Direct-forcing Fictitious Domain Research Articles

Numerical study of collective motion of microswimmers in Giesekus viscoelastic fluids

Few simulations currently explore the dynamics of microswimmers swimming through viscoelastic environments. In this study, we employ a direct-forcing fictitious domain method to investigate the collective behavior of spherical squirmers within viscoelastic fluids at low Reynolds numbers. Our findings reveal clear differences between pusher and puller swimmers: puller swimmers exhibit a tendency to aggregate into clusters, particularly noticeable in suspensions with high concentrations, which increases the average speed of the swimmers. Through an analysis of the cluster-size distribution function, we observe the larger-scale clusters of puller swimmers with increasing concentration. Moreover, the presence of fluid elasticity significantly reduces both the average swimming speed of squirmers and the fluid’s kinetic energy.

Swimming of an inertial squirmer and squirmer dumbbell through a viscoelastic fluid

We investigate the hydrodynamics of a spherical and dumbbell-shaped swimmer in a viscoelastic fluid, modelled by the Giesekus constitutive equation. The ‘squirmer’, a model of a micro-swimmer with tangential surface waves at its boundaries, is simulated utilizing a direct-forcing fictitious domain method. We consider the competitive effects of the fluid inertia and elasticity on the locomotion of the swimmers. For the neutral squirmer, its speed increases monotonically with increasing Reynolds number in the Giesekus fluid, in contrast to holding a constant speed in a Newtonian one. Meanwhile, the speed of the finite inertial squirmer increases monotonically with fluid elasticity, as measured by the Weissenberg number. Regarding the dumbbell-shaped swimmers in the Giesekus fluid, we find the pusher–puller (the puller is in front of the pusher) dumbbell swimming to be significantly faster than other dumbbells, providing practical guidance in assembling the complex micro-swimming devices. We further consider the squirmer's (or the dumbbell-shaped squirmer's) energy expenditure and hydrodynamic efficiency, finding that swimming in viscoelastic fluids expends less energy than its Newtonian counterpart, and the neutral squirmer expends more energy than a pusher or puller. The energy expenditure and hydrodynamic efficiency are relevant to steady swimming speeds, in which a faster counterpart squirmer (dumbbell-shaped squirmer) expends less energy but has higher hydrodynamic efficiency.

Interface-resolved simulations of particles in active nematics

An accurate coarse-grained simulation of an active fluid is invaluable as a tool to understand its hydrodynamic behaviors. The study on the dynamics of particles immersed in an active fluid also requires accurate resolution of the fluid–particle interaction. In this paper, we propose a robust direct forcing fictitious domain method to study the dynamics of suspended particles in an active fluid modeled by “active nematics.” This method serves as both a coarse-grained approach and an accurate model of fluid–particle interaction. We first validate the method by computing the kinetic energy spectrum for the bulk active nematics and find that it accurately reproduces the scaling laws reported theoretically and experimentally. By utilizing these interface-resolved simulations, we illustrate that the model's activity parameter cannot be simply considered as the concentration of bacterial suspensions. Moreover, we find that the diffusion coefficient DT of an individual disk is relevant to the length scale lc of the active nematics, following a power-law scaling DT ∼ lc−1.5. Regarding collective dynamics, we discover a self-organized length scale of approximately 7.5 times the disk's diameter in the active nematics. Additionally, the disks modify the kinetic energy spectrum of the active nematics at both the self-organized length scale and the individual disk's diameter scale, respectively.

Cargo carrying with an inertial squirmer in a Newtonian fluid

We numerically investigate the hydrodynamics of a spherical swimmer carrying a rigid cargo in a Newtonian fluid. This swimmer model, a ‘squirmer’, which is self-propelled by generating tangential surface waves, is simulated by a direct-forcing fictitious domain method (DF-FDM). We consider the effects of swimming Reynolds numbers (Re) (based on the radius and the swimming speed of the squirmers), the assembly models (related to the cargo shapes, the relative distances (ds) and positions between the squirmer and the cargo) on the assembly's locomotion. We find that the ‘pusher-cargo’ (pusher behind the cargo) model swims significantly faster than the remaining three models at the finite Re adopted in this study; the term ‘pusher’ indicates that the object is propelled from the rear, as opposed to ‘puller’, from the front. Both the ‘pusher-cargo’ and ‘cargo-pusher’ (pusher in front of the cargo) assemblies with an oblate cargo swim faster than the corresponding assemblies with a spherical or prolate cargo. In addition, the pusher-cargo model is significantly more efficient than the other models, and a larger ds yields a smaller carrying hydrodynamic efficiency η for the pusher-cargo model, but a greater η for the cargo-pusher model. We also illustrate the assembly swimming stability, finding that the ‘puller-cargo’ (puller behind the cargo) model is stable more than the ‘cargo-puller’ (puller in front of the cargo) model, and the assembly with a larger ds yields more unstable swimming.

