Effects of finite spatial and temporal resolution in direct numerical simulations of incompressible isotropic turbulence

As computing power has grown and turbulent flows at increasing Reynolds numbers are being computed by direct numerical simulations, conventional assumptions on adequate spatial and temporal resolutions are being continually challenged. We perform a systematic study of the resolution effects via numerical simulations at various spatial and temporal resolutions to clarify the proper scaling of dissipation and enstrophy (vorticity squared). Results show that inadequate resolution in space and/or time leads to overestimation of the likelihood and intensity of extreme fluctuations in dissipation and enstrophy. In order to capture rare events accurately, for instance, one needs to have not only grid resolutions that are increasingly smaller fractions of the Kolmogorov scale as the Reynolds number increases but also the Courant number that becomes increasingly smaller than that assumed to be adequate previously. Some comparisons are made with results obtained from an alternative approach where a stricter criterion for truncation in wave-number space allows aliasing errors to be removed completely. In contrast to prior work, the present data do not support the notion that dissipation and enstrophy probability density functions (PDFs) approach each other in the far tails at high Reynolds number. However the two PDFs are remarkably similar in form, being well described by stretched exponentials.

Figure 1

PDFs of normalized dissipation rate (red) and enstrophy (blue) in (a) linear-log scales and (b) log-log scales, at $R_{\lambda }\approx 650, {4096}^3, k_{max}\eta \approx 2.8, C=0.6$. Results are computed from velocity fields filtered at $k_c/k_{max}=1$, 0.75, and 0.5, in the direction of the arrows.

Figure 2

PDFs of normalized dissipation rate (red) and enstrophy (blue) in (a) linear-log scales and (b) log-log scales, obtained from unfiltered velocity fields in simulations at $R_{\lambda }\approx 650, C=0.6$. Grid resolutions ${2048}^3, {4096}^3$, and ${8192}^3$ corresponding to $k_{max}\eta \approx 1.4$, 2.8, and 5.6, increasing in the directions of the arrows.

Figure 3

Standardized PDFs of longitudinal (red) and transverse (blue) velocity gradients as well as vorticity (green), obtained from 17 snapshots in simulation with $R_{\lambda }\approx 650, {4096}^3, C=0.6$. Resolution levels $k_c/k_{max}=1$ and 0.5 (decreasing in the directions of the arrows).

Figure 4

(a) Longitudinal (red) and transverse (blue) 1D spectra, and (b) the ratio $E_{11}\left(k_1\right)/E_{22}\left(k_1\right)$ (magenta for actual data, green for estimate at $k_1$ close to $k_c$), with truncation levels $k_c/k_{max}=1$, 0.75, and 0.5 in the direction of the arrows. The plots are based on the same simulation datasets as in Figs. 1 and 2.

Figure 5

Comparison of peak $\epsilon /\langle \epsilon \rangle$ (left) and $\mathrm{\Omega }/\langle \mathrm{\Omega }\rangle$ (right) for results obtained at different Courant numbers, $R_{\lambda }\approx 390$, over time spans in the order 10 ${\tau }_{\eta }$. Different colors represent $C=$ 0.6 (red), 0.3 (green), and 0.15 (blue). From top to bottom $k_{max}\eta \approx 1.4$, 2.8, and 5.6 (grid resolutions ${1024}^3, {2048}^3, {4096}^3$). In many places lines in green and blue are almost coincident (with only blue being directly visible).

Figure 6

Same data as Fig. 5, but with data for different $k_{max}\eta \approx$ 1.4 (red), 2.8 (green), 5.6 (blue) shown in the same frames at each given choice of the Courant number. From top to bottom: $C=0.6$, 0.3 and 0.15.

Figure 7

Similar to Fig. 6, but at $R_{\lambda }\approx 650$, at grid resolutions ${2048}^3, {4096}^3, {8192}^3$ for $k_{max}\eta \approx$ 1.4 (red), 2.8 (green), 5.6 (blue). From top to bottom: $C=0.6$, 0.3 and 0.15.

Figure 8

PDFs of $\epsilon /\langle \epsilon \rangle$ (left) and $\mathrm{\Omega }/\langle \mathrm{\Omega }\rangle$ (right) for results obtained at $C=0.15$. Top row: $R_{\lambda }\approx 390$; middle and bottom rows: $R_{\lambda }\approx 650$, on linear-log and log-log scales respectively. Different colors indicate different levels of spatial resolution: namely $k_{max}\eta \approx$ 1.4 (red), 2.8 (green), 5.6 (blue).

Figure 9

Comparison of peak dissipation (left) and peak enstrophy (right), between (blue lines) DNS results at $C=0.15$ on an $N^3$ grid with truncation at $\sqrt{2}N/3$ and (red lines) test results from alias-free simulation at $C=0.6$ on a ${\left(2N\right)}^3$ grid also truncated at $\sqrt{2}N/3$. Frames (a) and (b): $R_{\lambda }\approx 390, N=1024$. Frames (c) and (d): $R_{\lambda }\approx 650, N=2048$.

Figure 10

PDFs of $\epsilon /\langle \epsilon \rangle$ (red) and $\mathrm{\Omega }/\langle \mathrm{\Omega }\rangle$ (blue) for results obtained at $C=0.15, k_{max}\eta \approx 5.6$: $R_{\lambda }\approx 390$ (left), $R_{\lambda }\approx 650$ (right). Dashed lines in green show the PDF of $2\epsilon /\langle \epsilon \rangle$. (Note the differences in range of $x$-axis between the two frames.)