advanced finite element methods for fluid dynamic analysis of ships

7/31/2019 Advanced Finite Element Methods for Fluid Dynamic Analysis of Ships

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ADVANCED FINITE ELEMENT METHODS

FOR FLUID DYNAMIC ANALYSIS OF SHIPS

E. Oate and J. Garca

International Center for Numerical Methods in Engineering (CIMNE)Universidad Politcnica de Catalua

Campus Norte UPC,08034 Barcelona,Spaine-mail:[email protected], web page:http://www.cimne.upc.es

Key words: Finite element, incompressible ows, free surface, nite calculus

Introduction

In recent years, the advent of advanced numerical schemes for the Navier Stokes (NS)equations and the rapid development of computer hardware have enabled a realisticprediction of fluid flow about ship forms. The troubles in accurately solving thisproblem are mainly due to the difficulty of solving numerically the incompressible fluiddynamic equations, which include significant non linearities and the obstacles in solvingthe constraint equation stating that at the free surface boundary the fluid particlesremain on that surface which position is in turn unknown.

Among the schemes developed over the last decade for the solution of theincompressible NS equations the fractional step schemes [1,2,3,4] yield highly accurate,pressure-stable [2] results by integrating in an explicit manner the advective terms of the

NS equations. However, in most of the cases of interest for the naval architecture thetime step imposed by the smallest elements may be orders of magnitude smaller than thetime step required to obtain time-accurate results (physical time step). In some cases thisimplies tens of thousands of time steps per simulation, rendering the schemesimpractical. Most of the artificial compressibility and preconditioned schemes sufferfrom the same shortcoming.On the other hand, the monolithic schemes treat, in general, the advective term in animplicit manner, which avoids the mentioned disadvantages. Nevertheless, thesemethods are very expensive from a computational point of view: the velocity andpressure discrete equations are coupled.

This paper presents advances in recent work of the authors to derive a fractional stepscheme based on the stabilized finite element method that allows overcoming the abovementioned problem. The starting point is the modified governing differential equationsfor the incompressible turbulent viscous flow and the free surface conditionincorporating the necessary stabilization terms via a finite calculus (FIC) proceduredeveloped by the authors [5,6,7,8].To reach the mentioned objective, an implicit and uncoupled second order fractionalstep method based on the scheme originally proposed by Soto [9] is presented.


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FIC formulation

We consider the motion around a body of a viscous incompressible fluid including afree surface. The stabilized finite calculus (FIC) form of the governing differentialequations [10] for the three dimensional (3D) problem can be written as

(1a)

(1b)

(1c)

where

In above ui is the velocity along the i-th global reference axis,p is the dynamic pressure, is the wave elevation and ij are the viscous stress tensor components.

Let ni be the unit outward normal to the boundary and denoting by an overbarprescribed values, the boundary conditions for the stabilized problem are:

(2a)

(2b)

(2c)

for t (t0,tf). The boundary has been considered split into three sets of disjointcomponents u, p and , the latter being the part where mixed conditions are

prescribed.Vectors gi and si span the space tangent to . Finally, u and p are the two disjointcomponents of where Dirichlet and Neumann boundary conditions for the velocityare prescribed. Initial conditions have to be appended to problem (1)-(2).

The underlined terms in eqs. (1)-(2) introduce the necessary stabilization for thenumerical solution. Additional time stabilization terms can be accounted for in eqs. (1)-(2) [5,6] although they have been found unnecessary for the type of problem solvedhere.

The characteristic length distances hj represent the dimensions of the finite domainwhere balance of momentum and mass is enforced. The characteristic distances hj ineq. (1c) represent the dimensions of a finite domain surrounding a point where thevelocity is constrained to be tangent to the free surface. Details of the derivation of eqs.

10 , 1,2,3

2

10 1,2,3

2

10 1,2

2

i

i

j

mm j

j

dd j

j

j

rr h on i j

x

rr h on j

x

rr h on j

x

= =

= =

= =

( )

1,2,3

1,2,3

i

ijim i j

j i j

id

i

ii

u pr u ut x x x

ur i

x

r u it x

= + +

= =

= + =

1 2

,

12

1 1,

2 2

i

i i

u

ij ij j j m p

nj j j ij i j j m i j ij i j j m i

u u on

p p and n h n r t on

u n u n g h n r g t and n s h n r s t on

=

= =

= = =


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(1)-(2) and recommendations for computation of the stabilization parameters can befound in [5,11,12].

Eqs. (1)-(2) are the starting point for deriving a variety of stabilized numerical methodsfor solving the incompressible Navier-Stokes equations with a free surface. It can beshown that a number of stabilized finite element methods and new meshless methods

allowing equal order interpolations for the velocity and pressure fields can be from themodified form of the momentum and mass balance equations given above [5-8].