A hybrid smoothed particle hydrodynamics coupled to a fictitious domain method for particulate flows and its application in a three-dimensional printing process

A hybrid smoothed particle hydrodynamics (SPH) coupled to a direct-forcing fictitious domain (FD) scheme is proposed for the simulation of particulate flows. The new method is effective in calculating the particle-fluid interaction forces and uses a kernel interpolation function to realize the momentum exchange between fluids and particles. A higher-order interpolation scheme is applied to preserve the consistency and accuracy of the kernel function. We first test our method by simulating the benchmark of flow past a cylinder at Reynolds number ranging from 20 to 200 and obtain the convergent results, including accurate lift and drag coefficients, Strouhal number, length of the recirculation bubble, and separation angle. Subsequently, we verify our new method for the typical particulate flows, including the motion of a neutrally-buoyant cylinder in Couette flow, the motion of a neutrally-buoyant cylinder in a planar Poiseuille flow, the sedimentation of a circular cylinder in a channel, and the motion of a neutrally-buoyant slender particle in Couette flow. A good agreement with the existing experimental data, with other numerical results, and with available theoretical solutions is obtained indicating the accuracy and the robustness of the new method. Subsequently, our SPH-FD is applied to simulate a slender fiber in a three-dimensional (3D) printing process, showing a good potential and advantages in simulating particulate flows of some industrial processes.

Turbulent channel flow of a binary mixture of neutrally buoyant ellipsoidal particles

A turbulent channel flow of a binary mixture of finite-size neutrally buoyant ellipsoidal particles is studied by using a parallel direct-forcing fictitious domain method at a friction Reynolds number of 180 and the particle aspect ratios of 1/3 (oblate particle) and 4 (prolate particle), respectively. The total particle volume fraction is fixed at 14.16% and 21.24% and the relative fraction of prolate and oblate particles are varied. The mean velocity profile normalized by the bulk velocity shows a strong difference between the single phase flow and particulate flow, while showing a small difference between mono-disperse and bi-disperse particulate flow cases. The lower fluid and particle Reynolds stress contributions in a high oblate particle volume fraction ratio cases lead to the drag reduction. Under moderate particle volume concentration, the oblate and prolate particle results show self-similarity of velocity profiles of solid phase, including particle velocity, concentration, orientation, and rotation. Both kinds of ellipsoidal particles tend to align their long axes with the streamwise direction in the whole channel.

Hydrodynamics of an inertial squirmer and squirmer dumbbell in a tube

We study the hydrodynamics of a spherical and dumbbell-shaped microswimmer in a tube. Combined with a squirmer model generating tangential surface waves for self-propulsion, a direct-forcing fictitious domain method is employed to simulate the swimming of the microswimmers. We perform the simulations by considering the variations of the swimming Reynolds numbers (Re), the blockage ratios (κ) and the relative distances (ds) between the squirmers of the dumbbell. The results show that the squirmer dumbbell weakens the inertia effects of the fluid more than an individual squirmer. The constrained tube can speed up an inertial pusher (propelled from the rear) and an inertia pusher dumbbell; a greater distance ds results in a slower speed of an inertial pusher dumbbell but a faster speed of an inertial puller (propelled from the front) dumbbell. We also illustrate the swimming stability of a puller (stable) and pusher (unstable) swimming in the tube at Re = 0. At a finite Re, we find that the inertia and the tube constraint competitively affect the swimming stability of the squirmers and squirmer dumbbells. The puller and puller dumbbells swimming in the tube become unstable with increasing Re, whereas an unstable–stable–unstable evolution is found for the pusher and pusher dumbbells. With increasing κ, the puller and puller dumbbells become stable while the pusher and pusher dumbbells become unstable. In addition, we find that a greater ds yields a higher hydrodynamic efficiency η of the inertial squirmer dumbbell.

