3.1 Subgrid modeling in the particle equation of motion
Instead of DNS also LES can be used for the description of the continuous phase in point-particle simulations. This implies that filtered fluid quantities are employed, usually defined by
$$ \bar{u}({\textbf{x}},t)={\int}_{V}G({\textbf{x}}-{\textbf{y}})u({\textbf{y}},t)d\textbf{y}, $$
(11)
where
G is a convolution kernel and
u a velocity component or the pressure. Possible filter functions are a top-hat filter, which is only unequal to zero inside a small rectangular domain or a Gaussian filter. The governing equations for the filtered fluid velocity are derived by applying the filter operation to the Navier-Stokes equation and to the continuity equation in case of incompressible flow [
40]. Due to the non-linearities in the Navier-Stokes equation some terms appear which cannot be expressed in filtered quantities only, but also contain unfiltered quantities, for example a product of two unfiltered velocity components. Such terms have to be closed by a model, which expresses them as functions of the filtered quantities.
The most important term that needs to be closed is the subgrid stress tensor, which is defined by
$$ \tau_{ij}=\overline{u_{i}u_{j}}-\bar{u}_{i}\bar{u}_{j} $$
(12)
for incompressible flow, where
u
i
is a component of the fluid velocity. Various subgrid models have been proposed. The best known is the Smagorinsky model [
95], which is based on the eddy-viscosity assumption, which renders the model dissipative. This model has later been improved by including corrections near a wall and by the dynamic procedure, which reduces the eddy viscosity in laminar regions [
39]. Other subgrid models are based on the scale-similarity assumption [
10], a Taylor series expansion [
26], or approximate deconvolution [
99]. For many types of turbulent flow LES has become a valuable simulation method, because of the significantly reduced computational costs as compared to DNS.
If LES is applied to the continuous phase in point-particle simulations of particle-laden turbulent flow, also the particle equations of motion contain terms that need to be closed. In particular, the pressure gradient force (
5), the added-mass force (
6), the drag force (
7) and the history force (
10) contain the unfiltered fluid velocity, which is unknown. In cases where the particle relaxation time is large compared to the typical time scales of the turbulent flow and to the smallest time scale resolved in the LES, the subgrid scales in the fluid velocity do not influence the particle motion significantly. In such cases a subgrid model in the particle equation of motion is not required and this approach has been followed in several papers [
108,
115,
118]. The
a priori and
a posteriori study of Armenio et al. [
5] showed that the subgrid scales have only a small effect on the particle motion, but they restricted this study to quantities and test cases where this is typically the case. It has later been shown [
53] that disregard of subgrid scales has a large effect on turbophoresis in LES of particle-laden turbulent channel flow, if the particle relaxation time is of the same order of magnitude as the Kolmogorov time.
Two different types of models have been developed to account for the missing subgrid scales in the particle equation of motion. In the first type a stochastic model is applied, either by adding broadband stochastic noise forcing to the Navier-Stokes equation for the continuous phase [
50], or by adding an additional velocity to the particle equation of motion [
94]. The second type uses approximate deconvolution of the filtered fluid velocity to approximate the unfiltered fluid velocity, which is then used in the particle equation of motion [
51].
Stochastic models for the subgrid contributions to the fluid velocity, which can be used as a subgrid model in the particle equation of motion, are often based on models that have originally been developed for the Reynolds-averaged Navier-Stokes equation. Some of them use a generalized Langevin model of the fluid velocity along the path of a particle and applied this to homogeneous isotropic turbulence [
94], while others use a transport equation for the subgrid scale kinetic energy in particle-laden turbulent channel flow [
116]. Shotorban and Mashayek [
91] performed
a priori and
a posteriori tests of particle-laden decaying homogeneous isotropic turbulence with a stochastic model for the subgrid contributions to the fluid velocity. The
a priori tests showed a very good agreement in the particle kinetic energy for lower Stokes numbers, whereas the use of the subgrid model appeared less necessary for particles with larger Stokes numbers. These results were confirmed by the
a posteriori simulations.
In [
3] three different stochastic models were considered in LES of particle-laden channel flow. The first is based on a transport equation for the subgrid scale kinetic energy, while the other two include the effects of temporal correlation in the subgrid scale contributions to the fluid velocity and the effects of anisotropy. It was shown that both additional effects are especially important in the near-wall region. The RMS of the particle velocity in wall-normal and spanwise direction is strongly influenced by the subgrid contributions to the fluid velocity.
