Experimental Lagrangian Acceleration Probability Density Function Measurement
Nicolas Mordant, Alice M. Crawford, Eberhard Bodenschatz
备注：
submitted for the special issue of Physica D: "Anomalous Distributions" 11 pages, 6 figures revised version: light modifications of the figures and the text
We report experimental results on the acceleration component probability distribution function at $R_λ= 690$ to probabilities of less than $10^{-7}$. This is an improvement of more than an order of magnitude over past measurements and allows us to conclude that the fourth moment converges and the flatness is approximately 55. We compare our probability distribution to those predicted by several models inspired by non-extensive statistical mechanics. We also look at acceleration component probability distributions conditioned on a velocity component for conditioning velocities as high as 3 times the standard deviation and find them to be highly non-Gaussian.
The statistical properties of the Lagrangian acceleration vector of passive tracers in statistically-steady rotating turbulence is studied by Particle Tracking Velocimetry. Direct effects of the background rotation are the suppression of high-acceleration events parallel to the (vertical) rotation axis, the enhancement of high-acceleration events for the horizontal acceleration, and the strong amplification of the auto-correlation of the acceleration component perpendicular to both the rotation vector Ω and local velocity vector u. The auto-correlation of the acceleration component in the plan…
Particles in turbulence frequently encounter extreme accelerations between extended periods of quiescence. The occurrence of extreme events is closely related to the intermittent spatial distribution of intense flow structures such as vorticity filaments. This mixed history of flow conditions leads to very complex particle statistics with a pronounced scale dependence, which presents one of the major challenges on the way to a non-equilibrium statistical mechanics of turbulence. Here, we introduce the notion of persistent Lagrangian acceleration, quantified by the squared particle acceleration…
We report experimental results on the three dimensional Lagrangian acceleration in highly turbulent flows. Tracer particles are tracked optically using four silicon strip detectors from high energy physics that provide high temporal and spatial resolution. The components of the acceleration are shown to be statistically dependent. The probability density function (PDF) of the acceleration magnitude is comparable to a log-normal distribution. Assuming isotropy, a log-normal distribution of the magnitude can account for the observed dependency of the components. The time dynamics of the accelera…
Lagrangian acceleration statistics in a fully developed turbulent channel flow at $Re_τ= 1440$ are investigated, based on tracer particle tracking in experiments and direct numerical simulations. The evolution with wall distance of the Lagrangian velocity and acceleration time scales is analyzed. Dependency between acceleration components in the near-wall region is described using cross-correlations and joint probability density functions. The strong streamwise coherent vortices typical of wall-bounded turbulent flows are shown to have a significant impact on the dynamics. This results in a st…
We investigate the response of large inertial particle to turbulent fluctuations in a inhomogeneous and anisotropic flow. We conduct a Lagrangian study using particles both heavier and lighter than the surrounding fluid, and whose diameters are comparable to the flow integral scale. Both velocity and acceleration correlation functions are analyzed to compute the Lagrangian integral time and the acceleration time scale of such particles. The knowledge of how size and density affect these time scales is crucial in understanding partical dynamics and may permit stochastic process modelization usi…
The Lagrangian approach is natural to study issues of turbulent dispersion and mixing. We propose in this work a general Lagrangian stochastic model including velocity and acceleration as dynamical variables for inhomogeneous turbulent flows. The model takes the form of a diffusion process and the coefficients of the model are determined via Kolmogorov theory and the requirement of consistency with the velocity-based models. It is shown that the present model generalises both the acceleration-based models for homogeneous flows and the generalised Langevin models for the velocity. The resulting…
The Lagrangian (LA) and Eulerian Acceleration (EA) properties of fluid particles in homogeneous turbulence with uniform shear and uniform stable stratification are studied using direct numerical simulations. The Richardson number is varied from $Ri=0$, corresponding to unstratified shear flow, to $Ri=1$, corresponding to strongly stratified shear flow. The probability density functions (pdfs) of both LA and EA have a stretched-exponential shape and they show a strong and similar influence on the Richardson number. The extreme values of the EA are stronger than those observed for the LA. Geomet…
We report experimental measurements of Lagrangian accelerations in the bulk of intense turbulent flows of dilute polymer solutions by following tracer particles with a high-speed optical tracking system. We observed a significant decrease in the acceleration variance in dilute polymer solutions. The shape of the normalized acceleration probability density functions, however, remain the same as in Newtonian water flows. We also observed an increase in the turbulent Lagrangian acceleration autocorrelation time in these dilute polymer solutions.
