LA-10612-MS

(!39

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This reportwaspreparedasan accountof worksponsoredbyanagencyof theUnited SUStesGovernment.Neiiherthe United SratesGovemmenl noranyagencythereof,noranyoftheiremployeea, makcsanysvarmnty,expressor implied.orassumesany legalliability or responsibilityfor theaccuracy,completeness.orusefuhressofanyinformation,apparatus.product.or processdisclosed,or representsthat itsusewouldnot infringeprivatelyownedrights.Referencehereintoanysptcificcommercialproduct,process.or servicebymadename.trademark.manufacturer,or otherwise,doesnotnecessarilyconstiusleorimplyitsendorsement,recommendation.or favoringbytheUnited SsaesGovernmentoranyagencythereof.Ilseviewsandopinionsof authorsexpressedhereindo no! necessarilystateor reflectthoseof the United StatesGovernmentoranyagencythereof.

LA-10612-MS

UC-32Issued: December 1985

NASA-VOF2D: A Computer ProgramforIncompressibleFlowswith Free Surfaces

MartinD.TorreyLawrenceD.CloutmanRaymondC.Mjolsness

C.W. Hirt*

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“FlowScienceInc.,1325TrinityDrive,LosAlamos,NM87544.

CONTENTS

ABSTRACT

I. INTRODUCTION

II. THE NUMERICALMETHOD

III. PROGRAMDESCRIPTION

IV. SAMPLEAPPLICATIONS

APPENDIXA. THE PARTIALCELLTREATMENT

APPENDIXB. PROGRAMLISTING

1

1

4

30

46

83

85

●

iv

NASA-VOF2D:A COMPUTERPROGRAMFOR INCOMPRESSIBLE

FLOWSWITHFREESURFACES

by

MartinD. Torrey,LawrenceD. Cloutman,

RaymondC. Mjolsness,and C. W. Hirt

ABSTIU4CT

We presentthe NASA–VOF2D two-dimensional,transient,free–surfacehydrodynamicsprogram. It has a varietyofoptions that provide capabilitiesfor a wide range ofapplications,and it is designedto be relativelyeasy touse. It is basedon the fractionalvolume-of-fluidmethodand allows multiple freesurfaceswith surfacetensionandwall adhesion. It alsohas a partial cell treatment thatallowscurvedboundariesand internalobstacles.This reportincludes a discussionof the numerical method, a codelisting,and a selectionof sampleproblems.

I. INTRODUCTION

The NASA-VOF2D program simulatesincompressibleflowswith freesurfaces1-3This techniqueis basedon theuseusingthevolume-of–fluid(VOF)algorithm.

of donor-acceptordifferencing4to trackthe freesurfaceacrossan Eulerian

grid. Similarproceduresweredevelopedcontemporarilyby other workers5–9 to

track materialinterfacesas wellas freesurfaces. A similarprogramhas been

writtenspecificallyfor simulating the draining of spacecraft propellant

tanks.10 The NASA-VOF2Dprogramcontainsseveralimprovementsover thesecodes,

and theseimprovementswillbe describedalongwith thegoverningequations,the

numerical algorithm,a programdescription,severalnumericalsolutions,and a

programlisting.

1

One of the chiefweaknessesof theearlierVOF codeswrittenfor the tank

drainingproblemis thatobstaclesand curvedboundariesare createdby blocking

out the appropriatecomputationalcells. This weakness means that curved

boundaries are representedby discretestair–stepapproximations.Our original

planwas to improve upon this situation by letting curved boundaries be

representedby a stringof cellsidesusingthearbitrarymesh capabilityof a11 The resultingtechniquewouldmodifiedversionof theALE codeCONCHAS-SPRAY.

12 but witha rezoningcapability.‘owever~have resembled theLINC technique,

theoccurrenceof a strong parasiticnode-couplingmode in tank-draining

problems forcedus to abandonthisapproach. It shouldbe notedthatthismode

is much lessseverewhen thecodeis run in the Lagrangianmode. It might

therefore be possible to utilize such a code in theLagrangianmodewith

infrequent,grossrezoningsuchas thatproposedby Horaket al.,13 Dukowicz,14

and Ramshaw.15In particular,thiswouldallowthe code to put the freesurface

alonga grid line,whichin turnwouldallowthesimpleand accuratecalculation

of surface-tensionand wall-adhesionforces. The main limitationwouldbe

difficultyin programmingthelogicneededto allowone regionof fluidto join

another region of fluid, such as we have in propellantreorientation

calculations.However,the techniquewouldbe straightforwardand accurate for

thesimpletank-drainingsolutionpresentedin thisreport.

The developmentof a partial-celltreatmentfor SOLA-VOFtypeprograms(for

example,as describedin Ref. 16) allowedUS tO return to this more robust

methodology. The basicconceptis to introducethe fractionof the cellvolume

open to fluid,the fractionof eachcellfaceopen to flow, and to use this

informationto modifythedifferenceequationsin thepartialcells. A concept17 Appendix A

similarto thiswas usedat leastas earlyas the FLIC program.

givesa justificationfor thepartialobstacletreatment.

A number of other improvementshavebeenmade to theoriginalSOLA-VOF

program. Theseincludefixes in the donor-acceptordifferencing,a better

algorithm for picking the fluid orientationflags, and an upgraded

surface-tensioncalculation.In addition,we haveaddedtheoption of solving

the pressure equation by a conjugateresidualmethodratherthansuccessive

over-relaxation(SOR)method.

The basic

incompressible

v“(e~) =

algorithm is based on the Navier-Stokesequations for an

fluid. The continuityequationis

1 a(re U) + a(ev) =0—● (1)

r ax ay

2

where x and y are thecoordinates,0 is thepartialcellparameterdefinedin

AppendixA, ~ is thevelocityvector,u and v are velocitycomponentsin the x-

and y–directionsrespectively,and r is a factorthatis unityin Cartesian

coordinatesor x in cylindricalcoordinates.In cylindricalcoordinates,y is

theaxialcoordinate.The momentumequationsare

~+ufi+v au 1 ap

[

a2uat ax

—=gx_i=+v ~+aY

$+, [;:->]]

and .

(2)

(3)

where p is thepressure,p is thedensity,

externalacceleration,t is a factorthatis

gx and gy are the componentsof the

zero or unity in Cartesian or

cylindricalcoordinatesrespectively,and v is thekinematicviscosity. These

equationsapplyat everypointinsidethe fluid.

The freesurfacesare treatedby introducinga function F(x,y,t) that is

defined to be unity at any pointoccupiedby fluidand zeroelsewhere. When

averagedover thecellsof a computingmesh,theaveragevalueof F in a cell is

equal to the fractionalvolumeof thecelloccupiedby fluid. In particular,a

unitvalueof F correspondsto a cell full of fluid, whereas a zero value

indicates thecellcontainsno fluid. Cellswith F valuesbetweenzeroand one

are partiallyfilledwith fluidand are eitherintersectedby a freesurface or

contain voids (bubbles)smallerthancellmeshdimensions.The F functionis

utilizedto determinewhichcellscontaina boundary and where the fluid is

located in those cells. Additionally,the derivativesof F can be used to

determinethemean localsurfacenormal,and usingalso the cell F value, to

construct a line cutting the cell that will approximatethe interface.

Sufficientinformationis then available to incorporatesurface-tensionand

wall-adhesionforces into the calculation.BecauseF is a step functionits

derivativesmustbe

The timedependence

computedin a specialway,as describedin thenextsection.

of F is governedby

3

aF aF aF ~z+uz+v~= “ (4)

This approach allows the fluidinterface(s)to be representedin a simpleand

elegantway in thedifferentialequationform. However,the fluxof F through

each cell face of our Eulerian grid will be required. Standard

finite-differencetechniqueswouldleadto a smearingof F, and the interfaces

would lose their definition.The factthatF is a step functionallowsus to

use a formof donor-acceptordifferencingthat preserves the discontinuous

natureof F.

The next section describes the numerical methodin detail. The third

section’isa user’smanualfor thecomputerprogram,and the finalsectionis a

selection of applicationsof NASA–VOF2D. A program listing is givenin

AppendixB .

II. THE NUMERICALMETHOD

This sectionprovidesa detaileddescriptionof the numerical algorithm

used in the NASA-VOF2Dprogram. It is essentiallyan extendedversionof the

SOLA-VOFmethod,zand our descriptionof thenumericalmethoddraws heavily on

the discussion inRef. 2. The programis Eulerianand allowsfor an arbitrary

numberof segmentsof freesurfacewith any reasonableshape. The complete

Navier-Stokesequations in primitivevariablesfor an incompressiblefluidare

solvedby finitedifferences,withsurfacetensionand walladhesionincluded.

A typicalcomputationalgrid is illustratedin Fig.1. The Eulerian mesh

comprises rectangularcells havingvariablesizes,&xi for the ith columnand

&yj for thejth row. Thereis a row of fictitiouscellsaroundthe entiremesh.

Figure2 showsthe locationof the velocity components,pressures, and

F-values within each cell. The pressurepi,jand fractionalvolumeof fluid

Fi ~ are cell-centeredquantities.The horizontalvelocity component l-li+l/2,j?

is definedon the rightcellface,and theverticalvelocitycomponentvi,j+l/2

is definedon the topof the cell. Also shown are the locations of the

geometricalquantitiesAR, AT, and AC, arisingfrom thepartial–celltreatment,

whichare the fractionsof the rightcellface,top cellface,and cell volume

open to flow. In cylindricalcoordinates,thesefractionsare multipliedby the

valueof 2KX evaluatedat theappropriatecellfaceor cellcenter.

4

k~xf+”Fig.2. location06 the vhblti in a -tgpicalmuh cdl.

A free–surfaceor interfacecell(i,j)is definedas a cell containinga

nonzero value of F and havingat leastone neighboringcell (i~l,j)or (i,j~l)

thatcontainsa zerovalueof F. In a surfacecell,we assume that the free

surface can have onlyone orientation(mainlyhorizontalor mainlyvertical),

and for thatorientationall theliquidis on one side or the other of the

surface. That is, the liquid is mainlylocatedadjacentto one of the four

sidesof the cell. Somesurface orientationsresult in cells with nonzero

values of F and no emptyneighbors(i~l,j)or (i,j+l). Theseare treatedas

fullcells. We keep trackof the free–surfaceorientationwith a cell flag

array,NF(i,j),as shownin TableI.

The basic procedure for advancingthesolutionthroughone timestep at

consistsof threesteps. First,from timen values,explicitapproximationsto

the momentum Eqs. (2)and (3) are used to findprovisionalvaluesof thenew

timevelocities.Second,an.iterativeprocedureis used to solve for advanced

time pressure and velocityfieldsthatsatisfyEq. (1) to withina convergence

criterion(EPSI)at thenew time.

Finally,theF-functionis updatedby solvingexplicitlyEq. (4), the new

free-surfaceorientationin eachsurfacece’11is determined,and surface–tension

forcesare computed. Suitableboundaryconditionsmustbe imposedat all mesh,

free-surface,or obstacle boundariesduringeachstage. Repetitionof these

threestepsallowsadvancingthesolutionthroughan arbitraryintervalof time.

Detailsof theseoperationsare presentedin theparagraphsbelow.

The first task in solvinga problemis to set up thecomputationalmesh.

Two kindsof informationare required. First, the horizontaland vertical

locations of cell faces ‘i+l/2 and yj+l/2are specified.The cellcenters(Xi,yj)are definedas

xi = 0.5 (xi-1/2+ ‘i+l/2)

and

Yj = 0.5 (Yj-1/2 + yj+l/2) “

(5)

(6)

Second,geometricalarrays~i+l/2,j9 ATi,j+l/29and ACi,jmustbe definedif an

obstacle is includedin themesh. Thesetasksare simplifiedby automaticmesh

and obstaclegeneratorsincludedin thecode.

6

TABLE1

VEFIN1TIONOF MF ARRAYVALUES

Freesurfacedoesnot bisectcell

NF = O fluidcell--novoidcellsadjacent

NF > 5 voidcell--nofluidin cell

Freesurfacemainly

NF = 1 surface

NF = 2 surface

Freesurfacemainly

NF = 3 surface

NF = 4 surface

vertical

cell--neighboringfluidcell to the left

cell--neighboringfluidcell to the right

horizontal

cell–-neighboringfluidcellon thebottom

cell--neighboringfluidcellon the top

Free–surfaceorientationindeterminate

NF = 5 surfacecell–-isolated,all adjacentcellsare void

The next taskis to select boundary conditions,to define the volume

occupied by fluid,and to set the initialvaluesof thevelocityand pressures

fields. Selectingboundaryconditionsis described in detail in the next

section and is accomplishedby settingfourinputboundaryflags. The other

tasksare problemdependentand mustbe implementedby the user unless the

simple default setuproutineis”adequate.Finally,physicalpropertiesof the

fluid,outputrequirements,and solutionoptions must be furnished as input

data.

The program is then readyto begina cycle. The firststep is to solve

explicitdifferenceequationsfor an estimateof thenew timevelocities. This

calculationis done only if thecellfaceon whichthevelocity(u or v) is

locatedis open to flow (thatis,ARi*l/2,j> 0 or ATi,j+l/2> 0) and some fluid

is adjacent to the appropriateboundary. Void/void and fluid/obstacle

boundariesare skippedforefficiency.We beginby defining the gradients of

the velocity components that are required for the finite-difference

approximationfor Eqs. (2)and (3). Theyare for thex-coordinate,

[1au ‘i+l/2,j- ‘i–l/2,jZ i,,j = ‘i+l/2- ‘i-l/2

(7)

7

and

[1

au ‘i+l/2,j+1 - ‘i+l/2,j . (8)* i+l/2,j+l/2= Yj+l - Yj

Similarequationswithappropriateindices are used to calculate u-velocity

derivativesat (i+l,j),(i+l/2,j-l/2)sincesecondderivativeswill be required.

Once the velocity gradients are calculated,theviscoustermin Eq. (2) is

approximatedby

‘1sxi+1,2j=v([[4i+l$j-[4d’(xi+l-xi)[[1au

+2~ i+l/2,j+l/2- [:li+l/2j-J(yj+l-yj-1)

.

(9)

In the precedingequations,factors such as (au/ax)i+1t2,j are obtained bY

linearinterpolationof cell-centeredquantities.

We use similardifferenceequationsfor theviscoustermsin Eq. (3). We

definethevelocitygradientcomponents

[1av ‘i,j+1/2- ‘i,j-1/2=~ i,j Yj+1/2- Yj-1/2

[1av ‘i+l,j+l/2- ‘i,j+l/2 .Z i+l/2,j+l/2= ‘i+l- ‘i

8

(lo)

(11)

The viscoustermin Eq. (3) is thenapproximatedby

‘lsyi,j+l/2‘v([[~]i,j+~- [#]i,j]/(yj+l-yj)

[[1av+2~ i+l/2,j+l/2- [;li-1/2)j+l/21’(xi+l-xi-1)

t

[[1

av‘q ~ i+l/2,j+l/2+ [gli-1/2jj+J} p

(12)

where the last termis a simpleaverage of derivativeslocated at the cell

corners.

We follow theprocedureof Nichols,Hirt,and Hotchkiss2in approximating

the convectiveterms. Thesetermsare calculatedexplicitlyat the same timeas

theviscoustermsusingthesamevelocityderivatives.Thus,we have

[1u au

{ [1‘i+l/2,j 6X au

[1

auS i+l/2, j = &Xa i 5 i+l, j

+ tix~+lK i,j

+ a sgn(ui+l/2,j)[’xi+l[%litj -’xi [:li+l$jl}~ (13)

where

&xa = 6X~ +&X~+l + a sgIl(Ui+l/2,j) (6x.i+l- axi) “ (14)

● Here &xi is thewidthof celli, a is thedonor-cellfraction,and sgn(u)means

the sign of u.1+1/2,j; that is, the convectivedifferencingis a linear

combinationof donor cell and centered differencing. When a = O, this

formulation reduces to a second-order-accurate,centered-difference

approximateion. Whena = 1, the firstorderdonor-cellformis recovered.This

particularformis thenat leastfirst-order-accurateforany a between these

limits. In order to obtain stabilitywhilemaximizingaccuracy,experience

9

indicatesthat0.25 < a < 0.5 may be utilized. In practicethe initialruns ofa problem are generallymadewitha= 1. Subsequentrunsare oftenmadewithreduceda. Similarly, the vertical convectionof horizontalmomentum is

approximatedby

[1v au = ‘i+l/2,j

((Yj+l

[1

au- Yj) ~ i+l/2,j_l/2

% i+l/2,j &ya

[1au+ (Yj- Yj-1) ~ i+l/2,j+l/2

[ [1au- Y.j) ~ i+l/2,j-l/2+ aSgn(Vi+l/2,j) (yj+l

‘(yj-yj-l)[3+1,2A+J9 (15)

where

&ya= @j+l/2 (16)+ Ayj-1/2+ a sgn(vi+l/2,j)1‘yj+1/2- “j-1/2] ‘

‘yj+1/2= yj+l - yj 9 (16.1)Ayj-1,2 = yj - yj-~ ‘

and wherewe use doublelinearinterpolationto compute

‘xi‘Vi+l,j+l/2+ ‘i+l,j–1/2)+ ‘Xi+l(vi,j+l/2+ ‘i,j-1/2)‘i+l/2,j = 2 (6x~+ ax~+~)

. (17)

Analogousdifferenceequationsare written for theconvectionof v.

The completemomentumequationsare

n+l‘i+l/2,j = ‘i+l/2,j

[+ at gx - (p~~~,j- p~~~)/(P(Xi+l- ‘i))

- [“~li+l/2$j - [v:li+l/2tj+vrsxi+1/2jl10

(18)

.

“n+l

[

n+li,j+1/2= ‘i,j+l/2+ &t g. - (p.1,j+1 - p~~~)/(P(Yj+l- Yj))

- [uEli fj+l/2- [v:;iyj+l/2+v1syij+1/21 9(19)

wherea superscriptn+l denotesan advancedtimequantity. Variableswithout a

superscriptare takenat theold timelevel,n.

The velocitiesfrom Eqs. (18) and (19)willnot satisfythe continuity

Eq. (l),whichis differenceas

[

n+l“ ~i+l/2,j

- Un+li-1/2,j ‘i-l/2,j‘i+l/2,J

tix~

n+l - “q+l+ ‘i,j+l/2‘Ti,j+l/2 1,j-1/2

1-‘Tifj–l/2 /Aci ~ = o

6yj

(20)

In thisequation,fractionalvolumes open to flow are denoted by integral

subscriptsand fractionalareas open to flow are denotedby half-integral

subscripts,i.e.tei,j and ‘i+l/2,j respectively.Moreover,as currently used

in thecode

‘i+l/2,j = ‘i+l/2,j $

‘Ti,j+l/2= ‘i,j+l/2 ~

and

ACij = ei,j .

In order to satisfyEq. (20)simultaneouslythroughoutthemesh, thepressures

and velocitiesmustbe adjustedin eachcomputationalcell occupied by fluid.

The solution may be obtainedby the followingiterativeprocess:Eqs. (18)and

(19) evaluatedwith timen quantitiesproducea provisionalvelocityfield used

as an estimateof theadvancedtimevelocities.In cellsthatcontainfluid,a

pressurecorrectionis calculatedfrom

. &p = -s/(as/ap) , (21)

where S in eachcellis thevalueof theleftsideof Eq. (20), the residual,

evaluated with themostupdatedvaluesof p, u, and v available.Equation(21)

is derivedby substitutingtherightsidesof Eqs. (23)through(26)belowinto.aEq. (20)

6=

and solvingfor 8P. The quantity@ = 2 isawap

p ACi,j

26t (Ai+l/2+ ‘i-l/2+ cj+U2 + Lj_l/2) ‘

where

‘i+l/2,j‘i+l/2= &xi(6xi+1+ ~xi) ‘

‘i-l/2,j~i-1/2= tix~(iixi+ axi-1) ‘

AT.Cj+l/2 =

l,j+l/26yj(6yj+l + ‘Yj) ‘

and

‘Ti,j-l/2<j-1/2= ~yj(&yj+ ‘Yj-1) “

12

The Vth iteratefor thepressureis

and thevelocitiesforall fourcellfacesare updatedwith

HUv N!v-1- /2,j= u -1/ ,j - at 6p/[P(Xi- ‘i-l)] ~

and

(22)

(23)

(24)

(25)

(26)

This procedureis modifiedfor cellscontaininga freesurface. For these

cells [pi,j] is obtainedfromlinearinterpolation(orextrapolation)between

the surfacepressure [ps], computed from surface-tensionand wall–adhesion

forces, and a neighboringpressure[pn]insidethe fluidin a directionmost

nearlyperpendicularto the freesurface. Specifically,

Pi,j= [1 - rl]pn+ n ps , (27)

where~ = de/d is theratioof thedistancebetweenthe cell centers and the

distance between the freesurfaceand thecenterof the interpolationneighbor

cellas shownin Fig. 3. A &p is obtainedfromthenew pn and the old pi,j

and, as explained below, the new pi,j is obtained iteratively by

under-relaxation.

