Hyperbolic method to explore multiplicity flow solutions in a four-sided lid-driven cavity

In this study, the hyperbolic method is adopted to explore the flow field states of incompressible flow in a four-sided lid-driven square cavity. In particular, we focus on the flow bifurcation obtained at the critical Reynolds number $R_e \simeq 130$. In the hyperbolic method, the diffusive term is transformed into an hyperbolic one by introducing a diffusion flux term, which is the solution of an additional equation. A classical Riemann-like solver with a finite-volume discretization is thus employed for the full flux (splitted into advective and diffusive parts), in order to solve the steady-state incompressible Navier-Stokes equation. The incompressibility of the flow is treated via the artificial pseudo-compressibility method. It is shown that our numerical code is able to detect the bifurcation, by the analysis of the residual term relaxation during the pseudo-time iteration procedure. Moreover, depending on the combination choice of slope limiters for the two spatial directions, our method is able to select the first or the second stable solution among the double flow field state obtained when the Reynolds number is higher than the critical value that is estimated to be $129.4$ in our study.