Continuous Random Walk Model Research Articles

An improvement to continuous random walk model for sediment diffusion in inhomogeneous turbulent flows

An improvement to continuous random walk model for sediment diffusion in inhomogeneous turbulent flows

Assessment of stochastic models for predicting particle transport and deposition in turbulent pipe flows

Particle dispersion and deposition in turbulent pipe flows were studied numerically. The mean flow solution was calculated via a Reynolds Stress Model. Three different stochastic models were used to evaluate the effect of instantaneous fluctuations on entrained particles including the standard Discrete Random Walk (DRW) model available in Ansys-Fluent and two Continuous Random Walk (CRW) schemes that solve a normalized Langevin equation to better account for the effects of inhomogeneous turbulence on particle entrainment. Special attention is paid to the inconsistent directional vector resulting from solving the Langevin equation in non-rectangular geometries in location-specific coordinate systems. Concentration, axial distributions of deposition velocity, and deposition velocity versus particle relaxation time were compared with existing experiments and direct numerical simulation (DNS) using two different geometries. The validity of DNS data for near wall corrections were discussed. A third geometry representative of the hot section of a gas turbine was employed to study the capability of DRW and CRW on modeling spatially-developing flows. Comparisons were drawn between different stochastic models with an all-stick condition and experimental results. It was shown that the classical DRW model tends to overpredict the effect of entrance and does not yield good agreement in the deposition statistics. The CRW model shows a promising capability of modeling particle dispersion and deposition. However, the accuracy of CRW relies heavily upon the fidelity of the underlying mean flow solution and root-mean-square turbulence statistics.

Efficient simulation of bubble dispersion and resulting interaction

In this work, an efficient model for simulating bubble dispersion and coalescence due to turbulence is developed in the Euler-Lagrange framework. The primary liquid phase is solved on the Euler grid with the RANS turbulence model. Bubble motion is computed with the force balance equations. One-way coupling between two phases is assumed and the framework is designed for the computation of disperse bubbly flows at low Eötvös number. The turbulent dispersion of the dispersed phase is reconstructed with the continuous random walk (CRW) model. Bubble-bubble collisions and coalescence are accounted for deterministically. To accelerate the time-consuming search for potential collision partners in dense bubbly flows, the sweep and prune algorithm is employed, which can be utilized in arbitrary mesh types and sizes. Validation against experiments of turbulent pipe flows demonstrates that the one-way coupled EL-CRW dispersion model can well reproduce the bubble distribution in a typical dense bubbly pipe flow. Good agreement of the bubble size distribution at the pipe outlet between the simulation and the experiment is obtained.

Transport and fate of human expiratory droplets-A modeling approach.

The transport and fate of human expiratory droplets play a key role in the transmission of respiratory infectious diseases. In this paper, we present a modeling approach to understand the fundamental dynamics of exhaled droplets in human respiratory activities. The model solves a series of governing equations of droplets and uses a continuous random walk model to simulate turbulent fluctuations in violent expiratory events. The validation of the model shows the improvement in the prediction of dispersion of median-sized droplets. We show that these droplets are sensitive to environmental conditions, including temperature, humidity, and ambient flows. Applying the model to a set of idealized conditions such as free-fall and continuous jets, we demonstrate significantly different impacts of environmental parameters on droplets with different sizes. Using a realistic droplet size distribution and cough duration, we quantify the transport and fate of droplets in the near field of source and the potential influences by ambient conditions. The model we developed from this study could be applied to study the mechanisms for airborne pathogens, e.g., influenza virus and new coronavirus.

Particles dispersion and deposition in inhomogeneous turbulent flows using continuous random walk models

The suitability of the normalized Langevin stochastic equation with appropriate drift correction for simulation of instantaneous fluctuation velocities in inhomogeneous turbulent flows was studied. The Reynolds Stress Transport turbulence model of the ANSYS-Fluent code was used to evaluate the inhomogeneous turbulent flow properties in a two-dimensional duct flow. The simulation results were then used in an in-house Matlab particle tracking code and the trajectories of about 2 × 105 randomly distributed particles in the channel were evaluated by solving the particle equation of motion including the drag and Brownian forces under the one-way coupling assumption. The performance of the Continuous Random Walk (CRW) stochastic model using the conventional nonnormalized, as well as the normalized Langevin equations without and with the drift term for predicting a uniform distribution for the fluid-tracer particles in an inhomogeneous turbulent flow was examined. The accuracy of these models in predicting the deposition velocities and distribution of solid particles with diameters ranging from 10 nm to 30 μm was also carefully examined. In addition, the effects of including the finite-inertia coefficient in the drift term and using the corrected root mean square normal velocity in the near-wall region on the accuracy of the results were emphasized. By exploring the concentration profiles and deposition velocities of particles resulting from different CRW models, it was concluded that the Normalized-CRW model including the appropriate drift term leads to the most accurate results.

