Monte Carlo Probability Density Function Research Articles

A new optimisation framework based on Monte Carlo embedded hybrid variant mean–variance mapping considering uncertainties

This study proposes a new optimisation framework based on Monte Carlo embedded hybrid variant mean–variance mapping (MVMO-SH) optimisation​ for planning Photovoltaic Distributed Generation (PVDG) in the urban Radial Distribution Network (RDN). The Active Power Loss (APL) index was calculated considering the risk of uncertain photovoltaic generation and urban load distributions. The Monte Carlo Probability Density Function method was initially used to manage uncertainties. The Monte Carlo-embedded MVMO-SH was then used to optimise PVDG in the urban RDN. Simulations were run for several scenarios in three load cases based on 288 segments: residential, commercial, and industrial urban loads. The MVMO-SH had the lowest APL index compared to genetic algorithm and particle swarm optimisation when the probabilistic power flow with PVDG was optimised under uncertainty. The APL indexes with three PVDG installations in the 33-bus RDN for residential, commercial, and industrial urban load models were 0.4094, 0.4811, and 0.4655, respectively. In the 69-bus RDN, the APL indexes with three PVDG installations for residential, commercial, and industrial urban load models were 0.3403, 0.3570, and 0.3504, respectively. For all load models examined, there was a significant reduction in the APL index for the case of three PVDGs compared to the system without PVDG. The findings showed that uncertainty significantly impacted the optimal location and size of PVDG in the RDN.

Cancer and non-cancer risk associated with PM10-bound metals in subways

Non-cancer and cancer health risks to humans associated with respirable particulate matter ≤10 µm (PM10) in an indoor microenvironment such as a subway cabin are currently a public concern. In this study, detailed investigations of human health risks due to PM10-bound metals in a subway cabin were conducted for the first time. Cancer risks (CRs) were estimated for inhalation exposure (CRinh) using a Monte Carlo probability density function and were compared with incremental lifetime CRs (ILCRs). Moreover, the percentage contributions of each metal to the risk levels were calculated to identify the elements potentially responsible for human health risks. The significant (>1) for non-CRs levels as HI (hazard index) was estimated for children and adults for all types of exposure (inhalation, ingestion, and dermal). Pb, Cr, and Ni were recognized as the foremost contributors to the HQ (hazard quotient) levels for all types of exposure. For subway commuters, the CRinh and ILCR levels for adults were marginally higher than the satisfactory maximum point of confinement of the lifetime carcinogenic risk level (1 × 10−5) where as CRinh for children was within the acceptable limit (1 × 10−6–1 × 10−5). Cr was identified as the predominant carcinogenic element, with 91% contribution to the total CR level in the subway cabin on the Seoul Metropolitan Subway.

Estimation of the probability density function of transmission loss in the ocean using area statistics

Predictions of acoustic transmission loss (TL) in the ocean and its uncertainty are useful for a variety of ocean and naval engineering applications. Previously, an ad-hoc technique, area statistics (AS), was developed as a computationally efficient method to estimate the probability density function (PDF) of TL in any uncertain ocean environment from a single range-depth TL-field calculation in that environment. Here, the performance of AS is reported for ten ocean environments having uncertain sound speed profile, bathymetry, and seabed properties. In each environment, a parabolic-equation-based prediction of TL from a point source was generated as a function of range and depth at frequencies of 100, 200, and 300 Hz using the most probable values of the uncertain environmental parameters. From these calculations, AS was used to estimate the PDF of TL at more than 3000 locations within the ten environments. These estimated PDFs are then quantitatively compared, via an L1-error norm, to PDFs generated from traditional Monte Carlo techniques for the same locations and frequencies. Current results show that the L1-error of the AS PDF is less than 0.5, indicating 75% or greater overlap between the AS and Monte Carlo PDFs, at 90% of the test locations. [Sponsored by ONR.]

