## Abstract

The presence of a wall near a rigid sphere in motion is known to disturb the particle fore and aft flow-field symmetry and to affect the hydrodynamic force. An immersed boundary direct numerical simulation (IB-DNS) is used in this study to determine the wall effects on the dynamics of a free-falling sphere and the drag of a sphere moving at a constant velocity. The numerical results are validated by comparison to the published experimental, numerical, and analytical data. The pressure and velocity fields are numerically computed when the particle is in the vicinity of the wall; the transverse (lift) and longitudinal (drag) parts of the hydrodynamic force are calculated; its rotational velocity is investigated in the case of a free-falling sphere. The flow asymmetry also causes the particle to rotate. The wall effect is shown to be significant when the dimensionless ratio of the wall distance to the particle diameter, L/D, is less than 3. The wall effects are more pronounced and when the particle Reynolds number, Re, is less than 10. Based on the computational results, a useful correlation for the wall effects on the drag coefficients of spheres is derived in the range 0.75 < L/D < 3 and 0.18 < Re < 10.

## 1 Introduction

Particulate flows and the advection of particles occur very frequently in industrial systems and the environment. Key to the understanding of particulate flows, the design of industrial equipment, and the prediction of biological and environmental particulate advection, is the knowledge of the forces acting on the particles and, in particular, of the drag and lift coefficients. Starting with the work of Stokes [1] for steady flow and Bousinesq [2] and Basset [3] for transient flow, for more than 150 years, scientists and engineers have diligently worked to understand and document the dynamics of particles within a flow field. The most informative works on particle dynamics and the drag coefficients of particles may be found in several books and reviews [4–7]. Among the data on the drag coefficients for steady flows, those by Schiller and Nauman [8] are expressed in the form of a simple and useful correlation. This correlation has endured the passage of time and is still widely used by many.

One of the more intriguing and less understood types of particulate flows is the transport of particles close to plane and curved solid walls. Wall effects on particles have diverse implications in particulate flows such as the dynamics of blood cells in arteries and microvessels; the transport of particulates in channel and pipe flows; the filtering of particles; particle deposition; and heat exchanger fouling. The wall effects were first investigated by Faxen [9] and are still referred to as *the Faxen problem.* The presence of the flow boundary exerts a transverse force on the particles and also affects the particle drag coefficients in the directions both parallel and perpendicular to the wall. The origin of the transverse force is a combination of asymmetry of the particle wake and lower pressure on the wall-sides of the particles, which is due to higher fluid velocities resulting from the flow acceleration in the particle-wall gap. Analytical expressions for the wall-induced forces have been presented for rigid spherical particles at low but finite Reynolds numbers by Cox and Hsu [10]; Vasseur and Cox [11]; and Happel and Brenner [12]. Among the other studies on the Faxen problem, experimental data for freely settling spheres in circular cylinders are available in Fidleris and Whitmore [13]; Sutterby [14]; and Ambari et al. [15].

In the experimental study by Ambari et al. [15], a magnetic field was used to maintain a fixed distance between the particle and the wall. The particle was forced to move through the fluid parallel to the wall and the gap was maintained constant by an optical measurement system that modulated an electric current, which generated the desired magnetic force. The results of this study were compared to the analytical solutions and it was determined that analysis and experiment agree well when the particle distance from the wall is less than 0.9 radii or the gap is less than 0.4 radii. More recently, Takemura [16] presented results for the migration and sedimentation velocities of particles freely settling in the presence of a plane wall at Re < 5. The results of this study demonstrated that the particle migration velocities are significantly lower than the sedimentation velocities and that the wall effects appear to be most significant at distances from the wall that are less than three particle diameters. Leach et al. [17] used optical tweezers to confirm the longitudinal drag coefficient corrections proposed by Faxen and presented a correction for the rotational drag on the particle. It was noted that the rotational drag correction was one order of magnitude less than the translational drag correction. This study provided useful results for the use of optical tweezers to measure viscosity at the pico-liter lengthscales.

