38C (Fluid Dynamics II)

Take

k > 0

; reflection in

x

supplies the result for negative wavenumber. Since each region has uniform base velocity, the perturbation is an incompressible, irrotational flow with a velocity potential satisfying the Laplace equation. Its decaying outer forms are

ϕ_{+} = A_{+} e^{- k (y - h)} e^{ik x + σ t}, ϕ_{-} = A_{-} e^{k (y + h)} e^{ik x + σ t} .

cosh (k y)

sinh (k y)

At the upper interface, the linearized kinematic boundary condition is

σ η = \partial_{y} ϕ_{+} (h), (σ + ik U) η = \partial_{y} ϕ_{in} (h) .

ϕ_{+} (h) = - σ η / k

. The second gives

ϕ_{in} (h) = \frac{σ + ik U}{k} η {coth (kh), tanh (kh), varicose, sinuous .

p^{'} = - ρ (\partial_{t} + U \partial_{x}) ϕ

σ^{2} + F (σ + ik U)^{2} = 0, F = {coth (kh), tanh (kh), varicose, sinuous .

σ = \frac{k U}{1 + F} (\pm F - i F) .

Equivalently,

σ_{var} σ_{sin} = \pm \frac{k U}{2} 1 - e^{- 4 kh} - \frac{ik U}{2} (1 + e^{- 2 kh}), = \pm \frac{k U}{2} 1 - e^{- 4 kh} - \frac{ik U}{2} (1 - e^{- 2 kh}) .

The plus sign is the unstable Kelvin-Helmholtz instability branch. In particular, the two modes have exactly the same growth rate,

σ_{R} (k) = \frac{k U}{2} 1 - e^{- 4 kh} .

Its limiting forms are

σ_{R} \sim U h k^{3/2} (kh \to 0), σ_{R} \sim \frac{U k}{2} (kh \to \infty) .

Thus neither type grows faster in this equal-density idealization, although their phase speeds differ. The unbounded large-

k

growth is the familiar short-wave pathology of an inviscid vortex sheet.

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