Implicit Fractional Step Formulation

Let us discretize in time the stabilized momentum equation (1a) using the trapezoidalrule (or method) as

(3)

the superscripts n and refer to the time step and to the trapezoidal rule discretizationparameter, respectively. For = 1 the standard backward Euler scheme is obtained,which has a temporal error of 0(t). The value = 0.5 gives the standard CrankNicholson scheme, which is second order accurate in time 0(t2).A classical implicit fractional step method can be simply derived by splitting eq. (3).The resulting continuous problem, omitting the boundary and initial conditions forbrevity, is as follows:

(4a)

(4b)

(4c)

where the terms denoted by an overbar are those calculated with the intermediatevelocity i, which is introduced to allow the momentum splitting.The error due to taking implicit advective and viscous terms in (4a) can be shown [9] tobe of the same order than the error of the stabilizing term and therefore, it has the sameorder of approximation than the original time discretization (3).

Monolithic Stabilized Scheme

At this point, it is important to introduce the associated matrix structure correspondingto the variational discrete form of (4) (see [5] for details of this derivation):

(5a)

(5b)

(5c)

By taking n+1 from (5c) and inserting the result in (5a)-(5b), the following system ofequations is obtained:

( )1 1

02

i

nnn n nmijn ni i

i j jj i j j

ru u pu u h

t x x x x

+++ +

+ +

+ + =

( )

( )

( )

1

2 *1

1 1

10

2

12

i

nnn n nmijn ni i

i j jj i j j

n n i dj

i i i j

n n ni i

i

ru u pu u h

t x x x x

u rt p p hx x x x

u u t p px

+++

+ +

+

+ +

+ + =

= +

=

( ) ( )

( ) ( )

( ) ( )

1

1 1

1 1 1

10

0

10

n n n n n

n n n n

n n n n

M U U K U U GPt

t L P P HU DU

U U P Pt

+ + +

+ +

+ + +

+ =

+ + =

=


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(6a)

(6b)

The term E(Un+) in (6a) is the error coming from the implicit treatment of the advective

and viscous terms, which is of order 0(t2

). However, such term can be eliminated as in(6a)-(6b) by writing the following analog monolithic scheme:

(7a)

(7b)

Basically, in this final formulation the convergence of the block uncoupled solution isenforced by the first term of (7b), while the pressure stability is attained by the secondterm of the same equation.

Finally, the wave elevation coupling effect is included in the monolithic scheme,obtaining the following system of equations:

(8a)

(8b)

(8c)

where the Uin+ is the vector of the velocity along the i-th global reference axis.

Application Example

The application example here presented is the parametric analysis of a 79 catamarandesigned by the company NAUTATEC. The main characteristics of this boat are listednext (see Figure 1):

Length overall 23.95 mLength between perpendiculars 23.35 mTotal breadth 11.75 m

Extreme breadth of every hull 1.65 mDraught with appendages 1.31 mHull depth 2.40 mDisplacement 26 730 KgMaximum sail area 222 m2Passengers 120

This study started from a fixed well-known geometry of the hulls, being the object ofthe study, the effect of the distance between hulls in the resistance and lateral forces.Three different alternatives of distance between hulls studied were:

Case NameDistance between hulls centerlines 10.1 m 79cata5_0Distance between hulls centerlines 9.1 m 79cata4_5

( ) ( ) ( )

( ) ( ) ( )

1

1 1 1

1

,

n n n n n n

n n n n n

M U U K U U GP E Ut

t L DM G P P H U P DU

+ + + +

+ +

+ =

=

( ) ( )

( ) ( ) ( )

1, , , 1, 1

1 1, 1, 1, 1,

10

,

n i n n i n i n i

n i n n i n i n i

M U U K U U GPt


+ + + +

+ + + +

+ =

=

( ) ( )

( ) ( ) ( )

( ) ( )

1, , , 1, 1

1 1, 1, 1, 1,

1, , , , ,3

10

,

1( , )

n i n n i n i n i

n i n n i n i n i

n i n n i n i n i n n i

M U U K U U GPt


M B B A U B R U B MUt

+ + + +

+ + + +

+ + + + +

+ =

=

+ =


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Distance between hulls centerlines 8.1 m 79cata4_0

Figure 1. NURBS based geometrical definition of the 79catamaran.Different alternatives studied.

The analyses were carried out for a range of ship speed from 8 kn to 28 kn, being thereal range from 8 kn to 16 kn. Regarding the drift angles, four cases were considered:0.0 , 1.0, 2.5 and 5.0, being the project range from 0.0 to 2.5.

The grid used for the analyses consisted of 450.000 linear tetrahedra. The fluid viscositytaken was 1.3 Kg/ms and the density 1024 Kg/m3. The k turbulence model with theextended law of the wall was used.

The basic results of these analyses are presented next:

Figure 2. Resistance and lift curves for the 79cata4_5 case.

Resistencia

0

5

1 0

1 5

2 0

2 5

3 0

3 5

4 0

4 5

8 1 0 1 2 1 4 1 6 1 8 2 0 2 2 2 4 2 6 2 8

N u d o s

KN

7 9 c a t a 4 _ 5 d e r 0

7 9 c a t a 4 _ 5 d e r 1

7 9 c a t a 4 _ 5 d e r 2 , 5

7 9 c a t a 4 _ 5 d e r 5

Sus te nta c in- 1 8 0

- 1 6 0

- 1 4 0

- 1 2 0

- 1 0 0

- 8 0

- 6 0

- 4 0

- 2 0

0

8 1 0 1 2 1 4 1 6 1 8 2 0 2 2 2 4 2 6 2 8

N u d o s

KN

7 9 c a t a 4 _ 5 d e r 0

7 9 c a t a 4 _ 5 d e r 1

7 9 c a t a 4 _ 5 d e r 2 , 5

7 9 c a t a 4 _ 5 d e r 5

Lift

Resistance


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Figure 3. Resistance curves for the different cases analyzed.

