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INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDSInt. J. Numer. Meth. Fluids (2007)Published online in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/fld.1706

1

Assessment of regularized delta functions and feedback forcingschemes for an immersed boundary method3

Soo Jai Shin, Wei-Xi Huang and Hyung Jin Sung∗,†

Department of Mechanical Engineering, Korea Advanced Institute of Science and Technology 373-1,5Guseong-dong Yuseong-gu, Daejeon 305-701, Korea

SUMMARY7

We present an improved immersed boundary method for simulating incompressible viscous flow aroundan arbitrarily moving body on a fixed computational grid. To achieve a large Courant–Friedrichs–Lewy9number and to transfer quantities between Eulerian and Lagrangian domains effectively, we combined thefeedback forcing scheme of the virtual boundary method with Peskin’s regularized delta function approach.11Stability analysis of the proposed method was carried out for various types of regularized delta functions.The stability regime of the 4-point regularized delta function was much wider than that of the 2-point13delta function. An optimum regime of the feedback forcing is suggested on the basis of the analysis ofstability limits and feedback forcing gains. The proposed method was implemented in a finite-difference15and fractional-step context. The proposed method was tested on several flow problems, including the flowpast a stationary cylinder, inline oscillation of a cylinder in a quiescent fluid, and transverse oscillation17of a circular cylinder in a free-stream. The findings were in excellent agreement with previous numericaland experimental results. Copyright q 2007 John Wiley & Sons, Ltd.19

Received 1 August 2007; Revised 30 October 2007; Accepted 6 November 2007

KEY WORDS: immersed boundary method; regularized delta function; feedback forcing; stabilityanalysis; finite-difference method21

1. INTRODUCTION

The immersed boundary (IB) method has received much attention in the field of numerical simu-23lations because it can easily handle viscous flows over or inside complex geometries in a Cartesiangrid system. In the IB method, no-slip boundary conditions are imposed at the IBs by introducing25a momentum forcing into the Navier–Stokes (N–S) equations. Hence, N–S solvers based on a

∗Correspondence to: Hyung Jin Sung, Department of Mechanical Engineering, Korea Advanced Institute of Scienceand Technology 373-1, Guseong-dong Yuseong-gu, Daejeon 305-701, Korea.

†E-mail: [email protected]

Contract/grant sponsor: Creative Research Initiatives of the Korea Science & Engineering FoundationContract/grant sponsor: Brain Korea 21

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S. J. SHIN, W.-X. HUANG AND H. J. SUNG

Cartesian grid system can be easily extended to complex flow geometries without the need for a1boundary-conforming grid. Depending on how the momentum forcing is applied, the IB methodis classified as either a discrete or continuous forcing approach [1].3

In the discrete forcing approach [2–6], the desired velocities at the IB are obtained by inter-polating neighboring points, and the momentum forcing is acquired directly from the discretized5equation of motion. Kim et al. [3] introduced a mass source/sink to satisfy the local continuity nearIB points. Huang and Sung [4] improved the accuracy near the IB by using more accurate mass7source/sink. Kim and Choi [5] developed an IB method adopting a non-inertial reference framethat is fixed to the body with an arbitrary motion. However, to simulate flow around a moving9body using a fixed Eulerian grid system, more complicated interpolation schemes are needed [6].

In contrast to the discrete forcing approach, the continuous forcing approach can be applied in11a straightforward manner when simulating flows with moving boundaries. The continuous forcingapproach, which was pioneered by Peskin [7] to simulate blood flow in the human heart, has13proved to be an efficient method for simulating fluid–structure interactions. In Peskin’s classicalIB method [7–11], a singular force distribution at arbitrary Lagrangian positions is determined15and applied to the flow equations in the fixed reference frame via a regularized delta function. Forimmersed elastic boundaries, the singular force is just the elastic force that can be derived by the17principle of virtual work or a constitutive law such as Hooke’s law. For rigid boundaries, however,the constitutive laws for elastic boundaries are not generally well posed. Goldstein et al. [12]19developed a virtual boundary method that employs a feedback forcing to enforce the no-slipcondition at the surface of a structure embedded in a fluid domain. Saiki and Biringen [13] modified21the virtual boundary formulation and proposed the so-called ‘area-weighted’ virtual boundarymethod. However, both Goldstein et al. and Saiki and Biringen reported that this method suffers23from a very severe time-step restriction since the amplitude of the feedback forcing needs tobe large for proper operation, resulting in a very stiff system. Lee [14] relieved the time-step25restriction by exploiting the stability characteristics of the virtual boundary method. Instead ofthe regularized delta function, in the virtual boundary method the velocities at the surface of a27structure are obtained using a bilinear interpolation, and the momentum forcing is extrapolatedback to the surrounding grid points by area-weighted averages. Under this approach, the total29amount of forcing is not conserved, in contrast to the conservation of the total forcing achievedusing the regularized delta function, as will be analyzed in detail later in this paper. Lai and31Peskin [9] considered the boundary to be elastic but extremely stiff for simulation of flow witha rigid body. In their method, IB points move according to the local flow velocity, while in the33virtual boundary method the boundary points do not move from the body surface. Accordingly, inPeskin’s IB method both the position and the velocity of the IB by feedback forcing have to be35compensated. On the contrary, in the virtual boundary method only the velocity of the boundaryhas to be corrected because the boundary remains on the body surface. As a result, the limitation37of the computational time step of the virtual boundary method is different from that of Peskin’sIB method due to differences of the forcing scheme. Uhlmann [15] retained the use of regularized39delta functions in his direct forcing method, which can be regarded as a special case of the feedbackforcing scheme. Huang et al. [16] proposed an IB formulation for simulating flexible inextensible41filaments in a uniform flow. In their method, the Eulerian fluid motion and the Lagrangian filamentmotion were solved independently and their interaction force was explicitly calculated using a43feedback law.

In this study, we sought to find an efficient IB formulation by accounting for the regularized45delta functions and feedback forcing schemes. To this end, we made a detailed comparison of

Copyright q 2007 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Fluids (2007)DOI: 10.1002/fld

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ASSESSMENT OF REGULARIZED DELTA FUNCTIONS 3

Peskin’s IB method and the virtual boundary method. Stability analysis of the feedback forcing1gains was performed for various types of regularized delta functions to relax the computationaltime-step restriction and to decrease the l2-norm error and avoid non-growing oscillations. The3analytical solution was compared with numerical results obtained using the proposed IB method.From the stability analysis, an optimum region of the feedback forcing gains was obtained. The5present method was validated by simulating three flow problems, namely flow past a stationarycylinder, inline oscillation of a cylinder in a quiescent fluid, and transverse oscillation of a circular7cylinder in a free-stream.

