No full-text available
To read the full-text of this research,
you can request a copy directly from the author.
Simulation Methods for Particulate Flows and Concentrated Suspensions
Abstract
Numerical simulations are extensively used to investigate the motion of suspended particles in a fluid and their influence on the dynamics of the overall flow. Contexts range from the rheology of concentrated suspensions in a viscous fluid to the dynamics of particle-laden turbulent flows. This review summarizes several current approaches to the numerical simulation of rigid particles suspended in a flow, pointing out both common features and differences, along with their primary range of application. The focus is on non-Brownian systems for which thermal fluctuations do not play a role, whereas interparticle forces may result in particle self-assembly. Applications may include the motion of a few isolated particles with complex shape or the collective dynamics of many suspended particles. Expected final online publication date for the Annual Review of Fluid Mechanics Volume 49 is January 03, 2017. Please see http://www.annualreviews.org/page/journal/pubdates for revised estimates.
... For small particles and large numerical grid cells, the particle momentum source is applied to the numerical grid cell that the center of the particle occupies, which essentially corresponds to a top-hat function and is known as the particle source in cell (PSIC) method (Crowe et al. 1977). As the particle size approaches the size of the numerical grid cell, the particle momentum source is regularized with a Gaussian or a polynomial approximation of a Gaussian (Maxey 2017;Maxey & Patel 2001;Evrard et al. 2020;Poustis et al. 2019;Keane et al. 2023), which can be theoretically justified in the volume-filtered framework under the assumption that the filter width is much larger than the particle. It is unknown, however, how large the filter width has to be and how the shape of the regularization changes if the filter width is smaller. ...
... Furthermore, we derive an analytical expression for the regularization of the particle momentum source in the Stokes limit and compare it to the commonly used Gaussian for different filter width to particle size ratios. A common simplification for dilute regimes, i.e., large fluid volume fractions or small particle volume fractions, is to assume a constant fluid volume fraction, such that the governing fluid equations are equal to the single-phase NSE with an additional particle momentum source term (Maxey 2017). With this assumption, an equation for the phaseaveraged fluid kinetic energy can be derived (Xu & Subramaniam 2007;Subramaniam et al. 2014;Mehrabadi et al. 2015), which we show to be an oversimplification for large particle volume fractions or small fluid volume fractions. ...
... We exclude the discussion of the modeling of the hydrodynamic force on the particle, F h,i . Instead, we discuss the regularization kernel, i.e., how s i varies in space, which is typically approximated with a Gaussian, a polynomial approximation of a Gaussian or a top-hat function (Crowe et al. 1977;Maxey 2017;Maxey & Patel 2001;Evrard et al. 2020;Poustis et al. 2019;Keane et al. 2023). It is commonly argued that if the filter width is much larger than the size of the particle, i.e., δ ≫ a, the filter kernel g varies insignificantly across the volume of the particle and can be treated as a constant for the integration (Capecelatro & Desjardins 2013): ...
The volume-filtering of the governing equations of a particle-laden flow allows for sim- ulating the fluid phase as a continuum and accounting for the momentum exchange between the fluid and the particles by adding a source term in the fluid momentum equations. The volume-filtering of the Navier-Stokes equations allows to consider the effect that particles have on the fluid without further assumptions, but closures arise of which the implications are not fully understood. It is common to either neglect these closures or model them using assumptions of which the implications are unclear. In the present paper, we carefully study every closure in the volume-filtered fluid momentum equation and investigate their impact on the momentum and energy transfer dependent on the filtering characteristics. We provide an analytical expression for the viscous closure that arises because filter and spatial derivative in the viscous term do not commute. An analytical expression for the regularization of the particle momentum source of a single sphere in the Stokes regime is derived. Furthermore, we propose a model for the subfilter stress tensor, which originates from filtering the advective term. The model for the subfilter stress tensor is shown to agree well with the subfilter stress tensor for small filter widths relative to the size of the particle. We show that, contrary to common practice, the subfilter stress tensor requires modeling and should not be neglected. For small filter widths, we find that the commonly applied Gaussian regularization of the particle momentum source is a poor approximation of the spatial distribution of the particle momentum source, but for larger filter widths the spatial distribution approaches a Gaussian. Furthermore, we propose a modified advective term in the volume-filtered momentum equation that consistently circumvents the common stability issues observed at locally small fluid volume fractions and identify inconsistencies in previous studies of the phase-averaged kinetic energy of the volume-filtered fluid velocity. Finally, we propose a generally applicable form of the volume-filtered momentum equation and its closures based on clear and well founded assumptions and propose guidelines for point-particle simulations based on the new findings.
... Besides external forces, the dynamics of particles is significantly impacted by hydrodynamic interactions (HIs) [10]. The incorporation of HIs poses challenges for simulations due to their influences across a wide range of spatial scales, spanning both short and long ranges, and the inherent many-body nature of HI, which renders pairwise interaction models inadequate [11,12]. In low Reynolds number regimes, the dynamics of particles is primarily governed by viscous effects due to the significantly shorter inertial relaxation time compared to the viscous relaxation time. ...
... The key to simulating particulate suspensions, i.e., evolving the dynamical equation (1), is the calculation of the mobility matrix M at each time step. Two types of traditional approaches exist for obtaining the mobility matrix M(X): directly solving the Stokes equations with prescribed boundary conditions on the particles' surfaces [12,[16][17][18]; and approximating M(X) through multipole expansions truncated at the stresslet level, as in SD and its variants [11,22,24,41]. However, both methods are computationally demanding, and the computed M(X) lacks transferability to other suspensions even with the same type of particles but different particle numbers or concentrations. ...
We present a fast and scalable framework, leveraging graph neural networks (GNNs) and hierarchical matrix (-matrix) techniques, for simulating large-scale particulate suspensions, which have broader impacts across science and engineering. The framework draws on the Hydrodynamic Interaction Graph Neural Network (HIGNN) that employs GNNs to model the mobility tensor governing particle motion under hydrodynamic interactions (HIs) and external forces. HIGNN offers several advantages: it effectively captures both short- and long-range HIs and their many-body nature; it realizes a substantial speedup over traditional methodologies, by requiring only a forward pass through its neural networks at each time step; it provides explainability beyond black-box neural network models, through direct correspondence between graph connectivity and physical interactions; and it demonstrates transferability across different systems, irrespective of particles' number, concentration, configuration, or external forces. While HIGNN provides significant speedup, the quadratic scaling of its overall prediction cost (with respect to the total number of particles), due to intrinsically slow-decaying two-body HIs, limits its scalability. To achieve superior efficiency across all scales, in the present work we integrate -matrix techniques into HIGNN, reducing the prediction cost scaling to quasi-linear. Through comprehensive evaluations, we validate -HIGNN's accuracy, and demonstrate its quasi-linear scalability and superior computational efficiency. It requires only minimal computing resources; for example, a single mid-range GPU is sufficient for a system containing 10 million particles. Finally, we demonstrate -HIGNN's ability to efficiently simulate practically relevant large-scale suspensions of both particles and flexible filaments.
... The latter approach is the basis of Immersed Boundary Methods [8] [9], which are frequently used when the (moving) boundaries have complex shapes. More information on fully resolved methods can be found in the review by Maxey [10]. ...
... For all the simulations in this paper, we use a collision time = 0.1Δ and a restitution coefficient = 1.0. The contribution Δ in equation (10) accounts for close-range hydrodynamic forces (called lubrication forces) which occur on a length scale on the order of the particle radius. For this reason, they cannot be captured by the fluid solver in an unresolved method such as the one presented here. ...
Many interesting particulate flow problems can only be studied using efficient numerical methods. We present a method based on a lattice Boltzmann fluid coupled to unresolved particles that interact with each other via the Discrete Element Method. Our method improves upon existing numerical schemes through the addition of a novel subcycling algorithm that guarantees momentum conservation during each DEM substep. The intended application is studying transport of solid particles in physiologic processes, although the method is generally applicable. We present in detail the development and (parallel) implementation of the model and show how intricacies of the coupling scheme must be considered to avoid unphysical behavior and instabilities. The scalability of the code is tested on two modern supercomputers. We demonstrate the method's applicability to biomedical applications by simulating the injection and distribution of particles in an idealized liver vasculature.
... We exclude the discussion of the modelling of the hydrodynamic force on the particle, F h,i . Instead, we discuss the regularization kernel, 996 A41-10 i.e. how s i varies in space, which is typically approximated with a Gaussian centred at the particle centre (referred to as Gaussian regularization kernel), a polynomial approximation of a Gaussian or a top-hat function (Crowe et al. 1977;Maxey & Patel 2001;Maxey 2017;Poustis et al. 2019;Evrard et al. 2020;Keane et al. 2023). It is commonly argued that if the filter width is much larger than the size of the particle, i.e. δ a, the filter kernel g varies insignificantly across the volume of the particle and can be treated as a constant for the integration (Capecelatro & Desjardins 2013): ...
... A very common simplification of the volume-filtered Euler-Lagrange framework is to use the following fluid momentum equation instead of (2.15) (Maxey & Patel 2001;Maxey 2017;Poustis et al. 2019;Evrard et al. 2020): ...
The volume filtering of the governing equations of a particle-laden flow allows for simulating the fluid phase as a continuum, and accounting for the momentum exchange between the fluid and the particles by adding a source term in the fluid momentum equations. The volume filtering of the Navier–Stokes equations allows consideration of the effect that particles have on the fluid without further assumptions, but closures arise of which the implications are not fully understood. It is common to either neglect these closures or model them using assumptions of which the implications are unclear. In the present paper, we carefully study every closure in the volume-filtered fluid momentum equation and investigate their impact on the momentum and energy transfer dependent on the filtering characteristics. We provide an analytical expression for the viscous closure that arises because the filter and spatial derivative in the viscous term do not commute. An analytical expression for the regularization of the particle momentum source of a single sphere in the Stokes regime is derived. Furthermore, we propose a model for the subfilter stress tensor, which originates from filtering the advective term. The model for the subfilter stress tensor is shown to agree well with the subfilter stress tensor for small filter widths relative to the size of the particle. We show that, in contrast to common practice, the subfilter stress tensor requires modelling and should not be neglected. For filter widths comparable to the particle size, we find that the commonly applied Gaussian regularization of the particle momentum source is a poor approximation of the spatial distribution of the particle momentum source, but for larger filter widths, the spatial distribution approaches a Gaussian. Furthermore, we propose a modified advective term in the volume-filtered momentum equation that consistently circumvents the common stability issues observed at locally small fluid volume fractions. Finally, we propose a generally applicable form of the volume-filtered momentum equation and its closures based on clear and well-founded assumptions. Based on the new findings, guidelines for point-particle simulations and the filter width with respect to the particle size and fluid mesh spacing are proposed.
