 Jet limit of the top-hat plume model

With zero source volume flux and zero buoyancy flux but positive source momentum flux, the axisymmetric top-hat plume model reduces to a turbulent round jet. For full-cross-section kinematic fluxes, put $c=2{\alpha }_e\sqrt{\pi }$. Entrainment gives $Q^{\prime }=c\sqrt{M}$, while momentum conservation gives $M=M_0$. Hence $Q=c\sqrt{M_0}z$, speed $w=\sqrt{M_0}/(cz)$ and radius $r=cz/\sqrt{\pi }$. A finite source flux replaces $z$ by $z+Q_0/(c\sqrt{M_0})$. The ideal zero-flux source has singular speed and is a far-field model, not a finite nozzle.