Eulerian Statistics Research Articles

Exact relation between Eulerian and Lagrangian velocity increment statistics

We present a formal connection between Lagrangian and Eulerian velocity increment distributions which is applicable to a wide range of turbulent systems ranging from turbulence in incompressible fluids to magnetohydrodynamic turbulence. For the case of the inverse cascade regime of two-dimensional turbulence we numerically estimate the transition probabilities involved in this connection. In this context we are able to directly identify the processes leading to strongly non-Gaussian statistics for the Lagrangian velocity increments.

Eulerian and Lagrangian statistics in stably stratified turbulent channel flows

Using direct numerical simulation, we investigate the Eulerian and Lagrangian characteristics of the near-wall turbulence under stable stratification. In the near-wall region, large-scale motions are suppressed by stable stratification, while small-scale motions are enhanced, which leads to the increase of intermittency. It is shown that the modification of the Reynolds shear stress is caused by the increased intermittency and anisotropy. Investigation of acceleration and vorticity reveals that the length scales and timescales of the near-wall coherent vortices are reduced by stratification. The role of baroclinic torque around the near-wall turbulent structures is also discussed. The Lagrangian statistics indicate that the scaled dispersion in the streamwise direction remains unchanged, while those in the spanwise as well as in the wall-normal direction are suppressed by stratification.

Eulerian and Lagrangian statistics in the Bluelink numerical model and AVISO altimetry: Validation of model eddy kinetics

We compare Eulerian and Lagrangian timescales, velocity autocorrelations, and spectra from 14 years of hindcast ocean model and satellite velocities with direct observations. Our main aim is to determine how well the model (Bluelink, version Bran 2.1) and satellite‐derived currents (Archiving, Validation and Interpretation of Satellite Oceanographic data (AVISO)) simulate mesoscale eddy variability, which is important in mixing heat, salinity, and momentum and in controlling dispersion. Ocean Eulerian observations are from Topex‐Poseidon (T/P) Crossover velocities, and Lagrangian observations are from Global Drifter Program (GDP) surface drifters. Eulerian spectra from Bluelink are similar to Eulerian spectra from Crossover velocities over the range of common frequencies (the ∼10‐day sampling from T/P limits the comparison to low frequencies), and Eulerian timescales from Bluelink are slightly shorter than from the observations. Part of this difference may be an effect of the T/P sampling rate. In contrast, Lagrangian spectra from Bluelink are considerably steeper than spectra from GDP drifters, and consequently, Lagrangian timescales from Bluelink are longer than from the observations. Both Eulerian and Lagrangian spectra from AVISO are steeper than from observations so that both Eulerian and Lagrangian AVISO timescales are longer than observed. The real ocean is an ocean where the Eulerian‐Lagrangian relationship described by Middleton (1985) is followed; the Tasman Sea averaged ratio of Lagrangian to Eulerian timescales is about 0.1 zonally and 0.15 meridionally. However, the Eulerian‐Lagrangian relationship described by Middleton (1985) is not obeyed by Bluelink, and we believe that this behavior in Bluelink reflects a loss of short‐wavelength eddies in the model.

Turbulent particle dispersion in arbitrary wall-bounded geometries: A coupled CFD-Langevin-equation based approach

A Lagrangian continuous random walk (CRW) model is developed to predict turbulent particle dispersion in arbitrary wall-bounded flows with prevailing anisotropic, inhomogeneous turbulence. The particle tracking model uses 3D mean flow data obtained from the Fluent CFD code, as well as Eulerian statistics of instantaneous quantities computed from DNS databases. The turbulent fluid velocities at the current time step are related to those of the previous time step through a Markov chain based on the normalized Langevin equation which takes into account turbulence inhomogeneities. The model includes a drift velocity correction that considerably reduces unphysical features common in random walk models. It is shown that the model satisfies the well-mixed criterion such that tracer particles retain approximately uniform concentrations when introduced uniformly in the domain, while their deposition velocity is vanishingly small, as it should be. To handle arbitrary geometries, it is assumed that the velocity rms values in the boundary layer can locally be approximated by the DNS data of fully developed channel flows. Benchmarks of the model are performed against particle deposition data in turbulent pipe flows, 90° bends, as well as more complex 3D flows inside a mouth-throat geometry. Good agreement with the data is obtained across the range of particle inertia.

