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A NEW METHOD FOR SOLVING THE PROBLEM OF A FLOW BY IDEAL LIQUID STREAM OF A COMPACT BODY OF REVOLUTION AND PERIODICALLY ROUGH SURFACE
Alexander G. Kyurkchan,
Head of the PT and AM Department, Moscow Technical University
of Communications and Informatics,
Kotel'nikov Institute of Radio Engineering and Electronics,
Fryazino Branch, Russian Academy of Sciences,
Central Research Institute of Communication, Moscow, Russia,
agkmtuci@yandex.ru
Sergey A. Manenkov,
Docent of the Mathematical Analysis Department,
Moscow Technical University of Communications and Informatics,
Moscow, Russia, mail4447l@mail.ru
This work was supported by Russian Foundation for Basic Research, Projects No. 14-02-00976, 16-02-00247.
Keywords: problem of a flow, discrete sources, analytical continuation, periodical Green's function.
In the paper the problem of a flow by a stream of ideal liquid of motionless periodically rough surface and a compact body of revolution is considered. Earlier for the solution of a flow problems the approach based on conformal mapping is offered. However one can apply this method only to two-dimensional problems, and it is necessary that the streamlined surface has rather simple geometry. In the real article the modified method of discrete sources (MMDS) which has been earlier applied to the wide class of boundary problems is used to the solution of the problem of the flow. This method possesses universality relative to geometry of a streamlined surface. In the paper MMDS is applied to the solution of the problem of the flow by ideal liquid of periodically sinusoidal surface and the surface in the form of cycloid. To control the accuracy of the obtained results the dependence of the residual for the problem of the flow of sinusoidal surface has been plotted. The problem of the flow of the motionless sphere, the prolate spheroid and Chebyshev particle has been also solved. In the case of the problem of the flow of the sphere comparison of the exact solution of the problem and the solution obtained by means of MMDS is carried out. The streamlines for all considered geometries and also dependences of pressure versus coordinate vertical relative to the periodic surface and radial coordinate for the body of revolution have been built.
Для цитирования:
Кюркчан А.Г., Маненков С.А. Новый метод решения задачи обтекания потоком несжимаемой жидкости компактного тела вращения и периодически неровной поверхности // T-Comm: Телекоммуникации и транспорт. - 2016. - Том 10. - №9. - С. 66-72.
For citation:
Kyurkchan A.G., Manenkov S.A. A new method for solving the problem of a flow by ideal liquid stream of a compact body of revolution and periodically rough surface. T-Comm. 2016. Vol. 10. No.9, рр. 66-72.
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Introduction
The problem of a flow by an ideal liquid stream of motionless compact body and periodically rough surface is considered in this study. As it is known die problem of the flow is reduced to the solution of the Laplace equation with a Neumann boundary condition on the rough surface or on the body border [11. The method of con formal mapping is applied to the solution of similar problems earlier [ 1 ]. This method is applied to the problem of the flow of a circular cylinder, a thin plate, etc. Thus, the method of conformal mapping is applicable in the case of the solution of the flow problem for the simple geometry bodies. In more difficult cases use of other approaches is required. In the present paper for the solution of the problem of the flow of compact body and periodically rough surface the modified method of discrete sources (MMDS) which is successfully applied earlier to the problems connected with the solution of Maxwell equations, Helmholtz and Laplace equations is used [2]-[10|. In the latter case the electrostatic problem for a body of revolution has been considered.
MMDS is based on two ideas. First, the integral equation to which the initial boundary problem is reduced, has the solution corresponding to the initial problem if and only if the auxiliary surface on which the unknown function is distributed, covers the set of singularities of analytical continuation of the field inside the area occupied by the body. Secondly, the choice of the auxiliary surface should be made by means of analytical deformation of the boundary of the body [2]-| 10]. In this work the problem of the flow is reduced to the solution of the integral equation of the first kind, relative to some unknown function distributed on the auxiliary surface which is taken by the way stated above. The integral equation is solved by the collocation method. Numerical results given in the paper belong to the problem of the flow of sinusoidal surface and the surface in the form of cycloid as well as the sphere, prolate spheroid and Chebyshev particle. It is of interest to consider the dependence of the pressure versus the vertical coordinate for the periodical surface or the dependence of the pressure versus radial coordinate for the body of revolution in the case of different geometries.
