IBM kang thesis

AN IMPROVED IMMERSED BOUNDARY METHOD FOR COMPUTATION OF

TURBULENT FLOWS WITH HEAT TRANSFER

A DISSERTATION

SUBMITTED TO THE DEPARTMENT OF MECHANICAL ENGINEERING

AND THE COMMITTEE ON GRADUATE STUDIES

OF STANFORD UNIVERSITY

IN PARTIAL FULFILLMENT OF THE REQUIREMENTS

FOR THE DEGREE OF

DOCTOR OF PHILOSOPHY

Seongwon Kang

June 2008

c Copyright by Seongwon Kang 2008All Rights Reserved

ii

I certify that I have read this dissertation and that, in my opinion, it is fully

adequate in scope and quality as a dissertation for the degree of Doctor of

Philosophy.

(Parviz Moin) Principal Co-Adviser



Philosophy.

(Gianluca Iaccarino) Principal Co-Adviser



Philosophy.

(Heinz Pitsch)

Approved for the University Committee on Graduate Studies.

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Abstract

The immersed boundary (IB) method is a technique to enforce boundary conditions on sur-

faces not aligned with the mesh in a numerical simulation. This method has been used as

a practical approach to model ow problems involving very complex geometries or moving

bodies. Our objective is to assess the accuracy and eciency of the IB method in simula-

tions of turbulent ows, where the ow dynamics in the near-wall region is fundamental to

correctly predict the overall ow. The rst part of this work focuses on the development of a

simulation tool based on the IB method that can correctly predict the wall temperature and

pressure uctuations in turbulent ows. In the second part, we illustrate the application of

the method to a multi-material heat transfer problem where convective heat transfer of the

uid and conductive heat transfer of the solid are handled simultaneously.

This work achieves sucient accuracy at the immersed boundary and overcomes decien-

cies in previous IB methods by augmenting the formulation with additional constraints a compatibility constraint relating the interpolated velocity boundary condition with mass

conservation and a decoupling constraint for the pressure. We derived an IB method with

a revised boundary interpolation and a strictly mass conserving scheme, which does not

show pressure oscillations near the immersed boundary. Although accurate, the complexity

of this method prompted the development of another variant the immersed boundary-approximated domain method (IB-ADM). This approach satises the pressure decoupling

constraint with an inexpensive computational overhead. The IB-ADM correctly predicts

the near-wall velocity, pressure and scalar elds in several example problems. The IB-ADM

is shown to successfully predict the ow around a very thin solid object for which incor-

rect results were obtained with previous IB methods. The IB-ADM has been successfully

validated through computation of the wall-pressure space-time correlation in DNS of a tur-

bulent channel ow. When applied to a turbulent ow around an airfoil, the computed ow

statistics the mean/RMS ow eld and power spectra of the wall pressure are in good

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agreement with a previous LES and experiment.

In order to establish the viability of the IB method as an ecient tool for LES/DNS

of conjugate heat transfer applications, the problem of a heated cylinder in a channel with

heating from below is considered. Here, the uid-solid interface is constructed as a collec-

tion of disjoint faces of control volumes associated to dierent material zones. Coupling

conditions for the material zones have been developed such that continuity and conservation

of the scalar ux are satised by a second-order interpolation. The local mesh renement

technique is crucial to accommodate the large dierence in length scales in the present ap-

plication (i.e., small heated cylinder in a large channel). In the region upstream of the

transition to turbulence, numerical predictions show a strong sensitivity to the mesh resolu-

tion and inlet condition. Predictions of the local Nusselt number show good agreement with

the experimental data. The eect of the Boussinesq approximation on this problem was also

investigated. Comparison with the variable density formulation suggests that, in spite of a

small thermal expansion coecient of water, the variable density formulation in a transi-

tional ow with mixed convection is preferable, since it does not involve the uncertainty in

the material properties required in the Boussinesq approximation.

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Acknowledgements

Financial support for this work was provided by the Department of Energy under the Ad-

vanced Simulation and Computing (DOE-ASC) program. Computational resources for this

research were provided by the Lawrence Livermore National Laboratory under the DOE-

ASC program. Parts of this work were done in collaboration with Dr. Frank Ham. The

authors acknowledge his helpful suggestions as well as his eorts in the development of

the CDP - ow solver. The authors gratefully acknowledge professors Heinz Pitsch, Javier

Jimenez, Ugo Piomelli, Meng Wang, and Dr. Stephane Moreau for their helpful comments

and suggestions. Also, comments and help from Dr. Prasun Ray, Dr. Lawrence Cheung, Ms.

Laurie Gibson, Mr. Edward Perry, and Mr. Shunn Lee in preparation of this manuscript

are greatly appreciated.

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Contents

Abstract iv

Acknowledgements vi

1 Introduction 1

1.1 Motivation and objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Accomplishments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2 Numerical methods 9

2.1 Description of the Navier-Stokes solvers . . . . . . . . . . . . . . . . . . . . . 9

2.2 Generation of locally rened meshes . . . . . . . . . . . . . . . . . . . . . . . 11

3 Treatment of immersed boundary 14

3.1 Basic equations of the IB method and general issues . . . . . . . . . . . . . . 14

3.2 Treatment of mass conservation near immersed boundary . . . . . . . . . . . 19

3.3 Treatment of velocity and scalar - standard reconstruction method . . . . . . 21

3.4 Issues in the standard reconstruction method . . . . . . . . . . . . . . . . . . 25

3.5 A revised reconstruction method with mass conservation for reshaped CV . . 31

3.5.1 Revised interpolation methods . . . . . . . . . . . . . . . . . . . . . . 31

3.5.2 Enforcing mass conservation for reshaped CV . . . . . . . . . . . . . . 35

3.5.3 Test cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.6 Immersed boundary-approximated domain method . . . . . . . . . . . . . . . 44

3.6.1 Enforcing mass conservation for approximated domain . . . . . . . . . 45

3.6.2 Implementation for the fractional step method . . . . . . . . . . . . . 49

3.6.3 Interpolation method for the velocity . . . . . . . . . . . . . . . . . . . 50

3.6.4 Accuracy test and comparison study . . . . . . . . . . . . . . . . . . . 50

vii

3.7 Implementation for a multi-material problem (conjugate heat transfer) . . . . 56

4 Verication and validation studies 62

4.1 Verication study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

4.1.1 Developing boundary layer over a thin plate . . . . . . . . . . . . . . . 62

4.1.2 Flow around a heated sphere . . . . . . . . . . . . . . . . . . . . . . . 64

4.2 DNS of a turbulent channel ow at Re=180 . . . . . . . . . . . . . . . . . . 65

4.2.1 Eect of the IB method on the DNS results . . . . . . . . . . . . . . . 71

4.3 LES of the turbulent ow around an airfoil . . . . . . . . . . . . . . . . . . . 73

4.4 A heated cylinder in a channel heated from below . . . . . . . . . . . . . . . . 81

4.4.1 Experimental conguration . . . . . . . . . . . . . . . . . . . . . . . . 82

4.4.2 Computational setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

4.4.3 Eect of grid resolution . . . . . . . . . . . . . . . . . . . . . . . . . . 85

4.4.4 Eect of inow condition . . . . . . . . . . . . . . . . . . . . . . . . . 89

4.4.5 Results with conjugate heat transfer . . . . . . . . . . . . . . . . . . . 94

4.4.6 Eects of the Boussinesq approximation . . . . . . . . . . . . . . . . . 97

5 Conclusions and future work 99

Bibliography 102

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List of Tables

3.1 Non-dimensionalized frequency of the vortex shedding. . . . . . . . . . . . . . 43

4.1 RMS wall pressure scaled by the wall shear stress. . . . . . . . . . . . . . . . 67

4.2 Mesh spacings in wall units for grids in the heated cylinder case. . . . . . . . 85

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List of Figures

1.1 Preparation steps for a simulation with the body-tted mesh and IB method. 2

1.2 Examples of a locally rened mesh and body-tted mesh. . . . . . . . . . . . 7

2.1 Examples of user-input keywords for local mesh renement. . . . . . . . . . . 13

3.1 A very thin solid object between two channels with ows in the opposite

directions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3.2 Examples of velocity eld mirrored across the IB: (a) collocated grid; (b)

staggered grid. A square denotes a grid. . . . . . . . . . . . . . . . . . . . . . 17

3.3 A local coordinate at the immersed boundary. . . . . . . . . . . . . . . . . . . 18

3.4 Dierent schemes for dening control volumes (CVs) for mass conservation

near IB: (a) standard scheme; (b) mass conservation for reshaped CVs; (c)

mass conservation for uid-side CVs. The shaded area denotes a control

volume where mass conservation is enforced. . . . . . . . . . . . . . . . . . . . 20

3.5 Conguration of the immersed boundary, grid and nodes in the linear inter-

polation method: , immersed boundary; , grid; , velocity node.uc is the reconstructed velocity. . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.6 Grid conguration for a simplied one-dimensional case. u denotes a velocity

node and p denotes a pressure node. . . . . . . . . . . . . . . . . . . . . . . . 24

