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Numerical and Analytical Investigations of a Stationary Flow past a Self-PropeDed Body N. P. Moshkin Institute of Theoretical and Applied Mechanics Novosibirsk, USSR V. V. P~lchnachov and V. L. Senn~tskii Lavrentyev Institute of Hydrodynamics Novosibirsk, USSR Abstract A stationary viscous incompressible liquid flow past a body is considered. The liquid velocity distribution on the surface of the body is prescribed so that the total discharge through it equals zero. The velocity vector tends to a nonzero constant vector at infini- ty. The following equations are to be fulfilled: = 0, (*) ~ = 0, (* *) Here S and T are, respectively, the momentum and the moment of momentum, transfered by the body to the liquid (for two-dimensional flows, S and ~ are the above-mentioned values, related to the unit length). These conditions form the boundary-value problem for the Navier-Stokes equations, which we call the problem of stationary flow past a self-propelled body. Though the problem of the flow past of a self-propelled bo- dy has a natural origin~self propulsion executes inhabitants,ships and airplan- es~and though it has a practical impor- tance,the number of works concerning it is very limited.This work contains the results of investigations of different models of a self-propulsion of a body in a liquid obtained by abalytical,nume- rical and experimental methods. 1. 1.1 Let be the bounded closed surfa- ce in Rue and Q be the domain exter- nal with respect to ~ . The problem of defining vector-function The difference between flow velocity at the point Y hand constant velocity at in- finity, en = `1, o, o' ~ and scalar function pox) (pressure) which satis- fy the stationary system of Navier-Sto- kes equations in the domain ~ 329 TV - Re av/Ox<-~vp _ Re v vv,`1 1 and the following boundary conditions: v = amp (1.2) v ~ O (1.3) when x ~ a, Here a is the vector-function satis- ~ a n dS = 0, (1.4) whilst still arbitrary in other respects' ~ is the nun' vector of the external normal to the boundary of the domain Q. Equations (1.1) have been written down in dimensionless variables, so that Re = V-1 ~ v is the Reynolds num- ber, v is the kinematic viscosity coefficient, l is the characteristic linear scale (e.g., l = diem. Q ). The value of V" (velocity of flow past) is a natural velocity scale, and the pressure scale is assumed equal to p Via V / l , where p is the liquid density. Further the surface of ~ will be as- sumed to belong to the Holder class C23=, 0~ <1, and the components of vector a - to belong to the class C 20^ ~ ~ ). Problem-~1.1~-~1.4) with the fixed function a was consi- dered in a great number of works. The most significant results were obtained by R.Fin~n [8] and K.I.Babenko 623. If a = - e' , we come to a classic prob- lem of a flow past a body with an unmo- vable impermeable boundary ~ . It is well-known that in this case the resis- tance force of the liquid exerted on the body differs from zero. Thus, to realize a stationary regime of flow past a body, it is necessary to cr~nfi- ne it to the liquid flow by external forces. The classic problem of flow

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pasr a body therefore should be called the problem of flow past a towered body. Here our particular emphasis will be p l ac e d on the p rob l em of momentum! e ss flow past a body, or problem ( 1. 1 ~ (1.4) with the additional conditioning. In the terms v, p the latter is exp- ressed by the equal ity if_ i[-P4.n ~ Revs ~ elf n]dE=0~1.5) Here F is the resistance force, PV is the stress tensor corresponding_,to the velocity and pressure fields ~ v and p , respectively ), having the el- ements (PV)ij =-P6ij + aV~/aXj + ~V`~/axi (i, j = 1, 2, 3~. It is clear that in the model under con- sideration condition ~ * ~ may be provid- ed only by the mobility and (or) the per- meability of the body boundary ~ . In the latter case it is natural to be ex- pec ted t hat the total discharge of a 1 i - quid through the body surface is equal to zero, that is expressed by ~ 1 . 4 ~ . The condition of momentumless flow past a body ~ ** ~ is not imposed here upon the solution . I t i s we l l-known that wi th f ixed a problem ~ 1 . 1 ~ - ~ 1.4 ~ has at least one solution for any Reynolds number, Redo ~2, 8, 14] . With low Re , its solution is unique [23. From here it follows that the problem of momentumless f low past a body, (1.1~-~1.5) is solvable, gene ral ly speak ing, only i f the_'addi t i o- nal conditions on the function a ~ x are fulfilled. In other words, this problem is to be considered as a prob- lem of a joint definition of functions Hi, p and a from the relations ( 1. 