= Confined plume with compensating return flow
{title2=$Q=b^2w,\quad v=-Q/(R^2-b^2)$}
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 $b^2w+(R^2-b^2)v=0$. An <entrainment coefficient> $\alpha$ based on the relative speed gives $Q'=2\alpha b(w-v)$. 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> $J=b^2w^2+(R^2-b^2)v^2$ obeys $J'=b^2g'$. Homogeneous unmodified ambient and no buoyancy loss give $b^2wg'=B_s$. These are leading integral-model closures; restoring the ambient dynamic pressure can change the singularity criterion.
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