Forced plume

A forced plume has both imposed momentum flux and positive buoyancy flux. Its near-source region behaves like a jet, while its far field approaches a pure plume. The crossover is measured by a jet length.

For source kinematic momentum flux $m_0$ and buoyancy flux $f_0>0$, dimensional analysis gives $L_J=m_0^{3/4}/f_0^{1/2}$. Under constant Batchelor entrainment, the actual turning length also contains an entrainment coefficient factor of order ${\alpha }^{-1/2}$.

An inclined forced plume curves upward under buoyancy. In a stationary homogeneous ambient, its horizontal momentum flux is conserved and its vertical momentum flux increases. For kinematic plume fluxes, $m_x^{\prime }=0$ and $m_z^{\prime }=fq/m$, where $m=\sqrt{m_x^2+m_z^2}$ and the derivative is with respect to centreline arc length.

For zero source volume flux, constant Batchelor entrainment, and a horizontal source, the near-source trajectory is cubic: $z\sim \alpha \sqrt{\pi }{f}_0{x}^3/(3m_0^{3/2})$. Far from the source the centreline approaches a vertical asymptote, with its horizontal distance from that asymptote proportional to $z^{-1/3}$.