A non-hybrid method for the PDF equations of turbulent flows on unstructured grids

In probability density function (PDF) methods of turbulent flows, the joint PDF of several flow variables is computed by numerically integrating a system of stochastic differential equations for Lagrangian particles. A set of parallel algorithms is proposed to provide an efficient solution of the PDF transport equation modeling the joint PDF of turbulent velocity, frequency and concentration of a passive scalar in geometrically complex configurations. In the vicinity of walls the flow is resolved by an elliptic relaxation technique down to the viscous sublayer, explicitly modeling the high anisotropy and inhomogeneity of the low-Reynolds-number wall region without damping or wall-functions. An unstructured Eulerian grid is employed to extract Eulerian statistics, to solve for quantities represented at fixed locations of the domain (i.e., the mean pressure and the elliptic relaxation tensor) and to track particles. All three aspects regarding the grid make use of the finite element method employing the simplest linear shapefunctions. To model the small-scale mixing of the transported scalar, the interaction by exchange with the conditional mean (IECM) model is adopted. An adaptive algorithm to compute the velocity-conditioned scalar mean is proposed that homogenizes the statistical error over the sample space with no assumption on the shape of the underlying velocity PDF. Compared to other hybrid particle-in-cell approaches for the PDF equations, the current methodology is consistent without the need for consistency conditions. The algorithm is tested by computing the dispersion of passive scalars released from concentrated sources in two different turbulent flows: the fully developed turbulent channel flow and a street canyon (or cavity) flow. Algorithmic details on estimating conditional and unconditional statistics, particle tracking and particle-number control are presented in detail. Relevant aspects of performance and parallelism on cache-based shared memory machines are discussed.

Eulerian and Lagrangian eddy statistics of the Tasman Sea and southwest Pacific Ocean

Ocean eddy variability dominates many important processes in the ocean at a range of length and timescales. We describe the eddy statistics in the Tasman Sea–New Zealand region using drifter data from the Global Drifter Program (GDP) to derive Lagrangian statistics and TOPEX/Poseidon (T/P) satellite data to derive Eulerian statistics. The ratio η of the Lagrangian to Eulerian timescales is also a measure of length scales in the ocean; when η is high, Lagrangian length scales are short compared with Eulerian length scales and vice versa. We compute η both in the time domain from temporal autocorrelations of drifter and satellite data, and in the frequency domain from drifter and satellite velocity spectra. The zonal component of this ratio appears to be constant in the Tasman Sea, whereas the meridional component shows a latitudinal dependence. We argue that if η is known, all Eulerian and Lagrangian timescales can be determined from relatively easy measurements of the Eulerian timescale and eddy kinetic velocity.

Eulerian-Lagrangian aspects of a steady multiscale laminar flow

One key feature for the understanding and control of turbulent flows is the relation between Eulerian and Lagrangian statistics. This Brief Communication investigates such a relation for a laminar quasi-two-dimensional multiscale flow generated by a multiscale (fractal) forcing, which reproduces some aspects of turbulent flows in the laboratory, e.g., broadband power-law energy spectrum and Richardson’s diffusion. We show that these multiscale flows abide with Corrsin’s estimation of the Lagrangian integral time scale, TL, as proportional to the Eulerian (integral) time scale, LE∕urms, even though Corrsin’s approach was originally constructed for high Reynolds number turbulence. We check and explain why this relation is verified in our flows. The Lagrangian energy spectrum, Φ(w), presents a plateau at low frequencies followed by a power-law energy spectrum Φ(w)∼w−α at higher ones, similarly to turbulent flows. Furthermore, Φ(ω) scales with LE and urms with α>1. These are the key elements to obtain such a relation [Φ(w)∼ϵw−2 is not necessary] as in our flows the dissipation rate varies as ϵ∼urms3∕LEReλ−1. To complete our analysis, we investigate a recently proposed relation [M. A. I. Khan and J. C. Vassilicos, Phys. Fluids 16, 216 (2004)] between Eulerian and Lagrangian structure functions, which uses pair-diffusion statistics and the implications of this relation on Φ(ω). Our results support this relation, ⟨[uL(t)−uL(t+τ)]2⟩=⟨[uE(x)−uE(x+Δ2¯(τ)e)]2⟩, which leads to α=γ∕2(p−1)+1. This Eulerian-Lagrangian relation is striking as in the present flows it is imposed by the multiscale distribution of stagnation points, which are an Eulerian property.

