Development of a novel immersed boundary lattice boltzmann method and its applications

LIST OF FIGURES Figure 2.5 Comparison of velocity profiles for 2D lid-driven cavity flow 48 simulation Figure 2.6 Comparison of velocity profiles on the plane of y = 0.5 for 49 3D lid-d

DEVELOPMET OF A OVEL IMMERSED

BOUDARY-LATTICE BOLTZMA METHOD AD

ITS APPLICATIOS

WU JIE

A THESIS SUBMITTED

FOR THE DEGREE OF DOCTOR OF PHILOSOPHY

DEPARTMET OF MECHAICAL EGIEERIG

ATIOAL UIVERSITY OF SIGAPORE

2010

My sincere appreciation will go to my dear family Their love, concern, support and continuous encouragement help me with tremendous confidence in solving the problems in my study and life

Finally, I would like to thank all my friends who have helped me in different ways during my whole period of study in NUS

WU JIE

1.3.4 Some efforts on improvement of computational efficiency 17

non-uniform mesh

Chapter 2 Development of Efficient Lattice Boltzmann Method on Non-Uniform 25

Cartesian Mesh

2.2 Taylor Series Expansion and Least Squares-base Lattice Boltzmann 30 Method

3.2.5 Drawback of conventional IB-LBM 59

3.4.2 Flows over an array of circular cylinders placed at the middle of 70 straight channel

Moving Boundary and Particulate Flow Problems

4.2.1 Brief review on moving boundary problems and computational 106 sequence

4.2.2.3 Unsteady flows at low Reynolds number flapping flight 115

4.2.2.4 Flows over a flapping flexible airfoil 117

4.3.1 Brief review on the study of particulate flow problems 120 4.3.2 Force, torque calculation on the particle and computational 124 sequence

4.3.3.1 A moving neutrally buoyant particle in linear shear flow 126

4.3.3.3 Particle suspension in a 2D symmetric stenotic artery 131

A One particle passes the stenosis throat with b = 1.75d 131

B Two particles pass the stenosis throat with b = 1.75d 132

Chapter 5 Application of New IB-LBM to Study Flows over a Stationary 158

Circular Cylinder with a Flapping Plate

5.3 Numerical Study of Flows over a Stationary Circular Cylinder with 162

a Flapping Plate

5.3.1 Flow patterns due to flapping plate 162

Chapter 6 Extension of New IB-LBM to Simulate Three-dimensional Flows 186

around Stationary/Moving Objects

6.2 Efficient Three-dimensional LBM Solver on Non-Uniform Cartesian 189 Mesh

Chapter 7 Conclusions and Recommendations 230

Summary

In recent years, the immersed boundary method (IBM) has been developed into a popular numerical technique in the community of computational fluid dynamics (CFD) As a successful example of non-boundary conforming methods, the Cartesian mesh is utilized for resolving flow field in IBM The effect of boundary is replaced by the body force density which influences flow phase around the boundary The governing equations with this force density are solved on the whole computational domain including the exterior and interior of the boundary On the other hand, as an alternative CFD tool, the lattice Boltzmann method (LBM) has gained wide range applications recently Since the Cartesian mesh is also employed in LBM, an efficient solver can be generated by combining IBM with LBM, which is called IB-LBM Some efforts have been made in this aspect and the achievement is obvious However, there are still some shortcomings in this newly developed approach In this work, two major improvements were made

Firstly, a new version of IB-LBM was proposed in order to strictly satisfy the non-slip boundary condition In the conventional IB-LBM, the non-slip boundary condition is not enforced, and is only approximately satisfied at the converged state

As a consequence, the accuracy of solution is reduced, and the situation of streamline penetration to solid boundary is present To overcome this drawback, a boundary condition-enforced IB-LBM was developed Different from the conventional IB-LBM

in which the body force is computed in advance, the unknown body force is employed

in present method Such force is resolved by enforcing the non-slip boundary condition Applying the developed approach, the two-dimensional (2D) stationary and moving boundary flow problems, as well as particulate flow problems, were simulated Since the non-slip boundary condition is enforced, no flow penetration happens and the accuracy of resolution is improved All the obtained numerical results are compared well with previous experimental and numerical results