... Discrete particles reveal a clearer shear banding behaviour in creep flow or quiescent flow conditions [24], because the cohesion between particles can be well defined by a dynamic adhesive force that dominates the thixotropic properties, time dependency and temporary build-up of bonds [31,55]. Among the rheological descriptions available in the literature [45,23,[27][28][29][30][31][32][33][34][35][36][37][38][39][40][41][42][43], there is a growing consensus that a reversible evolution of interparticle bonds mostly comes from non-contact interactions of cement particles, which should be responsible for the variation in the macroscopic flow behaviour. However, this assumption has not been attached great importance so far due to the lack of a quantifiable and applicable model that can replicate the physical performance of cement suspension flows. ...
... It is well known that longer-range hydrodynamic forces, such as the drag force, will play a dominant role in pressure-driven flow [32,59]. However, given the adopted rotation speed of 3.3 rpm, which is equivalent to a velocity of 3.4Á10 -4 m/s at the blade tips, the drag forces arising from the velocity difference between the spheres and the fluid will be negligible compared to the rigid resistance of the interparticle bonds [33]. It is therefore reasonable to simplify the effect of fluid flow under low applied shear rate. ...
Thixotropy of cementitious materials is a crucial intrinsic property that determines the flowability and workability of cement-based grout. A novel virtual bond model of cement particles is developed in this paper to depict the thixotropy of cement grout. A particulate description of the reversible and erasable interparticle bonds is established based on experimental observations with a focus on the non-contact interactions mainly contributed in practice by calcium silicate hydrates (C–S–H). The structural breakdown of the cement network is realized through bonds breakage under applied motion, and the bonding network recovers with regeneration of interparticle connections that involve reversible hydrate reactions in the mixture. The balance between bond rupture and rebuilding can be tuned by assigning different strength limits for bond breakage. We have implemented this model in the open-source code Yade to carry out 3D discrete element method simulations of a rotational vane system filled with spherical particles, and the results show good agreement with experimental data. The modelling results reveal the transition from a solid-like structure to a fluid-like medium within cement suspensions caused by the evolution of broken interparticle bonds. The results also provide a distinct view of thixotropic variation upon disturbance. This model is extendable to other cohesive materials providing an explicit physical definition of the interparticle interactions. It also provides a theoretical explanation for the empirical estimations of thixotropy common in engineering industries and a potential means of measuring cementitious granular flow that may be useful in future studies.
... For an overview of methods for solving the Stokes mobility problem, see Maxey [16], and the PhD theses of Bagge [17] and Peláez [18]. In the literature on approximate methods, such as the rigid multiblob method and Stokesian dynamics, the mobility problem is solved via saddle-point linear systems, where the given net forces and torques on each particle appear as additional constraints. ...
... Using this as the matrix-vector multiply, the preconditioned vector γ is solved for iteratively, then the proxystrengths are recovered by (30). Finally, from the latter one can extract the charges via (17), or use (16) to evaluate the solution anywhere in the domain, including on boundaries. ...
The method of fundamental solutions (MFS) is known to be effective for solving 3D Laplace and Stokes Dirichlet boundary value problems in the exterior of a large collection of simple smooth objects. Here we present new scalable MFS formulations for the corresponding elastance and mobility problems. The elastance problem computes the potentials of conductors with given net charges, while the mobility problem -- crucial to rheology and complex fluid applications -- computes rigid body velocities given net forces and torques on the particles. The key idea is orthogonal projection of the net charge (or forces and torques) in a rectangular variant of a "completion flow". The proposal is compatible with one-body preconditioning, resulting in well-conditioned square linear systems amenable to fast multipole accelerated iterative solution, thus a cost linear in the particle number. For large suspensions with moderate lubrication forces, MFS sources on inner proxy-surfaces give accuracy on par with a well-resolved boundary integral formulation. Our several numerical tests include a suspension of 10000 nearby ellipsoids, using 26 million total preconditioned degrees of freedom, where GMRES converges to five digits of accuracy in under two hours on one workstation.
... Acoustic wave analysis of fluid suspensions filled with solid particles finds numerous industrial applications and is the subject of a large number of existing studies [1][2][3][4][5][6]. Earlier works aimed to develop various analytical models and numerical schemes on wave scattering due to solid particles and wave dispersion, and extensive numerical calculations are always requested and different models often give quite different results for attenuation and phased velocity, with typically 20-30% or even larger relative errors [7][8][9][10][11][12][13][14][15][16][17][18]. ...
... Fig. 6 Dependency of effective phase velocity (39) with δ 0.2 on a the mass density ratio (ρ S /ρ f ) with the bulk modulus ratio (B S /B f ) 20; b the bulk modulus ratio (B S /B f ) with the mass density ratio (ρ S /ρ f ) 2 ...
A direct method is developed to study acoustic waves in a viscous fluid filled with randomly distributed hard spherical particles. The present method is based on the assumption that the relative shift of the velocity field of immersed hard particles from the host fluid is responsible for dynamic behavior of the suspension, and its role can be formulated by substituting the inertia term of governing equations by the acceleration field of the mass centre of the representative unit cell. Compared to existing models based on rather complicated mathematical formulation and numerical calculations, the present model enjoys conceptual and mathematical simplicity and the generality. Explicit formulas are derived for the attenuation coefficient and effective phase velocity of plane compression waves and shear waves. The efficiency and accuracy of the model are demonstrated by quantitatively good agreement between the predicted results and known data for a wide range of material and geometrical parameters. The proposed model could offer a relatively simple general method and easy-to-use explicit formulas to study acoustic wave propagation in hard particle-viscous fluid suspensions.
... Despite the broad usage of the point-particle approach, the accuracy, convergence and stability of the method are affected by its numerical treatment. The computational approximation of point-particles requires interpolation between the Lagrangian point tracer and what is usually a grid based approximation of the Eulerian carrier phase model (Balachandar and Eaton, 2010;Fox, 2012;Maxey, 2017;Elghobashi, 2019;Brandt and Coletti, 2022;Bec et al., 2023). Particles move freely through the domain in locations different from the computational grid points of the flow, whose tractability in parallel computing is involved and requires the use of interpolation (Yeung and Pope, 1988;Balachandar and Maxey, 1989;Jacobs et al., 2007). ...
... The point-particle assumption leads also to convergence issues related to a strong grid dependence because the forcing is modeled by averaging in the volume of the computational cell unless the number of particles per cell exceeds a threshold (Gualtieri et al., 2015). These drawbacks have inspired the development of different alternative approaches for the simulation of particle-laden flows (Koumoutsakos, 2005;Maxey, 2017). Some of the recent research includes the volume averaged method (Capecelatro and Desjardins, 2013;Ireland and Desjardins, 2017;Shallcross et al., 2020;Fuster et al., 2014;Hsiao et al., 2017;Bryngelson et al., 2019), the modeling of interparticle forcing by the pairwise interaction extended point-particle (PIEP) model (Akiki et al., 2017a,b;Moore et al., 2019;Balachandar et al., 2020), the microstructure-informed probabilitydriven point-particle (MPP) model (Seyed-Ahmadi and Wachs, 2020), the exact regularized point particle (ERPP) method (Gualtieri et al., 2015;Battista et al., 2018) to tackle convergence issues, the use of discrete Green's functions to find the undisturbed velocity and correct the particle's self-force , and machine learning (ML) models to find closures to reduced descriptions (He and Tafti, 2019;Seyed-Ahmadi and Wachs, 2022). ...
... However, at present, we know that the collective dynamics of small particles or bubbles can affect the underlying flow significantly through various numerical and experimental studies (Hetsroni & Sokolov 1971;Gore & Crowe 1989;Sun & Faeth 1986;Lance et al. 1991) and the most work has been done for isothermal systems (Balachandar & Eaton 2010;M. Kuerten 2016;Maxey 2017). The effect of the suspended particles on the convective heat transfer, particularly Rayleigh-Bénard convection, is gaining importance in recent times and is the main theme of the current work. ...
We investigate the effect of inertial particles on Rayleigh-B\'enard convection using weakly nonlinear stability analysis. In the presence of nonlinear effects, we study the limiting value of growth of instabilities by deriving a cubic Landau equation. An Euler-Euler/two-fluid formulation is being used to describe the flow instabilities in particle-laden Rayleigh-B\'enard convection. The nonlinear results are presented near the critical point (bifurcation point) for water droplets in the dry air system. It is found that supercritical bifurcation is the only type of bifurcation beyond the critical point. Interaction of settling particles with the flow and the Reynolds stress or distortion terms emerge due to the nonlinear self-interaction of fundamental modes, breaking down the top-bottom symmetry of the secondary flow structures. In addition to the distortion functions, the nonlinear interaction of fundamental modes generates higher harmonics, leading to the tendency of preferential concentration of uniformly distributed particles, which is completely absent in the linear stability analysis. It is shown that in the presence of thermal energy coupling between the fluid and particles, the difference between the horizontally averaged heat flux at the hot and cold surface is equal to the net sensible heat flux advected by the particles. The difference between the heat fluxes at hot and cold surfaces is increased with an increase in particle concentration.
