Esler, JG; Rump, OJ; Johnson, ER; (2007) Non-dispersive and weakly dispersive single-layer flow over an axisymmetric obstacle: the equivalent aerofoil formulation. J FLUID MECH , 574 209 - 237. 10.1017/S0022112006003910.

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Abstract

Non-dispersive and weakly dispersive single-layer flows over axisymmetric obstacles, of non-dimensional height M measured relative to the layer depth, are investigated. The case of transcritical flow, for which the Froude number F of the oncoming flow is close to unity, and that of supercritical flow, for which F > 1, are considered. For transcritical flow, a similarity theory is developed for small obstacle height and, for non-dispersive flow, the problem is shown to be isomorphic to that of the transonic flow of a compressible gas over a thin aerofoil. The non-dimensional drag exerted by the obstacle on the flow takes the form D(Gamma)M-5/3, where Gamma = (F - 1)M-2/3 is a transcritical similarity parameter and D is a function which depends on the shape of the 'equivalent aerofoil' specific to the obstacle. The theory is verified numerically by comparing results from a shock-capturing shallow-water model with corresponding solutions of the transonic small-disturbance equation, and is found to be generally accurate for M less than or similar to 0.4 and vertical bar Gamma vertical bar less than or similar to 1. In weakly dispersive flow the equivalent aerofoil becomes the boundary condition for the Kadomtsev-Petviashvili equation and (multiple) solitary waves replace hydraulic jumps in the resulting flow patterns.For Gamma greater than or similar to 1.5 the transcritical similarity theory is found to be inaccurate and, for small M, flow patterns are well described by a supercritical theory, in which the flow is determined by the linear solution near the obstacle. In this regime the drag is shown to be c(d)M(2)/(F root F-2 - 1), where c(d) is a constant dependent on the obstacle shape. Away from the obstacle, in non-dispersive flow the far-field behaviour is known to be described by the N-wave theory of Whitham and in dispersive flow by the Kortewegde Vries equation. In the latter case the number of emergent solitary waves in the wake is shown to be a function of A = 3M/(2 delta(2) root F-2 - 1), where delta is the ratio of the undisturbed layer depth to the radial scale of the obstacle.

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