The iteration is continueduntilall theS/s are madesufficientlysmall

such thatthevelocityfieldis within accuracy requirements;that is, the

magnitude of S < EPSI,whichis chosento be approximately10-3timesa typical

absolutevalueof 3u/3xor 3v/3y. The lastiteratedquantitiesof velocity and

pressureare takenas theadvancedtimevalues.

13

Surfaceml

f

y--xSurface

‘r

[interpolation--r

In most cases, convergenceof the SOR iterationsmay be acceleratedby

multiplyingthe &p fromEq. (21)by an over-relaxationparameter(A)(OMG in the

program). A meshwiseconstantvalueof 1.8 is oftenused,but theoptimalvalue

is flow–dependent.A valueof twoor greatergivesan unstableiteration.This

parameteris incorporatedintotheBETAarrayin thecode.

We have found empiricallythat the free-surfacecondition,Eq. (27),

requiresthatthe interpolationneighborcellof any surfacecell must have a

modified u parameter. We employ thecorrectionof Eq. (15)of the SOLA-VOF

report,2 notingthatthe! of that report is our pp-l, and that further

correctionsto thedenominatorof Eq. (15)(includedin thepresentcode)occur

when some interpolationneighborcellhas severalsurfacecells as neighboring

cells. This stabilizes the coupling between the surface cell and the

interpolationneighborcellpressures.Thismodifiedrelaxationis incorporated

into the pressureiterationthroughthePETAarray.

A second procedure for finding theadvancedtimevelocityand pressure

fieldsmay be invokedby switchingthe inputvariableISORfroma valueof unity

to zero. For mostproblemsthisalternateprocedureis moreefficientthan the

successive-over-relaxation(SOR)methodjustdescribed. It is based on the18 for solvinglinearsystemsof equations.conjugateresidualsolutiontechnique

14

First, Eqs. (18)-(20)are

advancedtimepressurefield. We

form:

where the viscous, body, and

converted into a Poisson equation for the

beginby rewritingEqs. (2)and (3) in vector

(28)

advective terms have beenabsorbedinto the

auxiliaryquantity~. The pressureequationis derivedby substitutingEq. (28)

into the continuityequation

v ● (e ~n+l)= o , (29)

whichyields

:Vv” (e v Pn+l) = v v “(q) , (30)

where V is the volume of the computationalcell (i,j). The cellvolumeis

requiredto insurethatthe linear system created by the finite-difference

approximationto Eq. (30) is conservativeand thereforesymmetric.The exact

formof thedifferenceoperatorsmay be seenby substitutingEqs. (18)and (19)

into (20), but it is not usefulto writethemout in detailhere. We merely

note that Eq. (30) leads to a sparse, symmetric linear system when the

free-surface,obstacle, and perimeter boundary conditionsare algebraically

incorporatedinto thesystemof

retained. We writethissystem

where A is

theunknown

equations in such a way that symmetry is

as

(31)

thesymmetriccoefficientmatrix,~ is thesolutionvector(thatis,

P~:;ts)tand ~ is the right–handsideof thelinearsystem. We use.#.

the conjugate residual technique to solve this system. Dalyand Torreyly

discussthemotivation for using this procedure in detail. However, one

important feature of the methodis that~ is founditerativelyin sucha way

that thevth iterate?(~)has an a~~ociatedresidualA !(”)- !?

v V“(eyn+l). Therefore, convergenceof the iterationon the

equivalentto drivingthedivergenceof thevelocityto zero,

method.

thatis equalto

linearsystemis

as in the SOR

There is one exception to therule thatV*(@) is drivento zero. In a

fluidcell (NF= O), the F-transportalgorithm can generate tiny spurious

bubbles, or voids, when thedivergenceis zero,and thesevoidspersistonce

generated. To suppress

as describedbelow,and

resultis an actualnet

net inflow of fluid

this,a smallpositivenumberis addedto S when NF = O,

themodifiedS is drivento zeroby the iterations.The

negativedivergencein thesecells(correspondingto a

duringtheF-transport).The velocityfieldis slightly

nonconservative,but theconservationof the total volume of fluid is not

affected.

After thevelocityand pressurefieldshavebeen foundfora fixedF-field,

we advanceF in time. We combineEqs. (1)and (4) to obtain

a(eF) 1 a(reFu)+ a (ew) =0at ‘i ax ay Y (32)

where0 is thepartialcellparameterdefinedin AppendixA, and r is a factor

that is unity in Cartesiancoordinatesor x in cylindricalcoordinates.This

conservativeformof Eq. (1)allowsus to writea differenceapproximationthat

conservesfluidvolume;namely,

+1 = FF~,j[[

&t 1ARi+l/2,ju~~~/2,jFi+l/2,ji,j ‘~—

i,j ‘xi

.Uq+l“Fi-1/2,j

-ARi-112,J1-1’2YJ1

1

[

n+l n+l‘~ ‘Ti,j+l/2vi,j+l/2Fi,j+l/,2- ‘Ti7j-l/2vi,j-l/2Fiij-l/2

II9

(32.1)

whichservesas thebasisfor the convectionof F.

16

The convectionalgorithm must (1)preservethesharpdefinitionof free

boundaries,which we denote here as fluid instances;(2) avoid negative

diffusiontruncationerrors;and (3)not fluxmore fluid,or morevoid,acrossa

computingcellinterfacethanthecelllosingthe fluxcontains.To accomplishthis, the SOLA-VOF technique adopts a type of donor-acceptorfluxapproximation,4 in whichboth the fluidvelocitiesand the F values of thecomputing cell boundaries,or interfaces,appearing in Eq, (32.1), are

redefined,as necessary.

It shouldbe noted

really be understood

really~(&tn+ 6tn+l).

firsthalf timestepof

in Eq. (32.1)thatFij (ourshorthand for F~j) should

as F~~l/2 and F~jlreallydenotesFn+3/2, while&t is

The codeignoresthechangein F thatoccursduring the

thecalculationand uses initialvaluesof Fii for F~{2.

The codegraphicsdisplaysF~j at tn-J

insteadof at themorecorrect tn+; i;n.

These imprecisionare not usuallysignificant:typicallythe fluidstartsfrom

rest,while the timestepis usuallyuniform,or it changes slowly if it is

being controlledby a timesteplimiter. However,in one testproblem(of slug

flow),it was necessaryto displayFij at the-correcttime to obtain precise

graphicalcorrespondencewithan analyticsolution.

At eachboundaryof eachcomputingcell,the twocellsimmediatelyadjacent

to the interfaceare distinguished,one becominga donorcelland the other an

acceptor cell, and cell quantitiesare given the subscriptsD and A,

respectively,e.g.,FD and PA. The labelingis accomplishedby means of the

algebraic sign of the fluidvelocitynormalto theboundary;thedonorcell is

alwaysupstream, and the acceptor cell downstream,

emphasize that the D and A labels are assigned

boundary. Thus,eachcomputationalcellwillhave four

D or A correspondingto eachof its cellboundaries.

denotedby FAD,whichappearsat the fourboundariesin

of the boundary. We

separatelyfor each cell

separateassignmentsof

The boundaryvalueof F,

Eq. (32.1), will be

either ‘D ‘r ‘A’ according to an algorithmchosento partiallyaccomplish

purposes(1)and (2)above. onceFAD is chosenPtheactualvalueof F used at

the boundaries in Eq. (32.1)is redefinedfurtherby a simplealgorithm,which

accomplishespurpose(3)and furtherassist’sin carryingout purposes (1) and

(2).. .

We denotethe finalvalueof theboundaryF by FAD. It is the finalFAD

that,when convectedwith the fluidvelocitynormalto the interface,will not

remove more fluid per unitarea,nor morevoidper unitarea,thanthedonor

cellcontains,evenwhen theboundaryis completelyopen to flow.

17

.We illustratetheredefinitionFAD,assuming that FAD has already been

determined at the right-handcell boundary, discussingfirsttheoriginal

SOLA-VOFalgorithm.2For thiscase

‘AD ‘!~~/2,j[1

&t = sgn ui~{~2,jMIN(FADIVXI+ CF, FD~xD) ~ (33)

where

Vx = Un+l.i+l/2,j‘t

and

CF = MAX([l.O- FAD]IVXI- (1.0- FD)6xD! 0.0) .

.

For numericallystablecomputations,i.e.,for IVxl< &xD,theMIN statement of

Eq. (33) prevents fluxing more fluid per unit area thanthe donorcell

contains,while theMAX statementpreventsfluxingof morevoid per unit area

than the donor cell contains, namely FD&xDand [1.0- FD]&xDrespectively,.

providedthatFAD is fluxed at the fluid normal velocity ui+l/2,j.

however, that it is theentireleft-handsideof Eq. (33)thatappears,.

right-hand cell boundary in Eq. (32.1). Thus, ‘AD never appears

independentlydefinedsymbolin thecode.

Note,

at the

as an

For the present code, theCF givenaboveand in the SOLA–VOFreport2 is

modifiedto be more restrictivein theamountof void fluxedby the operationof

the CF term. This is done to improvetheaccuracyof the fluidconvectionand

to suppresstheappearanceof spurioussmallwispsof fluidin thevoid cellsof

the computingmesh. We definetheDM cell to be thecellupstreamof the donor

celland thenspecify

CF = MAX([<F>- FAD]IVXI- [<F>- FD]6xD,0.0) , (34)

where

<F> = MAX(FD,FDM,0.1) ●

18

(34.1)

The firsttwooptionsfor <F>.increasetheaccuracyof therepresentationof the

fluid convectionprocess, while the thirdoptionpreventstheconvectionof

fluidby theCF termuntileithertheD or theDM cellbecomesat least0.1 full

of fluid,whichsuppressesthespuriouswispsof fluid.

TO determinewhetherFAD is FD or FA, we use FD when theD cell is a fluid

cell, which is automaticallyincorporatedinto a standard donor-acceptor

algorithm.For surfacecells,we determinetheprimaryorientationof thedonor

cell fluidand of the fluidinterfacefrom NFD and determine by inspection

whether convectionof donor cellfluidacrossthecomputingcellboundaryis

primarilynormalor primarilytangentialto the fluid interface. For most

cases, we set ‘AD . FA when theconvectionis primarilynormalto the fluid

interfaceand FAD = FD when it is primarilytangentialto the fluid interface.

However, if the A cellor theDM cellis empty,thenwe alwaystakeFAD = FA.

This requiresthata donor cell must be nearly full before any fluid is

convectedto a downstreamemptycell.

The complete fluxingalgorithmis appliedcompletelyindependentlyat the

fourcomputingcellboundariesappearing in Eq. (32.1). When the necessary

fluxeshavebeencomputed,F is advancedthroughone timestepusingEq. (32.1).

The full operation of the F-convectionalgorithm at a computingcell

boundary,namely,thechoiceof FAD and theapplicationof Eqs. (33),(34), and

(34.1), is equivalent to the implicit constructionin the donorcellof

intra-cellinterfacesthatgoverntheconvectionof fluid into the acceptor

cell. The interfacesare independentlydeterminedfor each of the four

computingcellboundaries,are not thesameas the localmean freesurface, and

may be of threegeneralshapes,twoof whichare illustratedin Fig. 4. The

intra-cellinterfacesmay be a singlehorizontalor vertical line, with all

fluid pushed towardthemainvolumeof adjacentfluid,as illustratedin Figs.

4b and 4c; theymay be a pair of lines, one horizontaland one vertical,

confining the fluidintothecellcorneradjacentto themain bodyof fluid,as

illustratedin Fig.4d; or they may be a step function whose height is

determinedby FA and FDM.

SometimestheVOF F-advectionalgorithmwillgeneratenonphysicalfilaments

of fluidthatpropagatethroughoutlargeregionsof void. The appearanceof 0.1

as a lower boundfor <F> in Eq. (34.1)helpssuppressthesefilaments.If a

surfacecell is tryingto fluxmaterialintoan emptycell,the fluxis set to

zerountilF is greaterthan0.1 in thesurfacecell. The limitingvalueof 0.1

may not be optimalforall problems. The original.SOLA-VOF schemes may be

recoveredby settingthe limitingvalueto zero.

(c) AD=A (d)AD=A

Next we applyseveralfine-tuningproceduresto thenew valuesof F. The

firstis an optional algorithm for suppressingnonphysicalfluid voids or

filaments. It is invokedby settingtheoff/onflagNPACKto unity. Used only

for problemsin whichthe fluidliesagainstthebottomof the tank and has a

simplesurfaceshape,thisschemeforcesthecode to startat thebottomof each

columnof cellsand to packall of the fluid down, producing a single–valued

surface–heightfunction. Unfortunately,thisschemeis not easy to generalize

to problems with a multivaluedsurface-heightfunction, such as fuel

reorientationproblems.

A second algorithm utilized to suppress spurious voids,mentionedin

discussingthe pressureiterationearlier,overcomes these restrictions. The

modifieddivergence(~i,j) thatis drivento zerois definedas

?i,j= si,j + MIN(ADEFM*EPSI,BDEFM*(l.O- Fi,j)/6t) ,

where ADEFM is typically 102 to 103, BDEFM is typically0.10, and the.

convergencecriterionEPSI is normally10-J. The first term represents the

maximum allowable incrementto be addedto thecomputeddivergenceon a single

timestep,while thesecondis a fractionof thecurrentdeficitthatwill tend

to zero as the cellfills. BDEFMis actuallythe reciprocalof thenumberof

timestepsin whichthedeficiencyis to be removed. The input variable IDEFM

is an off/onflagfor thisoption(O = off, 1 = on).

20

The nextoptionalalgorithmis controlledwith theoff/onflagIDIV,and it

usuallyimprovestheaccuracy and robustnessof the program. This scheme

recognizes that the divergenceof yn+l (s=v. (e~n+l)) is not exactlyzero

becauseof the finitenumberof pressureiterationsand roundingerrors,and it

adjusts the value of F in eachcellproportionalto theactualvalueof its

divergence.If IDEFM= 1, this algorithm must be skipped. The algorithms

controlledby IDEFM,NPACK,and IDIVare referredto as ‘defoamersvbecausethey

are sometimesneededon problemsto preventvoidregions from filling with a

frothof tinyF’s or the formationof spuriousvoidswithinthe fluidinterior.

Truncationerrorsand roundingerrorscan causetheF-valuesdeterminedby

theaboveprocedureto occasionallyhave values slightly less than zero or

slightly greater than unity. Therefore,aftertheadvectioncalculationhas

beencompleted,a pass is made throughthemesh to resetvaluesof F less than

zero back to zero and valuesof F greaterthanone back to one. Accumulated

errorsin fluidvolumeintroducedby theseadjustmentsare recordedand may be

printedat any time.

There is one final adjustmentneeded in F so thatit may be used as a

boundarycellflag. Boundarycellshavevaluesof F lyingbetweenzeroand one.

However, in a numerical solution, F-values cannot be testedagainstexact

numberslikezeroand one becauseroundingerrorswouldcausespurious results.

Instead, a cell is definedto be emptyof F whenF is less thanCF (EMFin the

program)and fullwhen F is greaterthan1 - EF,where CF is typically 10-6.

If, after advection,a cellhas an F valueless than&F, thisF is set to zero

and all neighboringfullcellsbecomesurfacecells by having their F-values

reducedfromunityby an amount1.1 &F. Thesechangesin F are also includedin

theaccumulatedvolumeerror. Volume errors after hundreds of cycles are

typicallyobservedto be a fractionof 1% of the totalF volume.

Following the calculationof the new F-valuesthecodesweepsthemesh

redefiningthenew cell typesand assigningappropriateflags NFi,j. At the

same time the approximateorientationof the fluidin eachsurfacecell is

determined,and a pressureinterpolationneighborcellis assigned.

Determinationof whetherthe fluidis mostlyhorizontalor vertical in a

surface cell relieson estimatingthe localslopeof the fluid/voidinterface.

We assumethatthe interfaceis a straightlinesegmentpartitioningthe cell.

The slopeis estimatedby introducingtwosurface-heightfunctionsY(x)and X(y)

basedon thevalueof F in thesurfacecelland its eight neighbors. A good

approximationto Y(x) is

21

Y~ =

where Y

cellsthe

[1

dYz

where 6xi-

Y(x~)= ‘i,j-l&Yj-l + Fi,j6yj+ ‘i,j+l~yj+l‘ (35)

= O has been takenas thebottomof thej-1 row of cells. For fluid

slopeis givenby

= [(yi+~- ‘i) ~xi-1/2/~xi+l/2i

+ (Yi– ‘i-l) ‘xi+l/2/&xi-l/2]/(&xi-l/2 + ~xi+l/2) ? (36)

.,,9- (6X++ &x4,,)/2.Interchangingthe rolesof x and y, we cant-.i.,.z A A-I-A

—

write a similarequationfor (dX/dy)j.WhenevaluatingtheX– and Y-functions,

one must be carefulto includeany obstacle volume in F that modifies the.

algorithm.

If ldY/dxlis smallerthan ldX/dyl,theboundaryis morenearlyhorizontal

thanvertical. Otherwise,it is more nearly vertical. The side of the

interface containing the fluid is then chosendependingon the signof the

largerderivative.For example,if”ldY/dxlis the smaller, the interface is

horizontal. If dX/dyis negative,fluidliesbelowthe interface,NF = 3, and

cell (i,j–1)is usedas the interpolationneighborin Eq. (27). If dX/dy iS

positive, the fluid lies at the topof thecell,NF = 4, and cell (i,j+l)is

chosenas the interpolationneighbor.

For a free-surfacecell thesurface-tensionand wall-adhesionforces are

evaluated and expressedas a surfacepressureps[PS(I,J)in the program]. The

surfacepressureentersthecalculationthroughEq: (27).

The surface–tension,wall-adhesionalgorithmmust takeaccountof the facts

that the codehas a very impreciseknowledgeof thenatureand the locationof

obstaclesand thatthe codeis usuallyusedat the limitsof flowresolution to

save computationalcosts. (Frequentlyflowsalongthe tankwallswill be only

one or two computationalcells wide.) Thus it is essential that the

surface–tension,wall–adhesionroutine be robust(thatis, shouldacceptvery

imprecisedata,shouldnot makelargeerrors in processing the data, should

yield smooth output, and should never cause a catastrophicfailureof the

calculation)and simple.

For each free-surfacecell,the program uses the array NF to specify

whether the fluidliesprimarilyat thebottom,top,left,or rightside of the

22

cell. We illustratehow thesurface-tension,wall-adhesionmodelworks in the

case of a free-surfacecellin whichthe fluidis locatedat thebottomof the

cell (thatis,NF = 3). The otherthreecasesare treatedby analogous means.

We alsoassumeCartesiancoordinates.The extensionto cylindricalcoordinates

will be discussedafterwards.

The surface-tensionforceactingacrossthe facesof the computationalcell

is computedfrom [20]

g=-a Ic ~ X d~ , (37)

where a is thesurfacetension,~ is theoutwardunitnormal(thatis, directed

into thevoid)to the fluid surface, and the path c denotes the counter

clockwise-traversedintersectionof the fluidinterfacewith the facesof the

computationalcell. For thisNF = 3, Cartesiancoordinateexample,we consider

the cell to be three-dimensionalwithunitdepthin the thirddirection.Thus,

the pathc is a closedlooparoundthe fourverticalcellfaces. This equation

is validif the freesurfacecontinuesbeyondthecurrentcell in all directions

and doesnot terminateon an obstacle wall in the current cell. Then the

surfacepressureps may be computedfromtheverticalforceas follows:

ps(I$J)&xi &Z = -$*Y= - a &z [sin(~) - sin(~)] ,

where 6 denotes the local angle thatthe tangent(in

interfacemakes(rotatedcounterclockwise)with thex-axis

normal makes with the y-axis), while thesubscriptE

(right)or west (left) boundary of cell (I,J). The

y-directionis denotedby y.

(38)

thex-y plane)to the

(or the angle the

or W denotesthe east

unit normal in the

For theNF = 3 case,theprogramdeterminesan averagefluidheightAVFR in

the columnto theeastof (I,J),an averagefluidheightAVFCX in cell (I,J),

and an averagefluidheightAVFL in thecolumnto thewestof cell (I,J)froma

weightingschemethatinvolvestheF valuesin (I,J)and its eight neighboring

cells. The (3anglesare determinedfromthedefiningrelations

tan(~) = 2 (AVFR- AVFCX)&Xi + &Xi+~

.

(39)

23

and

tan(~) = 2 ‘AVFCX- ‘vFL)6Xi-~+ 6Xi

(40)

afterwhichwe use relationsof the form

tan(~)sin(~) =

[1 + tan2(~)]1/2(41)

to evaluatequantitiesappearingin theps formulas.

