Hydrodynamic interaction in rotational flow

Abstract

The interaction of water waves with arrays of vertical cylinders problem is studied using diffraction of water waves and addition theorem for bessel functions. The linear boundary value problem which is derived from physical assumptions is used as the approximate mathematical model for time-harmonic waves. Linearization procedure is described for the nonlinear boundary conditions on the free surface. The problem is solved by using Addition theorem for Bessel functions. Limiting case, k 0, known as long wave approximation, is analysed using limiting forms of Bessel functions. Vortex-cylinder interaction is analyzed using a similar technique involving Laurent series expansions of complex velocity and the Circle Theorem. But this method failed to work. Further analysis is necessary. Vortex dynamics is analysed in annular domains, which can conformally be mapped into infinite domain with two cylinders, using the terminology of q-calculus. Finally, the result of vortex-cylinder interaction in annular domain is transformed into the infinite domain with two cylinders using conformal mapping. Image representation clearly shows the mechanism of inverse images which accumulate at zero and infinity in the w-plane and a and 1/a in the z-plane.