Behavior of the Two-Dimensional Viscous Flow over Two

Behavior of the Two-Dimensional Viscous Flowover Two Circular Cylinders with Different Radii

Surattana Sungnul and Ekkachai Kunnawuttipreechachan

Abstract—In this paper, we study numerical simulation oftwo-dimensional viscous flow over two circular cylinders withdifferent radii. The flow structure depends on the rate of rota-tion, the gap-spacing and the Reynolds number. The algorithmused to simulate the numerical solutions is based on the conceptof projection method. A mathematical model describing the flowover the two rotating cylinders is applied by the cylindricalbipolar coordinate system. The main objective is to investigatethe characteristics of the fluid flow. This investigation gives a setof numerical simulations for the hydrodynamic characteristics,which can be applied to other related problems.

Index Terms—numerical simulation, cylindrical bipolar co-ordinate, projection method.

I. INTRODUCTION

THE interaction of the flow over two cylinders is atopic of prime scientific interest with many engineering

and real life applications. Most of researches studied ontwo cylinders were concerned with two rotating and non-rotating cylinders of an identical diameter (see for example[1]- [4] and literature cited there). There are two differenttypes of stationary motion of bodies in a fluid. The firsttype is a towed body which in the stationary motion regimeexternal forces must affect the body. The second type isself-propelled body. Self-propelled means that a body movesbecause of the interaction between its boundary and thesurrounding fluid and without the action of an externalforce. In the present work, flow structures were calculatedbetween two circular cylinders (the left cylinder is non-rotating and the right cylinder is rotating in counterclockwiseangular velocity) with different radii and uniform stream flowdirected perpendicular to the line connecting the cylinderscenters. We show some results of numerical simulations atfixed Reynolds number and moderate gap spacing and rateof cylinders rotation.

II. MATHEMATICAL MODELLING

The governing equation is the Navier-Stokes equationswritten in cylindrical bipolar coordinate. The coordinatesystem is moving together with the cylinders. The cylindricalbipolar coordinate system can be defined by the followingequation (1)

x =a sinh η

cosh η − cos ξ, y =

a sin ξ

cosh η − cos ξ, (1)

Manuscript received December 28, 2017; revised January 25, 2018. Thisresearch was financially supported by the Faculty of Applied Science,King Mongkut’s University of Technology North Bangkok (Contract no.5642110).

Surattana Sungnul is a lecturer at the Department of Mathematics, KingMongkut’s University of Technology North Bangkok, 10800, THAILANDand a researcher in Centre of Excellence in Mathematics, CHE, Bangkok,10400, THAILAND e-mail: [email protected]

Ekkachai Kunnawuttipreechachan is a lecturer in the Department ofMathematics, King Mongkut’s University of Technology North Bangkok,10800, THAILAND e-mail: [email protected]

where ξ ∈ [0, 2π), η ∈ (−∞,∞), a is a characteristiclength in the cylindrical bipolar coordinate system whichis positive. This transformation maps the xy− plane (formwhich the domain occupied by the cylinders is excluded)into the rectangle η2 ≤ η ≤ η1, 0 ≤ ξ < 2π and η2 < 0,η1 > 0. The surfaces of the cylinders are located at η = η2and η = η1. The cylinder’s radii r1, r2 and the distance oftheir centers from the origin d1, d2 are given by ri = acsch|ηi|, di = a coth |ηi|, i = 1, 2. The center to centerdistance between the cylinders is d = d1 + d2.The Navier-Stokes equations (2)-(4) in the cylindrical bipolarcoordinate system (ξ, η) [5] are

∂vξ∂t

+1

h

(vξ∂vξ∂ξ

+ vη∂vξ∂η

)−

−1

a

(sinh η(vξvη)− sin ξ(vη)

2)

= − 1

h

1

ρ

∂p

∂ξ+

+ν

h

{1

h

(∂2vξ∂ξ2

+∂2vξ∂η2

)− 2

a

(sinh η

∂vη∂ξ− sin ξ

∂vη∂η

)−(

cosh η + cos ξ

a

)vξ

}, (2)

∂vη∂t

+1

h

(vξ∂vη∂ξ

+ vη∂vη∂η

)+

+1

a

(sinh η(vξ)

2 − sin ξ(vξvη))

= − 1

h

1

ρ

∂p

∂η+

+ν

h

{1

h

(∂2vη∂ξ2

+∂2vη∂η2

)+

2

a

(sinh η

∂vξ∂ξ− sin ξ

∂vξ∂η

)−(

cosh η + cos ξ

a

)vη

}, (3)

1

h2

(∂(hvξ)

∂ξ+∂(hvη)

∂η

)= 0, (4)

where vξ and vη are the physical components of velocityvector v = (vξ, vη), p is the pressure, ρ is density, ν is thekinematic viscosity of the fluid and h =

a

(cosh η − cos ξ).

