Higher Order Statistics Research Articles

Study of Stable Stratification in HiRJET Facility With Direct Numerical Simulation

Stable density stratification in a large enclosure could significantly hamper the effectiveness of natural convection cooling in pool-type liquid metal or gas-cooled advanced reactors. In addition, accurate prediction of stratified front behavior remains to be a challenging task for turbulence modeling. With the rapid growth of high-performance-computing capabilities in recent years, conducting high-fidelity simulations for a large-timescale transient has become more affordable and hence a valuable data source to support turbulence modeling as well as to gain further physical insights. In this work, direct numerical simulation is performed at experiment-consistent conditions to simulate the density stratification transient High-Resolution Jet (HiRJET) facility. Specifically, we focus on the case where an injected aqueous sugar solution has 1.5% density higher than that in the enclosure. In the early stage of the transient, the impingement of the denser jet to the bottom surface of the enclosure promoted turbulent mixing locally. This rendered the establishment of the mixture layer, formation and swift upward propagation of the stratified front, and elevation with (locally) the highest vertical concentration gradient. As the front rose, the diminishing turbulent mass flux slowed down the propagation, and a larger vertical concentration gradient was established. In this stage, both the velocity and the concentration scalar showed large-timescale fluctuation behavior around the stratified front. For the concentration time signal, the characteristic frequency in the power spectral density was found to agree well with the Brunt-Väisällä frequency. The preliminary validation endeavor showed that the stratified front location and the corresponding concentration gradient magnitude in the simulation agreed well with the experiment data. Further validation will mainly revolve around benchmarking against high-resolution density measurement and high-order flow statistics.

An Assessment of Second Moment Closure Modeling for Stratified Wakes Using Direct Numerical Simulations Ensembles

Abstract Buoyant wakes encountered in the ocean environment are characterized by high Reynolds (Re) and Froude (Fr) numbers, leading to significant space–time resolution requirements for turbulence resolving CFD models (i.e., direct numerical simulations (DNS), large eddy simulations (LES)). Therefore, Reynolds-averaged Navier–Stokes (RANS) based models are attractive for these configurations. The inherently complex dynamics of stratified systems render eddy-viscosity-based modeling inappropriate. RANS second-moment closure (SMC) based modeling is more suitable because it accounts for flow anisotropy by solving the transport equations of important second-moment terms. Accordingly, eleven transport equations are solved at the SMC level, and a range of submodels are implemented for diffusion, pressure strain and scrambling, and dissipation terms. This work studies nonstratified and stratified towed wakes using SMC and DNS. Submodels in the SMC are evaluated in terms of how well their exact Reynolds averaged form impacts the accuracy of the full RANS closure. An ensemble average of 40 and 80–100 DNS realizations are required and conducted for these temporally evolving nonstratified and stratified wakes, respectively, to obtain converged higher-order statistics. SMC over-predicts wake height by over a factor of 2, and under-predicts defect velocity, wake width, and turbulent kinetic and potential energies by factors ranging from 1.3 to 3.5. Also, SMC predicts a near isotropic decay of normal Reynolds stresses (a33→−0.25), in contrast to the anisotropic decay (a33→−0.64) returned by DNS. The DNS data also provide important insights related to the inaccuracy of the dissipation rate isotropy assumption and the non-negligible contribution of pressure diffusion terms. These results lead to several important recommendations for SMC modeling improvement.

Higher-order turbulence statistics and multiscale characterization of morphodynamics in a riverbank section with an upstream mining pit

Channel dredging has become a common phenomenon across several fluvial systems. Pits dredged in the riverbank can influence the downstream turbulence and affect the bank slope and central channel morphodynamics. Erodible bed experiments were conducted in a laboratory flume having a riverbank cross section with three different bank slopes, i.e., 25°, 31°, and 40° with and without a mining pit. Flow over the bank slope and near the bank toe was most affected by the pit excavation at the channel upstream. Turbulence levels were amplified within the flow over the slope and near the bank toe. The logarithmic scaling range of higher-order fluctuations at the bank toe is greater for with pit case. The complex interaction of simultaneous processes like pit migration, sliding failure and bank erosion, and fluvial erosion was studied at multiple length scales and time scales. A wavelet cross correlation analysis was used to calculate the celerity of bedform migration at the bank toe along the flow. Statistical celerity of bedforms with smaller scales (up to 15 mm length scales) is significantly higher due to pit interaction. The study reveals that instream mining has notable effects on the inherent nature of higher-order turbulence statistics, especially near the bank slope and toe, as well as the multiscale morphological structures.

