# Reynolds analogy

The **Reynolds Analogy** is popularly known to relate turbulent momentum and heat transfer.[1] That is because in a turbulent flow (in a pipe or in a boundary layer) the transport of momentum and the transport of heat largely depends on the same turbulent eddies: the velocity and the temperature profiles have the same shape.

The __main assumption__ is that heat flux q/A in a turbulent system is analogous to momentum flux τ, which suggests that the ratio τ/(q/A) must be constant for all radial positions.

The complete Reynolds analogy* is:

Experimental data for gas streams agree approximately with above equation if the Schmidt and Prandtl numbers are near 1.0 and only skin friction is present in flow past a flat plate or inside a pipe. When liquids are present and/or form drag is present, the analogy is conventionally known to be invalid.[1]

In 2008, the qualitative form of validity of Reynolds' analogy was re-visited for laminar flow of incompressible fluid with variable dynamic viscosity (μ).[2] It was shown that the inverse dependence of Reynolds number (*Re*) and skin friction coefficient(*c*_{f}) is the basis for validity of the Reynolds’ analogy, in laminar convective flows with constant & variable μ. For μ = const. it reduces to the popular form of Stanton number (*St*) increasing with increasing *Re*, whereas for variable μ it reduces to *St* increasing with decreasing *Re*. Consequently, the Chilton-Colburn analogy of *St*•*Pr*^{2/3} increasing with increasing *c*_{f} is qualitatively valid whenever the
Reynolds’ analogy is valid. Further, the validity of the Reynolds’ analogy is linked to the applicability of Prigogine's Theorem of Minimum Entropy Production.[3] Thus, Reynolds' analogy is valid for flows that are close to developed, for whom, changes in the gradients of field variables (velocity & temperature) along the flow are small.[2]

## References

- Geankoplis, C.J.
*Transport processes and separation process principles*(2003), Fourth Edition, p. 475. - Mahulikar, S.P., & Herwig, H., 'Fluid friction in incompressible laminar convection: Reynolds' analogy revisited for variable fluid properties,'
*European Physical Journal B: Condensed Matter & Complex Systems*,**62(1)**, (2008), pp. 77-86. - Prigogine, I.
*Introduction to Thermodynamics of Irreversible Processes*(1961), Interscience Publishers, New York.