Linear vortex-sheet growing modes

For $x,t\in \mathbb{R}$, assume the fields are independent of a third spatial coordinate. Let $H_{+}=[0,\mathrm{\infty })$ and $H_{-}=(-\mathrm{\infty },0]$. The real-valued interface displacement is a $C^{\mathrm{\infty }}$ map $\eta :{\mathbb{R}}_x\times {\mathbb{R}}_t\to \mathbb{R}$, and the real-valued scalar potentials are maps ${\varphi }_{\pm }:{\mathbb{R}}_x\times H_{\pm }\times {\mathbb{R}}_t\to \mathbb{R}$. Each ${\varphi }_{\pm }$ is $C^{\mathrm{\infty }}$ on the open half-domain; ${\varphi }_{\pm }$ and their first partial derivatives ${\mathrm{\partial }}_x{\varphi }_{\pm }$, ${\mathrm{\partial }}_y{\varphi }_{\pm }$, and ${\mathrm{\partial }}_t{\varphi }_{\pm }$ admit continuous one-sided traces at $y=0$, locally in $(x,t)$. On the open half-domains use ordinary pointwise Euclidean derivatives in $(x,y)$ and set $\mathrm{\Delta }={\mathrm{\partial }}_{xx}+{\mathrm{\partial }}_{yy}$. Let $U_{+},U_{-}\in \mathbb{R}$ be constant horizontal base velocities, and let the two half-planes $y>0$ and $y<0$ have the same constant density ${\rho }_0>0$. The potentials satisfy $\mathrm{\Delta }{\varphi }_{\pm }=0$ and, for every fixed $(x,t)$, ${\varphi }_{\pm }$, ${\mathrm{\partial }}_x{\varphi }_{\pm }$, ${\mathrm{\partial }}_y{\varphi }_{\pm }$, and ${\mathrm{\partial }}_t{\varphi }_{\pm }$ tend to zero as $y\to +\mathrm{\infty }$ and $y\to -\mathrm{\infty }$ respectively. The stipulated linear interface equations, interpreted by one-sided traces at $y=0$, are $({\mathrm{\partial }}_t+U_{\pm }{\mathrm{\partial }}_x)\eta ={\mathrm{\partial }}_y{\varphi }_{\pm }$ and continuity of the defined pressure perturbations $p_{\pm }^{\mathrm{\prime }}(x,t)=-{\rho }_0[{\mathrm{\partial }}_t{\varphi }_{\pm }+U_{\pm }{\mathrm{\partial }}_x{\varphi }_{\pm }{]}_{y=0^{\pm }}$. Gravity and surface tension are absent. Complexify this real linear system for the following normal-mode calculation.

For every real $k>0$, every nonzero complex amplitude $H$, and every $A_{+},A_{-},\lambda \in \mathbb{C}$, consider a mode of the form $\eta =H{e}^{ikx+\lambda t}$, ${\varphi }_{+}=A_{+}{e}^{-ky}{e}^{ikx+\lambda t}$, ${\varphi }_{-}=A_{-}{e}^{ky}{e}^{ikx+\lambda t}$. Set $\bar{U}=(U_{+}+U_{-})\mathrm{/}2$ and $\mathrm{\Delta }U=U_{+}-U_{-}$. The mode equations require

$\lambda =-ik\bar{U}\pm \frac{k\mathrm{\mid }\mathrm{\Delta }U\mathrm{\mid }}{2}\mathrm{.}$

If $\mathrm{\Delta }U\mathrm{\ne }0$, one root has positive real part $k\mathrm{\mid }\mathrm{\Delta }U\mathrm{\mid }\mathrm{/}2$, and its real or imaginary part gives a real exponentially growing solution of the stated linear system.

For the $2\pi$-periodic interface, require the fields to be periodic in $x$ and set ${\mathbb{T}}_{2\pi }=\mathbb{R}\mathrm{/}(2\pi \mathbb{Z})$. For a complex-valued trigonometric polynomial $\eta$ on this torus, define ${\widehat{\eta }}_j=(2\pi {)}^{-1}{\int }_0^{2\pi }\eta (x)e^{-ijx}dx$ and, for $m\ge 0$, $\mathrm{\parallel }\eta {\mathrm{\parallel }}_{H^m}^2={\sum }_{j\in \mathbb{Z}}(1+j^2{)}^m\mathrm{\mid }{\widehat{\eta }}_j{\mathrm{\mid }}^2$. Let $H^m({\mathbb{T}}_{2\pi };\mathbb{C})$ be the completion of the complex-valued trigonometric polynomials in this norm; real-valued fields are viewed as elements of its real subspace. If $\mathrm{\Delta }U\mathrm{\ne }0$, then for every fixed $t>0$ this exact mode family has bounded initial $H^m$ norm but unbounded evolved $H^m$ norm as its integer wavenumber $k\to \mathrm{\infty }$. Thus no uniform estimate of the form $\mathrm{\parallel }\eta (t){\mathrm{\parallel }}_{H^m}\le C_t\mathrm{\parallel }\eta (0){\mathrm{\parallel }}_{H^m}$ holds for this family. This is a linear interface-model conclusion, not a theorem of nonlinear turbulent transition, viscous shear-layer behavior, or physical instability in a particular experiment.

Tong, Chapter 4, §5.1.1, zero-gravity/zero-surface-tension vortex-sheet dispersion

MIT OpenCourseWare 6.642, Kelvin-Helmholtz Instability, pp. 1-4