Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 345 / 3 / a / Solution

Batchelor entrainment models a turbulent plume as drawing ambient fluid inward at a speed proportional to its characteristic vertical velocity. In a top-hat axisymmetric plume, letbe volume, momentum, and buoyancy flux. The Boussinesq approximation uses a common density in inertia and volume conservation while retaining the small density difference in $g^{\prime }=g({\rho }_0-\rho )/{\rho }_0$. It requires $g^{\prime }\ll g$ and becomes inaccurate for very hot source fluid or near openings with large density changes.

For a point source in an unstratified lower layer, the integral plume equations arewhere $\alpha$ is the entrainment coefficient. Their pure-plume solution iswith