Dynamic Similarity in Superfluid Flows

• Dynamic Similarity in Superfluid Flows is a framework that uses dimensionless groups, such as the superfluid Reynolds number, to explain universal behavior in quantum turbulence and vortex dynamics.
• The methodology bridges classical fluid mechanics with quantum effects by replacing viscosity with quantum circulation and incorporating thresholds for vortex nucleation.
• Experimental and numerical studies confirm that these scaling laws predict drag coefficients, wake transitions, and interface instabilities, providing actionable insights for superfluid research.

Dynamic similarity in superfluid flows is the principle that certain classes of superfluid dynamics exhibit universal behavior when characterized in terms of appropriate dimensionless parameters, analogously to the Reynolds similitude in classical hydrodynamics. The complexity of superfluidity—including quantized vortices, two-fluid behavior, mutual friction, and phase separation—necessitates a careful identification of the relevant nondimensional groups governing such universality. Both experimental and theoretical studies across a range of geometries and regimes now confirm that the appropriate formulation of dynamic similarity underpins quantum turbulence, vortex shedding, drag laws, interface instabilities, and phase separation in quantum fluids.

1. Dimensionless Groups and Their Physical Interpretation

The cornerstone of dynamic similarity in superfluid flows is the replacement and augmentation of the Reynolds number. In classical fluids, dynamic similarity is achieved when flows share the same Reynolds number, $\mathrm{Re} = UL/\nu$, where $U$ is a characteristic velocity, $L$ a length scale, and $\nu$ the kinematic viscosity. In the inviscid limit of pure superfluids, $\nu\to0$, so $\mathrm{Re}\to\infty$ becomes meaningless. Instead, a superfluid Reynolds number is constructed by replacing the viscosity with the quantum of circulation, $\kappa=\hbar/m$, and introducing a velocity threshold for vortex nucleation, $u_c$. The canonical dimensionless control parameter thus takes the form

$\mathrm{Re}_s = \frac{(u-u_c)\,D}{\kappa},$

where $u$ is the bulk flow velocity, $D$ the effective obstacle diameter, and $u_c$ the critical velocity for vortex shedding (Reeves et al., 2014, Kwon et al., 3 Feb 2026, Christenhusz et al., 2024, Takeuchi, 2023).

Associated dimensionless groups in two-fluid systems include the mutual-friction parameter $\varphi$, which quantifies the interplay between vortex-induced drag and normal-fluid viscous smoothing:

$$\varphi = \frac{g\rho_s\kappa\alpha L D^2}{\eta_n},$$

where $\rho_s$ is the superfluid density, $\alpha$ the mutual friction coefficient, $L$ the vortex-line density, $D$ the channel half-width, and $\eta_n$ the normal-fluid viscosity (Yui et al., 2017).

Other system-specific dimensionless parameters include:

• Strouhal number $\mathrm{St} = fD/u$ (oscillation frequency scaling) (Reeves et al., 2014, Kwon et al., 3 Feb 2026)
• Drag coefficient $C_D=2F_d/[\rho(u^2-u_c^2)D]$ (Christenhusz et al., 2024)
• Domain-size scaling exponent $z$ and energy decay exponents in phase-separating mixtures (Ito et al., 7 May 2025)
• Interface similarity parameter $\alpha=d_v/\xi$ for binary superfluids, where $d_v$ is inter-vortex separation and $\xi$ the vortex core size (An et al., 2024).

2. Dynamic Similarity in Quantum Turbulence and Superfluid Wakes

The universality of turbulent spectra in superfluids is evident from experiments, computational Gross–Pitaevskii modeling, vortex-filament and two-fluid (HVBK) models, as well as holographic duality approaches. For homogeneous isotropic turbulence in He II and atomic BECs, the energy spectrum $E(k)\propto k^{-5/3}$ emerges robustly at large scales ($k\ll\ell^{-1}$, with $\ell$ the mean intervortex spacing), provided the relevant Reynolds numbers are large and the system is in the inertial range (Barenghi et al., 2013, Adams et al., 2012). Deviations from this scaling appear only at the quantum-crossover scales where Kelvin-wave cascades and vortex discretization dominate.

In quantum wake flows past obstacles, dynamic similarity is demonstrated through the collapse of drag and Strouhal curves for different obstacle sizes and flow velocities when plotted versus the appropriately defined $\mathrm{Re}_s$. The drag coefficient exhibits classical Stokes-like scaling ($C_d\sim\mathrm{Re}_s^{-1}$) at low Reynolds numbers, transitions to nonlinear regimes around $\mathrm{Re}_s\approx 0.7-2$, and saturates to a finite value (dissipative anomaly) at high $\mathrm{Re}_s$ (Reeves et al., 2014, Christenhusz et al., 2024, Kwon et al., 3 Feb 2026).

Key features of dynamic similarity in superflow past obstacles include:

• Thresholdless drag below $u_c$, linear scaling for $0<\mathrm{Re}_s\lesssim0.3$, and turbulent cluster shedding above $\mathrm{Re}_s\sim0.7$ (Reeves et al., 2014, Christenhusz et al., 2024)
• Universal drag law for both hard-wall and penetrable (Gaussian) obstacles, provided the effective diameter is set by the Mach-1 contour of the supersonic region (Kwon et al., 3 Feb 2026)
• Empirical fits for $C_d(\mathrm{Re}_s)$ and $\mathrm{St}(\mathrm{Re}_s)$ closely matching classical forms, e.g.

$$\mathrm{St}(\mathrm{Re}_s) = \mathrm{St}_\infty\left(1-\frac{\alpha}{\mathrm{Re}_s+\beta}\right)$$

with fitted parameters reflecting obstacle softness and quantum effects (Reeves et al., 2014).

