In this paper we focus on the existence of a weak solution to a system describing a self-propelled motion of a single deformable body in a viscous compressible fluid that occupies a bounded domain in the three-dimensional Euclidean space. The governing system considered for the fluid is the isentropic compressible Navier–Stokes equation. We prove the existence of a weak solution up to a collision.

We consider the relativistic Euler equations governing spherically symmetric, perfect fluid flows on the outer domain of communication of Schwarzschild space-time, and we introduce a version of the finite volume method which is formulated from the geometric formulation (and thus takes the geometry into account at the discretization level) and is well-balanced, in the sense that it preserves steady solutions to the Euler equations on the curved geometry under consideration. In order to formulate our method, we first derive a closed formula describing all steady and spherically symmetric solutions to the Euler equations posed on Schwarzschild spacetime. Second, we describe a geometry-preserving, finite volume method which is based from the family of steady solutions to the Euler system. Our scheme is second-order accurate and, as required, preserves the family of steady solutions at the discrete level. Numerical experiments are presented which demonstrate the efficiency and robustness of the proposed method even for solutions containing shock waves and nonlinear interacting wave patterns. As an application, we investigate the late-time asymptotics of perturbed steady solutions and demonstrate its convergence for late time toward another steady solution, taking the overall effect of the perturbation into account.

In multi physics computations where a compressible fluid is coupled with a linearly elastic solid, it is standard to enforce continuity of the normal velocities and of the normal stresses at the interface between the fluid and the solid. In a numerical scheme, there are many ways that velocity- and stress-continuity can be enforced in the discrete approximation. This paper performs a normal mode stability analysis of the linearized problem to investigate the stability of different numerical interface conditions for a model problem approximated by upwind type finite difference schemes. The analysis shows that depending on the ratio of densities between the solid and the fluid, some numerical interface conditions are stable up to the maximal CFL-limit, while other numerical interface conditions suffer from a severe reduction of the stable CFL-limit. The paper also presents a new interface condition, obtained as a simplified characteristic boundary condition, that is proved to not suffer from any reduction of the stable CFL-limit. Numerical experiments in one space dimension show that the new interface condition is stable also for computations with the non-linear Euler equations of compressible fluid flow coupled with a linearly elastic solid.

Une étude de l'écoulement dans une soupape de décharge de turbocompresseur de suralimentation, est présentée dans ce travail. Une modélisation numérique tridimensionnelle de l'écoulement turbulent à travers la soupape a été entreprise en s'appuyant sur la résolution des équations de Navier-Stokes et d'énergie couplées à un modèle de turbulence de type k−ε. Cette modélisation a permis de fournir une connaissance fine des structures de l'écoulement à travers la soupape, et de prédire ses courbes caractéristiques de fonctionnement. À partir des résultats locaux de l'écoulement, le coefficient de débit de la soupape a été déterminé en fonction de sa levée. De même, à partir des données géométriques de la soupape, une modélisation analytique a été établie, en vue de donner le débit en fonction du taux de détente. Pour valider aussi bien le calcul par voie de simulation numérique que celui fourni par l'approche analytique proposée, nous avons utilisé un banc d'essai équipé d'un turbocompresseur à soupape de décharge pour automobile. La comparaison des courbes caractéristiques de fonctionnement obtenues à partir de notre travail avec celles des travaux expérimentaux antérieurs, a montré une bonne concordance.