1~- -~1.5~. In this case condition (1.5), equiva] ent to three scalar relations, admits a wide Arbitrariness -in choosing the function a , which may be consi- dered as a determining function. It should be noted that by virtue of the law of momentum, condition ( 1. 5! is equivalent to the equal i ty to zero of the liquid pulse flow through any cont- rol surface covering , in particu- lar, an infinitely distanced co-,1;rol surface t183. 1. 2. The existence of the solution to the p roll em of morrientuml e ss f l ow pas t a body has not been considered so far. The ex i steno e p rob l em may be so lved ra- ther effectively in the simplest case when the Reynolds number is equal to zero 619] . Let us denote system ( 1.1 ) when Re = 0 ( Stokes system) by ( 1.1 ) O and the relation ( 1. 5 ) when Re = 0 by (1.5)o. System (l.l)o, conditions (1.2) - ( 1. 4) and equality ( 1. 5 ~ O form the problem which will be denoted by ( 1.1 ) O -~1 . 5) o and called the problem of momen- tumless flow past a body in the Stokes approximation . In contrast to ~ 1. 1 ~ - ~ 1. 5 ), problem (l.l~o-~1.5)o is linear, and the Stokes operator generated by system ( 1.1 ) O is a self-conjugated one t143. The above- mentioned circumstances make it possib- le to find an effective solution of problem ( 1.1 ) o-( 1.5 ~ o in terms of eigen- functions of some spectral problem. It may be formulated as follows. It is required to' find number ~ and vector-function up ~ x ~ ~ O. determi- ned on the surface 2,which satisfies con- dition (1.4), from the relation Here ~ denotes a linear operator, which gluts in correspondence with func- tion cp the valye of the trace of the stress vector PV on the surface I, calculated from the solution v' p_,of problem ~ 1. 1 ~ 0 - ~ 1. 5 ~ O. where a = (p Operator A , initially determined on the functions tp ~ ~~ ~ ~( ~ ~ , admits a self-conj~agated expansion up to the operator, acting to H-~ (E ~ from ~ ^/2 ~ ~ ~ A tonality of the trace of the functions u ~ H ~ ~ Q ~ on the surface are denoted by H 4~2( ~ ~ and the closure in the Dirichlet integ- ral norm of a set; of solenoidal vector- -functions, smooth in Q , and equal to zer o at sufficiently high values of | x ~ is denoted by ~4 ~ Q ). The spa- ce ~-~/2 ~ ~ ~ is conjugate with res- pect to p - 2( ~ ). Using the theory, de- veloped in t15], it is possible to find the following properties of operator A: ~ i ~ it has an inverse A ~ , which is c ant i nuous and se l f - c on j ugat e d; ~ i i ~ the spectrum of operator ~ is discrete and finite-fold; ~ iii ~ all the eigenvalues \~<~2<~3 are positive; ~ iv) Woo when k ~ Go; the eigenfunctions (Pk and (p3 corresponding to the eigenva- lues As and \~ ;` Ak , are orthogo- nal both in metrix I: ~ ~ ~ and that gene rated by a sc al ar produc t ~ AM ~ ~2 The totality of eigenfunction~ { epic } forms a full system both in ~2.( ~ ~ and Hi/= (E ). Now it is not difficult to formulate the solution algorithm for the problem of moment:umless flow past a body~in~the Stokes approximation. Let as = (a, ~~) denote the Fourier coefficients of ex- pansion of the function a (x) in basis { Ilk } , orthonormalized to t~ ~ ~ ~ Then the condition (1.5)o may be written as follows: 330

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oo ~ ( Aka g Kid 2) - O (1 6) where Ins . is; the ith ~omnonent of vector ' Ok . Condition (1.4) means that oo ~ (ak i ~~ n d Z) = 0. (1.7) k=1 ~ Let us choose an arbitrary element () satisfying conditions (1.6), (1.7) and then solve problem (l.l)o - - (1.5)o. In so doing the velocity vec- tor ~ is defined one-valued, and the pressure p is defiend accurate to the additive constant. Thus, in the so- lution of (1.1)o~~1~5)o being She prob- lem of determining functions v, p and ~ , there is a functional arbitra- riness. Therefore, determining the fun- ction ~ from the condition of mini- mum power functional is considered to be natural: J = ~ ~ ~ ud Z. The value of J is equal to the work expended per unit line in sustaining a stationary self-motion regime. In the general case the problem of de- termining minimum J under certain n~- tural restrictions upon the function a is solved by the method of Lagrange in- definite coefficients. Let us consider the examples of such restrictions:,the function ~ + Alphas the support ~ which does not coincide with the whole surface of ~ (physically the case when the region ~ c is sufficient- ly small is of interest); function em has a zero normal component (the boundary of a self-mov~ng body is im- permeable); function al + em has a ze- ro tangential component (the body sur- face is unmovable). If the