Specific volume coupling and convergence properties in hybrid particle/finite volume algorithms for turbulent reactive flows

We investigate the coupling between the two components of a Large Eddy Simulation/Probability Density Function (LES/PDF) algorithm for the simulation of turbulent reacting flows. In such an algorithm, the Large Eddy Simulation (LES) component provides a solution to the hydrodynamic equations, whereas the Lagrangian Monte Carlo Probability Density Function (PDF) component solves for the PDF of chemical compositions. Special attention is paid to the transfer of specific volume information from the PDF to the LES code: the specific volume field contains probabilistic noise due to the nature of the Monte Carlo PDF solution, and thus the use of the specific volume field in the LES pressure solver needs careful treatment. Using a test flow based on the Sandia/Sydney Bluff Body Flame, we determine the optimal strategy for specific volume feedback. Then, the overall second-order convergence of the entire LES/PDF procedure is verified using a simple vortex ring test case, with special attention being given to bias errors due to the number of particles per LES Finite Volume (FV) cell.

Charged-particle thermonuclear reaction rates: II. Tables and graphs of reaction rates and probability density functions

Numerical values of charged-particle thermonuclear reaction rates for nuclei in the A = 14 to 40 region are tabulated. The results are obtained using a method, based on Monte Carlo techniques, that has been described in the preceding paper of this issue (Paper I). We present a low rate, median rate and high rate which correspond to the 0.16, 0.50 and 0.84 quantiles, respectively, of the cumulative reaction rate distribution. The meaning of these quantities is in general different from the commonly reported, but statistically meaningless expressions, “lower limit”, “nominal value” and “upper limit” of the total reaction rate. In addition, we approximate the Monte Carlo probability density function of the total reaction rate by a lognormal distribution and tabulate the lognormal parameters μ and σ at each temperature. We also provide a quantitative measure (Anderson–Darling test statistic) for the reliability of the lognormal approximation. The user can implement the approximate lognormal reaction rate probability density functions directly in a stellar model code for studies of stellar energy generation and nucleosynthesis. For each reaction, the Monte Carlo reaction rate probability density functions, together with their lognormal approximations, are displayed graphically for selected temperatures in order to provide a visual impression. Our new reaction rates are appropriate for bare nuclei in the laboratory. The nuclear physics input used to derive our reaction rates is presented in the subsequent paper of this issue (Paper III). In the fourth paper of this issue (Paper IV) we compare our new reaction rates to previous results.

An efficient algorithm for scalar PDF modelling in incompressible turbulent flow; numerical analysis with evaluation of IEM and IECM micro-mixing models

In this paper, we investigate the application of probability density function (PDF) Monte Carlo methods to scalar release from small sources in a turbulent flow spanning a large physical domain. This is a typical situation encountered when modeling the dispersion of a gaseous substance in the atmosphere. Monte Carlo PDF methods have recently been applied to atmospheric modeling responding to the need for predicting the higher statistics and the concentration PDF generated by the continuous release of reactive and non-reactive substances. In this work we introduce some optimized numerical techniques based on the paradigm that the main field of interest is the scalar field and not the fluid dynamic field; the scalar are considered dynamically passive and the statistical characteristics of the turbulence velocity field are assumed known. These techniques are a block-structured grid coupled with a particle splitting/erasing algorithm and a localized time stepping. The proposed technique is different from others presented before since the particle splitting and erasing is done in a more straightforward and consistent manner. This method has been applied to the study of scalar dispersion from localized line sources in a canopy generated boundary layer. The line source has been treated as a point source in a two-dimensional space but the extension to three dimensions is straightforward. Our framework allows for an evaluation of the effects induced by different levels of discretization in the velocity space of the involved micro-mixing model, starting from the interaction by the exchange with the mean (IEM) toward the more physically consistent interaction by exchange with the conditional mean (IECM). Therefore, aside from the algorithm description and a complete numerical analysis of the code, a comparison between the IEM and IECM micro-mixing models has been investigated.

