Smoothed Profile Method: Error Analysis, Verification and

Application in Dielectric Problems

by

Xiaohe Liu

B. S, Fudan University, China, 2015

Thesis

Submitted in partial fulfillment of the requirements for the

Degree of Master of Science in the School of Engineering at Brown University

Providence, Rhode Island

May 2017

This dissertation by Xiaohe Liu is accepted in its present form

by the School of Engineering as satisfying the

thesis requirements for the degree of Master of Science

Date

Martin Maxey, Ph.D, Advisor

Approved by the Graduate Council

Date

Andrew Campbell, Dean of the Graduate School

ii

Abstract of “Smoothed Profile Method: Error Analysis, Verification and

Application in Dielectric Problems”

by Xiaohe Liu, M. S., Brown University, May 2017

We study the smoothed profile method (SPM) first proposed by Nakayama &

Yamamoto (2005) and extended by Luo et al. (2009), which is a numerical method

for particle-laden flows. SPM represents each particle with a smoothed indicator

function, and solves the Navier-Stokes equations for incompressible flow on the whole

domain on a fixed computational mesh, while applying a local momentum impulse at

each time step of the iterative solver to adjust the velocity field in the particle domain

so that it matches a target velocity field for rigid-body motion.

First we use SPM as a direct forcing method, and quantify the accuracy of SPM for

several prototype flows including steady Couette flow and unsteady oscillating Stokes

layer flow. Further verification of SPM is presented by simulations of a spherical

particle settling in a channel at finite Reynolds number. We found that the modeling

error of SPM depends on the ratio of√ν∆t/ξ, and the most accurate results occur

at√ν∆t/ξ equals to 0.75 to 1; the exact optimum ratio depends on the time step-

ping scheme. Comparisons with the analytic solutions showed that SPM is resolving

accurately the far-field flows and the particle forces, while allowing some error locally

at the particle-fluid interfaces.

Subsequently, we extended SPM to simulate electro-rheology flows allowing for

spatially varying dielectric coefficient. We solve the Poisson equation in electrostatics

for the whole domain and verify the method for prototype problems of dielectric beads

in non-conducting fluid, where the modeling error is quantified. We also proposed

an alternative indicator function based on a fourth order polynomial, which showed

similar effect in simulations as the traditional tanh indicator function, but since it

has a more definite start point and end point of the smoothed layer, it can provide

better control of the smoothed thickness.

Acknowledgements

I would first like to thank my thesis advisor Professor Martin Maxey of the Depart-

ment of Applied Mathematics at Brown University. The door to Professor Maxey’s

office was always open whenever I ran into a trouble spot or had a question about

my research or writing. He consistently allowed this paper to be my own work, but

steered me in the right the direction whenever he thought I needed it.

I am grateful to Professor Nathaniel Trask, Professor Kenneth Breuer and Pro-

fessor Jennifer Franck for their continuing guidance in my research and extremely

helpful advice in academic and non-academic related activities.

I would also like to thank, in chronological order, Lu Lu, Xiongfeng Yi, Seungjoon

Lee, Amanda Howard and Xiaoyue Zhao for their enlightening conversations and

email discussions.

Finally, I must express my very profound gratitude to my parents and to my

boyfriend for providing me with unfailing support and continuous encouragement

throughout my study and through the process of researching and writing this thesis.

This accomplishment would not have been possible without them.

Thank you.

Xiaohe Liu

April 30, 2017

iv

Contents

Acknowledgements iv

List of Tables vii

List of Figures viii

Chapter 1. Introduction 1

1.1. Background 1

1.2. Objectives 3

1.3. Outline 4

Chapter 2. Formulation of the Smoothed Profile Method 5

2.1. Smoothed Profile for Particles 5

2.2. Time Stepping Scheme 6

2.3. Approximation of Particle Boundary Conditions 8

Chapter 3. Prototype Flows: Verification and Error Analysis 9

3.1. Planar Couette Flow 9

3.2. Oscillating Stokes Layers 15

3.3. Sphere Settling in Finite Reynolds Flows 18

Chapter 4. Application in Dielectric Problems 23

4.1. Dipole Problem: Sphere in a Uniform External Electric Field 23

4.1.1. SPM simulation 25

4.1.2. Polynomial Smoothed Profile 28

4.1.3. Interpolate 1/eps 29

4.2. Electrostatic Force on a Dielectric Sphere in Non-uniform Field 33

4.2.1. Numerical Simulation and Force Calculation 35

v

4.2.2. Correction Methods by Retracting Boundary 37

4.3. Periodic Dielectric Spheres in a Uniform Electric Field 41

Chapter 5. Conclusion and Future Work 45

Appendix A. Calculating Wall Force Using Fourier Coefficients 47

Bibliography 48

vi

List of Tables

4.1 Error of force evaluated at a large Gauss surface (r = a + 5ξ). a = 1,

ε1/ε2 = 2, using SPM tanh indicator function. 37

4.2 Result of force calculation with and without the correction methods using

polynomial smoothed profile. ε1/ε2 = 2. Gauss surface at r = a+ ξ/2. 40

vii

List of Figures

3.1 Sketch of the planar Couette flow problem. 9

3.2 Couette flow: velocity profile of un, u∗ and the exact solution (locally

enlarged at the interface). ξ = 0.1, dt = 0.005625, and ν = 1. 10

3.3 Couette flow: SPM force density profile. ξ = 0.1, and dt = 0.005625. 11

3.4 Couette flow: L2 error of the velocity profile against ∆t. ξ = 0.05. 12

3.5 SPM error of Couette flow simulation: a. L2 error against√ν∆t/ξ. b.

Wall stress error against√ν∆t/ξ. ξ = 0.05, ν = 1. 12

3.6 Couette flow option 1 v.s. standard SPM: velocity profile (locally enlarged

around the interface). ξ = 0.1. 14

3.7 Couette flow option 1: L2 error of the velocity profile against ξ. 14

3.8 Couette flow option 2: velocity profile (left) and force density profile

(right) at ν1/ν0 = 100 and ξ = 0.1. 15

3.9 Couette flow option 1: L2 error of the velocity profile against ξ (left) and

against ν1/ν0 (right). 16

3.10 Sketch of the SPM simulation geometry of the oscillating Stokes problem. 17

3.11 Stokes layer: velocity profile at t = 3/2T with ∆t = 0.0019 and ξ = 0.05. 17

3.12 Stokes layer: wall stress data from 1/2T to 3/2T . 18

3.13 Stokes layer: relative error of the amplitude of τw (left) and phase shift

comparing to the exact solution (right). 18

3.14 Stream-wise fluid velocity profiles against x2 axis at locations in the

wake region (left), and in the upstream region (right). ξ = 0.1 ≈ 1.5∆x,

∆t = 0.015 20

viii

3.15 Stream-wise fluid velocity along the centerline of the duct agains x1 axis. 21

3.16 Stream-wise fluid velocity contour plot at the plain perpendicular to x2

axis across the center of the sphere. Flow is from left to right. 21

3.17 Wall shear stress (gradient of u1) profile on one side wall. 22

4.1 Sketch of the dielectric sphere problem. 24

4.2 Numerical F (r) and D(r) profile compared to the exact solution. Here

D(r) = εdFdr

. 25

4.3 SPM simulation: relative error of B and effective radius aeff against ξ.

a = 1. 26

4.4 SPM simulation: relative error of B and effective radius aeff against ε1. 27

4.5 SPM simulation: similarity relationship of aeff . 28

4.6 Polynomial indicator function φ(x− 0.1). ξ = 0.1. 29

4.7 Polynomial smoothed profile: relative error of B and effective radius aeff . 30

4.8 Polynomial smoothed profile: similarity relationship of aeff . 31

4.9 Interpolating 1/ε v. s. interpolating ε: ε(r) profile. ξ = 0.1, ε1 = 100. 31

4.10 Error of interpolating 1/ε and interpolating ε for ξ = 0.1 (left) and ξ = 0.2

(right). 32

4.11 Error against r (left axis, solid line) plotted with ε(r) (right axis, dotted

line) when using a polynomial smoothed profile with ξ = 0.2 (left) and

SPM tanh indicator function with ξ = 0.05 (right). ε1 = 2. 36

4.12 Relative error of force against ξ in regular scale (left) and in log scale

where Error is converted to its abstract value (right). ε1 = 2. ξ ranges

from 0.05 to 0.32. Use polynomial smoothed layer; error calculated at

r = a+ ξ/2. 37

4.13 Dipole problem: achieve second order of accuracy by retracting boundary.

Use polynomial smoothed profile; ξ ranges from 0.01 to 0.3, with ε1 set to

2, 20 and 40 respectively. 38

ix

4.14 Error of force against ξ, calculated using correction method with ε1 = 2

in log scale. The order of accuracy is not clear. 39

4.15 Result of ψ at the x3 = 0 plane (left) and at x2 = x3 = 0 (right). L = 4,

a = 1, ξ = 0.05, ε1 = 2, ε2 = 1, E∞ = 1. N = 256, the 5th iteration.

