
tively independent parts. It enables to introduce two pa-
rameters of thinness – separately for cavitator

and cavity

(for disk
  
and for infinitely thin body
0 
). De-
pending on ratio
 
problem on determination of thin
cavities form turns out problem of regular [2] at
 
1O  
or special at
 
1o  
perturbations. In the first case in
the same thin cavitator and cavity, boundary conditions in
points of jets separation in the limit are kept and can be used
in solving. Loss in the limit of boundary conditions in
points of jets separation is characteristic for the second case
of essentially distinguishing parameters of thinness. Sizes
of segment with larger parameter of thinness at
0  
become in the limit infinitely small what allows also in this
case consider cavity to be thin body.
Integro-differential equation for cavity form. Boundary
problem for axial-symmetric body detached flow by infinite
stationary flow takes the form:
 
2 2
2 2
2
10,
ln (1) (1)
r r
x r
    
  

 
 
(1)
 
 
( )
2
( )
2
,
(1) (1) ln
,
(1) (1) ln




 
 
 
 
 
 
 
 
r T x
r R x
dT dT
V
r dx x dx
dR dR
V
r dx x dx
(2)
 
 
2 2
( )
12
( ),
1 1 ( ) ,
2 2
ln ln (1) (1)
r R x
T T x
V P x
r x x



  
   
      
   
  
   
  
(3)
, , 0,r x    
(4)
2 2 2 2 2 2
2 2
2 2
, ,
x xox x x x
o o
dR dT d R d T
R T
dx dx dx dx
 
  
(5)
Here
,r x
- are the coordinates of point in cylindrical coor-
dinate system;
 
T T x
,
 
R R x
- is the equation of
meridian cavitator and cavity;
   
c
P x P P

  
- is the
difference between pressure in undisturbed and disturbed
flow at cavity surface;
o
x x
- is the coordinate of points
of jets separation. The last condition from (5) is satisfied
only in case of free jets separation.
It is necessary to find an approximate solution to the prob-
lem (1) ÷ (5) for thin cavity form. Boundary conditions (2)
and (3) are satisfied at surface of a thin body (near axis). So
approximate solution to the problem will be sought in
agreement with passage to the limit in term of parameter of
thinness satisfied for points close to surface in its middle
part (internal – over
r
, external – over
x
solution [2]) un-
der conditions

.
.
2
(1), (1),
(1), (1), (1).
O V O
x r
L O O O
L L

  
  

(6)
To estimate orders of smallness of terms of boundary prob-
lem equations asymptotic representations were used:
1) for cavitator and cavity –
 
,T T x  
 
,R R x  
where
   
1 ,T x O

   
1 ;R x O

2) for potential of velocity near surface – known from
the theory of thin bodies [9] –
2
2
( )
1,
2
( )
1,
2
T
k
d T x
V ln r
dx
d R x
V ln r
dx


 

 
 
 

 
 






where
 
1 .
r
r O 


Substitution of these representations
taking into account conditions of passage to the limit (6)
will determine orders of smallness of terms of problem
equations (1) ÷ (5). Result of estimation in form of “rela-
tive” (with respect to the largest term in the equation) orders
is written below each equation of the problem.
Neglecting small values
2ln 
we receive more simple
problem:
2
2
10,
r r
r
  
 


( ) ( )
, ,
r T x r R x
dT dR
V V
r dx r dx
 
 
 
 
 
(7)
2
( ,)
1
( ), ( )
2r R x
T T x V P x
r x

 
   
     
   
 
   
2 2
2 2
2 2 2 2
2 2
, ,
.
x xox xo
x xo
dR dT
R T
dx dx
d R d T
dx dx


 

In this problem boundary conditions at infinity appear to be
lost. Derived expression for potential in form sources along
the axis and using its expansion for small
r
[9], solution to
the problem (7) is reduced to solving integro-differential
equation for the cavity form (
0x
corresponds to nose
point of cavitator):
 
   
     
