Generalized Pohlhausen integral method - Mehmood, Ahmer; Awais, Muhammad; Dar, Amanullah (European physical journal plus, 2024)
Approximate methods constitute an important class of analytic methods to calculate boundary-layer flows. The last few decades have observed remarkable progress in computational fluid dynamics tools and several efficient solution techniques have been developed due to tremendous developments in numerical methods and the widespread use of computing machines. With the invention of such modern computing tools, the progress in the subject of fluid dynamics is incremental and substantial. However, it is a matter of fact that all such developments are achieved at the cost of the huge loss of theoretical research as no prominent contributions have been made toward the development of approximate methods during these years. Moreover, all such approximate methods are only applicable to the flows over the surfaces of finite length (of the type of Falkner–Skan flow) and do not apply to the flows due to moving continuous surfaces (of the type of Sakiadis’ flow). Therefore, in this study attention has been given to developing an approximate method for the calculation of boundary-layer flows due to moving continuous surfaces in a quiescent fluid. Such kinds of flows essentially involve no pressure-gradient term in contrast to the flows over surfaces of finite lengths, such as Falkner–Skan flow, which essentially involve a pressure-gradient term whenever the external potential velocity is non-constant. The classical Pohlhausen integral method is already available in the literature for the boundary-layer flows over the surfaces of finite lengths, which in general involve a nonzero pressure-gradient term. It is interesting to note that the existing Pohlhausen method is applicable to the flows over finite surfaces only, and gives wrong results for the flows due to moving continuous surfaces. Therefore, the existing Pohlhausen integral method has been extended to boundary-layer flows due to moving continuous surfaces. A combination of the two (the existing Pohlhausen integral method and the current extended Pohlhausen integral method) is referred to as the generalized Pohlhausen integral method applicable to both kinds of boundary-layer flows, namely, the flows over surfaces of finite lengths and the flows due to moving continuous surfaces. The extended Pohlhausen integral method has been examined for both similar and non-similar flow scenarios to assess its accuracy. The findings have been presented through tabular and graphical representations, and comparisons have been given with the existing/exact results. It is noted that in the case of a similar flow problem, specifically for the Sakiadis flow (AIChE J 07(2):221-225, 1961), the momentum thickness and skin-friction coefficient obtained using the extended Pohlhausen integral method were observed to be approximately 4% lower than those obtained from the exact solution. The integral value of the displacement thickness exhibits a reduction of nearly 13% compared to the exact value. Nevertheless, it is imperative to acknowledge that the proposed method provides a substantial underestimation, approximately 26%, of the boundary-layer thickness in comparison to the exact value. The extended Pohlhausen integral method is further simplified by suggesting some new parameters and a quadrature for momentum thickness has been derived. The non-similar flow problem, considering the linearly diminishing wall velocity, is also investigated using the newly proposed quadrature. The momentum thickness and displacement thickness at the separation position are both found to be nearly 4% lower than the exact values. It is noteworthy that the separation position is found by using quadrature as 0.4033, while the exact value is 0.4200, providing strong evidence of the method's accuracy and validity.