Self-similar Subgrid-scale Models for Inertial Range Turbulence and Accurate Measurements of Intermittency

TL;DR

This paper introduces self-similar spectral subgrid models for turbulence that extend the inertial range and improve measurements of intermittency, revealing limitations of existing phenomenological models.

Contribution

It proposes a new class of reversible, self-similar subgrid models that significantly extend the inertial range and enhance intermittency measurement accuracy.

Findings

Extended inertial range by roughly one order of magnitude.

Intermittency for high order moments not captured by many existing models.

Improved measurement of anomalous exponents in turbulence.

Abstract

A class of spectral subgrid models based on a self-similar and reversible closure is studied with the aim to minimize the impact of subgrid scales on the inertial range of fully developed turbulence. In this manner, we improve the scale extension where anomalous exponents are measured by roughly one order of magnitude, when compared to direct numerical simulations or to other popular subgrid closures at the same resolution. We found a first indication that intermittency for high order moments is not captured by many of the popular phenomenological models developed so far.

Tables

Table 1. Table 1: Simulations: N 𝑁 N : number of collocation points in each spatial direction; k c subscript 𝑘 𝑐 k_{c} : smallest wavenumber where the SGS closure acts. k m ​ a ​ x subscript 𝑘 𝑚 𝑎 𝑥 k_{max} : maximum wavenumber evolved by the dynamics. ε 𝜀 \varepsilon : mean energy injection; ν 𝜈 \nu : kinematic viscosity. R ​ e = ε 1 / 3 ​ L 4 / 3 / ν 𝑅 𝑒 superscript 𝜀 1 3 superscript 𝐿 4 3 𝜈 Re=\varepsilon^{1/3}L^{4/3}/\nu : Reynolds number with L = 2 ​ π 𝐿 2 𝜋 L=2\pi . T 𝑇 T : duration of simulations in units of the eddy turn over time ε − 1 / 3 ​ L 2 / 3 superscript 𝜀 1 3 superscript 𝐿 2 3 \varepsilon^{-1/3}L^{2/3}

	$N$	$k_c$	$k_{max}$	$\epsilon$	$\nu$	$T$	$Re$
SGSM-sharp	$1024$	340	512	3.0	$8.0\cdot {10}^{-5}$	8.5	$2.1\cdot {10}^5$
SGSM-smooth	$1024$	340	512	3.0	$4.0\cdot {10}^{-5}$	8.5	$4.2\cdot {10}^5$
DNSx1	$1024$	…	340	2.5	$8.0\cdot {10}^{-4}$	12	$2.0\cdot {10}^4$
DNSx8	$8192$	…	3861	1.5	$4.4\cdot {10}^{-5}$	3.4	$3.0\cdot {10}^5$

Table 2. Table 2: Δ ¯ n + n / 2 subscript ¯ Δ 𝑛 𝑛 2 \bar{\Delta}_{n}+n/2 obtained as a fit of ξ n ​ ( r ) / ξ 2 ​ ( r ) subscript 𝜉 𝑛 𝑟 subscript 𝜉 2 𝑟 \xi_{n}(r)/\xi_{2}(r) for r ∈ [ 0.03 : 0.9 ] L r\in[0.03:0.9]\,L for SGSM-smooth and DNSx8, and for r ∈ [ 0.15 : 0.9 ] L r\in[0.15:0.9]\,L for DNSx1. Errors for the numerical data refer to the sum among statistical fluctuations and the variations considered by fitting in the first or second half of the scaling range (see the Supplemental Material for more details). The last three columns give the prediction from She-Leveque (SL) She and Leveque ( 1994 ) , Yakhot (Ya) Yakhot ( 2017 ) and Eling-Oz (EO) Eling and Oz ( 2015 ) models, where the last two have been fitted to have the value for n = 4 𝑛 4 n=4 identical to the smooth SGSM case. Errors in the Ya-EO models are estimated by fixing their free parameter to match either the maximum SGSM value 1.843 + 0.015 1.843 0.015 1.843+0.015 or the minimum, 1.843 − 0.015 1.843 0.015 1.843-0.015 , for n = 4 𝑛 4 n=4 , see Supplemental Material.