In the application of IB-LBM, the non-uniform mesh is usually employed in order to improve the computational efficiency To apply LBM on the non-uniform mesh, many variants of LBM can be chosen A simple way is to use Taylor series expansion and least squared-based LBM (TLLBM) Its final form is an algebraic formulation, in which the coefficients only depend on the coordinates of mesh points and lattice velocity As compared to the standard LBM, the drawback of TLLBM is that additional memory is required to store the coefficients Due to the limitation of virtual memory, it is not easy to apply TLLBM in three-dimensional (3D) simulations

To overcome this difficulty, an efficient LBM solver based on the one-dimensional interpolation was developed As compared to TLLBM, much less coefficients are calculated Combing with this efficient LBM solver, the new IB-LBM was easily extended to 3D simulation The 3D flows around complex stationary and moving boundaries were simulated The obtained numerical results are agreed well with the results and findings in the literature

LIST OF TABLES

Table 3.1 Comparison of forces f and c f for flows over an array of w 83

circular cylinders placed at the middle of a straight channel

Table 3.2 Comparison of drag coefficient, length of bubbles and separation 84

angle for flows over a circular cylinder with previous studies

Table 3.3 Comparison of drag coefficient, lift coefficient and Strouhal 85

number for flows over a cylinder with previous studies

Table 3.4 Comparison of drag coefficient, lift coefficient and Strouhal 86

number for flows over a pair of side-by-side cylinders at Re = 100

with previous studies

Table 3.5 Comparison of time-averaged drag coefficient, lift coefficient 86

and Strouhal number for flows over a pair of side-by-side

cylinders at Re = 200 with previous studies

Table 3.6 Comparison of drag coefficient, lift coefficient and Strouhal 87

number for flows over a pair of tandem cylinders at Re = 100

with previous studies

Table 5.1 Parametric study of flows over a stationary cylinder with a flapping 177

plate at Re = 100 for θ0 =100 and St = f 0.2

Table 6.1 Comparison of drag coefficients for flows over a sphere at Re = 100 210

and 200

Table 6.2 Drag coefficients of flows over a torus 210

LIST OF FIGURES

Figure 2.5 Comparison of velocity profiles for 2D lid-driven cavity flow 48

simulation

Figure 2.6 Comparison of velocity profiles on the plane of y = 0.5 for 49

3D lid-driven cavity flow simulation

Figure 2.7 Comparison of consumed CPU time for lid-driven cavity flow 50

Figure 3.2 Configuration of boundary points and their surrounding fluid points 88 Figure 3.3 Convergence of numerical error versus mesh spacing for decaying 89

vortex problem

Figure 3.4 Streamlines for flows over a cylinder at Re = 20 and 40 89

Figure 3.5 Flow field for flows over a cylinder at Re = 100 and 200 90 Figure 3.6 Evolution of drag and lift coefficients for flows over a cylinder at 91

Figure 3.9 Evolution of drag and lift coefficients for flows over a pair of side- 93

by-side cylinders (T = 1.5D) at Re = 100 and 200

Figure 3.10 Vorticity contours and streamlines for flow over a pair of side-by- 94

side cylinders (T = 3D) at Re = 100 in a cycle

Figure 3.11 Vorticity contours and streamlines for flow over a pair of side-by- 95

side cylinders (T = 3D) at Re = 200 in a cycle

Figure 3.12 Evolution of drag and lift coefficients for flows over a pair of side- 96

by-side cylinders (T = 3D) at Re = 100 and 200

Figure 3.13 Vorticity contours and streamlines for flow over a pair of tandem 97

Figure 3.20 Distribution of pressure coefficient along the boundary of 103

NACA0012 airfoil at Re = 500

Figure 3.21 Vorticity contours and streamlines for flow over a NACA0012 104

airfoil at Re = 1000 and AOA = 10° in a cycle

Figure 4.1 Streamlines for moving and stationary cylinder cases at Re = 40 135

Figure 4.2 Vorticity distribution on the surface of cylinder at Re = 40 for 136

moving and stationary cylinder cases

Figure 4.3 Pressure profile on the surface of cylinder at Re = 40 for 136

moving and stationary cylinder cases

Figure 4.4 CPU time required for stationary and moving cylinder cases 137

at Re = 40

Figure 4.5 Comparison of vorticity contours for flow over a rotationally 137

oscillating cylinder when St f=St at Re = 100 n

Figure 4.6 Vorticity contours at Re = 100 for flows around a rotationally 138

oscillating cylinder with different Strouhal numbers

Figure 4.7 Time histories of the lift and drag coefficients for two non lock-on 139

