A Review of the Homogenized Lattice Boltzmann Method for Particulate Flow Simulations: From Fundamentals to Applications

Abstract

The homogenized lattice Boltzmann method (HLBM) has emerged as a flexible computational framework for studying particulate flows, providing a monolithic approach to modeling pure fluid flows and flows through porous media, including moving solid and porous particles, within a unified framework. This paper presents a thorough review of HLBM, elucidating its underlying principles and highlighting its diverse applications to particle-laden flows in various fields as reported in literature. These include studies leading to new fundamental knowledge on the settling of single arbitrarily shaped particles as well as application-oriented research on wall-flow filters, hindered settling, and evaluation of the damage potential during particle transport. Among the strengths of HLBM are its monolithic approach, which allows seamless simulation of different fluid-solid interactions, and its ability to handle arbitrary particle shapes, including irregular and concave geometries, while resolving surface interactions to capture local forces. In addition, its parallel scheme based on the lattice Boltzmann method (LBM) results in high computational efficiency, making it suitable for large-scale simulations, even though LBM requires small time steps. Important future development needs are identified, including the addition of a lubrication force correction model, performance enhancements, such as support for hybrid parallelization and GPU, and the extension of compatible contact models to accommodate concave shapes. These advances promise expanded capabilities for HLBM and broader applicability for solving complex real-world problems.

1. Introduction

The modeling and simulation of particulate flows are important in several academic and industrial areas within and beyond process engineering. For example, a better understanding of particulate flow dynamics contributes to advances in aerosol drug delivery [1], optimization of flow conditions for combustion processes to improve engine efficiency [2], and particle classification [3]. Process engineering research aims to understand the transformation processes of substances into other states, often described and modeled as particulate flows.

1.1. Overview of Particle-Laden Flows Characteristics

Particle-laden or particulate flows are typically modeled as two-phase or two-component fluid flows characterized by the presence of two distinct dynamic phases or components. The first component, called the carrier fluid, forms a continuous phase, while the second component, called the dispersed phase or component, exists as discrete and separate entities [4]. Depending on the states of matter of the carrier and the dispersed components, several distinctions are made:

• Aerosols occur when the carrier component is a gas and the dispersed component is either a liquid (fog, mist) or a solid (dust, soot).
• Bubble flows occur when the carrier component is a liquid and the dispersed component is a dilute gas.
• Suspensions occur when the carrier component is a liquid and the dispersed component is a solid.
• Emulsions occur when both components are in a liquid state.

In foams, the gas phase is typically immobilized by a surrounding liquid or solid, resulting in a partly dynamic character. Therefore, foams are not classified as particulate flows.

If all particles within a dispersion have the same characteristic properties, the system is called monodisperse. Otherwise, it is referred to as polydisperse. These properties may be of physical, chemical, or organic origin. The Feret diameter is a common parameter that represents the maximum distance between two parallel tangential planes perpendicular to a given direction [5]. Occasionally, an average is calculated over several directions or a specific direction is chosen to give the minimum or maximum Feret diameter depending on the application.

Another size measure is the equivalent diameter, which is determined by comparing a particular property of an irregular particle to that of a regularly shaped particle. There are several types of equivalent diameters depending on the property chosen for comparison. The spherical equivalent diameter is particularly important in process engineering, where an irregularly shaped particle, such as a grain of sand, is related to a sphere of equivalent value by the diameter of the sphere. Other relevant properties in dynamic particle systems include the aspect ratio, volume, sphericity, or convexity [6]. Distributions of these shape parameters, along with other physical, chemical, or organic parameters, help to characterize a particle system and its dynamics.

Using a size parameter, such as the averaged Feret diameter, which considers the average size of the system across all or specific directions, allows for the differentiation between

• molecularly dispersed dissolved particles having a size of less than 1 nm,
• colloidally dispersed dissolved particles having a size between 1 nm and 1 μm,
• coarsely dispersed dissolved particles having a size greater than 1 μm.

Interactions among particles and between particles and the carrier component are commonly observed and must be considered. For colloidal and coarsely dispersed dissolved systems, the relevant scale ranges from ${10}^{-9}\mathrm{m}$ to ${10}^{-4}\mathrm{m}$, which corresponds to the nano- and microscale. Since process engineering deals with transformation processes relevant to industrial applications [7], investigations are performed at process scales. Typical length scales in this domain range from $1\mathrm{m}\mathrm{m}$ to $100\mathrm{m}$. Consequently, sophisticated models are required to describe nano- or microscale interactions, such as agglomeration, separation, clustering, or phase changes, that affect process-scale suspension dynamics. The following example illustrates challenges and limitations.

In solid-liquid separation, effective separation of fine particle systems ranging from $100\mathrm{n}\mathrm{m}$ to $10\mu \mathrm{m}$ remains a formidable challenge. Experiments at such scales are difficult to perform, and the lack of multiscale models and simulation tools makes experiments even more difficult. Despite efforts in various processes, such as centrifugal air classification [8] or cross-flow filtration [9], separation efficiencies remain insufficient. Within the size range considered, hydrodynamic, inertial, and particle interaction forces approach the equilibrium, rendering conventional methods ineffective.

Larger particles (>$10\mathsf{\mu }\mathrm{m}$) allow for the use of inertial forces, as seen in centrifugal separators. Conversely, particles smaller than $100\mathrm{n}\mathrm{m}$ show the dominance of diffusion and van-der-Waals forces. Here, techniques, such as gas-expanded liquids [10,11], are suited for particle separation. Although some processes are applicable in this size range [12], their separation efficiency tends to be lower compared to the above methods due to the absence of predominant diffusion or inertia effects.

Chromatography [13] or ultra- and microfiltration [14,15] are only suitable for minute quantities. The maximum throughput of cyclones is significantly limited by the particle diameter to be separated, e.g., for a particle diameter of $10\mathsf{\mu }\mathrm{m}$ the maximum cyclone radius is $0.15\mathrm{m}$ [16].

Ensuring adequate separation efficiency, especially when dealing with oversized particles, is critical for many applications, such as wafer manufacturing. In chemical mechanical planarization, the presence of oversized particles in the polishing suspension renders the wafer unusable. The limitations inherent in the classification of fine particle systems underscore the critical need for a fundamental understanding of the interactions within these suspensions.

Complex models are essential for a detailed understanding of the dynamics of fine particle systems. These models serve as the basis for both physical and numerical experiments, providing insight into areas inaccessible by direct measurement. The study of particle-laden flows requires careful consideration of individual particle shapes, since they significantly influence the flow behavior [17,18]. Moreover, practical applications typically involve polydisperse suspensions. Therefore, a comprehensive model capable of capturing different particle properties, in particular shape characteristics, as well as a simulation method and its implementation are crucial.

The following key aspects need to be addressed: First, flow simulations must strike a balance between detailing the influence of different particle shapes and ensuring computational efficiency. Second, realistic models and algorithms are needed to accurately describe particle dynamics. Third, thorough understanding, modeling, and accurate simulation of the coupling between the fluid and particle components are essential [19].

1.2. Overview of Modeling Approaches and Their Limitations

As discussed above, effective simulation of particulate flows requires a model that consists of at least three distinct submodels. The first submodel addresses the behavior of the carrier fluid, the second focuses on the dispersed component, and the third accounts for their interaction. Each subproblem is characterized by mathematical equations that are discretized and then translated into algorithms for computational implementation.

Within the length scales considered, i.e., >$100\mathrm{n}\mathrm{m}$, continuous Eulerian descriptions such as the Navier–Stokes, Stokes or Euler equations, are often used to describe the behavior of the carrier fluid.

At various scales, including nano-, micro-, and process scales, Lagrangian descriptions are commonly applied to characterize the dispersed component. This approach models discrete particles according to Newton’s law of motion. At the atomistic scale, with diameters around $0.1\mathrm{n}\mathrm{m}$, the interaction between neutral pairs of atoms is typically described by the Lennard–Jones potential [20] or alternatively by the DLVO theory [21] which integrates van-der-Waals attraction, electrostatic repulsion, and Born repulsion phenomena. Beyond the atomic scale, molecular potentials become negligible. Instead, properties such as recoil forces due to deformation and frictional forces become important. These effects are often captured by the widely used spring-dashpot model [22].

In certain cases, the dispersed component can be effectively described by a continuous Euler formulation. In this approach, advection-diffusion equations are frequently adopted as the primary governing equations [1]. For monodisperse systems, a single equation proves adequate, whereas polydisperse systems require multiple equations to accurately represent the diverse particle sizes and distributions. This strategy is typically used to model diffusion or when the use of a Lagrangian approach becomes computationally prohibitive. The latter may occur when the number of suspended particles is too large to simulate individually or when the exact trajectory of each particle is not of primary interest. The computational cost associated with an Euler scheme scales primarily with the numerical resolution of the computational domain rather than with the number of particles. However, when considering nano- or microscale particles in process-scale applications, the number of particles to be simulated can easily exceed billions. In filtration, for example, scenarios assuming typical values for cake height and nutsche geometry can result in simulations involving approximately 30 billion particles. Despite their computational efficiency, Eulerian models often fail to capture finer details at the nano- or microscale because they rely on averaged quantities. They tend to neglect several potentially important properties, such as size or shape distributions, as well as inter-particle interactions.

If only the influence of the fluid on the particles is considered, the model is called one-way coupled. A model is classified as two-way coupled, if it assumes mutual influence between the fluid and the particles, and vice versa. It thus introduces back-coupling. Four-way coupled models consider interactions between the particles themselves as well as particle-wall interactions, providing a more comprehensive representation of the dynamic interactions within the system. The coupling approaches vary depending on the model used for the dispersed component.

In Lagrangian models, hydrodynamic forces are typically computed as integral forces for fully resolved particles or they are modeled using drag laws, such as Stokes, Kaskan, Kürten, or Martin, for unresolved particles of simpler shapes, including spheres and ellipsoids [23]. These models often assume a dilute regime, implying the absence of other particles. Sophisticated models for denser situations rely on heuristics and measurements and are usually only applicable to simpler scenarios, such as swarm sedimentation of spheres [24]. These latter approaches are known as subgrid-scale methods, where particles are described as points rather than volumetric domains, while transporting properties, such as mass, momentum, energy, and others.

