Past exam of the mathematics course of the University of Cambridge / 2019 / ii / Paper 3 / 38A / a

Solution

Use the continuity equation to write the axial momentum equation in conservative form:

\frac{\partial}{\partial z} (r w^{2}) + \frac{\partial}{\partial r} (r u w) = ν \frac{\partial}{\partial r} (r \frac{\partial w}{\partial r}) .

Integrating from the axis to infinity, regularity at

r = 0

and decay of

u, w, w_{r}

r \to \infty

\frac{d M}{d z} = 0.

Let

W (z)

be the centre-line velocity scale and

δ (z)

the jet width. Conservation gives

M \sim W^{2} δ^{2}

, hence

W δ \sim M^{1/2}

. The axial inertial and viscous terms scale as

\frac{W ^{2}}{z} \sim \frac{ν W}{δ ^{2}},

W δ^{2} \sim ν z

. Combining the two relations gives

δ (z) \sim \frac{ν z}{M ^{1/2}}, W (z) \sim \frac{M}{ν z} .

Thus the width grows linearly and the centre-line speed decays as

z^{- 1}

. The volume flux scales as

Q (z) = \int_{0}^{\infty} r w d r \sim W δ^{2} \sim ν z,

so it increases downstream. By fluid entrainment, the extra flux must be ambient fluid drawn radially into the jet.

There are no discussions about this article yet.

There are currently no matching articles.