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ADVANCES IN APPLIED MECHANICS. VOLUME 28

StabilizedFinite ElementFormulations forIncompressibleFlow ComputationstT. E. TEZDUYAR

Department of Aerospace Engineering and Mechanicsand

Minnesola Supercomputer InstituteUniversity of M innesotaMinneapolis, Minnesoto

I. lntroduct ionI I . T he C ov e r n i ngEqua t i ons

II l . The Space-TimeFormulat ion and the Galerk in/Least-Squa res tabi l izat ion.. . . . . .A . T he M ethodB. Numer ical Examples

IV . T he F o r m u l a t i ons i t h t he SU PC and PSPG S tab i l i z a t i ons . . . . . . . . . . . . .A. The MethodB. Numer ical Examples

V. Appl icat ion to Moving Boundar iesand Inter laces:The DSD,UST rocedure.. . . . . .A. The MethodB . N um er i c a lEx am p l es

V I . C onc l ud i ngR em ar k s . . . . . . . . .References

I. Introduction

I7o9t 2

l 5l )1 926ztJ293 842

Finite element computation of incompressible flows involves two mainsourcesof potential numerical instabil it iesassociatedwith the Galerkinformulation of a problem. One source s due to the presence f advectionterms in the governingequations,and can result n spurious node-to-nodeoscillations primarily in the velocity field. Such oscillations become more

rThis researchwas sponsored by NASA-Johnson SpaceCenter (under grant NAG 9-449),NSF (under grant MSM-8796352), U.S. Army (under contract DAAL03-89-C-0038), and theUniversi tvof Par is VI .

(1992) p. 1-44

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T. E. Tezduyorapparent for advection-dominated (i.e., high Reynolds number) flows andflows with sharp ayers n the solution. The other source of instability is dueto using inappropriate combinations of interpolation functions to representthe velocity and pressure fields. These instabilities usually appear asoscillations primarily in the pressure ield. In fact, there is not much abouteither of thesenumerical instabilities hat could be considered o be inherentto the finite element formulation. Such instabilities appear also in thestandard versionsof other discretization techniquessuch as finite differenceand finite volume methods.

This chapter consists of a review of the stabilized finite elementformulations designed o prevent the potential numerical instabilities justdescribed. The stabilization techniques hat are reviewed more extensivelythan others are the Galerkin,/least-squares GLS), streamline-upwind/Petrov-Galerkin (SUPG), and pressure-stabllizing/Petrov-Galerkin(PSPG) formulations. All these formulations are consistent n the sensethat, for reasons o be explained soon, an exact solution still satisfies hestabilized formulation. The descriptions of the stabilized formulationsemphasized n this chapter, and the numerical examplespresented,have al lbeen extractedfrom recent papersby Tezduyar et ql . (1990c,d, e) and Liouand Tezduyar(1990).The SUPG stabilization for incompressible lows is achievedby addirtg tothe Galerkin formulation a seriesof terms, each n the form of an integralover a different element. These ntegrals nvolve the product of the residualof the momentum equation and the advective operator acting on the testfunction. This formulation was introduced by Hughes and Brooks (1979).A comprehensivedescription of the formulation, together with variousnumerical examples, can be found in Brooks and Hughes (1982). Theimplementation of the SUPG formulation in Brooks and Hughes (1982)wasbased on QlP0 (bilinear velocity/constant pressure)elementsand one-steptime-integration of the semi-discrete equations obtained by using suchelements. The SUPG stabilization for the vorticity-stream functionformulation of incompressible flow problems, including those withmultiply-connected domains, was introduced by Tezduyar et al. (1988).

It is relevant o mention that the SUPG stabilization has beensuccessfullyapplied to not only incompressible flows, but also compressible lows. Infact, there has been alwayssome exchangeof technology between hese woapplication areas.The SUPG stabilization for hyperbolic systemsn generaland compressibleEuler equations in particular was first introduced in aNASA report by Tezduyar and Hughes (1982). This report includes a

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StobilizedFinite Element Formulationsdetailed stability and accuracy analysis and several one- and two-dimensional examples.The method was also presented n an AIAA paperby Tezduyar and Hughes (1983).The journal version of the NASA reportwas published with some additional numerical examples (Hughes andTezduyar, 1984).The stabilization techniques ntroduced in Tezduyar andHughes (1982) constituted a pilot work for compressible flows. Forexample, the Taylor-Galerkin stabilization method, which appeared n anarticle by Donea (1984) s very similar (under certain conditions identical) toone of the stabilization methods introduced in Tezduyar and Hughes(1982).Another example is the SUPG stabilization for compressible lowsin the entropy variables ormulation (Hugheset al.,1987). Among otherswith interest in SUPG stabil ization, Johnson and his group (see, e.g.,Johnson and Saranen, 1986)has been perhapsone of the most involvedones.

Becausen the SUPG stabilization the stabilizing terms added involve theresidual of the momentum equation as a factor, when an exact solution issubstituted nto the stabilized ormulation, theseadded erms vanish. and asa result the stabi lized formulation is satisfied by the exact solution in thesame way as the Galerkin formulation is satisfied. It is because of thisproperty of the SUPG stabilization (and the other stabilization approachesemphasized in this chapter) that numerical oscillations are preventedwithout introducing excessive umerical diffusion (i.e., without "over-stabilizing"), and therefore without compromising the accuracy of thesolution.

Two other stabilization techniques hat becamequite known in the pastseveralyears should be mentioned here. One is the selectivemass umpingmethod of Kawahara et al. (1982).This method has been successfullyusedparticularly in solving flow problems governed by the shallow waterequations. It can be shown that there is a close relationship between thismethod and adding isotropic numerical diffusion to the governingequations. In fact, although currently the selectiveumping parameter usedin this method is determined empirically, some theoretical guidelines indetermining this parameter can be provided basedon this relationship. Theother stabilization technique is the balanced tensor diffusivity (BTD)method of Greshoet ul. (1984).In this method, a streamlinediffusion termis added to the differential equations o compensate or the time truncationerror corresponding o the forward Euler time-integration. It was shown byGresho (1990) and Gresho and Chan (1990) hat the BTD method exhibitsthe symptoms of excessive iffusion for certain test problems, particularly

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4 T. E. TeTduyarfor the problem involving an inviscid vortex for which the initial conditionis also an exactsolution hat a reliablealgorithm is expectedo maintain asaccuratelyaspossible.Greshopredicted hat the suPG stabil izationwouldexhibit similar symptoms. It was essentially his prediction that motivatedthe work leading to the article by Tezduyar et al. (1990a).Basedon numerical experimentswith the inviscid vortex and the unsteadyflow past a cylinder at Reynolds number 100, it was shown inTezduyar etql. (1990a\ that:

(a) the SUPC stabilization for the vorticity-stream function formulationexhibitsno symptomsof excessive iffusion;(b) the SUPG stabil ization,as mplemented n Brooks and Hughes 1982)with the Q1P0 element and the one-step time-integration, doesexhibit symptomsof excessive iffusion;(c) this situation can be improved signif icantly if the one-step ime-integrationscheme s replacedby the multi-stepT6 scheme roposedin Tezduyar et ol. (1990a), and in which the SUpG stabilization isapplied only to the advectionstep; this schemeshows virtually nosymptomsof excessive iffusion.