Self-Organizing Interval Type-2 Fuzzy Neural Network With Adaptive Discriminative Strategy

Covariate shift is a critical issue of interval type-2 fuzzy neural networks (IT2FNNs) due to the distribution discrepancy between training and testing samples. In this situation, IT2FNNs usually struggle to identify potential features from samples with explicit inductive biases. To address this problem, a self-organizing IT2FNN with an adaptive discriminative strategy (ADS-SOIT2FNN) is developed to maintain the identification performance in the presence of covariate shift. First, a granularity-based metric (GM), using higher-order statistics of local samples, is designed to distinguish the distribution discrepancy caused by covariate shift. The multiple kernels incorporated into GM are able to cover the sample features of the whole Hilbert space. Second, a self-organizing strategy, associated with GM-based discriminative information, is presented to alleviate the structural bias by growing and pruning fuzzy rules. Then, a compact structure of ADS-SOIT2FNN is achieved to adapt to the covariate shift of samples and further strengthen its inductive ability. Third, an adaptive risk mitigation learning algorithm (RMLA) is introduced to update the parameters of ADS-SOIT2FNN. RMLA can regulate the derivatives of parameters with arbitrary distribution samples, which is beneficial for maintaining the global accuracy by relieving the risk of parameter biases. Finally, the effectiveness of ADS-SOIT2FNN is verified by some experiments for identifying nonlinear systems with covariate shift.

Contribution of direct numerical simulations to the budget and modelling of the transport equations for passive scalar turbulent fields with wall scalar fluctuations

We perform direct numerical simulations (DNS) of the fully developed turbulent channel flow with a passive scalar subjected to constant time-averaged scalar fluxes at the wall. The first case (case I) considers a constant time-averaged scalar flux $\left \langle q_w \right \rangle$ along with the specific condition of a non-fluctuating scalar $\theta _w$ imposed at the wall implying zero wall scalar fluctuation $\theta '_w=0$ and zero variance (isoscalar boundary condition) whereas the second case (case II) accounts for a constant instantaneous scalar flux $q_w$ in time and space leading to zero gradient of scalar fluctuation along the normal to the wall $(\partial \theta '/ \partial x_n)_w = 0$ (isoflux boundary condition) implying a non-zero variance. The friction Reynolds number takes on the value $R_\tau =395$ and the molecular Prandtl numbers considered are $P_r=0.01$ , 0.1, 1 and 10. The purpose is to investigate the effect of the wall scalar fluctuations on the scalar field. Emphasis is put on the mean passive scalar $\left \langle \theta \right \rangle$ , the half-scalar variance $k_\theta$ , the turbulent scalar fluxes $\tau _{i \theta }$ , the correlation coefficients $R_{i \theta }$ , the passive scalar to dynamic time-scale ratio $\mathcal {R}$ , the turbulent Prandtl number $Pr_t$ and higher-order scalar statistics. Systematic comparisons between these two scalar fields are undertaken. As a result of interest, it is found that the mean scalar remains almost the same whatever the type of the boundary layer condition but not the scalar variance. The budgets of transport equations for the half-scalar variance and turbulent fluxes reveal that some scalar quantities such as the dissipation-rate and the molecular diffusion terms are highly modified in the near-wall region but not really the production, diffusion and the scalar-pressure gradient correlation terms in a first approximation. Visualization of the instantaneous scalar fields indicates that the topology of the structures is strongly modified as the Prandtl number increases from $P_r=0.01$ up to $10$ . Finally, it is shown how to use this DNS database to devise and calibrate scalar flux equation models in the framework of both first and second moment closures. This study suggests that accounting for wall scalar fluctuations should be considered in the simulation of turbulent flows involving fluid and solid combinations at the interface and provides a useful high resolution DNS database.

