Buoyant Motion of a Turbulent Thermal

TL;DR

This paper derives analytical expressions for the buoyant acceleration of thermals, linking it to their shape and buoyancy, and validates these with numerical simulations of turbulent thermals.

Contribution

It introduces a novel analytical framework connecting thermal shape, buoyancy, and acceleration, validated by direct numerical simulations.

Findings

Analytical acceleration for spherical thermals is 2/3 of buoyancy.

Ellipsoidal thermals' acceleration depends on aspect ratio.

Results accurately predict initial thermal motion in turbulent flows.

Abstract

By introducing an equivalence between magnetostatics and the equations governing buoyant motion, we derive analytical expressions for the acceleration of isolated density anomalies, a.k.a. thermals. In particular, we investigate buoyant acceleration, defined as the sum of the Archimedean buoyancy $B$ and an associated perturbation pressure gradient. For the case of a uniform spherical thermal, the anomaly fluid accelerates at $2B/3$, extending the textbook result for the induced mass of a solid sphere to the case of a fluid sphere. For a more general ellipsoidal thermal, we show that the buoyant acceleration is a simple analytical function of the ellipsoid's aspect ratio. The relevance of these idealized uniform-density results to turbulent thermals is explored by analyzing direct numerical simulations of thermals at $\textit{Re}=6300$. We find that our results fully characterize a thermal's initial motion over a distance comparable to its length. Beyond this buoyancy-dominated regime, a thermal develops an ellipsoidal vortex circulation and begins to entrain environmental fluid. Our analytical expressions do not describe the total acceleration of this mature thermal, but still accurately relate the buoyant acceleration to the thermal's mean Archimedean buoyancy and aspect ratio. Thus, our analytical formulae provide a simple and direct means of estimating the buoyant acceleration of turbulent thermals.