The boundary conditions are a no-slip requirement on cylin-ders

vξ = ωiri, vη = 0, on η = ηi, ξ ∈ [0, 2π), i = 1, 2, (5)

where ωi, i = 1, 2 are constant angular velocities of thecylinders rotation. Positive values of ωi, i = 1, 2 correspondto counterclockwise rotation. Upstream and downstreamboundary conditions at infinity are

vx = 0, vy = U∞, as r2 = x2 + y2 →∞, (6)

where vx and vy are components of velocity vector in xand y directions respectively and U∞ is the oncoming free

Proceedings of the International MultiConference of Engineers and Computer Scientists 2018 Vol I IMECS 2018, March 14-16, 2018, Hong Kong

ISBN: 978-988-14047-8-7 ISSN: 2078-0958 (Print); ISSN: 2078-0966 (Online)

IMECS 2018

stream velocity. The net force and torque exerted by fluid onan immersed body with surface Σ are

F =

∫Σ

τ dS, M =

∫Σ

[r× τ ]dS,

where n is the unit vector normal to the Σ that points outsidethe region occupied by the fluid. The force per unit areaexerted across a rigid boundary element with normal n in anincompressible fluid is defined by

τ = −p n− µ(n× ω)

where ω is vorticity defined as ω = curl v and µ is thecoefficient of viscosity. If Fxi and Fyi , i = 1, 2 are the liftand drag on the cylinders, the lift and drag coefficients aredefined by

CLi=

Fxi

ρU∞D, CDi

=Fyi

ρU∞D, i = 1, 2, (7)

where D is a diameter of right cylinder and each consistsof components due to the friction forces and the pressure.Hence

CL = CLf + CLp, CD = CDf + CDp. (8)

The problem of self-motion is to find solution of theNavier-Stokes equations (2)-(4) with boundary conditions(5)− (6) and additional constraints

F = M = 0. (9)

Equation (9) determines the basic distinction between sta-tionary flow over self-propelled and towed bodies. Thenumerical simulation of the flow past self-moving bodiesbecomes more complicated as a result of the nonlocality ofconstraints like (9). For such flows, the results depend notonly on the Reynolds number, Re, but also depend on thenon-dimensional gap spacing between the two cylinders, g,and parameters, αi representing the ratios of the rotationalvelocities of the cylinder walls to the oncoming flow velocity

Re =U∞ D

ν, αi =

Dωi2U∞

, i = 1, 2, and g =d− r1 − r2

D/2.

III. NUMERICAL ALGORITHM AND VALIDATION

The algorithm of the problem solution is based on theconcept of projection methods (Chorin, 1968) [6]. The in-termediate velocity components vξ, vη are computed in afirst step by solving a finite difference approximation of themomentum equations. Intermediate velocity vector v (whichis not solenoidal) is then decomposed into divergence freeand rotational free vector fields by solving Poisson equationwith homogeneous Neumann boundary conditions. The finalapproximation of the v and p at time tn+1 can be found andthe steady-state computed solution is defined by

‖θn+1 − θn‖4t‖θn+1‖

≤ ε,

where θ = (vξ, vη, CD, CL); 4t is the time step and θn

refers to the numerical approximation at time n4t.To validate the present numerical algorithm, the uniformflow past rotating circular cylinders with Re = 20, 0.1 ≤α1(= α2) ≤ 2.0 and a large gap between cylinder surfacesg = 14 have been calculated and the results comparedwith simulation data for flow past a single cylinder. All the

TABLE IDRAG COEFFICIENT OF FLOW OVER A ROTATING CIRCULAR CYLINDER

AT Re = 20 WITH GAP SPACING g = 14

ContributionCD

α = 0.1 α = 1.0 α = 2.0

Present 2.119 1.887 1.363Badr et al. [8] 1.990 2.000 —Ingham and Tang [7] 1.995 1.925 1.627Chung [9] 2.043 1.888 1.361

