Validity of the self-similar subgrid-scale closure at higher Reynolds numbers and the high-order intermittency discrepancy in homogeneous isotropic turbulence

A self-similar, time-reversible spectral subgrid-scale (SGS) closure was shown by Biferale, Bonaccorso, Buzzicotti and Iyer [Phys. Rev. Lett. 123, 014503 (2019)] to reproduce the anomalous scaling of velocity structure functions in 1024^3 large-eddy simulations at Reynolds number Re ≈ 4×10^5, matching the inertial-range extension of an 8192^3 direct numerical simulation at roughly 1000× lower cost, and to reveal that the 10th-order moment is more intermittent than the fitted She–Lévêque, Yakhot, and Eling–Oz predictions. Whether this closure remains quantitatively accurate at still higher resolutions and/or Reynolds numbers, and whether the order-10 excess intermittency is genuine asymptotic physics or a finite-resolution/closure artifact, remains open: no published work reports a test of the closure beyond 1024^3/Re ≈ 4×10^5, and no statistically controlled adjudication of the order-10 discrepancy.

Background

In fully developed turbulence, the statistics of velocity increments across scales are characterized by structure functions $S_n(r)=\langle[\delta_r u]^n\rangle\sim (r/L)^{\zeta_n}$ , where $\delta_r u$ is the longitudinal velocity increment over separation $r$ , $L$ is the integral scale, and $\zeta_n$ are scaling exponents. Kolmogorov's 1941 (K41) phenomenology predicts self-similar scaling $\zeta_n=n/3$ ; experiments and simulations robustly show deviations ("anomalous scaling" or intermittency), driven by rare, intense small-scale events. No first-principles derivation of the $\zeta_n$ exists, so they must be measured. Accurate measurement requires both a wide inertial range — the range of scales between the forcing scale and the dissipation scale, whose width grows with the Reynolds number Re — and huge statistical samples, since high-order moments are dominated by rare events. Direct numerical simulations (DNS) resolving all scales at Re high enough for a wide inertial range (e.g. $8192^3$ grids) are extremely expensive, and standard large-eddy simulations (LES), in which scales below a cutoff are modeled by a subgrid-scale (SGS) closure such as the Smagorinsky eddy viscosity, which represents the unresolved scales by an effective viscosity proportional to the local resolved strain rate, corrupt the inertial-range statistics of high-order moments. As a result, the field still lacks the accuracy needed to discriminate among the leading phenomenological intermittency models of $\zeta_n$ : the She–Lévêque (SL) log-Poisson model, which pictures the energy cascade as a multiplicative process dominated by rare, intensely dissipative vortex filaments; the Yakhot (Ya) mean-field model; and the Eling–Oz (EO) model, which derives the exponents from the roughness properties of random surfaces.

Biferale, Bonaccorso, Buzzicotti and Iyer [Phys. Rev. Lett. 123, 014503 (2019)] introduced a class of spectral SGS closures designed to leave the inertial range untouched. Building on earlier constrained-dynamics ideas of She–Jackson and Jimenez, they replace the viscous term of the Fourier-space Navier–Stokes equations in a buffer window near the ultraviolet cutoff with a reversible, nonlinear term implemented via Lagrange multipliers $\lambda_k(t)$ , which imposes a self-similar Kolmogorov-like $k^{-5/3}$ spectrum for $k_c<k<k_{\max}$ while leaving the Fourier phases untouched. Two variants ("sharp" and "smooth" projector $\gamma_k$ ) were studied at $1024^3$ resolution with $\mathrm{Re}=2.1$ – $4.2\times10^5$ ( $\mathrm{Re}:=\varepsilon^{1/3}L^{4/3}/\nu$ ). The closure extends the range over which anomalous exponents can be measured by roughly one order of magnitude relative to other closures at the same resolution: the scale-by-scale ratio $\xi_n(r)/\xi_2(r)$ of the (absolute) structure-function local exponents — the exponent with which $S_n(r)$ grows as a function of $S_2(r)$ , a normalization in the spirit of the Extended Self-Similarity (ESS) idea that cancels finite-Reynolds-number drift — develops a plateau over $r\in[0.03,0.9]L$ as extended as an $8192^3$ DNS, at an estimated $\sim1000$ -fold saving. For orders $n=4,6,8$ the plateau values agree with DNS and experiments and with SL/Ya/EO (the latter two having one free parameter fixed to match the $n=4$ value 1.843). At the largest achievable order, $n=10$ , the data are more intermittent than all three models: the SGSM fit gives $\xi_{10}/\xi_2=3.504\pm0.081$ , versus SL 3.727, Ya 3.730(96), EO 3.875(83); the $8192^3$ DNS reference was itself statistically inadequate at this order. The authors call this "a first indication that intermittency for high order moments is not captured by many of the popular phenomenological models".

Problem Statement

Determine whether the self-similar, time-reversible spectral subgrid-scale (SGS) closure of Biferale, Bonaccorso, Buzzicotti and Iyer [Phys. Rev. Lett. 123, 014503 (2019)] remains quantitatively accurate for inertial-range intermittency measurements at resolutions and/or effective Reynolds numbers beyond the published $1024^3$ runs at $\mathrm{Re}\approx4\times10^5$ , where an increase beyond the published settings in either one suffices, and decide whether the excess intermittency observed at the 10th-order moment relative to the She–Lévêque, Yakhot and Eling–Oz predictions ( $\xi_{10}/\xi_2=3.504\pm0.081$ versus $3.727$ , $3.730(96)$ , $3.875(83)$ ) is genuine asymptotic behavior of homogeneous isotropic turbulence or a finite-resolution/closure artifact. Concretely, an answer should: (i) assess the closure (sharp and/or smooth projector variants) at higher resolution and/or effective Reynolds number with statistically converged high-order moments — from new simulations, or from reanalysis of existing runs and published data — quantifying the scale-by-scale exponents $\xi_n(r)/\xi_2(r)$ for $n$ up to at least 10 over the extended inertial range with controlled error bars; (ii) validate against matched references — fully resolved DNS at comparable effective Reynolds number and/or high-Reynolds-number experimental data — to the statistical accuracy those references support, bearing in mind that the original $8192^3$ DNS lacked statistical accuracy at order 10; and (iii) state a verdict on closure accuracy (does the anomalous-scaling plateau extend and agree with the references within errors, or does the closure degrade?) and on the order-10 discrepancy (genuine or artifact), including which, if any, of the She–Lévêque, Yakhot and Eling–Oz phenomenologies remain consistent with the data at high orders.

References

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