Initial evolution of three-dimensional turbulence en route to the Kolmogorov state: Emergence and transformations of coherent structures, self-similarity, and instabilities

In this paper, we study numerically the temporal evolution of an initially random large-scale velocity field governed by the hyperviscous incompressible Navier-Stoke equations. Three stages are clearly observed during the evolution. First, the initial condition development is characterized by a spectrum evolving in a self-similar way with a wave number front $k_{*}\left(t\right)$ propagating toward high values exponentially in time. This evolution corresponds to the formation and shrinking of vortex pancakes exponentially in time, as previously reported in simulations of the incompressible Euler equations. At the second stage, pancakes become unstable, rolling up on the edges and breaking up in the middle, leading to the emergence of vortex ribs—quasiperiodic arrangements of vortex filaments. Those filaments then twist, creating structures akin to ropes. At the last stage, a fully developed turbulent state is observed, characterized by a Kolmogorov energy spectrum and exhibiting a decay law compatible with standard turbulent predictions in the case of an integral scale saturated at the scale of the box.

Figure 1

Temporal evolution of the energy and energy dissipation. Run at resolution ${1024}^3$

Figure 2

Visualizations of the enstrophy field at $t=0$ (top row) and $t=1.2$ (bottom row). The left panels show 3D visualizations of the field, whereas the center and right columns displays cuts of the $xy$ and $xz$ planes. Run at resolution ${1024}^3$.

Figure 3

Temporal evolution of energy spectra and fits. Fit Eq. (6) is plotted in magenta and Eq. (7) in cyan. Run at resolution ${2048}^3$.

Figure 4

Evolution of the fitting parameters in Eqs. (6) and (7). Run at resolution ${2048}^3$.

Figure 5

Temporal evolution of $k^{*}t$. The maximum wave number of the simulation is $k_{\max }=2048/3$.

Figure 6

Self-similar form Eqs. (5) for different times before $t*$. Error function (arbitrary normalization) ]Eq. (10)] as a function of the scaling parameter $\gamma$. Bounds of the integrals are set to ${\zeta }_{\mathrm{i}}=0.0126, {\zeta }_{\mathrm{f}}=1.12$ (green dashed vertical lines) and $T_0=.7475, T_1=1.2275$. Run at resolution ${2048}^3$.

Figure 7

Visualizations of the enstrophy field at $t=3.6$ (top row) and $t=4.8$ (middle row). The left panels of the top and middle rows show 3D visualizations of the field, whereas the center and right columns display cuts of the $xy$ and $xz$ planes. The bottom row shows several consecutive frames zoomed into a region with a typical vortex sheet roll-up and formation of the vortex ribs. Run at resolution ${1024}^3$.

Figure 8

Energy spectra at intermediate times. Run at resolution ${1024}^3$.

Figure 9

Visualizations of the enstrophy field at $t=7.05$. The left panel shows a 3D visualization of the field, whereas the center and right columns display cuts of the $xy$ and $xz$ planes. Run at resolution ${1024}^3$.

Figure 10

Temporal evolution of compensated energy spectrum at larger times. Run at resolution ${1024}^3$.

Figure 11

Temporal evolution of compensated energy spectrum at larger times. Run at resolution ${1024}^3$.