When two-way coupling is required, adjustments to the fluid dynamics equations are essential. In fully resolved simulations, the fluid domain changes dynamically with time due to particle motion. For subgrid-scale models, modeling back-coupling typically involves extending the Navier–Stokes equations to the volume-averaged Navier–Stokes equations (VANSE) [25], incorporating a dynamic solid volume fraction. The challenges in both back-coupling approaches arise primarily in the numerical implementation rather than in the modeling phase.

The coupling of two Eulerian models follows an approach similar to that of subgrid-scale Lagrangian models. In addition, a drag model is required that imposes similar restrictions on simpler shaped particles in a dilute regime as described above. Back-coupling can also be achieved using VANSE, but requires the provision of diffusion and convection coefficients. Currently, closure approaches still rely on measurements or heuristics and models are accessible only for very simple suspensions. A representation for complex particle shapes may be lacking [19].

1.3. Overview of Numerical Solution Approaches and Their Limitations

A widely used numerical approach for Lagrangian models, especially for simulations involving large numbers of particles (>1000), is the discrete element method (DEM). In its conventional form, DEM treats particles as ideal spheres. A comprehensive review of DEM can be found in the works of Zhu et al. [26,27]. This subgrid-scale approach treats particles as mass points whose dynamics are governed by applied forces, such as drag, according to Newton’s second law. These methods typically use explicit time discretization schemes, such as forward Euler, which require small time steps. Consequently, the simulation involves a separation of free flight and collisions, where the forces acting on the particles adjust their velocity for the next time step. Extensions of the DEM have been developed to accommodate ellipsoidal [28,29,30,31] and polygonal [32,33,34] particle shapes. However, as the complexity of the particle shapes increases, so does the computational complexity of the simulations.

More sophisticated DEM approaches combine convex shapes. For example, the multi-sphere approach approximates arbitrary particle shapes by assembling multiple spheres [35,36]. Other convex shapes can also be used [37]. In such approaches, forces acting on each element associated with the particle are computed and aggregated to characterize the behavior of the entire object. However, accurately representing arbitrarily shaped particles requires a significant number of spheres or complex subelements, resulting in significant computational requirements.

In recent years, the Immersed Boundary Method (IBM) has received considerable attention as a numerical technique for integrating fully resolved Lagrangian particles with Eulerian fluid descriptions. This method uses a discrete set of Lagrangian points to represent the surface of particles within an Eulerian mesh [38,39], allowing the simulation of particles with arbitrary shapes. However, this approach requires a sufficiently fine resolution of the computational grid to accurately capture interactions between the Eulerian mesh and the Lagrangian points by interpolation. The combination with modern collision detection algorithms [31,40] leads to high computational costs.

In Lagrangian methods, the treatment of particle contact is a major challenge. While computationally efficient spring-dashpot models, i.e., the Hertz-Mindlin model, are typically applicable to spherical particles only [22], handling more complex shapes leads to higher computational costs. Common models are limited to polygonal [28,41,42] or convex [40] particles. Only recent developments allow for the modeling of more complex shapes [37,43,44,45], but their application to large-scale problems with thousands of particles and four-way coupling remains challenging.

In the case of IBM, back-coupling is inherently integrated, while in the case of DEM, it is not. Zhang et al. [46] introduced an IBM-based approach that relies on pulse exchange using the lattice Boltzmann method (LBM). However, this method requires further calibration through additional simulations and has been shown to lack convergence for single sphere sedimentation problems [47]. Another LBM-based approach proposed by Blais et al. [25] solves the VANSE, but it also exhibits convergence problems in dynamic scenarios, although these could potentially be solved using recently proposed methods [48,49].

Another numerical approach to simulating particle-laden flows uses Eulerian models in which the particles are represented as distributions. Discretization methods based on LBM include the multi-phase/component methods developed by Shan and Chen [50] and methods that discretize the advection-diffusion equation [51,52]. The latter uses a one-way coupling approach and incorporates a Stokes drag law. This method has been extended to a two-way coupling approach by introducing an LBM-based discretization for the VANSE [48,49]. While these models offer efficient computation, they are limited by their relatively low level of detail. Currently, these models allow for the inclusion of population balances and for the capture of particle size distributions, inertial forces, and particle interactions [53,54]. However, Eulerian models for determining particle shape properties or their distribution are not yet available.

In conclusion, there is currently a lack of methods capable of simulating large numbers of arbitrarily shaped particles. Some approaches lack the required level of detail regarding shape effects, while others struggle with their computational demands. Therefore, there is a need to develop, test, and implement new methods that meet these criteria [19].

1.4. Aims of the Homogenized Lattice Boltzmann Method

The aim of this paper is to comprehensively review the homogenized lattice Boltzmann method (HLBM) [55,56], which was introduced specifically to address the above challenges. HLBM’s strengths lie in its monolithic nature, which seamlessly accommodates pure fluid dynamics, porous media, and moving solid and porous particles. It excels at representing arbitrary particle shapes, provides surface-resolved simulations including detailed surface force calculations, and operates efficiently in parallel, thus accelerating computations. By providing a detailed investigation of HLBM and its capabilities of simulating complex particle-laden flows, this review aims to contribute to the advancement of numerical methods in this field.

1.5. Structure of This Review Paper

Section 2 provides details on the underlying models, while Section 3 explains the numerical methods used. Section 4 presents reported applications of HLBM, so that the strengths and weaknesses as well as future development needs can be discussed in Section 5. Finally, Section 6 summarizes and concludes the review.

2. Modeling

This section describes the numerical study of particulate flows by applying a fully resolved Eulerian-Lagrangian approach to the problem. The essential parts are the fluid (denoted by f), a particle set (denoted by P) consisting of individual particles $\mathrm{p}\in \mathrm{P}$, and the interaction between these components. The approach is introduced in the following sections, where we consider a fluid domain ${\mathsf{\Omega }}_{\mathrm{f}}\left(t\right)$ and particle domains ${\mathsf{\Omega }}_{\mathrm{P}}\left(t\right)={\bigcup }_{\mathrm{p}\in \mathrm{P}}{\mathsf{\Omega }}_{\mathrm{p}}\left(t\right)$. The entire computational domain is thus given by $\mathsf{\Omega }={\mathsf{\Omega }}_{\mathrm{f}}\left(t\right)\cup {\mathsf{\Omega }}_{\mathrm{P}}\left(t\right)$. Note that both the fluid and particle domains are time-dependent, while $\mathsf{\Omega }$ remains constant.

2.1. Fluid Dynamics

The fluid dynamics model is based on the conservation laws of classical hydrodynamics. Firstly, the principle of mass conservation reads [57]

$$\begin{array}{c}\hfill {\displaystyle \frac{\partial {\rho }_{\mathrm{f}}}{\partial t}}+\mathsf{\nabla }·\left({\rho }_{\mathrm{f}}{\mathit{u}}_{\mathrm{f}}\right)=0,\end{array}$$

(1)

with the fluid density ${\rho }_{\mathrm{f}}$, velocity ${\mathit{u}}_{\mathrm{f}}$, and dynamic viscosity $\eta$. Secondly, the conservation of momentum is described using Cauchy’s equation of motion [58]

$$\begin{array}{c}\hfill {\displaystyle \frac{\partial {\rho }_{\mathrm{f}}{\mathit{u}}_{\mathrm{f}}}{\partial t}}+{\rho }_{\mathrm{f}}({\mathit{u}}_{\mathrm{f}}·\mathsf{\nabla }){\mathit{u}}_{\mathrm{f}}=\mathsf{\nabla }·\sigma +{\mathit{F}}_{\mathrm{f}},\end{array}$$

(2)

where $\sigma$ denotes the Cauchy stress tensor [58] representing surface forces and ${\mathit{F}}_{\mathrm{f}}$ is the total of all volume forces acting on the fluid.

In case of a Newtonian fluid with uniform density ${\rho }_{\mathrm{f}}$ and temperature-independent volume forces ${\mathit{F}}_{\mathrm{f}}$, the above conservation laws simplify to the incompressible Navier–Stokes equations [57]

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \mathsf{\nabla }·{\mathit{u}}_{\mathrm{f}}& =0,\hfill \\ \hfill {\rho }_{\mathrm{f}}\left({\displaystyle \frac{\partial {\mathit{u}}_{\mathrm{f}}}{\partial t}}+({\mathit{u}}_{\mathrm{f}}·\mathsf{\nabla }){\mathit{u}}_{\mathrm{f}}\right)& =-\mathsf{\nabla }p+\eta \mathsf{\Delta }{\mathit{u}}_{\mathrm{f}}+{\mathit{F}}_{\mathrm{f}},\hfill \end{array}\end{array}$$

(3)

with the pressure p.

Coupling from a solid component to the fluid is introduced by adding a linear body force term according to Spaid and Phelan [59]

$$\begin{array}{c}\hfill \mathit{F}(\mathit{x},t)=-\eta {\mathbf{K}}^{-1}(\mathit{x},t){\mathit{u}}_{\mathrm{f}}(\mathit{x},t),\end{array}$$

(4)

which introduces the influence of a porous medium with the permeability $\mathit{K}$. By assuming isotropy and periodicity, $\mathit{K}$ reduces to its only eigenvalue K. This leads to the homogenized Navier–Stokes equations [19,60], in which the momentum equation becomes

$$\begin{array}{cc}\hfill {\rho }_{\mathrm{f}}\left({\displaystyle \frac{\partial {\mathit{u}}_{\mathrm{f}}}{\partial t}}+({\mathit{u}}_{\mathrm{f}}·\mathsf{\nabla }){\mathit{u}}_{\mathrm{f}}\right)& =-\mathsf{\nabla }p+\eta \mathsf{\Delta }{\mathit{u}}_{\mathrm{f}}-\eta K^{-1}{\mathit{u}}_{\mathrm{f}}+{\mathit{F}}_{\mathrm{f}}.\hfill \end{array}$$

(5)

The permeability-dependent term in the applied body force follows

$$K^{-1}(\mathit{x},t)=\left\{\begin{array}{cc}{K}_{\mathrm{p}}^{-1},\hfill & \mathrm{if}\mathit{x}\in {\mathsf{\Omega }}_{\mathrm{p}}\left(t\right)\hfill \\ 0,\hfill & \mathrm{otherwise}\hfill \end{array}\right.,$$

(6)

which means that the force disappears outside of particles or depends on the permeability $K_{\mathrm{p}}$ of the particle intersecting the position.

