Rayleigh-Taylor

The Rayleigh-Taylor instability happens whenever a heavier fluid rests on top of a lighter fluid under gravity or an equivalent downward acceleration. The easiest physical example people often give is having water on top of oil in a cup. Gravity wants to drag the water down to the floor, while the lighter oil has to squeeze upward to get out of the way. If the boundary between them were mathematically flat with zero motion, it might sit there in an unstable balance. In the real world, even a microscopic ripple will trigger the flow to develop.

As the heavy fluid drops and the light fluid rises, they form distinctive shapes. The falling heavy fluid punches downward in narrow fingers called spikes, while the rising light fluid floats upward in rounded structures called bubbles. As these features slide past each other, the speed difference along their edges creates shear. That shear triggers secondary rollups, which curl the tips of the spikes into a recognizable mushroom (think mushroom clouds from explosions). Instabilities are the onset of turbulence, so as it runs it will turn into a chaotic mess. Rayleigh-Taylor instabilities are seen in everything from astrophysics, meteorology, and even bioflows, and are really important for understanding the movement and mixing of fluids.

This tool lets you set up an initial domain of the instability and watch it evolve in real time. During the early stage of the instability, the height of the ripple grows exponentially according to linear stability theory.

η(t) = η₀ exp(√ A g k  t)

The starting disturbance height is η₀, and the term inside the square root controls how fast that ripple explodes. Here is how the main settings tie into that growth and the simulation grid.

• Atwood Number (A) sets the density difference between the materials. A value of 0 means both fluids share the same density, which gives zero growth. Higher values increase the growth rate and make the flow asymmetric, lower values have a decreased growth rate and are more symmetric.
• Acceleration (g) controls the downward force pulling on the fluids. Stronger acceleration directly speeds up the instability growth.
• Waves and Domain Width determine the wavenumber k = 2π / λ. Setting more waves packs shorter wavelengths into the domain. The linear growth equation shows that shorter wavelengths with higher k grow the fastest.
• Amplitude sets the initial height (η₀) of the ripple. Larger values skip past the early linear predictions almost immediately, and quicken growth.
• Interface Width softens the density jump across several cells. This naturally dampens ripples that are smaller than the transition layer itself.

Hit Run to start the calculation, or use Next Snapshot to step forward by your chosen save interval. You can scrub through recorded snapshots using the slider to inspect earlier stages of the mixing layer, or switch back to Live and continue the run.

• Wikipedia page for the Rayleigh-Taylor instability
• Review paper of RTI Studies
• Review on Rayleigh-Taylor Theory
• Hydrodynamic and Hydromagnetic Stability by Chandrasekhar
• Introduction to Hydrodynamic Stability by Drazin