The sphere is centered at the origin and the lattice cell extends from

x1 = −L/2 to x1 = L/2. 42

4.16 Result of the x1 component of ∇ψ (left) and the x2 component of ∇ψ

(right). L = 4, a = 1. 43

4.17 Result of ψ at the x3 = 0 plane (left) and at x2 = x3 = 0 (right). L = 8,

a = 1, ξ = 0.05, ε1 = 2, ε2 = 1, E∞ = 1. N = 256, the 5th iteration. 43

4.18 Result of the x1 component of ∇ψ (left) and the x2 component of ∇ψ

(right). L = 8, a = 1. 44

x

CHAPTER 1

Introduction

1.1. Background

Particle-laden flow is very common in natural phenomena and industrial appli-

cations such as suspensions in a viscous fluid and particle transport, and it has be-

come important to understand the rheology and dynamics behind them. Research in

particle-laden flows has developed significantly over the years in the aspects of exper-

iments, numerical simulations and theoretical modeling. Numerical simulations are

powerful in that they can provide details that are not detectable from experiments,

and give insightful understanding of the physical processes. Recent developments in

numerical simulations include results for suspensions of rigid particles from zero to

moderate Reynolds numbers and various particle shapes.

Many different approaches have been proposed to simulate particulate flows. The

choice of a specific approach depends on the type of flow in question, the character-

istics of the particles and their movement, and the balance between computational

complexity and available resources. Some of the most used methods include the full

arbitrary Lagrangian Eulerian (ALE) method [1, 2, 3] which adopts a Lagrangian

description of the particles and a computational mesh that follows the particles, and

several fictitious domain methods that provide an approximately resolved simulation

of particles.

A recent review [4] points out that in fictitious domain methods, a fixed com-

putational mesh is generated for the whole domain, including the particle domain,

and body force density f(x, t) is set to represent the rigid-body-like movement of the

fictitious fluid inside each particle. Navier-Stokes equations for incompressible flow

are solved for the whole domain. In some of the methods, the forces are localized

to the body surface (immersed boundary methods), while others apply the forces

1

throughout the particle volume (immersed body methods); the methods also differ in

how the forces are determined.

The force coupling method (FCM) [5, 6, 7] falls into this category, which repre-

sents each particle in the flow by a low-order expansion of force multipoles and applies

the forces as a distributed body force on the flow. It can capture the far-field solution

accurately with comparably low computation cost, but it does not resolve the flow

field close to the particle surfaces.

Another method worth mentioning is the distributed Lagrange multiplier (DLM)

method developed by Glowinski et al. (1999) [8], which is a type of immersed body

scheme.

Among the fictitious domain schemes, the direct forcing methods aim to apply a

local momentum impulse at each time step of the iterative solver to adjust the fluid

velocity for each particle and ensure that the total velocity field u matches a target

velocity field for rigid-body motion. One of the implementations is the immersed

boundary method of Peskin (2002) [9], which employs a regular Eulerian mesh for the

solvent over the entire domain, together with Lagrangian markers for the immersed

boundary. Breugem (2012) [10] modified and extended the method so that the results

for the particle velocity reaches second-order accuracy while the original method is

only first order accurate owing to local errors near the body surface. Another option

is to implement the direct forcing with an immersed body method, among which is

the smoothed profile method (SPM) proposed by Nakayama and Yamamoto (2005)

[11] and extended by Luo et al. (2009) [12].

In SPM, each particle is associated with a smoothed indicator function φ(α) that

is 1 inside the particle, 0 outside and has a smooth transition over a short distance

ξ across the surface. A combined indicator function is defined as the sum of the

individual functions for non-overlapping particles as φ =∑φ(α).

The advantage of SPM is that it allows one to separate more easily the algorithm

from the flow solver. For example, [11, 12] apply a pseudo-spectral method, and [13]

uses a staggered finite-difference scheme, where some level of forcing is maintained

2

from one time-step to the next. Moreover, SPM provides a tighter control on the

interior fluid motion within the particle. It may be shown that SPM satisfies the

relevant no-slip and no-penetration flow boundary conditions on the particle surface

within the limits of the smoothing length scale ξ.

Luo et al. conducted error analysis for SPM and showed that these depend on

∆t as well as the smoothing scale ξ [12]. Here with direct forcing, 1/∆t is a penalty

or control parameter, and in principle a closer match of the local fluid velocity to

the target rigid body motion should result from smaller time steps. The accuracy of

the flow does not improve monotonically but instead reaches an optimal value when

ξ2 = O(ν∆t) and then diverges. The precise value depends on the time stepping

scheme. At this value of ξ, the shear stress is correctly estimated. The impulses

required to sustain the rigid body response generate a viscous Stokes layer that is no

longer resolved if it is thinner than ξ.

SPM has its limitation as it does not work very well for low Reynolds number

flows. At lower Reynolds number, a longer simulation time is required for viscous

diffusion of vorticity to bring the system to equilibrium and this can be slow with

direct forcing.

We are also interested in electro-rheology fluids, specifically dielectric beads in

non-conducting fluids. In an electric field such beads will form chains, and inter-

particle forces may result in particle self-assembly. SPM has been used to study the

dynamics of particles in electro-kinetic flows [14, 15] and suspensions of paramagnetic

particles in a magnetic field [16]. However, few numerical simulations can be found

to apply SPM in solving dielectric systems.

1.2. Objectives

The main objective of the thesis is to quantify the errors of SPM by simulating

prototype flows and comparing the results with exact solutions. We want to find out

how to use SPM as a flow solver with the direct forcing method, and how SPM works

for high and low Reynolds number flows.

3

A second objective is to develop a new strategy for using SPM to solve Poisson-

type dielectric problems. We aim to analyze the modeling error of SPM in dielectric

problems, and seek ways to improve accuracy. Furthermore, we want to compare the

effect of using different type of indicator functions in this context.

A third objective is to construct a 3D finite volume SPM flow solver using Open-

FOAM, and apply the solver to complex flows with moving boundaries.

1.3. Outline

The thesis is organized as follows.

In Chapter 2, we introduce the formulation of SPM and the time stepping scheme

based on a projection method for incompressible flow.

In Chapter 3, we quantify the accuracy of SPM for several prototype flows includ-

ing steady Couette flow and unsteady oscillating Stokes layer flow. Further verifica-

tion of SPM is presented by simulations of a spherical particle settling in a channel

at finite Reynolds number.

In Chapter 4, we extend the application of SPM to dielectric problems. We

use SPM representation for dielectric beads in non-conducting fluid and solve the a

Poisson equation. We show that the simulated problems agree well with analytical

solutions.

We conclude in Chapter 5 by a summary and some future work.

4

CHAPTER 2

Formulation of the Smoothed Profile Method

2.1. Smoothed Profile for Particles

In the Smoothed Profile Method (SPM), each particle is represented by a smoothed

profile, which is an indicator function. The function equals 1 inside the particle do-

main, equals 0 in the fluid domain, and changes smoothly from 1 to 0 across the

interface. Several analytical forms of the indicator function have been proposed in

[11], and the most commonly used is the following one:

φi(x) =1

2

[1 + tanh

(−di(x, t)

ξi

)], (2.1)

where di(x, t) is the signed distance to the ith particle surface with negative value

inside the particle and positive value outside, and ξi is the smoothed thickness for the

ith particle.