Note that Guo and Zhao [61] criticize the lack of a nonlinear inertial and convective term. Therefore, they include a Forchheimer term leading to

$$\begin{array}{c}\hfill \mathit{F}(\mathit{x},t)=-{\displaystyle \frac{\eta }{K}}{\mathit{u}}_{\mathrm{f}}(\mathit{x},t)-{\displaystyle \frac{ϵ}{\sqrt{K}}}{\displaystyle \frac{1.75}{\sqrt{150{ϵ}^3}}}\left|\right|{\mathit{u}}_{\mathrm{f}}(\mathit{x},t)\left|\right|{\mathit{u}}_{\mathrm{f}}(\mathit{x},t),\end{array}$$

(7)

with the porosity $ϵ$. However, in previous studies of particulate flows, the model of Spaid and Phelan [59] has led to accurate results as shown in Section 4.

2.2. Particle Dynamics

For the particle component, Newton’s second law of motion is used to describe the particles’ translation

$$\begin{array}{c}\hfill m_{\mathrm{p}}{\displaystyle \frac{\partial {\mathit{u}}_{\mathrm{p}}}{\partial t}}={\mathit{F}}_{\mathrm{p}},\end{array}$$

(8)

and rotation

$${\mathit{I}}_{\mathrm{p}}{\displaystyle \frac{\partial {\omega }_{\mathrm{p}}}{\partial t}}+{\omega }_{\mathrm{p}}\times \left({\mathit{I}}_{\mathrm{p}}·{\omega }_{\mathrm{p}}\right)={\mathbf{T}}_{\mathrm{p}}.$$

(9)

The symbols $m_{\mathrm{p}}$, ${\mathit{I}}_{\mathrm{p}}$, ${\mathit{u}}_{\mathrm{p}}$, and ${\omega }_{\mathrm{p}}$ represent the particle’s mass, moment of inertia, velocity, and angular velocity, respectively. $\mathit{F}\mathrm{p}$ and ${\mathit{T}}_{\mathrm{p}}$ denote the total force and torque acting on the particle, respectively. They refer to the particle’s center of mass ${\mathit{X}}_{\mathrm{p}}$.

The forces acting on the particle can also be divided into surface forces and volume forces. The latter are caused by gravity and buoyancy, which are commonly included via their net force ${\mathit{F}}_{\mathrm{p},\mathrm{V}}=({\rho }_{\mathrm{f}}-{\rho }_{\mathrm{p}})V_{\mathrm{p}}\mathit{g}$, with the particle density ${\rho }_{\mathrm{p}}$, the particle volume $V_{\mathrm{p}}$, and the gravitational acceleration vector $\mathit{g}$. The surface forces consist of contact forces ${\mathit{F}}_{\mathrm{p},\mathrm{c}}$ and hydrodynamic forces ${\mathit{F}}_{\mathrm{p},\mathrm{h}}$, meaning

$$\begin{array}{c}\hfill {\mathit{F}}_{\mathrm{p}}={\mathit{F}}_{\mathrm{p},\mathrm{V}}+{\mathit{F}}_{\mathrm{p},\mathrm{c}}+{\mathit{F}}_{\mathrm{p},\mathrm{h}}={\mathit{F}}_{\mathrm{p},\mathrm{V}}+{\mathit{F}}_{\mathrm{p},\mathrm{c}}+{\int }_{{\mathsf{\Gamma }}_{\mathrm{p}}}\sigma ·{\mathit{n}}_{\mathrm{p},\mathsf{\Gamma }}d{\mathsf{\Gamma }}_{\mathrm{p}},\end{array}$$

(10)

where the hydrodynamic forces are related to the above stress tensor $\sigma$ [57,62]. The particle’s surface is denoted by ${\mathsf{\Gamma }}_{\mathrm{p}}$ and surface normal vectors by ${\mathit{n}}_{\mathrm{p},\mathsf{\Gamma }}$.

Similarly, the total torque reads

$$\begin{array}{c}\hfill {\mathbf{T}}_{\mathrm{p}}={\mathbf{T}}_{\mathrm{p},\mathrm{c}}+{\mathbf{T}}_{\mathrm{p},\mathrm{h}}=\sum _{p\ne q}({\mathit{x}}_{\mathrm{c},\mathrm{pq}}-{\mathit{X}}_{\mathrm{p}})\times {\mathit{F}}_{\mathrm{c},\mathrm{pq}}+{\int }_{{\mathsf{\Gamma }}_{\mathrm{p}}}\left(\left({\mathit{x}}_{\mathsf{\Gamma }}-{\mathit{X}}_{\mathrm{p}}\right)\times \sigma \right)·{\mathit{n}}_{\mathrm{p},\mathsf{\Gamma }}d{\mathsf{\Gamma }}_{\mathrm{p}},\end{array}$$

(11)

which uses the above relationship between the hydrodynamic force and the stress tensor $\sigma$, the center of mass of the particle ${\mathit{X}}_{\mathrm{p}}$, and the individual contact forces ${\mathit{F}}_{\mathrm{c},\mathrm{pq}}$ with their corresponding contact points ${\mathit{x}}_{\mathrm{c},\mathrm{pq}}$ as well as the surface points ${\mathit{x}}_{\mathsf{\Gamma }}$. See Section 2.3 below for details on the formulation of the contact forces.

2.3. Contact

A common approach to account for solid-solid interactions is to use overlaps to model deformations that occur in reality, which is only valid for very small overlaps. The total contact force

$$\begin{array}{c}\hfill {\mathit{F}}_{\mathrm{p},\mathrm{c}}=\sum _{p\ne q}{\mathit{F}}_{\mathrm{c},\mathrm{pq}}\end{array}$$

(12)

acting on a particle p is the sum of all separate forces caused by individual contacts between that particle and any other solid q. Here, the subscript q represents each distinct solid object other than particle p. The individual contact forces usually consist of a normal and a tangential component, i.e.,

$$\begin{array}{c}\hfill {\mathit{F}}_{\mathrm{c},\mathrm{pq}}={\mathit{F}}_{\mathrm{c},\mathrm{pq},\mathrm{n}}+{\mathit{F}}_{\mathrm{c},\mathrm{pq},\mathrm{t}}.\end{array}$$

(13)

Nassauer’s and Kuna’s model [44] provides equations for both normal

$${\mathit{F}}_{\mathrm{c},\mathrm{pq},\mathrm{n}}=k{E}_{\mathrm{pq}}^{*}\sqrt{V_{\mathrm{pq}}{d}_{\mathrm{pq}}}\left(1+c_{\mathrm{pq}}{\dot{d}}_{\mathrm{pq},\mathrm{n}}\right){\mathit{n}}_{\mathrm{c},\mathrm{pq}},$$

(14)

and tangential contact forces

$${\mathit{F}}_{\mathrm{c},\mathrm{pq},\mathrm{t}}=-{\displaystyle \frac{{\mathit{u}}_{\mathrm{pq},\mathrm{t}}\left({\mathit{x}}_{\mathrm{c},\mathrm{pq}}\right)}{\left|\right|{\mathit{u}}_{\mathrm{pq},\mathrm{t}}\left({\mathit{x}}_{\mathrm{c},\mathrm{pq}}\right)\left|\right|}}\left(\left(2{\mu }_{\mathrm{pq},\mathrm{s}}^{*}-{\mu }_{\mathrm{pq},\mathrm{k}}\right){\displaystyle \frac{a_{\mathrm{pq}}^2}{a_{\mathrm{pq}}^4+1}}+{\mu }_{\mathrm{pq},\mathrm{k}}-{\displaystyle \frac{{\mu }_{\mathrm{pq},\mathrm{k}}}{a_{\mathrm{pq}}^2+1}}\right)\left|\right|{\mathit{F}}_{\mathrm{c},\mathrm{pq},\mathrm{n}}\left|\right|,$$

(15)

with

$${\mu }_{\mathrm{pq},\mathrm{s}}^{*}={\mu }_{\mathrm{pq},\mathrm{s}}\left(1-0.09{\left({\displaystyle \frac{{\mu }_{\mathrm{pq},\mathrm{k}}}{{\mu }_{\mathrm{pq},\mathrm{s}}}}\right)}^4\right),$$

(16)

and

$$a_{\mathrm{pq}}={\displaystyle \frac{\left|\right|{\mathit{u}}_{\mathrm{pq},\mathrm{t}}\left({\mathit{x}}_{\mathrm{c},\mathrm{pq}}\right)\left|\right|}{u_k}}.$$

(17)

The normal force considers factors like the effective Young’s modulus

$$E_{\mathrm{pq}}^{*}={\left({\displaystyle \frac{1-{\nu }_{\mathrm{p}}^2}{E_{\mathrm{p}}}}+{\displaystyle \frac{1-{\nu }_{\mathrm{q}}^2}{E_{\mathrm{q}}}}\right)}^{-1},$$

(18)

a constant k, overlap volume $V_{\mathrm{pq}}$, indentation depth $d_{\mathrm{pq}}$, damping factor $c_{\mathrm{pq}}$, and relative velocity ${\dot{d}}_{\mathrm{pq},\mathrm{n}}$ along the contact normal ${\mathit{n}}_{\mathrm{c},\mathrm{pq}}$.

