# Effect of rough wall on drag, lift, and torque on an ellipsoidal particle in
a linear shear flow

Atul Manikrao Bhagat  Partha Sarathi Goswami<EMAIL_ADDRESS>Chemical
Engineering Department, Indian Institute of Technology Bombay, Mumbai - 400076
(India)

###### Abstract

The present study provides a detailed description of the forces on an
ellipsoidal particle in the vicinity of the rough wall. Three-dimensional
numerical simulations are performed using body-fitted mesh to estimate the
drag, lift, and torque coefficients. A large number of simulations are
conducted over a range of parameters such as shear Reynolds number ($10\leq
Re_{s}\leq 100$), orientation angle ($$$\leq\theta\leq$$$), and wall-particle
separation distance ($0.1\leq\delta\leq 2.0$) to get a comprehensive
description of variation of the above coefficients. Using the simulation
results, we develop the correlations for the drag and lift coefficients to
describe the effect of rough wall, inclination angles, and particle Reynolds
numbers on the hydrodynamic coefficients. The proposed correlations can be
used for two phase flow simulation using Eulerian-Lagrangian framework.

###### keywords:

Drag, lift, torque, non-spherical particle, rough wall, linear shear flow

††journal: Powder Technology

##  1 Introduction

The flow of suspensions containing nonspherical particles is ubiquitous in
natural, industrial, and biomedical processes. Industrial processes like,
fluidization [1], pneumatic transport of solids [2, 3], processing in paper
industry [4, 5, 6], and wastewater treatments [7] deal with the flow of
nonspherical particles. Raw materials used in all the processes deviate a lot
from the ideal spherical structure. Transport and deposition of nonspherical
particles in biological flows are also important issues to be addressed [3].
Therefore, understanding the forces and dynamics of the nonspherical particles
helps to design and develop these processes. Most of the earlier studies deal
with the spherical particles [8, 9, 10] where symmetry plays an important role
to simplify the problem unless the particles are placed near the solid
surfaces. But, nonspherical particles increase the degrees of freedom based on
their orientation with respect to the ambient flows.

An attempt to describe the dynamics of a nonspherical particle in viscous flow
initiated almost a century back with the phenomenal work by Jeffery and Filon
[11]. They described the motion of an ellipsoid in the creeping flow limit.
Brenner [12, 13, 14] investigated the forces on particles with arbitrary shape
in the Stokes flow limit using analytical theory and provided the expression
for drag and lift forces. Later, an expression for the lift force on a
nonspherical particle in shear flow was derived by Harper and Chang [15] using
asymptotic expansions. A comprehensive discussion on the dynamics of elongated
particles for a wide range of flow is reported by Lin et al. [16].

A number of investigations that analysed the hydrodynamic forces on the
particles placed near the wall at very low Reynolds number have been reported
[17, 18, 19, 20, 21, 22, 23]. The boundary integral method was used to
estimate the force and velocity of nonspherical particles in shear flow in the
Stokes flow limit [17]. It was reported that the wall effect is very localized
near the wall, and there is a decrease in the effect of wall on hydrodynamic
forces and torque as the non-sphericity of the particle increases. A spectral
boundary element method was used by Pozrikidis [18] to describe the motion of
spheroidal particles for very low particle Reynolds number. A modification of
the Jeffery orbit due to the presence of wall was reported by the author. Kim
et al. [20] studied the dynamics of a cylindrical particle near a stationary
wall in linear shear flow at different Stokes numbers. They have reported that
there is an increase in drag with the aspect ratio of the particle. However,
the torque is independent of wall-normal distance.

During the last decade, a number of notable works to provide correlations to
calculate drag and lift are reported for finite Reynolds numbers [24, 25, 26,
27, 28, 29]. Majority of those are obtained through data fitting and satisfy
the theoretical formulation at the limiting cases. Using the experimental data
reported in the literature, Holzer and Sommerfeld [24] reported a correlation
to calculate drag coefficient, which includes the effect of particle shape,
orientation angle, and particle Reynolds number. Their correlation
successfully predicts the drag coefficient with a maximum deviation of $14\%$.
Immersed Boundary Method (IBM) was used by Zastawny et al. [25] to estimate
drag and lift forces on the nonspherical particles. The parameters used in
their correlations are exclusive functions of aspect ratio. Andersson and
Jiang [30] also used IBM to analyse the forces on ellipsoid with a very high
aspect ratio in uniform flow at very low Reynolds number. They outlined the
challenges in computation at such a low Reynolds numbers for various
inclination angles and computation domains. Ouchene et al. [31, 26] computed
drag, lift, and torque coefficients for uniform flow over ellipsoidal particle
at moderate Reynolds number using body-fitted mesh and presented the
correlations. Sanjeevi and Padding [32] and Sanjeevi et al. [27] analyzed the
drag on a nonspherical particle at different inclination angles in an uniform
flow and for different Reynolds numbers. Forces on the spherical particle in
the vicinity of a rough wall for liner shear flow have been estimated by Lee
and Balachandar [8]. Frohlich et al. [28] have also performed a number of
simulations using the Cartesian cut cell method for an ellipsoid in an
unbounded uniform flow. They have reported that beyond certain aspect ratio
($>3$), the flow is not influenced much with the aspect ratio. Zarghami and
Padding [22] examined the hydrodynamic forces and torque acting on a particle
placed near the wall with 2D numerical simulations using Lattice Boltzmann
method. A significant variation of forces based on wall separation and
orientation angle was observed. The presence of a wall was found to change the
directions of lift and torque acting on the particle [22]. Palmer and Smith
[23] modeled the partially immersed finite body in viscous layers near a wall
in a 2-dimensional scenario. Their study showed the oscillating motion of
particles based on the center of mass of the particle. This motion of body
directs the fluid displacement in between wall and particle, which results in
non-linear pressure responses of a wall. All of these works were carried out
either at Stokes flow regime or at a 2-D scenario. Very recently Fillingham et
al. [29] developed correlations for the drag, lift, and torque coefficients on
an ellipsoid particle placed on a smooth wall in a linear shear flow. They
carried out a number of simulations at several combinations of parameters such
as incident angle (in the plane perpendicular to the plane of shear), Reynolds
number of flow, and aspect ratio of particle. The data from simulations were
used to formulate the correlations, which reproduce the results for a
spherical particle [10] as a limiting case. They have reported a maximum
deviation of $16\%$ as predicted by the correlation.

In the present study, we analyse the forces acting on an ellipsoidal particle
in a linear shear flow. The particle is placed in the vicinity of a rough
wall. 3D finite element method with body-fitted meshes is used for the
simulations. The ellipsoidal particle is assumed to be fixed at a predefined
distance away from the rough wall with an inclination angle. The objective is
to estimate the forces acting on a particle at different distances from the
rough wall and at various orientations with flow direction in the plane of
shear. The effects of variation of the above parameters and the flow Reynolds
number on the drag, lift, and torque coefficients acting on the particle are
reported in detail. For all parameters investigated here, corresponding
studies are carried out for the domain with a smooth wall. The results for the
rough wall have been compared with the results for the smooth wall. Semi-
empirical correlations are reported to predict the drag and lift coefficients
for different inclination angles and a wide range of Reynolds numbers.

##  2 Numerical method

This section outlines the numerical technique used for the simulations. First,
we describe the system configuration followed by the governing equations.
Then, a validation study is presented for the flow conditions reported in the
literature. Finally, an assessment for the flow development and finding
particle location to obtain steady forces are presented.

