Development of immersed boundary methods for isothermal and thermal flows 3

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Chapter 3 Stream Function-Vorticity Formulation-based IBM
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Chapter 3

Stream Function-Vorticity Formulation-based
Immersed Boundary Method
1


In this chapter, a novel and efficient stream function-vorticity
formulation-based immersed boundary method is presented. The main idea of
the method is to accurately satisfy both the governing equation and boundary
condition, which is realized through velocity correction and vorticity
correction procedures. Since the physical boundary condition is usually for
velocity, we enforce the no-slip boundary condition through a velocity
correction process. For the vorticity correction, unlike adding complicated
source terms into the vorticity transport equation as suggested by Wang et al.
(2009), we do not add any source term in the vorticity transport equation.
Instead, the vorticity correction is directly evaluated from the first order
derivatives of velocity correction. In this work, two ways are proposed to
evaluate the vorticity correction. One is based on finite difference
approximation of velocity-correction derivatives, and the other is based on
derivative expressions of Dirac delta function and velocity correction. These
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1
Parts of materials in this chapter have been published in
W.W. Ren, J. Wu, C. Shu, W.M. Yang, Int. J. Numer. Meth. Fluids, 70 (2012) 627-645.

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Chapter 3 Stream Function-Vorticity Formulation-based IBM
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two ways and the whole solver are tested by their applications to simulate
flows over a circular cylinder for both steady and unsteady states. Examples of
moving boundary problems, like flow over a left moving circular cylinder,
flow over an inline oscillating circular cylinder, sedimentation of a circular
particle between two parallel walls, are also provided for further validation.
Compared to the previously proposed stream function-vorticity
formulation-based immersed boundary method, the present method is more
efficient and attractive for two-dimensional (2D) incompressible flows.

3. 1 Methodology
3.1.1 Governing equations
The two-dimensional computational configuration shown in Fig. 2.1 is
recalled here. To take into account the effect of immersed boundary on the
flow field, an additional term, which plays the role of vorticity correction, is
introduced in the stream function formulation in this work. As a result, the
governing equations describing the fluid motion with the use of IBM can be
expressed in the stream function-vorticity formulation as

)()(
2
2
2
2
yxy
v

x
u
t ∂
∂
+
∂
∂
=
∂
∂
+
∂
∂
+
∂
∂
ωω
μ
ωωω
ρ
(3.1)

Θω
ϕϕ
+=
∂
∂
+
∂
∂

2
2
2
2
yx
(3.2)
which is subject to the no-slip boundary condition (2.3), where

x
and
y

denote the components of the Eulerian coordinate
x
,
u
and
v
correspond
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Chapter 3 Stream Function-Vorticity Formulation-based IBM
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
to the components of the velocity vector
u
in the
x
and
y
directions

respectively.
ϕ
and
ω
denote the stream function and vorticity. The
additional term
Θ
is the vorticity source. In addition, the vorticity is defined
as

uv
y
x
ω
∂∂
+Θ= −
∂∂
(3.3)
and the stream function is related to the velocity by

y
u
∂
∂
=
ϕ
(3.4)

x
v

∂
∂
−=
ϕ
(3.5)
Note that
ω
+Θ
in Eq. (3.3) is the physical vorticity. Eqs. (3.1) to (3.5)
together with Eqs. (2.3) and (2.5) provide a complete description of stream
function-vorticity formulation for incompressible viscous flows in the entire
computational domain
Ω
involving the immersed boundary
Γ
.

Note that the vorticity source
Θ
in Eq. (3.2) is to consider the effect of the
immersed boundary

Γ
. The solution of equation systems (3.1)-(3.2) can be
obtained by the following two steps. In the first step, the normal stream
function-vorticity formulation without the vorticity source term (that is, the
effect of immersed boundary is ignored) is solved,

)()(
2

2
2
2
yxy
v
x
u
t ∂
∂
+
∂
∂
=
∂
∂
+
∂
∂
+
∂
∂
ωω
μ
ωωω
ρ
(3.6)

ω
ϕϕ
=

∂
∂
+
∂
∂
2
2
2
2
yx
(3.7)
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Chapter 3 Stream Function-Vorticity Formulation-based IBM
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By solving the above equations, the predicted vorticity
*
ω
and stream
function
*
ϕ
can be obtained. Then from Eqs. (3.4) and (3.5), the predicted
velocity field
*
u
is calculated. The basic solution procedure is described as
follows. At first, we apply the Euler forward scheme to discretize the time
derivative in Eq. (3.6) and obtain the predicted vorticity
*

ω
by

*
2
()
n
nn n
t
ωω
ρ
ρωμω
−
=− ⋅∇ + ∇
Δ
u
(3.8)
Then the predicted stream function field
*
ϕ
is solved from the following
Poisson equation

2* *
ϕ
ω
∇=
(3.9)
For the spatial derivatives, the well-known central difference scheme is
employed for numerical discretization.