The problem of a flow of a periodically rough surface
Let's consider the mathematical statement of the problem of the flow of periodically rough surface. We introduce the scalar potential ii as follows:
v = Vw, (1)
where v is the distribution of the velocity of ideal liquid in the whole space. Then the required potential is satisfied the two-dimensional Laplace equation
Au = 0 (2)
and the boundary condition at the periodically rough surface S, which is defined by the equation y = f{x),
lim 1 Vu1 |=0-
(4)
cu dn
= 0.
(3)
Using the fact that the considered problem is two-dimensional, it is expedient to pass to the conjugate problem (see the equation for the streamlines) [ 1 j. The new unknown function satisfies to Laplace equation
Avji = 0 (5)
and the boundary condition at the periodical surface S
= + y = f{x). (6)
The constant C is defined from the condition at infinity
tty
lim
-0 ■
(7)
The latter problem we solve by use MMDS [2]-[10]. For this aim we consider two unknown functions (x,_y) and
V|/2(jc, y) which satisfy the following boundary conditions and
W2(x,y) = lt y = f(x). (9)
Then on the strength of linearity of the considered problem the required function is vfj = \_(Jf1 Ci|/2 ■ For the solution of the
problems (8) and (9) we write the functions ^(a",^) and \j/-,(jr,j!/) in the form
da
VP(x,y)= J jp{xt,y^{xyytxz>yz)dxii P = h2,
-d! 7
((x^eZ), (10)
where j are the unknown functions, G(x,y,x',y') is the periodic Green's function (see below), 2 is the auxiliary surface which is chosen as follows. Let's introduce the complex variable
[5], [&]:
|(x) = t-iS+ if(t-;'S), te[ d/2,d/2], (11) where § is the positive parameter. If the value § is equal to zero then the where CQ is the some contour in the complex
plane £. C0 corresponds to the contour S of the periodical surface and congruent to this contour. When we increase the parameter S the contour C0 is deformed and is located in the domain lying under the contour of the surface 5. Thus we get the new contour, which can be chosen as the auxiliary contour S. Note that such deformation is possible so long as the mapping £(jc) = a" + if(x) remain biunique. The points in which this condition is disturbed satisfy the equation [5], [8], [11]:
f\x) = i. (12)
Eq, (12) permits us to define the critical value of the parameter 8 that is the degree of the deformation of the initial contour of the rough surface, in particular for the sinusoidal surface which is determined by the equation
We suppose that f(x + d) — f{x), where d is the period of
the surface. In the formulas (I) - (3) u = z/J -l-ii1 is the full potential, and u° = v(1x is the primary potential (v0 is the velocity
of the incident flow of the liquid). At infinity the gradient of the secondary potential tends to zero
y = asm
2jdt
the critical value of the parameter 5 is
8,™*=i,Yin(y + \/"r+i)' y — dj (2na)-
(13)
(14)
VL/
After the deformation parameter 5 is found one can define the equation of the auxiliary contour £ in the parametric form as follows
xs=KeZ0, JV-Im^a te[-d/2,d/2]. (15)
Let's consider the Green's function of the problem. The Green's function satisfies the equation
AG=£S(x-x'-«</)80>-/)
(16)
and the condition at infinity which lies in the fact that the derivatives of the Green's function are limited at infinity. To derive the Green's function we write the equality 2
(57)
8(x - x' - nd) =— + — V cos f —^ ( a: - x') | ■
d dt! I d J
Then we present the function G(x,y,x',y') as the series
(18)
Substituting the formulas (17) h (18) into equation (16), it is easy to obtain the equality
G(x,y,x',y') = -\y-y'\ + 2d
(19)
otj
exp| ~-~\y-y' ij cos(^
2n n
d v J \ d Summarizing the series in (19), we get G(x,y,x',y') =
1 i, , 1 ,
= —I ln2+—In 2n{. 2
(x-x') .
(20)
As a result of substitution (10) into the formulas (8) and (9) the initial problem is red need to solving of two integral equations of the first kind
da
J"y~Ax)> i2l)
-d/2
d/2
j =1, y=f(x). (22)
-da
After solving the equations (21) and (22) we find the constant C from the equality
da
J Mxz,yT)dxz
C = -^2-
(23)
da
-da
cal system of coordinates. Direct the axis z along the axis of the rotation of the body. Assume that the stream of liquid has the speed v() = y(1i_, where / is a single vector along the axis 2. As it
stated above we introduce the scalar potential u — a0 + ul like in the formula (1). Here ua — v{)z is the primary potential. As a
result the problem is reduced to the solution of three-dimensional Laplace equation with Neumann boundary condition for the full potential at the surface S of the body. Hereafter it is convenient to introduce the orthogonal coordinate system of revolution (a, ß,(p) connected with the body [6J, [7J. Let S is described by
the equation a —a(ß) in these coordinates.
