Overview

In this edition - and linking to Lagragian Particle Tracking in CFD Edition, we will talk about the concept of Particle Sub-Grid Scale (SGS) modeling. After reading this article, you should have an idea about the relevance of this type of modeling and when to apply it in CFD. We will discuss the key concepts and parameters in Particle SGS modeling and describe the different SGS models used in LES and RANS. Finally, I'll show a case study that highlights the application of Particle SGS modeling in a turbulent channel flow.

Particle SGS Modeling

As we mentioned before, turbulent flows are characterized by chaotic and unpredictable behavior, and most importantly by having a large spectrum of energy levels. This complexity is further compounded when a discrete phase (hereinafter referred to as particles) is present in the flow, as the behavior of these particles is influenced by the underlying turbulent flow structures. Therefore, understanding the interaction between particles and turbulent flows is crucial in many industrial, medical, and environmental applications such as combustion, chemical reactions, patient diagnoses through blood samples, sediment transport, and air pollution.

To properly simulate the particle motion, one has to solve the carrier fluid accurately (fair enough?). The only catch is: solving a highly turbulent flow in an "accurate" fashion usually requires very fine meshes (translating into high CPU requirements). For this and for other practical reasons, in most applications, the majority of the flow is modeled in space or time, or both. By "modeled" here I mean that the continuous phase is actually not resolved (e.g. turbulent structures are not represented well). This is "fine" as far as the fluid flow is concerned, BUT when you're advancing particles on top of that fluid flow, we need to account for the missing part of the picture (I hope that's still fair 😊). Here's where the Sub-grid scale (SGS) modeling comes into play.

It is worth mentioning that it used to be a rule of thumb to neglect the effect of sub-grid scales on particle motion. This was the mainstream approach until Armenio et al., debunked that belief.

"In many previous LES particle-laden studies, the effect of the SGS fluid velocity fluctuations on the particles was assumed to be negligible, and therefore the filtered fluid velocity was used in the particle equation of motion. However, Armenio et al. [1] showed that neglecting SGS effects can lead to inaccurate particle statistics, particularly when the particle relaxation time is small and the LES resolution is coarse (a large filter size). Furthermore, the investigation of Marchioli et al. [2] revealed that neglecting the SGS effects causes errors in particle clustering (preferential concentration) statistics and produces inaccurate wall depositions in turbulent channel flows. Therefore, numerous particle SGS models which account for the influence of the SGS fluid field on the particles have been proposed in the literature." - Cernick et al, 2015

Today, SGS modeling is an essential tool used in particle-laden turbulent flows to study the behavior of small particles, especially ones that are much smaller than the computational mesh used in numerical simulations. The main essence of an SGS model in particulate flows is to capture the effects of these unresolved turbulent flow structures on the behavior of small-to-mid-sized particles, which are critical in understanding the dynamics of particle advancement. - Fair? = Fair and square, but how can we judge particle size (when is a particle characterized as a small, medium-sized, or inertial particle)? I got you -> keep reading 📖

Key Parameters in Particle SGS Modeling

The accuracy of the Particle SGS model depends on the choice of the model parameters, which need to be carefully selected to ensure the model captures the relevant physics. Here I mention the key parameters which are used every particulate flow simulation:

• Particle Relaxation Time (tau_p): The particle relaxation time is the time required for a particle to adjust its velocity to the local fluid velocity. Or the time required for a particle to "relax" its velocity to a new condition of the carrier flow. It is a critical parameter in Particle SGS modeling, as it determines the degree of coupling between the particles and the fluid. The particle relaxation time is typically defined as:

• Kolmogorov Length Scale (tau_k): The Kolmogorov length scale is the smallest length scale in a turbulent flow, and it is critical in Particle SGS modeling. It determines the size of the smallest turbulent eddies and the degree of resolution required to capture the relevant physics. The Kolmogorov length scale is typically used to define the sub-grid scale for Particle SGS modeling.
• Particle Stokes Number (St): The Particle Stokes Number is a dimensionless parameter that characterizes the behavior of particles in turbulent flows. It is defined as the ratio of the particle response time to the characteristic time scale of the turbulent flow. It is a key parameter in Particle SGS modeling, as it determines the degree of particle clustering and the degree of coupling between the particles and the fluid. In simple terms, a high stokes number (>>1) can be pictured as a stone where a small stokes number (<<1) can be seen as a bead of sand or a particle of dust.

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Now let's have a quick look on the main SGS modeling approaches.

Types of SGS modeling in particle-laden flows

Particle SGS models can be broadly classified into two categories (or at least that's how I see it): Discrete Random Walk (DRW) like models and Continuous Random Walk (CRW) models. To have a clear view of how they models work, keep in mind that the main goal here is to supplement particles with the "lost" part of turbulent kinetic energy due to the coarseness of either the spatial field or the temporal time scale or both. Now what are these random walk models and how do they work?

Discrete Random Walk (DRW)

The DRW (sometimes called as Eddy Interacction Model "EIM") model [3] is a common Lagrangian SGS model that models the behavior of individual particles using a stochastic process.  developed by Gosman and Ioannides (1983) and used in most CFD codes including Fluent. In this model, particles are made to interact with the instantaneous velocity field U + u′(t), where U is the mean velocity and u′(t) is the fluctuating velocity. It is clear that DRW is based on reconstructing the instantaneous field from the local mean values of velocity and turbulent intensity.