In the second step, the velocity field should be corrected so that the no-slip
condition (2.3) on the immersed boundary
Γ
can be satisfied. It should be
indicated that although the stream function-vorticity form is used, we still
prefer to make velocity correction. This is because the physical condition at
the immersed boundary is usually the condition for velocity rather than for
stream function. In addition, for some multiply-connected domain problems,
the value of stream function at the surface of immersed body is an unknown
constant. This unknown constant may vary with time, and must be determined
and updated at each time step during the computation. This brings difficulty in
its numerical implementation. To make velocity correction, the corrected
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Chapter 3 Stream Function-Vorticity Formulation-based IBM
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velocity field
u
is written as

*
=+Δuu u
(3.10)
where
Δu
is the velocity correction, and
*
u
is the predicted velocity field

derived in the predictor step. Correspondingly, the corrected vorticity can be
written as

*
ωω
=+Θ
(3.11)
Here,
*
ω
is the predicted vorticity field, and
Θ
is the vorticity correction.
*
ω
in Eq. (3.11) is actually the same as
ω
in Eq. (3.2) since they are from
the same governing equation (vorticity transport equation without considering
the effect of immersed boundary). It is only for convenience that we note the
solution of Predictor step as
*
ω
. In this sense,
Θ
in Eq. (3.11) is indeed the
vorticity source in Eq. (3.2). Substituting Eqs. (3.10) and (3.11) into Eq. (3.3)
and using Eq. (3.9) gives

() ()uv

y
x
∂Δ ∂Δ
Θ= −
∂
∂
(3.12)
It should be noted that the key step in the proposed stream function-vorticity
formulation-based IBM is the calculation of velocity correction. Once it is
determined, the corrected velocity and vorticity can be calculated from Eqs.
(3.10) and (3.11) respectively.

3.1.2 Velocity correction procedure
The velocity correction aims to enforce the no-slip boundary condition (2.3).
The basic idea is that the velocity
( (,),)st tuX
at the boundary (Lagrangian)
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Chapter 3 Stream Function-Vorticity Formulation-based IBM
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point interpolated from the corrected velocity
(,)tux
at the surrounding fluid
(Eulerian) points should be equal to the given boundary velocity
((,),)
B
st tUX
, i.e.,

()
(,), (,) ( (,))
B
st t t st dV
δ
Ω
=−
∫
UX ux xX
(3.13)
Noticing that the corrected velocity
u
is contributed from the predicted
velocity
*
u
and the velocity correction
Δ
u
, Eq. (3.13) can be reformulated
as

()
()
*
(,), (,) (,) ( (,))
B
st t t t st dV
δ
Ω

=+Δ −
∫
UX ux ux xX
(3.14)
It is known that the velocity correction is introduced due to the presence of the
immersed boundary, so it is reasonable to assume that the velocity correction
stems from the virtual boundary flux
Δ
P
through the delta function
interpolation

(,) ( (,),) ( (,))tsttstds
Γ
δ
Δ=Δ −
∫
ux PX x X
(3.15)
Employing a similar Eulerian and Lagrangian mesh discretization as those in
Section 2.4.2, Eq. (3.14) and (3.15) can be approximated in their spatial
discrete forms

()
()
*2
,()()
ij
Bi j j h
j

tDh=+Δ
∑
UX ux ux
(3.16)
and

()
( , ) , ( 1,2, , ; 1,2, , )
ij
jihi
i
ttDsiMjNΔ=Δ Δ= =
∑
ux P X ""
(3.17)
Substituting Eq. (3.17) into Eq. (3.16), an equation system for the virtual
boundary flux
Δ
P
would be formed and written in the matrix form as

[
]
[
]
[
]
PP P
=AX B
(3.18)