In order to evaluate de efficiency of the implicit algorithm, the CPU time needed toachieve the pseudo-steady state using both the explicit and implicit algorithms wasevaluated. This test was carried out for three different grids. Using these, one singlepoint of the 79_cata5_0 geometry analysis was run. The time step chosen for theimplicit algorithm was 0.01L/V, being L the length of the ship and V the speed of theanalysis. For this time step, no significant differences were found between the resultsobtained with the explicit or implicit algorithm.The results of this comparison are shown next.

Grid characteristics Explicit / Implicit algorithmtime ratio

150 000 tetrahedra elements 1.1 : 1.0300 000 tetrahedra elements 1.3 : 1.0450 000 tetrahedra elements 2.2 : 1.0600 000 tetrahedra elements 4.1 : 1.0

Conclusions

An implicit second-order accurate monolithic scheme, based on the FIC formulationwas presented to solve incompressible free surface flow problems. The final system ofequations resulting from the time and space discretization is solved in each time step inan uncoupled manner.

The numerical experience indicates that the formulation is very efficient for freesurfaces flows, when the critical time step of the problem is some orders of magnitude

Resistencia

0

5

10

15

20

25

30

8 10 12 14 16 18 20 22 24 26 28Nudos

KN

79cata5_0der079cata4_5der079cata4_0der0

Resistencia

0

5

10

15

20

25

30

35

8 10 12 14 16 18 20 22 24 26 28Nudos

KN

79cata5_0der179cata4_5der179cata4 0der1

Resistencia

0

5

10

15

20

25

30

35

8 10 12 14 16 18 20 22 24 26 28Nudos

KN

79cata5_0der2,579cata4_5der2,5

79cata4 0der2 5

Resistencia

0

5

10

15

20

25

30

35

40

45

8 10 12 14 16 18 20 22 24 26 28Nudos

KN

79cata5_0der579cata4_5der5

79cata4 0der5

Resistance Resistance

Resistance Resistance


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smaller than the time step required to obtain time-accurate results (physical time step),and that its time accuracy is excellent.

Acknowledgements

The authors want to thank the support of NAUTATEC ([email protected]) forthis work.

The authors also thank Dr. O. Soto for many useful discussions.

References

[1] Chorin. On the convergence of discrete approximation to the Navier-Stokesequations. Math. Comput., 23, 1969.[2] R. Codina. Pressure stability in fractional step finite element methods forincompressible flows. J. Comp. Phys., Submitted for publication, 2000.[3] J. Garca, E. Oate, H. Sierra, C. Sacco y S. Idelsohn.A stabilised numerical method

for analysis of ship hydrodynamics. ECCOMAS 98 (Vol. II). Atenas 1998.[4] R. Lhner, C. Yang, E. Oate, and S. Idelsohn. An unstructured grid-based, parallel

free surface solver. AIAA-97-1830, 1997.[5] E. Oate and J. Garca. Finite Element Analysis of Incompressible Flows with FreeSurface Waves using a Finite Calculus Formulation. ECCOMAS 2001. Swansea UK2001.[6] E. Oate.A stabilized finite element method for incompressible viscous flows using a

finite increment calculus formulation. Comp. Meth. Appl. Mech. Engng. 182, 1-2, 355-370, 2000.[7] E. Oate and J. Garca. A methodology for analysis of uid-structure interactionaccounting for free surface waves. European Conference on Computational Mechanics(ECCM99). Munich, Germany, September 1999.

[8] E. Oate and J. Garca. A stabilized finite element method for analysis of fluidstructure interaction problems involving free surface waves. Proceedings of FluidStructure Interaction Conference, Trondheim (1999).[9] O. Soto, R. Lhner, J. Cebral and R. Codina. A Time Accurate Implicit-MonolithicFinite Element Scheme for Incompressible Flow. ECCOMAS 2001. Swansea UK 2001.[10] E. Oate, J. Garca and S. Idelsohn, An alpha adaptive approach for stabilized

finite element solution of advective-diffusive problems with sharp gradients, New Adv.In adaptive Comp. Met. In Mech., P. Ladeveze and J.T. Oden (Eds.), Elsevier (1998).[11] E. Oate, Derivation of stabilized equations for advective-diffusive transport and

fluid flow problems, Comput. Meth. Appl. Mech. Engng., Vol. 151, 1-2, pp. 233-267(1998).

[12] E. Oate, J. Garca and S. Idelsohn, Computation of the stabilization parameter forthe finite element solution of advective-diffusive problems with sharp gradients , Int. J.Num. Meth. Fluids, Vol. 25, pp. 1385-1407 (1997).

advanced finite element methods for fluid dynamic analysis of ships

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