It is the belief of this author that the symptomsof excessive iffusion isnot due to the SUPG stabil ization n general,but the combination of theSUPG stabil ization, he QlP0 element,and the one-step ime-integration.The pressurefunction spaceis too poor for the discrete formulation tobenefit from the consistency property of the SUPG stabilization. Thesituation improves significantly in the T6 formulation because he SUpGstabilization is applied only to the advection step, and that step does notinvolve any pressure erms. In fact, there s more evidence o support thisbelief. For higher-order elementssuch as Q2Pl (biquadratic velocity/linearpressure)and pQ2Pl (pseudo-quadratic ersion of Q2pl), for which thepressure unction space s richer, it was shown by Tezduyar et al. (1990b)that, for the sameset of test problems used n Tezduyar et al. (1990a), theexcessdiffusion exhibited by the one-step and T6 schemes are quitecomparable and very small. Furthermore, it was shown inTezduyar et al.(1990c) that for the stabil ized QlQl (bil inear velocity and pressure)element, which has just a richer pressure unction space han the elp0element, the excess iffusion exhibited by the one-stepand T6 schemes re,again, quite comparable and very small.It is quite well-known that, without any kind of stabilization, for reliablecomputations, appropriate combinations of interpolation functions must be

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Stabilized Finite Element Formulstions 5used o representhe velocityand pressure.Elements hat do not satisfy heBrezzicondition (Brezzi,1973), et look attractive or some eason,shouldbe handled with care. For example, he QlP0 element s one that doesnotsatisfy this condition, yet it has always been a very popular element.Nevertheless t is an element that can potentially yield unstable computa-tions. The Q2Pl and pQ2Pl elements, n the other hand, are known to beamong the quadrilateral elementssatisfying the Brezzi condition, and havebeen successfully mplemented with the SUPG stabilization (Tezduyar etal.,1990b) o be used or high Reynoldsnumber flows. Recently,Pironneauand Rappaz (1989) and Bristeau et al. (1990) showed that inappropriatecombinationsof interpolation functions can lead to numericaloscil lationsalso in some compressible low problems. Furthermore, hey showed hatcombinationssimilar to those known to be stable or incompressiblelowscan be successfully sed for compressible lows.

It was shown that (seeBrezzi and Pitkaranta, 1984,and Hughes e/ a/.,1986),with proper stabil ization,elements hat do not satisfy the Brezzicondition can be used for Stokes flow problems. The Petrov-Galerkinstabilization proposed in Hughes et al. (1986) is achieved, ust like in theSUPG stabil ization, by adding to the Galerkin formulation a seriesofintegralsover elementdomains.Again, these erms involve the residua lofthe momentum equation as a factor, and therefore the stabilizedformulation is consistent. Several researchershave been actively involvedwith stabil ization echniques or Stokes lows, and many articleson thissubject appeared n the recent literature (or about to appear soon); to givea few examples:Hughes and Franca (1987);Franca and Hughes (1988);Franca and Dutra do Carmo (1989);Douglasand Wang (1989);FrancaandStenberg 1990);and Silvester nd Kechkar (1990).

The PSPG stabilization term proposed in Tezduyar et a/ . (1990c) s ageneralization, to finite Reynolds number flows, of the Petrov-Galerkinstabil ization erm proposed n Hughes et ol. (1986) or Stokes lows. Thecoefficients n the PSPG stabilization terms vary with the Reynoldsnumber(based on a global scaling velocity) very much as the coefficients in theSUPG stabil ization erms do. In the zeroReynoldsnumber imit, the PSPGstabilization term reduces o the one proposed in Hughes et al. (1986). InTezduyar et al. (1990c), the SUPG and PSPG stabilizations are usedtogether with both one-step (T1) and multi-step (T6) time-integrationschemes.With the Tl scheme. he SUPG and PSPG stabil izationsareapplied simultaneously . As will be explained soon, another way to arrive atthis combined SUPG/PSPG stabilization is by considering the GLS

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T. E. Tezduyarstabilization for the steady-stateequations of incompressib le flows. Withthe T6 scheme,on the other hand, the SUPG stabilization is applied only tothe steps nvolving the pressure erms. Both schemeswere implemented inTezduyar et al. (1990c)basedon the QlQl and PlPl (l inear velocity andpressure)elements,and were successfullyapplied to a set of nearly standardtest problems.Also, recently, Lundgren and Mansour (1990)applied thistype of stabilization techniques o Lagrangian finite element computationof viscous ree-surfacelows.

The GLS stabilization is a more general stabilization approach thatincludes the essence f the SUPG and PSPG type stabil izations.Thisapproach has been successfully applied to Stokes flows (Hughes andFranca, 1987),compressiblelows (Hugheset al. 1989,and Shakib, 1988),and incompressible flows at finite Reynolds numbers (Hansbo andSzepessy, 990,Tezduyaret sl., 1990d,e,and Liou and Tezduyar, 1990). nthe GLS stabilization of incompressible lows, the stabilizing terms addedare obtained by minimizing the sum of the squared residual of themomentum equation integrated over each element domain. Consequently,just l ike in the SUPG and PSPG stabil izations, ecausehe stabil izing ermsinvolve the residual of the momentum equation as a factor, the stabilizedformulation is consistent.For time-dependent problems, a strict implementation of the GLSstabilization techniquenecessitatesinite elementdiscretization n both spaceand time, and therefore leads o a space-timefinite element formulation ofthe problem. The space-time finite element formulation has recently beensuccessfullyused, in conjunction with the GLS stabilization, for variousproblems with fixed spatial domains. This author is most familiar withreferences ugheset al. (1987),Hughesand Hulbert (1988),Shakib(1988),and Hansbo and Szepessy1989).The basicsof the space-time ormulation,its implementation, and the associated tability and accuracyanalysiscan befound in these references. t is important to realize that the finite elementinterpolation functions are discontinuous in time, so that the fully discreteequations are solved one space-timeslab at a time, and this makes he com-putations feasible. Still, the computational cost associatedwith the space-time finite element formulations using piecewise inear functions in time isquite heavy. For large-scaleproblems, it becomes mperative to employefficient iteration methods to reduce he cost involved. This was achieved nLiou and Tezduyar (1990) by using the generalized minimal residual(GMRES) iteration algorithm (Saad and Schultz, 1983) with rhe clusteredelement-by-element CEBE) preconditioners (Liou and Tezduyar, 1990).

