Section: New Results

Boundary conditions for the Navier-Stokes equations in the transitional regime

Participants : Céline Baranger, Pietro Marco Congedo, Giorgio Martalo, Julien Mathiaud, Luc Mieussens [Corresponding member] .

In reentry flows at high altitude, the parietal fluxes along the boundary of a space vehicle are computed by solving the Boltzmann equation of the gas kinetic theory. It depends on the way particles are reflected by the solid wall. In this kind of applications, the reflection is usually supposed to be 80% diffuse (the particle are re-emitted in a random direction in thermal equilibrium with the wall), and 20% specular.

In lower altitude, it is possible to use the compressible Navier-Stokes equations, but the standard boundary conditions do not take into account the specular part. These equations are hence not very accurate in the transition regime (in which the Knudsen number is around 0.01).

By using an asymptotic boundary layer analysis, we have derived boundary conditions for the compressible Navier-Stokes equations that formaly make the fluid model a second order approximation of on the Boltzmann equation of the kinetic gas theory (with respect to the Knudsen number), and that can take into account the effect of the specular reflections. These boundary conditions include a slip velocity at the wall and a temperature jump, with some coefficients that depend on some auxiliary half-space problems. An existing numerical method has been extended to solve these problems and give numerical values for these coefficients. This make our boundary conditions practically usable into any Navier-Stokes code.