B,15 • POWER LAWS 



(Eq. 13-9) with a = 0.75, and n = ^ based on the power law of incom- 

 pressible skin friction to compare with experiments. It is seen that more 

 experiments at higher Mach numbers are needed in order to understand 

 skin friction better and to formulate better theories. The data of Fig. 

 B,14a are plotted according to [1 8,29, 3£, 34,38, 39, 42, 62, 63, 641 



Fig. B,14b, B,14c, and B,14d illustrate the variations of the experi- 

 mental skin friction coefficients C/ and Cf with the Reynolds numbers, 

 based on the free stream conditions and the momentum thickness. It is 

 interesting to see whether the experiments would follow some semi- 

 empirical laws of skin friction. For this purpose, Eq. 13-11 is plotted for 

 n = 2.58, A = 0.455, and a = 0.75 in Fig. B,14b, and we see that Eq. 

 13-11 is in quite good agreement with experiments. 



CHAPTER 4. GENERAL TREATMENT OF 

 INCOMPRESSIBLE MEAN FLOW ALONG WALLS 



B,15. Power Laws. In attempting to deal with turbulent flows con- 

 fined within pipes and channels or bounded on one side by a wall, much 

 attention was given in the older literature to power laws. These were 

 found to be very useful in that they could be made to approximate ob- 

 served mean velocity distributions and to yield resistance laws that were 

 reasonably correct over a limited Reynolds number range. These laws 

 are, of course, purely empirical, but they have not lost their usefulness 

 when one wishes to express the general character of a velocity profile in 

 a pipe or boundary layer, or wishes to make an estimate of skin friction. 

 We should, however, be mindful of their limitations. 



The power laws stem from the Blasius resistance formula for smooth 

 straight pipes of circular cross section [68]. They were found, however, 

 to be transferable to two-dimensional channels with parallel walls and 

 two-dimensional boundary layers, if the radius of a pipe, the half-width 

 of a channel, and the thickness of a boundary layer were regarded as 

 equivalent dimensions, and if velocities were referred to those at the 

 center or free stream. In all cases the walls are assumed to be smooth. 

 Since the detailed development is available elsewhere [69], only the main 

 steps are given here. Because the condition of incompressibility has been 

 imposed, the physical properties of the fluid are independent of the flow 

 and constant for any set of conditions. Hence we may simply denote the 

 density by p, the viscosity by n, and the kinematic viscosity by p. 



Since it is not necessary to distinguish among the flows in pipes, 

 channels, and boundary layers in bringing out the elemental aspects of 

 power formulas, the distance from the wall is expressed by y, the velocity 

 at the center or in the free stream by Ue, and the value of y where the 

 velocity is C/e by 8. The only constraint on the flow considered is the 



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