Wind forces on bridges — numerical vs. experimental methods

Journal of Wind Engineering and Industrial Aerodynamics, 32 (1989) 145-159

145

Elsevier Science Publishers B.V., A m s t e r d a m - - P r i n t e d in T h e N e t h e r l a n d s

WIND FORCES ON BRIDGES

-

NUMERICAL VS. EXPERIMENTALMETHODS

H.P. SANTO1 and F.B. BRANCOI ZCMEST - Center f o r Structural Mechanics and Engineering of the Technical University of Lisbon, I n s t i t u t o Superior T~cnico, 1096 Lisboa Codex, Portugal

ABSTRACT The present paper reports the preliminary stages of a research project which aims p r i m a r i l y at developing p r a c ti c a l numerical tools to assess the performance of structures under wind actions. I t is argued t hat , as in other f i e l d s of engineering, the f i n i t e element method can constitute a competitive and r e l i a b l e a l t e r n a t i v e to t r a d i t i o n a l experimental analyses. INTRODUCTION Until the f i r s t

decades of this century the actions of wind on bridge struc -

tures were taken i n t o account by the sole consideration of s i m p l i f i e d s t a t i c forces to simulate the pressure of a i r . The disaster of the ( f i r s t ) rows Bridge in 1940, due to aerodynamic i n s t a b i l i t y ,

Tacoma-Nar-

led to the beginning of a

new era in the analysis of bridges with wind being considered appropriately a dynamic action, represented by an average speed, associated with gust effects. This c h a r a c t e r i s t i c of the wind may i n t e r f e r e with the dynamic response of the structure causing aerodynamic i n s t a b i l i t y to occur. This f a c t is especially important in slender, long suspension or cable-stayed bridges, usually with natural frequencies under 0.5 Hz. For medium and short-span frame bridges

the

dynamic e f f e c t is not important and wind actions can be regarded as equivalent s t a t i c forces, determined from the pressure d i s t r i b u t i o n . In practice, the aerodynamic (or, more precisely, a e r o s t a t i c , in the present case) behavior

of bridges is estimated by drag, l i f t

and moment c o e f f i c i e n t s

from which an acceptable load d i s t r i b u t i o n can be obtained. These c o e f f i c i e n t s are given by Codes of Practice f o r simple standard cases. When uncommon crosssections are present, professionals w i l l have to e i t h e r consider s i m p l i f i e d geometries to match the shapes included in the guidelines of the codes - with a corresponding decrease in the accuracy of the analysis - , or resort to wind-tunnel tests, which are hardly cheap or easy to perform, and normally not a v a i l a b l e and/or f e a s i b l e . The purpose of this paper is to report the development of an a l t e r n a t i v e f o r designers to evaluate with enough precision wind actions on structures, partic__u l a r l y bridge decks. The method of analysis considers the u t i l i z a t i o n of state-

0167-6105/89/$03.50

© 1989 Elsevier Science Publishers B.V.

146

- o f - t h e - a r t f i n i t e element formulations of f l u i d mechanics. Reference 1 docu ments the outstanding research e f f o r t s regarding the application of the F i n i t e -Element Method (FEM) to f l u i d mechanics problems. While the FEM has proven to be general, appropriate and r e l i a b l e in a l l f i e l d s of engineering, i t has only recently begun to be seriously applied to the aerodynamics of structures(Ref.9 1 2 ) , a f i e l d thathasalwaysbeenviewed as s t r i c t l y experimental. The preliminary stage of the authors' research project aims p r i m a r i l y therefore at providing immediate practical use of the outcome of the recent academic i n v e s t i g a t i o n s , in the assessment of wind loads and c a l c u l a t i o n of the associated shape c o e f f i cients. The numerical results presented compare f a i r l y well with the corre sponding experimental data but, though encouraging, they also reveal that deeper research is needed in a few but fundamental areas. The present

work can

be regarded as an extension and complement of previous papers (Refs. 2,3,4). THE NUMERICAL MOOEL The solution of the generalized Navier-Stokes (N-S) equations, in terms of the p r i m i t i v e variables (velocity-pressure), forms the basic mathematical model adopted. At present the simplest formulation, based on the Galerkin weighted residual method, in association with 8-node isoparametric elements is used throughout. This approach is described in d e t a i l in the l i t e r a t u r e ( e . g . , Ref.5,6,7) so only a summary is outlined below. For a Newtonian, incompressible viscous f l u i d , the general 2-D steady-state N-S equations reduce to ~k + v ~x