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StabilizedFinite ElementFormulationsWith a slightly more liberal implementation of the GLS stabilization,

computation of time-dependent incompressible flow problems can beachievedby using the finite elementdiscretization n spaceonly, rather thanin both spaceand time. To do this, we first consider the GLS stabilizationfor the steady-state equations of incompressible flows. Then, in thedefinition of the stabilizingterms, we replace he residualof the steady-stateequationswith the time-dependentone. Thesestabilizing erms are addedtothe Galerkin formulation of the time-dependent equations. The stabilizedformulation obtained this way is, of course,sti l l consistent.Furthermore,this stabil ization s very close o the combined SUPG/PSPG stabil izationmentionedpreviously.

Perhapsone of the most strikingapplications f the stabil ized pace-timefinite element formulation is, as it was first pointed out by Tezduyaret qt. (1990d, ), n computingmoving boundariesand nterfaces. he DSD/ST (Deforming-Spatial-Domain,/Space-Time) procedure introduced byTezduyaret al . (1990d,e) serveshis purpose,and was successfully pplied toseveralunsteady ncompressible low problems nvolving moving boundariesand interfaces,suchas ree-surface lows, liquid drops, two-liquid flows, andflows with drifting cylinders. In the DSD/ST procedure, the finite elementformulation of a problem is written over ts space-timedomain, and thereforethedeformation of thespatialdomain with respecto time is aken nto accountautomatically. Furthermore, in the DSD/ST procedure the frequency ofremeshing is minimized. Here, we define remeshing as the process ofgeneratinga new mesh, and projecting the solution from the old mesh o thenewone. Since emeshing,n general, nvolvesprojectionerrorS,minimizingthe frequency of remeshing esults n minimizing the projection errors.

The outline of the rest of this chapter is as follows. In Section I, thegoverning equations of the unsteady incompressible flows are reviewed.The review of the space-timeand GLS formulations is presented n SectionIII. The SUPG and PSPG stabil izationsare reviewed n Section IV. InSection V, as an application to moving boundaries and interfaces, theDSD/ST procedure s reviewed.Sections II, IV, and V includenumericalexamples or the methods reviewed n those sections.Concluding remarksare given n SectionVI.

II . The Governing EquationsLet C),C Rn'dbe the spatial domain at t ime I e (0,7'), where n,o is the

number of space dimensions. Let Il denote the boundary of O,. We

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8 T. E. Tezduyarconsider the following velocity-pressureformulation of the Navier-stokesequations governing unsteady incompressible lows:

V ' u : 0on f) , v/ e (0,Z), (2.1)on O, vt e (0,Z), (2.2)

wherep and u are he densityand velocity,and o is the stressensorgivenaso(p, u) : -pl + 2pt(u), (2.3)with

e(u) : j lvu + (vu)tl (2.4)Here, p and p are the pressureand the dynamic viscosity, and I is theidentity tensor. The part of the boundary at which the velocity is assumedto be specified s denotedby (Il)r:

u : g on (1,)gv/ e (0,?.). (2.s)The"natural" boundary onditionsssociatedith (2.1)are heconditionson the stress omponents, nd theseare the conditions ssumedo beimposed t the remaining art of theboundary:

n . o : h o n 1 , ) r , v t e ( 0 , ? . ) . (2.6)The homogeneous ersion of (2.6), which corresponds o the "traction-free" (i.e., zeronormal and shearstress) onditions, s often imposedat theout flow boundaries. As initial condition, a divergence-freevelocity fielduo(x) is specified over the domain f), at I : 0:

u(x, 0) : uo(x) on Qo. (2.7)Let us now consider two immiscible fluids, A and B, occupying thedomain C),. Let (Qr)o denote the subdomain occupied by fluid A, and (Il)odenote he boundary of this subdomain.Similarly, et (er)s and (Il)u be thesubdomain and boundary associatedwith fluid B. Furthermore, let (Il)oube the intersectionof (1,)a and (Il)u, i.e., the interfacebetween luids Aand B.The kinematical conditions at the interface (|,)es are based on thecontinuity of the velocity field. The dynamical conditions at the interface,for two-dimensional problems, can be expressed y the following equation:

t re 'oe * ns . 68 : n^y / R^ on ( | , )en Vt e (0 , 7 ) , (2 . 8 )

/ 0 u \, ( " * u ' v u / - v ' o : 0

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Stabilized Finite Element Formulations 9where no and n" are the unit outward normal vectors at the interface, oAand os are the stress ensors,y is the surface ension coefficient,and Ro isthe radius of curvature defined to be positive when no points towards thecenterof curvature.The condition (2.8) s formally applicablealso for free-surface flows (i.e., when the second luid does not exist), provided thatsubdomain(Q,)o is the one assigned o be occupiedby the fluid.

III. The Space-Time Formulation and the Galerkin/Least-SquaresStabilization

A. Tsr MprnorLet us first assume hat the spatial domain is fixed in time. Under this

assumption, he subscript is dropped from the symbolsC),and f,. In thespace-time finite element ormulation, the time interval (0, I) is partitionedinto subintervalsn : ( tn, t , * , ) , where tnand /n*r belong o an orderedseries f t ime levels0 : /o < /r < ... ( l,v : ?".The space-timeslab Qn sdefined as the space-time domain e) x In. The lateral surface of Q, isdenoted by P,; this is the sur facedescribedby the boundary f, as I traverses1n. Similar to the way it was represented y Eqs. (2.5) and (2.6), P, isdecomposed into (P,)s and (P,)n with respect to the type of boundarycondition being imposed.

Finite element discretization of a space-time slab Q, is achieved bydiv id ing t into elements "" ,e : 1,2, . . . , (nrr )n,where n. )n s the numberof elements n the space-time slab Qn. Associatedwith this discretization,for each space-time slab we define the following finite element interpola-tion function spaces or the velocity and pressure:

(sl), : lon un e lHth(e)]',a, uh gh on (PJ*], (3 .1 )V5, : {,on ttn e lH'n(Q)l""ir, wh = 0 on (P,)r}, (3.2)(s : )" : v | ) " : lqnlqn e H'h(Q)] . (3.3)

Here Hrh(Qn) represents the finite-dimensional function space over thespace-time slab Q". This space is formed by using, over the parent(element) domains, first-order polynomials in space and time. It is alsopossible to use zeroth-order polynomials in time. In either case,globally,the interpolation functions are continuous in space but discontinuous intime.

