Probability Density Function Method Research Articles

Impact of carbon pricing on the integrated green supply chain of the semiconductor industry

To better respond to the strategic goals of "carbon peaking" and "carbon neutrality,” businesses are collaborating with suppliers and customers to jointly promote green and environmentally friendly practices to increase economic benefits, enhance social reputation, and improve environmental competitiveness while achieving environmental performance. Currently, some theoretical achievements in the field of green supply chain integration have been made. Nevertheless, systematic retrospective research needs to be improved to help us fully understand the status of research in China and elsewhere. Therefore, this article analyzed the concept of green supply chain integration using eight years of data (from 2014 to 2021) from a small and medium-sized environmental company1 and divided the concept into four systems: the water cycle system (H2O), renewable energy system (CaCl2), carbon peak system (CO2), and carbon-neutral system (CO2). These four systems were analyzed with regard to internal green integration, green supplier integration, and green customer integration, and the Scheffé method was used to analyze the impact of these four systems on corporate ecological impact and financial and social sustainability. The study also verified that the intensity of sunlight on solar panels is significantly affected by environmental temperature (ET), carbon neutrality, and the electricity generated by solar panels. The impact of carbon pricing was also explored using the probability density function (PDF) method. The contribution of this research lies in the fact that it provides evidence that solar energy can help reduce the use of fossil fuels, reducing demand for the carbon quota market. This can help lower carbon prices, thereby encouraging more businesses and countries to participate in carbon neutrality and emission reduction actions. In addition, solar power systems can replace renewable energy sources, making the carbon quota market more diverse and stable. Carbon pricing options can be used to manage the risks and uncertainties related to carbon pricing policies, and policy-makers can use them to develop appropriate risk management strategies.

N-Euler mathematical modelling and numerical simulation of liquid–gas–solid flows

Three phase flows are omnipresent in industrial as well as natural configurations. Understanding the behaviour of solid particles interacting with liquid–air flows can be of great interest for both topics. In this paper we present a new liquid–gas–solid modelling technique based on the hybrid method combining liquid–gas two-fluid phase-average approach and probability density function (PDF) methods for solid dispersed phases. The PDF approach is based on single particle dynamic equation taking into account its interaction with more than one fluid and separate stochastic Lagrangian equations for each fluid phase velocity seen by the particles. This method does not assume the carrier phases typology. The solid particles can interact with bubbles, droplets, pockets and stratified flow through drag only (other interaction forces are supposed to be negligible for our cases). In this work, it is assumed that the total drag applied to the particle is a linear combination of the drags computed if the particles were interacting separately with each fluid phases. Finding the linear combination coefficient represent the main modelling step. Particle collisions and particle turbulent dispersion are taken into account in the model. Dispersed bubbles can undergo coalescence and fragmentation. The effect of all these sub-models is studied in the comparison with experimental data.The fluid and particle equations are first derived before introducing the appropriate coupling terms. Then, the closure terms for turbulence and multiphase drag are introduced. The model is implemented in neptune_cfd, a finite volume solver developed by EDF, CEA, IRSN and Framatome which allows for the numerical resolution of separate Eulerian equations (mass, momentum and energy) for n phases coupled by interfacial transfer terms. The model is then compared against experimental data obtained in a bubble column charged with particles. The results obtained with it are satisfactory and enable us to study more in details its hypothesis and to confirm comments made in the first section. Eventually, the model is tested on an industrial case which shows its efficiency.

Machine learning tabulation of thermochemistry for turbulent dimethyl ether (DME) flames

