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

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

Outside, pressure is hydrostatic with slope $-{\rho }_{0}g$. Inside it has the same slope below $h_1$ and the shallower slope $-{\rho }_{1}g$ above $h_1$. Thus $\Delta p=p_{\mathrm{in}}-p_{\mathrm{out}}$ is a negative constant below the interface, rises linearly above it with slope ${\rho }_0{g}_{10}^{\prime }$, and is positive at the ceiling. The floor vent admits air and the ceiling vent exhausts it.

Let $C_d$ be a common discharge coefficient and define the effective opening areaThe ventilation flow isSteady volume and buoyancy balances requireTherefore $h_1=H/2$ precisely when the effective area is chosen asThe individual areas must additionally realize this $A^{\ast }$; their ratio fixes how the total pressure drop is divided between the floor and ceiling vents.

The weaker plume detrains into the middle layer at $h_2$, while the stronger plume passes through it and feeds the upper layer at $h_1$. Middle-layer volume balance isUsing $Q_i(z)=C_Q{F}_i^{1/3}{z}^{5/3}$ gives

The middle layer receives buoyancy flux $F_2$ and loses volume $Q_2(h_2)$ by entrainment into the strong plume. The upper layer ultimately receives both source buoyancy fluxes and loses volume $Q_1(h_1)$ through the ceiling. Hence

The interior-minus-exterior pressure is constant below $h_2$, rises with slope ${\rho }_0{\hat{g}}_{10}^{\prime }$ in the middle layer, and with slope ${\rho }_0{\hat{g}}_{20}^{\prime }$ in the upper layer. PutEqual ideal openings split the total pressure difference equally, and the common ventilation rate isThus the required equal vent areas are