Orszag-Tang Vortex

The Orszag-Tang vortex is an initial value problem in a square periodic domain (length L). It has been investigated by [66] and [64] amongst others . It demonstrates that turbulent scales can result from a coherent initial condition with only two spatial frequencies. The initial conditions are:

An appropriate choice of initial magnetic stream potential is:

$$\phi = \frac{L}{2 \pi} cos(\frac{2 \pi y}{L}) - \frac{L}{4 \pi} cos(\frac{4 \pi x}{L})$$

 

Parameter Value

Dimension 2d

S_v 100

S_r 100

$$\Delta t$$ 0.001

N-Range 10 to 25

K_Quad 64

Method Galerkin

As the solution evolves in time we see that the initial vortex splits into two vortices. The primitive variable fields are shown in figure 6.4 at t=1. We see that sharp gradients build up and it took 64 quadrilateral elements at N=25 to resolve these features well. We see in figure 6.4 that the vorticity is smooth and well resolved, and that the velocity divergence is small. The vorticity exhibits complex small scale structures. Also in figure 6.4 we see that the divergence free conditions for the velocity field is met to 10^-4 and the magnetic field is divergence free to 10^-8. A summary of the simulation parameters is given in table 6.3.

  

Figure 6.4: Incompressible Orszag-Tang vortex (t=1, instantaneous fields). Top Left: x component of velocity, Top Right: y component of velocity, Middle: Pressure, Bottom Left: x component of magnetic field, Bottom Right: y component of magnetic field.

  

Figure 6.5: Incompressible Orszag-Tang vortex (t=1, instantaneous fields). Top Left: Velocity streamlines, Top Right: Magnetic streamlines, Middle Left: Vorticity, Middle right: Divergence of velocity, Bottom Left: Curl of magnetic field, Bottom Right: Divergence of magnetic field.