3.7 An example of control volumes crossed by the IB in a staggered mesh. . . . . 26

3.8 Flow eld and grid conguration of laminar ow over a wedge: (a) streamlines

and pressure contours; (b) conguration of the grid lines and IB. . . . . . . . 27

3.9 Time trace of u, u, v, v, and pressure around an IB cell shown in Fig-ure 3.8 (b) of the steady laminar ow over a wedge: , u/U0; ,u/U0; M, v/U0; O, v/U0; , pressure. . . . . . . . . . . . . . . . . . . . . . . 27

x

3.10 Contours of the x-velocity and pressure with the standard reconstruction

method. Bold lines denote boundaries of the channels. . . . . . . . . . . . . . 29

3.11 Contours of the x-velocity with the standard reconstruction method with the

periodic B.C. and a constant momentum forcing term. . . . . . . . . . . . . . 30

3.12 Proles of the x-velocity at x=0 with the standard reconstruction method

with dierent B.Cs.: , the exact solution; , with the parabolic velocityB.C. at x/=-10 and 10; , with the periodic B.C. and a constant

momentum forcing term. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.13 Conguration of the immersed boundary, grid and nodes in the quadratic+momentum

interpolation method: , immersed boundary; , grid; , velocitynode. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.14 Treatment of the divergence of the velocity for a control volume crossed by

the IB. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3.15 Grid and IB conguration for a decaying vortex problem: (a) IB lines aligned

on the grid lines; (b) IB lines inclined by 45 with respect to the grid lines;

(c) IB line of a circular shape. . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.16 Maximum error in u1 at t=0.2 with the dierent settings for mass conserva-

tion in the rotated square IB: , with the standard mass conservation and

the exact solution at the boundary outside of the IB; , with the stan-

dard mass conservation and the exact solution at t=0 left unchanged at the

boundary outside of the IB; , with the approximate mass conservation;

, with the strict mass conservation. The LIM and the rotated square

IB geometry are used. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3.17 Maximum error in u1 and p at t=0.2 with the linear interpolation schemes:

, LIM; , RLIM; , QMIM. The lines with symbols denote

pressure. In square IB, dashed line is masked by solid line. . . . . . . . . . . . 40

3.18 Instantaneous pressure contours from the dierent interpolation methods: (a)

LIM; (b) RLIM; (c) QMIM. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.19 Maximum error in u1 and p at t=0.2 with the dierent interpolation meth-

ods: , RLIM; , RLIM with the strict mass conservation; ,

QMIM; , QMIM with the strict mass conservation. The lines with

symbols denote pressure. The circular IB geometry is used. . . . . . . . . . . 41

xi

3.20 Time-trace of the wall pressure at = 70: , LIM; , mixed

LIM; , RLIM; , QMIM. = 0 and = 180 correspond to the

stagnation and base points, respectively. . . . . . . . . . . . . . . . . . . . . . 43

3.21 Time-averaged wall pressure coecients around a circular cylinder: , Parket al. (1998); , Case A; , Case B; , Case C; , Case D. 44

3.22 Examples of the approximated domain and boundary. . . . . . . . . . . . . . 46

3.23 Examples of the interpolated velocity components on the approximated bound-

ary a in dierent arrangements of the velocity variables. . . . . . . . . . . . 46

3.24 Maximum error in u1, p and T at t=0.2 with the IB-ADM: , u1; ,

p; , T . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

3.25 Contours of the x-velocity with the IB-ADM and standard reconstruction


3.26 Contours of the pressure with the IB-ADM and standard reconstruction


3.27 Proles of the x-velocity at x=0 with the IB-ADM and standard reconstruc-

tion method: , the exact solution; , IB-ADM; , standard re-construction method (SRM); , SRM with doubled the number of mesh

points in x- and y-directions. . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

3.28 Locally rened mesh for a laminar ow around a circular cylinder. . . . . . . 54

3.29 Time-averaged wall-pressure coecients around a circular cylinder: , Parket al. (1998); , IB-ADMwith CFL=1.6; , IB-ADMwith CFL=0.4;

, standard reconstruction with CFL=1.6; , standard reconstruc-

tion with CFL=0.4: (a) near the stagnation point (b) the whole range. . . . . 55

3.30 Time-averaged wall-pressure coecients around a circular cylinder: ,

IB-ADM; , stair-step approximation; , stair-step approximation

with a coarse mesh (doubled grid spacings). . . . . . . . . . . . . . . . . . . . 56

3.31 Schematic diagrams for interface treatment between dierent materials: (a)

true uid-solid interface; (b) construction of approximated boundaries facing

each other (fluid and solid); (c) computation of interpolation coecients

from the projected boundaries. . . . . . . . . . . . . . . . . . . . . . . . . . . 57

3.32 A conjugate heat transfer problem of a rotating ow between two coannu-

lar cylinders. Thick lines denote boundaries where temperature boundary

condition is imposed. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

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3.33 Maximum errors in the velocity and temperature for the conjugate heat trans-

fer problem between two coannular cylinders: , x-velocity; , tem-

perature. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

4.1 Locally rened mesh for developing laminar boundary layer over a thin plate. 63

4.2 The skin-friction coecient in laminar boundary layer: , Blasius solu-

tion; , present result. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 634.3 Locally rened mesh for a heated sphere. . . . . . . . . . . . . . . . . . . . . . 64

4.4 Results of the case with Gr = 104: (a) contours of the y-velocity and tem-

perature; (b) heat transfer coecients. , Jia & Gogos (1996); , the presentstudy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

4.5 Locally rened meshes for a turbulent channel ow. . . . . . . . . . . . . . . . 66

4.6 Mean streamwise velocity proles in wall units: , Kim et al. (1987); ,inclined (IB) case; , body-tted case. . . . . . . . . . . . . . . . . . . . 67

4.7 RMS velocity proles in wall units: , Kim et al. (1987); , inclined (IB)case; , body-tted case. . . . . . . . . . . . . . . . . . . . . . . . . . . 68

4.8 Energy spectra of the streamwise velocity and pressure at y+ = 5 in the

streamwise (x) and spanwise (z) directions: , Kim et al. (1987) (velocity),Moser et al. (1999) (pressure); , inclined (IB) case; , body-tted

case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

4.9 Wall-pressure power spectra (t): , Choi & Moin (1990); , inclined(IB) case; , body-tted case; , Cartesian collocated 10M mesh. . 69

4.10 Wall-pressure power spectra (t) with dierent time periods and a Cartesian

staggered 10M mesh: , Choi & Moin (1990); , the rst run; ,the second run; , the third run; , the fourth run. . . . . . . . . . . 70

4.11 Convection velocity Uc(rx) scaled by the centerline velocity U0: , Choi &Moin (1990); +, inclined (IB) case; , body-tted case; M, Cartesian collo-cated 10M mesh; O, Cartesian staggered 10M mesh. . . . . . . . . . . . . . . 714.12 Mean streamwise velocity proles in wall units: , inclined (IB) case; ,stair-step approximation with y+ = yu/; , stair-step approximation

with the down-shifted coordinate y+ = (y k)u/. . . . . . . . . . . . . . . 724.13 Geometry of the full domain RANS simulation (left) and sub-domain LES

calculation (right) and contours of the streamwise velocity. . . . . . . . . . . . 73

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4.14 The locally rened mesh used in the present study and a reference Cartesian

mesh for the airfoil. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

4.15 Contours of the instantaneous x-velocity and x-vorticity. . . . . . . . . . . . . 75

4.16 Averaged velocity proles at several x-locations: , Wang et al. (2004);, IB method; , body-tted. . . . . . . . . . . . . . . . . . . . . . . . . 76

4.17 Averaged wall-pressure coecients: , Roger & Moreau (2004); , Wang et al.(2004); , IB method; M, body-tted. . . . . . . . . . . . . . . . . . . . . . . 774.18 RMS velocity proles at several x-locations: , Wang et al. (2004); , IBmethod; , body-tted. . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

4.19 Contours of instantaneous x-velocity with meshes: (a) Wang et al. (2004); (b)

body-tted; (c) IB method. A gray circle indicates a region where velocity

wiggles are observed. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

4.20 Wall-pressure power spectra at the trailing edge: , Roger & Moreau (2004);, Wang et al. (2004);, IB method; - - - -, body-tted. . . . . . . . . . . 804.21 A streamline over the airfoil and contours of the convection velocity along

this streamline. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

4.22 Schematic diagram of a heated cylinder inside a channel heated from below. . 82

4.23 The locally rened mesh for the heated cylinder case. . . . . . . . . . . . . . . 84

4.24 Contours of the instantaneous temperature using dierent grids in Table 4.2. . 86

4.25 Proles of the averaged streamwise velocity at 5cm upstream of the cylinder

using dierent grids in Table 4.2: , experiment (Laskowski et al., 2007); ,RANS (Laskowski et al., 2007); , Grid #1; , Grid #2; ,

Grid #3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

4.26 The time-averaged heat ux at the outer cylinder using dierent grids in Ta-

ble 4.2: , experiment (Laskowski et al., 2007); , RANS (Laskowski et al.,2007); , Grid #1; , Grid #2; , Grid #3. 0 and 90 corre-

spond to the forward stagnation point and the top of the cylinder. . . . . . . 87