We also propose a fourth polynomial function as an alternative indicator function

in Chapter 4.1.2, and compare the result with that of using the tanh indicator function

(2.1).

In this paper, we mainly focus on two-phase flow with a spherical particle inside

the flow. For this case, we can use the following simplified indicator function:

φ(x, t) =1

2

[1 + tanh

(a− |x−R|

ξ

)], (2.2)

where a is the radius of the particle, R is its position vector, and ξ is the smoothed

thickness.

The particle velocity field is constructed by the indicator function and the rigid

motion of the particle up(x, t). The particle is located at R in the fluid, and has a

translational velocity of V = dRdt

and an angular velocity of Ω. The particle velocity

5

field is then

φ(x, t)up(x, t) = V(t) + Ω(t)× [x−R(t)] φ(x, t) (2.3)

The total velocity field is then constructed based on the particle velocity field up

and the fluid velocity field uf :

u(x, t) = φ(x, t)up(x, t) + (1− φ(x, t))uf (x, t). (2.4)

The total velocity field satisfies that u = up inside the particle domain where φ = 1,

while u = uf in the fluid domain where φ = 0. In the interfacial domain where

0 < φ < 1, u smoothly varies from up to uf .

SPM solves for the total velocity field u in the entire domain D. It solves the

Navier-Stokes equations with an extra force density term added to the momentum

equation:

∂u∂t

+ (u · ∇)u = −1ρ∇p+ ν∇2u + g + fs,

∇ · u = 0,(2.5)

where ρ is the density of fluid, p is the pressure field, ν is the kinematic viscosity of

fluid, g is the net external forces on the fluid, fs represents the body force caused

by the interactions between the particles and the fluid. For a time stepping scheme,

generally we have f = φ (up − u∗) /∆t; explanations will be given in the next section.

2.2. Time Stepping Scheme

In the time stepping for the SPM scheme, we first move the particle to its new

position at t = tn+1, and then apply a fractional time-step to (2.5) for the whole

domain D but without the extra forcing term fs initially. We obtain a provisional

flow us(x, t) through a Backward-Euler step:

us − un

∆t+ un · ∇us = ν ∇2us + g, (2.6)

where g may be an external body force or an imposed external pressure gradient.

Appropriate viscous no-slip boundary conditions are applied on the outer boundary

of D.

6

Then we apply a projection step for the pressure, and generates a provisional

pressure field p∗(x, t) and incompressible flow u∗(x, t):

∇2p∗ = ∇ ·(

us

∆t

), (2.7)

u∗ = us −∆t · ∇p∗. (2.8)

On the outer boundary of D, common boundary condition for the pressure can be

used. For example, ∂p∂n

= 0 is appropriate for higher Reynolds number flow. But in low

Reynolds number flows, Karniadakis et al. [17] suggest that appropriate boundary

conditions will depend on specific time-stepping scheme and viscosity.

In the next stage, fs is applied as a momentum impulse to bring the total velocity

field into alignment with the rigid body motion within the particles so that fs satisfies:

∫ tn+1

tn

fs dt = φ (up − u∗) . (2.9)

This is combined with another projection step for the pressure to ensure that the

final flow un+1(x, tn+1) is incompressible:

∇2q = ∇(φ (up − u∗)

∆t

), (2.10)

un+1 = u∗ −∆t · ∇q + φ (up − u∗) , (2.11)

pn+1 = p∗ + q. (2.12)

From fs, which represents the interaction between the fluid and the particle, it is

possible to evaluate the resultant fluid force and fluid torque on the particle. These

are used in the equation of motion for the particle to give the new particle velocity

and angular velocity. The time-stepping then repeats.

Except for the time-stepping scheme proposed in this thesis, other schemes are

available for SPM, such as the stiffly-stable backward differencing scheme used by

Luo in [12].

7

2.3. Approximation of Particle Boundary Conditions

The smoothed representation of SPM approximates the non-slip and no-penetration

constraints on particle boundaries.

In the limit of the smoothed thickness ξ → 0, the gradient of the indicator function

is perpendicular to the particle surface at the particle interface, i. e. surface normal

n = ∇φ|∇φ| .

The no-slip constraint can be expressed as n×(up−uf ) = 0, or (∇φ)×(up−uf ) =

0. Taking the curl of the total velocity field yields

∇× u = ∇× [uf + φ(up − uf )] = φ(∇× up) + (1− φ)(∇× uf ) + (∇φ)× (up − uf ).

If we assume that the vorticity of the total flow field is an addition of the vorticity of

the fluid velocity and the vorticity of the particle velocity:

∇× u = φ(∇× up) + (1− φ)(∇× uf ),

then the no-slip constraint is imposed. This assumption has been verified numerically

in a simulation example by Luo et al. in [12].

The condition of no fluid penetration of the particle surface is tied to the constraint

of incompressible flow. For the particle domain, we have ∇·up = 0 due to rigid body

motion. The fluid is incompressible: ∇ · uf = 0. Then take the divergence of the

total velocity field:

∇ · u = ∇ · [uf + φ(up − uf )] = (∇φ) · (up − uf ).

As we will impose incompressible condition ∇·u = 0 in our simulations, we will have

(∇φ) ·(up−uf ) = n ·(up−uf ) = 0, which indicates there is no penetration on particle

surfaces.

8

CHAPTER 3

Prototype Flows: Verification and Error Analysis

3.1. Planar Couette Flow

We simulate the steady laminar flow of a viscous fluid between two parallel plates,

as shown in Figure 3.1. The upper wall is located at y = h, and is moving with a

constant velocity V = (V, 0, 0). The lower wall is located at y = 0 and is fixed. Here

we specify V = 1, h = 3.

Figure 3.1. Sketch of the planar Couette flow problem.

First, we want to verify that SPM captures the exact solution of this problem:

uexact = (V y/h, 0, 0) for y ∈ [0, h], and (V, 0, 0) for y ∈ [h, 2h].

We derive the SPM scheme as follows: The top wall ( h ≤ y ≤ 2h ) is considered

as the particle, and its boundary at y = h is treated as a smoothed profile with the

indicator function

φ(x, t) =1

2[tanh(

y − hξ

) + 1] (3.1)

derived from Eq. (1b). We use the time stepping scheme for SPM which is given

by Eqs. (2.6), (2.7), (2.8), (2.10) and (2.12). We can obtain an ODE boundary

9

problem from it by assuming that: (1) all the field quantities are only y-dependent;

(2) the velocity field has a nontrivial component only in the x direction, i.e., u(y) =

(u(y), 0, 0); (3) steady state is achieved, i.e., un+1 = un; (4) the pressure is uniform

everywhere, i.e., ∇p∗ = ∇pp = (0, 0, 0). Then we can obtain a reduced system for u∗

and un: ν∆t∂2u∗

∂y2− φu∗ = −φup,

un = u∗ + φ(up − u∗).(3.2)

The above reduced system can be regarded as a boundary value ODE problem,

and can be solved on the domain 0 ≤ y ≤ 2h using a very accurate fourth-order

Runge-Kutta solver called ‘bvp4c’ in MATLAB. Solving the problem as an ODE

decouples the truncation errors of SPM from the numerical errors of a flow solver.

This demonstrates the underlying accuracy or errors of SPM found with an ideal flow

solver and spatial discretization.

We expect local error at y = h in velocity profile. Figure 3.2 shows how the

velocity profile un and u∗ matches with the exact solution at most places, but the

mismatch at the sharp corner is still obvious. However, the main concern is that we

can get correct wall stress from the simulation with correct choice of parameters.

y2.6 2.8 3 3.2 3.4 3.6

u

0.85

0.9

0.95

1 u*

un

Exact u

Figure 3.2. Couette flow: velocity profile of un, u∗ and the exact

solution (locally enlarged at the interface). ξ = 0.1, dt = 0.005625, and

ν = 1.

