Lagrangian Autocorrelation Research Articles

Unraveling the Non‐Homogeneous Dispersion Processes in Ocean and Coastal Circulations Using a Clustering Approach

AbstractDispersion processes in environmental flows have been traditionally studied under the strong assumption of homogeneous, isotropic and stationary turbulence. To overcome this limitation, we propose a new approach that combines autocorrelation analysis of simulated Lagrangian trajectories together with unsupervised clustering. To test the approach, we consider several dynamic scenarios around a coastal gulf, subject to different forcing, in order to compare our method with other approaches. Lagrangian trajectories forced by the varying coastal circulation exhibited different behaviors, looping and non‐looping paths, and produced a variety of Lagrangian autocorrelation functions. Our approach proves to be able to reveal spatio‐temporal variations in ocean dispersion processes without any a priori knowledge of the character of the trajectories. Clusters based on the autocorrelation functions are associated to different inhomogeneous dispersion processes. Finally, we propose a new stochastic model capable of predicting the different forms of autocorrelations.

In situ cloud particle tracking experiment

The collision-coalescence process of inertial particles in turbulence is held responsible for the quick growth of cloud droplets from ∼15 to ∼50 µm in diameter, but it is not well understood. Turbulence has two effects on cloud droplets: (1) it brings them closer together, preferentially concentrating them in certain parts of the flow, and (2) it sporadically creates high accelerations, causing droplets to detach from the underlying flow. These turbulence-cloud droplet interactions are difficult to study numerically or in the laboratory due to the large range of scales involved in atmospheric turbulence, so in situ measurements are needed. Here, we present a Lagrangian particle tracking (LPT) experimental setup situated close to the summit of Mt. Zugspitze at an altitude of 2650m above the sea level on top of the environmental research station Schneefernerhaus. Clouds naturally occur at this location about a quarter of the time. The LPT experiment probes a volume of ∼40 × 20 × 12mm3, has a spatial resolution of 5 µm and a temporal resolution of 0.1ms, and measures accelerations to within 0.1ms-2. Furthermore, the experiment can slide over a set of rails, driven by a linear motor, to compensate for the mean wind. It can slide up to 7.5ms-1. By doing so, the average residence time of the particles in the measurement volume increases. The mean wind compensation allows us to study various dynamical quantities, such as the velocity autocorrelation, or the dynamics of clustering. Moreover, it is beneficial for particle tracking, in general, since longer particle tracks allow us to apply better filtering to the tracks and thus increase accuracy. We present the radial distribution function, which quantifies clustering, the longitudinal relative velocity distribution, and the Lagrangian velocity autocorrelation, all computed from cloud droplet trajectories.

Turbulence-obstacle interactions in the Lagrangian framework: Applications for stochastic modeling in canopy flows

Lagrangian stochastic models are widely used to predict and analyze turbulent dispersion in complex environments, such as in various terrestrial and marine canopy flows. However, due to a lack of empirical data, it is still not understood how particular features of highly inhomogeneous canopy flows affect the Lagrangian statistics. In this work, we study Lagrangian short time statistics by analyzing empirical Lagrangian trajectories in sub-volumes of space that are small in comparison with the canopy height. For the analysis we used 3D Lagrangian trajectories measured in a dense canopy flow model in a wind-tunnel, using an extended version of real-time 3D particle tracking velocimetry (3D-PTV). One of our key results is that the random turbulent fluctuations due to the intense dissipation were more dominant than the flow's inhomogeneity in affecting the short-time Lagrangian statistics. This amounts to a so-called quasi-homogeneous regime of Lagrangian statistics at small scales. Using the Lagrangian dataset we calculate the Lagrangian autocorrelation function and the second-order Lagrangian structure-function, and extract associated parameters, namely a Lagrangian velocity decorrelation timescale, $T_i$, and the Kolmogorov constant, $C_0$. We demonstrate that in the quasi-homogeneous regime, both these functions are well represented using a second-order Lagrangian stochastic model that was designed for homogeneous flows. Furthermore, we show that the spatial variations of the Lagrangian separation of scales, $T_i/\tau_\eta$, and the Kolmogorov constant, $C_0$, cannot be explained by the variation of the Reynolds number, $Re_\lambda$, in space, and that $T_i/\tau_\eta$ was small as compared with homogeneous turbulence predictions at similar $Re_\lambda$. We thus hypothesize that this occurred due to the so-called "wake production", and show empirical results supporting our hypothesis.

