Development of immersed boundary methods for isothermal and thermal flows 2

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Chapter 2 Governing Equations and Boundary Condition-Enforced IBM
29
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Chapter 2

Governing Equations and Boundary
Condition-Enforced Immersed Boundary Method

A new version of boundary condition-enforced IBM, which is given under the
framework of NS solver in primitive variable form, is presented in this chapter.
It aims at extending the LBM solver-based IBM of Wu & Shu (2009) to the
NS solver-based IBM for an accurate evaluation of the body force. The present
boundary condition-enforced IBM is established based on the fractional step
technique while the body force in the modified momentum equation is
implicitly determined in a way that the no-slip condition on the immersed
boundary is accurately satisfied. The performance of the new version of IBM
is carefully examined, firstly through the classical problem of flow over a
single stationary circular cylinder, and then the flow interference between two
side-by-side circular cylinders. Results from moving boundary problems such
as vortex-structure interaction around a transversely oscillating cylinder and
vortex-induced-vibration of an elastically mounted circular cylinder are also
provided as a further validation.

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Chapter 2 Governing Equations and Boundary Condition-Enforced IBM
30

2.1 Governing equations

Let us begin by stating the mathematical expression of the present IBM.
Consider an incompressible viscous flow in a two-dimensional domain
Ω

which contains an immersed object in the form of a closed curve
Γ
, as shown
in Fig. 2.1. With the use of the IBM, the immersed object is modeled as
localized body forces acting on the surrounding fluid. As a result, the IBM
formulation for the incompressible viscous flow involving immersed
objects/boundaries is expressed in the primitive variable form as

2
(()) p
t
ρμ
∂
+
⋅∇ =−∇ + ∇ +
∂
u
uu uf
(2.1)

0∇⋅ =u
(2.2)
subject to the no-slip boundary condition (2.3) on
Γ

((),)
B
st=uX U
. (2.3)
The fluid pressure
p
and velocity vector
u
are the dominating flow
variables.
B
U
is the prescribed velocity of the immersed boundary
Γ
.
ρ

and
μ
are the fluid density and viscosity. Note that a forcing term
f
is
added to the right hand side (RHS) of the momentum equation (2.1) to
represent the effect of immersed object
Γ
. The forcing term
f
is the
localized body force density at the fluid (Eulerian mesh) point, which is
distributed from the surface force density

F
at the immersed boundary
(Lagrangian) point and can be expressed as

∫
−=
Γ
δ
dststst )),(()),((),( XxXFxf
(2.4)
Here
x
and
X
denote the Eulerian and Lagrangian coordinates describing
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Chapter 2 Governing Equations and Boundary Condition-Enforced IBM
31

the fluid domain and immersed boundary respectively.
((,))
s
t
δ
−
xX
is the
Dirac delta function responsible for the interaction between fluid and the
immersed boundary. In addition, the velocity at the immersed boundary
(Lagrangian) point in IBM can be interpolated from the velocity at the fluid

(Eulerian mesh) points as

((,)) ()( (,))st st dV
δ
Ω
=−
∫
uX ux x X
. (2.5)
In summary, Eqs. (2.1)-(2.5) build the complete set of governing equations for
an incompressible viscous flow system involving immersed
objects/boundaries, among which Eqs. (2.1)-(2.2) are the familiar
Navier-Stokes equations and Eqs. (2.4)-(2.5) represent the interaction between
the fluid and the immersed objects/boundaries.

2.2 Solution procedure
In IBM, the solution to Eqs. (2.1)-(2.5) is frequently accomplished by making
a good use of the fractional step algorithm. In a time-discrete form, the
fractional step procedure is written as:
(1) Predictor step:
Solve the normal Navier-Stokes equation for a predicted velocity field
*
u
by
disregarding the body force terms in Eq. (2.1),

[]
*
2
()

n
p
t
ρρ μ
−
+⋅∇=−∇+∇
Δ
uu
uu u
. (2.6)
(2) Corrector step:
Take the effect of body forces into consideration and update the predicted
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Chapter 2 Governing Equations and Boundary Condition-Enforced IBM
32

velocity field to the physical one

1*
1
n
n
t
ρ
+
+
−
=
Δ
uu

f
(2.7)
which satisfies the no-slip boundary condition (2.8)

(
)
11
()
nn
B
dV
Ω
δ
++
=−
∫
UX u xX
. (2.8)

In the predictor step, Eq.(2.6) is advanced to the predicted velocity field
*
u

under the divergence-free constraint (2.2) which couples the velocity and
pressure. This constraint is the major difficulty in solving the incompressible
Navier-Stokes equations and could be successfully overcome by the popular
and well-established projection method. The details on the projection method
and its implementation will be given in Section 2.3. The corrector step, as
shown in Eq. (2.7), involves evaluating the unknown body force
1n+

f and
updating the velocity field
*
u
to the desired one
1n
+
u
. Therefore, the
evaluation of body force poses as a crucial issue and may embody the unique
feature of the IBM. The corresponding technique to determine the body force
will be illustrated in details in Section 2.4.

