Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 345 / 3 / c / Solution

For plume radius $r$, top-hat speed $U$, and plume reduced gravity $g_B^{\prime }$, define the volumetric flow rate, momentum flux, and buoyancy fluxThe integral balances for a steady axisymmetric pure plume in a uniform lower layer areThe source is at the plume's virtual origin $z=0$. Solving these equations givesIt is useful to defineThen the remaining similarity laws take the compact formThe second relation also follows immediately from conservation of buoyancy flux, $B=Q_B{g}_B^{\prime }$.