An Analysis of Flow around a Propeller

using Fictitious Domain Finite Element Method

Kazutaka Harada and Mutsuto Kawahara

Department of Civil Engineering, Chuo University, Kasuga 1-13-27, Bunkyou-ku, Tokyo 112-8511, Japan ABSTRACT In this paper, an analy is of 3-dimensional finite element method based on the mixed bubble function interpolation using a fictitious domain method is presented. Following the present approach, the computation of the moving boundary problems can be solved successfully. How to use a fictitious domain method based on the Navier-Stokes equation is explained. Then, the incompressible viscous fluid around rotating propeller is solved. It is shown that the flow around the rotating propeller can be solved successfully by the present method.

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KEYWORDS finite element method, fictitious domain method, distributed Lagrange multiplier, Navier-Stokes equation,

1. INTRODUCTION Purpose of this research is to analyze the moving boundary problems in the incompressible viscous fluid. Most of the natural phenomena which are dealt with in engineering field consist of the moving boundary problems. Thus, it is important to analyze the moving boundary problems to solve the natural phenomena.

Experimental study of the complicated moving boundary problems of the incompressible viscous fluid is one of the problems. The finite element method is useful tool for the analysis. To compute the moving boundary problems by the conventional finite element method, there are some disadvantages. One is an increase of the computational time and the other is a shortage of computational memory because the moving boundary problem is more and more complicated and large-scaled owing to the developments of the finite element method and of the computer. The most important feature is to avoid the remeshing in the conventional finite element method. Therefore, the moving boundary problems using the finite element method with the fictitious domain method [1]~[3] is employed in this research. To solve the finite element equation, the mixed interpolation for velocity and pressure based on the bubble function formulation is employed.

The basic idea of this method is simple. It is assumed that the inside of an objective domain is filled with fluid. An objective action is reflected in fluid. Moreover, objective boundary condition can be expressed as the constraint condition using the distributed Lagrange multiplier. Therefore, remeshing is not necessary because both objective and computational domains are introduced. The computation is much better because moving boundary problems can be analyzed with the moving mesh of the objective domain. The mixed interpolation [7]~[10] is employed with the bubble function interpolation for velocity and linear function interpolation for pressure to obtain the stable finite element computation.

For the computational study, the falling particulate flow has already been presented in the papers [3]~[7]. In this paper three dimensional moving boundary problem of moving propeller is computed.

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2. COMPUTATIONAL SCHEME 2.1. Basic equation As the basic equation of the incompressible viscous fluid, the incompressible Navier-Stokes equations is used, which is expressed as,

fuupuuu T =∇+∇∇−∇+∇+ )(ν& in ω\Ω , (1)

0=∇u in ω\Ω , (2)

0gu = on 1Γ , (3)

1gu = on γ , (4)

tnuup T =⋅∇+∇+− ])([ ν on 2Γ , (5)

Γ=ΓΓ 21 U , (6)

φ=ΓΓ 21 I , (7)

ω

Figure 1: Computational D

where and u p denote the velocity and p

domain, ω is the bounded domain contain

boundary of ω ,respectively. The domain ω

values and are boundary conditio

vector of outward normal to , respectivel

0g 1g

Γ 2.2. Finite element interpolation The weighted residual equation can be desc

[∫∫∫ ΩΩΩ+−⋅∇+Ω∇⋅+Ω pwudwuduw &

Ω

omainΩ and Objective Domainω .

ressure,ν is viscosity, and Ω is a computational

ed in Ω , Γ is the boundary of , and Ω γ is the

is referred to as the objective domain. The given

ns. Traction is denoted by and is the unit

y.

t n

ribed as follows;

,])(2

2∫∫ ΓΩΓ⋅+Ω⋅=Ω∇+∇ tdwfdwduu Tν (8)

2

0=Ω⋅∇∫Ω udq , (9)

where and are weighting functions. The fractional step projection method [11] is

applied in this formulation. As for the spatial discretization, the finite element method based on the bubble function interpolation [8]~[10] for the velocity and the linear interpolation for kinematic pressure are applied and expressed as follows. The linear interpolation can be denoted as;

w q

,44332211 ppppp Ψ+Ψ+Ψ+Ψ= (10)

,,,, 44332211 LLLL =Ψ=Ψ=Ψ=Ψ (11)

where are nodal values of pressure and are area coordinate. The Bubble

function interpolation can be represented as;

41 ~ pp 41 ~ LL

,~5544332211 iiiiii uuuuuu Φ+Φ+Φ+Φ+Φ= (12)

)(41~

432155 iiiiii uuuuuu +++−= , (13)

,,,, 44332211 LLLL =Φ=Φ=Φ=Φ ,256 43215 LLLL=Φ (14)

where shows the components of iu u, and are nodal values. Nodes are shown in

Fig.2. The inside node is taken at the center of gravity, at which value can be eliminated.