Fede and Simonin [
37] studied the effects of the subgrid scales on particle motion in forced homogeneous isotropic turbulence. They both performed DNS to study the small-scale fluid velocity fluctuations seen by the particle, and solved the particle equation of motion with explicitly filtered fluid velocity fields to evaluate the effect of the subgrid scales on particle statistics. Not surprisingly, they found that particle dispersion and kinetic energy are only affected by the filtering if a significant part of the turbulence kinetic energy is removed by the filter. However, particle accumulation and collision rates are significantly influenced when the particle relaxation time is of the same order or smaller than the subgrid Lagrangian integral time scale measured along particle paths.
Bini and Jones [
14] focused on a stochastic model for the subgrid contributions to the fluid velocity that is able to reproduce the far from Gaussian behavior of the particle acceleration observed in experiments and direct numerical simulation. They achieved this through a nonlinear stochastic differential equation for the relative velocity between particle and fluid. In a later paper they applied this idea to a droplet-laden spatially developing mixing layer [
15].
Pozorski and Apte [
80] investigated the effect of using the filtered velocity field in the particle equation of motion on the particle motion in homogeneous isotropic turbulence. They especially considered and quantified the changes in preferential concentration patterns of particles. They proposed a stochastic Langevin model for the subgrid contributions to the fluid velocity based on the subgrid scale turbulence kinetic energy, which results in a correct reconstruction of the particle turbulence kinetic energy.
The first paper on approximate deconvolution as a subgrid model [
53] considered LES of particle-laden turbulent channel flow at Re
τ
=150 and studied in particular the question whether LES is capable of predicting turbophoresis. They used the dynamic eddy-viscosity model [
39] to describe the fluid flow and inverted the (implicitly defined) filter in Fourier space in the two periodic directions and numerically in the wall-normal direction. Approximate deconvolution (ADM) is only able to recover the energy in the resolved scales. The scales beyond the cut-off wave length cannot be retrieved in this way. Especially in the region of intermediate Stokes numbers, when the particle relaxation time is of the same order of magnitude as the Kolomogorov time, application of the filtered fluid velocity in the particle equation of motion results in significantly under-predicted turbophoresis. The subgrid model based on approximate deconvolution gives a significant improvement. This research was later extended to a higher Reynolds number [
51] and by considering also the approximate deconvolution model [
99] as a subgrid model in the fluid equation.
Shotorban and Mashayek [
90] developed a similar idea and applied this to LES of homogeneous shear flow where the dynamic eddy-viscosity model [
39] was applied to the fluid equation. They studied the effect of the subgrid model on the particle turbulence kinetic energy and turbulence diffusivity and also found a substantial improvement compared to simulations without subgrid model in the particle equation of motion. In a later paper [
92] the same authors also obtained a beneficial effect of the subgrid model in the prediction of preferential concentration of particles in homogeneous shear flow.
Marchioli et al. [
62] studied the effects of ADM in particle-laden channel flow at Re
τ
=150 for particles at several Stokes numbers and for coarse and fine LES. Only in the fine LES, ADM yields improved results compared to a simulation without subgrid model in the particle equation of motion. In the coarse LES the prediction of the filtered fluid turbulence kinetic energy is too high and since ADM adds the kinetic energy lost in the filtering, ADM only makes the results worse. They also considered a subgrid model based on fractal interpolation, but this did not give any improvement of the results. The same authors studied the effects of different filter widths and Stokes numbers in an
a priori and
a posteriori study of the same flow [
63]. Although for coarser LES resolutions the filtered fluid velocity fluctuations are over-predicted, the particle wall accumulation and local segregation always appear to be under-predicted.
Gobert [
42] presented an analytical method to assess the various subgrid models for the particle equation of motion, focussing on ADM [
51] and two stochastic models [
91,
94]. His conclusion was that the stochastic models are able to predict first and second moments of particle velocity accurately for smaller values of the Stokes number, whereas ADM performs better at higher Stokes numbers. For homogeneous isotropic turbulence Gobert and Manhart [
43] proposed a special way of interpolation to retrieve the correct spectral properties of the fluid velocity at the particle position. In this way they obtained better results for statistical properties of the particles than with ADM, both for velocity fluctuations and for preferential clustering. A prerequisite of this model is that a model spectrum should be available, which is not as easy for inhomogeneous turbulent flows.