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Chapter 3 Stream Function-Vorticity Formulation-based IBM
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
where
[]
11 12 1 11 12 1
12
21 22 2 21 22 2
2
12
12 1 2
12
MN
hh h h h hM
MN
hh h h h hM
P
NN NMM M MN
hh h h h hM
DD D DsDs Ds
DD D DsDs Ds
h
DD D DsDs Ds
⎛⎞⎛ ⎞
ΔΔ Δ
⎜⎟⎜ ⎟
ΔΔ Δ
⎜⎟⎜ ⎟
=

⎜⎟⎜ ⎟
⎜⎟⎜ ⎟
⎜⎟⎜ ⎟
ΔΔ Δ
⎝⎠⎝ ⎠
A
""
""
##%# # #% #
""

(3.19)

[]
11 21 1 *
1
1
12 22 2 *2
2
2
12 *
N
hh hB
N
hh h
B
P
MM NM
M
hh h NB

DD D
DD D
h
DD D
⎛⎞⎛⎞
⎛⎞
⎜⎟⎜⎟
⎜⎟
⎜⎟⎜⎟
⎜⎟
=−
⎜⎟⎜⎟
⎜⎟
⎜⎟⎜⎟
⎜⎟
⎜⎟
⎜⎟⎜⎟
⎝⎠
⎝⎠⎝⎠
u
U
uU
B
u
U
"
"
##%###
"
(3.20)

[]
1
2
P
M
Δ
⎛⎞
⎜⎟
Δ
⎜⎟
=
⎜⎟
⎜⎟
Δ
⎝⎠
P
P
X
P
#
(3.21)
and
(1,,)
i
iMΔ=P "
are the abbreviations for
(,)
i
t

Δ
PX
. Similar to
[]

F
A
,
the elements of coefficient matrix
[
]
P
A
are only related to the coordinates of
Lagrangian boundary points and their adjacent Eulerian points.

After solving the equation system (3.18) and obtaining
i
Δ
P
at all Lagrangian
points, they are substituted into Eq. (3.17) to obtain the velocity correction
j
Δu
, which are further substituted into Eq. (3.10) to get the corrected velocity
),,1( Nj
j
"=u
.

3.1.3 Vorticity correction procedure
The vorticity correction procedure is straightforward and easier as compared
to that of velocity correction. As shown in Eq. (3.12), once the velocity
correction has been obtained, the vorticity correction can be calculated from
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Chapter 3 Stream Function-Vorticity Formulation-based IBM
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the first order derivatives of velocity correction. Then from Eq. (3.11), the
corrected vorticity can be easily computed. In the vorticity correction
procedure, the key step is to approximate the derivatives of
()u
y
∂Δ
∂
and
()v
x
∂Δ
∂
at Eulerian points. In this work, we propose two ways to do the
approximation. They are denoted as Method 1 and Method 2. Method 1 is very
simple. It directly approximates
()u
y
∂
Δ
∂
and
()v

x
∂
Δ
∂
by finite difference
schemes. Method 2 avoids numerical approximation of derivatives. As shown
in Eq. (3.15), the velocity correction
Δ
u
is a function of physical location x,
and the Dirac delta function
st
δ
−
(x X( , ))
is also the function of x.
Therefore, we have

x
ut st
P
st t ds
yy
δ
Γ
∂Δ ∂ −
=Δ
∂∂
∫
((x,)) (xX(,))

(X( , ), )
(3.22a)

y
vt st
P
st t ds
xx
δ
Γ
∂Δ ∂ −
=Δ
∂∂
∫
((x,)) (xX(,))
(X( , ), )
(3.22b)
where (
uΔ
,
vΔ
) are components of
Δ
u
, and (
x
P
Δ
,
y

P
Δ
) are components of
ΔP
. Substituting Eq. (3.22) into Eq. (3.12) and approximating the integral by
summation, we can obtain,

(,)( )
( , ) ( )
xi h i i
i
yi h i i
i
PtD s
y
PtD s
x
∂
⎛⎞
Θ= Δ − Δ
⎜⎟
∂
⎝⎠
∂
⎛⎞
−Δ −Δ
⎜⎟
∂
⎝⎠
∑

∑
XxX
XxX
(3.23)
which can be further written as
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Chapter 3 Stream Function-Vorticity Formulation-based IBM
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

2
2
1
(,) ( ) ( )
1
( , ) ( ) ( )
ii
x
ih h i
i
ii
yi h h i
i
xX yY
Pt s
hhyh
xX yY
Pt s
hx h h
δδ

