Bimodal Probability Density Function Research Articles

Stochastic Dynamics of Fusion Low-to-High Confinement Mode (L-H) Transition: Correlation and Causal Analyses Using Information Geometry.

We investigate the stochastic dynamics of the prey-predator model of the Low-to-High confinement mode (L-H) transition in magnetically confined fusion plasmas. By considering stochastic noise in the turbulence and zonal flows as well as constant and time-varying input power Q, we perform multiple stochastic simulations of over a million trajectories using GPU computing. Due to stochastic noise, some trajectories undergo the L-H transition while others do not, leading to a mixture of H-mode and dithering at a given time and/or input power. One of the consequences of this is that H-mode characteristics appear at a smaller input power Q<Qc (where Qc is the critical value for the L-H transition in the deterministic system) as a secondary peak of a probability density function (PDF) while dithering characteristics persists beyond the power threshold for Q>Qc as a second peak. The coexisting H-mode and dithering near Q=Qc leads to a prominent bimodal PDF with a gradual L-H transition rather than a sudden transition at Q=Qc and uncertainty in the input power. Also, a time-dependent input power leads to increased variability (dispersion) in stochastic trajectories and a more prominent bimodal PDF. We provide an interpretation of the results using information geometry to elucidate self-regulation between zonal flows, turbulence, and information causality rate to unravel causal relations involved in the L-H transition.

P-Bifurcation Analysis of a Quarter-Car Model With Inerter-Based Pendulum Vibration Absorber: A Wiener Path Integration Approach

Abstract In this theoretical study, a recently developed inerter-based pendulum vibration absorber (IPVA) coupled with energy harvesting capabilities is applied to the quarter car model with class C road conditions (ISO 8608). The impact of varying the pendulum length parameter on power harvesting, ride comfort (sprung mass acceleration), and road handling is investigated. It is discovered that P-bifurcation of the probability density function (PDF), can simultaneously occur with enhanced output power (40% improvements), low sprung mass acceleration (60% improvements), and better road handling (60% improvements) when compared with the linear benchmark system. To predict this bifurcation, a Wiener path integration (WPI) method coupled with curvature checking is developed for the PDF. An efficient bifurcation detection algorithm is developed which leads to the prediction of monomodal, bimodal, and rotation PDF regions in the noise intensity-electrical damping plane. Using Monte Carlo simulations (MCS), the performance metrics were then compared against the optimal linear benchmark for varying driving speed on a class F road while varying the electrical damping so that the system is at or near P-bifurcation. Energy transfer into the electrical domain and power harvested are shown to be up to 43% and 20% higher than for the optimized linear system, respectively. Electrical efficiency considerations show that generator selection is also a factor. Ride comfort and road handling still saw improvements of at least 59%. Finally, the new algorithm effectively reduces an exhaustive MCS for various parameter configurations when qualitative changes in the PDF are linked to performance.

Coupling particle image velocimetry and digital image analysis to characterize cluster dynamics in a fast fluidized bed

Particle clusters affect interphase interaction and dominate the overall performance of gas-solid fluidized beds. In this work, the particle image velocimetry, particle tracking velocimetry and digital image analysis were coupled to characterize clustering dynamics in a lab-scale riser. The clusters were identified by the Otsu algorithm based on grayscale distribution, and the cluster properties such as equivalent size, solids concentration and velocity distribution were extracted and analyzed under various operating conditions. It was found macroscopic operating conditions affect cluster characteristics significantly whereas the effect of particle composition is secondary. A bimodal or skewed probability density function was observed for not only the time-averaged hydrodynamic velocity but also the particle velocity around the cluster interface, indicating the coexistence of dilute and dense phases and the breaking of equilibrium assumption. The nonuniform distributions for laminar and Reynolds-stress-like granular temperatures were further discussed, which show remarkable anisotropy at both the microscopic and mesoscopic scales.

