Extended Self-Similarity in the Dissipation Range of Fully Developed Turbulence

Benzi, Roberto; Ciliberto, S.; Baudet, Christophe; Chavarría, Gerardo Ruiz; Tripiccione, R.

doi:10.1209/0295-5075/24/4/007

Cited by 162 publications

(114 citation statements)

References 6 publications

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“…1) with a constant energy flux for k ∈ [5, 20] and maximally helical vortex tubes are found, as predicted in [10] and shown in [11,12]. Finally, the anomalous exponents of longitudinal structure functions are in excellent agreement with previous studies [1] up to order p = 8 (see Table I), including analysis without using the extended self-similarity (ESS) hypothesis [13].…”

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confidence: 88%

Imprint of Large-Scale Flows on Turbulence

2005

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We investigate the locality of interactions in hydrodynamic turbulence using data from a direct numerical simulation on a grid of 1024 3 points; the flow is forced with the Taylor-Green vortex. An inertial range for the energy is obtained in which the flux is constant and the spectrum follows an approximate Kolmogorov law. Nonlinear triadic interactions are dominated by their non-local components, involving widely separated scales. The resulting nonlinear transfer itself is local at each scale but the step in the energy cascade is independent of that scale and directly related to the integral scale of the flow. Interactions with large scales represent 20% of the total energy flux. Possible explanations for the deviation from self-similar models, the link between these findings and intermittency, and their consequences for modeling of turbulent flows are briefly discussed.PACS numbers: 47.27. Eq,47.27.Ak,47.65.+a Flows in nature are often in a turbulent state driven by large scale forcing (e.g. novae explosions in the interstellar medium) or by instabilities (e.g. convection in the sun). Such flows involve a huge number of coupled modes leading to great complexity both in their temporal dynamics and in the physical structures that emerge. Many scales are excited, for example from the planetary scale to the kilometer for convective clouds in the atmosphere, and much smaller scales when considering microprocesses such as droplet formation. The question then arises concerning the nature of the interactions between such scales: are they predominantly local, involving only eddies of similar size, or are they as well non-local? It is usually assumed that the dominant mode of interaction is the former, and this hypothesis is classically viewed as underlying the Kolmogorov phenomenology that leads to the prediction of a E(k) ∼ k −5/3 energy spectrum; such a spectrum has been observed in a variety of contexts although there may be small corrections to this power-law due to the presence in the small scales of strong localized structures, such as vortex filaments [1].Several studies have been devoted to assess the degree of locality of nonlinear interactions, either through modeling of turbulent flows, as is the case with rapid distortion theory (RDT) [2] or Large Eddy Simulations (LES) [3], or through the analysis of direct numerical simulations (DNS) of the Navier-Stokes equations (see e.g. [3, 4, 5]), and more recently through rigorous bounds [6]. The spatial resolution in the numerical investigations was moderate, without a clearly defined inertial range and the differentiation between local and non-local interactions was somewhat limited. Thus, a renewed analysis at substantially higher Reynolds numbers in the absence of any modeling is in order; we address this issue by analyzing data stemming from a newly performed DNS on a grid of 1024 3 points using periodic boundary conditions.The governing Navier-Stokes equation for an incompressible velocity field v, with P the pressure, F a forcing term and ν = 3 × 10 −4 the...

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confidence: 88%

Imprint of Large-Scale Flows on Turbulence

2005

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“…In this sense, ESS enables one to extend the scaling range to the complete range of separation scales examined, as shown empirically by [28][29][30] and references therein, and provides another way to characterize the dependence of the scaling exponent ξ (q) on q. Extended self-similarity is described by the following power-law relationship between sample structure functions of different orders [31,32] …”