cases in rotationally oscillating cylinder simulation

Figure 4.8 Variations of drag and lift coefficients versus St in rotationally f 140

oscillating cylinder simulation

Figure 4.9 Vorticity contours of a flapping wing in one cycle for φ = π 4 141 Figure 4.10 Time histories of lift and drag coefficients of a flapping wing 142

shear flow

Figure 4.21 Comparison of lateral migration of particle with previous data 151 Figure 4.22 Comparison of particle translational velocities with previous data 152 Figure 4.23 Instantaneous vorticity contours of one particle sedimentation at 152

different time stages

Figure 4.25 Sedimentation of two particles in a channel at different time stages 154 Figure 4.26 Instantaneous vorticity contours of two particles sedimentation at 154

different time stages

Figure 4.28 Longitudinal coordinates of two particle centers 155

Figure 4.32 Trajectories of two particles symmetric to the centerline initially 157 Figure 4.33 Trajectories of two particles with the initial position asymmetric 157

to the centerline with asymmetry d/4000

Figure 4.34 Enlarged part of x- direction coordinates with respect to the time 157

when two particles pass through the throat

Figure 5.2 Flow patterns vary with flapping frequency at constant flapping 178

amplitude θ0 =200

Figure 5.3 Flow patterns vary with flapping amplitude at constant flapping 179

frequency St = f 0.4

Figure 5.5 Variations of overall drag coefficients on the cylinder and plate 180

Figure 5.7 Evolution of near-wake structure as a function of flapping 182

frequency at constant flapping amplitude 0

θ =Figure 5.8 Evolution of near-wake structure in a flapping period at θ0 =200 183

and St = f 0.2

Figure 5.9 Evolution of near-wake structure in a flapping period at θ0 =20 184

Figure 6.2 Comparison of recirculation length L for flows over a sphere s 212

Figure 6.5 3D vortical structures of sphere for planar symmetric flows 214

Figure 6.11 Streamlines for flow over a rotating sphere at Re = 300 with 219

Figure 6.14 Phase diagram of C and y C for flows over a rotating sphere z 222

at Re = 300

Figure 6.17 Time histories of the force coefficient for fish swimming 224

at Re = 4000

Figure 6.18 Streamlines with vorticity contours for fish swimming 225

at Re = 4000

Figure 6.22 Time histories of x-, y-, and z-direction force coefficients for 228

dragonfly flight at Re = 500

Figure 6.23 Evolution of pitching angle and lift coefficient at one hindwing 229 Figure 6.24 3D vortical structures for dragonfly flight at four different stages 229

f Vortex shedding frequency

eq

gα Post-collision state of distribution function

DLM/FD Distributed Lagrange multiplier/fictitious domain

The field of CFD has a wide range of applicability in engineering It ranges from the study of air flow around an airplane, wind past tall buildings, blood flow through heart valves, to river sediment resuspension and transport, cell deformation and transport, fish swimming, insect and bird flight and so on Broadly speaking, such problems can be generally divided into three groups: (1) the object does not move in the fluid; (2) the object undergoes prescribed motion in the fluid; (3) the object moves freely in the fluid Among the three groups, the simulation of flows around moving objects is still a challenging area in CFD