The present work applies a machine learning tabulation methodology for the thermochemistry of dimethyl ether (DME), a typical biodiesel fuel, to accelerate the real-time computation of a large-size DME mechanism in turbulent non-premixed combustion. This approach is known as the hybrid flamelet/random data and multiple multilayer perceptrons (HFRD-MMLP) method (Ding et al., 2021). The essence of the HFRD-MMLP method lies in the generation of training data using the HFRD approach, which enhances the capacity of generalisation for the reactive composition space encountered in practical turbulent combustion by using the random process to expand the training dataset from laminar flamelets. The MMLP artificial neural networks (ANNs) are then trained to predict different composition states, aiming to improve the accuracy of ANN predictions. To validate the effectiveness of the ANNs, they are initially tested on 1-D laminar flame simulations with varying strain rates. Subsequently, a test is conducted on Sandia non-premixed turbulent DME series flames D and F, with an increasing jet Reynolds Number. The results regarding species mole fractions and temperature show overall excellent agreement with the direct integration method. The HFRD-MMLP method achieves speed-up factors of over 16 and 10 for the reaction step, and total computational costs, respectively, in the LES-PDF simulation with the DME mechanism in this work, surpassing the speed-up factors of approximately 12–14, and below 5 for the reaction step, and total time costs, respectively, in previous works on GRI-1.2 mechanism for CH4 (Ding et al., 2021) and CH4/H2 (Ding et al., 2022) combustion. This indicates that the HRFD-MMLP method is highly effective for mechanisms of large size in reducing the computational cost by avoiding a significant number of highly stiff ordinary differential equations (ODEs) integration. This approach can be applied in real-time calculations of reaction source terms in methods including Direct Numerical Simulation (DNS), Probability Density Function (PDF) methods, unsteady flamelet, Conditional Moment Closure (CMC), Multiple Mapping Closure (MMC), Linear Eddy Model (LEM), Thickened Flame Model, the Partially Stirred Reactor (PaSR) method (as in OpenFOAM) and laminar flame computation.

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.

Stochastic modeling of turbulent mixing based on a hierarchical swapping of fluid parcels

AbstractTurbulent mixing is an omnipresent phenomenon that constantly affects our everyday life and plays an important role in a variety of industrial applications. The simulation of turbulent mixing poses great challenges, since the full resolution of all relevant length and time scales is associated with an immense computational effort. This limitation can be overcome by only resolving the large‐scale effects and completely model the sub‐grid scales. The development of an accurate sub‐grid mixing model is therefore a key challenge to capture all interactions in the sub‐grid scales. At this place, the hierarchical parcel‐swapping (HiPS) model formulated by A.R. Kerstein [J. Stat. Phys. 153, 142–161 (2013)] represents a computationally efficient and scale‐resolving turbulent mixing model. HiPS mimics the effects of turbulence on time‐evolving, diffusive scalar fields. In HiPS, the diffusive scalar fields or a state space is interpreted as a binary tree structure, which is an alternative approach compared to the most common mixing models. Every level of the tree represents a specific length and time scale, which is based on turbulence inertial range scaling. The state variables are only located at the base of the tree and are treated as fluid parcels. The effects of turbulent advection are represented by stochastic swaps of sub‐trees at rates determined by turbulent time scales associated with the sub‐trees. The mixing only takes places between adjacent fluid parcels and at rates consistent with the prevailing diffusion time scales. In this work, the HiPS model formulation for the simulation of passive scalar mixing is detailed first. Preliminary results for the mean square displacement, passive scalar probability density function (PDF) and scalar dissipation rate are given and reveal the strengths of the HiPS model considering the reduced order and computational efficiency. These model investigations are an important step of further HiPS advancements. The integrated auxiliary binary tree structure allows HiPS to satisfy a large number of criteria for a good mixing model. From this point of view, HiPS is an attractive candidate for modeling the mixing in transported PDF methods.

Micro-flow structure at regime transition from bubbling to turbulent fluidization in a fluidized bed

Gas-solid fluidized beds have found extensive utilization in frontline manufacturing, in particular as low-velocity beds. The fluidization status, the bubbling or turbulent flow regime and the transition in between, determine the system performance in practical applications. Though the convoluted hydrodynamics are quantitively evaluated through numerous data-processing methodologies, none of them alone can reflect all the critical information to identify the transition from the bubbling to the turbulent regime. Accordingly, this study was to exploit a coupling data processing methodology, in the combination of standard deviation, power spectrum density, probability density function, wavelet transform, and wavelet multiresolution method, to jointly explain the micro-flow structure at the regime transition from bubbling to turbulent fluidization. The transient differential pressure fluctuation was measured for the evaluation in a fluidized bed (0.267 m i.d. × 2.5 m height) with FCC catalysts (d¯p=65μm, ρp=1780kg/m3) at different superficial gas velocities (0.02–1.4 m/s). The results show that the onset of turbulent fluidization starts earlier in the top section of the bed than in the bottom section. The wavelet decomposition displays that the fluctuation of differential pressure mainly concentrates on the sub-signals with an intermediate frequency band. These sub-signals could be synthesized into three types of scales (micro-scale, meso-scale, and macro-scale), representing the multi-scale hydrodynamics in the fluidized bed. The micro-scale signal has the characteristic information of bubbling fluidization, and the characteristic information of turbulent fluidization is mainly represented by the meso-scale signal. This work provides a systematic comprehension of fluidization status assessment and serves as an impetus for more coupling analysis in this sector.

