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

Abstract

Spatially-periodic channels are increasingly attracting attention as an efficient alternative to packed columns for a number of analytical and engineering processes. In incompressible flows, the periodic geometry allows to compute the flow structure by solving the Navier-Stokes (NS) equations in the minimal periodic cell of the structure, however large the pressure gradient. Besides, when gas flow under large pressure drops are dealt with, the velocity field is not periodic because of the density dependence on pressure. In this case, the momentum balance equations must be solved numerically on the entire channel, thereby requiring massive computational effort. Based on the marked separation of scales between the length of the periodic cell and the overall channel length characterizing many applications we develop a general method for predicting both the large-scale pressure and velocity profiles, and the small-scale flow structure. The approach proposed is based on the assumption that the local dimensionless pressure drop as a function of the Reynolds number, say $g({\rm Re})$, can be estimated from the solution of the incompressible NS equations within the minimal periodic cell of the channel. From the knowledge of $g({\rm Re})$, the pressure profile $P(Z)$ vs the large scale axial coordinate $Z$ is derived analytically by quadratures. We show how qualitatively different profiles $P(Z)$ can be obtained depending on the equation of state the gas. The approach is validated by comparing the predicted profiles with the full-scale numerical solution of the compressible NS equations in different axially-symmetric periodic channel geometries.