With the vast advancement of computational power and numerical methods, numerical results on particulate flow have become the focus of recent work. Zeng et al. [18,19] undertook an extensive numerical study on the drag coefficients of fixed spheres in the proximity of plane walls that cover a wide range of particle Reynolds numbers, 2<Re < 250. This study showed that the particle lift coefficient decreases with the Reynolds number and the wall gap. It also demonstrated that the rotation of the sphere has only a very small effect on the particle lift and drag forces. Sugioka and Tsukada [20] also used a direct numerical simulation method to derive results for the drag and lift forces on a spherical bubble traveling vertically upward near a wall. They showed an increase of the drag coefficients in the vicinity of the wall.

It must be noted that the effect of walls on particle movement has very important ramifications in several practical applications including the deposition of particles on the wall and the formation of scaling in both laminar and turbulent flows [21]; the flow of blood and the formation of blood clots in arteries [22]; the effect of thermophoresis on the deposition and collection of particles and especially of nanoparticles [23]; and the hindered diffusion of macromolecules, micro- and nanoparticles in narrow pores [24]. It must also be noted that when particles are at very small distances from the wall, van der Waals type forces—often called DLVO forces that act at submicron distances—become significantly higher than the hydrodynamic forces and dominate the movement of the particles and either attract them to wall (deposition) or repel them (resuspension). A great deal about the nature and magnitude of the DLVO forces may be found in the literature of particle deposition [25,26].

This study makes use of an immersed boundary-based direct numerical simulation (IB-DNS) method to examine the effects of vertical plane walls on rigid spheres that fall freely or move at a prescribed velocity in the vicinity of the wall, at moderate Reynolds numbers. When the spheres fall freely, they are allowed to move longitudinally and laterally from their original position, as it would happen in actual sedimentation processes. The migration velocity and sedimentation velocity are compared to the experimental work of Takemura [16] and the comparisons show very good agreement. Furthermore, the effects of the wall on the rotational motion have been investigated. Extensive simulations of a sphere translating at a constant velocity, parallel to the wall, are conducted and a simple correlation is derived, which may be used in computer simulation codes to determine the particle drag coefficients in terms of the Reynolds numbers and the particle distance from the wall.

## 2 The Immersed Boundary-Direct Numerical Simulation Method

The immersed boundary (IB) method for single-phase fluid computations was introduced by Peskin [27] to model the blood movement in the heart. Feng and Michaelides were the first to extend the method for the movement of particles in combination with the Lattice Boltzmann method [28] and later with a direct forcing scheme [29]. Shortly thereafter, Uhlman [30] independently combined the method with a solver based on finite differences. The fundamental premise of the IB method is that solid particles within the flow field are represented by the system of forces they exert on the surrounding fluid. The actual particles are substituted in the fluid domain by a body force density function, which is incorporated into the momentum equation of the fluid [28,29]. The IB method uses a fixed Cartesian mesh for the fluid, which is composed of Eulerian nodes and another, Lagrangian grid for the solid particles that are immersed in the fluid, with the boundaries of the solid particles being represented as points in this Lagrangian grid. The details of constructing the Lagrangian grid for a sphere may be found in Ref. [31]. The Lagrangian points of the particle surface are advected as the solid particles move, according to the rules of fluid–solid interactions, and are projected in the Eulerian grid points of the fluid domain as forces [28–31]. This representation of the solid particle boundaries in the combined Eulerian/Lagrangian approach avoids remeshing at each time-step, a significant advantage of the numerical method.

The particle motion is tracked using the rigid body equations of motion. The momentum interactions between the fluid and particles are represented by a force density function, which is applied to the Lagrangian nodes using a direct forcing scheme in this study. The force density function ensures the no-slip boundary condition at the fluid–particle interface. One of the principal advantages of this numerical scheme is that it is derived from first principles and does not need to make use of empirical closure equations for the determination of fluid–particle interactions.

