Keywords: finite element, finite volume, incompressible flows, heat transfer, two-fluid flows, shallow water flow, GMRES.

In this chapter we present our implicit hybrid finite element/volume solvers for three-dimensional incompressible flows coupled with energy equation, three-dimensional two immiscible fluids free surface flows, and two-dimensional shallow water flows. Our general approach is to solve the Navier Stokes equations using the segregated pressure correction or projection method on staggered unstructured hybrid meshes. An intermediate velocity field is first obtained by solving the momentum equations with the matrix-free implicit cell-centered finite volume method. Then a pressure Poisson equation is solved by the node-based Galerkin finite element method for an auxiliary variable. The auxiliary variable is used to update the velocity field and the pressure field.

The incompressible flows we present here are governed by the Navier-Stokes equations with the Boussinesq buoyancy approximation coupled to the thermal energy transport equation. For turbulence we have incorporated the detached eddy simulation (DES) turbulence model to compute the eddy viscosity. The finite volume method is used for both temperature and eddy viscosities. For the two-fluid free-surface flow solver, the fluid-interface is captured by solving an advection equation for the volume fraction of one of the fluids. The same matrix-free finite volume method as the one used for momentum equations is used to solve the advection equation. We use the interface sharpening strategy to minimize the smearing of the interface over time. We have developed and implemented a global mass conservation algorithm which enforces the conservation of the mass for each fluid.

In our hybrid shallow water flow solver, we solve the momentum equation using the cell-center finite volume method and the wave continuity equation using node-based finite element method. To enhance the stability of the hybrid method around discontinuities, we introduce a new shock capturing scheme which will act only around sharp interfaces without sacrificing the accuracy elsewhere.

Several test problems are presented to demonstrate the robustness and applicability of our numerical methods. The parallelization is based on the Message Passing Interface (MPI) and our flow solvers are extremely well scalable on a parallel platform.

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