Multi-scale physics of cryogenic liquid helium-4: Inverse coarse-graining properties of smoothed particle hydrodynamics

TL;DR

This paper explores the multiscale physics of cryogenic helium-4 using two-fluid models and SPH, revealing inverse scale transformations and potential subgrid-scale modeling insights for quantum vortex interactions.

Contribution

It demonstrates the connection between classical and quantum hydrodynamics via scale transformations and proposes SPH as a tool to reproduce microscopic fluctuations at macroscopic scales.

Findings

SPH can reproduce microscopic fluctuations at macroscopic scales.

Scale transformations link classical and quantum two-fluid models.

Condiff viscosity acts as a potential subgrid-scale model for quantum vortex interactions.

Abstract

Our recent numerical studies on cryogenic liquid helium-4 highlight key features of multiscale physics that can be captured using the two-fluid model. In this paper, we demonstrated that classical and quantum hydrodynamic two-fluid models are connected via scale transformations: large eddy simulation (LES) filtering links microscopic to macroscopic scales, while inverse scale transformation through SPH connects macro back to microscales. We showed that the spin angular momentum conservation term, introduced as a quantum-like correction, formally corresponds to a subgrid-scale (SGS) model derived from this transformation. Moreover, solving the classical hydrodynamic two-fluid model with SPH appears to reproduce microscopic-scale fluctuations at macroscopic scales. The amplitude of these fluctuations depends on the kernel radius. This effect may arise from truncation errors from kernel smoothing, which can qualitatively resemble such fluctuations. However, this resemblance lacks first-principle justification and should be viewed as a speculative analogy rather than a physically grounded effect. Our theoretical analysis further suggests that the Condiff viscosity model can act as an SGS model, incorporating quantum vortex interactions under point-vortex approximation into the two-fluid framework. These findings provide new insight into the microscopic structure of cryogenic helium-4 within a multiscale context. Notably, the normal fluid can be understood as a mixture of inviscid and viscous fluid particles. While molecular viscosity renders the normal fluid at microscopic scales, its small magnitude contributes little to the large-scale effective viscosity, which includes both molecular and eddy viscosities; therefore, in laminar regimes where eddy viscosity is negligible, the normal fluid may be effectively treated as inviscid at large scales if molecular viscosity is sufficiently small.