Nonlinear steady two-layer interfacial flow about a submerged point vortex

Two-dimensional, two-layer steady interfacial flow about a point vortex is studied in a uniform stream for each layer. The upper layer is of finite depth with a rigid lid on the upper surface, and the depth of the lower layer is assumed infinite. The point vortex is located in lower-layer fluid. We study this problem using not only a linear analytical method but also a nonlinear numerical method. A linear solution is derived in terms of a complex exponential integral function. The fully nonlinear problem is formulated by an integro-differential equation system. The equation system is solved using Newton’s method to determine the unknown steady interfacial surface. The numerical results of the downstream wave are provided by a linear solution and fully nonlinear solution. A comparison between linear solutions and nonlinear solutions shows that the nonlinear effect is apparent when the vortex strength increases. The effects of point vortex strengths, Froude numbers, and density ratios on the amplitudes of the downstream waves are studied. We analyze the effects of point vortex strengths, Froude numbers, and density ratios on the wavelengths of the downstream waves.