surface is a sphere, the minimization problem of functional J for the second and third type of the above-mentioned re- strictions is solved explocitly with the use of the results obtained in t10~. In both cases the minimum of J is achieved on the functions a , cor- responding to the regime of potential flow past a body. It is unknown if this property of extremals of functional J holds for an arbitrary surface ~ . The matter of solvability of problem (1.1~-~1.5) when Re > 0 is rather com- plicated. However, it is hoped that its solution may be positive for low Reynolds numbers. The solution asymptotics for a classical problem of flow past a body ~ ~ - -elf when Be ~ O to] is the basis for such an optimism. 331 1.3. Now let, us consider an asymptoti- cal behaviour of the solution to prob- lem (1.~-~1.5) when r = |x | ~ ~ . For now, let us ignore the additional con- dition (1.~) and recall that the exis- tence theorem for (1.~-(1.4) "as a whole" is solid in the class of vector- -functions v having Dirichlet fini- te integral t14], v : pvdx < (1.8) Q In 62] it was established that any so- lution of the above-mentioned problem satisfying the inequality (1.8) admits the estimate Iv~xyl< Cr-1/2- when r ~ ~ (1.9) with some positive constants C and As was shown in t8], any solution of problem (1.1~-~1.4) Ratifying inequa- lity (1.9) has an asymptotical beha- viour v (x) = ~ ~ (x) + ~ (x). (1.10) Here ~ is constant vector determined by the formula (1.5), and ~ (~) is the remaining term, for which the fol- lowing estimate has been obtained ~ r is high): 1: 1 < c1r~3/2+ ~ ~1 ~ S) .~1.11) Here ~ = r - x , ~~ is arbitrarily small, and Cal = const. > 0. symbolE(x) denotes the fundamental tensor of Oseen system, corresponding to (1.1~. The ex- pressions for the elements of tensor E may be found in t83. It follows from (1.10),(1.11) that the- re is a paraboloidal region of the wake in the direction of e' , inside of which v- Or. Beyond any circular cone having the axis directed along ~ , = 0(r~9 ).The field asymptotics terms vex) of the order of r - ~ /2 obtained in 63] on the assumption of the colli- nearity of vectors ~ and en . (This assumption is fulfilled, for example. in the case of axisymmetrical flow past a surface of revolution ~ , thereby the condition (**) being also fulfilled). In 03, 6] the velocity vortex behaviour at great distances from the body is in- vestigated, and the vortex is shown to decrease exponentially outside the wake Formula (1.10) means that far from the towed body, the velocity field distur- bance (accurate to the smalls of higher order) will be as that of Oseen flow "flowing past" a concentrated force ~ .

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The velocity field asymptotics in the problem of momentumless flow past a bo- dy proved to be determined by a much higher number of functionals characterizing both the body shape and the way of realization of self-mo- tion regime. The basic result may be formulated as follows :203. Let A, p be the solution of problem (1.1~-~1.4) from the class (1.~! satis- fying the additional condition (1.5~. Then when I. - ~ the asymptotical repre- sentation of vex) in the form ~~) - R ~ DE(x) + (1.12) is valid. Here R = (Rz: ), Q = (Qi~)are the constant tensors ~ i,j = 1,2,3~. The elements of tensor R are express- e~d explicitly in the terms of function aid) . Symbols DE and VE denote the third-rank tensors having the ele- ments (DE)ijk ~ 2 (a x + a ~ ~ ' and (V E) _ `~ , i,j,k = 1,2,3. Summarizing in consoli- dation R : DE , Q : ~ E is made with respect to indices i and a . Func- tion q(~) admits the estimate 1~ (x)1 ~ Car 2+ (s + 1~-1/2 `1 13' when r ~ = , where ~ and C2 are the positive constants, ~ being arbitra- rily small, s= r -x . Formula (1.12) is derived on the basis of integral reperentation of problem (1.1~-~1.3) obtained in [9]. To estima- te the volume integral ~(x) = /~) v~y) = E(X - y) by Q when r ~ = , the results of t3] have been used. According to (1.12),(1.13), in the re- gime of momentumless flow past a body we have ~ - O(r~3/23in a paraboidal re- gion of the trail and ~ = 0 (r~s/~) beyond any cone having the axis e ~ . Thus, a quicker decrease in velocity disturbance at great distances from a self-moving body, as compared to towed one, is evident. Representation (1.12) also means that (at least over the wake region) the main terms of veloci- ty field asymptotics in the problem of momentumless flow past a body are cha- racterized by 18 parameters, they are the elements of tensors R and ~ . In an axisymmetrical case the number of these parameters decreases to eight. Identification of the elements of