Equations relating structure functions of all orders

Hill, R. J.

doi:10.1017/s0022112001003949

Cited by 116 publications

(161 citation statements)

References 32 publications

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“…As shown in Fig. 3, experimental and numerical data do not support a plateau of 2 around its peak value, a fact directly related to the shape of the acceleration autocorrelation.…”

Section: A Acceleration Auto-correlationmentioning

confidence: 80%

“…Such correlation functions are expressible in terms of products of velocity differences of the velocity v at different times, and are invariant under t → −t. In contrast, the example involving two Lagrangian points cited in the introduction [1][2][3] shows that measurements with more than one Lagrangian point permit to distinguish time irreversibility in the system. …”

Section: A Time-irreversibilitymentioning

confidence: 98%

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On Lagrangian single-particle statistics

Falkovich

Pumir

et al. 2012

Physics of Fluids

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The length distribution of streamline segments in homogeneous isotropic decaying turbulence Phys. Fluids 24, 045104 (2012) Conditional vorticity budget of coherent and incoherent flow contributions in fully developed homogeneous isotropic turbulence Phys. Fluids 24, 035108 (2012) Near-field investigation of turbulence produced by multi-scale grids Phys. Fluids 24, 035103 (2012) Reynolds number effect on the velocity increment skewness in isotropic turbulence Phys. Fluids 24, 015108 (2012) Spectral approach to finite Reynolds number effects on Kolmogorov's 4/5 law in isotropic turbulence Phys. Fluids 24, 015107 (2012) In turbulence, ideas of energy cascade and energy flux, substantiated by the exact Kolmogorov relation, lead to the determination of scaling laws for the velocity spatial correlation function. Here we ask whether similar ideas can be applied to temporal correlations. We critically review the relevant theoretical and experimental results concerning the velocity statistics of a single fluid particle in the inertial range of statistically homogeneous, stationary and isotropic turbulence. We stress that the widely used relations for the second structure function, D 2 (t) ≡ [v(t) − v(0)] 2 ∝ εt, relies on dimensional arguments only: no relation of D 2 (t) to the energy cascade is known, neither in two-nor in three-dimensional turbulence. State of the art experimental and numerical results demonstrate that at high Reynolds numbers, the derivative d D 2 (t) dt has a finite non-zero slope starting from t ≈ 2τ η . The analysis of the acceleration spectrum A (ω) indicates a possible small correction with respect to the dimensional expectation A (ω) ∼ ω 0 but present data are unable to discriminate between anomalous scaling and finite Reynolds effects in the second order moment of velocity Lagrangian statistics. C 2012 American Institute of Physics.

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“…As shown in Fig. 3, experimental and numerical data do not support a plateau of 2 around its peak value, a fact directly related to the shape of the acceleration autocorrelation.…”

Section: A Acceleration Auto-correlationmentioning

confidence: 80%

Section: A Time-irreversibilitymentioning

confidence: 98%

On Lagrangian single-particle statistics

Falkovich

Pumir

et al. 2012

Physics of Fluids

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“…Since X 2 F (X) = A 1,2 = 1/2, one ends up with the relation hA 1,2 = −bA 0,2 . The important fact is that this expansion is consistent with Gaussianity of transverse fluctuations and also gives a reasonable account for pressure contributions in the structure function equations [7,15,16].…”

supporting

confidence: 52%

“…Furthermore, by integrating (3) over U one obtains G(X) = −hXF (X) − bX. Apparently this is a two parameter expansion, however the constraint V P y,v = 0 [7,8] implies hX 2 F (X) = −bX 2 . Since X 2 F (X) = A 1,2 = 1/2, one ends up with the relation hA 1,2 = −bA 0,2 .…”

