Tail Of Probability Density Function Research Articles

Non-stationary probabilistic analysis of non-linear ship roll motion due to modulated periodic and random excitations

This paper is about the study on the non-stationary probability density function (PDF) solution of the non-linear ship rolling system under modulated periodic and random excitations. The quadratic and cubic damping terms are introduced to describe the non-linear damping moment and the power terms in state variables are adopted to describe the righting moment in the ship rolling system, respectively. The relevant Fokker–Planck–Kolmogorov (FPK) equation of the ship rolling system is constructed and the non-stationary probabilistic solutions are obtained by the presented solution procedure. In the solution process, the time variable t is introduced to the traditional exponential-polynomial-closure (EPC) method to obtain the non-stationary PDF solutions of the ship rolling system. The obtained non-stationary PDF solutions of the non-linear ship rolling systems demonstrate that the presented solution procedure can give high solution accuracy even in the PDF tails when they are compared with Monte Carlo simulation (MCS). Moreover, owing to the strong system nonlinearity and the interaction of the periodic excitation and random excitation, the achieved non-stationary PDF solutions of the system responses can deviate significantly from the Gaussian PDF. It is also found that the PDFs of responses are not even symmetrical about the means of responses due to the interaction of the periodic excitation and the random excitation even if the system only contains odd nonlinear terms. In addition, the computational efficiency is increased by over five hundred times by the presented solution procedure compared to MCS.

Kinematic Effects on Probability Density Functions of Energy Dissipation Rate and Enstrophy in Turbulence.

Direct numerical simulation and theoretical analyses showed that the probability density functions (PDFs) of the energy dissipation rate and enstrophy in turbulence are asymptotically stretched gamma distributions with the same stretching exponent, and both the left and right tails of the enstrophy PDF are longer than those of the energy dissipation rate regardless of the Reynolds number. The differences in PDF tails arise due to the kinematics, with differences in the number of terms contributing to the dissipation rate and enstrophy. Meanwhile, the stretching exponent is determined by the dynamics and likeliness of singularities.

The added value of high‐resolution EURO‐CORDEX simulations to describe daily wind speed over Europe

AbstractIn the context of the CORDEX project, an ensemble of regional climate simulations (RCMs) of high resolution on a 0.11° grid has been generated for Europe with the objective of improving the representation of regional to local‐scale atmospheric phenomena. However, such simulations are computationally expensive and do not always reveal added value. Here, a recently proposed metric (the distribution added value [DAV]) is used to determine the added value of all available EURO‐CORDEX high‐resolution simulations at 0.11° for daily mean wind speed compared to their respective coarser‐gridded 0.44° counterparts and their driving fields, hindcast and historical experiments. The analysis consists in comparing the degree of similarity between normalized wind probability density function (PDF) of simulations and observations. In addition, the use of a normalized PDF allows for a direct spatial comparison among the different regions and time periods. Results show that RCMs add value to their reanalysis or forcing global model, but the nature and magnitude of the improvement on the representation of wind speed may vary depending on the model, region and season. We found most RCMS at 0.11° to outperform models at the 0.44° resolution in terms of their quality in capturing the measured wind speed PDF. When looking at the upper tail of the wind speed PDF, the benefits of downscaling are generally larger. At the regional scale, added value is obtained for 0.11° with respect to 0.44° resolution for all subdomains studied, particularly over the Mediterranean, the Iberian Peninsula and the Alps. When analysing the added value dependence on altitude, runs at 0.11° models represent better the locations below 50 m and above 350 m of altitude, and the 0.44° increasingly under‐perform for higher altitudes. Overall, DAV are larger at 0.11° than at 0.44° resolution, due to a better performance of local‐scale feedbacks at high resolution.