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l0 T. E. TezduyarThe space-time ormulation of (2.1)-(2.7)canbe written as ollows: Startwith

(uh;; : (uo;h; (3.4)s equen t i a l l yo r Qr ,Qr , . . . , eN_r , g i v en (uh ) , , f i nd uh e (S l ) , andph e 1sj),, such hat vwh e (zuh)n nd vqh e 1vj),,

[ * ' t ( + * u n . v u n ) d O + \ c ( w h ) : o ( p n , u h )O - \ w h . h d p,\e, \ dI / . lO, J t&r,i r '* )n.n v . uhdQ + .lo(wh);.((uh); 1uh;;; o

* ' f l "I Jr(+ * un.v,on)v. . r" = r J e i t \ d / / " ( q n ' n n ) )l / a r n \ II r ( ; + u h ' v u h / v ' o ( p h . " n t ) a e : . ( 3 . s y

In the variational formulation givenby (3.5), the following notation is beingused:(3.6)

(3 .7)

(3 .8)

(un)ot : ]in1un{r" * d),I r f. ln . t " ' rn ) , . ,1nr" .redt.l r rI t r d P : \ l r . - t a r a , .. r Pa . t r , J r

Remarksl If we were in a standard finite element formulation, rather than aspace-time one, the Galerkin formulation of (2.1)-(2.7) would haveconsisted of the first four integrals (their spatial versions of course)appearing in Eq. (3.5). In the space-time formulation, because theinterpolation functions are discontinuous in time, the fifth integral in Eq.(3.5)enforces,weakly, the continuity of the velocity in time. The remainingseriesof integrals in Eq. (3.5) are the least-squares erms added to theGalerkin variational formulation to assurethe numerical stability of thecomputations. The coefficient r determines he weight of suchadded terms.

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Stabilized Finite Element Formulations l12. This kind of stabilization of the Galerkin formulation is referred to

as the Galerkin,/least-squaresGLS) procedure, and can be consideredas ageneralizationof the stabilization based on the streamline-upwind/Petrov-Galerkin (SUPG) procedure employed for incompressible lows. lt is withsuch stabilization proceduresthat it is possible to use elements hat haveequal-order interpolation functions for velocity and pressure,and that areotherwiseunstable.

3. It is important to realize hat the stabilizing terms added involve themomentum equation as a factor. Therefore, despite hese additional terms,an exact solution is still admissible o the variational formulation given byEq. (3 . 5 )

The coefficient ? used in this formulation is obtained by a simple multi-dimensional generalizationof the optimal z given in Shakib (1988) or one-dimensional space-time formulation. The expression or the r used in thisformulation is

| / 2 \ ' /2 l lun l l \ ' /4u \ ' .1 - ' "1 : l r - l + r - " - " l * ( ; : ) | ( 3 . 9 ) [ \ n t / \ h / \ , , / jwherev is the kinematic viscosity, and Ar and h are he temporal and spatial"elernent lengths." For steady-statecomputations, a different definitionfor r is used:

(3 . 0 )For derivation of r for higher-order elements,seeFranca et al. (1990).

Remqrk4. Becausehe finite element nterpolation functions are discontinuous n

time, the fully discreteequationscan be solvedone space-timeslab at a time.Still, the memory needed or the global matrices nvolved in this method isquite substantial. For example, n two dimensions, the memory needed orspace-time formulation (with interpolation functions that are piecewiselinear in time) of a problem is approximately four times greater comparedwith using the finite element method only for spatial discretization.However, iteration methods can be employed o substantiallyreduce he costinvolved in solving the linear equation systemsarising from the space-timefinite elementdiscretization. It was shown in Liou and Tezduyar (1990) hat

':[(#)'.#)']-'"

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I2 T. E. Tezduyarthe clustered element-by-element (CEBE) preconditioners (Liou andTezduyar, 1990), ogether with the generalizedminimal residual (GMRES)method (Saadand schultz, 1983)can be effectivelyused for this purpose.

B. NuuEnrcar ExaMprssIn this section, numerical examples are presentedfrom the space-timefinite element omputationsbasedon the cEBE/GMRES iterationmethod.The interpolation functions used for velocity and pressureare piecewisebilinear in spaceand piecewise inear in time. Thesecomputations involve

no global coefficient matrices, and therefore need substantially lesscom-puter memory and time compared to non-iterative solution of the fullydiscreteequations seeRemark 4). By using very large ime stepsizes e.g.,100,000)he steady-stateolutionsare obtained n a few time steps.For thedescriptionof the terationmethod, ts performancecharacteristics,nd thedetailsof the numericalexamples, eeLiou and Tezduyar (1990).The lid-driven cavity flow ot Reynolds number 1000

In this problem, the cavity has a squareshape, and the Reynolds numberis basedon the sizeof the cavityand the velocityof the id. A uniform meshwith 64 x 64 elements and 4225 nodes is employed. Every time step,approximately 25,000equations are solvedsimultaneously. Figure I shows,for the steady-statesolution, velocity components along the vertical andhorizontal centerlines,pressure,vorticity, and stream function."Steady-state" solution for flow past a cylinder at Reynolds number 100

In this test problem, the dimensions of the computational domain,normalized by the cylinder diameter, are 30.5 and 16.0 in the flow andcross-flow directions, respectively.The free-stream velocity is 0.125.Reynolds number is basedon the free-streamvelocity and the diameter ofthe cylinder. symmetry conditions are imposed at the upper and lowercomputational boundaries,and the traction-free condition is imposed at theoutflow boundary. A meshwith 5400elementsand 5510 nodes s employed.Every time step approximately 33,000equations are solved simultaneously.Figure 2 shows, for the "steady-state" solution, pressure,vorticity, streamfunction and stationary stream function.

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Stubilized Finite Element Formulotionsu ( a t x - O . S ) v ( a t y : 0 . 5 )

4 . 4 4 . 2 0 . 0 c . 2 o . 4 0 . 8 0 . 0 4 . 2 0 . 4 1 . 0

l 3

{

L

1III

I

a:

I

ciI,,']I

o

I

' \ ) / / \ \ \^N.// \'

: (- . . )

' \ \/ /

\ /

0 . 6 1 . 0 0 . Epressure

vort ic i ty stream funct ion

Frc. 1. Steady-statesolution for the lid-driven cavity flow at Reynolds number 1000:velocity components along the vertical and horizontal center l ines, pressure, vorticity, andstream function (Liou and Tezduyar, 1990).

' I ' - + *

I

t /f= il

@

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I 4 T. E. Tezduyor

vort ic i ty

Ftc. 2. "Steady-state" solution for flow past a cylinder at Reynolds number 100:pressure,vorticity, stream function, and stationary stream function (Liou and Tezduyar, 1990).

pressure pressure

vort ic i ty

stream funct ion s t r eam func t i on

s ta t i ona r V t r eam f unc t i on stat ionarv stream funct ion

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Stabilized inite ElementFormulotions 15IY. The Formulationswith the SUPG and PSPGStabilizations

A. TnB MrrnorIn this section, the variational formulations with the SUPG and PSPG

stabilization terms are described. These formulations are based on finiteelementdiscretizationn spaceonly, rather than in both spaceand time.