Investigating the Number of Monte Carlo Simulations for Statistically Stationary Model Outputs

The number of random fields required to capture the spatial variability of soil properties and their impact on the performance of geotechnical systems is often varied. However, the number of random fields required to obtain higher-order statistical moments of model outputs has not yet been studied. This research aims to investigate the number of Monte Carlo simulations needed to achieve stationary higher-order statistics of a pore pressure head in an unsaturated soil slope under steady-state infiltration. The study recommends using at least 500 Monte Carlo samples for the probabilistic analysis of geotechnical engineering models. A more conservative choice for up to second-moment analysis is 1000 samples. The analysis reveals significant variations in skewness, which become stationary for all mesh grids when the number of samples exceeds 15,000. Kurtosis stabilizes only when the number of samples reaches 25,000. The pore pressure head in the unsaturated zone is less uncertain. Additionally, the probability density function of the pore pressure head follows a leptokurtic distribution.

Exploiting the higher-order statistics of random-coefficient pure states for quantum information processing

Exploiting the higher-order statistics of random-coefficient pure states for quantum information processing

Higher-order statistics and intermittency of a two-fluid Hall–Vinen–Bekharevich–Khalatnikov quantum turbulent flow

The Hall–Vinen–Bekharevich–Khalatnikov (HVBK) model is widely used to numerically study quantum turbulence in superfluid helium. Based on the two-fluid model of Tisza and Landau, the HVBK model describes the normal (viscous) and superfluid (inviscid) components of the flow using two Navier–Stokes type of equations, coupled through a mutual friction force term. We derive transport equations for the third-order moments for each component of velocity involving the fourth-order moments, which are classical probes for internal intermittency at any scale, and revealing the probability of rare and strong fluctuations. Budget equations are assessed through direct numerical simulations of the HVBK flow. We simulate a forced homogeneous isotropic turbulent flow with Reynolds number of the normal fluid (based on Taylor's microscale) close to 100. Values from 0.1 to 10 are considered for the ratio between the normal and superfluid densities. For these flows, an inertial range is not discernible and the restricted scaling range approach is used to take into account the finite Reynolds number (FRN) effect. We analyse the importance of each term in budget equations and emphasize their role in energy exchange between normal and superfluid components. Some interesting features are observed: (i) transport and pressure-related terms are dominant, similarly to single-fluid turbulence; and (ii) the mathematical signature of the FRN effect is weak despite the low value of the Reynolds number. The flatness of the velocity derivatives is finally studied through the transport equations and their limit for very small scales, and it is shown to gradually increase for lower and lower temperatures, for both normal fluid and superfluid. This similarity highlights the strong locking of the two fluids. The flatness factors are also found in reasonable agreement with classical turbulence.

Modelling of wind turbine wakes over forests along the diurnal cycle

This work presents a methodology for the Large-Eddy Simulation (LES) of the continuous transition of atmospheric stability over forests along the diurnal cycle and its effect on the turbulence characteristics of wind turbine wakes. The forest is modelled as a porous surface where temperature changes, transferred to the air via sensible transport, are caused by the variation of net radiation and in proportion to the tree height and leaf density. The flow is driven by a pressure gradient including Coriolis forcing to allow for the development of nocturnal inertial oscillations. An actuator disk is employed to model the wake of a wind turbine located in Ryningsnäs, Sweden, for which metmast measurements are available to carry out a comparison. Results show a good prediction of the inflow and wake characteristics during daytime whereas turbulence fluctuations seem to be overestimated during night periods, attributed to a combination of an excess in geostrophic velocity and coarse mesh resolution. Observations of velocity, heat flux, potential temperature, velocity spectra and other higher order statistics are used to characterize the diurnal variations both in the inflow and across the wake. The results show that the model is capable of representing the turbulence flow dynamics during the diurnal stability transition, hence laying the ground to future studies to assess the performance of wind parks over forested areas.