Study of the positional and orientational contributions to the Helmholtz free energy of a finite hard-disk system. A molecular dynamics analysis of its hexatic transition

Based on an equilibrium thermodynamic scheme, this work studies the separation of the Helmholtz free energy of a hard-disk system into its positional and orientational contributions. For the positional part, an accurate two dimensional version of the Carnahan-Starling equation of state is proposed, while a simple ansatz of the one particle partition function for the orientational part is proposed. The latter expression, which contains information about the probability of the norm of the six-fold orientational order parameter, is obtained through molecular dynamics simulations. Perhaps due to finite size effects, the ordering of the fluid is not a sudden phenomenon: the fluid-hexatic phase transition is preceded by a linear increase in the orientational contribution to the pressure of the fluid. A similar result is obtained for the hexatic-solid phase transition. Thus, the hexatic first-order phase transition is located between two linear ordering processes, consistent with the KTHNY theory. The proposed separation also predicts the existence of first order fluid-hexatic phase transition by using two procedures: the first, based on a purely bimodal probability density function, produces a pressure with a van del Waals-like first-order phase transition, while the second, considering density fluctuations, results in a flattened pressure phase transition.

Effects of body force on the statistical behaviour and modelling of scalar variance in turbulent premixed flames

The effects of body force/external pressure gradient on the statistical behaviours of the reaction progress variable variance and the terms of its transport equation have been investigated for different turbulence intensities using DNS data of statistically planar flames. Since the extent of flame wrinkling increases with the strengthening of body force promoting unstable stratification, the scalar variance has been found to decrease under strong body force promoting stability. This trend is particularly strong for low turbulence intensities where the probability density function of the reaction progress variable cannot be approximated by a bimodal distribution. Therefore, an algebraic relation for the reaction progress variable variance, derived based on a presumed bimodal probability density function of reaction progress variable, cannot be used for general flow conditions. The contributions of chemical reaction and scalar dissipation rates in the scalar variance transport equation remain leading order source and sink, respectively for all cases irrespective of the strength and direction of the body force. The counter-gradient type transport is found to weaken with increasing body force magnitude when the body force is directed from the heavier unburned gas to the lighter burned gas side of the flame brush, and vice versa. Although a scalar dissipation rate-based reaction rate closure can be utilised to model the reaction rate contribution to the scalar variance transport accurately, the dissipation rate contribution due to the gradient of the Favre-averaged reaction progress variable cannot be ignored and it plays a key role for large magnitudes of body force promoting stable stratification. An algebraic closure of the scalar dissipation rate, originally proposed for high Damköhler number combustion, has been modified for the thin reaction zones regime combustion by incorporating the effects of Froude number. This model has been shown to predict the scalar dissipation rate accurately for all cases considered here.

Monte Carlo Simulation of Stochastic Differential Equation to Study Information Geometry.

Information Geometry is a useful tool to study and compare the solutions of a Stochastic Differential Equations (SDEs) for non-equilibrium systems. As an alternative method to solving the Fokker–Planck equation, we propose a new method to calculate time-dependent probability density functions (PDFs) and to study Information Geometry using Monte Carlo (MC) simulation of SDEs. Specifically, we develop a new MC SDE method to overcome the challenges in calculating a time-dependent PDF and information geometric diagnostics and to speed up simulations by utilizing GPU computing. Using MC SDE simulations, we reproduce Information Geometric scaling relations found from the Fokker–Planck method for the case of a stochastic process with linear and cubic damping terms. We showcase the advantage of MC SDE simulation over FPE solvers by calculating unequal time joint PDFs. For the linear process with a linear damping force, joint PDF is found to be a Gaussian. In contrast, for the cubic process with a cubic damping force, joint PDF exhibits a bimodal structure, even in a stationary state. This suggests a finite memory time induced by a nonlinear force. Furthermore, several power-law scalings in the characteristics of bimodal PDFs are identified and investigated.

Soot particle size distribution reconstruction in a turbulent sooting flame with the split-based extended quadrature method of moments

