Numerical Investigation of Thermal Counterflow of He II Past Cylinders

We investigate numerically, for the first time, the thermal counterflow of superfluid helium past a cylinder by solving with a finite volume method the complete so-called two-fluid model. In agreement with existing experimental results, we obtain symmetrical eddies both up- and downstream of the obstacle. The generation of these eddies is a complex transient phenomenon that involves the friction of the normal fluid component with the solid walls and the mutual friction between the superfluid and normal components. Implications for flow in a more realistic porous medium are also investigated.

Figure 1

Variation of the dimensionless numbers with heat flux and bath temperature, (a) densities ratio $a$, (b) Reynolds number $\Re$, (c) mutual friction over the advection of the normal component $\gamma$, and (d) mutual friction over the advection of the superfluid component $a\gamma$.

Figure 2

Streamlines of the normal fluid component past a cylinder at early time steps.

Figure 3

Streamlines of the normal fluid component. (a) $T_b=2.0\mathrm{K}$, $\dot{q}=25\mathrm{kW}/{\mathrm{m}}^2$, $\gamma =620$, $a=0.68$; (b) $T_b=1.8\mathrm{K}$, $\dot{q}=25\mathrm{kW}/{\mathrm{m}}^2$, $\gamma =300$, $a=1.88$; (c) $T_b=1.9\mathrm{K}$, $\dot{q}=37\mathrm{kW}/{\mathrm{m}}^2$, $\gamma =633$, $a=0.97$; (d) $T_b=1.6\mathrm{K}$, $\dot{q}=25\mathrm{kW}/{\mathrm{m}}^2$, $\gamma =288$, $a=3.1$.

Figure 4

Velocity vectors and streamlines for the normal component. (a) At low heat flux the flow pattern is similar to laminar flow in classical hydrodynamics. (b) For high heat flux, large vortices appear in the pore space.