G ■ COOLING BY PROTECTIVE FLUID FILMS 



influence of a pressure gradient in transpiration-cooled turbulent bound- 

 ary layer theory is still uncertain; however, it is believed that the influ- 

 ence is less on a turbulent layer than on a laminar one. 



Another important application of transpiration cooling is in reducing 

 the aerodynamic heating problem in high speed flight. Since both heat 

 transfer and drag coefficients are known to be lower for laminar than for 

 turbulent flows, it is more advantageous to have a laminar boundary 

 layer than the turbulent type. The solution of a transpiration-cooled 

 boundary layer on a flat plate can be employed here with reasonably 

 good approximation. 



The treatments in the subsequent articles are divided into approxi- 

 mate methods for the solution of the laminar boundary layer, exact solu- 

 tions of the laminar boundary layer, and approximate solutions of the 

 turbulent boundary layer. The stream fluid and the injected fluid are 

 assumed to be homogeneous. 



Approximate Methods for the Solution of Heat Transfer in 

 THE Laminar Boundary Layer. The heat transfer in the laminar 

 boundary layer of a transpiration-cooled wall in a flow can be solved by 

 the von Karman momentum and energy equations for the boundary layer. 

 The basic derivation of these equations is treated at length in Vol. IV. 

 For two-dimensional compressible flow with a pressure gradient and a 

 uniform injection (or suction) at the wall, the momentum and energy 

 equations for the boundary layer are given, respectively, as 



(4-1) 



pu{ue — u)dy + -^ / (peUe — pu)dy = p^u^v^ '^ \^'^) 



- I pucAT. - T)dy + ^^ I u(Pe - ,)dy + j^ . ^^ dy 



= p^v^CpiT. - T^) + (k ^j (4-2) 



where the subscript e represents quantities at the outer edge of the 

 laminar layer and w quantities at the wall. The other symbols are stand- 

 ard and a sketch of velocity and temperature fields within the boundary 

 layer along a transpiration-cooled wall is shown in Fig. G,4a. 



Incompressible boundary layer on a porous flat plate. It was men- 

 tioned in the previous article that in many of the applications of heat 

 transfer the boundary layer is turbulent. Nevertheless, it is interesting 

 and important to understand the mechanism of heat transfer qualita- 

 tively, as well as to clarify some physical quantities involved which can- 

 not be easily interpreted in complicated cases. The following analysis is 

 made by the investigation of the flow of a hot gas over a porous flat plate 

 under the condition of uniform gas injection from the bottom of the plate. 

 The assumptions made in the present investigation are: (1) the mass 



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