A stochastic formulation for the drag force based on multiscale numerical simulation of fluidized beds

Highlights

• Micro-scale particle resolved and meso-scale DEM-CFD simulations of a small size fluidized bed,

• Characterization of the drag coefficient as a probability density function for a pair of (Reynolds, porosity),

• New stochastic drag model with parameters extracted from micro-scale particle-resolved simulations,

• Improvement of classical DEM-CFD models using a Gaussian filtering kernel and a new stochastic drag model to recover the right level of particles fluctuations.

Abstract

In Esteghamatian, A., Bernard, M., Lance, M., Hammouti, A. and Wachs, A., 2017, Micro/meso simulation of a fluidized bed in a homogeneous bubbling regime, International Journal of Multiphase Flow, 92, 93–111, we performed a one-to-one comparison of micro-scale Particle-Resolved Simulation (PRS) results and meso-scale two-way coupled Euler-Lagrange (a. k. a. DEM-CFD for Discrete Element Method-Computational Fluid Dynamics) simulation results in a homogeneous bi-periodic liquid/solid fluidized bed. These data showed an acceptable agreement between micro- and meso-scale predictions for integral measures as, e.g., pressure drop and bed height. However, particles fluctuations are markedly underpredicted in DEM-CFD simulations, especially in the direction transverse to the main flow. The filtered DEM-CFD porosity appears to be a coarse descriptor of the local microstructure and hence is the primary reason for the inability of the meso-scale DEM-CFD model to fully recover the actual features of the flow. In this paper, we explore two different directions in improving the meso-scale model: (a) we adopt an improved inter-phase coupling scheme and (b) we introduce a stochastic formulation for the drag law derived from our PRS results. The new stochastic drag law, which incorporates information on the first and second-order moments of PRS results, shows promises to recover the appropriate level of particles fluctuations.

Introduction

Owing to their wide presence in environmental phenomena, pharmaceutical industry, energy sector, chemical processes, biological flows, etc., particle-laden flows have been subject of extensive analytical, numerical and experimental studies. Thanks to substantial advances in the computing resources, numerical approaches attracted much attention in the past decades. Accordingly, more care is given to the development of more advanced numerical models with higher levels of accuracy.

The nonlinear and multi-scale nature of fluid/particle interactions in such systems highlights the rich dynamics of the system and calls for well-adapted numerical models. A variety of numerical models depending on the required computational resources and desired level of accuracy have been proposed in the literature (Van der Hoef, van Sint Annaland, Deen, Kuipers, 2008, Tenneti, Subramaniam, 2014). At the most refined end of the spectrum, micro-scale models based on PRS are able to resolve the flow at the scale of the particle boundary layer. Meso-scale DEM-CFD models filter the fluid field at scales at least one order of magnitude larger than the particle volume. Closure laws are hence required to estimate the momentum transfer between particle phase and fluid phase. Traditionally, this filtering operation is implicit as the volume of fluid grid cells is usually much larger than the particle volume. As contributions from a Lagrangian particle are simply distributed to the Eulerian fluid grid cells intersected by this particle, the filter size implicitly spans a maximum of two fluid grid cells in each direction. Over the past few years, a new and enhanced approach that decouples the filtering kernel from fluid grid cells has been developed (Capecelatro, Desjardins, 2013, Link, Cuypers, Deen, Kuipers, 2005, Pepiot, Desjardins, 2012). This methodology provides the model with the flexibility of using fluid grid cells with volumes comparable to the particle volume, or even smaller. The filter size is typically a few particle diameters and spans the required number of fluid grid cells as controlled by the particle diameter to fluid grid size ratio. At the coarsest end of the spectrum, macro-scale models (either Two-Fluid models or moments based methods, see Fox (2012) for more details) consider both particle phase and fluid phase as continuous media described by Eulerian transport equations. In the particular framework of Two-Fluid models, the kinetic theory of granular media is employed to consider particle-particle and particle-wall interactions.

Meso-scale models have gained a large popularity in simulation of dense fluid-particle systems during the past two decades (Pepiot, Desjardins, 2012, Tsuji, Kawaguchi, Tanaka, 1993). In contrast to Two-Fluid models, a major source of inaccuracy is eliminated by directly taking into account particle-particle and particle-wall interactions. Also, employing closure laws for fluid-particle momentum transfer leads to a drastic reduction of the computational load (by several orders of magnitude) as compared to micro-scale models. The reasonable compromise between level of accuracy and required computational effort favors this type of models in the simulation of meso-scale systems, i.e., systems that comprise 10³ to 10⁹ particles and the upper limit is increasing with advances in parallel algorithms, High Performance Computing and computational resources.