(a)

(b)

Figure 1: (a) Domain for numerical analysis (3D) (b) Ellipsoid particle in
domain

A schematic of the domain, along with coordinate system used, is shown in Fig.
1. An ellipsoidal particle with semi-major axis $a$, semi-minor axis $b$
(aspect ratio $AR=a/b$), is placed at a location of $L$ from the horizontal
base wall. In the present study, all kinds of motion of ellipsoid such as
rolling, tumbling, and linear movement are not considered; essentially forces
on the stationary particle are analysed. The origin of a coordinate system is
located on the base wall. A rough base plane is constructed using
hemispherical particles of diameter $d$. These hemispheres are arranged in a
triangular array fashion and in contiguity to generate a compact structure.
The location of these hemispheres on the base plane is calculated using the
method given by Lee and Balachandar [8] as,

$\displaystyle x_{ij}=$ $\displaystyle\ \begin{cases}x_{o}+j&\text{if i is
odd}\\\ x_{o}+(j-\frac{1}{2})&\text{if i is even}\end{cases}$ (1)
$\displaystyle y_{ij}=$ $\displaystyle\ \ y_{o}+sin\
\bigg{(}\frac{\pi}{3}\bigg{)}(i-1)$ $\displaystyle z_{ij}=$ $\displaystyle\ \
0$

Here $x$, $y$, and $z$ are scaled with equivalent diameter of particle
$d_{p}$. Hemispheres are arranged in rows ($i$) and columns ($j$). The first
row of hemispheres is kept at 5 particle diameters away from the entrance (at
$x_{0}=5$). The value of $y_{o}$ ($\approx-2.165$) is maintained such that
there should be equal numbers of rows of hemispheres across ellipsoidal
particle in channel cross-section.

Due to the presence of roughness on the wall, the effective clearance between
the ellipsoid and the base plane is different as compared to a smooth wall. In
case of the smooth wall, clearance (wall separation) is the distance from base
plane to closest point on the ellipsoid in vertical direction. On the other
hand, for rough wall, the minimum vertical distance changes with location of
particle due to non-uniform surface. In that case, two different base levels
are needed to be defined. First one is the actual wall base where the
hemispheres are centered, and the other one is the bed height. Based on two-
dimensional configuration the bed height is defined as minimum vertical
distance between base level and lowest point on the ellipsoid, when it is
seating on the bed. For the present configuration the bed height $h_{BL}$ is
taken to be $0.3165d_{p}$. Further, the clearance of the ellipsoid from the
bed height ($\delta$) and orientation angle ($\theta$) are the key parameters
considered in the present study. $\delta$ is scaled with volume equivalent
spherical particle diameter. The rotation of ellipsoid defined through
orientation angle ($\theta$), is about $y$-axis, and anticlockwise rotation is
assumed to be positive.

A linear shear flow is considered to advance towards the rough surface with
mean a velocity component in $x$-direction only and no flow in the span-wise
direction. The velocity profile can be expressed as $U_{x}(z)=G_{x}z$ with
$G_{x}=dU_{x}/dz$, a shear rate of approaching flow. Here, suffix $x$,
represents inlet flow in $x-$direction. The shear Reynolds number based on the
applied shear can be defined as,

$Re_{s}=\frac{G_{x}d_{p}^{2}}{\nu}.$ (2)

Here, $d_{p}$ is related to major and minor axes of ellipsoid as
$r_{p}=2\times(ab^{2})^{1/3}$. The hydrodynamic coefficients for drag, lift,
and torque based on the volume equivalent spherical diameter and ambient fluid
velocity ($U_{x}$) at the center of mass location of ellipsoid($z^{\prime}=L$)
are defined as

$C_{D}=\frac{F_{d}}{\frac{1}{2}\rho
U_{x}^{2}\pi\frac{d_{p}^{2}}{4}},C_{L}=\frac{F_{L}}{\frac{1}{2}\rho
U_{x}^{2}\pi\frac{d_{p}^{2}}{4}},C_{M}=\frac{F_{T}}{\frac{1}{2}\rho
U_{x}^{2}\pi\frac{d_{p}^{3}}{8}}.$ (3)

###  2.1 Governing equations

Flow of incompressible Newtonian fluid in the described domain is governed by
the continuity and Navier-Stokes equations as,

$\rho\bigg{(}\frac{\partial\textbf{u}}{\partial
t}+\textbf{u}\cdot\nabla\textbf{u}\bigg{)}-\nabla\cdot\sigma=0\hskip
28.45274pt\mathrm{on}\ \Omega(\textbf{x},t)$ (4)

and

$\nabla\textbf{u}=0\hskip 28.45274pt\mathrm{on}\ \Omega(\textbf{x},t).$ (5)

Here, $\sigma$ is a stress tensor, and is expressed as,

$\sigma=-p\textbf{I}+\textbf{T}$ (6) $\textbf{T}=2\mu\varepsilon(\textbf{u})$
(7)

and $\varepsilon(\textbf{u})$, the deformation tensor can be expressed as,

$\varepsilon(\textbf{u})=\frac{1}{2}\bigg{[}(\nabla\textbf{u})+(\nabla\textbf{u})^{\mathrm{T}}\bigg{]}.$
(8)

$p$ and $t$ are the pressure and time respectively. The simulations are done
using COMSOL Multiphysics. The no-slip boundary condition is applied on the
surface of ellipsoid and on the rough wall. Fully developed linear shear flow
profile is imposed at the inlet. Free slip condition with no-flow across it,
is set for the top wall. The domain is discretized using non-uniform
tetrahedral grids. An extremely fine grid size is imposed on the surface of
rough wall and ellipsoid, in order to capture the flow gradient correctly. A
large number of simulations have been carried out using different angles of
inclination of the particle and the wall normal separation distances.
Hydrodynamic forces on the particle are calculated and results obtained are
discussed in the next section.

###  2.2 Validation

Before analyzing the results related to forces on an ellipsoid, the validation
is conducted using the data from other simulations reported in the literature
and with theoretical results, if available. The validations have been
performed for two different cases. In the first case, the drag and the lift
coefficients for ellipsoid (aspect ratio AR = 2.5) in uniform flow at low
Reynolds number ($Re_{p}=0.1$) is computed. The results have been compared
with DNS results of Zastawny et al. [25], as shown in the Figs. 2(a) and 2(b).
The figures show a variation of the coefficients with angle of inclination.
Drag and lift coefficients predicted by the present simulations show a good
agreement with that predicted theoretically by Happel and Brenner [33]. There
is an insignificant difference (about $0.3\%$) in drag coefficient as
predicted by Zastawny et al. [25] using Immersed Boundary Method (IBM). Our
results on lift coefficient, show a deviation of less than $5\%$ from that
predicted by Zastawny et al. [25] at an angle of $\theta=$$$.

(a)

(b)

Figure 2: Force coefficients for ellipsoid particle in uniform flow, compared
with results of Zastawny et al. [25]

The second set of validations is performed by analysing the flow over a
spherical particle near the rough surface. The results (Fig. 3) are compared
with Lee and Balachandar [8] who investigated hydrodynamic forces acting on
spherical particle in an uniform shear flow over rough wall using IBM for a
shear Reynolds number $Re_{s}\leq 100$. Figs. 3(a) and 3(b) show the drag and
lift coefficients for particle at different location above the rough wall at
$Re_{s}=100$. It is observed that the results are in good agreement with the
results of Lee and Balachandar [8]. A small difference is observed when the
particle is placed very close to the wall. In any of the cases, maximum
difference is always less than $5\%$. The results from the correlation by Zeng
et al. [10] are plotted with similar conditions and found to under predict the
value. It is to be noted that results reported by Zeng et al. [10] are for the
case of smooth wall.