Characteristics of transitional multicomponent gaseous and drop-laden mixing layers from direct numerical simulation: Composition effects

Transitional states are obtained by exercising a model of multicomponent-liquid (MC-liquid) drop evaporation in a three-dimensional mixing layer at larger Reynolds numbers, Re, than in a previous study. The gas phase is followed in an Eulerian frame and the multitude of drops is described in a Lagrangian frame. Complete dynamic and thermodynamic coupling between phases is included. The liquid composition, initially specified as a single-Gamma (SG) probability distribution function (PDF) depending on the molar mass, is allowed to evolve into a linear combination of two SGPDFs, called the double-Gamma PDF (DGPDF). The compositions of liquid and vapor emanating from the drops are calculated through four moments of their PDFs, which are drop-specific and location-specific, respectively. The mixing layer is initially excited to promote the double pairing of its four initial spanwise vortices, resulting into an ultimate vortex in which small scales proliferate. Simulations are performed for four liquids of different compositions, and the effects of the initial mass loading and initial free-stream gas temperature are explored. For reference, simulations are also performed for gaseous multicomponent mixing layers for which the effect of Re is investigated in the direct-numerical-simulation–accessible regime. The results encompass examination of the global layer characteristics, flow visualizations, and homogeneous-plane statistics at transition. Comparisons are performed with previous pretransitional MC-liquid simulations and with transitional single-component (SC) liquid-drop-laden mixing layer studies. Contrasting to pretransitional MC flows, the vorticity and drop organization depend on the initial gas temperature, this being due to drop/turbulence coupling. The vapor-composition mean molar mass and standard deviation distributions strongly correlate with the initial liquid-composition PDF. Unlike in pretransitional situations, regions of large composition standard deviation no longer necessarily coincide with those of large mean molar mass. The rotational and composition characteristics are all liquid-specific and the variation among liquids is amplified with increasing free-stream gas temperature. The classical energy cascade is found to be of similar strength, but the smallest scales contain orders of magnitude less energy than SC flows, which is confirmed by the larger viscous dissipation for MC flows. The kinetic energy and dissipation are liquid-specific and the variation among liquids is amplified with increasing free-stream gas temperature. The gas composition, of which the first four moments are calculated, is shown to be close to, but distinct from, a SGPDF. Eulerian and Lagrangian statistics of gas-phase quantities show that the different observation framework may affect the perception of the flow.

Lagrangian dispersion in turbulent channel flow and its relationship to Eulerian statistics

Lagrangian and Eulerian statistics were obtained from the direct numerical simulation of a turbulent channel flow. The Eulerian integral time scales and time microscales were compared to their Lagrangian equivalents. It is found that the Lagrangian time scales equal the Eulerian time scales at the wall. Otherwise, these proved to be consistently larger than the Eulerian time scales, and the latter are also found to be scaled by the propagation velocity rather than the mean velocity. The near-wall behavior of the ratio of the Lagrangian to Eulerian integral time scales is explained by the difference between the mean velocity and the propagation velocity of the turbulent structure. The ratio is proportional to the inverse of the turbulence intensity ( T L/ T E = βU/ σ). β for the streamwise component is nearly a constant value of 0.6 (20 < y + < 70), in agreement with atmospheric data and analyses. The ratio of the Lagrangian to the Eulerian time microscales is fairly constant away from the wall ( y + > 40). After period over the integral time scale elapsed, the effect of shear on turbulent diffusion appears and the mean-square dispersion is almost proportional to t 3 in agreement with the predictions of Riley and Corrsin [The relation of turbulent diffusivities to Lagrangian velocity statistics for the simplest shear flow. J. Geophys. Res. 79 (1974) 1768–1771] for homogeneous shear flow. After particles are distributed fairly uniformly between two walls, the mean-square dispersion becomes proportional to t and in agreement with theory of Taylor’s longitudinal dispersion.