Testing multiple mapping conditioning mixing for Monte Carlo probability density function simulations

Mitarai et al. [Phys. Fluids 17, 047101 (2005)] compared turbulent combustion models against homogeneous direct numerical simulations with extinction/recognition phenomena. The recently suggested multiple mapping conditioning (MMC) was not considered and is simulated here for the same case with favorable results. Implementation issues crucial for successful MMC simulations are also discussed.

Testing of mixing models for Monte Carlo probability density function simulations

Testing of mixing models widely used for Monte Carlo probability density function simulations of turbulent diffusion flames is performed using the data obtained from direct numerical simulations (DNS) that are specifically designed for the study of local flame extinction and reignition. In particular, the interaction by exchange with the mean (IEM) [J. Villermaux and J. C. Devillon, “Représentation de la coalescence et de la redispersion des domaines de ségrégation dans un fluide per modéle d’interaction phénoménologique,” in Proceedings of the Second International Symposia on Chemical Reaction Engineering (ISCRE, Netherlands, 1972), p. B1], the modified Curl [J. Janicka, W. Kolbe, and W. Kollmann, J. Non-Equilib. Thermodyn. 4, 47 (1979)], and the Euclidean minimum spanning tree (EMST) [S. Subramaniam and S. B. Pope, Combust. Flame 115, 487 (1998)] mixing models are tested. The tests are designed to examine the mixing model performance when implemented in both Reynolds-averaged simulations and large-eddy simulations. The exact value of the mixing frequency is taken from the DNS, so that the model performance can be more accurately determined. It is found that, in general, the EMST mixing model yields much better results than the IEM and the modified Curl mixing models.

Finite-rate chemistry effects in turbulent opposed flows: comparison of Raman/Rayleigh measurements and Monte Carlo PDF simulations

This study reports results from experimental and numerical investigations of a partially premixed turbulent opposed methane/air jet flame. Experimentally determined properties of the scalar and the flow field are compared to the results from a Monte Carlo simulation. One-dimensional spatially resolved Raman/Rayleigh scattering serves to quantify the mean species concentrations and temperature, whereas laser Doppler velocimetry is used to measure axial and radial velocity components. The simulation is simplified by using a one-dimensional formulation. It includes a Reynolds-stress turbulence model and a Monte Carlo simulation of the joint scalar probability density function (PDF). A non-uniform Monte Carlo particle distribution is used to minimize stochastic errors. The flame is operated close to extinction with strong interactions between turbulence and chemistry. Comparisons between experimental and numerical results reveal a good agreement of mixture fraction profiles along the centreline. However, species scatter plots and mixture fraction PDFs show discrepancies between experiment and simulation. Numerical simulations over-predict the extinction limits and therefore under-predict the intermittent nature of turbulence and mixing of the scalars.

ON THE IMPORTANCE OF CHEMISTRY / TURBULENCE INTERACTIONS IN SPRAY COMPUTATIONS

This article presents results from spray computations performed as a part of the National Combustion Code (NCC) development activity. The NCC is being developed with the aim of advancing the prediction tools used in the design of advanced technology combustors based on multidimensional computational methods. The solution procedure combines the novelty of the application of the scalar Monte Carlo probability density function (PDF) method to the modeling of turbulent spray flames with the ability to perform the computations on unstructured grids with parallel computing. A validation summary is provided for two different cases: one is a reacting spray without swirl, and the other is a similar spray but non-reacting. The comparisons involving both gas-phase and droplet velocities, droplet size distributions, and gas-phase temperatures show reasonable agreement with the available experimental data. The comparisons involve both the results obtained from the use of the Monte Carlo PDF method as well as those obtained from the conventional computational fluid dynamics (CFD) solution. The detailed comparisons clearly highlight the importance of chemistry / turbulence interactions in the modeling of reacting sprays. The results from the PDF and non-PDF methods were found to be markedly different, and the PDF solution is closer to the reported experimental data. The PDF computations predict that most of the combustion occurs in a predominantly diffusion-flame environment and the rest in a predominantly premixed-flame environment. However, the non-PDF solution predicts incorrectly that the combustion occurs in a predominantly vaporization-controlled regime. The Monte Carlo temperature distribution shows that the functional form of the PDF for the temperature fluctuations varies substantially from point to point. The results bring to the fore some of the deficiencies associated with the use of assumed-shape PDF methods in spray computations. Finally, we end the article by demonstrating the computational viability of the present solution procedure for its use in 3-D combustor calculations by summarizing the results of parallel performance for a 3-D test case with periodic boundary conditions. For the 3-D case, the parallel performance of all the three solvers (CFD, PDF, and spray) has been found to be good when the computations were performed on a 24-processor SGI Origin workstation.

