Skimming impacts and rebounds of smoothly shaped bodies on shallow liquid layers
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Skimming impacts and rebounds of smoothly shaped bodies on shallow liquid layers Ryan A. Palmer
· Frank T. Smith
Received: 4 May 2019 / Accepted: 31 July 2020 © The Author(s) 2020
Abstract Investigated in this paper is the coupled fluid–body motion of a thin solid body undergoing a skimming impact on a shallow-water layer. The underbody shape (the region that makes contact with the liquid layer) is described by a smooth polynomic curve for which the magnitude of underbody thickness is represented by the scale parameter C. The body undergoes an oblique impact (where the horizontal speed of the body is much greater than its vertical speed) onto a liquid layer with the underbody’s trailing edge making the initial contact. This downstream contact point of the wetted region is modelled as fixed (relative to the body) throughout the skimming motion with the liquid layer assumed to detach smoothly from this sharp trailing edge. There are two geometrical scenarios of interest: the concave case (C < 0 producing a hooked underbody) and the convex case (C > 0 producing a rounded underbody). As C is varied the rebound dynamics of the motion are predicted. Analyses of small-time water entry and of water exit are presented and are shown to be broadly in agreement with the computational results of the shallow-water model. Reduced analysis and physical insights are also presented in each case alongside numerical investigations and comparisons as C is varied, indicating qualitative analytical/numerical agreement. Increased body thickness substantially changes the interaction structure and accentuates inertial forces in the fluid flow. Keywords Aircraft icing · Fluid–body interactions · Shallow water skimming
1 Introduction The notion of an object skimming across the surface of a liquid layer is likely to be familiar to many. A main point of reference is the childhood game of ducks and drakes or stone skimming, where smooth stones are thrown across a body of water in an attempt to cause the stone to bounce off of the surface. Thrown with a significant horizontal velocity, the stone, as it skips, descends initially into the liquid layer with a small downward velocity. As it impacts, pressure on the body from the liquid layer increases producing a positive lift, carrying the stone back out of the R. A. Palmer (B) · F. T. Smith Department of Mathematics, UCL, Gower Street, London WC1E 6BT, UK e-mail: [email protected] R. A Palmer Present address: School of Mathematics, University of Bristol, Fry Building, Woodland Road, Bristol BS8 1UG, UK
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R. A. Palmer, F. T. Smith
water. Subsequent bounces may occur thereafter. In addition to this toy example, there are several other scientific and industrial occurrences of such dynamics, many noted in [1]. One key motivation is aircraft icing, a real-world application that admits many avenues for mathematical research. This includes the analytical study of fluid–body interaction in boundary-layer flow, such as [2,3] and the references therein, and a wealth of experimental, scienti
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