... For decades, numerical methods for particleladen flows have been greatly developed, and numerical simulation has become a powerful tool in both fundamental research and practical processes. Depending on the different levels of assumptions, the numerical methods for particleladen flows can be roughly classified as the Eulerian two-fluid method, the Lagrangian particle method, and the fully resolved method (Drew, 1983;Elghobashi, 1994;Crowe, Troutt and Chung, 1996;Loth, 2000;Eaton, 2009;Wang, 2009;Balachandar and Eaton, 2010;Yu and Shao, 2010;Subramaniam, 2013;Tenneti and Subramaniam, 2014;Kuerten, 2016;Maxey, 2017;Elghobashi, 2019;Brandt and Coletti, 2022;Fox, 2024;Schneiderbauer, 2024). The prediction accuracy of particle-laden flows is critically dependent on the fluid force models used in numerical simulations (Michaelides, 1997(Michaelides, , 2003Loth and Dorgan, 2009). ...
Predicting particle-laden flows requires accurate fluid force models. However, a reliable particle force model for finite-size particles in turbulent flows remains lacking. In the present work, a fluid force model for a finite-size spherical particle in turbulence is developed by simulating turbulent flow past a fixed spherical particle using particle-resolved direct numerical simulation (PRDNS). Our simulation demonstrates that turbulence increases the mean drag force of the particle, which is consistent with previous studies. By correlating the DNS data as functions of the Reynolds number of particles, the ratio of the particle-to-turbulence scale, and the intensity ratio of the turbulence, an empirical correlation for the mean drag force is obtained. Furthermore, we find that the fluctuations of both the drag and lateral forces follow the Gaussian distribution. Consequently, the temporal variations of the fluctuating drag and lateral forces are modeled using a stochastic Langevin equation. Empirical correlations of the fluctuation intensities and time scales involved in the stochastic model are also determined from the DNS data. Finally, we simulate the movement of a finite-size particle in turbulence and the dispersion of particles in a turbulent channel flow to validate the proposed model. The proposed fluid force model requires the mean flow velocity, the kinetic energy of the turbulence, and the dissipation rate of the turbulence as inputs, which makes it well suited for combination with the Reynolds-averaged Navier-Stokes (RANS) approach.
... Hasta ahora, no hay una teoría con base en el medio continuo que explique lo observado de forma experimental [22], es decir, un análogo de lo que las ecuaciones de Navier-Stokes representan para la dinámica de fluidos simples. De modo que se requiere describir la física que ocurre en la meso-escala (interacciones partícula-partícula y partícula-matriz) [24,25]. Para este fin, se utilizan herramientas computacionales predictivas que imiten al sistema real. ...
En esta revisión se describe y se muestra la compleja reología que exhiben las suspensiones concentradas de partículas sólidas no coloidales. Su presencia en la industria y en la vida cotidiana hace que sea relevante describir y explicar el origen de su comportamiento reológico. Para conocer qué son, cómo se comportan, cómo se diferencian de los fluidos Newtonianos y qué estrategias se usan para reproducir su flujo, esta revisión se divide de la siguiente manera: en la sección I se describe la composición y los tipos de suspensiones en función de la fracción de sólidos; en la sección II se describe la reología, que se encarga de caracterizar y describir las propiedades materiales de los fluidos, en este caso, las suspensiones concentradas; en la sección III se exhibe el comportamiento de las suspensiones concentradas, como son: el adelgazamiento al corte, el engrosamiento al corte continuo y discontinuo, y el flujo bandeado en la dirección de la vorticidad; en la sección IV se describen cómo se modelan y cuáles son los mecanismos propuestos que buscan reproducir la reología de las suspensiones concentradas, y en la sección V se muestra el trabajo por hacer en otros tipos de suspensiones.
... This requires to capture the velocity and temperature gradients at the interface between the fluid and the particle. For a review of PR-DNS methods, the reader is referred to Tenneti and Subramanian [16], Maxey [17] and more recently Marchelli et al. [18]. The Eulerian grid resolution associated with PR-DNS is then more than 40 meshes per diameter for viscous flows and increase with the Reynolds number [19]. ...
Particle-Resolved Simulations (PRS) of fluid-solid particles are conducted to study fluid-particle heat transfers and wall-to-bed heat transfers in an anisothermal liquid-solid fluidized bed. An overview of existing PRS methods to study anisothermal fluid-solid flows is presented. In the framework of fluidized bed simulations, the collision detection method is optimized using Verlet tables. The overall computation time is reduced by 20%. An original Lagrangian method to compute the fluid-particle heat transfers is presented for an isolate particle. A parametric study on the fluid-particle heat transfer is performed to assess well-known correlations of the literature for a settling particle in a quiescent fluid. 77 PRS are performed for Reynolds numbers between 1 and 32 and Prandtl number between 0.1 and 10 for a grid resolution of 20 meshes per particle diameter. For two of three correlations considered, the predicted heat flow is within 10% error. An anisothermal fluidized bed of 2 134 particles is finally studied. Four fluidization velocities are considered for a solid fraction comprised between 0.13 and 0.35. Three grid resolutions are carried out to assess the sensitivity of the mesh for the lowest fluidization velocity (12, 24 and 36 meshes per particle diameter). Results show that the macroscopic behavior of the bed is well retrieved even with a coarser grid as the solid fraction is well predicted. However, strong effects of the grid resolution are observed on the fluid-particle and the wall-to-bed heat transfers. The study of the velocity-temperature correlation shows that the parietal heat transfer is driven by the turbulent heat flow near the wall (+ ∼ 30).
... The flexibility-induced alteration of the dense fiber suspension introduces complexities, making numerical simulation of such systems a challenging problem due to their multi-scale nature, involving varying length and time scales [6,7]. At the smallest scale, particle-resolved simulations are usually employed, capturing the boundary layer at the surface of each deformable particle while tracking individual particle deformations [6]. ...
Understanding the sedimentation behavior of flexible fiber suspensions is crucial for various industrial and environmental applications. In this study, we investigate the settling dynamics of flexible fibers in a quiescent fluid environment through numerical simulations. The Navier-Stokes equations govern the fluid phase, while particles are modeled using an incompressible hyperelastic Mooney-Rivlin material description. The immersed boundary method was employed to couple the fluid and particle phases. We compare the behavior of flexible and rigid fibers under gravity-induced sedimentation across different Cauchy numbers, revealing significant differences in deformation patterns and settling dynamics. The hindered velocity effect is observed exclusively for rigid fibers. Additionally, hydrodynamic interactions lead to distinct regions of low and high fiber concentration, with flexible fibers forming denser clusters compared to rigid fibers, affecting overall clustering dynamics. Notably, increased flexibility enhances cluster formation and influences settling velocities. Our results demonstrate a clear correlation between flexibility (Cauchy number) and sedimentation velocity, with increased flexibility leading to enhanced sedimentation dynamics.
... The dynamics of the free fall motion of solid particles in a fluid can be studied by analyzing the drag force experienced by the particles (Maxey, 2017;Liu & Yu, 2022). Particle diameter, fluid velocity, fluid density, and fluid viscosity affect the magnitude of the drag force. ...
Investigating the motion of solid particles in fluids analyzes the drag force experienced by the particles, depending on parameters such as particle diameter, fluid velocity, density, and viscosity. The Reynolds number, which expresses a fluid's inertia relative to its viscosity, governs the dimensionless drag coefficient, which is critical to understanding drag forces. Terminal velocity, achieved when the force of gravity equals the buoyancy and drag forces of the fluid, is a critical concept often analyzed using the Stokes model. However, differences between theoretical and experimental terminal velocities arise due to oversight of the model's application conditions. Numerical experiments offer controlled conditions to address this, helping predictions align with theoretical models. This research explores the influence of density ratio and particulate diameter on terminal velocity, aiming to support research-based learning for teachers and conceptual understanding for students. Numerical experiments designed by Arbie et al. (2021) investigated two-dimensional particulate configurations, allowing controlled manipulation of parameters. The results show a strong influence of the density ratio and diameter to the terminal velocity, with larger parameter values influencing the Reynolds number and giving rise to differences between theoretical and experimental values. Therefore, careful parameter selection is essential for viscosity experiments, aligning with the objectives and comparability of theoretical models.
... The DNS techniques solve the equations of motion for each individual particle. The review studies by Maxey (2017) for the non-Brownian and Larson (2021) for the Brownian suspension flows mainly categorize these techniques based on how the fluid medium is meshed and how the flow within this medium is simulated. While these methods are robust for studying particle pair interactions, suspension microstructure, or bulk rheological properties such as shear viscosity, the high computational demand limits their application to realistic suspension flows. ...
The present study simulates shear-induced migration (SIM) in semi-dilute pressure-driven Stokes suspension flows using a multi-fluid (MF) model. Building on analysis from a companion paper (Harvie, 2024), the specific formulation uses volume-averaged phase stresses that are linked to the binary hydrodynamic interaction of spheres and suspension microstructure as represented by an anisotropic, piece-wise constant pair-distribution function (PDF). The form of the PDF is chosen to capture observations regarding the microstructure in sheared suspensions of rough particles, as reported in the literature. Specifically, a hydrodynamic roughness value is used to represent the width of the anisotropic region, and within this region the concentration of particles is higher in the compression zone than expansion zone. By numerically evaluating the hydrodynamic particle interactions and calculating the various shear and normal viscosities, the stress closure is incorporated into Harvie's volume-averaged MF framework, referred to as the MF-roughness model. Using multi-dimensional simulations the roughness and compression zone PDF concentration are then globally optimised to reproduce benchmark solid and velocity distributions reported in the literature for a variety of semi-dilute monodisperse suspension flows occurring within rectangular channels. For comparison, two different versions of the phenomenological stress closure by Morris and Boulay (1999) are additionally proposed as fully tensorial frame-invariant alternatives to the MF-roughness model. Referred to as MF-MB99-A and MF-MB99-B, these models use alternative assumptions for partitioning of the mixture normal stress between the solid and fluid phases. The optimised solid and velocity distributions from all three stress closures are similar and correlate well with the experimental data.
... We therefore confine ourselves here to a brief overview of the field, identifying some of the main approaches and their recent variants in order to highlight the broad diversity of methods available and to provide a wider context within which to place our cut-cell method. We justify the brevity of this overview by noting that comprehensive reviews of immersed boundary methods are available by Mittal & Iaccarino [1], Sotiropoulos & Yang [2] and recently by Maxey [3]. ...
