Theoretical and Numerical Modeling Problems of the Free Surface Flow of Potential Fluid Using Boundary Element Method

7/27/2019 Theoretical and Numerical Modeling Problems of the Free Surface Flow of Potential Fluid Using Boundary Element

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Theoretical and Numerical Modeling Problems of the Free

Surface Flow of Potential Fluid Using

Boundary Element Method

Marius Popa

Romanian Lloyd

SUMMARY

The flow of potential fluid with free surface is investigated in a 2D basin.

The problem is theoretically split in cinematic and dynamic problems.

The Laplace equation, cinematic problem, is solved using numeric

modeling with Boundary Element Method. The free surface time

advancing solution, dynamic problem, is realized by an Eulerian-Lagrangian method. Waves are generated by a fan wing excitant which

avoid the contradictions on boundary conditions reported for other

excitement. Results are compared with bibliographic reference and

theoretical model.

Introduction

The needs of modern design lead to thenecessity of approaching with real phenomenaby using of numeric simulating instead of thetheory what use approximations.

In this way, in last time, hydrodynamicproblems have the benefit of once of the mostmodern numeric simulating methods. In thiscase numeric simulating became a studymethod more feasible than experimentalmethod for hypothesis or results validation.In present, Numerical Modeling of the FreeSurface Flow is made on base of twofundamental hypotheses: potential fluidhypothesis and viscous fluid hypothesis.The final goal of the author is the study offluid-structure interactions, so in this paper itis use potential fluid hypothesis. This

hypothesis preserves the composition of stressinduced by fluid in structure and also bringssignificant savings of hardware resource. Inthis way study became possible with P.C.possibilities.Bibliography and also authors studiesemphasis B.E.M. as the most adequatemethod for this numeric modeling.

This paper studies the Flow of the Fluid withFree Surface without floating body and can bethe first step for fluid-structure interactionstudy.

BEM versus FEM

The same problem was tackle in a previouswork [3] but using Finite Element Methodand obtaining satisfactory results. However,in view of fluid-structure interaction study,the author concluded that B.E.M. offer bettermeans for fluid with Free Surface andeventual floating body, domain modelation,especially for cases when F.S. has strongdistorsions and/or body has great amplitudemovements. Thus, in some extreme studiedcases, it was catched the starting of

overturning phenomena, respectively wavesharpening and heightening followingbending in propagation sense. Thefundamental hypotesis of those studied wasntproper and also the grid wasnt enoughrefined for this purpose but starting of thisphenomen correlating with the increasing ofinput energy cant be catch with F.E.M. Also,


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against to F.E.M., B.E.M. uses variables mostintuitive as velocity so the resultinterpretation is more facile.Other pro-BEM arguments from body-fluiddomain modelation will be presented in a

future works dedicated of this problem.At this time the author conclusion is thatBEM is an adequate method for the study offluid with F.S. in or without presence floatingbody.

B.E.M. equation numerical modeling

Function described potential flow of fluidon D R2 or R3 domain where D is boundedby a smooth segments outline . Potentialfunction satisfy following equations on D and

his boundary : ( ) ,02 = x for xD

(ruling equation); ( ) knewxu = for x 1;knew

n=

for x 2 and 1 + 2= . The

ruling equation solving it is make in this casewith weighting residue method. Weightingfunction is the solution of equation

( ) ( )xxxw = 2 where ( )xx isDirac function and x is a fixed point

respectively ( )rw ln= for 2D,r

w1

= for 3D

where r is the distance between x and x.Thus, for x boundary point, ruling equationbecame

( ) 0'* =

+

wdn

dn

wxc

where

( )'* xc is main Cauchy values of integrals. cis a constant values depends of boundaryshape in x.This work makes the study in 2D.The author uses a B.E.M. with nodescorresponding with collocation points. and

alson

have a linear variation between

nodes. Geometricaly this element represent alinear segment from boundary. This elementis used in [2].The normal to boundary is governed by righthand rule, taking z-axis normal to 2D domainand positive counterclockwise for covers theboundary.

