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On the problem of motion of a tangential discontinuity in an ideal incompressible liquid
ON THE PROBLEM OF MOTION OF A TANGENTIAL DISCONTINUITY IN AN IDEAL INCOMPRESSIBLE LIQUID* I. D.
SOFRONOV
Moscow (Received 24 November 1961) 1. A space is filled with an ideal incompressible liquid. The motion is two-dimensional i.e. it is sufficient to consider the picture only in the plane xy. At the initial moment of time the curve L is given with vortices distributed along it, the intensity r of the vortices being a function of a parameter a, for example, the length of arc at the initial moment. The problem consists in finding out the positions of the curve L at successive times. 2. Let us consider the integral
(1) L
where zOis a fixed point in space, and z = z(u) is a current point belonging to L. If zO does not lie on the line L, the function &zO) expresses the complex velocity induced by vortices distributed on L (see [I]). To determine the significance of v(zJ for points lying on the line of integration we transform the expression under the integral. Since
therefore
(2) Now let us suppose that the point 5, not lying on the line of integration, approaches it along a certain path. Let v+(z,) and v-(z,,) be the limiting values of integral (2). when the point zO approaches L from one direction and the other. The formulae of Sokhotskii-Plemel’-Privalov [2] are true for boundary values:
-
Gf(z(J - w- (ZO)= - r (0)
w+(zO)+~-(zO)
= -$s
F(u) L
dz -&,
di
dz -dz z-z0
'
It follows from the first of these equations that only the tangential components of the velocity suffer a discontinuity, while the normal components remain continuous, i.e. we are in fact dealing with a tangential discontinuity. * Zh. vych. mat. 2; No. 3, 494499,
1962. 518
Motion of tangential discontinuity
519
in ideal liquid
It follows from the second of the equations that the velocity of points lying on the line of discontinuity is equal to half the sum of the velocities of boundary particles of the liquid i.e. y(qJ
=
aZ,
at -
In the last equation the partial derivative is taken, since z,, is a function of two variables t and or,. It follows from what has been said that we can write the equation of motion of points of the line of tangential discontinuity:
az, at
-=
-_
1 2ni
r(u)do z-z,.
s
L
From Helmholtz’s theorem [l] on the conservation of vortices it follows that the quantity r(a)du remains unchanged in time (we shall bear this in mind later on). Equation (4) is a singular integro-differential equation, and the integral on the right hand side must be taken as the principal value of Cauchy [3]. Separating the real parts of equation (4) from the imaginary we obtain a system of two real equations:
h __=_ a
1 2~ sL
(v - YO)F(a) (x-x0)”
(Y - vo)* +
ay, __=___ at
das
1 (X - x0) T(a) 23r sL (y - yo)” + (x -X0)” da.
(5)
3. It is possible to find an exact solution of this system only in very special cases, for example in the case of rectilinear or circular tangential discontinuity with a constant intensity r. In the first case if the discontinuity is directed along the axis X, we obtain
s_dx=O
r
ah -=--
at
2nLx-xx,
’
i.e. the points of the line of discontinuity are at rest, the liquid above the discontinuity the left, and that below the discontinuity to the right.
moves to
In the second case we assume that the discontinuity has the form of a unit circle with its centre at the origin of coordinates. Writing x = cos u, y = sin a and substituting jn system (5), we obtain after transformation
ax0 = -sina
__ at
’
ah ,1=cosu9
i.e. the absolute value of the velocity at all points of the discontinuity is constant, and the direction of the velocity coincides with the direction of the tangent to the circle. The liquid inside the circle is at rest, and outside the circle the motion of the liquid is the same as in the case: of a central vortex with the corresponding intensity (see [l]). 4. If the initial data are periodic, More precisely let x(u+2n.k)
= x(cr)+Zs.k,
the region of integration
y(fJ+27C.k)
= Y(U),
can be reduced to one period.
r(u+2n.k)
= r(u),
where 2s is the period with respect to x. Then expressing the integral on the right hand side of (4) as the sum of the integrals with respect to the periods and interchanging the order of integration and summation, we obtain the following equation:
520
D.