tensor Q being some functionals of the so- lution to problem (1.1~-~1.5) is of par- ticular interest. This problem has not been solved so far. One of paradoxical results, related to the classsical problem of flow past a body for the Navier-Stokes equations, is as follows. Let in (1.2) a ~ - em (This means immobility and impermeability of the broody boundary). Then for any solut- ion v, p of problem (1.1~-~1.3) satis- fying the condition (1.8) we have i 1~1'dx Q Such a statement related to the energy of disturbed motion in the problem of viscous flow past a body first appeared in t73. It is obvious that a self-moving cannot contribute such a great disturbance in- to a flow. An appropriate exact formu- lation is as follows. Let V, p be the solution of problem (1.1~-~1.4) satis- fying the additional conditions (1.5), (1.8~. Then ~ 1 V12 Ox ~ ~ (1.14) Q The property of the solution of the problem of momentumless flow past a bo- dy expressed by inequality (1.14-) dis- tinguishes it among all the possible solutions of problem (1.1~-~1.4) if function a entering it is varied. We hope that this property may be used to investigate the existence of the solu- tion of problem (1.1~-~1.5), if the lat- ter is considered as some optimization problem. The two notes are to be made in connec- tion with the velocity field asympto- tics at great distances from a self-mo- ving body. The first of them deals with the velocity vortex behaviour with in- creasing the distance from a body. When r ~ ~ , the vorticity in the solution of problem (1.1~-~1.5) decreases quicker than in a calssical problem of flow past a body both inside and outside of the wake r63. In the exceptional cases the wake may be absent. In [16] such a si- tuation is illustrated by a plane poten- tial viscous flow past a self-moving "body". Its boundary consists of two symmetric coupled components, on each of them a normal velocity component be- ing equal to zero, and a tangential one being constant. 332

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The second note is connected with the velocity asymptotics in a turbulent flow regime. In this case, based on the con- sideration of ~203 and additional assum- ptions related to the Reynolds stress behaviour when r ~ ~ (for a self-moving body these assumptions are discussed in [53), it is possible to obtain represen- tation (1.12) for the averaged velocity field. However, here the elements of tensors R and Q are the functionals of the solution of unclosed system of Reynolds equations. These equations are treated as the Navier-Stokes equations with the density of external mass for- ces ~ =-2.Re . div ~ where ~ is the Reynolds stress tensor having the ele- ments USA =',i'v~' (i, j = 1,2,3~. Here condition (*) changes its form in comparison with (1.15) and is as follows: t-Pi' ~ + Be' (v + edgy Ii + 2. ~ = (l, o, o);Re - aV-/~ is the Reynolds number ~ V is the kinematic coefficient of the liquid viscosity); ~ is a function of Re; f is an add function~of the angle ~ between the vectors ~ wand X ~ + Y ~ ~ ~ = (0, 1, o); ~ -,.f,,' sin ~ 6; fit are constants; t~ ~ O; f2 ~ 0~; ~ - _ ~ ~ ~ = (0, 0, 1~; ~ is the unit external normal to the cylinder boundary). The dependences f on ~ and ~ on Re are prescribed so that ~ = 0. In [213 the problem (2.1) was solved approximately for low Re . The fol- lowing asymptotical formula for the li- quid velocity at great distances from the body was obtained: rat ~ 2 ~1f2 it a v1/2 ' ~ ' L, ~ r ~ V4/2 Z;~/ - 1 ~ + Re II rid did = 0. _ x, jeep (- 2 v ~ 4v~ 2.1. Liquid flows at great distances from self-propelled and non-self-propel- led bodies can be significantly diffe- rent. Let us consider the problems of a sta- tionary flow of a viscous incompressib- le liquid past self-propelled bodies [21,223. a). The body is a circular cylinder with a moving body. The cylinder axis coinci- des with the axis Z of the system of the rectangular coordinates X, Y. Z. The liquid flow is plane and symmetri- cal in relation to the axis X in the plane X, Y. The body boundary moves so that the correlation (*) is fulfilled. The correlation (**) is fulfilled by the reason of the symmetry of the flow in relation to the axis X. The Navier-Stokes and the continuity equations and the conditions on the cy- linder boundary and at infinity have the following form: 1 (7 V)~ 2 ~ P + R 67' V. ~ = 0, e (2.1) ~ = Off for r = 1, vet ~ for r ~ oo, where V = V/V ~ V is the liquid ve- locity; V is the X-component of ]be liquid velocity Vex at infinity ~ r = _ ~ V ,o, O) ;V04 so; = P/~V~) ~ P is the pressure in the liquid: _p is the liquid density); r = FAX + Y7a a is the radius of the cylinder); 