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 we used were:
where C fixes the initial background mach speed and p is the instantaneous pressure for the equivalent incompressible flow. A summary of the simulation parameters is given in table 8.2, and figure 8.3 shows the mesh used for this case.
| Parameter | Value |
| Dimension | 2d |
| Sv | 100 |
| Sr | 100 |
| A (Alfven Number) | 1. |
| Mach | 0.5 |
| N | 19 |
| KQuad | 64 |
| Method | Discontinuous Galerkin |
In figure 8.4 we show iso-contours of the conserved variables at t=1. We see that using an expansion order of nineteen shows very smooth and symmetric fields.
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We now consider the effect of expansion order and using an unstructured grid on accuracy. Vorticity is a good indicator of noise in the solution of unsteady simulations. Low resolution will show up as non-smoothness in vorticity since it is the derivative of the computed fields. In figure 8.7 we compare the vorticity at time t=1 for a 132 triangle mesh (shown in figure 8.6) run with N=4 and N=6. The top figures show how the vorticity varies across the lower-left to top-right diagonal. The vorticity in this direction should be symmetric about the mid-point. We see that at N=4 the profile is noisy, the peeks are not well resolved and the symmetry is not very well represented. At N=6 the symmetry is better but the profile is still quite noisy. In the next figure 8.8 we see at N=10 that the symmetry is very good but the slopes are still a little rough. Finally, at N=16 we see that the symmetry is perfect and the profile is very smooth. We also see that the smoothness of the iso-contours improve with each increase in expansion order.
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