Apply MMDS to the solution of the problem. Present the secondary potential as follows
Ar)= \j{rz)G{r,rz)dsz, (25>
where
4tt I r - K
(26)
As a result of calculation one can plot the streamlines which determined by the equation
\j;(jf,jf) + v0 v = const, (24)
where the function \j/(A\y) is defined by the algorithm described above.
The problem of a flow of a body of revolution
We consider the problem of a flow of a motionless body of revolution by a stream of ideal liquid. We introduce the cylindri-
!n the formula (25) X is the auxiliary surface located inside the body, j(K ) is the unknown function. To choose the auxiliary surface we express the Cartesian coordinates of the point (x, y,z) in terms of the coordinates (a, ß,(p)
* = p(a,ß)cos<p, = p(a,ß)sintp, z-z{a,ß), (27) where (p,(p,z) are the cylindrical coordinates. Then for the auxiliary surface L, the following equations are satisfied [6], [7|:
Xv = Im^cosq), yL - Im^sintp, = (28)
where ^ is a certain function of the variable r[(ß) = a(ß + /6) + /(ß t /5) (see below). The coordinates az H ßv ol the "image1 of the point f (/. ß,(pj on the initial surface S of the body are determined from the relations
ay = Rer|, ßy = Imr| - (29)
In the expression for the function r|{ß), 5 (like in the previous section) is the positive parameter which determines the degree of deformation of the surface of the body. The choice of the parameter 8 is described in [6], [7]. Note that, as 8 increases, the points of the auxiliary surface move inside the body along orthogonal trajectories. In this paper, spherical, spheroidal (prolate and oblate), coordinates are used. In these coordinates, the following formulas relating complex variables and r) are valid [6]: |(ß) = exp(Ti(ß)), ße[0,7i], (30)
for spherical! coordinates;
S(ß)=/ch(rKß», {£<ß)=/sh(n<ß))), ߀[0,*]- (31)
for prolate (oblate) spheroidal coordinates. Note that, in the case of spherical coordinates we denote a = In r, ß — 0.
Substituting the formula (25) into the boundary condition at the body surface we get the following integral equation of the first kind
-dsE = — dn dn
(32)
where djdn is the derivative along the outward normal to the surface of the body. Using the axial symmetry of the problem we
pass from the equation (32) to the following one-dimensional equation [6], [7|
0
(33)
dn
dn
where J — jjj /rv^l + (av j is the new unknown function. Here h? and =hti = h^ are Lame coefficients of the corresponding coordinate system at the point (otv,ßv,<J>) 011 'he auxiliary surface X. The point in the formula for function / means the derivative respect to the variable ß . In the formula (33) we denote
s0 - 1 J cly (34)
Note that the function and its derivatives can be expressed by the following integrals:
c =)-
cos my
(35)
„(a-bcosyy
where «>¿>0, v = l/2, 3/2, m = 0,l- The integrals (35) arc calculated by the formulas [6]:
fn =-I-I r^lfhTO-r0 V (36)
2 1 71 =-(aI°-I° )■
f , \V V JV r \ V 1 V-l )
\a + b) O
where
I_m = E( k), /„.2 = K(K), lm = --7£(K)
1 -K"
(37)
In the formulas (37) K( k), E( k) are the complete elliptic integrals of the lirst and second kinds and K —
2b a + b Numerical results
We solve the equations (21), (22) and (33) by collocation method. For this aim we divide the intervals [-d / 2,d / 2] and [0,tt] of the parameters a n p by the points
or
fc-fi-i)-1-*-
(38)
(39)
and replace integrals in formulas (21), (22) h (33) with Riemann sums. Further we equate the obtained expressions in the collocation points which we choose at x~xr or p = p,. Thereby we
pass from the integral representation of the field to the discrete sources located on the auxiliary contour £.
To test the method we plot the graph of the residual at the contour of sinusoidal periodic surface (13). The residual is determined by the formula
A = J j\ Uv, Xi( X, y,xz, Vv )d\\_ +
-d/2 d/2
+C I j2 (av , Vv YJi-x, y, -h, Vv )dxz + v0y - C,
(40)
where y = f (x) - The value j: is chosen halfway along between
the collocation points. The parameters of the problem are as follows: the velocity of the stream v(, = 5. the amplitude and period
of the surface a = d — 5, the number of the discrete sources (the number of collocation points) A' = 50 (dashed curve) n A/ = 100 (solid curve). As is seen from the figure the maximum level of the residual does not exceed 5 ■ 10 4 when N - 50 and 5-10 h when A' = 100. This fact approves the correctness of the proposed algorithm.
