The Reynolds similitude, a key concept in hydrodynamics, states that two phenomena of different length scales with a similar geometry are physically identical. Flow properties are universally determined in a unified way in terms of the Reynolds number $\mathcal{R}$ (dimensionless, ratio of inertial to viscous forces in incompressible fluids). For example, the drag coefficient $c_D$ of objects with similar shapes moving in fluids is expressed by a universal function of $\mathcal{R}$. Certain studies introduced similar dimensionless numbers, that is, the superfluid Reynolds number ${\mathcal{R}}_s$, to characterize turbulent flows in superfluids. However, the applicability of the similitude to inviscid quantum fluids is nontrivial as the original theory is applicable to viscous fluids. This Letter proposes a method to verify the similitude using current experimental techniques in quantum liquid He II. A highly precise relation between $c_D$ and ${\mathcal{R}}_s$ was obtained in terms of the terminal speed of a macroscopic body falling in He II at finite temperatures across the Knudsen (ballistic) and hydrodynamic regimes of thermal excitations. The Reynolds similitude in superfluids proves the quantum viscosity of a pure superfluid and can facilitate a unified mutual development of classical and quantum hydrodynamics; the concept of quantum viscosity provides a practical correspondence between classical and quantum turbulence as a dissipative phenomenon.

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• Vortices in superfluids
• Classical fluids
• Helium-4 superfluids
• Quantum fluids & solids
• Superfluids