In the present paper, the two-dimensional problem of the wave motion produced in a heavy fluid of finite depth by the horizontal rectilinear and uniform motion of a solid body of arbitrary shape immersed under the surface of the fluid is considered by the method of N. E. Kochin

Haskind, M D

English

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NATIONAL aMTISORY COJXMITTEE FOR AERONBUTICS 
TECHNICAL MFMOl3UDuM 1345 
!lBANSMTIONAL MOTION OF BODIES UNDER THE FlEiEE SURFACE OF A 
FIEAVY FLUID OF FIEITE DEPTH* 
By Me D, Haskind 
In reference 1, entitled "The Two-Dimensional Problem of the 
Vibration of Bodies under the Surface of a Heavy Fluid of Finite Depth," 
the pr~blem was to determine the wave motion of a heavy fluid excited 
by the periodic vibrations of a body of arbitrary shape situated under 
the free surface of the fluid of finite depth; the method of N. E. Kochin 
(reference 2) was used, 
In the present paper, the two-dimensional problem of the wave 
motion produced in a heavy fluid of finite depth by the horizontal 
rectilinear and uniform motion of a solid body of arbitrary shape 
immersed under the surface of the fluid is considered by the same 
method . 
1. Statement of the Problem 
The problem of the translatory motion of a solid body under the 
free surface of a heavy incompressible fluid of finite depth will be 
considered. The case in which the motion of the body occurs with con- 
stant horizontal velocity c will be studied, The motion of the fluid 
w i l l  be defined with reference to a moving system of coordinates Oxy 
fixed to the body, the x-axis coinciding with the undisturbed level of 
the fluid and directed along the direction of motion of the body, and 
the y-axis directed vertically upward. 
It will be assumed that the motion of the fluid is potential and 
steady relative to the body. From the integral of Lagrange for the 
pressure within the fluid, 
*"O postupatelnom dvizhenii tel pod svobodnoi poverkhnostlyu 
tyazheloi zhidkosti konechnoi glubiny, Prikladnaya Matematika i 
Mekhanika," vol. IX, Sept. 1945, pp. 67-78. 
where po is the 'atmospheric pressure, p the density of the fluid, 
g the acceleration of gravity. c~(x,~) the potential of the absolute 
motion of the fluid, and v = (grad 91 the magnitude of the absolute 
velocity of the fluid. 
The function c~ (x,y) is determined from the boundary conditions; 
the flow condition on the wetted contour of the body, 
where n is the outer normal to the contour C; 
on the free boundary p = po, and hence 
on the bottom of the channel for y = -ho, the following condition 
applies 
According to the theory of waves of small amplitude, condi- 
tion (1.3) may be linearized. For this purpose the boundary condi- 
tion (1.3) is referred to the x-axis and the tern v2/2 neglected. In 
place of condition (1.32, 
It is easily seen that on the free surface the following relation 
holds 
cy(x) = cp + const (1.6) 
where Jr is the stream function. In fact, when the stream function of 
the motion of the fluid relative to the body is denoted by Jro, there 
is obtained 
From this relation, equation (1.6 f follows, since the bow- of 
the fluid in the relative motion is represented by stream lines on which 
$ is constant, For the free surface, it may be assumed that $O = 0, 
Hence, on the free surface, 
and therefore boundary condi'tion (1.5) assumes the form 
a9 & - " $ = O  for y - 0  
where 
From condition (1.5) it is seen that the equation of the free sur- 
face will be 
2. Fundamental Formulas of the Problem 
The problem may be mathematically formulated as follows. It is 
required to determine the characteristic function w(z) = tp + i$ 
(z=x+iy; i =  ), satisfying the conditions: 
1. For O>y>-% in the region occupied by the fluid, the deriv- 
ative dw/dz is finite and at infinity for x -+ +* , the derivative 
dw/dz vanishes. 
2. On the contour C, the smooth flow condition applies 
3. On the free surface for y = 0, the linearized condi$ion holds 
with regard to the constancy of the pressure 
~e(dw/dz + ivw) = 0 
4. On the bottom of the channel for y = -hg, the following condi- 
tion holds 
In the region occupied by the fluid, the point z is taken and 
two contours C1 and C, are drawn, of which C, contains both the 
point z and the contour C,  while the C1 contains the contour C, 
but not the point z (fig 1 ) By the formula of Cauchy for a single- 
valued function dw/dz =. P(z) , 
where the bar over a letter indicates, as usual, the transition to the 
complex conjugate value, The following notation is introduced 
It is evident that TTl(z) is a holomorphic function in the entire 
plane of the complex variable outside the contour C1, having at infin- 
ity the order z-1 and capable of being continued analytically in the 
entire part of the complex variable plane which lies outside the con- 
tour C, while V2(z) is a holomorphic function within the contour C, 
by the extension of which an analytical continuation of this function 
may be obtained over the entire strip 0 > y> -be 
The function v~(z) may be represented in another form, For this 
it is possible to find a function w(z), which in the strip O>y> -hg 
has a single pole of the first order ( = E + iq with residue ~/23ri 
and which satisfies conditions 1, 3, and 4. 
