Assessing the nonequilibrium of decaying turbulence with reversed initial fields

Recently, it was observed that in nonequilibrium turbulent flows the normalized dissipation rate depends in a fairly universal way on the Reynolds number of the flow. To assess theoretical explanations of this observation, we consider here the nonequilibrium properties of freely decaying turbulence from initial conditions where the velocity field is reversed in every point in space. This test-case allows us to manipulate a turbulent flow differently from the usual way, where nonequilibrium is induced by modification of large-scale forcing mechanisms. It is shown that it is possible to obtain a different Reynolds-dependent scaling of the dissipation rate, which can be derived directly as a perturbation around the equilibrium state.

Figure 1

(a) Relation between dissipation coefficient $C_{\varepsilon }$ and Reynolds number ${\mathrm{Re}}_{\lambda }$ in different evolution stages. Stages are marked as $I, \mathit{II}$, III, and $\mathit{IV}$, respectively. (b) Relation between $L/\lambda$ and Reynolds number ${\mathrm{Re}}_{\lambda }$ in different evolution stages. Stages are marked as $I, \mathit{II}$, III, and $\mathit{IV}$, respectively. The dash-dotted lines corresponds to $L/\lambda \propto {\mathrm{Re}}_{\lambda }$, and $L/\lambda \propto {\mathrm{Re}}_{\lambda }^{-1}$. Arrows indicate the time directions.

Figure 2

(a) Temporal evolution of the skewness of longitudinal velocity derivative $S_k$. The subfigure is the zoom of stage $I$. (b) Variation of $C_{\varepsilon }$ with $S_k$ in stage $I$.

Figure 3

Temporal evolution of (a) the kinetic energy $k$, (b) the integral scale $L$, (c) the turbulent energy dissipation rate $\varepsilon$, and (d) the dissipation coefficient $C_{\varepsilon }$. Insert: short-time evolution of each statistic.

Figure 4

Dissipation coefficient $C_{\varepsilon }$ against Reynolds number ${\mathrm{Re}}_{\lambda }$ in stage $I$. The dash-dotted line represents the relation in Eq. (16).

Figure 5

Comparison between energy flux $-\mathrm{\Pi }({\kappa }_c,t)$ and dissipation ${\varepsilon }^{>}({\kappa }_c,t)$, with ${\kappa }_c$ fixed and its value is shown in Fig. 6. (a) NN, the inset shows $-\mathrm{\Pi }/{\varepsilon }^{>}$; (b) RR, the inset shows a zoom view on the early evolution of the quantities.

Figure 6

Compensated energy spectra for (a) NN case and (b) RR case, respectively. The vertical dashed lines indicate the location of ${\kappa }_c$ used in Fig. 5.

Figure 7

Energy flux $-\mathrm{\Pi }(\kappa ,t)$ at different times. (a) NN; (b) RR. The vertical dashed lines indicate the location of ${\kappa }_c$ used in Fig. 5.

Figure 8

(a) Temporal evolution of the energy flux $-\mathrm{\Pi }(\kappa ,t)$ at fixed wave numbers. The filled squares denote the maximum value of $\mathrm{\Pi }(\kappa ,t)$, defining the time $t=\mathrm{\Theta }\left(\kappa \right)$. (b) The relationship between the characteristic time $\mathrm{\Theta }\left(\kappa \right)$ and the wave number $\kappa$ in reversed turbulence. The dash-dotted line indicates a power-law behavior proportional to ${\kappa }^{-2/3}$. The Kolmogorov scale ${\eta }_0:=\eta (t=0)$ is used for normalization.

Figure 9

(a) Temporal evolution of the energy flux $-\mathrm{\Pi }(\kappa ,t)$ at fixed wave numbers. The filled squares denote the maximum value of $\mathrm{\Pi }(\kappa ,t)$, defining the time $G\left(t\right)={\mathrm{\Theta }}_G\left(\kappa \right)$. (b) The relationship between the characteristic time ${\mathrm{\Theta }}_G\left(\kappa \right)$ and the wave number $\kappa$ in reversed turbulence. The dash-dotted line indicates a power-law behavior proportional to ${\kappa }^{-2/3}$. The Kolmogorov scale ${\eta }_0:=\eta (t=0)$ is used for normalization.