= Non-Boussinesq top-hat plume equations
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 $F=\pi b^2wg(\rho_0-\rho)$. With non-Boussinesq <Batchelor entrainment> $v_e=\alpha w\sqrt{\rho/\rho_0}$, their balances are
$$
\partial_t(Q^2/M)+\partial_zQ=2\alpha\sqrt{\pi\rho_0M},\qquad
\partial_tQ+\partial_zM=QF/M,\qquad
\partial_t(QF/M)+\partial_zF=0.
$$
Here $Q^2/M$ 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. https://doi.org/10.1017/S0022112006001212[Scase, Caulfield, Dalziel and Hunt's time-dependent plume model] derives the equivalent system with the common factor $\pi$ suppressed.
Back to article page