In fact, for a vortex of strength r ,  located at the complex point 
[ = 5 + iv, an expression for the complex velocity was obtained by 
Tikhonov (reference 3) 
sh XO("r7 +1 
rw sin O(Z - < + iho) 
v ho - ch2 ~ I Q  
where A. is the real and positive root of the equation 
v sh X% = x ch m0 (2.4) 
For c2< in all cases where the funchion to be integrated 
has a singularity, the principal value in the sense of Cauchy is taken 
under the integral. 
For c2 > , equation (2.4) has only imaginary roots and the 
fourth term of formula (2.31, which determines the hresence of free 
waves, is absent. 
For a source of strength Q located at the colnplex point 5 = E + iq, the expression of the complex velocity may be obtained in 
the same manner as in the case of a vortex. Without the computations, 
the final result is 
X(Y + ho) sin ~ ( z  - 5 + ihg) 
v sh Ahg - A ch Xhg dA - 
ch A g h  + ho) Qv cos $(z - 5 + ihg) 
who - ch2 XOk 
By the use of expressions (2.3) and (2.59, to obtain the. function 
w(z) may be obtained without difficulty. For 'this purpose, since 
A = I' + iQ, the follqwing expression Ps obtained after simple 
tr&sf omnations : 
- 
A sin X(Z - z + 2%) - A sin X(Z - 5 )  
vsh hhg - Xch Xhg dX + 
- 
v A cos %(z - 1 + 2ih0) - A cos hg(z - 5 )  
- 
2 i v k  - ch2 bb (2.6) 
Here, as in the preceding formulas, the fourth term, which deter- 
mines the presence of free waves, is present only if c2< 
- 
When A = v(<) d( is substituted in the previous f oMnula and 
integration is carried out over the contour C1, 
sin X(Z - 1 + 2 i k )  
d X  - 
sin X z - 
nv +A$ e n 1  i i l ) { $ ; v + * l  eq(- a ~ )  v s h  a. ! *ci))&o d X  - 
If both points z and are situated in the strip O > y >  -%, 
the following equation holds 
With this equation taken in account, it is found from equation (2.7) 
that the function vZ(z) can be represented in the form 
(2 .9)  
The conjugate complex functions are introduced for real h 
By an interchange in equation (2.9) of the order of integration, 
and by simple transformations, there is readily obtained 
H(A) exp iAz + H(- 1) exp(-  ikz))  
n iv  
9 
2(vh  - ch2 hgk) (fi(- hg) exp i k 0 ( z  + 2i.I + B(x0) e q  
~ ( 1 0 )  exp i ~ o z  - H(- hg) e r n ( -  i b l ) )  (2.11) 
It is of interest to find the character of the waves that remain 
behind the moving body. For this purpose the aspnptotic expression of 
the complex velocity is first obtained for x - r -  w in the case of a 
vortex and source. In reference 3, the asymptotic expression of the 
coqlex velocity in the case of a vortex is of the form 
In a similar manner, the asymptotic expression of the complex 
velocity is obtained in the case of a source. Without the co~~putations, 
the final result is 
For the function ~(z), having a polarity with residue ~/2ni, the 
following asymptotic expression is obtained: 
- 
A cos X0(z - f + 2ihg) - A cos 10(z - 5 )  
- - (a)x+-- - iw (2.14) vb - ch2 Xoho 
- 
Setting A = v(5) d[ and integrating over the contour C1 yields 
- 
the asymptotic expression of the function v(z) = dwhz: 
- 
H ( x ~ )  exp [- a0(z + 2i1qJJ - ~ ( 1 ~ )  exp fiOz - a(- lo) exp(- fioz 
Finally, from the formula 
it is readily found that for x -t - - sinusoidal waves of length 2n/A0 
are formed behind the amplitude of which, after some s-imple transforma- 
tions, may be represented in the form 
3, Formulas fo r  Determining the Forces 
The forces acting on the contour C are  now computed, The l i f t  
force of the contour i s  denoted by P, the resistance by R, aad the  
moment of the forces on the contour about the origin by M. These 
forces w i l l  be computed by the formulas of Chaplygin-Blasius: 
where C2 i s  an arbi t rary contour, si tuated i n  the region O>y> -% 
and containing the contour C; and vo(z) i s  the complex velocity i n  
the re la t ive  motion obtained by superposing on the absolute flow a 
uniform motion of the f l u i d  with velocity c i n  the direction of the 
negative x-axis* Thus, 
where the contour C1 i s  chosen t o  l i e  between C and C2. 
