Reynolds analogy

The Reynolds analogy is an approximate equivalence between the transport of momentum and the transport of heat in a turbulent fluid flow. Formulated by Osborne Reynolds in 1874, it asserts that the mechanisms that carry momentum across a shear layer — turbulent eddies mixing fluid from high-velocity regions with fluid from low-velocity regions — are structurally identical to the mechanisms that carry heat across a temperature gradient. The analogy implies that the dimensionless coefficients describing these transports — the skin friction coefficient and the Stanton number — are proportional to each other, with the proportionality constant determined by the ratio of momentum diffusivity to thermal diffusivity, known as the Prandtl number.

The Reynolds analogy is not a derived theorem but an empirical observation with a theoretical rationale. In turbulent flow, the molecular transport processes — viscosity and thermal conduction — are overwhelmed by the eddy transport processes. The eddies that carry momentum also carry heat, and if the Prandtl number is close to 1 (as it is for most gases), the two transport processes proceed at similar rates. The analogy breaks down when the Prandtl number deviates significantly from 1, as in oils (Pr >> 1) or liquid metals (Pr << 1), and it breaks down near walls where the turbulent mixing is suppressed and molecular transport dominates.

The significance of the Reynolds analogy extends beyond engineering heat transfer calculations. It is a specific instance of a broader principle in transport phenomena: that different conserved quantities, when transported by the same turbulent mechanism, develop similar statistical structures. This principle has been generalized to mass transfer, to the transport of scalar quantities in geophysical flows, and to the transport of information in complex networks. The analogy is not merely a computational convenience; it is evidence that turbulence imposes a universal structure on the transport of any quantity that is advected by the flow.