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Comparison of ${\u03f5}^{\left(\text{2}\right)}$ Formulations for Artificial Dissipation

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Two formulations for ${\u03f5}^{\left(\text{2}\right)}$, which is required for artificial dissipation, are compared.

The first formulation, which will be labeled "Maximum Nu", is as follows, $${\u03f5}_{i+\frac{1}{2}\text{,}j\text{,}k}^{\left(\text{2}\right)}={\kappa}^{\left(\text{2}\right)}\mathrm{max}\left\{{\nu}_{i+1}\text{,}\nu \right\}$$ The second formulation, which will be labeled "Average Nu", is as follows, $${\u03f5}_{i+\frac{1}{2}\text{,}j\text{,}k}^{\left(\text{2}\right)}=\frac{{\kappa}^{\left(2\right)}}{2}\left({\nu}_{i+1}+{\nu}_{i}\right)$$ The following plot shows the convergence of both formulations for a NACA 0012 Euler run at Mach 1.4 and an angle of attack of 4 degrees. The following contour plot shows the L2 norm of the Δq vector for each grid point for the maximum nu formulation at the last time step.