Coherent Structures and Extreme Events in Rotating Multiphase Turbulent Flows

By using direct numerical simulations (DNS) at unprecedented resolution, we study turbulence under rotation in the presence of simultaneous direct and inverse cascades. The accumulation of energy at large scale leads to the formation of vertical coherent regions with high vorticity oriented along the rotation axis. By seeding the flow with millions of inertial particles, we quantify—for the first time—the effects of those coherent vertical structures on the preferential concentration of light and heavy particles. Furthermore, we quantitatively show that extreme fluctuations, leading to deviations from a normal-distributed statistics, result from the entangled interaction of the vertical structures with the turbulent background. Finally, we present the first-ever measurement of the relative importance between Stokes drag, Coriolis force, and centripetal force along the trajectories of inertial particles. We discover that vortical coherent structures lead to unexpected diffusion properties for heavy and light particles in the directions parallel and perpendicular to the rotation axis.

Popular Summary

Rotating, turbulent flows are ubiquitous in nature and often occur in geophysical and astrophysical problems (e.g., oceans, the Earth’s atmosphere and inner mantle, gaseous planets, and the formation of planetesimals) and engineering (e.g., turbomachinery and chemical mixers). These flows are characterized by the presence of strong coherent vortical structures and small-scale chaotic fluctuations, affecting both the long- and short-term diffusion properties of momentum, energy, and mass. Here, we present results from a series of direct numerical simulations at an unprecedented turbulence intensity. We seed the flow with millions of particles (lighter and heavier than the carrying fluid), and we provide, for the first time, a direct measurement of the statistical properties of both turbulent fields and the spatial distributions of particles.

We investigate how turbulence is affected by rotation using simulations in a cubic domain. In particular, we study the effect of different forces (e.g., centrifugal, Coriolis, and Stokes), and we show that particles of different mass are extremely well differentiated by rotation. We recover unexpected multiscale extreme fluctuations for the velocity field and highly sensitive preferential concentration in turbulent “cyclones” and anisotropic diffusion for the inertial particles. In particular, we show that heavier particles diffuse more easily perpendicular to the axis of rotation; light particles diffuse mostly vertically. This “elevator effect” may have several industrial applications.

We expect that our findings will inform future efforts to model particle dispersion in environmental and geophysical applications.

Figure 1

Top: 3D rendering of a turbulent flow at Rossby number $\mathrm{Ro}=0.25$ and rotation rate $\mathrm{\Omega }=10$. An inverse energy cascade is present in the turbulent dynamics. The stationary behavior is characterized by the formation of three cyclonic coherent columnar vortices emerging from the background of 3D turbulent fluctuations. Bottom: Vortical structures parallel to the rotation axis. Note the turbulent fluctuations exist also inside the core of each vortex. Color scale is based on the velocity amplitude.

Figure 2

3D rendering of the evolution of two different puffs of particles, one light (blue) and one heavy (black), released in a turbulent flow at Rossby number $\mathrm{Ro}=0.25$. Particles are injected on the same rotation axis and with a velocity equal to that of the underlying fluid. The dispersion dynamics follows two different evolutions: light particles get trapped by the nearest columnar vortex and diffuse mainly vertically, while heavy particles tend to avoid the columnar structures and diffuse mainly horizontally. In the bottom plane, we show the intensity of the vertical vorticity averaged along the rotation axis. Bottom panel shows an enlargement of the top panel close to one intense vertical structure.

Figure 3

Kinetic energy evolution for a typical run with large rotation rate, in the presence of an inverse energy cascade. We show the thermalization regime when rotation is not applied (black continuous line), the inverse cascade regime after rotation is switched on (blue dashed line), and the stationary regime obtained by the application of a large-scale friction (red dotted line). Inset: Kinetic energy flux measured at the two stationary regimes: without rotation (black continuous line) and with rotation and large-scale friction (red dashed line), in the DNS with large rotation rate $\mathrm{\Omega }=10$.

