rejecting terras involving small orders of magnitude. For steady incompress- 

 ible flow with negligible longitudinal curvature, the turbulent -boundary-layer 

 equations of motion are found to be 



6x + v Sy " p dx + p ay W l J 



for < y < 6 and < u < U. Here u and v are the x- and y-components of the 

 velocity within the boundary layer, parallel and normal to the surface of the 

 body respectively, U is the velocity at the outer edge of the boundary layer, 

 d is the thickness of the boundary layer, p is the density of the fluid, and 

 r is the total shearing stress within the fluid. It is to be noted that the 

 velocities and pressures indicated in Equation [1 ] represent time averages of 

 the turbulent quantities in the flow. In addition to two-dimensional flow, 

 Equation [1 ] is also applicable to axisymmetric flow when the boundary-layer 

 thickness 6 is small relative to the transverse radius of the solid boundary 



V 



The accompanying equation of continuity for two-dimensional flow is 



ax- + ey =0 t2a] 



and for axisymmetric flow it is 



Su , dv , u w . ro , i 



al + ey- + ^air= ° [2b l 



for 6 « r w . 



Prom the boundary-layer equation of motion, Equation [1], and the 

 equation of continuity, Equation [2], the von Ka'rman momentum equation is de- 

 rived by integrating each term of [1] with respect to the y-coordinate. The 

 von Karman equation represents the basic working equation for calculating the 

 growth of boundary layers in the presence of pressure gradients. For purposes 

 of analysis the momentum equation is best expressed in terms of the displace- 

 ment thickness 6*, the momentum thickness 6 and the shape parameter H which 

 are defined as 





and 