Migration and Alignment of Three Interacting Particles in Poiseuille Flow of Giesekus Fluids

Effect of rheological property on the migration and alignment of three interacting particles in Poiseuille flow of Giesekus fluids is studied with the direct-forcing fictitious domain method for the Weissenberg number (Wi) ranging from 0.1 to 1.5, the mobility parameter ranging from 0.1 to 0.7, the ratio of particle diameter to channel height ranging from 0.2 to 0.4, the ratio of the solvent viscosity to the total viscosity being 0.3 and the initial distance (y0) of particles from the centerline ranging from 0 to 0.2. The results showed that the effect of y0 on the migration and alignment of particles is significant. The variation of off-centerline (y0 ≠ 0) particle spacing is completely different from that of on-centerline (y0 = 0) particle spacing. As the initial vertical distance y0 increased, the various types of particle spacing are more diversified. For the off-centerline particle, the change of particle spacing is mainly concentrated in the process of cross-flow migration. Additionally, the polymer extension is proportional to both the Weissenberg number and confinement ratio. The bigger the Wi and confinement ratio is, the bigger the increment of spacing is. The memory of shear-thinning is responsible for the reduction of d1. Furthermore, the particles migrate abnormally due to the interparticle interaction.

Turbulence modulation by finite-size heavy particles in a downward turbulent channel flow

Interface-resolved direct numerical simulations of downward particle-laden turbulent channel flows are performed by using a direct-forcing fictitious domain method. The effects of the particle settling coefficient, the density ratio (2, 10, and 100), and the particle size on fluid-turbulence interactions are investigated at a bulk Reynolds number of 5746 and a particle volume fraction of 2.36%. Our results indicate that the significant particle-induced reduction in the turbulence intensity does not take place for the downflow at a low density ratio of 2, and the turbulence intensity generally increases with an increasing particle Reynolds number at the same other control parameters, unlike the upflow case. The total turbulent kinetic energy (TKE) in the channel is larger for the downflow than for the upflow at the same particle Reynolds number, whereas the TKE at the channel center is roughly independent of the flow direction when the particle inertia is very large. For a density ratio of 2, the particles aggregate and are preferentially located in the low-speed streaks in the near-wall region, whereas for a density ratio of 10, the particles migrate toward the channel center, similar to the zero-gravity case. The flow friction increases with an increasing settling coefficient for the same density ratio and particle size, and the friction at the density ratio of order (10) is smallest. The pair distribution function shows the transition from the turbulence-dominated feature to the sedimentation-dominated feature, as the settling coefficient increases.

Effects of the collision model in interface-resolved simulations of particle-laden turbulent channel flows

The effects of the particle collision model in a direct-forcing fictitious domain method on the fluid and particle statistics of a fully developed turbulent channel flow laden with finite-size neutrally buoyant particles are numerically investigated. The particle collisions are described by a combination of the discrete element method and the lubrication force correction model. We first validate our code via several benchmark tests, including the normal particle–wall collisions at different impact Stokes numbers and the oblique collisions with varied incidence angles. Subsequently, the effects of the lubrication correction and the particle stiffness on the fluid and particle statistics of the particle-laden turbulent flows are examined. The results show that the lubrication force correction has an important effect on the particle pair statistics at the near-contact regime. Both the lubrication force between the particles and the decrease in the particle stiffness result in the decrease in the flow friction mainly due to the increase in the fluid Reynolds stress. The flow friction is always larger for smaller particles at the same particle volume fraction irrespective of the lubrication correction. The particle–particle lubrication force decreases the near-wall particle concentration, whereas the particle–wall lubrication force has the opposite effect.

