On random walk models for simulation of particle-laden turbulent flows
Introduction
The availability of an accurate model for evaluation of dispersion and deposition of micro- and nano-particles in turbulent flows is of vital importance to the computer simulations of a wide range of industrial, environmental, and biomedical processes. Pneumatic conveying, ventilation systems, cloud formation, precipitation (Devenish et al., 2012; Warhaft, 2008), air pollution, sand and dust storms (Luo et al., 2016; Rahman et al., 2016; Sajjadi et al., 2016), and transport and deposition of inhaled particles in human respiratory system (Fan and Ahmadi, 2000; Cheng, 2003; Matida et al., 2004; Zamankhan et al., 2006; Longest et al., 2008; Shi et al., 2008; Longest and Vinchurkar, 2009; Tian and Ahmadi, 2012; Tavakoli et al., 2012; Tian and Ahmadi, 2013; Yazdani et al., 2014; Tavakol et al., 2015; Tavakol et al., 2017) are a few examples of the processes involving particle-laden turbulent flows. Numerical simulation of these flows requires an accurate evaluation of turbulence characteristics and their interactions with particles.
The very first step for simulating turbulent flows is the selection of an appropriate method. Currently, there are three main approaches for simulating turbulent flows. The most advanced approach is the direct numerical simulation (DNS) in which all scales of turbulence down to the Kolmogorov scale are resolved. While the DNS approach is quite accurate, it is computationally expensive. The prohibitive computational cost of the DNS approach has restricted its application to large-scale industrial and environmental problems. Therefore, the applications of DNS have been limited to simple flow passages and are typically done for fundamental research studies. In the next level of accuracy is the large eddy simulation (LES) approach that resolves the details of turbulent flows larger than the grid cell size while the subgrid-scale fluctuations are modeled (Rogallo and Moin, 1984; Lesieur et al., 2005; Sagaut, 2006). The LES requires less computational resources compared to the DNS, but still, it is computationally demanding for simulating flows in complex configurations at high Reynolds numbers of industrial and environmental interests.
The approach that is more commonly used for practical applications is the Reynolds-averaged Navier–Stokes (RANS) model. While quite economical, the RANS approach requires the use of a turbulence model and evaluates only the mean velocities and turbulence statistics. Due to the relative simplicity and computational efficiency of the RANS models, considerable attention has been given to developing appropriate turbulence models. While typically the two–equation models (e.g., k-ε, k-ω, etc.) in conjunction with the eddy viscosity assumption are used, there are also the Reynolds stress transport models (RSTM) that directly evaluate the components of Reynolds stresses and account for the anisotropy of turbulence fluctuations (Hanjalić and Launder, 1972; Durbin, 1993; Pope, 2000).
The RSTM provides the time-averaged velocity and turbulence properties; however, for certain problems such as the one involving particle dispersion and deposition, knowledge of the instantaneous turbulence fluctuations is required. In these cases, a pseudo-turbulence fluctuation along the particle trajectories is generated based on the RANS evaluation of the turbulence statistical properties. Accurate evaluation of instantaneous velocity fluctuations is required for realistic evaluation of turbulent diffusion effects for accurate predictions of particle dispersion and deposition on surfaces (Loth, 2000; Bocksell and Loth, 2006). When the LES approach is used for simulating particle-laden turbulence flows, the subgrid scales (SGS) fluid fluctuating motions seen by particles should also be modeled properly for an accurate description of particle dispersion. Although the effects of SGS on particle motion were shown to be significant in several investigations (Kuerten and Vreman, 2005; Kuerten, 2005; Marchioli et al., 2008; Salmanzadeh et al., 2010; Innocenti et al., 2016), they were totally neglected in some other studies (Yeh and Lei, 1991; Uijttewaal and Oliemans, 1996; Wang and Squires, 1996; Jayaraju et al., 2008; Afkhami et al., 2015). Additional information regarding the state-of-the-art of SGS models were reported by Minier (2015), Pozorski (2017), and Marchioli (2017).
