Landwebev 



(rt - s2)p2 + K[t(l +p2) + r(l +q2) - 2pqs] p + K^ = (11) 



where 



K = [ 1 + p2 + q2p. 



Other relations between the geometric parameters of the 

 orthogonal coordinate system based on the lines of principal curva- 

 ture are given in [ 26] » 



VIII. EQUATIONS OF VORTICITY IN A BOUNDARY LAYER 



Lighthill [ l] makes a convincing case for the primary im- 

 portance of vorticity in a boundary layer. If the vorticity is known, 

 the velocity field cam be calculated by the Biot-Savart law. Secondly, 

 vorticity is diffused and convected more gradually than other fluid 

 properties and hence is more readily determinable numerically. 

 From the mathematiczil point of view, Lighthill implies that it is 

 easier to solve the diffusion equation for vorticity than the boundary- 

 layer momentum equations governed by an outer irrotational flow, 



Sherman [ 30] has also been impressed by Lighthill's views, 

 euid has contributed a more mathemiatical discussion of "sources of 

 vorticity, " Neither he nor Lighthill, however, have formulated the 

 vorticity equations for a three-dimensional boundary layer. This 

 will now be undertaken. 



The Navier-Stokes equations for an incompressible fluid may 

 be written in the vector form 



~ - V X cj + grad (-^ v • v +^ + gzj = - v curl co (12) 



where v is the velocity at a point ol the fluid, w = curl v Is the 

 vorticity, t denotes time, p is the pressure, p the mass density, 

 g the acceleration of gravity, z is a vertical coordinate, positive 

 upwards, and v is the kinematic viscosity. An immediate conse- 

 quence of (12), obtained by applying the nonslip condition at the wall 

 surface S, Is 



g 



rad(-2 + gz j = - V curl co on S (13) 



which relates the vorticity at the wall to the pressure gradients of 

 the flow outside the boundary layer. By taking the curl of the mem- 

 bers of Eq, (12) we obtain the Helmholtz vorticity diffusion equation 



460 