	SGSM	DNSx1	DNSx8	SL	Ya	EO
n=4	$1.843\left(15\right)$	$1.828\left(25\right)$	$1.824\left(18\right)$	$1.839$	$1.843\left(15\right)$	$1.843\left(15\right)$
n=6	$2.537\left(38\right)$	$2.501\left(78\right)$	$2.485\left(39\right)$	$2.555$	$2.563\left(38\right)$	$2.586\left(35\right)$
n=8	$3.092\left(30\right)$	$3.034\left(147\right)$	$2.982\left(56\right)$	$3.176$	$3.186\left(66\right)$	$3.257\left(58\right)$
n=10	$3.504\left(81\right)$	$3.440\left(230\right)$	$-$	$3.727$	$3.730\left(96\right)$	$3.875\left(83\right)$

Table 3. Table 3: Parameters of simulations. N 𝑁 N : number of collocation points in each spatial direction; k c subscript 𝑘 𝑐 k_{c} : smallest wavenumber where the SGS closure acts. k m ​ a ​ x subscript 𝑘 𝑚 𝑎 𝑥 k_{max} : maximum wavenumber evolved by the dynamics. Δ ¯ 4 + 2 subscript ¯ Δ 4 2 \bar{\Delta}_{4}+2 : fit of the exponents ξ 4 ​ ( r ) / ξ 2 ​ ( r ) subscript 𝜉 4 𝑟 subscript 𝜉 2 𝑟 \xi_{4}(r)/\xi_{2}(r) shown in Fig. 6 in the range r ∈ [ 0.03 : 0.9 ] L r\in[0.03:0.9]\,L

	$N$	$k_c$	$k_{max}$	${\overline{\Delta }}_4+2$
SGSM-sharp	$1024$	340	512	1.837 (0.013)(0.013)
SGSM-sharp-dealias	$1024$	230	340	1.838 (0.006)(0.010)
SGSM-smooth	$1024$	340	512	1.843 (0.006)(0.009)

Equations

$$S_n(r)=⟨{\left[{\delta }_{r}u\right]}^n⟩\sim {\left(\frac{r}{L}\right)}^{{\zeta }_n}$$

$$({\partial }_t+\nu k^2){\hat{\mathbf{u}}}_{\mathbf{k}}(t)={\hat{\mathbf{T}}}_{\mathbf{k}}(t)+{\hat{\mathbf{f}}}_{\mathbf{k}}(t)$$

$$E_k(t)={\left(k/k_c\right)}^{-\frac{5}{3}}{E}_{k_c}(t);k_c\le k\le k_{max},$$

$${\partial }_t{\hat{\mathbf{u}}}_{\mathbf{k}}(t)={\hat{\mathbf{T}}}_{\mathbf{k}}(t)+{\hat{\mathbf{f}}}_{\mathbf{k}}(t)-{\gamma }_{\mathbf{k}}{\lambda }_k(t){\hat{\mathbf{u}}}_{\mathbf{k}}(t)$$

$${\lambda }_k(t)=\frac{1}{2}\frac{T_k(t)-{\left(k/k_c\right)}^{-5/3}{T}_{k_c}(t)}{E_k(t)},$$

$${\gamma }_{\mathbf{k}}=\{\begin{array}{l}0\text{ if }k<k_c\\ 1\text{ with prob. }{P}_k=\frac{k-k_c}{k_{max}-k_c}\text{ if }{k}_c\le k<k_{max}.\end{array}$$

$${\xi }_n(r)=\frac{d\log S_n(r)}{d\log (r)}$$

$$F_n(r)=\frac{S_n(r)}{[S_2(r){]}^{n/2}}\sim [S_2(r){]}^{\left(\frac{{\xi }_n(r)}{{\xi }_2(r)}-\frac{n}{2}\right)}.$$

$$\{\begin{array}{l}{\zeta }_n=n/9+2(1-(2/3{)}^{n/3});\text{(SL),}\\ {\zeta }_n=[(\gamma -1)+\sqrt{(\gamma -1{)}^2+4\gamma n/3}]/(2\gamma );\text{(EO)},\\ {\zeta }_n=n(1+3\beta )/(3+3\beta n));\text{(Ya).}\end{array}$$

