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A COMPUTATIONAL METHGC FOR FREE SURFACE HYDRODYNAMICSC. U. Hlrt and B. D. NicholsTheoretical Division, Group T-3

University of CaliforniaLos Alamos Scientific Laboratory

Los Alamos, NM 87545

ABSTRACTThere are numerous flw phenomena in pressure vessel and olping sys-

tms that involve the dynamics of free fluid surfaces. For example, fluidinterfaces must be considered during the draining or filling of tanks, in

the fomlation and collapse of vapor bubbles, and in seismically shaken ves-sels that are partially filled. To aid in the analysis of these types offlow phenomena, a new technique has been developed for the computation ofcomplicated free-surface motions. This technique is based on the conceptof d local average volume of fluid (VOF) and is emijodiedin a computer pro-gram for two-dimensional, transient fluid flow called SOLA-VOF. The basicapproach used in the V(IFtechnique is briefly described, and compared toother free-surface methods. Specific capabilities of the SCLA-VOF programare illustrated by generic examples of bubble growth and collapse, flows ofimmiscible fluid mixtures, and the confinement of spilled liquids.

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1.

most

INTRODUCTIONNumerical simulations of fluid flows undergoing large deformations areeasily performed using Eulerian representations for the flow vari-

ables. That is, the flow isvolumes that remain fixed infree boundaries are present,

computed relative to a grid of small controls ace. However, where free surfaces or otherspecial techniques must be devised to track

these surfaces in the Eulerian grid. The need for a special treatment canbe readily understo~d from the following argument. After one time step ofcalculation all fluid elements that find themselves in a given cell of thegrid must be averaged together to define cell values needed for the nexttime step. This averaging procedure introduces a smootning of all varia-tions in thu flow variables. In particular, surfaces of dlscontinuit.ysuchas free surfaces can be smoothed to the point of being unrecognizable.

To overcome the numerical smoothing of Interfaces a special interfacetracking method is needed that satisfies three basic requirements. First,it must provide a numerical description of the location and shape of theboundary. Second, there must.be an algorithm for advancing the boundarydescription in time. Finally, a scheme must be provided for imposing thedesired boundary conditions on fluid In the surrounding computational grid.

A variety of interface tracking methods satisfying the above require-ments are available, but most have limitations of one sort or another. Forexample, the d~flnition of a surface by Its height above some referencelevel (y u h(x,t) In two dimensions) is a simple definition requiring a min-imum of stcred information, hut It is limited to single-valued surfaces.

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Lagrangian marker particles llnked together by straight line segments canbe use to define (in two dimensions) arbitrary surfaces. The disadvan-tage of this nthod is that the intersection of surfaces becomes a diffi-cult computational problem and such intersections are usually allowed onlywith special,particles canboundaries of

problem dependent, logic statements. Alternatively, markerbe used to mark all fluid-occupied regions rather than thethe regions. In this way the inters~ction of surfaces is no

problem. Unfortunately, this method needs considerably more storage andcunputational time than the other methods because it requires several mark-er particles in each cell occupied by fluid and each particle must be movedevery time step. To determine where a boundary Is located also requireskeeping track of all particles in a cell so that an average surface loca-tion can be computed.

In this paper anotlermethnd is described that is simple and yet in-corporates all the desirable features of che other methods. This new nwth-od, referred to as the volume of fluid (VOFJ method, is based on a functionwhose value is unityThe average value offractional volume of

at any point occupied by fluid and zero elsewhere.this function, F, +n u grid cell then represents thethe cell occupfcd by fluid. Thus, a unit value of F

Indicates the cell Is fullcell. Cells with F valuesary. Computer storage for

of fluid, while a zero value indicates an emptybetween zero and one must then contafn a bound-the VOF method Is a minimum at one word per

cell, which Is equivalent to the storage requirement for all other flowvariables. Because It follows flulclregions rather than surfaces It diS()works for a~bitrarlly Interscctlllgsurfaces. In addition, the F dlstribu-

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tion used in the VOF method has all the remaining properties desired of aninterface tracking scheme. Surface locations, slopes, and curvatures areeasily computed fsr the setting of boundary conditions, and the F distribu-tion can be advanced In time by advection through the Eulerian grid. How-ever, to avoid the type of numerical smoothing noted earlier it is neces-sary to use special advection algorithms. In the SOLA-VOF program, de-scribed in the next section, a type of donor-acceptor fluxing is used tocompute the advection of F. This technique is simple and works quite wellfor most applications.

