Numerical simulations of thermal counterflow in the presence of solid boundaries Andrew Baggaley Jason Laurie Weizmann Institute Sylvain Laizet Imperial College London

Overview ① Introduction/motivation ② Numerical method ③ Profile of vortex line density as a function of temperature and counterflow velocity ④ Modeling the effect of turbulence in the normal fluid

The system Three regimes of turbulence TI – low flow rate (heat flux), normal fluid assumed to be laminar TII – larger flow rates, normal fluid turbulence? Very large aspect ratio channels – only one regime TIII Vinen model : Tough, 1982 Babuin et al. 2012

Motivations Number of interesting results from computational models in 2π periodic domain. However this gives us no information about the structure of the tangle in experimental systems. Early simulations from Aarts and de Waele (Physica B, 1994) suggested the vortex line density peaks near the boundary. We shall attempt to answer a few simple questions 1.What is the dependence on the structure of the tangle on T or v ns ? 2.Is the structure of the tangle and the dynamics independent of the channel width. 3.What happens if the normal fluid is turbulent?

Numerical approach Vortex filament method Biot-Savart Integral Model reconnections algorithmically ‘cut and paste’ Exact contribution Average contribution Barnes & Hut, Nature, 196

Split the velocity of the filament of the i th vortex point into a local contribution (near the singularity) and non-local contributions,

Mutual friction Normal viscous fluid coupled to inviscid superfluid via mutual friction. Superfluid component extracts energy from normal fluid component via Donelly- Glaberson instability, amplification of Kelvin waves. Kelvin wave grows with amplitude Counterflow Turbulence

Donnelly-Glaberson Instabillity

System configuration Periodic boundaries: Solid (smooth boundaries): Moving to a more ‘realistic’ setup

Parameter space Work at 3 temperatures Vary counterflow velocity to have approximately constant range of vortex line densities. Later (if there’s time!) we will vary the size of the wall normal direction:

Babuin et al. 2012

Course graining Course-grained fields averaged temporally and spatially over the two periodic directions, leaving quantities purely as a function of position normal to the boundaries:

Profile seems to be approximately independent of the counterflow velocity, only temperature dependent. Vortex density profiles Take additional averaging over v ns As pointed out by Tough (1982) Schwarz’s homogeneous theory is worse in TI state (compared to TII) and so the TI state may not be homogeneous or isotropic.

Understanding the temperature dependence

The rate of change of the vortex line length in our system can be expressed as After a little manipulation this can be shown equivalent to Hence we define a local stretching rate as Where is vortex density generated?

What does this tell us about the tangle?

From this result we can surmise that in counterflow turbulence the quantized vortices are very organised in the centre of the channel and expand due to the mutual friction force. However within the boundary layer region close to the solid boundaries the tangle is much more random, and so the effect of mutual friction is to contract the vortices, as one would observe for a single vortex ring propagating within a stationary normal fluid. Let’s revisit the structure of the vortex density again given this insight.

In the centre of the channel quantized vortices are slave to the mutual friction and are pushed out to the boundaries of channel. They arrive at the solid boundary and reconnect creating a random tangle on the boundary. More vortices are pushed to boundary which reconnect with vortices at the walls creating a turbulent boundary layer where vortex density peaks in our simulations. The structure of the tangle is very different in centre of channel when compared to that at the boundaries, these boundary layers being areas of approximately homogeneous isotropic turbulence. Close to the boundaries, we have an almost random tangle that will diffuse out a distance with the rate given by Can we understand this powerlaw?

Energy from normal fluid flow is transferred to quantized vortices via the Donnelly-Glaberson instability, where Kelvin waves of a given wave number are destabilized with a growth rate, If we seek the balance of these two competing effects then this would provide a natural length scale for the width of the boundary layer: Plugging in the numbers at T=1.6K we predict and we observe the peak in the density at 0.01cm

Validity of assumption of laminar normal fluid Parabolic profile assumed for normal fluid, some experimental evidence of such behaviour at low flow velocities. We compare magnitude of viscous term to mutual friction term in the equation for v n. Vinen, JLTP, 2014

What about turbulence in the normal fluid? We ‘cheat’, use a frozen snapshot from a DNS simulation

TI TII

Jump in vortex line density supported for given heat flux

Corresponding increase in γ

In the laminar (TI) regime the vortex density is strongly peaked at the boundaries with a temperature dependent profile. Supporting evidence that TI-TII transition is associated with transition to turbulence in normal fluid. Homogenisation of the tangle appears to be associated with normal fluid turbulence. More work needs to be done on the theoretical aspects of the transition. Conclusions