(a)

(b)

Figure 3: Force coefficients for spherical particle in uniform shear flow,
compared with results of Lee and Balachandar [8]

###  2.3 Flow development analysis

(a)

(b)

(c)

Figure 4: Ambient velocity u variation with all normal direction z for
$Re_{s}=50$, at (a) $y^{\prime}$ = 0.144 and several x-locations, (b)
$y^{\prime}$ = 0.433 and various x-locations, (c) $x^{\prime}$ = 15 and
different y-locations

A fully resolved 3D simulations with body fitted fine mesh is computationally
very much expensive. In the present study, the bottom wall is made rough using
hemispheres of equivalent diameter of the ellipsoid. Such an arrangement also
adds up to the overall computational cost. One of the major reason for this is
the requirement of fine mesh on these hemispheres as well to capture the
boundary layer on the particle. Therefore, to optimize the computational cost,
only a small section of the bottom wall is made rough with an ordered
arrangement of the hemispheres as shown in 1(a). First, simulations are
carried out for flow over rough wall in the absence of particle. The rough bed
is constructed arranging hemisphere in 10 rows and 20 columns. The shear
Reynolds number is varied between 10 to 100. The flow profile obtained from
the simulations are plotted at different span-wise locations in domain. Figs.
4(a), 4(b) and 4(c) show ambient flow profiles as a functions of wall normal
direction ($z$) at different $x^{\prime}$ and $y^{\prime}$ locations over
rough bed. It is observed that the flow remains developed for all the cases.
4(a) shows the velocity profile at $y^{\prime}=0.144$, which represents the
deep cavity (along trough) in to the rough surface. Similarly 4(b) shows
velocity profile at $y^{\prime}=0.433$ that is the top position (along crest)
on surface of hemispheres. In both the Figs. 4(a) and 4(b), the velocity
profiles are shown at different stream-wise locations ($x^{\prime}$ = 10, 11,
12, 13, 14, and 15). The figures depict that the flow becomes developed after
$x^{\prime}$ = 14, that is after $9^{th}$ row of hemispheres. Further
investigation are carried out at $x^{\prime}$ = 15 and $y^{\prime}$ = 0.144,
0.722, and -1.010, which represent a deepest cavities along $y^{\prime}$ =
0.144 column. The corresponding profiles are shown in 4(c). Here, velocity
profiles for all the considered positions along $y^{\prime}$ are seen to be
merged into a single line, give the confidence that the velocity is almost
identical along that column of hemispheres around $x^{\prime}$ = 15.

(a)

(b)

Figure 5: (a) Drag and (b) lift coefficients variation over several x-position
on rough bed ($\delta=0.1$) for $y^{\prime}$ = 0.144, $Re_{s}=10$ and AR = 2.
Figure 6: Torque coefficient variation over several x-position on rough bed
($\delta=0.1$) for $y^{\prime}$ = 0.144, $Re_{s}=10$ and AR = 2.

The next set of simulations is conducted to explore the location on the rough
bed where the hydrodynamic forces on the particle are not be influenced by the
edge effect of the rough bed. Three different cases are simulated based on the
the Reynolds numbers ($Re_{s}$) 10, 50, and 100. Since presence of the wall
affects the hydrodynamic forces acting on particle [10, 8, 22], it is
important to carry out these test cases with particle positioned near the
wall. Here we place the particle at $\delta=0.1$ in $z$-direction, which
presents the situation where the particle just touches the rough bed. The
span-wise position, $y^{\prime}=0.144$ and the particle position is varied in
$x$-direction ($x^{\prime}$ = 10, 11, 12, 13, 14, 15, 16, and 17). The drag,
lift, and torque coefficients are then calculated and are shown in Fig. 5. The
figure shows that the variation of drag coefficient is only $0.5\%$ for
position higher than $x^{\prime}=15$. In case of the lift coefficient (5(b)),
the variation is small and it remains almost constant with an approximate
change of $0.5\%$ over entire range of the particle positions. Fig. 6 shows
the variation of torque coefficient with particle position in stream-wise
direction. An insignificant change is observed for particle position
$x^{\prime}<15$ and for $x^{\prime}>15$ the coefficient appears to be constant
having a maximum deviation of $0.2\%$. The above analysis confirms that a
stream-wise distance $x^{\prime}=15$ or higher and a span-wise location
$y^{\prime}=0.144$ can be considered to obtain hydrodynamic forces which are
free of edge effects of rough bed.

###  2.4 Grid independence study

(a)

(b)

(c)

Figure 7: Hydrodynamic coefficients variation with number of elements on
ellipsoid surface at $Re_{s}=10$ and $\theta=$$$

A proper grid resolution is a must to calculate accurate forces on the
particle. Either the CFD results are to be validated with experiments to get
an idea of proper resolution or in case of unavailability of the experimental
data, grid-independent results are to be produced through different levels of
grid refinements. There is no standard protocol yet to set-up the required
grid resolution to analyse the flow over an object when it is placed near a
wall with finite roughness [8]. Therefore, a grid independence study for the
present work is of foremost importance. Use of an extremely fine mesh
throughout the domain will also be computationally very expensive. Thus, an
extremely fine mesh has been employed on the rough-surface using 3 layers of
boundary layer meshing. The number of elements on the ellipsoid is then
systematically varied in order to obtain a converge solution of the flow. The
effect of variation of number of mesh-elements on the drag, lift, and torque
coefficients is shown in the Fig. 7. Although it is observed that there is an
insignificant variation ($\approx 1\%$) of the coefficients with the change in
the number of elements considered here, we have considered 120 elements on the
ellipsoid to perform all the simulations reported here.

##  3 Results and discussions

###  3.1 Flow structure around ellipsoid

The effects of shear Reynolds number $Re_{s}$ and wall-normal distance along
with orientation angle of the ellipsoid are shown with the roughness and
particle induced disturbed flow field. There are significant contribution from
the particle-wall separation distance and orientation of the particle to
decide the flow characteristics around the ellipsoid. Fig. 8 shows streamlines
across the ellipsoid at the plane of center (i.e. $y^{\prime}$ = 0), at
$Re_{s}=100$ for different value of wall-normal distances
$\delta=0.1,0.5,1.0,1.5$ and orientation angle $\theta=$$,$$,$$,$ and $None$.
The streamline plots show that as the $\delta$ increases, the effect of wall
diminishes rapidly. For $\delta>0.5$, at all orientation angles, only the
effect of ambient shear but not the wall roughness is observed. A similar
observation is followed for all simulations carried out at lower $Re_{s}$’s
(not shown here). Appearance of the wake behind the ellipsoid is mainly
observed for higher values of $Re_{s}$’s. The structure of wake is also found
to vary with separation from wall and the orientation angle. The location of
the wake on either top (8(h)) or bottom (8(p)) part of the ellipsoid depends
on the orientation angle to which it is exposed to the background flow. In the
case of $\theta=$$$, when a particle is away from the wall, the asymmetric
wake is only due to ambient shear of flow.

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

(k)

(l)

(m)

(n)

(o)

(p)

Figure 8: Streamlines across particle at $Re_{s}=100$, $\theta=$$$ (first
row), $None$ (second row), $None$ (third row) and $\theta=$$$ (fourth row) and
columns represents constant $\delta\ (=0.1,0.5,1.0,1.5)$ from left to right

The effect of Reynolds number on the development of wake behind the ellipsoid
is visualised through the streamline plots Fig. 9. The figure shows the
streamlines for $\delta=0.1$, $\theta=$$$, and at $Re_{s}$ 10, 50, 100. For
$\delta=0.1$, the ellipsoid is sitting in the deepest cavity of rough wall.
The flow between ellipsoid and wall is mainly due to channeling through the
cavities on the rough wall. The accelerated flow observed in the gap for
$\delta>0.1$ is absent here, as ellipsoid acts as a barrier to the flow.
Another important parameter is orientation angle which decides the location of
the appearance of the recirculating zones.

(a)

(b)

(c)

Figure 9: Flow streamlines around the ellipsoid for AR = 2, $\theta$ = $None$,
at (a) $Re_{s}=10$ (b) $Re_{s}=50$ (c) $Re_{s}=100$

Fig. 9 shows that at lower $Re_{s}$, the flow is mostly streamlined over the
ellipsoid, and a small recirculating zone is seen in the cavity of the rough
wall. As $Re_{s}$ increases (to 50), two counter-rotating vortices appear near
the wall and flow separation happens at the bottom (near wall) part of the
ellipsoid.Finally, for highest value of $Re_{s}$ (i.e., 100) considered in the
present study, the flow from bottom and top merges in the down stream and
separation happens from the top of the ellipsoid.