2.3 Calculation of Predicted velocity field – Projection method
Projection method was introduced decades ago by Chorin (Chorin 1968) and
later independently by Temam (1969), as an efficient numerical device to
compute incompressible Navier-Stokes equations in primitive variable
formulation where the pressure is only present as a Lagrangian multiplier for
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Chapter 2 Governing Equations and Boundary Condition-Enforced IBM
33

the incompressibility/ divergence-free constraint (2.2). Based on the Hodge
decomposition, projection method efficiently decouples the computation of
velocity and pressure in a time-splitting scheme and avoids solving the
momentum equation (2.6) and incompressibility constraint (2.2)
simultaneously. Projection method proceeds in the first step to compute an
intermediate velocity field
u


 by using the momentum equation (2.6) and
ignoring the pressure gradient term and the incompressibility constraint (2.2).
In the second step, the intermediate velocity field
u

is projected onto the
space of incompressibility field to obtain the pressure and divergence-free
velocity field. To be specific, its implementation is as follows:
Firstly, solve for the intermediate velocity
u

through Eq. (2.9)

[]
()
1/2
2
()
2
n
n
n
t
μ
ρρ
+
−
+⋅∇ =∇+
Δ
uu

uu uu


(2.9)
by approximating Eq. (2.6) using the trapezoidal rule and dropping the
pressure gradient term. The convective term, which appears in Eq. (2.9), can
be approximated using the 2
nd
-order explicit Adams-Bashforth formula

[] [][]
1/2 1
31
() () ()
22
nnn+−
⋅∇ = ⋅∇ − ⋅∇uu uu uu
(2.10)
Then, the immediate velocity
u

is corrected to the predicted velocity
*
u

through

*
1n
p

t
ρ
+
−
=−∇
Δ
uu

(2.11)
Before the use of Eq. (2.11), the pressure field should be calculated first by
solving an elliptical Poisson equation (2.12), which is deduced by taking the
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Chapter 2 Governing Equations and Boundary Condition-Enforced IBM
34

divergence on both sides of Eq. (2.11) and letting
*
u
be subjected to the
continuity constraint

21n
p
t
ρ
+
∇⋅
∇=
Δ
u


(2.12)
Substituting the solution of pressure equation (2.12) into Eq. (2.11) will finally
produce the predicted velocity field
*
u
.

2.4 Evaluation of Body force
The evaluation of the body force has long been the key issue for the IBM and
a number of notable strategies have been developed.

2.4.1 The Conventional IBM
Early remarkable methods to calculate the body force include the well-known
penalty force scheme, feedback forcing scheme and direct forcing scheme
which are generally known as “conventional IBM”. These conventional IBMs
have played an important role in the early and current development of the
IBM.

2.4.1.1 Penalty force scheme
The penalty force scheme was originally proposed by Peskin (1972) to deal
with elastic boundaries on the basis of Hooke’s law and was later utilized by
Lai & Peskin (2000) to calculate the singular Lagrangian force density on
solid objects. In the penalty force scheme, it is assumed that the boundary
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Chapter 2 Governing Equations and Boundary Condition-Enforced IBM
35

points
X

of the immersed object are being attached to their equilibrium
positions
e
X by a spring with high stiffness
κ
. When the boundary moves
and deviates from its equilibrium location, a restoring force
F
will be
generated according to the Hook’s law

111
(, ) (( ) ( ))
nnen
ttt
κ
+++
=−FX X X
(2.13)
so that the boundary points will stay close to their target boundary positions.
To impose the no-slip condition on the immersed boundary accurately, a large
value of stiffness
κ
is often required which, unfortunately, would render a
stiff system of equations and lead to a severe stability constraint. However, if a
lower value of
κ
is utilized, the spurious elastic effects such as an excessive
deviation from the equilibrium location may arise.

