Past exam of the mathematics course of the University of Cambridge / 2004 / iii / Paper 75 / 6

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.

Two source scales obtained from dimensional analysis areThe jet length $L_M$ measures how far source momentum remains important relative to buoyancy-generated momentum. The source-volume length of a turbulent plume $L_Q$ measures how far imposed source volume remains important relative to entrained volume. Numerical constants and entrainment coefficient factors depend on the chosen normalization. From the equations here the useful exact volume-growth scale is ${\ell }_Q=[Q_0^3/(20{\alpha }^4{B}_0){]}^{1/5}$; the pure-similarity height whose flux equals $Q_0$ is $5{\ell }_Q/3$. A momentum crossover likewise contains a factor of order ${\alpha }^{-1/2}{L}_M$. The source radius $Q_0/\sqrt{M_0}$ is another useful geometrical combination, but it does not replace the two buoyancy/source crossover comparisons.

Integrate the differential relation from part (a) with finite source values:DefineThis plume flux-balance invariant rearranges toUse it to express $\sqrt{M}=[5B_0{Q}_0^2/(8\alpha ){]}^{1/5}[(Q/Q_0{)}^2-c{]}^{1/5}$, and substitute into $Q^{\prime }=2\alpha \sqrt{M}$. The coefficient satisfies $(2\alpha /Q_0{)}^5[5B_0{Q}_0^2/(8\alpha )]=20{\alpha }^4{B}_0/Q_0^3$, givingWith positive source fluxes, $c<1$. For $c<0$ the source has excess momentum relative to pure plume balance, so it is a momentum-dominated forced plume. For $c=0$ it is already in pure plume balance, and the exact solution is the pure similarity translated to a finite plume virtual origin. For $0<c<1$ the source has deficient momentum relative to its buoyancy and volume fluxes, a lazy plume. Here “forced” in the broad sense merely means finite source fluxes; this sign classification distinguishes its jet-like and lazy branches. At large height, finite $c{Q}_0^2/Q^2$ vanishes and every positive branch approaches pure plume balance.

Differentiate $b=Q/\sqrt{M}$ and use both balances:Substituting $Q^2=8\alpha M^{5/2}/(5B_0)+c{Q}_0^2$ givesFor $c<0$, the radius grows faster than the pure-plume slope $6\alpha /5$ and its growth rate decreases toward that slope as $M$ increases. For $c=0$, it grows at that slope exactly. For $0<c<1$, it grows more slowly, possibly initially contracting, and its slope increases toward $6\alpha /5$. Since $M^{\prime }=B_{0}Q/M>0$ and $M\to \infty$, all branches satisfy $db/dz\to 6\alpha /5$ as $z\to \infty$.

Initial necking requiresThus a sufficiently lazy plume initially narrows as buoyancy accelerates its fluid faster than fluid entrainment supplies additional volume. It still has increasing volume and momentum fluxes; shrinking radius does not imply volume loss. At equality the initial slope is zero, and for $c>3/5$ the slope later becomes positive once $M^{5/2}=5c{B}_0{Q}_0^2/(12\alpha )$. This gives a finite minimum radius before the eventual pure plume regime.