Sınır tabaka denklemlerinin sayısal çözümlemesi
Numerical solution of boundary layer equations
- Tez No: 21972
- Danışmanlar: DOÇ. DR. TANER DERBENTLİ
- Tez Türü: Yüksek Lisans
- Konular: Makine Mühendisliği, Mechanical Engineering
- Anahtar Kelimeler: Sayısal çözüm yöntemleri, Sınır tabaka denklemleri, Numerical solution methods, Boundary layer equations
- Yıl: 1992
- Dil: Türkçe
- Üniversite: İstanbul Teknik Üniversitesi
- Enstitü: Fen Bilimleri Enstitüsü
- Ana Bilim Dalı: Belirtilmemiş.
- Bilim Dalı: Belirtilmemiş.
- Sayfa Sayısı: Belirtilmemiş.
Özet
ÖZET Bu tezde, sınır tabaka akışları ele alınarak akışı karakterize eden korunum diferansiyel denklemlerinin çeşitli dönüşüm ve kabuller altında çözümleri gerçekleştirilmiştir, öncelikle, bugüne kadar elde edilmiş olan analitik çözüm yöntemleri incelenmiş, daha sonra ve ağırlıklı olarak da Patankar'ın boyutsuz akım fonksiyonu kav ramından hareketle geliştirdiği genel sayısal çözüm yön temi üzerinde durulmuştur. Akış alanındaki çok küçük bir kontrol hacmine korunumun genel ilkesi uygulanarak elde edilen sınır tabaka denklemleri için genel bir analitik çözüm yöntemi mevcut olmadığından özel akış ve geometriler düzeyinde çözümler aranmıştır. Söz konusu çözümler, paralel iki levha arasındaki Couette akışı, aniden hareketlendirilmiş levha üzerinde akış (Stokee'un 1. problemi), düzlemsel bir levha üzerindeki sürekli akış için Blasius'un benzerlik çözümlemesi ve yine düzlemsel bir levha üzerindeki sürekli akış için Von Karman* m momentum- integral yöntemi şeklin dedir. Sayısal çözüm yöntemi olarak, türbülanslı akış problemlerine de uygulanabilen Patankar Yöntemi ele alınmış, yönteme ait matematiksel bağıntılar ayrıntılı olarak çıkartılmış ve FORTRAN programlama dili ile verilen SINTAB bilgisayar programı açıklanmıştır. Ayrıca somut bir fi kir vermesi açısından da, yöntem üç farklı akış problemi ne uygulanmıştır.
Özet (Çeviri)
SUMMARY NUMERICAL SOLUTION OF BOUNDARY LAYER EQUATIONS In this thesis, first laminar boundary layer equations were derived by using the general principle of conservation, then various solution methods were given for these equations. The solution methods of interest here are analytical and mostly numerical. Since a general analytical solution for the boundary layer equations has not been found yet, the analytical solutions studied are for the special flow problems having special geometries. The numerical method studied is due to Patankar and known as Patankar's procedure or algorithm. In the flow of a viscous, in other words real, fluid over a solid body, because of the effects of viscosity, a very thin region in which the normal velocity gradients are very high, becomes near the surface of the body. This region whose thickness increases along the surface is called boundary layer. The influence of the viscous forces in the boundary layer is assumed to be directly proportional to the velocity gradient where Z is the viscous-shear stress, y is the dynamic viscosity of the fluid and öu/öy is the normal velocity gradient. Derivation of the boundary layer equations is made for Newtonian fluids under the assumption above. This is because the assumption of interest makes the derivation and solutions possible. Besides the velocity gradients, in the flow field, if there exists temperature differences, the temperature gradients occur in the boundary layer. These gradients cause a thermal boundary layer akin to the hydrodynamic boundary layer to develop over the surface. viii -The mathematical analysis of a fluid flow consists of the derivation of the conservation equations and the solution of them. Inside the scope of the thesis, these processes were both done successively. The derivation starts with the selection of the coordinate system. The equations are Lagrangian in nature and utilize the derivative d & ~3F ~ ~5F + (V.v) where V is the velocity field vector in the cartesian coordinates. The boundary layer equations are considered as the three laws of conservation for physical systems and these are - Conservation of mass (continuity equation) - Conservation of momentum (Newton's second law) - Conservation of energy (conservation of enthalpy, first law of thermodynamics) In this thesis, furthermore, conservation of chemical species was handled and solved in the numerical method of Patankar ' s. Derivation of the boundary layer equations, as mentioned in the first paragraph of the summary, is made by applying the general principle of conservation to a very small particle (elemental control volume) in the flow field (laminar boundary layer). The complete form of the boundary layer equations obtained is very complicated. Therefore, it is very difficult to solve them. Even at high Reynolds numbers, in other words in turbulent flows it is impossible to solve them by any means, due to the boundary conditions becoming randomly time-dependent. But, by making some boundary layer assumptions, the equations in elliptic form can be simplified