Convergence of a Vector Penalty Projection Scheme for the Navier-Stokes Equations with moving body

TL;DR

This paper analyzes a vector penalty projection scheme for simulating incompressible viscous flows with moving bodies, proving stability and convergence to weak solutions of the Navier-Stokes equations as parameters tend to zero.

Contribution

It introduces a convergence analysis of a vector penalty projection scheme for fluid-structure interaction problems with moving bodies, including stability and weak limit results.

Findings

Scheme is stable under certain parameter conditions.

Velocity and pressure converge to a limit as penalty and time step tend to zero.

Weak solutions of Navier-Stokes are obtained in the limit.

Abstract

In this paper, we analyse a Vector Penalty Projection Scheme (see [1]) to treat the displacement of a moving body in incompressible viscous flows in the case where the interaction of the fluid on the body can be neglected. The presence of the obstacle inside the computational domain is treated with a penalization method introducing a parameter $\eta$. We show the stability of the scheme and that the pressure and velocity converge towards a limit when the penalty parameter $\epsilon$, which induces a small divergence and the time step $\delta$t tend to zero with a proportionality constraint $\epsilon$ = $\lambda$$\delta$t. Finally, when $\eta$ goes to 0, we show that the problem admits a weak limit which is a weak solution of the Navier-Stokes equations with no-sleep condition on the solid boundary. R{\'e}sum{\'e} Dans ce travail nous analysons un sch{\'e}ma de projection vectorielle (voir [1]) pour traiter le d{\'e}placement d'un corps solide dans un fluide visqueux incompressible dans le cas o` u l'interaction du fluide sur le solide est n{\'e}gligeable. La pr{\'e}sence de l'obstacle dans le domaine solide est mod{\'e}lis{\'e}e par une m{\'e}thode de p{\'e}nalisation. Nous montrons la stabilit{\'e} du sch{\'e}ma et la convergence des variables vitesse-pression vers une limite quand le param etre $\epsilon$ qui assure une faible divergence et le pas de temps $\delta$t tendent vers 0 avec une contrainte de proportionalit{\'e} $\epsilon$ = $\lambda$$\delta$t. Finalement nous montrons que leprob{\`i} eme converge au sens faible vers une solution des equations de Navier-Stokes avec une condition aux limites de non glissement sur lafront{\`i} ere immerg{\'e}e quand le param etre de p{\'e}nalisation $\eta$ tend vers 0.