A. S. Iberall 



Because of the low Mach number and the long isothermal wall system, the ef- 

 fects of any minor cross-channel temperature distribution will be disregarded. 



The boundary condition commonly invoked— see, for example, Laufer [8] — 

 relates the pressure gradient to the shearing stress at the wall. Typically, this 

 is achieved by integration of the appropriate NS momentum equation written in 

 the form of Reynolds stresses. Assuming, for parallel-plate flow, that the mean 

 values of the quadratic terms involving the fluctuating components have no axial 

 variation, it is first shown that the pressure gradient g has no cross-channel 

 variation and then that the first integral of the equation of motion is 



X + = U. ,. W 



dx 



( I) ( 1) 



Thus, at the walls, 



dRo 

 dx 



at X = +1 



In addition, there is a second condition which is not commonly noted. The 

 mean momentum equation in the z direction is 



g + 





W 

 "'(1) 



dx^ 



Thus, a second boundary condition is 



g + =0 at X = ±1 . 



dx2 



Laufer demonstrates quite satisfactorily (Ref . [8], Figs. 8 and 19) that the 

 shear stress obtained from the velocity gradient and the pressure gradient are 

 in accord with the first boundary condition, and consistent with the known devi- 

 ations near the wall of the von Karman logarithmic velocity law ([8], Fig. 7). 

 The relation 



is approached at the wall (the common normalization based on the friction ve- 

 locity which is computed from the wall shear). Another experimental study [9] 

 presents more detail on the flow field near the wall. 



Consistent with Laufer 's data and the universal von Karman curves. Fig. 1 

 depicts the character of the mean velocity distribution near the wall in terms of 

 the properties of qp, qp', and cp" in an attempt to clarify the boundary conditions. 



This may be transformed into the more familiar parameters of the loga- 

 rithmic presentation. In terms of the variables of this paper, the variation 

 near the wall was estimated to be 



710 