Simulation of turbulent premixed flames with machine learning - tabulated thermochemistry

The numerical integration of the differential equations describing chemical kinetics consumes the majority of computational time in combustion simulations that involve direct coupling of chemistry and flow, such as transported probability density function (PDF) methods, direct numerical simulation (DNS), conditional moment closure (CMC), unsteady flamelet, multiple mapping closure (MMC), thickened flame model, linear eddy model (LEM), partially stirred reactor (PaSR) as in OpenFOAM and laminar flame computation. This step can be accelerated by tabulation, and artificial neural networks (ANNs) have recently emerged as a powerful technique in this domain. To be applicable to a wide family of problems, an ANN tabulation approach must be based on data generated by an abstract process, rather than from the turbulent flame to be simulated. In the present work, the hybrid flamelet/random data and multiple multilayer perceptrons (HFRD-MMLP) method (Ding et al., Combust. Flame 231, 111493, 2021) for non-premixed flames is taken as a basis to develop a thermochemistry tabulation method for premixed flames. In the spirit of maintaining an essentially random data set that still originates in meaningful composition states, a set of one-dimensional premixed flame simulations is employed to generate data that are used as starting points for a random data generation process and subsequently discarded. The approach is applied to large eddy simulations (LES) of the Cambridge/Sandia swirl burner in configurations five and six, with the transported PDF method employed to provide closure for the filtered reaction source terms and the stochastic fields method used for numerical solution. Very good agreement in both major and minor species is observed between the LES-PDF simulations using direct integration of the reaction source term and the ones with the ANNs. Furthermore, the average time taken for reaction source term computations is reduced by fourteen times, while memory requirements constitute only 1.4 MB.

A DNS evaluation of three MMC-like mixing models for transported PDF modelling of turbulent nonpremixed flames

Transported probability density function (TPDF) methods are suitable for modelling turbulent reactive flows. One of the main challenges is to accurately model the molecular mixing terms. In TPDF mixing models, it is desired that the principle of localness is satisfied so that the molecular mixing is performed locally in both physical and composition spaces. The multiple mapping conditioning (MMC) mixing model can ensure mixing localness without violating other desired principles. The present study examines three MMC-like mixing models, including the original MMC (OMMC) mixing model, shadow-position mixing model (SPMM) and a conceptually simplified multiple mapping conditioning (SMMC) mixing model. Three direct numerical simulation (DNS) datasets modelling turbulent nonpremixed ethylene flames with increasing levels of extinction are used for model evaluation. The DNS datasets are also used to provide both initial conditions and inputs needed over the course of TPDF runs to remove the uncertainties caused by turbulence closure, allowing the study to focus on the molecular mixing model. The mixing model coefficients are specified analytically by reference to a canonical mean scalar gradient (MSG) flow in order to achieve specifiable dissipation rate and user-controllable localness. The results show that the MMC-like mixing models yield similar prediction of flame extinction and reignition if the coefficients are properly specified. The MMC-like mixing models can also be tuned to achieve a desired level of localness. For conditional statistics, the MMC-like mixing models can yield correct level of conditional variances. By changing localness, the MMC-like models can yield solutions resembling the interaction by exchange with the mean (IEM) or Euclidean minimum spanning tree (EMST) mixing model for extreme parameter choices. The models differ in their abilities to specify unconditional dissipation rates, at least in the flow considered, and in their ease of implementation.