10

In order to find the shear stress on the wall, first we find the force density by

f = φ(up − u∗)/∆t, as shown in Figure 3.3. The density is concentrated at the

location of the wall. We then integrate the force density over y to find the total wall

shear stress τw, and compare it with the exact answer, τ0 = 1/h = 1/3.

y0 1 2 3 4 5 6

f

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

Figure 3.3. Couette flow: SPM force density profile. ξ = 0.1, and

dt = 0.005625.

We vary ∆t, ξ and ν and find the L2 error of the velocity profile, which is plotted

in Figure 3.4. We note that the L2 error does not only depend on ∆t, but also on the

smoothing scale ξ.

In fact, with a scaling argument of the parameters, we find that the error is a

function of√ν∆t/ξ, which is shown in Figure 3.5-a. The accuracy of the flow reaches

an optimal when√ν∆t/ξ reaches an optimal value, in this case, 0.75.

Similar results are also found in the relative error wall shear stress, defined as

(τw− τ0)/τ0; see Figure 3.5-b. A physical explanation of the error behavior is given in

[12], namely that the optimal result is reached when the Stokes layer induced by the

penalty force term is well balanced with the smoothed profile thickness and spatial

resolution.

We also solve the reduced system Eq. (3.2) using other finite difference schemes,

and find that the optimal√ν∆t/ξ ratio for the simulation depends on the time

11

Figure 3.4. Couette flow: L2 error of the velocity profile against ∆t.

ξ = 0.05.

Figure 3.5. SPM error of Couette flow simulation: a. L2 error against√ν∆t/ξ. b. Wall stress error against

√ν∆t/ξ. ξ = 0.05, ν = 1.

stepping scheme. For a backward Euler scheme, the optimal ratio is the same as

above, 0.75; for a Crank-Nicolson scheme, the optimal ratio is 0.81.

Note that the force density profile is skewed. For example, for the force density

profile in Figure 3.3 generated with dt = 0.005625, and ξ = 0.1, we view f(y) as a

probability density function, and find that the skewness is not zero and positive:

< y3 > − < y >3= 0.1251,

12

which indicates that f(y) has a longer tail on the right hand side. The center of f(y)

is < y >= 2.9991, which is smaller than the expected center at y = h = 3. The

variance is about the same size as ξ: σ = (< y2 > − < y >2)1/2 = 0.1177.

With the previous SPM simulation, we can find optimal ∆t. But we have not

demonstrated how do errors change with ξ. We then construct two alternative options

using the SPM tanh profile Eq. (3.1) to study the spatial accuracy.

Alternative Option 1

The first alternative option is that we construct a symmetrical force density profile

using the SPM tanh profile:

f = Fdφ

dy. (3.3)

In contrast, that previous force density f(y) is skewed in Figure 3.3. The wall layer

forcing will set the wall stress, and we want to find u by solving:

0 = ν∂2u

∂y2+ f. (3.4)

With the boundary conditions τw = 0 at y = h+, and τw = τ0 at y = h−, we find the

analytical solution to Eq. (3.4) to be

u(y) =∫ y0

1ντ0(1− φ(y′))dy′

= 12νh

[y − ξ log(cosh(y−hξ

)) + ξ log(cosh(−hξ

))].(3.5)

Note that here u depends on ξ but not ∆t.

Figure 3.6 shows the velocity profile obtained from this option compared to the

exact solution. As expected, there is a local error around y = h, but we could get the

correct velocity gradient ∂u∂y

away from the interface at [h− 4ξ, h+ 4ξ].

In order to study the spatial errors of the different options, we keep√ν∆t/ξ = 0.75

in the standard SPM to use the optimal ∆t for each ξ value. We then calculate the

L2 error of un and wall force with various ξ values and compare with the results from

the time dependent scheme. Figure 3.7 shows the spatial accuracy of the two options.

The error depends on ξ for both, but the error from the reduced scheme Eq. (3.2)

changes more slowly with respect to ξ.

Alternative Option 2

13

y2.6 2.8 3 3.2 3.4 3.6 3.8

u0.86

0.88

0.9

0.92

0.94

0.96

0.98

1

numerical uexact u

Figure 3.6. Couette flow option 1 v.s. standard SPM: velocity profile

(locally enlarged around the interface). ξ = 0.1.

Figure 3.7. Couette flow option 1: L2 error of the velocity profile

against ξ.

The second option is that we construct a smoothed viscosity profile using the SPM

function:

ν = ν0(1− φ) + ν1φ, (3.6)

where ν0 is the viscosity of the flow, and ν1 is set to a large value to simulate the stiff

wall. Then we find u by solving the boundary value problem:

0 =∂

∂y

(µ(y)

∂u

∂y

), (3.7)

14

with u = 0 at y = 0, and u = V = 1 at y = 2h. We set h = 3, ν0 = 1. First we look at

the velocity profile and force density profile, as shown in Figure 3.8. The numerical

velocity reaches the maximum of u = 1 before y = h. For ν1/ν0 = 100, it reaches a

maximum at about y = h − 3ξ , and for ν1/ν0 = 1000, at y = h − 4ξ. If ν1/ν0 gets

too large, the flow stress τ becomes very inaccurate, which is between 0.35 and 0.45,

larger than the expected value 1/3.

Figure 3.8. Couette flow option 2: velocity profile (left) and force

density profile (right) at ν1/ν0 = 100 and ξ = 0.1.

In order to study the spatial error, we vary the smoothed thickness ξ and also

ν1/ν0 to see how the L2 error of velocity changes. We find that the L2 error has

an linear relationship with ξ while it keeps a linear relationship with ν1/ν0 only on

a logarithm scale; see Figure 3.9. We also check the relative error of flow stress:

τ = ν du/dy, and find the same relationship.

3.2. Oscillating Stokes Layers

Next we study an unsteady problem to analyze the performance of SPM in such

cases. We simulate the unbounded Stokes flow with an infinite oscillating plate. The

plate is located at y = h and is oscillating in the x direction with an angular frequency

ω; the flow is at rest at far field. The exact solution to the problem in complex form

is

u(y, t) = eY+i(Y+ωt), (3.8)

15

Figure 3.9. Couette flow option 1: L2 error of the velocity profile

against ξ (left) and against ν1/ν0 (right).

where Y =√

ω2ν

(hy).

For the SPM simulation, we conduct on the domain y ∈ [0, 2h], where y ∈ [0, h]

is considered as the solid ‘particle’ modeled by SPM with the concentration field of

φ(x, t) = 12[tanh(h−y

ξ) + 1], and the fluid flow is in the interval y ∈ [h, 2h]. We treat

the boundary at y = 2h as a Dirichlet boundary using the exact solution; we set

u = eiωt at y = 0. For the simulation, we specify h = 3, ω = 1, ν = 1. The geometry

of the simulation is sketched in Figure 3.10. We then solve the reduced system Eq.

(3.2) using a second order Crank-Nicolson time stepping scheme.

The velocity profile is changing with respect to time, and we find that the velocity

profile un and u∗ to be a good match with the exact solution at each time step with

some local error at the wall. Figure 3.11 shows how the velocity profiles agree with

exact solution at t = 3/2T with ∆t = 0.0019 and ξ = 0.05.

We find the shear stress τw on the wall at every time step by integrating the

force density f = φ(up − u∗)/δt over y, as we did in Section 2.1. Now that we have

the data of wall stress at every time step, we can compare it with the exact answer

τ0 = − sin(ωt− π4). We discard the first 1/2 cycle as a transient result, and study the

wall stress from 12T to 3

2T , as plotted in Figure 3.12.

16

Figure 3.10. Sketch of the SPM simulation geometry of the oscillating

Stokes problem.

y0 1 2 3 4 5 6

u

-1.2

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

u*

un

exact

Figure 3.11. Stokes layer: velocity profile at t = 3/2T with ∆t =

0.0019 and ξ = 0.05.

By checking the Fourier coefficients of the numerical wall stress, we can find the

amplitude and phase shift of τw comparing to τ0. The detailed process is described in

Appendix A. And then we study the error of the amplitude and phase shift. Figure

3.13 shows the percentage error of force estimation. Again, we recognize the error

behavior that the accuracy of force amplitude reaches an optimal when√ν∆t/ξ

reaches a certain value, in this case, around 0.85. However, the phase shift is not

17

Figure 3.12. Stokes layer: wall stress data from 1/2T to 3/2T .

necessary the smallest at the same condition. The optimal√ν∆t/ξ ratio for phase

shift to reach zero is smaller.