~u _ i 3y p

(F x

9v + v

~v _

I

(F

9x

~y

p

}P) + V_k_ (~2___~+ g2____~u ) 9x P ~x 2 3y 2

(1)

where

_ ~_~)

y

+ ~_k_ ( ~2v + 32.v)

~y

p

~x 2

~y2

p = mass density (constant) u = v i s c o s i t y (constant) ~ , F y = body forces in x and y directions p = pressure u , v = v e l o c i ~ e s i n x and y d i r e c t i o n s ,

and the local stresses become ax

-p+2~

3x ~v dy

y

= ~xy = Tyx

~u --

( ~u 14 ~ y +

(2) ~v ) . ?x

147

From the s o l u t i o n of the system of equations, obtained from the m a t r i x assembly of ( I ) ,

the r e s u l t i n g forces on the cross-section are computed by i n t e -

g r a t i n g the pressure and f r i c t i o n

components along the boundary Bbythe f o l l o w -

ing r e l a t i o n s FPx = - IB Cx P db

F fx = ~ IB (2 Cx ax + Oy Ty x) db

D =F p +F f X

(3)

X

F pY = - IB %

F f = ]JIB (Cx T + 2 C a ) db y xy y y

p db

L =F p +F ~ Y Y

where

Cx,

C

Y

= d i r e c t i o n cosines

Fp , Fp = total x y

pressure forces in x and Y

F f , Ff = total x y

friction

forces in x and y

= drag force = lift

force.

The t o t a l moment with respect to the center of r o t a t i o n C is given by

where

dx , dy = p

,

fy , ff,

ff

=

distances from the nodal forces to C

pressure and f r i c t i o n

nodal forces in x and y.

The a e r o s t a t i c c o e f f i c i e n t s are then c a l c u l a t e d by Drag

:

CD -

-

2D -

pv2L Lift

:

c~

-

2

y

L

p v 2oo L x

Moment :

CM =

2 Mc p v2 L2 x

where

v Lx, Ly

= field =

velocity

reference lengths.

THE DRAFTYBRIDE SYSTEM To support the present research p r o j e c t a special-purpose CAD system, enti t l e d DRAFTYBRIDE (from DRAg, LiFT and Moment AerodYnamic C o e f f i c i e n t s of BRldge DE_cks) has been created (Ref.2).

I t incorporates the above model and a command-

148

-structured,problem-oriented language,tentatively named WASP (Wind Actions

on

Structures, Problem-Oriented Language), which is being developed to c o n s t i t u t e a p r a c t i c a l , complementary tool f o r s t r u c t u r a l aerodynamics studies• The system is p a r t i c u l a r l y useful at the early stages of design, when adequate cross-sections and overall geometries have to be defined,and pertinent whenever

scale

models are to be b u i l t f o r subsequent wind-tunnel tests, since a l o t of time and cost

can

be saved i f nearly optimum shapes are achieved from the s t a r t .

At the present stage of development the system comprises three main modules, described below. The p . o . l . WASP, which can be regarded as a fourth module, l i n k i n g a l l routines, is not introduced herein, however, for i t is the subject of a forthcoming paper (Ref.8). • Module I : Pre-processing This

module includes a 2-D mesh generation sub-system, based on the super -

element isoparametric mapping concept, being very f l e x i b l e in this context , allowing for several types of elements and node or element optimal numbering. Figure 6 shows a mesh generated by the algorithm. The boundary data are also automatically set by t h i s module from the control parameters. • Module 2 : Processing The processing i t s e l f is mainly the s o l u t i o n of the system of equations estab lished from the equations ( I ) , by a f r o n t a l method solver f o r unsymmetric matrices. • Module 3 : Post-processing The processed data from module 2 (namely the calculated forces and

coeffi-

cients) can be presented numerically or g r a p h i c a l l y . The user-selectable re sults include the following standard information: numerical output input data; generated mesh data; v e l o c i t y

and pressure values in the whole

domain; absolute pressure, i t s components, the dimensionless pressure c o e f f i c i e n t and the stresses on a l l cross-section nodes; the pressure, f r i c t i o n and t o t a l drag, l i f t

and moment c o e f f i c i e n t s .

graphical output v e l o c i t y f i e l d (Figure 7), including d e t a i l i n g when required; pressure f i e l d on the cross-section; streamlines.