Let us discretize he domain O by subdividing it into elements O",e : 7,2, . . . , f ret ,where z . , s the numberof e lements . ssoc iated i th th isdiscretization,we define he following finite element nterpolation unctionspaces or the velocity and pressure:

s,l : fun uh e 1,a1h1elr1n,o,h= gh on lsl,Zoh fwh wh e 1111h1fl)1n"o,h= 0 on lrl,sI : v; : lqnlqhe r lrhlo; ; ,

where 11'n(f,)) represents the finite-dimensional function space over thespatialdomain O. This spaces formed by using,over the elementdomains,first-order polynomials in space. The stabilized Galerkin formulation of(2.1)-(2.7)can be written as follows: Find uh e S| andpn e Sj'such that,vwh Zuhand vqh e V!,

(4 .1 )(4.2)(4.3)

(4.4)As it can be seen rom Eq. (4.4), wo stabil izing ermshave been added othe standard Galerkin formulation of (2.1)-(2.7); the one with 6h is theSUPG term, and the one with eh s the PSPG (pressure-stabil izing/Petrov-Galerkin) term. The Petrov-Galerkin functions 5h and ehare defined as

* un vun) ao + J,'c(wh): (ph, h) ow h . h d r + J n a n v

# * " . v") - v. o(pn,u' l ]o o.

6h : Ts upc h ' v wh ." l, , TpspcpYq, , ,

[,*'' '(q#-J",[ , (. uh f, *

"!, Jr"(6h+ eh)

(4.5)(4.6)

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t 6where

T. E. Tezduyar

rsupcffi.G.,;,?pspcffi ze.fll.

Here, Reuand Re{' are the elementReynolds numbers, which are based,respectively, n the ocal velocityuh and a global scaling elocityU. That is,

(4.7)

(4.8)

(4.e)(4 .10)

l lunl lrR e " : t ; '

- l lu l l r#R e i : ; .The "element length" /t is computedby using the expression

/ ' r n \ - lh : 2 l I l t ' v r u , l\ a = 1 /

z ( R e ) : f n e z l ' o c R e < 3 '( l , 3 < R e .

( 4 . 1 l )where r.n is the number of nodes n the element,No is the basis unctionassociatedwith node a, and s is the unit vector in the direction of the localvelocity.The "element length" hn. on the other hand, is defined o be equalto the diameter of the circle which is area-equivalento the element.Thefunction z(Re) used in Eqs. (4.'7) and (4.8) is defined as

(4.r2)

The spatial discretizationof Eq. (4.4) leads o the following set of non-linear ordinary differential equations.( M + M u ) a + N ( v )+ N 6 ( v ) ( K + K u ) t - ( G + G u ) p : F + F 6 ) , ( 4 . 1 3 )G r v + M . a + N . ( v ) + K . v * G . p : E + E . , ( 4 . 1 4 )wherev is the vector of unknown nodal values of uh, a is the time derivativeof v, and p is the vector of nodal valuesof ph. The matricesM, N, K andG are derived, respectively, rom the time-dependent, dvective,viscous,and pressure erms. The vector F is due to the boundary conditions (2.5)and (2.6) (i.e., the g and ft terms), whereas he vector E is due to theboundary condition (2.5). The subscripts6 and e identify the SUPG andPSPC contributions, respectively.

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Stabilized Finite Element FormulationsFirst, consider he time-integrationof Eqs. (4.13)and (a.14)by a one-stepgene ra l i z edrapez o ida lu le : . e . .g i v en uh ) , , i nd (uh )n * ,and (pn )n_ , rh i s

will be referred to as Tl formulation). When written in an incrementalform, the Tl formulation leads o

t 7

where

(4 .15)(4 .16)

(4.r7)

(4.1e)(4.20)(4.2r)

M * A a - G * A P : 1 1 ,(Gr)*aa + G. Ap : Q,

R : F * Fo - [(M + Mu)a+ N(v) + N6(v)+ (K + Ku)v (G + Gu)pl,Q : E * E. - [G 'v + M.a + N. (v )+ K.v + G.p ] , (4 .18)

M* : M + Mu+ "or (H. * + K + K6) ,G * : G * G o ,

M , + o ^ ' ( * + K , + G r ) .The parameter cvcontrols the stability and accuracy of the time integrationalgorithm.Remark

5. The systems 4.15)and (4.16)can be solvedby treating the velocityexplicit ly n the momentum equation. Since he SUPG and PSPG supple-ments are applied to all terms in the momentum equation, in explicitcomputations the coefficient matrix of the pressureequation is generallynot symmetric.All explicit Tl computations reported in this section arebased on the symmetrization of the coefficient matrix of the pressureequation,and the resultsare obtainedwith two passes er time step. n suchcomputations,M*, G* and (Gr)* are replacedwith

(Gt)* :

M * : M r ,G * : G ,

( G t ) * : . ' A / G r ,where M. is the lumped version of the mass matrix M.

(4.22)(4.23)(4.24)

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1 8 T. E. TezduyarTo write he T6 formulation Tezduyar t al.,1990a)of (2.1)-(2.7), eneed o slightlymodify the definitionof the solutionspace or velocity:

(S,l),*" lun uh e [r11h1f2r1n,o,h Gn)n*"on I"]; (4.2s)the definitionsof the other function spacesemainas they weregivenbyEqs.(a.2)and (4.3).We now summarizehe T6 formulation.

l. Find (uh)f*,e (S,l),*a uch hat, vwh Zuh,i , "n r ( tuh) , *_e (uh) ,+ (uh) - v run t - )ao. l a ' \ a \ r " /

- i [ , r . , ( (un) ' - t :tun l '

+ (un)\

(4 .26 \e | ' r o " 1 d A l ' ' v ( u ' \ ^ ) d a : 0 '2. Find (uh)n*u (S,l),*oand (ph)n*u Sj such hat, vwhe Zuh ndvqhe V],[ * r . p [ ( uh ) , - q- ( uh t ; - u ] r i o+ | e ( wh ) :6 h ) n _ 0 d e. l o 0 L , t . l n

i { '- I w h . ( f t h ) n * o a r +| q h v . 1 u h \ n * u d e.l r" .l oilet r' ( pl,rUh)n*a - (on);* ul )+ | I . n . - v . ( o h ) n * o {a e i : o . ( 4 . 2 7 1" = r . j " ( d A l )

3. Find (uh);+1 o e (S,l)n*r-osuch that, vwh e Zuh,

[ * r . p [ (uh ) r_ * r d__ -!uh) , *d ] rn * [ e (wh) : 1oh ; n * rdo. J n ' 0 - 2 0 ) L t . l oI- I ton. f tn)n*, f : o. (4.28)J f r

4. Find (uh)n*,-o e (S,l) ,*r-e such that, vwh e Zoh,i , f { ' u h ) n * r , - ( u n ) n - * ' - d h . \

Jnn ,\-"i=r;f"- + (uh),*,_ev(uh),*_u)aa

, e r [ ' / t u , h ) n+ r _o - ( on ) i ' * r _ , , h . _ , h . \ . ^* " 1 , l r "0 ' p ( t . r - z B l, + ( u h ) ' * ' - ' '( u h ) n *- u ) o: 0. (4.29)