4.27 Proles of the square root of the kinetic energy at 5cm upstream of the

cylinder using dierent grids in Table 4.2: , experiment (Laskowski et al.,2007); , RANS (Laskowski et al., 2007); , Grid #1; , Grid #2;, Grid #3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

4.28 Contours of the instantaneous streamwise velocity and temperature using

Grid #3 in Table 4.2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

xiv

4.29 Proles of the averaged streamwise velocity at x = 0 using dierent inlet

velocity proles: , interpolated from Laskowski et al. (2007); ,

uniform (Uin); , recycled from x=36cm. . . . . . . . . . . . . . . . . . . 90

4.30 Proles of the averaged streamwise velocity at 5cm upstream of the cylinder

using dierent inlet velocity proles: , experiment (Laskowski et al., 2007);, RANS (Laskowski et al., 2007); , interpolated from Laskowski et al.(2007); , uniform (Uin); , recycled from x=36cm. . . . . . . . . . 91

4.31 Proles of the square root of the kinetic energy at 5cm upstream of the

cylinder using dierent inlet velocity proles: , experiment (Laskowski et al.,2007); , RANS (Laskowski et al., 2007); , interpolated from Laskowskiet al. (2007); , uniform (Uin); , recycled from x=36cm. . . . . . . 91

4.32 Contours of the instantaneous streamwise velocity using dierent inlet veloc-

ity proles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

4.33 Contours of the instantaneous temperature using dierent inlet velocity proles. 93

4.34 The time-averaged heat ux at the outer cylinder using dierent inlet velocity

proles: , experiment (Laskowski et al., 2007); , RANS (Laskowski et al.,2007); , interpolated from Laskowski et al. (2007); , uniform

(Uin); , recycled from x=36cm. 0 and 90 correspond to the forward

stagnation point and the top of the cylinder. . . . . . . . . . . . . . . . . . . . 93

4.35 The time-averaged heat ux at the outer cylinder: , experiment (Laskowskiet al., 2007); , RANS (Laskowski et al., 2007); , with solid conductionand the interpolated inow; , with solid conduction and the recycled

inow; , without solid condition and the interpolated inow. 0 and 90

correspond to the forward stagnation point and the top of the cylinder. . . . . 94

4.36 RMS temperature at the outer cylinder for the cases with solid conduction:

, with the interpolated inow; , with the recycled inow. 0 and

90 correspond to the forward stagnation point and the top of the cylinder. . 95

4.37 Contours of the instantaneous temperature with solid conduction (side view

(x y)). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 954.38 Contours of the instantaneous temperature with solid conduction (top view

(x z) at y=1.43cm). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

xv

4.39 Proles of the averaged streamwise velocity and the square root of the kinetic

energy at 5cm upstream of the cylinder: , experiment (Laskowski et al.,2007); , Boussinesq approximation; , variable density formulation. 97

4.40 Contours of the instantaneous streamwise velocity using dierent reference

temperatures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

xvi

Chapter 1

Introduction

1.1 Motivation and objectives

Simulations based on CFD (Computational Fluid Dynamics) have expanded their role in

both scientic research and engineering analysis. Accurate CFD simulations are regarded

as a cost-eective alternative to the experiments in various industries.

As CFD becomes an important tool in industrial applications, mesh generation has

become a key issue. Workstation performance as well as numerical algorithms for CFD

have evolved rapidly; as a result, simulation time has decreased steadily. However, the time

required for mesh generation has not decreased substantially. Often mesh generation is

the major bottleneck, especially for realistic and complex machinery components. Several

factors contribute to this situation. First, the mesh-generation process is still largely manual.

Automatic generation of a body-tted mesh is not always possible. Moreover, there are

many situations where the quality of an automatically generated mesh is not satisfactory.

Because a good mesh is optimized for the solution of a specic problem, expert knowledge is

often necessary throughout its development. Although adaptive mesh approaches have been

developed, mesh generation aided by an engineer's experience still produces the best results.

The second reason for the diculties in mesh generation is related to the fact that it is largely

a serial process. Recent speed-up of CFD simulations has benetted greatly from evolution

of the parallel computing environment. The steps involved in generating a body-tted mesh

are (i) construction of the surface mesh from a CAD design, and (ii) generation of the

volume mesh from the surface mesh. In many cases, the preparation of the surface mesh is

the most time-consuming phase because the imperfections present in the CAD model must

1

CHAPTER 1. INTRODUCTION 2

Body-fitted mesh

1. Geometry

IB method

1. Mesh

2. Mesh 2. Geometry

Body-fitted mesh

1. Geometry

IB method

1. Mesh

2. Mesh 2. Geometry

Figure 1.1: Preparation steps for a simulation with the body-tted mesh and IB method.

be xed in order to guarantee water-tightness. Also, this phase often requires trimming

or approximation of tiny parts dicult to resolve with the mesh. Generation of a large

mesh can be parallelized by generating meshes in sub-domains. However, mesh structure

and resolution in a sub-domain are often strongly non-local (i.e., aected by the other sub-

domains), and therefore most algorithms remain sequential. These two factors lead to an

imbalance in time devoted to simulation versus grid generation. It is easy to nd an example

where generating a body-tted mesh takes a few months, whereas the corresponding RANS

simulation only requires a few days.

The immersed boundary (IB) method has emerged as an alternative, since it can relax the

diculty and time requirement of mesh generation. The hallmark of the IB method is that

it does not require the computational mesh to conform to the physical boundaries. Instead,

the solution algorithm is locally modied to enforce the desired boundary conditions. This

feature is attractive for very complex geometries, because a very simple mesh structure, such

as a Cartesian mesh, can be used. Body-tted and the IB meshes are notionally dierent

in the preparation step: for the former, the geometry is specied rst, followed by mesh

generation (Figure 1.1); in IB methods, a mesh is easily constructed, then the eect of the

geometry is imposed in the solution step.

The IB method is largely advantageous in three situations. First, it provides an alter-

native to body-tted mesh for very complex geometry; in this case, the amount of human

work and time for mesh generation can be reduced by using a simpler mesh structure. The

second advantage relates to problems associated with moving geometries. Since the eect


of geometry is imposed in the solution step, dynamically updating geometric changes in the

solution is relatively straightforward. Another advantage presents itself in multi-phase or

multi-material problems. More specically, the interface between dierent materials can be

regarded as an immersed boundary. The IB method is then equivalent to the imposition of

physical conditions at the interface.

The IB method was rst introduced by Peskin (1982) for computing blood ow in the

cardiovascular system. Subsequently, there have been numerous eorts to enhance the ac-

curacy, stability and range of applicability of the IB method. A short review on existing

IB methods is given here. Readers can refer to articles by Mittal & Iaccarino (2005) and

Iaccarino & Verzicco (2003) for further information on the previous studies.

In the work of Peskin (1982), the IB method was specically designed to handle deform-

ing (elastic) boundaries and low Reynolds number ows. The basic idea is to determine

localized forces for the Navier-Stokes equations at a set of Lagrangian points distributed on

the deforming boundaries in order to enforce the physical conditions. The deforming bound-

ary is modeled as elements with elastic (spring) links, and the forcing term is a function of

the deformation and elastic properties of the boundary. The singular forces at Lagrangian

points are transferred to Eulerian mesh points via a regularized Dirac delta function. Gold-

stein et al. (1993) generalized this concept as a two-mode feedback control with spring and

damping constants in order to enforce the boundary condition at the IB. These methods,

along with related studies (e.g., Grith & Peskin, 2005; Uhlmann, 2005; Taira & Colo-

nius, 2007), are based on a nite-width, regularized Dirac delta function and referred to

as continuous forcing techniques. This technique is notionally independent of the spatial

discretization and easy to implement in a Navier-Stokes solver. However, there are a few

drawbacks. Using a regularized Dirac delta function results in the boundary eect diusing

into the uid region, thus decreasing the solution accuracy. This issue motivated subsequent

studies (e.g., Grith & Peskin, 2005) which attempted to improve accuracy. Another issue

is that the spring and damping constants are problem-dependent, and often result in severe

restrictions on the time step, which then requires specic treatment (e.g., Lee, 2003; Newren

et al., 2007). Many continuous forcing techniques are also referred to as explicit forcing

techniques, because the forcing term is explicitly evaluated during computation.

There is another family of IB methods referred to as the direct forcing techniques, rst

developed by Mohd-Yusof (1997). In this method, the forcing term is computed so that

it directly compensates for the errors between the calculated velocities and the desired


velocities on the IB. Since the direct forcing technique is based on specic discretization

schemes, it enforces the exact boundary condition on the IB. This method has no problem-

dependent parameter, and does not suer from severe restriction on the time step. In

the ghost-cell method (Majumdar et al., 2001; Tseng & Ferziger, 2003), the method of

Mohd-Yusof (1997) was simplied by replacing the forcing in the Navier-Stokes equation

with a simple linear interpolation. Fadlun et al. (2000) introduced a direct forcing method

in which the velocities at the rst grid points into the uid region are reconstructed by

interpolating from the boundary condition and neighboring velocities. This method is also

called the reconstruction method (Gilmanov et al., 2003; Gilmanov & Sotiropoulos, 2005).

