Measure downward distance from the <plume virtual origin> by
$$
s=z_o-z>0,
$$
and let $\mathcal B$ be the magnitude of the cold plume's <buoyancy flux> per unit span. A one-sided <wall line plume> with width $b$, downward speed $W>0$, and reduced-gravity magnitude $g'$ satisfies
$$
\frac d{ds}(bW)=\alpha W,
\qquad
\frac d{ds}(bW^2)=bg'=\frac{\mathcal B}{W},
\qquad
bWg'=\mathcal B.
$$
The <pure plume> solution is
$$
b(z)=\alpha(z_o-z),
\qquad
W(z)=\left(\frac{\mathcal B}{\alpha}\right)^{1/3},
\qquad
g'(z)=
\left(\frac{\mathcal B}{\alpha}\right)^{2/3}
\frac1{z_o-z}.
$$
Imposing $b(z_V)=H'/2$ fixes
$$
\boxed{z_o=z_V+\frac{H'}{2\alpha}}.
$$
The constant speed is a special feature of a pure top-hat line plume; its width and <volumetric flow rate> grow linearly with downward distance while entrainment dilutes its density anomaly like $(z_o-z)^{-1}$.

Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 345 / 3 / d / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2024/iii/paper-345/3/d/solution