δδ
⎛⎞
⎛⎞
−−
∂
Θ= Δ Δ
⎜⎟
⎜⎟
∂
⎝⎠
⎝⎠
⎛−−⎞
∂
⎛⎞
−Δ Δ
⎜⎟
⎜⎟
∂
⎝⎠
⎝⎠
∑
∑
X
X
(3.24)
In Eq. (3.24), the continuous kernel functions take the form of (2.21), whose
derivatives can then be analytically given as
2
2
2

2
sgn( ) 2( )
sgn( )
0 | | 1
4
14| |4| |
3sgn( ) 2( )
sgn( )
()
1 | | 2
4
712| |4| |
0
ii
i i
ii
ii
i
i i
h
ii
xX xX
xX xX
hh
hh
xX xX
hh
xX xX
xX
xX xX

hh
xh
hh
xX xX
hh
δ
−−
−
−−
−+ ≤≤
−−
+−
−−
−
−
∂
−−
=
−+ <≤
∂
−−
−+ −
| | 2
i
xX
h
⎧
⎪
⎪
⎪

⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
−
⎪
>
⎪
⎩
(3.25)
2
2
2
2
sgn( ) 2( )
sgn( )
0 | | 1
4
14| |4| |
3sgn( ) 2( )
sgn( )
()
1 | | 2
4

712| |4| |
0
ii
ii
ii
ii
i
ii
h
ii
yY yY
yY yY
hh
hh
yY yY
hh
yY yY
yY
yY yY
hh
yh
hh
yY yY
hh
δ
−−
−
−−
−+ ≤≤
−−

+−
−−
−
−
∂
−−
=
−+ <≤
∂
−−
−+ −
| | 2
i
yY
h
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
−

⎪
>
⎪
⎩
(3.26)
With above formulations, Method 2 evaluates the vorticity correction directly
from the velocity correction without numerical approximation of derivatives.


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Chapter 3 Stream Function-Vorticity Formulation-based IBM
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3.1.4 Computational sequence
The computational sequence of the present solver can be summarized as below.
To march solution from time level
n
to
1n
+
,
1)
Use
n
u and
n
ω
as initial flow field to solve Eqs. (3.8) and (3.9) and
obtain predicted vorticity
*

ω
and stream function
*
ϕ
;
2)
Substitute
*
ϕ
to Eqs. (3.4) and (3.5) to calculate the predicted velocity
field
*
u
;
3)
Compute the elements of matrix
[
]
P
A
;
4)
Use equation system (3.18) to calculate the virtual boundary fluxes
i
ΔP

(
1, ,iM= "
) at all Lagrangian points and then substitute them into Eq.
(3.17) to get the velocity correction

Δ
u
;
5)
Correct the fluid velocity at Eulerian points using Eq. (3.10);
6)
Use either Method 1 or Method 2 described in Section 3.1.3 to calculate
the vorticity correction
Θ
;
7)
Correct the vorticity at Eulerian points using Eq. (3.11);
8)
Use the corrected vorticity and velocity as the initial conditions, and
repeat steps 1 to 7 for the computation of next time level. The process
continues until a converged solution is achieved (steady case) or the given
time is reached (unsteady case).

3.2 Results and Discussion
The proposed stream function-vorticity formulation-based IBM, using velocity
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Chapter 3 Stream Function-Vorticity Formulation-based IBM
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
correction and vorticity correction technique, will be validated in this section
by applying it to simulate flow over a stationary circular cylinder, flow over a
left moving circular cylinder, flow over a inline oscillating circular cylinder in
a fluid at rest and sedimentation of a single circular particle between two
parallel walls.

3.2.1 Flow over a stationary circular cylinder
The classical problem of flow over a stationary circular cylinder is taken as the
first case to examine the performance of the proposed
stream-function-vorticity formulation-based IBM solver, where numerical
simulations are performed for both steady (
40Re
=
) and unsteady (
100Re =
)
states using different vorticity correction methods. Flow characteristics in
terms of the drag coefficient
D
C
, lift coefficient
L
C
(for unsteady case),
recirculation length
w
L
(for steady case) behind the cylinder, as well as
streamlines and vorticity patterns are presented and compared with previous
results in the literature. The computational domain and dimensionless
parameters are defined the same as those in Section 2.6.1. The top and bottom
boundaries are confined with boundary conditions of
const
=
ϕ
and

0=
ω
.
At the inflow boundary, stream function
)(yf
=
ϕ
and vorticity
0=
ω
are
specified. At the outflow boundary, the homogeneous Neumann boundary
conditions of
0=
∂
∂
=
∂
∂
xx
ω
ϕ
are applied. Non-uniform meshes with a fine
resolution of
/50xyhD
Δ
=Δ = = near the cylinder have been used for the
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Chapter 3 Stream Function-Vorticity Formulation-based IBM
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
following discussions. For the unsteady case
100Re
=
, a time step size of
0.001tΔ= is used.