(a)

(b)

(c)

Figure 10: Flow streamlines around the ellipsoid for AR = 2, $\theta$ =
$None$, at (a) $Re_{s}=10$ (b) $Re_{s}=50$ (c) $Re_{s}=100$

It is shown in Fig. 10 that at $\theta=$$$ the streamlines are completely
different to those which are observed for $\theta=$$$. At $Re_{s}=10$ (i.e.
lowest value considered in study), a small vortex appear at upstream rather
than downstream as in case of $\theta=$$$. Strength of circulation increases
with increase in $Re_{s}$. Further, the detachment of boundary layer at high
$Re_{s}$ also observed at lower half of ellipsoid, distinctly different from
the case of $None$. This has significance in deciding the lift and torque
acting on particle. As we will see in later sections, the lift and torque are
found to change the direction based on detachment of the boundary layer, which
in turn depends on orientation angle of the ellipsoid with the incident flow.

(a)

(b)

Figure 11: 3D view of flow streamlines around the ellipsoid for AR = 2, Re =
50, $\theta$ = $None$, $\delta=0.1$ from (a) side and (b) top view

In the above, all streamlines are shown in x-z 2D plane. A simultaneous 2-D
view of the streamlines in x-z and x-y are shown for $\delta=0.1d_{p}$ in at
plane of particle center. Figs. 11(a) and 11(b). The figures show the side and
top view of 3D streamlines for $Re_{s}=50$ and $\theta=$$$. As discussed
earlier, the ellipsoid at this location acts as a barrier to flow. The
channeling of flow through the rough cavity can be observed. Due to the
roughness pattern on a surface and weak flow below ellipsoid, the
recirculating vortices appear in deep cavities.

###  3.2 Drag coefficient

(a) $Re_{s}=10$

(b) $Re_{s}=50$

(c) $Re_{s}=100$

Figure 12: Drag coefficient ($C_{d}$) as a function of wall normal distance
$\delta$ at several Reynolds number. A dashed line with same color represent
the corresponding values of $C_{d}$ for smooth wall condition.

In this section, we present the drag coefficient as a function of wall-
particle separation distance ($\delta$) for different orientation angles
($\theta$) and shear Reynolds numbers ($Re_{s}$), ranging between 10 and 100.
In each of the cases, we have compared the results obtained for a rough wall
with that of a smooth wall. Symbols with different shapes show $C_{D}$ values
obtained from the simulations with the rough wall, and dashed lines represent
the $C_{D}$s for a smooth wall. For all Reynolds numbers, drag coefficients
are much higher when the particle is placed near the wall (Fig. 12). There is
a distinct difference in the effect of roughness on $C_{D}$ at different
Reynolds numbers. At low $Re_{s}$, if the particle is placed horizontally near
the wall ($\delta<0.5$), $C_{D}$ for a rough wall is almost two times higher
than that for the smooth wall. With an increase in wall-normal distance, the
differences between two drag coefficients decrease for all the Reynolds
numbers reported here. For $\delta>1.0$, the effect of roughness becomes
significant. Another notable observation at low Reynolds number is the
significantly higher $C_{D}$ values for horizontally aligned ellipsoid
compared to the other orientation angles. This difference eventually vanishes
at elevated shear Reynolds number. It is worth noting that for higher wall-
particle gap ($\delta>1.0$) drag coefficient in case of a vertically aligned
particle is higher ($\approx 50\%$ for $Re_{s}$ = 100) than the horizontal
arrangement, even though the undisturbed velocity interpolated at particle
center is higher in case of the former. At low Reynolds number ($Re_{s}=10$),
the difference between drag coefficients for different angles of orientation
decreases for the separation distance ($\delta$) greater than 0.5. This
observation motivates us to investigate the variation of drag coefficient as a
function of angle of inclination with approaching fluid stream.

(a) $Re_{s}=10$

(b) $Re_{s}=50$

(c) $Re_{s}=100$

Figure 13: Drag coefficient ($C_{d}$) as a function of orientation angle
$\theta$ at several Reynolds number. A dashed line with same color represent
the corresponding values of $C_{d}$ for smooth wall condition.

The second important parametric effect to study is the orientation angle
($\theta$) of the particle with the flow direction. Fig. 13 shows the
variation of drag coefficient with $\theta$ for different wall particle
separation distance distance ($\theta$) and Reynolds number ($Re_{s}$). In the
case of a smooth wall, the variation of $C_{D}$ with $\theta$ at higher
separation distance follows a similar trend reported by Happel and Brenner
[33] for uniform flow over an ellipsoid particle. At low Reynolds number
$Re_{s}=10$, there is an insignificant variation of $C_{D}$ with angle of
inclination, but with increase in Reynolds number, $C_{D}$ is maximum when
$\theta=$$$. The effect of roughness becomes prominent when wall-particle
separation ($\delta$) is less than 0.5. At low shear Reynolds number
($Re_{s}=10$), for the minimum separation distance reported here, drag
coefficient is minimum when the particle is vertically aligned. For the
horizontal alignment, the drag coefficient is almost 1.5 times higher than the
vertical alignment. As shear Reynolds number increases, the above mentioned
difference decreases. One important point to note here is that the undisturbed
fluid velocity at the center of mass of the particle is higher in case of a
vertically aligned particle than that of a horizontally aligned one. To
address this ambiguity, we have reported the variation of drag force as well
besides drag coefficient, which is shown in Fig. 14.

(a)

(b)

Figure 14: Drag force $F_{D}$ and drag Coefficient $C_{D}$ as a function of
$\theta$ for $Re_{s}$ = 10 at $\delta$ = 0.1

13(a) shows that, away from the wall, there is a significant increase in the
drag coefficient at $\theta=$$$. Similar to it, there is an increase in the
undisturbed velocity, as mentioned earlier. To understand the contribution of
viscous and pressure drag, we have computed these parameters separately and
presented them in Fig. 15. We have restricted this exercise only for
$\delta=0.1$ and for two Reynolds numbers $Re_{s}=10$ and $100$. It is
observed that at low Reynolds number, a viscous component of drag always
dominates over the pressure drag for both the smooth and rough walls. Effect
of wall roughness plays a more prominent role in case of the viscous drag for
horizontally aligned particle. At high shear Reynolds number ($Re_{s}=100$),
the pressure drag coefficient is almost 1.5 times higher compared to the
viscous counterpart for vertically aligned particle. This could be related to
the formation of a wake in the rear end.

(a) $Re_{s}=10$

(b) $Re_{s}=100$

Figure 15: Components of drop coefficient ($C_{d,p}$, $C_{d,v}$) as a function
of orientation angle $\theta$ and, $\delta=0.1$. A dashed line with same color
represent the corresponding values of variable for smooth wall condition.

One of the objectives of the present work is to express drag coefficient as a
function of particle Reynolds number ($Re_{p}$), wall-normal distance, and
angle of inclination of the major axis with the flow direction. It is to be
noted that in the present work, main emphasis is given to the effect of wall
roughness on the hydrodynamic force coefficient. Here, we have computed all
the coefficients using only one aspect ratio. To include the effect of aspect
ratio, we have used the shape factor $k$ based upon theoretical formulation
given in [34] and relevant modifications suggested by Ouchene et al. [26],
which is applicable in the present scenario. It is observed from the
simulations that the drag coefficients evolve as a function of $sin(\theta)$
between angle $\theta=$$$ and $None$ irrespective of the particle Reynolds
numbers. Effect of particle Reynolds number is included in the drag
coefficient of $\theta=$$$ and $None$. Effect of aspect ratio has also been
included in the expression for drag coefficients for $\theta=$$$ and $None$
[35, 26]. Separation distance from the wall ($\delta$), one of the important
parameters of major focus in the present study, is found to contribute non-
linearly to the drag coefficient. A similar observation for the drag on a
spherical particle has been reported by Zeng et al. [10] in an earlier study.
With this described approach, the drag coefficient can be expressed by the
following correlations.

