The jnomentum thickness is defined by: 



i/Uoo) Cl-u/Uoo) dz 



= r CW 



z=0 

 The loss of momentiam of the fluid in the boundary layer is given by: 



P u2 6^ • 

 Calculations by the Blasius (1910) theory show that for viscous flow: 



6 J = 1.7208 (vx/u„)^ , 



and 



&m = 0.664 (vx/uoo)' 



Schlichting (1968) presented experimental data which indicated that 

 the Blasius solution is satisfied within the limits of measurement. The 

 Blasius theory showed that the C^ of equation (1) must decrease with 

 increasing fetch because an increase in thickness of the boundary layer 

 leads to a decrease in the velocity shear near the boundary. 



Analytic solution of the steady-state boundary layer equations for 

 unstratified air have also been obtained for turbulent flow near a smooth 

 plate. Uoo is the airspeed just outside the boundary layer. These solu- 

 tions are reviewed by Schlichting (1968, ch. 21). The momentum thickness, 

 6^, for turbulent flow is given by: 



&„ = 0.036x(UooX/v)"-^'^^ . (13) 



Turbulence actually occurs in bursts and the instantaneous thickness 

 of the turbulent flow is variable. Thus, the boundary layer thickness 

 described here, is a meaningful concept only when the average over some 

 finite time is considered. Analytic solutions are not available for rough 

 surfaces, but niamerical techniques are possible. Wade and Debrule (1973) 

 used the method developed by Truckenbrodt and presented by Schlichting 

 (1968, ch. 22) to integrate the equations governing the turbulent flow near 

 ijoth smooth and rough boundaries. The numerical solutions agree with 

 equation (12) in indicating that the turbulent boundary layer thickness 

 increases nearly linearly (0.8 power) with x and decreases very slowly 

 as Uoo is increased. Presumably equation (12) , established analytically 

 for laboratory flows and flows past objects of finite size, breaks down 

 in the atmosphere when 6 reaches the value at which the pressure 

 gradient and Coriolis acceleration must be considered, e.g., at a value 



27 