Figure 3.13. Stokes layer: relative error of the amplitude of τw (left)

and phase shift comparing to the exact solution (right).

3.3. Sphere Settling in Finite Reynolds Flows

In order to verify SPM for finite Reynolds number flows, we study the problem

with a single spherical particle settling in a duct formed by four rigid no-slip walls at

Reynolds number 10. The particle with a radius of a = 1 is settling vertically in a

long duct between two pairs of parallel walls. The width of the duct is L2 = L3 = 7,

18

and the length of the duct in stream-wise direction is L1 = 20. Periodic boundary

conditions are specified in x1 direction, and no-slip boundary conditions are imposed

in both x2 and x3 directions. The dimensionless viscosity and density are µ = 1 and

ρ = 1. The net external force acts in the negative x1 direction such that the particle

has a terminal velocity of V1 = 5.0. Therefore, the particle Reynolds number scaled

by the particle diameter is Re = V1(2a)/ν = 10.0.

In the SPM simulation, the particle is fixed at center of the duct with coordinates

x1 = x2 = x3 = 0, and each side wall is moved upward with constant velocity

5.0. The Navier-Stokes equations with SPM force correction (2.5) is solved with the

OpenFOAM finite volume code based on the time stepping scheme proposed in Section

2.2. Standard OpenFOAM boundary conditions on the flow velocity and pressure are

used. We run on a structured coarse mesh initially with a resolution of 128×45×45,

and then map the result to a fine mesh with resolution of 256× 90× 90, and run to

convergence. In the coarse mesh, the SPM smoothed thickness is ξ = 0.3 ≈ 2∆x,

the time step is ∆t = 0.05, and we run from t = 0 to t = 50. In the fine mesh, we

first use ξ = 0.15 ≈ 2∆x, ∆t = 0.02, and run from t = 50 to t = 80; and then use

ξ = 0.1 ≈ 1.5∆x, ∆t = 0.015, and run from t = 80 to t = 98.

We study the result when it reaches steady state at t = 98. Figure 3.14 shows the

stream-wise fluid velocity profiles between the walls against x2 axis at locations in

the wake and the upstream regions. Figure 3.15 shows the stream-wise fluid velocity

profile along the x1 axis. Figure 3.16 shows the stream-wise fluid velocity contour

plot at the plane perpendicular to x2 axis across the center of the sphere. In the

particle region, the velocity should be zero, but there is some modeling error due to

the second projection step. In Figure 3.14, at the center of the the sphere, Ux = 0.16,

and in Figure 3.15, Ux = 0.3 at about x1 = 9 and Ux = 0.068 at about x1 = 11 (with

ξ = 0.1). The result with the coarse mesh and ξ = 0.3 gives Ux = 0.40 at the center;

the result with the fine mesh and ξ = 0.15 gives Ux = 0.20 at the center. We find that

the error of Ux is linked with ξ. For better accuracy, we can refine the mesh and use

19

smaller ξ, but a smaller time step must be used, which may increase the computation

cost.

Liu et al. [6] simulated a similar problem using both DNS, with the Nektar

spectral element code, and the force coupling method (FCM). They studied a longer

duct, L1 = 30, L2 = L3 = 7. Their results agree with our SPM results in general

trend, but they have a slightly larger settling velocity of the sphere, which is expected

for a longer duct.

Next, we check if the total wall shear stress matches with the force on the particle,

which is expected for steady flow. We integrate the force density fs of Eq. (2.9), and

find the particle force in x1 direction to be 145.46. Then calculate the wall stress

based on the gradient of u1; wall stress profile on one side wall is shown in Figure

3.17. Integration of the wall stress on all side walls yields −145.46, which matches

well with the SPM particle force.

Figure 3.14. Stream-wise fluid velocity profiles against x2 axis at

locations in the wake region (left), and in the upstream region (right).

ξ = 0.1 ≈ 1.5∆x, ∆t = 0.015

20

Figure 3.15. Stream-wise fluid velocity along the centerline of the

duct agains x1 axis.

Figure 3.16. Stream-wise fluid velocity contour plot at the plain per-

pendicular to x2 axis across the center of the sphere. Flow is from left

to right.

21

Figure 3.17. Wall shear stress (gradient of u1) profile on one side wall.

22

CHAPTER 4

Application in Dielectric Problems

In this chapter, we develop a new strategy to use the ideas of SPM to simulate

problems with dielectric beads in a non-conducting fluid. In this context, the fluid

is an insulator with no free charges; for example, silicon oil and the particles have

a different dielectric coefficient. We are looking at larger beads, 10 microns in scale

for which Brownian motion and colloidal forces are not significant. The electrostatic

potential field is governed by Eq. (4.1), which is a prototype Poisson problem.

∇ · (ε∇ψ) = 0. (4.1)

The dielectric coefficient ε varies across the particle-fluid boundary but we have the

matching conditions at the surface of the sphere.

ψf = ψp; εfEf · n = εpEp · n. (4.2)

The electric field in the fluid needs also to match some far field conditions.

We use the continuous profile across the interface:

ε = εpφ+ εf (1− φ), (4.3)

where φ is the SPM tanh function, εp is the dielectric coefficient of the dielectric bead

(particle), and εf is the dielectric coefficient of the fluid.

4.1. Dipole Problem: Sphere in a Uniform External Electric Field

First, we study the problem of a dielectric sphere with radius a placed in a uniform

external electric field E∞ = E∞x, as shown in Figure 4.1. The far field potential is

then ψ∞ = −E∞r cos θ.

23

Figure 4.1. Sketch of the dielectric sphere problem.

An analytic result can be obtained using spherical polar coordinate.

ψ = (−E∞r +Br−2) cos θ , for r > a

ψ = −Ar cos θ , for r < a(4.4)

The matching conditions give:

A = E∞(

1− ε1−ε2ε1+2ε2

)B = E∞a3

(ε1−ε2ε1+2ε2

) (4.5)

It is important to calculate the dipole moment correctly in this problem, and we

can find its analytic solution [18]. The dipole moment p is defined as

p = lim|h→0|

(Q h),

where h is the displacement between two point charges Q and −Q. The electric

potential ψ can be found through the Maxwell’s equations for electrostatics and the

definition of electric potential in free space with permittivity ε0:

ψ(x) =Q

4πε0

(1

|x− x0 − h|− 1

|x− x0|

),

where x0 is the position vector of the negative point charge. Taking the limit of

h → 0, we obtain the relationship between the ψ and p in Cartesian coordinates,

and change the expression to polar coordinates considering E∞ is only non-zero in x1

direction.

ψ =Qh · x4πε0r3

=p · x

4πε0r3=p (r cos θ)

4πε0r3=

p

4πε0r−2 cos θ. (4.6)

24

Comparing Eq. (4.6) to Eq. (4.4), we find that the dipole moment is linked with the

dielectric coefficient B:

B =p

4πε0. (4.7)

4.1.1. SPM simulation. We develop the SPM simulation by separating the

variables: ψ = −F (r) cos θ, and Eq. (4.1) can be simplified to an ODE about F (r):

d

dr

εr2

d

drF (r)

− 2εF (r) = 0, (4.8)

where ε is represented by the SPM profile Eq. (4.3). The simulation domain is

r ∈ [r1, r2], and set a = 1, r1 = 0.5, r2 = 2.0 and ε2 = 1. We use the exact solution at

the inner point as initial conditions, and solve the ODE problem using the MATLAB

solver ‘ode45’. Figure 4.2 shows the numerical results compared to the analytical

solution Fexact. We can see that SPM does not resolve the near surface field around

r = a exactly because of the smoothing layer. However, we can aim to get the dipolar

far field and the dipole moment correct.

Figure 4.2. Numerical F (r) and D(r) profile compared to the exact

solution. Here D(r) = εdFdr

.