333 (2.2) for Pa ~ + ~ and constant Y /(aX),Re. In accordance with (2.2) for the plane flow past a self-propelled body, the disturbance of the livid velocity va- nishes by the law X (for the pla- ne flow past a body which receives from the liquid per unit length per unit ti- me a non-zero momentum the disturbance of the liquid velocity vanishes by the law A/ rs~ my. The formula (2.2) has such a form as the formula for the velocity of plane liquid flow far from a self-propelled body founded in ~5~. b). The body is a ball with a liquid- -permeable boundary. The ball centre coincides with the origin of the coor- dinates X, Y. Z. The liquid flow is sym- metrical in relation to the axis X and is non-swirling around it. On the body boundary the liquid velocity component normal to it is distributed so that the liquid mass flux through this boun- dary equals zero and the correlation (*) is fulfilled. The correlation (**) is fulfilled by the reason of the non- -swirling of the flow around the axis X. The Navier-Stokes and the continuity equations and the conditions on the ball boundary and At infinity have the following form ( - V V);~= -ha+ ale if' of' ~ ~ = 0' (2.3) v= Afn for r=1, vat i for rat Oo, where r= ~ x2+y2+ zc /a ~ a is the radius of-the ball); ~ is a function of Re ; f is a function-~of the angle ~ between the vectors ~ and

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Z~ + YJ + Zig (f =~ ~ '~ (cos ~ ); fig ~ are~constants; I1 ; f2 /0; Pm are the polynomials of Legendre). The dependences f on ~ and ~ on Re are prescribed so that ~ = 0. In C22] the problem (2.3) was solved approximately for low RO . The follo- wing asymptotical formula for the li- quid velocity at great distances from the body was obtained: 2 ~ ~ 3 foal y ~ z2 V ~ V [1 + 2 (1 - V" 4 Vie )x x em (- V 00 - 4 j ] ~ (2.4) fo/ (a ~ + ~ and constant(~2+Z2~/ In accordance with (2 .4 ~ for the axi- symmetrical flow past a self-propelled body the disturbance of the liquid ve- locity vanishes by the law X~2 (for the axisymmetrical flow past a body which receives from the liquid per unit time a non-zero momentum the disturban- ce of the liquid velocity vanishes by the law ~-1 t5J'. 2.2. A propeller work can significant- ly affect a liquid flow near a body at distances which do not exceed its several transverse dimensions. This may be used to organize required liquid flow around a body (for example, a flow with closed streamlines). Let us consider the problem of a sta- tionary flow of viscous incompressible liquid past a pair of rotating cylin- ders (a pair of identical parallel cir- cular cylinders rotating around their axes with opposite angular velocities) 6232. The axes of the cylinders are parallel to the axis Z and intersect the plane X, Y at points X = 0, Y = h and X = 0, Y = -h. The liquid flow is plane and symmetrical in relation to the axis X in the plane X, Y. The Navier-Stokes and the continuity equations and the conditions on the boundaries of the cylinders and at in- finity have the following form: ( - v.~79v = PUP+ Re ~ or, v V :: 0, v_ _k~ r+ for I r+ I =1 ~ A_ k x r ( 2 . ~ ) for Or ~ =1, v ~ i for r ~ ~ , where ~ = V/(Q a) ( Q is the mo- dule of the angular velocities of the cylinders; a is the radius of the 334 cylinders); p = ~ ~ pa 2a2~ ; r r - ~~ + ~ +21/~2 ~) i, = ~ (~) ; Be - a Q is the Rey- nolds number. In :23] the problem (2.5) was solved approximately for small ~ . It was ascertained that in the considered ap- proximation the pair of rotating cylin- ders is a self-propelled body. ,9 _: 1~] it: ~= Fig. ~ - x :~ ~ In Fig. l the pattern of streamlines of the flow around the cylindersis dis- played(the sections of the cylinders by the plane X , By are represented by the points X = o, y = l/2 and SX = 0, By = -l/2; the cylinders are surrounded by the liquid layer which is streamed continuously by the liquid mo- ving from infinity) This flow around the cylinders is realized by the motion of their boundaries, i.e. by the propel- ler work. The problem of a stationary flow of a viscous incompressible liquid past a pair of rotating cylinders was conside- red in 623] in connection with the prob- lem of decreasing the energy required for a body to propel in a liquid. In connection with the latter problem the motion of a pair of rotating cylinders in a liquid was also investi.g~.ted ex- pe-rimentally 524, 253. The measurements showed that the self-propelling of the cylinders was energetically non-prof.i.- `;able and a reduction of the energy which was necessary for their propelling