For segment (j, j + 1), respectivelyn

for

an intermediate point on segment are:

( ) ( )ii

i

i

iL

+= +1 ;

( )

+

=

+ jjj

j

j nnLnn

1

where is position parameter

jjj L+ (Lj is the length of the

segment). Introduct this relation in rulingequation and is obtain for 2D case:

( ) ( )( )

( ) ( )( )

+

+=

j

jj

j

j

j

j

rnL

xxc ln* 1

( )( )

( ) ( ) ( )jjj

j

j

j

darnnLn

+

+

ln1

where ddaj = . With notation

jjjL+=+ 1 equation became:

( ) ( )( )

( )( )

+

=

+

j j

j

j drnL

xxc

j

ln*1

( )( ) +

+

drnL

j j

j

j ln1

( )( )

( )( )

+

+

+

+

dr

Lndr

Lnjj

j

j

jj

j

j

lnln1

1

respectively:( ) ( ) ( ) ( )( ++= +

j

b

jj

a

jj xTxTxxc *** 1

( ) ( )

+

+

xTn

xTn

d

j

j

c

j

j

**1

Setting null Newmann condition on boundary,

0=

n

on , it is obtain banal solution or

= ct. that means ( ) ( ) ( )( ) +=j

b

j

a

j xTxTxc .

This relation gives a useful possibility tocompute c(x).For the coefficients computing a localcoordinate system it is considered for eachsegment. This system has the origin in theintersection of normal from i-point to segmentwith segment and positive direction (j,j+1).


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Second direction is the normal to segment asis also described. Those are obtainingcoordinates j , j +1 and i used inexpressions. With this coordinates the T-coefficients computing can be done without

Gauss quadrature integration.First of all

( )( )

n

r

lnmust be computed.

Because 222 +=i

r and dn=di it is

obtain( )( ) ( )( )

22222ln

2

1ln

rn

r i

i

ii

i

=

+=+

=

Finally coefficient formulae are:

+

+

+=

+++

i

j

i

j

j

j

ji

ji

j

ia

j arctgarctgLL

T

1122

21

2

ln*2

+

+=

++

i

j

i

j

j

j

ji

ji

j

ib

j arctgarctgLL

T

122

2 12ln*2

+

+=

+++

i

j

i

j

j

ij

ji

ji

j

ic

j arctgarctgLL

T

1122

21

22

ln*4

( )[ ] ( )[ ]{ }1ln1ln*4

1 22221

221 +++ ++ jijjij

jL

( ) ( ) 12212

12

21 ln

*2ln

*2 ++

+

+++++ jji

j

jj

ji

j

j

LL

+

+

+=

++

i

j

i

j

j

ij

ji

ji

j

id

j arctgarctg

LL

T

122

21

22

ln

*4

( )[ ] ( )[ ]{ }1ln1ln*4

1 22221

221 ++ ++ jijjij

jL

( ) ( ) jjij

j

ji

j

jj

LL

+++ +

+ 222

21

21 ln*2

ln*2

When point i coincide with node j or j+1,singularities are obtaining. In these cases,despite singularities, integrals have finitevalues because:- for i = 0: integrals including as parameter

( )( )22

ln

+=

i

i

n

r

are null and those which

have i as limit are computed in base of( ) 0lnlim

0=

xx

x

respectively

( ) ( )( ) ( )aaxxxdxxa

a

lnlnln0

0

== .

Thus 0=ajT ; 0=b

jT

- for i=0 and j=0,

( )[ ]{ }1ln*4

1 21

21 += ++ jj

j

c

jL

T

( ) 12

1

2 1 ln*2 +++

+ jjj

j

L ;

( )[ ]{ }1ln*4

1 21

21 = ++ jj

j

d

jL

T

and the same for i=0 and j+1=0.

Problem structure. Time advancing

Thus it is also underline in bibliography ([1],[2], [5], [6]) and in the author work [3], [4],[5], the problem of flow of fluid with freesurface, emphasize two problems: cinematic problem or pure spatial

problem consist in solving of rulingequation in D domain, respectively solvinga differential equation's system with onlyspatial variables. This problem has theadvantage that D domain has well knownfixed geometry.

dynamic problem or boundary shapeproblem consist in computing of the newgeometry of D domain boundary conditionon free surface, respectively solving adifferential equations system with time-

space variables.Cinematic Problem consist in Laplaceequation = 0 solving using Dirichletboundary conditions on free surface (=knew on 1), respectively Newmann

boundary conditions on solid boundary (n

=

knew on 2), all of this for a given time t anda fixed D domain. It is important to beemphasis that also time t and D domainrepresent components of time integrationprocess. Thus t is the following time of initial

time to and D represent one of theapproximation of final domain, correspondingto t.Dynamic Problem consists in the timeadvancement of the solution using boundaryconditions on free surface. These conditionsare: cinematic equation on free surface