The Method of Moments (MOM) has largely been applied to investigate sooting laminar and turbulent flames. However, the classical MOM is not able to characterize a continuous particle size distribution (PSD). Without access to information on the PSD, it is difficult to accurately take into account particle oxidation, which is crucial for shrinking and eliminating soot particles. Recently, the Split-based Extended Quadrature Method of Moments (S-EQMOM) has been proposed as a numerically robust alternative to overcome this issue [Salenbauch et al., “A numerically robust method of moments with number density function reconstruction and its application to soot formation, growth, and oxidation,” J. Aerosol Sci. 128, 34–49 (2019)]. The main advantage is that a continuous particle number density function can be reconstructed by superimposing kernel density functions (KDFs). Moreover, the S-EQMOM primary nodes are determined individually for each KDF, improving the moment realizability. In this work, the S-EQMOM is combined with a large eddy simulation/presumed-probability density function flamelet/progress variable approach for predicting soot formation in the Delft Adelaide Flame III. The target flame features low/high sooting propensity/intermittency and comprehensive flow/scalar/soot data are available for model validation. Simulation results are compared with the experimental data for both the gas phase and the particulate phase. Good quantitative agreement has been obtained especially in terms of the soot volume fraction. The reconstructed PSD reveals predominantly unimodal/bimodal distributions in the first/downstream portion of this flame with particle diameters smaller than 100 nm. By investigating the instantaneous and statistical sooting behavior at the flame tip, it has been found that the experimentally observed soot intermittency is linked to mixture fraction fluctuations around its stoichiometric value that exhibits a bimodal probability density function.

A kinetic approach to studying low-frequency molecular fluctuations in a one-dimensional shock

Low-frequency molecular fluctuations in the translational nonequilibrium zone of one-dimensional strong shock waves are characterized for the first time in a kinetic collisional framework in the Mach number range 2≤M≤10. Our analysis draws upon the well-known bimodal nature of the probability density function (PDF) of gas particles in the shock, as opposed to their Maxwellian distribution in the freestream, the latter exhibiting two orders of magnitude higher dominant frequencies than the former. Inside the (finite-thickness) shock region, the strong correlation between perturbations in the bimodal PDF and fluctuations in the normal stress suggests introducing a novel two-bin model to describe the reduced-order dynamics of a large number of collision interactions of gas particles. Our model correctly predicts two orders of magnitude differences in fluctuation frequencies in the shock vs those in the freestream and is consistent with the small-amplitude fluctuations obtained from the highly resolved direct simulation Monte Carlo computations of the same configuration. The variation of low-frequency fluctuations with changes in the conditions upstream of the shock revealed that these fluctuations can be described by a Strouhal number, based on the bulk velocity upstream of the shock and the shock-thickness based on the maximum density gradient inside the shock, that remains practically independent of Mach number in the range examined.

Analytical prediction of low-frequency fluctuations inside a one-dimensional shock

Linear instability of high-speed boundary layers is routinely examined assuming quiescent edge conditions, without reference to the internal structure of shocks or to instabilities potentially generated in them. Our recent work has shown that the kinetically modeled internal nonequilibrium zone of straight shocks away from solid boundaries exhibits low-frequency molecular fluctuations. The presence of the dominant low frequencies observed using the direct simulation Monte Carlo (DSMC) method has been explained as a consequence of the well-known bimodal probability density function (PDF) of the energy of particles inside a shock. Here, PDFs of particle energies are derived in the upstream and downstream equilibrium regions, as well as inside shocks, and it is shown for the first time that they have the form of the noncentral Chi-squared (NCCS) distributions. A linear correlation is proposed to relate the change in the shape of the analytical PDFs at a specified upstream number density and temperature as a function of Mach number, within the range \(3 \le M \le 10\), with the DSMC-derived average characteristic low-frequency of shocks, as computed in our earlier work. At a given Mach number \(M=7.2\) and upstream number density \(n_1=10^{22}\,\hbox {m}^{-3}\), it is shown that the variation in DSMC-derived low frequencies is correlated with the change in most-probable-speed inside shocks at the location of maximum bulk velocity gradient for upstream translational temperature in the range \(\sim 90 \le T_{tr,1}/(K) \le 1420\). Using the proposed linear functions, average low frequencies are estimated within the examined ranges of Mach number and input temperature and a semi-empirical relationship is derived to predict low-frequency oscillations in shocks. Our model can be used to provide realistic physics-based boundary conditions in receptivity and linear stability analysis studies of laminar-turbulent transition in high-speed flows.