2 2 2 2
2 2
22 2 2
11
21
0
1 1
2 2 2 2 2 2
2 2
0
112
1
1 1 1
2 ln 4
ln (1) ln
( )
2
2
ln ln ln (1)
xo
x x
Lx x x L x
xo
d T d R
dx dx
dR d R R dx
dx x L x x x
dx
d R d R dR dT
dx dx dx dx P x
dx
x x L x x V

 
 

  

    
   
 
 


  
  
  


(8)
at boundary conditions in separation points
o
x x
2 2
2 2 , ,
x xox xo
dR dT
R T
dx dx

 
(9)
condition for determining coordinates of these points and
condition for determining
L
2 2 2 2
2
2 2 , 0 .
x L
x xo
d R d T R
dx dx 

 
(10)
“Ring” model of flow near surfaces of thin bodies and
cavities. Introducing a part of terms under the logarithm
sign the equation (8) may receive the following form:
22 2
2 2
( )
ln ,
2
dR d R P x
dx R
dx V
 
   
  
 
(11)
which enables to interpret the equation by the simple
method (Fig. 1). Essence of this coming from the equations
(11) consists in the following. Pressure at surface of thin
axial-symmetric body or the law of expansion of lateral sec-
tions of thin axial-symmetric cavity, with the accuracy up
to small values
2ln 
, occurs the same as if flow near
their surface would be modeled by layer of cylindrical ex-
panding fluid rings with external radius
 
x  
, internal
–
 
.R R x
Value
 
x  
characterizes fluid amount
which disturbs body and cavity in every section. Fluid par-
ticles, being outside surface
 
x  
, with the accuracy
up to
2ln 
, do not have effect on surfaces of thin bodies
and cavities. Value
ln R

 
characterizes inertial proper-
ties of fluid ring with respect to applied along its surface
pressure difference
 
P x
and may be conditionally named
by coefficient of added mass of cavity ring. External sur-
face of ring layer is depicted as an example in case of ellip-
soid cavity in Fig. 1. The equation of this surface takes the
form

.
.
3
1
2
21
2 1 ,
2
x
a
xe
a

 
 
 
 
   
 
where
a
- is the half-width of cavity (longitudinal half-axis
ellipsoid).
At cavitator and cavity thinness parameter tending to zero
main perturbation in flow – inertia force energy of fluid
particles is accumulate near their surface. In this sense flow
of ideal fluid near surfaces of thin bodies and cavities turns
out similar boundary layer or wake. The of expansion of
lateral sections of thin cavity turns out in the limit not de-
pending on actual form of function
 
x  
and is deter-
mined only by elements of section in question. Variable
coefficient
 
x  
turns out constant in the limit depend-
ing on surface lengthening. It is easy to show by differenti-
ating the equation (8) with respect to
x
and neglecting
small values
 
1
ln .


As a result ultimate theory of thin
cavities may be expressed by the equation
2 2
2 2
( ) 0,
2
d R P x
dx V

  

(12)
where
const 
is determined by the expression
ln  
;

- is the value of external ring radius in some middle sec-
tion of cavitator and cavity surfaces, related to half-length;

- is the lengthening of this surface.
Ultimate dependences. Form of thin cavity behind thin
cavitator may be determined in the first approach from the
equation (12) and boundary conditions in separation points.
Condition for finding coordinates of separation points at
free jets department in the limit takes the following form:
2 2
2 2
( ) 0 .
2x xo
d T P x
dx V


 

(13)
This condition in some cases is useless in particular don’t
allow determining coordinate of separation points for a cav-
ity at
P 
const
behind ellipsoid. The equations (12) and
(13) are the more applicable for calculation the thinner cavi-
tator and cavity, and the less distinctions in their thinness
parameters. The last condition can be written in the form
 
2ln 1O   
and was obtained in the work [3]. Solu-
tion to the equation (12) at
P 
const
coincides with solu-
tions to equations differed from (12) and found in the works
[1, 4]. This solution is ellipsoid of rotation
2 2
22
1,
2
cc
R x
RR
 


(14)
where
2
,
c c c
R L R


-are the half-axises. As compari-
son with experimental data shows deviation of cavity ex-
perimental contour from ellipsoid is within the range 5÷7 %
and the less the thinner cavity is. In case of thin cavity be-
hind not thin cavitator
2
ln ln ,

     
(15)
approximate value

may be found by the formula
2
1
1 2
ln , 1.
2
n n o
 

    
 

 
(16)
Dependence of cavitation number on lengthening in the
limit takes the form
2
2ln .  