I.
a;;i,
z cot
s
1
-zzz
4si
at
SOFRONOV
“(‘2;
“)
r(a)da.
--A
As before, separating the real and imaginary parts, we obtain a system of two real equations:
s 31
ax,
I
-=-
at
4~
n(Y-Yo)
sh
s
Z(Y -n) s
-,ch
n
at
(7)
n(x--0)
sh
1 4s s _,ch-
ah __=_-
r(c)dda,
_ cos n(x3x,) s s
F(a)da.
cos n(x--0)
7z(Y - YO) S
S
5. Let us consider a special case. Let the discontinuity have a constant intensity and the form of a sinusoid with a small amplitude E at the initial moment: y = E. sinx. With an accuracy up to 9 we obtain x = u.
s =z, Neglecting small quantities
of the second order we find from (7)
ax,
at -4n
x s
E
sinx - sinx, 1 -cos(x-x0)
--n
aY, ~=_
1 4n
%t
x s.
-7c
dx = - f
sin(x -x0) 1 -cos(x -x0)
=
sinx,,
0,
i.e. the line of tangential discontinuity is deformed at the initial moments in such a way that its points move parallel to the x-axis. 6. As stated above, there is little chance of finding an exact solution of systems (5) and (7) in cases of any importance, so that it is necessary to devklop an approximate method for their solution. Let us consider system (7). We divide up the entire period [-n, n] into 2n parts, for example, at equal intervals of u. Then for the chosen nodes we can write the following system of differential equations : n-1
axi -=-
at
sh
c
4sn
II-1
n
ari
4sn
c
k=-n
C’ denotes that the summation We replace the derivatives we shall obtain finally
-
Yi)
S
’
kc-n
_zze at
n (Yk
I
n
ch
rk, ~(Yk-Yyi) s
_
cos
-
xi)
S n (xk
sin --
,
-
xi)
S rk;
_cosn(xk-~xi)
n(yk-Yi) ch
7Z (xk
S
S
is carried out only over k # i. with respect to time by a first difference
with interval 7. Then
Motion of tangential discontinuity n-1
&YLYyf)
,
s
*; = $ + 25
4sn c
k=-n
521
in ideal liquid
&
NY:
- Yi)
-rk,
cos rc (xl, - d) s
s
(8)
sinn(X-xt)
n-1
S
rk. ,n(y’;yt)
__-‘,S
72(x’k - xf) _~ S
The upper index shows to what moment the quantity under it belongs.
I
a.1
I
I
I
2 FIG. 1
FIG. 2
FIG. 3
I. D. SOFRONOV
522
System (8) is implicit. It must be solved iteratively, choosing as the first approximation the position of the discontinuity at the preceding moment. The approximate system for (5) in the case of a closed discontinuity is written in exactly the same way. We shall not give it here. 7. A number of examples were calculated by the method described. The results of the calculations are expressed in Figures. As examples we chose discontinuities which had a simple geometric form at the initial moment: sinusoid (Fig. l), elliptic (Figs. 2, 3), square (Figs. 4, 5, 6). All the
-I FIG. 5
FIG. 6
Motion of tangential discontinuity
in ideal liquid
523
discontinuities chosen had a plane of symmetry, so that it was sufficient to consider only one half of the discontinuity, the one depicted. The initial and final calculated forms are shown in all the graphs. In addition, the trajectories of points of the line of discontinuity are given in Fig. 1. The first and last Figures have periodic discontinuities (with period 2n), although only one half of the periods are shown. Discontinuities with constant initial intensity are given in Figs. 1, 2 and 4. In the rest the initial intensity of the variable r(x) = -x. I take this opportunity to thank N. A. Dmitriev, who drew my attention to the subject, for discussing the problems arising in this connection. Translated by PRASENIITBASU REFERENCES 1. KOCHIN, N. E., et al., Teoreticheskaya gidromekhanika. (Theoretical hydromechanics.) Gostekhizdat, Moscow-Leningrad, 1955. 2. MARKUSHEVICH, A. I., Teoriya analiticheskikh funktsii. (Theory of analytical functions.) Gostekhizdat, Moscow-Leningrad, 1950. 3. MUSKHKLISHVILI, N. I., Singulyarnye integral’nye uravneniya. (Singular integral equations.) Gostekhizdat, Moscow-Leningrad, 1946.