and take parabolic form. Some examples for the assumptions of interest are as follows. - Incompressible flow, in other words the density is constant - Transport and some thermodynamic properties are constant - The pressure gradient in the y direction is nearly zero - Only the Su/öy portion of the dissipation function in the energy equation is important, so the remaining portion is negligible. - The diffusion in the x direction is neglected since it is much smaller compared with the other terms - Outside the boundary layer, Bernoulli's equation is valid Under these assumtions, the equations were simplified for two dimensional flows and combined in a general form as - ix --J^p^) + div(pW>) » div(rgrad*>) + S where the first term on the left-hand side is the unsteady term, the second is the convection term, the first term on the right-hand side is the diffusion term and the second is the source term. Furthermore, <p is the general dependent variable such as velocity component, enthalpy, temperature, mass fraction of chemical species and defined by 4> = <Mx,y,t) where x,y and t are the independent variables. In the general equation, r stans for the diffusion coefficient such as (J İn the case 4> - u, k/c in the case <p = i, D in the case 4> - m. The general equation obtained is but still elliptic in this form because of the nonlinear convection term. As for all diferantial equations, there are three ways of solving this equation. - Analytical - Numerical - Graphical The third one is outside the scope of this thesis. Any analytical (exact) solution does not exist yet. But, some particular solutions are available. Four of these solutions were studied in this thesis. These are Couette flow between parallel two flat plates, suddenly accelerated plane wall (Stokes' first problem), two dimensional steady flows over a plane wall (Blasius' similarity solution and Von Karman' s momentum- integral method). In the first of the particular solutions mentioned above, Couette flow between the two infinite plates of which the upper one moves at constant speed U relative to the lower one, were handled. For different values of the dimensionless pressure coefficient (B) and the multiplication of PrEc, the velocity and temperature profiles were plotted. In the second solution, the motion of the fluid over a suddenly accelerated plane wall was studied. This problem is known as Stokes1 first problem. At different time values, the profiles of velocity were plotted. - x -In the third solution, for two dimensional steady flows over a plane wall, Blasius' similarity solution which is based on the stream function concept and its dimensionles form, was studied. By using the similarity between the velocity and boundary layer profiles, the similarity variables are formed. Then the boundary layer equations are converted to a simplier form and this equation is integrated by the shooting and Runge-Kutta method. The velocity profiles belonging to the flow were plotted. In the fourth and last solution, for the flow considered in Blasius“ solution, Von Karman1 s momentum-integral method was studied. In this method, the conservation of momentum is derived by using a control volume enclosing the boundary layer, not an elemental control volume. The velocity profiles obtained from Blasius* and Von Karman' s solutions were compared on a figure. Further, various formulas which give the boundary layer thickness, the coefficient of fiction, the shear stress and the force acting on the wall were derived for both solutions. As the numerical solution for the boundary layer equations, a procedure given by Patankar was studied. In this solutioun technique developed for two dimensional flows, the general equation for boundary layer equations is considered to be in the form of PU^F ?v^y ^ ^ycrl ^y^ > + s where r is the distance of the calculation point to the symmetry axis. This means the procedure solves both the flow with plane geometry and the flow with axi-symmetric geometry. The essence of the procedure is to set a grid on the boundary layer and to integrate the foregoing equation over the control volumes within the grid in the downstream direction. In order to apply Patankar' s procedure to a special problem. The general equation in the x-y coordinates must be defined in the x-w coordinates. Here, w is the dimensionless or normalized stream function and given by V - Vj. v - Vj. CO) = \p - w M> where the subscripts I an E stand for the imaginary Internal and external surfaces bounding the region of interest successively, y is the stream function at any x forward step. - XIThe reason for the conversion of coordinate is to keep the number of the grid points in the layer constant at each x forward step. But, according to the procedure, the equation in the x-y coordinates must first be defined in the x-y coordinates (Von Mises plane). After the conversion, the equation in the x-y coordinates is obtained as ”ox“ ~ ~§y tP^ r ”oy" > + pu