41 ~ ii uu

4 2

1

5

Figure 2: Nodes of el 2.3. Fictitious Domain formulation The basic idea of the fictitious domain method is simand solid body is denoted by ω and is referred to as tassumed to be filled with fluid. An objective reactionobjective boundary condition can be expressed as con

3

3 ement.

ple. The fluid domain is denoted by Ω he objective domain. The inside of ω is is reflected in the fluid. Moreover, the straint condition using the distributed

Lagrange multipliers. Denoting the distributed Lagrange multiplier by λ , waiting functions

by and QW , µ , respectively, the formulation of the fictitious domain method can be

expressed as follows,

∫∫ ΩΩ∇+∇⋅∇+∇⋅+

∆−

ωω

ν\\

)(2

)(~

dxUUWdxUUtUU

W Tnh

nhh

nh

nh

nhh

h ,\ ∫∫ ⋅+⋅=

Ω ωωλ dxWdxFW hhhh

(15)

∫∫ Ω

+

Ω

+

∇+∇+−⋅∇+∆−

ωων

\

1

\

1

)(21

~ dxUUPWdx

tUU

W Tnh

nh

nhh

hn

hh ,

2∫∫ ⋅+⋅=

Γ ωλ dxWtdxW hhh

(16)

0\

=∇⋅∫Ω ωdxUQ hh , (17)

,0)( 1 =−∫ω µ dxgUhh (18)

where are interpolated functions by the bubble function and hh QW , hU~ is the intermediate

velocity. In order to integrate more easily the last terms in eqs.(15) and (16) added by the fictitious domain method easily, the following Dirac’s delta function is introduced.

)(0)(

)(i

ii XXif

XXifXX

≠=∞

=−δ , (19)

The integral form is expressed as:

∫ ≠=

=−ωδ

)(1)(0

)(i

ii XXif

XXifdxXX , (20)

The interpolation of λ and µ which are expressed by hλ and hµ , can be expressed as

follows.

∑=

−=dN

iiih XX

1),(δλλ (21)

∑=

−=dN

iiih XX

1),(δµµ (22)

where is the number of the nodes of the domain of dN ω and the integration over the

domain ω can be written as follows,

,)(1∑∫=

=dN

iihihh Xvdv λωλ

ω (23)

.))()(()(1

11 ∑∫=

−=−dN

iihihh XgXUdgU µωµ

ω (24)

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Eqs.(15)~(18) can be solved at each time cycle. The simultaneous equation can be solved by the element by element conjugate gradient method [12]. 2.4. Movement of propeller The motion of the propeller is caused by the fluid force, of which the momentum equations of the Newton second law is expressed by

pp

p Tdt

dI =

ω, (25)

where pω is the angular speed, is the moment of inertia, is the moment force

imposed on the propeller by the fluid, respectively, in which subscripted

pI pT

p expresses the

quantity is concerned with the propeller. The force and the moment imposed on the propeller by the fluid are described as follows,

∫= γσndxFp , (26)

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡ −=

1000cossin0sincos

np

np

np

np

nR θθθθ

, (27)

∫ −=γ

σ dxnRGxT np

np )()( . (28)

where is coordinate of gravity center, is stress tensor, pG )( TuupI ∇+∇+−= νσ x is

coordinate of point and n is outward unit normal to the boundary γ ,respectively. The

velocity of propeller is computed as follows,

p

npn

pnp I

Tt∆+=+ ωω 1 . (29)

p

npn

pnp I

Tt 21 ∆+=+ θθ . (30)

where pθ in eq.(30) is the angular displacement. The movement of propeller can be expressed

by the following procedure. A nodal coordinate of a points on the surface of the domain ω is

expressed by in eq.(31). Then it can be moved rotating by multiplying . Those

procedures are schematically illustrated in Fig.3.

ωX TnR )( 1+

[ ]ωωωω zyxX = , (31)

5

θ

y x

ωX ωXR Tn )( 1+

Figure 3: Image for rotational movement. 3. NUMERICAL STUDY 3.1. Numerical model To show the efficiency of the fictitious domain method, the simulation of the flow around the propeller in the channel ))0.100.4()0.20.2()0.20.2(( ×−××××−=Ω is given as the numerical study as shown in Fig.4. The Reynolds number is 1000. Time increment is taken 0.01 in this numerical computation.

u

Figure 4: Numerical model of propeller. 3.2. Finite element mesh The finite element meshes for and Ω ω are shown in Figs.5 and 6, respectively.

Figure 5:Mesh for computational domain.

6

Figure 6: Mesh for propeller.

