Structure functions in homogeneous and non-homogeneous turbulent flows

The scaling behaviour of the longitudinal velocity structure functions $\langle (\unicode[STIX]{x1D6E5}_{r}u)^{2p}\rangle ^{1/p}$ (where $2p$ represents the order) is studied for various wall-bounded turbulent flows. It has been known that for very large Reynolds numbers within the logarithmic region, the structure functions can be described by $\langle (\unicode[STIX]{x1D6E5}_{r}u)^{2p}\rangle ^{1/p}/U_{\unicode[STIX]{x1D70F}}^{2}\approx D_{p}\ln (r/z)+E_{p}$ (where $r$ is the longitudinal distance, $z$ the distance from the wall, $U_{\unicode[STIX]{x1D70F}}$ the friction velocity and $D_{p}$, $E_{p}$ are constants) in accordance with Townsend’s attached eddy hypothesis. Here we show that the ratios $D_{p}/D_{1}$ extracted from plots between structure functions – in the spirit of the extended self-similarity hypothesis – have further reaching universality for the energy containing range of scales. Specifically, we confirm that this description is universal across wall-bounded flows with different flow geometries, and also for both the longitudinal and transversal structure functions, where previously the scaling has been either difficult to discern or differences have been reported when examining the direct representation of $\langle (\unicode[STIX]{x1D6E5}_{r}u)^{2p}\rangle ^{1/p}$. In addition, we present evidence of this universality at much lower Reynolds numbers, which opens up avenues to examine structure functions that are not readily available from high Reynolds number databases.

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A dissipative random velocity field for fully developed fluid turbulence

Journal of Fluid Mechanics ◽

10.1017/jfm.2016.166 ◽

2016 ◽

Vol 794 ◽

pp. 369-408 ◽

Cited By ~ 19

Author(s):

Rodrigo M. Pereira ◽

Christophe Garban ◽

Laurent Chevillard

Keyword(s):

Energy Transfer ◽

Velocity Field ◽

Turbulent Flows ◽

Free Parameter ◽

Structure Functions ◽

Gradient Tensor ◽

Fluid Turbulence ◽

Fully Developed Turbulent Flow ◽

Dissipative Case ◽

Velocity Increments

We investigate the statistical properties, based on numerical simulations and analytical calculations, of a recently proposed stochastic model for the velocity field (Chevillardet al.,Europhys. Lett., vol. 89, 2010, 54002) of an incompressible, homogeneous, isotropic and fully developed turbulent flow. A key step in the construction of this model is the introduction of some aspects of the vorticity stretching mechanism that governs the dynamics of fluid particles along their trajectories. An additional further phenomenological step aimed at including the long range correlated nature of turbulence makes this model dependent on a single free parameter,${\it\gamma}$, that can be estimated from experimental measurements. We confirm the realism of the model regarding the geometry of the velocity gradient tensor, the power-law behaviour of the moments of velocity increments (i.e. the structure functions) including the intermittent corrections and the existence of energy transfer across scales. We quantify the dependence of these basic properties of turbulent flows on the free parameter${\it\gamma}$and derive analytically the spectrum of exponents of the structure functions in a simplified non-dissipative case. A perturbative expansion in power of${\it\gamma}$shows that energy transfer, at leading order, indeed take place, justifying the dissipative nature of this random field.

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Signatures of non-universal large scales in conditional structure functions from various turbulent flows

New Journal of Physics ◽

10.1088/1367-2630/13/11/113020 ◽

2011 ◽

Vol 13 (11) ◽

pp. 113020 ◽

Cited By ~ 15

Author(s):

Daniel B Blum ◽

Gregory P Bewley ◽

Eberhard Bodenschatz ◽

Mathieu Gibert ◽

Ármann Gylfason ◽

...

Keyword(s):

Turbulent Flows ◽

Structure Functions

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Effects of fluctuating energy input on the small scales in turbulence

Journal of Fluid Mechanics ◽

10.1017/jfm.2013.575 ◽

2013 ◽

Vol 737 ◽

pp. 527-551 ◽

Cited By ~ 10

Author(s):

Chen-Chi Chien ◽

Daniel B. Blum ◽

Greg A. Voth

Keyword(s):