A Continuous Random Walk Model With Explicit Coherence Regularization for Image Segmentation.

Random walk is a popular and efficient algorithm for image segmentation, especially for extracting regions of interest (ROIs). One difficulty with the random walk algorithm is the requirement for solving a huge sparse linear system when applied to large images. Another limitation is its sensitivity to seeds distribution, i.e., the segmentation result depends on the number of seeds as well as their placement, which puts a burden on users. In this paper, we first propose a continuous random walk model with explicit coherence regularization (CRWCR) for the extracted ROI, which helps to reduce the seeds sensitivity, so as to reduce the user interactions. Then, a very efficient algorithm to solve the CRWCR model will be developed, which helps to remove the difficulty of solving huge linear systems. Our algorithm consists of two stages: initialization by performing one-dimensional random walk sweeping based on user-provided seeds, followed by the alternating direction scheme, i.e., Peaceman-Rachford scheme for further correction. The first stage aims to provide a good initial guess for the ROI, and it is very fast since we just solve a limited number of one-dimensional random walk problems. Then, this initial guess is evolved to the ideal solution by applying the second stage, which should also be very efficient since it fits well for GPU computing, and 10 iterations are usually sufficient for convergence. Numerical experiments are provided to validate the proposed model as well as the efficiency of the two-stage algorithm.

Continuous-time random walks and Fokker-Planck equation in expanding media

We consider a continuous random walk model for describing normal as well as anomalous diffusion of particles subjected to an external force when these particles diffuse in a uniformly expanding (or contracting) medium. A general equation that relates the probability distribution function (pdf) of finding a particle at a given position and time to the single-step jump length and waiting time pdfs is provided. The equation takes the form of a generalized Fokker-Planck equation when the jump length pdf of the particle has a finite variance. This generalized equation becomes a fractional Fokker-Planck equation in the case of a heavy-tailed waiting time pdf. These equations allow us to study the relationship between expansion, diffusion and external force. We establish the conditions under which the dominant contribution to transport stems from the diffusive transport rather than from the drift due to the medium expansion. We find that anomalous diffusion processes under a constant external force in an expanding medium described by means of our continuous random walk model are not Galilei invariant, violate the generalized Einstein relation, and lead to propagators that are qualitatively different from the ones found in a static medium. Our results are supported by numerical simulations.

Evaluation of the turbulence models for gas flow and particle transport in URANS and LES of a cyclone separator

Gas-solid separators have played an important role for process or emission control of many facilities including fluidized bed reactors. The flow field and collection efficiency were investigated for a Stairmand- type cyclone separator by Reynolds stress transport model (RSTM) and large eddy simulation (LES) in this work. RSTM and LES both showed good agreement with measured mean tangential velocity, whereas LES reproduced unsteady flow in the core with better agreement of the mean axial velocity profile. The particle dispersion model is required to obtain an accurate overall collection efficiency in both RSTM and LES. Anisotropy in velocity fluctuation should be taken into account, whereas its effect is more significant in RSTM than in LES. The continuous random walk (CRW) model gave more accurate results than the discrete random walk (DRW) model in RSTM, whereas there was no significant difference in LES resolving most large scale eddies. LES made better prediction of the overall particle collection efficiency than RSTM, although further investigation may be required for deviation in the maximum inlet flow case of the cyclone separator.

