SUMMARYIn this paper, we describe an implicit hybrid finite volume (FV)/element (FE) incompressible NavierStokes solver for turbulent flows based on the Spalart-Allmaras detached eddy simulation (SA-DES). The hybrid FV/FE solver is based on the segregated pressure correction or projection method. The intermediate velocity field is first obtained by solving the original momentum equations with the matrix-free implicit cell-centered FV method. The pressure Poisson equation is solved by the node-based Galerkin FE method for an auxiliary variable. The auxiliary variable is closely related to the real pressure and is used to update the velocity field and the pressure field. We store the velocity components at cell centers and the auxiliary variable at vertices, making the current solver a staggered-mesh scheme. The SA-DES turbulence equation is solved after the velocity and the pressure fields have been updated at the end of each time step. The same matrix-free FV method as the one used for momentum equations is used to solve the turbulence equation. The turbulence equation provides the eddy viscosity, which is added to the molecular viscosity when solving the momentum equation. In our implementation, we focus on the accuracy, efficiency and robustness of the SA-DES model in a hybrid flow solver. This paper will address important implementation issues for high-Reynolds number flows where highly stretched elements are typically used. In addition, some aspects of implementing the SA-DES model will be described to ensure the robustness of the turbulence model. Several numerical examples including a turbulent flow past a flat plate and a high-Reynolds number flow around a high angle-of-attack NACA0015 airfoil will be presented to demonstrate the accuracy and efficiency of our current implementation.

SUMMARYA hybrid numerical scheme based on finite element and finite volume methods is developed to solve shallow water equations. In the recent past, we introduced a series of hybrid methods to solve incompressible low and high Reynolds number flows for single and two-fluid flow problems. The present work extends the application of hybrid method to shallow water equations. In our hybrid shallow water flow solver, we write the governing equations in non-conservation form and solve the non-linear wave equation using finite element method with linear interpolation functions in space. On the other hand, the momentum equation is solved with highly accurate cell-center finite volume method. Our hybrid numerical scheme is truly a segregated method with primitive variables stored and solved at both node and element centers. To enhance the stability of the hybrid method around discontinuities, we introduce a new shock capturing which will act only around sharp interfaces without sacrificing the accuracy elsewhere. Matrix-free GMRES iterative solvers are used to solve both the wave and momentum equations in finite element and finite volume schemes. Several test problems are presented to demonstrate the robustness and applicability of the numerical method.

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