In sphere–half-space and sphere–sphere contacts, the constant k is set to $4/\left(3\sqrt{\pi }\right)$ [44]. Conversely, for a cylindrical flat punch, k has the value $2/\sqrt{\pi }$ [43]. The effective Young’s modulus is calculated from the Young’s modulus E and Poisson’s ratio $\nu$ of the objects p and q, while the damping factor $c_{\mathrm{pq}}$ correlates with the signed coefficient of restitution $e_{\mathrm{pq}}$ based on Carvalho’s and Martins’ relationship [63]

$$c_{\mathrm{pq}}=\left\{\begin{array}{cc}1.5{\displaystyle \frac{(1-|e_{\mathrm{pq}}\left|\right)(11-|e_{\mathrm{pq}}\left|\right)}{(1+9|e_{\mathrm{pq}}\left|\right)u_0}}\hfill & \mathrm{if}{u}_0>0\hfill \\ 0\hfill & \mathrm{if}{u}_0=0\hfill \end{array}\right.,$$

(19)

using the magnitude of the initial relative velocity $u_0\ge 0$ [64].

Tangential forces influenced by static ${\mu }_{\mathrm{pq},\mathrm{s}}$ and kinetic ${\mu }_{\mathrm{pq},\mathrm{k}}$ friction coefficients are described by Equation (15), involving the relative tangential velocity between the objects p and q denoted by ${\mathit{u}}_{\mathrm{pq},\mathrm{t}}$ at the contact point ${\mathit{x}}_{\mathrm{c},\mathrm{pq}}$, while the model parameter $u_k$ quantifies the velocity at which the shift from static to kinetic friction takes place.

3. Numerical Methods

In the next section, the numerical methods used to solve the above models are presented. Note that all values discussed in Section 3.1 and Section 3.2 are expressed in lattice units.

3.1. Lattice Boltzmann Method

The lattice Boltzmann method (LBM) [65,66] is a well-established and robust option to solve the incompressible Navier–Stokes equations. LBM operates on a mesoscopic level, discretizing both spatial and temporal domains by employing fluid particle distributions to characterize fluid flow dynamics. These distributions, symbolized as $f_i(\mathit{x},t)$, depict the likelihood of encountering particles possessing discrete velocities ${\mathit{c}}_i$ at a given position $\mathit{x}$ and time t. Various velocity sets have been proposed and extensively discussed in existing literature [65]. Studies using HLBM typically use the D2Q9 velocity set for two-dimensional setups [55,67], which contains 9 discrete velocities in a two-dimensional space, while for a three-dimensional setup [56,68], D3Q19 is used, where the space is three-dimensional and the velocity set contains 19 velocities.

LBM computationally determines fluid flow by iteratively updating fluid particle distributions using two steps. Firstly, the collision step manages local interparticle interactions. It is expressed as

$$f_i^{*}(\mathit{x},t)=f_i(\mathit{x},t)+Q_i(\mathit{x},t)+S_i(\mathit{x},t).$$

(20)

Here, $f_i^{*}$ denotes the post-collision distribution, $Q_i$ the collision operator, and $S_i$ an optional source term. Secondly, the propagation step distributes the fluid particles to neighboring lattice nodes and is given by

$$f_i(\mathit{x}+\mathsf{\Delta }t{\mathit{c}}_i,t+\mathsf{\Delta }t)=f_i^{*}(\mathit{x},t),$$

(21)

where $\mathsf{\Delta }t=\mathsf{\Delta }x=1$ represent the time step size and grid spacing, respectively.

A variety of collision operators exist [65]. In combination with HLBM, however, the Bhatnagar–Gross–Krook (BGK) collision operator [69] is used exclusively, which reads

$$Q_i(\mathit{x},t)=-{\displaystyle \frac{1}{\tau }}(f_i(\mathit{x},t)-f_i^{\mathrm{eq}}({\rho }_{\mathrm{f}},{\mathit{u}}_{\mathrm{f}})),$$

(22)

where $\tau$ is the relaxation time that governs the relaxation towards an equilibrium distribution $f_i^{\mathrm{eq}}$. The state of equilibrium is characterized by the Maxwell–Boltzmann distribution and expressed as [70]

$$f_i^{\mathrm{eq}}({\rho }_{\mathrm{f}},{\mathit{u}}_{\mathrm{f}})=w_i{\rho }_{\mathrm{f}}\left(1+{\displaystyle \frac{{\mathit{c}}_i·{\mathit{u}}_{\mathrm{f}}}{c_s^2}}+{\displaystyle \frac{{\left({\mathit{c}}_i·{\mathit{u}}_{\mathrm{f}}\right)}^2}{2c_s^4}}-{\displaystyle \frac{{\mathit{u}}_{\mathrm{f}}·{\mathit{u}}_{\mathrm{f}}}{2c_s^2}}\right).$$

(23)

The associated weights $w_i$ are obtained from a Gauss-Hermite quadrature rule and remain constant for the chosen velocity set. Meanwhile, the constant lattice speed of sound $c_s$ also relies on the specific velocity set selected for the simulation and is additionally used to determine $\tau$ from the fluid’s viscosity via $\tau =(1/c_s^2)(\eta /{\rho }_{\mathrm{f}})+0.5$.

Utilizing the above fluid particle distributions allows for the derivation of macroscopic quantities [65], including the fluid density ${\rho }_{\mathrm{f}}(\mathit{x},t)={\sum }_i{f}_i(\mathit{x},t)$, which is related to the pressure via $p={\rho }_{\mathrm{f}}{c}_s^2$ and momentum ${\rho }_{\mathrm{f}}{\mathit{u}}_{\mathrm{f}}(\mathit{x},t)={\sum }_i{\mathit{c}}_i{f}_i(\mathit{x},t)$.

3.2. Homogenized Lattice Boltzmann Method

To account for the moving porous media, i.e., to solve the homogenized Navier–Stokes equations, see Equation (5), HLBM is used. It couples porous particles to the fluid by introducing a continuous model parameter $B(\mathit{x},t)\in [0,1]$, which is mapped onto the whole computational domain using a trigonometric level set function [43]

$$B(\mathit{x},t)=\left\{\begin{array}{cc}0,\hfill & \mathrm{if}{d}_s\le -\epsilon /2\hfill \\ {cos}^2\left({\displaystyle \frac{\pi }{2}}\left({\displaystyle \frac{d_s}{\epsilon }}-{\displaystyle \frac{1}{2}}\right)\right),\hfill & \mathrm{if}\epsilon /2>d_s>-\epsilon /2\hfill \\ 1,\hfill & \mathrm{if}{d}_s\ge \epsilon /2\hfill \end{array}\right..$$

(24)

Here, $\epsilon$ is a smoothing parameter, while $d_s$ is the signed distance to the particle’s surface. Figure 1 illustrates this concept using a circular boundary, showing the smooth transition region with its width defined by $\epsilon$, which is of the scale of the lattice spacing $\mathsf{\Delta }x$. For $\epsilon \to 0$, the transition region would vanish, resulting in a sharp boundary [55].

Note that STL files are also usable for particle shape description, for example from the online particle database PARROT [71]. For such files, the signed distance is then cached in a second mesh with a predefined spacing smaller than or equal to $\mathsf{\Delta }x$, which is then used to interpolate the dynamic signed distances [56,72].

In the context of HLBM, this level set function is also called confined permeability and is related to the permeability of the medium K by

$$\begin{array}{c}\hfill B(\mathit{x},t)=1-\tau {\displaystyle \frac{\eta }{{\rho }_{\mathrm{f}}K}}.\end{array}$$

(25)

Here, $\tau$ is the relaxation time, which also appears in the collision operator in Equation (22). This leads to a dependence on the numerical setup and constrains the resolvable permeability K to the range $\left[\tau \eta /{\rho }_{\mathrm{f}},\infty \right)$ [73].

The confined permeability is used to couple the particle to the fluid, by the convex combination of velocities

$$\mathsf{\Delta }{\mathit{u}}_{\mathrm{f}}(\mathit{x},t)=(B(\mathit{x},t)-1)\left({\mathit{u}}_{\mathrm{f}}(\mathit{x},t)-{\mathit{u}}_{\mathrm{p}}(\mathit{x},t)\right)$$

(26)

in an adaptation of the exact difference method (EDM) by Kupershtokh et al. [74], which introduces the source term

$$\begin{array}{c}\hfill S_i(\mathit{x},t)=f_i^{\mathrm{eq}}({\rho }_{\mathrm{f}},{\mathit{u}}_{\mathrm{f}}+\mathsf{\Delta }{\mathit{u}}_{\mathrm{f}})-f_i^{\mathrm{eq}}({\rho }_{\mathrm{f}},{\mathit{u}}_{\mathrm{f}})\end{array}$$

(27)

in Equation (20) [68].

In previous publications, coupling of the particle was also considered part of HLBM, for which the momentum exchange algorithm (MEA) of Wen et al. [75] has shown to provide the most accurate results [68]. Using the MEA, the hydrodynamic force is calculated per lattice node using

$$\begin{array}{c}\hfill {\mathit{F}}_{\mathrm{p},\mathrm{h}}^{\mathrm{local}}(\mathit{x},t)=\sum _i({\mathit{c}}_i-{\mathit{u}}_{\mathrm{p}}(\mathit{x},t))f_i(\mathit{x}+\mathsf{\Delta }t{\mathit{c}}_{\overline{i}},t)-({\mathit{c}}_{\overline{i}}-{\mathit{u}}_{\mathrm{p}}(\mathit{x},t))f_{\overline{i}}(\mathit{x},t).\end{array}$$

(28)

$\overline{i}$ denotes particle distributions characterized by the discrete velocity ${\mathit{c}}_{\overline{i}}=-{\mathit{c}}_i$, indicating that the distribution $f_{\overline{i}}$ has an opposite orientation to $f_i$. The sum of all forces on nodes inside the particle, which are identified by the position ${\mathit{x}}_{\mathrm{b}}$, leads to the total hydrodynamic force acting on the particle

$$\begin{array}{c}\hfill {\mathit{F}}_{\mathrm{p},\mathrm{h}}\left(t\right)=\sum _{{\mathit{x}}_{\mathrm{b}}}{\mathit{F}}_{\mathrm{p},\mathrm{h}}^{\mathrm{local}}({\mathit{x}}_{\mathrm{b}},t).\end{array}$$

(29)

This results in the torque

$$\begin{array}{c}\hfill {\mathit{T}}_{\mathrm{p},\mathrm{h}}\left(t\right)=\sum _{{\mathit{x}}_{\mathrm{b}}}({\mathit{x}}_{\mathrm{b}}-{\mathit{X}}_{\mathrm{p}})\times {\mathit{F}}_{\mathrm{p},\mathrm{h}}^{\mathrm{local}}({\mathit{x}}_{\mathrm{b}},t).\end{array}$$

(30)

Using the above equations for hydrodynamic force and torque, along with the contact forces, see Section 2.3 and Section 3.3, gravity, and buoyancy, the total force and torque are calculated with the help of Equations (10) and (11). The forces are converted into accelerations according to Newton’s second law of motion, which are then used in the velocity Verlet algorithm [76] to solve Equations (8) and (9) and to obtain the particle velocity ${\mathit{u}}_{\mathrm{p}}$ and then the position of the particle’s center of mass ${\mathit{X}}_{\mathrm{p}}$.

