Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 345 / 3

Taking the curl of the Boussinesq momentum equation and using incompressibility gives the vorticity equationThe second term on the right is baroclinic vorticity generation. It is nonzero where a horizontal density gradient crosses the vertical gravitational acceleration.

In steady inviscid two-dimensional flow the spanwise vorticity obeysIntegrating this equation over the front-frame control volume converts the left side to vorticity flux through its upstream and downstream faces. For a sharp interface, the baroclinic source integrates to the circulation generated by the hydrostatic pressure jump, $g^{\prime }h$. With plug flow downstream, the resulting balance isVolume conservation in the front frame gives $u_{1}H=u_2(H-h)$. The undisturbed indoor air is stationary in the laboratory, so $u_1$ is the front speed:

With a long uniform corridor, constant depth, and negligible entrainment, the pressure head $g^{\prime }h$ driving the gravity current does not change as the nose advances. There is no growing geometric length in the local front balance, so dimensional analysis gives the constant velocity $U_f\propto \sqrt{g^{\prime }h}$ after the short release transient.

The energy-conserving full-depth lock-exchange flow is symmetric between the cold lower current and warm upper return current, soPart (b) then gives $u_2=\sqrt{g^{\prime }H}$ and the laboratory front speedDuring one opening, the cold current displaces the volumeThe same volume of warm air exits in the upper layer.

For an ideal gas in the Boussinesq limit,Each opening removes ${\rho }_0{c}_p{V}_{\mathrm{ex}}(T-T_0)$ of heat. With $n$ openings per hour, the mean removal rate is $(n/3600)$ times this quantity. Equating it to $\dot{q}$ and writing $\Delta T=T-T_0$ givesTherefore

Mechanical mixing destroys the sharp density interface that supports efficient displacement ventilation. It entrains warm air into the incoming cold current and cold air into the outgoing current, reducing both reduced gravity and the heat removed per exchanged volume. Stopping fans or other mixing while the door is open therefore preserves the two-layer lock-exchange flow and improves cooling efficiency.