ON THE EQUATION-OF-MOTION OF A THIN-LAYER OF UNIFORM VORTICITY

A higher order extension to Moore's equation governing the evolution o f a thin layer of uniform vorticity in two dimensions is obtained. The equation, in fact, governs the motion of the center line of the layer and is valid for consideration of motion whereby the layer thickness is uniformly small compared with the local radius of curvature of the center line. It extends Birkoff's equation for a vortex sheet. The equ ation is used to examine the growth of disturbances on a straight, ste ady layer of uniform vorticity. The growth rate for long waves is in g ood agreement with the exact result of Rayleigh, as required. Further, the growth of waves with length in a certain range is shown to be sup pressed by making this approximate allowance for finite thickness. How ever, it is found that very short waves, which are quite outside the r ange of validity of the equation but which are likely to be excited in a numerical integration of the equation, are spuriously amplified as in the case of Moore's equation. Thus, numerical integration of the eq uation will require use of smoothing techniques to suppress this spuri ous growth of short wave disturbances.