3.3. Discrete Contacts

The discrete contact model [43] consists of three fundamental mesh-based procedures: Rough contact detection, subsequent correction, and the computation of resultant forces.

While mapping $B(\mathit{x},t)$ onto the domain, initial rough contact detection is performed. For this, the signed distance of particles is evaluated at each lattice node. If the signed distance between two particles at a node is less than half the diagonal of a cell, i.e., $d_{\mathrm{s}}<\sqrt{0.5}\mathsf{\Delta }x$ in two dimensions and $d_{\mathrm{s}}<\sqrt{0.75}\mathsf{\Delta }x$ in three dimensions, a potential contact is detected. This preliminary estimate will produce an approximate cuboid that encapsulates the contact area, forming a bounding box.

Due to the relatively large lattice spacing, however, the overlap region is insufficiently resolved, resulting in an imprecise bounding box. To overcome this limitation, a subsequent correction step is implemented to refine the bounding box and align it more accurately with the actual overlap. This correction involves iteratively traversing the surface of the initial approximation using a predefined number of points along each spatial direction, known as the contact resolution $N_{\mathrm{c}}$. At each point, discrete directional computations are performed to measure the distance to the true contact, thereby increasing the accuracy of the bounding box.

In the final step, the contact resolution $N_{\mathrm{c}}$ is used again for systematic iteration over the entire overlap region to determine the overlap volume, contact point, contact normal, indentation depth, and contact force.

For a full understanding of the details of the algorithm, readers are encouraged to consult the relevant publications [43,77].

4. Applications

Several publications extended HLBM or used it for studies in the past. A chronological overview is given in Table 1, starting with the oldest publication. As can be seen, these publications generally follow a very logical order, starting with relatively simple cases, i.e., 2D simulations [55], and then continuously adding complexity.

Table 1. Overview of publications on HLBM.

Publication	Spatial Dimensions	Particle Shapes	Particle Number per Simulation	Coupling
Krause et al. [55]	2	Circles, squares, and triangles	1–24	Two- and four-way
Trunk et al. [56]	3	Spheres and limestone CT scans	1–15	Two-way
Dapelo et al. [78]	3	Spheres	1	Two-way
Haussmann et al. [67]	2	Circles	1	Two-way
Bretl et al. [79]	3	Symmetric regarding the principal planes	1	Two-way
Trunk et al. [68]	2 and 3	Circles and spheres	1–1865	Two-way
Trunk et al. [6]	3	Superellipsoids	1	Two-way
Hafen et al. [73]	3	Cubic disks	48–240	Two-way
Marquardt et al. [43]	3	Spheres	1	Four-way
Hafen et al. [80]	3	Cubic disks	1	Two-way
Hafen et al. [81]	3	Cubic disks	1–480	Two-way
Hafen et al. [64]	3	Cubic disks	240–720	Four-way
Marquardt et al. [82]	3	Spheres	166–991	Two-way
Marquardt et al. [77]	3	Spheres and cubes	191–1934	Four-way
Marquardt et al. [83]	3	Cubes	394–784	Four-way

Table 1. Overview of publications on HLBM.

Publication	Spatial Dimensions	Particle Shapes	Particle Number per Simulation	Coupling
Krause et al. [55]	2	Circles, squares, and triangles	1–24	Two- and four-way
Trunk et al. [56]	3	Spheres and limestone CT scans	1–15	Two-way
Dapelo et al. [78]	3	Spheres	1	Two-way
Haussmann et al. [67]	2	Circles	1	Two-way
Bretl et al. [79]	3	Symmetric regarding the principal planes	1	Two-way
Trunk et al. [68]	2 and 3	Circles and spheres	1–1865	Two-way
Trunk et al. [6]	3	Superellipsoids	1	Two-way
Hafen et al. [73]	3	Cubic disks	48–240	Two-way
Marquardt et al. [43]	3	Spheres	1	Four-way
Hafen et al. [80]	3	Cubic disks	1	Two-way
Hafen et al. [81]	3	Cubic disks	1–480	Two-way
Hafen et al. [64]	3	Cubic disks	240–720	Four-way
Marquardt et al. [82]	3	Spheres	166–991	Two-way
Marquardt et al. [77]	3	Spheres and cubes	191–1934	Four-way
Marquardt et al. [83]	3	Cubes	394–784	Four-way

The initial proposal of the method [55] was followed by allowing the consideration of arbitrarily shaped three-dimensional particles for two-way coupled simulations [56] and later by adding a compatible contact model for convex particles [43] in order to also consider higher particle volume fractions, as previous studies showed the need for an explicit contact model in these cases [68].

After the introduction of the necessary models, together with extensive validations, HLBM has been used to study industrially relevant processes. For example, studies covered a shape-dependent drag correlation [6,79], the application to wall-flow filters [64,73,80,81], hindered settling [77,82], as well as the damage potential of particles during transport through cross-sectional constrictions in pipes [83].

As research using HLBM is extensive, we focus on the most recent and relevant studies using HLBM and their results to provide an overview of the applicability of HLBM below.

This includes the flow around a cylinder in Section 4.1, the settling of a single sphere in Section 4.2, and shape-dependent single particle settling in Section 4.3. Furthermore, we demonstrate the incorporation of the above discrete contact model by showing the results of a cylinder-wall impact in Section 4.4 and a spherical particle rebound in Section 4.5. Finally, we consider the results of the complex wall-flow filter in Section 4.6, the investigation of the hindered settling phenomena in Section 4.7, and mechanical stress during transport through cross-section constrictions in Section 4.8.

All of the above publications and the applications shown below used the open source software OpenLB [72,84].

4.1. Flow Around a Cylinder

To ensure convergence of the hydrodynamic forces ${\mathit{F}}_{\mathrm{h}}$, Trunk et al. [56] considered a flow around a cylinder following Schäfer et al. [85]. The corresponding simulation setup is shown as a projection onto the x-y plane in Figure 2. The cylinder fixed at the beginning of the channel is modeled with HLBM and has a length of $0.41$ $\mathrm{m}$, which is equal to the channel length in the z-direction. The lattice spacing is $\mathsf{\Delta }x=0.01\mathrm{m}$/N and the fluid flows from the left to the right.

The walls were modeled with a bounce back boundary. The inlet on the left ($x=0$ $\mathrm{m}$) was set as a regularized velocity boundary, for which the velocity profile is given by [85]

$$\begin{array}{c}\hfill {\mathit{u}}_{\mathrm{f}}(y,z)=7.2yz{\displaystyle \frac{(0.41-y)(0.41-z)}{0.{41}^4}}.\end{array}$$

(31)

At the outlet on the right, a regularized pressure boundary was used. Details of the boundary conditions can be found in literature [86]. The considered fluid had the viscosity $\eta =1$ $\mathrm{Pa}\mathrm{s}$ and density ${\rho }_{\mathrm{f}}=1000$ $\mathrm{k}\mathrm{g}/{\mathrm{m}}^3$. Using the average inflow velocity ${\overline{u}}_{\mathrm{f},0}$, the drag coefficient is defined as

$$\begin{array}{c}\hfill C_{\mathrm{d}}={\displaystyle \frac{2F_{\mathrm{h},x}}{0.041{\rho }_{\mathrm{f}}{\overline{u}}_{\mathrm{f},0}^2}},\end{array}$$

(32)

and the lift coefficient as

$$\begin{array}{c}\hfill C_{\mathrm{l}}={\displaystyle \frac{2F_{\mathrm{h},y}}{0.041{\rho }_{\mathrm{f}}{\overline{u}}_{\mathrm{f},0}^2}}.\end{array}$$

(33)

Table 2. Drag and lift coefficients computed in [56] for a flow around a cylinder.

	Drag Coefficient ${\mathit{C}}_{\mathbf{d}}$					Lift Coefficient ${\mathit{C}}_{\mathbf{l}}$ ($\times {10}^{-2}$)
	$\mathbf{N}$ = 1	$\mathbf{N}$ = 2	$\mathbf{N}$ = 3	$\mathbf{N}$ = 4	$\mathbf{N}$ = 5	$\mathbf{N}$ = 1	$\mathbf{N}$ = 2	$\mathbf{N}$ = 3	$\mathbf{N}$ = 4	$\mathbf{N}$ = 5
$\epsilon =2\mathsf{\Delta }x$	$9.83$	$7.57$	$7.05$	$6.81$	$6.68$	$0.00$	$1.04$	$1.11$	$1.11$	$1.09$
$\epsilon =1\mathsf{\Delta }x$	$8.15$	$6.90$	$6.63$	$6.51$	$6.43$	$0.00$	$0.66$	$0.81$	$0.87$	$0.90$
$\epsilon =0.5\Delta x$	$6.93$	$6.44$	$6.35$	$6.30$	$6.27$	$0.00$	$0.77$	$0.88$	$0.91$	$0.92$
$\epsilon =0\mathsf{\Delta }x$	$6.90$	$6.38$	$6.24$	$6.22$	$6.19$	$0.00$	$0.53$	$0.66$	$0.73$	$0.76$
Schäfer et al. [85]	$6.05$–$6.25$					$0.80$–$1.00$

lists the resulting drag and lift coefficients for different resolutions N and smooth boundary sizes $\epsilon$ along with the range provided by Schäfer et al. [85]. For all $\epsilon$, it is noticeable that the higher N is, the better the HLBM results fit the range of Schäfer et al. [85]. In general there is good agreement, but for $N=1$ there are large discrepancies, suggesting underresolution. Looking at the drag coefficient, this is particularly noticeable for high $\epsilon$, as it increases the object size depending on the lattice spacing. This highlights the dependence of the results on $\epsilon$, as we also find that using $\epsilon =2\mathsf{\Delta }x$ leads to an overestimation of the drag and lift coefficients. Drag is even overestimated with $\epsilon =1\mathsf{\Delta }x$, suggesting that $\epsilon <\mathsf{\Delta }x$ gives the best results.