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Stabilized Finite Element Formulations5. Find (uh);*, e (S,l),*r such hat, vwh Zoh,

t 9

, \ , ' ' ' (@#i't * (uh)n+,'vlun;r* , -u)ao* (uh)o+r-0.1uny,* , -u)e iun) i* r - (uh)o+r-o0 L t

6. Find (uh),*, e (S,l),*, and (ph)n*, e Sj such that,vqh e V!,

n e t i /I I o ' ' p (e - l . l 0 e \0. (4.30)

vwh e Zuh and

[ * '. r op [ (uh ) , -r - (uh ) ; * r ] do0 L r + {n etron)1oh;n*,oiI.l fr

n e lrwh (fth)n*,a * J, qhv . 1uh;n*,o[ . ' I1 e ( v . ton) , . , ]ao :[(un),*r (un);*rl 0. (4.31)0 L t

Remarks6. The parameter 0 is the one used in the d-scheme Bristeauet ol.,

1987); or the numericalexamples o be reported n this section, t is set o ].7 The matrix forms corresponding to Eqs. (4.26), (4.28), (4.29), and(4.30) can be solved implicitly or explicitly as described nTezduyar et al.(1990a).The matrix form of the two "Stokes substeps," .e., Eqs. (4.27)and (4.31 , are quite similar to the matrix form of the T I formulation; theycan be solved implicitly or by treating the velocity explicitly. The resultsreported in this section are based on the explicit treatment of all substeps.The numbers of passesused in the substepsare 4-2-2-2-4-2.

B. Nuunnrcar ExeuprEsTo have a better basis of comparison among the solutions obtained by

using different elements, meshes generated with different elements arerequired to have the same distribution of the velocity and pressurenodes.The nodal values of the stream function and vorticity are obtained by theleast-squares nterpolation. For the meshes generated with the PlPl

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20 T. E. Tezduyqrelements, hesequantities are computed from the velocity field by using themeshes eneratedwith the QlQl element.For detailsof the computationsand the performance characteristics,seeTezduyar et al . (1990c).

Unsteady low post a cylinder at Reynolds number 100The problemset-up n this cases the sameas t was or the "steady-state"caseof Section II. However, this time we are interested n the unsteadybehavior.The meshused or Q I Q 1consists f 5240elements,while he num-ber of elements or P I P 1 s 10,480.Both meshes ontain5350velocitynodes.The periodicsolution s computedby introducinga short-termperturbationto the symmetricsolution. We have observed,at least or small perturba-tions, that theperiodicsolution s ndependent f the mode of perturbation.Strouhal number and the time history of the lift and drag coefficientsare shown in Figs. 3 and 4. Compared to the Tl formulation, the T6o ' lo 1 / T 1 o 1 0 1 / T 6

4

aoo.o

P 1 P 1 / T 1 P 1 P 1 / T 6

Ftc. 3. Periodic olution obtainedwith various ormulations)or flow pasta cylinderatReynolds umber100:Strouhalnumberand he imehistoryof the ift coefficientTezduyaret a|. .1990c\.

n i ti l

! ii l l I I1 l L tL.0.16t t

l lir i i : lL, i llj i/i i1firl

I I I l ti l 1J t l- 1 v l If i,0.170l J+l]{+ljl f/ \ '

,lI

+]]I tII Irl]llItI fII Ir I fIul l l 1 I

ti \j)r=u. rbo

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o 1 0 1 / T 1

I /i tI.,--a

Stabilized Finite Element Formulutions

P1 P , 1T 1Il ",iffi \A

t I500.0 too.o 1000,0 1200.0 0.0 400,0 !000.0 1200.0

Frc. 4. Periodic solution (obtained with various formulations) for flow past a cylinder atReynolds number 100: time history of the drag coefficient (Tezduyar et al., 1990c).

formulation gives a slightly higher Strouhal number. Also, the Q1Q1element gives a Strouhal number abouL 2Vo higher than what the PlPlelement gives. Although the lift and drag coefficients show no significantdifferenceamong different formulations, the QlQl elementgivesa slightlyhigherdrag coefficient han the PlPl element,and the T6 formulation givesa slightly higher drag coefficient than the Tl formulation.

The periodic solution flow patterns correspondingto the crest value ofthe lift coefficient are shown in Figs. 5-8. The patternscorresponding o thetrough value of the lift coefficient are simply the mirror images, withrespect o the horizontalcenterline,of the patternsshown n Figs. 5-8. Thesolutions obtained with different formulations are very similar. However, itcan be seen, upon close comparison, that the T6 formulation is lessdissipative han the Tl formulation and that the QlQl element s lessdissipative han the PlPl element.On comparing thesesolutionswith the

21o 1 0 1 / T 6

- a

P1 P1 I T 6

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22 T. E. Tezduyar

vort ic i ty

@4OOoovort ic i ty

stream funct ion

stat lonary stream f unct ion stat ionarv stream funct ion

Frc. 5. Periodic solution (obtained with QlQl/Tl) for flow past a cylinder at Reynoldsnumber 100: flow patterns corresponding to the crest value of the lift coefficient (Tezduyaret o1..1990c\.

stream funct ion

pressure pressure

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Stabilized Finite Elemen t Formulations Z J

vortrcr ty

@l,Qo0o0vort ic i ty

stream funct ion s t r eam func t i on

s ta t i ona r ys t r eam f unc t i on stat ionary stream f unct ion

Frc. 6. Periodic solution (obtained with QlQl/T6) for flow past a cylinder at Reynoldsnumber 100: flow patterns corresponding to the crest value of the l ift coefficient (Tezduyaret al . .1990c\.

1 \\\V' \I \ \ \ ' -\ \--\.....-:-

pressure

\Ao^o'^cv U ( \ u - t )

pressure

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z+ T. E. Tezduyar

vort ic i ty

@pOOoov o r t i c i t y

s t r eam unc t i on s t r eam func t i on

stat ionary stream f unct ion

pressue

Frc. 7. Periodic solution (obtained with P1P1/Tl) for flow past a cylinder at Reynoldsnumber 100: flow patterns corresponding to the crest value of the lift coefficient (Tezduyaret al . .1990c\.

s t a t i ona r ys t r eam f unc t i on

pressure

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Stqbilized Finite Element Formulations 25

vort ic i ty

@@0O0oovort ic i ty

s t r eam func t i on stream funct ion

Frc.8. Periodic solution (obtained with PlPl/T6) for flow past a cylinder at Reynoldsnumber 100: flow patterns corresponding to the crest value of the l ift coefficient (Tezduyaret a| . .1990c) .

s ta t i ona r ys t r eam func t i on

\ \\\-/ v/ \r \ 5 - /Y

stat ionary stream f unct ion

pressure

0'6*c

pressure

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26 T. E. Tezduyarones reported in Tezduyar et al. (1990b), it can be observed hat thesolutionsobtainedwith the QlQl and PlPl elements re very close o theones obtainedwith the pQ2P1 and QlP0/T6 elements.