An explicit moving boundary method for the numerical solution of time-dependent hyperbolic conservation laws on grids produced by the intersection of complex geometries with a regular Cartesian grid is presented. As it employs directional operator splitting, implementation of the scheme is rather straightforward. Extending the method for static walls from Klein et al., Phil. Trans. Roy. Soc., A367, no. 1907, 4559-4575 (2009), the scheme calculates fluxes needed for a conservative update of the near-wall cut-cells as linear combinations of standard fluxes from a one-dimensional extended stencil. Here the standard fluxes are those obtained without regard to the small sub-cell problem, and the linear combination weights involve detailed information regarding the cut-cell geometry. This linear combination of standard fluxes stabilizes the updates such that the time-step yielding marginal stability for arbitrarily small cut-cells is of the same order as that for regular cells. Moreover, it renders the approach compatible with a wide range of existing numerical flux-approximation methods. The scheme is extended here to time dependent rigid boundaries by reformulating the linear combination weights of the stabilizing flux stencil to account for the time dependence of cut-cell volume and interface area fractions. The two-dimensional tests discussed include advection in a channel oriented at an oblique angle to the Cartesian computational mesh, cylinders with circular and triangular cross-section passing through a stationary shock wave, a piston moving through an open-ended shock tube, and the flow around an oscillating NACA 0012 aerofoil profile.
... Thus, there is a clear need for developing continuum models and improving existing ones to enable reliable predictions of the flow behavior of inertial suspensions. Additionally, fully resolved multiparticle numerical simulations [47,48] could be utilized to investigate the mechanism underlying the destabilizing effect of the particles. Such ...
Although inertial particle-laden flows occur in a wide range of industrial and natural processes, there is both a lack of fundamental understanding of these flows and continuum-level governing equations needed to predict transport and particle distribution. Towards this effort, the Taylor-Couette flow (TCF) system has been used recently to study the flow behavior of particle-laden fluids under inertia. This article provides an overview of experimental, theoretical, and computational work related to the TCF of neutrally buoyant non-Brownian suspensions, with an emphasis on the effect of finite-sized particles on the series of flow transitions and flow structures. Particles, depending on their size and concentration, cause several significant deviations from Newtonian fluid behavior, including shifting the Reynolds number corresponding to transitions in flow structure and changing the possible structures present in the flow. Furthermore, particles may also migrate depending on the flow structure, leading to hysteretic effects that further complicate the flow behavior. The current state of theoretical and computational modeling efforts to describe the experimental observations is discussed, and suggestions for potential future directions to improve the fundamental understanding of inertial particle-laden flows are provided.
... During the last years, various methods have been proposed to perform interface-resolved direct numerical simulations (DNS) of particulate flows. The state of art and the different principles and applications have been recently documented in the comprehensive review article by Maxey (2017). In the present study, the immersed boundary method (IBM) originally proposed by Uhlmann (2005) and modified by Breugem (2012) has been used to simulate suspensions of finite-size neutrally-buoyant spherical particles in turbulent square duct flow. ...
We study the turbulent square duct flow of dense suspensions of neutrally-buoyant spherical particles. Direct numerical simulations (DNS) are performed in the range of volume fractions , using the immersed boundary method (IBM) to account for the dispersed phase. Based on the hydraulic diameter a Reynolds number of 5600 is considered. We report flow features and particle statistics specific to this geometry, and compare the results to the case of two-dimensional channel flows. In particular, we observe that for and 0.1, particles preferentially accumulate on the corner bisectors, close to the duct corners as also observed for laminar square duct flows of same duct-to-particle size ratios. At the highest volume fraction, particles preferentially accumulate in the core region. For channel flows, in the absence of lateral confinement particles are found instead to be uniformily distributed across the channel. We also observe that the intensity of the cross-stream secondary flows increases (with respect to the unladen case) with the volume fraction up to , as a consequence of the high concentration of particles along the corner bisector. For the turbulence activity is strongly reduced and the intensity of the secondary flows reduces below that of the unladen case. The friction Reynolds number increases with in dilute conditions, as observed for channel flows. However, for the mean friction Reynolds number decreases below the value for .
... Several techniques have been developed in the past few decades for simulating the hydrodynamics of multiple spherical particles including the Stokesian dynamics approach (e.g., [4,11,47]), multipole methods (e.g., [8,37]), fictitious domain methods (e.g., [33]) and boundary integral methods (e.g., [1,9,34]). We refer the reader to [29] for a recent review on the broader topic of simulation methods for particulate flows. The present work combines features from both the multipole methods (spectral representations) and the boundary integral methods (second-kind formulations, fast algorithms) to arrive at a fast, spectrally accurate numerical method. ...
We show that the standard boundary integral operators, defined on the unit sphere, for the Stokes equations diagonalize on a specific set of vector spherical harmonics and provide formulas for their spectra. We also derive analytical expressions for evaluating the operators away from the boundary. When two particle are located close to each other, we use a truncated series expansion to compute the hydrodynamic interaction. On the other hand, we use the standard spectrally accurate quadrature scheme to evaluate smooth integrals on the far-field, and accelerate the resulting discrete sums using the fast multipole method (FMM). We employ this discretization scheme to analyze several boundary integral formulations of interest including those arising in porous media flow, active matter and magneto-hydrodynamics of rigid particles. We provide numerical results verifying the accuracy and scaling of their evaluation.
... The force and torque balances yield a low Reynolds number mobility problem for the translational and angular motion of the segments that is identical to that for a collection of rigid particles. We solve this mobility problem using a regularized multipole approach for Stokes flows known as the force-coupling method (FCM) [36,37,38,35,39]. In the limit of large N flag , FCM provides an approximation of the hydrodynamics consistent with a regularized version of slender body theory [40,41,42,43,44,31,33]. ...
Swimming cells and microorganisms are as diverse in their collective dynamics as they are in their individual shapes and propulsion mechanisms. Even for sperm cells, which have a stereotyped shape consisting of a cell body connected to a flexible flagellum, a wide range of collective dynamics is observed spanning from the formation of tightly packed groups to the display of larger-scale, turbulence-like motion. Using a detailed mathematical model that resolves flagellum dynamics, we perform simulations of sperm suspensions containing up to 1000 cells and explore the connection between individual and collective dynamics. We find that depending on the level of variation in individual dynamics from one swimmer to another, the sperm exhibit either a strong tendency to aggregate, or the suspension exhibits large-scale swirling. Hydrodynamic interactions govern the formation and evolution of both states. In addition, a quantitative analysis of the states reveals that the flows generated at the time-scale of flagellum undulations contribute significantly to the overall energy in the surrounding fluid, highlighting the importance of resolving these flows.
... Below we provide a short overview of recently developed numerical methods that model driven colloidal suspensions at low Reynolds number. An extensive recent review for suspensions at finite Reynolds number is given by Maxey [120]. Here, we only focus on methods that explicitly solve the inertia-less Stokes equations for the incompressible fluid phase. ...
In this review article, we focus on collective motion in externally driven colloidal suspensions, as well as how these collective effects can be harnessed for use in microfluidic applications. We highlight the leading role of hydrodynamic interactions in the self-assembly, emergent behavior, transport, and mixing properties of colloidal suspensions. A special emphasis is given to recent numerical methods to simulate driven colloidal suspensions at large scales. In combination with experiments, they help us to understand emergent dynamics and to identify control parameters for both individual and collective motion in colloidal suspensions.
... The particle motion is then described by (van der Hoef et al. 2008;Maxey 2017) ...
We studied the transport and deposition behaviour of point particles in Rayleigh–Bénard convection cells subjected to Couette-type wall shear. Direct numerical simulations (DNSs) are performed for Rayleigh number ( Ra ) in the range with a fixed Prandtl number , while the wall-shear Reynolds number ( ) is in the range . With the increase of , the large-scale rolls expanded horizontally, evolving into zonal flow in two-dimensional simulations or streamwise-oriented rolls in three-dimensional simulations. We observed that, for particles with a small Stokes number ( St ), they either circulated within the large-scale rolls when buoyancy dominated or drifted near the walls when shear dominated. For medium St particles, pronounced spatial inhomogeneity and preferential concentration were observed regardless of the prevailing flow state. For large St particles, the turbulent flow structure had a minor influence on the particles’ motion; although clustering still occurred, wall shear had a negligible influence compared with that for medium St particles. We then presented the settling curves to quantify the particle deposition ratio on the walls. Our DNS results aligned well with previous theoretical predictions, which state that small St particles settle with an exponential deposition ratio and large St particles settle with a linear deposition ratio. For medium St particles, where complex particle–turbulence interaction emerges, we developed a new model describing the settling process with an initial linear stage followed by a nonlinear stage. Unknown parameters in our model can be determined either by fitting the settling curves or using empirical relations. Compared with DNS results, our model also accurately predicts the average residence time across a wide range of St for various .
... Accurate and efficient simulation of particulate flows is significant in many industrial fields, such as chemical, metallurgy, energy, and microfluidics [1][2][3][4]. Understanding the transport behaviors of particulate flows has attracted considerable attention in the past decades. Conventional computational fluid dynamics (CFD) methods, such as the finite volume method (FVM) and the finite element method (FEM), have been successfully applied to simulate particulate flows [5][6][7][8]. ...
A volumetric lattice Boltzmann (LB) method is developed for the particle-resolved direct numerical simulation of thermal particulate flows with conjugate heat transfer. This method is devised as a single-domain approach by applying the volumetric interpretation of the LB equation and introducing a solid fraction field to represent the particle. The volumetric LB scheme is employed to enforce the nonslip velocity condition in the solid domain, and a specialized momentum exchange scheme is proposed to calculate the hydrodynamic force and torque acting on the particle. To uniformly solve the temperature field over the entire domain with high numerical fidelity, an energy conservation equation is first derived by reformulating the convection term into a source term. A corresponding LB equation is then devised to automatically achieve the conjugate heat transfer condition and correctly handle the differences in thermophysical properties. Theoretical analysis of this LB equation is also performed to derive the constraints to preserve the numerical fidelity even near the solid-fluid interface. Numerical tests are first performed to validate the present volumetric LB method in various aspects. Then, the sedimentation of a cold particle with conjugate heat transfer in a long channel is investigated. It is found that the sedimentation process can be divided into the accelerating, decelerating, and equilibrium stages. As a further application to dense particulate flows, the sedimentation of 2048 cold particles with conjugate heat transfer in a square cavity is simulated. The particulate Rayleigh-B\'{e}nard convection is successfully captured in this particle-resolved simulation.