The current numerical methods for simulating flows around moving objects can

Chapter 1 Introduction

be roughly classified into two major categories: (1) boundary conforming method; and (2) non-boundary conforming method For the boundary conforming method, the body-fitted mesh is often used This means that the boundary points coincide with mesh points Hence, the physical boundary condition can be imposed at the boundary point directly To simulate the flow over a moving object under the boundary conforming framework, one choice is to use the time-dependent coordinate transformation technique (Tsiveriotis and Brown, 1992; Mateescu et al 1996; Dimakopoulos and Tsamopoulos, 2003; Luo and Bewley, 2004) In this approach, the physical domain is transformed into a computational domain All the numerical computations are easily performed in the computational domain The most popular boundary conforming method is perhaps the arbitrary Lagrangian-Eulerian (ALE) approach (Hirt et al 1974; Hu et al 2001; Anderson et al 2004; Chew et al 2006) In many applications, the unstructured finite element (FE) mesh is employed in the ALE method In this category, a scheme named space-time finite element method (Tezduyar et al 1992a, 1992b) can be viewed as the general form of ALE finite element scheme In the boundary conforming method, since the implementation of boundary condition is straightforward, accurate solutions can be achieved The main difficulty of boundary conforming method is the complexity involved in regenerating the mesh to conform to the object boundary at all times, especially when the boundary surface is very complex Therefore, it would be desirable to decouple the boundary from the computational mesh which is used to solve governing equations of the fluid flow This motivates the development of non-boundary conforming method In this

Chapter 1 Introduction

category, the flow field is solved on a fixed Cartesian mesh, and the boundary points

do not need to coincide with the mesh points The effect of boundary on the flow field

is accounted through the proper treatment of the solution variables at the mesh points

or cells around the boundary As compared to the boundary conforming method, the non-boundary conforming method eliminates the requirement of tedious grid adaptation, which makes the simulation of flows over complex/moving objects be more straightforward

The primary purpose of this thesis is to develop efficient computational approaches in the category of non-boundary conforming method for accurate simulation of flows over stationary and/or moving objects In the present work, the flow field is obtained by the lattice Boltzmann method (LBM) In the following, we will firstly give a literature review on the non-boundary conforming method and the LBM, and then describe the objectives of this research and layout of the thesis

1.2 on-Boundary Conforming Method

Up to now, a plethora of prominent methods have been developed to handle flows over objects based on the fixed Cartesian mesh Unlike the boundary conforming method which is fairly standardized, the non-boundary conforming method demonstrates great differences in implementation details, depending on the problems and the definition of boundary In terms of treatment of boundary conditions, the

Chapter 1 Introduction

non-boundary conforming method falls into two distinct types: (1) sharp interface approach; and (2) diffuse interface approach

1.2.1 Sharp interface approach

For the sharp interface type, the boundary is tracked and its thickness is negligible The effect of boundary is demonstrated by directly modifying the computational stencil around the boundary One popular example of sharp interface type is the immersed interface method (IIM) Instead of adding forcing terms in the governing equations, the jump conditions are directly incorporated into the finite difference approximation of governing equations near the boundary

The immersed interface method was originally proposed by LeVeque and Li (1994) to solve elliptic equations In IIM, the Navier-Stokes (N-S) equations for flow field are discretized on a Cartesian mesh The basic idea of IIM is to explicitly introduce the jump conditions for the pressure and velocity across the interface Due

to the use of jump conditions, the IIM holds the second-order accuracy Following the original IIM, lots of variants have been developed and diverse applications have been made LeVeque and Li (1997) applied the IIM to simulate the Stokes flow with elastic boundaries or surface tension Li and Lai (2001) coupled IIM with the projection method for the full incompressible N-S equations to simulate fixed and moving interface flow problems In their work, the level set function is used to represent the interface, which simplifies the algorithm Lee and LeVeque (2003) introduced the

Chapter 1 Introduction

discrete delta function into the IIM to study the immersed elastic membranes In this work, the force density is decomposed into tangential and normal components The normal part is coupled with the jump conditions for pressure And the tangential part

is distributed into nearby Cartesian mesh points through the delta function By employing the IIM, Le et al (2006) successfully simulated flows around both rigid and elastic moving objects Xu and Wang (2006a) systematically derived all the necessary jump conditions for simulating flows around moving boundaries with the second-order spatial and temporal discretization The jump conditions have been incorporated into IIM in two-dimensional (2D) and three-dimensional (3D) numerical simulations (Xu and Wang, 2006b; 2008) The IIM has also been used to solve the streamfunction-vorticity equations on irregular domains (Calhoun, 2002; Li and Wang, 2003; Russell and Wang, 2003; Linnick and Fasel, 2005) Similar to IIM, there is an approach called ghost fluid method (GFM) which also uses the jump conditions in the solution (Fedkiw et al 1999; Liu et al 2000) This method is simpler than IIM and the governing system of equations is symmetrically positive definite However, only the low order accuracy of solution can be obtained nearby the interface