A method similar to Fadlun et al. (2000) and Tseng & Ferziger (2003) imposes the forcing

eect implicitly and is alternatively referred to as an implicit forcing technique. The direct

forcing technique is often referred to as the discrete forcing technique, since the forcing term

is closely related to the discretized Navier-Stokes equation as opposed to the continuous

forcing technique.

There are other methods in the discrete forcing family. The immersed interface methods

(Xu & Wang, 2006; Linnick & Fasel, 2005, among others) were developed to achieve a

high order accuracy with a formal discretization and a jump condition across the immersed

interface. The methods referred to as the Cartesian grid method (e.g., Coirier & Powell,

1993; Aftosmis et al., 1998) or cut-cell method (e.g., Ye et al., 1999; Tucker & Pan, 2000;

Udaykumar et al., 2001; Kirkpatrick et al., 2003) are based on constructing body-tted,

irregular grid cells near the IB. Then, a modied discretization similar to that used for

unstructured meshes is applied to enforce the boundary conditions. The term sharp interface

method has been used for most discrete forcing techniques. As opposed to the diuse nature

of the boundary condition in the continuous forcing techniques, the sharp interface methods

result in the boundary condition enforced as a step function across the boundary. The

sharp interface method in multi-phase ows denotes a method allowing discontinuity in the

solution across an interface between phases.

There are methods referred to as the ctitious domain methods which employ an op-

timization technique to enforce the physical boundary conditions. The ctitious domain

methods are categorized as explicit forcing techniques. Some of them (e.g., Bertrand et al.,

1997) employ the continuous forcing techniques, while the others (e.g., Yu & Shao, 2007)

employ the direct forcing techniques. These methods have been popular for particulate

ows.


So far, IB methods have been applied to a wide range of applications: compressible ows

(Ghias et al., 2007; Liu & Vasilyev, 2007), particulate ows (Uhlmann, 2005; Yu & Shao,

2007), micro-scale ows (Atzberger et al., 2007), interaction with solid bodies (Gilmanov &

Sotiropoulos, 2005; Zhao et al., 2008, among others), multi-phase ows (Ge & Fan, 2006),

conjugate heat transfer (Iaccarino & Moreau, 2006; Yu et al., 2006), environmental ows

(Smolarkiewicz et al., 2007), bio-uids (Fauci & Dillon, 2006), etc. However, only a small

number of studies have been published for cases with high Reynolds numbers.

Although many previous studies have reported theoretical improvements and encour-

aging results, several issues remain to be addressed. These issues are often very dierent

in each case, especially for the discrete forcing methods. In the cut-cell method, the nu-

merical complexity and the small-cell problem (the discretized governing equations become

sti when the volume of a cut-cell CV is very small) have been long-standing issues (Ghias

et al., 2007). In the immersed interface methods, restricted stability (Newren et al., 2007)

and aliasing errors (Zhao et al., 2008) have been reported. In the direct forcing techniques

(Mohd-Yusof, 1997; Fadlun et al., 2000; Tseng & Ferziger, 2003), issues related to enforcing

mass conservation have been raised. As an example, in some approaches, mass conservation

at the IB is satised by the velocity elds both in the uid and solid regions. Although

these methods have been classied as sharp interface methods, both pressure and pressure

gradient are continuous across the IB, which implies that the articial ow eld in the solid

region may aect the uid region. This issue can become more serious in the reconstruction

methods, since treatment for the velocities at the rst grid points into the solid region is

notionally undened. The velocity eld in the solid is important because it contributes to

the computation of the divergence of the grid cell on the IB and can aect the pressure

near the IB. Both Gilmanov et al. (2003) and Choi et al. (2007) employed interpolation of

the pressure near the IB. Theoretically, the pressure interpolation is not necessary because

the pressure accuracy is guaranteed by correctly satisfying mass conservation. Iaccarino &

Verzicco (2003) observed that dierent treatments of the solid velocity eld do not aect

the ow eld in the uid region in their numerical experiments; this is tested in the present

study. Further issues on the general IB method and the reconstruction method will also be

discussed later.

In the literature, several IB methods have been applied to LES/DNS of various turbulent

ows: ow over a wavy wall (Tseng & Ferziger, 2003; Lee, 2003; Yang & Balaras, 2006),

ow over a sphere (Yun et al., 2006), ow inside a piston engine (Verzicco et al., 2000),


ow in an impeller-stirred tank (Verzicco et al., 2004), ow over a mannequin (Choi et al.,

2007), inclined channel (Ikeno & Kajishima, 2007), ow in a nuclear rod-bundle (Ikeno &

Kajishima, 2007), ow over a building (Smolarkiewicz et al., 2007), stator-rotor interaction

(Tyagi & Acharya, 2005), etc. Most of these studies have focused on the ow eld away

from the wall. In some cases, the distribution of the time-averaged pressure is shown, but

there is no previous study which presents high-order statistics of the wall pressure at the IB.

Thus, it is very important to assess the ability of the IB method to predict wall variables

correctly in turbulent simulations.

Another important issue arising in LES/DNS of turbulent ows is generation of a suf-

ciently resolved mesh. While the combination of a Cartesian mesh and the IB method

has attractive features, it has a limitation in practical cases, especially in complex turbu-

lent ows. The Kolmogorov scale is proportional to Re3/4; thus, the smallest mesh size

decreases as the Reynolds number increases. In order to reduce the total number of mesh

points, the mesh size in a region with a small velocity gradient (e.g., in the far-eld) needs

to remain large. Thus, the ratio of the largest and smallest mesh sizes increases with the

Reynolds number in turbulent simulations. As a result, a ow with a high Reynolds number

may not be handled eciently with a purely Cartesian mesh.

A solution to this problem is a mesh structure that allows easy local renement/coarsening

while remaining simpler than an unstructured body-tted mesh. The local mesh renement

technique in the present study uses a Cartesian hexahedral mesh as the basic element. Local

mesh resolution is increased by recursively dividing a hexahedral mesh cell in one or more

directions, which allows both isotropic and anisotropic renements. This recursive mesh

renement maintains the hexahedral shape of mesh cells throughout the domain with the

capability of local renement or coarsening (Aftosmis et al., 1998). Figure 1.2 shows ex-

amples of a locally rened mesh and body-tted mesh. Two geometric features of a locally

rened mesh are (i) hanging nodes, which enable intensive local mesh renement, and (ii)

Cartesian hexahedral mesh, which makes the development of an automatic mesh-generation

algorithm easy. The local mesh renement is discussed in Section 2.2.

The rst objective of the present study is to assess the eectiveness of the IB method to

correctly predict the wall temperature and wall pressure uctuations in turbulent ows. In

order to accomplish this, the pressure uctuations are computed for a channel ow and a ow

around an airfoil. Then, wall pressure RMS uctuations as well as the spectra are computed

and compared with the previous studies. In addition, we explore a turbulent conjugate heat


(a) A locally rened mesh (b) A body-tted mesh

Figure 1.2: Examples of a locally rened mesh and body-tted mesh.

transfer problem where the modes of convective and conductive heat transfer are handled

simultaneously. For this application, the ability of the IB method to handle multi-material

problems with ease becomes useful. Thus, the second objective is to assess the eectiveness

of the IB method for a turbulent conjugate heat transfer problem.

In the next chapter, the Navier-Stokes solver and the local mesh renement technique

are described. In Chapter 3, various aspects of the IB method are discussed. In order to

resolve identied issues and implement capability for multi-material problems, revisions to

existing IB methods are devised. In Chapter 4, results of the verication and validation

study are presented, followed by the conclusion in Chapter 5.

1.2 Accomplishments

The following list summarizes important contributions of this work:

A modied mathematical formulation of the IB method, including the pressure decou-pling constraint.

Analysis of the incompatibility issue between the interpolated velocity boundary con-dition and the continuity equation.

Modied interpolation methods with explicit contribution from the local pressure gra-dient.

Development of a new IB method with a revised interpolation method and mass con-servation for reshaped CV. It does not introduce pressure wiggles and satises the


pressure decoupling constraint.

Development of a new IB method called immersed boundary-approximated domainmethod. It satises the pressure decoupling constraint with an inexpensive computa-

tional overhead.

Validation of the IB method with space-time correlation of the wall-pressure from achannel DNS and power spectra of the wall-pressure from LES of an airfoil.

First application of LES/DNS with the IB method to a turbulent conjugate heattransfer problem.

Development of a 20,000-line add-on program for the IB method. It can import aCAD geometry and is merged seamlessly into the user interface of the comprehensive

ow solver package, CDP.

Chapter 2

Numerical methods

2.1 Description of the Navier-Stokes solvers

The Navier-Stokes equation and the continuity equation for unsteady incompressible viscous

ow in Cartesian coordinates are (in non-dimensional form):

uit

+uiujxj

= pxi

+1Re

2uixjxj

, (2.1)

uixi

= 0, (2.2)

where t is the time, Re = U0L/ is the Reynolds number, U0 is a reference velocity, L is a

reference length, is the kinematic viscosity, ui is the velocity component in the i direction,

p is the pressure non-dimensionalized by U20 , and is the density.

