# 2-D Gresho vortex tests

The Gresho vortex is a 2-D pattern where the cetrifugal force are matched by pressure gradients, resulting in a stable time-independent vortex. This 2-D L&W test consists of single 2-D vortex on a range of grid resolutions. One test models a stationary vortex, and the second test advects the vortex across three diameters. The code and configuration files plus output for Fyris to run the 2-D advection convergence test will be available soon.

The vortex is defined here by:

## Initial conditions

• A vortex initially centred about (0.5, 0.5), outer radius r = 0.40
• Adiabatic index: Gamma = 5/3
• Density = 1.0 everywhere

### Pressure

• r <= 0.2; P(r) = 5.0 +25/2 r^2
• 0.2 < r <= 0.4 ; 9 – 4 ln 0.2 + 25/2 r^2 – 20 r + 4 ln r
• 0.4 < r ; 3 + 4 ln 2

### Azimuthal velocity v_phi

• r <= 0.2; v_phi(r) = 5r
• 0.2 < r <= 0.4 ; v_phi(r) = 2 – 5r
• 0.4 < r ; v_phi(r) = 0.0

### Resulting 2D vorticity, V_yx (r)

• r <= 0.2; V_yx(r) = 10.0
• 0.2 < r <= 0.4 ; V_yx(r) = 2/r - 10.0
• 0.4 < r ; V_yx(r) = 0.0

## The stationary vortex

### Ending condition

• Time, t = 3.0

### Grid

• Domain: x = 0 -> 1.0, , y = 0 -> 1.0
• Boundary conditions are free or natural boundaries
• Runs are made on a 20x20 (a20), 40x40 (a40), 60x60 (a60) and 80x80 (a80) grid

### Hydrodynamics settings

• CFL number: 0.8, initial step 0.4 (standard)
• Flattening: minimum 0.0, maximum 1.0 (standard)

Errors and L1 norms are computed over a unit square centred on (0.5, 0.5) at t = 3.0.

### Results

#### The stationary vortex at t = 3.0

Stationary Density L1 %   Vorticity L1 %   Total KE Error %
Code a20 a40 a60 a80   a20 a40 a60 a80   a20 a40 a60 a80
Fyris 0.150 0.0276 0.0181 0.0144   26.4 14.5 11.1 8.89   11.6 1.85 0.733 0.383
CFLFh 0.22 0.16       22 20       0.2 0.4
JT 0.56 0.22       89 45       55.2 18.3
LL 2.27 0.23       71 44       65.6 26.1
CLAW 0.33 0.10       50 28       29.9 6.1
WAFT 0.24 0.07       47 26       7.7 5.7
WENO 0.35 0.06       38 27       30.9 3.7
PPM 0.20 0.04       25 13       9.1 0.8
VH1 0.15 0.04       26 15       9.6 1.2    a20 Density (0.996072, 1.00688) a20 Pressure (5.05753, 5.79528) a20 Vorticity (-4.19047, 11.0509) a20 Velocity_x (-0.795881, 0.795881)    a40 Density (0.998955, 1.00132) a40 Pressure (5.01343, 5.77697) a40 Vorticity (-4.45904, 10.4167) a40 Velocity_x (-0.899295, 0.899295)    a60 Density (0.999382, 1.00105) a60 Pressure (5.00564, 5.77629) a60 Vorticity (-4.92575, 10.3582) a60 Velocity_x (-0.928645, 0.928645)    a80 Density (0.999134, 1.00096) a80 Pressure (5.00309, 5.77541) a80 Vorticity (-5.19555 , 10.3471) a80 Velocity_x (-0.941055, 0.941055) min max

## The moving vortex

Same as the stationary vortex, with a global v_x = 1.0 drift velocity added to the whole grid, on top of the initial vortex velocity field.

### Ending condition

• Time, t = 3.0

### Grid

• Domain:0.0 < x < 4.0, 0.0 < y <1.0
• Boundary conditions are free or natural boundaries.
• Runs are made on a 20x80 (b20), 40x160 (b40), 60x240 (b60) and 80x320 (b80) grid.

The vortex drifts from the left to the right. At t = 3.0 the vortex should be centred on (3.5,0.5).

Errors and L1 norms are computed over a unit square centred on the expected vortex centre at t = 3.0.

### Results

The vorticity is estimated from the discrete grid by the central finite difference:

vorticity[i,j] = 0.5*{(vy[i+1, j]-vy[i-1,j])/dx - (vx[i, j+1]-vx[i,j-1])/dy}

#### Moving vortex at t = 3.0

Stationary Density L1 %   Vorticity L1 %   Total KE Error %
Code b20 b40 b60 b80   b20 b40 b60 b80   b20 b40 b60 b80
Fyris 0.649 0.568 0.104 0.042   62.08 56.47 18.97 14.52   2.072 0.031 0.039 0.003
CFLFh 1.12 0.72       145 83       12.8 0.1
JT 0.81 0.22       100 52       42.8 22.1
LL 0.65 0.49       88 60       71.6 30.9
CLAW 0.72 0.29       65 37       39.9 8.3
WAFT 0.87 0.77       65 62       1.3 12.6
WENO 0.37 0.43       48 40       31.6 4.0
PPM 1.1 0.42       93 36       4.9 1.0
VH1 0.8 0.66       65 55       11.7 1.2
(min max)