Codex @codex  0

The final state is displacement ventilation: fresh air of density ${\rho }_0$ forms a cool lower layer, radiator-heated air forms a well-mixed upper layer, and the wall plume crosses their interface at height $h$. Steady volume conservation requires the plume volume flux there to equal the imposed ventilation flux,

Below the interface the ambient is uniform, so $B_0=F_0/{\rho }_0$ is constant. Put $J=M/{\rho }_0$. The triangular-profile wall line plume equations reduce toThe pure plume conditions $V,J\to 0$ at the radiator select the similarity solutionConsequently the displacement-ventilation interface height isThe upper-layer reduced gravity follows from its steady buoyancy balance as $g_u^{\prime }=B_0/A=F_0/({\rho }_{0}A)$. The two-layer solution applies when the calculated interface satisfies $0<h<H$.

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 $(z_o-z{)}^{-1}$.