Interface‐capturing finite element technique for transient two‐phase flow

2003 ◽  
Vol 20 (5/6) ◽  
pp. 725-740 ◽  
Author(s):  
Nahidh Hamid Sharif ◽  
Nils‐Erik Wiberg
Keyword(s):  
Finite Element ◽  
Porous Media ◽  
Stokes Equations ◽  
Time Dependent ◽  
Navier Stokes ◽  
Porous Media Flow ◽  
Advection Equation ◽  
Fluid Interfaces ◽  
Navier Stokes Equations ◽  
Two Fluid

A numerical model is presented for the computation of unsteady two‐fluid interfaces in nonlinear porous media flow. The nonlinear Forchheimer equation is included in the Navier‐Stokes equations for porous media flow. The model is based on capturing the interface on a fixed mesh domain. The zero level set of a pseudo‐concentration function, which defines the interface between the two fluids, is governed by a time‐dependent advection equation. The time‐dependent Navier‐Stokes equations and the advection equation are spatially discretized by the finite element (FE) method. The fully coupled implicit time integration scheme and the explicit forward Eulerian scheme are implemented for the advancement in time. The trapezoidal rule is applied to the fully implicit scheme, while the operator‐splitting algorithm is used for the velocity‐pressure segregation in the explicit scheme. The spatial and time discretizations are stabilized using FE stabilization techniques. Numerical examples of unsteady flow of two‐fluid interfaces in an earth dam are investigated.

A fully-mixed finite element method for the Navier–Stokes/Darcy coupled problem with nonlinear viscosity

2017 ◽  
Vol 25 (2) ◽  
Author(s):  
Sergio Caucao ◽  
Gabriel N. Gatica ◽  
Ricardo Oyarzúa ◽  
Ivana Šebestová
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Porous Media ◽  
Stokes Equations ◽  
Mixed Finite Element ◽  
Mixed Finite Element Method ◽  
Transmission Conditions ◽  
Navier Stokes ◽  
Porous Media Flow ◽  
Element Method

AbstractWe 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

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