For simulating particle-laden flows, the Eulerian–Eulerian and Eulerian–Lagrangian approaches are typically used. The Eulerian–Lagrangian approach, which is more physical as it accounts for the discrete nature of particles, is used in the present study. (Taylor, 1920) was the pioneer in utilizing the Lagrangian approach, where he employed a stochastic model to simulate fluid-particle (material point) dispersion in homogeneous turbulent flows from a point source. He reported that the standard deviation of fluid-particle distances from their initial position varies linearly with time initially and then becomes proportional to the square root of time for large times. Since then, several Lagrangian stochastic models were introduced. The discrete random walk (DRW) and continuous random walk (CRW) stochastic models are widely used in CFD codes where velocities are assumed to be the summation of the mean fluid velocity and turbulence fluctuations. In the DRW and CRW models, which are not strictly derived from the Navier–Stokes equation, the velocity fluctuations are considered as Markov processes that are generated with zero mean and variances corresponding to those of turbulence velocities and appropriate time scale (Shirolkar et al., 1996; Bocksell and Loth 2001). Also, there is a Probability Density Function (PDF)-based stochastic differential equation (SDE) that is used for evaluating the fluid fluctuation velocity (Haworth and Pope 1987; Minier and Peirano 2001; Minier et al., 2004; Minier 2015; Pozorski 2017). The PDF approach was also extended for simulating the subgrid fluctuation in the LES, which is referred to as the Filter Density Function (FDF) (Givi 1989; Colucci et al., 1998; Wacławczyk et al., 2008; Innocenti et al., 2016).
In the DRW model, it is assumed that a particle interacts with an eddy for an interaction time interval tint; at the end of the interaction time interval a new random fluctuation independent of the previous one is introduced to account for the interaction time with a new turbulence eddy.
In the CRW model, a stochastic differential equation is used to find the turbulent fluctuations seen by fluid-point particles. The corresponding Langevin equation is given aswhere α and λ are coefficients, is the turbulence fluctuating velocity component and ξi(t) is a Gaussian white noise process. The first term of the right-hand side (RHS) of Eq. (1) represents the persistence of the fluid motion that generates a correlation between successive fluctuations, and the second term includes the random variation of the fluctuations. The coefficients α and λ were evaluated by Legg and Raupach (1982) as 1/τi and where σi and τi are the RMS velocity fluctuations and the Lagrangian time scale in the i-direction.
The stochastic model proposed by Taylor and extended by others was originally established for homogenous turbulent flows. However, most practical turbulent flows, like the atmospheric boundary layer flows and flows in ducts, are inhomogeneous with spatially varying vertical root-mean-square (RMS) velocity, σ2, and Lagrangian time scale, τ2. In this regard, several researchers examined the performance of the conventional DRW and CRW models for simulating the fluctuation fields in inhomogeneous turbulent flows and proposed needed improvements for their applications (MacInnes and Bracco, 1992; Bocksell and Loth, 2001).
The studies conducted on the CRW model showed that this model could predict reasonable results for concentration profiles of fluid-particles in turbulent Atmospheric Boundary Layer (ABL) flows with an inhomogeneous vertical time scale (Hall, 1975; Reid, 1979; Wilson et al., 1981b; Legg, 1982). Wilson et al. (1981a), however, found that using the stochastic models for ABL-flows with inhomogeneous velocity fluctuations generates an unphysical inhomogeneous concentration profile. They suggested that these models lead to (where c is concentration), which was also reported by Thomson (1984), leading to a concentration gradient of fluid (material) point-particles in inhomogeneous flows. To rectify this defect and obtain a homogenous fluid-particle (material point) concentration in flows with variable RMS vertical velocity fluctuations, Wilson et al. (1981a) introduced a drift velocity given asas a correction for the inhomogeneous flows.
Legg and Raupach (1982) and Ley and Thomson (1983) also noted the necessity of including a correction term to a mean pressure gradient associated with the missed Reynolds stress terms. They concluded that ignoring this correction leads to a spurious mean drift of particle trajectories. Legg and Raupach suggested using as the mean drift velocity and proposed a modified Langevin equation in the y-direction (perpendicular to the wall) for turbulent flows with variable vertical velocity variance. That is,
There seems to be a factor of 2 difference between the correction mean drift velocity of Wilson et al. (1981a) and that of Legg and Raupach (1982). Eq. (3) is referred to as a non-normalized Langevin equation (Non-normalized CRW model).