Numerical solution of the Fokker–Planck equation using physics-based mixture models

The Fokker–Planck equation governs the uncertainty propagation of dynamical systems driven by stochastic processes. The solution of the Fokker–Planck equation is a time-varying Probability Density Function (PDF) that is usually of high dimension with unbounded support. There are also properties that must be conserved in time for the joint and any marginal solution PDF (i.e., a time-varying PDF is a non-negative function that must integrate to unity at any given time.) Satisfying these properties poses a significant challenge to the numerical solution of the Fokker–Planck equation. Another challenge is to capture the tail behavior of the solution PDF with unbounded support, required to predict the probability of rare events like a system failure. Satisfying the conditions of a proper PDF with unbounded support limits the application of traditional grid-based numerical methods, like finite difference and finite element methods. This paper develops a novel numerical method based on physics-based mixture models for the transient and steady-state solutions of the Fokker–Planck equation. In the proposed numerical method, the trial function space consists of the convex combinations of some parametric PDFs (i.e., mixture models) that trivially satisfy the necessary conditions of the solution. Estimating the unknown parameters of the mixture model is via Bayesian inference while considering the constraints on model parameters. Bayesian inference facilitates integrating data on the responses of dynamical systems (e.g., from simulations) with the governing Fokker–Planck equation to estimate the unknown parameters. Since the solution PDF is not observable, combining it with observable data on system responses is far from trivial based on current numerical methods. The formulation of Bayesian inference also introduces a weighting function to reduce the estimation error in specific regions of interest like the tail of the solution PDF. To reduce the computational demand, the paper develops an importance sampling algorithm that generates a small set of collocation points at which the residual of the Fokker–Planck equation is evaluated. The performance of the proposed numerical method is demonstrated with several benchmark problems. The obtained results are compared with analytical solutions when available and otherwise with simulations.

Analysis of ENSO-Driven Variability, and Long-Term Changes, of Extreme Precipitation Indices in Colombia, Using the Satellite Rainfall Estimates CHIRPS

Climate change includes the change of the long-term average values and the change of the tails of probability density functions, where the extreme events are located. However, obtaining average values are more straightforward than the high temporal resolution information necessary to catch the extreme events on those tails. Such information is difficult to get in areas lacking sufficient rain stations. Thanks to the development of Satellite Precipitation Estimates with a daily resolution, this problem has been overcome, so Extreme Precipitation Indices (EPI) can be calculated for the entire Colombian territory. However, Colombia is strongly affected by the ENSO (El Niño—Southern Oscillation) phenomenon. Therefore, it is pertinent to ask if the EPI’s long-term change due to climate change is more critical than the anomalies due to climate variability induced by the warm and cold phases of ENSO (El Niño and La Niña, respectively). In this work, we built EPI annual time series at each grid-point of the selected Satellite Precipitation Estimate (CHIRPSv2) over Colombia to answer the previous question. Then, the Mann-Whitney-Wilcoxon test was used to compare the samples drawn in each case (i.e., change tests due to both long-term and climatic variability). After performing the analyses, we realized that the importance of the change depends on the region analyzed and the considered EPI. However, some general conclusions became evident: during El Niño years (La Niña), EPI’s anomaly follows the general trend of reduction -drier conditions- (increase; -wetter conditions-) observed in Colombian annual precipitation amount, but only on the Pacific, the Caribbean, and the Andean region. In the Eastern plains of Colombia (Orinoquía and Amazonian region), EPI show a certain insensitivity to change due to climatic variability. On the other hand, EPI’s long-term changes in the Pacific, the Caribbean, and the Andean region are spatially scattered. Still, long-term changes in the eastern plains have a moderate spatial consistency with statistical significance.

Transition of fluctuations from Gaussian state to turbulent state.

Variation of the statistical properties of an incompressible velocity, passive vector and passive scalar in isotropic turbulence was studied using direct numerical simulation. The structure functions of the gradients, and the moments of the dissipation rates, began to increase at about from the Gaussian state and grew rapidly at in the turbulent state. A contour map of the probability density functions (PDFs) indicated that PDF expansion of the gradients of the passive vector and passive scalar begins at around , whereas that of the longitudinal velocity gradient PDF is more gradual. The left tails of the dissipation rate PDF were found to follow a power law with an exponent of 3/2 for the incompressible velocity and passive vector dissipation rates, and 1/2 for the scalar dissipation rate and the enstrophy; they remained constant for all Reynolds numbers, indicating the universality of the left tail. The analytical PDFs of the dissipation rates and enstrophy of the Gaussian state were obtained and found to be the Gamma distribution. It was shown that the number of terms contributing to the dissipation rates and the enstrophy determines the decay rates of the two PDFs for low to moderate amplitudes.This article is part of the theme issue ‘Scaling the turbulence edifice (part 1)’.