Numerical Simulation of Soot Formation in a Turbulent Flame with a Monte-Carlo PDF Approach and Detailed Chemistry

The paper presents an original work in which a hybrid turbulent combustion model, based upon a stochastic evaluation of the Joint Scalar Probability Density Function (PDF), is used in conjunction with a skeletal soot model and a detailed kinetic mechanism for fuel oxidation. First, the Probabilistic Eulerian Lagrangian turbulent combustion model, which is theoretically able to describe chemical reactions occurring in a turbulent flow for a wide range of Damkohler numbers, is presented and justified. Then, the chemistry model which accounts for soot formation and oxidation is exposed and validated in rich ethylene premixed laminar flames. This modelling approach coupling turbulence and chemistry is eventually applied to predict soot levels in a turbulent jet diffusion flame of ethylene burning in still air. Results are in good agreement with experimental data and the peak value of soot volume fraction on the centreline is fairly well described even though an accurate radiative heat transfer model is still necessary to be more predictive on mean temperature levels.

Modeling of turbulent mixing in opposed jet configuration: One-dimensional Monte Carlo probability density function simulation

A one-dimensional numerical model of turbulent mixing in an opposed jet configuration is presented with Monte Carlo simulation of joint scalar probability density function (PDF). The formulation is based on axisymmetric flowfield along its centerline. Due to the one-dimensionality, one can afford a large number of stochastic particles in the PDF Monte Carlo simulation such that statistical uncertainty is reduced to a negligible level. With this accurate numerical representation, we investigate the importance of specification of mixing time (or frequency) in two commonly used mixing models, namely, the modified Curl's mixing model and the interaction by exchange with the mean (IEM) model (or the linear mean square estimation model). The effect of mixing time on the predicted results is investigated by varying the ratio between the turbulence turnover time and the turbulent mixing time. For the experimental data of Sardi et al. (1998), excellent agreement between predictions and the measurements is achieved by using a constant time ratio. A similarly good agreement is also seen in the PDF distributions. An analysis of the computed results reveals a strong correlation between the PDF shape and the peak level of the variance of the mixture fraction. However, it has not been possible for the numerical model to match simultaneously both the variances and the PDF shapes for the experimental results of Mastorakos et al. (1994). Further experiments on the detailed mixing properties in the opposed jet configuration are warranted to establish a comprehensive set of data for validation of numerical models.

SCALAR MONTE CARLO PDF COMPUTATIONS OF SPRAY FLAMES ON UNSTRUCTURED GRIDS WITH PARALLEL COMPUTING

The article describes the extension of the scalar Monte Carlo probability density function (PDF) approach to the modeling of spray flames for application with unstructured grids and parallel computing. The solver is designed to be massively parallel and accomodates the use of an unstructured mesh with mixed elements comprised of triangular, quadrilateral, and or tetrahedral type. The ability to perform the computations on unstructured meshes allows representation of complexgeometries with relative ease. Several numerical techniques are outlined for overcoming some of the high computer time and storage limitations associated with the Monte Carlo simulation of practical combustor flows. The application of this method to a confined swirl-stabilized spray flame shows reasonable agreement with the available drop velocity measurements. The parallel performance of both the PDF and Computational Fluid Dynamics (CFD) computations has been found to be excellent but the results are mixed for the spray module, showing reasonable performance on massively parallel computers such as the Cray T3D but poor performance on workstation clusters. In order to improve the parallel performance of the spray module, two different domain decomposition strategies were developed, and the results from both strategies are summarized.