In Eq. (7), $q(t)$ is the unit quaternion for the rotation matrix of the particle; and $\Xi ij$ is the rotational tensor associated with the angular velocity $\omega S(t)$.

It must be noted that the above equations may be used to describe the motion and rotation of all types of particles, including irregularly shaped particles. The spherical particles are a special case in which $Ii(t)$ is described as a constant diagonal matrix with its diagonal element equal to $25msa2$.

## 3 Results and Discussion

### 3.1 Validation—Drag Coefficient of a Sphere Without Wall Effects.

*D*settling under gravity or moving at a prescribed constant velocity along the centerline of a large flow domain, which was chosen to be a rectangular box with a width of $12D$, a depth of $12D$, and a height ranging from $24D$ to $60D$ depending on the particle Reynolds numbers (defined as $Re=\rho UD\mu $, where

*U*is the prescribed constant velocity or the particle terminal velocity). The calculations have shown that lower particle Reynolds numbers required larger domain sizes than higher Reynolds numbers in order to achieve results that are independent of domain sizes. The grid spacing is uniform and one diameter of the sphere was outlined by 12–20 grid steps depending on the Reynolds numbers. Periodic boundary conditions are used at the left and right surfaces as well as at the front and back surfaces. The stress-free boundary condition is used at the top surface; and zero velocity is prescribed at the bottom surface. In the particle sedimentation case, a set of different fluid viscosities are used to achieve the desired variation of the terminal velocity and Reynolds numbers. Figure 1 shows the simulation results of the drag coefficient of the sphere at Reynolds numbers ranging from 0.18 to about 100. These results agree very well with the Schiller–Nauman correlation [8] of drag coefficient of an isolated sphere in an unbounded flow

The same or similar numerical parameters such as the grid step and domain size are then used in the rest of numerical simulations.

### 3.2 Lateral Migration Velocity of a Sphere Near a Vertical Wall.

Takemura [16] experimentally studied the migration velocity of a particle near a vertical wall at a range of low Reynolds numbers (Re < 5). The experiments were conducted in a 500-mm long glass channel with a $60\xd760\u2009$mm cross section filled with a viscous fluid. A vertical wall of 450 mm long and 40 mm wide was inserted in the middle of the glass channel. Four silicone oils with kinematic viscosities 10.5, 19.8, 28.6, and 47.1 mm^{2}/s were used. The spherical particles in their study were made of nitride silicon and glass with densities 3200 and 2500 kg/m^{3}, respectively; and diameters were 0.8 mm. In this study we numerically simulated the same setup and extended the Reynolds numbers to about 100.

The simulation domain and the parameters used in this study are shown in Fig. 2. The particle diameter, $D$, is 0.8 mm to match that of the experimental study by Takemura [16]. The dimension, *L*, is defined as the distance from the particle center to the wall and this implies that the wall gap—the closest distance from the wall to the particle surface—is equal to $\delta =L\u2212a$. The ratio, $L/D$, is a dimensionless measure of the proximity of the particle to the channel wall and its minimum value is 0.5 when the particle actually touches the wall. The simulation domain size is chosen such that the computational domain boundaries other than the vertical wall have negligible effect to the simulation results. A wide range of fluid viscosities are used to achieve different sedimentation velocities and Reynolds numbers in the simulations. Depending on the particle Reynolds numbers, the grid step varies from $D/12$ to $D/20$.

Figure 3 shows typical numerical results for the free-fall sedimentation velocity of a particle at several different Reynolds numbers, ranging from 4.3 to 86.1 and the comparison of the numerical results with the experimental data by Takemura [16]. The simulations were terminated when the particles get close to the bottom of the computational domain. For this reason, the faster particles (higher Re) appear to end closer to the wall. It may be seen that, when the particles move vertically near the wall, they also gradually migrate away from the wall; their sedimentation velocity increases and asymptotically approaches the limit of the terminal velocity when the particles are at distances further than 1.5 diameters from the wall. The comparison with the experimental data in Fig. 3 also shows that there is very good agreement of the numerical results with the experimental data by Takemura [16], which pertain to Re = 4.3.

