Stokes akımlarında viskoz direncin teorik olarak hesaplanması ve direnci minimum olan formların belirlenmesi

Abstract

ÖZET Bu çalışmada, serbest su yüzeyinden sonsuz derinlikte sabit hızla ilerleyen bir çift-modelin viskoz direncini. belirle mek amacıyla, Stokes yaklaşımı çerçevesinde matematiksel bir yöntem geliştirilmiştir. I. Bölümde konunun önemi vurgulanmış ve ilgili çalışmalar sıralanarak, temel denklemlere sınır koşulları ortaya konulmuştur. II. Bölümde direncin sonlu kalması gerektiği fiziksel gerçeğinden hareketle, orijinal problemin kısmi türevli diferansiyel denklemleri, Hilbert uzayında açılım yoluyla sonsuz boyutlu adi türevli diferansiyel denklem takımına indirgenmiştir. Ayrıca, sınır koşulları da geliştirilen yönteme uygun olarak düzenlenerek, problemin seri çözümlerinin yakınsama kriterlerine ilişkin önermeler ispatlanmıştır. III. Bölümde sonlu boyutlu alt uzaylarda seri çözümlerin nasıl bulunacağı gösterilmiş ve bir boyutlu alt uzaydaki çözümler kapalı formda elde edilmiştir. Çalışmanın ilgili literatüre getirdiği en önemli yenilik, cismin geometrisine bağlı olmaksızın süreklilik ve dönel simetri koşulları altında, her türlü cisim için geçerliliğidir. Cismin yüzeyini karakterize etmek üzere önerilen şekil fonksiyonunun parametrik yapısı nedeniyle yöntem, belirli dizayn şartlarını sağlatmanın dışında, parametrelerin uygun şekilde seçilmesiyle direnci minimum olacak formların belirlenmesi için de elverişli dir. Bir boyutlu alt uzaydaki kapalı çözümler kullanılarak bulunan sayısal sonuçlar, literatürle karşılaştırıldığında oldukça yeterli olup paket bilgisayar programları yardımıyla hesaplanmış optimum formlar da gerçekçi ve kabul edilebilir niteliktedir. Çalışmanın, bundan sonra, hangi yönlerde geliştirileceği konusundaki görüş ve öneriler ise son bölümde tartışılmıştır. Buna göre, sonlu alt uzaylarda yakınsaklığı gösterilmiş çözümler kullanılarak geliştirilecek bilgisayar programıyla, sadece bir boyutlu alt uzay için verilmiş sayısal sonuçlar uzayın boyutu artırılarak test edilebilecektir, öte yandan uzayın boyutu sonsuza götürüldüğünde, çözüm serilerinin her biri teker teker yakınsak olmakla birlikte, bunların lineer bileşiminden oluşacak genel ' çözüm sonsuz eleman içereceğinden yakınsaması garanti değildir. Gelecekte çalışmaların sürdürüleceği bir diğer alan ise, sabit hacim koşulu halinde çıkacak optimum formlar dışında başka dizayn koşullarının da dikkate alınarak optimizasyon tekniklerinin geliştirilmesi olacaktır. -iv-

