The Reynolds Analogy is popularly known to relate turbulent momentum and heat transfer. 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 wall heat flux q ˙ w {\displaystyle {\dot {q}}_{w}} in a turbulent system is analogous to momentum flux, or wall shear stress, τ w {\displaystyle \tau _{w}} , which suggests that the ratio τ w q ˙ w {\displaystyle {\frac {\tau _{w}}{{\dot {q}}_{w}}}} must be constant for similar geometry. A common example for "similar geometry" is every radial position around a pipe at some cross-section with circular shape. If the wall shear stress ( τ w {\displaystyle \tau _{w}} ) is known, a relation for the turbulent heat flux ( q ˙ w {\displaystyle {\dot {q}}_{w}} ) can be found (or vice versa), making it a useful analogy for pipe flows and hypersonic applications where τ w {\displaystyle \tau _{w}} may be more easily estimated than q ˙ w {\displaystyle {\dot {q}}_{w}} . This is often used to estimate wall heat flux on a body in a flow:
q ˙ w ≈ τ w c p ( T w − T ∞ ) u ∞ {\displaystyle {\dot {q}}_{w}\approx {\frac {\tau _{w}c_{p}(T_{w}-T_{\infty })}{u_{\infty }}}}
where:
T w , T ∞ {\displaystyle T_{w},T_{\infty }} are the temperatures of the wall and freestream (or bulk, depending on application), respectively;
c p {\displaystyle c_{p}} is the specific heat capacity at constant pressure for the fluid;
u ∞ {\displaystyle u_{\infty }} is the freestream (or bulk, depending on application) fluid velocity.
The complete Reynolds analogy (in a form more often seen in relation to pipe flows) is:
f 2 = h c p G = k c ′ v avg {\displaystyle {\frac {f}{2}}={\frac {h}{c_{p}G}}={\frac {k'_{c}}{v_{\text{avg}}}}}
where:
f {\displaystyle f} is the Fanning friction factor;
h {\displaystyle h} is the heat transfer coefficient;
c p {\displaystyle c_{p}} is the specific heat at constant pressure;
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