We have shown in Eq. (4.7) that the dipole coefficient B in Eq. (4.4) is closely

related to the dipolar moment, so we choose to use the relative error of B to examine

the accuracy of the simulations. At r = r2, comparing to the analytical solution Eq.

25

(4.4), we have

F2 = E∞r2 − Br22,

E2 = E∞ + 2Br23.

(4.9)

And by solving the above equation system, we can compute the numerical B and E∞:

B = − r22

3(F2 − r2E2) ,

E∞ = 13

(2r2F2 + E2

).

(4.10)

Then we compare it to the exact B:

B0 = E∞a3(ε1 − ε2ε1 + 2ε2

). (4.11)

We find that the error is a function of the smoothed thickness ξ and the ratio of

the dielectric coefficients inside and outside ε1/ε2. Figure 4.3 shows that the error in

dipole moment is first order in ξ for fixed ε1. In the case of ε1/ε2 = 100, the relative

error is 13.07% for ξ/a = 0.1, and 5.31% for ξ/a = 0.05. Figure 4.4 shows how error

changes against ε1 at different ξ values.

Figure 4.3. SPM simulation: relative error of B and effective radius

aeff against ξ. a = 1.

We can also examine the level of error by finding the effective radius aeff , which

is the sphere radius of the exact dielectric sphere problem that matches with the

simulation. We can find aeff by

B = E∞a3eff

(ε1 − ε2ε1 + 2ε2

), (4.12)

26

Figure 4.4. SPM simulation: relative error of B and effective radius

aeff against ε1.

27

where B and E∞ are the numerical results. The effective radius also depends on ξ

and ε1/ε2, as shown in Figure 4.3 and Figure 4.4. In general when ε1 > ε2, SPM

simulation gives too large a dipole moment, corresponding to a larger effective radius

of the sphere, and the reverse for ε1 < ε2.

We find that there are similarity relations among the aeff - ε1 plots. By scaling

with aeff = 1 + kξf(ε1), where k is a constant, we find that the aeff data can be

matched for different ξ. The overlaid f(ε1) function for different ξ values is plotted

in Figure 4.5.

Figure 4.5. SPM simulation: similarity relationship of aeff .

4.1.2. Polynomial Smoothed Profile. In order to compare with the SPM

tanh indicator function Eq. (2.1), we develop another method by switching the SPM

indicator function to one based on a fourth polynomial function:

φ(z) =1

ξ4(z + ξ)2(z − ξ)2, 0 ≤ z ≤ ξ, (4.13)

where ξ is the smoothed thickness at the interface. The polynomial indicator function

varies from φ = 1 at z = 0 to φ = 0 at z = ξ, and is plotted in Figure 4.6. We aim

to provide well defined transition points with the polynomial indicator function, i. e.

dφdz

= 0 at z = 0 and z = ξ, while the SPM tanh indicator function can not provide

such well defined transition points.

28

x0 0.05 0.1 0.15 0.2 0.25 0.3

Po

lyn

om

ial φ

(x)

0

0.2

0.4

0.6

0.8

1

1.2

Figure 4.6. Polynomial indicator function φ(x− 0.1). ξ = 0.1.

The dielectric coefficient ε is then represented by

ε(r) =

ε1 , for 0 < r < 1− ξ

2

(ε1 − ε2)φ(r − (1− ξ

2))

+ ε2 , for 1− ξ2≤ r ≤ 1 + ξ

2

ε2 , for r > 1 + ξ2.

(4.14)

We also find that the relative error of B and effective radius aeff are functions of

the smoothed thickness ξ and the ratio ε1/ε2, as shown in Figure 4.7. The pattern

of the functions are not exactly the same as that of the SPM simulation, but we can

also find the similarity relationship in aeff (ξ, ε1), as shown in Figure 4.8. Assuming

aeff = 1+kξf(ε1), we could find function f(ε1) overlays for simulations with different

ξ values.

4.1.3. Interpolate 1/eps. We find from the previous simulations that the error

is large when the dielectric coefficient of the sphere ε1 is much larger than ε2. For

these cases, we can interpolate 1/ε instead of ε so that we can limit the ‘contrast’ at

the interface. This is a common method used for such situations.

We use the SPM tanh smoothed profile to interpolate 1/ε:

1

ε=

1

ε1φ+

1

ε2(1− φ) (4.15)

An example of ε(r) profile obtained by interpolating 1/ε instead of ε is shown in

Figure 4.9. In regular SPM, the average ε value, ε = (ε1 + ε2)/2 occurs at the sphere

29

Figure 4.7. Polynomial smoothed profile: relative error of B and

effective radius aeff .

30

Figure 4.8. Polynomial smoothed profile: similarity relationship of aeff .

surface r = a, but by interpolating 1/ε, the average ε occurs at some point inside the

actual sphere surface r < a, which has the same effect as retracting aeff , and this

will help reduce the error for large ε1 values.

r0.5 1 1.5

φ

0

10

20

30

40

50

60

70

80

90

100

interpolate ǫinterpoalte 1/ǫ

Figure 4.9. Interpolating 1/ε v. s. interpolating ε: ε(r) profile. ξ =

0.1, ε1 = 100.

As expected, we find that this method works best when ε1 ε2. Figure 4.10

shows the relative error of B against ε1 when interpolating ε and 1/ε respectively.

When ε1/ε2 is larger than 2, interpolating 1/ε gives a more accurate simulation. And

the larger ε1/ε2 is, the more accurate the result is. For ε1 < ε2, the regular SPM is

fine.

31

Figure 4.10. Error of interpolating 1/ε and interpolating ε for ξ = 0.1

(left) and ξ = 0.2 (right).

32

4.2. Electrostatic Force on a Dielectric Sphere in Non-uniform Field

In this section, we look into a problem of a dielectric sphere with radius a in an

external electric field that also has a slow variation. The sketch of the problem is the

same as Figure 4.1, but the E∞ field is the addition of a uniform field and a linearly

changing field:

E∞ = E∞ x + EA2(−2x x + y y + z z),

where EA2, is the linear coefficient of the increasing field. The far field potential is

thus

ψ∞ = −E∞r cos θ + EA2r21

2

(3cos2θ − 1

). (4.16)

An analytic result can be obtained by assuming with an additional term for second

spherical harmonics to represent the linear variation of E∞.

ψ = −A1r cos θ + A2r2 12

(3cos2θ − 1) , for r < a,

ψ = (−E∞r +B1r−2) cos θ +

(EA2r2 +B2r

−3) 12

(3cos2θ − 1) , for r > a,(4.17)

where A1 and B1 are the dipole coefficients, given in Eq. (4.5), and A2 and B2 are

the quadrupole coefficients, given by the matching conditions:

A2 = EA2(

1− 2(ε1−ε2)2ε1+3ε2

)B2 = −EA2a5

(2(ε1−ε2)2ε1+3ε2

) (4.18)

We are interested in calculating the electrostatic force in the field correctly using

different versions of our simulation methods. We obtain the exact force by

F = p · ∇E∞

at x = 0, where p is the dipole moment of the sphere. Since p only has a non-zero

component in the x direction and since only the x component of the electric field

varies with x, the force should only have the x component too. The x component of

the gradient of the E field is

(∇E∞)1 = −2EA2.

33

In response to the imposed uniform electric field, the x component of the dipole

moment p1 is uniform inside the sphere. The field due to EA2 will not generate a

dipole only a quadrupole. Finally, we find that the force in x direction is

F1 = 2EA2 · 4πa3ε2E∞ε1 − ε2ε1 + 2ε2

. (4.19)

This is used as the exact force, and is compared with numerical results in the next

sections.

An alternative approach to calculate the force is by integrating the Maxwell stress

tensor [19]. In electrostatics, the Maxwell stress tensor is used to represent the inter-

action between electrostatic forces and mechanical momentum, and has the following

form:

τEij = ε(EiEj −1

2EkEkδij). (4.20)

In Cartesian form, the force on the dielectric sphere can be obtained by integrating

the Maxwell stress tensor outside the sphere surface:

F1 =

∮r=a(1+α)

τEij nj ds, (4.21)

where nj =xjr

and α 1. The surface r = a(1 + α) is known as the Gauss surface.

