Non-dispersive and weakly dispersive single-layer flow over an axisymmetric obstacle: the equivalent aerofoil formulation

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(Γ) M5/3, where Γ = (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≲0.4 and |Γ| ≲ 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 Γ ≳ 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 $c_d M^2/(F\sqrt{F^2-1})$, where cd 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 Korteweg–de Vries equation. In the latter case the number of emergent solitary waves in the wake is shown to be a function of ${\cal A}= 3M/(2\delta^2 \sqrt{F^2-1})$, where δ is the ratio of the undisturbed layer depth to the radial scale of the obstacle.

Wave patterns generated by an axisymmetric obstacle in a two-layer flow

“…Fig. 9 highlights the increase of energy transferred to gravity waves around the critical regime F r = 1 or equivalently Γ = 0, as it is found in the case of a single layer flow from the consideration of the wave drag [11]. Moreover, there exists a range of Γ for which the value of the potential energy is significantly increased which can be associated with the transcritical regime.…”

“…A fourth parameter, Γ = (F r − 1)M −2/3 , emerges from a previous study in the case of a single layer flow [11] and is known as the transcritical similarity parameter. The range of parameters 1 As obstacles are towed upside-down, they cannot be fully immersed, so effective h 0 m is slightly smaller than the values given above and is 7.5 cm (resp.…”

“…Moreover, there exists a range of Γ for which the value of the potential energy is significantly increased which can be associated with the transcritical regime. This specific regime delimited by the subcritical regime and supercritical regime has been mostly studied in the case of single layer flows [11]. In the set of experiments considered here, it it is difficult to quantitatively delineate the different regimes.…”

“…1). These experiments were inspired by recent theoretical works [16,17,11] predicting the wave structure in similar flows where dispersive effects dominate over dissipative effects, as in many geophysical situations. Under certain conditions, for example when the flow speed is close to the gravity waves speed, these results lead to different predictions than previous work in the non-dispersive limit (e.g.…”

Swimming obstructed by dead-water

“…Swimming speed, however, is often much higher than c′, ranging from 0.75 m·s −1 for recreational up to 2 m·s −1 for competitive swimmers (Toussaint and Truijens 2005). This discrepancy in speeds makes it unlikely that the swimmer's body, seen as a displacement hull, experiences additional drag by generating interfacial waves, due to lack of coupling (Esler et al 2007). This might explain the absence of any measurable effect in a genuine swimming pool experiment, where the upper layer was less than 0.40 m (Maas and van Haren 2006), despite the fact that the swimmer's body (chest depth approximately 0.3 m (Vennell et al 2006)) was well within the 0.7-m distance over which a surface or interface might be disturbed by a moving object in its vicinity (Vennell et al 2006;Costill et al 1992).…”