The VOF-based program, SOLA-VOF, that is described in Sec. 11 is ageneral purpose solution algorithm for a wide class of fluid dynamics prob-lems. Originally the program was developed to solve time-dependent prob-lems involving an incompressible Navier-Stokes fluid containing free sur-faces. In its present form, however, SOLA-VOF is also applicable to prob-lems involving two immiscible fluids. Additionally, It has an option forincluding surface tension with wall adhesion, an option for limited com-pressibility effects, and it has an internal obstacle capability. SOLA-VOFis an easy to use program because of rlumerousautomatic features. For ex-ample, it has a flexible grid generator, built-in time step controls, andsome self-testing features that automatically detect numerical stabilityproblems and correct them.

A variety of sample calculations illustrating the power and usefulnessof the SOLA-VOF program are presented in Sec. 111. These examples cover awide range of fluid phenomena associated with pressure VCSCPIS and pipingsystems.

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JI. THE S!3LA-VOFPROGRAMThe governing differential equations are the Navier-Stokes momentum

equations [1],

Ue+gx+v[a%+&+~_.--L+ u $+ v ~= -- (1 atiat ay P ax 2X2 ajf2 x ax Xz)]

(1)

Fluid pressure is here denoted by p. Velocity components (u,v) are in theCartesian coordinate directions (x,y) or axisymmetric coordinate directions(r,z). The choice of coordinate system is controlled by the value of L,where C = O corresponds to Cartesian and E = 1 to axisymmetric geometry.Body accelerations are denoted by (gx,gy), v is the coefficierlLof kine-matic viscosity, and P is the fluid density.

If the fluid is to have limited compressibility the appropriate masscontinuity equation is [2]

(2)

where C Is the adiabatic speed of sound in the fluid. For incompressiblefluids l/C2 is set to zero. In the limited compressibility model densitychanges are assumed to be small (say less than 10%) and the p appearing inth~ Fressuretwo immlscibthe constant

gradier~ttoms In Eq. (1) can be treated as constant. [Whene fluids are present this p is an appropriate lccal mixture ofp values fcr each fluid,]

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Equations (1} and (2) are discretized with respect to an Eulerian gridof fixed rectangular cells. Grid cells may have variable sizes, say 6x,.ifor the ith column and dyi for the jth row, as shown schematically in Fig.1. Dependent variables are located at the staggered grid locations indi-cated for a typical cell In Fig. 2.

The basic procedure for advancing a solution through one increment intilw, t, consists of three steps:

(1) Explicit finite difference approximations of Eq. (1) are used tocompute first guesses for the new time-level velocities. In this step theinitial dependent variable values, cr the values from the prevfous time-lev-el, are used to evaluate all advective, pressure, and viscous accelera-tions.

(2) To satisfy the continuity equation, Eq. (2), pressures are iter-atively adjusted in each cell. As each pressure value is changed the ve-locities dependent on this pressure are also changed. This pressure itera-tion is continued until Eq. (2) is satisfied to a prespeclfied level of ac-

curacy.(3) Finally, the F function defining fluid regions is updated to give

the new fluid configurtitlon. After all necessary bookkeeping adjustmentsare completed, including data output, this three-step process can be re-started for the n~xt time-level ca!cu?ation. At each step, of course,suitable boundary conditions must be Imposed at all boundaries.

The actual finite difference approximatlon~ used in SOLA-VOF for Eqs.(1) and (2) are not a crucial part of the algorithm. That is, various ap-proximations could bc used without affecting the basic solution procedure.

