16 



is the friction velocity. The functional relationship satisfy- 



where u T = 



ing the turbulent portion of the inner flow is 



u T y, 



■= c, + c. 



log(-J-) 



[29] 



For simplicity of analysis, the above logarithmic formulation can be closely 

 approximated by the general power law 



i- *¥? 



[30] 



By virtue of the law of the wall, C and n Q are independent of pressure gradi- 

 ent and depend only on the Reynolds number. The applicability of the power- 

 law formulation is demonstrated in Figure 8 where a logarithmic plot of u/U 

 against y/0 is shown for various pressure gradients. It is seen that the 

 curves are straight and parallel up to about y = 9 . This situation can be* ex- 

 plained analytically from the power law as follows. Substituting y = 6 and 

 u = u into the power law gives 



u.e v 



Jii = c(— ) 



U T \ V I 



Dividing Equation [30] by [31] produces 



u. _ /_y\"° 



[31 



[32; 



log^= log^ +n o log-2 



[33] 



Since n does not depend on pressure gradient, the curves of log u/U should be 

 straight and parallel up to y = 6. 



Ludwieg and Tillmann demonstrated experimentally that the law of the 

 wall is also valid for analyzing the variation of the wall shearing stress in 

 a pressure gradient. From test data in the range of Reynolds number 

 1 x 10 3 ^R < 4 x 10 4 , they obtained the following expression for the varia- 

 tion of the local-skin-friction coefficient in a pressure gradient 



Jj* _ 0.0290 v1 



PU £ 



.0.268 



705 



where 



2.333 x 10 



-0.398H 



[34] 



[35: 



