A Lagrangian–Eulerian compressible model for the trans-critical path of near-critical fluids

•A Lagrangian–Eulerian compressible model for the Navier–Stokes equations.•Propagation of sound waves in a supercritical fluid.•Modeling of phase-separation from monophasic to diphasic states near the critical point.

The main objective of the present work is to model the trans-critical path from supercritical to subcritical states near the critical point. The model is based on full compressible sets of equations. The pressure, temperature and density fields are determined in a Lagrangian form through the divergences of velocity and heat flux and advected afterwards in the Eulerian step. Three cases have been considered and are discussed: (i) an isothermal atmosphere in order to validate the model for a perfect gas, (ii) the propagation of sound waves under piston effect timescales for supercritical fluids and the comparison with experimental data from literature and (iii) the third case which constitutes the basis of the present work corresponds to a numerical simulation of the separation of phases from supercritical to subcritical states near the critical point.