Numerical study of advanced dispersion models in particle-laden swirling flows

In this study, the dispersion phenomenon in a poly-dispersed gas-solid confined swirling jet is investigated numerically using an Eulerian–Lagrangian framework. Three continuous random walk models, including Sommerfeld’s model, Moissette et al. (MOB) model, and probability density function (PDF), and the conventionally used eddy life time (ELT) model are incorporated and compared. The results manifest that the important features of the flow, such as the penetration of the medium-sized particles into the corner recirculation zone and the distribution of the light particles downstream of the internal recirculation zone, can only be captured by a proper dispersion modeling even in an unsteady URANS approach resolving the precessing vortex core. Moreover, the ELT model has the largest errors for all particle statistics, especially for the fluctuating velocities (about 60% in average) while other models, especially MOB and PDF, bring about a satisfactory agreement with the experiment. MOB is the most accurate model in predicting the cross-stream particle velocity fluctuations with less than 20% error. Finally, for such a complex practical flow, the spatial decay of concentration profile is a key factor which is much more sensitive to the choice of dispersion model than the mean and fluctuating velocities.

RANS modeling of fluid flow and dust deposition in nuclear pebble-beds

In the nuclear pebble-beds, occurrence of localized hot-spots in the fluid flow, as well as accumulation of abraded graphite dust are two of the major concerns during a potential accident. The present paper deals with the Reynolds Averaged Navier Stokes modeling of the fluid flow and dust deposition in a chosen set of pebble-bed configurations. A standard k–ε model with enhanced wall treatment is used for modeling the flow turbulence. For the dust phase, a continuous random walk model, coupled with a carefully chosen near-wall anisotropic model is used to predict the dust transport and deposition. The chosen standard k–ε model is first validated by applying it to study the flow behavior in a so-called single cubic pebble-bed arrangement where q-DNS data is available for validation. The validated model is then applied to analyze the complex flow behavior in a structured and an unstructured pebble-bed arrangement. The chosen dust deposition model is first validated across five different test-cases where experimental data is available for validation. The validated model is then applied to analyze the complex three dimensional deposition patterns in a structured and an unstructured pebble-bed configurations.

Particle deposition modeling in the secondary side of a steam generator bundle model

A steam generator (SG) tube rupture (SGTR) model is studied in this paper. This model based on a experimental facility called Aerosol Trapping In a Steam Generator (ARTIST), which is a model of a scaled steam generator tube bundle consisting of 270 tubes and a guillotine tube to address aerosol deposition phenomena on two different scales: near the tube break, where the gas velocities and turbulence are very intensive, and far away from the break, where the flow velocities are three orders of magnitude lower.Owing to complexity of the flow, 3D simulations with highly resolved computational mesh near the break were done. First, the flow inside an isolated tube with a guillotine tube break has been studied in the framework of Reynolds Averaged Navier Stokes (RANS) approach. The next part is devoted to the simulation of an inclined gas jet entering the SG tube bundle via the guillotine tube breach with more advanced CFD tools. In particular, Detached Eddy Simulation (DES) and RANS are applied to tackle the wide range of flow scales. Flow field velocity comparison showed that DES results are reproducing wavy structure of the flow field in far field from the break observed in experiment.Particle transport and deposition is modelled by Lagrangian continuous random walk (CRW) model, which has been developed and validated previously. It is found that the DES combined with the CRW to supply fluctuating velocity components predicts deposition rates that are generally within the scatter of the measured data. Monodisperse, spherical SiO2 particles with AMMD = 1.4μm were used as aerosol particles in simulations.To be economically feasible, the computations were made with the open source CFD code OpenFOAM. Comparison of the calculated flow with the experimental axial velocity distribution data at different vertical levels has been performed.

RANS modeling for particle transport and deposition in turbulent duct flows: Near wall model uncertainties

Dust accumulation in the primary system of a (V)HTR is identified as one of the foremost concerns during a potential accident. Several numerical efforts have focused on the use of RANS methodology to better understand the complex phenomena of fluid–particle interaction at various flow conditions. In the present work, several uncertainties relating to the near-wall modeling of particle transport and deposition are addressed for the RANS approach. The validation analyses are performed in a fully developed turbulent duct flow setup. A standard k−ɛ turbulence model with enhanced wall treatment is used for modeling the turbulence. For the Lagrangian phase, the performance of a continuous random walk (CRW) model and a discrete random walk (DRW) model for the particle transport and deposition are assessed. For wall bounded flows, it is generally seen that accounting for near wall anisotropy is important to accurately predict particle deposition. The various near-wall correlations available in the literature are either derived from the DNS data or from the experimental data. A thorough investigation into various near-wall correlations and their applicability for accurate particle deposition predictions are assessed. The main outcome of the present work is a well validated turbulence model with optimal near-wall modeling which provides realistic particle deposition predictions.

