Transonik dış akımın sonlu hacimler yöntemiyle çözülmesi
Transonic external flow calculations using a finite volume method
Yazar: FIRAT OĞUZ EDİS
Danışman: DOÇ.DR. ÜLGEN GÜLÇAT
İstanbul Teknik Üniversitesi / Fen Bilimleri Enstitüsü
Derece: Yüksek Lisans, 1991
Konu: Uçak Mühendisliği = Aircraft Engineering
Dizin: Sonlu hacimler yöntemi = Finite volumes method ; Transonik dış akım = Transonic external flow

Özet:
ÖZET Profil etrafında transonik akış problemi, yüksek sesaltı hızlarda uçan uçakların tasarımı açısından önemli bir problemdir. Bu problemin çözümünde viskoz etki lerin çepere çok yakın dar bir bölgede etkin oldukları kabul edilirse, akış Euler denklemleri kullanılarak çözülebilir. Euler denklemlerinin korunumlu şekillerinin çözümü Rankine-Hugoniot bağıntılarını da içerir. Bu denklemler profil üzerinde oluşacak şokun yeri ve şiddeti hakkında kasin bilgi verir. Euler denklemlerinin daimi şekilleri sesaltı hızlarda parabolik sesüstü hızlarda hiperbolik karakterdedir. Bu ne denle hernekadar daimi çözümler aransa da, transonik akış probleminde daimi ol mayan denklemlerin kullanılması uygun olur. Bunların çözümleri Zaman Adımlama yöntemleri kullanılarak yapılır. Kullanılan denklemler korunum ifadeleri olduğu için bunların diferansiyel şekilleri yerine integral şekillerinin ayrıklaştırılması daha uy gundur. Bu tip bir ayrıklaştırma için en doğal yol ise Sonlu Alanlar (Hacimler) yöntemidir. Bu çalışmada, profil üzerindeki transonik akımı inceleyebilmek amacıyla, Eu ler denklemlerini transonik akış hali için çözen bir bilgisayar programı geliştirilmiştir. Geliştirilen bilgisayar programı, N.Kurtuluş ve T.Arts tarafından "Von Karman In stitute For Fluid Dynamics" de hazırlanan bir bilgisayar programım temel almıştır. Adı geçen program kaskat akışı çözebilecek şekilde hazırlanmıştır. Bu çalışmada ise program, Kurtuluş ve Arts tarafından kullanılan yönteme bağlı kalınarak dış akım problemini çözebilecek şekilde değiştirilmiştir. Kullanılan yöntemler, zaman adımlama yöntemi olarak Düzeltilmiş Viskozite Şeması ve uzaysal ayrıklaştırma yöntemi olarak Sonlu Alanlar Yöntemi'dir. Uzaysal ayrıklaştırmada, kontrol yüzeyi olarak akım doğrultusunda örtüşen bitrapezoidal e- lemanlar kullanılmıştır. Geliştirilen program önce kalibrasyon amacıyla NACA 0012 profil üzerindeki transonik akışın çözülmesinde kullanılmıştır. Daha sonra modifiye edilmiş NACA 6548 profili üzerinde akımın çözümü yapılmıştır. Çözümler sonucunda yöntemin ve buna bağlı programın akınım incelenmesi, şokun yerinin ve şiddetinin bulunmasında başarılı olduğu anlaşılmıştır. vii

Abstract:
SUMMARY TRANSONIC EXTERNAL FLOW CALCULATIONS USING A FINITE VOLUME METHOD The role of Computational Fluid Dynamics in Aerodynamics researchs have become more and more strong, since mid of 1960's, when the Time-Dependent tech niques were introduced and the computation coasts began to decrease due to the rapid development of the computer hardware. Since then, the CFD methods can be considered as a third dimension in Fluid Dynamics besides the two other, anlytical and experimental methods. In the present study, by employing Finite Area Method for spatial discretiza tion and Corrected Viscosity Scheme for time discretization, the Euler equations for transonic external flow problem over an airfoil are solved numerically. The feature of the problem is that both subsonic and supersonic flow regions exist together within the same domain in the case of the transonic flow and the Euler equations are of different character for these kind of flow regimes. The code applied here is based on the work of N.Kurtuluş and T.Arts at the VKI, that they developed this code for cascade flow calculations. The principle of a time marching method is to consider the solution of a sta tionary problem as the solution after a long calculation time of instationary equations describing this problem. The physical interpretation is as follows: the computation vuiincreased from the trailing edge to the outlet plane. The space between the pseudo stream lines is increased uniformly from the profile to the upward and downward boundaries as well. So there can be benefit from the computational expenses. For spatial discretisation, a Finite Area method is used. It is based on a numerical