Statistical Equilibrium of Large Scales in Three-Dimensional Hydrodynamic Turbulence

We investigate experimentally three-dimensional (3D) hydrodynamic turbulence at scales larger than the forcing scale. We manage to perform a scale separation between the forcing scale and the container size by injecting energy into the fluid using centimetric magnetic particles. We measure the statistics of the fluid velocity field at scales larger than the forcing scale (energy spectra, velocity distributions, and energy flux spectrum). In particular, we show that the large-scale dynamics are in statistical equilibrium and can be described with an effective temperature, although not isolated from the turbulent Kolmogorov cascade. In the large-scale domain, the energy flux is zero on average but exhibits intense temporal fluctuations. Our Letter paves the way to use equilibrium statistical mechanics to describe the large-scale properties of 3D turbulent flows.

Figure 1

3D power spectrum density $E(k)$ (green) derived from the 1D spectra of the longitudinal velocity $E_{uu}(k_x)$ (red), and transverse velocity $E_{vv}(k_x)$ (blue), [Eq. (4)]. Dashed line: $k^2$ power law illustrating the large-scale statistical equilibrium regime. Dot-dashed line: $k^{-5/3}$ power law illustrating the inertial range of the turbulent cascade. The vertical dashed line corresponds to the inverse of the integral scale $k_i/2\pi =1/L_i$ and separates the large-scale domain ($k<k_i$) from the inertial range ($k>k_i$). The PIV measurements are performed at $F=20\mathrm{Hz}$, $B=290\mathrm{G}$ and $N=55$. Inset: power spectrum densities (PSD) at different $\epsilon \in [1.1,3.2]\times {10}^{-4}{\mathrm{m}}^2/{\mathrm{s}}^3$.

Figure 2

Effective temperature $T_{\exp }$ of the statistical equilibrium regime of the large scales for different energy injection rates $\epsilon$. The energy injection rate $\epsilon$ is measured using $\epsilon =2\nu {\int }_{2\pi /L}^{\infty }{k}^{2}E(k)dk$. $T_{\exp }$ is measured from the 3D spectra shown in Fig. 1 using Eqs. (2) (blue circle) and (5) (red cross). The solid dashed line corresponds to Eq. (6) using $k_i/2\pi =13.3{\mathrm{m}}^{-1}$. Insets: structure functions ${\mathcal{S}}_2(r)$ (top) and ${\mathcal{S}}_3(r)/(4\epsilon /5)$ (bottom) for different $\epsilon$ (same colors as in the inset of Fig. 1). Solid lines correspond to $r^{2/3}$ and $-r$, respectively.

Figure 3

Probability density functions (PDFs) of the normalized fluid velocity fluctuations $u/\sqrt{⟨u^2⟩}$ of (diamond) large-scale modes, (square) all modes, and (circle) small-scale modes for ${\sigma }_u=0.6\mathrm{cm}/\mathrm{s}$. The cutoff value of the filter is equal to $k/2\pi =9.4({\mathrm{m}}^{-1})$. The black dashed line represents a Gaussian distribution. Inset: kurtosis ($K=⟨u^4⟩/⟨u^2{⟩}^2$) of the large-scale modes as a function of $\epsilon$ (${\sigma }_u\in [0.6,1.3]\mathrm{cm}/\mathrm{s}$).

Figure 4

Time-averaged energy flux spectrum $\overline{\mathrm{\Pi }}(k)$. At large scales ($k<k_i$), zero-mean energy flux is measured (equipartition). In the inertial range ($k>k_i$), the energy flux is positive and implies a direct cascade of energy. Insets: (top) standard deviation of the energy flux spectrum ${\sigma }_{\mathrm{\Pi }}(k)$. (bottom) PDFs of the temporal fluctuations of the energy flux $\mathrm{\Pi }/{\sigma }_{\mathrm{\Pi }}$ for three values of the wave number $k$ (see colored bullets in the main figure). The green line represents a Gaussian distribution.

Figure 5

Temporal power spectrum density of the horizontal velocity $E_u(f)$. The dashed line represents a $f^0$ power law and the dot-dashed line represents a $f^{-5/3}$ power law. The forcing parameters are identical to Fig. 1. $f_i$ indicates the beginning of the inertial range, i.e., the typical correlation frequency of the flow. Inset: Horizontal velocity $u(t)$ low-pass filtered at 2 Hz (blue), 0.2 Hz (red), and 0.02 Hz (black), as illustrated by the colored arrows in the main figure.