𝑞-Eulerian polynomials: Excedance number and major index

In this research announcement we present a new q q -analog of a classical formula for the exponential generating function of the Eulerian polynomials. The Eulerian polynomials enumerate permutations according to their number of descents or their number of excedances. Our q q -Eulerian polynomials are the enumerators for the joint distribution of the excedance statistic and the major index. There is a vast literature on q q -Eulerian polynomials that involves other combinations of Eulerian and Mahonian permutation statistics, but this is the first result to address the combination of excedance number and major index. We use symmetric function theory to prove our formula. In particular, we prove a symmetric function version of our formula, which involves an intriguing new class of symmetric functions. We also discuss connections with (1) the representation of the symmetric group on the homology of a poset introduced by Björner and Welker; (2) the representation of the symmetric group on the cohomology of the toric variety associated with the Coxeter complex of the symmetric group, studied by Procesi, Stanley, Stembridge, Dolgachev, and Lunts; (3) the enumeration of words with no adjacent repeats studied by Carlitz, Scoville, and Vaughan and by Dollhopf, Goulden, and Greene; and (4) Stanley’s chromatic symmetric functions.

Determination of the coefficients of Langevin models for inhomogeneous turbulent flows by three-dimensional particle tracking velocimetry and direct numerical simulation

A promising and, in terms of computer power, low-cost way of describing flow properties such as turbulent diffusion is by Langevin models. The development of such models requires knowledge of Lagrangian statistics of turbulent flows. Our aim is to determine Lagrangian statistics of inhomogeneous flows, as most turbulent flows found in practical applications are inhomogeneous. The present paper describes how a Lagrangian measurement technique, three-dimensional particle tracking velocimetry, has been developed and applied to the most common example of inhomogeneous flows: turbulent pipe flow. A new direct numerical simulation (DNS) code has been developed and experimental results have been compared with results of this DNS code. The results concern Eulerian and Lagrangian velocity statistics at two Reynolds numbers. Based on these, coefficients of the Langevin model have been determined and physical consequences for Langevin modeling and turbulent dispersion have been explained.

Self-similar two-particle separation model

We present a new stochastic model for relative two-particle separation in turbulence. Inspired by material line stretching, we suggest that a similar process also occurs beyond the viscous range, with time scaling according to the longitudinal second-order structure function S2(r), e.g.; in the inertial range as ε−1∕3r2∕3. Particle separation is modeled as a Gaussian process without invoking information of Eulerian acceleration statistics or of precise shapes of Eulerian velocity distribution functions. The time scale is a function of S2(r) and thus of the Lagrangian evolving separation. The model predictions agree with numerical and experimental results for various initial particle separations. We present model results for fixed time and fixed scale statistics. We find that for the Richardson-Obukhov law, i.e., ⟨r(t)2⟩=gεt3, to hold and to also be observed in experiments, high Reynolds numbers are necessary, i.e., Reλ&amp;gt;O(1000), and the integral scale needs to be large compared to initial separation, i.e., L∕r0&amp;gt;30 and d∕L&amp;gt;3 need to be fulfilled, where d is the size of the field of view. Removing the constraint of finite inertial range, the model is used to explore separation dynamics in the asymptotic regime. As Reλ→∞, the distance neighbor function takes on a constant shape, almost as predicted by the Richardson diffusion equation. For the Richardson constant we obtain that g→0.95 as Reλ→∞. This asymptotic limit is reached at Reλ&amp;gt;1000. For the Richardson constant g, the model predicts a ratio of gb∕gf≈1.9 between backwards and forwards dispersion.

Intermediate level Lagrangian subsurface measurements in the northeast Pacific: Isobaric RAFOS float data

Isobaric RAFOS floats have been used to track the California Undercurrent and to investigate its continuity since 1992. The data include 61 quasi‐Lagrangian subsurface trajectories sampled for the most part between 150 and 600 m. The data set allows estimates of Eulerian and Lagrangian statistics for the region, studies of mesoscale eddy activity, and analysis of seasonal variability of circulation patterns off Central California. A browsable Web‐based inventory and a Graphical User Interface have been developed to provide access to this data set, including interactive manipulation of the data.

