Shallow flow simulation on dynamically adaptive cut cell quadtree grids

Abstract

SUMMARYA computationally efficient, high-resolution numerical model of shallow flow hydrodynamics is described, based on dynamically 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. Cartesian cut cells are 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 and overcome the time step restriction that would otherwise apply. The numerical model is validated through simulations of reflection of a surge wave at a wall, a low Froude number potential flow past a circular cylinder, and the shock-like interaction between a bore and a circular cylinder. The computational efficiency is shown to be greatly improved compared with solutions on a uniform structured grid implemented with cut cells.

An hp‐adaptive discontinuous Galerkin method for shallow water flows

“…Hence, for practical computations in engineering sciences typically error indicators rather than error estimators are used. A frequently used indicator is the gradient of the water depth or the free surface [3]. Another indicator introduced by Tate et al [27] is based on computing the residual of the continuity equation.…”

“…The most predominant numerical method used for modelling the SWE over the last decade is the finite volume method, e.g. [1][2][3]. Lately, there is a growing body of studies concerned with the application of discontinuous Galerkin (DG) methods to the SWE.…”

Boussinesq cut‐cell model for non‐linear wave interaction with coastal structures

“…Hence, we use the ghostcell immersed boundary method (GCIBM) proposed by Tseng and Ferziger [36] to treat the small cut cells. The flow information at the small cut cells is interpolated from the boundary conditions and neighbouring cells as explained by Liang et al [20].…”

Inflow–outflow boundary conditions along arbitrary directions in Cartesian lake models

“…Model coding and grid definition in this grid-system is much simpler than in others. Grid generation, for example, in unstructuredgrid models is not a completely automatic process, requiring separate grid creation software, and user intervention is often need to produce a grid of satisfactory quality (Liang et al, 2007), especially if complex topographic features are present. It is also computationally expensive.…”

“…Berger and Oliger, 1984;Ham et al, 2002;Gibou et al, 2007;Peng et al, 2010, and references therein). Cut cells can also be used for the solution of the shallow water equations (Causon et al, 2000;Liang et al, 2007), and in this case, boundary contours are cut out of a background Cartesian mesh and cells that are partially or completely cut are singled out for special treatment. Other approaches such as the immerse boundary method of Peskin (1972Peskin ( , 2002, the virtual Contents lists available at ScienceDirect journal homepage: www.elsevier.com/locate/cageo boundary method (Saiki and Biringen, 1996) or the Brinkman penalization method (e.g.…”