Interface-resolved direct numerical simulations of the interactions between spheroidal particles and upward vertical turbulent channel flows

The interactions between finite-size spheroidal particles and upward turbulent flows in a vertical channel are numerically simulated with a direct-forcing fictitious domain method at two particle settling coefficients $u_{s}$ (the ratio of the particle Stokes free-fall velocity to the bulk velocity) of 0.1 and 0.3, a bulk Reynolds number of 2873, a ratio of the particle equivalent diameter to the channel width of 0.05, a particle volume fraction of 2.36 % and particle aspect ratios of $1/3$ , 1 and 2. Our results show that the flow friction is largest for the case of a sphere, and smallest for the oblate case when the particle sedimentation effect is weak ( $u_{s}=0.1$ ), whereas the flow friction is smallest for the case of a sphere, and largest for the oblate case when the particle sedimentation effect is moderately strong ( $u_{s}=0.3$ ). The reason for the lower flow friction of the spherical particles is that the large-scale vortices are more strongly attenuated by the spherical particles than by the non-spherical particles in the case of $u_{s}=0.3$ . The settling particles tend to migrate towards the channel centre due to the Saffman effect, and the migration is strongest for the spherical particles. The non-spherical particles tend to align their long axes with the streamwise direction in the near-wall region, and perpendicular to the streamwise direction in the bulk region due to the significant settling effect.

A fictitious domain method for particulate flows of arbitrary density ratio

In this paper, our previous direct-forcing fictitious domain method is modified for particle-laden flows of arbitrary density ratio. The instability of original scheme for low density ratio is circumvented by a simple modification in the particle motion equations, with the key idea of increasing the coefficient of particle inertial term. Three schemes are proposed, which are shown to be equally accurate. The modified DF/FD method is extensively validated with various tests, and it is shown that the accuracy of our method is comparable to previous methods. In addition, our method is applied to the direct numerical simulations of the motion of finite-size particles (or bubbles) in a vertical turbulent channel flow of the density ratio 0.001. Our results indicate that the effects of particle-fluid density ratio on turbulent channel flow are negligibly small when the density ratio is smaller than unity and the gravity effect is not considered. The fluid Reynolds shear stress and the wall friction are enhanced in upflow where the particles are aggregated near the wall, whereas they are reduced in downflow where the particles move towards the channel center.

Migration of spherical particles in a confined shear flow of Giesekus fluid

The lateral migration of a spherical particle in a sheared Giesekus fluid is numerically studied by the direct-forcing fictitious domain method. The model is first validated by comparing the simulation results with the available data in the literature. Effects of the viscosity ratio, shear thinning, Weissenberg number, and wall confinement on the particle migration are examined. Results show that the particle migrates toward the wall, irrespective of whether the fluid is shear thinning. The wall confinement, shear thinning, and high Weissenberg number could respectively facilitate the particle lateral migration. While the effect of viscosity ratio on the particle migration is not monotonic, a separatrix value is found which divides the viscosity ratio into two ranges. Moreover, effects of rheological properties on the particle angular velocity and its variation throughout the migration pathway are also explored, and the position where the angular velocity starts to decrease is affected by fluid rheological properties.

Direct-forcing fictitious domain method for simulating non-Brownian active particles.

We present a direct-forcing fictitious domain method for simulating non-Brownian squirmer particles with both the hydrodynamic interactions and collisions being fully resolved. In this method, we solve the particle motion by distributing collocation points inside the particle interior domain that overlay upon a fixed Eulerian mesh. The fluid motions, including those of the "fictitious fluids" being extended into the particle, are solved on the entire computation domain. Pseudo-body forces are used to enforce the fictitious fluids to follow the particle movement. A direct-forcing approach is employed to map physical variables between the overlaid meshes, which does not require additional iterations to achieve convergence. We perform a series of numerical studies at both small and finite Reynolds numbers. First, accuracy of the algorithm is examined in studying benchmark problems of a free-swimming squirmer and two side-by-side squirmers. Then we investigate statistic properties of the quasi-two-dimensional collective dynamics for a monolayer of squirmer particles that are confined on a surface immersed in a bulk flow. Finally, we explore the physical mechanisms of how a freely moving short cylinder interacts with a monolayer of active particles, and find out that the cylinder movement is dominated by collision. We demonstrate that a more directional migration of cylinder can be resultant from an inhomogeneous distribution of active particles around the cylinder that has an anisotropic shape.