(17)
Comparison with experimental data gives value
0.54 0.64. 
Dependence for
 
    
 
 
at
0.61 
is presented in Fig. 2.
Infinite smallness in the limit of cavitator compared to cav-
ity and also property of limiting independence of it sections
expansion enables to consider formation of thin axial-
Fig. 1. Ring model
Fig. 2. Dependence
[ ( )]    
at
P const 

.
.
4
symmetric cavity which occurs in the following way. In
motionless fluid infinitely small field of non-uniformity
(“point”) moves. Threading fluid this field of non-
uniformity gives rise to cavity ring which after receiving
some initial velocity of expansion at zero radius begins to
expand independently. In the limit the surface
 
x  
may be considered jet surface of which total energy of each
ring at
P 
const
is kept. All energy of ring by thickness
dx
is potential in midship. Writing energy conservation
equation in the form
 
2,
o k c
W dx P P R dx

  
(18)
we receive a known formula for the largest radius. At initial
moment all ring energy is kinetic what leads to the equation
2
22
2 2
o
VdR
W dx dx
dx
 

 
 
 
(19)
and enables to find expression for initial velocity of section
expansion
2
0x
dR
dx 
valid also at
 
.P P x  
Using this
expression ultimate model of thin cavity formation may be
described by the equations
2 2
2 2
2
2
02
0
( ) 0,
2
0 , 2 .
o
x
x
d R P x
dx V
W
dR
R
dx k V



  

  
(20)
The equations are similarly derived for non-stationary cav-
ity
2 2
2
2
2
( , )
2 0,
0 , 2 ,
o
x xk
t t x x
ok
t to
R P x t
t
W
R
R
t k



 
  



 
 
(21)
where
c
x x
- is the section coordinate;
o
t t
corresponds
to starting moment of expansion; correction
k
is similar
correction in the known formula for the largest cavity radius
[5]; value

due to its independence is taken also the some
as for cavity at
P 
const
. The equations (20) and (21)
can be considered as simplified equations of known princi-
ple of cavity “expansion independence” [5]. These equa-
tions giving at
P 
const
results coinciding in middle part
of cavity with results of calculations by existing equations
[5] simplify cavity calculation in case of variable pressure
difference
 
, .P x t
Asymptotic solutions. Using dependences (17) and (18) it
is not difficult to show that in case of thin cavity behind not
thin cavitator, cavitator sizes compared to cavity length are
small
 
2ln .O 
It allows to neglect in the equation (8)
term taking into account cavitator form. Selecting cavity
half-length as a scale and relating coordinate system to its
middle section we write the boundary problem for this
equation in the following form:
2
2 2 2 2
2 2
2 2 2 2
2 2
1
1
1
1
1
2 2
1 1
2 2 2 2
1 1,
0
1ln 4(1 )(1 )
2
2 ( ),
1 1
0 , , 0
x x
x x
x x
x
dR d R R
dx x x
R dx
d R d R
dx dx
dx
x x
dR dR
dx dx
x
x x
R R R


 
 

   
 
   
 

 

   
 
   


(22)
where
   
2
, .
2c c
V
x P x R L


    

Assuming that
 
,x 

are known solutions for
 
2 2 ,R R x 
and
 
   
were sought in the form of asymptotic serieses
1 2
2 2 2 2 2
1 2
2 2
1
2
1 1
2 2
1 1
ln ln ... ,
1 1
ln ln ...
o
o
R R R R
 
 


 
   
    
 
   
 
   
 
 
 
   
        
 
   
 
   
 
 