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xDt

Dx

=

;

yDt

Dy

=

and dynamic

condition on free surface

( )

2

*2

+= yg

Dt

D. Time integration of

dynamic conditions offers Dirichlet conditionfor Cinematic Problem. This must beemphasis because represent for author of thework the key for this study. Confirmation forthis can be finds in bibliography ([2], [5],[6]). Cinematic Problem solving offers thevelocities on free surface that integrated withcinematic equation of free surface lead to thenew geometry of D domain.Time advancing of the solution it isconsidered solved when two successiveapproximations of D domain have difference

in accepted error range.

Numeric simulations. Results

Triangular wave in a containerThis study is already make by author usingF.E.M. in [3]. Bibliography reference forresult evaluation is [2]. In this work it is usedB.E.M. and it is find a better agree with linerspresented in [2] respectively time-elevationsline and time-errors line. For both lines the fitwith bibliography is also for values range andshapes. These results dont disabled the use ofF.E.M. but emphasis that the author use aconfirmed B.E.M. For this example all thesimulation date are adimensionalised so g=1.The container dimensions are 2.0 length and1.0 depth. The initial wave height is 0.2 andwidth 1.0. A grid with NO= 21 regulatespaced horizontal points and NV= 11 regulatespaced vertical points. Time step is 0.05.

Free surface shape at time tt= 0.0 t= 0.5 t=1.0

t=1.5 t=2.0 t=2.5

t=3.0 t=3.5 t=4.0


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t=4.5 t=5.0 t=5.5

Time-error diagram in percents from initial fluid mass

Time-elevation diagram for x= 0 (initial elevation 0) and x=1 (initial elevation 0.2)

Fan wave makerAlso this study is already make by authorusing F.E.M. in [3]. This case allowed to

present the advantages of B.E.M. using versusF.E.M.This study differs by [3] about geometry and


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excitement modality. These modifications areintroduced taking account to the experiencefrom [3] and the future objective.The basin has 20 m length and 1.0 m depth.Gravitational field is normal so g= 9.81 m/s2.

The basin length is correlate with generatedwavelength in purpose to avoid the wavereflection in downstream wall. The depth is incorrelation with another authors fluid-structure study, in order to be used asreference for those.In this case the excitement is make by a wingarticulated in his lower point, which has aharmonically movement. This kind ofexcitement theoretically offers a better fluidmass preservation as triangular flow of fluidwith harmonically amplitude, used in [3].This excitement simulates a real phenomenon

thus avoiding problems on free surface -upstream boundary intersection point reportedin [6]. Modeling of triangular flow excitementby F.E.M. is easiest than fan modeling bysame method. Using of B.E.M. make fanexcitement not only easiest but also normalfor approaching with real phenomena.The wing is fixed into articulation defined byNART point from solid left boundary(upstream). This wing has an angularmovement with rule: Angle between wing andhorizontal = /2 + angle_amplitude*

sin(2**t/T). Thus is emphasis in [3], angleamplitude, position of articulation point(which determined wing length) and timeperiod T must be correlated for avoids waveoverturning.In this work articulation point is theintersection point between bottom line and

left boundary and time period is T= 2 s. Theangle amplitude is 0.14889 rad means approx.8.5 for first example and 2*0.14889 radmeans approx. 17 for second example. Inideea of proper energy input, it is obtaining an

overturning phenomenon during first timeperiod for angle amplitude 2.45*0.14889.Grid has: 64 equal spaced points on freesurface, 40 equal spaced points on bottom and8 equal spaced points on eachupstream/downstream boundary. Time step isT/40 except where overturning is surprisedwhere dt= T/80. Acceptable error for distancefor successive position of a free surface point,during time advancing cyclical process, is1mm.For each case are presented: free surface shape at time t; time-error diagram in percents from initial

fluid mass; time-elevation diagram for a fixed x.From analysis result that: generated wave has a 2 second period as is

well because it appear as a result of apermanent 2 second harmonicallyexcitement;

wave length is close by theoretical length(respectively 5.216 m);

deviations from theoretical wavelength orperiod are gathered in time, space portions

where phenomenon is transitory.For wave generated with higher energy it canbe observed that wave tend to be sharper, sobecome trohoidal. This result is in accordancewith theory.

