The virtual source method (VSM) has been developed to simulate water waves based upon the solution of Laplace's equation for the velocity potential integral equations with full nonlinear surface conditions. The basis of the method is the use of specific Green's functions for a rectangular ‘virtual domain’ which is an extension of the physical domain. The solution variables are frequency components of the velocity potential at the upper virtual boundary and these are found by specifying appropriate conditions on the physical boundaries (i.e. wavemaker, walls and wave surfaces). The authors have shown that the model successfully simulates both linear and nonlinear standing waves and simple sloshing problems and is more effective and efficient than simple boundary element methods for these problems. In this paper, we develop the VSM to generate nonlinear progressive waves in a numerical wave tank. In order to remove the transmitted energy of the waves and so reduce the reflection from the right wall of the tank, an artificial damping term is added to the free surface boundary condition. The VSM results are compared with those from both second order Stokes theory and from a boundary element method (BEM).
The numerical simulation of nonlinear water waves using the potential flow studies has been started in the 1970's. Longuet-Higgins and Cokelet (1976) were the first to simulate an asymmetric overturning deep water wave. They used a conformal mapping approach with space periodic boundary conditions and an artificial pressure distribution to enforce the breaking. Since then, many researchers have taken up the topic of nonlinear wave simulation, often in combination with wave-body interaction. Kim et al. (1999) reviewed the research on the development of numerical wave tanks (NWT's). More recently, a review on the topic of numerical wave modelling is given by Thomas and Dwarakish (2015). Most numerical methods for potential flow wave simulation are based on the mixed Eulerian-Lagrangian method to separate the elliptic boundary value problems (BVP) from the dynamic equations at the free surface. The vast majority of available numerical schemes that simulate free surface potential flow, approximate the BVP using a Boundary Integral (BI) formulation. Recently, Langfeld et al. (2016) introduced the virtual source method (VSM) for solving free-surface potential flow problems and applied it to simulate standing waves in 2D and 3D. Al-Tameemi et al (2018) illustrated the the energy and volume conservation of the VSM method for nonlinear standing wave simulation. In this paper we develop the VSM to simulate nonlinear progressive waves in a numerical wave tank. The results are compared with second order Stokes wave theory as well as boundary element method (BEM) numerical results.
