u* = V^oTp » tne shear stress velocity, 



T = the shear stress 



p = the density 



K = 0.4 - the universal constant of turbulence 



z = the roughness height, (cm). 



In = the logarithm to the base e, and log is the logarithm to 

 to the base ten. 



Over rough walls, the value z is directly related to the measurements 

 of wall-roughness. Thus, Prandtl gives, for sand roughness, z = k/30, 

 where k represents the average diameter of the sand grains glued on it. 



For flow over smooth walls, the transition between the smooth wall 

 and the turbulent air flow is not abrupt, bur rather is tempered by a 

 laminar boundary layer of a thickness of approximately 1 mm, and there 

 exists according to v. Karman^ 7 ^ a somewhat modified form of the logarithmic 

 law of wind velocity, which however could be transformed into Equation (1) 

 if we defined roughness value z , which is proportional to the inverse of 

 the shear stress velocity u* (z° = 1/u*). These roughness values are 

 several orders of magnitude smaller than the usual ones for rough walls. 



Motzfeld investigated in a wind channel the flow over various wave 

 shapes (sine, trochoid, and waves with sharp crests) and drew the follow- 

 ing conclusions. 



Over wave profiles with sharp crests the flow separates from the 

 boundary at the crests; eddies are formed at the leeward side of the 

 wave. The average vertical distribution of wind velocity is represented 

 by the equation: 



u = 5.75 u* log (7.25 z/h); z = h/7.25 (2) 



The roughness parameter, z , is proportional to the wave height h. We 

 observe thus a relationship similar to the case of rough walls. 



At wave profiles with round crests (sine, trochoid) Motzfeld found 

 no separation of the flow; rather there resulted an average distribution 

 of wind velocity, which is similar to that of a smooth surface 



tan a 

 o o 3u* 



y 3 17 l!i_ 



u s 5.75 u* log (z/z ), where z = -r-j- 10 * u* (3) 



wherein V is the kinematic viscosity of the air and tan a the maximum 

 slope of the wave profile. Since the variation of the exponent remains 

 small, the roughness z is essentially inversely proportional to the shear 

 stress velocity u*, where the coefficient is dependent upon the wave form 



(tan a ). To be sure, this influence of the wave form is relatively small. 



m 



This result permits the conclusion that for wave forms with round 

 crests the frictional component of the flow resistance (caused by tangential 