Another example in the family of sharp interface type is the Cartesian/cut-cell method In this method, the fixed Cartesian cells are cut by the boundary interface which goes through the cells Using appropriate interpolation techniques, the variables

on the cut-cells can be computed in accordance with the boundary condition The Cartesian/cut-cell method was motivated by the work of Noh (1964) named Coupled Eulerian-Lagrange (CEL) method Based on the idea of CEL, the Cartesian/cut-cell

Chapter 1 Introduction

method has been successfully applied to simulate the two-dimensional inviscid and viscous flow problems (DeZeeuw and Powell, 1993; Udaykumar et al 1997) Due to the requirement to cut cells, various possible shapes of irregular interface-cells would appear To implement the boundary condition in such cells, it is necessary to use different special treatments, which would unavoidably lead to tedious and complex coding logistics To overcome this drawback, Quirk (1994) applied the cell merging technique to simulate shock wave over arbitrarily complex bodies Later, Ye et al (1999) combined the cell merging technique with finite volume scheme to study the unsteady incompressible viscous flows with complex geometries In this method, the reconstruction of control volumes where the interface passes is required and the integration of weak form of governing equations over the modified control volumes is also needed Recently, the Cartesian/cut-cell method has also been extended to simulate the 2D and 3D moving boundary flow problems (Udaykumar et al 2001; Marella et al 2005) and solidification problems (Udaykumar and Mao, 2002; Yang and Udaykumar, 2005)

1.2.2 Diffuse interface approach

Different from the sharp interface type, the boundary in the diffuse interface type has finite thickness and it is smeared across some surrounding Cartesian mesh points Mostly, this category of methods uses the body force density to replace the boundary effect on the surrounding fluid As a consequence, it is easy for the diffuse interface

Chapter 1 Introduction

approach to implement boundary conditions, as compared with the sharp interface approach in which the jump conditions are usually complicated to implement The most well-known example of the diffuse interface approach is the immersed boundary method (IBM) It has been applied successfully for simulating various flow problems (Mittal and Iaccarino, 2005)

1.2.2.1 Immersed boundary method

The immersed boundary method was first proposed by Peskin (1977) to study the cardiac mechanics and associated blood flows In IBM, the N-S equations for flow field are discretized on the fixed Cartesian (Eulerian) mesh, and the boundary is represented by a set of Lagrangian points The basic idea of IBM is to treat the immersed boundary as deformable with high stiffness A small distortion of boundary will yield a force which tends to restore the boundary into its original shape Therefore, the effect of boundary is depicted by such restoring force Using the discrete delta function, the balance of such force is distributed into the surrounding Eulerian points The N-S equations with the force density are then solved on the whole domain including exterior and interior of the object Based on the work of Peskin (1977), various modifications and refinements were made and a great number

of variants of this approach have been proposed Fogelson and Peskin (1988) extended the IBM to simulate the rigid particle suspension in the Stokes flow Beyer (1992) presented a two-dimensional computational model of the cochlea by

Chapter 1 Introduction

modifying and extending Peskin’s IBM The projection method with the second-order accuracy is used to solve N-S equations and the second-order transfer of boundary force to fluid phase is employed By employing the feedback forcing term to represent the solid body, Goldstein et al (1993) developed a model termed virtual boundary method to simulate laminar and turbulent flows This method was also adopted by Saiki and Biringen (1996) to simulate the flow over a cylinder with the use of high-order finite difference schemes By generating appropriate binding forces to model the cell-cell aggregation and the cell-substratum, Dillon et al (1996) employed the IBM to study the complex biofilm system on a microscale level Zhu and Peskin (2002) applied the IBM to simulate a flapping flexible filament in a flowing soap film

In their work, the filament mass and elasticity, gravity, air resistance, and the two wires that bound the flowing soap film are considered By enforcing the boundary condition via a ghost cell method, Tseng and Ferziger (2003) presented an efficient IBM to study the turbulent flows in complex geometries Balaras (2004) took the direct forcing scheme to perform the large-eddy simulation (LES) around complex boundaries