3. Two-Fluid Hydrodynamics and Mutual-Friction Scaling

Superfluid $^4$He in the thermal counterflow regime exhibits two strongly coupled components: inviscid superfluid and viscous normal fluid, interacting via mutual friction. The coupled dynamics are described by the vortex-filament model for the superfluid and Navier–Stokes equations for the normal fluid, with mutual-friction forces dependent on vortex-line density and velocity difference (Yui et al., 2017).

The deformation of the normal-fluid velocity profile is fully governed by two dimensionless groups:

1. The normal-fluid Reynolds number, $\mathrm{Re}=\rho_n V_0 D/\eta_n$,
2. The mutual-friction parameter $\varphi$.

For fixed geometry and typical experimental heat flux, $\mathrm{Re}$ is often held constant, making $\varphi$ the primary similarity variable that organizes the family of steady-state velocity profiles from classical Poiseuille (low $\varphi$) to plug-flow (high $\varphi$). Experiments and simulations at fixed $(\mathrm{Re},\varphi)$ exhibit dynamic similarity, with profiles collapsing for normal fluids in channels of different widths, temperatures, and heat currents (Yui et al., 2017).

4. Dynamic Similarity in Interfaces and Phase Separation

Dynamic similarity principles extend to the interface instabilities of binary superfluids and to the coarsening dynamics of phase-separating superfluid mixtures. In strongly-coupled binary superfluids, the interface instability is governed by the dimensionless ratio $\alpha=d_v/\xi$ of inter-vortex separation to core size. The fastest-growing mode of the interface, $k_0$, becomes a universal function of $\alpha$, yielding a master curve for instability spectra independent of temperature and interaction strength. This dynamic similarity is a direct consequence of the universality of the ratio of macroscopic to microscopic scales (An et al., 2024).

In phase-separating superfluid mixtures, as in ferromagnetic Bose gases, the vorticity structure factor and energy spectrum evolve following dynamic scaling laws with a growing domain size $l(t)\propto t^{2/3}$ in the inertial hydrodynamic regime. The spectrum $S_\omega(k,t)$ and the kinetic energy $K(t)$ collapse onto universal curves when lengths are rescaled by $l(t)$, up to a microscopic cutoff at the healing length $\xi$. This collapse identifies the dynamic similarity class as that of binary-fluid universality, distinct from classical (single-fluid) turbulence (Ito et al., 7 May 2025).

5. Emergent Universal Drag Laws and Experimental Protocols

The universal scaling of drag coefficients and Strouhal numbers in superfluid wakes underpins the practical utility of dynamic similarity. Recent studies confirm that, for objects of similar geometry, the drag coefficient as a function of superfluid Reynolds number matches classical empirical laws (after accounting for $u_c$ where appropriate), both in numerical studies and in proposed precision experiments using falling bodies in He-II (Takeuchi, 2023, Christenhusz et al., 2024). For penetrable obstacles as used in cold-atom experiments, dynamic similarity is preserved when the obstacle diameter is replaced by the width of the Mach-1 region, reflecting the "active" scale for vortex nucleation (Kwon et al., 3 Feb 2026).

Practical protocols for verifying dynamic similarity include:

• Measuring terminal velocity of falling spheres in He-II and computing reduced drag curves versus $\mathcal{R}_s$ (Takeuchi, 2023)
• Extracting drag by momentum-flux integration across a control volume in atomic BECs, avoiding the need for near-obstacle field measurements (Christenhusz et al., 2024)
• Visualization of interface waves and patterns in immiscible condensates at fixed $\alpha$ (An et al., 2024)
• Monitoring the collapse of normalized velocity profiles or spectra for varying channel sizes, temperatures, obstacle strengths, and interaction parameters (Yui et al., 2017, Kwon et al., 3 Feb 2026, Ito et al., 7 May 2025).

6. Physical Origin, Regimes of Validity, and Limitations

Dynamic similarity in superfluids arises from the scale separation between macroscopic flow structures and microscopic quantum scales (e.g., circulation quantum, healing length, vortex core size). It holds provided that collective vortex or interface dynamics, rather than single-vortex or microscopic phenomena, dominate the observed flow behavior. Breakdown of similarity occurs at sufficiently low Reynolds numbers ($\mathrm{Re}_s\ll1$, single-vortex regime), near quantum critical points where dissipation mechanisms change, or when strong boundary-layer or pinning effects introduce new relevant length scales (Takeuchi, 2023, Adams et al., 2012).

A practical implication is that protocols and scaling laws validated in superfluid helium can be transferred to cold-atom systems, Fermi superfluids, or even neutron star matter once the relevant dimensionless groups are matched. Conversely, experimental and numerical results can be meaningfully compared across systems of vastly different absolute scales, provided the organizing parameters for dynamic similarity are respected.

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In summary, dynamic similarity in superfluid flows is established by identifying and employing appropriate dimensionless parameters, primarily the superfluid Reynolds number, mutual-friction parameter, and system-specific scaling exponents. These groups control the universality of turbulence spectra, drag laws, wake transitions, interface instabilities, and phase-separation kinetics across superfluid helium, atomic condensates, and quantum turbulent systems, enabling cross-platform correspondence and predictive modeling (Yui et al., 2017, Kwon et al., 3 Feb 2026, Christenhusz et al., 2024, Reeves et al., 2014, An et al., 2024, Barenghi et al., 2013, Adams et al., 2012, Takeuchi, 2023, Ito et al., 7 May 2025).