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could be achieved when the cylinders simultaneously rotated and were af- fected by an external force (the pro- peller worked but not in a self-propel- ling regime). 2,0 The power N which is necessary for a l8 pair of cylinders to proper is the sum of two quantities: ~ _ FQ ~ W. 1,6 1,4 where ~ is the external force acting onto the cylinders; Q is the veloci- ty of the pair of cylinders; W is the power which is necessary for the cylin- lo ders to rotate. ],6 ],4 i,2 i,0 0~6r 0,41 C em _ ~ ~ 6 _~ 6 ~'D, ED ~ ~ O . 1 1 V ~> D D ID D ID ~ ~ ID ~ ~e . 0 0,8 I,6 2,4 3,2 4,0 4,B i 0 3 (D 5 ,,9 me 40 0,2 Fig. 2 In Figs. 2, 3 the data obtained by mea- surements on the dependences C = F/~2La ~2) and w = W/ (2 ~ a p Q3) on u = Q ~ Q are displayed ~ L is the length of the cylinders; symbols 1-5 correspond to = 0.45;0.4~0.35; 0.3;0.25) :253. In accordance with the- se data, the most energy profitable propelling is realized at u % 1.3.The power which is necessary for this pro- pelling of the cylinders is approxima- tely, 80% of the power required for the propelling of the non-rotating cylinders. I,2 0,6 0,4 0,2 o 3 335 W I ~ 2 e 3 ~ 4 0 5 0 0,8 1,6 2,4 3,2 4,0 4,8 Fig. 3 3.1. Let us come over to the problem of numerical modelling of momentumless flow past a body. It is based on the Navier-Stokes equations with the use of numerical methods t1,4,11,132. The two cases are considered when relation (*) is fulfilled. In one of them (down- stream), there is a surface S behind the ball (a part of sphere with a grea- ter radius). The liquid flows over S and gets thereby an additional pulse. In the other case the ball surface is permeable. On one its part, between two cones with the divergence semi-angles Q' , Q2 and a mutual axis 0 = ~ , the liquid is sucked in , and on the other its part, "cut" by a cone Q3<65 the same quantity of the liquid is re- turbed to the flow. The other way to fulfil the condition (*) are also possible, for example, by prescribing a self-consistent distribu- tion of volume forces, localized in a small region behind the body. A numeri- cal solution of the problem of momentum- less flow past an extended ellipsoid of

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revolution was obtained in this state- ment in t123. The solutions of the problem of flow past a self-moving body becomes more complicated due to nonlocality of con- dition Gil. In the present paper a sta- tiona.ry solution was found when solving the nonstationary Navier-Stokes equa- tions. In the spherical coordinate sys- tem they have the form: (u-cos6)aU + (~+sin6)au v(~+sinO) r2E2 ~ [flu ~ ~ r 2 Bin ~ at X x (sin ~ V) ] , a~U-cos~' a TV u(~+sin6) 2,~2ct~3~ ~ aP~ _ ~ _r __ _ ~ ~~6 r ~ r r as (3.1) +ReL69~ rid (3 - ae ], at +(U-cS6)aa~+ Using ~'u-cosOw.(v~sin6)W Ct'Z:6 1 x X(6w~ Use E3 )' r12 arfr U)''rsinSaG resin E3 v)=0, Re avV; r= Va a, (A r2 ~3~(r ar) + r' sin (3 aa x x-~ninO4~), HereU, v and W are the projections of the velocity vector to r , ~ and ~ directions, respectively, a is the sphere radius, V. is the modulus of inflowing flow velocity, ~ is the characteristic value of swirling by the propeller model. Equations (3.1) have been written down in dimensionless v~- riables. The values of a , Vat , pV-, a ~ have been chosen as the scales of length, velocity, pressure and swirling. The boundary conditions are prescribed as follows: The sphere surface r = 1; u = uO (e ~ + cos v = - sin 6, W = WO(O ~ The axis of symmetry ~ = 0, ~ = a: an/ as = o, v = o, w = o; (3.3) The conditions on the sphere having a large enough radius r = r : ~ = 3, W = 0, p = 0; (3.4) The conditions on the surface S ~ (r, S): r = rS, 6~< 0 < A}. p+- p~_ [p], W~~ ~'V~- W (r 0 ~ (3'5) Here an upper index + or - states to the side of the surface S (e.g. (rs' 0 ) ==llm+Ov (r, any. The considerations dealing with the va- lidity of such a statement can be 336 found in Ply. S.~. To solve numerically the model problem under consideration Ill;, the methods such as [~AC, SUMAC and other similar ones were generalized ~l,4,ll]. The main differences are due to a spe- cial way of prescribing conditions (3.4) when r=r" and the presence of surface S having the pressure jump. The unknown values were calculated in the nodes of displaced network ~ u , v in the middles of the cell sides; P , ~ at the centers of them). The radi- al velocity component u on the sphe- re with a large radius was determined from a difference analogy of the con- tinuity equation. Surface S passed through the centers of the cells. The velocity components v and W pres- cribed on S were found with the use of (3.5), i.e. the equation at these points was not used. When the problem with surface S is considered, two pressures P+ and P , are