B,8 • FUNDAMENTAL EQUATIONS OF MOTION 



experiments will be given in Art. 14. Since no unique theory has evolved, 

 the emphasis will be placed on the description and discussion of the bases 

 and assumptions underlying the theoretical treatments rather than their 

 detailed analysis. Experimental data will be compared with theories. 

 This method of approach seems best to show the present state of the 

 subject and to serve as a guide to future theoretical and experimental 

 investigations. 



B,8. Fundamental Equations of Motion of a Compressible 

 Boundary Layer. When applying the hydrodynamic equations of Art. 

 4, 5, and 6 to the boundary layer developed on a flat plate with steady 

 free stream velocity, certain simplifying approximations may be made. 

 First of all, the mean flow is assumed two-dimensional with mean veloci- 

 ties denoted by U and V in the x and y directions respectively, where x is 

 the coordinate parallel to the plate, measured from the leading edge, and 

 y is normal to the wall. The turbulence is still three-dimensional, with 

 components u, v, and w in the x, y, and z directions. 



We now consider the order of magnitude of terms involved in the 

 hydrodynamic equations. If U is taken as a magnitude of standard order 

 0(1), and the thickness of the boundary layer 5 is small compared to the 

 distance x, it follows that d/dt, d/dx, d^/dx^ '^ 0(1), and d/dy ^^ 0(5"^), 

 d^/dy^ /^ 0(5~^). Also we assume that V '^ 0(5), the mean density p is 

 0(1), and the total energy content per unit mass E is 0(1). If the viscous 

 term of Eq. 4-5 is to be at most of the same order as the remaining terms, 

 then it follows that ju is at most of the order of 6^. By the same reasoning, 

 the correlations involving M, v, p', T', such as uv, uT'; vT' ; p'u, p'v, p'T', are 

 at most of the order of 5, while the triple correlation p'uv will be at most 

 of the order of 6^. 



Retaining the predominant terms of the same order of magnitude, we 

 can easily reduce the dynamic equations (Eq. 4-5, 4-6, and 6-4a) respec- 

 tively to the following forms: 



(8-1) 

 -|-4(.^) = (8-2) 



t + l (^^) + Ty (^^^ + ty (?^) = « ^«-3> 



(89) 



