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

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