to be determined at points of this surface, one of ~hem~be- ing excluded with the help of P - P = Opt. In the Poisson difference equation for pressure, which is derived in a usual way, in its righthand part there appears an additional term const UP] differing from zero only at the points near surface S. The pressure equation was solved by the method of upper re- laxation. Fulfillment of (3.5) was ac- hieved by correction of the pressure jump En], prescribed on S. or the volu- me of liquid flowing over the body sur- face. The pressure jump or discharge variations were made within the general iteration process. The flow region was mapped onto the rectangle by transfor- mation r = exp Z. ~ _ 6~. The dimensio- nal network was introduced in plane ~ Z. 0' ~ 3.3. We made some calculations in or- der to compare the basic hydrodynami- cal characteristics for towed or self- -moving sphere. The numerical experi- ments were made under the following boundary conditions. In the problem with surface S we have no ~ ~ ~ - O. WOL 6) - O. (3.6) Ws~rs,(3~~ (1 e~33 R2Si=2 63~/(R . sinO) (3^ ~ 6) ~ TC, (3.7) O. for other ~ (here Rs is the dimensionless radius of surface S). If self-motion is model- led by the liquid flow over the sphere surface, conditions (3.5) on S are not used. Functions use ~ ~ and WOL e ~ were

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Pig. 4a . . . . - ~ )~ ea -` ~ ~=~ ! n ~ Fig. id prescribed in the following way: -E f God, Q USA- E1f1~. ~30 for other 0, E'= ~ (Jo Si;E~0 d6) 1 2 ~ (r God) sin E3 d6)-' ~3 where f1~43) =~63-61)~632-6 )'2~) ~ 3), W (~3)~(1-e 63)/Si~1 ~ ~ 63< ~ < i, ~ O. for other ~ . ( 9 (3.8) (3.7), (3.9) mean that the flow is swir- ling only after it has passed through the unit which models a propeller. Presented in Fig. 4 are the isolines of the stream function, swirling and pres- sures for Re = 50. Fig. 4a illustrates the towed sphere, and Figs.4b,c illust- . . at' <'=~ ~ Fig. 4b 337 \ \~ / Fig. 4c rate the self-moving sphere. The propel- ler is modelled by the surface having the pressure jump Rae = 1.4; 6~=162. Fig.4b illustrates the flow rotation when ~ = 0; C P ~ = 3.00. Fig. 4c cor- responds to ~ = 15, C P ~ = 4.38. Figs. 4d,e represents a self-moving sphere with a permeable surface. The case when ~ = 0, 6' = 108; 62= 132; 63= = 156 is shown in Fig.4d. Fig.4e corres- ponds to the case when ~ = 20; 61 = = 1125 ; 62 = 135; 63 = 157 The calculations were made with the use of different networks. The most detailed network consisted of 40 nodes in a ra- dial direction and 60 nodes in an angle. In all the calculations r = 12.1 ~ Z = = 2.5~. The number of iterations requi- red for a stationary distribution of pa- rameters to be achieved was dependent on the choice of initial values Of r P ~ and U . When the rest state was assu- med to be an initial state, not more than 1500 iterations were to be found. Contuniation in the number Re reduced the required number of iterations by a factor of 1.S or 2. 3.4. The problem of stationary motion of an arbitrary body with a constant ve- locity is tightly connected with the es- timate of the consumption of energy ne- cessary for providing the motion regime under consideration. In the case of to- wed body the required power ~ is simp- ly expressed through the resistance ~ ;

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N = ~ V . In the case of momentum- less flow ~ - O. One can judge about the value of required power by the los- ses of mechanical energy, associated with dissipation, A- 2 ~ V r D: Ddx Q where D is the strain velocity tensor; 5~ is the flow region which is unbound in the case of external problem of flow past a body. Fig. 5 gives the values of dimensionless dissipation ~ ~ ~ - =~2~pV3a3~) for different cases of stationary motion of a sphere having the radius a with a constant velocity V" in the liquid having the density p for different Reynolds numbersRe_~/ Ah. Fig. He The solid line denotes the data for the towed sphere ([ = Cd ~ 4) . A dashed line denotes the data for a linear prob- lem (Stokes approximation). The calcula- ted values of dissipation for different rotations for the problem with a per- meable surface of the sphere, O' =112.5. 62 = 135; 03 = 157.5 are denoted by rectangles. The problem with the sur- face S ( rs = 1.4; 6~ = 162) is deno- ted by circles. Non-shaded circles cor- respond to the problem with zero rota- tion. It is easily seen that the flow rotation amplification is accompanied by the increase in dissipation of mecha- nical energy, and, consequently, the increase in required power. For the problem with surface S the re- sistivity coefficients C p and Cop may be calculated in the usual way. Fig.6 presents the values of C p , C ~ for different Re . It is interesting to no- te that if Cop > Cp for a towed sphere, Cop may be less than Cp for a self- -moving one ire the Re range under 338 t.