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The Batchelor entrainment hypothesis takes the inward mean edge velocity of a turbulent plume to be $v_e=\alpha w$ in the Boussinesq approximation. Its density-corrected non-Boussinesq form is $v_e=\alpha w\sqrt{\rho /{\rho }_0}$, where $\rho$ and ${\rho }_0$ are plume and ambient mass densities and $\alpha$ is the entrainment coefficient.

In an impermeable cylinder, an axisymmetric turbulent plume with negligible source volume flux requires a downward return flow. With uniform upward plume velocity $w$ and ambient velocity $v$, volume conservation gives $b^{2}w+(R^2-b^2)v=0$. An entrainment coefficient $\alpha$ based on the relative speed gives $Q^{\prime }=2\alpha b(w-v)$. A specified pressure closure is essential for momentum conservation. If the departure from ambient hydrostatic pressure and wall force are neglected in the whole-section integral, the total specific momentum flux $J=b^2{w}^2+(R^2-b^2)v^2$ obeys $J^{\prime }=b^2{g}^{\prime }$. Homogeneous unmodified ambient and no buoyancy loss give $b^{2}w{g}^{\prime }=B_s$. These are leading integral-model closures; restoring the ambient dynamic pressure can change the singularity criterion.

A filling box model describes a turbulent plume rising through a confined fluid and spreading beneath its ceiling. Fluid entrainment transfers initially unmodified ambient fluid into the plume; the discharged fluid forms a descending filling-box first front. A maintained heat source can have negligible source volume flux while maintaining nonzero buoyancy flux. A finite injected fluid flux requires an equal exhaust if the container volume is fixed.

For a turbulent plume in a homogeneous ambient of density ${\rho }_0$, define actual sectional mass flux, momentum flux and density-weighted buoyancy flux by $Q=\pi b^2\rho w$, $M=\pi b^2\rho w^2$ and $F=\pi b^{2}wg({\rho }_0-\rho )$. With non-Boussinesq Batchelor entrainment $v_e=\alpha w\sqrt{\rho /{\rho }_0}$, their balances areHere $Q^2/M$ is sectional mass, $QF/M$ is sectional density-deficit buoyancy, and the entrained ambient initially has zero vertical momentum. The last equation follows by subtracting mass conservation from ${\rho }_0$ times volume conservation. Scase, Caulfield, Dalziel and Hunt's time-dependent plume model derives the equivalent system with the common factor $\pi$ suppressed.

Use $G=g\beta (T-T_0(z))$ for the plume reduced gravity, and take $z$ upwards. For a symmetric line plume define fluxes per unit source length,The integrals span the plume; its edges can move horizontally with $z$. The fluid entrainment rate at one moving edge is the inward normal transport relative to that edge. For an edge $x=b(z)$ it is $E=w{b}^{\prime }-u$; symmetry supplies equal transport through the other edge. The Batchelor entrainment hypothesis closes its mean value as $E=\alpha w_c$, where $w_c$ is a representative vertical velocity and $\alpha$ is the entrainment coefficient. The integrated volume conservation equation is therefore $q^{\prime }=2\alpha w_c$.

Using volume conservation, the vertical acceleration equation has conservative form ${\partial }_x(uw)+{\partial }_z(w^2)=G$. Incoming ambient has zero vertical momentum; the integrated momentum conservation equation is consequently $m^{\prime }=\int Gdx$. For the temperature anomaly, $T_0$ depends on height andAmbient fluid enters with $G=0$, giving the buoyancy flux balance. The integral model isHere $N$ is the buoyancy frequency when $N^2>0$; the same algebra permits unstable temperature gradients with $N^2<0$.

These are mean integral balances. The advective equations printed in the question omit explicit turbulent stresses and heat fluxes; a turbulent plume interpretation requires the usual entrainment closure for that unresolved transverse transport. Assuming vanishing edge stresses and heat fluxes, and negligible additional axial turbulent-flux contributions, gives the displayed bulk model. It would be inconsistent to treat a discontinuous top-hat profile as a smooth exact pointwise solution of the inviscid temperature-advection equation.