Based on the transformed coordinates used by Wilson et al. (1981a), Thomson (1984) and Durbin (1984, 1983) proposed a normalized Langevin equation given as
Later, Bocksell and Loth (2006) suggested that normalizing the Langevin equation is necessary to de-correlate the successive fluctuations generated for fluid (material) point-particles in flows with severe inhomogeneity.
More recently, Minier et al. (2014) described a more rigorous derivation of the Langevin equation based on the PDF approach of Pope (Haworth and Pope, 1987; Ahmadi and Hayday 1988; Minier, 2015; Pozorski, 2017). Minier et al. (2014) also pointed out that if it is assumed that , where C0 is a constant, the normalized Langevin equation for homogenous flows is identical to the Simplified Langevin Model (SLM) developed by Pope for turbulent reactive single-phase flows (Pope, 1985; Haworth and Pope, 1986). They noted that this equation does not have the exact invariance properties of inhomogeneous flows, which makes it inconsistent with the Reynolds-stress equation. Earlier, however, Iliopoulos and Hanratty (1999) and Iliopoulos et al. (2003) found a satisfactory agreement between the results obtained from Eq. (4) and the DNS results for dispersion of fluid (material) point particles and inertial solid particles in a fully developed inhomogeneous turbulent flow.
Bocksell and Loth (2006) also noted that the drift correction term for inhomogeneous flows is valid for fluid velocity fluctuations seen by fluid-particles that follow the flow. However, if the modified Langevin equation is used for generating fluid velocity fluctuations seen by inertial particles, a factor of , where Stk (Stokes number) is the ratio of the particle relaxation time to the local turbulent integral time scale, should be included as a coefficient for the drift correction term.
Comparisons of the results predicted by Bocksell and Loth with the DNS data for different Stokes numbers revealed that including the correction factor significantly improves the estimated concentration profiles of inertial particles. Dehbi (2008, 2010) and Jayaraju et al. (2015) examined the performance of this approach for predicting the deposition velocity and dispersion of particles in inhomogeneous turbulent flows and found reasonable agreement with the experimental data and DNS results. However, Jayaraju et al. (2015) showed an overestimation of the deposition velocity of the Brownian particles in a channel flow using the Bocksell and Loth CRW model and the (ANSYS-Fluent code, 2011). They suggested that the defect of the Brownian model of the ANSYS-Fluent code (2011) was the cause of the overestimation of deposition velocity. Recently, however, Mofakham and Ahmadi (2019) compared different versions of the CRW models and showed that the CRW model based on the normalized Langevin equation, including the drift correction term of Bocksell and Loth (2006) (Normalized-CRW), predicts reasonably accurate results.
It should be pointed out that the Simplified Langevin Model (SLM) is similar to the Non-normalized-CRW model. Several recent studies showed that the SLM (the Non-normalized-CRW) overestimates the deposition velocities of micro- and nano-size particles (Mofakham and Ahmadi 2019; Chibbaro and Minier 2008; Guingo and Minier 2008; Jin et al., 2015, 2016). To improve the performance of the Simplified Langevin model, Chibbaro and Minier (2008) and Guingo and Minier (2008) had to include some ad hoc boundary conditions that they argued are in connection with the influences of the sweep and ejection events and the near-wall coherent structures. Jin et al. (2015) also tried to improve the prediction of the SLM by introducing a quadrant analysis of normal velocity fluctuations to account for the effects of the sweep and ejection events. Later, Jin et al. (2016) studied the effects of including the lift forces and the near-wall corrections of the drag and lift forces to rectify the underestimation of their earlier model for deposition velocities of small size particles but found only slight improvements.