... This quantity profoundly impacts the spatial distribution, spreading rate, collision probability and gravitational drift of the dispersed phase (Balachandar & Eaton 2010;Pumir & Wilkinson 2016;Mathai, Lohse & Sun 2020;Bec, Gustavsson & Mehlig 2024). Moreover, u s contributes to defining the flow regime around the particles, features in the formulation of surface forces exerted on them by the fluid and is key for turbulence modification (Bellani & Variano 2012;Ling, Parmar & Balachandar 2013;Maxey 2017;Oka & Goto 2022;Balachandar, Peng & Wang 2024). In the context of numerical simulations, u s is also a primary parameter to select the appropriate computational approach (Balachandar 2009;Tenneti & Subramaniam 2014). ...
Predicting the magnitude of the slip velocity of non-tracer particles with respect to the surrounding fluid is crucial to address both fundamental and practical questions involving dispersed turbulent flows. Here we derive an analytical model to predict the slip velocity of spherical particles in homogeneous isotropic turbulence. We modulate the particle equation of motion according to the inertial filtering framework, and obtain closed-form expressions for the mean slip velocity magnitude as a function of the governing parameters. These are compared against laboratory measurements and direct numerical simulations, demonstrating close agreement for both light and heavy particles, both smaller and larger than the Kolmogorov scales. The predictive value of the model and its implications are discussed, as well as the range of validity of the underlying assumptions.
... This non-linear coupling between particle and fluid dynamics poses fundamental difficulties for modeling. One way out is to rely on particleresolving numerical simulations (Tenneti & Subramaniam 2014, Maxey 2017. For turbulent suspensions with many particles, this is still very challenging. ...
When very small particles are suspended in a fluid in motion, they tend to follow the flow. How such tracer particles are mixed, transported, and dispersed by turbulent flow has been successfully described by statistical models. Heavy particles, with mass densities larger than that of the carrying fluid, can detach from the flow. This results in preferential sampling, small-scale fractal clustering, and large collision velocities. To describe these effects of particle inertia, it is necessary to consider both particle positions and velocities in phase space. In recent years, statistical phase-space models have significantly contributed to our understanding of inertial-particle dynamics in turbulence. These models help to identify the key mechanisms and non-dimensional parameters governing the particle dynamics, and have made qualitative, and in some cases quantitative predictions. This article reviews statistical phase-space models for the dynamics of small, yet heavy, spherical particles in turbulence. We evaluate their effectiveness by comparing their predictions with results from numerical simulations and laboratory experiments, and summarise their successes and failures. Annu. Rev. Fluid Mech. 56: In press. DOI: 10.1146/annurev-fluid-032822-014140.
... The dynamics of the free fall motion of solid particles in a fluid can be studied by analyzing the drag force experienced by the particles (Maxey, 2017;Liu & Yu, 2022). Particle diameter, fluid velocity, fluid density, and fluid viscosity affect the magnitude of the drag force. ...
Students can understand the terminal velocity of solid particles by analyzing their free-fall motion in a fluid. We show it using a program designed by (Arbie et al., 2019)(Arbie et al., 2021). We use two parameters, namely theparticle-fluid density ratio and the particle diameter, to determine their effect on terminal velocity. The experimental results show that the terminal velocity increases linearly with changes in the magnitude of the two parameters. Then, we also show the effect of the Reynolds number on the magnitude of the error value produced in the experiment.
... In first-principles, fully resolved direct numerical simulations (DNS) of a particle-laden flow, the flow around each particle needs to be resolved explicitly (Balachandar & Eaton 2010;Maxey 2017) (so-called particle-resolved DNS, PR-DNS). While this approach is free from modelling assumptions for the dispersed phase dynamics, it is computationally expensive due to the explicit imposition of no-slip and no-penetration boundary conditions at the surface of many particles moving in a turbulent medium. ...
In particle-laden turbulent wall flows, lift forces can influence the near-wall turbulence. This has been observed recently in particle-resolved simulations, which, however, are too expensive to be used in upscaled models. Instead, point-particle simulations have been the method of choice to simulate the dynamics of these flows during the last decades. While this approach is simpler, cheaper and physically sound for small inertial particles in turbulence, some issues remain. In the present work, we address challenges associated with lift force modelling in turbulent wall flows and the impact of lift forces in the near-wall flow. We performed direct numerical simulations of small inertial point particles in turbulent channel flow for fixed Stokes number and mass loading while varying the particle size. Our results show that the particle dynamics in the buffer region, causing the apparent particle-to-fluid slip velocity to vanish, raises major challenges for modelling lift forces accurately. While our results confirm that lift forces have little influence on particle dynamics for sufficiently small particle sizes, for inner-scaled diameters of order one and beyond, lift forces become quite important near the wall. The different particle dynamics under lift forces results in the modulation of streamwise momentum transport in the near-wall region. We analyse this lift-induced turbulence modulation for different lift force models, and the results indicate that realistic models are critical for particle-modelled simulations to correctly predict turbulence modulation by particles in the near-wall region.
... In a first-principles, fully-resolved direct numerical simulation (DNS) of a particle-laden flow, the flow around each particle needs to be explicitly resolved (Balachandar & Eaton 2010;Maxey 2017) (so-called particle-resolved DNS, PR-DNS). While this approach is free from modelling assumptions for the dispersed phase dynamics, it is computationally expensive due to the explicit imposition of no-slip and no-penetration boundary conditions at the surface of many particles moving in a turbulent medium. ...
In particle-laden turbulent wall flows, lift forces can influence the near-wall turbulence. This has been recently observed in particle-resolved simulations, which, however, are too expensive to be used in upscaled models. Instead, point-particle simulations have been the method of choice to simulate the dynamics of these flows during the last decades. While this approach is simpler, cheaper, and physically sound for small inertial particles in turbulence, some issues remain. In the present work, we address challenges associated with lift force modelling in turbulent wall flows and the impact of lift forces in the near-wall flow. We performed direct numerical simulations (DNS) of small inertial point particles in turbulent channel flow for fixed Stokes number and mass loading while varying the particle size. Our results show that the particle dynamics in the buffer region, causing the apparent particle-to-fluid slip velocity to vanish, raises major challenges for accurately modelling lift forces. While our results confirm that lift forces have little influence on particle dynamics for sufficiently small particle sizes, for inner-scaled diameters of order one and beyond, lift forces become quite important near the wall. The different particle dynamics under lift forces results in the modulation of streamwise momentum transport in the near-wall region. We analyze this lift-induced turbulence modulation for different lift force models, and the results indicate that realistic models are critical for particle-modelled simulations to correctly predict turbulence modulation by particles in the near-wall region.
... In [1] various numerical simulation methods for studying the motion of rigid particles suspended in fluid are considered. The choice of the method should account for the fluid to solid density ratio, more about that in [2]. ...
In this research, we extend our studies of the extraction process from diverse plant materials, introducing advancements to our previous models. Our framework considers dynamic elements by taking into account the motion of the particles, departing from the statical particle assumption in prior articles. Several methods such as moving geometries or modified equations for a system with moving particles in Lagrangian coordinates are introduced to boost the precision of our simulations, taking into account the complex dynamics of the solvent and its interaction with the plant material. Expanding beyond our focus on supercritical carbon dioxide (scCO2), our research is addressing some different applications. Besides the traditional solvent-based extractions, we consider potential applications in filtration, wood industry processes, etc. This allows our model to adapt to diverse industrial contexts with varied extraction mediums. Our coupled system of equations contains fluid dynamics equations for solvent flow, reaction-advection-diffusion equations for solute, and equations governing remaining solute concentration in biomass. The exchange of active material between solid and fluid is modelled by the Langmuir law. Applying finite volume techniques and implemented in the Octave/Matlab environment, our model captures the temporal evolution of two and three dimensional solute distribution and solvent velocity field. This modular framework facilitates the integration of tailor-made laws to represent diverse plant materials, ensuring versatility across applications. Through our simulations, we present the analysis of our modified model’s performance and discuss its advantages and limitations. This research is a slight step forward in understanding and optimising extraction processes, offering valuable insights for industries involved in functional foods, nutraceuticals, pharmaceuticals, cosmetics, filtration, and wood processing.
... When the particle Stokes number St is large (indicating large particle inertia), the particle cannot readily follow the surrounding fluid, leading to notable differences in their velocities. Additionally, the fluid stress force acting on the particle results from the acceleration of the fluid (Du/Dτ ), encompassing the pressure gradient and viscosity terms of the fluid [48]. ...
The preferential accumulation of particles in turbulent flows occurs in many engineering and environmental applications. Recent research reveals that particle preferential accumulation is attributed to multiscale vortex structures in turbulent flows, which affect particle motion and cause particles to remain trapped in particular regions. In this study, the primary goal is to further investigate the mechanisms of particle trapping with emphasis on the effect of the vortex on particle motion. We specifically investigate particle trapping in a 2D unequal-strength counterrotating vortex pair (CVP) using a one-way coupled Euler–Lagrangian method. The small, rigid, spherical, dilute, and heavy particles are considered with the assumption that the particle Reynolds number is low. Using a geometric singular perturbation method to solve the particle motion, we first identify a particle-attracting ring in the potential CVP flow with a circulation ratio γ ∈ (−0.65, 0). The particle-attracting ring provides a simple mechanism to explain the occurrence of particle trapping, which is represented by a particle-clustering ring (PCR) in the CVP. We then conduct numerical simulations of particle motion for viscous CVP flows. In the simulations, the CVP is created through vortex–wall interaction, where a primary vortex induces the separation of the new discrete counterrotating vortex from the wall boundary, ultimately leading to the formation of the CVP. Particle trapping in the CVP flow is shown to be robust in the presence of viscosity. The trapping of particles in these viscous simulations has the same dynamical origin as the trapping phenomenon studied for potential CVP flows. The results of this research may help to further comprehend the mechanisms driving the preferential accumulation of particles in turbulent flows.