V. Application to Moving Boundaries and Interfaces:The DSD/ST Procedure

A. Tns MrrHoo

It was first shown in Tezduyar et al. (1990d,e) that the stabilizedspace-time finite element formulation described in Section III can beeffectively applied to fluid dynamics computations involving movingboundaries and interfaces. The variational formulation associatedwith theDSD/ST (Deforming-Spatial-Domain/Space-Time) procedure is onlyslightly different than the one givenby Eq. (3.5) n Section II. Becausehespatial domains are now time-dependent, the subscript I that was droppedfrom the symbols such asf), and Il needs o be reinstated.Furthermore, welet On : O,, and l, : 1,,, and define the space-time slabQn as the domainenclosed y the surfacesOn, On*1 andPn(seeFig. 9). The v ariational for-mulation replacing he onegivenby Eq. (3.5)can hen bewritten as ollows:

Jn,*n r(# * on.von) O In.e(wh)o1ph,nh1g- J*" whhdP J,*,^" h noy/RodP

.i-J,,'[,(#."[ , ( # * u n ' v u n )- v

+ | qh v . uh dO + \ (ton);. ((uh); - (uh);) aetJ Q " J O "or " t ) - v .o tqn ,wn) ]

. o(ph,onl]aO : o, (5.r)where (P)as is the space-timesurface describedby the boundary (1,)as as/ traverses he time interval (t,, tr+r).

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fn+1SrabilizedFin re Elemen Formulat ons

t = t n + 1

mL(Vr , V2,@,U)

27

tlt/:.= t nf.

Frc. 9. The space-time slab for the DSD/ST formulation (Tezduyar et al.,1990d).

Remarks8. The kinematical conditions at the interface (Il)ou are automatically

satisfiedbecausehe discretized ubdomains O,)o and (O,)u share he samenodesat th is nter face.

9. The additional term (i.e., the fourth integral) n Eq. (5.1) enforcesthe dynamical conditions associatedwith the interfaces and free-surfaces nthe presence f surface ensioneffects. f the interface s to be interpretedas the free-surfaceof a single fluid, then the fluid is assumed o occupy sub-domain (Q,)o. This variational formulation can of course be easilyextended to more than two fluids.

10. For two-liquid flows, the solution and variational function spacesfor pressure hould nclude he functions that are discontinuousacross heinterface.

As a special caseof drifting solid objects, let us now consider a driftingcylinder. The cylinder moves with unknown linear velocity components Z1and V, and angular velocity @. The temporal evolutions of theseadditionalunknowns depend on the flow field and can be described by writing theNewton's law for the cylinder:

D(4, Vz ,@,U)4dtdV,dt

(s.2)(5 .3 )

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28 T. E. TezduyardO_ T(n,vz ,6 ,u)d t J (s.4)

whereD, L , and 7"are the drag, lift and torque on the cylinder, respectively,while m and J are ts massand polar moment of inertia. The vector of nodalvaluesof velocity and pressure s denoted by u. Temporal discretization ofEqs. (5.2)-(5.4) eads o a set of equationswhich, in an abstract orm. canbe written as V - V - = A r D ( V - ,V , U ) . (5 .5)Here, V (unknown) and V (known) are vectors epresentinghe motion ofthe cylinder, respectively, nside the current space-time slab and at the endof the previousone. The current slab hickness r+t - t, is L,t. For l inear-in-time interpolation, Eq. (5.5) takes he form

: A t (5.6)

Based n thegeneral xpression5.5),wecanwrite he ncrementalorm of(5.6)as - ^, (*) ou* f r - a, *)l AV R"u, ). (s.7)\ a u l L - \ d v , z rEquation (5.7) is of course coupled with the incremental form of thediscreteequationsystem esulting rom (5.1):

(Mfiu)AU + (Milv)AV : Ru(U,V). (s.8)In computationseportedn thissection,hesystem5.7)-(5.8)s solved y

(V ) " * t

(V ) , * t

(@),*r

(V,)T(v);(ox

(4);(v),(@);

(n;(v)"(@);

+ Dr * )

+ L " * t )

+ T " * t )- D" * t )

- L i * t )

- T r * t )

I rlnt";t l| * t t ;I 'I u-Q;It ll*r";t 1l*r ' ;t l| 6 r Q ;

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Stabilized Finite Element Formulstions 29a block iteration scheme n which the term (AD/AV) is neglected.Duringeach teration, Eq. (5.8) s solved or AU only, using he valueof V from theprevious teration; and then V is updatedby (5.7)while U is held constant.However, he full system an, n principle,be solvedsimultaneouslyo takeadvantageof larger time stepsafforded by a fully implicit method. Iteratingon the solutionwill sti l l be needednot only because f the nonlinear natureof (2.1) but also because f the dependence f the elementdomains Qi onthe vector V.Remark

11. In the DSD/ST procedure, o facil itate he motion of free-surfaces,interfaces, nd solid boundaries,we need o move the boundary nodeswiththe normal componentof the velocityat thosenodes.Except or this restric-t ion, we have he freedom o move all the nodesany way we would l ike to.With this freedom,we can move the mesh n such a way that we only needto remesh when it becomesnecessary o do so to prevent unacceptabledegrees f meshdistortion and potential entanglements. y minimizing thefrequencyof remeshing,we minimize the projection errors expected o beintroducedby remeshing. n fact, for some computations,as a by-productof moving the mesh, we may be able to get a limited degreeof automaticmesh refinement, again with minimal projection errors. For example, amesh moving schemesuitable or a single cylinder drift ing in a boundedflow domain is described n Tezduyar et sl. (1990e).

B. Nuurnrcnr ExeuprBsAl l solutions presented n this sectionwere obtained with linear-in-time

interpolation functions. For the details of the computations, seeTezduyaret al. (1990e1.Free-surface wave propogot ion

This is a problem that was considered n Hughes et ol. (1981). Initially,the fluid is stationary, and occupiesa long rectangular region with dimen-sionsL x D, where L : 949.095and D: 10. The flow is assumed o beinviscid, and both the densityand the gravity are set o 1.0. The meshcon-sists of 320 elements,with two elements hrough depth. The wave isgeneratedby prescribing the velocity along the left-hand boundary of the

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30 T. E. Tezduvardomain according to the expressiona, : (Hc/D) sech2(crct/D 4), wherec : IS(D + H)lt" and rc : (3H/4D)t/2, with 8 : I and 11 : 0.86. Thetime step size s 1.789.Figure l0 shows he solutionsobtained at varioustime steps . After 160 time steps, the wave retains 94.4V0of its init ialamplitude. This solution compares well with those presented n Hughes ela/ . (1981).

