Onset of turbulence in channel flows with scale-invariant roughness

Abstract

Using 3D direct numerical simulations of the Navier--Stokes equations, we study the effect of a self-affine wall roughness on the onset of turbulence in channel flow. We quantify the dependence of the turbulent intensity (proportional to the mean-squared velocity fluctuations) on the Reynolds number $\mathrm{Re}$ for different roughness amplitudes $A$. We find that for sufficiently high amplitudes, $A>{A}_{b}$, the transition changes its nature from being subcritical (as is known at $A=0$) to supercritical, i.e., the boundary roughness renders the flow unstable for $\mathrm{Re}>{\mathrm{Re}}_{l}$, where the critical ${\mathrm{Re}}_{l}$ decays nontrivially with increasing $A$. The dependence of the friction factor on $\mathrm{Re}$ is found to follow a generalized Forchheimer law, which interpolates between the laminar and inertial asymptotes. The transition between these two asymptotes occurs at a second critical ${\mathrm{Re}}_{\text{c}}$ which is comparable in magnitude to ${\mathrm{Re}}_{l}$. This implies that transitional flow is an integral part of flow in open fractures when $\mathrm{Re}$ is sufficiently high, and should be accounted for in effective modeling approaches.