Waves generated by a fan wing with time period T= 2 s and an angular amplitude 0.14889 rad

Free surface shape at time tt= 0.5 s t= 1.0 s


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t= 1.5 s t= 2.0 s

t= 2.5 s t= 3.0 s

t= 4.0 s t= 5.0 s

t= 7.0 s



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Time-elevation diagram for a fixed x

x= 2.5 m x= 5.0 m

Waves generated by a fan wing with time period T= 2 s and an angular amplitude 2*0.14889 rad

Free surface shape at time t

t= 1.0 s t= 2.0 s

t= 3.0 s t= 5.0 s

t= 7.0 s


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Time-elevation diagram for a fixed x

x= 2.5 m x= 5.0 m

Conclusion

This method for wave generator seems tooffer proper results so can be used forexcitement generating in fluid-structure study.

References

[1] Berkvens, P. J., Zandbergen P. J.Nonlinear Reaction Forces on OscillatingBodies by a Time - Domain Panel Method,

Journal of Ship Research, vol. 40, 4, 1996, pp.288-302.[2] Medina Daniel E., Loggett James A.,Birchwood R.A., Torrance K.E. Aconsistent boundary element method for freesurface hydrodynamics calculations,International Journal for Numerical Methodsin Fluids, vol.12, 1991, pp. 835-897.

[3] Popa Marius, Ionas Ovidiu Theoreticaland Numerical Modeling Problems of theFree Surface Flow of Potential Fluid, TheAnnals of Dunarea de Jos University ofGalati, 1997[4] Popa Marius Theoretical, numericaland experimental modeling problems of thecomputing of loads on the ships hull, FirstPh. D. Report, Dunarea de Jos University,

Galati, 1997[5] Rusu Eugen Mecanica analitica amediilor continue cu aplicatii in tehnologiamarina, Sumary of Ph. D. Dissertation,Dunarea de Jos University, Galati, 1997.[6] Sen D. Numerical Simulation of Two-Dimensional Floating Bodies, Journal ofShip Research, vol. 37, 4, 1993, pp. 307-330.


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PROBLEME TEORETICE SI NUMERICEIN MODELAREA CURGERII CU SUPRA-FATA LIBERA A FLUIDULUI POTEN-TIAL CU AJUTORUL ELEMEN-TULUI DEFRONTIERA

In aceasta lucrare este studiata curgereafluidului potential cu suprafata libera intr-unacvatoriu 2D. Problema este impartita teoreticin subproblema cinematica si subproblemadinamica. Ecuatia Laplace reprezentindsubproblema cinematica, este rezolvata prinmodelarea numerica cu Metoda Elementuluide Frontiera. Avansarea in timp a suprafeteilibere respectiv subproblema dinamica esterealizata printr-o metoda Eulerian-Lagrangiana. Valurile sint generate de catre oaripa evantai care elimina contradictiile intre

conditiilor la limita aparute in cazul altorexcitatori. Rezultatele sint comparate cureferinte bibliografice si cu modelul teoretic.

PROBLMES THORIQUES ETNUMRIQUES DANS LE MODELAGE DELCOULEMENT LIBRE SUPERFICIEDU FLUIDE POTENTIEL LAIDE DE LAMTHODE DE LLMENT DE

FRONTIREDans cet ouvrage il sagit de ltude delcoulement du fluide potentiel libresuperficie dans un bassin 2D. Thoriquement,le problme se spare en deux: le sous-problme cinmatique et le sous-problmedynamique. Lcuation Laplace representantle sous-problme cinmatique est rsolue parle modelage numrique la mthode dellment de frontire. Lavancement entemps de la superficie libre, cest dire lesous-problme dynamique, est ralis par une

mthode Eulerian-Lagrangian. Les flots sontengendrs par une aile-ventail qui limine lescontradictions entre les conditions la limitesurgies au cas des autres excitateurs. Lesrsultats sont compars aux rfrencesbibliographiques et au modle thorique.

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Theoretical and Numerical Modeling Problems of the Free Surface Flow of Potential Fluid Using Boundary Element Method