Steady inertial flow of a compressible fluid in a spatially periodic channel under large pressure drops: A multiscale semi-analytical approach

Spatially-periodic micro-channels are increasingly attracting attention as an efficient alternative to packed columns for a number of analytical and engineering processes. In incompressible flows, the velocity field swiftly attains a periodic structure, thereby allowing to predict the flow structure by solving the Navier-Stokes equations in the single periodic unit of the structure, however large the pressure gradient. In contrast, gas flows never settle onto a spatially-periodic behavior because of the density dependence on pressure. As a consequence, the Navier-Stokes equations must be solved numerically on the entire channel, hence requiring massive computational effort. Based on the marked separation of scales between the length of the periodic cell and that of the channel characterizing many applications, we develop a general method for predicting the large-scale pressure/density/velocity profiles, and the small-scale flow structure. The approach proposed is inspired by Landau's approximation to the isothermal (Poiseuille) flow of an ideal gas through a straight open capillary, where the axial dependence of the gas velocity is obtained by assuming locally incompressible flow. This approach is here extended to periodic geometries by assuming that the local dimensionless pressure drop across the cell as a function of the Reynolds number, say g(Re), can be estimated from the solution of the incompressible NS equations within the minimal periodic cell of the channel. From the knowledge of g(Re), the axial profiles of pressure and density are derived analytically by quadratures. Results are validated by comparison with the full-scale solution of Navier-Stokes equations.