Turbulent Flows ◽

Energy Input ◽

Large Scale ◽

Similarity Theory ◽

Three Dimensional ◽

Structure Functions ◽

Temporal Variations ◽

Small Scale ◽

Scale Turbulence ◽

Kolmogorov Constant

AbstractIn the standard cascade picture of three-dimensional turbulent fluid flows, energy is input at a constant rate at large scales. Energy is then transferred to smaller scales by an intermittent process that has been the focus of a vast literature. However, the energy input at large scales is not constant in most real turbulent flows. We explore the signatures of these fluctuations of large-scale energy input on small-scale turbulence statistics. Measurements were made in a flow between oscillating grids, with ${R}_{\lambda } $ up to 262, in which temporal variations in the large-scale energy input can be introduced by modulating the oscillating grid frequency. We find that the Kolmogorov constant for second-order longitudinal structure functions depends on the magnitude of the fluctuations in the large-scale energy input. We can quantitatively predict the measured change with a model based on Kolmogorov’s refined similarity theory. The effects of fluctuations of the energy input can also be observed using structure functions conditioned on the instantaneous large-scale velocity. A linear parametrization using the curvature of the conditional structure functions provides a fairly good match with the measured changes in the Kolmogorov constant. Conditional structure functions are found to provide a more sensitive measure of the presence of fluctuations in the large-scale energy input than inertial range scaling coefficients.

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Relative Behaviour of Velocity and Scalar Structure Functions in Turbulent Flows

Variable Density Fluid Turbulence - Fluid Mechanics and Its Applications ◽

10.1007/978-94-017-0075-7_7 ◽

2002 ◽

pp. 167-200 ◽

Cited By ~ 1

Author(s):

P. Chassaing ◽

R. A. Antonia ◽

F. Anselmet ◽

L. Joly ◽

S. Sarkar

Keyword(s):

Turbulent Flows ◽

Structure Functions ◽

Scalar Structure

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Investigation of the influence of combustion-induced thermal expansion on two-point turbulence statistics using conditioned structure functions

Journal of Fluid Mechanics ◽

10.1017/jfm.2019.128 ◽

2019 ◽

Vol 867 ◽

pp. 45-76 ◽

Cited By ~ 9

Author(s):

V. A. Sabelnikov ◽

A. N. Lipatnikov ◽

S. Nishiki ◽

T. Hasegawa

Keyword(s):

Thermal Expansion ◽

Turbulent Flow ◽

Turbulent Flows ◽

Three Dimensional ◽

Structure Functions ◽

Single Step ◽

Small Scale ◽

Order Structure ◽

Constant Density ◽

Local Velocity

The second-order structure functions (SFs) of the velocity field, which characterize the velocity difference at two points, are widely used in research into non-reacting turbulent flows. In the present paper, the approach is extended in order to study the influence of combustion-induced thermal expansion on turbulent flow within a premixed flame brush. For this purpose, SFs conditioned to various combinations of mixture states at two different points (reactant–reactant, reactant–product, product–product, etc.) are introduced in the paper and a relevant exact transport equation is derived in the appendix. Subsequently, in order to demonstrate the capabilities of the newly developed approach for advancing the understanding of turbulent reacting flows, the conditioned SFs are extracted from three-dimensional (3-D) direct numerical simulation data obtained from two statistically 1-D planar, fully developed, weakly turbulent, premixed, single-step-chemistry flames characterized by significantly different (7.53 and 2.50) density ratios, with all other things being approximately equal. Obtained results show that the conditioned SFs differ significantly from standard mean SFs and convey a large amount of important information on various local phenomena that stem from the influence of combustion-induced thermal expansion on turbulent flow. In particular, the conditioned SFs not only (i) indicate a number of already known local phenomena discussed in the paper, but also (ii) reveal a less recognized phenomenon such as substantial influence of combustion-induced thermal expansion on turbulence in constant-density unburned reactants and even (iii) allow us to detect a new phenomenon such as the appearance of strong local velocity perturbations (shear layers) within flamelets. Moreover, SFs conditioned to heat-release zones indicate a highly anisotropic influence of combustion-induced thermal expansion on the evolution of small-scale two-point velocity differences within flamelets, with the effects being opposite (an increase or a decrease) for different components of the local velocity vector.

