Abstract
An analytical series method is presented for steady, two-dimensional, irrotational flow of a single layer of constant-density fluid over topography. This problem is formulated as a Laplacian free-boundary problem with fully nonlinear boundary conditions. The method is an iterative scheme that allows the calculation of analytical series solutions for supercritical, transcritical and subcritical flow regimes over arbitrary topography. By an appropriate choice of the free-boundary representation, exponential convergence of the series solution is achieved. With this accuracy, the issue of apparent dual transcritical/subcritical solutions previously obtained by boundary-integral-equation methods (BIEM) is resolved. Results are compared with solutions previously obtained using BIEM, and solutions are presented for flow over asymmetric and arbitrarily shaped obstacles.
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References
Kelvin W. (1886). On stationary waves in moving water. Phil. Mag. 22:353–357
Lamb H. (1932). Hydrodynamics, 6th ed. Cambridge University Press, Cambridge, 738 pp.
Long R.R. (1953). Some aspects of the flow of stratified fluids, I A theoretical investigation. Tellus 5:42–58
Wehausen J.V., and Laitone W.V. (1960). Surface waves. In: Flügge S. (eds). Handbuch der Physik, Vol 9. Springer, Heidelberg, pp. 446–778
Forbes L.K., and Schwartz L.W. (1982). Free-surface flow over a semicircular obstruction. J. Fluid Mech. 114:299–314
Forbes L.K. (1982). On the wave resistance of a submerged semi-elliptical body. J. Engng. Math. 15:287–298
King A.C., and Bloor M.T.G. (1987). Free surface flow over a step. J. Fluid. Mech. 182:193–208
Dias F., and Vanden-Breock J.M. (1989). Open channel flows with submerged obstructions. J. Fluid. Mech. 206:155–170
King A.C., and Bloor M.T.G. (1989). A Semi-inverse method for free surface flow over a submerged body. Quart. J. Mech. Appl. Math. 42:183–202
Belward S.R., and Forbes L.K. (1995). Interfacial waves and hydraulic falls: some applications to atmospheric flows in the lee of mountains. J. Engng. Math. 29:161–179
Belward S.R. (1999). Fully nonlinear flow over successive obstacles: hydraulic fall and supercritical flows. J. Austral. Math. Soc. Ser. B. 40:447–458
Read W.W., and Volker R.E. (1993). Series solutions for steady seepage through hillsides with arbitrary flow boundaries. Water Resources Res. 29:2871–2880
Gill A.W., and Read W.W. (1996). Efficient analytic series solutions for two-dimensional potential flow problems. Int. J. Numer. Meth. Fluids 23:415–430
Shen S.S.P. (1992). Forced solitary waves and hydraulic falls in two-layer flows. J. Fluid Mech. 234:583–612
Read W.W. (1993). Series solutions for Laplaces equation with nonhomogeneous mixed boundary conditions and irregular boundaries. Math. Comput. Model. 17:9–19
Read W.W., Belward S.R., and Higgins P.J. (2003). An efficient iterative scheme for series solutions to Laplacian free boundary problems. ANZIAM J. 44:C644–C663
Gilbarg D., and Trudinger M.S. (1983). Elliptic Partial Differential Equations of Second Order, 2nd ed. Springer, Berlin, 530 pp.
Belward S.R., Read W.W., and Higgins P.J. (2003). Efficient series solutions for non-linear flow over topography. ANZIAM J. 44:C96–C113
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Higgins, P.J., Read, W.W. & Belward, S.R. A Series-Solution Method for Free-Boundary Problems Arising from Flow Over Topography. J Eng Math 54, 345–358 (2006). https://doi.org/10.1007/s10665-006-9039-0
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DOI: https://doi.org/10.1007/s10665-006-9039-0