관성 입자와 성층화 난류 사이의 상호작용에 대한 직접수치모사

Direct numerical simulation is conducted to observe the modification between marine inertial particles and stationary stratified turbulence in the ocean. Particle behavior in isotropic and stratified turbulence is different as the horizontal motion in stratified turbulence is usually enhanced. However, due to the mutual interaction between inertial particles and stratified turbulence, turbulent structures are affected by inertial particles and in return the behavior of them also changes. In this study, it is obtained that the strength of stratification is expressed by horizontal froude number, that of turbulence is done by taylor-micro scale reynolds number using kolmogorov scales and turbulent scales in the realistic ocean using direct numerical simulation. The range of the stokes number with considering settling velocity is consistent with inertial particles in the realistic ocean. Objectives are the investigation for modification of vortical structures, turbulent spectrum affected by inertial particles and lagrangian single dispersion and velocity autocorrelation of them by modified stratified turbulence in the horizontal direction.

Reconciling Lagrangian Diffusivity and Effective Diffusivity in Contour-Based Coordinates

AbstractThe present study reconciles theoretical differences between the Lagrangian diffusivity and effective diffusivity in a transformed spatial coordinate based on the contours of a quasi-conservative tracer. In the transformed coordinate, any adiabatic stirring effect, such as shear-induced dispersion, is naturally isolated from diabatic cross-contour motions. Therefore, Lagrangian particle motions in the transformed coordinate obey a transformed zeroth-order stochastic (i.e., random walk) model with the diffusivity replaced by the effective diffusivity. Such a stochastic model becomes the theoretical foundation on which both diffusivities are exactly unified. In the absence of small-scale diffusion, particles do not disperse at all in the transformed contour coordinate. Besides, the corresponding Lagrangian autocorrelation becomes a delta function and is thus free from pronounced overshoot and negative lobe at short time lags that may be induced by either Rossby waves or mesoscale eddies; that is, particles decorrelate immediately and Lagrangian diffusivity is already asymptotic no matter how small the time lag is. The resulting instantaneous Lagrangian spreading rate is thus conceptually identical to the effective diffusivity that only measures the instantaneous irreversible mixing. In these regards, the present study provides a new look at particle dispersion in contour-based coordinates.

Lagrangian Investigation of Wave-Driven Turbulence in the Ocean Surface Boundary Layer

AbstractTurbulent processes in the ocean surface boundary layer (OSBL) play a key role in weather and climate systems. This study explores a Lagrangian analysis of wave-driven OSBL turbulence, based on a large-eddy simulation (LES) model coupled to a Lagrangian stochastic model (LSM). Langmuir turbulence (LT) is captured by Craik–Leibovich wave forcing that generates LT through the Craik–Leibovich type 2 (CL2) mechanism. Breaking wave (BW) effects are modeled by a surface turbulent kinetic energy flux that is constrained by wind energy input to surface waves. Unresolved LES subgrid-scale (SGS) motions are simulated with the LSM to be energetically consistent with the SGS model of the LES. With LT, Lagrangian autocorrelations of velocities reveal three distinct turbulent time scales: an integral, a dispersive mixing, and a coherent structure time. Coherent structures due to LT result in relatively narrow peaks of Lagrangian frequency velocity spectra. With and without waves, the high-frequency spectral tail is consistent with expectations for the inertial subrange, but BWs substantially increase spectral levels at high frequencies. Consistently, over short times, particle-pair dispersion results agree with the Richardson–Obukhov law, and near-surface dispersion is significantly enhanced because of BWs. Over longer times, our dispersion results are consistent with Taylor dispersion. In this case, turbulent diffusivities are substantially larger with LT in the crosswind direction, but reduced in the along-wind direction because of enhanced turbulent transport by LT that reduces mean Eulerian shear. Our results indicate that the Lagrangian analysis framework is effective and physically intuitive to characterize OSBL turbulence.