Temporal-Spatial Evolution of Kinetic and Thermal Energy Dissipation Rates in a Three-Dimensional Turbulent Rayleigh-Taylor Mixing Zone.

In this work, the temporal–spatial evolution of kinetic and thermal energy dissipation rates in three-dimensional (3D) turbulent Rayleigh–Taylor (RT) mixing are investigated numerically by the lattice Boltzmann method. The temperature fields, kinetic and thermal energy dissipation rates with temporal–spatial evolution, the probability density functions, the fractal dimension of mixing interface, spatial scaling law of structure function for the kinetic and the thermal energy dissipation rates in 3D space are analysed in detail to provide an improved physical understanding of the temporal–spatial dissipation-rate characteristic in the 3D turbulent Rayleigh–Taylor mixing zone. Our numerical results indicate that the kinetic and thermal energy dissipation rates are concentrated in areas with large gradients of velocity and temperature with temporal evolution, respectively, which is consistent with the theoretical assumption. However, small scale thermal plumes initially at the section of half vertical height increasingly develop large scale plumes with time evolution. The probability density function tail of thermal energy dissipation gradually rises and approaches the stretched exponent function with temporal evolution. The slope of fractal dimension increases at an early time, however, the fractal dimension for the fluid interfaces is 2.4 at times t/τ ≥ 2, which demonstrates the self-similarity of the turbulent RT mixing zone in 3D space. It is further demonstrated that the second, fourth and sixth-order structure functions for velocity and temperature structure functions have a linear scaling within the inertial range.

Effects of dimensional wall temperature on velocity-temperature correlations in supersonic turbulent channel flow of thermally perfect gas

Direct numerical simulations of temporally evolving supersonic turbulent channel flows of thermally perfect gas are conducted at Mach number 3.0 and Reynolds number 4800 for various values of the dimensional wall temperature to study the influence of the latter on the velocity-temperature correlations. The results show that in a fully developed turbulent channel flow, as the dimensional wall temperature increases, there is little change in the mean velocity, but the mean temperature decreases. The mean temperature is found to be a quadratic function of the mean velocity, the curvature of which increases with increasing dimensional wall temperature. The concept of “recovery enthalpy” provides a connection between the mean velocity and the mean temperature, and is independent of dimensional wall temperature. The right tails of probability density function of the streamwise velocity fluctuation grows with increasing dimensional wall temperature. The dimensional wall temperature does not have a significant influence on the Reynolds analogy factor or strong Reynolds analogy (SRA). The modifications of SRA by Huang et al. and Zhang et al. provide reasonably good results, which are better than those of the modifications by Cebeci and Smith and by Rubesin.

Subdiffusion and ergodicity breaking in heterogeneous environments subject to Lévy noise

The anomalous diffusion and ergodicity breaking of Brownian particles in heterogeneous environments (space-dependent diffusion coefficient D(x)=D0|x|α) subject to Lévy noise are investigated, by both the analytically theory and the numerically simulating schemes. Within the tailored parameter, the competition of the power-law exponent α and the Lévy index μ ensures the long tail of probability density function with the form P(x)∼1|x|μ∕p+1 with p=22−α, falling rapidly when μ∕p>2. Meanwhile, we observe the subdiffusive motion that is characterized by the scale of ensemble averaged mean-square displacement (MSD), 〈x2(t)〉∝tκ with κ=2pμ<1. Furthermore, we also observe the weak ergodicity breaking which means the inequivalence between individual time averaged MSD and the corresponding ensemble averaged MSD. We quantify the weak ergodicity breaking behavior in terms of the dimensionless parameters EB and EB. For 1≤μ<2 (except for μ=2), the ergodicity breaking enhances as the α increases, but weakens as μ or D0. Overall, the EB follows the from, EB∝(Δ∕T)(12−2pμ), in the large time. Additionally, the anomalous diffusion above might be particularly relevant for ultracold atoms diffusion in heterogeneous media.