Figure 4

Log-log plots of the energy spectrum. Top: Spectra for the runs at $N=2048$. Data with $\mathrm{Ro}=0.76$ and $\mathrm{\Omega }=4$ (squares) and data with $\mathrm{Ro}=0.25$ and $\mathrm{\Omega }=10$ (circles). Inset: Effect of rotation on the forward energy cascade is highlighted in terms of the ratio of the two spectra, $E_{\mathrm{\Omega }=4}(k)/E_{\mathrm{\Omega }=10}(k)\sim k^{1/3}$. Bottom: Spectrum for $N=4096$, $\mathrm{Ro}=0.25$, and $\mathrm{\Omega }=10$; the expected scaling behaviors $\propto k^{-2}$ and $\propto k^{-5/3}$, above and below the Zeman wave number $k_{\mathrm{\Omega }}$, respectively, are also plotted. Inset: Spectrum for $N=4096$ and $\mathrm{\Omega }=10$ compensated with $k^{-2}$ and with $k^{-5/3}$.

Figure 5

Log-log plot of the second- $S_{\perp }^{(2)}(r)$ and fourth-order $S_{\perp }^{(4)}(r)$ Eulerian transverse structure functions. Left: Case with $N=2048$ and $\mathrm{\Omega }=4$. The dimensional scaling predictions $\propto r^{p/3}$ according to the K41 isotropic scaling are also plotted. Right: Case with $N=4096$, 2048 and $\mathrm{\Omega }=10$. The dimensional scaling prediction $\propto r^{p/2}$ is also plotted.

Figure 6

Log-log plot of the second-order Eulerian transverse structure function in the plane perpendicular to the rotation axis, $S_{\perp }^{(2)}(r)$, for $N=2048$ and $\mathrm{\Omega }=10$. Whole field $\mathit{u}(x,y,z)$ (squares), to the 2D3C ${\mathit{u}}_{2\mathrm{D}}(y,z)$ component (empty circles), and to the fluctuating 3D component ${\mathit{u}}^{\prime }(x,y,z)$ (filled circles).

Figure 7

Left: Log-log plot of the fourth-order flatness $K_{\perp }^{(4)}(r)$ derived from the Eulerian structure functions transverse to the rotation axis, for data with $N=2048$. Data from DNS with large rotation rate, in the presence of an inverse cascade ($\mathrm{\Omega }=10$, $\mathrm{Ro}=0.25$): full field $\mathit{u}$ (empty squares); 2D3C ${\mathit{u}}_{2\mathrm{D}}$ component (empty circles); fluctuating 3D component ${\mathit{u}}^{\prime }$ (filled circles). Data from DNS with the direct cascade only, and no-columnar vortices ($\mathrm{\Omega }=4$, $\mathrm{Ro}=0.76$): full field $\mathit{u}(x,y,z)$ (filled squares). We also superpose the power -law prediction $\propto r^{-0.15}$ obtained from independent measurements of isotropic turbulence without rotation and the best fit for the power law measured with intense rotation in the present data $\propto r^{-0.35}$. Right: The same for the sixth-order flatness $K_{\perp }^{(6)}(r)$ (same symbols). Error bars are estimated from different velocity snapshots and shown for a representative subset of points.

Figure 8

Log-log plot of the time evolution of the contribution of the different forces to the rms particle acceleration, at resolution $N=2048$ and for high rotation (low Rossby number). Left: Inertial heavy particles, with $\beta =0.4$ and $\mathrm{St}=0.7$ (family $H2$ in Table 2). Right: Inertial light particles, with $\beta =1.6$ and $\mathrm{St}=0.7$ (family $L8$ in Table 2).

Figure 9

A measure of the inertial particles’ preferential sampling of the vorticity regions at low, $\mathrm{\Omega }=4$, and high, $\mathrm{\Omega }=10$, rotation rates for two different families: a heavy one $H2$ and a light one $L8$. Inset: For the case with $\mathrm{\Omega }=10$, the probability density function of the vertical vorticity ${\omega }_x$ normalized to its standard deviation, as measured at the positions of light particles $L8$ (blue), heavy particle $H2$ (black), and tracers $T0$ (red). Data refer to DNS at resolution $N=2048$.

Figure 10

Absolute dispersion of inertial particles in the direction parallel (left) and perpendicular (right) to the rotation axis. The mean-square displacement of inertial particles is normalized with that of tracers. Labels refer to the four families of heavy particles $H1–H4$ and to the five families of light particles $L5–L9$. See Table 2 for details. Data refer to the DNS with $N=2048$ and $\mathrm{\Omega }=10$.