Self-truncation and scaling in Euler-Voigt-$\alpha$ and related fluid models

A generalization of the 3D Euler-Voigt-$\alpha$ model is obtained by introducing derivatives of arbitrary order $\beta$ (instead of 2) in the Helmholtz operator. The $\beta \to \infty$ limit is shown to correspond to Galerkin truncation of the Euler equation. Direct numerical simulations (DNS) of the model are performed with resolutions up to ${2048}^3$ and Taylor-Green initial data. DNS performed at large $\beta$ demonstrate that this simple classical hydrodynamical model presents a self-truncation behavior, similar to that previously observed for the Gross-Pitaeveskii equation in Krstulovic and Brachet [Phys. Rev. Lett. 106, 115303 (2011)]. The self-truncation regime of the generalized model is shown to reproduce the behavior of the truncated Euler equation demonstrated in Cichowlas et al. [Phys. Rev. Lett. 95, 264502 (2005)]. The long-time growth of the self-truncation wave number $k_{\mathrm{st}}$ appears to be self-similar. Two related $\alpha$-Voigt versions of the eddy-damped quasinormal Markovian model and the Leith model are introduced. These simplified theoretical models are shown to reasonably reproduce intermediate time DNS results. The values of the self-similar exponents of these models are found analytically.

Figure 1

Temporal evolution (indicated by arrows) of the energy spectrum $E_{\alpha }\left(k\right)$ for $\beta =4$ and $k_{\alpha }=80$. Resolution ${1024}^3$ $\left(k_{\max }=342\right)$. The dashed lines respectively display the Kolmogorov $k^{-5/3}$ and the equipartition $k^2$ scaling. The self-truncation wavenumber is indicated by the small vertical arrow.

Figure 2

Energy spectrum $E_{\alpha }\left(k\right)$ versus $k$ for $\beta =2$ and $k_{\alpha }=50$, 100, and 200 at $t=10.2$. (b) Same conditions than in (a) but $\beta =4$. (c) Same conditions as in (a) but $\beta =6$. The dashed-black lines correspond to Kolmogorov scaling $E_{\alpha }\left(k\right)\sim k^{-5/3}$ and energy equipartition $E_{\alpha }\left(k\right)\sim k^2$. The thermalization wave number $k_{\mathrm{th}}$ and self-truncation $k_{\mathrm{st}}$ are indicated by arrows in (c).

Figure 3

Temporal evolution of (a) $k_{\mathrm{th}}$, (b) $E_{\mathrm{th}}$, (d) $\epsilon =\frac{\partial E_{\mathrm{th}}}{\partial t}$, (e) $k_{\mathrm{th}}/k_{\mathrm{d}}$ with $k_{\mathrm{d}}$ estimated based on the self-truncation wave number $k_{\mathrm{st}}$; see Eq. (10).

Figure 4

(a) Exponential decay of $E_{\alpha }\left(k\right)$ at $t=7.8$. (b) Time evolution of energy spectrum fit parameter $\delta$ of Eq. (11): Horizontal lines correspond to $\delta k_{\text{max}}=2$ (dashed black line) and $\delta k_{\alpha }=2$ (dot-dashed black line). Exponential law (dot black line) $\delta =2.7exp(-t/0.56)$ from Ref. [23].

Figure 5

Time evolution of $k_{\mathrm{st}}\left(t\right)$ at resolution ${1024}^3$ (see Table 1). (a) Different values of $\beta$ for $k_{\alpha }=20$. The crosses show the Euler evolution of $k_{\mathrm{st}}$ $\left(\alpha =0\right)$. (b) $\beta =4$ and different values of $k_{\alpha }$. The horizontal dashed lines represent the values of $k_{\alpha }/k_{\max }$.

Figure 6

Temporal evolution of the self-similar function $\mathrm{\Psi }[k/k_{\mathrm{st}}\left(t\right)]=E_{\alpha }(k,t)k_{\mathrm{st}}\left(t\right)$ [see Eq. (13)] for $\beta =4$ and $k_{\alpha }=4$. Data from direct numerical resolution of Eq. (2) at resolution ${512}^3$. The inset shows the temporal evolution of $k_{\mathrm{st}}\left(t\right)/k_{\max }$ for different values of $\beta$.

Figure 7

(a) Energy spectra $E_{\alpha }\left(k\right)$ at $t=3.5$ of the $\alpha \mathrm{V}$-EDQNM model, for different values of $\beta$, obtained with $k_{\alpha }={10}^4$ and $k_{\text{max}}=43918$ (corresponding to resolution $131072$, with $14.2$ points per octave). (b) Temporal evolution of the self-similar function $\mathrm{\Psi }[k/k_{\mathrm{st}}\left(t\right)]=E_{\alpha }(k,t)k_{\mathrm{st}}\left(t\right)$ for $\beta =4$ and $k_{\alpha }=5$ and $k_{\text{max}}=692$ $(28.4$ points par octave). The inset (b.1) shows the temporal evolution of $k_{\mathrm{st}}\left(t\right)$ for different values of $\beta$. The inset (b.2) shows the $\alpha \mathrm{V}$-EDQNM (red dashed line) and $\alpha \mathrm{V}$-Leith (solid green lines) theoretical prediction Eqs. (19) and (26) for the self-truncation exponent $\eta$ and numerical data from Table 2.

Figure 8

(a) Energy spectra $E_{\alpha }\left(k\right)$ of the $\alpha \mathrm{V}$-Leith model Eq. (24) for different values of $\beta$ obtained with $k_{\alpha }=400$ and $k_{\text{max}}=4000$. (b) Temporal evolution of the self similar function $\mathrm{\Psi }[k/k_{\mathrm{st}}\left(t\right)]=E_{\alpha }(k,t)k_{\mathrm{st}}\left(t\right)$ for $\beta =4$ and $k_{\alpha }=2$. The inset shows the temporal evolution of $k_{\mathrm{st}}\left(t\right)$ for different values of $\beta$.