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Linear vortex-sheet growing modes — worked interpretation

Example

On the periodic interface $x \in T_{2 π}$ , where $T_{2 π} = R / (2 π Z)$ , with half-planes $y > 0$ and $y < 0$ , take equal density $ρ_{0} = 1$ , base velocities $U_{+} = 1$ and $U_{-} = - 1$ , and no gravity or surface tension. For $t \geq 0$ , define $η (x, t) = e^{t} \cos x, ϕ_{+} (x, y, t) = e^{t - y} (\sin x - \cos x), ϕ_{-} (x, y, t) = e^{t + y} (\cos x + \sin x) .$ The potentials are harmonic and decay at their respective infinities; at $y = 0$ they satisfy $(\partial_{t} + U_{\pm} \partial_{x}) η = \partial_{y} ϕ_{\pm}$ and continuity of $p_{\pm}^{'} = - (\partial_{t} + U_{\pm} \partial_{x}) ϕ_{\pm}$ . Thus these fields give an exact real growing mode of the stipulated linear interface equations.

Facts & Assumptions

Given: The periodic interface equations, equal density, base speeds, mode fields and pressure definitions in [A1].

[A1]

$x \in T_{2 π}$ , $y > 0$ above and $y < 0$ below, $t \geq 0$ , $ρ_{0} = 1$ , $U_{+} = 1$ , $U_{-} = - 1$ , $Δ ϕ_{\pm} = 0$ , decay of $ϕ_{\pm}$ and their first derivatives at the corresponding infinities, $(\partial_{t} + U_{\pm} \partial_{x}) η = \partial_{y} ϕ_{\pm}$ at $y = 0$ , and $p_{\pm}^{'} = - (\partial_{t} + U_{\pm} \partial_{x}) ϕ_{\pm}$ with $p_{+}^{'} = p_{-}^{'}$ ; gravity and surface tension are absent.

[F1]

Linear vortex-sheet growing modes: for equal positive densities and the stated harmonic linear interface conditions without gravity or surface tension, a normal mode with $k > 0$ has $λ = - i k \overline{U} \pm k ∣ Δ U ∣ / 2$ ; nonzero velocity jump gives the positive real branch.

Verification

1.1givenA1algebra

The $x$ -dependence of each potential is a sine or cosine and its $y$ -dependence is $e^{- y}$ above or $e^{y}$ below, so $\partial_{x x} ϕ_{\pm} = - ϕ_{\pm}$ and $\partial_{y y} ϕ_{\pm} = ϕ_{\pm}$ . Hence $Δ ϕ_{\pm} = 0$ , and the exponential factors give the required decay.

1.2givenA1algebra

At $y = 0$ , $(\partial_{t} + \partial_{x}) ϕ_{+} = 2 e^{t} \sin x$ and $(\partial_{t} - \partial_{x}) ϕ_{-} = 2 e^{t} \sin x$ . Therefore $p_{+}^{'} = p_{-}^{'} = - 2 e^{t} \sin x$ , so pressure continuity holds.

2.1givenA1step 1.1algebra

At $y = 0$ , $(\partial_{t} + \partial_{x}) η = e^{t} (\cos x - \sin x) = \partial_{y} ϕ_{+}$ , while $(\partial_{t} - \partial_{x}) η = e^{t} (\cos x + \sin x) = \partial_{y} ϕ_{-}$ . Thus both kinematic conditions hold.

3.1givenA1F1algebra∎

Here $k = 1$ , $\overline{U} = 0$ , and $Δ U = 2$ . The growing branch in [F1] has $λ = 1$ , agreeing with the explicit factor $e^{t}$ in $η$ ; this is one exact linear-model example only.

Depends on

Linear vortex-sheet growing modes

Linear vortex-sheet growing modes — worked interpretation

Example

Facts & Assumptions

Verification

Depends on

Dependency tree · 2 levels