Opportunities for use of exact statistical equations

Hill, R. J.

doi:10.1080/14685240600595636

Cited by 19 publications

(23 citation statements)

References 29 publications

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“…respectively, and is energy dissipation rate per unit mass (Mann, Ott & Andersen 1999;Falkovich, Gawȩdzki & Vergassola 2001;Pumir, Shraiman & Chertkov 2001;Hill 2006). We also found this in our experiment, where the polystyrene particles behaved as tracers (Gibert et al 2010).…”

Section: Experimental Methods and Observationmentioning

confidence: 52%

Where do small, weakly inertial particles go in a turbulent flow?

Gibert

Bodenschatz

2012

J. Fluid Mech.

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We report experimental results on the dynamics of heavy particles of the size of the Kolmogorov scale in a fully developed turbulent flow. The mixed Eulerian structure function of two-particle velocity and acceleration difference vectors was observed to increase significantly with particle inertia for identical flow conditions. We show that this increase is related to a preferential alignment between these dynamical quantities. With increasing particle density the probability for those two vectors to be collinear was observed to grow. We show that these results are consistent with the preferential sampling of strain-dominated regions by inertial particles

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Section: Experimental Methods and Observationmentioning

confidence: 52%

Where do small, weakly inertial particles go in a turbulent flow?

Gibert

Bodenschatz

2012

J. Fluid Mech.

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“…When the relative motion of particles in a pair is considered, the physical meaning of the crossed velocity-acceleration structure function S au = δ r a · δ r u is however somehow subtler. S au can indeed be analytically related to the third order velocity structure function (and hence to the energy cascade accross scales) directly from Navier-Stokes equation, such that under local stationarity and homogeneity assumptions (see for instance Mann et al (1999);Hill (2006)) : 2 δ r a · δ r u = ∇ · δ r uδ r u · δ r u .…”

Section: A Ballistic Cascade Phenomenologymentioning

confidence: 99%

Turbulent pair dispersion as a ballistic cascade phenomenology

Bourgoin

2015

J. Fluid Mech.

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Since the pioneering work of Richardson in 1926, later refined by Batchelor and Obukhov in 1950, it is predicted that the rate of separation of pairs of fluid elements in turbulent flows with initial separation at inertial scales, grows ballistically first (Batchelor regime), before undergoing a transition towards a super-diffusive regime where the meansquare separation grows as t 3 (Richardson regime). Richardson empirically interpreted this super-diffusive regime in terms of a non-Fickian process with a scale dependent diffusion coefficient (the celebrated Richardson's "4/3rd" law). However, the actual physical mechanism at the origin of such a scale dependent diffusion coefficient remains unclear. The present article proposes a simple physical phenomenology for the time evolution of the mean square relative separation in turbulent flows, based on a scale dependent ballistic scenario rather than a scale dependent diffusive. It is shown that this phenomenology accurately retrieves most of the known features of relative dispersion for particles mean square separation ; among others : (i) it is quantitatively consistent with most recent numerical simulations and experiments for mean square separation between particles (both for the short term Batchelor regime and the long term Richardson regime, and for all initial separations at inertial scales), (ii) it gives a simple physical explanation of the origin of the super diffusive t 3 Richardson regime which naturally builts itself as an iterative process of elementary short-term-scale-dependent ballistic steps, (iii) it shows that the Richardson constant is directly related to the Kolmogorov constant (and eventually to a ballistic persistence parameter) and (iv) in a further extension of the phenomenology, taking into account third order corrections, it robustly describes the temporal asymmetry between forward and backward dispersion, with an explicit connection to the cascade of energy flux across scales. An important aspect of this phenomenology is that it simply and robustly connects long term super-diffusive features to elementary short term mechanisms, and at the same time it connects basic Lagrangian features of turbulent relative dispersion (both at short and long times) to basic Eulerian features of the turbulent field : second order Eulerian statistics control the growth of separation (both at short and long times) while third order Eulerian statistics control the temporal asymmetry of the dispersion process, which can then be directly identified as the signature of the energy cascade and associated to well known exact results as the Karman-Howarth-Monin relation.

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“…, which is known to asymptote to the value ε −2 in the inertial range (see e.g. [28][29][30]), has no known analytical expression in the dissipation range. By use of reasonings in the same vein as Kolmogorov's, we anticipate, however, Δ Δ 〈 〉 u a · to scale as r 2 in the dissipative range of scales.…”

Section: The Structure Functionmentioning

confidence: 99%

“…. Mann et al [28,29] and Hill [30] have shown that Δ Δ ε 〈 〉 = − v a · 2 within the inertial range, thereby explaining the negative behavior. The simulation results presented here show not only the temporary sub-ballistic excursion, but the eventual transition to super-ballistic behavior, presumably due to still higher-order terms in the Taylor expansion.…”

Section: Mixed Structure Functionmentioning

confidence: 99%

Turbulent pair dispersion in the presence of gravity

2015

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Turbulent pair dispersion of heavy particles is strongly altered when particles of two different Stokes numbers (bidisperse) are considered, and this is further compounded when a uniform gravitational acceleration is present. Lagrangian trajectories of fluid tracers, and bidisperse heavy particles with and without gravity were calculated from a direct numerical simulation of homogeneous, isotropic turbulence. Particle pair dispersion shows a short-time, ballistic (Batchelor) regime and a transition to super-ballistic dispersion that is suggestive of the emergence of Richardson scaling. A simple equation of motion for inertial, sedimenting particles captures the essential features of the pair dispersion at very short time and length scales. Kolmogorov scaling arguments are able to qualitatively describe the competition between gravity-induced and fluid-induced relative motion in modifying the amount of time the heavy particles spend in the ballistic regime. The transition from ballistic to super-ballistic dispersion for fluid tracers and monodisperse inertial particles exhibits a pronounced sub-ballistic behavior that can be attributed to the mixed velocity-acceleration structure function. The sub-ballistic behavior is strongly suppressed for bidisperse particles, both in the presence or absence of gravity, primarily because of a reduction in the correlation between velocity and acceleration increments. inherent aesthetic and intellectual merit. The parameter space corresponding to the inertial particle dispersion problem is enormous, so we use atmospheric clouds as a context for restricting the region of study within that space: particles small compared to the Kolmogorov length scale, heavy compared to the surrounding fluid, and obeying to good approximation a linear (Stokes) drag law. Significantly, from the point of view of this study, clouds consist of a population of condensing or evaporating water droplets advected by turbulent air flow in a thermodynamically varying environment, with the result that the size distribution of droplets can be quite broad [14]. There is considerable computational and experimental evidence already that different sizes and therefore different inertial response, can lead to significant changes in the behavior of inertial particles. For example, the tendency to cluster weakens in the case of inertial particles with a broad size distribution (e.g. [15,16]). Furthermore, the dynamics of this polydisperse population of particles can be significantly modified by the presence of a gravitational acceleration that results in particle decoupling from the fluid motions, and perhaps more importantly, a relative velocity of different-sized particles falling at terminal speed [17,18]. Simple Kolmogorov dissipative range scaling arguments (e.g. [19]) suggest that the mean turbulent acceleration in clouds is in the approximate range of 0.1-1 m s −2 , much less than gravitational acceleration. This has the consequence that gravity very likely has an immediate effect on the dispersion of polydispe...

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Opportunities for use of exact statistical equations

Cited by 19 publications

References 29 publications

Where do small, weakly inertial particles go in a turbulent flow?

Where do small, weakly inertial particles go in a turbulent flow?

Turbulent pair dispersion as a ballistic cascade phenomenology

Turbulent pair dispersion in the presence of gravity

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