TKE Model Involving the Distance to the Wall—Part 1: The Relaxed Case

We are considering a steady-state turbulent Reynolds-Averaged Navier–Stokes (RANS) one-equation model, that couples the equation for the velocity-pressure mean field with the equation for the turbulent kinetic energy. Eddy viscosities vanish at the boundary, characterized by terms like \(d(x, \Gamma )^\eta \) $d{(x,\mathrm{\Gamma })}^{\eta }$ and \(d(x, \Gamma )^\beta \) $d{(x,\mathrm{\Gamma })}^{\beta }$ , where \(0< \eta , \beta < 1\) $0<\eta ,\beta <1$ . We determine critical values \(\eta _c\) ${\eta }_c$ and \(\beta _c\) ${\beta }_c$ for which the system has a weak solution. This solution is obtained as the limit of viscous regularizations for \(0< \eta < \eta _c\) $0<\eta <{\eta }_c$ and \(0< \beta < \beta _c\) $0<\beta <{\beta }_c$ .