AĞIRLIKLI ARTIK YÖNTEMLERİN SINIR TABAKA PROBLEMLERİNE UYGULANMASI

Year 2020, , 36 - 42, 29.04.2020

Utku Cem Karabulut

https://doi.org/10.46387/bjesr.638116

Abstract

Falkner-Skan denklemi, akışkan içerisindeki bir levha üzerinde gelişen sınır tabaka akışını ifade eden üçüncü dereceden non-lineer bir sınır değer problemidir. Denklemin, baskın non-lineer bir yapıya sahip olması, başlangıç koşullarına yüksek derecede hassas olması ve yarı sonsuz bir tanım kümesine sahip olması dolayısı ile birçok araştırmacının ilgisini çekmiştir.Bu çalışmada, ağırlıklı artık bir yöntem kullanılarak Falkner-Skan denklemi yaklaşık olarak çözülmüştür. Artıklar en küçük kareler tekniği kullanılarak minimize edilmiştir. Sunulan prosedür sınır tabaka problemlerinin çözümü için oldukça basit ve kullanışlıdır. Çalışmanın ana amacı uygulanan yöntemin başarısını ortaya koymaktır. Sadece bir bilinmeyen ile en basit yaklaşımın bile sınır tabakasındaki hız profili için oldukça doğru sonuçlar verdiğini gözlemlenmiş ve ek olarak, bilinmeyen katsayı sayısı artırılarak istenen herhangi bir doğrulukla daha iyi sonuçlar elde edilebilmiştir. Ayrıca, bu yöntem tüm alan için geçerli olan analitik çözümler sunmaktadır.

Keywords

Sınır Tabaka Problemleri, Falkner ve Skan Denklemi, Ağırlıklı Artık Yöntemler, En Küçük Kareler Yöntemi, Nonlineer Diferansiyel Denklemler

Supporting Institution

Bandırma Onyedi Eylül Üniversitesi, Bilimsel Araştırma Projeleri Koordinatörlüğü

Project Number

BAP-18-DF-1009-074

AN APPLICATION OF THE METHOD OF WEIGHTED RESIDUALS TO THE BOUNDARY LAYER PROBLEMS

Abstract

Falkner-Skan equation is a third order non-linear boundary value problem which describes the laminar boundary layer flow developing on a plate. The strong non-linear characteristics of the problem, sensitivity of the equation to the initial conditions and the semi-infinite domain of the problem have attracted many researchers.In this paper, the method of weighted residuals is used to solve Falkner-Skan equations. The residuals are minimized by the least squares approach. The procedure is very simple and suitable for solving boundary layer problems. The main aim of this paper is to demonstrate the success of the proposed method. We observe that even the simplest approach with only one unknown provide quite accurate results for the velocity profile in the boundary layer. Additionally, better results with any desired accuracy can be obtained by increasing the number of unknown coefficient. Moreover, this method provides analytical solutions which are valid for whole domain.

Keywords

Boundary Layer Problems, Falkner and Skan Equation, Method of Weighted Residuals, Least Squares Method, Nonlinear Differential Equations

Project Number

BAP-18-DF-1009-074

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