Evolution of , which is the dilatational to solenoidal dissipation ratio, for unsplit and split forcing. When a single forcing term is used [Eq. (7)] there is no control over , and it often continues to grow throughout the simulation. With a split solenoidal/dilatational forcing [Eq. (17)] quickly adjusts to the imposed value (solid horizontal lines). For clarity, single forcing term data are averaged over a 5 s window. The light gray is unaveraged data, and shows high variability in the single term forcing results.
The ratio of dilatational to solenoidal kinetic energy reaches the equilibrium value about the same time as ; however, the specific value is different than and depends on the turbulent Mach number.
Evolution of the total dissipation for unsplit and split forcing. The forcing coefficients can be cast in a form that allows the specification of the dissipation (solid horizontal lines) and, thus, the Kolmogorov microscale. The split forcing method adheres closely to the imposed dissipation (solid horizontal lines) unlike the single term forcing method.
Time variation of the turbulent kinetic energy for unsplit and split forcing. The increase in for unsplit forcing causes the turbulent kinetic energy to increase as well.
The energy flux is equal to the dissipation for a range of wavenumbers for low- forcing but not for full spectrum linear forcing, which requires larger resolutions to develop an inertial range. The results correspond to the parameters from run 1c except as noted.
The energy content at low wavenumbers is lower when full spectrum linear forcing is used, compared to low- forcing, which results in a lower overall Reynolds number. The results correspond to the parameters from run 1c.
Compensated kinetic energy spectra obtained from low- split forced simulations at different and values: (a) total kinetic energy, (b) solenoidal, and (c) dilatational parts of the kinetic energy.
Third-order structure function from low-wavenumber split linear forcing (thick lines, black on-line) and full spectrum split linear forcing (thin lines, red on-line) for series 1c. At a particular resolution, low- forcing produces a higher peak than full spectrum forcing.
Structure function curves for the high resolution, nearly incompressible simulation 2a, with grid cells and , using low- split forcing. All three curves peak near the theoretically expected values (horizontal lines). Bottom: 4/15 law, ; middle: 4/5 law, ; and top: 4/3 law, .
Isotropy relation at third order for simulation 2a, with grid cells and using low- split forcing. Solid line (red on-line): , dashed line (blue on-line): .
Error in average statistics when using separately each of three realizations (thin lines) or the average of the three (thick line). The horizontal axis is the total model time required to compute the average. This comparison shows that computing time is better spent measuring statistics over one simulation, rather than taking the average of an ensemble of simulations.
Error in the modified wavenumber when the sixth-order compact finite difference scheme is on the same grid as a spectral method (a) and on a finer grid (b) with grid spacing contracted by a factor of .
Parameter values of simulations, where series 1 was used for long-time simulations on a mesh (Figs. 1–6) and series 2 was used for high-resolution simulations on a mesh (Figs. 5–10). All simulations use and . For the split forcing described below, the initial values of and target are close to their values in the stationary state.
Low- forcing achieves nearly double the Taylor Reynolds number as that of full spectrum forcing at the same resolution with the same criterion for all runs.
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