Supposethe fluidsurfaceendson an obstacleat theeastwall,its tangent

makingthecontactangle@c with theeastwall insteadof continuingon into the

east column of cells, as assumed in thepreviousformulafor ps. Then the

verticalforceat theeastwallneedsto be modifiedto allowfor the fact that

the true force is a wall-adhesionforceratherthana surface-tensionforce.

Specifically,assumethatthesurfacecell (I,J)is a partial flow cell whose

right, or east, boundaryis a linesegmentrunningfromits bottompoint~b =

(Xb, Yb) to its toppointEt = (xt,yt),whereyt > yb? and the fluid lies to

the left,or west,of thisboundary.Thenwe definetheangle~ thatthiseast

wall makeswith theverticalby

‘t - Xbtan(~) =

Yt - yb “(42)

That is, ~ > 0 if (~t- ~b) is in the firstquadrant,while~ < 0 if (Et - ~b)

is in the second quadrant. Then the vertical component of the east

wall-adhesionforceon the fluidis

fZE = tY6zCSANG*u&zcos(~ +@c) ,

and the surfacepressureis properlygivenby

24

(43)

[‘1p~ = - ~ [CSANG-sin(~)] ,i

(44)

Similarly,if Et and ~b definea linesegmentrepresentingthewest wall of a

partial flow cell, with the fluidlyingto theeastof thisboundary,thenwe

definetheangle~ thatthewestwallmakeswith theverticalby

tan(~) . - ‘t - ‘bYt - yb “

(45)

That is, in thiscase~ > 0 if (Et- ~b) iS in thesecondquadrantand ~ < 0

if (~t - ~b) ‘s in the firstquadrant.The verticalcomponentof thewest

wall–adhesionforceis stillgivenby

fZw = u 62 CSANG3 u 62 COS(~ + @c) , (46)

and thesurfacepressureis givenby

[1Ps = $ [CSANG- sin(~)] .i

(47)

Unfortunately,the implementationof the above physically correct

determination of Ps requires, in general, an expensive and complex

front-trackingprocedureto determinepreciselywhetherit is wall adhesion or

surface tensionthatis appropriateat any surfacecellboundaryat any timein

the calculation.In partialflowcellsin cylindricalgeometry, it is also

important to know on exactlywhichcellwalland at exactlywhat radiuseach

surfaceforceis delivered.Suchan increasein calculationalcomplexitywould

be incompatiblewithour goalof a simpleand robustalgorithm.Thuswe give up

the attempt at precise front tracking and precise evaluation of the

surface-tension,wall-adhesionforceand evaluatetheappropriatesurfaceforce

on eachwall of the computationalcellby meansof an algorithm,which has the

genericformin Cartesiangeometrytypifiedby

25

fzE = ~ ~z [AFEsin(~) + (1 - AFE)CSANGI ● (48)

It is theselatterexpressionsthatare used in theprogramto evaluateps.

When the codeescapesthegeneralsurfaceforcealgorithmsby settingthe

flagINW to INW = O (designedto providethe capabilityto treat walls with

porous baffles in them)then,forexample,AFE is thefractionalarea open to

flowand thesurfaceforceis taken to be a linear combinationof surface

tension and walladhesion. Even thoughsuchan expressionwillnot be correct

at any momentin any particularflow,it will represent the behavior of an

ensemble average (and, in somecases,a timeaverage)of flows. Thismode of

treatmentof AFE is

tension/walladhesion

cut at the complexities

Otherwise,INW = 1

and we willhaveeither

algorithmswill force

the simplest reasonabletreatmentof the surface

dichotomy,and is wellsuitedfora first,inexpensive!

of theporousbaffleoption.

(thiswillbe thenormalcasein a partial flow cell)

surfacetensionor walladhesion.That is, the computer

a choice AFE = 1.0 or AFE = 0.0. This forces any

wall-adhesionforce to appear on theappropriatewalland ignoresthe exact

radiallocation,and thus the exact numerical magnitude,of the actually

occurring wall adhesion. This permitsus to reducethecomplexitiesof front

trackingto a manageablelevel. In addition,we make the physicallyvery

reasonableapproximationthatec = O, whichfurtherreducesthenumberof cases

thatmustbe analyzed.

The basicdecisionalgorithmsare implementedby introducinganother cell

index NW(I,J) for partialflowcells,which,togetherwithNF and the F values

of theneighboringcells,determinesAFE and theotheranalogous quantities on

the othercellwalls. The indexNW labelsthegeometryof theobstacleto flow

in a partialflowcell. If NW = O, the ceil volumeis fullyopen to flow, but

the cellsurfacesmay be partiallyor whollyclosedto flow. WesetINW=Oin

thiscase,with the consequencesdescribedpreviously. Otherwise INW = 1, we

havea partialflowcelland NW describesthenatureof theobstacleto flow.

A seemingly unnatural orderingof thenumericalvaluesof NW is adopted,

whichsimplifiesthe teststo be madeon NW and NF. Specifically,we adopt:

NW= 1;

NW. 2;

when therightwall is closedand the top and bottom walls are

partiallyopen

when the left wall is closed and the topand bottomwallsare

partiallyopen

26

Nw = 3; when the topwall is closed and the left and right walls are

partiallyopen

NW = 4; when the bottom wall is closedand theleftand rightwallsare

partiallyopen

NW . 5; when the topand therightwallsare closed

Nw = 6; when thebottomand the rightwallsare closed

NW . 7; when the topand therightwallsare partiallyopen,but the other

wallsare open

Nw = 8; when the bottom and therightwallsare partiallyopen,but the

otherwallare open.

We returnto thediscussionof how AFE and theother

set when INW . 1. We startwithdefaultvaluesof 1.0

putssurfacetensiononlyon theappropriatecellwalls.

discussionof how AFE is changedto 0.0 (givingwall

This happensforNW . 2. It happensforNW . 3 or 4 when

cell wall flags are

for theseflags,which

We specialize to the

adhesion)whenNF = 3.

thecell to the right

is empty. It happensforNW . 5 when thecellbelowis empty,but for NW . 6

when the cellaboveis empty. It does not happen for M = 7. Finally, it

happens for NW = 8 when thecell to therightis empty. Analogousrulesare

givenfor theothercellwall flagsAFW,AFN,and AFS; other NF values also

generateappropriatevaluesforall fourcellflags.

In cylindricalgeometrytheaboveprescriptionsfor slabgeometrymust be

supplementedby an additionalcontributionto ps. In terms of the principal

radiiof curvaturein an axisymmetriccase,

[1ps . - a — 1

c‘q ‘

(49)

where R “Xy ~s theprincipalradiusof curvaturein thex-y planeand RT is the

principalradiusof curvaturein the~ direction.The expressionswe have just

described correspond t. the~y termonly. The RT termis evaluatedthe same

way as in NASA SOLA-VOF.10This termis of simpleuniversalform (thatis, it is

independentof the shape of thecurvein thex–y plane)and is unmodifiedby

wall-adhesioneffects. The near-horizontaland near-vertical cases are

consideredseparately. For a near-horizontalsurface,thesurfaceslopePFX =

tan(rs),where-rsis theanglethesurfacemakeswith thehorizontal.Then RT .

xi/sin(rs). The sign of RT is opposite that of pFX. For near-vertical

surfaces,PFY = tan(~s);onlynow Ts is theanglebetweenthe surface and the

27

vertical. ThenRT = (Xi+l/2 : Fi,j &xi)/cos(~s). The topsignsof the choices

y and T are takenif fluidis to therightof the freesurface,and the bottom

signs are chosenif the fluidis to theleftof the interface.The signof RT

is determinedb;.?FXas in thehorizontalcase.

In the code,a pressurepsatis addedto the ps calculatedabove. This

pressurerepresentsthevoidpressureappropriateto theproblembeingrun. For

example,in a passivetank-drainingproblem,psatis the liquidvapor pressure,

whileunderforcedtankdrainingconditions,psatis the totalforcingpressure.

In the lattercase,psatmay be takento be a functionof time?but thismustbe

done by hand.

For certain problems,namely those in which an initialsteady–state

meniscusis requiredin a rightcircularcylindricalsectionof a tank(such as

we have in the twosampleproblems),SUBROUTINEEQUIBcan be calledto generate

the initialsurfaceshape. We startwitha flatinterface,and the subroutine

then calculates the deviationsfrom flatness,T. Both T and the radial

coordinateC are scaledin unitsof the tankradius. The surface displacement

is calculatedby numericallysolving

Id

[

< (dT/dc).—1

Bo ~- 2 COS(TC)= O ,~ d< (1 + (dT/d~)2)1’2-

(50)

whereBo is theBondnumber. The boundary

dT/dC= tan(~c)at ~= 1. The constraint

2nI: C T dc = O

is enforced. The numerical method is

Hotchkiss.10

conditionsare dT/dC= O at C = O and

(51)

essentiallythe same as that of

For convenience,NASA-VOF2Dhas fivedifferentboundaryconditionoptions

on each sideof themesh. Theseoptionsare controlledby setting four flags,

KL, KR, KB, and KT, thatcontroltheleft,right,bottom,and top boundaries,

respectively.The choicesforKL are as follows:KL = 1, rigidfree-slipwall;

KL = 2, rigid no-slip wall;KL = 3, continuativeboundary;KL = 4, periodic

boundary(KRmustequalfouras well);and KL = 5, specified pressure outflow

boundary. The sameoptionsapplyto theotherthreeboundaries.The detailsof

how theseare implementedare describedin theSOLA-VOFreport,2or theymay be

28

gleaned from SUBROUTINEBC with littleeffort,so theywillnot be repeated

here. Othertypesof boundaryconditions,suchas the tank-drainhole in the

numerical example in Sec. III, must be suppliedby theuserat theend of

SUBROUTINEBC.

The finalconsiderationof thenumerical algorithm is stability. Since

many termsare evaluatedexplicitly,thatis,at timeleveln, the timestep 6t

must be smallerthana certaincriticalvalueto preventtheunboundedgrowthof

parasiticsolutionsof thedifferenceequations.Once themeshhas beenchosen,

severalrestrictionsare placedon &t to insurethatit is below the critical

value. First, material cannot move more thanone cellwidthper timestep.

Therefore,

(52)

where theminimumis takenovereverycellin themesh. Typicallyht is chosen

to be some fairly small fraction,say 0.25, of the minimum. Centered

differencing(a = O) is unconditionallyunstable,and it is necessaryto make a

at least approximately1.2 to 1.5 timeslargerthanthevalueof theminimum

functionin Eq. (52)dividedintotheactualat.

The explicitdifferencingof theviscoustermsalso limitsthe time step.

We require

For this stability limit,at can be quitecloseto thecriticalvalue,unlike

the convectivelimit[Eq. (52)].

Finally,theexplicittreatmentof thesurface-tensionforcesrequiresthat

capillary waves not travelmore thanone cellwidthin one timestep. A rough

estimatefor thislimitis

(J tit2<p 6X;

4 (1 + <) ‘ (54)

29

where&xm is theminimumcellwidthin eitherdirectionanywherein themesh.

All of the above time step controls except the choice of a are

automaticallysatisfied by theprogramif theuserselectstheautomatictime

step controlby settingAUTOT= 1.0. SettingAUTOT= 0.0 will permita constant

6t to be used. The automaticoptionalsoadjuststhe timestep to keep pressure

iterationsat approximately25 per cycle or less. In situationswhere the

pressure iterationshave not convergedafter 1000 iterations,or when an

advectivefluxexceedsmore thanhalf thevolumeof a cell, the time step is

automaticallycut in half independentof theAUTOTselection.Further,in the

caseof an excessiveadvectiveflux,thecycleis restartedwith the reduced&t.

Examplesof theuse of thenumerousoptionsof thiscodeare presented and

discussed in thecontextof severalspecificproblemsat theend of Sec. III,

PP” 45-47,and in Sec. IV.

III. PROGRAMDESCRIPTION

The NASA-VOF2Dprogramis highlystructuredso that individualcomponents

may be easily modified to fit specifi”cproblem requirementsor to accept

subsequentcodeupgrades. Thisapproachallowsgreaterefficiencyin operation,

simplifies problem setup,and facilitatescodemodification.WhileNASA-VOF2D

can run many problemsas is, in general a specific applicationwill require

changes to the logic for special inflow or outflow ports~ complicated

geometries,or unusualinitialconditions.Seldom does anything have to be

deleted; rather themodificationswillbe additionsto theexistingstructure.

Exampleswill be givenof how to easilyaccomplishchangesutilizingthe UPDATE

utility.

Figure5 displaysthecodelogicalsequence,whichis containedprincipally

in themain programroutineIIWI. The FILMSETblockcomprisesa seriesof calls

to system–dependentroutineswhichdefineand initializegraphicsoutputunits.

The user must substitutehis own comparableroutines or eliminate these

statements.SubroutineSETUPis calledto read theinputdata. Problemrestart

capabilityhas been incorporatedinto the code. If the input parameter

NDUMP> 0, SUBROUTINE

transferredback to the

initializationoccur.

meshand specifyzoning

interpret input data

30

TAPIN is called, dump tape7 is read,and controlis

maincalculationalloop. Otherwise,problem setup and

The meshgeneratoris calledto constructthe computing

variables.Next

and to produce

theobstacle generator is called to

the cellareaand volumeopen-to-flow

QSTART

I

! FILMSE”r I

I MESHSET \

-/

SETUPt=cycle=O

L

C!!4’1BC

PREssIT

EEL!oPriljIDEI-TADJ]

P““’’0;’0”%31

f:t*~lcycle=cycle● I

Fig. 5. ~.tOWduvd @h NASA- VOF2V

variablesthatare used throughoutthecalculationto representsolidor porous

interior boundaries.The fluidflowportionof themesh is theninitialized,a

freesurfaceconstructedif indicated,and variousparametersor variablearrays

set. Control is returnedto themain calculationalloop. Boundaryconditions

are implementedby SUBROUTINEBC, whichinterpretsthe cell BETA values, the

mesh input boundary flags, and thecellNF arrayvaluesand setsappropriate

quantitiesintointeriorobstaclecells,the fictitiousmesh perimeter cells,

and void cellsadjacentto freesurfaces.

Each cycle is started by calling SUBROUTINETILDE to calculatethe

provisionaltimen+l velocitiesusingtimen quantitiesfor advective fluxes,

pressure gradient accelerations,and ViSCOUS or body forces. Becausethe

velocityfieldwas changed,SUBROUTINEBC is calledagainto updatethe boundary

cells. The block PRESSIT representseither of the twopressureiteration

31

solutionmethodsavailable.SUBROUTINEVFCONVis nextcalled to advect fluid

utilizing the new velocitiesdeterminedfollowingconvergenceof the pressure

field. Boundaryvaluesare againupdatedwith thenew F’s . Because the the

fluidconfigurationhas changed,SUBROUTINEPETACALis calledto redeterminethe

locationand orientationof thesurfaceor surfaces,to identifysurface cells,

and to calculate the pressurein thesecellsdue to surfacetension. At the

same time,pressureinterpolationneighborsare designatedand the interpolation

coefficientscalculated. This completesthe timeadvancement,all quantities

now beingat timen+l. Outputindicatorsare tested,followedby an adjustment

of the time step and testingof problemend parameters.The problemtimeis

advancedalongwith thecyclecounterand controlreturnsto the beginning of

the calculationalloop.

It should be notedthatalthoughthisprogramevolvedfromSOLA-VOF,not

all of the capabilitiesof thelattercodeare available. In particular,the

particle-tracking scheme, the two-material capability, and limited

compressibilityhavenot yet beenmadecompatiblewithpartialflowareas.

A

IIWI

ASET

BC

CAVOVO

briefdescriptionof eachsubroutineappearsbelow.

(main program) Exercises logical control over the sequence of

calculations, initiatesoutput, and tests for problem termination

conditions.

(interior

parameters,

(boundary

boundaries

conditions;

obstacle generator) Generates the problem variables,

and flagsto implementthepartialflowareasfeature.

conditions)Setstheappropriatefluidvariablesat themesh

and interior”obstacles to produce specified boundary

sets velocitieson free surface and adjacentvoid cell

boundariesin theabsenceof realpressuregradients. Proper place to

incorporatespecialboundaryconditionsnot builtintocode.

(calculatesvoidvolume)Computesthevolumeof disjointvoid (F = 0.0)

regions.

DELTADJ(timestep

adjusts At

to converge

method.

DRAW (generates

adjustment)Computes

consideringstability

pressures;recomputes

maximum allowable at for stability;

limitandnumber of iterationsrequired

relaxationfactors(BETA)usedwith SOR

graphical output) Draws velocity vector plotswith free

surfaces;providesmeshplot;and callssystemutilityfor contour plots

of F and P.

DRWOBS (drawobstacles)Drawslinesaroundall obstacles.

32

DRWVEC (draw a vector)Providesa system-dependentcallopportunityto drawa

linebetweenpoints(Xl,yl)and (X2,Y2);plotsthesymmetric form of a

givenlineif requested.

DVCAL (additionalboundaryconditions)Providesfree-slipboundaryconditions

for all obstaclesurfaceflowcells.

EQUIB (generatesequilibriumsurface)Solves the two point boundary value

problem to obtain the equilibriumshapeof the freesurfacefor time

zero.

EXITLDC(exitmessage)Writesmessageupon terminationindicatingreason and

location.

FRAME (graphicsroutine)Drawsa framearoundcomputingmesh.

LAVORE (label void regions) Detects and labels(NFvalues> 5) all disjoint

(F = 0.0) regions.

MESHSET(meshgenerator)Generatesthecomputingmesh from NAMELIST/MSHSET/data

Providesthenecessarygeometricvariablesfor the code.

PETACAL(surfacetension)Determinesthelocalfree-surfaceorientation(slope),

the surfacecell type[NF(I,J)],interpolationcoefficients[PETA(I,J)]

and thesurfacepressure[PS(i,j)]exertedby surfacetensionforces.

NF = O fluidcell--novoidcellsadjacent

NF = 1 surfacecell--neighboringfluidcell to the left

NF = 2 surfacecell--neighboringfluidcell to the right

NF = 3 surfacecell--neighboringfluidcellon the bottom

NF = 4 surfacecell--neighboringfluidcellon the top

NF = 5 surfacecell--isolated,all adjacentcellsare void

NF > 5 voidcell--nofluidin cell

PLTPT (plot a point)Providesa systemdependentcall to graphicsroutineto

plota singlepoint(xl,yl).

PRESCR (conjugateresidualsolution) Utilizestheconjugateresidualmethodto

bringtheadvancedtime,pressure,andvelocityfieldsintoagreement.

PRESIT (successiveover-relaxationsolution)Utilizesthe SOR methodto bring

theadvancedtime,pressure,and velocityfieldsintoagreement.

PRTPLT (printsand plots)Outputroutinefor paperprints, film data prints,

and filmplots.

SETUP (problem initialization)Sets necessary constants,computesscaling

factorsand otherparametersforgraphics,calls mesh generator, calls

obstacle generator, initializescell variables,and computes the

relaxationfactorsusedwith SOR solutionand as cell-typeidentifiers.

TAPIN (restart)

continuea

TAPOUT (restart)

subsequent

Readsproblemvariablesand controlparametersfrom tape7 to

previousproblem.

Writesproblemvariablesand controlparametersto tape7 for

restartof problem.

TILDE (temporaryvelocity calculation)calculatesexplicitly provisional

valuesof thevelocitiesfromtimen advectionfluxes,pressures,viscous

stresses,and bodyforces.