2. NUMERICAL APPROACH9

The computational configuration considered in the present work is a circular solid object embeddedin a fluid, as shown in the schematic diagram in Figure 1. The fluid motion is defined on an11Eulerian–Cartesian grid, and the fluid–solid interface is discretized using a Lagrangian grid fixedon the body. The fluid–solid interface �S is evenly distributed by NL Lagrangian points, which13are denoted by

Xl ∈�S, 1�l�NL (1)15

Each Lagrangian point has a discrete volume �Vl with thickness equal to the mesh width h. NL isselected such that a discrete volume is comparable with a finite volume of the Eulerian grid, i.e.17�Vl ≈hm , where m is the number of space dimensions. Although we illustrate only a circular solidobject in Figure 1, the present method can be applied to objects of arbitrary shape.19

The governing equations are the N–S equations and the continuity equation, which arediscretized as21

un+1−un

�t+Nun+1=−Gpn+1/2+ 1

2Re(Lun+1+Lun)+fn (2)

Dun+1=0 (3)23

where u is the velocity vector, p is the pressure, Re is the Reynolds number, and f is the momentumforcing applied to enforce the no-slip boundary condition along the IB. In Equations (2) and (3), N25is the linearized discrete convective operator, G is the discrete gradient operator, L is the discrete

27

lS∂X l

lV∆

S

Figure 1. Schematic diagram of a circular solid object S in a fluid domain.


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Laplacian operator, and D is the discrete divergence operator. Here, n denotes the nth time step and1�t denotes the time increment. The discrete spatial operators N,G, L , and D are evaluated usingthe second-order central finite-difference scheme [17]. The present method is based on the N–S3solver adopting the fractional-step method and a staggered Cartesian grid system. A fully implicittime advancement is employed, in which both convection and diffusion terms are advanced using5the second-order Crank–Nicholson scheme.

The present numerical algorithm, including the fluid–solid interaction, is formulated as follows:7

Fn =�∫

(Un(Xl)−Ud(Xl))dt+�(Un(Xl)−Ud(Xl)) (4)

fn =∫

Fn�h(x−Xl)ds (5)9

1

�tu∗+Nu∗− 1

2ReLu∗ = 1

�tun−Gpn−1/2+ 1

2ReLun+fn (6)

�t DG�p=Du∗ (7)11

un+1=u∗−�tG�p (8)

pn+1/2= pn−1/2+�p (9)13

Un+1=∫

un+1�h(x−Xn+1l )dx (10)

where Ud(Xl) is the desired velocity of the Lagrangian point and u∗ is the intermediate fluid15velocity. In the present method, the Lagrangian point is required to move in concert with the rigid-body motion of a solid object. This is achieved by means of the feedback forcing in Equation (4),17which is calculated so as to make the velocity at the Lagrangian point equal to the desired velocity.Note that the feedback forcing points are located in a staggered fashion like the velocity components19defined on a staggered grid. Next, the feedback forcing in the Lagrangian domain is transferred tothe Eulerian domain by using the regularized delta function (Equation (5)). In Equations (6)–(9),21we solve un+1 and pn+1/2 using the fractional-step method with the momentum forcing calculatedexplicitly. Velocity and pressure are decoupled by approximate factorization which is based on23block LU decomposition. To calculate the intermediate velocity u∗ in Equation (6), the approximatefactorization is further extended to the velocity components. It results in a significant reduction in25computation time and memory by avoiding the inversion of a large sparse matrix. The pressurePoisson equation (Equation (7)) is then solved by a direct method using fast Fourier transform or a27multigrid method. The details of the present N–S solver can be found elsewhere [17]. The Eulerianvelocity at the new time step is interpolated to the Lagrangian points by using the regularized delta29functions as kernels in the transfer step (Equation (10)). The Lagrangian velocity obtained is thenused to calculate the feedback forcing in the next time step. This procedure is repeated.


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The regularized delta function is employed to transfer quantities between Lagrangian and Eule-1rian locations in Equations (5) and (10), respectively,

U(Xl)=∑u(x)�h(x−Xl)h

3, 1�l�NL (11)3

f(x)=NL∑l=1

F(Xl)�h(x−Xl)�Vl (12)

In the domain where quantities are transferred between Lagrangian and Eulerian locations, a5uniform grid with mesh width h is used. In this study, three types of regularized delta functionsare chosen:7

�h(x)= 1

h3�

( x1h

)�

( x2h

)�

( x3h

)(13)

�(r)={1−|r |, |r |�1

0 otherwise(14)

9

�(r)=

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

13(1+

√−3r2+1), |r |�0.5

16(5−3|r |−

√−3(1−|r |)2+1), 0.5�|r |�1.5

0 otherwise

(15)

�(r)=

⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

18 (3−2|r |+

√1+4|r |−4r2), 0�|r |�1

18 (5−2|r |−

√−7+12|r |−4r2), 1�|r |�2

0 otherwise

(16)

11

�(r)=

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

61112 − 11

42 |r |− 1156 |r |2+ 1

12 |r |3+√3

336 (243+1584|r |−748|r |2

−1560|r |3+500|r |4+336|r |5−112|r |6)1/2, 0�|r |�1

2116 + 7

12 |r |− 78 |r |2+ 1

6 |r |3− 32�(|r |−1), 1�|r |�2

98 − 23

12 |r |+ 34 |r |2− 1

12 |r |3+ 12�(|r |−2), 2�|r |�3

0 otherwise

(17)

Equation (14) shows the 2-point regularized delta function (linear interpolation in each direction),13which is similar to the ‘area-weighted’ virtual boundary method of Saiki and Biringen [13].Equation (15) shows the 3-point regularized delta function introduced by Roma et al. [8], and15Equations (16) and (17) are the 4- and 6-point regularized delta functions introduced by Peskin [10]Copyright q 2007 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Fluids (2007)

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r

φ(r

)

-3 -2 -1 0 1 2 3

0

0.2

0.4

0.6

0.8

1

4-point

2-point3-point6-point

Figure 2. Four types of regularized delta functions.

and Griffith et al. [11], respectively. Note that all four types of delta functions have the property1 ∑�h(x−X)h3=1 (18)

which is the discrete analogue of the basic property of the Dirac delta function. In Figure 2, the3value of �(r) is maximum at r =0 for all functions, and the value of �(0) decreases as the pointsof the delta function increase, except for the 6-point delta function. These properties are kernels5of the stability analysis in the following section.