Extreme estimation of wind pressure with unimodal and bimodal probability density function characteristics: A maximum entropy model based on fractional moments

The complicated probability density function (PDF) characteristics of wind pressure, including its highly skewed, highly leptokurtic, and bimodal characteristics induced by sophisticated architecture, call for improved extreme-estimation models. Most existing methods are based on a unimodal PDF's underlying assumption and fail in highly non-Gaussian cases. Owing to the complete probabilistic information used in cumulative distribution function (CDF)-mapping methods, such methods have the potential to deal with complicated PDF-type wind pressures. To yield better curve-fitting of the parent distribution, which is the critical step of CDF-mapping, a maximum entropy model based on fractional moments is combined with CDF-mapping in this work. By using the Legendre-Gauss quadrature rule, the computational efficiency of fitting parent distribution is improved. The model's performance is benchmarked against typical long-term wind pressure data obtained from wind tunnel tests conducted at a long-span airport terminal model. By considering the captured probability properties, four typical taps are selected to provide empirical wind-pressure extreme in the 57% fractile for detailed model assessment. The confidence intervals of the estimated results and errors for all taps are also calculated. Compared with existing polynomial-fitting- and CDF-mapping-based translation methods, the extreme estimated by the proposed method are shown to be more robust and stable. • A novel approach was proposed for extreme estimation of wind pressures with unimodal and bimodal cases. • The Legendre-Gauss quadrature rule was adapted for improving the model computational efficiency. • The model performance was verified with long-term measured data from the wind tunnel test of a long-span building. • A detailed comparison against existing translation-based methods was conducted.

Stochastic and spectra contents of detonation initiated by compressible turbulent thermodynamic fluctuations

A canonical system that contains the basic elements of detonation initiated by compressible turbulence thermodynamic fluctuations was proposed by Towery et al. [“Detonation initiation by compressible turbulence thermodynamic fluctuations,” Combust. Flame 213, 172–183 (2020)] and successfully tested using direct numerical simulation (DNS). In the present study, the DNS dataset is used to assess the applicability of a few compressible turbulence theories to the problem of detonation and to investigate some statistical aspects of the problem and the spectra of the turbulence kinetic energy (TKE) and reactive scalar fields within the context of detonating and deflagrating compressible flows. It has been found that scaling of the dilatational and solenoidal components of compressible turbulence as reported in the literature does not apply to the case of detonation or deflagration. Also, the detonating cases were found to have significantly lower Karlovitz numbers compared to the nondetonating ones and to conform to the Bray–Moss–Libby premixed combustion (bimodal probability density function) model for intense turbulence and large Damkhöler number. The nondetonating cases show distributed pdfs. The stronger combustion in the detonating cases shows effects on the spectra of the TKE and reactive scalars, particularly when the wavenumber is scaled with the Kolmogorov length and the Schmidt number.

Zonal flow reversals in two-dimensional Rayleigh-Bénard convection

We analyse the nonlinear dynamics of the large scale flow in Rayleigh-B\'enard convection in a two-dimensional, rectangular geometry of aspect ratio $\Gamma$. We impose periodic and free-slip boundary conditions in the streamwise and spanwise directions, respectively. As Rayleigh number Ra increases, a large scale zonal flow dominates the dynamics of a moderate Prandtl number fluid. At high Ra, in the turbulent regime, transitions are seen in the probability density function (PDF) of the largest scale mode. For $\Gamma = 2$, the PDF first transitions from a Gaussian to a trimodal behaviour, signifying the emergence of reversals of the zonal flow where the flow fluctuates between three distinct turbulent states: two states in which the zonal flow travels in opposite directions and one state with no zonal mean flow. Further increase in Ra leads to a transition from a trimodal to a unimodal PDF which demonstrates the disappearance of the zonal flow reversals. On the other hand, for $\Gamma = 1$ the zonal flow reversals are characterised by a bimodal PDF of the largest scale mode, where the flow fluctuates only between two distinct turbulent states with zonal flow travelling in opposite directions.

