Here’s an interesting plot of the dissipation rate of kinetic energy of the Taylor-Green vortex problem computed with our Discontinuous Galerkin Spectral Element Method workhorse code “Strukti”. The Reynolds number of the problem is 800, Mach number is 0.1 and all computations were done with a Lax-Friedrichs formulation for the inviscid flux and BR1 for the viscous fluxes. All simulations were done on 64 Nehalem cores, the times shown in the plots are the total (overall) wall times of each computation.

“N” in the figures above denotes the degree of the polynomial approximation inside each grid cell, the DNS solution from Brachet (384 DOF per direction, pseudo-spectral code) is shown as square symbols for comparison.

Two related issues are obvious from the plot above:

1) While both formulations (O(2) and O(16)) converge to the DNS as expected, the O(2) solution requires significantly more degrees of freedom and prohibitively more wall time to reach an acceptable agreement with the DNS data.

2) Conversely, for the same number of DOF (64³, clear underresolution for DNS), the O(16) scheme produces an acceptable result close to the DNS, while the O(2) solution is too dissipative to capture the physics of the flow and to allow transition to turbulence.

1) While both formulations (O(2) and O(16)) converge to the DNS as expected, the O(2) solution requires significantly more degrees of freedom and prohibitively more wall time to reach an acceptable agreement with the DNS data.

2) Conversely, for the same number of DOF (64³, clear underresolution for DNS), the O(16) scheme produces an acceptable result close to the DNS, while the O(2) solution is too dissipative to capture the physics of the flow and to allow transition to turbulence.