Reynolds Analogy

Another useful engineering relation for the analysis of aerodynamic heating is Rey­nolds analogy, which can easily be introduced within the context of our discussion of Couette flow. Reynolds analogy is a relation between the skin friction coefficient and the heat transfer coefficient. The skin friction coefficient c/ was first introduced in Section 1.5. In our context here, we define the skin friction coefficient as

tPeUi

Let us define the Reynolds number for Couette flow as

PeUeD

Then, Equation (16.53) becomes

Equation (16.54) is interesting in its own right. It demonstrates that the skin friction coefficient is a function of just the Reynolds number—a result which applies in general for other incompressible viscous flows [although the function is not necessarily the same as given in Equation (16.54)].

Now let us define a heat transfer coefficient as

In Equation (16.55), Ся is called the Stanton number; it is one of several different types of heat transfer coefficient that is used in the analysis of aerodynamic heat­ing. It is a dimensionless quantity, in the same vein as the skin friction coefficient.

For Couette flow, from Equation (16.24), and dropping the absolute value signs for convenience, we have

. fi (he – hw + 5 Ргм^ 4w ~ Pr ( D

Inserting Equation (16.39) into (16.56), we have for Couette flow

/г (haw hu

D

Inserting Equation (16.57) into (16.55), we obtain

„ (tx/?mhaw-hw)/D] /r/Pr 1 ,.жи1

CH = ————————– = ——– = —— [16.58]

Pe^eiPaw ^w’) Pe^e^ Pi*

Equation (16.58) is interesting in its own right. It demonstrates that the Stanton number is a function of the Reynolds number and Prandtl number—a result that applies generally for other incompressible viscous flows [although the function is not necessarily the same as given in Equation (16.58)].

We now combine the results for Cf and CH obtained above. From Equa­tions (16.54) and (16.58), we have

Ch	( 1 ї	1 Re
cf ~	i Re Pr )	’ 2

Equation (16.59) is Reynolds analogy as applied to Couette flow. Reynolds analogy is, in general, a relation between the heat transfer coefficient and the skin friction coefficient. For Couette flow, this relation is given by Equation (16.59). Note that the ratio Сн/сf is simply a function of the Prandtl number—a result that applies usually for other incompressible viscous flows, although not necessarily the same function as given in Equation (16.59).