Numerical study of bubbly flow in a swirl atomizer

In this work, we extend our previous research on swirl nozzles by introducing bubbles at the nozzle inlet. A large-scale hollow cone pressure-swirl atomizer is studied using scale-resolving simulations. The present flow conditions target a Reynolds number range of 600 ≤ Re ≤ 910 and gas-to-total volumetric flow rate ratios between 0.07 ≤ β ≤ 0.33 with β = 0 as an experimental and computational reference. The computational setup has relevance to high-viscosity bio-fuel injection processes. The flow rate ratio and bubble diameter sweeps are carried out to study their effect on the inner-nozzle flow and the liquid film characteristics outside the nozzle. The present flow system is shown to pose highly versatile physics, including bubble coalescence, bubble–vortex interaction, and faster liquid film destabilization relative to β = 0 case. The main results are as follows: (1) β is found to have a significant effect on the bimodal bubble volume probability density function inside the swirl chamber. In addition, the total resolved interfacial area of the near-orifice liquid film increases with β. (2) At the representative value of β = 0.2, the exact bubble size at the inlet is observed to have only a minor effect on the swirl chamber flow and liquid film characteristics. (3) The bubble-free (β = 0) and bubbly (β &amp;gt; 0) flows differ in terms of effective gas core diameter, core intermittency features, and spray uniformity. The quantitative analysis implies that bubble inclusion at the inlet affects the global liquid film characteristics with relevance to atomization.

Seizure activity classification based on bimodal Gaussian modeling of the gamma and theta band IMFs of EEG signals

In this manuscript, EEG signals of seizure and non-seizure activities have been discussed and classified into five groups on the basis of seizure onset, seizure action and brain signal recording location. EEG signals consisting of gamma-band (40–80 Hz) and theta-band (4–8 Hz) oscillations have been captured for performing empirical mode decomposition (EMD). Dominant intrinsic mode functions (IMFs) have been selected from the consequences of EMD and a statistical model is employed upon the IMFs to summarize the information on those. Bimodal Gaussian statistical model has been found most effective to prepare feature set taking the modeling parameters of its probability density function (PDF). Plotting together bimodal Gaussian PDF and empirical PDF for pictorial scrutiny; cumulative distribution function (CDF) in probability–probability (p–p) plot and goodness of fit K–S test result justified the effectiveness of proposed bimodal Gaussian statistical model. Hence, aforementioned statistical modeling parameters have been sent to numerous classifiers and rationalization of goodness of features has been shown through inter-class separability and intra-class compactness parameters. Extensive varieties of simulations are performed using a well-established dataset. The suggested strategy reveals the capability of making higher values of sensitivity, specificity and accuracy compared to that made by some cutting-edge methods utilizing the same EEG dataset.

Anomalous diffusion and nonergodicity for heterogeneous diffusion processes with fractional Gaussian noise.

Heterogeneous diffusion processes (HDPs) feature a space-dependent diffusivity of the form D(x)=D_{0}|x|^{α}. Such processes yield anomalous diffusion and weak ergodicity breaking, the asymptotic disparity between ensemble and time averaged observables, such as the mean-squared displacement. Fractional Brownian motion (FBM) with its long-range correlated yet Gaussian increments gives rise to anomalous and ergodic diffusion. Here, we study a combined model of HDPs and FBM to describe the particle dynamics in complex systems with position-dependent diffusivity driven by fractional Gaussian noise. This type of motion is, inter alia, relevant for tracer-particle diffusion in biological cells or heterogeneous complex fluids. We show that the long-time scaling behavior predicted theoretically and by simulations for the ensemble- and time-averaged mean-squared displacements couple the scaling exponents α of HDPs and the Hurst exponent H of FBM in a characteristic way. Our analysis of the simulated data in terms of the rescaled variable y∼|x|^{1/(2/(2-α))}/t^{H} coupling particle position x and time t yields a simple, Gaussian probability density function (PDF), P_{HDP-FBM}(y)=e^{-y^{2}}/sqrt[π]. Its universal shape agrees well with theoretical predictions for both uni- and bimodal PDF distributions.

Correlated power time series of individual wind turbines: A data driven model approach

Wind farms can be regarded as complex systems that are, on the one hand, coupled to the nonlinear, stochastic characteristics of weather and, on the other hand, strongly influenced by supervisory control mechanisms. One crucial problem in this context today is the predictability of wind energy as an intermittent renewable resource with additional non-stationary nature. In this context, we analyze the power time series measured in an offshore wind farm for a total period of one year with a time resolution of 10 min. Applying detrended fluctuation analysis, we characterize the autocorrelation of power time series and find a Hurst exponent in the persistent regime with crossover behavior. To enrich the modeling perspective of complex large wind energy systems, we develop a stochastic reduced-form model of power time series. The observed transitions between two dominating power generation phases are reflected by a bistable deterministic component, while correlated stochastic fluctuations account for the identified persistence. The model succeeds to qualitatively reproduce several empirical characteristics such as the autocorrelation function and the bimodal probability density function.