1.2.2.2 Force calculation in IBM

One of the key issues in the IBM application is the calculation of the restoring force Basically, there are three ways to compute it The first way was proposed by Peskin (1977) in his original work of IBM, where the restoring force is considered as a spring

Chapter 1 Introduction

force and is calculated by the Hook’s law This way is also called the penalty force method In this method, the free parameters are required to be specified by the user Therefore, it would have a significant effect on the computational efficiency and accuracy The second way is called the direct forcing scheme, which was proposed by Fadlun et al (2000) This scheme computes the restoring force by applying the governing momentum equations of the flow field at the boundary points Similar to the direct forcing scheme, Lima E Silva et al (2003) proposed a version named physical virtual model to simulate an internal channel flow and the flow around a circular cylinder This model is very similar to the work of Peskin (1977) except that the restoring force is calculated by applying the momentum equations at the boundary points Based on the direct forcing scheme, Uhlmann (2005) presented an improved IBM, which greatly suppresses the force oscillations, to investigate the flow around suspended rigid particles An enhanced version of direct forcing scheme was recently proposed by Luo et al (2007) to simulate spherical particle sedimentation A nonlinear weighted technique and boundary point classification strategy at the immersed boundary are introduced to modify the velocity near the body The third way is termed as the momentum exchange scheme, which was proposed by Niu et al (2006) This way is only applicable to the lattice Boltzmann method (LBM) In the LBM, at the boundary points, some particles come along the incoming directions while some other particles leave along the outgoing directions So, overall, there is a momentum exchange at the boundary point According to the Newton’s second law, this momentum exchange can be used to calculate the restoring force at the boundary

Chapter 1 Introduction

point

1.2.2.3 Advantages and disadvantages of IBM

The major advantage of IBM is its simplicity and easy implementation This is attributed to the decoupling of the solution of governing equation with the boundary

In other words, the governing equation can always be solved on a regular domain without consideration of embedded boundary in the flow field The effect of boundary

on the flow field is through the introduction of force density in the governing equation

In this sense, if one has a code for simulation of flows around a sphere, this code can

be easily applied to simulate the flows around any complex geometry such as aircraft and submarine This is probably the reason why the IBM is becoming more and more popular in the CFD community

On the other hand, IBM also suffers from some drawbacks One of them is the low order accuracy caused by the use of discrete delta function interpolation To improve the accuracy of IBM results, Lai and Peskin (2000) proposed a so-called second-order IBM to simulate flows over a circular cylinder However, since they still use the first-order delta function interpolation, their model does not truly have the second-order accuracy Recently, an extended IBM named immersed finite element method (IFEM) was proposed by Wang and Liu (2004) and Zhang et al (2004) By applying the finite element technique to both fluid and object domains, the immersed body can be handled more appropriately and accurately On the other hand, the

of the non-slip boundary condition in IBM is in fact due to pre-calculated restoring force Using the fractional step technique, they concluded that, adding a body force in the momentum equations is equivalent to make a correction in the velocity field To enforce the non-slip boundary condition, the velocity correction (or restoring force) should be considered as unknown, which is determined in such a way that the velocity

at the boundary interpolated from the corrected velocity field satisfies the non-slip boundary condition In the work of Shu et al (2007), the velocity correction is made

at adjacent points to the boundary along the horizontal and vertical mesh lines The approach is very simple However, it only has the first-order of accuracy and the computed forces at the boundary have some oscillations The reason may be that the linear relationship is applied along the horizontal/vertical mesh lines and the smooth Dirac delta function is not used Its implementation process is more complicated than

Chapter 1 Introduction

the conventional IBM Practically, there is a demand to further develop a new version

of IBM which can be implemented easily, and in the meantime, can satisfy the non-slip boundary condition This demand motivates the present work

1.3 Lattice Boltzmann Method

In recent years, as an alternative and promising computational technique to the N-S solvers, the lattice Boltzmann method has achieved a great success since the 1980’s It has attained wide popularity in handling various flow problems (Chen and Doolen, 1998)

1.3.1 Features of LBM

Unlike the N-S solvers which are based on the macroscopic continuum assumption, the LBM is based on the microscopic models and mesoscopic kinetic equations It consists of two sequential sub-steps: streaming process and collision process The dependent variables in LBM are the density distribution functions The macroscopic variables, such as velocity and pressure, are computed from the moments of density distribution functions at the lattice nodes Using the Chapman-Enskog expansion, the LBM can recover the incompressible N-S equations with the second-order accuracy