Analysis of Regional Ambient Seismic Noise in the Chukchi Sea Area in the Arctic Based on OBS Data from the Ninth Chinese National Arctic Scientific Survey

Ambient noise plays a crucial role in influencing the observation quality at seismic stations. By studying the distribution patterns of ambient noise, we can gain initial insights into the noise conditions within a specific research area. This paper investigates the properties of ambient noise in different frequency bands under environmental settings in the Chukchi Sea region, utilizing data collected from ocean bottom seismometers (OBSs) deployed during the Ninth Chinese National Arctic Scientific Survey. The probability density function (PDF) method is used to reveal the distinctive features of ambient noise. In addition, by comparing the crowed number values of ambient noise in the Chukchi Sea area with the global new low-noise model (NLNM) and new high-noise model (NHNM), a more comprehensive understanding of the patterns, distribution characteristics, and sources of ambient noise in the Arctic Chukchi Sea area is gained. The study suggests that the overlying sea ice in the Arctic Chukchi Sea area can suppress the microseismic band ambient noise, and the overall level of ambient noise in the Chukchi Sea area lies between the land seismic ambient noise level and the ambient noise level in the middle- and low-latitude sea areas. Meanwhile, an abnormal power spectrum caused by different levels of natural earthquakes is observed. This study fills the gap by using seafloor seismic instruments to investigate ambient noise in the Chukchi Sea area.

Assessment of the wind energy potential and economic viability of selected sites along Nigeria’s coastal and offshore locations

Wind energy is a promising sector in the power generation industry because it is renewable and globally available. In this research work, the wind energy potential and the economic viability of using wind turbines to generate electricity in some selected sites along Nigeria’s coastline and offshore locations were evaluated. Using the statistical two-parameter Weibull probability density function method, wind speed data retrieved from an indigenous oceanography company and global information system (GIS) were analyzed for wind energy harvest. The energy output, unit cost of electricity generated by three commercially available wind turbine models (3 MW, 4 MW, and 6 MW), net present value (NPV), and payback period were evaluated. Levelized cost of electricity (LCOE) sensitivity to the discount rate, foundation cost, and turbine lifespan were also examined. The findings from the study showed that the offshore sites have four times greater wind power potential than the coastal sites. The offshore sites can be categorized as “class IIIb” wind sites, making the locations suitable for wind energy harvest. The techno-economic analysis showed that the net gains from investing in a 60-MW wind farm in the region can be as high as $62,000,000.00, while the project payback time can be as low as 5.74 years. Two of the offshore sites are recommended for the development of an offshore wind farm in the country because of their relatively low LCOE (0.04 $/kWh), higher NPV, and lower investment payback time. The Vesta-117 model wind turbine is the most suitable wind turbine system and recommended for use in the region because of its low cut-in speed (3 m/s). Sensitivity analysis showed that the LCOE of offshore-01 was reduced by 31% when the lifespan of the V117 turbine was increased from 20 to 25 years. The results also showed that reductions in the discount rate and foundation cost positively affect the LCOE.

A pairwise mixing model with kernel constraint and its appraisal in transported PDF simulations of ethylene flames

Modeling micro-mixing remains a primary challenge for the transported probability density function (TPDF) method. In this study, the pairwise mixing with kernel constraint (KerM) model is proposed and evaluated in TPDF simulations of a turbulent non-premixed ethylene flame. The new model features controllable localness by selecting mixing particle pairs with a probability that depends on their distance in composition space, normalized by a characteristic kernel size. To evaluate the model in a controlled setting, TPDF simulations were carried out with the mean velocity, turbulent diffusivity and scalar mixing frequencies extracted from a direct numerical simulation (DNS) dataset. The predictions of the mean statistics and the conditional PDFs illustrate that the KerM model exhibits asymptotic behavior to the modified Curl (MC) and the Euclidean minimum spanning trees (EMST) model, respectively, depending on the kernel size. For the flame considered, results for KerM approach those for EMST when the kernel size is less than 0.01, while they approach those for MC when the kernel size is larger than 1.0. Given a kernel size of 0.1, KerM yields a notably better prediction of the overall combustion process than the conventional MC model. The predictions of the mean temperature and species mass fractions by KerM are similar to those by EMST, while KerM better predicts the conditional PDF of temperature than EMST that overdamps the conditional fluctuations. In addition, its complexity with the number of particles per cell is instead of O(Np2) for EMST. By incorporating different mixing timescales among species, the KerM-DD model better predicts the mean temperature field at the time instant close to local extinction than the KerM and EMST models. This work demonstrates that the KerM model, apart from being simple and robust in terms of implementation, features the advantages of being flexible in localness and being capable of incorporating differential diffusion.

