A Unified Wake Topology Map for He II Counterflow Past a Cylinder

Thermal counterflow of superfluid $^4$He past a cylinder produces quasi-steady eddies not only downstream but also anomalously upstream. However, the mechanism and organizing principles behind the observed multistable wake topologies (0-, 2-, 4-, and 6-vortex states) have remained unclear. We show that the full spectrum of reported normal-fluid wake states is captured numerically with a two-fluid model coupled to Vinen’s vortex-line-density equation. Our simulations further reveal that the superfluid component can also develop anomalous upstream eddies, a feature not previously reported. We trace these behaviors to a self-organized zone of enhanced mutual-friction dissipation near the cylinder shoulders that reshapes the effective obstacle, drives upstream eddies in both components, and suppresses intrinsic wake oscillations in the normal fluid. Guided by this mechanism, we perform systematic parameter scans and construct a unified phase diagram in terms of the normal-fluid Reynolds number $Re_n$ and a dimensionless interaction number $N$, separating inertia- and mutual-friction-controlled transitions and delineating the parameter windows for the discrete wake topologies. These results turn a striking phenomenology into a predictive map and establish mutual-friction feedback as a robust route to unusual wake structures in quantum fluids.

💡 Research Summary

In this paper the authors investigate the wake topology of superfluid helium‑II (⁴He) under thermal counterflow past a circular cylinder. Using a two‑fluid continuum model (Landau‑Tisza) coupled with Vinen’s vortex‑line‑density equation, they perform two‑dimensional numerical simulations in a channel of length Lc and height H containing a cylinder of diameter D. The normal fluid velocity vₙ and superfluid velocity vₛ are solved together with the vortex‑line density L, which evolves according to a balance of production, decay, and a term proportional to the relative velocity vₙₛ = vₙ − vₛ. Mutual friction is modeled as Fₙₛ = ρₙρₛ B_L(T) L vₙₛ, where B_L(T) is the temperature‑dependent mutual‑friction coefficient. At the inlet the normal‑fluid speed is set by the imposed heat flux q (|vₙ,0| = q/(ρₛT)), while the superfluid speed is chosen to enforce zero net mass flux (vₛ,0 = −(ρₙ/ρₛ)vₙ,0). No‑slip is imposed for the normal fluid on solid boundaries, and the superfluid is taken to be impermeable (vₛ·n̂ = 0).

The simulations reproduce the full spectrum of experimentally observed normal‑fluid wake states: vortex‑free (0‑vortex), a steady downstream pair (2‑vortex), an anomalous four‑vortex configuration with an upstream pair, and, for small block‑ratio B = D/H and sufficiently large heat flux, a symmetric six‑vortex pattern. Remarkably, the superfluid component also develops upstream eddies, a feature not reported before, despite its inviscid nature; these eddies arise from the coarse‑grained vorticity associated with the dense quantum‑vortex tangle.

The key physical mechanism identified is the formation of a highly dissipative zone of enhanced mutual‑friction near the cylinder shoulders. As the incoming flow is forced around the cylinder, the relative velocity vₙₛ becomes large at the shoulder regions, causing the vortex‑line density L to increase sharply (L ∝ |vₙₛ|² in local equilibrium). The resulting high‑L belt acts as a “mutual‑friction barrier,” effectively thickening the obstacle and redirecting the normal‑fluid streamlines. This barrier closes a recirculation cell upstream of the cylinder, thereby sustaining the anomalous upstream vortex pair. Simultaneously, the strong mutual‑friction damping suppresses the shear‑layer instability that would otherwise generate the classical Kármán vortex street, explaining the observed stability of the downstream pair up to Reynolds numbers Reₙ ≈ 1500.

To organize the transitions, the authors introduce two dimensionless numbers. The normal‑fluid Reynolds number Reₙ = ρₙvₙ,0D/ηₙ quantifies the balance between inertial advection and viscous diffusion, governing the 0‑ to 2‑vortex transition. The interaction number

N = (ρₙ³/ρₛ³) B_L κ (α_v/β_v)² vₙ,0D

measures the ratio of mutual‑friction forcing to inertial forcing (using the Vinen equilibrium scaling L₀ ≈ (α_v/β_v)² vₙₛ,0²). The authors find a nearly constant critical Reₙ,c ≈ 300 for the onset of vortex shedding, largely independent of temperature and block ratio. The second transition, from the 2‑vortex to the 4‑vortex state, occurs when N exceeds a critical value N_c ≈ 33. Increasing the block ratio B (i.e., making the cylinder occupy a larger fraction of the channel) lowers the heat‑flux thresholds for both transitions because stronger confinement enhances the shoulder‑induced mutual‑friction barrier.

By systematically sweeping heat flux q, block ratio B, and temperature T (1.9–2.1 K), the authors construct a unified phase diagram in the (Reₙ, N, B) space. The diagram delineates three regimes: (i) viscosity‑dominated potential flow (0‑vortex), (ii) inertia‑dominated steady wake (2‑vortex), and (iii) mutual‑friction‑dominated regime where upstream vortex pairs appear (4‑vortex, and at extreme parameters, 6‑vortex). Experimental data from previous works overlay neatly onto the predicted boundaries, confirming the predictive power of the model.

The study thus provides a comprehensive mechanistic explanation for the multistable wake topologies observed in He II counterflow, highlights the crucial role of self‑organized mutual‑friction dissipation, and demonstrates that even an inviscid superfluid can exhibit large‑scale eddy structures through coarse‑grained vortex dynamics. The unified phase diagram offers a practical map for future experiments and suggests that mutual‑friction feedback may be a generic route to unconventional wake patterns in other quantum‑fluid systems.