Coupled Lagrangian Monte Carlo PDF–CFD Computation of Gas Turbine Combustor Flowfields With Finite-Rate Chemistry

A coupled Lagrangian Monte Carlo Probability Density Function (PDF)-Eulerian Computational Fluid Dynamics (CFD) technique is presented for calculating steady three-dimensional turbulent reacting flow in a gas turbine combustor. PDF transport methods model turbulence-combustion interactions more accurately than conventional turbulence models with an assumed shape PDF. The PDF transport equation was solved using a Lagrangian particle tracking Monte Carlo (MC) method. The PDF modeled was over composition only. This MC module has been coupled with CONCERT, which is a fully elliptic three-dimensional body-fitted CFD code based on pressure correction techniques. In an earlier paper (Tolpadi et al., 1995), this computational approach was described, but only fast chemistry calculations were presented in a typical aircraft engine combustor. In the present paper, reduced chemistry schemes were incorporated into the MC module that enabled the modeling of finite rate effects in gas turbine flames and therefore the prediction of CO and NOx emissions. With the inclusion of these finite rate effects, the gas temperatures obtained were also more realistic. Initially, a two scalar scheme was implemented that allowed validation against Raman data taken in a recirculating bluff body stabilized CO/H2/N2-air flame. Good agreement of the temperature and major species were obtained. Next, finite rate computations were performed in a single annular aircraft engine combustor by incorporating a simple three scalar reduced chemistry scheme for Jet A fuel. This three scalar scheme was an extension of the two scalar scheme for CO/H2/N2 fuel. The solutions obtained using the present approach were compared with those obtained using the fast chemistry PDF transport approach (Tolpadi et al., 1995) as well as the presumed shape PDF method. The calculated exhaust gas temperature using the finite rate model showed the best agreement with measurements made by a thermocouple rake. In addition, the CO and NOx emission indices were also computed and compared with corresponding data.

APPLICATION OF SCALAR MONTE CARLO PROBABILITY DENSITY FUNCTION METHOD FOR TURBULENT SPRAY FLAMES

Abstract The objective of the present work is twofold: (1) extend the coupled Monte Carlo probability density function (PDF)/computational fluid dynamics (CFD) computations to the modeling of turbulent spray flames, and (2) extend the PDF/SPRAY/CFD module to parallel computing in order to facilitate large-scale combustor computations. In this approach, the mean gas phase velocity and turbulence fields are determined from a standard turbulence model, the joint composition of species and enthalpy from the solution of a modeled PDF transport equation, and a Lagrangian-based dilute spray model is used for the liquid-phase representation. The PDF transport equation is solved by a Monte Carlo method, and the mean gas phase velocity and turbulence fields together with the liquid phase equations are solved by existing state-of-the-art numerical representations. The application of the method to both open as well as confined axisymmetric swirl-stabilized spray flames shows good agreement with the measured data. Pre...

Monte Carlo PDF method for turbulent reacting flow in a jet-stirred reactor

A stochastic algorithm for the solution of the modelled scalar probability density function (PDF) transport equation for single-phase turbulent reacting flow is described. Cylindrical symmetry is assumed. The PDF is represented by ensembles of N representative values of the thermochemical variables in each cell of a non-uniform finite-difference grid and operations on these elements representing convection, diffusion, mixing and reaction are derived. A simplified model and solution algorithm which neglects the influence of turbulent fluctuations on mean reaction rates is also described. Both algorithms are applied to a selectivity problem in a real reactor.