To illustrate the capability of proposed two methods for evaluation of vorticity
correction, results obtained using Method 1 and Method 2 are compared. Fig.
3.1 shows the streamlines and vorticity patterns in the vicinity of the cylinder
when flows reach their steady state at
40Re
=
. A pair of symmetric eddies
are formed behind the cylinder. As can be observed clearly from Fig. 3.1,
there is no penetration of streamline through the cylinder surface, indicating
that the no-slip condition on the immersed boundary is accurately satisfied.
Table 3.1 shows the drag coefficient
D
C
and the recirculation length
/
w
LD
,
together with those from references (Dennis & Chang 1970; Fornberg 1980;
He & Doolen 1997; Niu et al. 2006; Wu & Shu 2009). It is obvious that the
drag coefficient and recirculation length obtained by the two methods are
close to each other. Basically, the present results agree well with the reference
data while Method 2 has a slightly better agreement with benchmark values

than Method 1. To illustrate the efficiency of the present stream
function-vorticity formulation-based IBM, the CPU time consumed by the
present method (Method 1 is used for the vorticity correction) is recorded and
compared with that consumed by the pressure-velocity formulation-based IBM
proposed in Chapter 2. Both runs are conducted on a HPC with 2.27GHz CPU
and 258 GB RAM, using the same
285 225
×
non-uniform mesh and setting
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Chapter 3 Stream Function-Vorticity Formulation-based IBM
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
16
110
nn+−
∞
−<×uu&&
as the convergence criteria. As shown in Table 3.2,
(5529.18 4669.2) / 5529.18 15.6%−= less CPU time is consumed by the
proposed stream function-vorticity formulation based IBM compared to the
pressure-velocity formulation-based IBM, indicating that the present method
is more efficient.

Fig. 3.2 shows the instantaneous streamlines and vorticity contours in the
vicinity of the cylinder for
100Re
=
. It is clear that the Karman vortex street
has been successfully revealed in both plots, which results in regular periodic

variations of drag and lift coefficients presented in Fig. 3.3. Table 3.3 provides
the drag and lift coefficients (in the form of average value and magnitude of
variation)
D
C
,
L
C
, Strouhal number
St
, together with those from Braza et
al. (1986), Liu (1998) and Ding et al. (2004) for this unsteady case. Once
again, we can see that the drag and lift coefficients as well as Strouhal number
given by the two methods are close to each other. The present results match
well with the reference data and Method 2 shows a slightly better agreement
with benchmark values than Method 1. So in the following simulations, we
will only use Method 2 to evaluate the vorticity correction.

3.2.2 Flow over a left moving circular cylinder
As a first test case of moving boundary problems, a circular cylinder moving
to the left at a velocity of
∞
U
in the stationary fluid is modeled. The same
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Chapter 3 Stream Function-Vorticity Formulation-based IBM
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
geometry as in the above example is used, except that the cylinder starts to
move at a distance of

12
to the right boundary. The boundary conditions are
imposed as

0=
∂
∂
=
x
ω
ϕ
on the left boundary

0
=
=
ω
ϕ
on the top and bottom boundary

0=
∂
∂
=
∂
∂
xx
ω
ϕ
on the right boundary

Note that the only difference between this left moving cylinder case and the
previous stationary cylinder case is the adjustment of reference frame. These
two problems should provide equivalent results.

Fig. 3.4 shows the adjusted streamlines at a non-dimensional time of
16
. Fig.
3.5 presents the vorticity patterns in the vicinity of the cylinder for both
stationary case and left moving cylinder case at the same moment. A
comparison between histories of drag coefficient and vorticity distribution
along the cylinder surface for both cases is also plotted in Fig. 3.6 and Fig. 3.7
respectively. It can be clearly seen that there are very little discrepancies
between the results for two cases.