In the top-hat plume model, vertical velocity $w(z)$ and reduced gravity $G(z)$ are uniform over $-b<x<b$. The ambient has zero vertical velocity, negligible ambient turbulence and a prescribed $T_0(z)$. Thenand the equations becomeEquivalently, the ordinary differential equations for the three unknowns areThe last equation displays dilution by fluid entrainment as well as loss of relative buoyancy while rising through a stably stratified ambient. The model assumes a steady slender two-dimensional turbulent plume, large Reynolds number, the Boussinesq approximation, constant $\beta$ and $\alpha$, negligible heat loss, negligible imposed crossflow, and negligible pressure departure from ambient hydrostatic pressure. The factors of two count both entraining edges; $b$ is a half-width. A finite source also requires its volume, momentum and buoyancy fluxes as boundary data.

The Batchelor entrainment hypothesis assumes that the mean ambient inflow normal to the edge of a turbulent turbulent plume is proportional to its representative axial velocity, $v_e=\alpha w$. With top-hat plume model fields and circumference $2\pi b$, the added volume per height is $2\pi b\alpha w=2\pi \alpha \sqrt{M}$. It is an integral turbulent mixing closure; it does not assert that every instantaneous eddy has the same inward speed. Use constant positive $\alpha$, uniform resting ambient, Boussinesq approximation, negligible drag and no buoyancy loss.

Since $B^{\prime }=0$, $B=B_0$. Eliminate $z$ between the volume and momentum balances:The zero-source-flux condition gives $M^{5/2}=5B_0{Q}^2/(8\alpha )$, the pure plume balance. Substitution in $Q^{\prime }=2\alpha \sqrt{M}$ and integration givesRecovering the top-hat fields from $b=Q/\sqrt{M}$, $w=M/Q$ and $g^{\prime }=B/Q$ yieldsThe paper's ratio is thereforeThis is the constant inverse-normalized ratio used here, not the catalog's normalized plume balance parameter, which is one for a pure plume. The zero-flux point source is a limiting similarity origin: the speed and density anomaly diverge there, so a physically finite source must regularize the near field. The expressions are valid in the region where the Boussinesq approximation and turbulent closure apply.

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.

Use the steady axisymmetric top-hat plume model, with plume radius $b(z)$, uniform mean upward speed $w(z)>0$ and reduced gravity $g^{\prime }(z)=g[{\rho }_a(z)-{\rho }_p(z)]/{\rho }_0$. The Boussinesq approximation uses constant reference density ${\rho }_0$ in inertia and retains the small density difference only in buoyancy. Assume a quiescent hydrostatic ambient, a slender fully turbulent plume, negligible molecular transport and an entrainment velocity $\alpha w$, with constant positive entrainment coefficient $\alpha$. Define the ambient buoyancy frequency by $N^2=-(g/{\rho }_0)d{\rho }_a/dz$; positive $N^2$ is stable stratification.

The integrated volume, momentum and buoyancy balances of a Boussinesq top-hat plume in a stratified ambient areThe first follows from side entrainment, the second from the integrated buoyancy force, and the third accounts for the changing ambient density. Common factors of $\pi$ are cancelled. Equivalently, the requested radius, speed and reduced-gravity equations areThese equations stop describing an upward light plume once $w$ vanishes, and their light-plume interpretation requires $g^{\prime }>0$.

For nonzero $C$, seek plume similarity in power-law stratification in the form $b=\beta z$, $w=A{z}^n$ and $g^{\prime }=G{z}^m$. The first balance gives $\beta =2\alpha /(n+2)$. Momentum gives $m=2n-1$ and $G=2(n+1)A^2$. The buoyancy balance then requires $3n=2+n+p$, soChoosing $A>0$ gives the exact power-law family, including its normalization:These are the amplitudes and physical branches of a power-law stratified plume. No source flux was specified, so an arbitrary finite-source solution is not determined by $C,p$ alone; the displayed family is an exact similarity solution, rather than the general solution for arbitrary initial fluxes.