Full text

Turbulence is ubiquitous in nature and in engineering applications and it is characterized by the presence of intense non-Gaussian fluctuations on a wide range of inertial scales and frequencies. The main mechanism to be controlled and, eventually, modeled is the energy transfer from the large-scale, $L$, where the flow is stirred, to the small-scale, $\eta$, where viscous effects are dominant Frisch (1995); Sreenivasan (1999); Pope (2000); Ishihara et al. (2009); Alexakis and Biferale (2018). The Reynolds number is a measure of the separation between the two scales, $Re\sim(L/\eta)^{4/3}$. For most applications, $Re$ is too large to allow the problem to be attacked by direct numerical simulations (DNS) Ishihara et al. (2009); Jiménez (2012). Similarly, fundamental problems connected to the presence of anomalous scaling Frisch (1995); Benzi et al. (2010); Iyer et al. (2017); Sinhuber et al. (2017) in the limit $Re\to\infty$ cannot be easily studied using numerical tools. In such a deadlock, the applied community resorts to Large Eddy Simulations (LES), a numerical approach that restricts the Navier-Stokes equations to a range of scales (or wavenumbers) larger (smaller) than a given cut-off, $r>r_{c}$ ($k<k_{c}$), and modeling all subgrid-scale (SGS) degrees of freedom with closures in configuration Smagorinsky (1963); Pope (2000); Meneveau and Katz (2000); Lesieur et al. (2005), or Fourier Kraichnan (1976); Chollet and Lesieur (1981); Baerenzung et al. (2008); Biferale et al. (2017) space. The aim is to achieve a good accuracy for the energy-containing modes, without paying too much attention to those (inertial) scales that are fully resolved, but also unavoidably affected by the subgrid-scale closure. As a matter of fact, most LES implementation reproduce successfully the large-scale dynamics, $k\ll k_{c}$, and are inaccurate for the highest resolved wavenumber modes, $k\sim k_{c}$. This fact, prevents the possibility to use LES models to improve our understanding of multi-scale velocity fluctuations and/or the feedback of small-scale fluctuations on global mean profiles. In particular, SGS models (SGSM) perform very poorly concerning the properties of the inertial-range scaling of velocity structure functions (SF):

[TABLE]

where we defined the longitudinal increments $\delta_{r}u=[{\boldsymbol{u}}({\boldsymbol{r}}+{\boldsymbol{x}})-{\boldsymbol{u}}({\boldsymbol{x}})]\cdot{\hat{\bf r}}$ and we have assumed isotropy and homogeneity. The exponents $\zeta_{n}$ in (1) are the key quantities to predict the asymptotic statistics for large Reynolds numbers, where $r/L$ can be arbitrarily small.

On one side, experiments and numerical simulations have provided many evidences that the scaling of $S_{n}(r)$ is anomalous, i.e. different from the Kolmogorov 1941 (K41) prediction $\zeta_{n}=n/3$ Benzi et al. (2010); Gotoh et al. (2002); Watanabe and Gotoh (2007); Sinhuber et al. (2017); Iyer et al. (2017). On the other hand, we do not have any first-principle derivations of the $\zeta_{n}$. Furthermore, it is extremely difficult to get accurate measurements of the exponents, due to the concurrent requirements of having a large scaling range and large statistical ensembles. As a result, we also lack the numerical and experimental accuracy to distinguish among different phenomenological models Kolmogorov (1962); Benzi et al. (1984); Meneveau and Sreenivasan (1987); She and Leveque (1994); Schumacher et al. (2007); Eling and Oz (2015); Yakhot (2017); Yakhot and Donzis (2017). Finally, few assessments exist of the robustness of the exponents with respect to the small-scale dissipative mechanism Falkovich (1994); Lohse and Müller-Groeling (1995); Frisch et al. (2008); Donzis and Sreenivasan (2010).