TABLE IILIFT COEFFICIENT OF FLOW OVER A ROTATING CIRCULAR CYLINDER AT

Re = 20 WITH GAP SPACING g = 14

ContributionCL

α = 0.1 α = 1.0 α = 2.0

Present 0.291 2.797 5.866Badr et al. [8] 0.276 2.740 —Ingham and Tang [7] 0.254 2.617 5.719Chung [9] 0.258 2.629 5.507

TABLE IIIDRAG COEFFICIENTS OF FLOW OVER TWO CIRCULAR CYLINDERS AT

FIXED Re = 20, α2 = 0.0 AND g = 1.0, 2.0, 3.0 FOR 1.0 ≤ α1 ≤ 6.0

g α1 CD CD1CD2

1.0

1.0 5.903 2.239 3.6642.0 3.662 0.581 3.0814.0 0.279 -1.603 1.8824.25 0.009 -1.550 1.5595.0 -0.417 -1.489 1.072

2.0

1.0 6.344 2.576 3.7682.0 4.656 1.212 3.4444.0 1.160 -1.111 2.2715.0 0.002 -1.434 1.4366.0 -1.185 -0.936 -0.249

3.0

1.0 6.217 2.678 3.5392.0 4.552 1.354 3.1984.0 1.343 -0.856 2.1995.5 −0.007 -1.157 1.1506.0 0.498 -0.473 0.971

simulations have been performed in a large domain so as toreduce the influence of the outer boundary. Tables I and IIlist drag and lift coefficients from our calculation and makesa comparison with (Ingham et al. 1990) [7], (Badr et al.1989) [8] and (Chung 2006) [9]. It can be seen that thedifferences are acceptable for CD and CL. The analysis ofthe data collected in Tables I and II indicate an acceptablelevel of agreement between our computational results andthe experimental and numerical data available in literature.

IV. NUMERICAL RESULTS

In this work, two cylinders are placed in a stream of theuniform speed U∞ at infinity. Numerical results have beenpresented into two parts. In the first part, the left cylinderwith radius 2 is non-rotating, whiles the right cylinder withradius 1 is rotating in anti-clockwise angular velocity. Theinfluence of the rotation rate α1 is demonstrated in TablesIII and IV. The values of drag and lift coefficients in case offixed Reynolds number, Re = 20 and various gap spacing,g = 1.0, 2.0, 3.0 for 1.0 ≤ α1 ≤ 6.0 are shown in Tables



IMECS 2018

TABLE IVLIFT COEFFICIENT OF FLOW OVER TWO CIRCULAR CYLINDERS AT FIXED

Re = 20, α2 = 0.0 AND g = 1.0, 2.0, 3.0 FOR 1.0 ≤ α1 ≤ 6.0

g α1 CL CL1CL2

1.0

1.0 2.805 5.104 -2.2992.0 6.204 8.322 -2.1184.0 16.944 17.770 -0.8264.25 18.526 19.100 -0.5745.0 24.130 24.110 0.020

2.0

1.0 4.070 4.610 -0.5402.0 7.035 8.342 -1.3074.0 17.059 17.950 -0.8915.0 24.461 24.720 -0.2596.0 28.728 28.410 0.318

3.0

1.0 4.924 4.343 0.5812.0 7.560 8.076 -0.5164.0 17.432 17.800 -0.3685.5 28.570 28.590 -0.0206.0 33.376 33.290 0.086

III and IV. The drag coefficients of both cylinders decreasewith increasing α1, at fixed g, (see in the third column ofTable III). The lift coefficients of both cylinders increasewith increasing α1, at fixed g, as shown in the third columnof Table IV. We found that in the case of fixed Reynoldsnumber, Re = 20, rate of rotation corresponding to zerodrag force (α1) increases when g increases. For example,Table III shows that at Re = 20, α1 corresponding to zerodrag force are α1 ≈ 4.25, (g = 1.0), α1 ≈ 5.0, (g = 2.0)and α1 ≈ 5.5, (g = 3.0). The streamline patterns fromthe simulations are shown in Fig.1- Fig.3 . The streamlinepatterns and pressure contours in the case rate of rotationcorresponding to zero drag force (α1) are occurred for g =1.0, 2.0, 3.0 with respectively (see in Fig. 2).In the second part, the left cylinder is rotating α2 = 1.0and the right cylinder is rotating in counterclockwise angularvelocity. The effect of the rotation rate α1 is demonstratedin Tables V and VI. The values of drag and lift coefficientsin the case of fixed Reynolds number, Re = 20 and variousgap spacing, g = 1.0, 2.0, 3.0 for 1.0 ≤ α1 ≤ 6.0 are alsoshown. The drag coefficients of both cylinders decrease withincreasing α1, at fixed g, (see in the third column of Table V).The lift coefficients of both cylinders increase with increasingα1, at fixed g, as shown in the third column of Table VI.The simulations of streamline patterns are shown in Fig.4- Fig.6. This work is a fundamental problem which can beapplied to some classes of obstructed flow.