where the term on the left-hand side is the convection in the downstream direction, the first term on the right- hand side is the diffusion normal to the downstream direction and the second is the source term. Thus, the convection normal to the downstream direction vanishes. After the second conversion mentioned above, the final general equation becomes where m is the mass fraction or the entrainment rate entering the layer through the surfaces and d is a coefficient including the source term. By integrating the final equation over the control volume by the control -volume formulation, the discretization equation is obtained in the form of a 4> - a 4> + a 4> +b p*p *tn e^s where the subscripts P, N, S stand for the calculation point, the point above P and the point below P respectively. During the discretization, Crank-Nicolson profile assumption is accepted for the variation of 4> with respect to x and t. Furthermore, the coefficients in the discretization equation are calculated by the power-law formulation. The power-law formulation gives nearly the actual values for the <f~Y variation. At each x forward step the discretization equations are solved by the Tri-Diagonal Matrix Algorithm (TDMA). The reason why this direct method is prefered, is that the equations have a particularly simple form. Since the general equation is parabolic and the downstream direction is one-way coordinate, it is essential to be known the initial values of all the variables before proceeding any integration step. For this reason, at the entrance of the boundary layer, the initial profiles known from experiments, empirical - xii -relations, approximate theories or guesswork must be entered. On the other hand, to solve the discretization equations, the boundary conditions must be known as well. The boundaries are considered to be of three types, - wall type of boundary - free boundary - symmetry line In the case of free boundary, the entrainment rates must be controlled. This is because the profile of the boundary changes with increasing or decreasing entrainment rate. But, the entrainment rates are known for the other types. For example, they are zero for the symmetry line. The rates are controlled by a standart formula at each x forward step. Sometimes to perform the integration step, the size of the forward step must be selected at each x forward step. As a result of the control -volume formulation, the stability is ensured even for large steps. But small steps are desirable for good accuracy. In some portions of the flow region, the small steps are needed because of the large gradients. For these reasons, sometimes the size of the forward step must be specified at each x forward step. A simple and approprate procedure to select the forward step is to take the step length proportional to the thickness of the boundary layer at the step of interest. An advantage of Patankar's procedure is that it is applicable to the turbulent flows. In this case, the fluctuating components are expressed in terms of the effective shear stresses and fluxes. The effective dynamic viscosity is calculated from Prandtl's mixing-length hypothesis given by, 2, du, tfff = p1 i nay- i where 1 is the mixing length. But, a disadvantage of this hypothesis is that it is invalid for zero velocity gradients. Sometimes Patankar's procedure is not applicable, although the given general equation is satisfied. For example, if reverse flow or seperation exists on the solid surface, the general dependent variable takes two values for only one value of the stream function and such a situation invalidates the boundary layer equations. - xiiiA computer program written by using FORTRAN language was given and its name is SINTAB. The procedure and the program were applied to three flow problems. These are - One dimensional and time-dependent heat conduction - Two dimensional free flow over a flat plate - One dimensional periodic steady state flow in a stone bed In the first problem, for various time steps, the temperature distributions and the heat fluxes from the boundaries were calculated. According to the figures plotted, although the initial velocity profile is nearly linear, in a short time the profile takes a parabolic form and the system reaches the steady state. In the second problem, two different flows with different Prandtl numbers were considered and the effects of these dimensionless numbers on the temperature profiles were studied. Furthermore, Nusselt numbers were compared with that obtained from Blasius' similarity solutions. In the third problem, one dimensional heat transfer in a stone bed or periodic steady state flow was studied. The bed is first heated up to a maksimum temperature and then cooled up to a minimum temperature. These two processes are periodically repeated. At various points of the bed, the temperature profiles of the stones were plotted with respect to time. xiv -
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