2.4.1.2 Feedback forcing scheme
Goldstein et al. (1993) generalized the penalty force model and provided a
two-mode feedback forcing scheme
[]
1
1 11
0
(, ) (,) (,) (, ) (, )
n
t
n nn
spring B damp B
tttdttt
αβ
+
+ ++
′′′
⎡
⎤
=−+−
⎣
⎦
∫
FX uX U X uX U X
(2.14)
which involves a spring constant
s
pring
α
and a damping constant

damp
β
for
the control of velocity condition at the immersed boundary. This forcing term
is a reflection of the velocity difference between the desired boundary value
B
U
and the interpolated
u
, and behaves in a feedback loop such that the
boundary velocity remains close to the desired one. In general, the method is
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Chapter 2 Governing Equations and Boundary Condition-Enforced IBM
36

successful for low Reynolds number flows but is confronted with similar
difficulties as the penalty force model in enforcing the boundary conditions.
Firstly, accurate satisfying of the boundary condition requires large values of
the spring and damping constants, which can result in numerical instability.
Secondly, these two constants are flow-dependent and have to be tuned in a
semi-empirical way. There is no general rule for their determination, thus
making the application of the method expensive (Fadlun et al. 2000).

2.4.1.3 Direct forcing scheme
To remove the annoying empirical constants, Mohd-Yusof (1997) suggested a
forcing evaluation approach in which the body force was directly derived from
the transformed momentum equation

1
1

(, ) (,)
(, ) RHS
nn
n
B
tt
t
t
ρ
+
+
−
=−
Δ
UX uX
fX
(2.15)
with
RHS
including convective and viscous terms as well as the pressure
gradient

()
2
RHS= p
ρμ
−∇−∇+∇uu ui
. (2.16)
The term
RHS

together with
(,)
n
tuX
constitute the interpolated velocity
1
(, )
n
t
+
uX
which is expressed as

1
(, ) (,) RHS
nn
t
tt
ρ
+
Δ
=+uX uX
. (2.17)
This method is frequently termed the direct forcing method. Essentially, the
discretized momentum equation is transformed such that the forcing term is
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Chapter 2 Governing Equations and Boundary Condition-Enforced IBM
37

calculated by compensating the difference between the interpolated velocities

1
(, )
n
t
+
uX
and the desired physical velocities
1
(, )
n
B
t
+
UX
on the boundary
points. In this way, the method is free from empirical parameters and no
longer suffers from the numerical stability limitation, thus showing substantial
improvements as compared to previous formulations. Although it was initially
suggested in a sharp interface method, the direct forcing scheme has been
successfully generalized into Peskin’s immersed boundary method by
Uhlmann (2005), who incorporated the regularized delta function into the
force calculation and spreading process. This strategy allows for a
straightforward and smoother transfer between Eulerian and Lagrangian
representations, therefore making the scheme more stable and easier to
implement.

However, these conventional IBMs generally compute
1n
+
f explicitly using

the information at time level
n
, and flow penetration to the surface of the
immersed object frequently occurs, i.e., the velocity condition on
Γ
is only
approximately satisfied. Therefore, special effort is required for an accurate
evaluation of the body force.

2.4.2 Boundary condition-enforced IBM
Recently, Wu & Shu (2009) proposed a novel velocity correction scheme
within the framework of LBM which is proven to be effective in guaranteeing
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Chapter 2 Governing Equations and Boundary Condition-Enforced IBM
38

the no-slip condition on the immersed boundary. They suggested that
introducing the body force
1n
+
f was equivalent to making a velocity
correction which should be determined implicitly in a way that the velocity
)),(( tsXu
at the boundary (Lagrangian) point interpolated from the physical
velocity
u
at the Eulerian points equals to the given boundary velocity
B
U
.

Therefore the basic idea of their velocity correction scheme may provide an
effective and accurate way to evaluate the body force. However, their velocity
correction procedure is proposed within the framework of LBM, and it would
be worthwhile to extend it into the framework of NS solver. Following the
idea in Wu & Shu (2009), the body force term
1n
+
f should be controlled by
Eq. (2.18)

()
1
11 * 1
()()
n
nn n
B
tdV
Ω
δ
ρ
+
++ +
=+Δ −
∫
f
UX u xX
, (2.18)
which is derived by substituting Eq. (2.7) into Eq. (2.8). Note that the force
density

f
at the Eulerian point, as shown in Eq. (2.4), is distributed from the
boundary force
F
through the Dirac delta function
)),(( tsXx −
δ

interpolation, Eq. (2.18) can be reformulated to be

()
11 1
11 * 1
()( )
()()
nn n
nn n
B
ds
tdV
Γ
Ω
δ
δ
ρ
++ +
++ +
−
=+Δ −
∫

∫
FX xX
UX u xX
.
(2.19)
As a result, the correlation between
1n
B
+
U
and
1n
+
f is now converted to the
correlation between
1n
B
+
U
and
1n
+
F , and the primary concentration in the
following would become the evaluation of the boundary force
1n+
F . Eq. (2.19)
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Chapter 2 Governing Equations and Boundary Condition-Enforced IBM
39


is in complex integral form, which can be numerically approximated by
algebraic equations as follows.