VFCONV (volume

from the

fractionconvection)Computesthe advective fluxes of F(I,J)

newlydeterminedvelocityfieldand updatestheF(i,j)array.

vARIABLEsLISTEDIN cOMON (EXCLUDINGINpuTpARAMETERS)

ADEFM

BDEFM

CYL

DTSFT

DTVIS

DXMIN

DYMIN

DUDB

DUDL

DUDR

DUDT

DVDB

DVDL

DVDR

DVDT

EHF

EMF1

EM6

EM6P1

EM1O

EP1O

FCVLIM

34

Defoamoptionparameterdefinedin SUBROUTINEPRESCR

Defoamoptionparameter

Meshgeometryindicator(= ICYL)

MaximumDELTvalueallowedby thesurface-tensionforces

stabilitycriterion(DELTis automaticallyadjusted)

MaximumDELTvalueallowedby theviscousforces

stabilitycriterion(DELTis automaticallyadjusted)

VariableusuallyequalingthelocalmeshsizeDELX(i)

VariableusuallyequalingthelocalmeshsizeDELY(j)

y-derivativeof u at bottomof cellat timeleveln

x-derivativeof u at leftof cellat timeleveln

x-derivativeof u at rightof cellat timeleveln

y-derivativeof u a topof cellat timeleveln

y-derivativeof v at bottomof cellat timeleveln

x-derivativeof v at leftof cellat timeleveln

x-derivativeof v at rightof cellat timeleveln

y-derivativeof v at topof cellat timeleveln

Smallvalue,typically1.Oe-06,used to negateround-off

erroreffectswhen testingf = 1.0or f = 0.0

=leO-emf

=1.Oe-06

=1.0+EM6

=1.Oe-10

=1.Oe+10

Variableis used to

as data,it is also

controlsizesof

resetinternally

velocities;entered

FLG

FLGX

FNOX

FVOL

I

IBAR

IBAR

IBAR2

IEQIC

IMAX

IM1

IPL

IPR

ITER

J

JBAR

JBAR

JBAR2

JMAX

Pressureiterationconvergencetestindicator(=0.0when

theconvergencetestis satisfied,=1.0when the

convergencetestis not satisfied)

Volumeof fluidfunctionconvectionlimitindicator

(DELTreducedand cyclestartedover if limit

is exceeded)

Pressureconvergencefailureindicator(=1.(),

convergencefailedand DELT is reduced,=0.0otherwise)

Volumeof fluidin computationalcell

Horizontal(radial)cellindex

Numberof realcellsin x-direction(excludesfictitious

cells)

Valueof the indexi at thenext to thelastrealcell

in thex-direction(=IMAX-2)

=IBAR+2,specifiedin parameterstatement

(=IBAR+3,if periodicin x-direction)

Flag indicatingwhethersubroutineforequilibriumsurface

shapeis to be called

Totalnumberof meshcellsin x-direction(=ibar+2)

(=IBAR+3,if periodicin x-direction)

Valueof the indexi at thelastrealcellin the

x-direction(=imax–1)

Leftmostpressureiterationindexin x-direction

(=3 for constantpressureboundarycondition,=2 for

all othercases)

Rightmostpressureiterationindexin x-direction

(=ibarfor constantpressureboundarycondition,=iml for

all othercases)

Pressureiterationcounter

Vertical(axial)cellindex

Numberof realcellsin y-direction(excludesfictitious

cells)

Valueof the indexj at thenext to the lastrealcell

in they-direction(=jmax-2)

=jbar+2,specifiedin parameterstatement

(=jbar+3,if periodicin y-direction)

Totalnumberof meshcellsin y-direction(=jbar+2)

(=jbar+3,if periodicin y-direction)

35

JM1

JPB

JPT

LITER

NCYX

NDUMP

NFLGX

NMAT

NOBS

NOCON

NP

NPRTS

NREQ

NVOR

MESHX

MESHY

PI

PSAT

RCSQ

RHOD

RPD

SF

T

TWPLT

Valueof the indexj at thelastrealcellin the

y-direction(=jmax-1)

Bottompressureiterationindexin y-direction

(=3 for constantpressureboundarycondition,=2 for

all othercases)

Top pressureiterationindexin y-direction

(=jbarfor constantpressureboundarycondition,=jmlfor

all othercases)

Numberof iterationson thepreviouscycle

Calculationaltimecyclenumber

Tape 7 dumpsequencenumber. Set to zero to

skip taperestart. Otherwise,it mustequalthe

sequencenumberon tape7 to successfullyrestart.

Numberof cyclesthevolume-of-fluidfunctionconvection

limit(flgc)is exceeded

(a fossil)The numberof materials,alwaysset to unity

Numberof obstaclefunctionsbeinginput

Numberof cyclespressureconvergencehas failed

Totalnumberof particlescomputedto be in mesh

Numberof particlesin mesh,specfiedin parameter

statement(usedto set arraysize-–mustbe O)

Numberof voidregionsgeneratedin calculation

Maximumnumberof voidregionsallowed,specifiedin

parameterstatement

Numberof submeshregionsin x-direction,specified

in parameterstatement

Numberof submeshregionsin y-direction,specified

in parameterstatement

=3.141592654

Constantvoidpressureaddedto surface-tensionpressure

(canbe convertedby hand to a time-dependentdriving

pressure)

= O (forthepresentreport)

= O (forthepresent,one-materialcapability)

Degreesto radiansconversionfactor

Plotscalingfactor

Problemtime

Problemtimeat whichthenextgraphicaloutputis made

36

TWPRT

TANGLE

VCHGT

VELMX1

XMAX

XMIN

XSHFT

YMAX

YMIN

YSHFT

ACOM(l)

AC(i,j)

AR(i,j)

AT(i,j)

BETA(i,j)

COSO(i,j)

DELX(i)

DELY(j)

F(I,J)

FN(i,j)

IP(k)

JP(k)

LABS(5)

NAME(10)

NF(I,J)

Problemtimeat whichthenextprintedoutputis made

Tangentof contactangle,cangle

Accumulatedfluidvolumechange

Velmxnormalizedto minimummeshcelldimension

Locationof right-handsideof mesh

Locationof left-handsideof mesh

Computedshiftalongtheplottingabscissato center

theplotframeon film

Locationof the topof themesh

Locationof thebottomof themesh

Computedshiftalongtheplottingordinateto centerthe

plotframeon film

ARRAYSIN COMMON(EXCLUDINGMESHSETUPPARAMETERS)

Firstword in common

Fractionalvolumeopen to flowin cell (i,j)

Fractionalareaopen to flowon right(east)

wall in cell (i,j)

Fractionalareaopen to flowon top (north)

wall in cell (itj)

Pressureiterationrelaxationfactorin cell (i,j)

Cosineof anglefluidmakesunderwall-adhesionin cell

(i,j)when it is partialflowcell. Anglemeasuredwith

respectto vertical

Meshspacingof the i-thcellalongthex-axis

Meshspacingof thej-thcellalongthey-axis

Volumeof fluidper unitvolumeof cell (1,3)at time

leveln+l

Volumeof fluidper unitvolumeof cell (i,j)at time

leveln

Cell indexfor particlek alongx-axis

Cell indexfor particlek alongy-axis

Temporarystoragelocationfor t, for internaluse

Problemidentificationline

Flagof surfacecell (1,3)indicatingthe location

of its neighboringpressureinterpolationcell

37

NR(k)

NU(I,J)

P(i,j)

PETA(i,j)

PN(i,j)

PR(k)

PS(i,j)

RDX(i)

RDY(j)

RX(i)

RXI(i)

RYJ(j)

SINO(i,j)

TANTH(i,j)

U(i,j)

LJN(i,j)

V( i , j )

VN(i,j)

VOL(k)

X(i)

XI(i)

XP(k)

Y(j)

YJ(j)

YP(k)

ZC(20)

Labelof voidregion,k > 5

Flagof partialflowcell(I,J)indicatinggeometryof

curvedsurfacewall

Pressurein cell(i,j)at timeleveln+l

Pressureinterpolationfactorforcell (i,j)

Pressurein cell (i,j)at timeleveln

Pressurein voidregionnr(k)

Surfacepressurein cell(i,j)computedfromsurface

tensionforces

Reciprocalof delx(i)

Reciprocalof dely(j)

Reciprocalof x(i)

Reciprocalof xi(i)

Reciprocalof yj(j)

Sineof anglefluidmakesunderwall-adhesionin cell (i,j)

when it is partialflowcell. Anglemeasuredwith respect

to vertical

Slopeof fluidsurfacein cell (i,j)

x-directionvelocitycomponentin cell (i,j)at time

leveln+l

x-directionvelocitycomponentin cell (i$j)at time

leveln

y-directionvelocitycomponentin cell(i,j)at time

leveln+l

y-directionvelocitycomponentin cell (i,j)at time

leveln

Volumeof voidregionnr(k)

Locationof theright-handboundaryof the i-thcell

alongthex–axis

Locationof thecenterof the i-thcellalongthe

x-axis

x-coordinateof particlek

Locationof the topboundaryof thej-th cellalongthe

y-axis

Locationof thecenterof thej-thcellalongthe

y-axis

y-coordinateof particlek

Storagefornumberof contourlinesto be plotted

38

PROBLEMINPUT

Problem input data are furnished the codeby theNAMELISTformat-free

option. The sourcedeckprologuetabulates the several sets and identifies

their function. Mosthavedefaultvaluesspecifiedby DATAstatementslocated

in themain routineIIWIand SUBROUTINESETUPjustbeforethe first executable

statements.Probleminputdatasetsneedonlyreferencevariablesor parameters

whosevaluesdifferfromthesedefaults.

NAMELISTXPUTparametersspecifythematerialpropertiesof the fluid, the

solution method options, problem-dependentcode parameters,and output

requirements.

INPUTPARAMETERS(NAMELIST/XPUT/)

ALPHA Controls amount of donor-cellfluxing (=1.0 for full donor-cell

differencing,=0.0for centraldifferencing)

AUTOT Automatic time stepflag(=1.0forautomaticDELTadjustment,=0.0 for

constantDELT)

CANGLE Contactangle,in degrees,betweenfluidand wall

CON C.F.L.condition––cellwidthfractionmovedin timestep

DELT Time step

DTCRMX MaximumDELTusingconjugateresidualsolutionmethod

EPSI Pressureiterationconvergencecriterion

FLHT Fluidheight,in y-direction

GX Bodyaccelerationin positivex-direction

GY Ilodyaccelerationin positivey-direction

ICYL Meshgeometryindicator(=1 for cylindricalcoordinates,=0 for plane

coordinates)

IDEFM Defoameroptionflagon = 1, off = O

IDIV Divergencecorrectionflagon = 1, off = O

IMOVY Movieindicator(=1 formoviefilmoutput,=0 for otherfilmoutput)

ISOR Pressureiterationsolutionmethod(conjugateresidual= O, SOR = 1)

ISURF1OSurface tension indicator (=1 for surfacetension,=0 for no surface

tension)

ISYMPLTSymmetryplot indicator(.1 for symmetryplot,=0 forno symmetryplot)

KB Indicatorfor boundaryconditionto be usedalongthe bottomof themesh

(=1 for rigid free-slipwall, =2 for rigid no-slip wall, =3 for

continuativeboundary,=4 forperiodicboundary,=5 for constantpressure

boundary)39

KL

KR

KT

NAME

NDUMP

NPACK

NPX

NPY

XNuOMG

PLTDT

PRTDT

QVOL

RHOF

RHOFC

SIGMA

TWFIN

UI

VI

VELMX

XPL

XPR

YPB

YPT

Indicatorfor boundary

Indicatorforboundary

Indicatorfor boundary

Problemidentification

conditionalongleftsideof mesh (seekb)

conditionalongrightsideof mesh (seekb)

conditionalongtopof mesh (seekb)

Dumpnumberfor problemrestart

Flag to activatepacking(on= 1, off = O)

Numberof particlesin x-directionin rectangularsetup

Numberof particlesin y-directionin rectangularsetup

Coefficientof kinematicviscosity

Over-relaxationfactorusedin pressureiteration

Time incrementbetweenplotsand/orprintsto be outputon film

Time incrementbetweenprintson paper

Availableto specifyflowrate(inflowor outflow)if required

Fluiddensity(forf = 1.0 region)

-rhof(forthepresent,one-material,capability)

Surface-tensioncoefficient

Problemtimeto end calculation

x–directionvelocityused for initializingmesh

y-directionvelocityusedfor initializingmesh

Maximumvelocityexpectedin problem,used to scalevelocityvectors

Locationof leftsideof rectangularparticleregion

Locationof rightsideof rectangularparticleregion

Locationof bottomof rectangularparticleregion

Locationof topof rectangularparticleregion

NAMELIST MSHSET is used to definethecomputationalregiondimensionsand

the zoningwithinthatregion. The computingmesh is constructedfroma number

of submeshesdefinedin eachcoordinatedirection.The namelistinputspecifies

the boundaries,thenumber of cells, the minimum cell dimension,and the

convergencepoint of each submesh. Variable spacings are accommodatedby

linkinga groupof submeshestogetherto achieve any desired distributionof

cell spacing. This is donein thesamemannerin bothdirections.The number

of cellsis specifiedin eachsubmeshon eachside (i.e.!to the leftand to the

right) of the convergencepoint. Both cells directly adjacent to the

convergencepointwillhavea cellspacingequalto theminimumvalue specified

in the inputas DXMNor DYMN. The cellspacingis thenexpandedquadratically

from thesecellsto the leftand rightedgesof thesubmeshin accordance with

the desired number of cells (NXR,NXLor NYR,NYL)in the inputlist. If the

numberof cellsspecifiedon theleft (right)shouldproducea uniformcellsize

40

that is less than the minimumspecifiedcellsizeDXMN (orDYMN),a uniform

spacingis thenusedon theleft (right).The numberof cellsto the left and

to the right of theconvergencepointneednot be equal,but theremust be at

leastone on bothsides.

When twoor moresubmeshesare linkedtogether,it is imperativethat the

location of the leftedgeof therightsubmeshbe thesameas the locationof

the rightedgeof the leftsubmesh. In addition, large disparitiesin cell

spacingshouldbe avoidedwithina submeshand goingfromone submeshto another

by adjustingthenumberof cellsin thevarioussubmeshes.As a general rule

the spacingof adjacentcellsshouldnot differby more than10-20%. The aspect

ratio(DELX/DELY)for a givencellshouldnot exceed1.5 and should be greater

than0.67.

An example of theproperformatto be used to specifya mesh spanningthe

x-dimensionLW < x < RW withn submeshesis

NKX=n, XL= LW, XL2,XL3, ..*!xLn~

xc = Xcl,XC2, ...xcn, XR = XL2, XL3, ...,XLn,Rw,

NXL = NL1,NL2, ...,NLn, NXR = NRl,NR2~ ...,%9DXMN= DXMN1,DXMN2,...,DXMNn

in whichNLi representsthenumberof cellsto the leftof Xci and NRi is the

numberof cellsto the rightof Xci in eachsubmeshi, i = 1, 2, ...,n.

References [2] and [10]containseveralexamplesof meshgenerationinput

and resultantcomputationalmeshes.

MESH SETUPINPUT(NAMELIST/MSHSET/)

DXMN(n)

DYMN(n)

NKY

NXL(n)

NXR(n)

NYL(n)

NYR(n)

Minimumspaceincrementin x-directionin submeshn

Minimumspaceincrementin y-directionin submeshn

Numberof submeshregionsin x-direction

Numberof submeshregionsin y-direction

Numberof cellsbetweenlocationsXL(n)and XL(n)in

submeshn

Numberof cellsbetweenlocationsXL(n)and XL(n+l)in

submeshn

Numberof cellsbetweenlocationsYL(n)and YC(n)in

submeshn

Numberof cellsbetweenlocationsYC(n)and YL(n+l)in

submeshn

41

XC(n) x-coordinateof theconvergencepointin submeshn

XL(n) Locationof theleftedgeof submeshn [NKX+lvalues

of XL(n)are necessarybecausetherightedge (XR)of

submeshn is determinedby theleftedgeof

submeshn+l]

YC(n) y-coordinateof theconvergencepointin submeshn

YL(n) Locationof thebottomof submeshn [NKY+lvaluesof

YL(n)are necessarybecausethe topedge (YR)of

submeshn is determinedby thebottomedgeof

submeshn+l]

NAMELISTASETINprovidesinformationto the interior-obstacle-generating

subroutine. Interior obstaclesare definedas any nonflowregionswithinthe

computationalmesh. As with themeshgenerator,it is convenientto have the

necessary flags and parametersproduced automaticallyby the code for

arbitrary-shapedobstacles.The generatorusesa seriesof conic sections to

defineobstaclesurfacesplusan additionalparameterthatdirectsthegenerator

to designatecomputationalcellsas eitherflowor nonflowregions. The conic

sections may overlap one anotherand thisfeatureis utilizedto formcomplex

surfaces.

The procedureis as follows:Coefficientsof thegeneralconicfunction

F(x,y)= a2x2+ alx + b2y2+ bly + C2XY+ c1

are chosensuch thatsomeportionof thedefinedsurfacecoincideswith that of

the desired internal obstacle. Computationalcellspartiallyor completely

insidethe conicsurface[i.e.,F(xc,yc)< 0.0 , xc and yc being cell boundary

coordinates]are flaggedclosedto flow,partiallyclosedto flow,or fullyopen

to flowdependingupon theadditionalparameterIOH (=1.0 closed, =0.0 open)

associatedwith each function.Generallyadditionalfunctionscan be utilized

to removeunwantedobstaclecellsaddedby otherfunctions.For example, given

reasonablezoning,a 1.O-x l.O-cmobstaclein thelowerrightcornerof a 2.0-

x Z.()-cm meshwouldresultfromfl = -x+l.0~ IOH1 = 1.0, which defines all

cells, regardlessof y value, to the’rightof x = 1.0 to be obstacles,and

followedby f2 = -Y+l.0,IOH2= 0.0,whichremovesall obstaclecellsabovey =

1.0.

42

After all setsof obstacledatahavebeenprocessed,the computingmesh is

sweptand individualcellareaand volumeopen–to–flowvariables are assigned.

Aside from geometric factorsfor cylindricalcoordinates,theyrangefrom0.0

(notopen to flow)to 1.0 (fullyopen to flow).

OBSTACLESETUPINPUT(NAMELIST/ASETIN/)

IOH(n) Indicatorto add obstaclesinsidefunctionn (=1)or

subtractobstacles(=0)

OA2(n) Coefficientof x2–termin functionn

OAl(n) Coefficientof x-termin functionn

OB2(n) Coefficientof y2-termin functionn

OBl(n) Coefficientof y-termin functionn

OC2(n) Coefficientof xy-termin functionn

OCl(n) Coefficientof constanttermin functionn

Problemoutputconsistsof dataprintson eitherpaper or film and film

plots of the velocity field including free surfacecontoursand obstacles

outlines. Meshgenerationdata,partialflowareadata,and cellvariablesdata

for cycles O and 1 are outputautomaticallyto bothmedia. Thereafterthe

frequencyof the celldataprintsor plots is determinedby print or plot

incrementsspecifiedin theprobleminputdataset. In addition,graphicsystem

packagesare utilizedto securecontourplots of F and P. Calls to these

routinesappearin SUBROUTINEDRAW.

Provision is availableto producea plotfilethatcan be processedintoa

movie. The inputparameterIMOVYsuppressesall referencesto the film file

otherthanthosefromvelocityvectorplotroutines.

The main taskin usingthecodeis to formulatetheproblemcorrectlyand

to presentthe codewith properinput. The theoreticalstudy or experimental

simulationshouldbe analyzedas to the regionsof principalinterestand a mesh

constructedof sufficientfineness to resolve the phenomena of interest.

Unconfined flows require attention to theselectionof appropriateboundary

conditionsand a mesh largeenough to minimize boundary perturbations. The

importanceof physicalprocessesnot specificallyaddressedin thecode should

be evaluated. Someof theproblemcontrolparametersare best determined by

experience.

Table2 is an exampleof a typicalinputdataset. The code is to simulate

a near-fullcylindricaltank with hemisphericalbottom draining under the

influence of reduced gravity,surface-tension,and wall–adhesionforces. The

43

TABLE 11

EXAMPLE INPUT

$XPUTNAME=55HH0TCHKIss CASE 1 4/28/85NDUMP=O, ISOR=O, NPACK=O. OVOL=3.139E01, IOEFM=I,DELT=.0002, GY=-14.72. ICYL=l. DTCRMX=0.0004,CANGLE=5. , SIGMA= 18.6, IsuRFlo=lt IEQIc=lwXNU=4.43E-03. RHOF=l. 58. EPSI=l.E-03*VELMX=1O. , TWFIN=2.301, PRTDT=I.E+IO,PLTDT=.05, FLHT=6. , ISYMPLT=I, AUTOT=I.$END$MSHSETNKX=2, XL=O.O, .4.2.. X:;:3261 .9. NxL=ft8. NXR=l, 1, DXMN=O.2.O.1.NKY=I, YL=O.0, 7.2, - . . NYL=18, NYR=18, DYMN=O.2,$END$ASETINNOBS=2.OA2(1 )=-IOBI(2)=-1$ENO

dischargeflow

. . OB2( 1)=-1., OBI( 1)=4., IOH(I)=l. ,

. . OC1(2)=2. , IOH(2)=0.

rateis specifiedand is constantwith time. Input and outputwill be in CGS units, although any consistentset of unitsis allowed.

Referringto the tabulatedinputparameterssection, note that the conjugate

residualmethodwas chosen(ISOR= O), thepackingoptionis on (NPACK= 1), and

thevolumetricflowratespecified(QVOL= 31.39). The initialfluidvolume in

the tank is specifiedby givingthe fluidheightabovethe tankbottom(FLHT=

6.0). The surface-tensionflagis on (ISURF1O = 1), cylindricalcoordinates

chosen (ICYL= 1), theequilibriumsurfacecalculationis requested(IEQIC= 1)

alongwithautomatictimestep”adjustment(AUTOT= 1). The initial DELT is

computed from the outflow rate, dischargeopeningarea,and mesh spacingto

satisfystabilityconstraints.The convergencecriterion (EPSI = 0.001) is

picked to be sufficientlysmallto assurethatnegligblepressureerrorsresult

fromnonzero(VOg)’s. Paperoutputis suppressedby thelargeprintinterval.