$C_{D}=C_{D,0}+(C_{D,90}-C_{D,0})sin^{2}\theta$ (9)

here,

$C_{D,0}=\big{(}\frac{24}{Re_{p}}\big{)}\big{(}k_{0}\alpha_{0}+\beta_{0}\omega^{-0.8}Re_{p}^{0.687}+\gamma_{0}(\omega-1)^{0.63}Re_{p}^{0.41}\big{)}$
$\alpha_{0}=1+0.001576exp(2\delta)+\frac{3.895}{1.611(1+2\delta)}$
$\beta_{0}=0.15-0.03(1+2.03\delta)exp(-0.392\delta)$
$\gamma_{0}=0.1497-0.01942(1+2.77\delta)exp(-0.2921\delta)$

and,

$C_{D,90}=\big{(}\frac{24}{Re_{p}}\big{)}\big{(}k_{90}\alpha_{90}+\beta_{90}\omega^{-0.54}Re_{p}^{0.687}+\gamma_{90}\omega^{1.043}(\omega-1)^{-0.17}Re_{p}^{0.41}\big{)}$
$\alpha_{90}=1+1.288exp(0.05953\delta^{2.804})+\frac{6.137}{6.479(1+0.05953\delta)}$
$\beta_{90}=0.15-0.3824(1+2.599\delta)exp(-0.344\delta)$
$\gamma_{90}=0.2516-0.247(1-0.11\delta)exp(\delta^{0.371})$

Here, the shape factor $k$ is presented by Ouchene et al. [26] as ,

$k=k_{0}=k_{90}=\frac{8}{3}\omega^{-1/3}\big{[}\frac{2\omega}{(\omega^{2}-1)}+\frac{2\omega^{2}-1}{(\omega^{2}-1)^{(3/2)}}ln\frac{\omega+\sqrt{\omega^{2}-1}}{\omega-\sqrt{\omega^{2}-1}}\big{]}^{-1}$
(10)

Above equation boils down to the correlation for the sphere reported by
Schiller and Naumann [36] for $\omega=1$ and $\delta>>1$. For $\omega=1$ and
in the limit of creeping flow and in the presence of a wall, the equation
follows the correlation of Goldman et al. [37] for all $\delta$, which was
further simplified by Zeng et al. [10]. Accuracy of the present non-trivial
correlation is evaluated by comparing it with the simulation results for a
wide range of parameters. Fig. 16 shows comparison of drag coefficients
obtained using correlation with that obtained from the simulations at two
$Re_{S}=10$ and $100$ and for different orientation angles $\theta$. In the
figure, the drag coefficients have been reported as a function of separation
distance from the wall ($\delta$). It is observed from the figures that the
prediction of the drag coefficients from the correlation matches well even for
a very low separation distance $\delta<0.5$ for both the shear-Reynolds
numbers. The maximum deviation of about $8\%$ between simulation and
prediction by the correlation is observed at around $\delta\approx 1.0$ and
$\theta=$$$ for all $Re_{s}$ considered in study. Whereas, for other
separation distances, the deviation does not exceed 3%. A similar trend is
also observed for other orientation angles ($\Theta$), considered in this
study. At higher $\delta$ values, this deviation is reduced to 1%. The overall
mean deviation for all simulated cases lies within 3.5%. Similarly, in Fig.
17, comparison of correlation prediction with simulation results is shown for
$C_{D}$ as function of $\theta$ at different $\delta$. The figure shows that
the correlation is fairly capable of producing accurate results from
$\delta=0.1$, where ellipsoid is placed very close to the wall to several
particle diameters away from the wall.

(a) $Re_{s}=10$

(b) $Re_{s}=100$

Figure 16: $C_{D}$ Vs $\delta$ for $Re_{s}$ = 10 and 100 at different
orientation angle ($\theta$). The dashed line in corresponding color
represents the fitted value from the correlation

(a) $Re_{s}=10$

(b) $Re_{s}=100$

Figure 17: $C_{D}$ Vs $\theta$ for $Re_{s}$ = 10 and 100 at different wall
separation distance ($\delta$). The dashed line in corresponding color
represents the fitted value from the correlation

A goodness-of-fit between simulation results and correlation prediction is
shown Fig. 18 for all shear Reynolds numbers, separation distances from the
rough wall $\delta$ (0.1, 0.5, 1.0, and 2.0) and orientation angles $\theta$
($$$,$$,$$,$$$). The figure shows that all the data collapse along the
diagonal line with a maximum error of $8\%$; confirms the applicability of the
above correlation for calculating drag on a ellipsoidal particle near a rough
wall.

Figure 18: Predicted $C_{D}$ plotted against $C_{D}$ from simulation data at
various $\theta$, and ($-\cdot-\cdot$) represent line of $y=x$

###  3.3 Lift coefficient

In this section, we report the lift coefficients for different particle-wall
separation distances at three different shear-Reynolds numbers ($Re_{s}$).
Fig. 19, shows the variation of lift coefficient ($C_{L}$) with $\delta$ at
different inclination angles ($None$ to $None$). In addition to the results
for rough wall, we have also presented the lift coefficients for a smooth
wall. At lower shear-Reynolds number ($Re_{s}=10$), the wall effects for both
types of wall are very prominent below $\delta=1.5$. At higher separation, the
lift coefficient becomes almost independent of the separation distance. It is
observed from the 19(a) that for all angles, the lift force act away from the
wall but for $\theta=$$$. At this angle, a particle is effectively pushed
towards the wall. Effect of rough wall is significant for a separation
distance up to $\delta=1.0$ at low shear-Reynolds number and up to
$\delta=0.5$ for higher shear Reynolds number as shown in Figs. 19(b) and
19(c). For a horizontally aligned ellipsoid very near the wall, lift induced
by a rough wall is almost 1.5 to 2 times higher than that exerted by a smooth
wall across all the Reynolds numbers.

(a) $Re_{s}=10$

(b) $Re_{s}=50$

(c) $Re_{s}=100$

Figure 19: Lift coefficient ($C_{l}$) as a function of wall normal distance
$\delta$ at several Reynolds number. A dashed line with same color represent
the corresponding values of $C_{l}$ for smooth wall condition.

The variation of lift coefficients with inclination angle ($\theta$) at
different shear Reynolds numbers is presented in Fig. 20. Here, the angle is
measured between the wall (or the direction of mean flow) and the major axis
of the ellipsoid in a counter clock-wise direction. The lift coefficients show
asymmetric nature at intermediate angles. It is shown in Fig. 20 that the
roughness plays a crucial role in deciding the asymmetry in lift coefficient
with change in angle at least when the particle is very close to the wall. It
is also clear from the figures that with angle up to $\theta=$$$, the
difference in lift coefficients between the smooth and rough wall is less than
that for angles higher than $\theta=$$$. Another important observation is that
the particle encounters a lift force towards the wall for an angle between
$None$ and $None$, which may be due to the creation of a low-pressure zone at
the bottom rear end of the ellipsoid. An analysis of the contribution of
viscous and pressure components towards the lift force provides more insight
into that, which is discussed below.

(a) $Re_{s}=10$

(b) $Re_{s}=50$

(c) $Re_{s}=100$

Figure 20: Lift coefficient ($C_{l}$) as a function of orientation angle
$\theta$ at several Reynolds number. A dashed line with same color represent
the corresponding values of $C_{l}$ for smooth wall condition.

