B • TURBULENT FLOW 



Similarly from Eq. 6-3a with the same order of approximations we 

 obtain 



li ipE') + ^ ipE^U) + I-, {-pEW) = A (^M ^ - P^ - 7~vE^ 



dt dx dy dy \ dy 



(^-i)f'^.(<=p^) 



+ ± 



dy \_\Pr / dy 



+ puv — (8-5) 



Except for the production term puvdU/dy, Eq. 8-4 and 8-5 have the same 

 form. In the following we shall be concerned with Eq. 8-4 rather than 

 Eq. 8-5. 



Eq. 8-1, 8-2, 8-3, 8-4, and 8-5 form a system of basic equations of the 

 compressible boundary layer. The effects of the density fluctuation are to 

 contribute an additional Reynolds stress, an apparent source, and an addi- 

 tional eddy conductivity respectively in the equations of momentum, con- 

 tinuity, and energy. 



B,9. Relationships between Velocity, Pressure, and Tempera- 

 ture Distributions. Some simple relations are now derived for velocity, 

 pressure, and temperature by integrating the momentum and energy 

 equations. This is done here without entering into the mechanism of tur- 

 bulence in the boundary layer. We consider a steady boundary layer, 

 with strictly parallel flow (all average quantities depend only on the 

 coordinate y). 



First by integrating the momentum equation (Eq. 8-2) from y to 8, 

 where 8 is the thickness of the boundary layer, we obtain 



P = Pe + PV^ 



= ^. (l + yMt §.^ (9-1) 



since Ml = peUl/ype. Here quantities without subscript are taken at the 

 coordinate y, while subscript e denotes the quantity at the edge of the 

 boundary layer. For future reference, superscript " denotes the total or 

 stagnation value, and subscript w denotes the value at the wall (y = 0). 

 The assumption of a constant pressure within the boundary layer is valid 

 if the free stream Mach number is of the order of 0(1), and if the previous 

 assumption of small turbulence level (jT^/U^ <$C 1) is made. 



For the derivation of energy relations, the following conventional 

 boundary conditions are used : 



Aiy = 0: U = 0, V ^ 0, u = 0, v = 0, w = 0, T = T^ (9-2a) 



At 2/ = 5: U = Ue,T = T, (9-2b) 



(90) 