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The reader Is referred to Refs. [3,4] for the particular approximationsuse : in the present program. This flexibility does not apply, however, tothe way in which the F distribution is advanced in time. Because F is ascalar quantity fixed in the fluid Its evolution Is governed by pure advec-t~on,

(3)

where r = x when & = 1 and r = 1 when E!= O. This equation Is strictlyvalid only for Incompressible flow, but Is also acceptable for the limitedcompressibility approximation. Nmerical approximations to Eq, (3) must bcconstructed with special care to avoid numerical smoothing of the F distri-bution. There are several ways to do this. SOLA-VOF employs a type of do-nor-acceptor fluxing using the fact that F values should be either one orzero. The basic idea can be grasped by considering the amount of F to befluxed across the right boundary of a cell during one time step. The totalvolume of both fluid @ void crossing the boundary, per unit cross-sec-tional area, Is V = u6t, where u Is the normal velocity at the boundary.The sign of u determines which cell Is loslng F (the donor) and whlsh isgaining F (the acceptor). The amount of F crossing the boundary depends onhow F Is distributed in the donor cell. Uhen the flux is primarily In thedirection normal to the F s~rface the fractional area of the flux boundaryacross which F Is flowlr,gis determined by the acceptor cell F value. Uhenthe flux is prlmarlly tangent to the surface the donor cell F value Isused. In both cases the amount of F flcjxedIs computed as the product ofthe cross-sectional area of the flux boundary times hF where,

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F= MIN{FADIVI + CF , FD 6XD}

and whereCF= MAX{ (l.O - FAO)IVI - i.O - FD) 3XD , 0.0} . (4)

Subscripts denote acceptor (A) and donor (D) cell values. The double sub-script (AD) is equal to A when the flux is normal to the free boundary andequal to D otherwise. The MINfeature lnEq. (4) prevents more F being ,fluxed than is available in the donor cell. The MAX feature accounts foran additional flux of F if more than the amount of void volume available inthe donor cell is f.uxed. Figure 3 illustrates these features for severaltypical cases. The fluid is assumed distributed in the donor andcells as shown depending on the orientation of the surface normalspect to the flux direction. In Fig. 3a the donor cell, acceptor

acceptorwith re-tell, and

the flux volume are defined. Then Fig. 3b il!listratesa situation in whichthe donor cell value of F is used to define the fractional area of the fluxboundary open for fluxing F. In case c of Fig. 3 the acceptor cell valueof F has been used to define the fractional area. In this case all the Fregion in the donor cell is fluxed, but it is less than the total flux pos-sible, which illustrates the use of theF!IN test in Eq. (4). Finally, inFig. 3d, more F than the amount determined by the acceptor cell definedarea must be fluxed. The extra flux contribution to F is the quantity CFdefined inEq. (4).

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The choice of the acceptor or donor cell F value to define a fluxarea, which depends on the orientation of the free boundary, is a featurenot used in other schemes of this type. It is essential to do this, how-ever, otherw~se boundaries advecting more or less parallel to themselveswill develop step ~rregularities.

Additional details of the SOL~,-VOFprogra~ ~elating to boundary condi-tions, numerical stability requirements, etc., can be found in Refs. [3,4].

The best way tn assess the strengths and weaknesses of the SOLA-VOFalgorithm is to examine the calculations tt can perform. This is done inthe next section, where several applications are used to illustrate itspower for a wide variety of difficult problems.III. SAMPLE APPLICATICNS

Pressure vessels and piping systems are subject to many kinds of com-plex flw phenomena. The following examples have been chosen to illustratehow some of these phenunena can be addressed with the new SOLA-VOF program.These ex:;,,Plesover problems involving the growth and collapse of vaporbubbles, problems associated with mixtures of intnisciblefluids. and prob-lems involving extreme deformations of free surface dominated flows.J. Bubble Gynamics

in pressurized systems for liquid transport it is sometimes possiblefor vapor bubbles to form. Under most circumstances bubbles are undesira-ble as their growth or collapse can result in significant pressure fluctua-tions andnamics isvolved.

local material damage. The theoretical prediction of bubble dy-complicated by the generally large free surface deformations ln-his, then, is an excellent area where the capabilities of the

