Abstract
This paper presents a new family of semi-implicit hybrid finite volume/finite element schemes on edge-based staggered meshes for the numerical solution of the incompressible Reynolds-Averaged Navier–Stokes (RANS) equations in combination with the k−ε turbulence model. The rheology for calculating the laminar viscosity coefficient under consideration in this work is the one of a non-Newtonian Herschel–Bulkley (power-law) fluid with yield stress, which includes the Bingham fluid and classical Newtonian fluids as special cases. For the spatial discretization, we use edge-based staggered unstructured simplex meshes, as well as staggered non-uniform Cartesian grids. In order to get a simple and computationally efficient algorithm, we apply an operator splitting technique, where the hyperbolic convective terms of the RANS equations are discretized explicitly at the aid of a Godunov-type finite volume scheme, while the viscous parabolic terms, the elliptic pressure terms and the stiff algebraic source terms of the k−ε model are discretized implicitly. For the discretization of the elliptic pressure Poisson equation, we use classical conforming P1 and Q1 finite elements on triangles and rectangles, respectively. The implicit discretization of the viscous terms is mandatory for non-Newtonian fluids, since the apparent viscosity can tend to infinity for fluids with yield stress and certain power-law fluids. It is carried out with P1 finite elements on triangular simplex meshes and with finite volumes on rectangles. For Cartesian grids and more general orthogonal unstructured meshes, we can prove that our new scheme can preserve the positivity of k and ε. This is achieved via a special implicit discretization of the stiff algebraic relaxation source terms, using a suitable combination of the discrete evolution equations for the logarithms of k and ε. The method is applied to some classical academic benchmark problems for non-Newtonian and turbulent flows in two space dimensions, comparing the obtained numerical results with available exact or numerical reference solutions. In all cases, an excellent agreement is observed.
Highlights
The study of the incompressible Navier–Stokes equations has started many decades ago, motivated by the necessity of modelling numerous industrial processes and natural phenomena
To solve the incompressible ReynoldsAveraged Navier–Stokes (RANS) equations in combination with the k − ε turbulence model, we extend the family of hybrid finite volume/finite element methods described in [27,28,29,30,31,32]
We have introduced a new family of semi-implicit hybrid finite volume/finite element methods on edge-based staggered simplex meshes and on edgebased staggered Cartesian grids for the numerical solution of the Reynolds-Averaged
Summary
The study of the incompressible Navier–Stokes equations has started many decades ago, motivated by the necessity of modelling numerous industrial processes and natural phenomena. More recently, with the growing development of high-performance supercomputers, numerical methods have become essential for its solution. Trying to keep the best of each approach, semi-implicit schemes have been developed, showing an excellent performance for the solution of many PDE systems using either FD [1,2,3,4,5,6], FV [6,7,8,9,10,11,12,13] or continuous and discontinuous Galerkin FE methods [14,15,16,17,18,19]
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