A Marker-and-Cell scheme for viscoelastic flows on nonuniform grids

Omar Mokhtari (1) , Yohan Davit (1) , Jean-Claude Latché (2) , Romain de Loubens (3) , Michel Quintard (1)

1 IMFT - Institut de mécanique des fluides de Toulouse
2 IRSN - Institut de Radioprotection et de Sûreté Nucléaire
3 Total E&P

In this paper, we develop a numerical scheme for the solution of the coupled Stokes and Navier-Stokes equations with constitutive equations describing the flow of viscoelastic fluids. The space discretization is based on the so-called Marker-And-Cell (MAC) scheme. The time discretization uses a fractional-step algorithm where the solution of the Navier-Stokes equations is first obtained by a projection method and then the transport-reaction equation for the conformation tensor is solved by a finite-volume scheme. In order to obtain consistency, the space discretization of the divergence of the elastic part of the stress tensor in the momentum balance equation is derived using a weak form of the MAC scheme. For stability and accuracy reasons, the solution of the transport-reaction equation for the conformation tensor is split into pure convection steps, with a change of variable from c to log(c), and a reaction step, which consists in solving one ODE per cell via an Euler scheme with local sub-cycling. Numerical computations for the Stokes flow of an Oldroyd-B fluid in the lid-driven cavity at We=1 confirm the scheme efficiency.

Mots clés: Viscoelastic flows; MAC scheme; Projection scheme

Domaines: Mécanique des fluides [physics.class-ph]

Omar Mokhtari, Yohan Davit, Jean-Claude Latché, Romain de Loubens, Michel Quintard. A Marker-and-Cell scheme for viscoelastic flows on nonuniform grids. FVCA 2020: Finite Volumes for Complex Applications IX, Jun 2020, Bergen, Norway. pp.645-653. ⟨hal-02950581⟩