Suppose that the immersed boundary is represented by a set of Lagrangian
points ( , )
iii
X
Y=X
),,2,1( Mi =
, and the fluid field is covered by a fixed
uniform Cartesian mesh
(, )
j
jj
x
y
=
x
),,2,1( Nj 
=
with mesh
spacing

x
yhΔ=Δ=
. Furthermore, let
)),(( tsXx
−
δ
be smoothly

approximated by a continuous kernel distribution

2
1
() ()()
ji ji
ij
hhji h h
x
X
y
Y
DD
hh h
δδ
−
−
=−=xX
(2.20)
where
()
h
r
δ
was proposed by Lai & Peskin (2000) as

2
2
1
(3 2| | 1 4| | 4 ) | | 1

8
1
( ) (5 2| | 7 12| | 4 ) 1 | | 2
8
0 | | 2
h
rrr r
rrrrr
r
δ
⎧
−++− ≤
⎪
⎪
⎪
=
−+−+ − <≤
⎨
⎪
>
⎪
⎪
⎩
. (2.21)
After performing the numerical integration, Eqs. (2.4) and (2.19) can be
approximated as

111
() ( ) ( 1, ; 1,2,,)
nnnij

jihi
i
Dsi Mj N
+++
=Δ==
∑
fx FX 
(2.22)
and

11 * 2
11
2
() ()
()
nn ij
Bi jh
j
nn
kj ij
k
hkh
jk
Dh
t
DsDh
ρ
++
++
=

Δ
+Δ
∑
∑∑
UX ux
FX
(2.23)

111(, ; , ; ,,)iMjNk M===

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Chapter 2 Governing Equations and Boundary Condition-Enforced IBM
40

where

i
sΔ
is the length of the
th
i
boundary segment. It can be observed that
Eq. (2.23) form a well-defined equation system for variables
1
(1, )
n
i
iM
+
=F  .

Particularly, the equation system (2.23) for the boundary force can be written
in the following matrix form as

[
]
[
]
[
]
FF
=AF B
(2.24)
where
[]
11 12 1 11 12 1
12
21 22 2 21 22 2
2
12
12 1 2
12
NM
hh h h h hM
NM
hh h h h hM
F
MM MNN N NM
hh h h h hM
DD D DsDs Ds
DD D DsDs Ds

t
h
DD D DsDs Ds
ρ
⎛⎞⎛ ⎞
ΔΔ Δ
⎜⎟⎜ ⎟
ΔΔ Δ
Δ
⎜⎟⎜ ⎟
=
⎜⎟⎜ ⎟
⎜⎟⎜ ⎟
⎜⎟⎜ ⎟
ΔΔ Δ
⎝⎠⎝ ⎠
A


  


(2.25)

[]
1
11 12 1 *
,1
1
1

21 22 2 *
,2
2
2
1
12 *
,
n
N
B
hh h
n
N
B
hh h
F
n
MM MN
BM
hh h N
DD D
DD D
h
DD D
+
+
+
⎛⎞
⎛⎞⎛⎞
⎜⎟

⎜⎟⎜⎟
⎜⎟
⎜⎟⎜⎟
=−
⎜⎟
⎜⎟⎜⎟
⎜⎟
⎜⎟⎜⎟
⎜⎟⎜⎟
⎜⎟
⎝⎠⎝⎠
⎝⎠
U
u
U
u
B
U
u





(2.26)

[]
1
1
1

2
1
n
n
n
M
+
+
+
⎛⎞
⎜⎟
⎜⎟
=
⎜⎟
⎜⎟
⎜⎟
⎝⎠
F
F
F
F

(2.27)
and
1
,
(1,,)
n
Bi
iM

+
=U 
,
1
(1, )
n
i
iM
+
=F  ,
*
(1,,)
j
j
N=u 
and
1
(1,2,,)
n
j
p
jN
+
= 
are the abbreviations for
11
()
nn
Bi
+

+
UX
,
11
()
nn
i
++
FX
,
*
()
j
ux
and
1
()
n
j
p
+
x
respectively. The elements of coefficient matrix
[
]
F
A
,
as shown in Eq.(2.25), are only related to the coordinate information of the
Lagrangian boundary points and their adjacent Eulerian points.

By solving the equation system (2.24) using a direct method or iterative

Chapter 2 Governing Equations and Boundary Condition-Enforced IBM
41

method, the unknown boundary force
1
(1,,)
n
i
iM
+
=F 
at all Lagrangian
boundary points are obtained simultaneously, which are then substituted into
Eqs. (2.22) and (2.7) to calculate the body force
1
()
n
j
+
fx
and the corrected
physical velocity
1
()( 1,,)
n
j
j

N
+
=ux 
.

It should be noted that although the suggested methodology is illustrated for
two-dimensional flows in the above, it may be naturally extended to
three-dimensional problems without modifications.

