Vorticity generation and conservation on generalised interfaces in three-dimensional flows

Abstract

This article presents a three-dimensional theory of vorticity creation on generalised interfaces, including both non-slip and free-slip boundaries, which generalises a previous two-dimensional formulation (Terrington et al., J. Fluid Mech., vol. 890, 2020, p. A5). Under this description, vorticity may be created on a boundary by the inviscid relative acceleration between fluid elements on each side of the boundary, driven by either tangential pressure gradients or body forces. Viscosity acts to transfer circulation between the vortex sheet representing the slip velocity on the interface, and the fluid interior, but is not responsible for the creation of vorticity on the interface. This formulation also describes a principle of vorticity conservation for interfacial and free-surface flows: in many flow configurations, the net generation of vorticity on the interface is zero, and the total circulation remains constant throughout flow evolution.

Vortex ring connection to a free surface

“…We remark that typically, vorticity and vorticity gradients exhibit a discontinuity across the interface (Terrington et al 2022). However, under the VOF method, this discontinuity is spread over a region near the interface where 0 < F < 1, which leads to difficulty in determining the boundary vorticity flux at the interface.…”

“…In several formulations of vorticity dynamics (Lundgren & Koumoutsakos 1999;Brøns et al 2014;Terrington et al 2020Terrington et al , 2022, a vortex sheet is included at the free surface, so that the total circulation is conserved. In three dimensions, Terrington et al (2022) define the local density of vector circulation in the interface vortex sheet above a free surface as…”

“…During the connection to the free surface, vorticity lost from the lower fluid is transferred into the interface vortex sheet ( § 4.2). The interface vortex sheet also satisfies a generalised solenoidal property (Terrington et al 2022)…”

“…Taking the curl of the momentum equation yields a transport equation for vorticity (the Helmholtz equation). For an incompressible Newtonian fluid, this equation is expressed as If this expression is integrated across a control volume (V), then these dynamical processes can be interpreted as boundary fluxes (Terrington et al 2022):…”

“…Recently, we have developed a new formulation for the generation and conservation of vorticity in three-dimensional interfacial flows (Terrington, Hourigan & Thompson 2022), based on an earlier two-dimensional description (Brøns et al 2014;Terrington, Hourigan & Thompson 2020). This formulation has two main advantages over existing descriptions of interfacial vorticity dynamics.…”