Influence of skewed three-dimensional sinusoidal surface roughness on turbulent boundary layers

The impact of roughness skewness (ksk) on turbulent boundary layer (TBL) flow with a zero pressure gradient over three-dimensional (3D) sinusoidal rough surfaces was experimentally investigated using a single hotwire anemometer. Nine 3D sinusoidal profiles were manufactured with positive, negative, and zero roughness skewness values. Measurements were taken at three different freestream velocities for each surface and compared with smooth wall TBL results. This study covered a range of friction Reynolds numbers (Reτ) from approximately 1000 to 4000, with δ/k≈20 ± 2, where δ represents the local boundary layer thickness, and k is the maximum height of the roughness, measured from the valley to peak. The results indicate that the wall-unit normalized streamwise mean velocity profiles for all rough surfaces exhibit a downward shift compared to the smooth wall profiles. Surfaces with positive roughness skewness produced the highest drag, leading to the largest downward shift. The friction coefficient (Cf) decreased as ksk decreased. The percentage increase in Cf and ΔU+ (the roughness function) was much larger when moving from negative to zero roughness skewness than when moving from zero to positive roughness skewness. The small differences in turbulence intensity profiles and higher-order turbulence statistics in the outer region of the TBL support the outer layer similarity hypothesis for the roughness considered in this study. The autocorrelation study revealed that surfaces with positive roughness skewness tend to shorten the average length of turbulence structures in the near-wall region.

Deep multi-similarity hashing technique based on multi-label semantics

A deep multi-similarity hashing technique based on multi-label semantics has been proposed in this paper as a means of a new controlled deep hashing system for multi-label image retrieval. Our proposed methodology comprises hash code learning and semantically-aware similarity matrix creation. To create a similarity matrix that considers semantic context, we integrate label-level and semantic-level similarity. Using the higher-order statistics of deep features as inputs, we meticulously craft the multi-similarity loss and quantization error loss for hash code learning, ensuring that the learned binary codes retain their high-ranking similarity. Our suggested strategy is successful, as shown by several testing on CIFAR-10 and NUS-WIDE.

Wind fields from atmospheric turbulence measurements

We present recent advances in the modeling of wind fields and their reconstruction from real-world atmospheric turbulence measurements. The proposed wind field model provides a statistically coherent framework for modeling the empirically observed occurrence of extreme atmospheric events. Furthermore, we demonstrate that the model is general enough to enmesh real-world measurement points, e.g., from meteorological masts or LIDAR, and thus provides a rather convenient method for the reconstruction of highly resolved, three-dimensional wind fields. Our methodology is based on the so-called superstatistics of Gaussian-distributed (or Mann-type) velocity fields with fluctuating covariances and exactly reproduces higher-order statistics due to extreme small-scale wind fluctuations. Therefore, it could lead to more realistic inflow conditions and might potentially enhance the current design standard of wind turbines. We further discuss potential applications of the wind field model in the context of LIDAR measurements as well as for wake situations.

Local–Global Decompositions for Conditional Microstructure Generation

Conditional microstructure generation tools offer an important, inexpensive pathway to constructing statistically diverse datasets for Integrated Computational Materials Engineering and Materials Informatics efforts. To provide this utility in practice, an ideal generative framework must be able to efficiently, systematically, and robustly generate microstructures corresponding to selected spatial statistics (e.g., 1- and 2-point statistics) while also producing realistic local features (e.g., shapes and sizes of individual phase constituents). Because of the austerity of these requirements, generative frameworks often target either statistical conditioning or visual realism, but not both. In this paper, we propose to bridge these two approaches by approximating a microstructure’s generating process (i.e., its stochastic microstructure function) using a two layer semi-directed probabilistic graphical model. The first layer – a Gaussian Random Field (GRF) – provides direct control of the 1- and 2-point statistics. The second layer – a Score Based Generative Deep Learning model – postprocesses GRF predictions to refine local features while preserving global patterning. To understand and evaluate our proposed framework, we apply it to generate statistically equivalent N-phase microstructures from experimental references, including a 2-Phase Nickel-based Super Alloy and a 3-phase α−β Titanium alloy. Through these two case studies, we demonstrate that our framework successfully matches both lower-order (1- and 2-point statistics) and several salient higher-order statistics. Additionally, we briefly explore the capacity of these models to extrapolate outside of their training data by varying the input 2-point statistics. We discuss the value of this ability towards systematically generating diverse microstructure datasets.