Formulas (3,l)  do not take in to  account the buoyancy 'force of 
Archimedes, equal t o  gpS, and it moment, equal t o  -gpSxc, where S 
i s  the area tha t  bounds the contour C, and xc i s  the abscissa of the 
center of gravity of t h i s  area. 
The following integral  i s  now computed: 
But the f irst  and second integrals  on the r ight  a re  equal t o  zero 
because the function %(z) i s  holomorphic outside the contour C2 
and has a t  in f in i ty  a zero of a t  l e a s t  the first order, while the func- 
t ion V2(z) i s  holomorphic within the contour C2. Hence, 
The veloci ty  circulat ion about any contour t h a t  contains the  con- 
tour C i s  denoted by I' , so that 
theref ore 
By the use of expressions (2.2) and (2 . l l ) ,  the following expression i s  
obtained 
( - 1 )  exp i ~ ( z  + 23%) + 
- 
.iv a(- l o )  exp ' iho(z + 2ih0)  + B(xo) exp [- iAO(z + 2%)] 
- + 2 v l ~  - chZ Xohg 
Since the  point (, which belongs t o  the contour C1, l i e s  within 
the  contour .C2, with an interchange i n  the order of integration and 
by the  following formula, 
There is  obtained 
niv a- lo) l 2  ern(- 210ho) + I H ( x ~ )  l 2  ern 2XOh0 - 2E(xo) a- lo) 
- 
2 v h g  - ch2 lo% 
- 3 )  
Hence, formula (2.3) assumes the form 
Separating the real and imaginary parts and adding to P the 
Archimedes force, not taken into account by the Chaplygin-Blasius for- 
mula, results in 
Formula (3.6) may be given another form, aamely 
It can be readily shown that the total resistance of the underwater 
wing consists only of the wave resistance. In fact, by the following 
well-known formula for computing the wave resistance in the case of a 
fluid of finite depth, 
, and with the value of the anrplitude a from formula (2.161, for- 
mula (3.7) is obtained after some transformations. 
The moment of the acting forces on the contour C is now com- 
puted. When the moment of the Archimedes force is taken into account, 
This expression is computed in an entirely similar manner to the 
computation of the expression P - iR. 
For very large absolute values of z the following expansion can 
be employed 
and, hence, 
Further, 
NACA 'I'M 1345 
and theref ore, 
M =; - gpSx, + Rep 4 z vl,v2 - I dz 
or, since the function v~(z) is holomorphic within the contour C2 
It is noted that 
The integrals in formula (3.10) are computed in the same manner as in 
the expression {3.3), and as a result there is obtained the formula 
V (af(- 10) $(- k) exp(- 2 7 ~ h . o )  - H f  (lo) %(kg) exp 2A0h0 - 
4 (vhO - .h2 hgho) 
NACA TM 1345 
Formulas (3,5), (3.71, and (3.11) in the limiting case for h~ + 
agree with the fornnilas obtained by Kochin in reference 2. 
The function H@) in formulas (3.51, (3.71, and ( 3 . ~ )  does not 
depend on the contour C1, and for example, the contour C or some 
other contour which contains the contour C may be taken for the con- 
tour of integration. Moreover, the value of the function H(X) does 
not change if, instead of the coqlex velocity ?(z) - of the absolute 
motion, the complex velocity of the relative motion vo(z) is taken, 
because these two functions differ by a constant c. The properties 
of the function H(X) will be used in the following section. 
In the preceding sections expressions were found in terms of the 
function H(X) of a number of important magnitudes, namely, the ampli- 
tude of the waves formed, the wave resistance, the lift force, and the 
moment of the forces acting on the contour. Thus, the function 
plays a fundamental part for the problem under consideration. In order 
to compute this function, it is necessary to know the expression for 
the complex velocity, i.e., the solution of the hydrodynamic problem. 
In case the relative depth of the submerged contour C is sufficiently 
large, howsver, a good approximation is obtained if, in place of the 
function v(z), there is substltuted in formula (4.1) the expression 
of the coqlex velocity which corresponds to the motion of the con- 
tour C in an infinite fluid. 