Stochastic modeling of particle diffusion in a turbulent boundary layer

Several Continuous Random Walk (CRW) models were constructed to predict turbulent particle diffusion based on Eulerian statistics that can be obtained with Reynolds-Averaged Navier Stokes (RANS) solutions. The test conditions included a wide range of particle inertias (Stokes numbers) with a near-wall injection ( y + = 4) in a turbulent boundary layer that is strongly anisotropic and inhomogeneous. To assess the performance of the models, the CRW results were compared to particle diffusion statistics gathered from a Direct Numerical Simulation (DNS). In particular, comparisons were made with transverse concentration profiles, root-mean-square of particle trajectory coordinates, and mean transverse particle velocity away from the wall. The results showed that accurate simulation required a modified (non-dimensionalized) Markov chain to handle the large gradients in turbulence near the wall as shown by simulations with fluid-tracer particles. For finite-inertia particles, an incremental drift correction for the Markov chain developed herein to account for Stokes number effects was critical to avoiding non-physical particle collection in low-turbulence regions. In both cases, inclusion of anisotropy in the turbulence model was found to be important, but the influence of off-diagonal terms was found to be weak. The results were generally good, especially for long-time and large inertia particles.

Flow Features in a Fully Developed Ribbed Duct Flow as a Result of MILES

The present contribution describes the topology associated with the turbulent flow in a square duct partially blocked by a rib of square section mounted on a single wall. The flow is simulated by means of a MILES method and the resulting velocity fields are analysed using the concepts of stream surface, vortex core detection, wall streamline and bifurcation line. Instantaneous and time averaged coherent structures are extracted applying the second scalar invariant of the velocity gradient tensor (so-called $Q$ criterion), respectively, to the instantaneous and time averaged velocity fields. This postprocessing reveals significant 3D effects induced by the geometry, namely the influence of the side walls, which is clearly identified. The combination of the different visualisation techniques offers a complement to the standard representation based on Eulerian statistics and contributes to a deeper understanding of this complex flow.

The effect of Doppler correction on measures of storm track intensity

In climate research, the intensity of a storm track is often measured by the band pass filtered eddy kinetic energy, geopotential variance or related variables. The interpretation of such fields disregards the influence the advection speed might have on these measures. In this paper the need for a Doppler corrected storm track measure is discussed. A Doppler corrected measure is defined and applied to 10 years of ECMWF reanalysis data, correcting the storm track measure for spatial and temporal variability of the advection speed. The storm track intensity is also calculated correcting only for the temporal variability of the advection speed. It is also related to fluctuations of the NAO teleconnection pattern. The Doppler correction suggests that (1) maximum and minimum baroclinic activity is found somewhat downstream of the locations indicated by non-corrected measures, (2) the storm track activity estimated by conventional measures is much too low in the areas of the eastern ends of the storm tracks, (3) the monthly mean time series of the strength of a storm track, as estimated by conventional measures, is strongly influenced by the variability of the advection speed at times, (4) the strength of the storm track seems to be less strongly connected with teleconnection patterns such as NAO or with the background mean flow speed than usually thought on the basis of conventional Eulerian statistics.

Lagrangian and Eulerian Statistics from Direct NumericalSimulations of Turbulent Channel Flow

Lagrangian and Eulerian statistics are obtained from the directnumerical simulation of a turbulent channel flow. The Reynolds number isobtained to be Reτ = 100 (based on friction velocity andchannel half-height). The Lagrangian time microscales are compared totheir Eulerian equivalents. It is found that the Lagrangian timemicroscales equal the Eulerian time microscales at the wall, but theyare consistently larger than the Eulerian away from the wall. TheEulerian time scales are also found to be scaled by the propagationvelocity rather than the mean velocity. The ratio of the Lagrangian tothe Eulerian time microscales is found to be nearly constant away fromthe wall (y+>40).

