Nonequilibrium ensembles for the three-dimensional Navier-Stokes equations 

At the molecular level fluid motions are, by first principles, described by time reversible laws. On the other hand, the coarse grained macroscopic evolution is suitably described by the Navier-Stokes equations, which are inherently irreversible, due to the dissipation term. Here, a reversible version of three-dimensional Navier-Stokes is studied, by introducing a fluctuating viscosity constructed in such a way that enstrophy is conserved, along the lines of the paradigm of microcanonical versus canonical treatment in equilibrium statistical mechanics. Through systematic simulations we attack two important questions: (a) What are the conditions that must be satisfied in order to have a statistical equivalence between the two nonequilibrium ensembles? (b) What is the empirical distribution of the fluctuating viscosity observed by changing the Reynolds number and the number of modes used in the discretization of the evolution equation? The latter point is important also to establish regularity conditions for the reversible equations. We find that the probability to observe negative values of the fluctuating viscosity becomes very quickly extremely small when increasing the effective Reynolds number of the flow in the fully resolved hydrodynamical regime, at difference from what was observed previously.

Figure 1

Energy spectrum of $\mathcal{I}$ (grayscale filled symbols) and $\mathcal{R}$ (colored open symbols) for $N_0=128$ and at changing $\nu$ (for $\mathcal{I}$) or $D$ (for $\mathcal{R}$). We observe a well developed hydrodynamic regime for $\nu ={10}^{-2}$ ($\mathrm{R}7◯$), a crossover regime for $\nu ={10}^{-3}(\mathrm{R}8▵$), and a quasithermalized regime for $\nu ={10}^{-5}$ ($\mathrm{R}10\square$); see Table 1 in Appendix pp3 for details on the runs.

Figure 2

Mean properties of $\alpha$. (a) The ratio ${\langle \alpha \left(\mathit{u}\right)\rangle }_D^{\mathcal{R},N}/\nu =1$ is a consistency check for equivalence conditions. (b) The ratio $\langle \mathrm{\Lambda }\rangle /\langle W\rangle$ from Eq. (7) as a function of $\nu$ for all $N_0$ considered. The data are for $\mathcal{R}$.

Figure 3

Probability distribution function of single mode kinetic energy $U\left(\mathit{k}\right)$ for chosen modes $\mathit{k}=(0,0,m), m=1,2,8,16$, showing the equivalence between $\mathcal{R}$ (red straight lines) and $\mathcal{I}$ (black dashed lines). Here $\nu ={10}^{-4}$ and $N_0=64$ ($\mathrm{R}4\square$), belonging to the quasithermalized regime.

Figure 4

Test of Conjecture 1 for single mode kinetic energy $U\left(\mathit{k}\right)$. (a) Averaged $\mathcal{R}/\mathcal{I}$ ratio of the mean of kinetic energy of considered modes, i.e., permutations of $(0,0,m), (1,0,m)$, and $(0,p,m)$, with $m=1,...,N$ and $p=$ 2, or 7, or 9. (b) Same for standard deviation ratio. The gray band indicates a 10% deviation from 1. Here $N_0=64$, and results from $\mathrm{R}3▵, \mathrm{R}4\square$ are shown.

Figure 5

Probability distribution function of energy spectra for $k=1,2,8,16$ showing the equivalence between $\mathcal{R}$ (red straight lines) and $\mathcal{I}$ (black dashed lines). Here $\nu ={10}^{-4}$ and $N_0=64$ ($\mathrm{R}4\square$) corresponding to the quasithermalized regime.

Figure 6

Test of Conjecture 1 for the energy spectrum. (a) $\mathcal{R}/\mathcal{I}$ ratio of mean energy spectrum at shell $k$ for $k=1,...,N$. (b) Same for standard deviation ratio. The gray band indicates a 10% deviation from 1. Here $N_0=64$, showing $\mathrm{R}3▵$ ($\nu ={10}^{-3}$, green triangles), $\mathrm{R}4\square$ ($\nu ={10}^{-4}$, red squares), and $\mathrm{R}5\square$ ($\nu ={10}^{-5}$, black circles). Note that we are here also inspecting the crossover regime (green triangles), hence outside the validity of Conjecture 1 [as shown in (b)]. Nonetheless, good agreement is still found in (a).

Figure 7

Probability distribution function of single mode kinetic energy $U\left(\mathit{k}\right)$ for chosen modes $\mathit{k}=(0,0,m), m=1,4,10,40$, showing the equivalence between $\mathcal{R}$ (red straight lines) and $\mathcal{I}$ (black dashed lines). Here $\nu ={10}^{-2}$ and $N_0=128$ ($\mathrm{R}7◯$), belonging to the hydrodynamic regime.

Figure 8

Test of Conjecture 2 for single mode kinetic energy $U\left(\mathit{k}\right)$. (i) Averaged $\mathcal{R}/\mathcal{I}$ ratio of mean kinetic energy of considered modes and (ii) standard deviation ratio, at (a) $\nu ={10}^{-2}$ and (b) $\nu ={10}^{-3}$. The gray band indicates a 10% deviation from 1. The red vertical bar indicates the order of magnitude of the Kolmogorov scale, estimated as $k_{\nu }={\left(\frac{\varepsilon }{{\nu }^3}\right)}^{1/4}$.

