The Reference Temperature Method
In this section we discuss an approximate engineering method for predicting skin friction and heat transfer for laminar compressible flow. It is based on the simple idea of utilizing the formulas obtained from incompressible flow theory, wherein the thermodynamic and transport properties in these formulas are evaluated at some reference temperature indicative of the temperature somewhere inside the boundary layer. This idea was first advanced by Rubesin and Johnson in Reference 80 and was modified by Eckert (Reference 81) to include a reference enthalpy. In this fashion, in some sense the classical incompressible formulas were “corrected” for compressibility effects. Reference temperature (or reference enthalpy) methods have enjoyed frequent application in engineering-oriented analyses, because of their simplicity. For this reason, we briefly describe the approach here.
Consider the incompressible laminar flow over a flat plate, as discussed in Section
18.2. The local skin friction coefficient is given by Equation (18.20), repeated below:
For the compressible laminar flow over a flat plate, we write the analogous expression
Evaluating Equation (18.54) at the reference temperature, we have
£-** ______ ^ w______
p* u*(haw hw)
Example 1 8.2 | Use the reference temperature method to calculate the friction drag on the same flat plate at the same flow conditions as described in Example 8.1b. Compare the reference temperature results with that obtained in Example 8.1b, which reflected the “exact” laminar boundary layer theory.
The reference temperature is calculated from Equation (18.53), where we need the ratio Tm/Te. For the present case, the flat plate is at the adiabatic wall temperature, hence we need the ratio Taw/Te. To obtain this, we use the recovery factor, which for a flat plate laminar boundary layer is given by Equation (18.47):
Also, the value of д* that corresponds to T* is obtained from Sutherland’s law, given by Equation (15.3)
_Д _ /rV,/2 T0 + 110 M о UJ T + 110
Recall: In Equation (15.3), до is the reference viscosity coefficient at the reference temperature T0. In Equation (15.3) T0 denotes the reference temperature, not the total temperature. Here we have a case of the same notation for two different quantities, but the meaning of T0 in Equation (15.3) is clear from its context. We will use the standard sea level conditions for the values of T0 and д0, that is,
до = 1-7894 x КГ5 kg/(m)(s) and T0 = 288 К
Hence, from Equation (15.3)
д* /Г*у1/2 T0+ 110 /612.7Л3/2 288 + 110
/У VW T* + 110 ~ V 288 / 612.7 + 110
д* = 1.709ДО = (1.709)(1.7894 x КГ5) = 3.058 x КГ5 kg/(m)(s)
From Equation (18.52) integrated over the entire chord of the plate, we have the same form as Equation (18.22), namely,
Hence, the friction drag on one side of the plate is
Df = p*VfSC*f = 5(0.574)(1000)2(40)(2.167 x 10“4) = 2844 N
The total friction drag taking into account both the top and bottom surfaces of the plate is
D = 2(2488) =
The result obtained from classical compressible boundary layer theory in Example 18.1b is D = 5026 N. The result from the reference temperature method used here is within one percent of the “exact” value found in Example 18.1b, a stunning example of the accuracy of the reference temperature method, at least for the case treated here.