We can also do the integral in polar form for convenience. The stress tensor

components in polar form are given by

τrr = (Er2 − 1

2(Er

2 + Eθ2))ε2,

τrθ = EθErε2,(4.22)

where the electric field components can be determined by the electric potential:

Er = −∂ψ∂r, Eθ = −1

r

∂ψ

∂θ.

And we evaluate the integral:

F1 =∮

r=a(1+α)

(τrr cos θ − τθr sin θ) ds

= 2π∫ π0r2 sin θ (τrr cos θ − τθr sin θ)dθ.

(4.23)

34

We can use the analytic ψ in Eq. (4.17) to calculate the force. Analytically the

force calculated by this approach should match with the force obtained using the first

approach Eq. (4.19). We did the integration Eq. (4.23) using a trapezoidal numerical

scheme in MATLAB with ε1 = 2, ε2 = 1, EA2 = E∞ = 1, α = 0.1, and a resolution

of 500, and find that F1 = 6.28319 using the first approach and F1 = 6.28299 using

the second approach, off by only 0.003%. This error is generated by the numerical

integration scheme of MATLAB.

In the next subsection, we use the second approach and numerical ψ to calculate

the force and compare with the exact force calculated with Eq. (4.19).

4.2.1. Numerical Simulation and Force Calculation. We develop the SPM

simulation by solving a dipole problem and a quadrupole problem numerically, and

add the results together. The simulation method for the dipole problem is the same

as that in Section 3.1.1 and 3.1.2, and we can obtain the numerical Fdipole(r) by

solving Eq. (4.8). To solve the quadrupole problem, we first separate the variables

with second spherical harmonics : ψ = Fquad(r) · 12 (3cos2θ − 1), and then Eq. 4.1 is

simplified to an ODE about F (r):

d

dr

εr2

d

drFquad(r)

− 6εFquad(r) = 0, (4.24)

where ε is represented by either the SPM profile Eq. (4.3) or the polynomial smoothed

profile Eq. (4.14). Set ε2 = 1, and a = 1. Using the same numerical ODE solver,

we obtain the numerical result of Fquad(r). For initial conditions, we use the exact

solution in Eq. (4.17) for the EA2 field inside the particle and set A2 from Eq. (4.18)

assuming EA2 = 1, but the actual value of EA2 can be a bit different depending on

each simulation, and is given by

EA2 =1

5

(Equad(r2)

r2+

3

r22Fquad(r2)

).

For the dipole problem, we aim for E∞ = 1, and the actual E∞ value is given by Eq.

(4.10). The total numerical potential is thus

ψ = Fdipole(r) · cos θ + Fquad(r) ·1

2

(3cos2θ − 1

). (4.25)

35

Then we calculate the numerical force by substituting the numerical ψ to Eq. (4.23)

and (4.22), and compare with the analytic results. We study the relative error of the

force: Error = (Fexact − Fnumerical)/Fexact.

First, we want to find out in each SPM simulation what spherical surface is good

to use as the Gauss surface when integrating the Maxwell stress tensor to evaluate

force. We choose the same ξ and ε1/ε2, and evaluate forces on different radius values

of Gauss surfaces. Figure 4.11 shows how the relative error of force change against

r compared to the ε(r) profile. The error remains the same when the SPM Gauss

surface is large enough so that ε(r) = ε2. For the polynomial smoothed layer, the

radius of Gauss surface can be r ≥ a + ξ/2; for the SPM tanh layer, the radius of

Gauss surface can be r ≥ a + 2.5ξ. Table 4.1 shows the exact force and the SPM

force evaluated at a large Gauss surface (r = a + 5ξ) for different choice of ξ. For

the next analysis, the force is calculated at a Gauss surface of r = a+ ξ/2 using the

polynomial smoothed profile, because the polynomial smoothed profile is more clear

with the location where ε(r) changes to ε2.

Figure 4.11. Error against r (left axis, solid line) plotted with ε(r)

(right axis, dotted line) when using a polynomial smoothed profile with

ξ = 0.2 (left) and SPM tanh indicator function with ξ = 0.05 (right).

ε1 = 2.

Next, we study how error changes with respect to ξ. We find that the order of

accuracy against ξ is about 1 for fixed ε1, as shown in Figure 4.12.

36

Table 4.1. Error of force evaluated at a large Gauss surface (r =

a+ 5ξ). a = 1, ε1/ε2 = 2, using SPM tanh indicator function.

ξ Exact Force SPM Force Error

0.05 6.16075 6.34617 -0.0301

0.10 6.06983 6.52032 -0.0742

0.15 6.00330 6.78126 -0.1296

0.20 5.95242 7.13116 -0.1980

Figure 4.12. Relative error of force against ξ in regular scale (left)

and in log scale where Error is converted to its abstract value (right).

ε1 = 2. ξ ranges from 0.05 to 0.32. Use polynomial smoothed layer;

error calculated at r = a+ ξ/2.

Then we study the relationship between error and ε1/ε2. The error behavior is

similar to that of the dipole problem, Figure 4.7. The error gets smaller as ε1 is closer

to ε2. We can also find a similarity relationship of the error v.s. ε1/ε2 for different ξ,

similar to Figure 4.8.

4.2.2. Correction Methods by Retracting Boundary. We found in Section

3.1.1 that when we are targeting at a dipole problem with a = 1, we actually get a

simulation result that matches with a problem with a = aeff (dipole). Similarly, if we

target at a quadrupole problem with a = 1, the simulation results actually matches

with that of a = aeff (quad). In order to improve accuracy of the simulation, we can

37

retract the particle boundary by targeting an a = 1 problem but setting the radius

as

a = 1/aeff

as a correction.

First we use the correction method for pure dipole problem Eq. 4.8. We test on

ξ ranging from 0.01 to 0.3, with ε1 set to 2, 20 and 40 respectively, and find that

the order of accuracy against ξ is improved to 2 by using this method, as shown in

Figure 4.13. Here error is defined as the relative error of B obtained by the correction

method against the exact B with the same simulation conditions but without the

correction.

Figure 4.13. Dipole problem: achieve second order of accuracy by

retracting boundary. Use polynomial smoothed profile; ξ ranges from

0.01 to 0.3, with ε1 set to 2, 20 and 40 respectively.

We also test the correction method on pure quadrupole problem. We also achieve

second order accuracy for the quadrupole coefficient B2 with ε1 = 2, 20 and 40

respectively, by retracting the boundary by 1/aeff (quad).

We move on to study how the correction method works for the force calcula-

tion of a dielectric sphere in non-uniform field. First, we retract the boundary by

1/aeff (dipole), and find that the relative error of force on the sphere is reduced for

larger ξ , but the method is not very effective for smaller ξ, as shown in Table 4.2.

The threshold for error less than 1% is at around ξ = 0.23. The order of accuracy of

is not clear with the correction method, as shown in Figure 4.14, because the problem

38

is a combination of a dipole and a quadrupole and we only use aeff (dipole) for the

correction.

Figure 4.14. Error of force against ξ, calculated using correction

method with ε1 = 2 in log scale. The order of accuracy is not clear.

We then use aeff (quad) for the correction method and calculate the force. The

results are recorded and compared in Table 4.2. We can see that errors are reduced,

but using aeff (quad) is not as effective as using aeff (dipole). However, the threshold

of ξ for error less than 1% is smaller, at around ξ = 0.2. Order of accuracy is not

clear either.

We also use aeff (dipole) for the dipole part of the problem and aeff (quad) for

the quadrupole part as correction. Results show that this method is about the same

effective as only using aeff (dipole), as shown in Table 4.2.

39

Table 4.2. Result of force calculation with and without the correction

methods using polynomial smoothed profile. ε1/ε2 = 2. Gauss surface

at r = a+ ξ/2.

