Efficient series solutions for non-linear flow over topography

S. R. Belward, W. W. Read, P. J. Higgins


Fluid flowing over topography occurs in many physical situations. As a consequence, study of flow over topography has been a research topic of prime interest for many decades. Formally, the problem can be modelled as a nonlinear free boundary problem. Although methods such as boundary integrals are typically used, analytic series methods have also been developed to solve some of these problems. Arguably the hardest problem to solve is the lee wave problem: when the flow conditions are suitable, waves form downstream of the obstacle. Wave solutions pose several problems for the analytic series methods. The solution method is iterative, and at each step the existing solution must be updated. For the iterative scheme to converge, very accurate series solutions must be obtained at each step. The convergence rate of the series solution itself is critical in this process, and depends to a large extent on the free boundary representation. In this paper, we compare and discuss the convergence rates for a variety of free surface representations. We show that spectral convergence is possible if the correct representation is used.

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DOI: https://doi.org/10.21914/anziamj.v44i0.674

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ANZIAM Journal, ISSN 1446-8735, copyright Australian Mathematical Society.