... When swarms of drops/bubbles are considered, the number of available investigations is more limited. For very small drops/bubbles, numerical investigations usually rely on the Lagrangian approach, in which drops/bubbles are assumed to have sub-Kolmogorov size and are treated as material points (Kuerten 2016;Maxey 2017;Chong et al. 2021;Wang et al. 2021a;Wang, Dalla Barba & Picano 2021b). When larger drops/bubbles are considered (i.e. ...
Heat transfer by large deformable drops in a turbulent flow is a complex and rich-in-physics system, in which drop deformation, breakage and coalescence influence the transport of heat. We study this problem by coupling direct numerical simulation (DNS) of turbulence with a phase-field method for the interface description. Simulations are run at fixed-shear Reynolds and Weber numbers. To evaluate the influence of microscopic flow properties, like momentum/thermal diffusivity, on macroscopic flow properties, like mean temperature or heat transfer rates, we consider four different values of the Prandtl number, which is the momentum to thermal diffusivity ratio: Pr=1 , Pr=2 , Pr=4 and Pr=8 . The drop volume fraction is for all cases. Drops are initially warmer than the turbulent carrier fluid and release heat at different rates depending on the value of Pr , but also on their size and on their own dynamics (topology, breakage, drop–drop interaction). Computing the time behaviour of the drops and carrier fluid average temperatures, we clearly show that an increase of Pr slows down the heat transfer process. We explain our results by a simplified phenomenological model: we show that the time behaviour of the drop average temperature is self-similar, and a universal behaviour can be found upon rescaling by . Accordingly, the heat transfer coefficient (respectively its dimensionless counterpart, the Nusselt number Nu ) scales as (respectively ) at the beginning of the simulation, and tends to (respectively ) at later times. These different scalings can be explained via the boundary layer theory and are consistent with previous theoretical/numerical predictions.
... Another example is that under the influence of air conditioning and the ventilation system, atmospheric pollutant particles (PM10 and PM2.5) originating from dust and smoke can be suspended in the air for an extended period, and they may spread to a wider area with the aid of airflow [4]. Numerical models for simulating particle-laden flows can be generally classified into two categories [5]: the point-particle model and the particle-resolved model. In the point-particle model, solid particles are treated as discrete masses that are much smaller than the mesh size of the computational grid. ...
... First, most of the works referenced above treat particles as material points, following the framework of Maxey & Riley (1983) and Gatignol (1983). This relies on restricting assumptions that limit its applicability, in particular when the particle Reynolds number is not small (Maxey 2017;Brandt & Coletti 2022). Specifically for wall-bounded flows, the high velocity gradients may result in significant lift forces (Costa, Brandt & Picano 2019) that are neglected in most point-particle simulations. ...
Inertial particles in wall-bounded turbulence are known to form streaks, but experimental evidence and predictive understanding of this phenomenon is lacking, especially in regimes relevant to atmospheric flows. We carry out wind tunnel measurements to investigate this process, characterizing the transport of microscopic particles suspended in turbulent boundary layers. The friction Reynolds number allows for significant scale separation and the emergence of large-scale motions, while the range of viscous Stokes number –870 is relevant to the transport of dust and fine sand in the atmospheric surface layer. We perform simultaneous imaging of both carrier and dispersed phases along wall-parallel planes in the logarithmic layer, demonstrating that streamwise particle streaks largely overlap with large-scale low-speed flow regions. The fluid–particle slip velocity indicates that with increasing inertia, the particle streaks outlive the low-speed fluid streaks. Moreover, two-point statistics show that the width of the particle streaks increases linearly with Stokes number, bounded by the size of the coherent flow structures. Finally, the particle-sampled flow topology suggests that particle streaks reside between the legs of hairpin packets. From these observations, we infer a conceptual view of the formation of particle streaks in the frame of the attached eddy model. A scaling for the particle streaks’ width is derived as a function of and , which reproduces the measured trends and predicts widths m in the atmospheric surface layer, comparable to aeolian streamers observed in the field.
When a solid particle in a fully developed turbulence is larger than the Kolmogorov length scale, the turbulent momentum, transmitted to the particle by random advection past particle of dissipative structures by eddies of order of the particle diameter, tends to reduce the relative velocity between the particle and the fluid. In this work, a corresponding model of the effective drag is discussed and numerically assessed with the experimental study. The velocity gradient in the fluid around the particle is the key-variable of this model, and consequently, the model highlights the role of the flow structure on the particle dynamics. In the simulation of the particle motion in the periodic box turbulence (the latter is resolved by Direct Numerical Simulation—DNS), the model reproduces fairly well the statistical properties known from measurements. First, in accordance with measurements, the simulation shows a universality of normalized distributions of the particle velocity and its fluctuation rate of the particle acceleration and its autocorrelation function—these distributions along the particle trajectory are almost insensitive to parameters of the particle inertia. Thereby, the typical correlation time of the particle acceleration is of the order of the Kolmogorov timescale, i.e., this correlation time is much less than the viscous relaxation time to the low-frequency solicitations in turbulence. In turn, the particle acceleration variance does depend on parameters of the particle inertia, and this dependency is predicted in the simulation consistently with the experiment. Second, the computed distributions of the particle acceleration as well as those of the particle velocity increments at small time lags expose the stretched tails. This is the way in which the intermittency of the flow structure is manifested: the intense velocity gradients in the fluid induce the strong fluctuations of the particle drag. Probability density functions of normalized Voronoï volumes indicate that increasing the particle density and, to a lesser extent, the particle size favors the preferential accumulation of particles above the Kolmogorov size.
A single particle representation of a self-propelled microorganism in a viscous incompressible fluid is derived based on regularised Stokeslets in three dimensions. The formulation is developed from a limiting process in which two regularised Stokeslets of equal and opposite strength but with different size regularisation parameters approach each other. A parameter that captures the size difference in regularisation provides the asymmetry needed for propulsion. We show that the resulting limit is the superposition of a regularised stresslet and a potential dipole. The model framework is then explored relative to the model parameters to provide insight into their selection. The particular case of two identical particles swimming next to each other is presented and their stability is investigated. Additional flow characteristics are incorporated into the modelling framework with in the addition of a rotlet double to characterise rotational flows present during swimming. Lastly, we show the versatility of deriving the model in the method of regularised Stokeslets framework to model wall effects of an infinite plane wall using the method of images.
Numerical simulations are widely used to study the behavior of suspension flows. Fully resolved simulations, in which the detailed flow around individual particles is computed, are accurate but computationally expensive. Unresolved methods reduce the computational cost significantly by only resolving the bulk flow and modeling the small-scale flow around particles. However, the degree to which modeling rather than computing the small-scale flow field information affects the predicted behavior of suspension flows is largely unknown. Here, we examine the steady homogeneous regime by simulating the pressure drop over a porous medium and the apparent viscosity of a sheared suspension as well as the transient heterogeneous regime by simulating a particle-induced Rayleigh–Taylor instability. From these simulations, we observe that unresolved methods are able to predict macroscopic quantities in steady state problems involving homogeneous suspensions but fail to capture particle entrainment, deformation, and breakup effects in transient problems involving heterogeneous suspensions. Our results suggest that the effect of the small-scale flow fields plays an important role in the onset and growth of instabilities in suspension flows which cannot be modeled in a trivial way. This has consequences for practical applications where such instabilities are essential, such as particle mixing. Further research into the mechanisms by which such instabilities are triggered as well as ways to include these effects in computationally inexpensive unresolved models is needed.
Particles suspended in air are often non-spherical shapes, giving rise to shape-dependent complex
dynamical processes. Suspended non-spherical particles are associated with a wide array of engineering
and scientific scenarios, embodying both their microscopic and macroscopic dynamical behaviors. A
comprehensive understanding of the dynamical behaviors of non-spherical particles in air hinges on the
accurate identification and description of particle shape, the development of shape-specific models for
the forces and torques acting on these particles, and the subsequent micro- and macroscopic phenomena
that emerge as a result. This review surveys the latest advancements in the field of non-spherical particles,
spanning from shape identification to the characterization of their dynamical properties. An
emphasis is placed on establishing a connection between the micro- and macroscopic dynamical behaviors
of non-spherical particles. The shape-induced features encompass periodic rotation and preferential
orientation, which result in an oscillating migration path and lead to distinctive macroscopic
characteristics. The macroscopic features of non-spherical particles are elucidated based on the preceding
analysis of forces, torques, and particle-flow interactions. The future perspectives are also discussed
in this review.
Interface-resolved direct numerical simulations (DNS) of clustered settling suspensions in a periodic domain are performed to study the filtered drag force for clustered particle-laden flows. Our results show that, for the homogeneous system, the filtered drag is independent of the filter size, whereas for the clustered particle-laden flows, the averaged drag becomes smaller than the homogeneous drag at the filter size above 4 particle diameters. The drag reduction saturates at the filter size being comparable to the cluster size in the horizontal direction in our simulations. A new correlation is proposed to account for the mesoscale effect on the filtered drag force by using drift velocity and variance of the solid volume fraction, based on the modification of existing subgrid drag models for the inhomogeneous system. The existing models for the drift velocity and the variance of the solid volume fraction are assessed using our DNS data. A new model for the drift velocity and the variance of the solid volume fraction is proposed, based on the combination and modification of the previous models. All mesoscale models considered can predict well the filtered drag with comparable accuracy, and are superior to the homogeneous drag model for the clustered system. Our models with the same parameter values obtained from the large-scale system can also predict well the filtered drag for smaller computational domain sizes.