Figure 9

Probability distribution function of energy spectrum showing equivalence between $\mathcal{R}$ (red straight lines) and $\mathcal{I}$ (dashed black lines), for $k=$ 1, 3. For (a) and (b) it holds $E_{\mathcal{R}}^{\left(1\right)}\left(k\right)/E_{\mathcal{I}}^{\left(1\right)}\left(k\right)\approx 1$ and $E_{\mathcal{R}}^{\left(2\right)}\left(k\right)/E_{\mathcal{I}}^{\left(2\right)}\left(k\right)\approx 1$, confirming full equivalence between $\mathcal{R}$ and $\mathcal{I}$ up to $k\sim 4$ with respect to $E\left(k\right)$ at $\nu ={10}^{-2}$. Here $\nu ={10}^{-2}$ and $N_0=128$ (R7$◯$) corresponding to the hydrodynamic regime, and the $y$ axis is in logarithmic scale.

Figure 10

Probability distribution function of energy spectrum showing equivalence between $\mathcal{R}$ (red straight lines) and $\mathcal{I}$ (dashed black lines), for $k=$ 2, 4, 6, 20. For (a)–(d) it holds $E_{\mathcal{R}}^{\left(1\right)}\left(k\right)/E_{\mathcal{I}}^{\left(1\right)}\left(k\right)\approx 1$ and $E_{\mathcal{R}}^{\left(2\right)}\left(k\right)/E_{\mathcal{I}}^{\left(2\right)}\left(k\right)\approx 1$, confirming full equivalence between $\mathcal{R}$ and $\mathcal{I}$ up to $k\sim 20$ with respect to $E\left(k\right)$ at $\nu ={10}^{-3}$. Here $N_0=512$ and $\nu ={10}^{-3}$ (R16$◯$) corresponding to the hydrodynamic regime, and the $y$ axis is in logarithmic scale.

Figure 11

Test of Conjecture 2 for the energy spectrum, considering the $\mathcal{R}/\mathcal{I}$ ratio of its mean at shell $k$ for $k=1,...,N$, at (a) $\nu ={10}^{-2}$ and (b) $\nu ={10}^{-3}$. The transparent region corresponds to the statistical error. The gray band indicates a 10% deviation from 1. $k_{\nu }$ shows the location of the Kolmogorov scale. The plots share the same $x$ axis.

Figure 12

Probability distribution function of energy spectrum for $\mathcal{R}$ (red straight lines) and $\mathcal{I}$ (dashed black lines) ($k=$ 10 and $80>k_{\nu }\sim 29$). For $k=10$ the two distributions have the same mean. Here $\nu ={10}^{-2}$ and $N_0=256$ (R12$◯$) corresponding to the hydrodynamic regime, and the $y$ axis is in logarithmic scale.

Figure 13

Test of Conjecture 2 for the energy spectrum, considering the $\mathcal{R}/\mathcal{I}$ ratio of its standard deviation at shell $k$ for $k=1,...,N$, at (a) $\nu ={10}^{-2}$ and (b) $\nu ={10}^{-3}$. In (a) the runs with $N_0=128$ and 256 are fully resolved and belong to the hydrodynamic regime, while in (b) only $N_0=512$ is fully resolved. The gray band indicates a 10% deviation from 1. $k_{\nu }$ shows the location of the Kolmogorov scale. The range of validity of Conjecture 2 increases as we decrease viscosity in the hydrodynamic regime, as expected. The plots share the same $x$ axis.

Figure 14

(a) A quantitative estimate of the locality cutoff scale $K$, defined as the maximum wave number where all observables used to check the equivalence (Conjecture 2) have the same value within $10\%$, versus $\nu$ at different $N_0$. Here filled points correspond to runs in the hydrodynamic regime and open points to the crossover regime. (b) The ratio $c_{\nu }=K/k_{\nu }$ for all runs in the hydrodynamic regime as a function of $\nu$, which is approximately constant. Both plots share the same $x$ axis in logarithmic scale.

Figure 15

Statistics of $\alpha$ for $\mathcal{R}$. (a) Temporal evolution of $\alpha /\nu$ as a function of time, which is rescaled by the Kolmogorov timescale ${\tau }_K$, at $\nu ={10}^{-3}$. In solid line, green (for $N_0=64$) and red (for $N_0=512$) is the running average of $\alpha /\nu$. The fact that $\langle \alpha /\nu \rangle =1$ in $\mathcal{R}$ is a rigorous prediction of Conjecture 2 [see also Eq. (14)]. (b) PDF of $\alpha /\nu$ obtained considering the whole available statistics for each $N_0$, at $\nu ={10}^{-3}$. The inset (i) is the energy spectrum, and inset (ii) is the PDF in logarithmic scale. (c) Same as (b) at $\nu ={10}^{-4}$. (d) Same as (b) at $\nu ={10}^{-3}$ showing successively small $N_0$ between 32 and 64.

Figure 16

Statistics of $\alpha$ for $\mathcal{I}$. (a), (c) Temporal evolution of $\alpha /\nu$ as a function of time, which is rescaled by the Kolmogorov timescale ${\tau }_K$, at (a) $\nu ={10}^{-2}$ and (c) $\nu ={10}^{-3}$. In solid line, green (for $N_0=64$) and red (for $N_0=512$) is the running average of $\alpha /\nu$. The fact that $\langle \alpha /\nu \rangle =1$ within 0.1% in $\mathcal{I}$ is remarkable (in spite of the difference in fluctuations with $\mathcal{R}$), as $\alpha$ is a nonlocal observable, hence not directly expected by Conjecture 2. (b), (d) Probability density function of $\alpha /\nu$ obtained considering the whole available statistics for each $N_0$, at (b) $\nu ={10}^{-2}$ and (d) $\nu ={10}^{-3}$. The inset is the PDF in logarithmic scale.