3. STABILITY ANALYSIS7

First, we consider the simple case in which the Lagrangian domain is a point located at x1+r(x2−x1) between two Eulerian points x1 and x2 in 1-D space, as shown in Figure 3(a). The velocity9at the Lagrangian point U(t) is obtained using the 3-point regularized delta function according toEquation (11):11

U(t)=�(−r)u1(t)+�(1−r)u2(t)+�(2−r)u3(t) (19)

where u1(t),u2(t), and u3(t) are velocities at Eulerian points x1, x2, and x3, respectively. The13value of r varies between 0.5 and 1.5, depending on the location of the Lagrangian point. Themomentum forcing is calculated by the feedback law (Equation (4)) and spreads back to the three15nearby Eulerian points using Equation (12) as follows:

f1(t)=�(−r)F(t), f2(t)=�(1−r)F(t), f3(t)=�(2−r)F(t) (20)17

Since only one Lagrangian point is considered, no summation is necessary. Stability analysis forthis method should be carried out using the largest Eulerian forcing as this represents the worst19case. When the Lagrangian point coincides with the Eulerian point x2 at r =1, the Eulerian forcingreaches its maximum, i.e.21

f2(t)��(0)F(t)= 23F(t) (21)

where �(0)= 23 for the 3-point regularized delta function.23


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F3f2f

2x 3x1x x

F

U U uu

2x r x+ ∆

F

uU

1f

3u1u U 2u

1f 3f2f

2x 3x1x x

(,

)u

xt

1x r x+ ∆

F

1x r x+ ∆(a) (b)

Figure 3. Virtual surface velocity and forcing in the 1-D case: (a) point, Lagrangian domain and (b) line,Lagrangian domain; •, Lagrangian grid point; ◦, Eulerian grid point.

1x

1y

2x 3x

2y

3y

F

F

F

f21f 3f

Figure 4. Virtual surface velocity and forcing in the 2-D case: line, Lagrangian domain; •, Lagrangiangrid point; ◦, Eulerian grid point.

Next, we consider the case in which the Lagrangian domain is a line immersed in a 1-D Eulerian1domain (Figure 3(b)). For simplicity, we assume that the adjacent Eulerian velocities are uniform.Hence, we haveU(t)=u(t) according to Equation (18), regardless of the position of the Lagrangian3point. The momentum forcing at the Eulerian point x2 is then obtained:

f2(t)=�(2−r)F(t)+�(1−r)F(t)+�(r)F(t)= F(t) (22)5

In Figure 4, this analysis is extended to the 2-D case (the thick line indicates the Lagrangiandomain). In this case, we again assume that the adjacent Eulerian velocities are uniform. The worstcase from the viewpoint of stability occurs when all Lagrangian points coincide with the Eulerianpoints. In that case, the maximum forcing at the Eulerian point f2 is acquired:

f2(t) = �(−1)�(0)F(t)+�(0)�(0)F(t)+�(1)�(0)F(t)

= �(0){�(−1)F(t)+�(0)F(t)+�(1)F(t)}= �(0)F(t) (23)

Similarly, we can obtain the maximum momentum forcing for other cases; the results are listed inTable I. In its general form, the maximum momentum forcing can be expressed as7

fmax(t)=CmaxF(t)=Cmax

(�∫ t

0u(t ′)dt ′+�u(t)

)(24)


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Table I. Maximum Eulerian forcing for 1-D, 2-D, and 3-D cases.

1-D 2-D 3-D

Point �(0)F(t) (�(0))2F(t) (�(0))3F(t)Line F(t) �(0)F(t) (�(0))2F(t)Area — F(t) �(0)F(t)

where Cmax is the coefficient of the maximum Eulerian forcing for each case. Here, we consider1a stationary problem for simplicity, i.e. Ud(t)=0 in Equation (4). Hence, the N–S equations canbe expressed as3

un+1−un

�t+Nun+1=−Gpn+1/2+ 1

2Re(Lun+1+Lun)+Cmax

(�

n∑i=0

ui�t+�un)

(25)

where∑n

i=0ui�t is an approximation to

∫ t0 u(t ′)dt ′. In the present method, since �(≈1/�t2) and5

�(≈1/�t) are much larger than the other terms, we can simplify Equation (25) as

un+1−un ≈Cmax�t

(�

n∑i=0

ui�t+�un)

(26)7

To obtain the recurrence formula for the stability analysis, the equation at the previous time stepis subtracted from the equation at the present time step, resulting in9

un+1−2un+un−1=�′un+�′(un−un−1) (27)

where �′ =Cmax��t2 and �′ =Cmax��t . Substitution of un =u0rn (r ≡ui+1/ui , i =0,1,2, . . .,n)11into Equation (27) leads to the equation for r

r2−(2+�′+�′)r+1+�′ =0 (28)13

Thus, the stability region abs(r)�1 is

−�′−2�′�4⇒−Cmax��t2−2Cmax��t�4 (29)15

From the above equation, we can see that the stability region is mainly affected by the coefficientCmax, which depends on the type of Lagrangian domain and Eulerian domain, as shown in Table I.17For example, for the case in which the Lagrangian domain is a line in a 2-D flow, Cmax=�(0),which is smaller than Cmax=1 for the case where the Lagrangian domain is a plane in a 2-D flow. In19other words, without applying the Lagrangian forcing to the interior of the solid object (Figure 1),the stability region relaxes for a given feedback forcing gain (�,�), as shown in Equation (29).21The stability region can also be affected by the time-advancing scheme. Although only the forwardEuler scheme is considered here, the stability regimes for various time-advancing schemes can be23obtained in a similar way [14].Copyright q 2007 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Fluids (2007)

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X

X X

X

X

X X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

-α∆t2

-β∆t

642 8 10 120

0

1

2

3

4

5

6

7

4-point

2-point

6-point

X

stable

unstable:

:

3-point

Figure 5. Stability regimes in 2-D flow for several types of delta functions. The dot/cross denotesstable/unstable cases in the simulation of a cylinder moving in a 2-D flow.

