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At steady state, the displacement-ventilation interface lies where the total plume volume flux equals the imposed ventilation flux. For one triangular-profile wall line plume with kinematic buoyancy flux $B_0$, entrainment coefficient $\alpha$, and ventilation flux $A$ per unit span,

Write $\hat{\rho }(z)$ for the ambient density far from the wall and $\rho (z)$ for the density at the wall. Across the plume, let $y=x/b$ and use the prescribed triangular profilesDirect integration gives the volume flux, mass flux, momentum flux, and density-weighted buoyancy flux, all per unit radiator length:These coefficients distinguish the triangular-profile wall line plume from a top-hat plume model.

The Batchelor entrainment hypothesis takes the inflow speed through the plume's exposed outer edge to be $\alpha W$, where $\alpha$ is the entrainment coefficient. A wall plume has only one such edge, so $dV/dz=\alpha W$.

The Boussinesq approximation replaces density by a constant reference value ${\rho }_0$ in inertia and mass flux while retaining the small density deficit in buoyancy. It requires $|\hat{\rho }-\rho |/{\rho }_0\ll 1$. A sufficiently hot radiator can violate this near the source, where thermal expansion is large and the developed-plume description may also fail.

Put $B_0=F_0/{\rho }_0$ for the kinematic buoyancy flux per unit length. Dimensional analysis for a line plume givesThe plume rise time is therefore $t_{\mathrm{rise}}\sim H/B_0^{1/3}$. Changing the room stratification requires a plume volume comparable with $XH$, so $t_{\mathrm{room}}\sim XH/V(H)\sim X/B_0^{1/3}$. HenceThe plume consequently follows the slowly changing ambient through a quasi-steady approximation when $X\gg H$.

Choose ${\rho }_0$ as a representative room density, for example the fresh-air density, and neglect relative density variations everywhere except in buoyancy. Withthe triangular profiles giveSolving these algebraic relations,

The Batchelor entrainment hypothesis, vertical momentum conservation, and mass conservation giveAn ascending parcel entrains ambient fluid from progressively lower ambient density. With the buoyancy frequencythe change of ambient reference density subtracts $Q{N}^2$ from its density-weighted buoyancy flux, so

The source supplies the kinematic buoyancy flux $B_0=F_0/{\rho }_0$. The unstratified line plume velocity scale is $B_0^{1/3}$. Comparing this with the stratification time scale $N_0^{-1}$ gives the stratified line-plume height scaleIf $z_s\gg H$, the initial plume is only weakly affected by the stable density stratification and reaches the ceiling much like an unstratified wall plume. If $z_s\sim H$, its buoyancy flux falls substantially during the rise; it approaches neutral buoyancy near the upper room, overshoots because of its momentum flux, and spreads as a horizontal buoyant intrusion. If $z_s\ll H$, the plume reaches neutral buoyancy low in the room and forms a low intrusion after a modest overshoot. In each sketch the plume widens by entrainment, while increasing $N_0$ lowers the neutral-buoyancy and overshoot heights.

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 $g_L^{\prime }$.

A pure plume is a similarity regime in which the source supplies buoyancy flux but no dynamically persistent source volume or momentum flux. Its finite source region is represented by a plume virtual origin.

A line plume of kinematic buoyancy flux $B_0$ rising through uniform buoyancy frequency $N$ has the dimensional height scaleIt measures the height over which ambient stratification removes an order-one fraction of the plume buoyancy before the plume reaches neutral buoyancy and spreads as a buoyant intrusion.

A top-hat plume model takes velocity and reduced gravity to be uniform across the plume and zero outside it. Integral volume, momentum, and buoyancy fluxes then depend only on plume width and these uniform values.