In the present study, the solution of the Navier-Stokes equations (2.1)-(2.2) is obtained

using two dierent solvers. The rst is a semi-implicit solver based on staggered-variable for-

mulation. Using the Crank-Nicolson method for diusion terms and the third-order Runge-

Kutta method for convective terms, the k-th sub-iteration of the multi-step procedure can

be written as:

uki uk1it

+ k

(uiujxj

)k1+ k

(uiujxj

)k2=

(k + k)pk

xi+

(k + k)2Re

[2ukixjxj

+2uk1ixjxj

]+O (t2) , (2.3)9

CHAPTER 2. NUMERICAL METHODS 10

where t is the time step and k and k are the discretization coecients. Second-order

central dierences are used for the spatial discretization of the dierential terms.

The second solver is called CDP

1

: a fully implicit LES solver based on an unstructured

collocated mesh. In this code, the time-staggered scheme of Pierce (2001) is employed.

Then, the discretized momentum equation at (n+ 1/2)-th time step is written as:

un+1i unit

+12

un+ 12j un+1ixj

+u

n+ 12

j uni

xj

=p

n+ 12

xi+

12Re

[2un+1ixjxj

+2unixjxj

]+O (t2) . (2.4)In CDP, the spatial derivatives are computed using a nite volume method (FVM) for

node-based collocated mesh. The ux at the face of a control volume is evaluated by

applying a second-order interpolation and mid-point rule for numerical integration. Further

details about CDP are available in Ham et al. (2006) and Ham (2007). It is more time-

consuming to use a solver for unstructured meshes with the IB method than to use a solver

specically developed for structured meshes. However, locally rened meshes (i.e., with

hanging nodes) can be treated without modication to the unstructured solver. Also, a

solver for unstructured meshes can easily support load-balanced computations in a massively

parallel computing environment.

To solve the discretized momentum (Eq. (2.3) or Eq. (2.4)) and continuity equations

(Eq. (2.2)) eciently, a variant of the fractional-step method (Kim & Moin, 1985) is em-

ployed. The overall procedure for Eq. (2.3) can be written as:[1

t (k + k)

2Re2

xjxj

]uki =

uk1it (k + k)p

k1

xi

k(uiujxj

)k1 k

(uiujxj

)k2+

(k + k)2Re

(2uk1ixjxj

), (2.5)

2

xjxj=

1(k + k)t

ukixi

, (2.6)

uki = uki (k + k)t

xi, (2.7)

1

CDP is named after Charles David Pierce (1969-2002)


pk = pk1 + , (2.8)

where u and are the intermediate velocity and pseudo-pressure.

In addition to incompressible ows, heat transfer problems with mixed convection are

considered. In this case, a variable-density formulation of the Navier-Stokes equations is

used:

uit

+uiujxj

= pxi

+

xj

[

(uixj

+ujxi

)]+ gi, (2.9)

t+uixi

= 0, (2.10)

h

t+ujh

xj=

xj

[kT

xj

], (2.11)

where is the density, is the molecular viscosity, and gi is the vector of the gravitational

acceleration. T is the temperature, h = cpT is the enthalpy, and k is the thermal conductiv-

ity. The same temporal and spatial schemes are used for these governing equations as are

used in the incompressible version. Details are available in Ham (2007).

2.2 Generation of locally rened meshes

As discussed in Section 1.1, the resolution requirements in a turbulent simulation can be

widely dierent within the computational domain, especially in complex geometries.

Locally rened mesh, sometimes referred to as adaptive mesh renement (AMR) has

been discussed in several previous studies (Berger & Oliger, 1984; Aftosmis et al., 1998;

Balsara & Norton, 2001; Iaccarino et al., 2004, among others). In the early studies (e.g.,

Berger & Oliger, 1984; Coirier & Powell, 1993), a hierarchical tree structure is used to

record the connectivity of mesh elements at dierent renement levels. For example, the

octree structure was used eciently with isotropic mesh renement. Several later studies

(e.g., Aftosmis et al., 1998; Ham et al., 2002) used the fully unstructured approach which

handles the elements with hanging nodes as polyhedra. Although the unstructured approach

uses more memory than the tree structure, it has the advantage of allowing anisotropic

(directional) renement. Another approach was presented in Iaccarino et al. (2004) which


uses an underlying structured grid to build the connectivity information for non-isotropic

renement. In the present study, the unstructured approach is naturally employed by the

ow solver.

Based on the features noted in the previous studies, locally rened mesh structure pro-

vides (i) easy control of the local resolution; (ii) a fast turn-around time with respect to

unstructured mesh generation for complex geometries; (iii) development of automatic mesh-

generation strategies. Another advantage, although not utilized in the present study, is the

possibility of dynamic change of mesh resolution during run-time. Drawbacks of the local

mesh renement are (i) requirement of more grid points compared to a body-tted mesh in

turbulent simulations, and (ii) potential introduction of numerical error due to the treat-

ment of interfaces between ne and coarse meshes (i.e., hanging nodes). Compared to a

body-tted mesh, a Cartesian locally rened mesh needs more mesh points at non-aligned

boundaries. In order to achieve the desired mesh resolution in the wall-normal direction

near a boundary, a body-tted mesh may require grid renement only in the wall-normal

direction. The other directions are naturally uncoupled. On the other hand, a Cartesian

locally rened mesh needs renement in all directions, if the boundary is not aligned with

one of the Cartesian coordinates. In the worst situation, the mesh at the wall needs to be

a cube of the size 1/

3 of the desired wall-normal mesh spacing.

Several approaches have been used in the previous studies for the treatment of hanging

nodes. Iaccarino et al. (2004) and Ham et al. (2002) used formal discretization methods

that treat hanging nodes as unstructured meshes and enforce the ux conservation. Durbin

& Iaccarino (2002) uses a simple interpolation formula to treat the hanging nodes. Pantano

et al. (2007) used a WENO (weighted essentially non-oscillatory) scheme for AMR meshes.

The method of Shari & Moser (1998) handles interfaces between ne and coarse meshes

by a mesh embedding technique. In all methods, the size of error at hanging nodes is larger

than that from standard mesh points. In the present study, meshes around hanging nodes

are treated as polyhedral meshes. The present ow solver employs a nite volume method

for unstructured meshes, and there is no need for special treatment for hanging nodes. The

ow solver achieves second-order accuracy for Cartesian meshes, but the accuracy is reduced

to rst order at hanging nodes. Even in this case, the L-2 norm of the error is observed to

be the second order, since the fraction of hanging nodes over total nodes is never large.

In the present study, the generation of meshes is based on a few simple algorithms that

control the local resolution. Figure 2.1 shows algorithms used in the examples presented in


Box window

(a)

Segment window

(b)

Surface refinement

(c)

Layer window

(d)

Figure 2.1: Examples of user-input keywords for local mesh renement.

this study. Figure 2.1 (a) shows a Box window, which enforces a prescribed mesh resolution

inside a box; (b) shows a Segment window, which enforces a prescribed mesh resolution

in the region within a distance from a line segment; (c) shows Surface renement, which

enforces a prescribed mesh resolution along the surface of a solid body (given in terms of the

normal and tangential resolutions); and (d) shows a Layer window (also called an Eikonal),

which enforces a prescribed mesh resolution within a distance from a solid body. Surface

renement is useful to achieve a desirable mesh resolution near the wall in LES and RANS.

A Layer window can be used to achieve a smooth mesh transition from the boundary to

the far-eld and is especially useful in LES.

The complexity of the mesh generation process can be measured by a combination of

two factors: the amount of human work and the time needed to build the mesh. With the

present tools, the amount of human work is directly proportional to the number of user

inputs required to control the mesh renement. We found that the algorithms illustrated

above become more eective as the complexity of a solid shape increases, thus the human

eort does not change substantially. Given the user inputs, the computational time required

for mesh generation varied, for the cases studied in the present work, from a few seconds to

20 minutes, depending on the number of grid cells and performance of the computer.

Chapter 3

Treatment of immersed boundary

This chapter discusses the issues of the IB method introduced in Section 1.1 in detail.

Mathematical formulations of the IB method are derived, and the corresponding issues are

discussed. As an example of the previous approaches, the standard reconstruction method

is presented. In order to address issues of this approach, two new IB methods are devised

and compared with the original method with numerical examples.

3.1 Basic equations of the IB method and general issues

In order to handle solid objects immersed in the uid, the IB method includes both uid and

solid regions in the physical domain. The governing equations, Eqs. (2.1)-(2.2) need to be

modied to enforce boundary conditions at the uid-solid interface in the physical domain.

From the work of Peskin (1982), enforcing the velocity boundary condition is expressed in

the additional forcing term to the original momentum equation:

uit

+uiujxj

= pxi

+1Re

2uixjxj

+ fi, (3.1)

fi = 0 in fluid= Fi in solid or IB

,

ui = ui,IB at IB, (3.2)

14

CHAPTER 3. TREATMENT OF IMMERSED BOUNDARY 15

uixi

= 0 in fluid + solid, (3.3)

where denotes a domain of uid or solid. IB denotes the interface between solid and

fluid. Enforcing the continuity equation for solid is discussed later in this study. The

forcing term Fi is determined such that the velocity boundary condition is satised at IB.