We investigate sources of error in acceleration statistics from Lagrangian Particle Tracking (LPT) data and demonstrate techniques to eliminate or minimise bias errors introduced during processing. Numerical simulations of particle tracking experiments in isotropic turbulence show that the main sources of bias error arise from noise due to position uncertainty and selection biases introduced during numerical differentiation. We outline the use of independent measurements and filtering schemes to eliminate these biases. Moreover, we test the validity of our approach in estimating the statistica…
We present in this article a novel Lagrangian measurement technique: an instrumented particle which continuously transmits the force/acceleration acting on it as it is advected in a flow. We develop signal processing methods to extract information on the flow from the acceleration signal transmitted by the particle. Notably, we are able to characterize the force acting on the particle and to identify the presence of a permanent large-scale vortex structure. Our technique provides a fast, robust and efficient tool to characterize flows, and it is particularly suited to obtain Lagrangian statist…
We study the Lagrangian velocity and acceleration statistics of light particles (micro-bubbles in water) in homogeneous isotropic turbulence. Micro-bubbles with a diameter of 340 microns and Stokes number from 0.02 to 0.09 are dispersed in a turbulent water tunnel operated at Taylor-Reynolds numbers (Re) ranging from 160 to 265. We reconstruct the bubble trajectories by employing three-dimensional particle tracking velocimetry (PTV). It is found that the probability density functions (PDFs) of the micro-bubble acceleration show a highly non-Gaussian behavior with flatness values in the range 2…
We outline a statistical theory of turbulence based on the Lagrangian formulation of fluid motion. We derive a hierarchy of evolution equations for Lagrangian N-point probability distributions as well as a functional equation for a suitably defined probability functional which is the analog of Hopf's functional equation. Furthermore, we adress the derivation of a generalized Fokker-Plank equation for the joint velocity - position probability density of N fluid particles.
We study the properties of various Eulerian contributions to fluid particle acceleration by using well-resolved direct numerical simulations of isotropic turbulence, with the grid resolution as high as $12288^3$ and the Taylor-scale Reynolds number $R_λ$ in the range between 140 and 1300. The variance of convective acceleration, when normalized by Kolmogorov scales, increases linearly with $R_λ$, consistent with simple theoretical arguments, but very strongly differing from phenomenological predictions of Kolmogorov's hypothesis as well as Eulerian multifractal models. The scaling of the local…
The Lagrangian velocity statistics of dissipative drift-wave turbulence are investigated. For large values of the adiabaticity (or small collisionality), the probability density function of the Lagrangian acceleration shows exponential tails, as opposed to the stretched exponential or algebraic tails, generally observed for the highly intermittent acceleration of Navier-Stokes turbulence. This exponential distribution is shown to be a robust feature independent of the Reynolds number. For small adiabaticity, algebraic tails are observed, suggesting the strong influence of point-vortex-like dyn…
We show that the probability densities af accelerations of Lagrangian test particles in turbulent flows as measured by Bodenschatz et al. [Nature 409, 1017 (2001)] are in excellent agreement with the predictions of a stochastic model introduced in [C. Beck, PRL 87, 180601 (2001)] if the fluctuating friction parameter is assumed to be log-normally distributed. In a generalized statistical mechanics setting, this corresponds to a superstatistics of log-normal type. We analytically evaluate all hyperflatnes factors for this model and obtain a flatness prediction in good agreement with the experim…
We present the extension of a modeling technique for Lagrangian tracer particles [B. Viggiano et al., J. Fluid Mech.(2020), vol. 900, A27] which accounts for the effects of particle inertia. Thereby, the particle velocity for several Stokes numbers is modeled directly by a multi-layered Ornstein-Uhlenbeck process and a comparison of key statistical quantities (second-order velocity structure function, acceleration correlation function, and root mean square acceleration) to expressions derived from Batchelor's model as well as to direct numerical simulations (DNS) is performed. In both approach…
We present a sweep-stick mechanism for heavy particles transported by a turbulent flow under the action of gravity. Direct numerical simulations show that these particles preferentially explore regions of the flow with close to zero Lagrangian acceleration. However, the actual Lagrangian acceleration of the fluid elements where particles accumulate is not zero, and has a dependence on the Stokes number, the gravity acceleration, and the settling velocity of the particles.
