AĞIRLIKLI ARTIK YÖNTEMLERİN SINIR TABAKA PROBLEMLERİNE UYGULANMASI

Year 2020, Volume: 2 Issue: 1, 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

References

• L. Prandtl, “Ueber die Fl¨ussigkeitsbewegung bei sehr kliner Ribung” In Verhandlungen des III. Internationalen Mathematiker-Kongress, Heidelberg, pp. 484-491, 1904.
• J.A. Schetz and R.D.W. Bowersox, “Boundary Layer Analysis”. Sec. Edition, New Jersey, Prentice-Hall Inc., 2011.
• T. Cebeci and H.B. Keller “Shooting and Parallel Shooting Methods for Solving the Falkner-Skan Boundary-Layer Equation,” Journal of Computational Phsics, vol. 7 pp. 289-300, 1971.
• H. Blasius, “Grenzschichten in Flüssigkeiten mit kleiner Reibung.” Z. Angew. Math. Phys., vol. 56, pp. 1–37, 1908.
• K. Hiemenz, “Die Grenzschicht an einem in den gleichförmigen Flüssigkeitsstrom eingetauchten geraden Kreiszylinder,” Dingler’s Polytech. J., vol. 324, pp. 321-324, 1911.
• V.M. Falkner and S.W. Skan, “Some Approximate Solutions of the Boundary Layer Equations,” Philosophical Magazine, vol. 12, pp. 865-896, 1930.
• C. Laine and L. Reinhart, “Further numerical methods for the Falkner-Skan equaitons: shooting and continuation techniques,” I. J. Num. Meth. Fluids, vol. 4, pp. 833–852, 1984.
• R. Fazio, “A novel approach to the numerical solution of boundary value problems on infinite intervals,” SIAM J. Num. Anal., vol. 33, no. 4, pp. 1473–1483, 1996.
• A. Asaithambi, “A finite-difference method for the Falkner-Skan equation,” Appl. Math. Comp., vol. 92, pp. 135–141, 1998.
• S.S. Motsa and P. Sibanda, “An efficient numerical method for solving Falkner-Skan boundary layer flows,” I.J. Num. Meth. Fluids, vol. 69, pp. 499–508, 2012.
• R. Fazio, “Blasius problem and Falkner-Skan model: T¨opfer’s algorithm and its extension,” Comp. Fluids, vol. 73, no. 15, pp. 202–209, 2013.
• C.S. Liu, “An iterative method based-on eigenfunctions and adjoint eigenfunctions for solving the Falkner-Skan equation,” Appl. Math. Lett., vol. 67, pp. 33–39, 2017.
• H. Bararnia, N. Haghparast, M. Miansari, and A. Barari, “Flow analysis for the Falkner-Skan wedge flow,” Curr.Sci., vol. 103, no. 2, pp. 169–177, 2012.
• B.I. Yun, “New approximate analytical solutions of the Falkner-Skan equation,” Journal of Applied Mathematics, vol. 2012, pp.1–12, 2012.
• A. Khidir, “A note on the solution of general Falkner-Skan problem by two novel semi-analytical technique,” Propul. Power Res., vol. 4, no. 4, pp. 212–220, 2015.
• B.A. Finlayson, “The Method of Weighted Residuals and Variational Principles With Applications in Fluid Mechanics,” Academic Press, New York and London, 1972.
• S.C. Chapra and R.P. Canale “Numerical Methods for Engineers,” 6th Edition, McGraw-Hill, New York, 2010.
• M. Abramowitz and I.A. Stegun, “Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables,” 9th printing. New York: Dover, 1972.
• D.R. Hartree, “On an equation occuring in Falkner and Skan’s approximate treatment of the equations of the boundary layer,” Proc. Cambridge Phil. Sot., vol. 33, pp. 223-239. (1937).
• A. Asaithambi, “A second-order finite-difference method for the Falkner–Skan equation,” Applied Mathematics and Computation, vol. 156, pp. 779–786, 2004.