Validation of short-term prediction of added resistance in head seas considering wave steepness nonlinearity with probability density function method

The added resistance in waves, which is an important component in the estimation of actual sea performance of ships, has been evaluated as the average value by short-term prediction methods. In this paper, a novel probabilistic method to account for non-linearities of added resistance with respect to wave height in short-term sea conditions is proposed using the joint occurrence probability distribution of wave height and period and non-linear added resistance response in the frequency region. It is a method of averaging the combination of crest and trough that ships encounter in irregular waves, which is considered as one wave of the regular waves, and the resulting added resistance is combined stochastically. The non-linearity with respect to wave height is empirically expressed as a “correction coefficient” obtained from the comparison of CFD data and potential-based linear calculation results in regular waves. The results of the proposed method have a reasonable agreement in the acceptable range of accuracy with results of RANSE based CFD numerical simulations in long crested irregular waves and the experimental model tests. As a result, the proposed method is applicable for obtaining added resistance in irregular head waves having the non-linearity with respect to wave height with a scope of further improvement in the method.

Estimating the state of health of lithium-ion batteries based on a probability density function

Accurate estimation of the state of health (SOH) of lithium-ion batteries is important to ensure safe operation. Although SOH models that extract health features (HFs) from incremental capacity (IC) curves have proven to be effective methods of evaluating battery SOH, the use of smooth IC curves that rely heavily on complicated algorithms reduces the reliability of SOH assessment to a certain extent. In this paper, a probability density function (PDF) method is integrated with Gaussian process regression (GPR) and used to build highly optimised SOH evaluation models for three different types of batteries. The PDF peak position and regional charging time are extracted from charging voltage data in the form of HFs using the PDF method. The proposed SOH estimation model shows good performance when these two HFs are both adopted. Our SOH estimation models for both cells and modules show good robustness for LiCoO2 (LCO), LiNi0.8Co0.15Al0.05O2 (NCA) and lithium iron phosphate (LFP) batteries.

On distributions of velocity random fields in turbulent flows

AbstractThe present paper aims to derive a partial differential equation (PDE) for the single‐time single‐point probability density function (PDF) of the velocity field of a turbulent flow. The PDF PDE is a highly non‐linear parabolic‐transport equation, which depends on two conditional statistical numerics of important physical significance. The PDF PDE is a general form of the classical Reynolds mean flow equation (O. Reynolds, Philos. Trans. R. Soc. Lond. Ser. A 186 (1895), 123–164. http://doi.org/10.1098/rsta.1895.0004) and is a precise formulation of the PDF transport equation (S. B. Pope, Turbulent flows, Cambridge University Press, Cambridge, 2000. https://doi.org/10.1017/CBO9780511840531.). The PDF PDE provides us with a new method for modelling turbulence. An explicit example is constructed. Though the example is seemingly artificial, it demonstrates the PDF method based on the new PDF PDE.

The impact of hydrogen substitution by ammonia on low- and high-temperature combustion