2.5 Computational sequence
The computational sequence of the IBM solver can be summarized below.
Assume that the flow information at time level
n
is known. To march the
flow solution from time level
n
to
1n
+
,
1)
Using
n
u
as the initial flow field, solve Eqs. (2.9), (2.12) and (2.11)
consecutively following the projection method described in Section 2.3 to
get the predicted velocity
*
u
;

2)
Compute the elements of matrix
[
]
F
A
based on Eq.(2.25);
3)
Solve equation system (2.24) to obtain the boundary force
1n
i
+
F

(
1iM= ,
) at all Lagrangian points and then substitute them into Eq.
(2.22) to get the body force
1n
+
f
at Eulerian points.
4)
Update the predicted velocity
*
u
to the physical velocity
1n+
u
using Eq.

(2.7);
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Chapter 2 Governing Equations and Boundary Condition-Enforced IBM
42

Now the flow field has been updated at time level
1n
+
. To continue the
computation, use the corrected velocity
1n
+
u
as the initial flow field, and
repeat steps 1) to 4) until a desired solution is achieved.

2.6 Results and Discussion
Numerical experiments conducted using the boundary condition-enforced
immersed boundary solver in primitive-variable formulation are demonstrated
in this section, to provide a clear view on the performance of the present
solver in solving complex flows. Examples including two stationary boundary
cases and two moving boundary cases are presented. They are respectively the
flow over an isolated stationary circular cylinder, flow interference between a
pair of side-by-side circular cylinders, vortex-structure interaction around a
transversely oscillating circular cylinder and vortex-induced vibration of an
elastically mounted circular cylinder.

2.6.1 Flow over an isolated stationary circular cylinder
Flow over an isolated stationary circular cylinder is a basic fluid dynamic
problem which exhibits vastly different patterns based on the Reynolds

number
μ
ρ
DU
∞
=Re
(
U
∞
is the free stream velocity,
D
is the cylinder
diameter). Abundant experimental (Tritton 1959; Roshko 1961; Grove & Shair
1964) and numerical results (Dennis & Chang 1970; Fornberg 1980; Braza
1986; He & Doolen 1997; Liu et al. 1998; Ye et al. 1999; Ding et al. 2004;

Chapter 2 Governing Equations and Boundary Condition-Enforced IBM
43

Niu 2006) have been accumulated in the literature and thus it has served as a
standard case to examine the capability of new numerical methods.

A schematic view of the problem is depicted in Fig. 2.2, where a stationary
circular cylinder of diameter
D
is immersed in a uniform free-stream. A
sufficiently large rectangular computational domain is used for the simulations,
whose top and bottom boundaries are equipped with slip conditions of
0
u

y
∂
=
∂

and
0v =
. The inflow boundary is specified with the prescribed free stream
velocity
(,) ( ,0)uv U
∞
=
, while on the outflow boundary, the natural boundary
conditions of
0
uv
xx
∂∂
==
∂∂
are applied. In the present study, numerical
simulations are carried out in both the steady and unsteady flow regimes. Two
Reynolds numbers of
40Re =
and
100Re
=
are respectively chosen as
their representatives. At each flow regime, streamlines and vorticity patterns
are first provided for a general visualization of different flow behaviors.

Quantitative flow characteristics like the drag coefficient
D
C
, lift coefficient
L
C
(for unsteady case), Strouhal number (for unsteady case), recirculation
length
/
w
LD
(for steady case) behind the cylinder are then calculated and
compared with published results in the literature. The drag and lift coefficients
are calculated based on the widely-used definitions

DU
F
C
D
D
2
2
1
∞
=
ρ
(2.28)
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Chapter 2 Governing Equations and Boundary Condition-Enforced IBM
44



DU
F
C
L
L
2
2
1
∞
=
ρ
(2.29)
where
D
F
and
L
F
are the drag and lift forces on the cylinder surface. For
unsteady flows, their time-mean values are calculated by

0
1
shed
T
DD
shed
CCdt

T
=
∫
(2.30)

0
1
shed
T
LL
shed
CCdt
T
=
∫
(2.31)
s
hed
T
is the vortex shedding period. The Strouhal number which is given as

s
f
D
St
U
∞
=
(2.32)
is a dimensionless parameter corresponding to the frequency

s
f
with which
the vortices are shed from the body. Unless otherwise specified, all the results
shown below are non-dimensionalized. Additionally, the dimensionless
parameters like Reynolds number, drag and lift coefficients in all the following
examples share the same definitions as in this case.