Some IB methods have non-zero Fi only at IB. Additional treatment for the pressure is

unnecessary as long as the momentum and continuity equations in fluid are satised. In

practical terms in the implicit forcing technique, the forcing term is never computed (e.g.,

Fadlun et al., 2000; Tseng & Ferziger, 2003); however, the mathematical formulation for

these methods still can be derived with the forcing term (Mittal & Iaccarino, 2005), so that

Eq. (3.1) does not loose generality. This set of governing equations has been used in most

of the previous studies and is referred to as the standard formulation of the IB method.

While Eqs. (3.1)-(3.3) have been used widely in IB methods, there are a few issues

worth clarifying. The rst is related to the pressure accuracy at the boundary IB. By

using Eqs. (3.1)-(3.3) for determining correct pressure, it is assumed that the unmodied

governing equations are satised in fluid, such that the pressure gradient satises following

relationship on the uid side of IB (namely, IBfluid):

p

xi

IBfluid

=

[u

fluidi

t u

fluidi u

fluidj

xj+

1Re

2ufluidixjxj

]IBfluid

withufluidixi

= 0,

(3.4)

where ufluidi denotes the velocity eld in fluid including the (eventual) boundary velocity.

By solving Eqs. (3.1)-(3.3), however, the resulting pressure gradient satises:

p

xi

IB

=[uit uiuj

xj+

1Re

2uixjxj

+ fi

]IB

withuixi

= 0. (3.5)

In practice, the terms in Eq. (3.5) are evaluated using velocity elds in both fluid and solid.

There is no evidence that the dierence between two pressure gradients in Eq. (3.4) and

Eq. (3.5) is always negligible, unless fi is zero at IB or the velocity gradient is discontinuous.

So, satisfying only Eqs. (3.1)-(3.3) may be insucient for an accurate prediction of the

pressure near IB; this might result in inaccurate prediction of the local velocity eld.


solid

Figure 3.1: A very thin solid object between two channels with ows in the opposite direc-

tions.

Another example demonstrating the need for an additional treatment of the pressure is

the interface problem shown in Figure 3.1. There is a thin solid boundary (interface) with

virtually zero thickness between two channels with steady laminar ows in the opposite

directions. The pressure at the boundary (IB) between two channels increases in the x-

direction for the lower channel and decreases for the upper channel. This example is inspired

by a ow around a thin airfoil where the pressure distribution on the pressure and suction

sides must remain decoupled. For the very thin interface in Figure 3.1, the solution requires

a discontinuous pressure prole across the interface. This can be achieved by enforcing

Eq. (3.4) at the boundary (IB) on each side of the interface, thus resulting in solutions

across IB independent of each other.

This isolation or decoupling process allowing discontinuous solutions across the interface

is similar to the Jump condition used in the immersed interface method (Lee & LeVeque,

2003; Xu & Wang, 2006, among others) and the ghost uid method (Fedkiw et al., 1999).

While multiple methods have a similar eect of enforcing Eq. (3.4), satisfying Eq. (3.4) is

also possible by enforcing a modied continuity equation at IB or in solid:

uixi

= s in solid or IB, (3.6)

where s is called the mass forcing term. This approach was used by Kim et al. (2001) in

order to remove unwanted coupling of the ow in the solid domain to the uid domain. In

the present study, Eq. (3.4) will be referred to as the decoupling constraint for the pressure.

This constraint is satised when the ow eld in the uid domain is decoupled from other

physical domains. In other words, both the momentum equation and the continuity equation

are satised using ow variables in the uid domain and the extrapolated variables inside


IBfluid

solidfluid

solid

(a)

fluid

solidIB

(b)

Figure 3.2: Examples of velocity eld mirrored across the IB: (a) collocated grid; (b) stag-

gered grid. A square denotes a grid.

the solid body. Then, Eq. (3.4) is automatically satised.

The second issue regarding Eq. (3.1) is the velocity in the uid region with a non-zero

forcing term. For some IB methods with the forcing term located on IB, the forcing term

is expressed as a Dirac delta function. If the Dirac delta function is not located at a grid

point, it is distributed to neighboring grid points. For certain IB methods (e.g., Fadlun et al.,

2000), Eq. (3.1) is replaced with a simpler relationship for easier computation in the uid

region, which is equivalent to a non-zero forcing term in the uid region. The forcing term

fi in the uid region is acceptable when the modied momentum equation is a convergent

approximation of the original momentum equation.

The third issue is related to the velocity boundary condition at IB (Eq. (3.2)). Up to

this point, the equations of the IB method have been written in continuous form. After

discretization, however, the positions where we want to enforce Eq. (3.2) are not necessarily

located on the grid points. A relationship that approximates Eq. (3.2) is then necessary.

One method of accomplishing this is to give the eect of Eq. (3.2) to nearby velocity

points in fluid by modifying their discrete momentum equation. One example is the method

by Fadlun et al. (2000) that employed a simple approximation to Eq. (3.1) that includes the

eect of Eq. (3.2). Another widely used method is to satisfy Eq. (3.2) using interpolation of

neighboring velocity points:

nb

wnbui,nb = ui,IB at IB, (3.7)

where wnb is the interpolation coecient and nb denotes the index of neighboring points. A

linear interpolation has been employed in several previous studies. In Mohd-Yusof (1997),

this method was described as mirroring the velocity eld across IB. An example is shown

in Figure 3.2 (a) where the velocity eld across the IB is mirrored such that the no-slip

boundary condition is satised using a linear interpolation.


fluid

mirror

IB

Figure 3.3: A local coordinate at the immersed boundary.

This mirroring method satises the velocity boundary condition with accuracy of the

interpolation method. However, the accuracy may be lower because of incompatibility with

the continuity equation Eq. (3.3). This issue was previously reported by Kim et al. (2001)

and schematically represented for a simple case in Figure 3.2 (b). Figure 3.3 shows tangential

and normal coordinates local to the immersed boundary IB. Let me assume that u (, )

and u(, ) are the tangential and normal velocity elds in fluid. The velocity eld in the

mirrored region mirror can be expressed as:u (, ) = u (,) + 2u (, 0)u(, ) = u(,) + 2u(, 0) in mirror. (3.8)We assume that the continuity equation is satised in fluid:

u (, ) +

u(, ) = 0 in fluid.

For simplicity, let us assume the no-slip condition (u (, 0) = u(, 0) = 0). Then, it is

easily proved that in the mirrored region,

u (, ) +

u(, ) =

u (,)

u(,)

= u

(,) + u

(,) 6= 0 in mirror.

The mirrored velocity eld across IB does not satisfy the continuity equation unless

u/ = 0 near IB. This is a direct consequence of mass conservation in case of the no-slip

condition. However, enforcing the mirrored velocity and continuity equation together can


reduce the degree of freedom for the velocity interpolation and aect the accuracy especially

in a coarse mesh. The accuracy is not aected when u/ 0 is satised, e.g., stream-lines are parallel near IB. In the worst situation, however, the accuracy is reduced to the

rst order. For example, the case in Figure 3.2 (b) requires the y-velocity components to

be zero in order to satisfy the no-slip x-velocity boundary condition, regardless of the grid

size. This situation typically occurs near a stagnation point. Kim et al. (2001) addressed

this issue by satisfying the continuity equation with a non-zero right-hand side (RHS) term,

which reduced the velocity error at least by 60%. The current analysis conrms that using

the mirrored velocity eld near the IB has a negative eect. However, it should be em-

phasized that u/ = 0 at IB is the correct condition, which implies that the accuracy

will recover as the mesh is rened. Interestingly, the mirrored velocity eld produces a less

serious problem in the collocated mesh. As shown in Figure 3.2 (a), the mirrored velocity

eld has more degrees of freedom in the collocated mesh, because mass-conserving velocity

components (located at the CV faces) are dierent from the primary velocity components

(located at the CV centers). The mass-conserving velocity is interpolated from the primary

velocity components. In the collocated mesh, both the mass-conserving velocity compo-

nent and the boundary condition (Eq. (3.7)) are interpolated, and they do not produce the

incompatibility as seriously as in the staggered mesh.

In the present study, a set of modied equations Eqs. (3.1)-(3.4) is referred to as the

modied formulation of the IB method. In order to address the issues of the decoupling

constraint for the pressure (Eq. (3.4)) and incompatibility between the interpolated velocity

boundary condition (Eq. (3.7)) and the continuity equation, special treatments have been

devised and are introduced in Sections 3.5 and 3.6.

3.2 Treatment of mass conservation near immersed boundary

In this section, a few options for treatment of mass conservation near the IB are discussed

and compared. Since the pressure in the Navier-Stokes equation is a Lagrange multiplier

for the continuity equation, satisfying mass conservation is very important for the accurate

prediction of the pressure. Figure 3.4 shows dierent grid congurations near the IB. The

shaded area is a control volume where mass is conserved in each conguration.