Particle resolved direct numerical simulation of a liquid–solid fluidized bed: Comparison with experimental data

Particle-resolved direct numerical simulations of a 3-D liquid–solid fluidized bed experimentally investigated by Aguilar-Corona (2008) have been performed at different fluidization velocities (corresponding to a range of bed solid volume fraction between 0.1 and 0.4), using Implicit Tensorial Penalty Method. Particle Reynolds number and Stokes number are O(100) and O(10), respectively. In this paper, we compare the statistical quantities computed from numerical results with the experimental data obtained with 3-D trajectography and High Frequency PIV. Fluidization law predicted by the numerical simulations is in very good agreement with the experimental curve and the main features of trajectories and Lagrangian velocity signal of the particles are well reproduced by the simulations. The evolution of particle and flow velocity variances as a function of bed solid volume fraction is also well captured by the simulations. In particular, the numerical simulations predict the right level of anisotropy of the dispersed phase fluctuations and its independence of bed solid volume fraction. They also confirm the high value of the ratio between the fluid and the particle phase fluctuating kinetic energy. A quick analysis suggests that the fluid velocity fluctuations are mainly driven by fluid–particle wake interactions (pseudo-turbulence) whereas the particle velocity fluctuations derive essentially from the large scale flow motion (recirculation). Lagrangian autocorrelation function of particle fluctuating velocity exhibits large-scale oscillations, which are not observed in the corresponding experimental curves, a difference probably due to a statistical averaging effect. Evolution as a function of the bed solid volume fraction and the collision frequency based upon transverse component of particle kinetic energy correctly matches the experimental trend and is well fitted by a theoretical expression derived from Kinetic Theory of Granular Flows.

Forcing homogeneous turbulence in direct numerical simulation of particulate flow with interface resolution and gravity

We consider the case of finite-size spherical particles which are settling under gravity in a homogeneous turbulent background flow. Turbulence is forced with the aid of the random forcing method of Eswaran and Pope [“An examination of forcing in direct numerical simulations of turbulence,” Comput. Fluids 16(3), 257-278 (1988)], while the solid particles are represented with an immersed-boundary method. The forcing scheme is used to generate isotropic turbulence in vertically elongated boxes in order to warrant better decorrelation of the Lagrangian signals in the direction of gravity. Since only a limited number of Fourier modes are forced, it is possible to evaluate the forcing field directly in physical space, thereby avoiding full-size transforms. The budget of box-averaged kinetic energy is derived from the forced momentum equations. Medium-sized simulations for dilute suspensions at low Taylor-scale Reynolds number Reλ = 65, small density ratio ρp/ρf = 1.5, and for two Galileo numbers Ga = 0 and 120 are carried out over long time intervals in order to exclude the possibility of slow divergence. It is shown that the results at zero gravity are fully consistent with previous experimental measurements and available numerical reference data. Specific features of the finite-gravity case are discussed with respect to a reduction of the average settling velocity, the acceleration statistics, and the Lagrangian auto-correlations.

The effect of flow and orography on the spatial distribution of the very short-term predictability of rainfall from composite radar images

Abstract. The spatial distribution and scale dependence of the very short-term predictability of precipitation by Lagrangian persistence of composite radar images is studied under different flow regimes in connection with the presence of orographic features. Data from the weather radar composite of eastern Victoria, Australia, a 500 × 500 km2 domain at 10 min temporal and 2 × 2 km2 spatial resolutions, covering the period from February 2011 to October 2012, were used for the analyses. The scale dependence of the predictability of precipitation is considered by decomposing the radar rainfall field into an eight-level multiplicative cascade using a fast Fourier transform. The rate of temporal development of precipitation in Lagrangian coordinates is estimated at each level of the cascade under different flow regimes, which are stratified by applying a k-means clustering algorithm on the diagnosed velocity fields. The predictability of precipitation is measured by its lifetime, which is derived by integrating the Lagrangian auto-correlation function. The lifetimes were found to depend on the scale of the feature as a power law, which is known as dynamic scaling, and to vary as a function of flow regime. The lifetimes also exhibit significant spatial variability and are approximately a factor of 2 longer on the upwind compared with the downwind slopes of terrain features. The scaling exponent of the spatial power spectrum also shows interesting geographical differences. These findings provide opportunities to perform spatially inhomogeneous stochastic simulations of space–time precipitation to account for the presence of orography, which may be integrated into design storm simulations and stochastic precipitation nowcasting systems.

