G,4 • TRANSPIRATION-COOLED BOUNDARY LAYER 

 temperature and the rate of coolant injection is determined as 



m n-i —Fr dlam 



Cp \Wlam / 



(4-31) 



Since experimental information on transpiration-cooled turbulent 

 boundary layers has not been achieved, the variation of laminar-sublayer 

 thickness with the velocity of injection remains unknown. It is assumed 

 that the flow in the turbulent case is not affected by the velocity of 

 injection, and hence that the shearing stress and the velocity Ui^^n. at the 

 edge of the core are the same as for flow in a smooth pipe. The thick- 

 ness of the laminar sublayer has been measured in smooth pipe and 

 found to satisfy the relation 



^h^ = y* (4-32) 



V 



where ul = r^/p and y* = 5.6 is taken by Prandtl after examination of 

 the velocity profile measured close to a wall. On the basis of Eq. 4-32 

 the following expressions are obtained : 



5u„ = vy^ 



(4-33) 



Wlam — y \j 



\ P 



where Tw = Cfpul/2. For the Reynolds number range 5000 < Re < 200,000 

 the friction coefficient C/ for smooth pipes satisfies the empirical relation 

 Cf = OMQ{Re)-K 



The relation between the wall temperature and the rate of coolant 

 flow is calculated from Eq. 4-31 and shown in Fig. G,4b. The result 

 reveals that for a designated wall temperature the rate of coolant re- 

 quired for transpiration-cooled turbulent flow is almost twice as much 

 as in the case of laminar flow. For an injected velocity equal to 1 per 

 cent of the hot fluid velocity, the heat transfer to the wall is reduced to 

 70 per cent of the value without transpiration cooling. The comparison 

 of this result with the result obtained under the same conditions in lami- 

 nar flow is shown in Fig. G,4c. 



Approximate Solution of Heat Transfer in Turbulent Bound- 

 ary Layer on a Flat Plate. The treatment above may be extended 

 to the case of the effect of coolant injection on the behavior of a com- 

 pressible turbulent boundary layer [28,29]. In order to simplify the analy- 

 sis, the following assumptions are made: (1) the coolant fluid is the same 

 as the boundary layer fluid, (2) the wall temperature in the direction of 

 flow is constant, and (3) the Prandtl number is equal to unity. 



< 455 ) 