Stability regimes of different types of delta functions are displayed in Figure 5 for the case1in which the Lagrangian domain is a line immersed in a 2-D flow. The flow is stable in theregion below the line and unstable above the line. Stability regimes are wider for the smaller3value of �(0) in the regularized delta functions. Compared with the 2-point regularized deltafunction, the stability region of the 4-point regularized delta function is twice as wide in each5direction (−��t2,−��t). To validate the numerical analysis, simulations of a moving cylinder ina 2-D flow using the 4-point regularized delta function were carried out for different −��t2’s and7−��t’s. Stable and unstable cases are denoted by circles and crosses, respectively, in Figure 5. Theanalytical solution is in good agreement with the numerical results obtained using the present IB9method. It is also confirmed that the assumption that the adjacent Eulerian velocities are uniformis reasonable for the above stability analysis. In fact, the difference between adjacent velocities is11negligible due to the small mesh width. Under this assumption, the stability analysis process issignificantly simplified compared with Lee [14]. Since the worst case is always considered in the13theoretical analysis, the actual stability regions of the numerical results will be slightly wider thanthose of the analytical predictions, as shown in Figure 5.15

4. COMPARISON BETWEEN PESKIN’S IB METHOD AND THE VIRTUALBOUNDARY METHOD17

4.1. Feedback forcing scheme

In Peskin’s IB method for solving the flow past a circular cylinder, the boundary is considered19to be elastic but extremely stiff, as shown in Figure 6(a). A feedback forcing is formulated asfollows:21

F(s, t)=�(Xbs(s, t)−Xib(s, t)) (30)


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vb bs 0− →U Uib bs 0− →X X

ibX

bsX

U∞ U∞

bs vb,U U

F

(a) (b)

Figure 6. Schematic diagram for comparison of the feedback forcing scheme: (a) Peskin’s IB method and(b) the virtual boundary method.

where Xib(s, t) and Xbs(s, t) denote the IB and its equilibrium location, and the stiffness coefficient1�1. The IB Xib(s, t) of the next time step moves with the local fluid velocity.

On the other hand,in the virtual boundary method the body surface Xbs(s, t) is considered as a3virtual boundary where the feedback forcing is applied on the fluid so that the fluid will be at reston the surface (see Figure 6(b)). Thus, the forcing term is expressed as5

F(s, t)=�∫ t

0(Uvb(s, t

′)−Ubs(s, t′))dt ′+�(Uvb(s, t)−Ubs(s, t)) (31)

where Uvb(s, t) denotes the velocity at the virtual boundary interpolated from the fluid velocity7field, and Ubs(s, t) denotes the velocity at the surface of the moving body. To apply the feed-back forcing defined in Equation (31), � and � should be large negative constants to enforce9the no-slip condition. The virtual boundary location Xvb(s, t) is considered as the body surfaceXbs(s, t).11

Comparing Figure 6(a) and (b), we can see that Peskin’s IB method uses a boundary whoseposition moves according to the local flow velocity, whereas the virtual boundary method uses13a boundary that always coincides with the body surface. Accordingly, in Peskin’s IB method,position compensation must be implemented to correct the position and the velocity of each point15on the boundary. In the virtual boundary method, however, velocity compensation is implementedto correct only the velocity of each point on the boundary, because the boundary remains on the17body surface. Such differences between the two methods require that |�| in Peskin’s IB method bemuch larger than |�| in the virtual boundary method to ensure accurate body surface conditions.19This suggests that the time-step restriction will be stricter for Peskin’s IB method than for thevirtual boundary method.21

4.2. Transfer of quantities between Lagrangian and Eulerian locations

In both methods, a transfer process between Lagrangian and Eulerian locations is necessary since23the virtual boundary points do not always coincide with the computational meshes in the discretegrid system. The transfer process of Peskin’s IB method is explained in Equations (11) and (12),25while the transfer process is a little different in the virtual boundary method. First, the velocity atthe virtual boundary point (Xvb) is obtained through a linear interpolation using the velocity at


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FLD 1706


the nearby mesh points (xi ):1

U(Xlvb)=

∑u(xi)�vb(xi −Xl

vb), 1�l�NL (32)

�vb(x)=�( x1h

)�

( x2h

)�

( x3h

), �(r)=

{1−|r|, |r|�1

0 otherwise(33)

3

where �(r) is the same as that of the 2-point regularized delta function, and NL is the numberof points distributed densely on the virtual boundary. After the feedback forcing is obtained, it is5extrapolated back to the nearby mesh points:

f(xi )= 1

NV

NV∑i=1

F(Xlvb)�vb(X

lvb−xi ), 1�l�NL (34)

7

where NV is the number of virtual boundary points Xlvb that influence the mesh point xi .

The velocity approximation is almost the same in the two methods, but the forcing approximationis not. There are two significant differences in the forcing approximation. One difference is thenumber of Lagrangian points. The virtual boundary method uses many points, leading to highcomputational overheads, and the rule for determining the number of points is ambiguous. Bycomparison, Peskin’s IB method uses fewer points; in this method, it is recommended to use thenumber of Lagrangian points that makes the volume of each forcing point equivalent to a finitevolume of the Eulerian grid (�Vl ≈hm , where m is the number of space dimensions) [15]. Thesecond major difference is the conservation of quantities between the Lagrangian and Euleriandomains. Peskin’s IB method uses the class of regularized delta function as kernel in the transfersteps between Lagrangian and Eulerian locations. Accordingly, the total amount of quantities is notchanged by the transfer step [10, 15]. The virtual boundary method, by contrast, uses an averageforcing and hence changes the total amount of quantities between the transfer steps. Figure 7displays 2-D cases for the two methods. We assume that all values of the forcing in the Lagrangiandomain are F . Figure 7(a) illustrates Peskin’s IB method using a 2-point regularized delta function.The Eulerian forcing f can be obtained using

f = �(1−rx)�(1−ry)F+�(1−rx)�(ry)F+�(rx)�(1−ry)F+�(rx)�(ry)F

= �(1−rx){�(1−ry)F+�(ry)F}+�(rx){�(1−ry)F+�(ry)F}= �(1−rx)F+�(1−rx )F, (�(1−r)+�(r)=1)

= F (35)

Since Peskin’s IB method conserves the forcing between the Lagrangian and Eulerian domains,9the drag and lift forces of a stationary cylinder can be obtained by integrating the Lagrangianforcings on the boundary �S [9]:11

FD=−∫�S

fx dx=−∫�S

Fx ds, FL =−∫�S

fy dx=−∫�S

Fy ds (36)


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FLD 1706


,yj

ry

∆

f

F

,x ir x∆

yry

∆f

F F

F F

xr x∆(a) (b)

Figure 7. Schematic diagram for comparison of the forcing transfer process in the 2-Dcase. Lagrangian domain: area (a) Peskin’s IB method (b) virtual boundary method; •,