Figure6 displaysthecomputingmeshgeneratedfrom MSHSET data and its

mirrorreflectionacrossthecylindricalaxis. This fullcrosssectionplot for

cylindricalgeometryis obtainedwhen theinput parameterISYMPLT = 1. The

x-coordinatehas twosubmesheswitha totalof 11 realcells,thespacingbeing

finernear the tankwall. The y-coordinatehas a singlesubmesh with 36 real

cells. The curved boundary at thebottomwas drawnfromcodevariablesand

parametersproducedby theobstaclegeneratorsubroutinefrom the ASETIN data

set. Computationalcellswhoseareasare bisectedby the curveare definedto

be partialflowcellswhoseboundariesare identifiedas partialflowareas and

treatedaccordinglythroughoutthecalculation.Curvedboundariesare simulated

44

withinthecodeby use of thesepartialflowareas. Computingmeshcellsto the

right and belowthecurveare designatedobstaclecells,while thoseaboveand

to the leftare fluidflowcells.

Figure7 showsthe timezerofluidconfigurationgeneratedfromXPUT data.

The initial surface shape is furnishedby SUBROUTINEEQUIB,whichsolvesthe

two-point boundary value problem for the equilibrium position of the

free-surface.The parametersof theequationare the contactangle(cangle)and

theBondnumber(pgR2/u,where R is the tank radius). This free surface

configurationis utilizedto specifyvoidcells,partiallyfilledfluidcells,

and completelyfullfluidcellsin thecomputationalmesh. The freesurface is

representedin the plot by the F = 0.5 contour, whichgenerallydoes not

correctlydisplaythe surfacemeniscusat thewalls.

Table3 containscodechangesnecessaryto run theexample. Programs for

maintainingand updatingsourcedecks(UPDATEor HISTORIAN)afforda convenient

way to incorporateproblem-dependentmodificationsinto the executable code.

These particularmodificationsarise since we wish to simulatea smallpipe

TABLE 111

EXAMPLE COVE MODIFICATIONS

*IDENT HOTCHI●D.COMMONI.2.L

c SET MESH DIMENSIONS - IF CR SOLUTION : IBAR2=IMAXc

PARAMETER(IBAR2=13 ,JBAR2=38,MESHX=3.MESHY=2.NVOR=25c●I,ASET. 171

LJBAR2=JMAX

NOBD=201

cc SET AREA ANO VOLUME OPEN TO FLOW FOR OESIGNATEO OUTFLOW CELLSc

DO 2400 1=2,2AT(I, l)=l.OAC(I, l)=l.OAC(I,2)=1.O

2400 CONTINUEc=I,BC. 164cc SET SPECIFIEO OUTFLOW VELOCITIES FOR DESIGNATED OUTFLOW CELLSc

DO 500 1=2,2U(I.1)=0.OV(I.1)=-OVOL/(PI=X(2)=*2)

500 CONTINUEc~I,PRESCR. 137cc INSERT SPECIAL TEST IN CONJUGATE RESIOUAL SOLUTION ALGORITHMc TO RECOGNIZE OUTFLOW CELL ANO CALCULATE PROPER COUPLINGc

IF(I.EO.2.ANO.J.EO.2) GO TO 40c*I,PRESCR.81

IF(F(I- l,d+l).LT.EMF.AND .13ETA( I-1.J+I).GT.O.0) GO TO 5IF(F(I- 1.J).LT.EMF.ANO .BETA(I- l,J).GT.O.0) GO TO 5IF(F(I- l,J-1).LT.EMF. AND.BETA(I -1.J-1).GT.O.0) GO TO 5IF(F(I, U-1 ).LT.EMF.ANO .6ETA(I, d-1).GT.O.0) GO TO 5IF(F(I+l .IJ-1).LT.EMF .ANO.BETA( I+l.J-1).GT .0.0) GO TD 5IF(F(I+l,J) .LT.EMF.ANO .BETA(I+l, .J).GT.O.0) GO TO 5IF(F(I+I,J+l ).LT.EMF .AND.BETA( I+l,J+l).GT.O.0) GO TO 5

45

locatedon the tankbottomat the cylindricalaxis, whereas the code lacks

provisionfor partiallyopenmeshboundaries.

First the PARAMETERstatement(no. 2 of COMMON1)is replaced,specifying

theminimumstoragerequirementsfor thevariousarrays. A commentis included

regarding theselectionof theconjugateresidualmethod. Next,statementsare

insertedat theend of SUBROUTINEASET to redefinethecellarea open to flow

variables for celli = 2, j = 1 so thata dischargeportis created. A second

insertionat theend of SUBROUTINEBC specifiesthe constant outflow velocity

for this port. Finallya changeis requiredin SUBROUTINEPRESCRto correctly

applyoutflowvelocityboundaryconditionsfor discharge port cells in the

pressureiteration.

Table 4 lists dimensionalvariables resultingfromthemeshgenerator.

Spacingin thex-directionis nonuniform,whereasthe y-directionis uniform.

Table 5 is a partiallistof theover-relaxationparameters,area,and volume

open-to-flowvariablesassociatedwitheachcell. BETA(i,j)= -1.0 identifies

cells completelyclosedto flow. Also tabulatedare thesineand cosineof the

angles(madewith thevertical)of the surface of the partial flow volumes

representingcurved walls. Table 6 tabulatesproblem control, mesh,and

obstacleinputdataor parameters.A completecycleprintis includedin Table

7 as an example of codeoutputand as a checkoutaid. This is the solution

afterone timestepwhen the flowfieldis establishedand the pressure and

velocity fieldsare consistent.Surfacepressuresare definedonly for surface

cells [0 < NF(i,j)< 6] and are negative indicatingthat the unbalanced

surface-tensionforcesare directedupwardand actingto straightenthesurface.

Cell i = 2, j = 30 has F = 0.111and NF = 3 showing that the surface passes

through and is nearly horizontal,and thatfluidis in its bottomportion.

Figure8 is thevelocityvectorplotat thistime.

Iv. SAMPLEAPPLICATIONS

In thissectionwe showresultsfor fourexamples. The firsttwocasesare

a fallingslugand a sloshingtankof water. Thesecaseshaveknownanalytical

solutionsand are usefulfor testingtheperformanceof theNASA-VOF2D program.

We thendiscussa tank-drainingproblemand a fuelreorientationproblem.

We first consider the one-dimensionalfalling slug problem. Both

cylindricaland Cartesiancaseswere runwith thesameresults. The problem is

to consider a slug of fluid well above the bottomof a tallcylindrical

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72

container. The voidbehavesas a vacuum. It starts from rest and undergoes

free fall untilit hits thebottomof thecylinder,whereit comesto restand

establishesthehydrostaticpressureprofile.

The computationalgrid is 3 zonesin theradialdirectionand 13 zones in

the axial direction. Initially thebottomof theslugof fluidis fourcell

heightsoff thebottomof thecylinder,and theslugis six cells deep. The

cells are squares 0.5 cm on eachside. The fluiddensityis 1.0g/cm3,while

viscosityand surfacetensionare neglected.The gravitationalaccelerationin

the axial directionis -980 cm/s2. The velocityand altitudeof thebottomof

the slugduringfreefallare

v = -980 t (55)

and

s = 2 - 490 t2 , (56)

where t is the timein seconds. The incompressibilitycondition requires the

velocityin theslug to be uniform.

Figure9 showsthevelocityof theslugas a functionof time. The problem

was runwith a = 1.0and,as discussedearlier,theresultsare plottedwith Fij

evaluated at n+l/2the correcttime(i.e.,F~j is reallyFij ). The solidline is

theNASA-VOF2Dsolutionwith IDEFM= 1, the finely dotted line is a second

numerical solution but with IDEFM = O, and thedashedlineis theanalytic

solution. The coarselydashedlinedisplaysthe performanceof a correction,

changing thedefinitionof a free-surfacecell,to thebasicSOLA-VOFalgorithm

addedby hand to accommodatethisparticularapplication.This curvestayswith

the analytic solutionverywell,but thealgorithmgivingthisresultdoesnot

have thegeneralityrequiredfor permanentinclusionin thecode. The numerical

solutions used the CR method to find the pressures.The threesolutions

coincidethroughcycle120,whichis 0.0531s. This is expectedas the program

shouldget theexactvelocitybecauseit is linear in t. However, the slug

shouldhit thebottomof thecylinderand come to a stopat 0.0639s ratherthan

on cycle121 as in theIDEFM= O solution.

In theIDEFM= 1 solution,the fluidslowsabruptlyas it entersthe bottom

row of cells,but thecellscontinueto fillslowly. This incorrectbehavioris

73

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—1

II

II

II

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–50

-700.00 0.02 0.04 0.06

L(s)

due to a peculiarityin theway theprogramdefinesa freesurface, and it is

found in all versions of thevariousVOF programs. A cellcontaininga free

surfacemust by definitionhaveat leastone emptyneighboringcell. When our

one-dimensionalslug firstcrossesintothebottomrow of realcells,theyhave

no emptyneighbors.Thus theprogramtreatsthemas if they were full, even

though they are onlypartiallyfull. This quirkof themethodwill not affect

many problems,suchas thenext twodescribed in this report, but it could

significantlyaffectresultsfor propellantreorientationproblemswherejets of

fluidimpacton dry tankwalls.

The IDEFM= 1 optionintroducedinto the pressure solution algorithm to

close small voids was modifiedto reducetheseverityof thisdifficulty.We

raisedthe product

minimum test to

celland its eight

changeis to force

function(i.e.,to

ADEFM*EPSIto a very largevalue(106) before applying the

determine if the cellbeingcomputedhad F < 1.0 and if that

surroundingneighborsall had NF . 0. The result of this

thoseparticularcellsto choosethe secondoptionin theMIN

closeup in l.O/BDEFMcycles). Otherwise, the gap in our

presentsolutionfillsfar tooslowly.

75

Figure10 showstheheightof thebottomof theslugas a functionof time.

Consideringthe large time steps, approximately0.0005s, the accuracy is

acceptable until the fluid stopsprematurely,for theIDEFM= O and 1 cases.

The coarselydashedlineoverlaysthe analytic curve. The behavior of the

program is consistentwith theexpectationthatthe time-marchingerrorsare

firstorderin &t.

The programdoesa verygoodjob of predictingthepressurefield in this

problem. During the free fall,thepressureis zero in the fluidbecausewe

assumethevoid pressureis zero. On cycle121of theIDEFM= O case,the fluid

must be broughtto a halt in one timestep. The waterhammereffectcausesthe

pressureat thecenterof thebottomrow of cellsto rise to 6.2 x 104 dyne/cm2.

On all subsequentcycles,thehydrostatichead is obtained,withagreementto at

leastfourdigitswith theanalyticalsolution

P=980d ,

0.00 0.02 0.04 0.06

t(s)

(57)

2

t5

1

05

0

.“...—..

76

whered is the fluiddepth. The conjugateresidualsolutionagreesto the six

digitsprintedby theprogram.

In theIDEFM= 1 solution,thepressurein thebottomrow of cellspeaksat

4.2 x 104 dyne/cm2on cycle121,dropsto 5000on cycle122,and relaxes toward

the correctsteadyvalueof 2695as thevoidcloses.

The second testcaseis thesloshingmotionof a tankof water. The tank

is 10.0cm wideand 6;5 cm deep,and it is coveredby a uniform20 by 13 mesh.

The wateris 5.0 cm deepon theaverage. The initialconditionis the fluidat

restwitha 0.5 cos(nx/10)perturbationof the surface. The gravitational

accelerationof 980 cm/s2startsthe fluidsloshingbackand forthacrossthe

tankin a mode in whichthedeviationof thesurface height from the average

depth is always proportionalto thecosinetermin the initialcondition,and

the sloshperiodis

P = 2n/(gk tanh(kD))l’2= 0.36417s ,

whereD = 6.5 cm is theaveragefluiddepth,g = 980 cm/s2is

of gravity, and k = rt/10is the horizontalwavenumber

disturbance.12

(58)

the acceleration

of the surface

Four typicalvelocityvectorand free–surfaceplotsare shownin Fig.

The upperleftpanelshowsthe initialcondition,and thenext threepanels

separatedin timeby 0.65s. The analyticalsolutionfor the free surface

marked by asterisks. The lower right panel(t = 1.95s) is just after

11.are

is

the

beginningof the sixthoscillation,and the computationalfree surface still

agrees as well with theanalyticalsolutionas it does for all earliertimes.

Thereis a slightdisagreementat all times,even t = 0.0,whichis an artifact

of the contour-plottingalgorithmused to plot the computationalfreesurface.

This artifactappearsto be the limitingfactor in how well the calculation

seemsto agreewith theanalyticalsolution.

The next example was Case 1 previouslycomputedby Hotchkiss.10 This

problemis the testcasedescribedin Sec. III. It is a drop-towertest of a

glassvesselwitha radiusof 2 cm in thecylindricalsection,theworkingfluid2is a freon,and thegravitationalaccelerationis –14.72cm/s . We find that

NASA-VO.F2D produces results in much better agreementwith theexperimental

measurementsof the locationof the freesurfacereportedby Symons21, as shown

in Fig. 12, thantheearlierresultfromNASA SOLAVOF.

77

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3.5

3.0

2.5

2.0

I .5

I .0

0.5

0 EXPERIMENTAL— CALCULATED

n’”b%~ \ WALL CONTACTo\

o~ POINT LOCATION

\

CENTERLINE 0LOCATION %

d10\ic

1-a ‘O 0.5 1.0 1.5 2.0 2.5Id> TIME

The fina1 example is a simula-

tionof a drop Lower fuel reorienta-

tionexperiment.A sma11 cylindrical1

container(r = 2 cm, 1 = 9 cm) with

hemispheric1 ends,half fullof liq-

uid (ethanol), and in free fa11 is

suddenly subjected to a sma11 con-

stant downward acceleration. The

liquid starts out at the top end of

the tank. The free surface assumes

an equilibriumshape under the influ-

ence of surface-tension and wa11-

adhesion forces. Following applica-

tion of the additional force, the

liquid flows along the sides of the

container to the opposite end, meets

at the axis, and jets upward. The

Bond number of the flow is 4.1, the

liquiddensity0.789 g/cm3, surface

tension coefficient22.33 dyneicm,

kinematic viscosity coefficient

0.001095cm2/a, and the appliedac-

celerationis 29.0cm/s2.

Table 8 liststheprobleminputdataset used to generateand executethis

problem. Table9 is a listof theUPDATEcodemodificationsnecessaryto adapt

the code logic for this application,as was shownearlierin the example

problem. Figure13 displays plots from the calculationat selected times

showingthe fluidflowand the free-surfaceconfiguration.Problemtime t = 0.0

correspondsto theexperimentaltimewhen the additionalaccelerationoccurs.

Sketches of the approximatefree-surfaceshapeat thesame timesare shownin

Fig. 14 takenfroma 16-mmmovieof thedrop-towerexperiment.Thesedatawere

obtainedwithoutdensitometeror computeranalysisof the filmand are therefore

very subjective.Howeverthe codesolutionroughlyreproducesthe actual flow

history,differingmainlyin thedetailsof thejet formation.

79

TABLE Vlll

FUEL REORIENTAT1Otd

%XP1lT

1hlPut-

ilAi4i=60HFUEL REORIENTATION PROBLEM BOND=4. I NASA-VOF2D SRCRPT SOURCE

D~LT=.o/soR;?~ .QSoNDUMP=O QVOL=O.O. IDEFM=I.

ICYL=I. IMOVY=O.OMG=I .?.CANG-E=O. ;5. SIGMA= 22:33. ISURCIO=I. IEQIC=l.XNU=~.52E-02. RHOF=O.?89, EP~l=i.E-c)3,VELMX=3. . TWFIN=f .0. PRTDT=I.E+IO,!JLTDT=.05. FLHT=4.5. ISYMPLT=I, AUTOT=I.SEND$MSHSETNKX=2. xL=O.O. .4.2.. XC=0.2,1 .9. NXL=I. lO. NxR=l. l. DxMN=o.2.0.10NKY=l. yL=o.o, 9.o, yc=4.5, NYL=29. NYR=20,DVMFJ=().225

$END$ASETINNOBS=4.OA2(1)= 0.0. 0.0. 1.0. 1.0. OA1(l)=O-O. 0-0. 0.0. O.o!OB2(1)= 0.0. 0.0. t.O. 1.0, OBl(l)=l -Os-l-o. -4.0. ‘14.o.OCI(I)=-2.O. 7.0. 0-0. 45.0. OC2(I)=0.O. 0.0. 0.0. 0.0,IOH( 1 )=1. 1,0.0.$ENC

TABLE 1X

FUEL REORIENTATlON CODE MQVIFICATIONS

oIDEN7 REOMOD=!J.COMMON:.2

PARAMETER (IBAR2=15.~BAR2=42 .MESHX=5.MESHV=5 .NVOR=200. N060=20)=i.EOIJIe.7

E~S=l.OE-8-G.OVCAL.5

OATA ISLIP 1_C,.IIb’i.2I6

old=<=C.iIWI.224

40 CONTINUE-0.SETUP.60,6 i

BONO=-0.0000Of-:.sETUP. ’17

DELREO=XIIM1 )iFLOAT(NEOU16-1 )=C.SETUP. 121 . 128

LuD=FLoAT(NEouIB-1 )=X( I).’XIIM1 )+1.000001LLOW=FLOAT( NEOUIB-t )*Xli-1 )/XIIMl1+1.000001LLOU=MAXO(LLOW, 1 )LUP=MINO( LUP.NEOUIB )H1=SFLHT-UI LLOW. 1 )-X(IM1 )H2=SFLHT-U(LUP, l)-X( IMt)IF(Y(LJ ).LT.HI.AN9.Y (LJ).LT.H2) GO TO 94IF(V(d- 11.GT.Ht.ANO. Y(J-1).GT.H2) GO TO 110SUM-O-O00 92 K=LLOW.LUPH3=SFLHT-U(K, 1)-X(IM1 )IF(H3.LT.VIJ-1)) H3=Y(IJ-t)SUM=SUM+fIMAXlf Y(U)-H3. 0.0 )

92 CONTINUEF(I.J)=SUM-OELREO*ROX( I )=RDY(J)GO TO ?1O

94 c(x.~)=o-o

=D.SETUP. 199P(:O 1 )=0.0

=O,SETUP.201 .204RHOYA=[AMINIIFI I.U-l ).O.5)=OELY(U-I )+AMAX1(C.O.F( 1.J)-O.5)=

! DEL} (J))-RHOFPl:.J)=P(I.~-l )+GV*RHOtA

80

t-o.o

Fig.

t=7. o

t=ll.30 t=o.so

t=Oe80

. . . . . . ...,... .. . . . . . , . . ....... . ! ! 1, . . . .n.... .....- -..

t=o .90

.FUEL REORIENTfl TION PROBLEM BOND NUMBER 4. t

t=O.60

t= 1.00

t=o.0 t=r-1,.3 t=o .5 t=O ,6

t “d.7 t =11s t=l],9 t= I ,0

DR~f’ TONER FILII i’l(3T9 BOND NUMBER4.1

.

The

perience

82

authors wish to thank R. S. Hotchkiss for sharing his insight and ex-

gained from similar studies.

APPENDIXA.

THE PARTIALCELLTREATMENT

The partialcell treatmentmay be motivated by consideringit to be a

special case of two–phaseflow. If we specializethe two-fluidequationsused

in theK-FIXprogram22,23to thecasewhereone of the fluidsis fixedin space

and time,and wherethedensityof theother,moving,fluidis constantin space

and time,thenthe continuityand momentumequationsfor themovingfluidare

v“(e:) so (A-1)

and

a(e~) Ku

[1

ea

at+vo(e~~) =e[g-v(plp)] -y+v”.y , (A-2)

where(3is thevolumefractionof theworkingfluid(thatis, the fraction of

the volume occupied by the motionless component),K is a couplingconstant

characterizingthedragbetweeninterpenetratingfluids,and : is the viscous

stress tensor. Equation (4) completes the set of equationsthatwe will

require.

The programis basedon nonconservativemomentumequations,a choice made

for SOLA-VOF2 on thebasisof an incompleteargumentbasedon truncationerror

analysis. This choicehas beenretainedin thepresentprogram,and it has the

fortuitousproperty of allowing us to omitany mentionof the partialcell

treatment(thatis, e) in themomentumequationsolver. If we makeuse of the

assumption that e is independentof time and use Eq. (A–1)to derivethe

nonconservativeformof Eq. (A-2),we obtain

(A-3)

Similarly,thenonconservativeF, Eq. (4),may be castinto conservativeform,

Eq. (32),by usingEq. (A-1)and the time-independenceof 0.

The connectionbetween the two-fluidequations and the partial-cell

treatmentis made by specializinge to a situationwhere0 is a step function

with valuesof O and 1 in theobstaclematerialand movingfluid,respectively.

