10 



and finally 



77 = 1 " [hThTT)]" [15] 



The curve for the relation just derived for 77(H) is drawn on Figure 4 and 

 shows close agreement with the test data of Gruschwitz . Likewise the curve 

 for the power law for the velocity profiles may be expressed in the form 



Curves drawn from this equation in Figure 5 show close agreement with the data 

 of von Doenhoff and Tetervin. From the experimental evidence just considered, 

 it is seen that the general power law is an excellent approximation for the 

 velocity profiles of turbulent boundary layers. 



After the particular parameter for the shape of the velocity profile 

 has been selected, its variation in a pressure gradient remains to be deter- 

 mined. It is now recognized that the rate of variation of this shape param- 

 eter and not the parameter itself is dependent upon the local-pressure- 

 gradient parameter and upon other local parameters of the boundary layer. 

 However, an early study by Buri 9 in 1931 on accelerated and retarded flows in 

 closed conduits considered the shape of the velocity profile to depend direct- 

 ly on the pressure gradient. 



The parameter used by Buri for the shape of the velocity profile is 



_ _ e dU R i/4 r ■, 



r= UoT R e [17] 



where R g = — and v is the kinematic viscosity of the fluid. Actually r is a 

 local parameter for the pressure gradient and hence can only roughly represent 

 the shape of the velocity profile whose development depends on the previous 

 history of the flow. The parameter /"as used by Buri in his analysis has 

 merit in providing a simplified method for calculating the momentum thickness 

 6 approximately. When r is substituted into the von Karman momentum equation, 

 Equation [6a], the following expression is obtained 



where f = — — R . From experimental evidence, Buri linearized this ex- 



P U 2 " 



expression to 



IX 4 *) = a " br M91 



