Permeability models affecting nonlinear stability in the asymptotic suction boundary layer: the Forchheimer versus the Darcy model

Abstract

The asymptotic suction boundary layer (ASBL) is used for studying two permeability models, namely the Darcy and the Forchheimer model, the latter being more physically correct according to the literature. The term that defines the two apart is a function of the non-Darcian wall permeability ${\hat{K}}_{2}$ and of the wall suction ${\hat{V}}_{0}$, whereas the Darcian wall permeability ${\hat{K}}_{1}$ is common to the two models. The underlying interest of the study lies in the field of transition to turbulence where focus is put on two-dimensional nonlinear traveling waves (TWs) and their three-dimensional linear stability. Following a previous study by Wedin et al (2015 Phys. Rev. E 92 013022), where only the Darcy model was considered, the present work aims at comparing the two models, assessing where in the parameter space they cease to produce the same results. For low values of ${\hat{K}}_{1}$ both models produce almost identical TW solutions. However, when both increasing the suction ${\hat{V}}_{0}$ to sufficiently high amplitudes (i.e. lowering the Reynolds number Re, based on the displacement thickness) and using large values of the wall porosity, differences are observed. In terms of the non-dimensional Darcian wall permeability parameter, a, strong differences in the overall shape of the bifurcation curves are observed for $a\gtrsim 0.70$, with the emergence of a new family of solutions at Re lower than 100. For these large values of a, a Forchheimer number ${{Fo}}_{\max }\gtrsim 0.5$ is found, where Fo expresses the ratio between the kinetic and viscous forces acting on the porous wall. Moreover, the minimum Reynolds number, ${{Re}}_{g}$, for which the Navier–Stokes equations allow for nonlinear solutions, decreases for increasing values of a. Fixing the streamwise wavenumber to α = 0.154, as used in the study by Wedin et al referenced above, we find that ${{Re}}_{g}$ is lowered from Re ≈ 3000 for zero permeability, to below 50 for a = 0.80 for both permeability models. Finally, the stability of the TW solutions is assessed using a three-dimensional linearized direct numerical simulation (DNS). Low-frequency unstable modes are found for both permeability models; however, the Darcy model is found to overpredict the growth rate, and underpredict the streamwise extension of the most unstable mode. These results indicate that a careful choice of the underlying permeability model is crucial for accurately studying the transition to turbulence of boundary-layer flows over porous walls.