Development of a 3D non-hydrostatic pressure model for free surface flows

Abstract

A three-dimensional, non-hydrostatic pressure, numerical model for free surface flows is presented. By decomposing the pressure term into hydrostatic and non-hydrostatic parts, the numerical model uses an integrated time step with two fractional steps. In the first fractional step, the momentum equations are solved without the hydrostatic pressure term using Newton's method in conjunction with the generalised minimal residual (gmres) method. This combined method does not require the determination of a Jacobian matrix explicitly but simply the product of the Jacobian and a vector, thereby reducing the amount of storage required and significantly decreasing the overall * Applied Mathematics, The University of Adelaide, South Australia, Australia. C624 computational time required. By using Newton's method, the numerical model can handle implicitly almost all variables, unlike many other numerical models. Hence numerical stability is achieved effectively. In the second fractional step, the pressure-Poisson equation is solved iteratively with a preconditioned linear gmres method. It is shown that preconditioning reduces the processing time dramatically. After the new pressure field is obtained the intermediate velocities, which are calculated from the previous fractional step, are updated and then these updated velocities preserve the local mass conservation. The newly developed model is verified against analytical solutions, with good agreement.

An AMG strategy for efficient solution of free-surface flows

“…From the literature, a number of advanced sparse linear solvers have been applied to FSM for the purpose of solving the pressure correction equation. These include successive over-relaxation [9], preconditioned GMRES [10,8], various conjugate-gradient type methods [11,12,13,14,15] and geometric multigrid methods [16,17,18]. Other attempts at solution acceleration include improved initial condition prediction [19] and algebraic multigrid (AMG) methods [20].…”

“…From the literature, a number of advanced sparse linear solvers have been applied to FSM for the purpose of solving the pressure correction equation. These include successive over-relaxation (Kurioka and Dowling, 2009), pre-conditioned GMRES (Lee et al, 2005;Oxtoby et al, 2015), various conjugate-gradient type methods (Van der Vorst, 1992;Koçyigit et al, 2002;Soto et al, 2003;Aubry et al, 2008;Štrubelj et al, 2009) and geometric multigrid methods (Farmer et al, 1994;Waltz and Löhner, 2000;Popinet, 2003). Other attempts at solution acceleration include improved initial condition prediction (Löhner, 2005) and algebraic multigrid (AMG) methods (Kim and Moin, 2009).…”