Firstwe note that~ =0 in theobstaclematerialand 0 is constant piecewise.

Therefore we neglect the termsin squarebracketsin Eq. (A-3)becausewe are

interestedonly in what is happeningin the fluid. Theseterms serve only toenforce y = O insidetheobstacleand eitherg = O or ~gt~~ = O on its surface.

The resultis thatEqs. (2)and (3)are themomentumequationsthatwe desire.

The finalpartof theconnectionis a set of rulesfor includinge in the

difference equations.We do thisby introducingthreearraysinto the program.

First,ACi,jis the fraction of cell (i,j) open to fluid, times 2nri in

cylindricalcases. For cases wherea cell-centeredvalueof rO is required,

AC. is used. Second,ARi+l/2,jl,j is thefractionof theareaof the right face

of cell (i,j) that is open to flow,times2~ri+1/2in the cylindricalcase.

Thisarrayis usedas thevalueof rO on the cell face. Finally, the array

‘Ti,j+l/2is the fractionof theareaof the top faceof cell (i,j)thatis open

tO flOW, times2nriin thecylindricalcase,and it is usedas rO for the top

cell face. These ruleshave thenice featurethatthe finite-differenceform

for thedivergenceof thevelocity,Eq. (20),exactlyconserves the volume of

fluidflowingthroughthecellfacesif one assumes~ is constantovereach cell

face. This procedureis admittedlynot rigorous,but we expect it to produce

the correctgeneralfeaturesof flowin nonrectangulargeometries.

The use of the partial cell variables AT, AR, and AC appearsto be a

powerfuland simple procedurefor embeddingthin screens and baffles as well as

solid obstacles in the grid. Baffles and screens can be approximatedas

zero-thicknesssurfaceson cellfaceswithappropriateflow losses, including

the K termin themomentumequation.However,additionalmodelingeffortswill

be requiredbefore this capabilitycan be added to the program. In addition,

code modifications will be necessary to correctly account, through the

fractionalarea/volumefunctions,forwallshearstresses(or theirabsence) on

curvedboundariesembeddedin thecomputinggrid.

84

APPEWIX B

PROGRAM LISTING

85

.COMDECK , COMMLINiPARAMETER (IBAR2=40. JBAR2=40.MESHX=20, MESHY=20,NVOR=25,NOED=20 )

cc =--= NOTE *==== ISOR=O (CON&lGATE RESIDUAL SOLUTION) REWIRES IBAR2c =IMAX ANO dBAR2=JMAX DUE TO USE OF CRAY SYSTEMc UTILITIES FOR VECTOR AND MATRIX OPERATIONS

PARAMETER (MSI-M=MESHX+I. MSHY=MESH’f+l)c

COMMON /FV/ ACOM( 1). UN(IBAR2.LJBAR2). vN(IBAR2.LJBAR2). pN(IBAE2, FN(IBAR2.JBAR2). U(IBAR2.&AR2), V(IBAR2.d3AR2). p(IBAR2

; ::;%, F(IBAR20dBAR2). PETA(IBAR2 .LJBAR2). BETA(13AR2, JBAR2), NF3 (IBAR2,J3AR2), PSI IBAR2,JBAR2), AR(IBAR20d3AR2).4 AT(IBAR2. LJBAR2),AC( IBAR2,d3AR2)

cCOMMON /ME/ X(IBAR2), XI(13AR2), RXI(IBAR2), OELX(IBAR2). ROX

1 (IBAR2), Rx(IBAR2), Y(dBAR2~, YJ(J13AR2). RYU(IJBAR2), DELy(JaAR2).2 IEOIC. NDUMp. (IVOL, CON, FCVLIM. ROY(J3AR21, XL(MSHX). XC(MESHX).3 OXMN(MESHX). NXL(MESHX), NXR(MESHX). YL(MSHY), YC(MESHy). OyMN4 (MEsHY). NYL(MESHY), NYR(MESHY). ZC(20), R(IBAR21. RI(IE3AR2).5 COSO(IBAR2,JBAR2), SINO(13AR2,JBAR2). NW(13AR2.JBAR2)

cCOMMON /PV/ NR(NVOR), PR(NVOR), VOL(NVOR), NAME( IO), FVOL

cCOMMON /IV/ IBAR, dBAR, IMAX, dMAX, IM1, LJMI, NKX. NKY, NCYC. DELT

1 T, AUTOT, PRTOT. TWPRT, PLTDT, TWPLT. TWFIN, FLHT. XNU.2 AHOF, NREG. VCHGT, ISURFIO3 , SIGMA, CANGLE, ICYL, CYL, GX, GY, UI, VI, OMG, ALPHA, KL4 , KR, KB. KT, ITER, EPSI. FLG. FLGC. FNOC, NOCON, NFLGC,5 ISYMPLT. IMOVY. VELMX, VELMXI, XSHFT. YSHFT, XMIN. XMAX. YMIN,6 YMAX, SF, YPB. YPT, IPL, IPR, JPB, dPT, DTVIS8 .l.d,OUOR.DUOL.OUDT, OUOB,DVDR,DVDL,DVDT .DVDB7 .OTSFT.OXMIN.OYMIN.PSAT .LITER. ISOR, IDEFM.NPACK. LABS(5)8 ~ADEFM1f30EFM.0TCRMX. IDIV

cCOMMON /CONST/ EMF, EMFI, EM6, EMIO, EPIO, PI, TPI, RPD, EM6PI

1CbMk%l/OBS/ NOBS, OA2(NOBO), OAI(NO13D), OB2(NOBD), OBI(NOBD), OC2

1 (NOBO), OCI(NOBO). IOH(NOBO)

PROGRAM IIWI (INpuT. TApE5=INPUT.OUTPUT ,TAPE6=ouTPuT.TAPEi, TAPE81 ,TTY.TAPE59=TTY)

cc ========-===-==*=’W== ============”-==’===’ ==--*-=-*=--==-==-====”=== =-C -===-=======””====”=” ==-==-=-===--==-======= =========*==.===.==== -=--=cco?)RIG1-iT. 1985. VIE REGENTS 9= TI-IE UNIVERSITY Or CALIFORNIA.c THIS S2FTWARE wAS !JRCDJCE2 UNDER & u- S- GOvERNM~NT CoNTRAC?c IW-7405-ENG-361 BY THE LOS ALAMOS NLTIONAL LABORATOR}. WHICH.. IS OPERATED 3V THE UNiVEi?51T~OF CALIFORNIA FOR THE u. S.

DEPARTMENT OF ENERGY. THE U. S. GOVERNMENT IS LICENSED TO USE,; REPROOUCE, ANO TO DISTRIBUTE THIS SOFTWARE. PERMISSION ISc GRANTEO TO THE PUBLIC TO COPY AND USE THIS SOFTWARE WITHOUTc CHARGE. PROVIDED THAT THIS NOTICE AND ANY STATEMENT OFc AUTHORSHIP ARE REPROOUCEO ON ALL COPIES.c

NEITHER THE GOVERNMENTNOR THE UNIVERSITY MAKES ANY WARRANTY. EXPRESS OR IMPLIED.

c OR ASSUMES ANY LIABILITY OR RESPONSIBILITY FOR THE USE OF THISc SOFTWARE.cc ======*===**==**===*” ==*===*===*=*========**="*==*="==*==*=w"=-==**=="c =======**===*==-=*=== *=*=======*==***==*==**=**==** ● =**==**==*==*=**==cc +++c +++c +++c +-+-+c +++cc +++c ++-+c +++cc +-++cccccccc

86

TAPE7 IS THE RESTART DUMPTAPE8 IS THE OUTPUT FILE FOR EOU13TAPE12 IS THE FILM OUTPUT

VOLUME OF FLUID METHOO

LIST OF PRIMARY VARIABLES

INPUT PARAMETERS (NAMELIST /XPUT/)

ALPHA CONTROLS AMOUNT OF DONOR CELL FLUXING (=1.0 FOR FULLDONOR CELL OIFFERENCING, =0.0 FOR CENTRAL DIFFERENCING)

AUTOT AUTOMATIC TIME STEP FLAG (=1.0 FOR AUTOMATIC OELTADJUSTMENT, =0.0 FOR CONSTANT OELT)

CANGLE CONTACT ANGLE, IN DEGRES, BETWEEN FLUID AND WALLCON C.F.L.CDNOITION - CELL WIOTH FRACTION MOVED IN TIME STEPDELT TIME STEP “

IItiI.2IIWI.3IIWI.4IIWI.511’n’I.61141.?IIwI.3IIWi.9IIwI.1OIIWI.!IIIWI.12IIWI.13IIWI.14IIWI.15IIWI.16IIWI.17IIWI.18IIWI.19IIWI.20IIWI.21IIWI.22IIWI.23IIWI.24IIWI.25IIWI.26IIWI.27IIWI.28IIWI.29IIWI.30IIWI.31IIWI.32IIWI.33IIWI.34IIWI.35IIWI.36IIWI.37IIWI.38IIWI.39IIWI.40IIWI.41IIWI.42

DTCRMXEPSIFLHTGXGYICYL

IDEFMI C I VI E Q I C

IMOVY

ISOR

ISURF1O

ISYMPLT

KB

KL

KR

KT

NAMENDUMPNPACKOMGPSAT

PLTDT

PRTDT

OVOL

RHOFSIGMATWFINU IV IVELMX

XNU

MAXIMUM DELT USING CONJUGATE RESIDUAL SOLUTION METHODPRESSURE ITERATION CONVERGENCE CRITERIONF L U I D H E I G H T , IN Y-DIRECTIONBODY ACCELERATION IN POSITIVE X-DIRECTIONBODY ACCELERATION IN POSIT IVE Y -D IRECTIONMESH GEOMETRY INDICATOR (=1 FOR CYLINDRICAL COORDINATES=0 FOR PLANE COORDINATES)DEFOAMER OPTION FLAG ON = 1 *** OFF = oDIVERGENCE CORRECTION FLAG 1 =ON 0=OFFF L A G U S E D T o A C TIV A T E E Q U I L I B R I U M F R E E S U R F A C ECALCULATION DURING SETUPM O V I E I N D I C A T O R ( = 1 F O R M O V I E F I L M O U T P U T , = 0 F O ROTHER FILM OUTPUT)PRESSURE ITERATION SOLUTION METHODC O N J U G A T E R E S I D U A L = O * * * * SDR = 1SURFACE TENSION INDICATOR (=1 FOR SURFACE TENSION,=0 FOR NO SURFACE TENSION)SYMMETRY PLOT INDICATOR (=1 FOR SYMMETRY PLOT.=0 FOR NO SYMMETRY PLOT)INDICATOR FOR BOUND ARY CONDITION TO BE USED ALONG THEBOTTOM OF THE MESH (=1 FOR RIGID FREE-SLIP WALL,=2 FOR RIGID N O - S L I P W A L L . = 3 F O R C O N T I N U A T I V EBOUNDARY, =4 FOR PERIODIC BOUNDARY, ‘5PRESSURE BOUNDARY)INDICATOR FOR BOUNDARY CONDITION ALONGMESH (SEE KB)INDICATOR FOR BOUNDARY CONDITION ALONGMESH (SEE KB)INDICATOR FOR BOUNDARY CONDITION ALONG(SEE KB)PROBLEM IDENTIF ICATIONDUMP NUMBER FOR PROBLEM RESTARTFLAG TO ACTIVATE PACKING O=OFF 1 =ON

FOR CONSTANT

LEFT SIDE OF

RIGHT SIDE OF

TOP OF MESH

OVER-RELAXATION FACTOR USED IN PRESSURE ITERATIONLIQUID SATURATION PRESSURE - IF NONZERO CODE SIMULATESPHASE CHANGE IN NF = 5 CELLSTIME INCREMENT BETWEEN PLOTS AND/OR PRINTS TO BE OUTPUTO N F I L MTIME INCREMENT BETWEEN PRINTS ON PAPER

AVAILABLE TO SPECIFY FLOW RATE (INFLOW OR OUTFLOW)IF REQUIREDFLUID DENSITY (FOR F= l .O REGION)SURFACE TENSION COEFFICIENTPROBLEM TIME TC END CALCULATIONXDoIRECTION VELOCITY USED FOR :NITIALIZING M E S HDIRECTION vELOCITY USED FOR INITIALIZING MESHMAXIMUM VELOCITY EXPECTED Ihj PROBLEM USED TO SCALEVELOCITY VECTORSCOEFF IC IENT OF K INEMATIC V ISCOSITY

C +++ MESH SETUP IIJPUT (NAMELIST /mSHSET/)

c DXMN(N) MINIMUM SPACE INCREMENT IN X-DIRECTION IN SUBMESH Nc DYMN(N) MINIMUM SPACE INCREMENT IN Y-DIRECTION IN SUBMESH Nc NKX NUMBER OF SUBMESH REGIONS IN X-DIRECTIONc N X L ( N ) NUMBER OF CELLS BETWEEN LOCATIONS XL(N) AND XC(N) INc SUBMESH Nc NXR(N)cc NYL(N)cc NYR(N)c

NUMBER OF CELLS BETWEEN LOCATIONS XC(N) AND XL(N+l) INSUBMESH NN U M B E R O F C E L L S BE T W E E N L O C A T I O N S Y L( N ) A N D YC (N ) I N

SUBMESH NNUMBER OF CELLS BETWEEN LOCATIONS YC(N) AND YL(N+l) INSUBMESH N

c X C ( N ) X-COORDINATE OF THE CONVERGENCE POINT IN SUBMESH Nc X L ( N ) LOCATION OF THE LEFT EDGE OF SUBMESH N (NKX+l VALUESc OF XL(N) ARE NECESSARY BECAUSE THE RIGHT EDGE (XR) OFc SUBMESH N IS DETERMINE BY THE LEFT EDGE OFc SUBMESH N+1 )

cc.Ccc +++cc

c YC(N) Y-COOROINATE OF THE CONVERGENCE POINT IN SUBMESH NL ( N ) LOCATION OF THE BOTTOM OF SUBMESH N (NKY+l VALUES OF

YL(N) ARE NECESSARY BECAUSE THE TOP EDGE (YR) OF -

SUBMESH N IS DETERMINED BY THE BOTTOM EDGE OFSUBMESH N+1)

NTERIOR OBSTACLE SETUP INPuT ( N A M E L I s T / A S E T I N / )

I I W I . 4 3I I W I . 4 4IIWI .45IIWI.46I I W I . 4 7I I W I . 4 8I I W I . 4 9IIWI. .50I I W 1 . 5 1I I w I . 5 2I I W I . 5 3I I W I . 5 4I I W I . 5 5I I W I . 5 6I I W I . 5 7I I W I . 5 8I I W I . 5 9I I W I . 6 0I I W I . 6 1I I W I 6 2I I w I . 6 3I I W I . 6 4I I W I . 6 5I I W I , 6 6I I W I . 6 7I I W I . 6 8I I W I . 6 9I I W I . 7 0I I W I . 7 1I I W I . 7 2I I W I . 7 3IIWI.74I I W I . 7 5I I w I . 7 6I I W I . 7 7I I W I . 7 8I I W I . 7 9IIWI.80IIWI.81

I I W I . 8 2I I W I . 8 3IIWi.84IIWI.85I I W I . 8 6I IWI . 87I I W . 8 8I I : W I . 8 9I I : W I . 9 0I I W I . 9 1I I W i . 9 2I I W I . 9 3I I W I . 9 4I I W I . 9 5I I W I . 9 6I I W I . 9 7I I W I . 9 8I I W I . 9 9IIWI .100IIWI. 1OlIIWI .102IIWI .103IIWI .104IIWI .105IIWI. 106IIWI .107IIWI. I08IIWI. 109IIWI .110IIWI .111I I W I . 1 1 2I I W I . 1 1 3I I W I . 1 1 4I I W I . 1 1 5I I W I . 1 1 6I I W I . 1 1 7I I W I . 1 1 8I I W I . 1 1 9

87

*CALL , COMMClh’1DATA EMF /1.oE-06/. EM6 /1.OE-06/, EMIO /1.oE-10/DATA EOIO /1.OE+10/, FiGC /0./, FLG /0./DATA T9UIT /60./DATA PI /3.14159265359/. RPD /0.0174532925/. EM6P1 /f.0000Ot/DATA TBEG /0.~, TPI / 6.283185307 /DATA EM61 / 0.999999 /

c +++ci’ +++ DEFAULT INPUT DATAc +++ NOTE USER MUST SUPPLY THE FOLLOWING REGARDLESSc oF DEFAuLTs: DELT,TWFIN.PRTDT.PLTOTc

DATA XNU 10.G/, ICYL /0/, EPSI /1.OE-03/, GX /0.0/. GY /0.0/, UI /1 0.0/, V: /0.0/. VELMX /1.0/, IMOVY /0/, OMG /1.7/. ALPHA /1.0/.2 KL /1/, KR /1/. KT /1/, KB /1/. AUTOT /1.0/, ISYMPLT /0/,3 ISURFIO /0/. SIGMA /0.0/, CANGLE /90.0/. RHOF /1.0/, FLHT /0.0/.4 PSAT /0.0/

c +++c +++c +++

c +++

c +++c +++c +++

cc +++c

cc +++c

c

cc +++c

10

c .-+c +.+

20

cc +++c

cc +++c

cc +++c

c

cc +++c

30

88

DATA IEOIC /1/

CALL SYSTEM-DEPENOENT ROUTINES TO INITIALIZE FILM FILE

CALL GpLOT (lHu.4HIIwI,4)CALL GRPHLUN (12)CALL LIB4020CALL GRPHCFTCALL SETFLSHSySTEM-DEPENDENT CALL TO GET THE LJOB TIME LIMITCALL GETdTL (TTL)

CALL THE INITIALIZATION ROUTINE

CALL SETUP

PRINT INITIAL INPUT DATA

CALL PRTPLT (1)

SET INITIAL BOUNOARY CONDITIONS

CALL BC

DTEND=O.002=OELTIF (NDUMD.LE. l) GC TO 39

START TIME CYCLE

CONTINUECALL SECOND (T5EG)

PROBLEM DEF5.Ni)EtJT - cALcuL&TioNAL sHuToFF FoR sMALL 71MEsTEp

IF(DELT.GT.DTEND 1 GO TC 20WRITE (59.150) N:YC.TWRITE ( 12, 150) NCYC,TTWFIN=T-0.999CONTINUEITER=OFLG=l.OFNOC=O.O

EXPLICITLY APPROXIMATE NEW TIME-LEVEL VELOCITIES

CALL TILDE

SET BOUNOARY CONDITIONS

CALL BC

ITERATIVELY AOJUST CELL PRESSURE AND VELOCITY

IF IISOR.EO.1) CALL PRESITIF (ISOR.EO.0) CALL PRESCRIF (FNOC.GT.0.5) CALL PRTPLT (4)

IF (FNOC.GT.0.5) GO TO 40

UPDATE FLUIO CONFIGURATION

CALL VFCONV

IIWI. 120IIWI. 121IIWI. 122IIWI. 123IIW?. 124IIWI. 125IIWI. 126IIWI. 127IIWI. 128IIWI. 129IIWI. 130IIWI. 131IIWI. 132IIWI. 133IIWI. 134IIWI. 135IIWI. 136IIWI. 137IIWI. 138IIWI. 139IIWI. 140IIWI. 141IIWI. 142IIWI. 143IIWI. 144IIWI. 145IIWI. 146IIWI. 147IIWI. 148IIWI. 149IIWI. 150IIWI. 151IIWI. 152IIWI. 153IIWI. 154IIWI. 155IIWI. 156IIWI. 157IIWI. 158IIWI. 159IIWI. 160IIWI. 161

IIWI .162IIWI. 163IIWI. 164IIWI. 165IIWI. 166IIh’I. 167IiWI. 16EIIWI. 169IiWI. 170IIWI .171IIWI .172IIWI .173IIWI. !74IIWI. 175IIWI .176IIWI .177IIWI .178IIWI .179IIWI. 180IIWI .161IIWI. 182IIWI. 183IIWI. 184IIWI.185IIwI.186IIWI.187IIWI.188IIWI.189IIWI.190IIWI.191IIWI.192IIWI.193IIWI.194IIWI.195IIWI.196IIWI.197IIWI.198

c

cc +++c

cc +.+c +*+c +++c

3234

cc +++c

40c

50cc +++c

cc +++c

c60

70cc +++c

c80

cc +++c

90

cc +++c

100cc +++c

cc +++c

IF (FLGC.GT.O.5) GO TO 100

SET BolJNDARy CONDITIONS

CALL EC

DETERMINE PRESSURE INTERPOLATION FACTOR AND NEIGHBORALSO DETERMINE SURFACE TENSION PRESSURES ANCWALL ADHESION EFFECTS IN SURFACE CELLS