As compared to the numerical methods which discretize N-S equations, the LBM

Chapter 1 Introduction

has three distinct advantages Firstly, the convection operator (streaming process) of LBM in phase space is linear In contrast, the convection terms in N-S equations are nonlinear Therefore, the computational efforts in LBM can be greatly reduced Secondly, the pressure in LBM can be directly calculated from the equation of state, which is different from the N-S solvers in which the pressure must be obtained from the Poisson equation Solving the Poisson equation usually produces numerical difficulties and some special treatments are required Thirdly, the LBM utilizes a minimal set of particle velocities in phase space In traditional Boltzmann equation with Maxwell equilibrium distribution, the phase space is a complete functional space The statistical process needs information from the whole phase space In LBM, on the other hand, since only limited number of speeds and a few moving directions are used, the transformation between the microscopic distributions and the macroscopic quantities is greatly simplified Only some simple arithmetic operations are used The LBM has been proven to be an efficient approach for various kinds of flow problems One of its attractive features is its simplicity, easy implementation, and parallel in nature

1.3.2 Historic development of LBM

The LBM is originated from the lattice gas cellular automata (LGCA), which was first introduced in 1973 by Hardy et al The LGCA is constructed as a simplified, fictitious molecular dynamic model in which space, time and particle velocities are all discrete

Chapter 1 Introduction

In 1986, Frisch, Hasslacher and Pomeau showed that LGCA with collisions that conserve mass and momentum, in the macroscopic limit, leads to the N-S equations when the underlying lattice guarantees isotropy However, the LGCA suffers from some drawbacks such as large statistical noise, non-Galilean invariance, unphysical velocity-dependent pressure and large numerical viscosities These shortcomings have greatly hampered its development as a good model in practical applications

To overcome the drawbacks of LGCA, LBM was developed The first LBM was introduced by McNamara and Zanetti in 1988 based on the LGCA Instead of using Boolean variables, it uses the continuous single-particle distribution which interacts locally and propagates after collision to the next neighboring node Fermi-Dirac distributions are used as the equilibrium distribution functions In 1989, Higuera and Jimenez firstly used the linearized collision operator in LBM, which can improve the numerical efficiency By using the single time BGK relaxation approximation (Koelman, 1991; Qian et al 1992), the collision operator, which is based on the rules

of LGCA, was further simplified These lattice BGK (LBGK) models mark a new level of abstraction: collisions are not defined explicitly anymore Since then, the LBGK model has been widely used

Succi et al (1989) used LBM to simulate the porous flow in a 3D random medium and confirmed Darcy’s law Gunstensen et al (1991) firstly developed a multi-component model using LBM In their work, two different fluids are represented by the red and blue particle distribution functions Ladd (1993) conducted

Chapter 1 Introduction

the study of the particle suspensions in fluid based on the LBM Dawson et al (1993) extended the LBM to describe a set of reaction-diffusion equations advected by the N-S equations Hou et al (1995) applied LBM to simulate 2D driven cavity flow covering a wide range of Reynolds numbers from 10 to 10,000 Benzi et al (1996) used LBM to study the 3D non-homogeneous turbulent flows such as the shear flow Due to its kinetic origin, the LBM has the potential to simulate micro-flows for which the continuum description is invalid It has been successfully applied to the pressure- driven micro-channel flow, mixing of binary fluids and electro-kinetic flow in micro- channels In addition, the LBM is also shown to be promising in several other applications, such as in viscoelastic flows, magneto hydrodynamics, and micro- emulsions

1.3.3 Implementation of boundary conditions in LBM

One of key issues in LBM is the implementation of boundary conditions The extensively used way to implement the non-slip boundary condition is the bounce back scheme, which was originally proposed by Wolfram (1986) In this scheme, the density distribution function just reverses its streaming direction when it contacts the boundary It is very simple for LBM to handle complex geometries in real applications However, this scheme has been found to have only the first-order accuracy at the boundaries (Ziegler, 1993)