Particle tracking modeling of sediment-laden jets

Abstract. This paper presents a general model to predict the particulate transport and deposition from a sediment-laden horizontal momentum jet. A three-dimensional (3-D) stochastic particle tracking model is developed based on the governing equation of particle motion. The turbulent velocity fluctuations are modelled by a Lagrangian velocity autocorrelation function that captures the trapping of sediment particles in turbulent eddies, which result in the reduction of settling velocity. Using classical solutions of mean jet velocity, and turbulent fluctuation and dissipation rate profiles derived from computational fluid dynamics calculations of a pure jet, the equation of motion is solved numerically to track the particle movement in the jet flow field. The 3-D particle tracking model predictions of sediment deposition and concentration profiles are in excellent agreement with measured data. The computationally demanding Basset history force is shown to be negligible in the prediction of bottom deposition profiles.

Inertial Range Scaling in Rotations of Long Rods in Turbulence

We derive a scaling relationship for the mean square rotation rate of rods with lengths in the inertial range in turbulence: <p(i)p(i)> ∝ l(-4/3). We present experimental measurements of the rotational statistics of neutrally buoyant rods with lengths 2.8<l/η<72.9, and find that the measurements approach the predicted scaling. The approach to inertial range scaling is shown to be more complex than anticipated with an overshoot and approach to the scaling from above. For all rod lengths, the correlation time of the Lagrangian autocorrelation of the rotation rate scales as the turnover time of the eddies of the size of the rod. Measuring rotational dynamics of single long rods provides a new way to access the spatial structure of the flow at different length scales.

Drifter-derived estimates of lateral eddy diffusivity in the World Ocean with emphasis on the Indian Ocean and problems of parameterisation

The Global Drifter Program data set, collected in the period 1979–2011, is divided in drogue-on and drogue-off subsets which are then used to develop 5°×5° bin estimates of lateral eddy diffusivity in the World Ocean's mixed and surface layers, respectively, by means of a modification of the Davis (1987. Journal of Marine Research, 430 (45), 635–666.; 1991. Deep Sea Research. 38, 432 (1), S531–S571.) approach. The analysis is geographically focused on the Indian Ocean, the region where such calculations have been lacking. As the Indian Ocean circulation is strongly determined by monsoonal winds, an effort was made to calculate the diffusivity relative to monsoon-driven mean currents. The drifter-derived estimates of the asymptotic lateral diffusivity at large time lags K∞, were found to be below 1×104m2s−1 almost everywhere in the World Ocean, while the maximum diffusivity Kmax, equivalent to integrating the Lagrangian velocity autocorrelation to the first zero crossing, exceeds K∞ in strong ocean currents by a factor of 2–4. At mid latitudes (20°≤|ϕ|≤50°), the estimates of K∞ in the mixed and surface layers are shown to follow the frozen field regime parameterisation K∞~EKE×Lbc, where EKE is the eddy kinetic energy and Lbc is the baroclinic Rossby deformation radius. In contrast, at low latitudes (|ϕ|<29°) in the surface layer, the fixed float regime parameterisation Kmax,K∞=EKE×T0, where T0=1.9 days and 1.3 days is the typical value of the Lagrangian integral timescale in the cases of Kmax and K∞, respectively, is preferable.

Rotation statistics of fibers in wall shear turbulence

In this paper, the rotation of rigid fibers is investigated for the reference case of turbulent channel flow. The aim of the study is to examine the effect of local shear and turbulence anisotropy on the rotational dynamics of fibers with different elongation and inertia. To this aim, statistics of the fiber angular velocity, Ω, are extracted from direct numerical simulation of turbulence at shear Reynolds number Reτ = 150 coupled with Lagrangian tracking of prolate ellipsoidal fibers with Stokes number, St, ranging from 3 to 100 and aspect ratio, λ, ranging from 1 to 50. Accordingly, the fiber-to-fluid density ratio ranges from \({S \simeq 7}\) (for St = 1, λ = 50) to \({S \simeq 3, 470}\) (for St = 100, λ = 1). Statistics are compared one to one with those obtained for spherical particles to highlight effects due to elongation. Results for mean and fluctuating angular velocities show that elongation is important for fibers with small inertia (St ≤ 5 in the present flow-fiber combination). For fibers with larger inertia, elongation has an impact on fiber rotation only in the streamwise and wall-normal directions, where mean values of Ω are zero. It is also shown that, in the center of the channel, the Lagrangian autocorrelation coefficients of Ω and corresponding rotational turbulent diffusivities match the exponential behavior predicted by the theory of homogeneous dispersion. In this region of the channel, the probability density function of fiber angular velocities is generally close to Gaussian, indicating that particle rotation away from solid walls can be modeled as a diffusion process of the Ornstein–Uhlenbeck type at stationary state. In the strong shear region (comprised within a distance of 50 viscous units from the wall in the present simulations), fiber anisotropy adds to flow anisotropy to induce strong deviations on fiber rotational dynamics with respect to spherical particles. The database produced in this study is available to all interested users at https://www.fp1005.cism.it.