SOFRONOV
Moscow (Received 24 November 1961) 1. A space is filled with an ideal incompressible liquid. The motion is two-dimensional i.e. it is sufficient to consider the picture only in the plane xy. At the initial moment of time the curve L is given with vortices distributed along it, the intensity r of the vortices being a function of a parameter a, for example, the length of arc at the initial moment. The problem consists in finding out the positions of the curve L at successive times. 2. Let us consider the integral
(1) L
where zOis a fixed point in space, and z = z(u) is a current point belonging to L. If zO does not lie on the line L, the function &zO) expresses the complex velocity induced by vortices distributed on L (see [I]). To determine the significance of v(zJ for points lying on the line of integration we transform the expression under the integral. Since
therefore
(2) Now let us suppose that the point 5, not lying on the line of integration, approaches it along a certain path. Let v+(z,) and v-(z,,) be the limiting values of integral (2). when the point zO approaches L from one direction and the other. The formulae of Sokhotskii-Plemel’-Privalov [2] are true for boundary values:
-
Gf(z(J - w- (ZO)= - r (0)
w+(zO)+~-(zO)
= -$s
F(u) L
dz -&,
di
dz -dz z-z0
'
It follows from the first of these equations that only the tangential components of the velocity suffer a discontinuity, while the normal components remain continuous, i.e. we are in fact dealing with a tangential discontinuity. * Zh. vych. mat. 2; No. 3, 494499,
1962. 518
Motion of tangential discontinuity
519
in ideal liquid
It follows from the second of the equations that the velocity of points lying on the line of discontinuity is equal to half the sum of the velocities of boundary particles of the liquid i.e. y(qJ
=
aZ,
at -
In the last equation the partial derivative is taken, since z,, is a function of two variables t and or,. It follows from what has been said that we can write the equation of motion of points of the line of tangential discontinuity:
az, at
-=
-_
1 2ni
r(u)do z-z,.
s
L
From Helmholtz’s theorem [l] on the conservation of vortices it follows that the quantity r(a)du remains unchanged in time (we shall bear this in mind later on). Equation (4) is a singular integro-differential equation, and the integral on the right hand side must be taken as the principal value of Cauchy [3]. Separating the real parts of equation (4) from the imaginary we obtain a system of two real equations:
h __=_ a
1 2~ sL
(v - YO)F(a) (x-x0)”
(Y - vo)* +
ay, __=___ at
das
1 (X - x0) T(a) 23r sL (y - yo)” + (x -X0)” da.
(5)
3. It is possible to find an exact solution of this system only in very special cases, for example in the case of rectilinear or circular tangential discontinuity with a constant intensity r. In the first case if the discontinuity is directed along the axis X, we obtain
s_dx=O
r
ah -=--
at
2nLx-xx,
’
i.e. the points of the line of discontinuity are at rest, the liquid above the discontinuity the left, and that below the discontinuity to the right.
moves to
In the second case we assume that the discontinuity has the form of a unit circle with its centre at the origin of coordinates. Writing x = cos u, y = sin a and substituting jn system (5), we obtain after transformation
ax0 = -sina
__ at
’
ah ,1=cosu9
i.e. the absolute value of the velocity at all points of the discontinuity is constant, and the direction of the velocity coincides with the direction of the tangent to the circle. The liquid inside the circle is at rest, and outside the circle the motion of the liquid is the same as in the case: of a central vortex with the corresponding intensity (see [l]). 4. If the initial data are periodic, More precisely let x(u+2n.k)
= x(cr)+Zs.k,
the region of integration
y(fJ+27C.k)
= Y(U),
can be reduced to one period.
r(u+2n.k)
= r(u),
where 2s is the period with respect to x. Then expressing the integral on the right hand side of (4) as the sum of the integrals with respect to the periods and interchanging the order of integration and summation, we obtain the following equation:
520
D.
I.
a;;i,
z cot
s
1
-zzz
4si
at
SOFRONOV
“(‘2;
“)
r(a)da.