In this letter, we introduce a class of subgrid models to minimize the impact of the SGS closure on the inertial-range: a sort of perfect energy-cascade sink that achieves a much higher effective numerical resolution to study scaling properties in turbulence. The idea was already presented in She and Jackson (1993); Jimenez (1993) but was never applied and developed in the way it is here. We introduce a self-similar buffer close to the highest resolved mode, such as to have an ultraviolet boundary condition for the energy cascade at high $k$ which is consistent with the existence of an infinitely extended inertial range. The advantages with respect to other closures are many. First, our model is time-reversible, allowing the formation of back-scatter events too. Second, it is a minor modification of the high-wave-number dynamics, without touching the Fourier-phases and therefore with a minimal impact on the formation of intense coherent events that are believed to be the responsible of anomalous scaling. Unlike in Jimenez (1993), here we focus on high Reynolds applications to assess the impact of the closure on the inertial range properties. Furthermore, we expand the protocol by considering also a new Fourier modulation where the closure is applied such as to improve its efficiency in absorbing the energy cascade.

In the following, we show that our LES protocol is able to obtain the same inertial-range extension of a fully resolved viscous DNS while saving roughly 1 order of magnitude of resolution. As a result, considering also the gain due to the possibility of relaxing the time step, the improvement in the computational resources is larger than a factor 1000, opening the way towards increased accuracy of measuring scaling exponents in turbulence, in both the scaling range extension and the statistical error. Moreover, we assess the universality issue with respect to the ultraviolet dissipation mechanism by comparing the scaling obtained with our SGS-model with the ones measured in DNS and experiments Gotoh et al. (2002); Sinhuber et al. (2017); Iyer et al. (2017). Another by-product is to have a LES that is accurate for small-scale evolution, something important engineering applications that control extreme non-Gaussian events close to the subgrid cutoff Stevens et al. (2014); Stevens and Meneveau (2017); Buzzicotti et al. (2018); Linkmann et al. (2018).

The model. Let us consider the Fourier-space evolution of the three dimensional Navier-Stokes equations in a periodic box of size $L=2\pi$ and resolved with $N$ grid points per direction and maximum wavenumber in all direction given by $k_{max}=N/2$:

[TABLE]

where $\nu$ is the viscosity, $\hat{{\boldsymbol{f}}}_{\boldsymbol{k}}(t)$ is the Fourier transform of the external forcing and $\hat{{\boldsymbol{T}}}_{\boldsymbol{k}}(t)=-i{\boldsymbol{k}}\cdot\left(\mathbb{I}-\frac{{\boldsymbol{k}}\otimes{\boldsymbol{k}}}{|k|^{2}}\right)\left[\sum_{{\boldsymbol{k}}^{\prime}}\hat{{\boldsymbol{u}}}_{{\boldsymbol{k}}^{\prime}}(t)\otimes\hat{{\boldsymbol{u}}}_{{\boldsymbol{k}}^{\prime}-{\boldsymbol{k}}}(t)\right]$ is the non-linear term. We follow Jimenez (1993) and we replace the viscous term on the lhs of (2) with a non-linear inertial closure that imposes a perfect self-similar Kolmogorov-like spectrum in a $k$-window close to the ultraviolet cut-off, $k_{max}$:

[TABLE]

where $E_{k}(t)=\frac{1}{2}\sum_{|{\boldsymbol{k}}|=k}|\hat{{\boldsymbol{u}}}_{\boldsymbol{k}}(t)|^{2}$. The LES equation for the resolved velocity field equipped with the fixed-spectrum SGS-model can be written using a Lagrangian multiplier $\lambda_{k}(t)$ She and Jackson (1993),

[TABLE]

where we have removed the viscosity and $\gamma_{\boldsymbol{k}}$ is a projector which selects the range of scales where the subgrid closure acts: $\gamma_{\boldsymbol{k}}=0$ if $k\leq k_{c}$ and $\gamma_{\boldsymbol{k}}=1$ if $k_{c}<k<k_{max}$ (SGSM-sharp). It is easy to realize that in order to satisfy (3) we can impose $dE_{k}/dt=(k_{c}/k)^{5/3}dE_{k_{c}}/dt$ and choose $\lambda_{k}(t)$ to be:

[TABLE]

where $T_{k}(t)$ is the transfer function: $T_{k}(t)=\sum_{|{\boldsymbol{k}}|=k}\hat{{\boldsymbol{u}}}_{\boldsymbol{k}}^{*}(t)\hat{{\boldsymbol{T}}}_{\boldsymbol{k}}(t)$. In order to mitigate the sharp transition across the SGS, $k_{c}$, we also explored another protocol where the percentage of constrained modes grows linearly from [math] at $k_{c}$ to $1$ at $k_{max}$. To do that, we define a (quenched) probability to apply the SGS model at any given wavenumber as follows (SGSM-smooth):

