A theory of three-dimensional interfacial vorticity dynamics

Abstract

A three-dimensional theory of vorticity dynamics on an incompressible viscous and immiscible fluid–fluid interface, or interfacial vorticity dynamics for short, is presented as a counterpart of the vorticity dynamics on an arbitrarily curved rigid wall [J. Fluid Mech. 254, 183 (1993)]. General formulas with arbitrary Reynolds numbers Re are derived for determining (1) how much vorticity exists on an interface S, (2) how much vorticity is created from S and sent into the fluid per unit area in per unit time, and (3) the force and moment acted on a closed interface by the created vorticity thereon. The common feature and fundamental difference between interfacial vorticity dynamics and its rigid-wall counterpart are analyzed. In particular, on a free surface, the primary driving mechanism of vorticity creation is the balance between the shear stress (measured by tangent vorticity) and the tangent components of the surface-deformation stress alone, which results in a weak creation rate of O (Re−1/2) at large Re. Therefore, the exact form of the theory with its full complexity is of importance mainly at low Reynolds numbers, especially in understanding the small-scale coherent structures of interfacial turbulence. The vorticity creation rate at high-Re approximations, including an interfacial boundary layer of finite thickness and the limit of Re→∞ (the so-called Euler limit), is also studied, both allowing for a rotational inviscid outer flow. While for the former this leads to a generalization of Lundgren’s theory [in Mathematic Aspects of Vortex Dynamics, edited by R. E. Caflish (SIAM, Philadelphia, PA, 1989), pp. 68–79] and amounts to solving a linear boundary-layer problem, for the latter the creation rate can be directly obtained from an inviscid solution, leading to a dynamic evolution equation of interfacial vortex sheet. In three dimensions, a vortex sheet may bifurcate into a normal vorticity field, upon which the dependence of the sheet velocity is determined. A few examples are examined to illustrate different aspects and approximation levels of the general theory.