Hall Vinen Bekharevich Khalatnikov Research Articles

Higher-order statistics and intermittency of a two-fluid Hall–Vinen–Bekharevich–Khalatnikov quantum turbulent flow

The Hall–Vinen–Bekharevich–Khalatnikov (HVBK) model is widely used to numerically study quantum turbulence in superfluid helium. Based on the two-fluid model of Tisza and Landau, the HVBK model describes the normal (viscous) and superfluid (inviscid) components of the flow using two Navier–Stokes type of equations, coupled through a mutual friction force term. We derive transport equations for the third-order moments for each component of velocity involving the fourth-order moments, which are classical probes for internal intermittency at any scale, and revealing the probability of rare and strong fluctuations. Budget equations are assessed through direct numerical simulations of the HVBK flow. We simulate a forced homogeneous isotropic turbulent flow with Reynolds number of the normal fluid (based on Taylor's microscale) close to 100. Values from 0.1 to 10 are considered for the ratio between the normal and superfluid densities. For these flows, an inertial range is not discernible and the restricted scaling range approach is used to take into account the finite Reynolds number (FRN) effect. We analyse the importance of each term in budget equations and emphasize their role in energy exchange between normal and superfluid components. Some interesting features are observed: (i) transport and pressure-related terms are dominant, similarly to single-fluid turbulence; and (ii) the mathematical signature of the FRN effect is weak despite the low value of the Reynolds number. The flatness of the velocity derivatives is finally studied through the transport equations and their limit for very small scales, and it is shown to gradually increase for lower and lower temperatures, for both normal fluid and superfluid. This similarity highlights the strong locking of the two fluids. The flatness factors are also found in reasonable agreement with classical turbulence.

Inertial particles in superfluid turbulence: Coflow and counterflow

We use pseudospectral direct numerical simulations to solve the three-dimensional (3D) Hall–Vinen–Bekharevich–Khalatnikov (HVBK) model of superfluid helium. We then explore the statistical properties of inertial particles, in both coflow and counterflow superfluid turbulence (ST) in the 3D HVBK system; particle motion is governed by a generalization of the Maxey–Riley–Gatignol equations. We first characterize the anisotropy of counterflow ST by showing that there exist large vortical columns. The light particles show confined motion as they are attracted toward these columns, and they form large clusters; by contrast, heavy particles are expelled from these vortical regions. We characterize the statistics of such inertial particles in 3D HVBK ST: (1) The mean angle Θ(τ) between particle positions, separated by the time lag τ, exhibits two different scaling regions in (a) dissipation and (b) inertial ranges, for different values of the parameters in our model; in particular, the value of Θ(τ), at large τ, depends on the magnitude of Uns. (2) The irreversibility of 3D HVBK turbulence is quantified by computing the statistics of energy increments for inertial particles. (3) The probability distribution function (PDF) of energy increments is of direct relevance to recent experimental studies of irreversibility in superfluid turbulence; we find, in agreement with these experiments, that, for counterflow ST, the skewness of this PDF is less pronounced than its counterparts for coflow ST or for classical fluid turbulence.

Global Regularity of the 2D HVBK Equations

The Hall-Vinen-Bekharevich-Khalatnikov (HVBK) equations are a macroscopic model of superfluidity at non-zero temperatures. For smooth, compactly supported data, we prove the global well-posedness of strong solutions to these equations in $\mathbb{R}^2$, in the incompressible and isothermal case. The proof utilises a contraction mapping argument to establish local well-posedness for high-regularity data, following which we demonstrate global regularity using an analogue of the Beale-Kato-Majda criterion in this context. In the appendix, we address the sufficient conditions on a 2D vorticity field, in order to have a finite kinetic energy.

Numerical Study on Entrance Length in Thermal Counterflow of Superfluid $$^4$$He

Three-dimensional numerical simulations in a square duct were conducted to investigate entrance lengths of normal fluid and superfluid flows in a thermal counterflow of superfluid $^4$He. The two fluids were coarse-grained by using the Hall-Vinen-Bekharevich-Khalatnikov (HVBK) model and were coupled through mutual friction. We solved the HVBK equations by parameterizing the coefficient of the mutual friction to consider the vortex line density. A uniform mutual friction parameter was assumed in the streamwise direction. Our simulation showed that the entrance length of the normal fluid from a hot end becomes shorter than that of a single normal fluid due to the mutual friction with the parabolically developed superfluid flow near the hot end. As the mutual friction increases, the entrance length decreases. Same as that, the entrance length of the superfluid from a cold end is affected by the strength of the mutual friction due to the parabolically developed normal fluid flow near the cold end. Aside from the entrance effect, the realized condition of a tail-flattened flow is discussed by parameterizing the superfluid turbulent eddy viscosity and the mutual friction.