Numerical modelling of horizontal sediment-laden jets

Sediment-laden turbulent flows are commonly encountered in natural and engineered environments. It is well known that turbulence generates fluctuations to the particle motion, resulting in modulation of the particle settling velocity. A novel stochastic particle tracking model is developed to predict the particle settling out and deposition from a sediment-laden jet. Particle velocity fluctuations in the jet flow are modelled from a Lagrangian velocity autocorrelation function that incorporates the physical mechanism leading to a reduction of settling velocity. The model is first applied to study the settling velocity modulation in a homogeneous turbulence field. Consistent with basic experiments using grid-generated turbulence and computational fluid dynamics (CFD) calculations, the model predicts that the apparent settling velocity can be reduced by as much as 30 % of the stillwater settling velocity. Using analytical solution for the jet mean flow and semi-empirical RMS turbulent velocity fluctuation and dissipation rate profiles derived from CFD predictions, model predictions of the sediment deposition and cross-sectional concentration profiles of horizontal sediment-laden jets are in excellent agreement with data. Unlike CFD calculations of sediment fall out and deposition from a jet flow, the present method does not require any a priori adjustment of particle settling velocity.

Three-dimensional Lagrangian Voronoï analysis for clustering of particles and bubbles in turbulence

AbstractThree-dimensional Voronoï analysis is used to quantify the clustering of inertial particles in homogeneous isotropic turbulence using data sets from numerics in the point particle limit and one experimental data set. We study the clustering behaviour at different density ratios, particle response times (i.e. Stokes numbers $\mathit{St}$) and two Taylor–Reynolds numbers (${\mathit{Re}}_{\lambda } = 75$ and 180). The probability density functions (p.d.f.s) of the Voronoï cell volumes of light and heavy particles show different behaviour from that of randomly distributed particles, i.e. fluid tracers, implying that clustering is present. The standard deviation of the p.d.f. normalized by that of randomly distributed particles is used to quantify the clustering. The clustering for both light and heavy particles is stronger for higher ${\mathit{Re}}_{\lambda } $. Light particles show maximum clustering for $\mathit{St}$ around 1–2 for both Taylor–Reynolds numbers. The experimental data set shows reasonable agreement with the numerical results. The results are consistent with previous investigations employing other approaches to quantify the clustering. We also present the joint p.d.f.s of enstrophy and Voronoï volumes and their Lagrangian autocorrelations. The small Voronoï volumes of light particles correspond to regions of higher enstrophy than those of heavy particles, indicating that light particles cluster in higher vorticity regions. The Lagrangian temporal autocorrelation function of Voronoï volumes shows that the clustering of light particles lasts much longer than that of heavy or neutrally buoyant particles. Due to inertial effects arising from the density contrast with the surrounding liquid, light and heavy particles remain clustered for much longer times than the flow structures which cause the clustering.

Lagrangian velocity and acceleration auto-correlations in rotating turbulence

Rotating turbulence has often been studied by numerical simulations and analytical models. Experimental data is scarce and purely of Eulerian nature. In the present study, experiments on continuously forced turbulence subjected to background rotation are performed by means of 3D Particle Tracking Velocimetry. The data is processed in the Lagrangian as well as in the Eulerian frame. The background rotation is confirmed to induce two-dimensionalisation of the flow field, and the large-scales are dominated by stable counter-rotating vertical tubes of vorticity. The auto-correlation coefficients along particle trajectories of velocity and acceleration components have been explored. We will discuss the effects of rotation on the Lagrangian temporal scales of the flow.

