We focus in this paper on the effect of the resolution of Direct Numerical Simulations (DNS) on the spatio-temporal development of the turbulence downstream of a single square grid. The aims of this study are to validate our numerical approach by com
paring experimental and numerical one-point statistics downstream of a single square grid and then investigate how the resolution is impacting the dynamics of the flow. In particular, using the Q-R diagram, we focus on the interaction between the strain-rate and rotation tensors, the symmetric and skew-symmetric parts of the velocity gradient tensor respectively. We first show good agreement between our simulations and hot-wire experiment for one-point statistics on the centreline of the single square grid. Then, by analysing the shape of the Q-R diagram for various streamwise locations, we evaluate the ability of under-resolved DNS to capture the main features of the turbulence downstream of the single square grid.
Superfluid helium is an intimate mixture of a viscous normal fluid, with continuous vorticity, and an inviscid superfluid, where vorticity is constrained to thin, stable topological defects. One mechanism to generate turbulence in this system is thro
ugh the application of a heat flux, so called thermal counterflow. Of particular interest is how turbulence in the superfluid responds to both a laminar and turbulent normal fluid in the presence of walls. We model superfluid vortex lines as reconnecting space curves with fixed circulation, and consider both laminar (Poiseuille) and turbulent normal fluid flows in a channel configuration. Using high resolution numerical simulations we show that turbulence in the normal fluid sustains a notably higher vortex line density than a laminar flow with the same mean flow rate. We exam Vinens relation, $sqrt{L}=gamma v_{ns}$, between the steady state vortex line density $L$ and the counterflow velocity $v_{ns}$. Our results support the hypothesis that transition to turbulence in the normal fluid is responsible for the TI to TII transition. We also consider the spectral properties of fluctuations of the superfluid vortices, which show a good agreement with previous experimental results.