To begin the simulations, domain independence study is first conducted. Three
different computational domains of sizes
24 16DD
×
, 48 32DD× and
72 48DD×
are examined, in which the cylinder is located at (8,8DD),
(
16 ,16DD) and ( 24 ,24DD), respectively. A convergence criteria of
16
110
nn+−
∞
−≤×uu
is set for
Re 40
=
while the simulations for Re 100=
are performed until a steady state of periodicity is achieved and lasts for at
least 20 cycles (unless otherwise specified, all other flow problems in the

Chapter 2 Governing Equations and Boundary Condition-Enforced IBM

45

thesis are simulated following similar criteria). The drag coefficients obtained
using each computational domain are presented for both
Re 40
=
and 100 in
Table 2.1 (for unsteady case
Re 100
=
, it is the time-mean value). While the
small domain
24 16DD× produces higher drag coefficients, the results
obtained using the intermediate and large domains are close to each other,
indicating that the intermediate domain
48 32DD
×
is sufficiently large to
provide domain-independent results. Therefore, it is chosen as our
computational domain for the following simulations and discussions.
Subsequently, mesh independence studies are performed. Five different
non-uniform Eulerian meshes are tested, with resolutions of
/20
xyhDΔ=Δ= = ,
/30D
,
/40D
,
/50D
and

/60D
, respectively,
around the cylinder. Results in Table 2.2 show that the calculated drag
coefficient well converges to stable values as the mesh is refined. A mesh size
of
/50hD= is fine enough to provide a reliable solution at both flow
conditions and will be used for the following simulations. For the unsteady
flow
Re 100= , time step size independence is also studied. Three time step
sizes
3
210t
−
Δ= ×
,
3
110
−
×
and
4
510
−
×
are examined. While the time step
3
210t
−
Δ= ×
leads to higher drag coefficient and lower lift fluctuation as

shown in Table 2.3, the results produced by time steps
3
110t
−
Δ=×
and
4
510
−
×
are close to each other. Therefore,
3
110t
−
Δ=×
will be selected for
the time integration.


Chapter 2 Governing Equations and Boundary Condition-Enforced IBM
46

The streamlines and vorticity patterns in the vicinity of the cylinder at
40Re =
and
100Re =
are shown in Figs. 2.3 and 2.4, respectively. While a
pair of symmetric eddies are firmly standing behind the cylinder in plots for
40Re =
, the eddies are shed alternatively from the cylinder at

100Re =

(Fig.2.4) and the famous Karman vortex street has been successfully captured.
As can be clearly observed from Figs. 2.3 and 2.4, there is no any penetration
of streamlines through the cylinder surface, indicating that no mass transfer
happens between the fluid inside the cylinder and that outside the cylinder. As
a comparison, the streamlines at
100Re
=
obtained using the conventional
IBM (Uhlmann 2005) are provided in Fig. 2.5 where streamline penetrations
across the cylinder surface are quite obvious, indicating that the no-slip
condition on the cylinder surface is severely violated. Table 2.4 shows the
drag coefficient
D
C
and the recirculation length
/
w
LD
, together with those
produced by the body-fitted methods (Dennis & Chang 1970; Shukla et al.
2007) and some previous immersed boundary methods (Russell & Wang 2003;
Lima E. Silva et al. 2003; Le et al. 2008). Particularly, the results from Shukla
et al. (2007) are calculated based on a very high-order scheme and the results
from the abovementioned immersed boundary methods are based uniform
meshes with resolutions of
/20hD
=
( Russell & Wang 2003), /10hD=

(Lima E. Silva et al. 2003) and
/18hD
=
(Le et al. 2008) respectively. It is
apparent that both the drag coefficient and recirculation length obtained by the
present method agree well with the benchmark values from the body-fitted

Chapter 2 Governing Equations and Boundary Condition-Enforced IBM
47

methods. It is also observed that for this steady case, the drag coefficients
derived by different immersed boundary methods are close to each other while
the recirculation lengths show relatively large discrepancy.

The time evolution of drag and lift coefficients for
100Re
=
is recorded in
Fig. 2.6 and shows regularly periodic oscillations, which implies the
periodicity of the flow field induced by the periodic vortex shedding from the
cylinder. Table 2.5 provides the time-mean and fluctuating values of drag and
lift coefficients and the Strouhal number
St
obtained in the present
simulation, together with those from the literature (Li et al. 1991; Liu et al.
1998; Lai & Peskin 2000; Uhlmann 2005; Ji et al. 2012). Once again, the
results from both the body-fitted methods (Li et al. 1991; Liu et al. 1998) and
some previously proposed immersed boundary methods which employ similar
mesh resolutions (Lai & Peskin 2000; Uhlmann 2005; Ji et al. 2012) are
presented for comparison. We can see that for the unsteady case, the drag

coefficient produced by the immersed boundary method is always a bit higher
than the body-fitted method due to the usage of the discrete delta function.
However, it is obvious that our result is closer to the reference ones (Li et al.,
1991; Liu et al., 1998). The lift coefficient given by the present method, as can
be seen, also matches quite well with the benchmark values provided. While
the Strouhal numbers produced by some previous immersed boundary
methods are 1%-4% higher than those from the body-fitted methods, our result

Chapter 2 Governing Equations and Boundary Condition-Enforced IBM
48

achieves a good agreement.