Kinetic energy transfer in compressible isotropic turbulence

Kinetic energy transfer in compressible isotropic turbulence is studied using numerical simulations with solenoidal forcing at turbulent Mach numbers ranging from 0.4 to 1.0 and at a Taylor Reynolds number of approximately 250. The pressure dilatation plays an important role in the local conversion between kinetic energy and internal energy, but its net contribution to the average kinetic energy transfer is negligibly small, due to the cancellation between compression and expansion work. The right tail of probability density function (PDF) of the subgrid-scale (SGS) flux of kinetic energy is found to be longer at higher turbulent Mach numbers. With an increase of the turbulent Mach number, compression motions enhance the positive SGS flux, and expansion motions enhance the negative SGS flux. Average of SGS flux conditioned on the filtered velocity divergence is studied by numerical analysis and a heuristic model. The conditional average of SGS flux is shown to be proportional to the square of filtered velocity divergence in strong compression regions for turbulent Mach numbers from 0.6 to 1.0. Moreover, the antiparallel alignment between the large-scale strain and the SGS stress is observed in strong compression regions. The inter-scale transfer of solenoidal and compressible components of kinetic energy is investigated by Helmholtz decomposition. The SGS flux of solenoidal kinetic energy is insensitive to the change of turbulent Mach number, while the SGS flux of compressible kinetic energy increases drastically as the turbulent Mach number becomes larger. The compressible mode persistently absorbs energy from the solenoidal mode through nonlinear advection. The kinetic energy of the compressible mode is transferred from large scales to small scales through the compressible SGS flux, and is dissipated by viscosity at small scales.

On the tails of probability density functions in Newtonian and drag-reducing viscoelastic turbulent channel flows

Direct numerical simulations (DNS) of Newtonian and drag-reducing viscoelastic turbulent channel flows generate crucial data to analyze and understand the nature of turbulence and its modification due to viscoelasticity. Such an understanding is necessary to enable better drag reduction technologies. Previously (Samanta et al., Phys. Fluids, 21, 115106, 2009) we have shown that viscoelasticity can significantly alter the non-Gaussian character of the probability density functions (PDFs) of many of the turbulent statistics in the flow. In this work, we investigate further the non-Gaussian characteristics of the PDFs generated in viscoelastic turbulence by focusing on their tail behavior. In particular, we show that many of the tails of the PDFs of the velocity and its derivatives obey a power law, as opposed to an exponential one exhibited by Gaussian PDFs, with power law indices that sometimes, for viscoelastic flows, are low enough to imply unbounded fourth- (kurtosis), or even third-(skewness) moments of the distribution, what we can call “strong fat tails” behavior. We use Hill's estimator to evaluate the power law index of the tails, as well as statistical inference methods to perform hypothesis testing for the appropriateness of the power law fit. We have found that fat tails corresponding to power law tail distribution behavior are observed in the PDFs for both Newtonian and viscoelastic cases. However, viscoelasticity leads to PDFs that exhibit consistently fatter tails than the Newtonian ones and it is only in the presence of viscoelasticity that turbulence leads to “strong fat tails” as defined above. We infer that drag-reducing viscoelastic turbulence involves more infrequent and more correlated and more extreme events which makes DNS of viscoelastic turbulent flows much more computationally demanding than Newtonian DNS, requiring finer spatial resolutions, longer integration times, and larger computational domain sizes.

