Numerical Simulation of Solitary Waves Using Velocity–Vorticity Formulation of Navier–Stokes Equations

Abstract

This paper presents the development of a novel numerical scheme to study solitary wave phenomena using velocity–vorticity formulation instead of the primitive variable (velocity-pressure) form for the Navier–Stokes equations. The free surface boundary conditions are accordingly modified to obtain the equations in the velocity and vorticity form without involving the pressure term. The kinematics of the fluid domain for the free surface flow is described using an arbitrary Lagrangian–Eulerian method. The proposed model is first validated with numerical and analytical solutions available from the literature for the following cases: (1) an oblique rigid-lid driven cavity flow and (2) an incident solitary wave. Then the versatility of the algorithm is demonstrated by obtaining numerical results for (3) periodic waves generated in a channel by a piston wavemaker, and (4) solitary waves generated by critical flow over a submerged solid structure. The salient characteristics of the propagation of solitary waves are well-illustrated in the present study.