2.6.2 Flow interference between a pair of side-by-side circular cylinders
Flow around a pair of side-by-side stationary circular cylinders has been found
to exhibit abundant flow patterns (Chang & Song 1990; Ravoux et al. 2003;
Kang 2003; Mizushima & Ino 2008). The vortices shed from the two cylinders
interact dynamically, generating much more complicated wake behaviors than
those behind an isolated cylinder. Therefore this topic is a favorable example
to further test the performance of our new method. Generally, the flow is
governed by two dimensionless parameters: the Reynolds number
Re
and
gap ratio
/GD
, with the latter one much more significant and sensitive. Here
G
denotes the gap distance between the cylinder centers. Depending on
various geometrical configurations of the two cylinders, the flow
characteristics can be quite different from each other. Our simulations here

choose several representative gap ratios and focus on a fixed Reynolds number
of
Re 100=
. The schematic diagram is plotted in Fig. 2.7, where

the
computational domain has a size of
50 40DD
×
. The two side-by-side
cylinders are located at (
15D , 20 / 2DG
−
) and ( 15D , 20 / 2DG+ ),
respectively. In our simulations, the whole domain is discretized by a
non-uniform mesh with a fine resolution of / 50
xyhD
Δ
=Δ = = inside a
local region around the cylinders. Meanwhile, a time step size of
0.001tΔ=
is used for the time intergration. The flow details in the near wake regions in

Chapter 2 Governing Equations and Boundary Condition-Enforced IBM
49

forms of streamlines and vorticity contours as well as the lift and drag
coefficients are presented and quantitatively compared with the
well-established results in the literature.

Fig. 2.8 plots the instantaneous streamlines and vorticity contours in the
near-wake region behind the cylinder-pair at
/3GD
=
. Consistent with the
observations in Chang & Song (1990) and Ding et al. (2007), the two vortex
streets developeing behind the cylinder pair are apparently symmetric, falling
in an anti-phase synchronization flow regime indicated by Kang (2003). The
symmetric feature also manifests itself from the time histories of drag and lift
coefficients presented in Fig. 2.9: while the drag coefficients of the upper and
lower cylinders are almost in the same phase, the lift coefficients of the two
cylinders have a
180
°
phase difference. The time-mean lift and drag
coefficients together with the Strouhal number are well calculated and listed in
Table 2.6
. Their comparison with the reported results of Chang & Song (1990)
derived using the body-fitted method and those of Ding et al. (2007) using the
mesh-free method exhibits good agreement.

With a systematic change of the gap ratio
/GD
, different flow patterns are
observed. Simulations are also carried out at four other gap ratios of
/1.2GD=
, 1.7, 2.5 and 4, as presented in the following. Instantaneous
streamlines and vorticity contours shown in Fig. 2.10 reveal four completely

Chapter 2 Governing Equations and Boundary Condition-Enforced IBM

50

different wake patterns. In the order of increasing gap ratio, they are single
bluff-body, flip-flopping, in-phase synchronized and anti-phase synchronized
flows. At the small gap ratio of
/1.2GD
=
(Fig. 2.10(a)), the two cylinders
are placed in such a close proximity that the vortices at the gap sides of the
cylinders are almost completely suppressed and neither individual vortex
street for the two cylinders exists. The single wake behind the cylinder pair
behaves as a normal Karman vortex street similar to that behind an isolated
cylinder. As seen in Fig. 2.11(a), the drag and lift coefficients evolve regularly
with time and imply a periodic vortex shedding from the cylinder system.
Therefore the flow pattern is called the “single bluff body”. When the gap
ratio increases to
/1.7GD=
, the individual Karman vortex street is captured
again behind each cylinder. The flow between the gap of the two cylinder
surfaces is biased (Fig. 2.10(b)), and the deflection direction randomly
flip-flops between upwards and downwards, resulting in a drastically irregular
vortex shedding behind the cylinder pair. Forces experienced by the two
cylinders (Fig. 2.11(b)), as a result, take an irregular variation as well. In this
regard, the flow in this pattern is described as “flip-flopping” by Kang (2003).
As a further increase in gap ratio to
/2.5GD
=
, the vortex shedding initially
appears in a symmetric pattern. After a short transitional period, the flow
switches to a steady in-phase vortex shedding, showing an anti-symmetric

pattern (Fig. 2.10(c)). Such a flow behavior can be clearly seen from the
dynamic force evolutions in Fig. 2.11(c), where the lift coefficients of the two

