Superfluid Turbulence Mystery Solved, Revealing Upstream Eddies for the First Time

Researchers investigating the complex behaviour of superfluid helium have mapped the complete range of wake states formed when the fluid flows past a cylinder. Yingxuan Hu, Wenling Huang, and Shihao Yang, all from the Institute of Refrigeration and Cryogenics at Zhejiang University, alongside Limin Qiu, Wei Guo, and Shiran Bao et al., demonstrate numerically how these states, ranging from zero to six vortices, arise from a self-organised zone of dissipation near the cylinder. This work clarifies the long-standing mystery surrounding the formation of quasi-steady eddies, even upstream of the obstacle, and importantly reveals analogous behaviour within the superfluid component itself. By systematically varying key parameters, the team constructed a unified predictive map relating the normal-fluid Reynolds number and a dimensionless interaction number, establishing mutual-friction feedback as a fundamental mechanism driving unusual wake structures in fluids.

Upstream vortex formation in superfluid helium flow around a cylinder

Researchers have uncovered a surprising phenomenon in the flow of superfluid helium, revealing that this quantum fluid forms anomalous upstream eddies, vortices that move against the primary flow, when passing around a cylinder. This behaviour sharply contrasts with classical fluids where eddies exclusively form downstream.
The work, detailed in a recent publication, demonstrates a comprehensive numerical model accurately captures the full spectrum of wake states observed in experiments, including configurations with zero, two, four, and six vortices. Simulations further reveal that the superfluid component itself develops these unusual upstream eddies, a previously unobserved feature.

This breakthrough stems from a self-organized zone of enhanced dissipation near the cylinder’s shoulders, reshaping the effective obstacle and driving eddy formation in both the normal and superfluid components. This dissipation suppresses inherent oscillations within the normal fluid, leading to the stable, multistable wake topologies.
By performing systematic parameter scans, the team constructed a unified diagram using the normal-fluid Reynolds number and a dimensionless interaction number, effectively separating regimes governed by inertia and mutual friction. The resulting map not only predicts the observed wake structures but also establishes mutual-friction feedback as a key mechanism driving these unusual fluid dynamics.

This research transforms a striking experimental observation into a predictive framework, offering insights into the behaviour of quantum fluids under thermal counterflow. The study employed a two-fluid model coupled with Vinen’s vortex-line-density equation to simulate the complex interplay between the superfluid and normal fluid components of helium-4 below 2.17 K. The computational domain, a two-dimensional channel containing a circular cylinder, was used to model the flow and observe the formation of these unique wake patterns.

Numerical implementation of a two-fluid model with vortex-line density dynamics

A coupled two-fluid model, incorporating Vinen’s vortex-line-density equation, underpinned the numerical simulations of superfluid helium-4 counterflow past a cylinder. This approach simultaneously solved the Landau, Tisza equations for both normal and superfluid velocity fields, alongside the evolution of the vortex-line density, denoted as L(r, t).

The Vinen equation governed the transport, growth, and decay of the vortex tangle, responding to the thermal counterflow driving force. Advection of vortices was accounted for by the term ∇· (vLL), utilising the local superfluid velocity, vs, as the vortex mean velocity, a standard practice in two-fluid/Vinen implementations.

Simulations employed a self-consistent evaluation of the mutual-friction force, derived from the locally determined vortex-line density. This ensured accurate representation of the interaction between the normal and superfluid components, crucial for capturing the observed wake topologies. The computational domain was discretised using a grid, with detailed governing equations, boundary conditions, and grid independence tests documented in the Supplemental Material.

This rigorous validation process confirmed the reliability and accuracy of the numerical scheme. Systematic parameter scans were then performed, varying the normal-fluid Reynolds number and a dimensionless interaction number to map the resulting wake structures. These scans allowed for the construction of a unified diagram delineating the parameter windows responsible for the observed 0-, 2-, 4-, and 6-vortex states.

The resulting map transforms a striking phenomenology into a predictive tool, establishing mutual-friction feedback as a key mechanism driving unusual wake structures in fluids. The work further revealed anomalous upstream eddies within the superfluid component, a previously unreported phenomenon linked to a self-organised zone of enhanced mutual-friction dissipation near the cylinder shoulders.

Mutual-friction governs wake topology and upstream eddy formation in superfluid helium counterflow

Simulations of thermal counterflow of superfluid helium past a cylinder reveal a full spectrum of normal-fluid wake states, encompassing 0-, 2-, 4-, and 6-vortex configurations. The research demonstrates that the superfluid component also develops anomalous upstream eddies, a previously unobserved phenomenon.

These behaviors are linked to a self-organized zone of enhanced mutual-friction dissipation near the cylinder shoulders, which effectively reshapes the obstacle and drives upstream eddies in both fluid components. This dissipation also suppresses intrinsic wake oscillations within the normal fluid. Systematic parameter scans were performed, constructing a unified diagram based on the normal-fluid Reynolds number and a dimensionless interaction number.

This diagram delineates transitions between inertia- and mutual-friction-controlled regimes and defines the parameter windows corresponding to each discrete wake topology. The study establishes mutual-friction feedback as a robust mechanism for generating unusual wake structures in quantum fluids. The computational domain employed a two-dimensional channel containing a circular cylinder with diameter D and channel width H.

The model utilizes two-fluid hydrodynamics, coupling the normal and superfluid components through a volumetric mutual-friction force, described by equations governing the evolution of velocities and densities. Specifically, the normal fluid density evolves according to a rate equation incorporating the mutual-friction force and normal fluid velocity.

The simulations accurately reproduce experimentally reconstructed streamline patterns for various temperatures, heat fluxes, and blockage ratios, including configurations at 1.94 K with heat fluxes of 50mW/cm2 and 170mW/cm2, 2.03 K at 1120mW/cm2, and 2.10 K at 167mW/cm2, with blockage ratios of 10% and 31.75%. This confirms the model’s ability to capture the complex interplay of superfluidity and thermal effects in the cylinder wake.

Mutual friction dictates wake state transitions and upstream eddy formation

Numerical simulations utilising a two-fluid model have successfully replicated the full range of wake states observed behind cylinders in counterflowing superfluid helium. These states include the previously documented 0-, 2-, 4-, and 6-vortex configurations, alongside the novel discovery of anomalous upstream eddy formation within the superfluid component itself.

The origin of these behaviours lies in a self-organised zone of increased mutual friction dissipation near the cylinder’s shoulders, effectively altering the shape of the obstacle and inducing upstream eddies in both the normal and superfluid fluids. This mechanism also suppresses inherent oscillations within the normal fluid wake.

Systematic parameter variations have enabled the construction of a unified diagram relating the normal-fluid Reynolds number and a dimensionless interaction number. This diagram delineates regions governed by inertial and mutual-friction forces, and accurately predicts the observed discrete wake topologies.

Consequently, a striking fluid dynamic phenomenon has been transformed into a predictive model, demonstrating that mutual-friction feedback provides a reliable pathway to unusual wake structures. The authors acknowledge limitations related to the numerical resolution required to accurately capture the sharp gradients in the mutual-friction force near the cylinder, necessitating a refined mesh for reliable results. Future research could explore the influence of varying cylinder shapes and flow conditions on the observed phenomena, potentially extending these findings to other systems exhibiting similar two-fluid behaviour.