The distribution of particle accelerations in turbulence is intermittent, with non-Gaussian tails that are quite different for light and heavy particles. In this article we analyse a closure scheme for the acceleration fluctuations of light and heavy inertial particles in turbulence, formulated in terms of Lagrangian correlation functions of fluid tracers. We compute the variance and the flatness of inertial particle accelerations and we discuss their dependency on the Stokes number. The closure incorporates effects induced by the Lagrangian correlations along the trajectories of fluid tracers…
In RB convection for fluids with Prandtl number $Pr\gtrsim 1$, rotation beyond a critical (small) rotation rate is known to cause a sudden enhancement of heat transfer which can be explained by a change in the character of the BL dynamics near the top and bottom plates of the convection cell. Namely, with increasing rotation rate, the BL signature suddenly changes from Prandtl--Blasius type to Ekman type. The transition from a constant heat transfer to an almost linearly increasing heat transfer with increasing rotation rate is known to be sharp and the critical Rossby number $Ro_{c}$ occurs t…
Accessing and characterizing a flow impose a number of constraints on the employed measurement techniques; in particular optical methods require transparent fluids and windows in the vessel. Whereas one can adapt apparatus, fluid and methods in the lab to these constraints, this is hardly possible for industrial mixers. We present in this article a novel measurement technique which is suitable for opaque or granular flows: an instrumented particle, which continuously transmits the force/acceleration acting on it as it is advected in a flow. Its density is adjustable for a wide range of fluids …
The Recent Fluid Deformation Closure (RFDC) model of lagrangian turbulence is recast in path-integral language within the framework of the Martin-Siggia-Rose functional formalism. In order to derive analytical expressions for the velocity-gradient probability distribution functions (vgPDFs), we carry out noise renormalization in the low-frequency regime and find approximate extrema for the Martin-Siggia-Rose effective action. We verify, with the help of Monte Carlo simulations, that the vgPDFs so obtained yield a close description of the single-point statistical features implied by the origina…
We analyze the vector nulls of velocity, Lagrangian acceleration, and vorticity, coming from direct numerical simulations of forced homogeneous isotropic turbulence at $Re_λ\in [40-610]$. We show that the clustering of velocity nulls is much stronger than those of acceleration and vorticity nulls. These acceleration and vorticity nulls, however, are denser than the velocity nulls. We study the scaling of clusters of these null points with $Re_λ$ and with characteristic turbulence lengthscales. We also analyze datasets of point inertial particles with Stokes numbers $St = 0.5$, 3, and 6, at $Re…
The angle between subsequent particle displacement increments is evaluated as a function of the timelag in isotropic turbulence. It is shown that the evolution of this angle contains two well-defined power-laws, reflecting the multi-scale dynamics of high-Reynolds number turbulence. The proba-bility density function of the directional change is shown to be self-similar and well approximated by an analytically derived model assuming Gaussianity and independence of the velocity and the Lagrangian acceleration.
We investigate the behavior of microscopic heavy particles settling in homogeneous air turbulence. The regimes are relevant to the airborne transport of dust and droplets: the Taylor-microscale Reynolds number is Re = 289 - 462, the Kolmogorov-scale Stokes number is St = 1.2 - 13, and the Kolmogorov acceleration is comparable to the gravitational acceleration (i.e., the Froude number Fr = O(1)). We use high-speed laser imaging to track the particles and simultaneously characterize the air velocity field, resolving all relevant spatio-temporal scales. The role of the flow sampled by the particl…
We report results from the first systematic Lagrangian experimental investigation in the regime of very light (air bubbles in water) and large particles (D/η >> 1) in turbulence. Using a traversing camera setup and particle tracking, we study the Lagrangian acceleration statistics of ~3 mm diameter (D) bubbles in a water tunnel with nearly homogeneous and isotropic turbulence generated by an active-grid. The Reynolds number (Re_λ) is varied from 145 to 230, resulting in size ratios, D/η in the range of 7.3--12.5, where η is the Kolmogorov length scale. The experiments reveal that gravity incre…
We report on simulations of two-dimensional turbulence in the inverse energy cascade regime. Focusing on the statistics of Lagrangian tracer particles, scaling behavior of the probability density functions of velocity fluctuations is investigated. The results are compared to the three-dimensional case. In particular an analysis in terms of compensated cumulants reveals the transition from a strong non-Gaussian behavior with large tails to Gaussianity. The reported computation of correlation functions for the acceleration components sheds light on the underlying dynamics of the tracer particles…
