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Recovery Factor
As a corollary to the above case for the adiabatic wall, we take this opportunity to define the recovery factor—a useful engineering parameter in the analysis of aerodynamic heating. The total enthalpy of the flow at the upper plate (which represents the
upper boundary on a viscous shear layer) is, by definition,
И2
ho = he + ~
(The significance and definition of total enthalpy are discussed in Section 7.5.) Compare Equation (16.45), which is a general definition, with Equation (16.39), repeated below, which is for the special case of Couette flow:
haw = he + Pr^ [16.36]
Note that haw is different from ho, the difference provided by the value of Pr as it appears in Equation (16.39). We now generalize Equation (16.39) to a form which holds for any viscous flow, as follows:
Similarly, Equation (16.40) can be generalized to
[16.46b]
In Equations (16.46a and b), r is defined as the recovery factor. It is the factor that tells us how close the adiabatic wall enthalpy is to the total enthalpy at the upper boundary of the viscous flow. If r = 1, then haw = ho – An alternate expression for the recovery factor can be obtained by combining Equations (16.46) and (16.45) as follows. From Equation (16.46),
haw he
Г = «2/2
From Equation (16.45),
и2
~f=ho~he
Inserting Equation (16.48) into (16.47), we have
where To is the total temperature. Equation (16.49) is frequently used as an alternate definition of the recovery factor.
In the special case of Couette flow, by comparing Equation (16.39) or (16.40) with Equation (16.46a) or (16.46b), we find that
[16.50]
For Couette flow, the recovery factor is simply the Prandtl number. Note that, if Pr < 1, then haw < h0; conversely, if Pr > 1, then haw > h0.
In more general viscous flow cases, the recovery factor is not simply the Prandtl number; however, in general, for incompressible viscous flows, we will find that the recovery factor is some function of Pr. Hence, the Prandtl number is playing its role as an important viscous flow parameter. As expected from Section 15.6, for a compressible viscous flow, the recovery factor is a function of Pr along with the Mach number and the ratio of specific heats.
