# 3-D Kelvin–Helmholtz test

This test consists of a dense central layer moving with respect to a lower density layer above and below the dense layer. Using periodiuc boundaries on all sides, a steady state can be established. Noise is added to the dense layer velocity field at the 1% level to seed the Kelvin-Helmholtz instability (KH instability). Note, the parameters here are identical to the 2-D version of this problem, excepting only the presence of the third z-axis.

There is not an analysitcal solution, but the growth of the y-component of the kinetic energy grows in essentially the same fashon as the test performed by the Athena code: (URL). There is a brief settling period, followed by exponential growth and finally saturation. The three resolutions excite increasing maximum wave number modes, and the higher wave number modes grow more quickly than the lower wave number modes, as expected.

## Initial conditions

• Adiabatic index: Gamma = 1.4
• Three layers
• Upper, 0.25 < y < 0.50, density, d = 1.0, pressure, P = 2.5, velocity, v_x0 = -0.5
• Middle, -0.25 < y < 0.25, density, d = 2.0, pressure, P = 2.5, velocity, v_x0 = +0.5
• Lower, -0.50 < y < -0.25, density, d = 1.0, pressure, P = 2.5, velocity, v_x0 = -0.5, same as upper
• Pressure is set to P = 2.5 everywhere
• Middle layer, velocity components v_x and v_y are modified with white noise, amplitude 0.01 peak to peak, ie +/- 0.005
• v_x = v_x0 + 0.01*(ran(1.0)-0.5), where ran(1.0) returns uniform random number 0.0 <= n < 1.0
• v_y = 0.01*(ran(1.0)-0.5)

## Grid

• 128 x 128x128, 256x256x256, 512x512x512 cells
• Domain: -0.5 < x < 0.5, -0.5 < y < 0.5, -0.5 < z < 0.5, (0.0, 0.0 in the centre of a unit square)
• All boundaries are periodic

## Algorithm settings

• CFL number: 0.8
• Striping correction on, artificial viscosity parameter: alpha = 0.001
• DivV threshold: 1.0, Scale: 1.0
• Flattening: minimum 0.0, maximum 1.0

## Ending condition

• Time, t = 5.0

## Results

### QuickTime movies

t = 0.0 - 5.0, 100 frames, one each for 512x512x512, 256x256x256 and 128x128x128 cell grids

### Central z = 0.0 plane images

#### 128 x 128 cells

min 0.8 2.2 max

#### 256 x 256 cells

min 0.8 2.2 max

#### 512 x 512 cells

min 0.8 2.2 max