mentioning

confidence: 98%

Closure of two-dimensional turbulence: The role of pressure gradients

2002

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Inverse energy cascade regime of two dimensional turbulence is investigated by means of high resolution numerical simulations. Numerical computations of conditional averages of transverse pressure gradient increments are found to be compatible with a recently proposed self-consistent Gaussian model. An analogous low order closure model for the longitudinal pressure gradient is proposed and its validity is numerically examined. In this case numerical evidence for the presence of higher order terms in the closure is found. The fundamental role of conditional statistics between longitudinal and transverse components is highlighted. PACS number(s) : 47.27.Ak, The existence of two simultaneous inertial ranges in twodimensional turbulence, as a consequence of coupled energy and enstrophy conservation, is one of the most important phenomena in statistical fluid mechanics [1]. At variance with 3D-turbulence, the energy injected into the system at scale ℓ f flows toward the large scales, while the enstrophy cascades down on the small scales. Because of the inverse energy cascade, the Navier-Stokes equations,which rule the evolution of an incompressible (∂ i u i = 0) velocity field, cannot reach a steady-state unless an energy sink at large scales is added. Alternatively one can consider an ensemble of solutions of (1) with a fixed energy value below the condensation level [2], i.e. with an integral scale L(t) (growing in time as t 3/2 ) still much smaller than the system size. Because of the scaling of the characteristic times, the small scales (inertial range) in the system ℓ f ≪ r ≪ L can be considered in a stationary state. One of the most challenging problems is to understand the statistics of velocity fluctuations ∆u(r) = u(x + r) − u(x) [3]. In homogeneous and isotropic turbulence it amounts to study the joint probability density function (PDF) P (U, V, r) of longitudinal U and transverse V velocity differences where ∆u = Ux + Vŷ andx = r r . Recently experimental [4] and numerical [5] investigations in two dimensions have shown that the probability distribution of the pure longitudinal P (U, r) and transversal P (V, r) velocity differences at inertial scales display a close-toGaussian statistics with undetectable intermittency corrections to structure function exponents. Although the establishment of normal scaling in all inverse cascades seem to be generic [6] nevertheless the Gaussianity of the statistics in inverse cascade of the forced two dimensional turbulence is remained to be understood. From (1), a set of equations for generic mixed structure functions, i.e. S n,m (r) ≡ U n V m = A n,m r ξn,m have been obtained [7,8]. In [8] those equations are elaborated from the joint PDF equation. Unfortunately, the PDF equation is not closed, resembling the well known closure problem in turbulence. In the inverse energy cascade regime, dissipative contributions can be neglected so the remaining unclosed terms are the longitudinal and transversal pressure gradients increments. Recently Yakhot [8,9] suggested a selfcon...

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“…Relations between structure functions become more and * Electronic address: peinke@uni-oldenburg.de; URL: http://www.uni-oldenburg.de/hydro more complicated with higher order, including also pressure terms [13,18].…”

mentioning

confidence: 99%

Different cascade speeds for longitudinal and transverse velocity increments of small-scale turbulence

Siefert

Peinke

2004

Phys. Rev. E

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We address the problem of differences between longitudinal and transverse velocity increments in isotropic small scale turbulence. The relationship of these two quantities is analyzed experimentally by means of stochastic Markovian processes leading to a phenomenological Fokker-Planck equation from which a generalization of the Kármán equation is derived. From these results, a simple relationship between longitudinal and transverse structure functions is found which explains the difference in the scaling properties of these two structure functions.PACS numbers: 47.27. Gs, 47.27.Jv, 05.10.Gg Substantial details of the complex statistical behaviour of fully developed turbulent flows are still unknown, cf. [1,2,3,4]. One important task is to understand intermittency, i.e. finding unexpected frequent occurences of large fluctuations of the local velocity on small length scales. In the last years, the differences of velocity fluctuations in different spatial directions have attracted considerable attention as a main issue of the problem of small scale turbulence, see for example [5,6,7,8,9,10,11,12,13]. For local isotropic turbulence, the statistics of velocity increments [v(x + r) − v(x)] e as a function of the length scale r is of interest. Here, e denotes a unit vector. We denote with u(r) the longitudinal increments (e is parallel to r) and with v(r) transverse increments (e is orthogonal to r) [40].In a first step, this statistics is commonly investigated by means of its moments u n (r) or v n (r) , the so-called velocity structure functions. Different theories and models try to explain the shape of the structure functions cf. [2]. Most of the works examine the scaling of the structure function, u n ∝ r ξ n l , and try to explain intermittency, expressed by ξ n l − n/3 the deviation from Kolmogorov theory of 1941 [14,15]. For the corresponding transverse quantity we write v n ∝ r ξ n t . There is strong evidence that there are fundamental differences in the statistics of the longitudinal increments u(r) and transverse increments v(r). Whereas there were some contradictions initially, there is evidence now that the transverse scaling shows stronger intermittency even for high Reynolds numbers [9,16].A basic equation which relates both quantities is derived by Kármán and Howarth [17]. Assuming incompressibilty and isotropy, the so called first Kármán equation is obtained:Relations between structure functions become more and * Electronic address: peinke@uni-oldenburg.de; URL: http://www.uni-oldenburg.de/hydro more complicated with higher order, including also pressure terms [13,18].In this paper, we focus on a different approach to characterize spatial multipoint correlations via multi-scale statistics. Recently it has been shown that it is possible to get access to the joint probability distribution p(u(r 1 ), u(r 2 ), . . . , u(r n )) via a Fokker-Planck equation, which can be estimated directly from measured data [19,20,21]. For a detailed presentation see [22]. This method is definitely more general than the...

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Equations relating structure functions of all orders

Cited by 116 publications

References 32 publications

On Lagrangian single-particle statistics

On Lagrangian single-particle statistics

Closure of two-dimensional turbulence: The role of pressure gradients

Different cascade speeds for longitudinal and transverse velocity increments of small-scale turbulence

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