Section: Scaling Of Statisticsmentioning

confidence: 93%

Statistical scaling of pore-scale Lagrangian velocities in natural porous media

et al. 2014

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We investigate the scaling behavior of sample statistics of pore-scale Lagrangian velocities in two different rock samples, Bentheimer sandstone and Estaillades limestone. The samples are imaged using x-ray computer tomography with micron-scale resolution. The scaling analysis relies on the study of the way qth-order sample structure functions (statistical moments of order q of absolute increments) of Lagrangian velocities depend on separation distances, or lags, traveled along the mean flow direction. In the sandstone block, sample structure functions of all orders exhibit a power-law scaling within a clearly identifiable intermediate range of lags. Sample structure functions associated with the limestone block display two diverse power-law regimes, which we infer to be related to two overlapping spatially correlated structures. In both rocks and for all orders q, we observe linear relationships between logarithmic structure functions of successive orders at all lags (a phenomenon that is typically known as extended power scaling, or extended self-similarity). The scaling behavior of Lagrangian velocities is compared with the one exhibited by porosity and specific surface area, which constitute two key pore-scale geometric observables. The statistical scaling of the local velocity field reflects the behavior of these geometric observables, with the occurrence of power-law-scaling regimes within the same range of lags for sample structure functions of Lagrangian velocity, porosity, and specific surface area.

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“…Though nonlinear variation of ξ(q) with q is also reproduced by the fractional Laplace model of Meerschaert et al (2004) (see Kozubowski et al, 2006;Ganti et al, 2009), the latter does not include cutoffs and thus fails to reproduce observed breakdown in power-law scaling at small and large lags. Benzi et al (1993aBenzi et al ( ,b, 1996 discovered empirically that the range s I < s < s II of separation scales over which velocities in fully developed turbulence (where Kolmogorov's dissipation scale is assumed to control s I ) scale according to Eq. (2) can be enlarged significantly, at both small and large lags, through a procedure they called Extended SelfSimilarity (ESS).…”

Section: Introductionmentioning

confidence: 99%

Extended power-law scaling of air permeabilities measured on a block of tuff

Siena

Guadagnini

Riva

et al. 2012

Hydrol. Earth Syst. Sci.

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Abstract. We use three methods to identify power-law scaling of multi-scale log air permeability data collected by Tidwell and Wilson on the faces of a laboratory-scale block of Topopah Spring tuff: method of moments (M), Extended Self-Similarity (ESS) and a generalized version thereof (G-ESS). All three methods focus on q-th-order sample structure functions of absolute increments. Most such functions exhibit power-law scaling at best over a limited midrange of experimental separation scales, or lags, which are sometimes difficult to identify unambiguously by means of M. ESS and G-ESS extend this range in a way that renders power-law scaling easier to characterize. Our analysis confirms the superiority of ESS and G-ESS over M in identifying the scaling exponents, ξ(q), of corresponding structure functions of orders q, suggesting further that ESS is more reliable than G-ESS. The exponents vary in a nonlinear fashion with q as is typical of real or apparent multifractals. Our estimates of the Hurst scaling coefficient increase with support scale, implying a reduction in roughness (anti-persistence) of the log permeability field with measurement volume. The finding by Tidwell and Wilson that log permeabilities associated with all tip sizes can be characterized by stationary variogram models, coupled with our findings that log permeability increments associated with the smallest tip size are approximately Gaussian and those associated with all tip sizes scale show nonlinear variations in ξ(q) with q, are consistent with a view of these data as a sample from a truncated version (tfBm) of self-affine fractional Brownian motion (fBm). Since in theory the scaling exponents, ξ(q), of tfBm vary linearly with q we conclude that nonlinear scaling in our case is not an indication of multifractality but an artifact of sampling from tfBm. This allows us to explain theoretically how power-law scaling of our data, as well as of non-Gaussian heavy-tailed signals subordinated to tfBm, are extended by ESS. It further allows us to identify the functional form and estimate all parameters of the corresponding tfBm based on sample structure functions of first and second orders.

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Extended Self-Similarity in the Dissipation Range of Fully Developed Turbulence

Cited by 162 publications

References 6 publications

Imprint of Large-Scale Flows on Turbulence

Imprint of Large-Scale Flows on Turbulence

Statistical scaling of pore-scale Lagrangian velocities in natural porous media

Extended power-law scaling of air permeabilities measured on a block of tuff

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