Whither the Zeroth Law of Turbulence?

TL;DR

The paper investigates whether the zeroth law of turbulence, i.e., finite energy dissipation as the Reynolds number Re tends to infinity, holds for homogeneous isotropic turbulence in a periodic box. Using direct numerical simulations of the incompressible Navier–Stokes equations, it computes the third-order structure function $S_3(l)$ and its inertial-range scaling to extract the exponent $\zeta_3^\nu$ and relate it to the bound $D[u] \leq C \mathrm{Re}^{\frac{3(1-\zeta_3)}{3+\zeta_3}}$. The results show $\zeta_3^\nu$ approaching values slightly above unity (roughly $\zeta_3 \approx 1.07$ in the extrapolated limit), which implies a weak dissipative anomaly with $D_{*}=0$ in the studied range, though a transition to $\zeta_3 \le 1$ at higher Re cannot be excluded. The Kolmogorov 4/5 law remains valid, with $-S_{3,\parallel}(l)/l$ approaching $(4/5)\varepsilon$ over an expanding inertial range as Re grows, indicating consistency between a weak anomaly and exact small-scale relations.

Abstract

Experimental and numerical studies of incompressible turbulence suggest that the mean dissipation rate of kinetic energy remains constant as the Reynolds number tends to infinity (or the non-dimensional viscosity tends to zero). This anomalous behavior is central to many theories of high-Reynolds-number turbulence and for this reason has been termed the "zeroth law". Here we report a sequence of direct numerical simulations of incompressible Navier-Stokes in a box with periodic boundary conditions, which indicate that the anomaly vanishes at a rate that agrees with the scaling of third-moment of absolute velocity increments. Our results suggest that turbulence without boundaries may not develop strong enough singularities to sustain the zeroth law.

Figures (3)

• Figure 1: Scaling of third-order absolute velocity structure function from the DNS of HIT in periodic cubes with edge-length $2\mathsf{L}_\mathsf{box}$. (A) Isotropic projections of $S_{3}({\boldsymbol{\ell}})$ normalized by cube of velocity scale, which is the root-mean-square velocity (see text), plotted against non-dimensional scale $\ell/\mathsf{L}$ with $\mathsf{L} = \mathsf{L}_\mathsf{box}$ for three different Reynolds numbers $\mathsf{Re} = {\mathsf{U} \mathsf{L}}/{\nu}$. Error bars corresponding to temporal variations of $S_{3}(\ell)$ in statistically steady-state are smaller than symbol sizes at higher $\mathsf{Re}$. At smallest scales $\ell/\eta \approx 1$, $\eta$ is the Kolmogorov length scale, $S_{3}(\ell)$ displays cubic scaling indicated by dash-dot lines, as expected. For the intermediate range of scales $\eta/\mathsf{L} \ll \ell/\mathsf{L} \ll 1,$ known as the inertial range (marked by filled symbols), a sub-cubic scaling range is seen, extending to increasingly smaller $\ell$ with increasing $\mathsf{Re}$. Dashed horizontal line is the power-law prefactor $C_2$ in \ref{['sfbndns']}. (B) Exponents $\zeta_3^\nu$ obtained by power-law fits to the inertial range are plotted as a function of $\mathsf{Re}$. Error bars correspond to standard deviation of $\zeta_3^\nu$ in statistically steady-state. Dotted line at unity is the naive Kolmogorov estimate kolmogorov1941local. Solid curve is the least-squares fit $\zeta_3^\nu = \zeta_3 + a {\mathsf{Re}}^b \ln {\mathsf{Re}}$ with $a=-0.284,$$b = -0.406$ (see Supplemental Information SIpaper for details); the asymptotic fit parameter $\zeta_3$ along with its $95\%$ confidence interval (shaded region) are shown, with a root mean square error of $0.0032$.
• Figure 2: Estimation of the asymptotic mean dissipation ${D_{*}}$ based on the asymptotic third-order absolute velocity exponent $\zeta_3$ from Figure \ref{['fig1']}. In the main panel the normalized mean dissipation $D[{\bf u}]$ is plotted as a function of $\mathsf{Re}$. Error bars computed from standard deviation of $D[{\bf u}]$ in statistically steady state are shown. Fits corresponding to vanishing asymptotic dissipation and non-zero asymptotic dissipation are given; also shown are the respective RMSE for both fits. Inset shows the logarithmic local slopes of $D[{\bf u}]$ which give the dissipation exponent $\alpha$ as a function of $\mathsf{Re}$ without any curve fitting. Solid horizontal line is the decay exponent from bound \ref{['expcond']} using $\zeta_3 = 1.066$ (from Figure \ref{['fig1']}), whereas dotted line at zero corresponds to $\zeta_3 \le 1$.
• Figure 3: Verification of the 4/5th law in 3D turbulence. Symbols correspond to the third-order longitudinal structure function compensated by scale-size (LHS of \ref{['K41precise.eq']}) at the three $\mathsf{Re}$ shown in Figure \ref{['fig1']}(A), plotted against non-dimensional scale $\ell/\mathsf{L}$. Horizontal lines show respective $(4/5) \varepsilon$ (RHS of \ref{['K41precise.eq']}) at the same three $\mathsf{Re}$. The ordinate is non-dimensionalized by ${\mathsf{U}}^3/\mathsf{L}$. Error bars in symbols and the shaded regions represent temporal standard deviations in $S_{3,\parallel}(\ell)$ and $\varepsilon$ respectively, in statistically steady state. With increasing $\mathsf{Re}$, $S_{3,\parallel}(\ell)$ develops a linear scaling or the ratio $S_{3,\parallel}(\ell)/\ell$ develops a plateau which matches up with $(4/5)\varepsilon$ over a widening scale-range extending towards smaller scales -- this is considered as the putative inertial range and is demarcated by filled symbols; consistent with decreasing $D[{\bf u}]$ with increasing $\mathsf{Re}$ (indicated by dotted arrow), the plateau height decreases in the direction marked by the solid arrow such that 4/5th law \ref{['K41precise.eq']} holds.