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Higher-order statistics and intermittency of a two-fluid HVBK 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 equations, coupled through a mutual friction force term. This feature makes the HVBK model very appealing in applying statistical tools used in classical turbulence to study properties of quantum turbulence. A large body of literature used low-order statistics (spectra, or second-order structure functions in real space) to unravel exchanges between the two fluids at several levels. The novelty in this study is to use a theoretical approach based on first principles to derive transport equations for the third-order moments for each component of velocity. New equations involve the fourth-order moments, which are classical probes for internal intermittency at any scale, revealing the probability of rare and strong fluctuations. Budget equations are assessed through Direct Numerical Simulations (DNS) of the HVBK flow based on accurate pseudo-spectral methods. 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 (RSR) 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; ii) the mathematical signature of the FRN effect is weak in the transport of the third-order moment, despite the low value of the Reynolds number; iii) for the normal fluid at very low temperatures, the mutual friction annihilates the effects of viscosity within the RSR. 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 the normal fluid and the superfluid. This similarity highlights the strong locking of the two fluids. The flatness factors are also found in reasonable agreement with classical turbulence.
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hal-04025306 , version 1 (12-03-2023)


  • HAL Id : hal-04025306 , version 1


Zhentong Zhang, Ionut Danaila, Emmanuel Lévêque, Luminita Danaila. Higher-order statistics and intermittency of a two-fluid HVBK quantum turbulent flow. 2023. ⟨hal-04025306⟩
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