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For a one-layer shallow water flow the Froude number is $F=u/\sqrt{g^{\prime }h}$. Supercritical flow has $F>1$, so both characteristic speeds $u\pm \sqrt{g^{\prime }h}$ point downstream. Subcritical flow has $F<1$, allowing information to travel upstream. At hydraulic control the flow passes through $F=1$ at a geometry-induced critical section; that condition fixes discharge in relation to upstream head when the exit is free.

For escaping light air, reflect the usual lower-layer picture vertically: a lowered doorway roof acts as an inverted broad-crested weir. The lower return flow can be ignored dynamically only while it has a sufficiently large available area and small velocity. If the warm interface lies far below the doorway top, the cool inflow occupies a substantial dynamically active fraction of the opening; its inertia and pressure cannot be replaced by a motionless deep ambient. A two-layer exchange model is then required. If the warm layer reaches below the doorway bottom, the assumed separate lower replacement path is lost entirely.

Outside a freely controlled doorway the outgoing current becomes supercritical and then rises rapidly. That upturn violates the small-vertical-acceleration shallow water approximation, but it does not alter the upstream discharge as long as it does not submerge or back-pressure the control. Information cannot return through the supercritical outflow to the critical section. A blocked conservatory or a drowned doorway is different and can move the control upstream.

The room's nearly motionless upper reservoir has interface $z_i=H-h$. A doorway roof at height $D$ has available light-layer head $\delta =D-z_i=h-(H-D)>0$. Evaluate the Bernoulli equation in the room and at a freely controlled doorway: $E=-g^{\prime }{z}_i$ upstream and $E=3g^{\prime }{h}_c/2-g^{\prime }D$ at its critical section. Therefore $h_c=2\delta /3$ andThis is the ideal broad-crested weir law, not a small-orifice law with $C_d$. A nonideal doorway would require its own measured loss correction.

When the floor heat source is on, volume conservation and buoyancy flux conservation still give $Q=B_0^{1/3}{z}_i^{5/3}/\gamma$ and $g^{\prime }=\gamma B_0^{2/3}{z}_i^{-5/3}$. Combine these with the doorway relation to obtain the plume-fed doorway ventilation equilibrium:There is one interface-height root in the indicated interval. The result is physically appropriate only if its upper outflow is thin enough for the one-layer doorway model to remain valid.

If $D<H$, air in the volume above the doorway is part of the upper thermal reservoir and can store heat and alter transients. But its extra depth $H-D$ does not add available head at the doorway: only $D-z_i$ enters the critical condition. In the steady ideal well-mixed model it shares the same density as the outflow. Treating the whole $h$ as the doorway head would overestimate discharge.

For a pure plume with $g_p^{\prime }=\gamma B^{2/3}{z}^{-5/3}$, a steady sharp interface at $z_i$ has $Q_p=B^{1/3}{z}_i^{5/3}/\gamma$ and warm-layer reduced gravity $g^{\prime }=B/Q_p$. A freely discharging upper doorway of height $D$ and width $W$ has available head $D-z_i$. Equating plume supply to the broad-crested weir flux gives the displayed implicit height relation, with $0<z_i<D$. Warm air above the doorway stores heat but adds no available head at its crest in this uniform-layer model.