[TABLE]

In this way, only a fraction of modes $(k-k_{c})/(k_{max}-k_{c})$ will be affected by the constraint for any given shell $k$, such that we move from fully unconstrained dynamics (for $k<k_{c}$) to a fixed spectrum dynamics (for $k=k_{c}$) with continuity (see inset of Fig. 1 for a graphical scheme of the Fourier space support of the projector $\gamma_{\boldsymbol{k}}$ for both sharp and smooth SGSM cases). We also anticipate that in order to minimize the transition across $k_{c}$ we will need to keep a small residual viscosity $\nu$ even when using the self-similar closure. This is unavoidable due to the fact that the closure acts on a finite range of scales and cannot mimic exactly the SGS dynamics at infinite Reynolds.

Results. We compare the LES data obtained at a resolution of $1024^{3}$ with the two different DNS resolutions: one identical to the LES (DNSx1) and one taken from a state-of-the-art study at $8192^{3}$ collocation points Iyer et al. (2017) denoted (DNSx8). All runs are forced with a white-in-time Gaussian forcing acting at $k_{f}\in[1,1.5]$ for DNSx1 and a $k_{f}\in[1,3]$ for DNSx8. More details on the numerical set up can be found in Table I. In Fig. 1 we show the spectral properties of all data. Our closure reproduces the same extension of the scaling range of DNSx8 and considerably extends the one obtained with DNSx1. We obtain an inertial behavior for all $k$ in the LES model without the viscous range of scales needed with standard viscosity in DNSx8.

Anomalous scaling of high order SF. To assess the scaling properties in a quantitative way, we measure the local scaling exponents:

[TABLE]

where in the presence of pure power-laws we must have $\xi_{n}(r)=const.=\zeta_{n}$.

By measuring where $\xi_{n}(r)$ is constant we have an unbiased definition of the inertial range extension and we can assess scale-by-scale the quality of our data. In particular, intermittency and scale-dependent corrections from a Gaussian behavior can be measured by the deviation from zero of $\Delta_{n}(r)=\xi_{n}(r)/\xi_{2}(r)-n/2$ as seen by expressing the generalized Flatness in terms of the 2nd order SF:

[TABLE]

In Fig. 2 we show $\xi_{2}(r)$ for our two SGSM closures and compare them with the same quantity measured on DNSx1 and on DNSx8.

As shown for the spectral case, LES data have a much larger extension of scaling then DNSx1, matching the DNS obtained with a 8-times larger resolution (DNSx8).

Despite the existence of a plateau for $\xi_{2}(r)$ for all data, the constraint $\xi_{2}(r)\to 2$ for $r\to 0$ makes the jump from inertial to viscous values too big and it is very difficult to quantitatively distinguish the Kolmogorov 1941 (K41) scaling from any intermittent phenomenological model as, e.g. the She-Leveque (SL) She and Leveque (1994), the Yakhot model Yakhot (2017) and the model proposed by Oz based on spontaneous symmetry breaking of dilation invariance and random geometry Oz and Oz (2018, 2018); Eling and Oz (2015). To be more accurate, in Fig. 3 we show the scaling of the generalized Flatness (inset) and of the scale-by-scale ratio $\Delta_{n}(r)+n/2=\xi_{n}(r)/\xi_{2}(r)$ (main panel) for $n=4,6$. Here, a Kolmogorov-like nonanomalous scaling corresponds to a constant value $n/2$ for all $r$. As one can see, the deviation from the Kolmogorov scaling is now evident and -much more importantly- our SGSM closures are able to develop an inertial range as extended as the DNSx8 case, if not even larger. Moreover, the SGSM-smooth closure is a bit better than the SGSM-sharp case. We consider these results a clear demonstration that the SGS model developed here can be considered a sort of infinite-Reynolds closure. Considering the fact that using the SGSM-smooth closure we can achieve the same accuracy for local exponents of a DNS with 8-time larger resolution, we estimate a gaining factor $8^{3}$ for the spatial grid, which together with the less stringent Courant-Friedrichs-Lewy (CFL) condition for the time integration, $\sim\varepsilon^{-1/3}k_{max}^{-2/3}$ leads to a total gain close to a factor 1000. In Table II we present a summary for the scaling properties of $F_{n}(r)$ from where it is clear that the SGS models agrees with the DNSx8 and with the prediction made by models SL-Ya-EO for moment, $n=6,8$ while for the largest achievable order, $n=10$, numerical data are more intermittent than all three phenomenological models (see also SM).