Exact Numerical Use a eff(dipole) Use a eff(quad) Use both

ξ Fexact F Error F Error F Error F Error

0.05 6.24759 6.34598 -0.0157 6.36382 -0.0428 6.36394 -0.0428 6.35423 -0.0412

0.08 6.22039 6.35012 -0.0209 6.20182 -0.0162 6.21201 -0.0179 6.20259 -0.0164

0.11 6.19451 6.36401 -0.0274 6.19622 -0.0153 6.17095 -0.0112 6.19668 -0.0154

0.14 6.18848 6.42707 -0.0386 6.20130 -0.0162 6.16945 -0.0109 6.19651 -0.0154

0.17 6.17990 6.45138 -0.0439 6.21291 -0.0181 6.17625 -0.0121 6.19964 -0.0159

0.20 6.16371 6.46873 -0.0495 6.18587 -0.0136 6.13947 -0.0060 6.17183 -0.0113

0.23 6.14404 6.50918 -0.0594 6.13758 -0.0057 6.07430 0.0046 6.13288 -0.0050

0.26 6.12672 6.55890 -0.0705 6.11985 -0.0028 6.03738 0.0107 6.12177 -0.0031

0.29 6.11356 6.61014 -0.0812 6.11419 -0.0019 6.01716 0.0140 6.11602 -0.0022

0.32 6.10265 6.64650 -0.0891 6.12197 -0.0032 6.01469 0.0144 6.12109 -0.0030

40

4.3. Periodic Dielectric Spheres in a Uniform Electric Field

In this section, we apply SPM to another problem of periodic dielectric spheres

with radius a in a uniform external electric field E∞ = E∞x. The spheres are arranged

in a periodic cubic lattice of length L. The simulation adopts a configuration of a = 1

and (i) L = 4 or (ii) L = 8.

Here ψ is the potential created by the periodic array of spheres in an otherwise

uniform electric field, i. e. E = E∞−∇ψ. Substituting this to the governing equation

∇ · (εE) = 0, we obtain the equation for the problem:

∇ · (ε∇ψ) = E∞ · ∇ε. (4.26)

And we solve the problem numerically by developing a SPM-FFT box code that

involves representing ε(x) with SPM tanh profile (4.3), and transferring the variables

back and forth from Fourier space.

Now we explain the SPM-FFT box code. Eq. (4.26) can be written as

∇2ψ = (E∞ −∇ψ) · ∇ (log(ε)) . (4.27)

We develop our iteration scheme based on this equation.

• To start with n = 0 and set ψ0 = 0.

• Then find ∇ψn by transferring ψn to Fourier space, computing ik ψn, and

transferring the value back to real space. Here k denotes the wave number

in FFT.

• Next we compute the right hand side of Eq. (4.27), RHS, in real space, and

transfer it to Fourier space, RHS.

• Then find ψn+1 = − RHSk2

and transfer it back to real space to get ψn+1. For

the case when k2 = 0, ψ is set to 0.

• Define the error of each tolerance as√∑

i,j,k

(ψn+1 ijk − ψn ijk)2, and iteration

ends when the error becomes smaller than a set tolerance.

The first simulation is run with a small periodic box L = 4, a = 1, with a resolution

of N = 256, and the SPM smoothed thickness ξ = 0.05. ε1 = 2, ε2 = 1 and E∞ = 1.

41

The error is reduced by about 10% after each iteration, so the result converges quite

fast. We use the result from the fifth iteration where the error is 0.0184. Figure 4.15

shows the result of ψ at the plane x3 = 0. We also study the electric field inside the

sphere by evaluating (−∇ψ), and find that it is dominated by the x1 component, and

is almost zero in x2 and x3 component, as shown in Figure 4.16. The electric field is

constant inside sphere. We take a box of side h = a inside sphere, and find that the

average of (−∇ψ)1 is 0.2311.

Figure 4.15. Result of ψ at the x3 = 0 plane (left) and at x2 = x3 = 0

(right). L = 4, a = 1, ξ = 0.05, ε1 = 2, ε2 = 1, E∞ = 1. N = 256, the

5th iteration. The sphere is centered at the origin and the lattice cell

extends from x1 = −L/2 to x1 = L/2.

The second simulation is run with a larger periodic box L = 8, a = 1, with

a resolution of N = 256, and the SPM smoothed thickness ξ = 0.1. The result

converges fast as the error is reduced by about 10% after each iteration. We study

the result from the fifth iteration where the error is 0.0147. Figure 4.17 shows the

result of ψ at the plane x3 = 0. Figure 4.18 shows the x1 component and x2 component

of (−∇ψ). Taking a box of side h = a inside sphere, and find that the average of

(−∇ψ)1 is 0.2348.

42

Figure 4.16. Result of the x1 component of ∇ψ (left) and the x2

component of ∇ψ (right). L = 4, a = 1.

Figure 4.17. Result of ψ at the x3 = 0 plane (left) and at x2 = x3 = 0

(right). L = 8, a = 1, ξ = 0.05, ε1 = 2, ε2 = 1, E∞ = 1. N = 256, the

5th iteration.

43

Figure 4.18. Result of the x1 component of ∇ψ (left) and the x2

component of ∇ψ (right). L = 8, a = 1.

44

CHAPTER 5

Conclusion and Future Work

In this thesis, we have analyzed the error behavior of the smoothed profile method

(SPM) for particle-laden flows. We found that the modeling error of SPM depends on

the ratio of√ν∆t/ξ, and the most accurate results occur at

√ν∆t/ξ equals to 0.75 to

1; the exact optimum ratio depends on the time stepping scheme. Comparisons with

the analytic solutions showed that SPM is resolving accurately the far-field flows and

the particle forces, while allowing some error locally at the particle-fluid interfaces.

Subsequently, we extended SPM to simulate electro-rheology flows allowing for

spatially varying dielectric coefficient (permittivity) ε. In this context, we solve the

Poisson equation in electrostatics (Gauss’s Law) for the whole domain including the

volume occupied by the dielectric beads. We also proposed an alternative indicator

function based on a fourth order polynomial, which showed similar effect in simu-

lations as the traditional tanh indicator function, but since it has a more definite

start point and end point of the smoothed layer, it can provide better control of

the smoothed thickness. We verified the method for prototype problems of dielectric

beads in non-conducting fluid, where the modeling error is also quantified. We also

constructed an iterative SPM solver with FFT for Poisson problems.

Furthermore, we have demonstrated in Chapter 3 how we used SPM as a finite

Reynolds flow solver with the direct forcing method, while in Chapter 4 we have shown

how to use SPM to solve elliptic problems, which also points out the possibility of

using SPM as a viscous flow solver.

The following improvements in the development and implementation of SPM are

recommended for future work.

• Comparison with direct numerical simulations (DNS). For problems without

analytic solutions, comparing SPM with DNS results will provide a better

45

idea of the low computation cost of SPM and its accuracy. Our research

suggests difficulties in implementing periodic boundary conditions on a DNS

unstructured mesh in OpenFOAM.

• Comparison with the force coupling method (FCM). FCM can be imple-

mented in the FFT box code in Chapter 4.3 as a direct forcing method and

compare with the SPM-FFT code.

• Developing a viscous flow solver based on SPM based on the implementation

of SPM on dielectric problems and the second alternative option for the

Planar Couette flow in Chapter 3.1, where we construct a smoothed viscosity

profile using the SPM tanh function.

• Simulation on a finer mesh for finite Reynolds number flow. We can refine

the mesh locally around the particle and use the original coarse mesh in the

far field, then we can take a smaller smoothed thickness ξ for better accuracy.

46

APPENDIX A

Calculating Wall Force Using Fourier Coefficients

We have the wall stress data τw(t), where T/2 ≤ t ≤ 3T/2 . The exact wall stress

should be τ0(t) = − sin(ωt− π4).

The Fourier coefficients of τw(t) can be found by

an = 1T/2

∫ 3T/2

T/2cos(nt) · τw(t)dt,

bn = 1T/2

∫ 3T/2

T/2sin(nt) · τw(t)dt.

We checked that all the Fourier coefficients except for a1 and b1 are zero.

The phase shift θ of τw(t) comparing to τ0(t) is found by

θ = arctan(b1/a1).

And then the amplitude of τw(t) is found by

τw =1

T/2

∫ 3T/2

T/2

Tw(t) · T0(t− θ)dt.

47

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