To visualize the change over resolutions, Figure 3 shows the relative error in the $L^2$ norm [65] of the drag (left) and lift (right) coefficients versus N for different $\epsilon$. For comparison, lines for the experimental orders of convergence (EOC) of 1 and 2 are also included. It is evident that the error of the baseline resolution of $N=5$ decreases with increasing N. It can also be seen that the trend follows $\mathrm{EOC}=2$ over all $\epsilon$ and for both drag and lift coefficients, suggesting quadratic convergence.

4.2. Settling Sphere

As a basic validation of moving particles, Trunk et al. [68] compare the simulation results of a simple single settling sphere with the experimental results of ten Cate et al. [87]. The corresponding simulation setup is illustrated in Figure 4. Here, a sphere with a diameter of $D_{\mathrm{s}}=0.015\mathrm{m}$ is placed in the center of a cubic container with dimensions of $0.1\mathrm{m}\times 0.1\mathrm{m}\times 0.16\mathrm{m}$ at a height of $0.12\mathrm{m}$. The spherical particle has a density of ${\rho }_{\mathrm{p}}=1120\mathrm{k}\mathrm{g}/{\mathrm{m}}^3$, while the density and viscosity of the surrounding fluid vary. The values used in the different cases are given in

Table 3. Values considered for the computation of a single settling sphere in [68].

Case	Fluid Density ${\mathit{\rho }}_{\mathbf{f}}$ in kg $m^{-3}$	Dynamic Viscosity $\mathit{\eta }$ in Pa s	Grid Spacing $\mathbf{\Delta }\mathit{x}$ in mm	Time Step Size $\mathbf{\Delta }\mathit{t}$ in ms
1	970	$0.373$	$1.671$	$0.3891$
2	965	$0.212$	$1.645$	$0.2410$
3	962	$0.113$	$1.610$	$0.1526$
4	960	$0.058$	$1.559$	$0.1010$

.

In the HLBM simulations, the walls are modeled using a no-slip halfway bounce-back condition [66]. Originally, Trunk et al. [68] considered several forcing and momentum exchange schemes as well as a wide range of discretization parameters. Since these studies found that a combination of the above MEA and the adapted EDM produced the most accurate results, however, the following studies using HLBM and the remainder of this paper also exclusively consider these as the superior reported schemes. We also limit the grid spacing and time step size to the values given in Table 3.

Figure 5 compares the HLBM simulation results with the experimental data for the cases mentioned in Table 3 by plotting the instantaneous particle velocity over time. The simulation and experimental data agree very well, hence, allow similar conclusions to be drawn. From case 1 to case 4, the density ratio increases, whereas the viscosity decreases. Therefore, the maximum settling velocity and acceleration also increase, resulting in shorter settling times as the particle contacts the bottom of the container more quickly.

Trunk et al. [68] also compare the predicted terminal velocities and drag coefficients of settling spheres for varying Reynolds numbers $\mathit{Re}$ with well-established drag correlations and conclude that the lattice velocity should not exceed $0.04$, while a lattice velocity of $0.01$ is considered most desirable. Furthermore, the authors report linear convergence, which differs from the convergence observed by [56], see Section 4.1, which is likely due to a less significant influence of the surrounding walls and the consideration of moving curved boundaries for which a staircase approximation is used [65,82,88].

4.3. Shape-Dependent Settling of Single Particles

After the above validation, Trunk et al. [6] employed HLBM to derive a shape-dependent drag correlation for single particle settling. To this end, the authors first identified three independent shape parameters, namely, the elongation ${\kappa }_{\mathrm{el}}$, the roundness ${\kappa }_{\mathrm{rnd}}$ [89], and the Hofmann shape entropy ${\lambda }_{\mathrm{H}}$ [90], using Pearson correlation coefficients [91]. These shape parameters are defined as

$$\begin{array}{cc}\hfill {\kappa }_{\mathrm{el}}& =a_{\mathrm{I}}/a_{\mathrm{L}},\hfill \end{array}$$

(34)

$$\begin{array}{cc}\hfill {\kappa }_{\mathrm{rnd}}& ={\displaystyle \frac{V_{\mathrm{p}}}{A_{\mathrm{p}}{\left(8a_{\mathrm{L}}{a}_{\mathrm{I}}{a}_{\mathrm{S}}\right)}^{1/3}}},\hfill \end{array}$$

(35)

$$\begin{array}{cc}\hfill {\lambda }_{\mathrm{H}}& =-{\displaystyle \frac{{\tilde{a}}_{\mathrm{S}}ln{\tilde{a}}_{\mathrm{S}}+{\tilde{a}}_{\mathrm{I}}ln{\tilde{a}}_{\mathrm{I}}+{\tilde{a}}_{\mathrm{L}}ln{\tilde{a}}_{\mathrm{L}}}{ln3}},\hfill \end{array}$$

(36)

where $V_{\mathrm{p}}$ and $A_{\mathrm{p}}$ are the particle’s volume and surface. $a_{\mathrm{S}}$, $a_{\mathrm{I}}$, and $a_{\mathrm{L}}$ correspond to the shortest, intermediate, and longest particle’s half-axis, respectively, which are normalized as ${\tilde{a}}_j=a_j/(a_{\mathrm{S}}+a_{\mathrm{I}}+a_{\mathrm{L}})$ for $j\in \{\mathrm{S},\mathrm{I},\mathrm{L}\}$.

To obtain the necessary data from simulations, the authors separately considered 200 superellipsoids in a domain similar to that in Figure 4. However, the width and depth were set to $7\mathrm{m}\mathrm{m}$, the height to $17.5\mathrm{m}\mathrm{m}$, and the particles were placed in the middle of the container with a distance of $9.554 \times {10}^{-1} \mathrm{m}\mathrm{m}$ from the top wall. The surrounding fluid had a density of ${\rho }_{\mathrm{f}}=998\mathrm{k}\mathrm{g}/{\mathrm{m}}^3$ and a viscosity of $\eta =9.98 \times {10}^{-4}\mathrm{Pa}\mathrm{s}$. In addition, the standard gravity $g=9.81\mathrm{m}\mathrm{s}$ was applied and the following correlation of drag coefficients for single particle settling with a coefficient of restitution of $0.96$ was reported [6]:

$$\begin{array}{cc}\hfill C_{\mathrm{d}}=& -0.386{\kappa }_{\mathrm{el}}-0.496{\kappa }_{\mathrm{rnd}}-0.575\mathit{Re}+0.097{\kappa }_{\mathrm{el}}{\kappa }_{\mathrm{rnd}}-0.322{\kappa }_{\mathrm{el}}\mathit{Re}\hfill \\ \hfill & +0.0497{\kappa }_{\mathrm{rnd}}^2-0.128{\kappa }_{\mathrm{rnd}}{\lambda }_{\mathrm{H}}+0.106{\kappa }_{\mathrm{rnd}}\mathit{Re}-0.296{\lambda }_{\mathrm{H}}\mathit{Re}+0.161{\mathit{Re}}^2,\hfill \end{array}$$

(37)

where the Reynolds number $\mathit{Re}={\rho }_{\mathrm{f}}{D}_{\mathrm{eq}}{u}_{\infty }/\eta$ was obtained from the diameter of a volume-equivalent sphere $D_{\mathrm{eq}}$ and the terminal velocity $u_{\infty }$.

Utilizing Equation (37), it is now feasible to include shape dependence on the subgrid scale and in Euler–Euler particle simulations to perform investigations at a process scale, which can also benefit from the above straightforward LBM parallelization. Particle simulations coupled with LBM have been demonstrated in Euler–Euler simulations of, for example, a bifurcation [51] and a high-gradient magnetic separation [52]. In addition, LBM was coupled with subgrid scale particle simulations to investigate the nasal cavity [92].

4.4. Cylinder-Wall Impact

To include dense suspensions in the possible applications of HLBM, Marquardt et al. [43] introduced and validated the discrete contact model by considering a cylinder-wall impact according to Park [93]. A sketch of the corresponding setup is shown in Figure 6. It displays the cylinder of density ${\rho }_{\mathrm{cyl}}\approx 1164\mathrm{k}\mathrm{g}/{\mathrm{m}}^3$, diameter $D_{\mathrm{cyl}}=8\mathrm{m}\mathrm{m}$, and length $L_{\mathrm{cyl}}=5.3\mathrm{m}\mathrm{m}$. The cylinder settles with a constant initial velocity $u_{\mathrm{cyl}}^{-}=-1\mathrm{m}/\mathrm{s}$ until it contacts the bottom wall, which means that there is no interaction between the fluid and the particle. The distance between the point of impact and the center of the cylinder is given by $r=\sqrt{{(D_{\mathrm{cyl}}/2)}^2+L_{\mathrm{cyl}}^2/4}$. The angle $\alpha$ is between the line from the center of the cylinder to the point of impact and the face of the cylinder, while the impact angle $\beta$ is between the bottom of the cylinder and the flat wall.