Lagrangian point; ◦, Eulerian point.

where FD and FL are the drag and lift forces, respectively. Figure 7(b) shows that the Eulerianforcing f is not equal to F by the virtual boundary method, i.e.

f = limNx ,Ny→∞

1

Nx

1

Ny

Nx∑i=1

Ny∑j=1

F(1−|rx,i |)(1−|ry, j |)

= limNx→∞

F

2

Nx∑i=1

(1−|rx,i |)�rx limNy→∞

1

2

Ny∑j=1

(1−|ry, j |)�ry

= F

2

∫ 1

−1(1−|rx |)drx 12

∫ 1

−1(1−|ry |)dry = F

4(37)

where �rx =2/Nx ,�ry =2/Ny , and Nx and Ny are the numbers of Lagrangian points in the x-1and y-directions, respectively. The virtual boundary method does not conserve the forcing in thetransfer step.3

5. RESULTS AND DISCUSSION

5.1. Stationary cylinder5

The flow past a stationary cylinder in a free-stream was simulated at two different Reynoldsnumbers (100 and 185) based on the free-stream velocity u∞ and the cylinder diameter. The first7case was simulated to compare the present method with other methods such as Peskin’s IB method.The second case was examined to investigate the effects of the feedback forcing gains (� and �)9and types of delta functions.

5.1.1. Stationary cylinder in a free-stream at Re=100. We used a computational domain of110�x, y�8 and a cylinder with diameter d=0.30 whose center is located at (1.85,4.0). A Dirichletboundary condition (u/u∞ =1,v=0) was used at the inflow and far-field boundaries, and a13convective boundary condition (�ui/�t+c�ui/�x=0, where c is the space-averaged streamwise


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FLD 1706


Table II. Comparison of drag coefficient, lift coefficient, and Strouhal number withthose obtained in previous studies.

h � or −� �t CD C ′L St CFL

Case 1 164 4.8×104 1.2×10−2 1.44 0.35 0.168 1.35

Case 2 164 4.8×104 6.0×10−3 1.44 0.35 0.168 0.7

Case 3 164 4.8×104 6.0×10−3 1.37 0.34 0.163 0.7

Lai and Peskin [9] 164 4.8×104 1.8×10−3 1.52 0.29 0.155 —

Lai and Peskin [9] 1128 9.6×104 9.0×10−4 1.45 0.33 0.165 —

Uhlmann [15] 1128 — 3.0×10−3 1.50 0.35 0.172 —

Kim et al. [3] — — — 1.33 0.32 0.165 —Linnick and Fasel [18] — — — 1.34 0.33 0.166 —Liu et al. [19] — — — 1.35 0.34 0.165 —Huang and Sung [4] — — — 1.36 0.33 0.167 —

velocity) was used at the outflow boundary. Table II shows the drag and lift coefficients, CD1and CL, obtained using the proposed method, as well as the Strouhal number defined from theoscillation frequency of the lift force. The drag and lift forces were obtained by integrating all3the momentum forcing applied on the boundary, as shown in Equation (36). Parameters such asthe mesh width h, time step �t , and feedback forcing gain � were selected to match the conditions5of Lai and Peskin [9] and a 4-point regularized delta function was employed [9]. To compare thepresent method with Peskin’s IB method, we used feedback forcing gains of �=−4.8×104 and7�=0. Table II indicates that the value of �=−4.8×104 used in the present method is large enoughto obtain reliable results. By contrast, the results of Lai and Peskin [9] using �=4.8×104 deviate9somewhat from the other results, especially those obtained in the same study using �=9.6×104.These findings are consistent with previous reports showing that compared with the value of −� in11the virtual boundary method, a larger value of the stiffness coefficient � in Peskin’s IB method isrequired to ensure accurate results for a rigid boundary problem [1]. Since we tested the stability13region of feedback forcing gains (�,�) with the 4-point regularized delta function (see Figure 5),we used a computational time step of 1.2×10−2(−��t2=6.912) to be consistent with −��t2<8.15As a consequence, the present results are in good agreement with those of Lai and Peskin [9],even though the computational time step of the present method (�t=1.2×10−2) is about an17order of magnitude larger than that of Lai and Peskin (�t=9.6×10−4). The maximum Courant–Friedrichs–Lewy (CFL) number in the present simulations exceeded 1 due to the adoption of the19feedback forcing scheme and the optimization of parameters by stability analysis. Uhlmann [15]also simulated the same problem with the same domain size. Uhlmann’s results showed some21deviation even though the computational time step and mesh size are smaller than those used inthe present work. When the domain size is enlarged to 0�x, y�16 (case 3 in Table II), the present23results agree better with those of other studies [3, 18–20].

Table III shows the influences of �X/h on drag and lift coefficients and Strouhal number using25the 4-point delta function at −��t2=3.9 and −��t=1.9. The mesh width h was 1

128 , and thecomputational time step was �t=0.006, giving a maximum CFL number of approximately 1.4.27Four cases were simulated using evenly distributed Lagrangian points and the last one was testedusing non-uniformly distributed Lagrangian points. In the non-uniform case, �X/h is increased29


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FLD 1706


Table III. Influence of the ratio of Lagrangian point distance to Eulerian grid width ondrag and lift coefficients and Strouhal number using the 4-point delta function.

�X/h NL CD C ′L St

2 60 1.45 0.36 0.1711 120 1.45 0.36 0.1710.5 241 1.45 0.35 0.1710.25 482 1.45 0.35 0.1710.25−2 142 1.45 0.36 0.171

from 0.25 to 2 by the ratio of approximately 1.015. The values resulting from the variations of1�X/h are very similar. Actually, the differences of CD,C ′

L, and St are within 3%, indicating thatthe number of Lagrangian points has a negligible effect on the results. This also implies that the3present method does not require even distribution of Lagrangian points. In this study, however,the Lagrangian points are selected such that a discrete volume is the same as the finite volume5of the Eulerian grid, i.e. �Vl ≈hm , for simplicity and computational efficiency of the solutionprocedure.7

5.1.2. Stationary cylinder in a free-stream at Re=185. Next we consider the case of a largecomputational domain −50d�x, y�50d, where d denotes the cylinder diameter. The number of9grid points in the streamwise (x) and transverse (y) directions was 352×192, respectively. Thirtygrid points in each direction were uniformly distributed inside the cylinder and the remaining grid11points were stretched outside the cylinder. The cylinder surface was made up of 94 Lagrangianpoints, and the Reynolds number Red=u∞d/� was set at 185. The computational time step was13�t=0.01, leading to a maximum CFL number of approximately 0.6. The boundary conditions atthe inflow, far-field and outflow boundaries were the same as those of the former case at Re=100.15