generated mesh (Figure 6) ; pressure contour lines and

The DRAFTYBRIDE System is subject of constant improvement, namely with the

149

introduction of more advanced FE formulations, as discussed in the last section of this paper and in Reference 8. Future developments also include the enhancement of the graphical features (e.g., complete simulation of the flow, with ani mation, taking advantage of the MOVIE.BYU capabilities - Ref.29), creation of a micro-computer version and extension to aeroelastic and tridimensional problems. The accuracy and v a l i d i t y of the mathematical model adopted, and the present FE formulation used, have been thoroughly assessed through a comparative study of the flow around the i n f i n i t e circular and square cylinders, reported in Reference 2. The a v a i l a b i l i t y of extensive reliable numerical and experimentaldata on these two classical problems of f l u i d mechanics, allowed to conclude for the very satisfactory efficiency of the model and the performance of the DRAFTYBRIDE System. CASE STUDIES Three bridge

decks were chosen to i l l u s t r a t e the preliminary studies carried

out, taking advantage of published numerical and/or experimental results. cover the

three

They

main approaches and sources of data: numerical analyses, ex-

perimental tests and code-of-practice guidelines. Trapezoidal Deck The standard trapezoidal deck represented in Figurelhas been analysed.

It

has been the subject of the extensive wind-tunnel tests and FE analysis reported in Reference I f . The results of this Reference are reproduced in Table I , along with the onesof the authors presented in previous papers (References 3 and 4). Their close relation is apparent.

L x - 12

Ly-2~__ I

"1

F i g . l . Trapezoidal Deck ( a l l dimensions in meters)

150

Experimental

Kawahara/Hirano

Authors

J

-5 o

(~

%

oo

_5 0

0o

+5 0

1.04

1.58

i. 24

1.04

I. 46

1.45

-0.044

0.56

0.52

-0.42

0.061

-0.047

1.46

C L -0.51

-50

+5 0

-0.605 -0.024

0o

+50

1.25

1.53

-0.017

0.45

i

@

0.024

0.039

0.026

0.024

0.0

0.042

0.146

TABLE I Box-Girder Deck Some r e s u l t s of the analysis of the section shown in Figure 2 are indicated in Table I I ,

along with data from an IABSE Box-Girder Bridge Manual (Ref. 13).

Again the numerical r e s u l t s compare well with those of the manual (derived from extensive experimental t e s t s ) . Lx-b

• 12

, -h-2

[ b2-2.4

L

b/h = 6

~,

bl-7.2

(geometric parameters of Ref.13)

bl/b2 = 3

Fig.2.

Box-Girder Deck

Schlaich/Scheef (IABSE)

Authors

1.56

1.49

0.9

1.01

TABLE I I Figure 3 represents the dimensionless pressure c o e f f i c i e n t

C~ in the zones

i n d i c a t e d t h e r e i n , from References 4 and 13. The agreement is very good, consid ering the r a t h e r coarse mesh used ( s i m i l a r to the one of Figure 6). The dimensionless pressure c o e f f i c i e n t is given by the f o l l o w i n g r e l a t i o n C

P

= 2(p

-

p~)

P v2

where

p = p~ =

pressure f i e l d pressure

151

®

CP

®I

® ®

I©

1.5

.5

0

I

® -.5

FEM -t.

,=I/ER. ~=d.UES(FEM) REF.13

-0.SS

: ~ :--.-: --

( I'~"~1 ) -- ( 0.5 )

Fig.3. Box-Girder Deck - Pressure C o e f f i c i e n t Guadiana Bridge Deck The project of the I n t e r n a t i o n a l Bridge over the Guadiana River has been recently concluded (Ref.14). I t is a cable-stayed bridge, with a pre-stressed concrete box-girder deck, spanning a t o t a l of 666 m (central span of 324 m), that w i l l be erected to l i n k Ayamonte and Vila Real de Santo Antonio, in the southern part of Spain and Portugal. The cross-section of the deck is represen~ ed in Figure 4. Extensive wind-tunnel tests were conducted (Ref.14) and Figure 5 reproduces the aerostatic

coefficient

curves

obtained

for

the

empty

deck. The authors performed the numerical analysis, using the mesh shown in Figure 6

along with the boundary conditions. The (smooth) flow parameters considered

were

152 p

=I.0

Voo=

1.0

V

-

p P

Poo

=

0.0

(cinematic viscosity)

- O. 0 6 2 5

which yielded R C

~-

voo Ly

=

40

(Reynolds number).

17.70

Q25

F

"1

!