To test the method we also calculate the dependence of the pressure versus the coordinate a for the problem of a flow of a motionless sphere. As is known this problem can be solved analytically [ 11- In the table we present the values of the pressure at several points of a . The values of the pressure are found by MMDS and by exact solution of the problem (see [1]). The velocity of the stream v0 = 5 and the radius of the sphere is
<3 = 2.5- The values of the pressure obtained for z = 0- When calculating we use the formula
p0v
p + —= const> 2
(41)
where p(i is the density of the liquid, p is the pressure and v is
the absolute value of the velocity at the point of the space. The number of discrete sources is JV =50. As is seen from the table the results obtained by MMDS and by exact solution practically coincide.
Table
The dependence of the pressure versus radial coordinate obtained by MMDS and by analytical solution for the problem of the How of the sphere
p Solution based on MMDS Exact solution
4.5 0.014774217646401 0.014774217645982
6.5 1.528686887134084 1.528686887133924
8,5 1.929943302285395 i.929943302285331
10.5 2.080712401870247 2.080712401870198
12.5 2.149800000000029 2.149800000000003
14.5 2.185852144194946 2.185852144194934
16.5 2.206483325971682 2,206483325971673
18.5 2.219133832684904 2.219133832684886
20.5 2.227318836564669 2.227318836564665
22.5 2,232847343355143 2.232847343355143
-d/2
Let's consider further results of calculations for various surfaces. The streamlines for the problem of the flow of the sinusoidal periodic surface (13) and the surface in the form of cycloid which is set by the equations
a=a{t + tcos/), ^ = 0<x< i, 0<t<2n- (42)
are given in Fig. 2 and 3. The velocity of the stream v0 = 5, the
parameter x = 0.8, the surface period d = 5 (the same for the sinusoidal surface and for the cycloid). The graphs of dependences of pressure versus vertical coordinate for the sinusoidal surface (solid curves) and for the cycloid (dashed curves) are given in Fig.4 and 5. The curves in Fig. 4 correspond to the case x = d / 4 {for the sinusoidal surface) and t — n! 2 (for the cycloid). Thus, the abscissa of the points of observation is chosen at the surface maximum. Apparently the pressure sharply in-
w
Fig, 6. The streamlines for the problem of the flow of the prolate spheroid
2■
' 1 ' 0 1 2 I 4
cog rd male !.
Fig. 7. The streamlines for the problem of the flow of the chebyshev particle
Note that at t = 0 the particlc pass to the spheroid and a = a + b is the sum of the semi-axes of this "unperturbed"
spheroid, f — ^ja2 — b2 is a half of interfocal distance of the
spheroid. In the formula (43) q —1,2,3,..- The parameter t
doesn't accept the values greater then |6|
|T|<T„= iJEf.
V a + b
since otherwise the contour of axial section of the panicle would have the self-crossing points. In calculating we choose the semi-axes of the spheroid a = 0.5, c = 2.5 (<' is the semi-axis along z), the
parameters of the particle arc: a=2.5, b=0.5, q=6, tniK/x=1.5. The velocity of the flow is v0 = 5 - As is seen from Figs. 6 and 7 the streamlines are symmetric relative to the plane z = 0 ■
In Fig, 8 the dependences of the pressure versus coordinate v for the spheroid (dashed curve) and for Chebyshev particle (solid curve) are presented. The coordinate y = 0- The parameters of the geometry of the bodies are the same as in the previous case. The value of coordinate z is chosen so that the observation point which lies on the surface of the body is located at the maximum of the Chebyshev particle axial contour (i.e. [i = jt/3) and is located at the maximum of the contour of spheroid (i.e. B — Tt/2). As is seen from the figure the dependence of the pressure increases significantly quicker in the case ofthe flow of
Chebyshev particle, it is obvious that the received results will be coordinated with the physical nature of the considered phenomenon since the velocity of the flow is higher near maximum of Chebyshev particle cross-section contour (where the streamlines are concentrated). In the case ofthe flow of spheroid which has smooth border the pressure dependence is closer to constant.