(23)
and reduced to solving succession of linear problems. Two
terms of cavity asymptotic series were found as an example
at
P 
const
:
 
 
 
 
 
2 2 2
1 1
2
1
ln 4 ln 1 1
2ln
c
x x
R R x
x x x
 

  


    


,
(24)
2
2ln ,
e

  
(25)
where
c
x
x
R

and value
c
R
remained unknown because
of loss boundary conditions in separation points may be
determined by the known formula [5]
 
1.
xo
c n
c
R R
k
 

(26)
If the formula (17) and (25) are compared then it turns out
that value
0.54 0.64 
obtained in experimental way

.
.
5
theoretically equals
10.605.
e
  
Comparison with
experimental data confirms well the solution found. For
example results of cavitations by expression (25) compared
to dependences [5] available are presented in Fig. 3.
In case of thin cavity behind thin cavitator
 
T T x  
it is
assumed that values
   
, ,T x x

thinness parameter of
cavitator
,
length of cavitator as long as separation points
,l
cavity length
c
l
– are known. Solution to the problem for
the equation (8) under conditions (9) and (10) was sought in
form of asymptotic serieses
1 2
2 2 2 2 2
1 2
2 2
1 1
2
1 1
2 2
1 1
ln ln ... ,
1 1
ln ln ...
o
o
R R R R
 
 


 
   
    
 
   
 
   
 
 
 
   
        
 
   
 
   
 
 
(27)
and reduced to solution of succession of linear problems.
After their solving cavitator length as far as separation
points
l
in the first approach can be found from the equa-
tion (origin of coordinates
0x
is placed in section of jets
separation)
2 2 2 2
2 2
0
,
o
x
d R d T
dx dx 

in the second approach – from the equation
1
2 2 2 2 2 2
1
2 2 2 2
0
1
ln
o
x
d R d R d T
dx dx dx



 
 
 

 

and so on.
For example two terms of asymptotic series were found in
case of cavity at
P 
const
behind cone
 
T x l  
in
the form
 
 
 
 
 

 
 
2 2
1
12
1
12
12
21
2
1
1
1
2
2 2
2
1 1 1
2 2
12
ln
ln 4
1
ln ,
1
ln ,
xL L x L
L L x l
x L x
o
o
R x L x L
x L x
x L x l
ex
a b
l
L x

  

  

 

 


 
 
  
 

 
     
 
 
 

 
  
 
 
 
 
 

 


  
 

 


 
   
    

 
   
   
 
 
  


 
      
 

 
(28)
where expressions for
1 2 2
, , , , o
L a b

 
are determined in
solving and value
L
should consider known. The solution
found is confirmed by experimental data the better the thin-
ner cavitator and cavity are and the less difference of their
thinness parameters.
It should be noted that in general case solution to the first
two problems of successions of linear problems at “arbi-
tary” cavitator form
 
T T x
and pressure difference
 
P P x  
is expressed in quadratures. Solution for cav-
ity at
P 
const
and for vertical cavities is obtained in
form of finite analytical expressions.
Finally we will note that theory of thin bodies was used for
calculation of axial-symmetric cavitational flows in the
works [3, 10] and later in the works [8, 11, 12]. The equa-
tions considered in the works [3, 10, 12] for stationary case
result from the equation (8) when neglecting a part of its
terms. Equations for cavity form which is considered as a
wake energy of which is kept in every lateral section were
derived in the works [1, 4, 5].
Asymptotic study per formed allows to determine these
works from the same point of view. Basis of solutions
which are found by the theory of thin bodies is property of
limiting independence of expanding cavity sections [5].
Consideration of cavity as a wake energy of which is kept
[1, 4] and application of the theory of thin bodies [3, 8, 10 ÷
12] result in similar results. Solutions derived from the
equation of energy conservation are the first terms of as-
ymptotic solution to equations found by the theory of thin
bodies.
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Fig. 3. Dependence
( )   
compared to data of the work [5]:
1 – by G.V. Logvinovitch; 2 – by L.A. Epshtein;
3 – by the formula [25].

.
.
6
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