Volume 25, Issue 2

We propose and analyze an augmented mixed finite element method for the coupling of fluid flow with porous media flow. The flows are governed by a class of nonlinear Navier–Stokes and linear Darcy equations, respectively, and the transmission conditions are given by mass conservation, balance of normal forces, and the Beavers–Joseph–Saffman law. We apply dual-mixed formulations in both domains, and the nonlinearity involved in the Navier–Stokes region is handled by setting the strain and vorticity tensors as auxiliary unknowns. In turn, since the transmission conditions become essential, they are imposed weakly, which yields the introduction of the traces of the porous media pressure and the fluid velocity as the associated Lagrange multipliers. Furthermore, since the convective term in the fluid forces the velocity to live in a smaller space than usual, we augment the variational formulation with suitable Galerkin type terms arising from the constitutive and equilibrium equations of the Navier–Stokes equations, and the relation defining the strain and vorticity tensors. The resulting augmented scheme is then written equivalently as a fixed point equation, so that the well-known Schauder and Banach theorems, combined with classical results on bijective monotone operators, are applied to prove the unique solvability of the continuous and discrete systems. In particular, given an integer k ⩾ 0, piecewise polynomials of degree ⩽ k , Raviart–Thomas spaces of order k , continuous piecewise polynomials of degree ⩽ k + 1, and piecewise polynomials of degree ⩽ k are employed in the fluid for approximating the strain tensor, stress, velocity, and vorticity, respectively, whereas Raviart–Thomas spaces of order k and piecewise polynomials of degree ⩽ k for the velocity and pressure, together with continuous piecewise polynomials of degree ⩽ k + 1 for the traces, constitute feasible choices in the porous medium. Finally, several numerical results illustrating the good performance of the augmented mixed finite element method and confirming the theoretical rates of convergence are reported.

The modified augmented Lagrangian preconditioner has attracted much attention in solving nondimensional Navier–Stokes equations discretized by the finite element method. In industrial applications the governing equations are often in dimensional form and discretized using the finite volume method. This paper assesses the capability of this preconditioner for dimensional Navier–Stokes equations in the context of the finite volume method. Two main contributions are made. First, this paper introduces a new dimensionless parameter that is involved in the modified augmented Lagrangian preconditioner. Second, with a number of academic test problems this paper reveals that the convergence of both nonlinear and linear iterations depend on this dimensionless parameter. A way to choose the optimal value of the dimensionless parameter is suggested and it is found that the optimal value is dependent of the Reynolds number, instead of the fluid’s properties, e.g., density and dynamic viscosity. The outcomes of this paper yield a potential rule to choose the optimal dimensionless parameter in practice, namely, correspondingly increasing with enlarging the Reynolds number.

We study numerically four regularization models with deconvolution for density currents, namely, Boussinesq- α , Boussinesq- ω , Boussinesq–Leray and Modified-Boussinesq–Leray. A Crank–Nicolson in time and finite element in space algorithm is proposed and proved to be unconditionally stable and optimally convergent, which is verified through convergence rates in simulations. Lastly, the regularized models are compared through the two-dimensional Marsigli’s flow benchmark for Re = 2000 and Re = 5000. We found that Boussinesq- α and Boussinesq–Leray models produced the most accurate solutions in the low Reynolds number test and, as expected, all regularized models had their solutions improved when deconvolution order was increased. On the other hand, in the high Reynolds number test the Boussinesq–Leray provided the best solution. Besides, the Boussinesq–Leray model is also more advantageous from the computational point of view because its momentum and filter equations are decoupled enabling to increase the deconvolution order with no significant increase in the computational cost.