With the specific flux convention $Q=b^{2}w$, $M=b^2{w}^2$ and $B=b^{2}w{g}^{\prime }$, their values areThe physical volume flux is $\pi Q$, mass flux is ${\rho }_0\pi Q$, momentum flux is ${\rho }_0\pi M$, and kinematic buoyancy flux is $\pi B$. Thus the “specific mass flux” is the mass flux divided by reference density, with or without the explicitly stated factor $\pi$.

The parameter restrictions follow directly from positivity. Positive radius needs $p>-6$. Positive $g^{\prime }$ and real positive $w$ further require $p>-4$ and $-C/(3p+8)>0$, givingFor stable $C>0$, buoyancy flux decreases with height, while volume and momentum fluxes increase. However, $B\propto z^{4+3p/2}$ diverges at $z=0$ throughout this interval. Also the ambient density gradient and plume speed are singular there. Thus this stable similarity branch can describe a region away from a virtual source, but cannot represent a finite-buoyancy-flux point source extending all the way to $z=0$. For ordinary regular stable stratifications, including positive $p$, a finite-source rising plume instead loses buoyancy and eventually reaches neutral buoyancy and a finite rise limit; the pure-power ansatz does not supply such a solution.

For unstable $C<0$ and $p>-8/3$, the buoyancy flux grows with height and tends to zero at the origin. The plume draws its increasing buoyancy from an unstable ambient rather than a finite positive source buoyancy flux. Such an ambient must be maintained or treated locally, since it is itself convectively unstable. If $p\le -1$, its density profile is singular at the source, while for $p>-1$ its increasing density contrast eventually violates the Boussinesq approximation at sufficiently large height. These are formal local similarity solutions with corresponding physical limits. For $C<0$ and $-6<p<-4$, the algebra also gives a real positive speed but negative $g^{\prime }$: this is a heavy upward flow, not the specified light plume.

At nonzero $C$, the values $p=-6,-4,-8/3$ do not give finite coefficients. Respectively they would require zero entrained volume-flux growth, zero momentum-flux growth despite nonzero buoyancy, or constant buoyancy flux despite nonzero ambient stratification. They are not omitted regular endpoints.

When $C=0$, the ambient is unstratified regardless of the nominal value of $p$. The nontrivial positive-buoyancy point-source solution is the Boussinesq point-source plume:Here $B_0>0$ is the constant specific buoyancy flux with $\pi$ removed. Its $Q\propto z^{5/3}$ and $M\propto z^{4/3}$ vanish at the point source, while $B=B_0$. Its near-source singularity is the usual ideal point-source limit; a finite vent replaces that limit by finite source data or a virtual origin.

Write $b=g(\rho -{\rho }_0)/{\rho }_0>0$ for the reduced gravity. The Boussinesq approximation requires $|\rho -{\rho }_0|\ll {\rho }_0$: replace density by ${\rho }_0$ in inertia and retain its difference in the gravitational force. A shallow-water approximation additionally requires a small aspect ratio, negligible leading-order normal acceleration and approximately hydrostatic pressure relative to the ambient fluid:Incompressibility makes the normal velocity smaller than the downslope velocity by the aspect ratio.

Thinness alone does not force the downslope velocity to be uniform across the layer. The additional top-hat plume model assumes that turbulent mixing makes both $u(x,t)$ and $b(x,t)$ approximately uniform inside it, with a sharp transition to the ambient fluid. Then the depth-integrated volume and momentum fluxes are $hu$ and $h{u}^2$. This is the same integral closure used for a turbulent plume, applied to a shallow layer; the hydrostatic pressure term is retained because gravity has a component normal to the channel.