The expected rebound angular velocity is given by [93]

$${\omega }_{\mathrm{cyl},y}^{+}={\displaystyle \frac{m{u}_{\mathrm{cyl},z}^{-}(1+e)rcos(\alpha +\beta )}{I_{\mathrm{yy}}+m{r}^2{cos}^2(\alpha +\beta )}},$$

(38)

where m is the mass of the cylinder, $I_{\mathrm{yy}}$ is the moment of inertia with respect to the y-axis, and $e=u_{\mathrm{cyl},z}^{+}/u_{\mathrm{cyl},z}^{-}$ is the signed coefficient of restitution with the rebound velocity [93]

$$\begin{array}{c}\hfill u_{\mathrm{cyl},z}^{+}={\omega }_{\mathrm{cyl},y}^{+}rcos(\alpha +\theta )-e{u}_{\mathrm{cyl},z}^{-}.\end{array}$$

(39)

In their simulations, the authors considered $e=-0.85$, $k=2/\sqrt{\pi }$, and resolved the cylinder diameter with $N=4$ cells for the mesh-based contact detection, while varying the contact region resolution $N_{\mathrm{c}}$. Figure 7 compares the computed rebound velocity (left) and rebound angular velocity (right) with the above solutions. In general, there is a strong dependence on the contact resolution $N_{\mathrm{c}}$, since for $N_{\mathrm{c}}=2$, the contact is clearly underresolved, leading to high discrepancies. With increasing $N_{\mathrm{c}}$, however, the accuracy improves, so that already for $N_{\mathrm{c}}=4$ a good agreement is visible, but the authors still suggest to use $N_{\mathrm{c}}\ge 8$, as this also leads to more accurate contact forces.

4.5. Sphere Rebound

For the validation of four-way coupled HLBM simulations, Marquardt et al. [43] considered the rebound of spherical particles in a viscous fluid according to Li [94]. In this case, a sphere was placed in the middle of a cubic box similar to the setup outlined in Figure 4. However, the container measured $42\mathrm{m}\mathrm{m}\times 42\mathrm{m}\mathrm{m}\times 50\mathrm{m}\mathrm{m}$ and the sphere with a density of ${\rho }_{\mathrm{p}}=7780\mathrm{k}\mathrm{g}/{\mathrm{m}}^3$ was placed at various distances, $5.5\mathrm{m}\mathrm{m}$, $19.6\mathrm{m}\mathrm{m}$, and $35.7\mathrm{m}\mathrm{m}$, from the bottom wall. Furthermore, the authors set the Young’s modulus of the particle to $20\mathrm{G}\mathrm{Pa}$ and that of the wall to $3\mathrm{G}\mathrm{Pa}$, while the Poisson’s ratios were $0.33$ and $0.24$, respectively. The fluid had a density of ${\rho }_{\mathrm{f}}=1230\mathrm{k}\mathrm{g}/{\mathrm{m}}^3$ and a viscosity of $\eta =50.2\mathrm{Pa}\mathrm{s}$.

Again, the surrounding walls were modeled using a halfway bounce-back condition [66]. The simulations used a lattice spacing of $\mathsf{\Delta }x=0.3\mathrm{m}\mathrm{m}$ and time step size of $\mathsf{\Delta }t=5\mu \mathrm{s}$, while the contact resolution $N_{\mathrm{c}}=16$, constant $k=4/\left(3\sqrt{\pi }\right)$, and the damping factor $c=0.12$ $\mathrm{s}/\mathrm{m}$ were chosen. To decouple the particle and fluid calculations, the authors introduced smaller time steps of $\mathsf{\Delta }t/1000$ for solving the equations of motion and calculating the contact forces. They also increased the particle size by $\mathsf{\Delta }x/7$ for the contact treatment only, as otherwise the viscous forces would become unphysically high due to the assumption of an incompressible fluid.

Figure 8 shows the distance of the sphere surface to the bottom wall over time for the three setups mentioned above. The lines correspond to the HLBM simulation results [43] and the points correspond to the experimental data [94]. In both cases, a similar trend can be seen: The higher the initial position is, the steeper is the curve, which means that the maximum velocities of the spheres increase with the distance. Also, the spherical particle rebounds more often when the distance increases, which is caused by the higher amount of potential energy that is converted into kinetic energy in these cases.

Comparing the HLBM simulation with the experimental data, we observe a good agreement, especially for the initial distances of $5.5\mathrm{m}\mathrm{m}$ and $19.6\mathrm{m}\mathrm{m}$. For an initial distance of $35.7\mathrm{m}\mathrm{m}$, however, there is a comparatively higher discrepancy after the first rebound, which is probably due to the use of a constant damping factor c, which, according to Equation (19), should actually depend on the initial relative velocity, which obviously changes among the cases, as already noticed before.

4.6. Wall-Flow Filters

Hafen et al. [64,73,80,81] considered the application to wall-flow filters, which is important because the exhaust from combustion engines contains particulate matter, which is a potential health hazard and therefore needs to be captured. To study the application numerically, the authors consider a single channel of a wall-flow filter as shown in Figure 9. Here, the walls and particles are made of porous media, which is modeled with HLBM. A constant velocity was set at the inlets and a constant pressure was set at the outlets. Periodic boundaries were applied perpendicular to the flow direction to represent real wall flow filters, which consist of hundreds of individual channels.

First, the flow without any particles was considered [73], showing superlinear grid convergence and good agreement with a reference solution [95]. It was also found that for non-spherical particles, lift forces are not the primary detachment forces, while the stopping distance is so long that back-end deposition is inevitable [80]. Fragment detachment occurs in successive stages, as the first row of fragments experiences the greatest hydrodynamic forces while shielding the following fragments, thus reducing the hydrodynamic forces and the likelihood of detachment for those fragments [81].

Finally, [64] dealt with particle detachment and plug formation as shown in Figure 9. The results indicate that a straightforward mean density methodology serves as a consistent method for measuring both the extent of a channel plug and its mean density, which closely matches values documented in the existing literature [96,97]. The study also shows that the topology of the fragment layer affects the final plug structure and, more significantly, the pressure drop. Larger layer heights caused by longer loading times or higher particle concentrations result in increased pressure drops in the plug state. The same effect is observed for larger fragment dimensions associated with a lower soot-to-ash ratio. The more particles are present after soot oxidation, the higher is the pressure drop. A transition to a particulate plug becomes advantageous when the fragmented layer limits the free substrate surface area in contact with the flow. Increasing the density of the particulate matter has been found to decrease the pressure drop because the higher inertia leads to a greater compactness of the plug. Following this reasoning, the authors mention that the use of additives to decrease the stiffness of the particles would also increase the compactness and therefore reduce the pressure drop. Removal of the first few rows of fragments significantly reduces the pressure drop, while adhesive forces noticeably affect detachment at the threshold of the first row only. When examining the relationship between gas velocity and momentum loss in the filter substrate, it was found that the dependence on pressure drop in the plugged state is less significant.

4.7. Hindered Settling

In order to improve the performance of HLBM, Marquardt et al. [82] introduced a particle decomposition and validated it using hindered settling [24]. However, the authors solely considered two-way coupling and found that especially at high particle volume fractions, four-way coupling is necessary to obtain accurate results. For this reason, the implementation was extended and a parallelization strategy for the discrete contact model was introduced, leading to further studies of the hindered settling phenomena [77].

The authors considered both spheres of $D_{\mathrm{s}}=3$ $\mathrm{m}\mathrm{m}$ in diameter and volume-equivalent cubes in a triple periodic domain with an edge length of $12D_{\mathrm{s}}$ that is filled with a fluid having the density of ${\rho }_{\mathrm{f}}=1000$ $\mathrm{k}\mathrm{g}/{\mathrm{m}}^3$. In both cases, the particles with a density of ${\rho }_{\mathrm{p}}=3300$ $\mathrm{k}\mathrm{g}/{\mathrm{m}}^3$ initially rest at exactly the same randomly generated positions and the standard gravity $g=9.80665$ $\mathrm{m}/\square \mathrm{s}$ is applied. The fluid viscosity is set depending on the considered Archimedes number $Ar\in \{500,1000,2000\}$, which is defined as

$$\begin{array}{c}\hfill \mathit{Ar}={\displaystyle \frac{g{D}_{\mathrm{s}}^3{\rho }_{\mathrm{f}}({\rho }_{\mathrm{p}}-{\rho }_{\mathrm{f}})}{{\eta }^2}}.\end{array}$$

(40)

In addition, the authors consider particle volume fractions ${\varphi }_{\mathrm{p}}$ between 10% and 30%, while a consistent lattice relaxation time $\tau =0.55$ is used. Furthermore, the authors vary the number of cells N per sphere diameter $D_{\mathrm{s}}$. A contact resolution of 8, $k=4/\left(3\sqrt{\pi }\right)$, $e=-0.926$, and $u_k=0.001 \mathrm{m}/\mathrm{s}$ were used.

Examining the results of the swarm of settling spheres with $N=18$, the authors find a good agreement with literature even without an explicit contact model for low particle volume fractions. However, as the particle volume fraction increases, the need for an explicit contact model becomes more apparent. The authors point out that for particle volume fractions ${\varphi }_{\mathrm{p}}>0.3$, a lubrication force correction model must be introduced to accurately capture the settling behavior.

Figure 10 shows the velocity field around the swarm of spheres (left) and cubes (right) at different times for a resolution of $N=18$, particle volume fraction of ${\varphi }_{\mathrm{p}}\approx 0.3$, and Archimedes number of $Ar=2000$. While voids with high fluid velocities form between the spheres, the fluid velocity is much more evenly distributed when cubes are considered instead.

This is also reflected by the average settling velocity of the particles, which is plotted over the particle volume fraction ${\varphi }_{\mathrm{p}}$ in Figure 11. For the normalization, the average settling velocity ${\overline{u}}_{\mathrm{p}}$ is divided by the terminal settling velocity of a single sphere, which reads

$$\begin{array}{c}\hfill u^{*}=\sqrt{{\displaystyle \frac{4g{D}_{\mathrm{s}}}{3C_{\mathrm{d}}}}\left({\displaystyle \frac{{\rho }_{\mathrm{p}}-{\rho }_{\mathrm{f}}}{{\rho }_{\mathrm{f}}}}\right)}.\end{array}$$

(41)

To calculate the drag coefficient, the authors use the correlation by Schiller and Neumann [98] given by

$$\begin{array}{c}\hfill C_{\mathrm{d}}={\displaystyle \frac{24}{\mathit{Re}}}\left(1+0.15{\mathit{Re}}^{0.687}\right).\end{array}$$

(42)

Figure 11 shows that in all cases the spheres settle faster than the cubes on the average, i.e., the cubes are 13% to 26% slower than the spheres. The authors reason that this is partly due to the spheres tending to build clusters more easily, as observed above, leaving voids for the fluid to pass through, thus reducing the velocity difference at the particle surface and, hence, the drag. Spheres are also rotationally invariant, which is why the rotation of the particle has little effect on the fluid. In cubes, on the other hand, the rotation changes the cross-sectional area and has a pronounced effect on the fluid.