Multiscaling in superfluid turbulence: A shell-model study.

We examine the multiscaling behavior of the normal- and superfluid-velocity structure functions in three-dimensional superfluid turbulence by using a shell model for the three-dimensional (3D) Hall-Vinen-Bekharevich-Khalatnikov (HVBK) equations. Our 3D-HVBK shell model is based on the Gledzer-Okhitani-Yamada shell model. We examine the dependence of the multiscaling exponents on the normal-fluid fraction and the mutual-friction coefficients. Our extensive study of the 3D-HVBK shell model shows that the multiscaling behavior of the velocity structure functions in superfluid turbulence is more complicated than it is in fluid turbulence.

Implications of Holm's Finite Temperature Corrections to the HVBK Equations

The macroscopic equations used to model helium II are the Hall–Vinen–Bekharevich–Khalatnikov (HVBK) equations. Recently Holm suggested a finite-temperature adjustment to these equations, which effects the mutual friction parameters. In this paper we investigate the effect of this adjustment on the linear stability of Couette flow. Using the original HVBK equations, good agreement between the predicted and observed Reynolds number at which Couette flow becomes unstable has been found, particularly at temperatures close to the lambda temperature. Performing the same test on Holm's finite temperature corrections, we find no such agreement. The stability curves all predict that long-wavelength axial disturbances are unstable. We conclude that with the current values of the mutual friction parameters, Holm's finite-temperature corrections do not constitute a good model of helium II. We also discuss the possibility that, since the method of experimentally determining the mutual friction parameters depends on the form of the mutual friction, our interpretation of these parameters needs to be amended.

End effects in rotating helium II

We numerically solve the two-fluid Hall–Vinen–Bekharevich–Khalatnikov equations to compute the steady flow of helium II confined inside a unit aspect ratio Couette annulus. The ends and outer wall of the cylinder are held fixed, while the inner cylinder is rotated. This simple flow configuration allows us to study how the vortex lines respond to a shear in the presence of boundaries which are both parallel and perpendicular to the natural axial direction of the vortex lines. The main result of our investigation is the anomalous motion of helium II when compared to the motion of a classical fluid. We also find that the normal fluid and the superfluid components of helium II can move in ways which are very different to each other, due to the tension in the vortex lines.

Evolution of coupled Gaussian vortex systems in helium II

We have investigated the evolution of the normal and superfluid flow in helium II using the two-fluid Hall–Vinen–Bekharevich–Khalatnikov equations. We found that closely matched Gaussian vorticity distributions form in both the superfluid and the normal fluid and decay together with a viscous time scale that is set by the total fluid density and not by the density of the normal fluid alone. This calculation suggests that the normal fluid and superfluid can have velocity fields which are strongly coupled together so that the two fluids behave as a single fluid.

The anomalous motion of superfluid helium in a rotating cavity

We numerically solve the nonlinear two-fluid Hall–Vinen–Bekharevich–Khalatnikov (HVBK) equations for superfluid helium confined inside a short Couette annulus. The outer cylinder and the ends of the annulus are held fixed whilst the inner cylinder is rotated. This simple flow configuration allows us to study how the vortex lines respond to a shear in the presence of boundaries. It also allows us to investigate further the boundary conditions associated with the HVBK model. The main result of our investigation is the anomalous motion of helium II when compared to a classical fluid. The superfluid Ekman cells always rotate in the opposite sense to a classical Navier–Stokes fluid due to the mutual friction between the two fluids, whilst the sense of rotation of the normal fluid Ekman cells depends on the parameter range considered. We also find that the tension of the vortex lines forces the superfluid to rotate about the inner cylinder almost like a rigid column.

SUPERFLUID HELIUM BETWEEN ROTATING CYLINDERS

We numerically solve the nonlinear two-fluid Hall–Vinen–Bekharevich–Khalatnikov equations for superfluid helium II confined inside a small aspect ratio Couette annulus. This simple flow configuration allows us to study how the vortex lines respond to a shear in the presence of boundaries. We find that the tension of the vortex lines forces the superfluid to move in a way which is very different from the motion of a classical fluid. We have considered rotation of the inner cylinder only and find that there is a higher density of vortex lines close to the inner cylinder than at the stationary outer cylinder.

Nonlinear Taylor–Couette flow of helium II

We solve the nonlinear two-fluid Hall–Vinen–Bekharevich–Khalatnikov equations of motion of helium II for the first time and investigate the configuration of quantized vortex lines in Taylor–Couette flow. The results are interpreted in terms of quantities which can be observed by measuring the attenuation of second sound. Comparison is made with existing experimental results.