$K$-$ϵ$-$L$ model in turbulent superfluid helium

Highlights

• One-fluid extended model of superfluid helium.

• Quantum turbulence is defined by the field L (vortex line length per unit volume).

• The $K$-$ϵ$ model of classical turbulence is applied to superfluid helium.

• How the turbulent coefficients depend on the fluctuations of fields involved.

Abstract

We generalize the $K-ϵ$ model of classical turbulence to superfluid helium. In a classical viscous fluid the phenomenological eddy viscosity characterizing the effects of turbulence depends on the turbulent kinetic energy $K$ and the dissipation function $ϵ$, which are mainly related to the fluctuations of the velocity field and of its gradient. In superfluid helium, instead, we consider the necessary coefficients for describing the effects of classical and quantum turbulence, involving fluctuations of the velocity, the heat flux, and the vortex line density of the quantized vortex lines. By splitting the several fields into a time-average part and a fluctuating part, some expressions involving the second moments of the turbulent fluctuations appear in the evolution equations for the average quantities. As in the $K-ϵ$ model, a practical way of closing such equations is to tentatively express such fluctuating terms as a function of the average quantities. In this context we propose how the turbulent coefficients so introduced could depend on the second moments of the fluctuations of $\mathbf{v}$, $\mathbf{q}$ and $L$ (respectively denoted as $K$, $K_q$ and $K_L$), and on their respective dissipation functions (related to the second moments of their gradients) $ϵ$, ${ϵ}_q$ and ${ϵ}_L$.

Introduction

The study of turbulence is an appealing topic which has interested many researchers in different fields, from the smallest scales to the largest ones, from nanosystems to universe. The methods and approaches used for dealing with all the phenomena involved into turbulence span from hydrodynamics to thermodynamics and statistics [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13].

Here we are interested in the $K-ϵ$ model as the zero-th order closure model in the hierarchy of the moments of the fluctuations of the main fields, usually applied in a viscous classical fluid [14]. In this paper we aim at generalizing this method to superfluid helium, where an additional different kind of turbulence occurs: quantum turbulence. Quantum turbulence is typically thought as a tangle of quantized vortex lines, which are characterized by a quantized circulation $\kappa =h∕m$ ($h$ and $m$ being the Planck’s constant and the helium atom mass, respectively), a fixed core of radius of the order of the size of the helium atom, and a vortex length density per unit volume $L$.

There are still many open questions about the existence and interaction of both kinds of turbulence (classical and quantum) in superfluid helium. Indeed, superfluid helium is usually thought as composed of two indistinguishable components, the normal component (a viscous fluid which carries the entropy and the viscosity of superfluid helium) and an inviscid superfluid component. According to the two-fluid model, proposed by Tisza and Landau [15], [16], quantized vortices are caused by a vorticity of the superfluid component. Recently, in [17] it was argued that the normal component could be also turbulent, and it would explain the presence of two regimes of turbulence in counterflow experiments. An alternative explanation could be that an inhomogeneous and locally polarized vortex tangle (regime TI) becomes an homogeneous state (regime TII) for enhanced applied heat flux, because of the breakdown of these localized polarizations [18]. The understanding of the phenomena is still open, but it points to different kinds of turbulence for the normal component with speed ${\mathbf{v}}_n$, the superfluid component with speed ${\mathbf{v}}_s$, and the vortex tangle itself. The last feature has been studied from the numerical point of view, and it is seen that at very low temperature (where the normal component is practically absent), quantized vortices may be gathered into bundles of vortices, which mimic classical eddies [19], [20], [21], [22].

From the thermodynamical point of view, an alternative model of superfluid helium was proposed by means of the Extended Thermodynamics [23], [24], which takes the heat flux $\mathbf{q}$ as further independent field than the usual mean velocity $\mathbf{v}$. In this paper we consider $\mathbf{v}$ and $\mathbf{q}$ (rather than ${\mathbf{v}}_n$ and ${\mathbf{v}}_s$) together with $L$ as the main independent quantities.

By splitting the several fields into a time-average part and a fluctuating part, some expressions involving the second moments of the turbulent fluctuations in ${\mathbf{v}}^{\prime }$, ${\mathbf{q}}^{\prime }$ and $L^{\prime }$ appear in the evolution equations for the average quantities $\overline{\mathbf{v}}$, $\overline{\mathbf{q}}$ and $\overline{L}$. In the usual $K-ϵ$ model for classical fluids, the way of closing the evolution equation for $\overline{\mathbf{v}}$ is to tentatively express the second moments of the fluctuating terms of ${\mathbf{v}}^{\prime }$ as a function of gradient of $\mathbf{v}$, in such a way that the final equations have a form analogous to the usual equations for $\mathbf{v}$, but with effective kinematic viscosity which is a sum of the molecular contribution and the turbulent contribution (the so-called eddy viscosity). In its turn, this turbulent contribution is expressed in terms of some quantities related to turbulent fluctuations, as the kinetic energy of fluctuations, $K$, and the dissipation function $ϵ$. In the context of the present paper, we analyze the equations, identify the turbulent contributions, propose for them some turbulent transport coefficients, and try to express them in terms of the second moments $\mathbf{v}$, $\mathbf{q}$ and $L$, namely $K$, $K_q$ and $K_L$, as well as of their corresponding dissipation functions $ϵ$, ${ϵ}_q$ and ${ϵ}_L$, related to the second-order moments of their gradients. Since some of the details may depend on the kind of flow, we do not pretend that this model will grasp all the complexity of turbulence, but it is logical to take it into consideration as a macroscopic starting point, in analogy to the $K-ϵ$ model in classical turbulence [1], [4], [8], [9], [10], [11], [14].

The paper is organized as follows: in Section 2 we deal with the basic equations of the one-fluid extended model of superfluid helium and their time-averaged expressions; in Section 3 we consider the zero-th order approximation closure and we propose some expressions to close the equations of the $K-ϵ-L$ model for $\mathbf{v}$ and $\mathbf{q}$; in Section 4 we deal with the equation for $L$; Section 5 is devoted to the results and the concluding remarks of the paper.