3.2.3 Flow over an inline oscillating circular cylinder in a fluid at rest
Flow over an inline oscillating circular cylinder in a fluid at rest has been
investigated both experimentally (Dütsch et al. 1998) and numerically (Wang
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Chapter 3 Stream Function-Vorticity Formulation-based IBM
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
et al. 2009; Yang & Balaras 2006; Choi et al. 2007; Guilmineau & Queutey
2002; Lee et al. 2011). Herein, it is studied to validate the proposed method
for solving moving boundary problems. The inline motion of the cylinder is
given by the harmonic oscillation

() sin(2 )
c
x
tA ft

π
=−
(3.27)
where
()
x
t
is location of the cylinder center in its oscillation direction
(x-direction),
A
and
c
f
are the oscillating amplitude and frequency of the
cylinder, respectively. The two key parameters characterizing this flow are
Reynolds number
max
Re
UD
ρ
μ
=
(
max
U
is the maximum velocity of the
cylinder) and Keulegan-Carpenter number
max
c
U

KC
f
D
=
, which are set as
Re 100=
and
5KC =
in the present investigation, according to the
experimental result of Dütsch et al. (1998). A dimensionless computational
domain of size
24 24×
is chosen, with the cylinder initially located at the
center of the domain. A uniform mesh of resolution
1/40h
=
with a time
step size of
0.002tΔ= is used for the simulation. On all four boundaries, the
natural boundary condition (Neumann type) is applied for both vorticity and
velocity.

Fig. 3.8 shows the velocity profiles in the oscillation and transverse direction
at four different
x
locations (
0.6x
=
−
,

0
,
0.6
,
1.2
) and three different
phase angles (
2180ft
φ
π
==°
,
210°
,
330°
). The corresponding experimental
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Chapter 3 Stream Function-Vorticity Formulation-based IBM
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
results of Dütsch et al.(1998) and numerical results of Wang et al. (2009) are
also displayed in these figures for comparison. A good agreement can be
observed from the comparison. Furthermore, a time evolution of the in-line
force
x
F
acting on the cylinder surface in one period is presented in Fig. 3.9.
The experimental result of Dutsch et al. (1998) is also included in Fig. 3.9 for
comparison. Once again, a good agreement is achieved. It is noted that for all
the cases tested, the present solver does not show any spurious force

oscillation. This may be attributed to the fact that the no-slip boundary
condition is enforced in the present approach and the velocity correction,
which is related to the boundary force, at all boundary points is obtained
simultaneously from the algebraic equation system (3.18).

3.2.4 Sedimentation of a single circular particle inside a box
Another problem we choose to further test the capability of present method in
solving moving boundary problems is the sedimentation of a single circular
particle inside a box. A single particle which settles inside a box has been
studied by several researchers (Hu 1996; Naury 1999; Hu et al. 2001;
Glowinski et al. 1999; Glowinski et al. 1999; Wan & Turek 2006). Specially,
Feng et al. (2004; 2005), Uhlmann (2005), Luo et al. (2007), and Wu & Shu
(2010) have used various versions of immersed boundary method to study this
problem.

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Chapter 3 Stream Function-Vorticity Formulation-based IBM
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In the present simulation, a domain with width
cm2
and height
cm6
(Fig.
3.10) is taken. The density
ρ
and viscosity
μ
of incompressible viscous
fluid filled in the domain are

3
/0.1 cmg
and
)/(1.0 scmg
⋅
, respectively. The
circular particle is rigid with density of
3
1.25 /
p
g
cm
ρ
=
and diameter of
0.25
p
dcm=
. Initially, the particle is released at
)4,1( cmcm
in static fluid,
and then falls down due to the gravity force.

A uniform mesh with resolution
/25
p
hd
=
and time step 0.005tΔ= is
used for the present simulation. The boundaries of the box are solid walls and

can be treated as a single streamline. The instantaneous vorticity patterns at
different moments of
st 2.0=
,
s4.0
,
s6.0
,
s8.0
are displayed in Fig. 3.11.
The variation of flow structure can be clearly observed. The time evolutions of
longitudinal coordinate
Y
, longitudinal velocity
V
, Reynolds number
Re

of particle center as well as the translational kinetic energy
T
E
are plotted in
Figs. 3.12-3.15. Here,
Re
and
T
E
are defined as
22
Re

pp
dU V
ρ
μ
+
=

and
22
0.5 ( )
Tp
E
mU V=+
, respectively, where
U
and
V
are velocity
components of particle center, and
p
m
is the mass of the particle. The results
of Wan et al. (2006) and Wu et al. (2010) are also included in the figures for
comparison. We can see clearly that present results match well with the
benchmark solutions. This shows that the present solver can be effectively
used to simulate moving boundary flow problems.

