K-ϵ-L model in turbulent superfluid helium

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 v, q and L (respectively denoted as K, Kq and KL), and on their respective dissipation functions (related to the second moments of their gradients) ϵ, ϵq and ϵL. •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.