Eksenel simetrik bir cisim etrafında üç boyutlu laminer sınır tabaka hesabı
Three-dimensional laminar boundary layer calculation on the bodies of revolution
- Tez No: 19413
- Danışmanlar: DOÇ.DR. M. ADİL YÜKSELEN
- Tez Türü: Yüksek Lisans
- Konular: Uçak Mühendisliği, Aircraft Engineering
- Anahtar Kelimeler: Belirtilmemiş.
- Yıl: 1991
- Dil: Türkçe
- Üniversite: İstanbul Teknik Üniversitesi
- Enstitü: Fen Bilimleri Enstitüsü
- Ana Bilim Dalı: Belirtilmemiş.
- Bilim Dalı: Belirtilmemiş.
- Sayfa Sayısı: 108
Özet
ÖZET Bu tezde bir elipsoid etrafındaki üç boyutlu sınır tabaka teorik olarak incelenmiştir. Sınır tabaka analizinin yapılabilmesi için ge rekli olan potansiyel akım çözümünde analitik bir m etod kullanılmıştır. Bu metodla cisim üzerindeki is tasyonlarda potansiyel akım alanına ait hız bileşen leri, es- potansiyel ve akım çizgileri ailesi elde edilmistir. Eksenel simetrik cismin ön ve arka durma nokta ları civarında özel bir eksen takımında yapılan bir çözüm ile sınır tabaka içindeki hız bileşenleri he saplanmıştır. Bulunan sonuçlar üç boyutlu çözümde kullanılacak olan akım çizgilerine bağlı bir koordi nat sistemine aktarılmıştır. Simetri düzlemlerinde yapılan bir çözümle ön durma noktasından itibaren laminer ayrılmanın gerçek leştiği noktaya kadar olan istasyonlardaki hız bile şenleri elde edilmiştir. Bu hesapta durma noktası civarında elde edilen sonuçlar başlangıç değerleri olarak kullanılmış ve Crank-Nicolson tipi bir sonlu fark metodu uygulanmıştır. üç boyutlu çözümde, akım çizgilerine bağlı bir koordinat sisteminde yazılan sınır tabaka denklemle rine üç boyutlu bir Crank -Nicolson metodu tatbik e dilmiştir. Hesaplama sırasında durma noktaları yakı nında ve simetri düzlemlerinde bulunan sınır tabaka hız bileşenlerinden başlangıç değerleri olarak yarar lanılmıştır. Yapılan çözümle cisim yüzeyi üzerindeki çeşitli istasyonlarda hız bileşenleri elde edilmiş, viskoz akım çizgilerinin koordinatları belirlenmiş tir. (vii)
Özet (Çeviri)
SUMMARY THREE-DIMENSIONAL LAMINAR BOUNDARY LAYER CALCULATION ON THE BODIES OF REVOLUTION The aerodynamic experiments for developing the air vehicles and some other industrial elements take excessive time and are very expensive- The computa tional aerodynamic methods, however, have supplied a very important support to the experimental works in the last 30 years. Although the present goal is to develop reliable, accurate computer codes to solve the full Navier - Stokes equations for the complex ae rodynamic configurations, more simple models are usu ally preferred for practical applications. In many aerodynamic problems on the aircraft or glider wings, propeller, helicopter or wind turbine blades, and alongated bodies of air vehicles the ef fects of the air viscosity can be considered to be confined to a thin layer. Thus the* flow field is di vided into two region : A boundary layer near the so lid surface in which the viscous effects are impor - tant, and a potential flow field out of the boundary layer in which the effects of viscosity are negligib le. The flow characteristics in these two regions a- re calculated seperately, then the results can be combined to obtain the real solution. As like as in any physical problem the physical behavior of the three-dimensional boundary layer and the mathematical nature of boundary layer equations are very important in view of the numerical procedure to be chosen or to be developed. The surface geomet ry and lateral pressure gradients play an important role on the development of the three-dimensional bo undary layers, making the physics of the flow and the numerical theory more complicated then the two dimen sional case. In the three- dimensional boundary layers two effects are apparent which were absent in two-dimen sional case. The first is the convergence or diver - gence of the external flow streamlines in the planes parallel to the surface. While the streamlines con verge to each other this results in a change in the boundary layer thickness different from the two- di mensional boundary layer development. The second ef fect is introduced by the lateral curvature of the external flow streamlines giving rise to a secondary (viii)flow in the boundary layer called usually a cross- flow (Figure 1). The cross-flow is defined as the flow component parallel to the surface but perpendi cular to the external streamline. The formulation of the secondary flow can be explained qualitatively as following. The lateral curvature in the external st reamlines is essentially a result of the lateral pressure gradients. Thus the pressure force on the the fluid particles on the external flow streamlines is balanced by the centrifugal force, since the pres sure is constant along the surface normal the lateral pressure force is constant in the boundary layer. However the centrifugal force decreases because of the decreased velocities within the boundary layer. Consequently, the transversal pressure force causes the fluid particles to move towards the concave side of the external streamlines. The