A simulation method has been developed to efficiently evaluate the motion of colloidal particles in a low-Reynolds-number confined microchannel flow using a Lagrangian-based approach. In this method, the background velocity within the channel, in the absence of suspended particles, is obtained from a fluid dynamics solver and is used to update the velocity at the particle centres using the Stokesian dynamics (SD) method, which incorporates multi-body hydrodynamic interactions. As a result, instead of computing the momentum of both the fluid and particles throughout the entire computational domain, the microscopic balance equation is solved only at the particle centres, increasing the computational efficiency. To accommodate complex boundary conditions within the SD framework, imaginary particles are placed on the channel walls, allowing the mobility relation to be reformulated to apply velocity constraints to immobilized wall particles. By employing this constrained SD approach, global mobility interactions that need to be computed at each time step are limited to the interior particles, resulting in a significant reduction in computational cost. The efficiency of this study is demonstrated through case studies on particulate flows in contraction and cross-flow microchannels. By using colloidal particles that incorporate Brownian motion and inter-particle attraction, observations through the entire stages of fouling dynamics are possible, from particle inflow to channel blockage. The fouling patterns observed in the simulations are consistent with experiments conducted under the same flow conditions. This study provides an efficient approach for analysing the effect of hydrodynamic interactions on particle dynamics in microfluidics and materials processing fields while allowing for predictions about structural changes over long-time scales, including complex phenomena such as clogging.
We investigate by direct numerical simulations the fluid–solid interaction of non-dilute suspensions of spherical particles moving in triperiodic turbulence, at the relatively large Reynolds number of . The solid-to-fluid density ratio is varied between 1.3 and 100 , the particle diameter D is in the range ( is the Kolmogorov scale) and the volume fraction of the suspension is 0.079 . Turbulence is sustained using the Arnold–Beltrami–Childress cellular-flow forcing. The influence of the solid phase on the largest and energetic scales of the flow changes with the size and density of the particles. Light and large particles modulate all scales in an isotropic way, while heavier and smaller particles modulate the largest scales of the flow towards an anisotropic state. Smaller scales are isotropic and homogeneous for all cases. The mechanism driving the energy transfer across scales changes with the size and the density of the particles. For large and light particles the energy transfer is only marginally influenced by the fluid–solid interaction. For small and heavy particles, instead, the classical energy cascade is subdominant at all scales, and the energy transfer is essentially driven by the fluid–solid coupling. The influence of the solid phase on the flow intermittency is also discussed. Besides, the collective motion of the particles and their preferential location in relation to properties of the carrier flow are analysed. The solid phase exhibits moderate clustering; for large particles the level of clustering decreases with their density, while for small particles it is maximum for intermediate values.
We investigate the stability of flows with low particle volume fractions in channels featuring porous walls. The particles, which are neutrally buoyant, interact with the carrier fluid through the Stokes drag force. Our study explores stability concerning particle relaxation time and mass fraction, employing different porous walls with varying permeabilities while maintaining a fixed porosity of 0.6. Our results reveal that in highly permeable porous walls, flow stability is mainly governed by the porous structure. The particle volume fraction and relaxation time exert relatively minor destabilizing and stabilizing effects, respectively. However, as porous wall permeability decreases, flow behavior becomes more sensitive to the particle volume fraction. In such cases, higher particle volume fractions and longer relaxation times contribute to stabilization. This suggests that particles and porous walls can effectively control flow, either maintaining laminar flow or inducing a transition to turbulence. We also analyze the impact of the momentum transfer coefficient at the porous surface, τ, on flow stability. Finally, we compare marginal stability curves obtained for various commonly used porous materials to conclude our study.
A single-field representation is derived to describe a multi-species gas flow seeded with perfectly rigid, non-porous and partially inert particles. Indeed, the latter are not involved in heterogeneous reactions with the gas phase (no consumption of the solid by the gas), but they are expected to potentially undergo adsorption and recombination of gaseous radicals on their surface, resulting in the desorption of stable species into the gas. In this work, integral balances are first established at the gas-solid interface, consistent with the local single-phase equations for reactive multi-species gas flow. This allows rigorous derivation of the jump equations that apply at the gas-solid interface in the presence of surface reactions. Then, the description in terms of local single-phase equations, completed by the jump equations at the interface, is recast into a set of equations for the full domain in the sense of distributions. The result extends Kataoka’s formalism (Kataoka, 1986) to reactive gas-solid flow in the presence of both homogeneous and surface reactions.
We generalize the multipole expansion and the structure of the Faxén operator in Stokes flows obtained for bodies with no-slip to generic boundary conditions, addressing the assumptions under which this generalization is conceivable. We show that a disturbance field generated by a body immersed in an ambient flow can be expressed, independently on the boundary conditions, as a multipole expansion, the coefficients of which are the moments of the volume forces. We find that the dualism between the operator giving the disturbance field of an nth order ambient flow and the nth order Faxén operator, referred to as the Hinch–Kim dualism, holds only if the boundary conditions satisfy a property that we call Boundary-Condition reciprocity (BC-reciprocity). If this property is fulfilled, the Faxén operators can be expressed in terms of the (m, n)th order geometrical moments of the volume forces (defined in the article). In addition, it is shown that in these cases, the hydromechanics of the fluid-body system is completely determined by the entire set of the Faxén operators. Finally, classical boundary conditions of hydrodynamic applications are investigated in light of this property: boundary conditions for rigid bodies, Newtonian drops at the mechanical equilibrium, porous bodies modeled by the Brinkman equations are BC-reciprocal, while deforming linear elastic bodies, deforming Newtonian drops, non-Newtonian drops, and porous bodies modeled by the Darcy equations do not have this property. For Navier-slip boundary conditions on a rigid body, we find the analytical expression for low order Faxén operators. By using these operators, the closed form expressions for the flow past a sphere with arbitrary slip length immersed in shear and quadratic flows are obtained.
Calculation of particle volume fraction in computational fluid dynamics-discrete element method simulation of particulate flows with coarse particles ABSTRACT Computational fluid dynamics-discrete element method is frequently used for modeling particulate flows due to its high efficiency and satisfactory accuracy. The particle volume fraction is a crucial parameter that significantly affects the computation accuracy. It may be extremely large when the particulate flows contain coarse particles because it is determined by the ratio of particle volume to cell volume. In this paper, the performance of different methods, such as the divided particle volume method (DPVM), the big particle method, and the diffusion-based method, for computing the particle volume fraction is thoroughly reviewed, implemented, and investigated. It turns out that the DPVM must not be used when the particle size is larger than cell size due to significant fluctuation of the particle volume fraction field. The big particle method is optimized for simulation accuracy and code implementation. The optimized big particle method is similar to the diffusion-based method by diffusing the particle effects to the surrounding cells. It demonstrates greater consistency with experimental observations compared to the diffusion-based method, primarily attributed to its incorporation of polydisperse effects.
The migration of neutrally buoyant finite sized particles in a Newtonian square channel flow is investigated in the limit of very low solid volumetric concentration, within a wide range of channel Reynolds numbers
Re = [0.07-120]. In situ microscope measurements of particle distributions, taken far from the channel inlet (at a distance several thousand times the channel height), revealed that particles are preferentially located near the channel walls at Re > 10 and near the channel center at Re < 1. Whereas the cross-streamline particle motion is governed by inertia-induced lift forces at high inertia, it seems to be controlled by shear-induced particle interactions at low (but finite) Reynolds numbers, despite the low solid volume fraction (<1%). The transition between both regimes is observed in the range Re = [1-10]. In order to exclude the effect of multi-body interactions, the trajectories of single freely moving particles are calculated thanks to numerical simulations based on the force coupling method. With the deployed numerical tool, the complete particle trajectories are accessible within a reasonable computational time only in the inertial regime (Re > 10). In this regime, we show that (i) the particle undergoes cross-streamline migration followed by a cross-lateral migration (parallel to the wall) in agreement with previous observations, and (ii) the stable equilibrium positions are located at the midline of the channel faces while the diagonal equilibrium positions are unstable. At low flow inertia, the first instants of the numerical simulations (carried at Re = O(1)) reveal that the cross-streamline migration of a single particle is oriented towards the channel wall, suggesting that the particle preferential positions around the channel center, observed in the experiments, are rather due to multi-body interactions.
Direct numerical simulations of freely falling and rising particles in an infinitely long domain, with periodic lateral boundary conditions, are performed. The focus is on characterizing the free motion of cubical and tetrahedral particles for different Reynolds numbers, as an extension to the well-studied behaviour of freely falling and rising spherical bodies. The vortical structure of the wake, dynamics of particle movement, and the interaction of the particle with its wake are studied. The results reveal mechanisms of path instabilities for angular particles, which are different from those for spherical ones. The rotation of the particle plays a more significant role in the transition to chaos for angular particles. Following a framework similar to that of Mougin and Magnaudet [“Wake-induced forces and torques on a zigzagging/spiralling bubble,” J. Fluid Mech.567, 185–194 (2006)], the balance of forces and torques acting on particles is discussed to gain more insight into the path instabilities of angular particles.
We study the effect of the Basset history force on the dynamics of small
particles transported in homogeneous and isotropic turbulence and show that
this term, often neglected in previous numerical studies, affects the particle
behavior even at high density ratios. The small-scale clustering typical of
inertial particles is reduced when the Basset history term is considered. We
examine the contribution of this force to the total particle acceleration and
demonstrate that, on average, it is responsible for about 10% of the total
acceleration and is particularly relevant during rare strong events.
We have performed direct numerical simulation of turbulent open channel flow over a smooth horizontal wall in the presence of finite-size, heavy particles. The spherical particles have a diameter of approximately 7 wall units, a density of 1.7 times the fluid density and a solid volume fraction of 5 × 10−4. The value of the Galileo number is set to 16.5, while the Shields parameter measures approximately 0.2. Under these conditions, the particles are predominantly located in the vicinity of the bottom wall, where they exhibit strong preferential concentration which we quantify by means of Voronoi analysis and by computing the particle-conditioned concentration field. As observed in previous studies with similar parameter values, the mean streamwise particle velocity is smaller than that of the fluid. We propose a new definition of the fluid velocity 'seen' by finite-size particles based on an average over a spherical surface segment, from which we deduce in the present case that the particles are instantaneously lagging the fluid only by a small amount. The particle-conditioned fluid velocity field shows that the particles preferentially reside in the low-speed streaks, leading to the observed apparent lag. Finally, a vortex eduction study reveals that spanwise particle motion is significantly correlated with the presence of vortices with the corresponding sense of rotation which are located in the immediate vicinity of the near-wall particles.