A simple stochastic quadrant model for the transport and deposition of particles in turbulent boundary layers

We present a simple stochastic quadrant model for calculating the transport and deposition of heavy particles in a fully developed turbulent boundary layer based on the statistics of wall-normal fluid velocity fluctuations obtained from a fully developed channel flow. Individual particles are tracked through the boundary layer via their interactions with a succession of random eddies found in each of the quadrants of the fluid Reynolds shear stress domain in a homogeneous Markov chain process. In this way, we are able to account directly for the influence of ejection and sweeping events as others have done but without resorting to the use of adjustable parameters. Deposition rate predictions for a wide range of heavy particles predicted by the model compare well with benchmark experimental measurements. In addition, deposition rates are compared with those obtained from continuous random walk models and Langevin equation based ejection and sweep models which noticeably give significantly lower deposition rates. Various statistics related to the particle near wall behavior are also presented. Finally, we consider the model limitations in using the model to calculate deposition in more complex flows where the near wall turbulence may be significantly different.

Numerical simulation of multilayer deposition in an obstructed channel flow

Simulation of multilayer deposition of dry aerosol particles in turbulent flows has gained a growing interest in various industrial and research applications. The multilayer deposition of carbonaceous aerosol particles in a turbulent channel flow obstructed by a succession of square ribs is here numerically investigated. The multilayer particle bed growth on the various wall surfaces affects the air flow, which in turn affects the overall deposition rate. An iterative numerical procedure is therefore suggested to simulate the evolution of the graphite layer. The iterative process used to reproduce the layer build-up is decomposed as follows: Reynolds-Avergared Navier Stokes is employed to generate the flow field. The turbulent dispersion of the particles is reproduced through the use of a continuous random walk model. After statistically sufficient deposition of particulate matter, the layer build-up is computed using mechanics of dry granular material. The layer build-up model shows good agreement with data obtained from experimental tests carried out on-site.

Model for diffusion at the microcanonical superheating limit from atomistic computer simulations

The diffusion statistics of atoms in a crystal close to the critical superheating temperature was studied in detail using molecular dynamics and Monte Carlo simulations. We present a continuous random-walk model for diffusion of atoms hopping through thermal vacancies. The results obtained from our model suggest that the limit of superheating is precisely the temperature for which dynamic percolation happens at the time scale of a single individual jump. A possible connection between the critical superheating limit and the maximization of the Shannon entropy associated with the distribution of jumps is suggested. As a practical application of our results, we show that an extrapolation of the critical superheating temperature (and therefore an estimation of the melting point) can be performed using only the dynamical properties of the solid state.

Distribution characteristics of exhaust gases and soot particles in a wall-flow ceramics filter

An inhomogeneous soot distribution in a diesel particulate filter may deteriorate its behavior and result in higher pressure drops and fuel consumption. This will cause mechanical stresses on the filter due to temperature gradients resulting from the non-uniformly burning of soot during regeneration. The purpose of this paper is to investigate the flow distribution of the exhaust gas entering into a diesel particulate filter, the turbulent motion of diesel soot particles in the inlet header, and their deposition and distribution in the front surface of a diesel particulate filter. A Lagranian continuous random walk (CRW) model is developed to simulate soot particulate motion, which considers a succession of uncorrelated random forcing and drift corrections. The effects of particle inertia, turbulent fluctuation, and lift on the particle motion and trajectory are analyzed. Correlations of the uniformity index of the exhaust gas and soot particles with the flow rate, soot loading, and inlet expansion angle are evaluated. The results show that there is a two-peak phenomenon in the soot distribution at the front entrance of the filter, which is comprised of a peak in the central area due to inertia and a second peak in the periphery owing to diffusion and recirculation action. Exhaust flow rates and the inlet expansion angle have a major influence on the flow uniformity and soot uniformity, while soot loading has a slightly smaller effect on soot uniformity.

