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

The Boussinesq approximation replaces density by a common reference value ${\rho }_0$ in inertia, mass conservation, and pressure acceleration, while retaining small density differences in the gravitational buoyancy term. It gives incompressible flow and is appropriate here wheneven though those small differences drive the room-scale motion. It would fail for order-one thermal density contrasts or strongly compressible ventilation.

The Batchelor entrainment hypothesis sets the mean inflow speed across a turbulent plume edge to $\alpha$ times a representative axial plume speed, where $\alpha$ is the entrainment coefficient. It closes integral plume balances by relating plume growth to its speed. Applied here, it produces an entraining axisymmetric warm plume above the floor source and a one-sided cold wall line plume below the vent. Treating both as turbulent top-hat plume models neglects source regions, detailed profiles, wall friction, interaction between the two plumes, and the finite thickness of the density interface; these are the principal modelling assumptions.

Define the indoor-to-outdoor reduced gravityBecause the whole opening lies above the interface, its indoor side contains upper-layer fluid of density ${\rho }_H$. Under hydrostatic pressure, the pressure difference varies linearly about a neutral pressure level. Equal opening geometry and equal discharge coefficients for inflow and outflow put that level at the vent midpoint. At vertical distance $y$ from it, the ideal Bernoulli equation gives speed $\sqrt{2g_V^{\prime }|y|}$.

Writing $Q_V$ for either one-way volumetric flow rate, integration over one half of the opening givesThe total unsigned exchange is $2Q_V$. The ideal sharp-edged inviscid model has $C_d=1$; an empirical $C_d<1$ represents contraction and losses.

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, $B=Q_B{g}_B^{\prime }$.

Measure downward distance from the plume virtual origin byand let $\mathcal{B}$ be the magnitude of the cold plume's buoyancy flux per unit span. A one-sided wall line plume with width $b$, downward speed $W>0$, and reduced-gravity magnitude $g^{\prime }$ satisfiesThe pure plume solution isImposing $b(z_V)=H^{\prime }/2$ fixesThe constant speed is a special feature of a pure top-hat line plume; its width and volumetric flow rate grow linearly with downward distance while entrainment dilutes its density anomaly like $(z_o-z{)}^{-1}$.

LetThe global steady heat, or buoyancy flux, balance equates the floor-source input to the buoyancy carried out by the one-way upper-layer exhaust:The corresponding upper-to-lower density jump is fixed by the warm plume crossing the interface,The first balance also shows that the descending cold plume has buoyancy-flux magnitude per unit wall length

At a steady interface, the upward axisymmetric-plume volume flux equals the total downward wall-plume volume flux. Using parts (c) and (d),After cancellation of $B^{1/3}$, the required implicit geometric relation isThus the ideal steady interface fraction is independent of the source strength: increasing $B$ multiplies both opposing plume volume fluxes by $B^{1/3}$. The floor area $A$ and room height $H$ affect the transient filling time and admissibility of the assumed ordering, but not this steady integral balance, provided $0<h<z_V$ and the plumes remain separated.

SetCombining the single-opening exchange flow relation $Q_V=K_V\sqrt{g_H^{\prime }}$ with $B=Q_V{g}_H^{\prime }$ givesIt also gives $g_H^{\prime }=(B/K_V{)}^{2/3}$, after which the second density balance determines $g_L^{\prime }$.

Assume the two identical openings and their wall plumes behave symmetrically and do not interact before reaching the interface. If $Q_V^{(1)}$ is the one-way rate through either vent, the global buoyancy balance isEach wall plume therefore has buoyancy flux per unit span $\mathcal{B}=B/(2L)$. The steady volume balance now includes two descending plumes:Hence the interface height is determined implicitly byFor completeness, if each opening retains the same single-opening exchange flow coefficient $K_V$, then