An overview of present understanding of microstructure in flowing suspensions is provided. An emphasis is placed on how the
microstructure leads to observable bulk flow phenomena unique to mixtures. The bridge between the particle and bulk scales
is provided by the mixture rheology; one focus of the review is on work that addresses the connection between microstructure
and rheology. The non-Newtonian rheology of suspensions includes the well-known rate dependences of shear thinning and thickening,
which have influence on bulk processing of suspensions. Shear-induced normal stresses are also measured in concentrated suspensions
and include normal stress differences, and the isotropic particle pressure. Normal stresses have been associated with shear-induced
migration, and thus have influence on the ultimate spatial distribution of solids, as well as the flow rate during processing;
a second focus is on these uniquely two-phase behaviors and how they can be described in terms of the bulk rheology. An important
bulk fluid mechanical consequence of normal stresses is their role in driving secondary flows.
Sedimentation behaviours of an ellipsoidal particle in narrow and infinitely long tubes are studied by a multi-relaxation-time lattice Boltzmann method (LBM). In the present study, both circular and square tubes with 12/13 <= D/A = 2.5 are considered with the Galileo number (Ga) up to 150, where D and A are the width of the tube and the length of major axis of the ellipsoid, respectively. Besides three modes of motion mentioned in the literature, two novel modes are found for the narrow tubes in the higher Ga regime: the spiral mode and the vertically inclined mode. Near a transitional regime, in terms of average settling velocity, it is found that a lighter ellipsoid may settle faster than a heavier one. The relevant mechanism is revealed. The behaviour of sedimentation inside the square tubes is similar to that in the circular tubes. One significant difference is that the translation and rotation of ellipsoid are finally constrained to a diagonal plane in the square tubes. The other difference is that the anomalous rolling mode occurs in the square tubes. In this mode, the ellipsoid rotates as if it is contacting and rolling up one corner of the square tube when it settles down. Two critical factors that induce this mode are identified: the geometry of the tube and the inertia of the ellipsoid. (C) 2014 AIP Publishing LLC.
Single-phase Stokes flow problems with prescribed boundary conditions can be formulated in terms of a boundary regularized integral equation that is completely free of singularities that exist in the traditional formulation. The usual mathematical singularities that arise from using the fundamental solution in the conventional boundary integral method are removed by subtracting a related auxiliary flow field, w, that can be constructed from one of many known fundamental solutions of the Stokes equation. This approach is exact and does not require the introduction of additional cutoff parameters. The numerical implementation of this boundary regularized integral equation formulation affords considerable savings in coding effort with improved numerical accuracy. The high accuracy of this formulation is retained even in problems where parts of the boundaries may almost be in contact.
The presence of finite-size particles in a channel flow close to the laminar-turbulent transition is simulated with the Force Coupling Method which allows two-way coupling with the flow dynamics. Spherical particles with channel height-to-particle diameter ratio of 16 are initially randomly seeded in a fluctuating flow above the critical Reynolds number corresponding to single phase flow relaminarization. When steady-state is reached, the particle volume fraction is homogeneously distributed in the channel cross-section (ϕ ≅ 5%) except in the near-wall region where it is larger due to inertia-driven migration. Turbulence statistics (intensity of velocity fluctuations, small-scale vortical structures, wall shear stress) calculated in the fully coupled two-phase flow simulations are compared to single-phase flow data in the transition regime. It is observed that particles increase the transverse r.m.s. flow velocity fluctuations and they break down the flow coherent structures into smaller, more numerous and sustained eddies, preventing the flow to relaminarize at the single-phase critical Reynolds number. When the Reynolds number is further decreased and the suspension flow becomes laminar, the wall friction coefficient recovers the evolution of the laminar single-phase law provided that the suspension viscosity is used in the Reynolds number definition. The residual velocity fluctuations in the suspension correspond to a regime of particulate shear-induced agitation.
A particle mesh Ewald (PME) Stokesian dynamics algorithm has been developed to model hydrodynamic interactions in suspensions of non-spherical dicolloidal particles. Dicolloids, which have recently been synthesized by a number of independent research groups), consist of two intersecting spheres of varying radii and centre-to-centre separation. One-body resistance tensors and disturbance velocity fields are computed for general linear flows using a superposition of Stokes singularities along the symmetry axis of the dicolloid particles. The coefficients and the locations of the singularities are optimized to minimize the norm of the velocity error on the particle surface. The one-body solution provides all coefficients required for the far-field many-body interactions in the Stokesian dynamics algorithm. These generalize the analytical results for spheres employed in the classic algorithm. Modified lubrication interaction tensors are developed for dicolloids for the singular near-field lubrication interactions. Accuracy of the one-body solutions and two-body generalized Stokesian dynamics solutions are validated by comparison with high-precision numerical solutions computed with the spectral boundary element method of G. P. Muldowney and J. J. L. Higdon [J. Fluid Mech. 298, 167–192 (1995; Zbl 0848.76066)]. The newly developed PME Stokesian dynamics algorithm was used to study transport properties in dicolloidal suspensions over a range of volume fractions (φ≤0·5). The effects of the degree of anisotropy on the properties of the suspension are discussed. For these mildly anisotropic particles, the transport properties remain close to those of spheres, however certain interesting trends emerge, with non-monotonic viscosity dependence as a function of increasing aspect ratio. The minimum viscosity in concentrated suspensions is lower than that for spheres with equal volume fraction over a range of volume fractions.
This paper reviews applications of the lattice-Boltzmann method to simulations of particle-fluid suspensions. We first summarize the available simulation methods for colloidal suspensions together with some of the important applications of these methods, and then describe results from lattice-gas and lattice-Boltzmann simulations in more detail. The remainder of the paper is an update of previously published work,(69,70) taking into account recent research by ourselves and other groups. We describe a lattice-Boltzmann model that can take proper account of density fluctuations in the fluid, which may be important in describing the short-time dynamics of colloidal particles. We then derive macro-dynamical equations for a collision operator with separate shear and bulk viscosities, via the usual multi-time-scale expansion. A careful examination of the second-order equations shows that inclusion of an external force, such as a pressure gradient, requires terms that depend on the eigenvalues of the collision operator. Alternatively, the momentum density must be redefined to include a contribution from the external force. Next, we summarize recent innovations and give a few numerical examples to illustrate critical issues. Finally, we derive the equations for a lattice-Boltzmann model that includes transverse and longitudinal fluctuations in momentum. The model leads to a discrete version of the Green–Kubo relations for the shear and bulk viscosity, which agree with the viscosities obtained from the macro-dynamical analysis. We believe that inclusion of longitudinal fluctuations will improve the equipartition of energy in lattice-Boltzmann simulations of colloidal suspensions.
In order to simulate the motion of bubbles, drops, and particles, it is often important to consider finite Reynolds number effects on drag, lift, torque, and history force. Herein, an equation of motion is developed for spherical particles with a no-slip surface based on theoretical analysis, experimental data, and surface-resolved simulations. The equation of motion is then extended to account for finite particle size. This extension is critical for particles which will have a size significantly larger than the grid cell size, particularly important for bubbles, and low-density particles. The extension to finite particle size is accomplished through spatial-averaging (both volume-based and surface-based) of the continuous flow properties. This averaging is consistent with the Faxen limit for solid spheres at small Reynolds numbers and added mass and fluid stress forces at inviscid limits. The finite Re
p
corrections are shown to have good agreements with experiments and resolved-surface simulations. The finite size corrections are generally fourth-order accurate and an order of magnitude more accurate than point-force expressions (which neglect quadratic and higher spatial gradients) for particles with size on the order of the gradient length-scales. However, further work is needed for more quantitative assessment of the truncation terms and the overall model robustness and accuracy in complex flows.
A direct boundary element method is formulated for the Stokes flow problem based on an integral equation representation for the components of traction. For problems in which the components of velocity are prescribed on the boundary of the domain, this new formulation results in a hypersingular Fredholm integral equation of the second kind. A method of regularization to evaluate the hypersingular integral is discussed. For certain problems involving flows about particles, the integral equation representation for the tractions is not unique because of the existence of rigid body eigenmodes. A method to constrain out these rigid body modes is also discussed. Several example problems are considered in which this new formulation is compared to more traditional boundary element formulations.
Cell motility in viscous fluids is ubiquitous and affects many biological processes, including reproduction, infection, and the marine life ecosystem. Here we review the biophysical and mechanical principles of locomotion at the small scales relevant to cell swimming (tens of microns and below). The focus is on the fundamental flow physics phenomena occurring in this inertia-less realm, and the emphasis is on the simple physical picture. We review the basic properties of flows at low Reynolds number, paying special attention to aspects most relevant for swimming, such as resistance matrices for solid bodies, flow singularities, and kinematic requirements for net translation. Then we review classical theoretical work on cell motility: early calculations of the speed of a swimmer with prescribed stroke, and the application of resistive-force theory and slender-body theory to flagellar locomotion. After reviewing the physical means by which flagella are actuated, we outline areas of active research, including hydrodynamic interactions, biological locomotion in complex fluids, the design of small-scale artificial swimmers, and the optimization of locomotion strategies. Comment: Review article
Several microvalves utilizing stimuli-responsive hydrogel
materials have been developed. The hydrogel components are fabricated
inside microchannels using a liquid phase polymerization process.
In-channel processing greatly simplifies device construction, assembly,
and operation since the functional components are fabricated in situ and
can perform both sensing and actuation functions. Two in situ
photopolymerization techniques, "laminar stream mode" and "mask mode,"
have been explored. Three two-dimensional (2-D) valves were fabricated
and tested (response time, pressure drop, maximum differential
pressure). In addition, a hydrogel/PDMS three-dimensional (3-D) hybrid
valve that physically separates the sensing and regulated streams was
demonstrated. Analytical modeling was performed on the 3-D valve.
Hydrogel-based microvalves have a number of advantages over conventional
microvalves, including relatively simple fabrication, no external power
requirement, no integrated electronics, large displacement (185 μm),
and large force generation (22 mN)