LI.~. 2.4o

11.10

.I, 2.40 1.Is c

Lx:t8.20

J,

Fig.4. Guadiana Bridge Deck

0.6.~ CD, CL,CM10

-10

-5

10

CD/10 • CL --+ CM

o

Fig.5, Guadiana Bridge Deck Aerostatic Coefficients (Ref,14)

e~ °

C

ss O=A

O=A

l I.=n

I.=n

l

Q

II

II

I.J_

o

154

Fig. 7. Guadiana Bridge Deck - angle = 0 deg. Velocity Field

Fig. 8. Guadiana Bridge Deck 'Smoke' Tunnel Flow Visualization

155

The v e l o c i t y f i e l d obtained in the v i c i n i t y of the section is shown in Figure 7, while Figure 8 shows a photograph ('smoke' tunnel, 1/200

taken during a flow v i s u a l i z a t i o n test

scale model)• There is a q u a l i t a t i v e s i m i l a r i t y between

both representations (to save space the computer drawn streamlines, which would allow a more adequate visual comparison, are not presented). Despite the very d i f f e r e n t flow conditions the pictures demonstrate the s u i t a b i l i t y of the nu medical model and the u t i l i t y

of the graphical features of DRAFTYBRIDE.

Table I I I compares the values of the a e r o s t a t i c c o e f f i c i e n t s • The o v er all v a l i d i t y of the assumptions and the results are discussed in the next section• Experimental

(Ref. 14)

This work

CD

0.655

0.744

CL

-0.085

-0.004

CM

0.130

0.168

a = 0 deg.

TABLE I I I General Remarks - Discussion of Results The foregoing results and the overall assumptions made, require the following j u s t i f y i n g comments: • the l i n e - l i k e geometry of bridge structures are s i m i l a r in nature to l o n g / i n f i n i t e cylinders; only the intermediate cross-sections are relevant to design and factors to account f o r the f i n i t e length can be derived; • the pertinence of the 2D analysis is reinforced by the f a c t that the wind d i rection i s , f o r practical purposes, horizontal and normal to the l o n g i t u d i n a l axis of the structure; • although natural flow conditions involve very high Reynolds numbers, t h e i r influence p r a c t i c a l l y vanish in view of the usual shapes of bridge cross-sections, with sharp-corners (and f o r the same reasons t h e i r roughness is i r r e l evant). This is demonstrated by the results above, obtained f o r very smooth flows and

CRe=40),

1.3×I0 s

w h i l e , in Reference I I ,

Re

is

10 s

f o r the experimental tests

f o r the numerical analysis, and in Reference 14, a

Re

of

2.2×10 s

is attained• These arguments stress the adequacy of the l a m i n a r / s u b - c r i t i c a l flow studies in the evaluation of wind loading on bridges; . the observed discrepancies in some f i g u r e s , e s p e c i a l l y regarding the very small values, can be mainly a t t r i b u t e d to the coarse mesh used. The

rezoning tech-

nique, combined with the refinement of the mesh in the v i c i n i t y of the corners (that c o n s t i t u t e s i n g u l a r i t i e s in the a n a l y t i c a l s o l u t i o n ) , as applied in References 2 and 5, would surely improve the results. Nevertheless, i t can be con cluded that even a 'rough' mesh may be acceptable f o r preliminary studies and

156

practical cases; • although the results seem to indicate that the steady-state analyses are suff i c i e n t l y precise, another improvement could c e r t a i n l y be gained by performing an unsteady or time-dependent (and time-consuming~) analysis and taking the f i n a l values of the shape c o e f f i c i e n t s as the average from the various time steps (as in References 9 and I0). The above remarks substantiate the general v a l i d i t y and u t i l i t y

of the nu-

merical model implemented• FURTHER RESEARCH The two main areas where f u r t h e r i n v e s t i g a t i o n is needed concern the develo~ ment or a p p l i c a t i o n of a compatible turbulence model and refinement of the

FE

formulations and techniques. For the l a t t e r goal, the f o l l o w i n g immediate work w i l l be carried out: selection and implementation of a penalty method scheme, most l i k e l y associat ed with the