V. CONCLUSION AND SUGGESTION

Numerical results show the flow structure over two rotatingcircular cylinders with different radii depend on the rateof rotation, the gap spacing and the Reynolds number. Wefound that at fixed gap spacing g, the drag coefficient of bothcylinders decrease with increasing α and the lift coefficientof both cylinders increase with increasing α. In additionwe obtain the rate of rotation corresponding to zero dragforce (α1) increases when g increases at fixed Re = 20.However these results (α1) are not the numerical solutionsof self-motion regime because its satisfy only CD ≈ 0but CL is not close to zero. Numerical solutions of flow

−6 −4 −2 0 2 4

−4

−2

0

2

4

6

8

0.126

−0.236

−0.598

0.126

0.488

0.85

0.126

−1.68

−0.15

−0.1

0.30

−0.15

−0.3

−0.05

−0.3

−0.05

0.3

1

α=0.0 α=2.0

−6 −4 −2 0 2 4

−4

−2

0

2

4

6

8

−0.317

0.0214

0.36

1.38

0.0214

−0.656

−1.33−0.656

−1.67

−0.994

−2.35

−1.67

1.38

−1.45

−0.1

−0.03

0.1

−0.03

α=0.0 α=2.0

−6 −4 −2 0 2 4

−4

−2

0

2

4

6

8 0.242

0.555

0.869

−0.0716

−0.0716

−1.01

−0.385

−2.27

−2.58

−1.95

−2.89

−1.95

−2.27

−1.640.242

−1.01

1.81

−3.21

−2.7

−0.025

0

−0.025

α=0.0 α=2.0

Fig. 1. streamline patterns of flow over two circular cylinders at Re =20, g = 1.0, 2.0, 3.0 and α1 = 2.0, α2 = 0.0 .

TABLE VDRAG COEFFICIENTS OF FLOW OVER TWO CIRCULAR CYLINDERS AT


g α1 CD CD1CD2

1.0

1.0 9.886 3.037 6.8492.0 6.958 1.343 5.6154.0 2.902 -0.924 3.8266.0 1.543 -0.324 1.867

2.0

1.0 8.528 2.344 6.1842.0 6.376 0.888 5.4884.0 2.106 -1.270 3.3766.0 0.994 -0.510 1.504

3.0

1.0 8.632 2.623 6.0092.0 7.432 1.482 5.9504.0 4.410 0.066 4.3446.0 1.565 -0.355 1.920

over multiple cylinders can be used the projection methodwith the finite difference method but the governing equations



IMECS 2018

TABLE VILIFT COEFFICIENTS OF FLOW OVER TWO CIRCULAR CYLINDERS AT


g α1 CL CL1CL2

1.0

1.0 5.988 5.974 0.0142.0 9.965 10.360 -0.3954.0 21.765 22.110 -0.3456.0 39.713 39.010 0.703

2.0

1.0 7.218 4.774 2.4442.0 10.396 8.795 1.6014.0 21.040 19.660 1.3806.0 37.257 35.780 1.477

3.0

1.0 10.069 4.677 5.3922.0 13.600 8.959 4.6414.0 24.659 20.570 4.0896.0 39.565 36.430 3.135