Thermal protection of a hypersonic vehicle by modulating stagnation-point heat flux

The heat flux at the stagnation-point is usually used to evaluate the heat transfer efficiency between high-temperature gas and a hypersonic flying vehicle. Controlling the stagnation-point heat flux is of critical important for the design of a thermal protection system for a hypersonic flying vehicle. In this work, we demonstrate that the nonequilibrium molecular dynamic (NEMD) method can well describe the shock wave and the stagnation-point heat flux. In particular, our results show that the heat flux is more sensitive to the change of the peak amplitude of the hypersonic flow in the bimodal velocity probability density function (PDF) at the stagnation point. Decreasing the peak amplitude of the hypersonic flow in the PDF can effectively reduce the stagnation-point heat flux. Three types of the outer shapes for the flying objects are employed to compare the heat flux at the stagnation-point. The NEMD simulation results confirmed that the heat flux can be decreased 16.8% for the spherical cavity outer shape compared with the flat shape in the leading edge.

Experimental study of hydrodynamics and bubble size distribution of an external loop airlift reactor

AbstractThe most important factors influencing the performance of an external loop airlift reactor (ELALR) are local hydrodynamics and bubble size distribution (BSD). In this work, we studied the hydrodynamic characters and BSD in a laboratory scale ELALR. The velocity field of the riser was captured using the particle image velocimetry (PIV) technique and the bubble size distribution (BSD) was determined by digital image analysis (DIA). The results of the PIV measurement show that the flow has an unsteady structure, and the wake oscillation phenomena of the bubble and the vortex shedding are periodically asymmetrical. The gas holdup calculated by DIA correlates with the experimental data when the superficial gas velocity is &lt; 0.02 m · s−1. With an increase in the superficial gas velocity, the probability density function (PDF) curve becomes flatter, changing from the unimodal to the bimodal. For the bimodal PDF curve, the peak of the small bubble is ∼1.5 mm, while the peak of lager bubble shifts from 4.5 to 5.5 mm with the increase in the superficial gas velocity. This work helps to provide insight into the mixing performance of the ELALR.

Information geometry in a reduced model of self-organised shear flows without the uniform coloured noise approximation

We investigate information geometry in a toy model of self-organised shear flows, where a bimodal PDF of x with two peaks signifying the formation of mean shear gradients is induced by a finite memory time of a stochastic forcing f . We calculate time-dependent probability density functions (PDFs) for different values of the correlation time and amplitude D of the stochastic forcing, and identify the parameter space for unimodal and bimodal stationary PDFs. By comparing results with those obtained under the uniform coloured noise approximation (UCNA) in Jacquet et al (2018 Entropy 20 613), we find that UCNA tends to favor the formation of a bimodal PDF of x for given parameter values and D. We map out attractor structure associated with unimodal and bimodal PDFs of x by measuring the total information length against the location x0 of a narrow initial PDF of x. Here represents the total number of statistically different states that a system passes through in time. We examine the validity of the UCNA from the perspective of information change and show how to fine-tune an initial joint PDF of x and f to achieve a better agreement with UCNA results.

Bimodality of hemispheric winter atmospheric variability via average flow tendencies and kernel EOFs

The topic of the existence of planetary winter circulation regimes has gone through a long debate. This article contributes to this debate by investigating nonlinearity in a 3-level quasi-geostrophic model and the Japanese JRA-55 reanalysis. The method uses averaged flow tendencies and kernel principal component (PC) analysis. Within two-dimensional (2D) kernel PCs the model reveals two fixed (or stationary) points. The probability density function (PDF) within this space is strongly bimodal where the modes match the regions of low tendencies in consistency with low-order conceptual models. The circulation regimes represent respectively zonal and blocked flows. Application to daily winter northern hemisphere sea level pressure and 500-hPa geopotential height yields strong bimodal PDFs. The modes represent respectively polar highs and lows with signatures of North Atlantic Oscillation. A clear climate change signal is observed showing a clear reduction (increase) of occurrence probability of polar high (low), translating into an increase of probability of zonal flow. Relation of the climate change signal to the polar amplification hypothesis is discussed.