We develop a stochastic model for Lagrangian velocity as it is observed in experimental and numerical fully developed turbulent flows. We define it as the unique statistically stationary solution of a causal dynamics, given by a stochastic differential equation. In comparison to previously proposed stochastic models, the obtained process is infinitely differentiable at a given finite Reynolds number, and its second-order statistical properties converge to those of an Ornstein-Uhlenbeck process in the infinite Reynolds number limit. In this limit, it exhibits furthermore intermittent scaling pr…
Inertial particles in turbulent flows are characterised by preferential concentration and segregation and, at sufficient mass loading, dense particle clusters may spontaneously arise due to momentum coupling between the phases. These clusters, in turn, can generate and sustain turbulence in the fluid phase, which we refer to as cluster-induced turbulence. In the present theoretical work, we tackle the problem of developing a framework for the stochastic modelling of moderately dense particle-laden flows, based on a Lagrangian formalism, which naturally includes the Eulerian one. A rigorous for…
We present high-resolution direct numerical simulations of turbulent three-dimensional Rayleigh-Benard convection with a focus on the Lagrangian properties of the flow. The volume is a Cartesian slab with an aspect ratio of four bounded by free-slip planes at the top and bottom and with periodic side walls. The turbulence is inhomogeneous with respect to the vertical direction. This manifests in different lateral and vertical two-particle dispersion and in a dependence of the dispersion on the initial tracer position for short and intermediate times. Similar to homogeneous isotropic turbulence…
We present measurements of fluid particle accelerations in turbulent water flows between counter-rotating disks using three-dimensional Lagrangian particle tracking. By simultaneously following multiple particles with sub-Kolmogorov-time-scale temporal resolution, we measured the spatial correlation of fluid particle acceleration at Taylor microscale Reynolds numbers between 200 and 690. We also obtained indirect, non-intrusive measurements of the Eulerian pressure structure functions by integrating the acceleration correlations. Our experimental data provide strong support to the theoretical …
We consider a superstatistical dynamical model for the 3-d movement of a Lagrangian tracer particle embedded in a high-Reynolds number turbulent flow. The analytical model predictions are in excellent agreement with recent experimental data for flow between counter-rotating disks. In particular, we calculate the Lagrangian scaling exponents zeta_j for our system, and show that they agree well with the measured exponents reported in [X. Hu et al., PRL 96, 114503 (2006)]. Moreover, the model correctly predicts the shape of velocity difference and acceleration probability densities, the fast deca…
We provide evidence that the small-scale statistics of the acceleration of a test particle in high-Reynolds number Lagrangian turbulence is correctly described by Tsallis statistics with entropic index q=3/2. We present theoretical arguments why Tsallis statistics can naturally arise in Lagrangian turbulence and why at the smallest scales q=3/2 is relevant. A generalized Heisenberg-Yaglom formula is derived from the nonextensive model.
We use an acoustic Lagrangian tracking technique, particularly adapted to measurements in open flows, and a versatile material particles generator (in the form of soap bubbles with adjustable size and density) to characterize Lagrangian statistics of finite sized, neutrally bouyant, particles transported in an isotropic turbulent flow of air. We vary the size of the particles in a range corresponding to turbulent inertial scales and explore how the turbulent forcing experienced by the particles depends on their size. We show that, while the global shape of the intermittent acceleration probabi…
The Lagrangian probability-density-function model, proposed in Part I for dense particle-laden turbulent flows, is validated here against Eulerian-Lagrangian direct numerical simulation (EL) data for different homogeneous flows, namely statistically steady and decaying homogeneous isotropic turbulence, homogeneous-shear flow and cluster-induced turbulence (CIT). We consider the general model developed in Part I adapted to the homogeneous case together with a simplified version in which the decomposition of the phase-averaged (PA) particle-phase fluctuating energy into the spatially correlated …
We report on probability-density-functions (PDF) of the mass density in numerical simulations of highly compressible hydrodynamic flows and the corresponding structure formation of Lagrangian particles advected by the flows. Numerical simulations were performed with $512^3$ collocation points and 2 million tracer particles integrated over several dynamical times. We propose a connection between the PDF of the Lagrangian tracer particles and the predicted log-normal distribution of the density fluctuations in isothermal systems.