_/\-

\

0< . 1' 60)3 oo . 1rhq)c' 69 nc -

80.o 100.ox/DFrc. 10. Free-sur facewavepropaga t ion:t imehistoryof hesurfacewave(Tezdrryaretal . ,

I 990e).

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Stabilized Finite Element Formulations

3 0 . 1 3 1 . 0 2 4 , A . a 1 2 0 . 0 4 : l i l u ' 1 8 0 . : l

Frc. 11. Pulsating drop: time history of the axial dimensionsof the drop (Tezduyar et al.,I 990e).

Pulsoting dropIn this problem, the drop is initially of elliptical shapewith axial dimen-

sions 1.25 (horizontal)and 0.80 (vertical).The density,viscosity,and thesurface ensioncoefficientare l0, 0 .001and 0.001, espectively. he effectof gravity s neglected. he number of elementss 380,and the time stepsizeis 1.0. Figure 1l shows he time history of the axial dimensions f the drop.Figures l2a, l2b, 12c, and l2d show the flow field and finite elementmeshcorresponding,approximately, wo points a, b, c, and d in Fig. 11.Large-amp I t ude s osh ng

This problem is similar to the one that was considered n Huerta and Liu(1988). Init ially, the fluid is stationary and occupies a 2.66'7 1.0rectangular region. The density and viscosity are 1.0 and 0.002,respectively.The gravity is 1.0, and the surface tension is neglected. Thewave is created by applying a horizontal body force of ,4 sin(rr.rl),whereA : 0.01 and a : 0.978.The Reynoldsnumber (basedon the heightof thefluid and the gravity) is 514. Inviscid boundary conditions are assumedatthe walls of the "tank." Compared to the problem consideredhere, the

3 1

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3 Z T. E. Tezduyarstream funct ion pressure

OFrc. 12. (a ) Pulsating drop: flow field and(approximately) to point a in Fig. 1l (Tezdtryar et al.,

s tream funct ion

veloci ty

Frc. 12. (b) Pulsaringdrop:(approximately)o point b in Fig. l1

mesh

flow field and finite element mesh(Tezduyar et al., 1990e).

finite element meshI 990e).

pressure

corresponding

corresponding

veloci ty @

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Stabilized Finite Element Formulationsstreamunction oressure

J J

4NN@mesh

Frc. 12. (c ) Pulsating drop: flow field and finite element mesh corresponding(approximately) o point c in Fig. l1 (Tezduyaret al . ,1990e).

stream funct ion Dressure

veloci ty

veloci ty

/-)L\\'7mesn

Frc. 12. (d ) Pulsating drop: flow field and finite element mesh corresponding(approximately) to point d in Fig. 1l (Tezduyar et al., 1990e).

O

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. AJ + T. E. Telduyar

0 . c 1 6 . 0 3 2 . 0 4 8 . 0 6 4 . C 8 0 . 0 9 6 . 0 11 2 . 0 2 8 . 0 1 4 4 . 0 6 J . 0 7 6 . 0 9 2 . 0t

c?r^-r,c J cI, O0 qo] -

NU)! ) oCJc ) ca -! Nc-), Ac - J q

c

oc

J ^ot t:( J c{ cO no] -Naa ac)) cO ca , ^O

o ' .o1 u 4 . 0 8 6 . 0 i u B . c 1 9 0 . r J : 9 2 . C

Frc. 13. Large-ampl i tude loshing: imestationary level) of the free-surface along(Tezduyar et al., 1990e).

1 9 4 . 0 1 9 6 . 0 L 9 B . i J 2 J U . 0 2 A 2 . : J 2 C 4 . JLhistory of the vertical location (relative to the

the left- and right-hand sides of the "tank"

Reynoldsnumber used n Huerta and Liu (1988) s 514,000.Furthermore,in Huerta and Liu (1988) the horizontal body force is removed after tencycles; n this case,on the other hand, this force is maintained during theentire computation. The number of elements s 441,and the time step sizeis 0.107.With thesevaluesof the frequencyand the time step size,a singleperiod of the forcing function takes 60 time steps.Figure l3 shows he timehistory of the vertical location (relativeto the stationary level of 1.0) of thefree-surfacealong the left- and right-hand sidesof the "tank." Figures 14a,

b dI\ ' IIT i i1l r/ il, lI\ /

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s t r eam func t i on

StabilizedFinite Element Formulations

Frc. 14. (a ) Large-amplitude sloshing: flow field and finite element mesh corresponding(approximately) o point a in Fig. l3 (Tezduyaret ol . , l '990e).

stream funct ion

----- -.\_..--- _=/

Frc. 14. (b ) Large-amplitude sloshing: flow field and finite element mesh corresponding(approximately) to point b in Fig. 13 (Tezduyar et al.,1990e).

35p essue

:--.. r:j t- -=:::-r.=.r= _=:_

v e l oc i t y m es n

pressure

v e l oc r l y mesh

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stream funct ion

iim@ "."==

36 T. E. Tezduyarp essue

Ftc' 14. (c ) Large-amplitude sloshing: flow field and finite element mesh correspondi ng(approximately) to poinr c in Fig. 13 (Tezduyar et al., 1990e).stream funct ion

Ftc. 14. (d ) Large-amplitude sloshing: flow field and finite element mesh corresponding(approximately) to point d in Fig. l3 ('Iezduyar et al., 1990e).

veloci ty mesn

pressure

-\N-._=\\\\\N-velocr ty

{ \P r .i r tr * *i s $i j $i j j j

mesn

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Stabilized Finite Element Formulationsl4b, l4c, and 14d show the flow field and finite element mesh correspond-ing, approximately, o points e, b, c, and d in Fig. 13.A cylinder drifting in a shear flow

This test problem involves a cylinder (with unit radius) drifting in a shearflow in a 6l x 32 bounded domain. The density and viscosity are 1.0and 0.005, respectively.The upper and lower walls move with velocit ies0.156 and 0.094. The upstreamvelocity profi le is assumed o be a linear

d L s p L o c e m e n L

3 /

: o "- l' l^ l,;l_ l: JI- l: ll

i . 0 2 5 . c 5 0 . 0 7 5 . 0 1 0 0 . 0 2 5 . cL

0 . 0 2 5 . c 5 0 . 0 7 5 . 0 1 0 0 . 0 2 5 . 0L

0 , 0 2 5 . 0 5 0 . 0 7 5 . 0 1 0 0 , 0 2 5 . 0L

0 . 3 2 5 . 0 5 0 . 0 7 5 . 0 1 0 0 . 0 2 5 . 0L

0 . 3 2 5 . 0 5 0 . 0 7 5 . 0 1 0 0 . 0 2 5 . 0L

v e e L L c o i v e L o c L t 3

q

c;

q

c;

ci

-