"By computing the paths of a large enough number of particles, the effects of the fluctuating flow field can be taken into account." - Dehbi A. 2008

In DRW models the turbulent dispersion of particles as a succession of interactions between a particle and eddies which have finite lengths and lifetimes. It is assumed that at t=0, a particle with velocity u_p is captured by an eddy which moves with the mean fluid velocity, augmented by a random “instantaneous” component which is piecewise constant in time. When the lifetime of the eddy is over or the particle crosses the eddy, another interaction is generated with a different eddy, and so forth.

Continuous Random Walk (CRW)

CRW is another class of models, which solves the normalized Langevin equation [4-7]. The model has been successfully applied to simulate particle deposition in inhomogeneous flow configurations such as mixing layer, and shear flows [8].

Unlike DRW models, CRW proved to be a more accurate approach as it better represents. This is due to the explicit modeling of the continuous movement of the flow seen by particles - which is carried to particle through terms that account for the retrieved TKE. The CRW model can be computationally more expensive than the DRW model but it captures the physical effects of turbulence on individual particles more accurately.

Now that we learned about these key parameters, let's see how we can quantify the impact of the SGS models on particles? Keep going…

How to measure the influence of a particle SGS model on my results?

Measuring the influence of a Particle SGS model on simulation results is crucial to evaluate the accuracy of the model and ensure that the results are reliable. Here are some methods that can be used to measure the influence of a Particle SGS model on simulation results:

• Sensitivity Analysis: This (as self-explanatory as it is) involves varying the input parameters of the model to evaluate the sensitivity of the results to changes in these parameters. In Particle SGS modeling, the input parameters can include the particle relaxation time, or the particle Stokes number. By varying these parameters and observing the changes in the simulation results, one can determine the sensitivity of the results to the Particle SGS model.
• Benchmarking: Involves comparing simulation results obtained using different Particle SGS models to experimental or numerical data. By comparing the results obtained using different models, one can evaluate the accuracy of the models and determine their influence on the simulation results.
• Error-based: Error analysis involves comparing the simulation results obtained using different Particle SGS models to evaluate the errors in the models. By quantifying the errors in the models, one can determine their influence on the simulation results.
• Grid Convergence: Grid convergence involves running the simulation on progressively finer meshes to evaluate the impact of mesh resolution on the simulation results. By comparing the results obtained on different meshes, one can determine the level at wich the spatial resolution impacts the Particle SGS model and hence the simulation results.
• Physical Insight: The target physical phenomenon can be used to evaluate the influence of the Particle SGS model on simulation results. By analyzing the behavior of the particles and the underlying turbulent flow structures, one can determine whether the model is capturing the correct (expected) physics and evaluate its influence on the simulation results.

Case Study: Testing Zeta-SGS model in a Channel Flow

For a non-exhaustive ready (if it was not challenging to reach here already 😃), I attach here the most self explanatory Figure I acquire for this study. I hope it is clear enough what point I attempted to get across. If not, feel free to have a read on this at bedtime. 🥱

Summary

In summary, Particle SGS modeling is an essential tool used in the study of dispersed flows (i.e. here the particulate-laden turbulent flows). It aims at correcting the behavior of small-to-mid size particles using additional sub-grid scale equations, while resolving or modeling the turbulent flow structures using LES or RANS equations respectively.

References

[1] V. Armenio, U. Piomelli, and V. Fiorotto, Effect of the subgrid scales on particle motion, Phys. Fluids 11 (1999), pp. 3030–3042.

[2] C. Marchioli, M. Salvetti, and A. Soldati, Some issues concerning large-eddy simulation of inertial particle dispersion in turbulent bounded flows, Phys. Fluids 20 (2008), pp. 040603.

[3] Gosman A.D., Ioannides, E., 1983. Aspects of computer simulation of liquid fuelled combustors. J. Energy 7, 482–490.

[4] A. Dehbi, Turbulent particle dispersion in arbitrary wall-bounded geometries: a coupled CFD-Langevin-equation based approach, Int. J. Multiph. Flow, (2008).

[5] A. Dehbi, A stochastic Langevin model of turbulent particle dispersion in the presence of thermophoresis, Int. J. Multiph. Flow, (2009).

[6] A. Dehbi et al., Validation of the Langevin particle dispersion model against experiments on turbulent mixing in a T-junction Powder Technol, (2011).

[7] A. Dehbi, Validation against dns statistics of the normalized langevin model for particle transport in turbulent channel flows, Powder Technol, (2010).

[8] T. Bocksell et al., Stochastic modeling of particle diffusion in a turbulent boundary layer, Int. J. Multiph. Flow, (2006)

[9] M.J. Cernick, S.W. Tullis & M.F. Lightstone (2015) Particle subgrid scale modelling in large-eddy simulations of particle-laden turbulence, Journal of Turbulence, 16:2, 101-135, DOI: 10.1080/14685248.2014.969888

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