The contribution from the pressure variation and viscous effect towards the
total lift is depicted in Fig. 21. The figures show the variation of $C_{L,v}$
and $C_{L,p}$ Vs $\theta$ at $\delta=0.1$ for $Re_{s}=10$ and $100$. We have
considered two different shear Reynolds numbers, $Re_{s}=10$ and $100$, for
this analysis. Results are shown for the case when a particle is placed very
near the wall ($\delta=0.1$). It is observed from the 21(a) that pressure
contribution towards the total lift is much higher compared to its viscous
counterpart even at low shear Reynolds number. Pressure lift works towards the
wall up to an angle of $None$, but at a higher angle, net pressure force is
exerted in the opposite direction. Maximum difference between the pressure to
viscous contribution happens at an angle $\approx$$$; the ratio is about 6. A
comparison with smooth wall at the lower shear Reynolds number (21(a)) depicts
that in the presence of a rough wall, both the viscous and pressure lift
increases dramatically. The overall lift force acts away from the wall when a
particle is very near the wall with an inclination angle of $>$$$. Such an
observation has huge implications in the area of research that addresses the
wall-deposition behaviour of nonspherical particles. It is shown in the figure
that the pressure contribution to the lift for a smooth wall is much lower
compared to the rough wall. The difference between those decreases at higher
Reynolds number (Figs. 21(a) and 21(b)). Another important outcome of the
present analysis is that in the presence of roughness, there is a significant
increase in viscous lift away from the wall when the particle is aligned
horizontally. Accepting the point that the scenario may change for the
particle moves parallel to the wall, the present results can be a motivation
for the wall modification to avoid particle deposition.

(a) $Re_{s}=10$

(b) $Re_{s}=100$

Figure 21: components of lift coefficient ($C_{l,p}$, $C_{l,v}$) as a function
of orientation angle $\theta$ and, $\delta=0.1$. A dashed line with same color
represent the corresponding values of variable for smooth wall condition.

Turning our attention to develop the correlation for lift coefficient, we
consider that the lift coefficient decreases exponentially with wall particle
separation distance $\delta$, magnitude of lift coefficient is a function of
shear Reynolds number $Re_{s}$, which is related to particle Reynolds number
$Re_{p}$ as a function of $\delta$, $\theta$, and $\omega$ (aspect ratio of
the ellipsoid). The effect of variation of angle is introduced through a
skewed function of $\sin\ \theta\cos\ \theta$.

Using all the above criteria, we reach to the functional form of the
correlation as,

$C_{L}=C_{L,s}\big{\\{}\boldsymbol{f}(\delta,Re_{s})+\boldsymbol{g}(\delta,Re_{s})\sin[\boldsymbol{h}(\delta,Re_{s})\theta]\cdot\cos[\boldsymbol{h}(\delta,Re_{s})\theta]\big{\\}}$
(11)

Where, $C_{L,s}$ represents the lift for ellipsoidal particle resting on wall
as reported earlier by Fillingham et al. [29].

$C_{L,s}=\frac{3.663}{(\omega^{2.021}Re_{p}^{2}+0.1173\omega^{1.559})^{0.22}}$
(12)

$\boldsymbol{f}(\delta,Re_{s})$ takes care of the variation of lift
coefficient including the change on direction of the lift.

$\boldsymbol{f}(\delta,Re_{s})=-0.043-0.00083Re_{s}+\frac{(0.93+0.0023Re_{s})}{e^{(0.98\delta+0.016Re_{s}\delta)}}$
(13)

$\boldsymbol{g}(\delta,Re_{s})$ is expressed as follows.

$\boldsymbol{g}(\delta,Re_{s})=\boldsymbol{g^{\prime}}(\delta)+\boldsymbol{g^{\prime\prime}}(\delta,Re_{s})$
(14)

Here, the first part is an exclusive function of $\delta$, while the second
part contains effect of variation of both the $\delta$ and $Re_{s}$.

$\boldsymbol{g^{\prime}}(\delta)=1.15e^{(-4.27\delta)}+0.8e^{(0.17\delta)}$
(15)
$\boldsymbol{g^{\prime\prime}}(\delta,Re_{s})=-\Big{[}0.285\delta+(0.001056Re_{s}\delta-3.2)Re_{s}\Big{]}\cdot
e^{-(6.91-\delta^{(0.16-0.0065Re_{s})})}$ (16)

$\boldsymbol{h}(\delta,Re_{s})$ contributes to capture the skewed variation of
$C_{L}$ with $\theta$.

$\boldsymbol{h}(\delta,Re_{s})=0.018Re_{s}+0.96$ (17)

where $Re_{s}$ is related to $Re_{p}$ as,

$Re_{s}=\frac{Re_{p}}{(0.3165+\delta)+\frac{1}{2\sqrt{\omega}}+\frac{\sqrt{\omega}}{2}(1-\frac{1}{\omega})\sin^{2}\theta}$
(18)

Performance of the developed correlation in predicting the lift coefficient is
assessed for different separation distance, angle of inclination, and for
different Reynolds numbers. Fig. 22 shows the comparison between lift
coefficient calculated using simulations and that predicted from the
correlation as function of $\delta$ at $Re_{s}=10$ (22(a)) and $Re_{s}=100$
(22(b)). It is observed that the predictions from the correlation match well
with the simulations results in majority of the cases. At $Re_{s}=10$, when
the particle is placed very near the wall at an $None$, a correlation over-
predicts the lift coefficient. In case of $Re_{s}=100$, at angle $None$,
correlation over-predicts the lift coefficients for all the separation
distances (22(b)). However, the expression can successfully reports the trends
of lift variation with wall separation distance at all shear Reynolds numbers.

(a) $Re_{s}=10$

(b) $Re_{s}=100$

Figure 22: $C_{L}$ Vs $\delta$ for $Re_{s}$ = 10 and 100 at different
orientation angle ($\theta$). The dashed line in corresponding color
represents the fitted value from the correlation

Lift coefficients calculated from simulation and predicted by correlation are
plotted as function of $\theta$ at several wall separation distance $\delta$,
as shown in Fig. 23. A good agreement is observed for both the shear-Reynolds
numbers. Most importantly, the asymmetry that occurs at a lower separation
distance is well captured by the correlation. It is observed that, at high
shear-Reynolds number ($Re_{s}=100$) when the wall separation distance
($\delta$) is $0.5$, correlation under-predicts the lift coefficients. To
obtain a comprehensive assessment of the performance of correlation, developed
in the present study, we have plotted the lift coefficient predicted by
correlation with that obtained from simulations for $\delta$ = 0.1, 0.5, 1.0,
and 2.0, and $\theta$ = $$$,$$,$$$, and $None$ in Fig. 24. Even though
majority of the data collapses along a diagonal line in Fig. 24, there is
significant deviation when the lift coefficient is less than $0.1$. At a lower
shear Reynolds number, the proposed correlation can predict the simulated
results fairly well with a maximum deviation of $20\%$ for all the orientation
angles but for $\theta=$$$. The error is maximum at those points where there
is a change is direction of the lift forces. At a higher shear Reynolds number
the lift becomes comparatively small and changes its direction as the particle
moves away from the wall at a particular orientation angle. The region (wall
separation distance) where lift is significantly small ($O(10^{-2}$)) and
changes sign, the proposed equation gives a large error as shown in the inset
of Fig. 24.

(a) $Re_{s}=10$

(b) $Re_{s}=100$

Figure 23: $C_{L}$ Vs $\theta$ for $Re_{s}$ = 10 and 100 at different wall
separation distance ($\delta$). The dashed line in corresponding color
represents the fitted value from the correlation Figure 24: Predicted $C_{L}$
plotted against $C_{D}$ from simulation data at various $\theta$, and
($-\cdot-\cdot$) represent line of $y=x$

###  3.4 Torque coefficient

For a nonspherical particle immersed in a shear flow, besides calculating drag
and lift, it is also essential to get an estimate of the torque. This section
analyses the torque coefficient around its center of mass point for three
different shear Reynolds numbers, different angles of orientation, and wall
separation distances. We compare the results for both the rough and smooth
walls. Fig. 25 shows a variation of torque coefficient $C_{M}$ with wall-
normal distance $\delta$ for four different orientation angles ($\theta$) and
at different Reynolds numbers. We use the right-hand coordinate system to
express the direction of torque. Legends used in the figures have same
significances that have been stated earlier. It is observed in the figure that
$C_{M}$ decreases exponentially as ellipsoid moves away from the wall, a
similar trend for the drag and lift coefficients.

