Fluid dynamics · interface stability

Kelvin–Helmholtz Shear-Layer Instability

When does a ripple between two fluid layers sliding past each other start to grow? When, for a ripple of a given wavelength, the shear feeds it faster than gravity and surface tension can pull the interface back flat, and viscosity cannot damp it out. Gravity resists through the density difference between the layers. The growth is exponential only at first; once the ripple rolls up into vortices, the motion turns nonlinear and the simple growth rate no longer applies.

Watch two counterflowing layers, the interface between them, the vorticity and the velocity vectors in two dimensions, next to a Fourier spectrum and a growth-rate plot. Five preset cases isolate the roles of density, gravity, surface tension, viscosity and the 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

Test the stability of a 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 the logarithm of the early amplitude against time. Its theory line ends where the plot marks the switch between models.

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. Once $|\eta_k|/\lambda_k\ge 0.075$ , the display switches visibly to a regularized Birkhoff–Rott sheet. This late-stage model does not resolve mixing, viscous diffusion or interface breakup.

Physics tutorial

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

BackgroundTwo incompressible layers flow in opposite directions along a horizontal interface. A small ripple with wavenumber $k$ is amplified by the velocity jump, while stable density stratification, gravity and surface tension resist the deformation. The Lab predicts the growth of each Fourier mode and visibly switches to a different model once the small-amplitude assumption fails.

Why it mattersCloud decks, ocean internal waves, jet edges and astrophysical plasmas can all show shear-driven billows. Looking alike does not mean one model fits them all. What you can test is the growth rate and the range where the model holds.

Start with the essentials

Focus question: When does amplification from the velocity difference overcome gravity, surface tension, and viscous damping?
One-sentence intuition: In the linear phase, each wavenumber evolves on its own. A positive rate $\gamma_k>0$ only tells you the perturbation grows exponentially. Once the interface rolls up, modes couple and the geometry turns nonlinear, so the linear prediction has to 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}

Under the square root, the first term is the restoring effect of stable stratification and surface tension; the second is the shear drive. When the quantity under the root turns negative, the frequency gains an imaginary part and the disturbance grows exponentially.

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]

The thickness factor and the viscous term are simple low-order teaching approximations, not a solution of the viscous Orr–Sommerfeld eigenvalue problem. They show how sensitive short waves are to thickness and dissipation.

Amplitude growth in the linear phase

\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 plot fits the slope of the log amplitude and compares it with the prediction. The theory line ends at the mark where the model switches.

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'

The Birkhoff–Rott velocity feeds the flow induced by the sheet back into the interface. The code smooths the singularity with a periodic vortex-blob kernel of finite core radius; that radius is a numerical choice, not a resolved physical scale.

Common difficulties

One growth rate does not describe the whole run

Typical misconceptionThe early growth is exponential, so the same rate should predict the later vortex size, mixing and breakup.

Better mental modelLinear theory needs the amplitude to be much smaller than the wavelength. When the amplitude-to-wavelength ratio reaches 0.075, the Lab ends the linear prediction and labels the later stage as a regularized vortex sheet.

A finite layer is not a sharp interface

Typical misconceptionIncreasing the thickness only blurs the picture.

Better mental modelA layer of finite thickness weakens the velocity jump that short waves feel. This Lab uses $\operatorname{sech}(k\delta)$ to show the trend; accurate values need an eigenvalue calculation for the actual velocity profile.

Run the experiment

01
Measure single-mode exponential growth
Choose Single-mode seed, pause, and press Step 30 several times. Compare the solid and dashed traces in the growth audit plot.
What to observe: While the amplitude is small, the log amplitude rises almost along a straight line and the fitted rate approaches the theory. After the model switch, the two traces are no longer required to agree.
02
Find the stability threshold
Choose Stable stratification, then raise the velocity jump or lower gravity while watching the fastest mode and the active phase.
What to observe: When the restoring forces win, the disturbance only oscillates and dies away. Past the threshold, a growing mode appears and the phase label changes to linear growth.
03
Let the spectrum pick a scale
Choose Noise spectrum, then increase the shear-layer thickness and the viscous damping one at a time while comparing the Fourier bars.
What to observe: Short waves get a stronger shear drive, but thickness, surface tension and viscosity also hold them back harder. The scale that wins comes out of this competition.

Conceptual inspiration: FluidMotion. Original simulations and graphics; the model limits are described above.