A few comments are now in order. First, it is useful to preserve a very small viscous term in (4) in order to have a smooth transition across $k_{c}$. This is implemented in our approach, keeping a term $\nu k^{2}\hat{{\boldsymbol{u}}}_{{\boldsymbol{k}}}(t)$ with a very small $\nu$ as shown in Table I. It is clear from Fig. 3 that even by optimizing $\nu$, there exists in the SGSM a pseudo-viscous range (extended over a few grid points) where scaling breaks down. This is unavoidable because our closure is acting in the Fourier space and does not enforce any pure scaling for the high order SFs. The existence of a small bump for the local slopes around the transition from viscous to inertial range is present also in experimental data at high Reynolds Sinhuber et al. (2017). On the other hand, the efficiency in extending the anomalous scaling-range is a good evidence that to capture intermittency the SGSM must maintain the correct phase-correlations Murray and Bustamante (2018), which is one of the main added value of (5). Second, the smooth projector recipe is not unique and one can imagine many different ways to enforce the transition from modes that evolve according to their Euler dynamics ($k<k_{c}$) to those that feel the spectral constraint. In particular, once the controlled buffer is introduced and it is large enough, one might imagine even avoiding the dealiasing protocol and keeping $k_{max}=N/2$ as done here. The effects of introducing a dealiasing are minor and discussed in Fig. 2 of the SM.

We now discuss the comparison with two other popular ways to enhance the effective Reynolds numbers. In Fig. 4 we compare the Flatness obtained from a DNS with hyper-viscosity Borue and Orszag (1995); Frisch et al. (2008) or from a Smagorinsky SGS model Smagorinsky (1963); Meneveau and Katz (2000); Linkmann et al. (2018) with the one proposed here. Notice that the hyperviscous data are only qualitatively as good as the SGSM-smooth as shown by the fact that the former has a less extended plateau wrt to the latter. There are no doubts that the closure (5) is superior to both Smagorinsky and hyperviscous models. Finally, we mention that from (4) one can define a Galilean-invariant Eyink and Aluie (2009) SGS energy transfer: $\Pi({\boldsymbol{x}})=\partial_{i}u_{j}({\boldsymbol{x}})\int d{\boldsymbol{k}}\,\gamma_{\boldsymbol{k}}\lambda_{k}e^{i{\boldsymbol{k}}\cdot{\boldsymbol{x}}}ik_{i}/k^{2}\hat{u}_{j,{\boldsymbol{k}}}$ which is non-positive definite and therefore able to reproduce back-scatter events.

Conclusions. We have shown that a self-similar SGS model is able to extend the anomalous scaling to almost the entire range of resolved scales. This protocol reduces the computational cost by a factor one thousand compared to a fully resolved DNS, with the same inertial range extension. The agreement between the scaling observed with the SGSM and that measured by DNS and experiments supports the universality of the inertial range dynamics with respect to the energy absorbing mechanism at small scales. Thanks to the unprecedented accuracy in the determination of the scaling properties we are able to find some small discrepancy between the numerical data and the predictions by some of the most popular phenomenological models She and Leveque (1994); Eling and Oz (2015); Yakhot (2017) for high order moments. It remains an open key question to check if our closure remains accurate also at higher resolution. If this is indeed the case, we have a chance to make a discontinuous improvement in the assessment of scaling properties in homogeneous and isotropic turbulence. Our model outperforms other common closures such as the Smagorinsky model or hyper-viscous DNS. Fully time-reversible models might of theoretical interest for the application of the chaotic hypothesis Gallavotti (1996). Beside the self-similar properties, another advantage of our SGS closure is that the phase dynamics is left untouched. Because of its generality, the closure can be applied to a broad set of other flow configurations such as rotating, stratified, or magnetohydrodynamic turbulence, including stiff problems as the kinematic dynamo in the limit of small Prandtl numbers Tobias et al. (2013). Similarly, one might imagine applications to wall bounded flows where small-scale anisotropy is negligible Biferale and Procaccia (2005), by imposing scaling laws on the spectral degrees of freedom in planes parallel to the wall (homogeneous directions), with properties dependent on the distance from the wall, eventually.