CALL PET&CAL—CALL BCIF(NCYC.NE .O.OR.ISOR .NE.0) GO TO 3400 32 1=2, 1!41PSAOO=O.O00 32 d=2,dMlJd=dM1-LJ+2IF(PS(I.UU).NE.O.0) PSAOO=PS(I,LJJ)-P(I,LM)P(I,LId)=P(I,tJJ)+PsAooCONTINUECONTINUE

PRINT TIME AND CYCLE DATA ON PAPER AND/OR FILM

CALL PRTPLT (2)

:F (NCYC.LE.0) GO T3 50IF (7+EM6. LT.TwPLT.ANo.T. LT.TWFIN) GO TO 60TWPLT=TWPLT+PLTOTCONTINUE

PRINT FIELO VARIABLE OATA ON FILM

CALL PRTPLT (3)

PLOT VELOCITY VECTOR, FREE SURFACE, MESH,

CALL ORAW

CONTINUEIF (NCYC.LE.0) GO TO 70IF (T+EM6.LT.TwPRT) GO TO so

TWPGT=TWPRT+PRTDTCONTINUE

PRINT FIELO VARIABLE DATA ON PAPER

CALL PRTPLT (4I

CONTINUEIF (T.GT.TWFIN)

SET THE AOVANCE

00 90 1=1, IMAx00 90 d=l,JMAXUN(I,LJ)=U(I,J)VN(I,J)=V(I.J)U(I,J)=O.OV(J,J)=O.OPN(I.J)=P(I,J)FN(I,d)=F(I,d)CONTINUENREGN=NREG

AOLJUST OELT

CALL OELTAOJ

AOVANCE TIME

T=T+OELT

AOVANCE CYCLE

GG TO 130

TIME ARRAYS INTO THE TIME-N ARRAYS

IIWI.199IIWI.200IIWI.201IIWI.202IIWI.203IIWI.204IIWI.205IIWI.206IIWI.207IIWI.208IIWI.209IIWI.21OIIWI.211IIWI.212IIWI.213IIWI.214IIWI.215IIWI.216IIWI.217IIWI.218IIWI.219IIWI.220IIWI.221IIWI.222IIWI.223IIWI.224IIWI.225IIWI.226IIWI.227IIWI.228IIWI.229IIWI.230IIWI .231IIWI.232IIWI.233IIWI.234IIWI.235IIWI.236IIWI.237IIWI.238IIWI.239IIWI.240IIWI.241IIWI .242IIWI.243IIWI .244IIWI .245IIWI .246IIWI.247IIWI.248IIWI.249IIWI.25OIIWI.251IIWI.252IIWI.253IIWI.254IIWI .255IIWI.256IIWI.257IIWI.258IIWI.259IIWI.260IIWI .261IIWI.262IIWI.263IIWI.264IIWI.265IIWI.266IIWI.267IIWI.268IIWI .269IIWI.270IIWI.271IIWI .272IIWI.273IIWI.274IIWI .275IIWI.276IIWI .277

IF (NFLGc.LT.5) GO TO 110WRITE (12,170) NCYC,TWRITE (59,170) NCYC.T

89

110

120

h++

c130

c140150160

CALL EXITLOC (8HIIWI PIT)CONTINUEIF (NOCON.LT.5) GO TO 120WRITE (12.160) NCYC,TWRITE (59.160) NCYC,TCALL EXITLDC (SHiIWIFCV)CONTINUENCYC=NCYC+ITERMINATE GRACEFULLY IF OUT OF TIMECALL SECOND (TIME)TLEFT=TTL-TIME-TOUITIF (TLEFT.LT.O. ) cALL TAPOUT (o)IF (MOO(NCYC.200) .NE.0) GO TO 10CALL SECONO (TENO)CALL SECONO (TENi)O)GRINO=Z 1000. =(TEhO+TEND-TENDO-TBEG)/FLOAT( IBAR*IJBAR)WRITE (59,140) NCYC,GRINO. ITERIF (IMOVY.EO.0) WRITE (12, f40) NCYC,GRINO.ITERWRITE (6,140) NCYC,GRINO.ITERGO TO 10

CONTINUECALL ExITLOC (4HIIwI)

FORMAT (IX.5HCYCLE, 17. IX,6HGRINO= ,IPE12.4,10H MS. ITER=FORMAT (IX,29HOELT LESS THAN OTENO ON CYCLE.16. IX.3HT =FORMAT (//lx.25HT00 MANy PREssIT Failures. 3x.5HNcyc=, 17

1 E14.6/jj’170 FORMAT (//lx.24HT00 MANy vFcONV Failures, 3x,5HNcyc=o 17,

1 14,6//)ENO

SUBROUTINE ASET~CALL,COMM.ONl

15)IPE15.7)1X,2HT=,1P

X,2HT=, IPE

c +++c +++c +++c +*+c -++

10

20

30

40

50

CONIC FCIQ=OA2=X=X+3AI =X+3E2=Y- Y+OBl=Y+GC2=Y.=Y&OC :INSI@E FCN=NEGATIVE VALUEIOH=I AOO CES INSIOE FCN, IOH=O SUBTRACT OBS INSIOE FCN

DIIKNSION IFLG(5), DIS(4), XM(5). }M(5)IF (NOPS.LE.01 GO 70 24000 230 K=I,NOBS00 220 d=2.dMl00 220 1=2.IMIROXDY=l .O/(DELX(I)-DELY( d))00 60 M=f.4GO TO (10,20.30,40), Mx1=x(I)Yl=Y(tJ-1)OIS(l)=OELY(d)GO TO 50Y1=Y(LI)X1=X(I)DIS(2)=OELX(I)GO TO 50X1=X(1-I)YI=Y(LJ)oIs(3)=DELY(d)GO TO 50YI=Y(J-1)X1=X(1-1)OIS(4)=DELX(I)IFLG(M)=OFcoNIc=oA21K )=xt=xI-oAI (K)=xI+062(K)=Yl Byl+0B 1(K)=y1+oc2(K)=x1=y 1

1 +OCI(K)IF (FCONIC.LE.O.0) IFLG(M)=IXV(M)=XIYM(Mj=Yl

60 CONTINUEIFLG(5)=IFLG(I )XM(5)=XM(1)YM(5)=YM(I)IFLGS=ODO 70 M=i,4

70 IfLGS=IFLGS+IFLG(M)BRIJ=O.OBTIJ=O.OIF (IFLGS.EO.0) GO TO 220IF (IFLGS.L7.4~ GO TO 80

IIWI.278IIWI .279IIWI.280IIWI.281IIWI.282IIWI.283IIWI.284IIWI.285IIWI.286IIWI.287IIWI.288IIWI.289IIWI.290IIWI.291IIWI.292IIWI.293IIWI.294IIWI.295IIWI .296IIWI.297IIWI.298IIWI.299IIWI.300IIWI .301IIWI.302IIWI.303IIWI.304IIWI .305IIWI.306IIWI.307IIWI.308

ASET.2ASET.3ASET.4ASET.5ASET.6ASET.7ASET.8ASET.?ASET.10ASET.11ASET.12ASET.13ASET. 14AsET.~5ASET.16ASET.17ASET.18ASET.19ASET.20ASET.21ASET.22ASET.23ASET.24ASET.25ASET.26ASET.27ASET.2BASET.29ASET.30ASET.31ASET.32ASET.33ASET.34ASET.35ASET.36ASET.37ASET.38ASET.39ASET.40ASET.41ASET.42ASET.43ASET.44ASET.45ASET.46ASET.47ASET.48

90

BIJ=I. OBRIJ=I.oBTId=l.OGO TO 200

80 iF (IFLG(I ).EQ. I.ANo. IFLG( 2).EQ.1) BRILI=l.OIF (IFLG(2) .EQ. I.AND.IFLG( 3).Eo.1) BTIIJ=I.ODO 160 M=I,4IF (IFLG(M) .EQ.IFLG(M+I) ) GO TO 160Xl=XM(M)Y1=YM(M)X2=XM(M+I)Y2=YM(M+1)IF (IFLG(M).Eo.0) Go To 90X2=XM(M)Y2=YM(M)XI=XM(M+l)YI=YM(M+I)

90 EPSIF=0.001.(ABS(X2-XI )+ABs(Y2-YI))SMN=O.OFMN=OA2(K).X2=X2+OAI (K)*X2+OB2(K) =Y2.Y2+OB

1 (K)Smx=l.oFMX=OA2(K).XI *XI+OAI (K).XI+OB2(K)=YI *YI+OB

1 (K)

(K)=y2+0c2(K)*x2=Y2+ocI

(K)=Yl+0c2(K).xI*YI+ocI

S=O.5100 XT=S=X1+(I.O-S)=X2

YT=S*YI+(l.O-S)*Y2Fs=oA2(K).xTIDxT+oA1 (K)=XT+OB2(K)=YT=YT+OBI (K)*YT+OC2(K).XT.YT+OCI

1 (K)

110

120

130

140

150160

170

180

J90

200

IF (ABS(FS). LT.EPSIF) GO TO 130IF (FS.GE.O.0) GO TO 110FDEN=ABS( FS-FMN)+I.OE- 10SE=S-FS*(S-SMN)/FOENIF (sE.GT.sMx) SE=SMXFMN=FSSMN=SGO TO 12aFOEN=ABS(FMX-FS)+l .OE-10SE=S-FS=( SMX-SJ/FOENIF (SE.LT.SMN) SE=SMNFM).=FSSMX=SSI=S-FS*(SMX-SMN )/(FMXS=O.5=(SE+SI)GO TO 100OIS(M)=SORT( (XT-X2 )==2.GO TO (140.150.160; 160BRId=OIS( l)/DELY(J)GO TO 160ETIbJ=oIs(2 )/DELx(I )

FMN)

(YT-Y2)=*2),M

CONTINUEM=oBId=O.OCONTINUEM=M+IIF (M.EO.5) GO TO 190IF (IFLG(M).EQ.0) GO TO 170MPI=M+IIF (MP1.EO.5) MPI=lMMI=M-IIF (MMI.EO.0)BILJ=BIJ+OIS(MIF (IFLG(MpI)DIS2=OIS(M)CONTINUEIF (IFLG(MMI)OIS1=DIS(MM1)GO TO 170CONTINUE

MMI=4*DIs(MM1EQ.I ) GO

EO.1) GO

TO 180

TO 170

IF (IFLGS.EO.3) BId=BILJ-01Sl*OIS2BId=o.5*BId*RoxoYIF (Bid.GT.l.0) BIJ=I.OCONTINUEIF (IoH(K).EO.0) GO TO 210BIJ=-BIJBRIJ=-BRIJBTIIJ=-9TILJ

ASET.49ASET.50ASET.51ASET.52ASET.53ASET.54ASET.55ASET.55ASET.57ASET.58ASET.59ASET.60ASET.61ASET.62ASET.63ASET.64ASET.65ASET.66ASET.67ASET.68ASET.69ASET.70ASET.71ASET.72ASET.73ASET.74ASET.75ASET.76ASET.77ASET.78ASET.79ASET.80ASET.81ASET.82AsET.e3ASET.84ASET.85A5ET.86ASET.87ASET.88ASET.89ASET.90ASET.91ASET.92ASET.93ASET.94ASET.95ASET.96ASET.97ASET.98ASET.99ASET.1OOASET.1OIASET. 102ASET. 103ASET. 104ASET. 105ASET. 106ASET. 107ASET. 108ASET. 109ASET. 110ASET. 111ASET. 112ASET. 113ASET. 114ASET. 115ASET. 116ASET. 117ASET. 118ASET. 119ASET. 120ASET. 121ASET. 122ASET. 123ASET. 124ASET. 125

91

210

220230240

270

280

290

300

310c ++.c +++c +++c +++ccc

ccccccc

cccc

cccccc

ccc

92

kC(I.d)=AC(I.LJ)+BIJIF(AC(I.LJ).GT.O.9999) AC(I,UJ=I.OIF(AC(I .LJ).LT.O.0001 ) AC(I.J)=O.OAR(I, J)=AR(I,d)+BRIdIF(AR(I .J).GT.O.9999 ) AR(I.d)=l.OIF(ARII .J).LT.O.0001 ) AR(I.d)=O.OAT(I,U)=AT( I,J)+5TIJIF(AT(I .J).GT.O.9999 ) LT(I,J)=l.O

ASETASETASETASETASETASETASETASET

IFiAT(I :LJi.LT.O.0001 ) AT(I,IJ)=O.OCONTINUE--CONTINUECONTINUE00 280 d=2,dMlIF (KL.GT.2) GOAR(l.J)=O.OAT(I,J)=O.OAC(1,J)=EM1OIF (KR.GT.2) GOAR(IMI.J)=O.OAR(IMAX,d)=O.OAT(IMAX.J)=O.OAC(IMAX;U)=EM1OCONTINUE00 300 1=2.IM1IF (KB.GT.2) GOAT(I.1)=0.OAR(I.1)=0.OAC(I.1)=EM1OIF (KT.GT.2) GOAT(I.dMl)=O.OAR(I,dMAX)=O.OAT(I,dMAX)=O.OAC(I,JMAX)=EM1OCONTINUEDO 310 d=2.dMl00 310 1=2.IMI

TO 270

TO 280

TO 290

TO 300

IF (AC(I.d).GT.EM6) GO TO 31oAR(I,d)=O.CAR(I-I,J)=O.OAT(:,u)=O.GAT(I.u-1)=0.OeET&(I,J)=-1-cCONTINUE

SEY AG ANC AT VALUES FOR IN ANO OUT ~LOWHERE AS UPOATE MODIFICATION OEPENOING 3N

BOUNDARY SEGMENTSAPPLICATION

START OF CURVEO WALL OPTION SECTION

00 330 d=2,JMl00 330 I=2,1M1

WE OEFINE ARRAY NW, WHICH LOCATES CURVEO WALL IN A PARTIAL FLOWCELL. DEFINITIONS TERMINATE AT: 315 CONTINUEWE OEFINE ANGLE PHI THAT CURVEO WALL MAKES WITH +Y AXIS. DEFINIT-IONS TERMINATE AT: 325 COSO...WE INTROOUCE OEFAULT VALUES FOR PHI ANO NW

PHI=CANGLENW(I,tJ)=O

FOR FLUID CELL OR OBSTACLE CELL USE OEFAULT VALUES FOR PHI ANO NWBYPASS ALL OF 00 LOOPS EXCEPT FOR OEFINING TRIG. FACTORS

IF(AC(I .J).LT.EM6.OR. BETA(I,J).LT.O.0) GO TO 325IF(AC(I, J).GT.EM61) Go To 325

SET NW FOR CELL

SET NW ARRAY FOR CELLS WITH TWO COMPLETELY CLOSEO

IF(AR(I, J).LT.EM6.ANO .AT(I,J).LT.EM6) NW(I,d)=5IF(AR(I, J).LT.EM6.ANO .AT(I, lJ-1).LT.E~6) NW(I.J)=6

BOUNDARIES

SET NW ARRAY FOR CELLS WITH EXACTLY ONE CLOSEO BOUNOARY

IF(AR(I, J).LT.EM6.ANO .Nw(I.J).LT.5) NW(I.IJ)=:IF(AR(I-l,J) .LT. EM6.ANO.AT(I,J) .GT. EM6.AN0.AT(l.J- 1).GT.

126127128129130131132133

ASET. 134ASET. 135ASET. 136ASET. 137ASET. 138ASET. 139ASET. 140ASET. 141ASET. 142ASET. 143ASET. 144ASET. 145ASET. 146ASET. 147ASET. 148ASET. 149ASET. 150ASET. 151ASET. 152ASET. 153ASET. 154ASET. 155AsET.156ASET. 157ASET.IE.8ASET. 159ASET. 160ASET. 161ASET. 162ASET. 163ASET. 164ASET. 165ASET. 166ASET, 167ASET. 16.2ASET. 16SASET. 17@ASET. 171ASET. 172ASET. 173ASET. 174ASET. 175ASET. 176ASET. 177ASET. 178ASET. 179ASET. 180ASET. 181ASET. 182ASET. 183ASET.1E4ASET. 185ASET. 186ASET. 187ASET. 188ASET. 189ASET. 190ASET. 191AsET. 192ASET. 193ASET. 194ASET. 195ASET. 196ASET. 197ASET. 198ASET. 199ASET.200ASET.201ASET.202ASET.203ASET.204ASET.205

L

cccc

305

315cccccc

cccc

ccc

cc

320

c

:c

cccc

c

:c

ccccc

325

330c

1 EM6) NW(I.J)=2IF(AT(I .J):LT.EM6.AND .AR(I,d) .GT.EM6.AND.AR( I-1

1 GT.EM6) NW(I.J)=3. . . -, —lF(k~(l :d).LT.5M6.AND .AR(l. d).GT.EM6.AND.I .EI).2IF(AT(I,J-1 ) .LT.EM6.AND.AR(I, J).GT.EM6.AN5.AR( I

1 EM6) NW(I,LJ)=4

SET Nti ARRAY FDR CELLS WHOSE NE OR S: CORNER ISBUT WHICH HAS NO COMPLETELY CLOSED BOUNDARY

d).

NW(I,d =31.J).GT

CLOSEO “O FLOW,

THIS CASEMORE INTRICATE NET OF IF STATEMENTS NEEOEO TO ESTABLISH

IF(AT(I, J-1) .LT.EM6. AND.AR( I,J).GT.EM6.ANo .I.EQ.2) NW(I,d) = 4IF(AR(I- l,u).LE.EM61 ) GO TO 315IF(AR(I, d).GT.EM6.ANo.AR( I,d).LT.EM61) GO TO 305GO TO 315IF(AT(I,J- 1).GT.EM61 .AND.AT( I,J).GT.EM6.AND .AT(I,Ll).LT.

1 EM61) NW(I.U)=7IF(AT(I .d).GT. EM61.ANO.AT( I, IJ-l).GT. EM6.ANO.AT

! EM61) NW(I.tJ)=8CONTINUE ‘-”-”

CALCULATE APPROPRIATE ANGLE FOR CURVED BOUNOARY

OEFINE FOR USE IN IF STATEMENTS: NON-INDEXEO V,

NWW=NW(I,d)

I,d-1).LT.