Efficient statistically accurate algorithms for the Fokker–Planck equation in large dimensions

Solving the Fokker–Planck equation for high-dimensional complex turbulent dynamical systems is an important and practical issue. However, most traditional methods suffer from the curse of dimensionality and have difficulties in capturing the fat tailed highly intermittent probability density functions (PDFs) of complex systems in turbulence, neuroscience and excitable media. In this article, efficient statistically accurate algorithms are developed for solving both the transient and the equilibrium solutions of Fokker–Planck equations associated with high-dimensional nonlinear turbulent dynamical systems with conditional Gaussian structures. The algorithms involve a hybrid strategy that requires only a small number of ensembles. Here, a conditional Gaussian mixture in a high-dimensional subspace via an extremely efficient parametric method is combined with a judicious non-parametric Gaussian kernel density estimation in the remaining low-dimensional subspace. Particularly, the parametric method provides closed analytical formulae for determining the conditional Gaussian distributions in the high-dimensional subspace and is therefore computationally efficient and accurate. The full non-Gaussian PDF of the system is then given by a Gaussian mixture. Different from traditional particle methods, each conditional Gaussian distribution here covers a significant portion of the high-dimensional PDF. Therefore a small number of ensembles is sufficient to recover the full PDF, which overcomes the curse of dimensionality. Notably, the mixture distribution has significant skill in capturing the transient behavior with fat tails of the high-dimensional non-Gaussian PDFs, and this facilitates the algorithms in accurately describing the intermittency and extreme events in complex turbulent systems. It is shown in a stringent set of test problems that the method only requires an order of O(100) ensembles to successfully recover the highly non-Gaussian transient PDFs in up to 6 dimensions with only small errors.

A simple method to assess the added value using high‐resolution climate distributions: application to the EURO‐CORDEX daily precipitation

ABSTRACTRegional climate models (RCMs) are used with increasing resolutions seeking to represent in an improved way regional to local‐scale atmospheric phenomena. The EURO‐CORDEX simulations at 0.11° and simulations exploiting finer grid spacing approaching the convective‐permitting regimes are representative examples. These climate runs are computationally very demanding and do not always show improvements, which depend on the region, variable and object of study. The gains or losses associated with the use of higher resolution in relation to the forcing model (global climate model or reanalysis), or to different resolution RCM simulations, are widely known as added value. Its characterization is a long‐standing issue, and many different added‐value measures have been proposed.In the current study, a new method is proposed to assess the added value of finer‐resolution simulations, in comparison to its forcing data or coarser‐resolution counterparts. This approach builds on a probability density function (PDF) matching score, giving a normalized measure of the difference between diverse resolution PDFs, mediated by the observational ones. The distribution added value (DAV) is an objective added‐value measure that can be applied to any variable, region or temporal scale, from hindcast or historical (non‐synchronous) simulations. The DAVs metric and its application to the EURO‐CORDEX hindcast daily precipitation data are presented here. Generally, the EURO‐CORDEX simulations at both resolutions (0.44° and 0.11°) display a clear added value in relation to ERA‐Interim, with maximum values around 30% in summer and 20% in the intermediate seasons. When both RCM resolutions are directly compared, only three of five models (0.11°) show added value, with a maximum of ∼10%. The regions with the larger DAVs are areas where convection is relevant, e.g. Alps and Iberia. When looking at the extreme PDF tail, the higher‐resolution improvement is generally greater than the low resolution for seasons and regions.

Flame thickness and conditional scalar dissipation rate in a premixed temporal turbulent reacting jet

The flame structure corresponding to lean hydrogen–air premixed flames in intense sheared turbulence in the thin reaction zone regime is quantified from flame thickness and conditional scalar dissipation rate statistics, obtained from recent direct numerical simulation data of premixed temporally-evolving turbulent slot jet flames [1]. It is found that, on average, these sheared turbulent flames are thinner than their corresponding planar laminar flames. Extensive analysis is performed to identify the reason for this counter-intuitive thinning effect. The factors controlling the flame thickness are analyzed through two different routes i.e., the kinematic route, and the transport and chemical kinetics route. The kinematic route is examined by comparing the statistics of the normal strain rate due to fluid motion with the statistics of the normal strain rate due to varying flame displacement speed or self-propagation. It is found that while the fluid normal straining is positive and tends to separate iso-scalar surfaces, the dominating normal strain rate due to self-propagation is negative and tends to bring the iso-scalar surfaces closer resulting in overall thinning of the flame. The transport and chemical kinetics route is examined by studying the non-unity Lewis number effect on the premixed flames. The effects from the kinematic route are found to couple with the transport and chemical kinetics route. In addition, the intermittency of the conditional scalar dissipation rate is also examined. It is found to exhibit a unique non-monotonicity of the exponent of the stretched exponential function, conventionally used to describe probability density function tails of such variables. The non-monotonicity is attributed to the detailed chemical structure of hydrogen-air flames in which heat release occurs close to the unburnt reactants at near free-stream temperatures.