−6 −4 −2 0 2 4

−4

−2

0

2

4

6

8

1.09

0.346

−0.0265

0.346

0.718

1.46

−0.0265

−0.771

−3

−0.399

−1.89

0

−0.01

−0.2

−0.1

−0.1

0.1

α = 0.0 α=4.25

-6 -4 -2 0 2 4

-4

-2

0

2

4

6

8

-2.28

-3.04

-6.08

-8.36

-3.04

-1.52

-0.76

-2.04

-0.5

-2.08

-2.12

-1.8

-1.6-1.6

-2.12

-2

α=4.25α=0.0

−6 −4 −2 0 2 4

−4

−2

0

2

4

6

8

1.37

0.306

−0.0479

−0.0479

−0.401

0.306

0.659

−0.0479

−0.401

−1.46

−1.81

−1.81

−0.01

0.1

−0.25

−0.01

α=5.0α=0.0

-6 -4 -2 0 2 4

-4

-2

0

2

4

6

8

1.34

0.458

-11.1

0.458

0.458

1.67

1.69

1.64

1.6

1.88

2

1.88

1.67

1.6

1.5

α=0.0 α=5.0

−6 −4 −2 0 2 4

−4

−2

0

2

4

6

8

0.999

0.638−0.0835

−0.445

−0.0835

0.277

0.638

−0.0835

−0.445

−1.89

−1.53

0−0.021

−0.021

−0.15α=0.0 α=5.5

-6 -4 -2 0 2 4

-4

-2

0

2

4

6

8

0.205

-0.814

-4.89

-1.83

0.205

0.205

0.5

0.6

0.7

0.74

0.76

0.8

0.78

0.74

1

0.85

0.76

0.76

0.4

α=5.5α=0.0

Fig. 2. Streamline patterns and Pressure contours of flow over two circularcylinders at Re = 20, α1 = 4.25 (g = 1.0), α1 = 5.0 (g =2.0), α1 = 5.5 (g = 3.0) and α2 = 0.0

have to transform to another coordinate system. In addition,numerical solutions of fluid flow over two rotating circular

cylinders for self-motion regime with different radii and incases of higher Reynolds numbers may be investigated forfuture works.

−6 −4 −2 0 2 4

−4

−2

0

2

4

6

8

1.43

0.679

−0.0772

−0.455

−0.833

−1.21

−0.0772

0.301

0.679

1.06

1.43

1.81

0.301

−0.0772

−0.455−1.59

0

−0.02

0.1

0.03

α=0.0 α=5.0

−6 −4 −2 0 2 4

−4

−2

0

2

4

6

8

2.06

2.06

0.0863

0.0863

−0.309

−1.1

−2.29

−1.5

0.0863

0.482

−0.309

−0.705

−1.5

−1.1

0

−0.12

0.03

α=0.0 α=6.0

−6 −4 −2 0 2 4

−4

−2

0

2

4

6

8

1.19

0.809

0.0567

−0.32

−1.07

−1.45

0.0567

0.433

0.809

−0.32

−0.696

−1.07 −1.82

0−0.06

−0.15

−0.06

α=0.0 α=6.0

Fig. 3. streamline patterns of flow over two circular cylinders at Re =20, α1 = 5.0 (g = 1.0), α1 = 6.0 (g = 2.0), α1 = 6.0 (g = 3.0)and α2 = 0.0

ACKNOWLEDGMENT

We would like to thanks the Faculty of Applied Science,King Mongkut’s University of Technology North Bangkokfor supporting us in doing this work.

REFERENCES

[1] M. M. Zdravkovich, “Review of Flow Interference between Two Cir-cular Cylinders in Various Arrangements,” ASME J. Fluids Eng., vol.99, pp. 618-633, 1977.

[2] S. Kang, “Characteristics of Flow over Two Circular Cylinders in aSide-by-Side Arrangement at Low Reynolds Numbers,” Phys. of Fluids,vol. 15, no. 9, pp. 486-498, 2003.

[3] S. Sungnul and N. P. Moshkin, “Numerical Simulation of Flow overTwo Rotating Self-Moving Circular Cylinders,” Recent Advances inComputational Sciences, World Scientific Publishing-Imperial CollegePress, pp. 278-296, 2008.