Prediction of Extrathoracic Aerosol Deposition using RANS-Random Walk and LES Approaches

Computational Fluid Dynamics (CFD) has emerged as a powerful and economical alternative to empiricism in the prediction of aerosol deposition inside the extrathoracic airways (ETA). In RANS-type treatments, a main difficulty is the specification of turbulent fluid fluctuations experienced by the particles, and hence recent research has concentrated on Large Eddy Simulations (LES) in conjunction with Lagrangian Particle Tracking (LPT). While providing close agreement with data, LES/LPT approaches are extremely time consuming, thus the motivation to investigate whether better Lagrangian stochastic models can help RANS-based treatments achieve comparable accuracy. In this study, the RANS-RSM model is used to obtain the mean carrier flow field, whereas turbulent fluid velocities are defined through a stochastic Continuous Random Walk (CRW) model based on the normalized Langevin equation. With extensive validation against flow field and particle deposition data, we demonstrate that RANS, combined with the Lang...

Validation of the Langevin particle dispersion model against experiments on turbulent mixing in a T-junction

The fluctuating fluid velocities seen by particles entrained in a turbulent fluid have recently been modeled using a stochastic model based normalized Langevin Continuous Random Walk (CRW). This model has been quite successful in predicting particle dispersion in mildly complex flows. In the present study, we aim at validating the CRW model further against data collected in a challenging 3D geometry. We consider turbulent fluid mixing downstream of a T-junction using a hybrid Euler–Lagrange approach whereby tracer particle trajectories are computed and mixing of the streams deduced from the relative concentration of particles originating from the two inlet branches of the Tee. In a first simulation, RANS Reynolds Stress Model (RSM) is used to obtain the mean flow field, whereas the fluid fluctuations are specified from a CRW. Simulation results are compared to experimental data on mixing of two isothermal streams consisting of tap and de-ionized water, respectively. It is found that RSM-CRW yields strong under-prediction of the mixing. Closer look at the results shows that the Reynolds stresses, which are required inputs to the CRW, are poorly predicted with RSM. Detached Eddy Simulations (DES) are subsequently performed to provide the mean flow field, and the DES-CRW model predictions are found to compare quite well with the experimental data.

A Systems Biology Approach to Understanding Alzheimer's Disease

A mathematical model for Alzheimer's disease (AD) has been developed using a systems biology approach. A cellular network of neurons, microglia and astrocytes has been created to model the levels of beta amyloid in the brain. The production and spatial distribution of beta amyloid, the key protein implicated in AD, has been modeled using the reaction-diffusion equation, where reaction rates have been modeled using stochastic functions. Neurons have been modeled using a previously developed McCulloch-Pitts neural network (Butz 2006) modified to account for neuronal cell death and loss of synaptic elements during high beta amyloid levels. Microglia are either in the ramified state (at rest) and modeled using a continuous random walk model, or in the activated state (actively moving towards a source of beta amyloid) and modeled using the Langevin equation of motion. Astrocytes are defined to set locations and contribute to removal of beta amyloid from the brain interstitial fluid. The roles that local cerebral blood flow, transport across the BBB, and local reactions play have also been modeled. Future work will look at the development of amyloid beta plaques in the cerebrovasculature and brain parenchyma, and their relationship to observed decreases in cerebral blood flow as the disease progresses.

A stochastic Langevin model of turbulent particle dispersion in the presence of thermophoresis

Direct numerical simulation (DNS) and experimental data have shown that inertial particles exhibit concentration peaks in isothermal turbulent boundary layers, whereas tracer-like particles remain well mixed in the domain. It is therefore expected that the interactions between turbulence and thermophoresis will be strong in particle-laden flows where walls and carrier fluid are at significantly different temperatures. To capture turbulent particle dispersion with active thermophoresis, a coupled CFD-Lagrangian continuous random walk (CRW) model is developed. The model uses 3D mean flow velocities obtained from the Fluent 6.3 CFD code, to which are added turbulent fluid velocities derived from the normalized Langevin equation which accounts for turbulence inhomogeneities. The mean thermophoretic force is included as a body force on the particle following the Talbot formulation. Validation of the model is performed against recent integral thermophoretic deposition data in long pipes as well as the TUBA TT28 test with its detailed local deposition measurements. In all cases, the agreement with the data is very good. In separate parametric studies in a hypothetical cooled channel flow, it is found that turbulence strongly enhances thermophoretic deposition of particles with dimensionless relaxation times τ + of order 1 or more. On the other hand, the thermophoretic deposition of very small inertia particles ( τ + < 0.2) in the asymptotic region far from the injection point tends to that which characterizes stagnant flow conditions, in agreement with the DNS results of Thakurta et al.