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Chapter 3 Stream Function-Vorticity Formulation-based IBM
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
3.3 Conclusions
In this chapter, a simple and efficient stream function-vorticity
formulation-based IBM is developed for simulating 2D incompressible
viscous flows. The effect of boundary on the flow field is considered through
velocity correction and vorticity correction. The velocity correction is
determined implicitly in such a way that the velocity at the immersed
boundary interpolated from the corrected velocity field satisfies the physical
boundary condition. Unlike adding complicated source terms into the vorticity
transport equation in the literature, in this work, the vorticity correction is
made through the stream function formulation. It is evaluated simply from the
first order derivatives of velocity correction. Furthermore, two ways are
presented to approximate velocity-correction derivatives. One is direct
approximation by the finite difference scheme. The other is based on
derivative expressions of the Dirac delta function and velocity correction. This
way does not involve numerical approximation of derivatives. Numerical
experiments show that both ways work very well, and the second way seems
to perform better.

The efficiency and capability of present method and two ways to evaluate the
vorticity correction are tested by their application to simulate both stationary
boundary and moving boundary problems. Numerical results show good
agreement with available data in the literature. It seems that the present
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Chapter 3 Stream Function-Vorticity Formulation-based IBM
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method has a promising potential for solving 2D incompressible viscous flows
with curved boundaries.


Chapter 3 Stream Function-Vorticity Formulation-based IBM
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Table 3.1 Comparison of drag coefficient
D
C
and recirculation length
/
w
LD

Cases References
D
C

/
w
L
D




Re=40
Dennis et al. (1970) 1.52 2.35
Fornberg et al. (1980) 1.50 2.24
He et al. (1997) 1.499 2.245
Niu et al. (2006) 1.589 2.26

Wu & Shu (2009) 1.554 2.3
Present
Method 1 1.552 2.37
Method 2 1.526 2.368


Table 3.2 Comparison of consumed CPU time
Method
CPU time ( s )
Stream function-vorticity formulation-based IBM 4669.2
Pressure-velocity formulation-based IBM 5529.18

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Chapter 3 Stream Function-Vorticity Formulation-based IBM
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

Table 3.3 Comparison of drag coefficient
D
C
, lift coefficient
L
C
, and
Strouhal number
St

Cases References
D
C

L
C

St



Re=100
Braza et al. (1986) 1.325
±
0.008
±
0.28 0.164
Liu et al. (1998) 1.350
±
0.012
±
0.339 0.164
Ding et al. (2004) 1.364
±
0.015
±
0.25 0.160
Present
Method 1 1.383
±
0.012
±
0.373 0.164

Method 2 1.335
±
0.011
±
0.356 0.164

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Chapter 3 Stream Function-Vorticity Formulation-based IBM
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


(a) Streamlines obtained by Method 1

(b) Streamlines obtained by Method 2

(c) Vorticity contours obtained by Method 1

(d) Vorticity contours obtained by Method 2
Fig. 3.1 Streamlines and vorticity patterns in the vicinity of circular cylinder
at
40Re
=

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Chapter 3 Stream Function-Vorticity Formulation-based IBM
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

)(a
Streamlines

)(b
Vorticity contours
Fig. 3.2 Streamlines and vorticity patterns in the vicinity of circular cylinder at
100Re
=





)(a
Method 1
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Chapter 3 Stream Function-Vorticity Formulation-based IBM
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


)(b
Method 2
Fig. 3.3 Time evolution of drag and lift coefficients at
100Re =



Fig. 3.4 Adjusted streamlines for flow over a left moving circular cylinder
(

40Re
=
)


Fig. 3.5 Vorticity patterns for flow over a left moving/stationary circular
cylinder at
40Re =
(solid line represents the result for left moving case, and
dashed line represents the result for stationary case)
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Chapter 3 Stream Function-Vorticity Formulation-based IBM
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







Fig. 3.6 Vorticity distribution on the surface of cylinder at
40Re =
for
stationary and left moving cylinder cases

Fig. 3.7 Evolution of drag coefficient at
40Re
=
for stationary and left
moving cylinder cases