Currently, the commercial CFD software is commonly used for extensively, including turbulent particle transport and dispersion in numerous industrial applications. In the present study, the performance of the default-DRW model of the ANSYS-Fluent 18.1 code for generating the instantaneous velocity fluctuations was carefully examined. It was found that the default-DRW model may lead to significant errors for the Reynolds-averaged Navier–Stokes (RANS) simulations of particle concentration and deposition. To improving the performance of the software, the Normalized-CRW was used. First, a fully developed turbulent airflow in a two-dimensional channel was simulated by the software using the RANS approach using the Reynolds Stress Transport model (RSTM). Then, the trajectories of a wide range of spherical particles from 10 nm to 30 µm were evaluated under the one-way coupling assumption by three approaches: (a) The discrete phase model (DPM) combined with the default random walk (DRW) stochastic model of the ANSYS-Fluent software was used. (b) The mean flow velocities, as well as root-mean-square (RMS) velocity fluctuations in different directions, were exported from the ANSYS-Fluent code and used in an in-house Matlab particle tracking code where the Normalized-CRW model was used to incorporate the effects of turbulence fluctuations on particle trajectories. (c) The Normalized-CRW model was implemented into the ANSYS-Fluent code by user-defined functions (UDFs) and the DPM model of ANSYS-Fluent was used to evaluate the particle trajectories. In each approach, a large number of particles of different sizes were tracked for long durations, and the corresponding time evolution of particle concentration profiles and deposition velocities were evaluated. The simulation results were compared with the experimental data, earlier RANS and DNS results, as well as with the empirical models. It was found that the simulation results of the in-house Matlab code and the ANSYS-Fluent software with the UDF showed that using the Normalized-CRW model for generating the fluid velocity fluctuation seen by particles markedly improved the accuracy of the model predictions for particle concentration, as well as, deposition in inhomogeneous turbulent flows.
Section snippets
Computational Domain
In this study, it is assumed that air is flowing in a channel at room temperature (288 K) with a kinematic viscosity of at a Reynolds number of 3329 based on an average velocity of 5 m/s and channel half-width. The corresponding shear Reynolds number is 219 based on a wall shear velocity of and channel half-width. Periodic velocity boundary conditions at the inlet and outlet and the no-slip boundary condition at the upper and lower walls of the channel were imposed.
Results and discussions
The flow is simulated by the ANSYS-Fluent CFD software using the RSTM model for a long time to reach a fully developed solution. It is assumed that the volume fraction of particles is sufficiently small so that the one-way coupling can be assumed. That is, the fluid carries the particles, but the effect of particles on the flow is small and can be ignored. The profiles of mean streamwise velocity, RMS streamwise and normal velocity fluctuations, turbulence kinetic energy, and turbulence
Conclusions
The mean flow velocities and RMS velocity fluctuations of a turbulent flow in a channel were evaluated using the RANS-RSTM turbulence model of the ANSYS-Fluent 18.1 code. Then, the trajectories of nano- and micro-particles were evaluated using the DPM of the ANSYS-Fluent code, as well as an in-house Matlab particle tracking code. To include the effect of instantaneous fluid velocity fluctuations on particle dispersion, the Default-DRW stochastic model of the ANSYS-Fluent code and the
Declaration of Competing Interest
This is to confirm that the authors do not have a conflict of interest.
References (115)
- et al.
Particle agglomeration and dispersion in fully-coupled turbulent channel flow using large eddy simulation and discrete element method
Procedia Eng.
(2015) - et al.
Stochastic modeling of particle diffusion in a turbulent boundary layer
Int. J. Multiph. Flow
(2006) - et al.
A new correlation for the aerosol deposition rate in vertical ducts
J. Colloid Interface Sci.
(1995) - et al.
Langevin PDF simulation of particle deposition in a turbulent pipe flow
J. Aerosol. Sci.
(2008) - et al.
A note on the consistency of hybrid Eulerian/Lagrangian approach to multiphase flows
Int. J. Multiph. Flow
(2011) Turbulent particle dispersion in arbitrary wall-bounded geometries: a coupled CFD-Langevin-equation based approach
Int. J. Multiph. Flow
(2008)Validation against DNS statistics of the normalized Langevin model for particle transport in turbulent channel flows
Powder Technol.
(2010)- et al.
Wall deposition of small ellipsoids from turbulent air flows—a Brownian dynamics simulation
J. Aerosol. Sci.
(2000) - et al.
A sublayer model for turbulent deposition of particles in vertical ducts with smooth and rough surfaces
J. Aerosol. Sci.
(1993) Model-free simulations of turbulent reactive flows. progress
Energy Combust. Sci.