Fig, fi. The dependence of the pressure versus coordinate x for the spheroid and chebyshev particle
Conclusions
Based on MMDS the numerical algorithm of the solution of the problem of a How of periodically rough surface and a body of revolution is developed. The initial boundary problem is reduced to the solution ofthe integral equation ofthe first kind relative to some unknown function distributed at the contour ofthe periodical surface or at the contour of axial section of the body of revolution. This integral equation is solved by the collocation method. To test the developed algorithm we have plotted the dependence of the residual on the contour of the sinusoidal surface. The computations show thai the maximum level of the residual does not exceed 10 As one more test the problem of the flow of the motionless sphere has been solved. Comparison of the exact solution of the problem and the solution received by means of MMDS is carried out. The results obtained by both methods coincide with high accuracy. The streamlines for the problem ofthe flow ofthe sinusoidal surface, the surface in the form of a cycloid, as well as the prolate spheroid and the prolate Chebyshev particle are plotted. The dependences ofthe pressure versus the vertical (radial) coordinate for the specified geometries are obtained. It is shown that when the vertical (radial) coordinate of the observation point increases the dependence of pressure has monotonously increasing character provided that the horizontal (axial) coordinate of the points of observation is located at the body surfacc maximum, ll is shown that the greater the streamlines are concentrated, the greater pressure sharply changes.
References
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МАТЕМАТИКА
ПУБЛИКАЦИИ НА АНГЛИЙСКОМ ЯЗЫКЕ
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НОВЫЙ метод решения задачи обтекания потоком несжимаемом жидкости
КОМПАКТНОГО ТЕЛА ВРАЩЕНИЯ И ПЕРИОДИЧЕСКИ НЕРОВНОЙ ПОВЕРХНОСТИ
Кюркчан Александр Гаврилович, зав. каф. ТВ и ПМ, д.ф.-м.н., Московский технический университет связи и информатики, ФИРЭ им. В.А. Котельникова РАН, ФГУП Центральный научно-исследовательский институт связи,
Москва, Россия, agkmtuci@yandex.ru
Маненков Сергей Александрович, доцент каф. Мат. анализа, к.ф.-м.н., Московский технический университет связи
и информатики, Москва, Россия, mail44471@mail.ru
Аннотация
Рассмотрена задача обтекания потоком несжимаемой жидкости неподвижной периодически неровной поверхности и компактного тела вращения. Ранее для решения задач обтекания предлагался подход, основанный на использовании конформных отображений. Однако этот метод применим лишь к двумерным задачам, причем обтекаемая поверхность должна иметь достаточно простую геометрию. Для решения задачи обтекания применен модифицированный метод дискретных источников (ММДИ), который ранее использовался для решения широкого класса краевых задач. Данный метод обладает универсальностью по отношению к геометрии обтекаемой поверхности. В работе ММДИ применен к решению задачи обтекания несжимаемой жидкостью периодически неровной поверхности в виде синусоидальной поверхности и поверхности в форме циклоиды. Для контроля точности получаемых результатов была построена зависимость невязки краевого условия для задачи обтекания синусоидальной поверхности. Также решена задача обтекания неподвижной сферы, вытянутого сфероида и чебышевской частицы. В случае задачи обтекания сферы проведено сравнение точного решения задачи и решения, полученного при помощи ММДИ. Построены линии тока для всех рассмотренных геометрий, а также зависимости давления от координаты, вертикальной по отношению к периодической поверхности и радиальной координаты для тела вращения.
Ключевые слова: задача обтекания, метод дискретных источников, аналитическое продолжение, периодическая функция Грина. Литература
1. Тихонов А.Н., Самарский А.А. Уравнения математической физики. - М., 1 999. - 798 с.
2. Кюркчан А.Г., Минаев С.А., Соловейчик А.Л. Модификация метода дискретных источников на основе априорной информации об особенностях дифракционного поля // Радиотехника и электроника, Т. 46, № 6, 2001. - С. 666-672.
3. Anyutin A.P., Kyurkchan A.G., Manenkov S.A., Minaev S.A. About 3D solution of diffraction problems by MMDS // Journal of Quantitative Spectroscopy & Radiative Transfer, Vol. 100, No. 1-3, 2006. Pp. 26-40.
4. Kyurkchan A.G., Manenkov S.A. The application of a modified method of discrete sources for solving the problem of wave scattering by group of bodies // Journal of Quantitative Spectroscopy & Radiative Transfer, Vol. 109, No. 8, 2008. Pp. 1430-1439.
5. Маненков С.А. Решение задачи дифракции на периодически неровной границе кирального полупространства // Радиотехника и электроника, Т. 56, № 8, 2011. - С. 923-931.
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