B • TURBULENT FLOW 



By integrating Eq. 24-1 from y = Qioy = h, using Eq. 24-2 to elimi- 

 nate V, and Bernoulli's equation to express p in terms of the local free 

 stream velocity U^, the Karman integral relation is obtained. The inte- 

 grals turn out to be the well-known expressions for 5* and Q. By intro- 

 ducing these and their ratio H, the Kdrman momentum equation is ob- 

 tained. It may be written as follows by again using Bernoulli's equation 

 to restore p: 



M ^ (H + 2) edp T^ . . 



dx 2 qdx^ 2q ^ ' 



where q = ^pUl. 



Eq. 24-3 is the starting point for most known methods. These pro- 

 ceed on the basis of some empirically determined form parameter for 

 the velocity profile. The earlier methods such as those of Buri [98] and 

 Gruschwitz [99] seem now to be mainly of historical interest. Gruschwitz's 

 method and his shape parameter, 



= 1 



WeA = 



found considerable use, but both have now been largely replaced by the 

 method of von Doenhoff and Tetervin [84], or variations of it, employing 

 H only. 



Von Doenhoff and Tetervin [84] made what appears to be the most 

 thorough search for a suitable form parameter. This resulted in the 

 adoption of the parameter H and the single parameter family of profiles 

 shown in Fig. B,19a. It is now clear from evidence previously cited that 

 all profiles do not fit this pattern, and that any method based on such 

 an assumption cannot be expected to give correct results under all con- 

 ditions. Nevertheless the method of von Doenhoff and Tetervin has had 

 certain successes and has appeared sufficiently promising to lead others 

 to attempt to improve upon it. 



The method is based on the assumption that it is only necessary to 

 determine 6 and H in order to establish the boundary layer character- 

 istics. Since the momentum equation (Eq. 24-3) alone is not sufficient for 

 this purpose, an auxiliary expression for H was set up. Recognizing that 

 a sudden change in pressure should not produce a discontinuity in the 

 velocity profile, it was assumed that the rate of change of H rather than 

 H itself would depend on local forces, t^ and dp/dx. When the ratio of 

 these forces was expressed by 



edq2q 



q dx T^ 



it was found that ddH/dx was a function of this ratio and also, to some 

 extent, of H itself, but it was independent of Reynolds number. Using 

 the Squire and Young formula (Eq. 18-10) for r^, thereby ignoring any 



< 154) 