Statistics of velocity and preferential accumulation of micro-particles in boundary layer turbulence

The distribution of inertial particles in turbulent flows is strongly non-homogeneous and is driven by the structure of the underlying carrier flow field. In this work, we use DNS combined with Lagrangian particle tracking to characterize the effect of inertia on particle preferential accumulation. We compare the Eulerian statistics computed for fluid and particles of different size, and observe differences in terms of distribution patterns and deposition rates which depend on particle inertia. Specifically, different statistics are related to the selective interaction occurring between particles and coherent flow structures. This selective response causes a preferential sampling of the flow field by the particles and eventually leads to the well-known phenomenon of long-term particle accumulation in the boundary layer. We try to measure particle preferential accumulation with a Lagrangian parameter related to the rate of deformation of an elemental volume of the particle phase along a particle trajectory. In the frame of the Lagrangian approach, this parameter is mathematically defined as the particle position Jacobian, J ( t ) , computed along a particle path. This quantity is related to the local compressibility/divergence of the particle velocity field. Lagrangian statistics of J ( t ) show that compressibility increases for increasing particle response times τ p + (up to τ p + = 25 and within the time span covered by the simulation).

Relating Eulerian and Lagrangian Statistics for the Turbulent Dispersion in the Atmospheric Convective Boundary Layer

AbstractEulerian and Lagrangian statistics in the atmospheric convective boundary layer (CBL) are studied by means of large eddy simulation (LES). Spectra analysis is performed in both the Eulerian and Lagrangian frameworks, autocorrelations are calculated, and the integral length and time scales are derived. Eulerian statistics are calculated by means of spatial and temporal analysis in order to derive characteristic length and time scales. Taylor’s hypothesis of frozen turbulence is investigated, and it is found to be satisfied in the simulated flow.Lagrangian statistics are derived by tracking the trajectories of numerous particles released at different heights in the turbulent flow. The relationship between Lagrangian properties (autocorrelation functions) and dispersion characteristics (particles’ displacement) is studied through Taylor’s diffusion relationship, with special emphasis on the difference between horizontal and vertical motion. Results show that for the horizontal motion, Taylor’s relationship is satisfied. The vertical motion, however, is influenced by the inhomogeneity of the flow and limited by the ground and the capping inversion at the top of the CBL. The Lagrangian autocorrelation function, therefore, does not have an exponential shape, and consequently, the integral time scale is zero. If distinction is made between free and bounded motion, a better agreement between Taylor’s relationship and the particles’ vertical displacement is found.Relationships between Eulerian and Lagrangian frameworks are analyzed by calculating the ratio β between Lagrangian and Eulerian time scales. Results show that the integral time scales are mainly constant with height for z/zi &amp;lt; 0.7. In the upper part of the CBL, the capping inversion transforms vertical motion into horizontal motion. As a result, the horizontal time scale increases with height, whereas the vertical one is reduced. Current parameterizations for the ratio between the Eulerian and Lagrangian time scales have been tested against the LES results showing satisfactory agreement at heights z/zi &amp;lt; 0.7.

Nonlinear formation of structures in rotating stratified turbulence

Nonlinear approaches such as direct numerical simulations (DNS) yield a characteristic structuring in homogeneous turbulence as a result of modified dynamics under the effect of rotation and stable stratification. The structures are elongated for dominant rotation or flat when stratification dominates. Nonlinearity is essential for constructing these anisotropic Eulerian features, which we quantify by single and two-point second-order statistics. Linear approaches such as rapid distorsion theory (RDT) and kinematic simulations (KS) do not reproduce these effects at all. However, when looking at Lagrangian statistics, both linear and nonlinear models seem to yield very similar anisotropic trends. In order to investigate this paradox, we consider statistically homogeneous turbulence with vertical stable stratification characterized by the Brunt–Väisälä frequency N and vertical system rotation with frequency Ω in the Boussinesq system of equations. For different values of the ratio 2 Ω/N, we compare Eulerian and Lagrangian statistics. The detailed dynamics of energy is studied by splitting the velocity in toroidal and poloidal modes, which we put in relation with the wave/vortex linear decomposition. From DNS, the results for dominant stratification show a large disequilibrium of anisotropy between the toroidal and poloidal parts. For dominant rotation, angular spectra show an equidistribution of energy between poloidal and toroidal parts, with non-isotropic angular distribution of the energy density down to the smallest scales. Regarding Lagrangian statistics, DNS results are compared with two linear models based on RDT and KS, respectively. The linear models reproduce extremely well the oscillations and confinement of vertical one-point dispersion when stratification is present. Horizontal diffusion laws compare well in a qualitative way. However, quantitative differences can be detected.