Total number of nodes of and Ω ω are 42345 and 4321, respectively. Total number of elements of and Ω ω are 244642 and 19528, respectively. 3.3. Numerical result The flow around the rotating propeller is analyzed. Numerical results are shown in Figs.7~15. The time history of the drag, lift and side forces are expressed in Figs.7, 8 and 9. Drag, lift and

side forces are fluid forces expressed in eq.(26) in the direction of x , and , respectively.

Getting to the stationary state, the periodic behavior can be seen in the figures. The pressure and streamlines are represented in Figs.10~15. in which streamlines, iso-surface of pressure and position of propeller are represented, respectively. It is clear that the rotating flow can be obtained behind the propeller. Looking at the figures, it is understood that the pressure at the point of the propeller receives stronger pressure than at the center of the propeller. Figs.16 and 17 express the pressure distribution behind the propeller. The influence distance to the rear side can be seen. Fig.18 is pressure distribution at the surface of the rotation side.

z y

7

Figure 7: Time history of drag force.

Figure 8: Time history of lift and side forces.

Figure 9: Figure 8 amplified for the period from 7000 to 8000.

8

Figure 10: t=0.05

Figure 11: t=1.0

9

Figure 12: t=2.0

Figure 13: t=3.0

10

Figure 14: t=4.0

Figure 15: t=5.0

11

Figure 16: t=3.0

Figure 17: t=5.0

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Figure 18: pressure distribution around propeller

4. CONCLUSIONS The fictitious domain method can be applied to the 3-dimensional flow around rotating propeller which has the complex body configuration and the moving boundary. These problems have to consider the interaction between fluid and solid. The mixed interpolation finite element method can be applied to the fictitious domain method with the bubble function interpolation for velocity and linear function for pressure. The element by element conjugate gradient method can be applied as the solver of the simultaneous equation. The fictitious domain finite element computation can be more stable and faster than the conventional finite element method in the computation of the moving boundary problems. REFERENCES [1] R.Glowinski, T-W. Pan and J.Periaux, ‘A fictitious domain method for Dilichlet problem and applications’, Compute. Methods Appl. Mech. Engrg., Vol.111, pp.283-303, 1994. [2] R.Glowinski, T-W. Pan and J.Periaux, ‘A fictitious domain method for externa incompressible viscous flow modeled by Navier-Stokes equation’,Comput. Methods Appl. Mech. Engrg., Vol.112,pp.133-148, 1994. [3] D. D. Joseph and R. Glowinski, ‘Interrogations of Direct Numerical Simulation of Solid-Liquid Flow’,http://www.aem.umn.edu./Solid-liquid_Flows/ [4] T-W. PAN, ‘Numerical Simulation of the Motion of a Ball Falling in an Incompressible Viscous Fluid’ ,http://math.uh.edu/particulate_flow/f98.ps [5] H.Kawarada and H.suito, ‘Numerical method for a free surface flow on the basis of the

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fictitious domain method’,East –West J.Numer. Math., Vol.5, No.1, pp.57-66,1997. [6] H. Okumura, H. Naya, N. Shimada and M. Kawahara, ‘A Distributed Lagrange multiplier / Fictitious domain method for Incompressible flows moving rigid bodies’, Proceedings of the 14th Symposium on Computational Fluid Dynamics, c06-4.pdf, 2000. [7] N. Shimada and M. Kawahara, ‘Analysis of particulate flows by fictitious domain method with Distributed Lagrange multiplier’, Proceedings of the First Asian-Pacific Congress on Computational Mechanics, Vol.1, 2001, pp.127-132. [8] A.Maruoka, J.Matsumoto and M.Kawahara, ‘Lagragian finite element method for incompressible Navier-Stokes equations using quadrilatera scaled bubble function’,Journal of Applied Mechanics, Vol. 44A, 1998, pp.383-390. (In Japanese). [9] J.Matsumoto and M.Kawahara, ‘Shape Identification for Fluid-Structure Interaction Problem using Improved Bubble Element’ International Journal of Computational Fluid Dynamics, 15, pp.33-45, 2001. [10] J.Matsumoto, T.Umetsu and M.Kawahara, ‘Incompressible Viscous Flow Analysis and Adaptive Finite Element Method Using Linear Bubble Function’, Journal of Applied Mechanics, vol.2, 223-232, 1999. [11] H. Okumura and K. Ohmori, ‘Mass conservative finite element method for immiscible two-fluid flow problems’, Journal of Applied Mechanics, J.S.C.E., Vol.7, 2004( In Japanese). [12] Zhang, S. L., ‘GPBi-CG: Generalized Product-type Method Based on Bi-CG for Solving Nonsymmetric Linear Systems’,SIAM J. Sci. Comput.,Vol.5, No. 4, 1995.

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