At the lower shear Reynolds number, reported here ($Re_{s}=10$) torque
coefficient ($C_{M}$) is positive even for horizontally aligned particle
placed near the wall. For an angle of inclination $None$, change in torque
coefficient is observed up to wall separation distance, $\delta\approx 1$.
With an increase in Reynolds number, the distance of influence decreases
gradually. In case of $Re_{s}=50$ (25(b)), the $\delta$ after which $C_{M}$
does not show any variation is $\delta\approx 0.7$, whereas for $Re_{s}=100$
that is $\delta<0.5$ (25(c)). A similar trend is observed for other angles as
well. The figures illustrate that there is a three to five-fold decrease in
torque coefficient for a ten times increase in Reynolds number for minimum
wall separation distance.

(a) $Re_{s}=10$

(b) $Re_{s}=50$

(c) $Re_{s}=100$

Figure 25: Torque coefficient ($C_{m}$) as a function of wall normal distance
$\delta$ at several Reynolds number. A dashed line with same color represent
the corresponding values of $C_{m}$ for smooth wall condition.

Fig. 26 show the variation of torque coefficient for different inclination
angles and at three different shear Reynolds numbers. It is to be noted that
the angle is changed in the plane of shear. It is shown in the 26(a) that the
rough wall has very prominent effect for the lowest wall-separation distance
($\delta=0.1$). With an increase in the distance from the wall, influence of
the nature of the wall becomes insignificant. Extent of asymmetry in $C_{M}$
increases with increase in Shear Reynolds number (Fig. 26). At shear-Reynolds
numbers, $Re_{s}=$ 50 and 100, if the particle-wall separation distance is
$\geq 0.5$, torque on the particle acts in the negative direction when the
inclination angle is between $>$$$ and $None$ (Figs. 26(b) and 26(c)).

(a) $Re_{s}=10$

(b) $Re_{s}=50$

(c) $Re_{s}=100$

Figure 26: Torque coefficient ($C_{m}$) as a function of orientation angle
$\theta$ at several Reynolds number. A dashed line with same colour represent
the corresponding values of $C_{m}$ for smooth wall condition.

##  4 Conclusion

Particle-laden flows with anisotropic particles find applications in
industrial and natural processes. The interactions between anisotropic
particles with fluid flow are complex in nature because the orientation of
particles changes the fluid flow behaviour. Presence of an uneven wall in the
vicinity of the particle adds additional complexity. Earlier efforts were
reported to estimate the forces in the presence of rough wall but for
spherical particles only. We have performed numerical simulations (with high
spatial resolution) to estimate the forces on an ellipsoidal particle placed
in the vicinity of a rough wall at different wall separation distances and
inclination angles. To create the wall roughness, we have followed the
approach already demonstrated by Lee and Balachandar [8]. Simulations have
been performed for a range of Reynolds numbers $Re_{s}$ ($10\leq Re_{s}\leq
100$), $\delta$ ($0.1\leq\delta\leq 2$), and $\theta$ ($$$\leq\theta\leq$$$).
Undisturbed (by the ellipsoidal) fluid velocity at particle center of mass
location is obtained using an empty channel simulation with wall roughness
only. This velocity is used to calculate the drag, lift, and torque
coefficients. Simulations with a smooth wall for the similar flow conditions
have also been performed and compared to quantify the effect of wall
roughness.

Streamline plots over the ellipsoid qualitatively depicts the appearance of
the flow separation and recirculating zone at the top edge for $\theta<90$ and
at the bottom edge for $\theta>90$. Such a flow separation plays an essential
role in deciding the direction of the lift force on the particle. If the
particle is placed near the wall at low shear Reynolds number, $C_{d}$ is
almost two times higher than the smooth wall. The effect of a wall-roughness
becomes insignificant away from the wall ($\delta>1.0$). The drag coefficient
for a vertically placed particle at a higher Reynolds number ($Re_{s}\approx
100$) is larger than the horizontally placed particle, even though a higher
undisturbed velocity occurs at the particle location. At low Re, when the
particle is placed very near the wall, a viscous component of the drag
dominates over the pressure component for both types of wall. If the particle
is vertically aligned at a higher Reynolds number, the pressure drag is almost
1.5 times higher than the viscous counterpart. In case of lift coefficient,
the effect of nature of wall is prominent below separation distance,
$\delta=1.5$. In all the cases, lift acts away from the wall except the angle
between $None$ to $None$. Effect of the rough wall is prominent up to a
separation distance ($\delta$) of 1.0. Irrespective of the nature of the wall,
an asymmetry is observed in the variation of lift coefficient as a function of
an angle of the particle’s major axis with the wall. At low wall separation
distance, the asymmetry is augmented by the wall roughness.

From the analysis of forces on the ellipsoidal particles, we develop and
present the semi-empirical correlations for the drag and lift coefficients for
different wall separation distances (including roughness) and different
inclination angles. The correlations are applicable for particle Reynolds
numbers up to 250. The accuracies of these correlations are verified by
comparing predictions by the correlations with the simulation results for a
wide range of parameters. Maximum relative deviation for drag coefficient is
found to be 8%. However, correlation of the lift suffers from the accuracy at
lower values of lift coefficients when there is a change in direction of the
lift.
One of the important limitations of the present study is that the simulations
have been performed for a single aspect ratio. Therefore, the developed
correlations are to be justified through further studies using ellipsoid with
other aspect ratios. The second important limitation is that the effect of
roughness has been demonstrated only for a single roughness pattern. A
detailed study to establish the effect of different roughness patterns will be
an interesting future scope.

## Acknowledgement

This work is supported by IRCC IIT Bombay (No: RD/0512-IRCCSH0-017).