Chapter 2 Governing Equations and Boundary Condition-Enforced IBM
51

cylinders eventually develop into an in-phase evolution. In the case
of
/4GD=
, the flow pattern and dynamics are the same as those
at
/3GD=
. As shown in Fig. 2.10(d), the two individual vortex streets are
symmetric to each other in an anti-phase synchronized pattern. The forces
experienced by the two cylinders (Fig. 2.11(d)) are fully periodic. While the
drag coefficients are almost in the same phase, the lift coefficients have a
180
°
phase difference. We can see that the flow patterns and their transitions
discovered in Kang et al. (2003) and Lee et al. (2009) are well captured in the
present study.


2.6.3 Flow around a transversely oscillating circular cylinder in a
free-stream
A single transversely oscillating cylinder in a cross flow has been extensively
investigated from a variety of perspectives (Bishop & Hassan 1964;
Koopmann 1967; Griffin & Votaw 1972; Zdravkovich 1982; Ongoren &
Rockwell 1988; Williamson & Roshko 1988; Gu et al. 1994; Morse &
Williamson 2009). This subsection is devoted to studying this moving

boundary problem in which the circular cylinder undergoes a prescribed
transverse harmonic motion

sin(2 )
c
yA ft
π
=
, (2.33)
where
A
is the cylinder oscillation amplitude and
c
f
is the oscillation
frequency. The present simulation follows the numerical experiments of

Chapter 2 Governing Equations and Boundary Condition-Enforced IBM
52

Guilmineau et al. (2002) who have studied the problem
at
Re 185=
,
/0.2AD=
and
/0.81.2
cs
ff
=

−
, where
s
f
is the vortex
shedding frequency corresponding to its stationary counterpart. A
computational domain of size
60 50DD
×
is selected for the simulations,
where the cylinder takes its equilibrium position at (
30D ,25D ). The domain
is discretized by a non-uniform mesh with a fine resolution of
/50
hxyD=Δ =Δ = around the cylinder. Meanwhile,
0.001t
Δ
=
is selected
for the time integration.

Fig. 2.12 presents the time evolution of drag and lift coefficients over the
range of
/
cs
f
f
considered. For
/1.0
cs

ff
≤
, the drag and lift behave fairly
periodic. The oscillating amplitude of both lift and drag increases with the
increase in
/
cs
f
f
. For
/
cs
f
f
greater than 1.0, the drag and lift exhibit the
beating form and the beating frequency increases as the vibration frequency
increases. These features are congruent with the findings reported by
Guilmineau et al. (2002), Lee et al. (2009) and Zhou & Shu (2011).

The variation of time-mean drag and r.m.s value of lift and drag versus
/
cs
f
f

also match well with the results from Guilmineau et al. (2002), Lee et al.
(2009) and Zhou & Shou (2011), as shown in Fig. 2.13. It is observed that the
time-mean drag peaks at
/1.0
cs

ff
=
and then decreases as
/
cs
f
f
increases.
The same behavior is also noted for the r.m.s value of the drag. The r.m.s. of

Chapter 2 Governing Equations and Boundary Condition-Enforced IBM
53

lift, on the other hand, shows a monotonic increase with increasing of
/
cs
f
f
.

The instantaneous wake structures at different oscillating frequencies are
plotted in Fig. 2.14, from which we can see that they all show a pattern similar
to the normal Karman vortex street. However, both the vortex length and
longitudinal spacing between adjacent vortices decrease as the oscillating
frequency
c
f
increases.

2.6.4 Vortex-induced vibration of an elastically mounted circular cylinder

in a free-stream
The proposed method is further tested by the vortex-induced vibration of an
elastically mounted circular cylinder in a free-stream, where the motion of the
cylinder is completely determined by the vortex-cylinder interaction, rather
than being given in advance as in the last example. The schematic diagram of
the problem is shown in Fig. 2.15. Modeling the cylinder as a two
degree-of-freedom mass-spring-damper system, its motion is governed by the
dimensionless equations:

2
22 2
2
D
rr r
x
xxC
UU m
ππ
ζ
π
⎛⎞⎛⎞
++=
⎜⎟⎜⎟
⎝⎠⎝⎠
 
(2.34)

2
22 2
2

L
rr r
yyyC
UU m
ππ
ζ
π
⎛⎞⎛⎞
++=
⎜⎟⎜⎟
⎝⎠⎝⎠
 
, (2.35)
where
ζ
is the damping ratio,
/
rsf
mmm
=
is the mass ratio between the
cylinder and the displaced fluid,
r
U
is the reduced velocity defined as