Bianco et al. [
13] quantified the filtering error in LES of particle-laden turbulent channel flow by means of an
a priori study. They considered filters of different type and width and particles of various Stokes numbers. In this way they determined the statistical properties of the filtering error as a function of the distance to the wall, which can be used to find the requirements a subgrid model in the particle equation of motion should satisfy. This research was extended by Geurts and Kuerten [
41] by considering channel flow at Re
τ
ranging between 150 and 950 and for particles of various Stokes numbers. They considered the use of a combination of ADM and a stochastic model and determined the properties of the stochastic part of the model by means of an
a priori study. A striking result is that the RMS of the stochastic contribution to the fluid velocity is independent of Reynolds number and Stokes number if considered at the same wall-normal position in wall units.
Michałek et al. [
70] used this result to develop a hybrid stochastic-deconvolution model for LES of particle-laden channel flow. In order to satisfy the well-mixed condition,
i.e, the property that passive tracers stay uniformly distributed, an additional term in the Langevin equation for the stochastic part of the fluid velocity turned out to be required, which is a function of the Stokes number. The resulting model showed better results for Re
τ
=950 than simulations based on ADM alone.
All subgrid models developed for LES of particle-laden flow have in common that the Stokes number has a significant effect on the quality of the model.
3.2 LES of one-way and two-way coupled particle-laden flow
Since 2005 quite some research has been devoted to subgrid modeling of the effects of the unresolved scales in the fluid velocity on the motion of particles in LES of particle-laden turbulent flow with one-way coupling. Unfortunately, similar efforts have not been made yet for LES with two-way coupling. It is known from DNS that particles may have an appreciable influence on the turbulence kinetic energy and this influence occurs mainly in the scales that are unresolved in an LES. The effects of this on the resolved scales should be taken into account in the two-way coupling force in LES of particle-laden flow.
Probably the first point-particle LES has been performed by Deardorff and Peskin [
29] for homogeneous shear flow. They do not mention the term large-eddy simulation at all in their paper, but they did consider the spatially averaged Navier-Stokes equation for the fluid and used a spatially end temporally variable small-scale eddy coefficient to model the effects of the subgrid scales. They tracked 48 passive particles in the flow. Effects of the subgrid scales on the particle motion were not taken into account. They calculated particle dispersion, velocity autocorrelation coefficients and statistical properties of particle separation. Uijttewaal and Oliemans [
108] performed point-particle DNS and LES to investigate particle dispersion and deposition in turbulent pipe flow at various Reynolds numbers. The particle equation of motion included only drag and lift forces and buoyancy. Due to the low particle volume fraction two-way coupling and particle collisions have not been applied here. Moreover, effects of subgrid scales of the fluid velocity have not been taken into account. It appeared that the particle relaxation time plays an important role in the deposition behavior of the particles.
Wang and Squires [
115] studied the deposition of particles in vertical turbulent channel flow by means of large-eddy simulation for particle volume fractions so low, that the effects of the particles on the turbulence and the effects of particle collisions are negligible. They used the dynamic eddy-viscosity model as a subgrid model in the fluid equations [
39]. They included the drag and lift force exerted by the fluid on the particles, but only the lift force due to the mean shear, which is in the wall-normal direction. They compared particle deposition rates with DNS results [
69] and found good agreement. They also studied the effects of the subgrid scale velocities on particle deposition by solving an additional transport equation for the subgrid scale turbulence kinetic energy and observed only a small effect.
Boivin et al. [
17] performed LES of gas-solid flows in forced homogeneous isotropic turbulence with two-way coupling. They applied several subgrid models in the fluid equations and several LES resolutions in both
a priori and
a posteriori tests. An increase of the particle mass loading results in a decrease in the dynamic constant in the dynamic eddy-viscosity model. An important conclusion of this paper is that at larger mass loadings a large part of the total dissipation in the flow is a result of interaction between fluid and particles. Therefore, the authors stipulate that modeling errors could have a smaller impact than in single-phase flows.
DNS results of a temporal mixing layer with evaporating droplets have been used to assess models for large-eddy simulation by Okong’o and Bellan [
76]. Various options for the fluid properties at the droplet locations are considered: the real fluid property, which is available in a DNS, the filtered value, a random model based on a Gaussian distribution with known mean value and standard deviation and a deterministic model based on approximate deconvolution. It was observed that the deterministic model gives in general the best agreement with the DNS results, while the random model performs worse than the simulations without model for the subgrid term. In a later paper [
55]
a posteriori tests of the same problem have been performed. Here, however, no subgrid model was applied in the equations for the droplets, but the filtered fluid quantities were used. In this LES computational droplets were used, where each computational drop represents a fixed number of physical drops. In this way the number of droplets tracked in the simulations could be reduced by a factor of 32 as compared to the DNS without reducing the accuracy, provided that accurate subgrid models are applied in the fluid equations.
