Space-local Navier–Stokes turbulence

We investigate the physical-space locality of interactions in three-dimensional incompressible turbulent flow. To that, we modify the nonlinear terms of the vorticity equation such that the vorticity field is advected and stretched by the locally induced velocity. This space-local velocity field is defined by the truncated Biot–Savart law, where only the neighboring vorticity field in a sphere of radius $R$ is integrated. We conduct direct numerical simulations of the space-local system to investigate its statistics in the inertial range. We observe a standard $E\left(k\right)\propto k^{-5/3}$ scaling of the energy spectrum associated with an energy cascade for scales smaller than the space-local domain size $k\gg R^{-1}$. This result is consistent with the assumption [Kolmogorov, Dokl. Akad. Nauk SSSR 30, 299 (1941)] made for the space locality of the nonlinear interactions. The enstrophy amplification is suppressed for larger scales $k\ll R^{-1}$, and for these scales, the system exhibits a scaling consistent with a conservative enstrophy cascade.

Figure 1

(a) Time evolution of energy spectrum $E(k,t)$ of space-local Navier–Stokes turbulence at $R=0.2L_f$. Time evolves from dark to light colors with an interval of $5T_f$ up to $70T_f$. The thick gray curve represents the time-averaged energy spectrum of the original Navier–Stokes turbulence. (b) Normalized instantaneous energy spectrum $E\left(k\right)$ at different values of $R$. Normalization is performed by the high-pass filtered energy dissipation rate ${\epsilon }^{>}$ (18) and the corresponding Kolmogorov length scale ${\eta }^{>}$ (19). Vertical dashed lines denote $2\pi /R$ normalized by the Kolmogorov length of the original turbulence. The darker (lighter) color represents smaller (larger) values of $R$: $R/L_f=0.1,0.2,0.4,0.6,0.8,1$, and the lightest color represents the original turbulence. For both panels, the red dashed and black dotted lines denote the $k^{-5/3}$ and $k^{-3}$ scalings, respectively.

Figure 2

Ratio of cumulative enstrophy amplification $V_{\omega }^{<}\left(k\right)$ and enstrophy flux ${\mathrm{\Pi }}_{\omega }\left(k\right)$. The horizontal dashed line denotes $V_{\omega }^{<}\left(k\right)={\mathrm{\Pi }}_{\omega }\left(k\right)$. The same colormap and the vertical dashed lines are employed as in Fig. 1.

Figure 3

(a) Enstrophy flux ${\mathrm{\Pi }}_{\omega }\left(k\right)$ normalized by the enstrophy input rate $P_{\omega }$. The red dashed line denotes $k^2$ scaling. (b) Compensated energy spectrum $E\left(k\right)$ based on the scaling (25). For both panels, the wave number is normalized by $R$ so that different $2\pi /R$ collapse onto $kR=2\pi$. Note that the spectrum of the original turbulence is not shown, since it corresponds to the $R\nearrow \infty$ limit. The horizontal dashed line denotes a plateau. The same colormap is employed as in Fig. 1.

Figure 4

A schematic of the energy spectrum associated with space-local Navier–Stokes turbulence. Dashed arrows denote $k^{-3}$ (plus logarithmic correction) and $k^{-5/3}$ scaling, respectively. Vertical dash-dotted lines denote the forcing wave number $k_f$, the intersecting wave number $k_{*}$ of the two scalings, and the Kolmogorov wave number $k_{{\eta }^{>}}$, respectively. Annotations (i)–(iv) correspond to different scaling regions divided by the characteristic wave numbers.

Figure 5

(a) Positive (red) and negative (blue) isosurfaces of the $x$ component of the forcing field, $f_x=\pm 0.5$. See (15) for the definition. (b) An instantaneous snapshot of vortical structures. Isosurfaces of vorticity magnitude in low-pass filtered $|{\mathit{\omega }}^{<}|=4$ for $k\le 3$ (red), band-pass filtered $|{\mathit{\omega }}^{\lessgtr }|=6$ for $3<k\le 6$ (yellow), and original $\left|\mathit{\omega }\right|=100$ (blue) are shown. Low-pass and band-pass filtering are applied in the Fourier-space velocity field.

Figure 6

Time series of (a) energy input rate $P\left(t\right)$ normalized by the energy dissipation rate $\epsilon \left(t\right)$ and (b) right-hand side of the energy equation (11) of space-local Navier–Stokes turbulence at different values of $R$. For both panels, time is normalized by $T_f$: the characteristic timescale of the forcing (17). The horizontal gray dashed line in panel (b) denotes $y=0$. See the caption of Fig. 1 for the parameter list for $R$ corresponding to each curve with a different color.

Figure 7

Snapshots of vortical structures in space-local Navier–Stokes turbulence at (a) $R=0.2L_f$ and (b) $R=0.6L_f$, respectively. Isosurfaces of the magnitude of low-pass filtered vorticity $|{\mathit{\omega }}^{<}|=8$ (red), band-pass filtered one $|{\mathit{\omega }}^{\lessgtr }|=3$ (yellow), and unfiltered one $\left|\mathit{\omega }\right|=50$ (blue) are shown. See the caption of Fig. 5 for the filtering wave-number ranges. Note that we employ smaller thresholds compared to the visualization of the original turbulent flow in Fig. 5, but they are common in panels (a) and (b). The gray spherical domain illustrates the size of the locality parameter $R$ at the center of the computational domain.

Figure 8

Three terms from the enstrophy budget equation (C5): enstrophy flux ${\mathrm{\Pi }}_{\omega }\left(k\right)$, cumulative enstrophy amplification $V_{\omega }^{<}\left(k\right)$, and cumulative enstrophy dissipation rate ${\epsilon }_{\omega }\left(k\right)$. Each spectrum and the wave number are normalized by the enstrophy dissipation rate ${\epsilon }_{\omega }^{<}=2\nu {\int }_0^{\infty }{k}^{4}E\left(k\right)dk$ and the Kolmogorov length scale $\eta$, respectively. Vertical solid and dashed lines denote $k_f$ and $k_{\eta }$, respectively.