Shallow Flow Simulation on Dynamically Adaptive Cut-cell Quadtree Grids

A computationally efficient, high-resolution numerical model of shallow flow hydrodynamics is described, based on boundary-fitted adaptive quadtree grids. The numerical model solves the two-dimensional non-linear shallow water equations by means of an explicit second-order MUSCL-Hancock Godunov-type finite volume scheme. Interface fluxes are evaluated using an HLLC approximate Riemann solver. The model is based on dynamically adaptive quadtree grids, with Cartesian cut cells used to improve the fit to curved boundaries. A ghost-cell immersed boundary method is used to update flow information in the smallest cut cells in order to overcome the time step restriction that would otherwise apply. The numerical model is validated for reflection of a surge wave at a standing wall, low Froude number potential flow past a circular cylinder, and the shock-like interaction between a bore and a circular cylinder. Excellent agreement is obtained between the numerical predictions using the present scheme with analytical solutions and other numerical results presented in the literature. It is demonstrated that the computational efficiency is greatly improved compared with solutions on a uniform structured grid implemented with cut cells.