The Lyman–Huggins interpretation of enstrophy transport

Abstract

The Lighthill–Panton and Lyman–Huggins interpretations of vorticity dynamics are extended to the dynamics of enstrophy. There exist two competing definitions of the vorticity current tensor, which describes the flow rate of vorticity in the fluid interior, and the corresponding boundary vorticity flux, which represents the local vorticity creation rate on a boundary. It is demonstrated that each definition of the vorticity current tensor leads to a consistent set of definitions for the enstrophy current, boundary enstrophy flux and the enstrophy dissipation term. This leads to two alternative interpretations of vorticity and enstrophy dynamics: the Lighthill–Panton and Lyman–Huggins interpretations. Although the kinematic evolution of the vorticity and enstrophy fields are the same under each set of definitions, the dynamical interpretation of the motion generally differs. For example, we consider the Stokes flow over a rotating sphere, and find that the flow approaches a steady state where, under the Lyman–Huggins interpretation, there is no enstrophy creation or dissipation. Under the Lighthill–Panton interpretation, however, the steady-state flow features a balance between the continuous generation and subsequent dissipation of enstrophy. Moreover, the Lyman–Huggins interpretation has previously been shown to offer several benefits in understanding the dynamics of vorticity, and therefore it is beneficial to extend this interpretation to the dynamics of enstrophy. For example, the Lyman–Huggins interpretation allows the creation of vorticity, and therefore enstrophy, to be interpreted as an inviscid process, due to the relative acceleration between the fluid and the boundary.

Effects of surface roughness on the drag coefficient of spheres freely rolling on an inclined plane

“…For non-zero Re, there is an additional contribution to the drag coefficient known as wake drag (C D,wake ), which is approximately independent of G/D (Houdroge et al 2023;Terrington et al 2023). Houdroge et al (2023) provide the following empirical expression for the wake drag coefficient, valid for 5 < Re < 300, based on numerical simulations performed at G/D = 0.005: C D,wake = 1.70 − 0.136(log 10 Re) − 0.0716(log 10 Re) 2 .…”

“…To effectively capture a value of comparable to experimental results, gap heights in the range of – are required, which are not computationally feasible to carry out over a large range of . To address these numerical difficulties, Terrington, Thompson & Hourigan (2023) proposed a combined analytic and numerical approach for a two-dimensional cylinder translating and rotating adjacent to a plane wall. Similar to the method used by Houdroge et al.…”