 Axisymmetric pure plume

Consider a steady, axisymmetric pure plume in a uniform ambient fluid, using the Boussinesq approximation and uniform top-hat cross-sectional profiles. If $r(z)$ is its radius, $U(z)>0$ its vertical velocity and $g^{\prime }(z)>0$ its reduced gravity, define the physical fluxesThey are respectively volumetric flow rate, momentum flux per reference mass density, and buoyancy flux. Only $B$ is conserved in this unstratified, source-free model. Fluid entrainment adds ambient volume to $Q$, and buoyancy increases $M$.

For constant entrainment coefficient $\alpha >0$, the Batchelor entrainment hypothesis gives an inward boundary speed $\alpha U$. Volume, vertical momentum and buoyancy balances therefore giveTo find a pure similarity solution, put $Q\propto z^p$ and $M\propto z^q$ relative to the plume virtual origin. The first balance gives $p-1=q/2$; the second gives $q-1=p-q$. Thus $p=5/3$, $q=4/3$, andIn particular $r=Q/\sqrt{\pi M}$ grows linearly while $U=M/Q$ decreases as $z^{-1/3}$. A finite source shifts the virtual-origin coordinate. Dividing all three fluxes by $\pi$ changes their balance coefficients, so pi-normalized and physical-flux formulas must not be mixed.