Four different forcing gains were simulated using a 3-point regularized delta function. For aquantitative comparison, an l2-norm error is defined as17

l2-norm error≡√

1

NL

NL∑k=1

(U(Xk, t))2 (38)

where NL is the number of Lagrangian points and U(Xk, t) is the velocity at the kth Lagrangian19point. l2-Norm errors of the four cases, displayed in Figure 8, show that the error converges to asmaller value for the larger value of −��t2, and the error decays more rapidly for the larger value21of −��t . In a stationary boundary problem, −��t influences only the initial behavior of the error.Two cases, using the same −��t2 value with different −��t values, show the same level of the23error, suggesting that −��t2 is a more critical parameter for enforcing the no-slip condition [14].However, −��t is also important as it influences the total computation time. For −��t=0, a long25computation time is required to converge to a small value. The present findings thus indicate thatboth −��t2 and −��t should be considered when optimizing the accuracy and efficiency of the27computation (Figure 9).

Four different types of the regularized delta functions were simulated with the largest forcing29gains in stability regimes. As expected, the error of the 4-point delta function (−��t2=4,−��t=2)is smaller than those of the 2-point (−��t2=2,−��t=1), 3-point (−��t2=3,−��t=1.5), and31


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FLD 1706


tU ∞ /d

l 2-n

orm

err

or

0 50 100 15010-5

10-4

10-3

10-2

10-1

100

−α∆t2=1,−β∆t=1.5

−α∆t2=0.01,−β∆t=1.5

−α∆t2=0.1,−β∆t=1.5

−α∆t2=0.1,−β∆t=0

Figure 8. l2-Norm error of the virtual surface velocity in the streamwise direction normalized by thefree-stream velocity u∞ for different forcing gains.

tU ∞ /d

l 2-n

orm

err

or

0 50 100 15010-5

10-4

10-3

4-point

2-point 3-, 6-point

Figure 9. l2-Norm error of the virtual surface velocity in the streamwise directionnormalized by the free-stream velocity u∞ for several types of delta functions

with the largest forcing gains in stability regimes.

6-point delta functions (−��t2=3.2,−��t=1.6). Table IV also shows that the result obtained1using the 4-point delta function is in better agreement with previous results [20, 21] than those ofthe 2-, 3-, and 6-point delta functions.3

5.2. Inline oscillation of a circular cylinder

We additionally simulated a periodic inline oscillation of a circular cylinder in a fluid at rest. The5Reynolds number is defined as Re=umd/� based on the maximum velocity um and the cylinderdiameter d. The Keulegan–Carpenter number is defined as KC=um/ f d based on the frequency of7the oscillation. The parameter set of the present simulation was Re=100 and KC=5, according to


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FLD 1706


Table IV. Comparison of drag coefficients, lift coefficients, and Strouhal numbers obtainedusing several types of delta functions with the largest forcing gains in stability regions.

CD CLrms St

2-Point 1.343 0.454 0.1903-Point 1.312 0.436 0.1904-Point 1.296 0.430 0.1906-Point 1.308 0.437 0.190Experimental results [21] 1.28 — 0.190Lu and Dalton [20] 1.31 0.422 0.195Guilmineau and Queutey [21] 1.287 0.443 0.195

x/d

y/d

-4 -3 -2 -1 0 1 2 3 4-3

-2

-1

0

1

2

3

Figure 10. Grid distribution near the cylinder in the simulation of the flowaround a cylinder undergoing inline oscillation.

the experimental and numerical results of Dutsch et al. [22]. We set the cylinder in time-periodic1motion:

xc(t)=−Am sin(2� f t) (39)3

where xc(t) is the position of the cylinder center at time t and Am is the amplitude of the oscillation.The computational domain was −50d�x, y�50d, and the number of grid points in the oscillatory5(x) and transverse (y) directions was 416×282, respectively. Sixty grid points in each directionwere uniformly distributed inside the cylinder and the remaining grid points were stretched outside7the cylinder, as shown Figure 10. Because the cylinder oscillates in the x-direction, the uniformlydistributed region had a rectangular form with longer length in the x-direction. Neumann boundary9conditions were used at all four far-field boundaries. The number of Lagrangian points NL aroundthe cylinder surface was set to 188 and the computational time step was �t=T/720 based on the11period of the oscillation, leading to a maximum CFL number of approximately 0.6. The 4-pointregularized delta function was selected and the feedback forcing gain was chosen as −��t2=3.913


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FLD 1706


and −��t=1.9, which is one of the largest cases in the stability region of the 4-point regularized1delta function.

In a moving boundary problem, the drag force is obtained by summing over the momentum3forcing on the boundary and other terms related to the acceleration of the moving boundary [15],as follows:5

FD=−∫�S

fx dx+ d

dt

∫Su dx (40)

where fx is the x-component of the force density f. The present case is the rigid-body motion on7the volume S:

d

dt

∫Su dx=V

ducdt

(41)9

where V is the volume of the rigid body and uc is the velocity at the cylinder center.In Figure 11, the time history of the drag coefficient in the oscillatory direction is in excellent11

agreement with that of Dutsch et al. [22]. Figure 12 shows the isolines of pressure and vorticityfor different phase angles of the oscillating cylinder motion. During the oscillation of the cylinder,13the flow is characterized by two counter-rotating vortices. These results are very similar to thoseof previous studies [21, 22]. To further compare the present findings with the results of Dutsch15et al. [22], we examined the velocity profiles at four locations (Figure 13). The results agree verywell with the numerical and experimental results of Dutsch et al. [22], indicating that the present17method accurately describes the velocity field.

5.3. Transverse oscillation of a circular cylinder19

Next, we simulated a periodic transverse oscillation of a circular cylinder in a free-stream:

yc(t)= Am cos(2� fet) (42)21

where yc is the position of the cylinder center, and Am and fe are the amplitude and frequencyof the oscillation, respectively. The detailed conditions were Re=185 and Am/d=0.2. fo is the23

t/T

CD

0 2 4 6 8-4

-2

0

2

4 presentDutsch et al.[22]

Figure 11. Time history of drag coefficient at Re=100 and KC=5.