4.8. Transport Through a Cross-Sectional Constriction

HLBM has been used to investigate the mechanical stress during transport through cross-sectional constrictions, as fully resolved particles are necessary to determine local forces on the surfaces [83]. In their study, the authors considered a setup, which is shown as a projection onto the x-y plane in Figure 12. At the inlet, the 500 $\mathrm{m}\mathrm{m}$ long geometry starts with a cylinder that has a diameter of 50 $\mathrm{m}\mathrm{m}$ and covers 70% of the length. After that, the geometry narrows down to 25 $\mathrm{m}\mathrm{m}$. The taper angle $\theta$ quantifies the rate at which the diameter of the pipe decreases along the conical transition. The non-Newtonian fluid is considered to be a Herschel–Bulkley fluid [99] with a density of 1100 $\mathrm{k}\mathrm{g}/{\mathrm{m}}^3$, a flow index of $0.42$, a consistency of $13.1$ $\mathrm{Pa}{\mathrm{s}}^{0.42}$, and a yield stress of $82.7$ $\mathrm{Pa}$. The cubic particles initially occupy 30% of the volume in the section with the big diameter and have the same density as the fluid. Particle edge lengths of $6.4$ and $8.1$ $\mathrm{m}\mathrm{m}$ are considered. The particles have Young’s moduli of 30 $\mathrm{k}\mathrm{Pa}$ and 60 $\mathrm{k}\mathrm{Pa}$. The walls have Young’s moduli of 190 $\mathrm{G}\mathrm{Pa}$ and the same Poisson ratio of $0.3$. The coefficient of restitution is set to $0.9$, the coefficient of kinetic friction to $0.1$, and the coefficient of static friction to $0.15$. A time interval of $1.1$ $\mathrm{s}$ is simulated. For the evaluation, the first $0.1$ $\mathrm{s}$ are neglected. The walls are considered no-slip boundaries using the Bouzidi boundary approach [100]. A constant pressure is set at the outlet. At the inlet, a constant velocity profile from prior simulations is used. While the setup applied for these simulations is the same as above, but the velocity is set to the average velocity over the entire inlet, the length is shortened to 300 $\mathrm{m}\mathrm{m}$, and no particles are considered. When the simulation converged, the velocity profile at $x=225$ $\mathrm{m}\mathrm{m}$ was saved and afterwards used as the boundary and initial condition for the following studies with particles.

Metzner–Reed Reynolds numbers [101] between 2 and 8 were studied and found have the greatest influence on the fluid- and contact-induced mechanical stress, along with particle properties, such as Young’s modulus and size. The taper angle $\theta$ showed no significant influence. In other words, as the pipe narrows, fluid-induced mechanical stress increases primarily because the Metzner-Reed Reynolds number increases in the smaller diameter sections. Contact-induced mechanical stress increases due to higher relative velocities at the initial contacts.

Results of an example with an edge length of $8.1$ $\mathrm{m}\mathrm{m}$, a particle Young’s modulus of 60 $\mathrm{k}\mathrm{Pa}$, and $\theta =45$ are given in Figure 13. Here, the average distance of the center of mass of the particles from the center of the pipe is shown over the length of the pipe (top left), where $x=0$ $\mathrm{m}$ corresponds to the beginning of the taper. The number of particle-particle and particle-wall contacts (top right), fluid-induced stress (bottom left), and contact-induced stress (bottom right) are also plotted over the length of the pipe. The plot displays the median induced stresses as a line, accompanied by an area encompassing the central 90% of the data, thus excluding the 5% lowest and 5% highest stresses to identify trends while reducing the influence of outliers.

The results show that the particles move towards the center of the pipe due to the cross-sectional constriction. As a result, interparticle contacts increase due to rearrangements on the way towards the constriction, while they remain at a constant level afterwards. Particle-wall contacts occur most frequently at the beginning of the constriction and before. They become less likely in the small-diameter section. Looking at the bottom plots, the normal mechanical stress is greater than the tangential component. While the fluid-induced stresses become smaller after the constriction, the contact-induced normal stresses remain on a similar level. The contact-induced tangential stresses increase, which is likely due to higher relative velocities.

5. Discussion

5.1. Strengths and Weaknesses

HLBM has several strengths that make it a versatile tool in fluid dynamics simulations. A key strength is its monolithic approach, which allows for seamless integration of various fluid-solid interaction scenarios in a unified framework. HLBM enables the simultaneous simulation of pure fluid flow, porous media, and moving porous particles, thereby streamlining the modeling process.

Another major advantage is its ability to handle arbitrary shapes, allowing for the consideration of a wide range of particle geometries. Specifically, HLBM accommodates arbitrary shapes with two-way coupling [6,56], while it efficiently handles convex shapes with four-way coupling [43,77], using the discrete contact model mentioned before. It accurately resolves particle shapes, thereby facilitating the extraction of forces acting on these particles—an essential aspect in various scientific investigations.

In addition, HLBM can be easily parallelized, which enhances its computational efficiency. Leveraging the strengths of the LBM, HLBM takes advantage of parallel computing architectures to optimize performance and speed up simulations [77,82]. Its robustness is underscored by extensive testing over the years, establishing a track record of reliability and accuracy in diverse applications, as outlined above. In particular, HLBM uniquely treats particles as porous media since the coupling to fluid is done using a locally confined permeability [73], a feature lacking in many alternative methods. This feature is particularly important in scenarios where the behavior of porous media significantly affects system dynamics [64,73,80,81] and it distinguishes HLBM as a comprehensive modeling approach.

Since HLBM is based on LBM, however, it requires small time steps due to diffusive scaling, i.e., $\mathsf{\Delta }t\propto \mathsf{\Delta }{x}^2$. Compared to alternative methods, such as subgrid scale or Euler–Euler particle simulations, HLBM has relatively high computational requirements so far, typical of DNS approaches. In addition, HLBM is currently limited to modeling rigid particles [55,56,68], which restricts its applicability in scenarios involving deformable or flexible entities.

In summary, HLBM is characterized by its ability to accommodate diverse particle shapes, resolving particle shapes, its parallelizability, and its treatment of particles as porous media. However, the need for small time steps, computational cost, and limitation to rigid particles are important considerations when applying this method to fluid dynamics simulations.

5.2. Outlook on Future Development

The future trajectory of HLBM is promising and there are several key areas needing advancement and refinement.

In general, performance improvements should be addressed in future developments. Optimization of algorithms and computational strategies can lead to improved efficiency, resulting in faster simulations without compromising accuracy.

Big improvements are to be expected by the implementation of hybrid parallelization techniques. The integration of hybrid parallelization techniques would significantly reduce communication overhead, a critical aspect in large-scale simulations. By efficiently distributing computational tasks across different computing resources while minimizing inter-node communication, this approach could significantly accelerate simulations, thereby extending the applicability of the method to more complex and larger systems.

Another key advancement would be the integration of graphics processing unit (GPU) support. Leveraging the computational power of GPUs could significantly accelerate HLBM simulations by taking advantage of their parallel processing capabilities. This improvement has the potential to revolutionize the efficiency of the method, allowing for faster and more scalable simulations across platforms.

Further advances in HLBM could also include extending the discrete contact model to include concave particles. Updating the contact detection mechanism to recognize and handle multiple contacts between pairs of particles would greatly increase the versatility of the method. This is because contact detection is currently unable to distinguish between connected and isolated parts of contacts. However, this could be added easily by running a second iteration over the rough detection and then splitting it into the separate contacts. Enabling HLBM to accurately model interactions involving concave particles is a critical step towards simulating more complicated real-world scenarios, thereby broadening its scope of applicability.

Another important future development is the addition of a lubrication force correction model. The introduction of this correction model could enable HLBM to accurately capture and account for lubricating forces between particles in close proximity. This addition would increase the accuracy of the method in scenarios where such forces play a critical role, thereby improving its predictive capabilities in simulating particle dynamics under varying conditions.

Deformable particles may potentially be supported by an integration of other LBM-based approaches that allow to solve nonlinear, transient, large displacement solid mechanics [102,103].

In summary, HLBM promises to meet upcoming scientific and industrial challenges. Performance improvements, hybrid parallelization, GPU support, extension of contact models to handle concave particles, integration of lubricating force corrections, and allowing for deformable particles are key challenges that would strengthen HLBM’s capabilities and enable more accurate, efficient, and versatile simulations in a wide range of applications.

6. Summary and Conclusions

In summary, HLBM is a versatile computational framework that is also widely used for the study of particulate flows. This paper has provided a comprehensive overview of HLBM, elucidating its fundamental principles and presenting its diverse applications in particle-laden flows as documented in the existing literature.

Notable strengths of HLBM include its monolithic nature, which allows for the seamless simulation of diverse fluid-solid interactions, and its ability to handle arbitrary particle shapes, even irregular and concave geometries, while accurately capturing local surface interactions. In addition, its parallel scheme based on LBM ensures high computational efficiency, making it suitable for large-scale simulations. However, being LBM-based comes with the drawback of requiring small time steps.

Future development needs focus on the underlying models and performance improvements. The discrete contact model needs to be extended by a lubrication force correction model to improve accuracy at particle volume fractions ${\varphi }_{\mathrm{p}}>0.3$ and to support concave shapes. Allowing for deformable particles would also be important to extend HLBM’s applicability. DNS, including HLBM, are computationally expensive. Increasing the performance e.g., by introducing hybrid parallelization techniques and GPU support would speed up simulations and also allow a much larger number of surface resolved particles to be considered.

These developments promise to greatly expand the capabilities of HLBM and increase its usefulness in solving complex real-world problems.