The Rayleigh-Taylor Instability

Problem Background

The Rayleigh-Taylor instability arises when a fluid pushes against another fluid of different density. The simplest way to imagine this situation is to consider a heavy fluid sitting atop a light fluid. The system is in hydrostatic equilibrium with gravity pointing downwards. A small perturbation at the interface between the two fluids will disturb this unstable equilibrium, and the heavy fluid will sink while the light fluid rises.

Initial Conditions

The problem setup is adapted from the Athena test suite page: ​http://www.astro.virginia.edu/VITA/ATHENA/rt.html

This page has information on both 2D and 3D simulations, Hydro and MHD, but the 2D Hydro case is the relevant one here. The value for g is not given on the Athena page, but the acceleration due to gravity used here is g = 0.1. Also, the grid resolution is different. Athena's test uses a fixed grid of 300 x 900. The resolution here varies for each simulation.

Simulations

All of the following simulation movies show density. (Red = 2 and Blue = 1)

This simulation is run with a fixed grid of 100 x 300.

• RT Movie

For this run, all parameters are the same. The grid is now 50 x 150, and there are 3 levels of AMR which yields an effective resolution that is 4x the previous run.

• RT Movie (Higher Resolution)​

At late times, the simulation should develop Kelvin-Helmholtz instabilities and other irregularities associated with the mixing of the two fluids. It became obvious that uniform gravity was not working properly in the first two simulations, so here is one in which it is working. Also, this simulation is a 50 x 150 grid with 2 levels of AMR.

• RT Movie (Gravity Working)​

The simulation should always be symmetric, but the previous simulation has asymmetry that is especially noticable in late time frames. After going back to the problem module, it was discovered that the perturbation had some z dependence which it should not since this is a 2D simulation. Fixing that yielded a nice symmetric simulation.

• RT Movie (symmetric)​

Tracking the Interface

In order to calculate the growth rate of the perturbation, the interface between the two fluids was tracked. To do this, a new bear2fix routine was used which calculates the position of the large density gradient (i.e the interface). To ensure that bear2fix tracked the interface accurately, the values that bear2fix calculated were plotted over the density map created in visit. The bear2fix data is represented by the black horizontal line, and the density map is zoomed in at the peak of the interface.

• Interface Movie​

Calculating the Growth Rate

With the accuracy of the tracking confirmed, the next step was to calculate the growth rate.

For small perturbations, a linear analysis is used to show that the position of the interface at x = 0 rises exponentially as: y(t)=y_o e^{\gamma t}.

Taking the log of both sides yields: \log{y(t)}=\log{y_o}+\gamma t.

Therefore, the slope of a \log{y(t)} vs. t plot would yield the growth rate \gamma. Said plot was created and used to determine which time period would be most appropriate for calculating the growth rate.

• Log(interface_position) Plot​

At early times (approximately t < 2.5), the simulation does not have enough resolution to measure such small changes in the interface position. At late times (approximately t > 4.0), the interface develops irregularities because the instability is entering the nonlinear regime. Therefore, this type of analysis only works between the two aforementioned times.

The plots were used to get a rough idea of when to calculate the growth rate. Bear2fix was then used to actually do the calculation directly from the data. The data does not actually need to be plotted to find the slope of a best fit line. Essentially, this slope (the growth rate) was calculated used the method of least squares.

Results