G • COOLING BY PROTECTIVE FLUID FILMS 



The basic equations, which represent the principles of conservation of 

 momentum, conservation of mass, and conservation of energy, for the 

 compressible turbulent boundary over a flat plate, can be expressed as 

 follows [Sec. B]: 



The momentum equation in the x direction is 



The continuity equation is 



^M + ^ = (4-35) 



dx dy 



The energy equation is 



— d{c^T) -d(c^T) d /, dT —jyjr\ , , , X {duV ,. „„. 



^^ -^ + ''^ ^T^ = a^ V^ aF " '''' ^)^^^ + ^^) \Ty) ^^-^^^ 



The quantities with bars represent time-average quantities, while primed 

 quantities represent instantaneous values of fluctuating quantities. The 

 specific heat is assumed to be a constant and the fluid properties of 

 density, viscosity, and thermal conductivity are considered to vary with 

 temperature. The boundary conditions are 



At ?/ = 0; T = T^', u = 



(4-37) 

 Kiy = b; ^ ^ 



For a Prandtl number equal to unity, the relation between T and u 

 can be expressed as in Eq. 4-9 in the following manner: 



T T 





In order to make the solution of Eq. 4-34 possible, at least for practi- 

 cal purposes, certain nonrigorous assumptions are made which are analo- 

 gous to those used in the case of low speed, incompressible turbulent flow 

 along a smooth wall without injection. The first simplification is to assume 

 that the dependency of variable quantities with respect to x is negligible 

 compared to their variations with respect to y in the neighborhood of a 

 permeable wall. Secondly, Prandtl's mixing-length hypothesis is assumed 

 to apply for the present problem. Then in the turbulent region, Eq. 4-34 

 can be simplified as follows: 



du dr d 



'^'-Ty = ry = ryl'^'y'' 



{ 456 ) 



(duY 



\dyj 



(4-39) 