3.3 LES of four-way coupled particle-laden flow
One of the first papers in which four-way coupling is applied in point-particle LES is by Yamamoto et al. [
117] for flow in a vertical channel at Re
τ
=644. They considered particle volume fractions up to 1.4×10
−4 and studied the influence of particle collisions. They included the drag force, the lift force due to particle rotation and due to shear and the gravity force on the particles, and they applied a deterministic particle collision model. They showed that particle collisions have a large effect on particle concentration and velocity statistics, even at the low volume fraction studied in this paper. This result is in agreement with later results where DNS is applied instead of LES [
54].
A higher particle volume fraction of 0.013 has been studied by Vreman et al. [
112] in LES of turbulent channel flow with four-way coupling. In this work only drag and gravity have been taken into account. The particles have a large effect on the turbulence. They result in a thinner boundary layer, increased gas velocity fluctuations in the streamwise direction and decreased in the other two directions.
Malloupas and Van Wachem [
60] performed LES of particle-laden channel flow at a particle volume fraction of 4.8×10
−4. They considered the effects of two-way and four-way coupling separately, compared the hard-sphere and soft-sphere collision models and studied the effects of various subgrid models in the LES. Moreover, they considered both smooth and rough walls. They compared their results with experimental data for a bulk Reynolds number of 42,000 based on the height of the channel. They observed that the differences between the Smagorinsky subgrid model [
95] and the dynamic model [
39] are very similar due to the relatively high resolution of the simulation. Particle collisions and the rough-wall model have a large effect on the results and lead to better agreement with experimental data. The difference between results of the hard-sphere and soft-sphere collision models is small.
Breuer and Aletto [
18] performed point-particle LES with four-way coupling, using an efficient deterministic collision search algorithm. They used the model of Pozorski and Apte [
80] as a subgrid model in the particle equation of motion. They investigated the effects of the particles and of the collisions in particle-laden flow in a turbulent channel at a bulk Reynolds number of 11,900, based on half the channel height and for a particle volume fraction of 7.3×10
−5. For this volume fraction the effect of the particles on the turbulence is very small, but the effect of particle collisions is large. This is in agreement with observations found for point-particle DNS of turbulent channel flow at a similar particle volume fraction [
54]. The results of this study are in good agrement with experimental results. In the same paper a second application is investigated of cold flow in a combustion chamber, where a jet of particle-laden flow mixes with an unladen annular flow. Two different mass loadings have been considered. For the highest mass loading of 110 % the particles have an appreciable influence on the turbulence. Also for this application good agreement with experimental results was obtained.
In a later paper [
2] the same authors studied four-way coupled point-particle LES of horizontal pipe flow at a bulk Reynolds number of 120,000 based on the diameter op the pipe. They considered poly-disperse particles and took into account particle rotation and apart from the drag and buoyancy forces also the Saffman lift force and the Magnus force due to particle rotation. They compared their results with experimental results in a smooth glass pipe and in a rough steel pipe, where the roughness is modeled in the simulations by using a wall-normal and tangential restitution coefficient unequal to 1, resulting in momentum loss and by considering the shadow effect, which becomes important for smaller impact angles. They especially focused on secondary flow, which appeared not te be driven by the particles.
Breuer and Almohammed [
19] included a model for particle agglomeration to four-way coupled point-particle LES of turbulent channel flow. They added a cohesive Van der Waals force between two colliding particles, which makes it possible that particles stick together after the collision and move with the same velocity. They implemented several models for the particle after agglomeration, all assuming a spherical shape. In one model the volume of the particles is conserved and the mass density constant, in a second model the moment of inertia around the center of mass is conserved, and in the third a closely-packed sphere is assumed. In the latter two models the mass density of the agglomerate is smaller than that of the original particles. The final results of the three models are only slightly different. The authors studied collision and agglomeration rates as functions of the wall-normal coordinate and investigated the effects of two-way coupling, the use of a subgrid model in the particle equation of motion and the use of lift force. These three together result in a significant reduction in the rates of both collision and agglomeration.