Disproportionate entrance length in superfluid flows and the puzzle of counterflow instabilities

Abstract

Systematic simulations of the two-fluid model of superfluid helium (He-II) encompassing the Hall-Vinen-Bekharevich-Khalatnikov (HVBK) mutual coupling have been performed in two-dimensional pipe counterflows between 1.3 and 1.96 K. The numerical scheme relies on the lattice Boltzmann method. A Boussinesq-like hypothesis is introduced to omit temperature variations along the pipe. In return, the thermomechanical forcings of the normal and superfuid components are fueled by a pressure term related to their mass-density variations under an approximation of weak compressibility. This modeling framework reproduces the essential features of a thermally driven counterflow. A generalized definition of the entrance length is introduced to suitably compare entry effects (of different nature) at opposite ends of the pipe. This definition is related to the excess of pressure loss with respect to the developed Poiseuille-flow solution. At the heated end of the pipe, it is found that the entrance length for the normal fluid follows a classical law and increases linearly with the Reynolds number. At the cooled end, the entrance length for the superfluid is enhanced as compared to the normal fluid by up to one order of magnitude. At this end, the normal fluid flows into the cooling bath of He-II and produces large-scale superfluid vortical motions in the bath that partly re-enter the pipe along its sidewalls before being damped by mutual friction. In the superfluid entry region, the resulting frictional coupling in the superfluid boundary layer distorts the velocity profiles toward tail flattening for the normal fluid and tail raising for the superfluid. Eventually, a simple analytical model of entry effects allows us to re-examine the long-debated thresholds of T 1 and T 2 instabilities in superfluid counterflows. Inconsistencies in the T 1 thresholds reported since the 1960s disappear if an aspect-ratio criterion based on our modeling is used to discard data sets with the strongest entry effects. Furthermore, it is observed that entry effects can spuriously reproduce the signature of a T 2 transition with a normal flow remaining laminar.

Ubiquity of particle–vortex interactions in turbulent counterflow of superfluid helium

“…Additionally, in the latter case, the investigated flow region included the channel walls, where quantized vortices tend to preferentially concentrate, as discussed, for example, by Baggaley & Laizet (2013) and La Mantia (2017), while the present results are obtained away from the walls, in the channel bulk region, but also at a distance from the heat source smaller than in the case reported by Mastracci & Guo (2018) – see, e.g. Bertolaccini, Lévêque & Roche (2017) and Švančara et al. (2018 b ) for discussions on the role of the entrance length in thermal counterflow.…”

Coupled dynamics of quantized vortices and normal fluid in superfluid He4 based on the lattice Boltzmann method

“…We developed a computational method for two-fluid coupled dynamics of superfluid 4 He at finite temperature and examined its performance by comparing it with numerical simulations performed with different numerical approaches [20,40]. The proposed computational approach for solving the normal-fluid flow is based on the LBM, which is widely adopted in CFD, yet rarely applied in the study of superfluid 4 He except for the recent study by Bertolaccini et al [17]. In their study, they adopted a two-dimensional LBM simulation for both normal fluids and superfluids.…”

“…The first numerical simulation of a flow of superfluid 4 He based on the LBM was performed by Bertolaccini et al [17]. The LBM considers the flow as a convection of particles with certain discretized momenta.…”

“…Although such preceding computational results have enabled us to understand the steady quantum turbulence quantitatively, in the majority of the early approaches, quantized vortices were considered as completely independent objects, i.e., the normal fluid velocity profile was prescribed and the vortices were influenced by the normal fluid through mutual friction, whereas the vortices did not influence the dynamics of the normal fluid. Several recent studies have addressed this issue by solving the Hall-Vinen-Bekarevich-Khalatnikov (HVBK) equations for normal fluid component [16][17][18][19][20][21][22]. The HVBK equations are essentially the Navier-Stokes (NS) equations with a forcing term ascribed to the mutual friction between the normal and superfluid components due to the motions of the quantized vortices [6].…”

“…In Ref. [17], Bertolaccini et al applied the LBM to both fluids (super and normal) and investigated the coupled dynamics under a thermal counterflow in a two-dimensional channel. Unlike their approach, we apply a three-dimensional LBM for only the normal-fluid component; we follow the dynamics of a superfluid based on the VFM such that the local superfluid flow information is not lost by the averaging processes.…”