Rayleigh-Taylor Instability

References

"Hydrodynamic and Hydromagnetic Stability", by S. Chandrasekhar. "A numerical study of Rayleigh-Taylor instability in magnetic fluids", by Jun, Norman, & Stone, ApJ 453, 332 (1995) "Comparison of Several difference schemes on 1D and 2D Test problems for the Euler equations" by Liska, R. & Wendroff, B.

Description

This instability results from perturbing a fluid of high density resting on top of a low-density fluid where the gravitational acceleration is downward. One can test that the linear growth rate of the perturbation agrees with analytic theory (a single mode perturbation is needed for this analysis). Eventually, the instability moves into the nonlinear regime, where the algorithm's ability to evolve nonlinear interactions can be examined. This problem also tests the implementation of a source term (gravity) to the fluid equations.

Set up

Domain 2D: -0.25 ≤ x ≤ 0.25, -0.75 ≤ y ≤ 0.75 3D: Same as in 2D, but with the z domain included: -0.25 ≤ z ≤ 0.25 (the y direction is still the direction of gravity.) Boundary conditions 2D: Periodic in x, reflecting in y 3D: Same as in 2D, but with periodic z boundaries as well Equation of state Adiabatic with γ = 1.4 Initial density ρ = 1 ifor y ≤ 0.0 and ρ = 2 for y > 0.0 Initial pressure The pressure is initialized to give hydrostatic equilibrium: P = 2.5 - ρgy. Initial velocity 2D: For the single mode perturbation, we perturb the y velocity with A [1+cos(2πx/Lx)] [1+cos(2πy/Ly)]/4 with Lx and Ly being the size of the x and y domains respectively, and A = 0.01. For the random perturbation, the y velocity is given a random value between -A/2 and A/2. 3D: Same as in 2D, but for the single mode perturbation, vy is A [1+cos(2πx/Lx)] [1+cos(2πy/Ly)] [1+cos(2πz/Lz)]/8 with Lz being the size of the z domain. MHD Components 2D: The MHD version of this test has Bx / (4π)1/2 = 0.25. 3D: The MHD version of this test has Bx / (4π)1/2 = 0.25/(2)1/2 and Bz / (4π)1/2 = 0.25/(2)1/2.

Results

2D Hydro - Single Mode Perturbation We ran the 2D simulations on a grid of Nx = 300 and Ny = 900. Plotted below are linear density maps ranging in value from 1.0 to 2.0 (higher values are redder). The left image is a frame of the single mode perturbation run at an early time of 5.0. The middle image is a frame from this simulation at time 7.5. One can see that secondary Kelvin-Helmholtz instabilities have begun to form along the edge of the plume. The right image is a frame from this simulation at the late time 11.0. Click on the this image to see an animation of the density over the whole simulation (size of animation: 3.3 MB).

2D Hydro - Random Perturbation The parameters are the same for this simulation as for the single mode perturbation except we initialized the instability with a random perturbation. The linear density maps range in value from 1.0 to 2.0. The linear density map on the left occurs at time 3.0, the middle image is from time 7.5, and the right image occurs at time 20.0. The right image is a link to an animation (size: 8.9 MB).

2D MHD - Random Perturbation The parameters are the same for this simulation as for the previous one, but an x magnetic field has been added with Bx / (4π)1/2 = 0.25. Again, we display linear density maps. The values of the density range from 1.0 to 2.0. The left image occurs at time 5.0, the middle image occurs at time 10.0, and the right image is at time 19.0. The right image is a link to an animation (size: 3 MB).

3D Hydro - Single Mode Perturbation We ran the 3D simulations on a grid of Nx = 96, Ny = 288, and Nz = 96. This particular simulation was initialized with a single mode perturbation. Plotted below are isosurfaces of the density at a value of 1.5. This value follows the boundary between the low and high density fluid rather well. The top image is the instability very early on (time = 3.5), and one can see the instability is still in the linear regime. The second image is from time 10.0. The maximum and minimum values for the density are shown in the legend in the left part of each image. Click on the second image to see an animation of the isosurface over the evolution of the instability (size: 961 KB).

3D Hydro - Random Perturbation The parameters are the same for this simulation as for the single mode perturbation except we initialized the instability with a random perturbation. In these images, multiple density isosurfaces are plotted, each with a different color. The value of the density corresponding to each isosurface is shown in the legend on the left. The top image corresponds to a time of 3.5, and the bottom image corresponds to a time of 12.0. The bottom image is a link to a movie (size: 4.3 MB).

Summary The growth of the instability from the linear regime to the nonlinear regime is observed for all simulations. The sharpness of the boundary between the high and low density material is well maintained (especially for the MHD simulations), which is a result of the relatively low diffusion of the scheme used (this is not particularly seen in the 3D plots given the use of the isosurfaces). The boundary is not as sharp in the hydro simulations where Kelvin-Helmholtz instabilities have formed along the boundary, causing some fluid mixing. In the MHD simulations, the magnetic field stabilizes the fluid against these secondary instabilities, and the density boundary is very sharp.

Return to Test Suite page.