The design and application of an instrumented particle for the lagrangian characterization of turbulent free surface flows is presented in this study. This instrumented particle constitutes a local measurement device capable of measuring both its instantaneous 3D translational acceleration and angular velocity components, as well as recording them on an embarked removeable memory card. A lithium ion polymer battery provides the instrumented particle with up to 8 hours of autonomous operation. Entirely composed of commercial off the shelf electronic components, it features accelerometer and gyr…
Conditional statistics of homogeneous isotropic turbulent flow is investigated by means of high-Reynolds number direct numerical simulations performed with $2048^3$ collocation points. Eulerian as well as Lagrangian velocity increment statistics under several conditions are analyzed and compared. In agreement with experimental data longitudinal probability density functions $P(δ^{||}_l u|ε_l)$ conditioned on a scale-averaged energy dissipation rate are close to Gaussian distributions over all scales within the inertial range of scales. Also transverse increments conditioned on either the dissi…
We review some of the properties of higher-dimensional superstatistical stochastic models. As an example, we analyse the stochastic properties of a superstatistical model of 3-dimensional Lagrangian turbulence, and compare with experimental data. Excellent agreement is obtained for various measured quantities, such as acceleration probability densities, Lagrangian scaling exponents, correlations between acceleration components, and time decay of correlations. We comment on how to proceed from superstatistics to a thermodynamic description.
Direct estimation of Lagrangian turbulence statistics is essential for the proper modeling of dispersion and transport in highly obstructed canopy flows. However, Lagrangian flow measurements demand very high rates of data acquisition, resulting in bottlenecks that prevented the estimation of Lagrangian statistics in canopy flows hitherto. We report on a new extension to the 3D Particle Tracking Velocimetry (3D-PTV) method, featuring real-time particle segmentation that outputs centroids and sizes of tracer particles and performed on dedicated hardware during high-speed digital video acquisiti…
The Navier-Stokes equations are considered by the use of the method of Lagrangians with covariant derivatives (MLCD) over spaces with affine connections and metrics. It is shown that the Euler-Lagrange equations appear as sufficient conditions for the existence of solutions of the Navier-Stokes equations over (pseudo) Euclidean and (pseudo) Riemannian spaces without torsion. By means of the corresponding (n-1)+ 1 projective formalism the Navier-Stokes equations for radial and tangential accelerations are found.
The behavior of the second-order Lagrangian structure functions on state-of-the-art numerical data both in two and three dimensions is studied. On the basis of a phenomenological connection between Eulerian space-fluctuations and the Lagrangian time-fluctuations, it is possible to rephrase the Kolmogorov $4/5$-law into a relation predicting the linear (in time) scaling for the second order Lagrangian structure function. When such a function is directly observed on current experimental or numerical data, it does not clearly display a scaling regime. A parameterization of the Lagrangian structur…
In Lagrangian turbulence one is faced with the puzzle that 2D Navier-Stokes flows are nearly as intermittent as in three dimensions although no intermittency is present in the inverse cascade in 2D Eulerian turbulence. In addition, an inertial range is very difficult to detect and it is questionable whether it exists at all. Here, we investigate the transition of Eulerian to Lagrangian probability density functions (PDFs) which leads to a new type of Lagrangian structure function. This possesses an extended inertial range similar to the case of tracer particles in a frozen turbulent velocity f…
Inpired by recent measurements of the velocity and acceleration statistics of Lagrangian tracer particles embedded in a turbulent quantum liquid we propose a new superstatistical model for the dynamics of tracer particles in quantum turbulence. Our model consists of random sequences S/N/S/... where the particle spends some time in the superfluid (S) and some time in the normal fluid (N). This model leads to a superposition of power law distributions generated in the superfluid and Gaussian distributions in the normal liquid, in excellent agreement with experimental measurements. We include mem…
We present a systematic study of the influence of different forcing types on the statistical properties of supersonic, isothermal turbulence in both the Lagrangian and Eulerian frameworks. We analyse a series of high-resolution, hydrodynamical grid simulations with Lagrangian tracer particles and examine the effects of solenoidal (divergence-free) and compressive (curl-free) forcing on structure functions, their scaling exponents, and the probability density functions of the gas density and velocity increments. Compressively driven simulations show a significantly larger density contrast, a mo…