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SOLA-VOF program can be put to use. For purposes of illustration we shallconsider what happens when steam is forced through a pipe submerged in a

poul of water. The pipe is located axisymmetrically in a cylindrical ves-sel approximately half filled with water (see Fig. 3). Experimental stud-ies [5 ] indicate that when sufficient steam IS injected into the pipe bub-bles may repeatedly form and collapse at the end of the pipe causing largepressurlstransients to be generated in the water pool. Presumably the in-crease in liquid surface area and stirring associated with the formation ofa bubble increases the condensation of steam to the point where it can nolonger support the bubble. Mhen this happens the bubble collapses and wa-ter rushes back into the pipe until sufficient steam pressure is againbui]t up to generate a new bubLle.

To simplify the problem we shall not attempt to model all the process-es associated with actual steam condensation, but shall use a simple pre-scribed pressure history for the steam. In particular, the steam pressureis approximately linearly increased until a bubble has been generated thenit is rapidly reduced to the saturation pr~ssure of the water in the pool.A one millisecond time interval was arbitrarily chosen for the depress~lri-zation time. The time at which the depressurization is started determinesthe size of the bubble transient. This crude model approximates what wouldhappen with a more detailed condensation model in which condensation pro-ceeds more rapidly than the inertial response time for the bubble. 1P anycase, It Is sufficient to illustrate how the SOLA-VOF program can be usedto study the complete history of bubble birth and death.

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Starting from an ?nitial vapor pressure of 1.1 psia in the pipe(0.146 ft 1.D.) th~ pressure is increased to about 3.16 psia over a periodof 190 ms then is ramped down in 1 ms to 0.38 psia. A sequence of computedvelocity plots and fluid configurations are shown in Fig. 4. The earliesttime shown corresponds to the time at which the vapor pressure is rampeddown to the liquid saturation pressure. Because of Inertia in the liquidthe bubble continues to grow (Fig. 4c). When the bubble begins to collapseit does so asymmetrically, pulling liquid in from the top of the pool.This causes a detached bubble to form shortly after Fig. 4d, which thendisappears some time between Figs. 4e and 4f. U~ter is seen to be movingrapidly up the pipe in the final frame.

Pressures computed at the center of the vessel floor are shown in Fig.5. The initial rapid increase in floor pressure occurs shortly after thepipe has been Cledred of water. At 0.19 s the pressure drops because ofthe decrease in vapor pressure at that time. A relatively violent pressuretransient develops when water reenters the pipe end and the detached bubblecollapses. This transient, denoted by the dashed line in Fig. 5, is showninan expanded scale in Fig. 6. To obtain this result the fluid must betreated as a compressible medium because the pressure transients have char-acteristic times short compared to the time needed for acoustic waves totravel across the pool. Except for this short, violent transient the watercan be treated as incompressible, but to correctly estimate the pressurepulse generated by the final bubble collapse requires compressibility. Thelimited compressibility model available in the SOLA-VOF program providesthis capability.

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B. r~is~ible Fluids,mixture of oil and water provides an excellent example of a two-

fluid system often encountered in practical situations. Because of theirslightly differing densities and the action of interracial surface tensionforces the mixture behaves dynamically quite different than either fluidseparately.

Using the two fluid and surface tension options in the SOLA-VOF pro-gram a variety of interesting mixture problems can be investigated. To il-lustrate, Fig. 7 presents results from a calculation of the passege of aliquid drop through a constriction in a tube. The drop has a density equalto 9/10 oi the density of the surrounding fluid. Surface tension at theinterface between the two fluids is such that the Weber n~mber (pV2r/o) isequal to 0.192, based on the drop radius and average flow rate through thetube. This nwans that surface tensio~ forces are more significant thanthose of inertia. In this example viscous for~es are also relativelystrong for the Reynolds number (Vr/v) was chosen to be 1.25, and it was as-sumed that both fluids have the same kinematic viscosity. Flow enteringthe flow channel is uniform, implying that the constriction is near thechannel entrance where boundary layers have had little tine to develop. Ifthis were an oil-water mixture, it wuld correspond to a small oil drop(r= 3x lC-4 cm) forced rapidly (V s 46.1 cm/s) through a hole in 3 thinplate.

