B • TURBULENT FLOW 



obtained from the equation of motion with only first order terms in the 

 boundary layer approximation, is known as the Karman momentum equa- 

 tion, and is expressed as 



ax 2 2 qdx 



where q is the dynamic pressure in the free stream where the pressure is p. 

 This equation gives a synoptic description of boundary layer development 

 and is independent of detailed processes. The relation between the various 

 quantities in the equation does, however, depend on the mechanics of the 

 turbulent diffusion process. 



When the pressure gradient is positive (adverse) and large, the second 

 term on the right-hand side of Eq. 19-3 may, and usually does, become 

 large compared to C//2. For this condition the growth of d with x depends 

 primarily on internal momentum losses resulting from the expenditure of 

 tangential forces against those portions of the stream which are retarded 

 by pressure gradient and which, by the action of the force, progress to 

 higher pressures but do not gain momentum equivalent to the forces ex- 

 pended. When a boundary layer exists, a pressure rise can be negotiated 

 only by the loss of momentum. A reduction of c/ by pressure gradient is 

 not an indication that drag is reduced. 



When dd/dx in Eq. 19-3 is due largely to the pressure gradient term, 

 it is obvious that c/ cannot be accurately determined from measurements 

 of dd/dx. It is now generally recognized that Eq. 19-3 is unsuited for this 

 purpose when pressure gradients assume appreciable values. Not only is 

 the accuracy poor but totally unreahstic values of c/ have been indicated. 

 Several explanations have been offered having to do with the neglected 

 terms in the equation of motion, but it now appears in the hght of 

 Clauser's experience [88] that departures of the flow from two-dimen- 

 sionality are largely responsible. 



The universal character of the law of the wall has suggested itseK as 

 a useful and reliable means of obtaining local skin friction coefficients 

 from measured velocity distributions. It seems that the first pubfished 

 recognition of this occurs in the paper by Clauser [83], who devised the 

 following procedure and used it in the analysis of his experimental results. 



Using Ur = Ue ■\/cf/2, the following expressions are written: 



T Ue\C/ V J/ \ 2 



With these and Eq. 19-1 he obtained the family of curves shown in Fig. 

 B,19d having C/ as the parameter. Application of the figure to a determi- 

 nation of Cf merely requires the placing of a measured velocity distribu- 

 tion thereon and reading off the value of C/, interpolating where necessary. 

 It is still necessary to measure velocities within a short distance of o, 



{ 134 > 