PIV Measurements of Open-Channel Turbulent Flow under Unconstrained Conditions

Many open-channel turbulent flow studies have been focused on highly constrained conditions. Thus, it is rather conventional to note such flows as being fully developed, fully turbulent, and unaffected by sidewalls and free surface disturbances. However, many real-life flow phenomena in natural water bodies and artificially installed drain channels are not as ideal. This work is aimed at studying some of these unconstrained conditions. This is achieved by using particle image velocimetry measurements of a developing turbulent open-channel flow over a smooth wall. The tested flow effects are low values of the Reynolds number based on the momentum thickness Reθ (ranging from 165 to 930), low aspect ratio AR (ranging from 1.1 to 1.5), and Froude number Fr (ranging from 0.1 to 0.8). The results show that the mean flow has an inner region with a logarithmic layer with a von Kármán constant of 0.40–0.41, and a log law constant ranging from 5.0 to 6.0. The friction velocity and coefficient of skin friction are predictable using the formulations of Fr and Reθ presented in this work. The outer region is also characterized by a dip location, which is predictable using an equation associated with Reθ. The higher-order turbulence statistics, on the other hand, show distinguishing traits, such as correlation coefficients ranging from −0.1 to 0.5. Overall, this work demonstrates that for the unconstrained conditions studied, friction evaluations associated with Reynolds shear stress and some notable turbulence modelling functions used in conventional open-channel flows are inapplicable.

Constraining single-field inflation with MegaMapper

We forecast the constraints on single-field inflation from the bispectrum of future high-redshift surveys such as MegaMapper. Considering non-local primordial non-Gaussianity (NLPNG), we find that current methods will yield constraints of order σ(fNLeq)≈23, σ(fNLorth)≈12 in a joint power-spectrum and bispectrum analysis, varying both nuisance parameters and cosmology, including a conservative range of scales. Fixing cosmological parameters and quadratic bias parameter relations, the limits tighten significantly to σ(fNLeq)≈17, σ(fNLorth)≈8. These compare favorably with the forecasted bounds from CMB-S4: σ(fNLeq)≈21, σ(fNLorth)≈9, with a combined constraint of σ(fNLeq)≈14, σ(fNLorth)≈7; this weakens only slightly if one instead combines with data from the Simons Observatory. We additionally perform a range of Fisher analyses for the error, forecasting the dependence on nuisance parameter marginalization, scale cuts, and survey strategy. Lack of knowledge of bias and counterterm parameters is found to significantly limit the information content; this could be ameliorated by tight simulation-based priors on the nuisance parameters. The error-bars decrease significantly as the number of observed galaxies and survey depth is increased: as expected, deep dense surveys are the most constraining, though it will be difficult to reach σ(fNL)≈1 with current methods. The NLPNG constraints will tighten further with improved theoretical models (incorporating higher-loop corrections and improved understanding of nuisance parameters), as well as the inclusion of additional higher-order statistics.

Constraining νΛCDM with density-split clustering

ABSTRACTThe dependence of galaxy clustering on local density provides an effective method for extracting non-Gaussian information from galaxy surveys. The two-point correlation function (2PCF) provides a complete statistical description of a Gaussian density field. However, the late-time density field becomes non-Gaussian due to non-linear gravitational evolution and higher-order summary statistics are required to capture all of its cosmological information. Using a Fisher formalism based on halo catalogues from the Quijote simulations, we explore the possibility of retrieving this information using the density-split clustering (DS) method, which combines clustering statistics from regions of different environmental density. We show that DS provides more precise constraints on the parameters of the νΛCDM model compared to the 2PCF, and we provide suggestions for where the extra information may come from. DS improves the constraints on the sum of neutrino masses by a factor of 7 and by factors of 4, 3, 3, 6, and 5 for Ωm, Ωb, h, ns, and σ8, respectively. We compare DS statistics when the local density environment is estimated from the real or redshift-space positions of haloes. The inclusion of DS autocorrelation functions, in addition to the cross-correlation functions between DS environments and haloes, recovers most of the information that is lost when using the redshift-space halo positions to estimate the environment. We discuss the possibility of constructing simulation-based methods to model DS clustering statistics in different scenarios.