In Figure 3.4 (a), mass is conserved for every control volume in the domain as it is

without the IB method. This has been a standard method for several IB methods with


IB

fluid

solid

(a)

IB

fluid

solid

(b)

IB

fluid

solid

(c)

Figure 3.4: Dierent schemes for dening control volumes (CVs) for mass conservation near

IB: (a) standard scheme; (b) mass conservation for reshaped CVs; (c) mass conservation

for uid-side CVs. The shaded area denotes a control volume where mass conservation is

enforced.

Eqs. (3.1)-(3.3) as their governing equations. One advantage in this situation is that no

additional treatment for mass conservation is necessary. For a control volume crossed by

IB, mass conservation is described by velocity values in both uid (fluid) and solid (solid)

domains. Since velocity elds in both domains aect the pressure at each point, the pressure

decoupling constraint Eq. (3.4) may not be satised. This limits the ability of handling a

very thin solid. If the mirroring scheme Eq. (3.8) is used to determine the local velocity eld

in solid, the incompatibility with mass conservation discussed in the previous section may

result in a decrease in the velocity accuracy. In the present study, this method is referred

to as standard mass conservation.

In Figure 3.4 (b), rectangular control volumes (CVs) crossed by IB are divided into

the ow region where solutions to the Navier-Stokes equations are desired, and boundary

regions (i.e., white area in the gure) where no solutions are needed. IB separates these

two regions. Mass conservation is satised for the reshaped CVs formed by existing CV faces

in fluid and IB. This method is closely related to the nite volume method (FVM) for an

unstructured mesh. Similar methods have been used by cut-cell approaches (Tucker & Pan,

2000; Udaykumar et al., 2001; Kirkpatrick et al., 2003, among others) and Kim et al. (2001).

This method has an advantage in that mass conservation (i.e., divergence of the velocity)

and the pressure gradient in fluid are independent from ow variables in solid. As a result,

the pressure decoupling constraint Eq. (3.4) is satised. This method obviously does not

suer from the incompatibility with the mirrored velocity, however, is very complicated.


As a preliminary step in the application of the IB method, it is necessary to determine

the portions of the grid that need a specic treatment, e.g., the CV crossed by immersed

surfaces (IB). These reshaped CVs can be identied in several ways. If the immersed

surfaces are simple and described analytically, the procedure is straightforward; otherwise,

increasingly complex computational geometry tools must be used (Iaccarino & Verzicco,

2003). Once the reshaped CVs are identied, new pressure/velocity discretization operators

for mass conservation are necessary. As noted by Kirkpatrick et al. (2003), another problem

is that the matrix condition number increases signicantly when the size of the reshaped

CV is very small. This method is discussed further in Section 3.5.

As a nal option, in Figure 3.4 (c), CVs crossed by IB and in solid are excluded from the

computational domain. By satisfying mass conservation only for CVs in fluid, this method

does not suer from the incompatibility with the mirrored velocity eld. The pressure

decoupling constraint is satised by properly computing the velocity at the boundary of

the shaded area in Figure 3.4 (c). Also, computational complexity is much lower than the

reshaped CV approach, since the original CV is used for mass conservation. This method is

discussed in greater detail in Section 3.6.

3.3 Treatment of velocity and scalar - standard reconstruction

method

In this section, a few discrete forcing methods for the velocity and scalar are presented, and

the most suitable method is chosen and discussed in more detail. The IB formulation devel-

oped is based on discrete forcing, because the boundary condition is enforced exactly and

the numerical treatment is more straightforward compared to continuous forcing techniques.

The direct forcing technique of Mohd-Yusof (1997) and the ghost-cell method of Ma-

jumdar et al. (2001) and Tseng & Ferziger (2003) are attractive because of simplicity, low

computational overhead and no severe restriction on the time step. Since these methods are

based on the mirrored velocity eld in the solid region, we expect a possible loss of accuracy

(as discussed in Section 3.1). The pressure decoupling constraint can be satised, under a

condition that the mirrored velocity is determined exclusively from the velocity eld in the

uid region. The ability to handle a very thin interface is also important in the present

study, since this feature will be very useful for the conjugate heat transfer problem. The

methods using the mirrored velocity eld are somewhat limited in this respect.


In the cut-cell methods (Udaykumar et al., 2001; Kirkpatrick et al., 2003, among oth-

ers), the governing equations (and solution procedures) are not modied, and only the

computational cells at the IB are altered to formally dene nite-volume or nite-dierence

operators. These methods are accurate, satisfy the pressure decoupling constraint and are

able to handle a very thin interface. However, their complexity and slow convergence due

to the small-cell problem are key disadvantages of these methods.

The method of Fadlun et al. (2000) is the starting point for the present IB method. In

the literature, this approach has been referred to as the reconstruction or the interpolation

method. The forcing term is not explicitly computed but rather, its eect is included in

an approximation to the momentum equation Eq. (3.1). As the result, an interpolation

formula replaces Eq. (3.1) near the IB. Its eect is to reconstruct the local velocity eld near

the IB. In addition to its simplicity, it has several advantages. Since the velocity boundary

condition is enforced with implicit forcing, there is no severe limit on the time step. The

velocity components from the regions across the IB are decoupled. And this approach does

not rely on the mirrored velocity eld in the solid region. It has an advantage in handling

a very thin interface. One concern is that the treatment of the velocities in the solid region

which contribute to mass conservation in the grid cells crossed by the IB is notionally

undened. Although Iaccarino & Verzicco (2003) observed that dierent treatments of the

solid velocity eld do not aect the ow eld in the uid region, arbitrary treatments might

have a negative eect on the pressure decoupling constraint. In the present study, revisions

to this approach are introduced with the objective of increasing the method's accuracy and

consistency.

In the rest of this section, we specify a linear interpolation method similar to Fadlun

et al. (2000) and show that the method has a second-order accuracy in space. A linear

interpolation formula is used to determine a velocity component that is in the uid region

and has one of the neighboring points in the discretization stencil outside of the uid region.

It is also assumed that velocity components are independent of each other and that each

component is determined by a separate interpolation. This is consistent with the assump-

tion of a linear prole. In Gilmanov & Sotiropoulos (2005) and Choi et al. (2007), higher

order interpolation methods were used. Choi et al. (2007) stated that a power-law based

interpolation is better suited to high Reynolds number ows than a linear interpolation.

Figure 3.5 shows two IB congurations commonly found in practical problems. A stag-

gered mesh is assumed. In Figure 3.5 (a), since the IB is exactly parallel to the grid lines, a


(a)

uIBuc

u2

(b)

uIB

uc u1

u2

Figure 3.5: Conguration of the immersed boundary, grid and nodes in the linear interpo-

lation method: , immersed boundary; , grid; , velocity node. uc is thereconstructed velocity.

linear interpolation stencil with uIB, uc and u2 is easily constructed along the x2 coordinate.

In Figure 3.5 (b), there are two velocity components (u1 and u2) nearest in the horizontal

and vertical directions to uc. A triangle is then identied by two adjacent velocity nodes

and a point on the IB (uIB) surrounding uc. A linear interpolation is then formulated be-

tween the IB point and vertices of the triangle. The resulting interpolation formula has the

following form:

uki,c = wi,1uki,1 + wi,2u

ki,2 + wi,IBu

ki,IB, (3.9)

where subscripts 1 and 2 denote the adjacent velocity nodes in the x1 and x2 directions,

and subscript IB denotes the point on the IB that is the boundary-normal projection of the

velocity node c. Superscript k denotes the next time step, and wi is an interpolation coe-

cient for the linear interpolation in the wall-normal direction. Assuming a local coordinate

whose center is located at a point on the IB, we can restate the linear interpolation method

as:

uki (x1, x2) = aki,1x1 + a

ki,2x2 + u

ki,IB, (3.10)

where ai,1 and ai,2 are coecients determined by the local IB geometry and velocity. Recall,

using the fractional-step method, ui after one time step is computed from the momentum

solution (ui) followed by projection onto a divergence-free eld. The intermediate velocity,

uki,c, is therefore determined by the interpolation formula. Extending this method to a


ul+1

pl1 pl pl+1pl2l

solid x

x+

ul1 u

Figure 3.6: Grid conguration for a simplied one-dimensional case. u denotes a velocitynode and p denotes a pressure node.

three-dimensional geometry is straightforward.

The present linear interpolation approach needs to be justied. When comparing Eq. (3.9)

with the discretized momentum equation Eq. (2.3), we observe that the linear interpolation

method does not include all the ow variables (e.g., pressure) in Eq. (2.3). However, there

are two velocity components, ui,1 and ui,2, that are determined by the discretized momentum

equation. So the eects of the excluded terms are implicitly included in the interpolation

formula.

A more systematic analysis can be carried out using a Taylor series expansion. To

this end, a simplied one-dimensional case is considered (as shown in Figure 3.6) with the

assumption that a velocity component at index l is determined by linear interpolation using

a boundary condition (ukl1) and a neighboring velocity component (ukl+1) determined by

the discretized governing equation.

