Stochastic Lagrangian Modeling for Large Eddy Simulation of Dispersed Turbulent Two-Phase Flows

Understanding the dispersion and the deposition of inertial particles in turbulent flows is a domain of research of utmost practical interest. With advances in computing resources, Direct Numerical Simulation (DNS) and Large Eddy Simulation (LES) have become powerful tools for the investigation of particle-laden turbulent flows with the hybrid Eulerian-Lagrangian approach playing a key role in predicting inertial particle dispersion and deposition. Computational intractability that arises due to the need of solving all the scales has restricted DNS to the very low Reynolds number turbulent flows that are not often of practical interest. LES, by solving only the large energy-containing eddies and modeling the small quasi-universal scales, is relaxed from this restriction. Thus, tackling high Reynolds number turbulent flow becomes possible. The use of large-eddy simulation has increased over the years as a promising tool to address these types of problems with the required accuracy at an affordable computing cost. In LES of dispersed turbulent multiphase flows, it has been common that tracking inertial particles in turbulent flows is carried out using only the filtered velocity field. This turned out to be inaccurate for cases dealing with very small, turbulence-responsive particles. For these cases, the timedependent velocity field seen by the inertial particles can be stochastically constructed in a Lagrangian framework. This can be achieved through the use of a stochastic diffusion process such as Langevin models.

For large eddy simulation of particle-laden turbulent flows, the accuracy with which the continuous phase is predicted is crucial for the simulation of the dispersed phase. In this chapter, issues that are important for LES of single-phase turbulent flow, including filtering, SGS modeling and numerics, are reviewed. Also, the advantage that LES holds over DNS and RANS to simulate practical single-phase turbulent flows is discussed.

The LES philosophy is based on the trivial observation that the loss of information incurred during filtering places a limit on the achievable accuracy of LES. Keeping that in mind, a broad array of SGS models for LES purposes have been developed using a variety of different approximations. Such models have been successful in many flows, yet they all have shortcomings.

There are many important issues in LES which make the validation of the results, and of the SGS models themselves, essential. The most obvious ones are those related to the physical modeling of the subgrid stresses, which is in many cases based on engineering approximations, justified by little more than dimensional considerations. Numerical issues are also important, since most models use as input the velocity gradients of the filtered field, which depend on the smallest scales resolved by the simulations, and therefore strongly influenced by numerical errors. Consistency with filtering of the physical models and numerical schemes adopted is as important as the previous issues.

In this chapter, the importance of the dispersed two-phase turbulent flows for industrial and environmental applications is highlighted. Both Eulerian and Lagrangian descriptions of particle-laden flows are reviewed. Since the book's focus is on the stochastic modeling for particle transport by subfilter motion in a Lagrangian framework, more emphasis is put on the Lagrangian description. Though the stochastic modeling is so far only tested in a context of one-way coupling between dispersed and carrier phases, the importance that subfilter motion may have in predicting turbulence modulation is also explained.

Success in simulating particle-laden turbulent flows relies heavily on a greater understanding of the interaction of the two phases. This can lead undoubtedly to increases in performance, reduction in cost and/or improved safety in systems where they are encountered. It also increases the quality of predictions of the effectiveness of natural flow phenomena for dispersing particulate pollutants to acceptable concentration levels.

In principle, the direct numerical simulation (DNS) of turbulent flows, involving a large number of particles, with appropriate boundary and initial conditions would describe completely the two-phase flows. Due to the high computational cost of DNS, both the velocity field of the carrier phase and trajectories of particles can be calculated through Large Eddy Simulations (LES). Yet another method, Stochastic Modeling (SM) coupled to RANS calculations can be used. The aim of RANS/SM is to reduce the computational effort through generating a synthetic turbulent flow field with statistical properties of interest identical to that of the real turbulent flow.

In this chapter, the adoption and development of a Langevin-type Lagrangian stochastic model for the transport of inertial particles by subfilter motion in LES is detailed. The different assumptions and numerical requirements needed for the implementation of the model are discussed.

The theoretical and numerical formulations of the Langevin model have been extensively discussed in the framework of particle-laden RANS. Its use is extended herein with the necessary modifications for the modeling of the fluid velocity seen by particles in LES framework. We introduce below the two formulations of Langevin equation used to model the time increment of the fluid velocity seen by inertial particles. For the first formulation, called standard formulation, closure of both drift and diffusion terms are similar to the one used for the fluid particle case. However, the SGS time scale with which inertial particles see the turbulence is modified to account for inertia and cross trajectories effects. For the second formulation, referred to as complete formulation, closure forms for the drift and the diffusion terms are described.

Numerical issues linked to the solution of the resulting stochastic differential equations as well as some important properties of the Langevin-type equations are discussed.

Numerical simulations using two approaches, namely RANS and LES, were conducted to compute inertial particle dispersion from a source point in a turbulent gas flow in a pipe at a high Reynolds number. Numerical predictions were compared to the experimental observations of Arnason (1982) and Arnason and Stock (1984). Stochastic modeling of the turbulent fluctuations seen by inertial particles along their trajectories has been used. In the framework of RANS, the aim is to reconstruct the whole turbulent field whereas in the context of LES, only modeling of SGS fluctuating velocities is sought.

Particle dispersion statistics such as particle concentration, radial velocity and the dispersion coefficient were computed for solid particles that have different inertia and drift. The use of a Langevin-type stochastic approach to model the sub-filter fluctuations has proven crucial for results concerning the small-Stokes-number particles. The stochastic model used has been extensively used in the framework of RANS. Its simplistic extension to predict the sub-filter fluctuations for LES has given very satisfactory results.