IMECS 2018

-6 -4 -2 0 2 4

-6

-4

-2

0

2

4

6

8

-1.53

-1.53

-1.76-1.07

-1.3

-1.07

-0.837

-1.53

-1.76

-1.3-0.146

-0.837

-0.837

-1.07

-0.607

-0.607

-2.22

-0.837

-2.45

-0.376

-1.47

-0.8

-1.47

-1.47

-0.8

α=1.0 α=2.0

-6 -4 -2 0 2 4

-6

-4

-2

0

2

4

6

8

-2.72

-2.51 -2.3

-2.09

-2.72

-2.93

-2.51

-2.51

-1.88

-2.09

-0.2

-0.831

-1.04

-0.621

-0.2

-1.25

-2.3

-2.09

-1.04

-3.14

-2.6

-2.6

-2.6

-2.6

-2.6

-1.47

α=2.0α=1.0

-6 -4 -2 0 2 4

-6

-4

-2

0

2

4

6

8

-4.5

-4.5

-4.74

-4.97

-0.237

-1.42

-3.79-0.71

-0.947

-0.474

-1.89

-4.03

-3.79

-4.03

-4.26

-0.71

-4.26

-4.6

-4.38

-0.8

-4.38

-0.8 -4.6

-4.38

-4.6

α=1.0 α=2.0


-6 -5 -4 -3 -2 -1 0 1 2 3 4

-6

-4

-2

0

2

4

6

8

-0.799

-1.06

-1.31

-1.06

-0.0277

-0.0277

-0.799

-2.09

-1.57

-2.6

-1.83

-1.31

-0.799

-0.542

-0.799

-0.542

-1.57-1.19

-1.47

-0.8

-1.19

α=1.0 α=4.0

-6 -4 -2 0 2 4

-6

-4

-2

0

2

4

6

8

-2.43

-2.91

-2.19

-1.94

-1.7

-0.243

-1.7

-0.972

-1.46

-0.972

-1.46

-1.21

-0.972

-2.43

-1.7

-3.16

-5.59

-2.3

-0.8

-0.8

α=4.0α=1.0

-6 -4 -2 0 2 4

-6

-4

-2

0

2

4

6

8

-4.2

-3.92

-3.64

-3.36

-3.08

-0.28

-0.839

-1.12

-1.68

-3.36

-2.52

-3.08

-2.8

-0.839

-5.6

-3.78

-3.22

-0.94

-3.22

-3.22

α=4.0α=1.0


[4] S. K. Panda, “Two-Dimensional Flow of Power-Law Fluids over a Pairof Cylinders in Side-by-Side Arrangement in the Laminar Regime,”Braz. J. Chem. Eng., vol. 34, no. 2, pp. 507-530, 2017.

[5] S. Sungnul, “On the Representation of the Navier-Stokes Equations

-6 -4 -2 0 2 4

-6

-4

-2

0

2

4

6

8

-1.02

-1.35

0.618

-1.02

-0.0393

-1.68

-2.01

-3.65

-2.67

-4.31

-3.98

-0.696

-1.02 -1.35

-0.0393

-1.19

-1.19

-1.47

-0.8

-1.47

-0.8

α=1.0 α=6.0

-6 -4 -2 0 2 4

-6

-4

-2

0

2

4

6

8

-1.47

-0.111

-1.13

-1.47

-2.15

0.23

-1.13

-1.81

-3.17

-3.85

-2.83

-0.791

-2.34

-2.34

-0.8

α=6.0α=1.0

-6 -4 -2 0 2 4

-6

-4

-2

0

2

4

6

8

-0.681

-0.341

-2.72

-2.72

-4.77

-1.02

-2.72

-1.02

-1.7

-2.38

-4.43

-4.09

-1.36

-5.11

-3.41

-2.5

-2.5

-1.2

-1.2

-2.5

α=6.0α=1.0


in Cylindrical Bipolar Coordinate System,” in Proceedings in AnnualNational Symposium on Computational Science and Engineering 2005,pp. 340-348.

[6] S. J. Chorin, “Numerical Solution of the Navier-Stokes Equations,”Math. Comp., vol. 22, pp. 745-762, 1968.

[7] D. B. Ingham and T. Tang, “A Numerical Investigation into the SteadyFlow past a Rotating Circular Cylinder at Low and IntermediateReynolds Numbers,” Journal of Computational Physics, vol. 87, pp.91-107, 1990.

[8] D. Badr, S. C. R. Dennis and P. J. S. Young, “Steady and UnsteadyFlow past a Rotating Cylinder at Low Reynolds Numbers,” Computerand Fluids, vol. 17, no. 4, pp. 579-609, 1989.

[9] M-H. Chung, “Cartesian Cut Cell Approach for Simulating Incompress-ible Flows with Rigid Bodies of Arbitrary Shape,” Computer and Fluids,vol. 35, no. 6, pp. 607-623, 2006.



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Behavior of the Two-Dimensional Viscous Flow over Two