One-dimensional versions of the linear interpolation formula (Eq. (3.9)) and the dis-

cretized governing equation (Eq. (2.3)) are written as:

ukl =x+

x + x+ukl1 +

xx + x+

ukl+1, (3.11)

[1

t (k + k)

2Re2

xx

]ukl =

uk1lt

+

[(k + k)

(p

x

)kl

k(uu

x

)k1l

k(uu

x

)k2l

+(k + k)

2Re

(2u

xx

)k1l

]= Sl. (3.12)

Here, both exact dierential and nite dierence relations are retained for simplicity, but

this does not aect the nal conclusion, assuming that the grid distribution and velocity

prole are smooth. It is also assumed that the governing equation is satised continuously


to the boundary and holds at l 1. When applying[

1t (k+k)2Re

2

xx

]to Eq. (3.11) and

using a Taylor series expansion, Eq. (3.11) becomes:[1

t (k + k)

2Re2

xx

]ukl

=x+

x + x+Sl1 +

xx + x+

Sl+1

=x+

x + x+

(Sl x

xSl +

x22

2

x2Sl +O(x3)

)+

xx + x+

(Sl + x+

xSl +

x2+2

2

x2Sl +O(x3+)

)= Sl +

xx+2

2

x2Sl +O(x3). (3.13)

This shows that the linear interpolation formula is a spatially O(x2) and temporallyO(t1) approximation (see the rst term in Sl in Eq. (3.12)) to the original equation.The largest dierence term ( xx+2 ) is regarded as a second-order error term in space.Fadlun et al. (2000) also discussed the temporal stability of this scheme. They state that

it does not suer from any time-step limitation. The maximum time step allowed for the

neighboring velocity components is therefore the only limitation. The additional computa-

tional cost for this procedure depends on the number of velocity components determined by

the IB method. In the test cases of the present study, this amounted to about 510% ofthe overall computation time.

Fadlun et al. (2000) did not include a special constraint on the mass conservation, thus,

the standard mass conservation (Figure 3.4 (a)) was used. We refer to the combination of the

linear interpolation method and standard mass conservation as the standard reconstruction

method (SRM).

3.4 Issues in the standard reconstruction method

The standard reconstruction method presented in the previous section has several advan-

tages, but also some shortcomings.

The rst issue is the decoupling of local velocity and pressure elds present in the inter-

polation method (Eq. (3.9)). A notable feature of the interpolation method is that there is

no explicit contribution of the velocity eld at the previous time step and of the pressure


IB1+iuiu

jv

1+jv

solid

Figure 3.7: An example of control volumes crossed by the IB in a staggered mesh.

gradient. Absence of the pressure gradient in an approximation to the momentum equa-

tion may create a problem; as an example in Figure 3.6, the pressure dierence between pl

and pl1 has no eect on the interpolation formula for ul, Eq. (3.11); this is similar to the

pressure-velocity decoupling observed in a non-staggered mesh method.

Assume that second-order central dierence is used for discretizing the pressure gradient.

In a staggered mesh, the interpolation formula, Eq. (3.9), may result in abnormal pressure

uctuations near the IB. Figure 3.7 shows an example of control volumes (CVs) crossed

by the IB. The velocities at two locations for u (ui and ui+1) and one for v (vj+1) are

determined by Eq. (3.9). If we set vj in the solid region to zero, it is impossible to determine

a solenoidal velocity eld because the pressure at the center of the CV is decoupled from all

other ow variables and cannot be used to satisfy mass conservation. If vj is determined by

the discretized momentum equation, then a solution may exist. However, the pressure may

not be physical because it is aected by the ow eld in the solid region.

With the fractional-step method in Section 2.1, we observe a pressure buildup around the

CVs crossed by the IB. Figure 3.8 shows the pressure contours of a two-dimensional laminar

Couette ow over a wedge. The no-slip condition is imposed at the IB and periodicity is

enforced in the streamwise direction. The Reynolds number is suciently low, and the nal

solution is steady. The overall ow pattern is reasonable, but the size of the pressure at or

below the CVs crossed by the IB is very large. In fact, the magnitude of the pressure in

this region is 10 to 1000 times the average value in the ow eld and, more importantly, it

grows unbounded. In the present fractional-step method, pk+1pk = , and is obtained bysolving the Poisson equation, Eq. (2.6). The observed pressure growth is therefore connected

to a nite value of u in the CVs where the interpolation is applied.Figure 3.9 shows the time history of the velocity, u and v, the intermediate velocity


0 0.2 0.4 0.6 0.80

0.2

0.4

0.6

0.8

1

(a)

x/L

y/L

u

v

=0vw

(b)

Figure 3.8: Flow eld and grid conguration of laminar ow over a wedge: (a) streamlines

and pressure contours; (b) conguration of the grid lines and IB.

-1.0e-4

4.0e-4

79 79.1 79.2 79.3 79.4

(a)

tU0/L

u/U0,v/U0

-13.66

-13.61

79 79.1 79.2 79.3 79.4

(b)

tU0/L

p/U20

Figure 3.9: Time trace ofu, u, v, v, and pressure around an IB cell shown in Figure 3.8 (b)of the steady laminar ow over a wedge: , u/U0; , u/U0; M, v/U0; O, v/U0;, pressure.


components, u and v, and pressure in a CV displayed in Figure 3.8 (b). Since there is nomass ux across the IB (vw = 0) and the grid shape is a square, u and v should be equal.Although the ow reached a steady state, u and v resulting from the linear interpolationmethod are dierent; on the other hand u and v are the same, and this conrms thatcontinuity is correctly enforced by the velocity-projection step. The observed dierence

between the predicted velocities u and v, although very small ( 104) is responsible forthe observed pressure buildup in time. Theoretically, this problem has no solution because

the pressure at the CV is decoupled from all other ow variables. When the fractional-

step method is applied, the pressure buildup is observed. This problem is resolved when

the contribution from the pressure gradient term is explicitly included in the interpolation

method Eq. (3.9). In the next section, Eq. (3.9) is modied to include the eect of the

local pressure gradient. Notably, the pressure buildup is not observed when the original

fractional-step method of Kim & Moin (1985) is used. In Kim & Moin (1985), the pressure

term is not included in the modied momentum equation, and the pressure pk+1 is computed

only from the velocity without the old pressure, pk.

In the case of a collocated mesh, the decoupling problem just discussed is much less

severe. Applying Eq. (3.9) does not aect the existence of the solution. With the collocated

mesh, the width of the discretization stencil of the gradient and divergence operators is

larger than that in staggered mesh. In this case, the pressure at the CV crossed by the IB

is used by the discretized momentum equation in a neighboring uid CV. This guarantees

the existence of the solution and signicantly reduces the size of the pressure wiggles. An

example is presented in Section 3.6.4.

The second shortcoming of the SRM is that the pressure decoupling constraint, Eq. (3.4),

is not satised. Although the linear interpolation method approximates the momentum

equation near the IB and results in the uid region decoupled from the solid region, the

standard mass conservation results in coupling between the solutions across the IB. Specif-

ically, the divergence operator of the velocity as well as the gradient of the pressure are

computed using the ow elds in both regions.

A simple example that illustrates the importance of the pressure decoupling constraint is

the case shown in Figure 3.1: two parallel channels separated by a very thin rigid interface

(notionally with zero thickness). Steady laminar ows in opposite directions in the two

channels are considered. Since the pressure drops in opposite directions, a pressure prole

normal to the interface shows a discontinuity across the thin wall. If the pressure decoupling


(a) x-velocityx/

y/

(b) Pressure

x/

y/

Figure 3.10: Contours of the x-velocity and pressure with the standard reconstructionmethod. Bold lines denote boundaries of the channels.

constraint (Eq. (3.4)) is satised at both sides of the thin wall, the pressure eld is correctly

predicted.

The Reynolds number based on the channel half-width and the centerline velocity Uo

is 100. The computational domain size is -10< x/


x/

y/

Figure 3.11: Contours of the x-velocity with the standard reconstruction method with theperiodic B.C. and a constant momentum forcing term.

velocity elds in both channels, which results in the coupling. In order to verify that the

reason of the distorted ow eld is the articial pressure link introduced, another test was

performed. Instead of imposing the parabolic prole at x/=-10 and 10, the periodic B.C.

and a constant momentum forcing term are used to drive the ow. In this case, the pressure

eld is supposed to have zero gradient in the entire domain, since the wall shear stress is

balanced by the momentum forcing term. Figures 3.11 and 3.12 show the x-velocity eld

using the standard reconstruction method. Unlike the previous case, there is no distor-

tion of the velocity eld. Agreement with the parabolic velocity prole is also good. The

conclusion is that satisfying the pressure decoupling constraint is important to numerically

decouple domains that are physically unrelated. The problem discussed above occurs also

for IB methods based on a staggered ow solver. The next section presents a solution to

this problem.

The third issue with the SRM arises only when this method is used in conjunction

with the fractional step method. The velocity boundary condition is exactly satised in

the momentum solution step, but a nite error is introduced during the projection to the

divergence-free velocity eld. This is because ui = ui is not enforced at the IB. In the

present fractional step method, ui ui = O(t2), and the error in the boundary conditionmay be ignored if the time step is suciently small. In the original fractional step method

(Kim & Moin, 1985), however, ui ui = O(t) which can lead to a large error. Taira& Colonius (2007) and Ikeno & Kajishima (2007) addressed this issue by modifying the

projection step, so that the interpolation formula is satised after the projection. This issue

is further discussed in Sections 3.5 and 3.6.

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