While meso-scale models have been widely applied to fluid-particle simulations since the 90s, only recently attention has been given to the accuracy of the model, particularly in terms of predictions of second-order particle statistics (Esteghamatian, Bernard, Lance, Hammouti, Wachs, 2017, Kriebitzsch, Van der Hoef, Kuipers, 2013, Subramaniam, Mehrabadi, Horwitz, Mani, 2014). Since the fluid field remains unresolved at the level of particles, particle-induced fluctuations and in turn particle agitations are suppressed to some extent by meso-scale models. Subramaniam et al. (2014) emphasized the underprediction of particles acceleration by point-particle models in decaying isotropic turbulent flow. Kriebitzsch et al. (2013) also pointed out the underestimation of the drag force in DEM-CFD simulations as compared to PRS. In our previous work (Esteghamatian et al., 2017a), we also observed the underestimation of the particles granular temperature particularly in the direction transverse to the mean flow.

In Esteghamatian et al. (2017b), we further analyzed the PRS of fluidization to have a better insight into the dominant mechanisms of particles motion. In short, we have so far drawn the following conclusions: (i) the meso-scale model partially resolves fluid fluctuations and underpredicts particles fluctuations, and (ii) local fluid fluctuations around the particles depend on the filter size and smoothly increase with the averaging control volume particularly in a homogeneous system. (i) and (ii) emphasize that any information transfer from PRS to the meso-scale model requires to take into account the effect of the filter size. Strictly speaking, fluctuations captured by the meso-scale model are limited by the size of the filter. In that sense, the meso-scale model can be considered as the equivalent of the Large Eddy Simulation (LES) approach in an analogy with single-phase turbulence modeling. However, contrary to single-phase turbulence where instabilities are generated at the resolved scales, small-scale instabilities - also called pseudo-turbulence - are generated at the particle scale which is inherently not accessible with meso-scale model. The lack of a self-similar pattern and the generation of instabilities at unresolved scales prevent us from developing a universal formalism that relates momentum transfer to mean values at the filter scale. This has been previously addressed in the literature and accordingly, closures based on PRS or experiments have been proposed to correlate the mean momentum transfer to integral measures (Di Felice, 1994, Ergun, 1952, Hill, Koch, Ladd, 2001). We hope that a similar procedure can be employed to recover momentum transfer fluctuations. Given that the PRS solution already accounts for all the relevant spatial scales of the system, it is technically possible to feed the meso-scale model with the missing information at the particle level in a multi-scale framework.

The scope of the present study is to answer the following questions: (a) to which extent the performance of meso-scale models are limited by the fluid field discretization?, (b) is there any methodology to quantify the sub-grid scale fluctuations from PRS results ? To answer (a), we perform a grid refinement study with our meso-scale model. A basic feature of many classical meso-scale models (including ours in Bernard, Climent, Wachs, 2016, Esteghamatian, Bernard, Lance, Hammouti, Wachs, 2017) is that the fluid grid cell is used as the fluid/particle averaging kernel. While pretty efficient when particles are much smaller than the fluid grid cell, it causes numerical stability issues when the size of particles are larger than (or even comparable to) the fluid grid cell. As suggested by Capecelatro and Desjardins (2013), it is technically possible to decouple the fluid/particle averaging kernel from the fluid grid cell. We have adopted this approach to attain the flexibility of using fine grid cells and perform a full grid refinement study. In an effort to answer (b), we have characterized the drag coefficient experienced by particles in PRS as a Probability Density Function, in contrast to the classical deterministic drag laws. This provides us with a basis for our stochastic formulation of the drag force. Stochastic approaches have been widely applied in turbulence modeling particularly involving reactive flows (Dreeben, Pope, 1998, Muradoglu, Jenny, Pope, Caughey, 1999, Pope, 1994). Sommerfeld and Zivkovic (1992) and Oesterle and Petitjean (1993) introduced a probabilistic approach to model inter-particle collisions inspired by the kinetic theory of gases. In spray modeling, Subramaniam (2000) proposed a statistical representation of a spray based on a stochastic point process model. Employing a stochastic approach to model the drag force is quite scarce in the literature, yet not nonexistent. In homogeneous isotropic turbulence and considering a one-way point-particle assumption, Fede et al. (2006) proposed a stochastic model to reconstruct the sub-filter fluid fluctuations based on a Langevin equation. DNS data were used to determine the constants of the model. The authors have shown that the correct level of particles kinetic energy can be attained by the LES of the fluid field and a discrete particle method incorporating a stochastic closure. A similar approach was adopted by Berrouk et al. (2007) for the LES of inertial particles in a turbulent shear flow. Andrews et al. (2005) employed an ad hoc approach to formulate a stochastic drag coefficient in DEM-CFD coarse-grid simulations of vertical risers. The authors used DEM-CFD simulations with relatively refined grid sizes as a reference to determine the model’s constants. In a more general approach, Tenneti et al. (2016) have recently proposed a stochastic formulation for the drag force based on PRS and the kinetic theory of granular flows.

The rest of the paper is organized as follows: In Section 2, we present the governing equations and numerical aspects of our numerical tool. In Section 3, we introduce the simulation parameters. Our stochastic formulation for the drag coefficient is detailed in Section 4. In Section 5, we present the main body of results in micro/meso simulations of low density ratio fluidization. A single test case of high density ratio fluidization is subsequently studied. Finally, the conclusions are summarized in Section 6.