Let $q_b$ be floor inflow and $q_t$ roof outflow. Use effective-area discharge $A\sqrt{\delta p/\rho }$, matching the factors in the supplied relations. With negligible source volume, both equal $q$. The lower region has exterior mass density while the warm upper layer has reduced gravity $g^{\prime }$ and depth $H-h$. hydrostatic pressure therefore supplies a total opening pressure head $g^{\prime }(H-h)$. Equal openings sharing equal discharge take half each, givingSteady lower-layer volume conservation sets this ventilation rate equal to the entrained turbulent plume flux at the interface, $q=\lambda B^{1/3}{h}^{5/3}$. Upper-layer buoyancy conservation gives $B=q{g}^{\prime }$. These are the required balances for displacement ventilation:Eliminating $g^{\prime }$ and $q$ gives the displacement-ventilation interface height equationThe left side is strictly increasing from zero to infinity, so there is one interface height. It depends on opening geometry and plume entrainment, not on $B$; the resulting $g^{\prime }=B^{2/3}/(\lambda h^{5/3})$ and throughflow do depend on $B$.

For finite source volume $Q$, conservation instead gives $q_t=q_b+Q$. The two pressure drops are no longer equal:Let $q_p(h;Q)$ be a physically entraining source plume: $q_p(0;Q)=Q$ and $q_p(h;Q)>Q$ for $h>0$. Its interface balance is $q_p=q_t=Q+q_b$. Therefore, as $q_b$ reaches zero, a positive-depth lower layer has no replenishment and cannot remain steady: this is no steady displacement layer without ambient supply. The limiting interface is $h=0$. At the threshold $q_t=Q$ and $g^{\prime }=B/Q$, and the entire hydrostatic head drives the roof discharge. HenceThis is source-volume blocking of displacement ventilation. The half-head factor of the negligible-source state must not be kept after the floor inflow vanishes. In the usual physical-area convention $q=A_{\mathrm{phys}}{C}_d\sqrt{2\delta p/\rho }$, the equivalent formula is $Q_c=(2C_d^2{A}_{\mathrm{phys}}^{2}BH{)}^{1/3}$.

The quoted pure-plume law cannot be used unmodified down to the origin for a source with nonzero volume: it would incorrectly give zero flux there. A compatible finite-source example uses a virtual origin,It recovers the supplied pure-plume limit when $Q\to 0$, and gives the same blocking threshold. The complete finite-$Q$ interface trajectory depends on that plume model, but the threshold needs only source-volume conservation and positive entrainment. For $Q>Q_c$, the assumed steady regime with a lower inflow no longer exists.

The Batchelor entrainment hypothesis models turbulent ingestion of ambient fluid through the plume edges. A self-similar turbulent plume has a spreading rate proportional to its axial velocity scale; the hypothesis represents the unresolved mixing by an inward edge speed proportional to that scale. For the prescribed centreline velocity, the Boussinesq approximation givesThe entrainment coefficient $\alpha$ is dimensionless and depends on the chosen velocity/profile convention and mixing regime. Here $e$ is the inward entrainment speed relative to the moving plume boundary, and the factor two counts its two sides.

A suitable density-corrected extension is $e=\alpha W\sqrt{\rho /{\rho }_0}$, representing the reduction in ambient ingestion for a much lighter non-Boussinesq plume. It reduces to the boxed closure when $|\rho -{\rho }_0|\ll {\rho }_0$. This is a phenomenological model, not an exact identity for arbitrary turbulent flow. Woods's non-Boussinesq plume model uses this density correction. The balances in part (c) can be written with either closure before taking their common Boussinesq limit.

Define $K=N_s^2{z}_s^{-\beta }>0$. The buoyancy frequency definition gives $F_z=-KQ{z}^{\beta }$. The powers in a plume similarity in power-law stratification satisfySolving givesA rising positive-buoyancy turbulent plume entrains, so $Q_z>0$; its momentum flux increases, so $M_z>0$; its buoyancy flux decreases in the stable ambient, so $F_z<0$. With positive coefficients, this requires $q>0$, $m>0$ and $f<0$. The strongest restrictions areIn this interval $1<q<5/3$, $0<m<4/3$ and $-2<f<0$.

Coefficient matching makes the positive similarity solution explicit:soIt is an ideal scaling for $z>0$, often a far-field description. As $z\to 0$, the prescribed ambient stratification and this model's buoyancy flux are singular; a finite physical source would require a regularized near-source region.

Turbulence is irregular fluid motion with interacting velocity fluctuations over a range of scales. It can mix a particle suspension and promote fluid entrainment into a turbulent plume.