To force the drop through the constriction requirw extra wrk to de-form the drop against its surface tension forces, Figs. 7a-7c. Much ofthis work, however, is recovered as the drop emerges from the constriction,Figs. 7d-7f.

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C. Larqe Distortion DynamicsThe advantage of formulating SOLA-VOF in terms of an Euleriar repre-

sentation is its ability to treat flows undergoing exireme deformations.To illustrde this capability consider what happens when a tank containingfluid collapses. Suppase a dike is to te constructed around tbe tank tocontatn the spilled fluid. The problem is how high to build the dike. Aspecific example is illustrated in Fig. 8. The fluid is initially a circu-lar column having a heig$t equal to its diameter. The dike is a low ax-isymmetric wail whose radius has been arbitrarily chosen to be 24 columnradii. As the column col?apses fluid rushes radially outward along theground. Upon striking the dike the leading edge of the fluid is deflectedupward, but if the dike isntmentum to splash down outsidein Fig. R where a signregion. When the dikeheight shown in Fig. 8the fluid is contained

high enough it retains sufficient radial mo-the dike. This is seen t.obe the situation

ficant amount of fluid has been lost from the dikeh~;qht is increased to approximat~ly 2 times theadditonal calculations indicate that virtually allwithin the dike.

Although this is a conceptually simple problem, it is obviously onethat involves highly complicated free surface dynamics. Nevertheless, theSOLA-VOF program does a remarkable job in representing the flow. Othervariations involving different initial fluid and obstacle configurationsare easily imagined. Of course, SOLA-VOF could also be used for similarproblems involving two immiscible fluids with or without interracial sur-face tcn5ion. !t is this flexibility, in fact, that makes the SOLA-iOFprogram such a powerful tool. With thoughtful use it provides a means ofinvestigating many previously intractable prublems associated with pressur-ized fluid systems.

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ACKNOWLEIXMENTSWe wish to thank R. S. Hotchkiss for his efforts in adding the surface

tension capability to the SOLA-VOF program and for running the second exam-ple problem. This work was supported by the Electric Power Research Insti-tute under contract RP-965-3.

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REFERENCES1, L. D. Landau and E. M. Lifshitz, Fluid Mechanics, Pergamon Press,London, 1959.2. C. W. Hirt and B. D. Nichols, Adding Limited Compressibility to In -

compressible Hydrocodes, J. Comp. Phys. 33, 1979.3. C. U. Hirt and B. D. Nichols, Yolume of Fluid (VOF) Method for theDynamics of Free Boundaries, submitted to J. Comp. Phys., 1979.4. B. Il.Nichols, C. W. Hirt, and R. S. Hotchkiss, SOLA-VOF: A SolutionAlgorithm for Transient Fluid Flow with PlultipleFree Boundaries, I.osAlamos Scientific Laboratory report in preparation.5. S. B, Andeen and J. S. Marks, A~alysis atidTesting of Steam Chugging

in Pressure Systems, Electric Power Research Institute report NP-908,1978.

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A

Fig. 1. (A) Mesh setup showingof dependent variables

+UdlJ ..-

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arc deflncflIn (a) where dashed line indicates left slhc of fluxvolumc. Cross-hatched regions In (b-d) are the amounts of 1ati-vecteci.

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Fig, 4. Veloclty vectors and fluid conflqurations at selected times ~ftcrInitiation of pressllrlzation. Times In seconds are (a) 0,19, (b)0.21, (c) 0.22, (d) 0.24, (c) 0.75, and (f) 0.30.

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*PM PRESSURE I JLTI TE 1s-o 0. I 0.2 0.3 0.4 0.5Tkm h)

Fig. 5. Floor pressure history computed during bubble growth and col-lapse, Dashed line region shown in exp]nded time scale ~n Fig,6.

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120 FLOOR PRESSU?E FWING EWBLE COLLAPSE

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Fig. (i . Computed floor pr~ssure on an expanded time scale follGwing huh-ble collapse.

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