Numerical investigation of the divergence-free constraint for the miscible fluids interaction in a lock-exchange arrangement

Abstract

The present work scrutinizes the implementation of the divergence-free constraint in miscible fluids flow algorithms. In general, the flow solvers handling the interaction of the miscible fluid enforce the zero divergence condition, considering the fluids to be incompressible. In these algorithms, however, a convection-diffusion equation for the mass fraction, in conjunction with the Navier-Stokes equation, is solved for incorporating the mixing effects. This equation leads to a non-zero velocity field in the numerical modeling and the derived velocity divergence is a function of Reynolds and Schmidt numbers. Therefore, in this work, a quasi-incompressible formulation is presented for such flows wherein weak compressibility effects exist, but Mach numbers are very small. This novel approach is tested extensively in the classical lock-exchange setup where the effects of Reynolds and Schmidt numbers are analyzed. The role of density ratios is also investigated to realize whether the quasi-incompressible formulation generates a different mixing behavior or is similar to the conventional incompressible approach. The results demonstrate the convincing differences in these two formulations subjected to the specific combinations of Reynolds number, Schmidt number, and density ratio. A lenient criterion is also suggested by investigating the numerical tests carried out in this work, which may predict the possibilities of generating consistent results in both formulations.