To enhance the accuracy for implementation of boundary conditions in LBM, a

Chapter 1 Introduction

number of efforts have been devoted Ziegler (1993) proposed a half-way bounce back scheme with the second-order accuracy In this scheme, the boundary is shifted into fluid field by one half mesh spacing Nobel et al (1995) presented a hydrodynamic boundary condition The unknown density distribution functions at the boundary are calculated under the conservation laws of mass, moment and energy Such kind of schemes can also be used in other boundaries such as the inlet and outlet boundary of channel flows Zou and He (1997) extended the bounce back scheme to

be used for the nonequilibrium part of density distribution functions By viewing the lattice Boltzmann equation (LBE), which is discrete in both velocity space and physical space, as a special finite difference approximation of the discrete Boltzmann equation (DBE), which is discrete in velocity space and continuous in physical space, Chen et al (1996) calculated the unknown density distribution functions at the boundary by using the second-order extrapolation Filippova and Hänel (1998) developed the boundary fitting condition for arbitrarily shaped boundaries Using the bounce back rule, the density distribution functions at the boundary are extrapolated with the second-order accuracy Bouzidi et al (2001) employed the quadratic interpolation scheme to implement the bounce back boundary condition The second-order accuracy is obtained Different from extrapolation, the treatment with interpolation is superior in terms of stability Recently, Chun and Ladd (2007) proposed an equilibrium interpolation lattice Boltzmann model for simulation of flows in narrow gaps Only the equilibrium part of density distribution function is interpolated at the boundary to achieve the second-order accuracy

Chapter 1 Introduction

1.3.4 Some efforts on improvement of computational efficiency

To improve the computational efficiency for solving lattice Boltzmann equation, Filippova and Hänel (1998) proposed a local mesh refinement LBM The simulation

is first carried out on the basic coarse mesh covering the whole domain In certain critical region, the fine mesh is employed The related variables on both meshes are advanced simultaneously The density distribution functions on the coarse mesh are calculated with the second-order interpolation in space and time at the boundary nodes of fine mesh Yu et al (2002) combined the multi-block technique with LBM

In their scheme, the flow field is divided into several blocks In each block, the mesh

is uniform with desired resolution The blocks with different mesh sizes do not overlapped with each other This differs from the local mesh refinement LBM The accurate and conservative interface treatment between connected blocks is adopted The continuity of mass, momentum and stresses across the interface is enforced However, due to existence of a few blocks, it is complicated and tedious to deal with the information on the interface, especially for the case of 3D simulation

1.3.5 Applications of LBM with curved boundary and/or non-uniform mesh

Due to restriction of lattice isotropy in the physical space, the standard LBM is limited

to the application on the uniform Cartesian mesh The computational domain must also be regular However, in practice, the flow problems often involve complex geometry, and the use of non-uniform mesh or unstructured mesh is preferred This is

Chapter 1 Introduction

because in the region near the boundary, we need to use very fine mesh to capture the thin boundary layer, but in the region far away from the boundary where the velocity gradient is small, we can use coarse mesh to improve the computational efficiency To apply LBM for simulation of flows with curved boundary and/or non-uniform mesh, various approaches have been proposed These approaches can be roughly classified into two categories One is based on the solution of partial differential equations The other is based on the interpolation

For the first category, the lattice Boltzmann equation can be converted into partial differential equations by using truncated Taylor series expansion in space and time up to the first order derivative terms These differential equations can then be solved by conventional CFD techniques such as finite difference (FD), finite element (FE) and finite volume (FV) methods Based on the higher-order time marching scheme, Cao et al (1997) developed a finite difference LBM (FDLBM) Mei and Shyy (1998) extended this method to curvilinear coordinates with the help of coordinate transformation Using a piecewise constant and piecewise linear interpolation to compute the flux, Succi et al (1995) first proposed a finite volume LBM (FVLBM) In order to minimize the numerical diffusion, a free parameter is introduced and adjusted in this method With the properly chosen forms of state flux functions, Chen (1998) offered a different finite volume scheme of LBM Since the topology of mesh is not arbitrary, the aforementioned versions of FVLBM with irregular meshes are not satisfactory From the point of view of modern finite volume method, Peng et al (1998) presented a new finite volume LBM by using arbitrary