Past exam of the mathematics course of the University of Cambridge / 2008 / iii / Paper 84 / 6

Use pi-normalized kinematic plume fluxes: physical volume, momentum and buoyancy fluxes are $\pi Q$, $\pi M$ and $\pi F$. For a top-hat plume model in a homogeneous ambient, constant entrainment coefficient $\alpha$ givesEliminating height between the first two equations yields $d(M^{5/2})/dQ=5FQ/(4\alpha )$. Hence the plume flux-balance invariant is $M^{5/2}-5F{Q}^2/(8\alpha )$. Pure plume balance means that this invariant vanishes, including at a finite-volume source:It does not mean that $Q_s$ and $M_s$ must individually be zero. The balanced solution has a plume virtual origin below the physical source:where $z_v=[Q_s/(C{F}_s^{1/3}){]}^{3/5}>0$. For given nonzero source fluxes, the displayed balance is a compatibility condition; they cannot be chosen independently and still describe an exactly balanced source.

Let $z_f(t)$ be the filling-box first front, with $z_f(0)=H$. Below it the original fluid has not yet been through the plume. In the filling box model, the turbulent plume adjusts rapidly and occupies a negligible fraction of the room's area. A point source with negligible source volume flux supplies $Q(z)=C{F}_s^{1/3}{z}^{5/3}$. All its volume crossing the front grows the processed region, so volume conservation givesPut $a_p=\pi C{F}_s^{1/3}/A_c$. Integrating $d(z_f^{-2/3})/dt=2a_p/3$ yieldsThe first front approaches the floor only as $t\to \infty$. A nonzero source volume flux would require an equal exhaust to keep the room volume fixed; it is treated in the following parts.

The balanced finite source has $Q(z)=C{F}_s^{1/3}(z+z_v{)}^{5/3}$ and $Q(0)=Q_s$. An ideal small opening exhausts the physical source volume $\pi Q_s$. When $z_f>h$, the opening is in the unprocessed lower region. Its outflow and the plume's entrainment both remove lower fluid, while all plume flux across the front enters the processed region. When $z_f<h$, the opening exhausts processed fluid, reducing the net growth of that region by $\pi Q_s$. Thus the finite-source filling-box front with a side opening obeysThe ideal exhaust switches fluid type as the front crosses its height. Above the opening the solution is $z_f=[(H+z_v{)}^{-2/3}+2a_{p}t/3{]}^{-3/2}-z_v$, and it reaches $h$ atBelow the opening, the implicit solution isThe model assumes an effective point opening with one-way source-volume exhaust; a tall opening supporting simultaneous density-driven exchange would need additional boundary-flow equations.

If $h>0$, the front enters the lower-opening branch before reaching the floor. Near $z_f=0$,The final approach is exponential and the time integral in part (c) diverges logarithmically. Thus no finite floor-arrival time exists. If the opening is at $h=0$, the first branch applies until arrival and has a finite nonzero downward speed at the floor. The arrival time isHence finite-time floor arrival occurs only for a floor opening, within the stated finite-source model. In the point-source limit $Q_s\to 0$, $z_v\to 0$ and this arrival time diverges, recovering part (b).

Assume both openings have positive area, no imposed wind pressure, a light buoyant plume, and a Boussinesq approximation. Write $S=\pi Q_s$ and $B_s=\pi F_s$ for physical source volume and buoyancy flux. Include discharge coefficients $C_H,C_0$; if ideal openings are intended, set these to one. In a steady two-layer displacement ventilation state with interface height $z_{\ast }$, let the floor inflow be $q_0$ and roof outflow $q_H$. Volume and buoyancy balances giveThe hydrostatic pressure head across the warm upper layer must equal the two opening losses:A positive unprocessed lower layer needs $q_0>0$. At the boundary where this layer disappears, $z_{\ast }=0$, the plume can entrain no lower ambient fluid, so $q_0=0$, $q_H=S$ and $g_u^{\prime }=B_s/S$. The critical condition is therefore $S^3=2C_H^2{A}_H^2{B}_{s}H$.

To determine the inequality, consider a fully processed room. Its steady reduced gravity is $g^{\prime }=B_s/S$. If $p_0$ is the interior-to-exterior pressure difference at the floor, source-volume blocking of fresh floor inflow requires $p_0\ge 0$. Both openings then exhaust, withThe right-hand side is increasing in $p_0$ and has its minimum at zero. Consequently a fully processed state exists exactly whenThis is source-volume blocking of displacement ventilation. At equality floor inflow vanishes; above it the floor can also exhaust and no fresh lower-layer supply is needed. Below it, the floor pressure must be negative, admitting fresh fluid and supporting a positive lower layer. In fact the steady interface equation has a unique positive root: its loss side increases with $z_{\ast }$ because $Q(z_{\ast })$ increases, while its buoyancy-head side decreases.

The floor-opening area affects the positive-interface state and its approach, but drops out of the blocking threshold because floor flux is zero at the threshold. Equal splitting of pressure head, or using a source-free effective opening area at this transition, would miss this distinction. If the floor opening is sealed, the fresh-inflow argument no longer applies: a finite source with only roof exhaust instead cycles the room asymptotically as in the ceiling-opening case of part (c).