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

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

(min  max)