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FLD 1706


pressure

φ=0 °

vorticity

φ=0 °

φ=96 ° φ=96 °

φ=192° φ=192°

φ=288° φ=288°

(a)

(b)

(c)

(d)

Figure 12. Pressure and vorticity isolines (negative values dashed) at Re=100 andKC=5 at different phase positions (�=2� f t).

natural shedding frequency from the stationary cylinder. The computational domain was 0�x, y�81and 1024×1024 grid points were uniformly distributed in the oscillatory (x) and transverse (y)directions, respectively. The cylinder diameter was set to d=0.30, and its center was located at3(1.85,4.0). The cylinder surface was composed by 120 Lagrangian points. A Dirichlet boundarycondition was used at the inflow and far-field boundaries, and a convective boundary condition5was used at the outflow boundary.

The behavior of the l2-norm error of the streamwise virtual surface velocity is shown in Figure 147for three different forcing gains with the 3-point regularized delta function. The error converges


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FLD 1706


y

-1

-0.5

0

0.5

1φ=180 ° φ=180 °

present

0.0

1.2

0.6

-0.6

x /d

y

-1

-0.5

0

0.5

1φ=210 ° φ=210 °

-1 0 1u

y

-1

-0.5

0

0.5

1φ=330 °

-1 0 1v

φ=330 °

(a)

(b)

(c)

Figure 13. Comparison of the velocity components between the present computation and the experimentalresults of Dutsch et al. [22] at four cross-sections for different phase positions (�=2� f t).

t/T

l 2-n

orm

erro

r

0 5 10 15 2010-5

10-4

10-3

10-2

10-1

−α∆t2=0.4,−β∆t=0

−α∆t2=0.4,−β∆t=1

−α∆t2=4,−β∆t=1

Figure 14. l2-Norm error of the virtual surface velocity in the streamwise direction normalized by thefree-stream velocity u∞ for three different forcing gains.


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FLD 1706


t/T

CD

0.1 0.2 0.3 0.4 0.5 0.61.2

1.25

1.3

1.35

1.4

−α∆t =4,−β∆t=0

−α∆t =0.004,−β∆t=1

−α∆t =4,−β∆t=1

(a) (b) t/T0.1 0.2 0.3 0.4 0.5 0.6

1.2

1.25

1.3

1.35

1.4

6-point

2-point

3-point

4-point

Figure 15. Time history of drag coefficient (a) for different forcing gains using the 3-point delta functionand (b) for different types of delta functions with −��t2=0.4 and −��t=1.

to a smaller value for larger −��t2, as observed above for the stationary problem, and the error1decays rapidly for larger −��t . Since the initial decays of the error are important in compensationof the boundary condition in moving boundary problems, the error also converges to a smaller3value for larger −��t . Accordingly, −��t2 and −��t should be as large as possible to decreasethe error.5

Figure 15(a) shows the time histories of the drag coefficient for three different forcing gains,determined using the 3-point delta function. Non-growing oscillations become smaller as −��t7is increased, but are increased for larger −��t2. The influence of −��t is greater than that of−��t2, in that when −��t is large the non-growing oscillations are not significant regardless of9the value of −��t2. The time history of the drag coefficient is illustrated in Figure 15(b) for fourtypes of delta functions with the same forcing gains. As the number of points in the regularized11delta function increases, the non-growing oscillations decrease. This suggests that the 4-pointregularized delta function with large forcing gains yields better results.13

To investigate the effect of delta functions on CPU time, we performed the first 100 time steps ofthe simulation using 100,1000, and 10 000 Lagrangian points for different types of delta functions.15These computations were performed on Intel 3.4GHz Pentium 4 Xeon processor. In Table V, CPUtime increases for more number of points in the regularized delta function, and this tendency is17evidently shown in 10 000 Lagrangian points. In this study, all cases were simulated in 2-D usingthe number of Lagrangian points of approximately 100; hence, the differences of CPU time among19different delta functions are relatively small. In 3-D cases, however, a k-point delta function issupported by k3 grid cells and much more Lagrangian points are used. Thus, the regularized delta21function can be an important factor on computational costs in 3-D cases.

For a further comparison of the present results with previous ones, we performed simulations23using a large computational domain of dimensions −50d�x, y�50d, with a grid size of 416×282.Sixty grid points were uniformly distributed in each direction inside the cylinder and the grids25were stretched outside the cylinder. The cylinder surface was made up of 188 Lagrangian points.Several cases were simulated for different oscillating frequencies in the range 0.8� fe/ fo�1.2.27The computational time step was chosen as �t=T/720 based on the period of the oscillation,leading to a maximum CFL number in the range of 0.7–1.1. The 4-point regularized delta function29was employed and the feedback forcing gains were −��t2=3.9 and −��t=1.9. Time historiesof the drag and lift coefficients for 0.8� fe/ fo�1.2 are illustrated in Figure 16. The drag and lift31


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FLD 1706


Table V. Effect of different types of delta functions and the numbersof Lagrangian points on CPU time (in seconds) required for the first

100 time steps of the simulation.

NL=100 NL=1000 NL=10000

2-Point 187.25 188.35 193.343-Point 187.58 188.65 197.534-Point 187.66 188.85 198.086-Point 188.20 190.55 213.24

CL

,CD

CL

,CD

CL

,CD

-2

-1

0

1

2

(a)

CL

CD

(d)

CL

CD

-2

-1

0

1

2

(b)

CL

CD

(e)

CL

CD

t0 50 100 150 200

-2

-1

0

1

2

(c)

CL

CD

t0 50 100 150 200

(f)

CL

CD

Figure 16. Time history of drag and lift coefficients for Re=185 and Ae/D=0.2 for fe/ fo values of(a) 0.80, (b) 0.90, (c) 1.00, (d) 1.10, (e) 1.12, and (f) 1.20.

coefficients behave regularly once vortex shedding is established. For fe/ fo>1.0, both the drag1and lift coefficients exhibit beat phenomena, with the beat frequency increasing with excitationfrequency. These results are in good agreement with those of Guilmineau and Queutey [21]. The3variations of the mean drag, rms drag, and lift fluctuation coefficients as a function of fe/ foare presented in Figure 17(a), and the phase angles between the lift coefficient and the vertical5position of the cylinder are shown in Figure 17(b). The mean drag coefficient increases with


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C D

C Drms

C Lrms

fe/fo

0.7 0.8 0.9 1 1.1 1.2 1.30

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

(a) fe/fo

0.7 0.8 0.9 1 1.1 1.2 1.3-20

0

20

40

60

80

100

120

140

(b)

presentKim and Choi [5]

φ (deg)

Figure 17. Variations of the force coefficients and phase angle with respect to fe/ fo: (a) meandrag coefficient and rms drag and lift fluctuation coefficients and (b) phase angle between CL

and the vertical position of the cylinder.

increasing fe/ fo up to a peak at fe/ fo=1.0, after which it decreases with further increases of1fe/ fo. The rms lift fluctuation coefficients and the phase angles show a change at fe/ fo=1.1,when vortex switching occurs [21]. The present results agree well with those of Kim and Choi [5].3The previous methods [5, 21] were based on a non-inertial reference frame that is fixed to thebody with an arbitrary motion, while in the present method we use a fixed reference frame and5the body is allowed to move across the grid line. Thus, the present method can be extended in amore straightforward manner to multi-body problems or flexible body problems, e.g. the studies of7Huang et al. [16].

