A novel approach of testing NSlet representation using the classical Blasius flow past a semi-infinite flat plate

Adamu, H A; Chadwick, E A; Borresen, J; Dang, B C; Darghoth, R

Abstract

Consider the new theory by Chadwick [1] that describe the incompressible Navier-Stokes equations by an integral distribution of Navier-Stokes fundamental solutions called NSlets. This paper tests this theory against the classical non-linear Blasius problem for two-dimensional steady flow past a semi-infinite flat plate. This problem has a numerical solution given by the shooting method. For this problem, the NSlets simplify under boundary layer approximations and the Oseen linearisation, which we call BL-lets. The boundary integral is along the half line describing the semi-infinite flat plate, and so reduces to the evaluation of the unknown strength function of a Wiener-Hopf type integral. The Fourier Transform of the BL-let is already analytic in the upper half plane, leading to a degenerate case of the Wiener-Hopf technique and the determination of the strength function going as the inverse square root. We present another approach of determining the strength function which we call Adamu's approach that transforms the kernel into a modified Bessel function and applied Wiener-Hopf technique. The results of the strength function obtained are compared with the results from the existing approaches such as Gautesen [13] and Chadwick approach[1]. The result agrees with the results from all the approaches. However, the BL-let representation is only valid in the far-boundary layer and so this solution is a first approximation and not accurate. By relaxing the boundary condition at the leading edge, equivalent to assuming the BL-let approximation breaks down there, enables us to consider a strength function expansion and therefore a velocity expansion such that the erf function is only the first term. The resulting velocity expansion is continued into the boundary layer and shown to converge on the solution after just five terms, and is compared to the numerical Blasius shooting method and Kusukawa's boundary layer expansion. Hence, the NSlet theory and representation successfully models this particular benchmark test.

Citation

Adamu, H. A., Chadwick, E. A., Borresen, J., Dang, B. C., & Darghoth, R. (in press). A novel approach of testing NSlet representation using the classical Blasius flow past a semi-infinite flat plate.

• Conference Name: 13th UK Conference on Boundary Integral Methods
• Conference Location: University of Aberdeen
• Start Date: Jul 10, 2023
• End Date: Jul 11, 2023
• Acceptance Date: Apr 1, 2023
• Deposit Date: Jul 3, 2023