cross- flow formati on in the three-dimensional boundary layer highly ef fects the domains of dependence and influence of the flow. inflexion point x2 i Reversed secondary flow velocity profile Figure 1 s Formation of the secondary flow in the three-dimensional boundary layer Before any numerical calculation it is very u- seful to understand the nature of the three-dimensio nal boundary layer equations. The three-dimensional boundary layer equations can be considered as an el liptic system in the direction which is normal to the body surface and a hyperbolic system in the other directions which are parallel to the body surface; while in all two- dimensional flows, these equations degenerate to a parabolic system. But it is usually preferred to take a unified view and consider all bo undary layer equations, two- or three- dimensional as parabolic. (ix)In the present work, the main purpose is to de velop a computer code for calculating the three-di mensional laminar boundary layers on alongated bodies with the aim of presenting a data base for future works. For this purpose elliptic bodies of revoluti- tion are preferred as a particular shape, since it contains all the fundamental three- dimensional parti cularities. Furthermore, it is easy to obtain the potential flow solution around this type of bodies. It is of significant practical interest and more cri tically subject to the cross flow effects (in cont rast to the flat wing type of problem). The exact solution so obtained may be used to compare and eva luate other approximate methods, also to help clarify the nature of three-dimensional seperation. The ang le of attack may vary from zero to 90 degrees, while the thickness ratio (minor axes / major axes) may vary from unity to near zero representing variations from a sphere to a slender body (Figure 2). - X Figure 2 s Elliptic body of revolution The boundary layer in the present case starts from an isolated stagnation point. Since the stagna tion point boundary layer solutions are used as part of initial values, it is generally convenient to a- dopt a coordinate system whose origin coincides al ways with the stagnation point so that the coordinate system moves as the incidence changes. One such cho ice naturally goes to the streamline coordinates (Fi gure 3). The advantage of using the stream coordina tes is usually connected to the simplification of small cross- flow, while in the present work motiva tion arises mainly from the initial value considera tion. In this work, the inviscid part of the problem includes the determination of streamline coordinates and metric coefficients. Due to the lack of explicit transformation between two sets of coordinates - the the spheroidal cordinates and the streamline coordi nates - this part of the problem is quite involved in spite of being a known problem. (x)Figure 3 : The streamline coordinates Existing solutions of three-dimensional boun dary layers usually invoke special assumptions of one kind or another. Among those commonly used are simi larity, rotational symmetry, small cross flow and in dependence principle so that the resulting problem becomes essentially two dimensional. Integral met hods can be applied to general three-dimensional problems, but provide only global quantities as does the two- dimensional counterpart. In the present work, a finite difference scheme of the Crank-Nicolson type is used to calculate the incompressible three-dimensional laminar boundary la yer equations. The basic idea of C- N scheme is that each term of the differential equations considered is evaluated at a central point. The process is impli cit in the direction along which the problem is of boundary-value nature (or elliptic), and explicit in other directions along which the problem is of initi al-value nature. However, the step sizes along two surface coordinates &re restricted according to the rule of the zone of dependence. To generate the initial values to be used in the three- dimensional calculation, two well-known li miting problems are involved, namely the boundary la yer at the stagnation point and that along the plane of symmetry. The former provides initial values a- long an equipotential line, the latter, itself is an inviscid streamline (Figure 4). A Crank-Nicolson finite-difference scheme is used for calculating the boundary layer along the plane of symmetry. (xi)surface X2=cons. surface Figure 4 : The initial value surfaces In the numerical applications of this work, an ellipsoid of revolution with a thickness ratio of 1/6 has been considered. After the potential solution, the numerical results near the stagnation points and along the symmetry surfaces have been obtained. With these results, the final solutions over the body have been calculated. In the calculations, the angle of attack has been taken as 10 degrees. The applicati ons are presented graphically. It is observed that the results are same of the other methods presented before. (>:ii)
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