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

Solution

r

, top-hat speed

U

g_{B}^{'}

Q_{B} = π r^{2} U, M_{B} = π r^{2} U^{2}, B = π r^{2} U g_{B}^{'} .

The integral balances for a steady axisymmetric pure plume in a uniform lower layer are

\frac{d Q _{B}}{d z} = 2 π α r U, \frac{d M _{B}}{d z} = π r^{2} g_{B}^{'} = \frac{B}{U}, \frac{d B}{d z} = 0.

The source is at the plume's virtual origin

z = 0

. Solving these equations gives

r (z) = \frac{6 α}{5} z,

U (z) = (\frac{25 B}{48 π α ^{2}})^{1/3} z^{- 1/3} .

C_{P} = \frac{6 α}{5} (\frac{9 α}{10})^{1/3} π^{2/3} .

Then the remaining similarity laws take the compact form

Q_{B} (z) = C_{P} B^{1/3} z^{5/3}, g_{B}^{'} (z) = C_{P}^{- 1} B^{2/3} z^{- 5/3} .

The second relation also follows immediately from conservation of buoyancy flux,

B = Q_{B} g_{B}^{'}

There are no discussions about this article yet.

There are currently no matching articles.