Effects of finite-size neutrally buoyant particles on the turbulent channel flow at a Reynolds number of 395

A direct-forcing fictitious domain (DFFD) method is used to perform fully resolved numerical simulations of turbulent channel flows laden with large neutrally buoyant particles. The effects of the particles on the turbulence (including the mean velocity, the root mean square (RMS) of the velocity fluctuation, the probability density function (PDF) of the velocity, and the vortex structures) at a friction Reynolds number of 395 are investigated. The results show that the drag-reduction effect caused by finite-size spherical particles at low particle volumes is negligibly small. The particle effects on the RMS velocities at Reτ=395 are significantly smaller than those at Reτ=180, despite qualitatively the same effects, i.e., the presence of particles decreases the maximum streamwise RMS velocity near the wall via weakening the large-scale streamwise vortices, and increases the transverse and spanwise RMS velocities in the vicinity of the wall by inducing smaller-scale vortices. The effects of the particles on the PDFs of the fluid fluctuating velocities normalized with the RMS velocities are small, regardless of the particle size, the particle volume fraction, and the Reynolds number.

Interface-resolved direct numerical simulations of the interactions between neutrally buoyant spheroidal particles and turbulent channel flows

A parallel direct-forcing fictitious domain method is employed to perform interface-resolved numerical simulations of the interactions between neutrally buoyant finite-size spheroidal particles and turbulent channel flows. The effects of the aspect ratio of the spheroidal particles on the turbulence modulation and the rotation of the particles are investigated at the friction Reynolds number of 180, with the ratio of the particle equivalent diameter to the channel width being 0.1, the particle volume fraction ranging from 0.79% to 14.16%, and the particle aspect ratio ranging from 1/3 to 8. Our results show that the flow friction decreases as the prolate particles become more slender or the oblate particles become flatter and is smaller than that of the single-phase flow for the aspect ratio being 1/3 and 8 at the particle volume fraction of 2.36%. Both effects of the low particle volume fraction in the near-wall region and the relatively small Reynolds stress are important to the occurrence of the drag-reduction by the non-spherical particles, and a lower flow drag for the oblate particles compared to the prolate particles at comparable aspect ratios (e.g., 1/3 vs 4) is mainly caused by a lower Reynolds stress contribution. The prolate particles preferentially align their symmetry axes with the streamwise direction, and the oblate particles preferentially align their symmetry axes with the wall-normal direction. However, the most probable orientation of the major axes of both prolate and oblate particles in the near-wall region is not exactly the streamwise direction but has a positive inclination angle with the streamwise direction. Generally the prolate particles have higher spinning velocities and lower tumbling velocities in the entire channel region, compared to the oblate particles.

Effects of finite-size neutrally buoyant particles on the turbulent flows in a square duct

Interface-resolved direct numerical simulations of the particle-laden turbulent flows in a square duct are performed with a direct-forcing fictitious domain method. The effects of the finite-size particles on the mean and root-mean-square (RMS) velocities are investigated at the friction Reynolds number of 150 (based on the friction velocity and half duct width) and the particle volume fractions ranging from 0.78% to 7.07%. Our results show that the mean secondary flow is enhanced and its circulation center shifts closer to the center of the duct cross section when the particles are added. The reason for the particle effect on the mean secondary flow is analyzed by examining the terms in the mean streamwise vorticity equation. It is observed that the particles enhance the gradients of the secondary Reynolds normal stress difference and shear stress in the near-wall region near the corners, which we think is mainly responsible for the enhancement in the mean secondary flow. Under a prescribed driving pressure gradient, the presence of particles attenuates the bulk velocity and the turbulent intensity. All particle-induced effects are intensified with increasing particle volume fraction and decreasing particle size, if other parameters are fixed. In addition, the particles accumulate preferentially in the near-corner region. The effects of the type of the collision model (i.e., if friction and damping are included or not) on the results are found significant, but not so significant to bring about qualitatively different results.

Effects of particle-fluid density ratio on the interactions between the turbulent channel flow and finite-size particles.

