The effect of air on solid body impact with water in two dimensions

Abstract In this paper we study the motion of wedge and catamaran shaped hulls initially in air, then impacting with water. We consider motion in two dimensions and simulate the impact using SPH in a form suitable for two-phase flow. This SPH algorithm has been tested for a variety of problems involving air and water with very good results. However, we supplement those tests by simulating the dynamics of bubbles, and the motion of piston driven by the pressure of air in a tube. The former confirms that our algorithm simulates the motion of two fluids with a high density ratio (in this case 1000) while the latter is relevant to the impact of a catamaran shaped hull where the trapped air cushions the impact. In both cases the results are in very good agreement with theory. When applied to simulate the dynamics of a wedge falling into a fluid the results are in good agreement with theory and confirm that the effect of the air on the motion is small because the air can escape easily as the wedge enters the water. When applied to simulate a model catamaran hull in two dimensions, where the air is trapped beneath the hull, the effect of air is pronounced. The simulations with the catamaran hull are not in close agreement with experiments because the hull in three dimensions does not trap air effectively as it does in two dimensions.

[1]  R. Collins.  A second approximation for the velocity of a large gas bubble rising in an infinite liquid , 1966 .

[2]  Holger Wendland,et al.  Piecewise polynomial, positive definite and compactly supported radial functions of minimal degree , 1995, Adv. Comput. Math..

[3]  Ashkan Rafiee,et al.  A simple SPH algorithm for multi‐fluid flow with high density ratios , 2013 .

[4]  J. Monaghan.  Smoothed Particle Hydrodynamics and Its Diverse Applications , 2012 .

[5]  S. Osher,et al.  A level set approach for computing solutions to incompressible two-phase flow , 1994 .

[6]  H. Wagner.  Über Stoß- und Gleitvorgänge an der Oberfläche von Flüssigkeiten , 1932 .

[7]  Benedict D. Rogers,et al.  Slam Modelling with SPH: The Importance of Air , 2013 .

[8]  Yoichi Sumi,et al.  On the water impact and elastic response of a flat plate at small impact angles , 2000 .

[9]  A. D. Young,et al.  An Introduction to Fluid Mechanics , 1968 .

[10]  Giles Thomas,et al.  Prediction of Slamming Behaviour of Monohull and Multihull Forms using Smoothed Particle Hydrodymamics , 2011 .

[11]  Rajamani Krishna,et al.  Rise velocity of single circular-cap bubbles in two-dimensional beds of powders and liquids , 2000 .

[12]  W. Dehnen,et al.  Improving convergence in smoothed particle hydrodynamics simulations without pairing instability , 2012, 1204.2471.

[13]  Stephen R. Turnock,et al.  Impact of a free-falling wedge with water: synchronized visualization, pressure and acceleration measurements , 2010 .

[14]  G. Oger,et al.  Two-dimensional SPH simulations of wedge water entries , 2006, J. Comput. Phys..

[15]  Paul W. Cleary,et al.  Modelling confined multi-material heat and mass flows using SPH , 1998 .

[16]  David Le Touzé,et al.  An Hamiltonian interface SPH formulation for multi-fluid and free surface flows , 2009, J. Comput. Phys..

[17]  Benedict D. Rogers,et al.  Numerical predictions of water–air wave slam using incompressible–compressible smoothed particle hydrodynamics , 2015 .

[18]  J. Monaghan.  Smoothed particle hydrodynamics , 2005 .

[19]  A. A. Korobkin,et al.  Initial Stage of Water Impact , 1988 .

[20]  P. Cleary,et al.  Conduction Modelling Using Smoothed Particle Hydrodynamics , 1999 .

[21]  G K Kapsenberg,et al.  Slamming of ships: where are we now? , 2011, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[22]  J. Ross Macdonald,et al.  Some Simple Isothermal Equations of State , 1966 .

[23]  Songdong Shao,et al.  Incompressible SPH simulation of water entry of a free‐falling object , 2009 .

[24]  Joseph John Monaghan,et al.  SPH and Riemann Solvers , 1997 .

[25]  Joe J. Monaghan,et al.  SPH particle boundary forces for arbitrary boundaries , 2009, Comput. Phys. Commun..

[26]  G. Taylor,et al.  The mechanics of large bubbles rising through extended liquids and through liquids in tubes , 1950, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences.

[27]  D. L. Pyle,et al.  The rising velocity of bubbles in two-dimensional fluidised beds , 1967 .

[28]  D. Yue,et al.  On the water impact of general two-dimensional sections , 1999 .

[29]  James Robert Whelan.  Wetdeck slamming of high-speed catamarans with a centre bow , 2004 .

[30]  Martin E. Weber,et al.  Bubbles in viscous liquids: shapes, wakes and velocities , 1981, Journal of Fluid Mechanics.