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:

${\displaystyle {\frac {f}{2}}={\frac {h}{C_{p}\times G}}={\frac {k'_{c}}{V_{av}}}}$

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]

See also

• Reynolds number
• Chilton and Colburn J-factor analogy

References

1. Geankoplis, C.J. Transport processes and separation process principles (2003), Fourth Edition, p. 475.
2. 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.
3. Prigogine, I. Introduction to Thermodynamics of Irreversible Processes (1961), Interscience Publishers, New York.