Fluid dynamics · interfacial stability workbench

Kelvin–Helmholtz Shear-Layer Instability

A two-dimensional workbench links counterflowing layers, interface, vorticity, and velocity vectors to a Fourier spectrum and growth-rate audit. Five cases separate the roles of density, gravity, surface tension, viscosity, and seed spectrum.

Shear time

t\,\Delta U/L

$0.000$

Mode gain

|\eta_k|/|\eta_k(0)|

$1.000$

Linear prediction

\gamma_{\mathrm{th}}

$0\,\mathrm{s}^{-1}$

Measured fit

\gamma_{\mathrm{fit}}

$0\,\mathrm{s}^{-1}$

Fastest resolved mode

n_*

$1$

Active model phase $\mathrm{linear}$

Stability experiment

Interrogate the shear layer

Velocity vectorsSample the same velocity field that advects the sheet markers.Linear envelopeDashed bounds disappear when the nonlinear model takes over.Velocity jump

\Delta U

0.56\,\mathrm{m\,s^{-1}}

Density ratio

\rho_u/\rho_\ell

0.92

Viscous damping proxy

\nu

6.0\times10^{-4}\,\mathrm{m^2\,s^{-1}}

Shear-layer thickness

\delta

0.034\,\mathrm{m}

Gravity

g

0.18\,\mathrm{m\,s^{-2}}

Surface tension

\sigma

8.0\times10^{-4}\,\mathrm{N\,m^{-1}}

Seed wavelength

\lambda_0

0.50\,\mathrm{m}

Seed amplitude

\eta_0

0.009\,\mathrm{m}

Panel C fits early log-amplitude against time. Its theory line stops at the visibly marked model switch.

References: Miles, JFM 3 (1957), DOI 10.1017/S0022112057000567; Krasny, JCP 65 (1986), DOI 10.1016/0021-9991(86)90210-X.

Early growth follows $\eta_k(t)=\eta_k(0)e^{\gamma_k t}$ and the gravity–capillary Kelvin–Helmholtz dispersion relation. At $|\eta_k|/\lambda_k\ge 0.075$ the display explicitly switches to a regularized Birkhoff–Rott sheet, which does not resolve mixing, viscous diffusion, or interface breakup.

Physics tutorial

Kelvin–Helmholtz Instability: from perturbation growth to sheet roll-up

BackgroundTwo incompressible layers counterflow along a horizontal interface. A small corrugation with wavenumber $k$ is amplified by the velocity jump while stable density stratification, gravity, and surface tension resist deformation. The workbench predicts each Fourier mode, then visibly changes models when the small-amplitude assumption fails.

Why it mattersCloud decks, ocean internal waves, jet edges, and astrophysical plasmas can all show shear-driven billows, but visual resemblance alone does not establish one model. Growth rate and validity bounds are the testable physics.

Start with the essentials

Focus question: When does amplification from the velocity difference overcome gravity, surface tension, and viscous damping?
One-sentence intuition: Each wavenumber competes independently in the linear phase. A positive rate $\gamma_k>0$ only establishes exponential perturbation growth. Roll-up introduces mode coupling and geometric nonlinearity, so the linear line must stop.

Core mathematical model

Two-layer gravity–capillary dispersion relation

\omega=k\frac{\rho_u U_u+\rho_\ell U_\ell}{\rho_u+\rho_\ell}\pm\sqrt{\frac{(\rho_\ell-\rho_u)gk+\sigma k^3}{\rho_u+\rho_\ell}-\frac{\rho_u\rho_\ell}{(\rho_u+\rho_\ell)^2}(\Delta U)^2k^2}

The first radicand term restores a stably stratified capillary interface; the second is shear drive. A negative radicand gives frequency an imaginary part and produces exponential growth.

Teaching corrections for thickness and damping

\gamma_k=\max\!\left[0,\sqrt{\frac{\rho_u\rho_\ell}{(\rho_u+\rho_\ell)^2}(\Delta U)^2k^2\operatorname{sech}(k\delta)-\frac{(\rho_\ell-\rho_u)gk+\sigma k^3}{\rho_u+\rho_\ell}}-2\nu k^2\right]

Thickness attenuation and viscous damping are transparent low-order teaching proxies, not a viscous Orr–Sommerfeld eigenvalue solution. They expose short-wave and dissipation sensitivity.

Linear amplitude audit

\eta_k(t)=\eta_k(0)e^{\gamma_k t},\qquad \gamma_{\mathrm{fit}}=\frac{\mathrm{d}}{\mathrm{d}t}\ln|\eta_k|

The growth panel fits the log-amplitude slope and compares it with prediction. The theory line terminates at the marked model switch.

Regularized vortex-sheet phase

\frac{\partial \overline{z}}{\partial t}=\frac{1}{2\pi i}\operatorname{PV}\!\int\frac{\gamma(\alpha')}{z(\alpha)-z(\alpha')}\,\mathrm{d}\alpha'

Birkhoff–Rott velocity feeds sheet-induced flow back into the interface. The implementation uses a periodic vortex-blob kernel and finite core radius rather than presenting a singularity as a resolved physical scale.

Common difficulties

Growth rate is not a formula for the whole animation

Typical misconceptionExponential early growth means the same rate predicts late vortex size, mixing, and breakup.

Better mental modelLinear theory requires amplitude much smaller than wavelength. At a ratio of 0.075, the workbench ends the linear prediction and relabels the late model as a regularized vortex sheet.

A finite layer is not a sharp interface

Typical misconceptionIncreasing thickness merely blurs the picture.

Better mental modelFinite thickness weakens the velocity jump seen by high wavenumbers. This Lab uses $\operatorname{sech}(k\delta)$ to show the trend; quantitative values require an eigenvalue problem for the specified velocity profile.

Run the experiment

01
Measure single-mode exponential growth
Choose Single-mode seed, pause, and press Step 30 repeatedly while comparing the solid and dashed growth traces.
What to observe: During small-amplitude evolution, log amplitude is approximately linear and the fitted rate approaches theory. The traces are not forced to agree after the switch.
02
Find the stability threshold
Choose Stable stratification, then raise velocity jump or lower gravity while watching the fastest mode and active phase.
What to observe: When restoration dominates, the perturbation oscillates and damps. Across the threshold, a growing mode appears and the phase label changes.
03
Let a spectrum choose its scale
Choose Noise spectrum, then increase interface thickness and viscous damping separately while comparing the Fourier bars.
What to observe: Short waves gain stronger shear drive but suffer stronger thickness, capillary, and viscous penalties. The winning scale emerges from competition.