Modelling turbulent flow of superfluid $^4$He past a rough solid wall in the $T = 0$ limit

We present a numerical study, using the vortex filament model, of vortex tangles in a flow of pure superfluid $^4$He in the $T = 0$ limit through a channel of width $D = 1$ mm for various applied velocities $V$. The flat channel walls are assumed to be microscopically rough such that vortices terminating at the walls are permanently pinned; vortices are liberated from their pinned ends exclusively through self-reconnection with their images. Sustained tangles were observed, for a period of 80 s, above the critical velocity $V_c \sim 0.20$ cm s$^{-1} = 20 \kappa/D$. The coarse-grained velocity profile was akin to a classical parabolic profile of the laminar Poiseuille flow, albeit with a non-zero slip velocity $\sim$ 0.20 cm s$^{-1}$ at the walls. The friction force was found to be proportional to the applied velocity. The effective kinematic viscosity was $\sim 0.1\kappa$, and effective Reynolds numbers within $\mathrm{Re'} < 15$. The fraction of the polarized vortex length varied between zero in the middle of the channel and $\sim$ 60% within the shear flow regions $\sim D/4$ from the walls. Therefore, we studied a state of polarized ultraquantum (Vinen) turbulence fuelled at short lengthscales by vortex reconnections, including those with vortex images due to the relative motion between the vortex tangle and the pinning rough surface.