Lagrangian velocity autocorrelations in statistically steady rotating turbulence

Lagrangian statistics of passive tracers in rotating turbulence is investigated by particle tracking velocimetry. A confined and steadily forced turbulent flow is subjected to five different rotation rates. The PDFs of the velocity components clearly reveal the anisotropy induced by background rotation. Although the statistical properties of the horizontal turbulent flow field are approximately isotropic, in agreement with previously reported results by van Bokhoven and coworkers [Phys. Fluids 21, 096601 (2009)], the velocity component parallel to the (vertical) rotation axis gets strongly reduced (compared to the horizontal ones) while the rotation is increased. The auto-correlation coefficients of all three components are progressively enhanced for increasing rotation rates, although the vertical one shows a tendency to decrease for slow rotation rates. The decorrelation is approximately exponential. Lagrangian data compare favorably with previously reported Eulerian data for horizontal velocity components but show a different behavior for the vertical velocity components at higher rotation rates.

Evaluation of a reduced model for investigating hurricane formation from turbulence

AbstractThis paper evaluates the adequacy of a reduced (three‐layer) model for understanding hurricane formation from turbulent initial conditions. The evaluation is based on a direct comparison to tropical cyclogenesis in a cloud‐system‐resolving (CSR) model that employs single‐moment warm rain microphysics. The reduced model has three cumulus parametrizations, referred to as the convergence‐based (CB), boundary layer quasi‐equilibrium (BLQ), and selective boost (SB) options. Regardless of which one is activated, the reduced model produces hurricanes on the same time‐scale as the CSR model. Generally speaking, the hurricanes emerge from turbulence through the coalescence and convective intensification of cyclonic vorticity. Moreover, in both the reduced and CSR models, the onset of ‘rapid intensification’ follows pronounced local growth of the η‐variable of Ooyama (1969), which is a combined measure of deep convective instability and middle tropospheric moisture. Eliminating the surface flux of moist entropy or surface friction in either model prevents or severely inhibits hurricane formation; however, hurricanes eventually form without surface friction in the BLQ or SB versions of the reduced model.Despite some measure of success, the reduced model has notable deficiencies that are apparent during the intermediate stage of genesis. Compared to the CSR model, rotational storms are less sporadic and their peak winds are less severe. In the intermediate mesoscale of the reduced model, the horizontal kinetic energy spectrum is relatively steep, and horizontal divergence is relatively weak. Furthermore, the Lagrangian autocorrelation time of vertical vorticity is relatively long. These discrepancies reflect a simplified (quasi‐two‐dimensional) form of rotational convective turbulence. The simplified turbulence has comparatively robust mesoscale vortices, and tends to produce more tropical cyclones than its counterpart generates in the CSR model. Copyright © 2011 Royal Meteorological Society

Scalar flux in a uniform mean scalar gradient in homogeneous isotropic steady turbulence

Statistics of a passive scalar flux in a uniform mean scalar gradient convected by homogeneous isotropic steady turbulence are numerically studied by using very high resolution direct numerical simulation. It is found that the Nusselt number increases in proportion to the Péclet number and that the one point probability density function of the scalar flux is negatively skewed and exponential, and is insensitive to the variation of the Péclet number. The scalar field is studied by visualization, and the ramp–cliff structure and the mesa–canyon structure are observed along the directions parallel and perpendicular to the mean scalar gradient, respectively. The probability density function of the scalar flux is theoretically computed and found to be in good agreement with the numerical results. A Lagrangian statistical theory for the scalar flux is developed, which predicts that the scalar transfer flux is given by the time integral of the Lagrangian velocity autocorrelation and increases in proportion to the Péclet number, which is consistent with the results of the direct numerical simulation. A physical explanation of the asymmetry of the scalar flux PDF is explored.

S0506-3-5 一様な平均勾配のもとでのパッシヴスカラー乱流の統計([0506-3]乱流における運動量,熱,物質の輸送現象(3))

Statistics of transfer flux of a passive scalar under the mean uniform scalar gradient convected by homogeneous isotropic steady turbulence are examined by using the high resolution direct numerical simulation. It is found that the Nusselt number is expressed by the time integral of the Lagrangian velocity autocorrelation in the direction of the mean scalar gradient and increases as ScR^2_λ which agrees well with the results of the direct numerical simulation, where Sc and R_λ are the Schmidt and Reynolds numbers, respectively. The one point probability density function (PDF) of the scalar flux is also examined and found to be negatively skewed exponential but insensitive to the variation of the Reynolds and Schmidt numbers. The PDF of the scalar flux is also theoretically examined and found to be in good agreement with the numerical results.