SUMMARY VISCOUS-DRAG EVALUATION VIA A THEORETICAL METHOD AND SHAPE OPTIMIZATION IN AX I SYMMETRIC STOKES FLOW The classical Stokes-flow problem describing the creeping motion of a single body, without rotation, has been studied for more than a century. The importance of solving the problem is its applications for instance drag evaluation of a double-ship model. Moreover, in some cases', the solution of Stokes equation is used as a first approximation of Navier-Stokes equation in full form. First explicit analytic solutions are due to Stokes [33] for a sphere, and Oberbeck [34] for an ellipsoid. General solution of the Stokes equation in spherical co-ordinates is given by Sampson [35] in terms of stream functions as an infinite series of Legendre polynomials by separation of variables. However, both analytical and numerical implementation of this exact solution is extremely difficult for a treatment of the problem of an arbitrary axisymmetrical body, except for some special geometries. For example, the classical theory has been developed for slightly deformed spheres [ 32], lens-shaped bodies [37] and spherical caps [38], mainly, based upon the use of seperability of a special orthogonal co-ordinate system. The Green's function technique can be applied in order to attain an alternative approach, which is an integral representa tion, for the solution of the problem [39], [40]. The main concepts of the theory and detailed discussions on the advantages of the solution in integral form may be found in the literature [41], [42], [43]. On the other hand, various numerical methods have been proposed for the approximate solutions of such problems using the general integral form or, in the axisymmetrical case of flow, the serial expansion of the stream function. In this work, we show an exact theory can be constructed to determine Stokes-flow past an axisymmetrical arbitrary body. In Section 1.4 the statement of the problem is given. The govern ing equations valid in the fluid domain, V, are Stokes equations, (SI) V2v=Vp -v-and the equations of continuity, (S2) V.v=0, where v and p are velocity vector and pressure function respect ively. For an axisymmetrical fluid motion, the mathematical problem is reduced to search for a scalar function called stream function, y, which must satisfy the fourth-order equation, e'^O. This may be. written in the form (S3) E2$=0, E2*^. Here E2 is a second order partial differential operator defined by (S4) E2= İİ + £İE6 _3_ ( _J__ _8_ } )t2 r2 86 sine 3* in spherical co-ordinates and then all flow characteristics can be expressed in terms of $ and ¥. If F(9) denotes the equation of axisymmetric body surface, the boundary conditions are (S5a) r=F(6) + v =v =0 r 8 (S5b) v =cos6, Vg=-sin8 as r-*°°, and drag will be evaluated by <s6> R-l //` lj<»i>drde V 3 0 F(6)SinG In chapter II, a general systematic method for solving -vi-the equations (S3) is presented. By performing the co-ordinate transformations (S7) n=cose, Ç=r-F(n), ne [-1,1], Çe.[0,«0 drag is expressible as (S8) r = 1 // *İ<£ıI!İ dçdn 1-n5 v 3 _i 0. `2 Physically acceptable value of drag must be finite, R «»..` This is an interesting point of view of the treatment of the problem. Because $ and y must be in the Hubert space of the square integrable functions, denoted by L2. Since the Gegenbauer polynomials generate an orthonormal basis set of the L2 space on the interval -'Uti^I, $ and y can be written as a linear combina tion of them, that is, <s9a) <Kç,n) = (1-n2) z B. (ç)C.(n) k=0 K K (S9b) *(ç,n)=(l-n2) z / (ç)C.(n) k=0 k K where A, and 8. are solely ç-dependent linear combination coefficients. By recalling some of the basic properties of the Hilbert space of the problem and the theory of differential equations, in addition to, choosing a function of the type, (s10) F(n)=a [1+ I a.Ti1] 0 i-1 x as the shape function, one can arrive at the following vector differential equations to determine A(Ç) and B(ç): (s11a) TB(Ç) = 0 -vii-(Sllb) T A(ç) = a2tç2l+ç*-q2lB(ç) ~ ~ o ~ ~ Here a. 's are design parameters and the ordinary differential operator, T, is 2 (S12) T=(C2IHA1+q1) - - A2 İ- -y dÇ2 '` dÇ wherein A1, A2, q1 and Y are (» x ») matrix coefficients. Changing the variable ç to ç, Ç=ç/a +1 ı tne body has been transformed to a circle with unit radius in the fluid domain and now, the effect of the shape or geometry of the body, is characterized by these matrices. Since the solution vector A(Ç) of (SI 1b) will include four arbitrary constant vectors, ~ boundary conditions were reconstructed to express them in terms of A(Ç) in Section 2.3. In Chapter III, we obtain the convergent series solution of the equations (S11) in a subspace ' dimension of N and an explicit analytic solution when N=1, too. By employing this approximate solution in closed form we also present certain numerical results. Since the flow quantities are functions of the shape parameters, they can be evaluated for all specified bodies which have continuous surfaces in the sense of mathemati cal concepts. Furthermore, in principle, the structure of the shape function gives us a flexibility to adjust some or all of the parameters in order to optimise a fluid mechanical problem, such as minimization of drag, especially, by using an available numerical method. Numerical results are seen to be consistent with the results in the literature and optimum forms are realis tic. In Chapter IV, we discuss how this work can be extended in the future. Although each series solution of the problem is valid for whole domain, on the interval 1^Ç<<», by Lemma 2 in Section 2.4, the convergence of the general solution, that is, the linear combination of the series as N-*», still remains as a conjecture. In this case where a finite-dimensional subspace is considered, truncated solutions for each N exactly satisfy both no-slip conditions and the conditions at infinity. It is highly probable that the proof of the conjecture may also yield a formulation of the truncation error in terms of N. Therefore, if an efficient computer program is developed to numerically imple ment the method, one will be able to obtain results to any desired accuracy by increasing the order of the truncation or the dimension N of vector space. However, the first approxima tion seems that it satisfactorily makes use of physical applica tions in a simple and concise manner. -viii-Consequently, an analytic and exact method has been present ed to determine the Stokes-flow past axisymmetrical bodies of arbitrary shape using the basis functions expansion of the space of the square integrable functions. The method is not so difficult to implement. The problem of streaming flow past a stationary sphere with unit radius, when setting F=1 in the equation (S10), is a parti cular case of our presentation which was originally treated and solved by Stokes in 1851. If this well-known problem is taken up for checking purposes then one can easily deduce that the present ed theory exactly yields the Stokes' solution. This is, of course, a desired property which gives an idea about the validity of the theory. Another well-known particular case of the classical Stokes theory is that of an ellipsoid. With the selection of the shape function of this type, we can not exactly expressed an ellipsoidal geometry by taking a finite number elements in the serial expan sion. As a result of this, the known exact solution of an ellip soid is not obtainable. It seems that this may be accomplished by employing the prolate or oblate spheroidal co-ordinates and so modified the shape function in order to describe an ellipsoid as a particular case. However, this is left to a future work. With these discussions; in perspective, our studies are continuing for completion of the subject. -IX-