It is apparent that the presence of the wall induces a lift force on the particles; and this causes the particles to migrate away from the wall. Figure 4 shows the numerical results for the migration velocity during the sedimentation process. Comparison with the experimental results by Takemura at Re = 4.3 again shows a very good agreement with the numerical results of this study. Figure 4 also shows that the lateral migration velocities are an order of magnitude lower than the sedimentation velocity of the particles. This implies that the migration away from the wall is significantly slower.

Figure 5 shows the velocity and pressure contours for the sphere settling near a wall at Re = 4.3 It is seen that the pressure is asymmetric with respect to the vertical plane of symmetry through the center of the sphere when the spherical particle is close to the wall. The figure also indicates that the asymmetry is more pronounced when the particle is very close to the wall (the upper two cases). The observed asymmetry induces a net lateral force in the direction away from the wall. As the sphere migrates away from the wall, the pressure asymmetry diminishes; the lateral force is reduced; and the particle migration velocity approaches zero as it is shown in Fig. 4.

It must be noted that the position, where the sphere is released, only affects the migration velocity and the terminal velocity at the beginning of the sedimentation process. The two velocities eventually merge into the same velocity pattern as it is demonstrated in Fig. 6 where a particle is released at three positions: *L/D* = 0.52, 0.6, and 1 (Re = 4.3 for all three cases).

### 3.3 Rotational Velocity Near the Wall.

When a sphere falls near a wall, and because of the asymmetry of the flow field, the sphere rotates in a direction that is parallel to the wall. It is of interest to know if the sphere rotates clockwise or counterclockwise. We investigated this by placing a sphere at an initial distance *L/D *=* *0.52, where the gap with the surface of the sphere and the wall is extremely small.

Figure 7 shows the evolution of the angular velocity at several Reynolds numbers. It is seen that at the lower Reynolds numbers Re = 3.2 and 4.3, the sphere rotates clockwise after being released. This happens because, at low Reynolds numbers, the viscous forces dominate, and the motion of the sphere is similar to a sphere rolling along the wall in the clockwise direction without sliding. The clockwise rotation diminishes as the sphere moves away from the wall, changes to a weak counterclockwise rotation and eventually stops. At the higher Reynolds numbers, the sphere rotates counterclockwise because the inertia effects are more significant than the viscous effects. In all cases, as the sphere moves away from the wall, its angular velocity gradually diminishes and the rotation stops.

We also investigated how the position, at which a particle is released, affects its rotational direction while it is settling. At Re = 4.3, the particle is released at 3 different positions: *L/D *=* *0.52, 0.6, and 1.0. The results are shown in Fig. 8 and indicate that the particle rotates clockwise when it is released at *L/D *=* *0.52 and at *L/D = *0.6. The rotation gradually changes its direction to counterclockwise as the particle moves away from the wall and, finally, the rotation stops. However, when the particle is released at *L/D *=* *1, the particle rotates counterclockwise after its release. It is apparent that the rotational direction of the particle at the first stages of its movement not only depends on the Reynolds number but also on the location of its release.

### 3.4 Lift Coefficient of a Sphere Near a Vertical Wall.

In all the simulation cases, the fluid is stagnant and the particle is traveling at a prescribed and constant velocity that is parallel to the wall and is at a given distance away from the wall. Numerical computations were conducted over a range of Reynolds numbers from 0.18 to 100 and at ten distinct *L/D* ratios, in the range 0.625 to 4.

