Universal drag and aerodynamics in a model superflow

The Reynolds law of dynamical similarity states that two objects of the same geometry, but at vastly different length scales, are hydrodynamically equivalent under the same Reynolds number leading to a universal drag coefficient law. Whether this behavior holds for superfluids has been a long open question. Here, we numerically study superfluid drag, building on earlier work by Reeves et. al on the Strouhal number [1]. Generating flow across a range of Reynolds numbers for the paradigmatic classical hard-wall obstacle and the Gaussian obstacle popular in experimental quantum hydrodynamics, we find a collapse of the superfluid drag coefficient C_dS onto a single curve. This illustrates the existence of the universal drag law, facilitated by the nucleation of quantised vortices. Our work suggests that all relevant dimensionless quantities are expressible as a universal function of the superfluid Reynolds number, Re_S, due to scale invariance -- analogous to classical hydrodynamics [2].

The existence of dynamical similarity provides a universal framework for understanding superfluid aerodynamics. A comparison between superfluid and classical drag coefficients for various obstacles suggests that streamlining is less effective in superfluids. Unlike in classical fluids, where a boundary layer mediates flow attachment and separation, superfluid obstacles lack such a mechanism. This raises a fundamental question: how do airfoils achieve streamlining in inviscid environments? Futhermore, we observe the critical vortex nucleation velocity differs significantly between a flat plate and an airfoil, suggesting that geometric effects play a crucial role in the emergence of superfluidic drag and lift. The connection between nucleation dynamics and flow behaviour at high superfluid Reynolds numbers provide new insights understanding superfluidic lift and drag.

[1] M. T. Reeves, T. P. Billam, B. P. Anderson, and A. S. Bradley, "Identifying a superfluid Reynolds number via dynamical similarity", Phys. Rev. Lett. 114, 155302 (2015).

[2] M. T. M. Christenhusz, A. Safavi-Naini, H. Rubinsztein-Dunlop, T. W. Neely and M. T. Reeves, "Emergent Universal Drag Law in a Model of Superflow", arXiv 2406.14049 (2024).