The combustion behaviour of ammonia has attracted intermittent interest with the original patent by Lyon (US3900554A) relating to its use for nitric oxide reduction through selective non-catalytic reduction (SNCR) a focal point. The recent interest in ammonia stems from its use as a hydrogen rich energy carrier with practical use requiring a much wider parameter space. The corresponding challenges (e.g. Kobayashi et al., Proc. Combust. Inst. 37 (2019) 109–133) include different flame dynamics and high emissions of oxides of nitrogen. The current paper explores the complex nature of ammonia oxidation and provides a reduced size reaction mechanism that enables application, without approximation, to the computation of turbulent flames through a joint-scalar transported probability density function (JPDF) method. Comprehensive validation suggests similar accuracy to a reference mechanism (Glarborg et al., Prog. Energy Combust. Sci. 67 (2018) 31–68) and highlights some uncertainties. The selected turbulent flame configuration features auto-ignition stabilised flames supported by a coflow of hot combustion products. The base case features a H2/N2 fuel jet that permits flame stabilisation at 1045 K corresponding to the onset of the SNCR temperature window. The impact of a gradual substitution of hydrogen by ammonia on flame stabilisation, emissions of oxides of nitrogen and the flame structure is quantified. It is shown that ammonia substitution leads to more prevalent local extinction, a more distributed flame structure and requires substantially increased coflow temperatures to achieve a similar flame stabilisation point. A lowering of the coflow temperature to operate within the SNCR regime substantially reduces NOx and leads towards a homogeneous/distributed reaction mode. The reduced fuel reactivity highlights the importance of turbulence-chemistry interactions leading to complexities in the design of practical devices.

The impact of ammonia addition on soot formation in ethylene flames

The present work investigates computationally the impact of blending ammonia with ethylene on soot formation in four laminar diffusion flames and the corresponding turbulent partially premixed flames. The conditions conform to experimental investigations (Bennett et al., Combust. Flame, 2020, and Boyette et al., Combust. Flame, 2021) with both sets of flames featuring identical fuel blends. A mass and number density preserving sectional method was applied to all cases with the soot surface growth and oxidation models updated to include reactions with oxides of nitrogen. The gas phase chemistry was extended to include a comprehensively validated ammonia sub-mechanism. For the laminar flames, the inception of soot particles is based on detailed chemistry with pyrene treated as representative of the PAH pool. The accuracy of fitting a global acetylene-based inception step is evaluated for the different fuel blends and subsequently applied to the turbulent flames using a fully coupled transported joint probability density function method featuring an 84-dimensional joint-scalar space. Results obtained for laminar flames show that detailed chemistry coupled with the sectional soot model provides excellent agreement with the measured suppression of the soot volume fraction with increased use of ammonia. Computed particle size distributions (PSDs) show an increase in smaller soot particles under such conditions, consistent with experimental observations. The experimentally observed reduced impact in turbulent flames is also reproduced computationally. The suppression of soot is principally caused by changes in the radical pool leading to reduced soot surface growth and, to a lesser extent, soot inception. The contribution of oxides of nitrogen to soot oxidation is modest. Computed PSDs in laminar and turbulent flames highlight the importance of differences in flame structures and flowfield timescales.

State of health estimation of lithium-ion batteries based on remaining area capacity

Accurate state of health (SOH) estimation can ensure the safe and reliable operation of the battery and prolong its service life. A new SOH evaluation method including the concepts of characteristic probability (CP) and remaining area capacity (RAC) are introduced in the framework of probability density function (PDF). Battery SOH evaluation models are respectively established for the lithium-iron phosphate (LFP) battery module and nickel-cobalt-aluminium (NCA) cell based on laboratory data at 1/3C-rate and 1C-rate at the sampling frequency of 1 min, and the effects of different CP values on the RAC - SOH models are investigated. The results show that there is a strong linear positive correlation between RAC and SOH whether charge and discharge conditions, while the model with the maximum PDF peak height as health factor is not ideal to assess battery SOH during discharging. The smaller the CP value, the better the effect of the RAC - SOH model. The R2 values of RAC - SOH models of NCA cell and LFP battery module are up to 0.99 at appropriate characteristic probability. The RMSE between the real and the estimated value of NASA battery SOH is not >1.2 % based on the RAC - SOH model. The improved PDF method has accurate and robust performance for the SOH estimation of different batteries.