9-node Lagrangian element. References 15 to 19 exemplify several

approaches• This formulation is equivalent to the one used in Reference lO and brings the following main advantages: e l i m i n a t i o n of the pressure as an expli~ i t v a r i a b l e , reducing the number of unknowns; the v i a b i l i t y to consider high Reynolds number, without the usual problems of convergence and accuracy; reduction of the t o t a l number of elements needed in view of the greater precision of the higher order 9-node element; • derivation and implementation of a suitable i n f i n i t e element to account f or the f a r - f i e l d boundary conditions• This w i l l allow an additional reduction in the number of unknowns f o r there w i l l be no need to extend the domain f a r upand down-stream: only a few i n f i n i t e elements w i l l do. References 20 and 21 represent outstanding contributions in this f i e l d • The above developments combined with the selection of a proper fast solver of l i n e a r systems of equations, w i l l provide in addition the groundwork f or an eff e c t i v e micro-computer version of the DRAFTYBRIDE System• An adequate simulation of turbulence constitutes a p a r a l l e l object of deeper research• References 22 to 25 summarize various possible techniques and models, Indeed, although laminar wind-tunnel tests are in general carried out f o r very known reasons (Refs.35 and 36), the e f f e c t of turbulence should not be neglected (References 27,32 to 34 and 37) i f one is to assess the overall aerodynamic/elastic behavior

of a bridge.

I t should be pointed out, however, th a t , in many cases, as far as the shape c o e f f i c i e n t s a r e concerned, the influence of turbulence is minimal (Ref.26). Reference 34 probably constitutes the ultimate answer or approach to these

157

questions. Lastly, i t should be noted that s t a t e - o f - t h e - a r t adaptive FE procedures are a must f o r an e f f e c t i v e aerodynamic CAD system. The continuing improvement of the DRAFTYBRIDE System w i l l incorporate, therefore, appropriate adaptive schemes , which Reference 28 summarizes. Other points that shall be considered in the future include the d e r i v a t i o n of parameters to take into account the l o n g i t u d i n a l e f f e c t of the structure, the simulation of t e r r a i n , water surface and the atmospheric boundary layer. CONCLUSIONS The concluding synthesis to be drawn from the foregoing examples and discussions is that the development of a s u i t a b l e numerical model to substitute, complement or support experimental analyses is a t r u l y viable and worthy prospect. The present research project corroborate s i m i l a r attempts in other areas and reveal that numerical methods can become a competitive a l t e r n a t i v e to t r a d i t i o nal experimental tests (outstanding contributions are represented byRefs9 to12). The advances of computational aerodynamics and f l u i d mechanics can only subs t a n t i a t e these remarks and contribute to the progress of wind engineering aerodynamics of structures themselves, and the development of cient tools to serve designers, researchers, educators and

and

new and e f f i aerodynamicists

a l i k e (Ref.30). The main idea of this paper can be summed up by the notion t hat , as in other f i e l d s of engineering, the FEM can become a competitive and r e l i a b l e tool

to

evaluate the behavior of structures subjected to wind loading. Numerical methods are not advocated per se but f o r being the answer to the

everyday p r a c t i t i o n e r .

Nowadays there seems to be no in-between: one can simply apply a Code-of-Pract i c e guideline or request an experimental test

from a s p e c i a l i s t or a labora-

tory. The gap can only be f i l l e d by special-purpose software, incorporating state - o f - t h e - a r t theories and techniques. The obvious analogy can be made with structural engineering. Many experimentaln~thods,ingeniousas theymay be(~g.,Ref.31),have been made obsolete by the advent of computers, matrix analysis and the FEM.

It

is expected, therefore, that by the s t a r t of the next century, when computer technology w i l l even permit the s o l u t i o n of the f u l l N-S equations (Ref.30), the wind engineering of today w i l l probably be a f a i n t memory. That's when the versus of the t i t l e

of this paper w i l l t r u l y have a meaning.

ACKNOWLEDGEMENTS The authors g r a t e f u l l y acknowledge the support of CMEST in the develoment of the project reported herein and preparation of this document, the competent typ ing job of Mrs. S i l v i a Santos and the ink drawings of Mr. Jorge Fernandes. Very special thanks are due to Eng9 J.L. Cancio Martins f o r granting