--A
As before, separating the real and imaginary parts, we obtain a system of two real equations:
s 31
ax,
I
-=-
at
4~
n(Y-Yo)
sh
s
Z(Y -n) s
-,ch
n
at
(7)
n(x--0)
sh
1 4s s _,ch-
ah __=_-
r(c)dda,
_ cos n(x3x,) s s
F(a)da.
cos n(x--0)
7z(Y - YO) S
S
5. Let us consider a special case. Let the discontinuity have a constant intensity and the form of a sinusoid with a small amplitude E at the initial moment: y = E. sinx. With an accuracy up to 9 we obtain x = u.
s =z, Neglecting small quantities
of the second order we find from (7)
ax,
at -4n
x s
E
sinx - sinx, 1 -cos(x-x0)
--n
aY, ~=_
1 4n
%t
x s.
-7c
dx = - f
sin(x -x0) 1 -cos(x -x0)
=
sinx,,
0,
i.e. the line of tangential discontinuity is deformed at the initial moments in such a way that its points move parallel to the x-axis. 6. As stated above, there is little chance of finding an exact solution of systems (5) and (7) in cases of any importance, so that it is necessary to devklop an approximate method for their solution. Let us consider system (7). We divide up the entire period [-n, n] into 2n parts, for example, at equal intervals of u. Then for the chosen nodes we can write the following system of differential equations : n-1
axi -=-
at
sh
c
4sn
II-1
n
ari
4sn
c
k=-n
C’ denotes that the summation We replace the derivatives we shall obtain finally
-
Yi)
S
’
kc-n
_zze at
n (Yk
I
n
ch
rk, ~(Yk-Yyi) s
_
cos
-
xi)
S n (xk
sin --
,
-
xi)
S rk;
_cosn(xk-~xi)
n(yk-Yi) ch
7Z (xk
S
S
is carried out only over k # i. with respect to time by a first difference
with interval 7. Then
Motion of tangential discontinuity n-1
&YLYyf)
,
s
*; = $ + 25
4sn c
k=-n
521
in ideal liquid
&
NY:
- Yi)
-rk,
cos rc (xl, - d) s
s
(8)
sinn(X-xt)
n-1
S
rk. ,n(y’;yt)
__-‘,S
72(x’k - xf) _~ S
The upper index shows to what moment the quantity under it belongs.
I
a.1
I
I
I
2 FIG. 1
FIG. 2
FIG. 3
I. D. SOFRONOV
522
System (8) is implicit. It must be solved iteratively, choosing as the first approximation the position of the discontinuity at the preceding moment. The approximate system for (5) in the case of a closed discontinuity is written in exactly the same way. We shall not give it here. 7. A number of examples were calculated by the method described. The results of the calculations are expressed in Figures. As examples we chose discontinuities which had a simple geometric form at the initial moment: sinusoid (Fig. l), elliptic (Figs. 2, 3), square (Figs. 4, 5, 6). All the
-I FIG. 5
FIG. 6
Motion of tangential discontinuity
in ideal liquid
523
discontinuities chosen had a plane of symmetry, so that it was sufficient to consider only one half of the discontinuity, the one depicted. The initial and final calculated forms are shown in all the graphs. In addition, the trajectories of points of the line of discontinuity are given in Fig. 1. The first and last Figures have periodic discontinuities (with period 2n), although only one half of the periods are shown. Discontinuities with constant initial intensity are given in Figs. 1, 2 and 4. In the rest the initial intensity of the variable r(x) = -x. I take this opportunity to thank N. A. Dmitriev, who drew my attention to the subject, for discussing the problems arising in this connection. Translated by PRASENIITBASU REFERENCES 1. KOCHIN, N. E., et al., Teoreticheskaya gidromekhanika. (Theoretical hydromechanics.) Gostekhizdat, Moscow-Leningrad, 1955. 2. MARKUSHEVICH, A. I., Teoriya analiticheskikh funktsii. (Theory of analytical functions.) Gostekhizdat, Moscow-Leningrad, 1950. 3. MUSKHKLISHVILI, N. I., Singulyarnye integral’nye uravneniya. (Singular integral equations.) Gostekhizdat, Moscow-Leningrad, 1946.