## References

## References

  * Kruggel-Emden and Vollmari [2016] Kruggel-Emden H, Vollmari K.  Flow-regime transitions in fluidized beds of non-spherical particles.  _Particuology_ 2016;29:1–15.  doi:https://doi.org/10.1016/j.partic.2016.01.004.
  * Mills [2003] Mills D.  Pneumatic conveying design guide.  Elsevier; 2003.
  * Kleinstreuer and Feng [2013] Kleinstreuer C, Feng Y.  Computational analysis of non-spherical particle transport and deposition in shear flow with application to lung aerosol dynamics—a review.  _Journal of Biomechanical Engineering_ 2013;135(2).  doi:10.1115/1.4023236.
  * Tadros [2011] Tadros TF.  Rheology of dispersions: principles and applications.  John Wiley & Sons; 2011.
  * Ogawa et al. [1990] Ogawa K, Yoshikawa S, Suguro A, Ikeda J, Ogawa H.  Flow characteristics and circular pipe flow of pulp-suspension.  _Journal of Chemical Engineering of Japan_ 1990;23(1):1–6.  doi:10.1252/jcej.23.1.
  * Lundell et al. [2011] Lundell F, Söderberg L, Alfredsson P.  Fluid mechanics of papermaking.  _Annual Review of Fluid Mechanics_ 2011;43:195–217.  doi:10.1146/annurev-fluid-122109-160700.
  * Cui and Grace [2007] Cui H, Grace JR.  Flow of pulp fibre suspension and slurries: A review.  _International Journal of Multiphase Flow_ 2007;33(9):921–34.  doi:https://doi.org/10.1016/j.ijmultiphaseflow.2007.03.004.
  * Lee and Balachandar [2017] Lee H, Balachandar S.  Effects of wall roughness on drag and lift forces of a particle at finite reynolds number.  _International Journal of Multiphase Flow_ 2017;88:116–32.  doi:10.1016/j.ijmultiphaseflow.2016.09.006.
  * Zeng et al. [2005] Zeng L, Balachandar S, Fischer P.  Wall-induced forces on a rigid sphere at finite reynolds number.  _Journal of Fluid Mechanics_ 2005;536:1–25.  doi:10.1017/S0022112005004738.
  * Zeng et al. [2009] Zeng L, Najjar F, Balachandar S, Fischer P.  Forces on finite-sized particle located close to a wall in a linear shear flow.  _Physics of Fluids_ 2009;21:033302.  doi:10.1063/1.3082232.
  * Jeffery and Filon [1922] Jeffery GB, Filon LNG.  The motion of ellipsoidal particles immersed in a viscous fluid.  _Proceedings of the Royal Society of London Series A, Containing Papers of a Mathematical and Physical Character_ 1922;102(715):161–79.  doi:10.1098/rspa.1922.0078.
  * Brenner [1963] Brenner H.  The stokes resistance of an arbitrary particle.  _Chemical Engineering Science_ 1963;18(1):1–25.  doi:10.1016/0009-2509(63)80001-9.
  * Brenner [1964a] Brenner H.  The stokes resistance of an arbitrary particle—ii: An extension.  _Chemical Engineering Science_ 1964a;19(9):599–629.  doi:https://doi.org/10.1016/0009-2509(64)85051-X.
  * Brenner [1964b] Brenner H.  The stokes resistance of an arbitrary particle—iii: Shear fields.  _Chemical Engineering Science_ 1964b;19(9):631–51.  doi:https://doi.org/10.1016/0009-2509(64)85052-1.
  * Harper and Chang [1968] Harper EY, Chang ID.  Maximum dissipation resulting from lift in a slow viscous shear flow.  _Journal of Fluid Mechanics_ 1968;33(2):209–25.  doi:10.1017/S0022112068001254.
  * Lin et al. [2003] Lin J, Shi X, Yu Z.  The motion of fibers in an evolving mixing layer.  _International Journal of Multiphase Flow_ 2003;29(8):1355–72.  doi:https://doi.org/10.1016/S0301-9322(03)00086-7.
  * Gavze and Shapiro [1997] Gavze E, Shapiro M.  Particles in a shear flow near a solid wall: Effect of nonsphericity on forces and velocities.  _International Journal of Multiphase Flow_ 1997;23(1):155–82.  doi:https://doi.org/10.1016/S0301-9322(96)00054-7.
  * Pozrikidis [2005] Pozrikidis C.  Orbiting motion of a freely suspended spheroid near a plane wall.  _Journal of Fluid Mechanics_ 2005;541:105–14.  doi:10.1017/S0022112005006117.
  * Lee and Hyun [2015] Lee SY, Hyun JY.  Analysis of forces acting on the non-spherical particle near a wall.  2015\.  doi:https://doi.org/10.1007/s13534-015-0205-z.
  * Kim et al. [2016] Kim DK, Hyun JY, Kim SC, Kim HS, Lee SY.  Inertial effects on cylindrical particle migration in linear shear flow near a wall.  _Microfluidics and Nanofluidics_ 2016;20(5):75.  doi:https://doi.org/10.1007/s10404-016-1742-1.
  * Smith [2017] Smith FT.  Free motion of a body in a boundary layer or channel flow.  _Journal of Fluid Mechanics_ 2017;813:279–300.  doi:10.1017/jfm.2016.706.
  * Zarghami and Padding [2018] Zarghami A, Padding J.  Drag, lift and torque acting on a two-dimensional non-spherical particle near a wall.  _Advanced Powder Technology_ 2018;29:1507–17.  doi:10.1016/j.apt.2018.03.019.
  * Palmer and Smith [2020] Palmer RA, Smith FT.  When a small thin two-dimensional body enters a viscous wall layer.  _European Journal of Applied Mathematics_ 2020;31(6):1002–28.  doi:10.1017/S0956792519000378.
  * Holzer and Sommerfeld [2008] Holzer A, Sommerfeld M.  New simple correlation formula for the drag coefficient of non-spherical particles.  _Powder Technology_ 2008;184(3):361–5.  doi:https://doi.org/10.1016/j.powtec.2007.08.021.
  * Zastawny et al. [2012] Zastawny M, Mallouppas G, Zhao F, Wachem BV.  Derivation of drag and lift forces and torque coefficients for non-spherical particles in flows.  _International Journal of Multiphase Flow_ 2012;39:227–39.  doi:10.1016/j.ijmultiphaseflow.2011.09.004.
  * Ouchene et al. [2016] Ouchene R, Khalij M, Arcen B, Tanière A.  A new set of correlations of drag, lift and torque coefficients for non-spherical particles and large reynolds numbers.  _Powder Technology_ 2016;303:33 – 43.  doi:10.1016/j.powtec.2016.07.067.
  * Sanjeevi et al. [2018] Sanjeevi SK, Kuipers J, Padding JT.  Drag, lift and torque correlations for non-spherical particles from stokes limit to high reynolds numbers.  _International Journal of Multiphase Flow_ 2018;106:325–37.  doi:10.1016/j.ijmultiphaseflow.2018.05.011.
  * Frohlich et al. [2020] Frohlich K, Meinke M, Schröder W.  Correlations for inclined prolates based on highly resolved simulations.  _Journal of Fluid Mechanics_ 2020;901:A5.  doi:10.1017/jfm.2020.482.
  * Fillingham et al. [2021] Fillingham P, Vaddi RS, Bruning A, Israel G, Novosselov IV.  Drag, lift, and torque on a prolate spheroid resting on a smooth surface in a linear shear flow.  _Powder Technology_ 2021;377:958–65.  doi:https://doi.org/10.1016/j.powtec.2020.09.042.
  * Andersson and Jiang [2019] Andersson HI, Jiang F.  Forces and torques on a prolate spheroid: low-reynolds-number and attack angle effects.  _Acta Mechanica_ 2019;230(2).  doi:10.1007/s00707-018-2325-x.
  * Ouchene et al. [2015] Ouchene R, Khalij M, Taniere A, Arcen B.  Drag, lift and torque coefficients for ellipsoidal particles: from low to moderate particle reynolds numbers.  _Computers and Fluids_ 2015;113:53–64.  doi:10.1016/j.compfluid.2014.12.005.
  * Sanjeevi and Padding [2017] Sanjeevi SKP, Padding JT.  On the orientational dependence of drag experienced by spheroids.  _Journal of Fluid Mechanics_ 2017;820:R1.  doi:10.1017/jfm.2017.239.
  * Happel and Brenner [1983] Happel J, Brenner H.  Low Reynolds Number Hydrodynamics with Special Applications to Particulate Media.  Springer Netherlands (Martinus Nijhoff Publishers, The Hague); 1983.  doi:https://doi.org/10.1007/978-94-009-8352-6.
  * Lamb [1993] Lamb H.  Hydrodynamics.  Cambridge university press; 1993\.
  * Loth [2008] Loth E.  Drag of non-spherical solid particles of regular and irregular shape.  _Powder Technology_ 2008;182(3):342 –53.  doi:https://doi.org/10.1016/j.powtec.2007.06.001.
  * Schiller and Naumann [1933] Schiller L, Naumann A.  Fundamental calculations in gravitational processing.  _Zeitschrift Des Vereines Deutscher Ingenieure_ 1933;77:318–20.
  * Goldman et al. [1967] Goldman A, Cox R, Brenner H.  Slow viscous motion of a sphere parallel to a plane wall—ii couette flow.  _Chemical Engineering Science_ 1967;22(4):653–60.  doi:https://doi.org/10.1016/0009-2509(67)80048-4.