B • TURBULENT FLOW 



with an insulated wall. We conclude that for Pr = 1, the relationships 

 between temperature and velocity in the turbulent boundary layer are 

 the same as those in the laminar boundary layer, which were first ob- 

 tained by Crocco [9]. 



In an insulated boundary layer at low speeds, Squire [10] and Acker- 

 man [11] have independently deduced the formula 



CpTw = CpTe +|Pr*t/2 (9-8) 



for Pr 5^ 1 or Pr = 1. Here T^ is the temperature at the wall. When 

 there is no heat transfer, T^ is sometimes called equilibrium temperature. 

 This formula may be expected to be not seriously in error at high speeds, 

 and includes Eq. 9-7 as a special case with Pr = 1. As Pr < 1 in general, 

 the temperature at the wall is accordingly smaller than the total free 

 stream temperature. 



In the light of Eq. 9-4, a more general formula for the case oi Pr 9^ 1 

 can be written as follows: 



CpTw = CpTe 4- ^uVl (9-9a) 



by introducing a factor re, called the recovery factor. The recovery factor 

 can then be considered as defined by Eq. 9-9a, and it then becomes 



^ Tw- Te 

 Using the adiabatic relation Tl/T, = 1 + (7 - l)Ml/2, 



(9-9b) 



00 + ^-0 



- 1 



-Ml 



(9-9c) 



Here M^ is the Mach number at the edge of the boundary layer, and 7 is 

 the ratio of specific heats. According to Eq. 9-8 and 9-9, the recovery 

 factor should not differ very much from the value 



re = Pr^ (9-10) 



The turbulent recovery factor, which shows a close agreement with 

 Eq. 9-10, has been measured by Mack [12] over the surface of a cone, 

 in the range of free stream Mach number from 1.33 to 4.50, to be 0.88 ± 

 0.01, as compared with the calculated value of Pr^ = 0.89, based on the 

 recovery temperature. Experiments for a flat plate have been made by 

 Stalder, Rubesin, and Tendeland [13] (re - 0.89 ± 0.01) at Mach num- 

 ber 2.4. Also the measurements of the laminar recovery factor show a 

 close agreement with the theoretical value of Pr^. The experimental 

 results of various investigations are summarized in Fig. B,9a and B,9b. 



In general the recovery factor depends on the Reynolds number. In 



<92> 



