Akişkan Yatakli Sistemde Bir Kanat Profili Üzerinde Sonlu Elemanlar Metodunu Kullanarak Navier-stokes Ve Difüzyon Denkleminin Çözümü

Abstract

Günümüzde teknolojinin çok hızlı bir atılım yapmasıyla birlikte bilimsel çalışmalar çok farklı bir konuma gelmiştir. Artık sayfalarca tutan hesaplar, ince ayrıntılı, çok hassas ve pahalı düzenekler yerlerini bilgisayarlara ve bilgisayar programlarına bırakmıştır. Böylece amandan paradan ve insan gücünden kazanılmış, bunlar yeni araştırma konularına farklı alanlara yöneltilebilmiştir. Yaşanan baş döndürücü bir hızdır, özellikle gelişmiş ülkeler bu durumun başının çekmektedir. Bizim gibi gelişmekte olan ülkeler içinse bu gelişmiş ülkeleri yakalamak için son firsattır. Çünkü sorun artık nitelikli insan gücünü geliştirmeye ve onlara ihtiyaçları olan koşullan yaratmaya kalmıştır. Bu çerçevede kompleks bir yapıya sahip olan ve hala çözülememiş öğeleri içinde barındıran akışkanlar dinamiği önemli bir alandır. Tez oluşturulmaya karar verildiğinde, tüm bunlar düşünülerek daha sonradan geliştirilmek üzere bir ön çalışma yapılarak bilgisayarla akışkanlar dinamiği çözümleme yöntemlerine giriş yapılmıştır. Bunun için yöntem olarak sonlu elemanlar metodu kullanılmıştır. Sayısal çözümleme yöntemlerinin bilgisayarlarla yaygın olarak kullanılmaya başlanması matematik, fizik gibi temel bilim dallarının yeniden gündeme gelmesinin nedenlerinden biridir. Bugün uçak yapımında kullanılan teknoloji ve malzeme "ileri teknoloji ürünleri 'nin başında gelir. Tez, bir kaplama tekniğinin "karbür kaplama" kanat profiline uygulanabilirliğini araştırmaktadır. Bunun için akışkan yataklı bir sistem kullanılmıştır. Belli bir sıcaklık ve hızda içine metal karıştırılmış hava kanat profili üzerinde bir sınır tabaka meydana getirir. Amaç bu sınır tabakayı çözmektir. Bunun için Navier-Stokes ve difuzyon denklemleri kullanılmıştır. Elde edilen tabakanın zamana bağlı değişimi gösterilecektir. Yöntemin uygulanabilirliği ayrıca incelenmelidir.

In recent years, new techonological ways of manufacturing new products has been advancing rapidly. Scientific research has reached a high level, where there is no need for piles of calculations,minute details and expensive experimental arrangements since specialized computer programs take care of all of these. Therefore, time, labor and money are economically spent and the method employed in this context may be expanded to cover other domains of technology forcing the limits of imagination. This era might be the last chance for the developing countries to catch up the developed ones. Because all that remains is to provide the proper conditions for qualified human powers to take place in technological improvements and develope innovations. Fluid dynamics subject occupies a complex domain of interest in mechanics where many problems still which many objects remain unsolved. This feature was heavily exercised in mind when choosing the topic of this thesis. An introdution for computational fluid dynamics is provided, upon which a further expolotation of the subject will take place. As a method, finite element method was chosensing it has found extensive use whenever computer aided numerical solutions needed in the fields of physics and mathematics or science in general. The technology and materials used in the manufacturing of aircraft components nowadays can be placed on the top of "Advanced Technology Products" listing. This thesis is concerned with the applicability of a new coating techique called "Carbur Coating" of various profiles including airfoils and circular cylinder. Here, an account is made of a system with fluidized bed system. If applied at a given velocity, temperature and metal powder content, there creates a boundary film along the profile where carbur diffusion will occur. In brief the title of this thesis is; "Solutions of the Navier-Stokes and Diffussion Equations via Finite Element Method Over an Airfoil Placed in a Fluidized Bed System " The Navier-Stokes equation can be derived form the law of conservation of momentum and the continuity equations. This equation is one of the two basic equations that we are going to deal with. Hence; fnu\ - - = F-Vp + iN2U \DtJ Mil Expanding the Navier-Stokes equation in the x and y direction we get the followings; dıı du du 1 dp "2 +M +V = -+vVu dt dx dy p dx dv dv dv 1 dp _, - + u - + v - = - + vV v dt dx dy pdy On the otherhand the stream function and vorticity is related by the following equations; V2\j/ = -w The vorticity transport equation is then; Dw _,_ = vvw Dt After finding the velocities u and v from the solution of Navier-Stokes equation and diffusion equation is ready to be solved. The diffusion equation gives us the level of concentration is the flow fled. Hence; 8C dw dC dy dC «-,»^, - +- - = tjV2C dt dy dx dx dy The numerical solution by use of the finite element method is sought. For the solution, triangular elements were used and in order to determine more accurate results especially around the profiles ends, the number of elements was increased. Employing Galerkin's Finite Element Method approch, the matrices that are determined provide us the solution to the problem which can be shown as: [*)-M»}-W»W [*]{*} +[4")]M + v[z>]M ? {G(C)} - {r,} = {0} M{c}+Hv*}+*>]M+{r,H<>} By the algorithm and program pronded in the literature w and C values can be calculated with At intervals. In order for the program to terminate, we execute the program until the flow field reaches a steady state conditions. After finding the concentration distribution on the surface of profile, the diffussion process IX was modelled to evaluate the unsteady concentration thicknesse inside the body. The concentration around the object is calculated according to the equation mentioned below: - = DV2C The approach applied to solve the above equation is "The Finite Difference Method". Thus the level of diffusion that results from the concentration at various time intervals may be computed. A setting of a cylinder and a wing profile placed in the fluidized bed is prepared. With enriched air charged into the setting the velocity vectors resulting from the air flow are calculated, then the change in the velocity vectors is examined with respect to three Re numbers. Moreover, various calculations are made regarding the stream function and vorticity, andvarious results are obtained according to miscellaneous Re values. Also, various observed changes that result from different Re values are examined on graphs. Consequently it was observed that the flow had increasingly approached turbulent level and the flow along the object deviated from steady flow. The concentration distribution was examined for various Re values at the point where the flow reached its developed stage. The value of the concentration was observed to be higher at the first point reached by the flow (attack side) compared to the last point where the flow departed the object (escape side), however eventually. The difference between concentration levels declines at further points away from the object but never reached to a nü! In conclusions the following may be said: By changing the mesh form or the constructions of the body defined in the theoretical part and algorithm are developed, an approach to various problems can be achieved Moreover the developed here approach, which is based on theoretical and simulation basis may be conveyed to serve industrial applications and thus a development of the "Carbur Coating" technique may be enhenced by the encounterment of theory and practice.