(1989)
Particle deposition in a nearly developed turbulent duct flow with electrophoresis
J. Aerosol. Sci.
A stochastic model for solid particle dispersion in a nonhomogeneous turbulent field
Int. J. Multiph. Flow
Large eddy and detached eddy simulations of fluid flow and particle deposition in a human mouth–throat
J. Aerosol. Sci.
RANS modeling for particle transport and deposition in turbulent duct flows: near wall model uncertainties
Nucl. Eng. Des.
The effects of near wall corrections to hydrodynamic forces on particle deposition and transport in vertical turbulent boundary layers
Int. J. Multiph. Flow
A numerical simulation of particle deposition in turbulent boundary layers
Int. J. Multiph. Flow
Deposition of aerosols on surfaces in a turbulent channel flow
Int. J. Eng. Sci.
Comparison of ambient and spray aerosol deposition in a standard induction port and more realistic mouth–throat geometry
J. Aerosol. Sci.
Inertial deposition of aerosols in bifurcating models during steady expiratory flow
J. Aerosol. Sci.
Numerical approaches for motion of dispersed particles, droplets and bubbles
Progress Energy Combust. Sci.
Influence of gravity and lift on particle velocity statistics and transfer rates in turbulent vertical channel flow
Int. J. Multiph. Flow
Improved numerical simulation of aerosol deposition in an idealized mouth–throat
J. Aerosol. Sci.
Statistical simulation of particle deposition on the wall from turbulent dispersed pipe flow
Int. J. Heat Fluid Flow
On Lagrangian stochastic methods for turbulent polydisperse two-phase reactive flows
Prog. Energy Combust. Sci.
The pdf approach to turbulent polydispersed two-phase flows
Phys. Rep.
Brownian diffusion of submicrometer particles in the viscous sublayer
J. Colloid Interface Sci.
PDF methods for turbulent reactive flows
Prog. Energy Combust. Sci.
Particle-turbulence interaction in a boundary layer
Int. J. Multiph. Flow
The transport of discrete particles in inhomogeneous turbulence
J. Aerosol. Sci.
Computational fluid dynamics (CFD) simulation of a newly designed passive particle sampler
Environ. Pollut.
Fundamental aspects of modeling turbulent particle dispersion in dilute flows
Prog. Energy Combust. Sci.
Stochastic dispersion of ellipsoidal fibers in various turbulent fields
J. Aerosol. Sci.
Deposition fraction of ellipsoidal fibers in a model of human nasal cavity for laminar and turbulent flows
J Aerosol Sci
Particle deposition in turbulent duct flows—comparisons of different model predictions
J. Aerosol. Sci.
A probability density closure model for turbulence
Acta Mech.
Some characteristics of small-scale turbulence in a turbulent duct flow
J. Fluid Mech.
Random walk models for particle diffusion in free-shear flows
AIAA J.
Turbulent deposition and trapping of aerosols at a wall
Phys. Fluids A Fluid Dyn.
Transfer of particles in nonisotropic air turbulence
J. Atmosp. Sci.
Aerosol deposition in the extrathoracic region
Aerosol. Sci. Technol.
Filtered density function for large eddy simulation of turbulent reacting flows
Phys. Fluids
Droplet growth in warm turbulent clouds
Q. J. R. Meteorol. Soc.
Probability density function and Reynolds‐Stress modeling of near‐wall turbulent flows
Phys. Fluids (1994-present)
A Reynolds stress model for near-wall turbulence
J. Fluid Mech.
Stochastic Differential Equations and Turbulent Dispersion
Comments on papers by Wilson et al.(1981) and Legg and Raupach (1982)
Boundary Layer Meteorol.
Turbulent normal velocity fluctuations close to a wall
Phys. Fluids
Cited by (41)
Particle tracking schemes for micron particle deposition in a 90° pipe bend and nasal airway geometry
2024, Journal of Aerosol ScienceNumerical study of single-cough generated droplets inhalation and deposition in an airway under realistic cyclic breathing in a quiescent air condition
2024, International Journal of Multiphase FlowA comprehensive approach to simulation of cartridge filtration using CFD
2023, Journal of Environmental Chemical EngineeringNumerical study of dry deposition of dust-PM<inf>10</inf> on leaves of coniferous forest
2023, Atmospheric Pollution Research