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On the universality of anomalous scaling exponents of structure functions in turbulent flows

Journal of Fluid Mechanics ◽

10.1017/jfm.2017.848 ◽

2018 ◽

Vol 837 ◽

pp. 657-669 ◽

Cited By ~ 13

Author(s):

E.-W. Saw ◽

P. Debue ◽

D. Kuzzay ◽

F. Daviaud ◽

B. Dubrulle

Keyword(s):

Turbulent Flows ◽

Large Scale ◽

Velocity Structure ◽

Swirling Flow ◽

Spatial Scales ◽

Structure Functions ◽

Inertial Subrange ◽

Scaling Exponents ◽

Von Karman ◽

Von Kármán

All previous experiments in open turbulent flows (e.g. downstream of grids, jets and the atmospheric boundary layer) have produced quantitatively consistent values for the scaling exponents of velocity structure functions (Anselmet et al., J. Fluid Mech., vol. 140, 1984, pp. 63–89; Stolovitzky et al., Phys. Rev. E, vol. 48 (5), 1993, R3217; Arneodo et al., Europhys. Lett., vol. 34 (6), 1996, p. 411). The only measurement of scaling exponents at high order (${>}6$) in closed turbulent flow (von Kármán swirling flow) using Taylor’s frozen flow hypothesis, however, produced scaling exponents that are significantly smaller, suggesting that the universality of these exponents is broken with respect to change of large scale geometry of the flow. Here, we report measurements of longitudinal structure functions of velocity in a von Kármán set-up without the use of the Taylor hypothesis. The measurements are made using stereo particle image velocimetry at four different ranges of spatial scales, in order to observe a combined inertial subrange spanning approximately one and a half orders of magnitude. We found scaling exponents (up to ninth order) that are consistent with values from open turbulent flows, suggesting that they might be in fact universal.

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CFHT Fabry-Perot 2D Spectroscopy in Hα of the Planetary Nebula NGC 40: Universal Multifractal Analysis and Turbulent Status

Symposium - International Astronomical Union ◽

10.1017/s0074180900209170 ◽

2003 ◽

Vol 209 ◽

pp. 395-396

Author(s):

Y. Grosdidier ◽

A. Acker ◽

S. Blais-Ouellette ◽

G. Joncas ◽

A.F.J. Moffat

Keyword(s):

Structure Function ◽

Turbulent Flows ◽

Planetary Nebula ◽

Quantitative Description ◽

Structure Functions ◽

Compressible Turbulence ◽

Order Structure ◽

Orion Nebula ◽

Continuous Scale ◽

Fabry Perot

Using CFHT/SIS Fabry-Perot interferograms of the planetary nebula NGC 40, we present an investigation of the statistical properties of fluctuating gas motions using structure functions traced by Hα emission-line centroid velocities (Grosdidier et al. 2002). We consider the structure functions 〈|Δv(r)|p〉 of order p, i.e. the spatially averaged moments of order p of the spatial velocity increments at projected spatial scale r of NGC 40's velocity field. In this poster, we test for i) Structure-function scaling related to turbulence in the nebula, 〈|Δv(r) |p〉 ~ rχ(p),ϛ and ii) Nonlinearity of the observed scaling exponents χ(p)s, as expected for intermittent turbulent flows. The first order structure function is indeed found to scale at the smallest scales with χ(1) = H ≈0.68. This value is larger than the value expected in the case of incompressible or compressible turbulence, for which one would obtain H = 1/3 or 1/2, respectively. This result suggests that the Hα layer is thick compared to the projected spatial separations (O'Dell & Castañeda 1987, and references therein). We can give a more quantitative description of the turbulent status of the PN NGC 40 through the examination of the structure function for different orders. Additionally we can discuss the nature of the turbulence in terms of Universal Multifractals, a continuous-scale limit of multiplicative cascades (Schertzer & Lovejoy 1987) and derive the level of intermittency in the nebula. The function χ(p) is indeed found to be nonlinear and well reproduced with the following Universal Multifractal parameters: α ≈ 1.90–2 (the field is highly multifractal with a gaussian generator) and C1 ≈ 0.08 (the field suffers significant intermittency, which is incompatible with monofractal additive stochastic models usually introduced in similar studies): ϛ(p) = p × H − C1(pα - p)/(α - 1). Fig. 1 shows the hierarchy of exponents χ(p) along with its best Universal Multifractal fit. Our results on the Wolf-Rayet ring nebula M 1–67 (Grosdidier et al. 2001) led to the same level of multifractality (α ≈ 1.9) but with smaller intermittency (C1 ≈ 0.04). Preliminary results on the Orion nebula (Grosdidier 2002) reveal a turbulent status relatively similar to that of the PN NGC 40. On the whole, such a study provides a way to quantify turbulence in PN, and more generally in other HII regions (with potential implications for the estimation of temperature fluctuations) and give insight on astrophysical turbulence for its own sake.

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