~T I.4. ~ n. . . _ 0.6 0.2 7.~ '.4~ i.0 0.4f up C', Cp consideration. 1. A. o o . ~~ \\\~ \ 10 30 50 Re Fig. 5 rl Polo C' ~ O + Cp c . ~ 0 \ \\ ~ 0 P ' _ _~ _ I I ~ 0 30 50 Fig. 6 References Re Amsden, A.A., Har]ow, F.H."A simpli- fied MAC technique for incompressib- le fluid flow calculations",J. of Comput. Vol.6, pp.322-325 (197C)~. abenko, K.I. "On stationary solu- tions of the problem of viscous in- compressible liquid flow past a bo- dy", Matematicheskii sbornik. Vol. 91~133), No.. 3. Babenko, K.I., Vasiliev, M.M. "An asymptotical behaviour of the solu- tion of the problem of viscous li- quid flow past a finite body'! Mos- cow - Reprint, Institute of Applied [mathematics, USSR Academy of Scien- ces, r`~O.84 (1971 ' r -do Belotserkovsky, O.M., Gushin, V.A., Shc. OCR for page 329
5. Birkhoff, G., Zarantonello, E.H. "Jets, wakes and cavities", New York Academic Press, (1957~. 6. Clark, D.C. "The vorticity at infi- nity for solutions of the stationa- ry Navier-Stokes equations in exte- rior domains", Indiana Univ.Nath.J. V]-~), inch 7; pp.~33-654 (1971~. 7. Finn, R. "An energy theorem for vis- cous fluid motions", Archive for Ra- tional Mech. and Anal., Vol ~9 No.5, 19. pp.371-381 (1960). 8. Finn, R. "On the exteriour stationa- ry problem for the Navier-Stokes equations, and associated perturba- tion problems", Arch. for Rational Mech. and Anal. Vol.19, No.5, pp.363- -406 (1965~. , ^..-., Hsiao, G.C., Wendland, W.L. "Singular perturbations for the exterior three-dimensional slow vis- cous flow problem", J.of Math.Anal. and Applications Vol.110, No.2, pp. 583-603 (1985). 10. Happel, T., Brenner, H. "Low Rey- nolds number hydrodynamics", Prentice-Hall (1965~. Harlow, F.H., Welch, J.E. "Numeri- cal calculation of timedependent viscous incompressible flow of fluid with free surface", Phys. of Fluids, Vol .8, No. 12, pp.2182-2189 (1965~. 12. Izteleuov, M.I. "Calculation of mo- mentumless flow past an ellipsoid", Problemy dinarniki vyazkoi zhidkosti, Novosibirsk (1985), (in Russian). 13. Kuznetsov, B.G., Moshkin, N.P. "Vis- cous flow in an angular cylindrical tube with flowing through an inner boundary", Problemy dinamiki vyazkoi zhidkosti:Trudy X Vsesoyuznoi konfe- rensii, Novosibirsk, pp.188-191 (1985), (in Russian). 14. Ladyzhenskaya, O.A. "Mathematical problems of viscous incompressible liquid dynamics", Moscow, Nauka Pub- lishers (1970), (in Russian). 15. Lions, J.-L., Magenes, E. "Problems aux limites non homogenes et appli- cations", Paris, Dunod (1968~. 16. Lugovtsov, A.A., I.ugovtsov, B.A. "The example of viscous incompres- sible flow past a body with a mov- ing boundary", Dinamika sploshnoi sredy, Novosi.birsk, vyp . 8 ( 1971 ) . 17. Moshkin, N.P., Pukhnachov, V.V. "A momentumless viscous incompressib- le flow over a body", Book of Ab- stracts, Soviet Union - Japan sym- posium on computational fluid dyna- ~r~ics (USSR, Novosibirsk, Sept. 9- -16, 1988), pp.84-85 (1988),~in Rus.) 18. Pukhnachov, V.','. "On some modifi- cation of flow past a body", Prob- lemy matematiki i mechaniki, Novo- sibirsk:Nauka Publishers (1983~. Pukhnachov, V.V. "Stokes approxi- mation in the problem of flow past a self-moving body", Boundary va- lue problems in mathematical phy- sics and their approximations, Kiev, Naukova Dumka Publishers (1989), (in Russian). 20. Pukhnachov, V.V."Velocity field asymptotics at great distances from a self-moving body", PMTF, No. 2, pp.52-60 (1989),~in Russian). 21. Sennitskii, V.L. "~.iq~id flow around a self-propelled body", Zhurnal Prikladnoi Mekhaniki i TekhnicI-~es- koi Fiziki, No.3, pp.78-83 (1978) (in Russian). 22. Sennitskii, V.L. "Example of flow of an axisymmetric liquid stream over a self-propelled body", Zhur- nal Prikladnoi Mekhaniki i Tekhni- cheskoi Fiziki, ?Io.4, pp.31-36 (1984), (in Russian). 23. Sennitskii, V.L. "Rotating cylinders in a viscous liquid", Part 1, Dina- mika sploshnoi sredy, Novosibirsk, Vol. 21, pp.70-83 (1975~; Part 2, Dinamika splosl-~noi sredy,Novosi- birsk, Vol. 23, pp.169-181 (1975) (in Russian). 24. Sennitskii, V.L. "On the propelling of a pair of rotating circular cy- linders in a liquid", Dinamika sploshnoi sredy, Novosibirsk, Vol. 47, pp.145-153 (1980) (in Russian). 25. Sennitskii, V.L. "On the drag for- ce acting on a pair of circular cylinders streamed by water", Di- namika sploshnoi sredy, Novosibirsk, Vol.52, pp.178-182 (1981) (in Rus- sian). 339

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