Numerical predictions show that, for the same flow, inertial particles with larger diameter (and hence larger Stokes number) can disperse faster than smaller particles (with smaller Stokes numbers). It was proved theoretically that this can be the case if the inertia parameter controls the dispersion. These findings back up the experimental observations of Arnason and Stock.

Comparison of RANS and LES results have shown that the RANS approach is unable to predict particle dispersion statistics as accurately as the LES in particular for inertial particles characterized by a Stokes number smaller than 0.5. For particles with Stokes number higher than 0.5, both LES and RANS predictions compare reasonably well with the experimental results.

LES was performed to study Aerosol deposition in a turbulent 90° bend flow with tubular crosssection. Numerical predictions were compared to the experimental observations of Pui et al. and the DNS-like work of Breuer et al. Due to the complexity of the turbulent flow in curved pipe characterized by curved streamlines and zones of recirculation and the lack of comparison studies of the same flow parameters, a good deal of care has been taken to ensure that the carrier phase is accurately simulated. Every effort was made to adapt the mesh to the dynamical features of the flow and boundary conditions were set such that the inlet and outlet conditions would not influence the turbulent flow in the bend.

The numerical predictions of the secondary flow and streamlines in the symmetry plan and in cross sections at different angle of deflection showed a good agreement with the DNS-like work of Breuer et al. A-posteriori estimation of the filtered-out kinetic energy demonstrated that the present LES is adequate according to the LES index of quality developed by Celik et al.

For the dispersed phase, a stochastic model that accounts for inertial particle transport by SGS motion was used. It was anticipated that such modeling should be crucial owing to the very small-Stokes-number particles tracked. An estimation of the time scale of the SGS fluctuations that are discarded by the filtering operation in LES showed clearly that particles with Stokes number smaller than 0.25 do sense the SGS turbulent fluctuations.

Numerical results concerning the deposition efficiency of inertial particles with Stokes number that range between 0.005 and 1.5 demonstrate the ability of the stochastic modeling to reproduce with good accuracy the SGS effects on small-Stokes-number particles. As it was expected the use of the filtered velocity field only to track particles with Stokes number smaller than 0.3 has proven inaccurate. The complete formulation of the stochastic model showed its superiority compared to the standard formulation. The latter was showed to produce an incorrect level of SGS turbulence.

It was shown clearly that the deposition efficiency of small inertial particles can be predicted with a very good accuracy in the framework of LES using a coarse numerical description. To achieve that, the effect the SGS motion has on inertial particle transport needs to be taken into account. The Langevin-type stochastic diffusion process has proven very adequate in this regard.

After the epidemic outburst of avian influenza and severe acute respiratory syndrome (SARS) in East and Southeast Asia, there has been a burgeoning research interest in investigating the control and transport's mechanisms of airborne bacteria and viruses indoors and in confined environments such as aircraft cabin. Dispersion of microorganism-laden aerosols exhaled from infected patients was admitted as a potential airborne transmission pathway. Consequently, competent understanding of aerosol dispersion and deposition is necessary to improve exposure assessment tools and models and endorse efficient ventilation strategies that can considerably reduce indoor particle concentrations and improve the indoor air quality.

This investigation attempts to provide a realistic simulation of the time-dependant flow field in a chamber inhabited with two heated manikins using the well resolved LES approach. The potential of both the Eulerian and the Lagrangian approaches coupled to LES of non-isothermal airflow to study dispersion characteristics of expiratory aerosols has been examined.

Experimental and numerical findings on the flow and temperature fields were recorded and show all good agreements. By taking into account the uncertainly of the particle counter used in the experimental work, globally the agreement between the experimental results and the computational modeling predictions on aerosols rates of decay is quite acceptable.

Stochastic Large eddy simulation emerges as a promising technique for dispersed turbulent two-phase flows. A stochastic Lagrangian model based on a Langevin-type stochastic diffusion process was described in this e-Book. The primary objective of such modeling is to account for the dispersion and deposition of inertial particles by subfilter or subgrid motion that is discarded by the filtering operation in LES. Particular attention was given to the testing of the model under standard and complete formulations in shear turbulent flows taking into account inertia effects that are caused by density difference between carrier and dispersed phases and the cross-trajectory effects due to gravity.

The stochastic modeling in particle-laden LES is motivated by the inability of RANS and LES using only the filtered velocity field to properly predict transport of inertial particles with Stokes numbers smaller than the smallest resolved turbulence scales. This modeling may be crucial also for LES characterized by an excessive filtering-out of kinetic energy due to the lack of spatial resolution in regions with high shear as such near the wall and zones of recirculation and boundary layer detachment.

This e-Book highlights the progress that has been made to date to improve the predicting capabilities of the large eddy simulation technique for two-phase turbulent flows. It represents only the start of the development and validation process that should be pursued in order to confidently apply this model to an increasing number of applications of industrial and biomedical nature. I hope this e-Book conveys the potential of stochastic large eddy simulation for predicting dispersed turbulent flows and will stimulate the use of the stochastic LES to many other challenging applications.

I have made an attempt to provide sufficient information to understand and to define the approach's potential and practicality. An attempt is also made to evolve general CFD guidelines which may be useful for solving practical engineering problems using stochastic LES. Adequate attention to the key issues mentioned in this e-Book and creative use of the stochastic LES model will make significant contribution to enhancing our understanding of the subject. New advances may be assimilated using the framework discussed in this e-Book.

Full text available

Full text available

Full text available

Full text available

Full text available