Measurements of scalar released from point sources in a turbulent boundary layer

Measurements of velocity and concentration fluctuations for a horizontal plume released at several wall-normal locations in a turbulent boundary layer (TBL) are discussed in this paper. The primary objective of this study is to establish a systematic procedure to acquire accurate single-point concentration measurements for a substantially long time so as to obtain converged statistics of long tails of probability density functions of concentration. Details of the calibration procedure implemented for long measurements are presented, which include sensor drift compensation to eliminate the increase in average background concentration with time. While most previous studies reported measurements where the source height is limited to, , where sz is the wall-normal source height and δ is the boundary layer thickness, here results of concentration fluctuations when the plume is released in the outer layer are emphasised. Results of mean and root-mean-square (r.m.s.) profiles of concentration for elevated sources agree with the well-accepted reflected Gaussian model (Fackrell and Robins 1982 J. Fluid. Mech. 117). However, there is clear deviation from the reflected Gaussian model for source in the intermittent region of TBL particularly at locations higher than the source itself. Further, we find that the plume half-widths are different for the mean and r.m.s. concentration profiles. Long sampling times enabled us to calculate converged probability density functions at high concentrations and these are found to exhibit exponential distribution.

Statistics of kinetic and thermal energy dissipation rates in two-dimensional turbulent Rayleigh–Bénard convection

We investigate the statistical properties of the kinetic$\unicode[STIX]{x1D700}_{u}$and thermal$\unicode[STIX]{x1D700}_{\unicode[STIX]{x1D703}}$energy dissipation rates in two-dimensional (2-D) turbulent Rayleigh–Bénard (RB) convection. Direct numerical simulations were carried out in a box with unit aspect ratio in the Rayleigh number range$10^{6}\leqslant Ra\leqslant 10^{10}$for Prandtl numbers$Pr=0.7$and 5.3. The probability density functions (PDFs) of both dissipation rates are found to deviate significantly from a log-normal distribution. The PDF tails can be well described by a stretched exponential function, and become broader for higher Rayleigh number and lower Prandtl number, indicating an increasing degree of small-scale intermittency with increasing Reynolds number. Our results show that the ensemble averages$\langle \unicode[STIX]{x1D700}_{u}\rangle _{V,t}$and$\langle \unicode[STIX]{x1D700}_{\unicode[STIX]{x1D703}}\rangle _{V,t}$scale as$Ra^{-0.18\sim -0.20}$, which is in excellent agreement with the scaling estimated from the two global exact relations for the dissipation rates. By separating the bulk and boundary-layer contributions to the total dissipations, our results further reveal that$\langle \unicode[STIX]{x1D700}_{u}\rangle _{V,t}$and$\langle \unicode[STIX]{x1D700}_{\unicode[STIX]{x1D703}}\rangle _{V,t}$are both dominated by the boundary layers, corresponding to regimes$I_{l}$and$I_{u}$in the Grossmann–Lohse (GL) theory (J. Fluid Mech., vol. 407, 2000, pp. 27–56). To include the effects of thermal plumes, the plume–background partition is also considered and$\langle \unicode[STIX]{x1D700}_{\unicode[STIX]{x1D703}}\rangle _{V,t}$is found to be plume dominated. Moreover, the boundary-layer/plume contributions scale as those predicted by the GL theory, while the deviations from the GL predictions are observed for the bulk/background contributions. The possible reasons for the deviations are discussed.

