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A Reynolds-uniform numerical method for Prandtl's boundary layer problem for flow past a wedge

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Butler, J., Miller, J., & Shishkin, G. (2004). A Reynolds-uniform numerical method for Prandtl's boundary layer problem for flow past a wedge. Journal of Computational Methods in Science and Engineering, 5(2), 387-402. doi:10.1142/S1465876304002459.

Cite as: http://hdl.handle.net/11858/00-001M-0000-0013-DB99-3
In this paper we consider Prandtl's boundary layer problem for incompressible laminar flow past a plate with transfer of fluid through the surface of the plate. When the Reynolds number is large the solution of this problem has a parabolic boundary layer. In a neighbourhood of the plate the solution of the problem has an additional singularity which is caused by the absence of the compartability conditions. To solve this problem outside nearest neighbourhood of the leading edge, we construct a direct numerical method for computing approximations to the solution of the problem using a piecewise uniform mesh appropriately fitted to the parabolic boundary layer. To validate this numerical method, the model Prandtl problem with self-similar solution was examined, for which a reference solution can be computed using the Blasius problem for a nonlinear ordinary differential equation. For the model problem, suction/blowing of the flow rate density is v0(x)=-vi2-1/2Re1/2x-1/2, where the Reynolds number Re can be arbitrarily large and vi is the intensity of the mass transfer with arbitrary values in the segment [-.3,.3]. We considered the Prandtl problem in a finite rectangle excluding the leading edge of the plate for various values of Re which can be arbitrary large and for some values of vi, when meshes with different number of mesh points were used. To find reference solutions for the the velocity components and their derivatives with required accuracy, we solved the Blasius problem using a semi–analytical numerical method. By extensive numerical experiments we showed that the direct numerical method constructed in this paper allows us to approximate both the solution and its derivatives Re–uniformly for different values of vi.