158

permission to use material from the Guadiana Bridge Project, to

Prof. J.Blessman~

for kindly commenting on our previous paper and c a l l i n g our a t t e n t i o n for problems related to turbulence,

to

Profo A.G. Davenport f o r providing us with Refer-

ences 33 and 34, and to Prof. M. Kawahara f o r References 9 to 12. REFERENCES 1 R.H. Gallagher, et a l . , e d s . , F i n i t e Elements in Fluids, v o l s . l to 6, John Wiley & Sons 2 H.P. Santo and F.B. Branco, DRAFTYBRIDE - A Computer System to Estimate the Aerodynamic Behavior of Bridge Decks, presented at the CIVIL-COMP 85 Confer ence, London, Dec.1985 3 H.P. Santo and F.B. Branco, Wind Loads on Bridge Decks by the F i n i t e Element Method, I I Simp. sobre Apl. del Met. El. Fin. en I n g . , Barcelona, June 1986. 4 H.P. Santo and F.B. Branco, AGoes Devidas ao Vento sobre Estruturas pelo M~to do dos Elementos F i n i t o s , VII Cong. Latino-Americano sobre Met. Comp. para Eng,, S~o Carlos, B r a s i l , Novembro 1986 5 D.K. Gartling and E.B. Becker, F i n i t e Element Analysis of Viscous Incompress i b l e Flow, Parts 1 & 2, Comp. Meth. App. Mech. Eng., 8, 51-60, 127-138, 1976. 6 C. Taylor and T.G. Hughes, F i n i t e Element Programming of the Navier-Stokes Equations, Pineridge Press, 1981 7 C. Cuvelier, A. Segal and A.A. Van Steenhoven, F i n i t e Element Methods and Navier-StokesEquations, Kluwer Academic Publishers, 1986. 8 H.P. Santo, A Problem-Oriented Language to Estimate Wind Actions on Structures, ClVIL-COMP 87, London, Sept. 1987. 9 H. Hirano, H. Hara and M. Kawahara, Two Step E x p l i c i t F i n i t e Element Method f o r High Reynolds Number Viscous Fluid Flow, in F i n i t e Element Flow Analysis, T. Kawai, ed.,Tokyo Press, 1982 I0 M. Kawahara, H. Hirano and T. Kodama, Two-Step E x p l i c i t F i n i t e Element Method f o r High Reynolds Number Flow Passed Through O s c i l l a t i n g Body, in F i n i t e Elements in Fluids, Vol.5,R.H. Gallagher, et a l . , e d s . , John Wiley & Sons,1984. I I M. Kawahara and H. Hirano, F i n i t e Element Analysis of Wind Force to Structures, B u l l e t i n of Chuo U n i v e r s i t y , 1985 12 M. Kawahara, H. Hirano and Y. I r i y e , A Three-dimensional Two-Step E x p l i c i t F i n i t e Element Method f o r High Reynolds Number Viscous Fluid Flow, in F i n i t e Elements in Fluids, Vol.6,R.H.Gallagher, et a l . , eds., John Wiley & Sons,1985 13 J. Schlaich and H. Scheef, Concrete Box-Girder Bridges, IABSE Document, 1982 14 ARj Borges, JCG Teles, DX Viegas, Report on Wind-tunnel Tests and Aeroelastic Studies, in P r o j . , International Bridge over the Guadiana River, J.L. Cancio Martins, Projectos de Estruturas, Lda., Lisbon, 1986 15 TJR Hughes, WK Liu and A. Brooks, F i n i t e Element Analysis of Incompressible Viscous Flows by the Penalty Function Formulation, Journal of Computational Physics, 30, 1-60, 1979 16 RL Sani, et al.,Consistent vs. Reduced Integration Penalty Methods for Incompressible Media Using Several Old and New Elements, I n t . Journ. Num. Meth. In Fluids, 2, 25-42, 1982 17 JN Reddy, On Penalty Function Methods in the Finite-Element Analysis of Flow Problems, I n t . Journ. Num. Meth. in Fluids, 2, 151-171, 1982. 18 G.F. Carey and R. Krishnan, Penalty F i n i t e Element Method for the Navier-Stokes Equations, Comp. Meth. App. Mech. and Engng., 42, 183-224, 1984. 19 PE A l l a i r e , MC Rosen and JG Rice, Simplex F i n i t e Element Analysis of Viscous Incompressible Flow with Penalty Function Formulation, F i n i t e Elem. in Anal. and Design, I , 71-78, 1985 20 P. Bettess and JA Betess, I n f i n i t e Elements f o r S t a t i c Problems, Engo Comput. I , 4-16, 1984 21LG Olson sid K-J Bathe, An I n f i n i t e Element f o r Analysis of Transient Fluid-Structure I n t e r a c t i o n s , Eng. Comput., 2, 319-329, 1985 22 C. Taylor, TG Hughes and K. Morgan, F i n i t e Element Solution of One-Equation Models of Turbulent Flow, Journ. of Comp. Physics, 29, 163-172, 1978

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