A parallel direct-forcing fictitious domain method is employed to perform fully resolved numerical simulations of turbulent channel flow laden with finite-size particles. The effects of the particle-fluid density ratio on the turbulence modulation in the channel flow are investigated at the friction Reynolds number of 180, the particle volume fraction of 0.84%, and the particle-fluid density ratio ranging from 1 to 104.2. The results show that the variation of the flow drag with the particle-fluid density ratio is not monotonic, with a larger flow drag for the density ratio of 10.42, compared to those of unity and 104.2. A significant drag reduction by the particles is observed for large particle-fluid density ratios during the transient stage, but not at the statistically stationary stage. The intensity of particle velocity fluctuations generally decreases with increasing particle inertia, except that the particle streamwise root-mean-square velocity and streamwise-transverse velocity correlation in the near-wall region are largest at the density ratio of the order of 10. The averaged momentum equations are derived with the spatial averaging theorem and are used to analyze the mechanisms for the effects of the particles on the flow drag. The results indicate that the drag-reduction effect due to the decrease in the fluid Reynolds shear stress is counteracted by the drag-enhancement effect due to the increase in the total particle stress or the interphase drag force for the large particle-inertia case. The sum of the total Reynolds stress and particle inner stress contributions to the flow drag is largest at the density ratio of the order of 10, which is the reason for the largest flow drag at this density ratio. The interphase drag force obtained from the averaged momentum equation (the balance theory) is significantly smaller than (but agrees qualitatively with) that from the empirical drag formula based on the phase-averaged slip velocity for large density ratios. For the neutrally buoyant case, the balance theory predicts a positive interphase force on the particles arising from the negative gradient of the particle inner stress, which cannot be predicted by the drag formula based on the phase-averaged slip velocity. In addition, our results show that both particle collision and particle-turbulence interaction play roles in the formation of the inhomogeneous distribution of the particles at the density ratio of the order of 10.

Effects of finite-size heavy particles on the turbulent flows in a square duct

A parallel direct-forcing fictitious domain method is applied in fully-resolved numerical simulations of particle-laden turbulent flows in a square duct. The effects of finite-size heavy particles on the mean secondary flow, the mean streamwise velocity, the root-mean-square velocity fluctuation, and the particle concentration distribution are investigated at the friction Reynolds number of 150, the particle volume fraction of 2.36%, the particle diameter of 0.1 duct width, and the Shields number ranging from 1.0 to 0.2. Our results show that the particle sedimentation breaks the up-down symmetry of the mean secondary vortices, and results in a stronger secondary-flow circulation which transports the fluids downward in the bulk center region and upward along the side walls at a low Shields number. This circulation has a significant impact on the distribution of the mean streamwise velocity, whose maximum value occurs in the lower half duct, unlike in the plane channel case. The flow resistance is increased and the turbulence intensity is reduced, as the Shields number is decreased. The particles accumulate preferentially at the face center of the bottom wall, due to the effect of the mean secondary flow. It is observed that the collision model has an important effect on the results, but does not change the results qualitatively.

Hydrodynamics characterization of a choanoid fluidized bed bioreactor used in the bioartificial liver system: Fully resolved simulation with a fictitious domain method

Choanoid fluidized bed bioreactors (CFBBs) are newly developed core devices used in bioartificial liver-support systems to detoxify blood plasma of patients with microencapsulated liver cells. Direct numerical simulations (DNS) with a direct-forcing/fictitious domain (DF/FD) method were conducted to study the hydrodynamic performance of a CFBB. The effects of particle–fluid density ratio, particle number, and filter screens preventing particles flowing out of the reactor were investigated. Depending on density ratio, two flow patterns are evident: the circulation mode in which the suspension rises along one sidewall and descends along the other sidewall, and the non-circulation mode in which the whole suspension roughly flows upward. The circulation mode takes place under non-neutral-buoyancy where the particle sedimentation dominates, whereas the non-circulation mode occurs under pure or near-neutral buoyancy with particle–fluid density ratios of unity or near unity. With particle–fluid density ratio of 1.01, the bioartificial liver reactor performs optimally as the significant particle accumulation existing in the non-circulation mode and the large shear forces on particles in the circulation mode are avoided. At higher particle volume fractions, more particles accumulate at the filter screens and a secondary counter circulation to the primary flow is observed at the top of the bed. Modelled as porous media, the filter screens play a negative role on particle fluidization velocities; without screens, particles are fluidized faster because of the higher fluid velocities in the jet center region. This work extends the DF/FD-based DNS to a fluidized bed and accounts for effects from inclined side walls and porous media, providing some hydrodynamics insight that is important for CFBB design and operation optimization.