The presence of the vertical wall in the vicinity of the sphere introduces asymmetries in the flow field that are very much pronounced at distances *L/D *<* *3. All the computations show that the asymmetry features are more evident at the lower range of the Reynolds numbers. Figure 9 demonstrates the flow field asymmetry for the values Re* *=* *0.18 and *L/D *=* *0.625, a case that corresponds to the most significant wall effect in this study. The pressure contours show that the highest pressure in the flow field occurs on the fore side of the sphere near the wall, whereas the lowest pressure occurs again at the side of the wall and on the aft side of the sphere. There is significant asymmetry of the pressure field, most predominantly on the side of the solid boundary. On the fluid velocity side, the results also show significant flow field asymmetry since the stationary wall, where the fluid is stagnant, retards the flow. Higher fluid velocities are observed on the side of the sphere away from the wall and lower velocities on the near-wall side, where the flow is constrained.

is also plotted in Fig. 10 as a solid line. The observed good agreement between our numerical results for the solid spheres and the experimental data of the contaminated bubbles shows that, just like the drag coefficient, a contaminated bubble has the same lift coefficient as a solid sphere.

### 3.5 Drag Coefficient of a Sphere Near a Vertical Wall.

The longitudinal drag coefficients of the sphere were also calculated from the simulation results conducted in Sec. 3.4. Figure 11 compares the drag coefficients from the present IB-DNS numerical method with other values from the literature that were obtained by numerical, analytical, and experimental methods. The drag coefficients using the IB-DNS method show good agreement (within −7.2% to 2.9%) with those obtained by Zeng et al. [18,19] for all Reynolds numbers in the range 0 to 10 and *L/D* in the range 0.75 to 4. In general, it was noted that the IB-DNS method tends to slightly underpredict the drag coefficients at lower Reynolds number and lower *L/D* ratio as compared to the data by Zeng et al., possibly due to the limitations of the use of a regular grid in the IB-DNS method. Figure 11 also depicts the Schiller–Nauman correlation [8], which is applicable to unbounded flows, and the results from the lubrication theory, which correspond to extremely low distances from the vertical wall (close to *L/D *=* *0.5) [35]. The IB-DNS results at *L/D *=* *4 demonstrate that the drag coefficients are very close to the data for unbounded flows. The results at lower values of *L/D* are significantly higher and converge toward the results of the lubrication theory limit as *L/D* approaches the limit 0.5. It is also apparent from all the results that the drag coefficients of rigid spheres moving very close to the boundaries are significantly higher than their values in unbounded flows. At low Reynolds numbers, where the lubrication theory applies, the drag coefficients close to boundaries lay between the two limits: the unbounded flow limit and the lubrication flow limit.

In the ranges 0.18 < Re < 10 and 0.75 < *L/D *<* *3, the accuracy of this correlation is between -9% and +7% [35]. The correlation is not recommended to be used for *L/D* > 4, where the Schiller–Naumann correlation (Eq. (10)) may be used with accuracy. A linear interpolation between Eqs. (10) and (12) is recommended in the range 3 < *L/D *<* *4.

## 4 Conclusions

The IB-DNS method has been used to study the dynamics of particles settling in a quiescent fluid near a vertical wall. The numerical results were validated with theoretical, experimental, and numerical data available in the literature. The study demonstrates that the particle proximity to the wall induces significant flow field and pressure asymmetries at the fore and aft sides of the particle. The asymmetries cause significant modifications on the particle longitudinal drag and the generation of a weaker lift force. A transient rotation of the particle was also observed when the particle falls in the vicinity of the wall and the direction of this rotation was investigated. As a result of the wall influence on the fluid velocity field, the particle experiences a higher drag force and an additional lift that drives it away from the wall. When *L/D *>* *4, the wall effects are not significant and the standard correlations for the drag of particles may be used. For wall distances *L/D *<* *3, an engineering correlation is presented for the drag modification in the vicinity of the wall.

## Funding Data

National Energy Technology Laboratory (Award No. DE-FE0031894; Funder ID: 10.13039/100013165).