Several examples of such an approximate solution of the problem 
will be considered 
1. The motion of a circular cylinder. - The circular cylinder of 
radius b, sTtuated at the depth h under the free surface of the 
fluid, is assumed to move with constant horizontal forward velocity c, 
since the circulation about the contour of the cylinder has a given 
value r. In this case, the characteristic function for the infinite 
fluid is known: 
cb + --: r ln(z + ih) 
= - z + hi 2111 
NACA TM 1345 
Hence, 
- 
v(z) 7 cb2 + r (, + hi12 2fii(z + ih) 
By formula (4.1) the function H(X) is now constructed: 
H(X) = cb2 + r 
+ hi)2 2xi(z + hi) exp - iXz dz 
Since the contour C contains one singular point z = -ih, there 
is obtained by the theorem on residues 
H(X) = (I' + 2ficb2X) e q  - Xh (4.3) 
With the use of formula (3.72, the expression for the wave resist- 
ance of the cylinder is obtained 
R = p v  
ch2 ho&, - vhg (4.4) 
and by the use of formula (3.5) the expression for the lift force of 
the cylinder is obtained 
(r2 + 4n2c2b412) sh 2X(h0 - h) + 4ncb21'X ch 2X(Z(bg - h) 
vsh % - X ch Xhg dX + gpS (4.5) 
The integral component of this formula may be computed by the 
method of mechanical quadratures. In the limiting cases v = 0 and 
v = =, this component can be very accurately computed, Moreover, if 
this integral component is considered as a function of the parameter 
a = 1/(~%)) = c2/(ghg), it can be shown that for a = I this component 
suffers a discontinuity. In the particular case when the radius b of 
the cylinder is taken equal to zero, i.e., when the motion of a vortex 
under a free surface is considered, formulas (4.4) and (4.5) lead to 
the expressions established by Tikhonov. It is noted further that for- 
mulas (4.4) and (4.5) have been derived on the assumption that c2< % 
For c2' gho, no free waves are formed behind the cylinder and the wave 
resistance R is equal to zero. 
For the moment of the forces  exerted by the f l u i d  on the cylinder, 
t h e  following expression i s  obtained by formula (3.11) : 
Pa0 p 4 = - -  4 s t ( -  101 H(- l o )  exp(- 2%) - H ~ ( x ~ )  H ( x ~ )  exp ZXA + 
vhg - ch2 Xoho 
But from equation (4.31, it i s  evident tha t  
H' (- X) = HI(- A) - 2xch2 exp hh 
Hence, a f t e r  simple transformations, 
The point of intersect ion with the y-axis of the resu l tan t  force 
on the body i s  determined by the  formula 
It i s  evident t h a t  f o r  R > O  t h i s  resul tant  never passes through 
the center of the cylinder. 
2. Motion of an e l l i p t i c  cyJinder, - An el l ipse,  having a center 
a t  the depth h and having axes 2a and 2J3 directed pa ra l l e l  t o  
the axes of coordinates x and y, i s  allowed t o  move with a constant 
velocity c i n  the direction of the x-axis. The circulat ion I' i s ,  
f o r  simplicity, taken equal t o  zero. In t h i s  case, the flow of an 
i n f i n i t e  f l u i d  about the contour C i s  determined with the a id  of an 
auxi l iary variable and the formula 
where r = and 1 us = r is the equation of the cir- 
cle in the u-plane which corresponds to the contour of the ellipse C. 
The exterior of this circle corresponds to the exterior of the ellipse, 
The following function is set up: 
When the substitution u = iv is made, there is obtained 
, H(X) = - 2 da- exp(- ~ h )  
But by the theory of Bessel functions it is lulown that 
hence, 
: From the formula 
and the value of r, %he following expression is obtained 
The computation is restricted to the wave resistance. By 
formula (3.7 ) , 
2 2 a + P  Ch2 lo(% - h) R = 43.c pgp -
" p ch2 ?@q - Y h g  (10 (4.9) 
From this formula, it f~llows that for certain Xo and, therefore, 
for 9 certain velocity c C , the wave resistance is equal to zero; 
i,e., the amplitude of the waves formed behind the mwing body becomes 
zero. This w i l l  be the case if the following relation is satisfied: 
where sk is the positive root of the Bessel function $(s). The 
first root of this function is 
Since the parameter W = g/c2 is connected with Xo by the 
equation 
the first velocity at which the wave resistance becomes zero is deter- 
mined by the formula 
Moreover, 
hence, 
In a similar manner a number of other examples may be considered. 
Moreover, as in reference 2, it is possible in this case to set up a 
functional equation for determining the function H ( X )  and the values 
of the circulation r from the condition of the finite velocity at the 
sharp edge. These equatiqns may be obtained by the same method, Their 
final form will be somewhat more complicated as compared with the case 
of the infinite fluid. 
Translated by S. Reiss 
National Advisory Committee 
for Aeromutics 
REFERENCES 
1. Khaskind, M. D.: Prikladnaya Matematika i Mekhanika, vol. VIII, 
no. 4, 1944. 
2. Kochin, N. E.: On the wave resistance and lift force of bodies sub- 
merged in a fluid. Reports of the conference on the theory of 
wave resistance. NAGI. M., 1937. 
3. Tikhonov, A. I,: Two-dimensional problem of the motion of wing 
under the surface of a heavy fluid of finite depth. Izvestia 
OTN AN SSSR, no. 4, 1940. 
Figure 1. 
NACA TM 1345 


Translational motion of bodies under the free surface of a heavy fluid of finite depth

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