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]