G,4 • TRANSPIRATION-COOLED BOUNDARY LAYER 



It indicates that for zero Mach number, unless the temperature difference 

 across the boundary layer is large, say T^/T^ > 3, the increase of the 

 rate of the coolant injection in order to maintain a predesignated wall 

 temperature is about 10 per cent over the case in which constant physical 

 properties of the gas are assumed. On the other hand, an appreciable in- 

 crease of the rate of the coolant injection for maintaining a predesignated 

 wall temperature is found between the case of M = 2 and M = 0. This 

 leads to the conclusion that for a flow of subsonic speed and in which the 

 temperature difference of the hot gas relative to the wall is not large, the 

 physical properties of the gas may be regarded as constant in the appli- 

 cation of transpiration cooling. 



As already mentioned, the relation between the rate of the coolant 

 injection and the wall temperature is based on the average value in a flow 

 of gas over a plate with a given Reynolds number. It must be borne in 

 mind that the boundary layer thickness increases almost linearly with the 

 length in the direction of flow, and the heat transfer to the wall decreases 

 proportionally from the leading edge of the plate to dov/nstream. This 

 results in a longitudinal temperature gradient along the transpiration- 

 cooled wall and, naturally, heat flow through the thermally conductive 

 plate occurs. For this reason, as far as the laminar flow is concerned, the 

 efficient method in transpiration cooling is to vary the rate of the coolant 

 injection along the plate in accordance with the local heat transfer at the 

 wall [11]. 



Compressible boundary layer on a porous wall with a pressure gradient. 

 Flows with pressure gradients (favorable and/or adverse) are of con- 

 siderable practical importance in connection with the transpiration cool- 

 ing of turbine blades or airfoil surfaces in high speed flow (aerodynamic 

 heating problem) [12]. The flow along a gas turbine blade is expected to 

 be laminar at least in the region around the nose of the blade, while at 

 supersonic speeds it may be possible to maintain a laminar boundary 

 layer along aircraft and missile surfaces. Since the presence of an adverse 

 pressure gradient has an effect similar to that of a normal injection mass 

 flow, i.e. they both tend to increase the boundary layer thickness, it is the 

 purpose of this article to determine the net effect of these parameters on 

 the flow over a transpiration-cooled surface. The present investigation is 

 based on the assumption that the coefficient of viscosity is linearly pro- 

 portional to the absolute temperature and the Prandtl number is unity. 



In order to solve the momentum equation (Eq. 4-1) and the energy 

 equation (Eq. 4-2) for the hydrodynamic and thermal boundary layer 

 thickness 5„ and 8h, respectively, it is convenient to replace the normal 

 distance y by the variable t, defined as follows : 



. = /'g)d< (4-13) 



< 445 ) 