6. CONCLUSIONS9

In this study, we compared the virtual boundary method and Peskin’s IB method, with a focus on twoaspects: the feedback forcing scheme and the transfer process of quantities between Lagrangian and11Eulerian locations. The feedback forcing scheme of the virtual boundary method made it possible touse a larger CFL number than was possible in Peskin’s IB method, thereby ensuring accurate body13surface conditions. However, the average forcing approach of the virtual boundary method was lesseffective than Peskin’s delta function approach for transferring quantities between Lagrangian and15Eulerian domains. We therefore combined the feedback forcing scheme of the virtual boundarymethod with Peskin’s regularized delta function approach to improve performance. The resulting17numerical method was implemented in a finite-difference and fractional-step context. We analyzedthe stability regimes of the feedback forcing gains in the proposed method for several types of19delta functions. The stability region of the 4-point regularized delta function was much wider thanthat of the 2-point delta function. The effects of regularized delta functions and feedback forcing21gains (�,�) were also investigated. For the regularized delta function supported by more points, itsnon-growing oscillations became smaller. On the other hand, the l2-norm error (a measure of the23no-slip condition along the IB) converged to a smaller value for larger −��t2 and decayed faster


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for larger −��t . In the stationary boundary problem, −��t influenced only the initial behavior1of the error, whereas in the moving boundary problem the error also converged to a smaller valuefor larger −��t . On the basis of the stability analysis of the present method, we can recommend3an optimum region of the feedback forcing gains that enables the use of a large CFL number anddecreases the l2-norm error and non-growing oscillations. The proposed method was applied to the5flow past a stationary cylinder, inline oscillation of a cylinder in a quiescent fluid, and transverseoscillation of a circular cylinder in a free-stream at large maximum CFL numbers (0.6–1.4). The7present findings are in excellent agreement with previous numerical and experimental results.

ACKNOWLEDGEMENTS9

This work was supported by the Creative Research Initiatives of the Korea Science & EngineeringFoundation and partially supported by Brain Korea 21.

REFERENCES

1. Mittal R, Iaccarino G. Immersed boundary methods. Annual Review of Fluid Mechanics 2005; 37:239–261.112. Fadlun EA, Verzicco R, Orlandi P, Mohd-Yusof J. Combined immersed-boundary finite-difference methods for

three-dimensional complex flow simulations. Journal of Computational Physics 2000; 161:35–60.133. Kim J, Kim D, Choi H. An immersed boundary finite-volume method for simulations of flow in complex

geometries. Journal of Computational Physics 2001; 171:132–150.154. Huang W-X, Sung HJ. Improvement of mass source/sink for an immersed boundary method. International

Journal for Numerical Methods in Fluids 2007; 53:1659–1671.175. Kim D, Choi H. Immersed boundary method for flow around an arbitrarily moving body. Journal of Computational

Physics 2006; 212:662–680.196. Yang J, Balaras E. An embedded-boundary formulation for large-eddy simulation of turbulent flows interacting

with moving boundaries. Journal of Computational Physics 2006; 215:12–40.217. Peskin CS. Numerical analysis of blood flow in the heart. Journal of Computational Physics 1977; 25:220–252.8. Roma A, Peskin C, Berger M. An adaptive version of the immersed boundary method. Journal of Computational23

Physics 1999; 153:509–534.9. Lai M-C, Peskin CS. An immersed boundary method with formal second-order accuracy and reduced numerical25

viscosity. Journal of Computational Physics 2000; 160:705–719.10. Peskin CS. The immersed boundary method. Acta Numerica 2002; 11:1–39.2711. Griffith BE, Hornung RD, McQueen DM, Peskin CS. An adaptive, formally second order accurate version of

the immersed boundary method. Journal of Computational Physics 2007; 223:10–49.2912. Goldstein D, Handler R, Sirovich L. Modeling a no-slip flow boundary with an external force field. Journal of

Computational Physics 1993; 105:354–366.3113. Saiki EM, Biringen S. Spatial simulation of a cylinder in uniform flow: application of a virtual boundary method.

Journal of Computational Physics 1996; 123:450–465.3314. Lee C. Stability characteristics of the virtual boundary method in three-dimensional applications. Journal of

Computational Physics 2003; 184:559–591.3515. Uhlmann M. An immersed boundary method with direct forcing for the simulation of particulate flows. Journal

of Computational Physics 2005; 209:448–476.3716. Huang W-X, Shin SJ, Sung HJ. Simulation of flexible filaments in a uniform flow by the immersed boundary

method. Journal of Computational Physics 2007; 226:2206–2228.3917. Kim K, Baek S-J, Sung HJ. An implicit velocity decoupling procedure for incompressible Navier–Stokes equations.

International Journal for Numerical Methods in Fluids 2002; 38:125–138.4118. Linnick MN, Fasel HF. A high-order immersed interface method for simulating unsteady incompressible flows

on irregular domains. Journal of Computational Physics 2005; 204:157–192.4319. Liu C, Zheng X, Sung CH. Preconditioned multigrid methods for unsteady incompressible flows. Journal of

Computational Physics 1998; 139:35–57.45


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20. Lu X-Y, Dalton C. Calculation of the timing of vortex formation from an oscillating cylinder. Journal of Fluids1and Structures 1996; 10:527–541.

21. Guilmineau E, Queutey P. A numerical simulation of vortex shedding from an oscillating circular cylinder.3Journal of Fluids and Structures 2002; 16(6):773–794.

22. Dutsch H, Durst F, Becker S, Lienhart H. Low-Reynolds-number flow around an oscillating circular cylinder at5low Keulegan–Carpenter numbers. Journal of Fluid Mechanics 1998; 360:249–271.


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