We acknowledge useful discussions with R. Benzi, M. Bustamante, M. Linkmann, C. Meneveau, Y. Oz, M. Sbragaglia, K.R. Sreenivasan, P.K. Yeung, M. Wilczek and funding from the European Union’s Programme (FP7/2007-2013) grant No.339032. L.B. acknowledges the hospitality of the Center for Environmental and Applied Fluid Mechanics at Johns Hopkins University where this work was started. Part of the simulations have been done at BSC-CNS (PRACE Grant No. 188513).

I Appendix: supplemental material ‘Self-similar subgrid-scale models for inertial range turbulence’

1 High order SF and error calculation

In this supplemental material (SM) we further extend the quantitative analysis of local scaling properties for high order generalized Flatness whose scaling properties are summarized in table II of the main body of the paper and in table I of this SM. To quantify the accuracy of the fit for $\bar{\Delta}_{n}$ we have measured the errors in two different ways. The first, consists in the maximum difference between the exponents averaged on the whole inertial range, $[r_{min}=0.03L:r_{max}=0.9L]$, with the values measured from the average on the first and the second half of the inertial range. The intermediate scale used in the evaluation of this error is $r_{int}=0.18L$ obtained as $\log(r_{int})=(\log(r_{min})-\log(r_{max}))/2$. The second is the root-mean-square error, hence it is the sum of squared deviations between the fitted value $\bar{\Delta}_{4}$ and the data points weighted by the number $N$ of measurements used in the fit, namely, $\sqrt{\frac{1}{N}\sum_{i=0}^{N}(\bar{\Delta}_{4}+2-\xi_{4}(r_{i})/\xi_{2}(r_{i}))^{2}}$.

In Table I of the paper we also show the prediction from three popular phenomenological models, the one proposed by She-Leveque (SL), Yakhot (Ya) and Eling-Oz (EO), which are given by the following expressions:

[TABLE]

All three expressions gives $\zeta_{3}=1$ independently of their free parameters. Both EO and Ya formula have one free parameter $\gamma$ and $\beta$, respectively. For each model the free parameter has been chosen such as to match the mean value $\bar{\Delta}_{4}+2=1.843$ of the SGSM data. In order to evaluate the sensitivity of the model prediction we give in Table II of the main paper the maximal variations obtained by fixing the free parameters to match either the maximum value measured within error bars by the SGSM model, $1.843+0.015$ or the minimum value $1.843-0.015$.

In Fig. 5 we show the equivalent of Fig. 3 of the main paper but for $n=8$ and $n=10$. In the main body of the figure we report the log-lin plot of the local scaling exponents, $\Delta_{n}(r)+n/2=\xi_{n}(r)/\xi_{2}(r)$, defining the scale-by-scale property of $F_{n}(r)$ and in the inset the log-log of the Flatness. Notice that the SGSM closure allow us to achieve good enough statistics up to $n=10$ (right panel).

2 dealiasing effect

As discussed in the main body of the paper, the SGSM-closure is not unique. One can play with (i) the ratio $k_{c}/k_{max}$, (ii) the absolute value of $k_{max}$ and the distribution of wavenumbers where the spectrum is fixed inside the window $[k_{c},k_{max}]$. In Fig. 6 we show the scale-by-scale ratio $\Delta_{n}(r)+n/2=\xi_{n}(r)/\xi_{2}(r)$ for $n=4$, for three different SGSM protocols (see table I in this SM): without dealiasing ($k_{max}=N/2$, $k_{c}/k_{max}=2/3$) for both SGSM-smooth and SGSM-sharp and with dealiasing ($k_{max}=N/3$, $k_{c}/k_{max}=2/3$) for SGSM-sharp. As we can see from Fig. 6, all simulations are in a very good agreement, suggesting that the aliasing errors are negligible in the estimation of the inertial-range scaling exponents with the self-similar SGSM closures explored here.