ProDMP: A Unified Perspective on Dynamic and Probabilistic Movement Primitives

Movement Primitives (MPs) are a well-known concept to represent and generate modular trajectories. MPs can be broadly categorized into two types: (a) dynamics-based approaches that generate smooth trajectories from any initial state, e. g., Dynamic Movement Primitives (DMPs), and (b) probabilistic approaches that capture higher-order statistics of the motion, e. g., Probabilistic Movement Primitives (ProMPs). To date, however, there is no MP method that unifies both, i. e. that can generate smooth trajectories from an arbitrary initial state while capturing higher-order statistics. In this paper, we introduce a unified perspective of both approaches by solving the ODE underlying the DMPs. We convert expensive online numerical integration of DMPs into position and velocity basis functions that can be used to represent trajectories or trajectory distributions similar to ProMPs while maintaining all the properties of dynamical systems. Since we inherit the properties of both methodologies, we call our proposed model Probabilistic Dynamic Movement Primitives (ProDMPs). Additionally, we embed ProDMPs in deep neural network architecture and propose a new cost function for efficient end-to-end learning of higher-order trajectory statistics. To this end, we leverage Bayesian Aggregation for non-linear iterative conditioning on sensory inputs. Our proposed model achieves smooth trajectory generation, goal-attractor convergence, correlation analysis, non-linear conditioning, and online re-planing in one framework. Our code can be found in <uri xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">https://github.com/ALRhub/ProDMP_RAL</uri> .

Starlet higher order statistics for galaxy clustering and weak lensing

We present a first application to photometric galaxy clustering and weak lensing of wavelet-based multi-scale (beyond two points) summary statistics: starlet peak counts and starlet ℓ1-norm. Peak counts are the local maxima in the map, and ℓ1-norm is computed via the sum of the absolute values of the starlet (wavelet) decomposition coefficients of a map, providing a fast multi-scale calculation of the pixel distribution, encoding the information of all pixels in the map. We employ the cosmo-SLICS simulations sources and lens catalogues, and we compute wavelet-based non-Gaussian statistics in the context of combined probes and their potential when applied to the weak-lensing convergence maps and galaxy maps. We obtain forecasts on the matter density parameter Ωm, the reduced Hubble constant h, the matter fluctuation amplitude σ8, and the dark energy equation of state parameter w0. In our setting for this first application, we consider the two probes to be independent. We find that the starlet peaks and the ℓ1-norm represent interesting summary statistics that can improve the constraints with respect to the power spectrum, even in the case of photometric galaxy clustering and when the two probes are combined.

Scaling Anisotropy with Stationary Background Field in Near-Sun Solar Wind Turbulence

The scaling of magnetic fluctuations provides crucial information for the understanding of solar wind turbulence. However, the observed magnetic fluctuations contain not only turbulence but also magnetic structures, leading to the violation of time stationarity. This violation would conceal the true scaling and influence the determination of the sampling angle with respect to the local background magnetic field. Here, to investigate scaling anisotropy, we utilize a simple but effective criterion ϕ < 10° to ensure the time stationarity of the magnetic field, where ϕ is the angle between the two averaged magnetic fields after cutting the interval into two halves. We study the scaling anisotropy using higher-order statistics of structure functions under the condition of stationarity for near-Sun solar wind turbulence for the first time based on measurements obtained from the Parker Solar Probe at 0.17 au. We find that the scaling indices ξ of the magnetic field show a linear dependence on the order p close to ξ(p) = p/4. The multifractal scaling of the magnetic-trace structure functions becomes monoscaling close to ξ(p) = p/3 with the local magnetic field perpendicular to the sampling direction and close to ξ(p) = p/4 with the local magnetic field parallel to the sampling direction when measured with the stationary background magnetic field. The scaling of the velocity-trace structure functions has similar but less significant changes. The near-Sun solar wind turbulence displays different scaling anisotropies from the near-Earth solar wind turbulence, suggesting the evolution of the nonlinear interaction process during solar wind expansion.

Moment quadrature method for uncertainty quantification of fatigue damage prognosis

The study proposes a new method for uncertainty quantification of fatigue damage prognosis using very few, typically 4, model evaluations, while achieving an accuracy comparable to the Monte Carlo method normally requires 106 model evaluations. Higher order statistics such as skewness and kurtosis can reliably be obtained using the method with 9 to 16 model evaluations. The proposed method provides a viable solution to problems where the computational demand is prohibitively high for conventional methods and/or real time computation is needed. Validations are made using several fatigue examples with realistic testing data, and results are compared with Monte Carlo simulations.