Past exam of the mathematics course of the University of Cambridge / 2023 / ii / Paper 4 / 38C (Fluid Dynamics II)

The unperturbed fluid wake is the horizontal strip $-h<y<h$: the fluid is stationary there and moves to the right with velocity $U{\mathbf{e}}_x$ above and below it. Its two shear layers are the straight lines $y=\pm h$. After the symmetric disturbance, draw the upper layer at $y=h+\eta (x,t)$ and the lower one as its reflection at $y=-h-\eta (x,t)$, so the local wake width is $2(h+\eta )$.

Write the velocity potentials in the upper, middle and lower domains as ${\varphi }_{+}$, ${\varphi }_0$ and ${\varphi }_{-}$. Irrotational flow and incompressible flow imply the Laplace equationThe far-field boundary conditions are

Every shear layer is a material interface. The exact kinematic boundary condition at $y=h+\eta$ isand at $y=-h-\eta$ it isHere each equation is evaluated on the indicated moving interface.

Finally, pressure continuity is the dynamic boundary condition for an inviscid interface. The Unsteady Bernoulli equation givesafter absorbing spatially constant functions of time into the potentials.

The displacement must have small relative amplitude and small slope,Write the potentials as perturbations of the base flow,with $|\nabla {\phi }_{\pm }|/U$ and $|\nabla {\phi }_0|/U$ small. A linearization evaluates the conditions at the unperturbed lines $y=\pm h$ and discards products of perturbations. At $y=h$ this giveswhereas at $y=-h$ it givesThe perturbation potentials remain harmonic functions, and their gradients decay at the corresponding far fields.

The linearized equations are homogeneous and have constant coefficients in $x$ and $t$. Consequently, spatial Fourier modes evolve independently, and the superposition principle reconstructs arbitrary sufficiently regular initial disturbances from them. It is therefore enough to seek the normal modesHere the real $k$ is the wavenumber, $\sigma$ is the possibly complex temporal exponent, and taking the real part produces a physical displacement.

Put $q=|k|$. Decay at infinity and the symmetry of the disturbance suggest the mode potentialsThe middle potential is an even function of $y$ because its vertical derivative must take opposite values at the two interfaces. The upper kinematic conditions giveThe lower conditions give the same two relations. The upper dynamic condition isEliminating $A$, $B$ and ${\eta }_0$ yields the dispersion relationWriting $C=\coth (qh)$, its two roots areFor every real $k\ne 0$, $C>0$ and one root has the positive growth rateThus every nonzero wavenumber is linearly unstable in this inviscid vortex sheet model; $k=0$ is neutral.

When $|k|h\gg 1$, the two shear layers are separated by many disturbance decay lengths. Since $\coth (|k|h)\to 1$, the roots reduce toThis is exactly the Kelvin-Helmholtz instability of one isolated vortex sheet separating equal-density streams of speeds $U$ and $0$: the pattern travels at their mean speed $U/2$, while its unstable amplitude grows at rate $|k|U/2$. The exponentially weak interaction between the two layers disappears in this short-wavelength limit.