On the impact of differential diffusion between soot and gas phase species in turbulent flames

The molecular diffusivities of larger PAHs and soot particles approach zero leading to differential diffusion with gas-phase species. The present work systematically quantifies the impact on soot moments, soot related statistical correlations and Particle Size Distributions (PSDs) using a fully coupled transported joint probability density function (JPDF) method featuring a 78-dimensional joint-scalar space, including enthalpy, gas phase species with the PSD discretised using 62 size classes via a mass and number density preserving sectional method. Differential diffusion of soot (DDS) is treated via a gradual decline of diffusivity among soot sections maintaining realisability and the expected exponential decay of variance. The solution of the flow field features a time-dependent second moment closure and an elliptic solver. The turbulent non-premixed Sandia C2H4 flame from the International Sooting Flame (ISF) data base was selected as a target along with the KAUST (C2H4/N2) variant of the same flame. Results show that reduced soot diffusion leads to a significant increase in the soot volume fraction RMS and that the correlation coefficient between soot volume fraction and temperature is further reduced in a particle-size-dependent manner. Similar observations are made for correlations between the soot volume fraction and the mass fractions of gas-phase species such as CO, OH, H and C2H2. The results suggest that computational methods that presume explicit (e.g. flame structure related) correlations between such scalars and with soot face leading order modeling challenges. It is also shown that the correlation between CO and soot increases due to oxidation of soot and that DDS leads to a modest downstream shift of PSDs towards larger particles.

Investigation of turbulent mixing using a stochastic hierarchical parcel swapping mixing model

AbstractTurbulent mixing plays an important role in a variety of applications ranging from astrophysics to combustion and even pollutant dispersion. The Direct Numerical Simulation (DNS) that resolves all scales is not feasible for most engineering applications since the flow has a wide range of length and time scales, which yields extremely high resolution requirements. Large Eddy Simulations (LES) overcome this limitation by modeling the sub‐grid scale effects. In transported Probability Density Function (PDF) methods, the key challenge is to develop an accurate mixing model. At this point, the Hierarchical Parcel‐Swapping (HiPS) model, introduced by A.R. Kerstein [J. Stat. Phys. 153, 142‐161 (2013)], is an attractive candidate. It is characterized by a computationally efficient representation of the effects of turbulence on a time‐evolving structure of diffusive scalar fields. The interpretation of the diffusive scalar fields or a state space as a binary tree structure is an alternative approach compared to the most common mixing models. The characteristic feature of HiPS is that every level of the tree corresponds to a specific length and time scale, which is based on turbulence inertial range scaling. The state variables only reside at the base of the tree and are understood as fluid parcels. The effects of turbulent advection are represented by stochastic swaps of sub‐trees at rates determined by prevailing turbulent time scales associated with the sub‐trees. The mixing of adjacent fluid parcels is done either instantaneously or at rates consistent with the corresponding diffusion time scales. In this work, HiPS is detailed for the simulation of passive scalar mixing first. Preliminary results for the scalar power spectra, mean square displacement and scalar dissipation rate are shown and reveal a reasonable agreement with experimental findings. Furthermore, the integrated binary tree structure allows to satisfy a large number of criteria for a good mixing model. Considering the reduced order and computational efficiency, HiPS is an attractive candidate for modeling the mixing in transported PDF methods.

A consistent scheme of the high-speed source term in probability density function methods for supersonic flows

To improve the particle energy accuracy of the probability density function (PDF) method in supersonic flows, a consistent numerical scheme for the high-speed source term in the particle energy equation is put forward. The proposed scheme is designed and computed based on the characteristic format, which shares the same right and left eigenvectors for the Jacobian of the convection term of the system. Therefore, this scheme is fully consistent with the spatial discretization of the inviscid term in the finite-difference solver of compressible flows. To show the advantages of the proposed scheme, a redundant energy equation is solved along with the Euler equation. The consistent scheme and the quasi scheme are numerically tested and compared in several canonical flows. The results show that the high-speed source term calculated by the consistent scheme generates much less numerical oscillation than the quasi scheme around discontinuities. Due to the accumulation of the numerical errors of the high-speed source term in every single iteration, the energy redundantly computed by the consistent scheme agrees better with the Euler results than the other scheme. Following this new high-speed source computation scheme, the large eddy simulation-PDF method is further developed and tested in a shock tube problem interacting with an isotropic turbulent flow and a supersonic temporally developing mixing layer. The results show that PDF with this consistent high-speed source scheme can improve the energy accuracy as well as turbulent combustion in supersonic flows.