The probability density function tail of the Kardar–Parisi–Zhang equation in the strongly non-linear regime

An analytical derivation of the probability density function (PDF) tail describing the strongly correlated interface growth governed by the nonlinear Kardar–Parisi–Zhang equation is provided. The PDF tail exactly coincides with a Tracy–Widom distribution i.e. a PDF tail proportional to , where w2 is the the width of the interface. The PDF tail is computed by the instanton method in the strongly non-linear regime within the Martin–Siggia–Rose framework using a careful treatment of the non-linear interactions. In addition, the effect of spatial dimensions on the PDF tail scaling is discussed. This gives a novel approach to understand the rightmost PDF tail of the interface width distribution and the analysis suggests that there is no upper critical dimension.

Characterisation of the Eulerian and Lagrangian accelerations in the intermediate field of turbulent circular jets

ABSTRACTA rich data-set of Lagrangian trajectories from 3D particle tracking velocimetry is used to study the structure of various acceleration components, vorticity, and strain in the intermediate field of a circular jet at Reynolds number Re = 6000. The total acceleration is decomposed into three distinctive sets: (1) streamwise–radial; (2) tangential–normal; and (3) local–convective components. Probability density function (PDF) and joint distributions of each set are characterised at various radial locations from the jet core within a streamwise band 16 ≤ x/dh ≤ 17, where dh is the diameter of the pipe. The PDF of the relative angle between the acceleration components and the velocity vector is also included to aid the characterisation. Results show that the acceleration components are described by two distinctive distributions: one of them exhibits symmetry and heavy tails, while the other is best fitted by a power-law type. The tails of acceleration PDFs are heavier with larger radial distance from the core. The increased departure from the Gaussian distribution with the distance from the core is a result of the increasing turbulence levels promoted by the mean shear. The variation of the third and fourth moments between the streamwise–tangential and the radial–normal accelerations indicate the anisotropy of the jet. Joint PDF of each acceleration decomposition exhibits distinctive distribution that appears to depend from the distance from the jet core. However, the vorticity and strain show similar PDF across radial distances. Finally, complementary analysis of a jet from a semicircular pipe shows the footprint of the nozzle geometry in the acceleration structure of jets.

Enhancement factor statistics of surface enhanced Raman scattering in multiscale heterostructures of nanoparticles.

Suitable metal nanostructures may induce surface-enhanced Raman scattering (SERS) enhancement factors (EFs) large-enough to reach single-molecule sensitivity. However, the gap hot-spot EF probability density function (PDF) has the character of a long-tail distribution, which dramatically mines the reproducibility of SERS experiments. Herein, we carry out electrodynamic calculations based on a 3D finite element method of two plasmonic nanostructures, combined with Monte Carlo simulations of the EF statistics under different external conditions. We compare the PDF produced by a homodimer of nanoparticles with that provided by a self-similar trimer. We show that the PDF is sensitive to the spatial distribution of near-field enhancement specifically supported by the nanostructure geometry. Breaking the symmetry of the plasmonic system is responsible for inducing particular modulations of the PDF tail resembling a multiple Poisson distribution. We also study the influence that molecular diffusion towards the hottest hot-spot, or selective hot-spot targeting, might have on the EF PDF. Our results quantitatively assess the possibility of designing the response of a SERS substrate so as to contain the intrinsic EF PDF variance and significantly improving, in principle, the reproducibility of SERS experiments.

Tuning of adaptive interacting particle system for rare event probability estimation

The interacting particle system (IPS) for rare event algorithm has been well mathematically formulated, with a wide variety of results on the estimation accuracy of the probability of rare event. Despite this theoretical point of view, the practical side of this algorithm has not been handled completely. Indeed, a tuning parameter has a significant influence on the effective algorithm performance. Moreover, the choice of a good parameter value often proves to be fastidious and may decrease the usefulness of the IPS algorithm. Therefore, we propose a statistical technique in order to make the IPS algorithm fully adaptive. We derive this strategy for threshold exceedance probability estimation and for the estimation of probability density function tail. The performances of the proposed method have been studied on a toy case and on two more complex estimation problems in optical fiber and financial engineering.