representation of conservative equations written in an integral form. The physical domain of the flow is divided into bitrapesoidal control surfaces. These control surfaces overlap in flow direction. The accuracy of the discretization by this kind of surfaces is of the first order. The different geometrical paramerters needed for an analysis of the con sistency and the order of accuracy of the discretization obtained with this kind of control surfaces were described in Fig. 4-3. In order to have a well-posed problem, the partial diferantial system of equations must be completed by a set of physcal boundary conditions. Along the solid boundaries an impermeability condition must be fullfilled. Since the solution procedure uses the transport terms explicitly, this condition will be verified by setting zero to the mass and energy fluxes, and by taking into account only the static pressure acting on the surface. The flow angle on the profile is obtained by this way. The stability of the numerical scheme is investigated using von Neumann stability analysis. The analysis of the Corrected Viscosity Scheme is handled in two satges. First, by taking into consideration that the artificial viscosity terms are constant during Nv time steps, amplification matrix is constructed for one time step only. By using this amplification matrix and taking the artificial viscosity terms into XIThe Corrected Viscosity Scheme is used for time-discretisation of these equa tions. This scheme is derived by adding a periodic correction term to the Lax scheme, which is spatially smoothed version of the Euler Scheme. The periodic correction term, which can be considered as an artificial viscosity term is updated every Nv iterations. For the two-dimensional Euler Equations, the scheme is as follows: c^1 = ^(Transport terms) \PÎ-ı,k + PÎ+ı,k + Ptk-i + Plk+i - 4P*,fc a = vc 1- The physical domain of the flow was described in Fig.6-2. It is made up of a profile region an extends several cord lenghts prependicular and parallel to the flow direction, The lines AB, DC and EF are solid walls across which no mass and energy transport can exist. The lines AD and BC are inlet and outlet boundaries respec tively. The space between the inlet plane and the leading adge is as twice or thrice much as chord lenght. The one between the trailing edge and the outlet plane is as thrice or five times much as the chord lenght. The reason of using such a long physical domain is that inlet and outlet conditions must be uniform in pitchwise direction. The numerical domain of the flow was shown in Fig. 6-3. It is made up of lots of pseudo stream lines, and pitchwise lines. The space between the succesive pitchwise lines is regularly decreased from the inlet plane to the leading edge, and xstarts from a rough approximation of the final solution, considered as a large per turbation and develops under certain boundary conditions until convergence. The conservative form of the stationary Euler equations is of parabolic character for sub sonic regions, and of hyperbolic character for supersonic regions. The adventage of the Time Marching methods in transonic flow calculations lies on the use of the insta- tionary Euler equations which are of the hyperbolic character for both flow regimes, supersonic and subsonic. By the use of the conservative Euler equations, it is also guaranteed that the shock will captured correctly without any shock fitting effort. In that present work, Corrected Viscosity Scheme is used as the Time Marching method. This scheme is first proposed by Couston. The vector form of the conservative instationary Euler equations is as follows: a = do | KM, W*) = 0 dt dx dy u p u p v pE m = p u pu2 +p p U V puİE + - m = p V p U V p v2 +p PVİE + - IXaccount, an other one is constructed for Nv time steps. When the eigenvalues of the amplification matrix are analysed, it is observed that the scheme is stable, if the Courant-Friedrichs-Lewy (CFL) condition is satisfied. The program is applied on two different airfoils: the NACA 0012 airfoil and teh modified NACA 6548 airfoil. Results are obtained for M=0.80 and 0.00 and 1.25 incidence on NACA 0012 airfoil, M=0.95, 0.00 incidence on NACA 6548 airfoil. The results are satisfactory. xu