An embedded strategy for large scale incompressible flow simulations in moving domains

Abstract

In this work we describe a methodology to approximate the incompressible Navier-Stokes equations in time dependent domains. To deal with the motion of the domain, we employ a fixed mesh method that we call fixed-mesh ALE. It consists of writing the equations in a moving ALE reference system but then projecting them onto a fixed background mesh. This implies that the boundaries of the elements do not necessarily coincide with the physical boundaries, and thus there is the possibility of badly cut elements. We use a Nitsche's type formulation to prescribe the boundary conditions and stabilise the bad cuts by introducing a term that penalises the gradient of the unknown orthogonal to the finite element space in a patch that contains the badly cut element. The flow formulation is a stabilised finite element method that allows one to treat convection dominated flows and to use equal velocity–pressure interpolation. Furthermore, this formulation can be shown to behave as an implicit large eddy simulation approach. A key issue is that the sub-grid scales on which the formulation depends are allowed to be time dependent; this fact has proved to be crucial for the robustness of the approach. The calculation of the velocity and the pressure is segregated by using a fractional step scheme designed at the pure algebraic level, and the conditioning of the pressure equation is improved by using an artificial compressibility technique. Finally, an adaptive mesh refinement strategy is described. All the algorithms are implemented in a parallel environment. The strategy described can be applied to complex flow problems, and in particular here we show the simulation of the air flow generated by a train moving inside a tunnel.