Kelvin-Helmholtz Instability

References

"Hydrodynamic and Hydromagnetic Stability", by S. Chandrasekhar. "The Magnetohydrodynamic Kelvin-Helmholtz Instability: A Two-dimensional Numerical Study", by Frank et al., ApJ 460, 777 (1996)

Description

The Kelvin-Helmholtz instability occurs when a perturbation is introduced to a system with a velocity shear. Here, we run this test problem to demonstrate the algorithm's ability to evolve a linear perturbation into nonlinear hydrodynamic turbulence. As a test of the linear regime, one can compare the growth rate of the instability with the analytic result before the instability becomes nonlinear. A single mode perturbation is needed for such a comparison.

Set up

Domain -0.5 ≤ x ≤ 0.5, -0.5 ≤ y ≤ 0.5 Boundary conditions Periodic everywhere Equation of state Adiabatic with γ = 1.4 Initial density ρ = 1 for |y| ≥ 0.25 and ρ = 2 for |y| < 0.25 Initial pressure P = 2.5 Initial velocity vx = 0.5 for |y| ≥ 0.25 and vx = -0.5 for |y| < 0.25. We perturb the x and y velocities. One can choose either a single mode perturbation, in which the perturbation is a sine wave with one wavelength in the x dimension, or a random mode perturbation. The amplitude of the perturbations is 0.01. MHD Components The MHD version of this test has Bx / (4π)1/2 = 0.5.

Results

Hydro - Single Mode Perturbation We ran the simulation on a grid of Nx = 600 and Ny = 600. Plotted below are linear density maps (higher values are redder). The left image is a frame from the single mode perturbation simulation at time 1.0 with density ranging from 0.8 to 2.1. The right image is a frame from this simulation at time 5.0 with density ranging from 0.5 to 2.2. Click on the right image to see an animation of the density over the whole simulation (size of animation: 42 MB).

Hydro - Random Perturbation The parameters are the same for this simulation except we initialized the instability with a random perturbation. The linear density map on the left occurs at time 1.0 and ranges in value from 0.8 to 2.1. The right image is a frame from this simulation at time 5.0 with density ranging from 0.5 to 2.2. Again, the right image is a link to an animation (size: 44 MB).

MHD - Random Perturbation The parameters are the same for this simulation, but an x magnetic field has been added with Bx / (4π)1/2 = 0.5. Again, we display linear density maps. Left image: Density at time 1.0 with values ranging from 0.9 to 2.1. Right image: Density at time 5.0 with values ranging from 0.9 to 2.1. The right image links to a movie (size: 18 MB).

Summary In the hydro case, a transition to nonlinear 2D turbulence is observed. The sharpness of the boundary between the high- and low-density material is well maintained. This property is a result of the relatively low diffusion of the algorithm. Notice also that the vortices formed from the turbulence slowly merge together since this turbulence is restricted to 2D. Finally, for the simulation with a magnetic field, the field suppresses the short wavelengths of the instability.

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