RIABLE NWW=NW(I,U)

IF NW = O OR 2, BYPASS REDEFINITION oF pHI AND coMpuTE sI~ ANoCOSO FROM DEFAULT VALUE: OTHERWISE REDEFINE PHI, STARTING AT 320

IF(NWW.EO.0) GO TO 325GO TO (320.325,320.320,320,320, 320,320). NWW

SPECIFY VARIABLES FOR USE IN OEFINING TAN(PHI) IN VARIOUS CASES,

INCLUDING DEFAULT TO AVOIO DIVISION BY ZERO

CRATIO=OELX(I )/OELV(LJ)cELoEN=AR(l, ~)-AR( I-I,J)IF(ABs( CEL2EN).LT.o.ool ) CELDEN=O.001

RECOMPUTE PHI IN FOUR STRAIGHTFOREWAR3 CASES: NW = :,6.3. OR ~FOR NW .NE. 1, CONVERT NEGATIVE PHI TO PI-II IN SECONO OLJAO!?A14T

IF(NW4.EQ.1) PHI=ATAN(CRATIO-(AT(I .J)-AT(I.J-l )))IF(Nww.EO.6) PHI=ATAN(CRATIO= (AT(I ,d)/AR(I-l.d)) )IF(NWW.EQ.3) PHI=ATAN(.CRATIO/CELDEN)IF(NWW.EO.4) PHI=ATAN( -CRATIO/CELOEN)IF(NWW.NE. I.ANO.PHI .LT.O.0) PHI=PI+PHI

WE NEEO WALL ADHESION OPTION EVEN FOR NW = 7 OR 8 CELLS: SORECOMPUTE PHI FOR THESE CASES HERE

IF(NWW.EO.8) PHI=ATAN(CRATIO*(( l.O-AT(I,J-1))/(1 .O-AR(I.J))))IF(Nww.Eo.7) PHI=ATAN2(cRATIO=( 1.O-AT( l,.J)), (AR(I,J)-I.0) )IF(NWW.NE.5) GD TO 325

SIMPLEST FORMAT FOR NW = 5 IS SLIGHTLY OIFFERENT FROM THAT OFOTHER CASES: SO TREAT NW = 5 SEPARATELY HERE

PHI=PI - ATAN(CRATID=(AT( I,J-1)/AR( I-led)))

EVALUATE TRIG. FACTORS USING EITHER THE DEFAULT PHI OR THERECALCULATED PHI, AS APPROPRIATE

CaSO(I,d)=CDS(PHI )SINO(I, J)=SIN(PHI)CONTINUE

RETURNENO

ASET.206ASET.207ASET.2C8ASET.209ASET.21CJASET.211ASET.212ASET.213AsET.214ASET.215ASET.216ASET.217ASET.218ASET.219ASET.220ASET.221ASET.222ASE7.223ASET.224ASET.225ASET.226ASET.227ASET.228ASET.229ASET.23OASET.231ASET.232ASET.233ASET.234ASET.235ASET.236ASET.237ASET.238ASET.239ASET.24OASET.241ASET.242ASET.243ASEi.244ASET.245

.&s~T.246AsET.2.4-&sE7.2AEASE-.24GASET.250ASET.251ASET.252ASET.253ASET.254ASET.255ASET.256ASET.257ASET.258ASET.259ASET.260ASET.261ASET.262ASET.263ASET.264ASET.265ASET.266ASET.267ASET.268ASET.269ASET.270ASET.271ASET.272ASET.273ASET.274ASET.275ASET.276ASET.277ASET.278

93

SUBROUTINE BC=CALL,COMMONI

EC.2BC.3

Lc +++ sET EOUN3ARY CONDITIONSc

00 100 d=l.d!~y.F11,LJ)=F[2.JIF(IMAX, ti)=F(IMf,J)P(l.J)=D[2,J)P(IMAX, JI=P(IM1,LJ)GO TO (10,20,30,40.30). K!-

10 U( 1.J)=O.OV( I,U)=V(2,J)GO TO 50

20

30

40

5060

70

80

90

100

110

120

130

140

150!60

170

180

U( I.d)=o.oV(l, IJ)=-V(2,1.1 )=OELX( 1)/oELx(2)GO TO 50IF (ITER.GT.o) GO TO 50U( 1.J)=U(2. d)*(X(2)-RX( l)=CYL+l.O-CyL)v(l,d)=v(2,d)GO TO 50u(I,J)=U(IBAR.J)v(l,d)=V(IBAR,J)F( I,IJ)=F(IBAR.J)GO TO (eO.70.80,90.80). KRU(IM1,J)=O.OV(IMAX, J)=V(IMl,d)SO TO 100U(IMI.J)BO.OV(IMAX, LJ)=-V( IMl,d)*oELx( IMAX)/OELx(IMl)GO TO 100IF (ITER.GT.0) GO TO 100U(IMl, J)=U(ISAR,J)=(x( IBAR)=RX( IMl)”c~L+l.o-cyL )v(IMAX, d)=V(IM~,Ll)GO TO 100u(IMl,d)=u(2.d)v(IMl,d)=v(2.lJ)P(IMl,d)=P(2,1J)PS(IM1, LJ)=PS(2,U)F(IM1,LJ)=F(2.J)V(IMAX,IJ)=V(3,J)F(IMAX, LJ)=F(3,LJ)C3NTINUE00 200 1=1.IMAXF(I.1)=F(I;2)F(I, JMAX)=F(I.JM1)P(I, I)=P(I .2)P(I. JMAX)=P(I,LJMI)GO TO (110. 120. 130s 140,130) , KTV(I.UMI)=O:OU(I. dMAX)=U(I.JMl)GO T3 150V(I,JMI)=O.Ou(I,dMAx)=-u(I,JMl )-OELy(dMAX)/OELY(JMl )GO TO 150IF (ITER.GT.0) GO TO 150V(I;JM1)=V(I,JBAR)U(I,JMAX)=U(I, JMI)GO TO i50V(I.dMl)=V(I,2)U(I,JMI)=U(I.2)P(I.dMl)=P(I,2)PS(I.IJMI)=PS(I .2)F(I.lJMl)=F(I.2)U(I;JMAX)=U(I?3)F(I.JMAX)=F(I.3)GO TO (160.170,180.190. 180). KBV(I.1)=0.OU(I. I)=U(I .2)GO-TO”200V(I.1)=0.Ou(I.l)=-u( I,2)*oELy(l )/0ELy(2)GO TO 200IF (ITER.GT.0) GO TO 200V(I!I)=V(I,2)U(I,1)=U(I,2)GO TO 200

190 V(I. l)=V(I.@AR)u(I.l)=U(I.JBAR)F(I.l)=F(I.LJBAR)

BC.4BC.5EC.3BC.7BC.8BC.9BC.ICBC.liBC.12BC.13BC.14BC.15BC.16BC.17BC.18BC.19BC.20BC.21BC.22BC.23BC.24BC.25BC.26BC.27BC.28BC.29BC.30BC.31BC.32BC.33BC.34BC.35BC.36BC.37BC.38BC.39BC.40BC.41BC.42BC.43BC.44BC.45BC.46BC.47BC.48BC.49BC.50BC.5~BC.52BC.53BC.54BC.55BC.56BC.57BC.5BBC.59BC.60BC.61BC.62BC.63BC.64BC.65BC.66BC.67BC.68BC.69BC.70BC.71BC.72BC.73BC.74BC.75EC.76BC.77BC.78BC.79BC.80BC.B1

94

200 CONTINUEcC +++ FREE SURFACE AND SLOPED BOUNDARYC

00 420 1=2 . IM1D O 4 2 0 J = 2 . J M lIF (BETA(I.J)I.GT.O.O) GO TO 210BMR=O.OBMT=0.OBML=C.OBMB=O.OF ( I , J ) = O . OP ( I . J ) = O . OI F ( B E T A ( I + 1 , JI F ( B E T A ( I . J + lI F ( B E T A ( I - I . JI F { B E T A ( I , J . IBMTOT=BMR+BMT=B[IF ( B M T O T . L E . O .

. G T . O . 0 ) B M R = l . O

. G T . O . 0 ) B M T = l . O

. G T . O . 0 ) B M L =GT.O.0 ) BMB=

ML+BMBO) GO TO 420

.0

.0

F ( I . J ) = ( B M R * F ( I + 1 , J ) + B M T * F ( IP ( I , J ) = ( B M R * P ( I + l . J ) + B M T * P ( IGO TO 420

210 CONTINUEc

CONDITIONS

J + I ) + B M L * F ( - 1 I , J ) + B M B = F ( I , J - 1 ) ) / B M T O TJ + I ) + B M L + P ( I - l , J ) + B M B - P ( 1 . d - 1 ) ) / B M T O T

I F ( N F ( I . J ) . E O . O . O R . N F ( I , J ) . G T . 5 ) G O T O 4 2 0I F ( I . E O . 2 . A N D . K L . E Q . 5 ) G O T O 3 8 0IF(I . EO. IM1.AND.KR. EQ.5 ) GO TO 380IF(J. EQ. IJMI.AND.KT. EQ.5 ) GO TO 380I F ( J . E Q . 2 . A N D . K B . E Q . 5 ) G O T O 3 8 0

cI F ( N F ( I + l . J ) . L T . 6 . O R . A R ( I . J ) . L T . E M 6 ) G O T O 2 4 0U ( I . J ) = U ( I . J ) * A R ( I - 1 , d ) * R ( I - 1 ) / ( A R ( I . J ) * R ( I ) )

2 4 0 I F ( N F ( I , J + l ) . L T . 6 . O R . A T ( I . L J ) . L T . E M 6 ) G 0 T O 2 5 0V ( I , J ) = V ( I , J - 1 )

2 5 0 I F ( N F ( I - l , d ) . L T . 6 . O R . A R ( I - 1 . d ) . L T . E M 6 ) G O T O 2 6 0U ( I - l . t J ) = U ( I . J ) * A R ( I . J ) * R ( I ) / ( A R ( I - l . I J ) * R ( I - 1 ) )

2 6 0 I F ( N F ( I , J - 1 ) . L T . 6 . O R . A T ( 1 . d - 1 ) . L T . E M 6 ) ” G 0 T O 2 7 0v ( I , d - l ) = V ( I , L J )

2 7 0 N F F = N F ( I , J )D I J = R D X ( I ) * ( A R ( I . J ) * R ( I ) * U ( I , t J ) - A R ( I - l , J )= R ( I - 1 ) * U ( I - 1 . d ) ) +

1 R D Y ( J ) = ( A T ( I . t J ) = R I ( I ) = V ( I . J ) - A T ( I . L I - I ) * R I ( I ) * V ( I . J - 1 ) )

3 0 03 1 0

3 2 0

3 3 0

3 4 0

3 5 0

cc + + +c

3 8 0

LOOP= 0G O T O ( 3 1 0 , 3 2 0 , 3 3 0 , 3 4 0 , 3 5 0 ) , N F FI F ( N F ( I + l . J ) . L T . 6 . O R . A R ( I , J ) . L T . E M 6 ) G O T O 3 5 0U I I , J ) ) = U ( I , J ) - D E L X ( I ) * D I d / ( A R ( I , L J ) * R ( I ) )GO TO 380I F ( N F ( I - l , J ) . L T . 6 . O R . A R ( I - l , d ) . L T . E M 6 ) G O T O 3 5 0U ( I - I , J ) = U ( I - l , J ) + D E L X ( I ) * D I J / ( A R ( I - 1 . J ) = R ( I - 1 ) )GO TO 380I F ( N F ( I . J + 1 ) . L T . 6 . O R . A T ( I . J ) . L T . E M 6 ) G O T O 3 5 0V ( I . J ) = V ( I , J ) - D E L Y ( I . J ) = D I J / ( A T ( I . J ) * R I ( I ) )GO TO 380I F ( N F ( I . J - 1 ) . L T . 6 . O R . A T ( l . J - l ) . L T . E M 6 ) G o T o 3 5 0V ( I . J - l ) = V ( I , J - l ) + D E L Y ( J ) * O I J / ( A T ( I . J - l ) * R I ( I ) )GO TO 380NFF=NFF + 1I F ( N F F . G T . 4 ) N F F = 1LOOP=LOOP + 1I F ( L O O P . L E . 4 ) G O T O 3 0 0

SET VELOCIT IES IN EMPTY CELLS ADJACENT TO PARTIAL FLUID CELLS

CONTINUEI F ( F L G . G T . 0 . 5 . A N 0 . 1 T E R . G T . O . A N D . I S O R . E O . l ) G O T O 4 2 0IF (F(I+1.J).GT.EMF) GO TO 390I F ( F ( I + 1 . J + l ) . L T . E M F . A N D . A T ( l + 1 , o ) . G T . E M 6 ) v ( = 1 + JI F ( F ( I 1 I , J - 1 ) . L T . E M F . A N D . A T ( I + 1 , J - 1 ) . G T . E M 6 ) V ( I+ 11 *F(I,J)

3 9 0 I F ( F ( I J + 1 ) . G T . E M F ) G O T O 4 0 0IF ( F ( I + 1 , J + l ) . L T . E M F . A N D O . A R ( l , J + 1 ) . G T . E M 6 ) U ( I . J + 1I F ( F ( I 1 I , J + 1 ) . L T . E M F . A N D . A R ( I - 1 , J + 1 ) . G T . E M 6 ) U ( I - 1

1 * F ( I , J )4 0 0 I F ( F ( I - 1 . J ) . G T . E M F ) G O T O 4 1 0

I F ( F ( I - 1 . J + 1 ) . L T . E M F . A N D . A T ( I - 1 , J ) . G T . E M 6 ) V ( I - 1 , JI F ( F ( I - 1 , J - 1 ) . L T . E M F . A N D . A T ( l - 1 , J - 1 ) . G T . E M 6 ) v ( I - 1

1 * F ( I , J )4 1 0 I F ( F ( I , J - 1 ) . G T . E M F ) G O T O 4 2 0

I F ( F ( I + 1 , J - 1 ) . L T . E M F . A t 4 D . A R ( I , J - 1 ) . G T . E M 6 ) U ( I , J - 1I F ( F ( I - 1 , J - 1 ) . L T . E M F . A N D . A R ( I - 1 , J - 1 ) . G T . E M 6 ) U ( I - 1

1 * F ( I - J )

= F ( I . J ) * V ( I . J )J - l ) = V ( I , L J - 1 )

= F ( I , J ) * U ( I . J )J + I ) = U ( I - 1 , J )

= F ( I , J ) * V ( I , J )J - l ) = V ( I , J - 1 )

= F ( I . J ) * U ( I , J )J - I ) = U ( I - I , 1 J )

B C . 8 2B C . 8 3BC .84B C . 8 5BC .86BC:.87B C . 8 8B C . 8 9B C . 9 0B C . 9 1B C . 9 2B C . 9 3B C . 9 4B C . 9 5BC .96B C . 9 7B C . 9 8B C . 9 9BC. 100BC. 101BC. 102BC. 103BC. 104Bc. 105BC. 106BC. 107BC. 108BC. 109B c . l l oB c . 1 1 1B C . 1 1 2B C . 1 1 3B C . l f 4B C . 1 1 5BC.1126B C . 1 1 7B C . 1 1 8B C . ! 1 9B C . 1 2 0EBC.121B C . 1 2 2B C . 1 2 3B C . 1 2 4B C . 1 2 5B C . 1 2 6B C . 1 2 7B C . 1 2 8B C . 1 2 9B C . 1 3 0B C . 1 3 1B C . 1 3 2BC. 133BC. 134Bc. 135BC. 136BC. 137BC. 138B C . 1 3 9BC. 140B C . 1 4 1BC. 142BC. 143BC. 144Bc. 145BC. 146Bc. 147BC. 148BC. 1498 C . 1 5 0B C . 1 5 1BC. 152Bc. 153B C . 1 5 4BC. 155BC. 156BC. 157BC. t58BC. 159BC. 160B C . 1 6 1

95

420 CONTINUEcc +++ SpECIA~ vELOC~Ty BOUNOARYCONOIT~ONSc

RETURNEND

SUBROUTINE CAVOVO=CALL,COMMON1cc ++-+cc +++c

13cc ++-+c

2030

40

c50

CALCULATE VOIO VOLUMES

zN~TIALIzE VOID VOLUMES

00 10 K=l.NVCRVOLIK)=O.O

COMPUTE VOIO REGION VOLUMES

00 30 d=2.dMl00 30 I=2,1MIINF=NF(I.d)IF (INF.GT.NvOR) GO TO 40IF (INF. EQ.,0.OR.BETA(I .d).LT.O.0) GO TO 30VOLA=( l.O-F(I, d) )=AC(I,LJ)=TPI =RI(I)=OELX( I)=OELY( d)IF (INF.GT.5) GO TO 20INFR=NF(I+l,J)INFT=NF(I,d+l)INFL=NF(I-l,d)INFB=NF(I.J-1)INF=MAXO(INFR; INFT,INFL,INFB)VOL(INF )=VOL(INF)+VOLACONTINUERETURNCONTINUEWRITE (59,50) I,d, INF,NVOR,NCYCWRITE (12,50) I.J,INF,NVOR.NCYCCALL EXITLOC (6HCAVOVO)

FORMAT (lXo5H===-- .lxo25HNVOR IS TOCI SMALL - I,U =,215, 1x,4HNF =1. 15,1X.6HNVOR =.15, 1X,7HCYCLE =,17)

END

SUBROUTINE DELTAOJ.CALL.COMk!ONl

OATA LITER / 15 /

cc +++c

10

20

96

OATA ITMIN /5!. ITMCIST /33/OAT& ITCdR / 100 I

OELT (TIME STEP; AOJUSTMEN7

DELTN=OEL7!F [FLGC.LT.O.5) GC TC 20T=T-OELTNCYC=NCYC-1o~LT=o.8.DELTDO 10 1=1. IMAXDO 10 d=l.JMAXPII,J)=PN(I,J)F(I.LJ)=FN(I,u)U(I,tJ)=O.OV(I.J)=O.OCONTINUEFLGC=O.ONFLGC=NFLGC+lCONTINUEIF (AUTOT. LT.O.5.ANO.FNOC .LT.O.5) GO TO 40DUMX=EMIODVMX=EM1OIF (FNOC.GT.o.5) OELT=OELTXO.800 30 1=1.IM1DO 30 J=2.LJMIUOM=ABS(UN(I,J)VOM=ABS(VN(I,d)OUMX=AMAXI(OUMXOVKX=AMAX1(OVMX

t(xI(I+t)-xI(I))~~;j(!J+l)-YJ(J))

VOM)

BC.162BC.163BC.164BC.IE5B:.166BC.167

CAVOVO.2CAVOVO.3CAVOVO.4CAVOVO.5CAVOVO.6CAVDVO.7CAVOV3.9CAVOVQ.9CAVOVO.10CAVOVO.11CAVOVO.12CAVOVO. 13CAVOVO.14CAVOVO.15CAVOVO. 16CAVOVO.17CAVOVO.18CAVOVO.19CAVOVO.20CAVOVO.21CAVOVO.22CAVOVO.23CAVOVO.24CAVOVO.25CAVOVO.26CAVDVO.27CAVOVO.28CAVOVO.29CAVOVO.30CAVOVO.31CAVOV0.32CAVOVO.33CAVOVO.34CAVOVO.35CAVOVO.36

OELTAOJ.4DELTAOJ.5oELTAod.6DELTAoJ.7OELTADd.812ELTAGti.9DEL7AOLJ.10oFLTADd. ~i

OELTADJ.12DELTAOJ. lSOELTADJ.14DELTADJ.15OELTAOJ.16OELTAOJ.17DELTAod.18DELTADd.19OELTAOJ.20DELTADL1.21OELTADJ.22OELTADIJ.23OELTADJ.24OELTAOJ.25OELTAOJ.26OELTAOJ.27oELTAolJ.28OELTADd.29DELTADJ.30OELTAOJ.31OELTAOd.32OELTADJ.33OELTADJ.34

3 0CONTINUEDTMP=l .01IF(ITER.LT.ITMIN) DTMP=l .02IF(ISOR.EO.0) GO TO 32IF(ITER.GT. ITMOST.ANO .LITER.GT.GO TO 34

32 IF(ITER.GT. ITCJR.ANO. LITER.GT.I.34 CONTINUE

DELTO=3ELT*DTk!P

OELTAOJ.35OELTADJ.36OELTAOLJ.37OELTADJ.38

TMOST) OTMP=O.99 OELTADJ.39oELTADd.40

CJR) OTMP=O.99 oELTADd.41DELTADJ.42DELTADJ.43

OELTO=AMIN1 (OELTO,OTCRMX )DELT=AMINl (OELTO,CON/OUMX .CON/OVMX,OTVIS .OTSFT JIF (IMovy.GT.o) DELT=AMINI (OELT,PLTOT)

40 IF (OELT.E().OELTN) GO TO 60OG 50 1=2.IW1DO 50 LJ=2,JM1IF (AC(I.d).LT.EM6) GO TO 50R!-IXR=RHOF*(DELX (I+l )+DELX(I))RHXL=RHOF-(DELX(I )+OELX(I-1))RHYT=RhOF- (DEL} (d+l )+DELY(d))RHYB=RHOF= (DELY(U)+DELY( J-l))XX=DELT=ROX( I)=(2.O=AR( I- l,d)-R( I-1 )/RHxL+2.O=AR( I,J)=R(I)/RHXR)+

1 OELT*ROY(J )*(2.O=AT( I,d)*RI (I)/RHYT+2.o=AT( I,d-l)*RI(I)/RHYB)XX=XX/(AC( I,J)*RI(I) )6ETA(I,d)=OMG/XX

50 CONTINUE60 CONTINUE

LITER=ITERRETURNEND

SUBROUTINE DRAW=CALL,COMMON1cc ++.+

c

c +++’c. ++.c +++

c

10

20

PLOT VELOCITY VECTOR. FREE SURFACE. MESH,

KT:M~s.I00 70 MESSUP=l,NTIMES

ORAW VELOCITY VECTOR PLOT

WRITE (59,170) T,NCYC.ITER,DELT

CALL AOV ( 1)IF (IMOVY.EO.1) CALL coLoR (o.)CALL FRAME (XMIN.XMAX. YMAX,YMIN)IF (IMOVY.NE.1) GO TO.10CALL COLDR (2. )ENCOOE (8,180,LA6s) TCALL OLCH (650, 100,8,LABS,3)CALL DLCH (650, 100,8,LABS,3)ENcOOE (15, 190,LA8s) FVOL

DELTADJ.44oELTAod.45DELTADJ.46OELTADJ.47CELTADJ.48OELTADJ.49oELTADIJ.50OELTADJ.51DELTADJ.52oELTADd.53OELTADJ.54OELTAOLJ.55DELTADd.56DELTAOJ.57DELTAGIJ.58DELTADLJ.59oELTADd.60oELTADd.61OELTAOd.62DELTADJ.63

DRAW.2ORAW.3ORAW.4DRAW.5DRAW.6DRAW.7ORAW.eDRAW.9DRAw.10ORAW.11ORAW.12ORAW.13ORAW.14DRAU.15ORAW.16DRAW.17ORAW.18DRAW.19ORAW.20DRAw.2~ORAW.22

CALL OLCH (200, IO5,10.1OHLOS ALAMOS,2) ORAW.23CALL OLCH (200,105,10, IOHLOS ALAMOS,2) DRAW.24CALL COLOR (O.) ORAW 25GO TO 20CONTINUECALL LINCNT (1)wRITE (t2,200) ”T,NcYc,NAMECONTINUECALL ORWOBSIF (IMOVY.EO.1) CALL COLOR (1.5)VELNEW=O.00 30 I=2,1M1ACELH=O. 5*CYL+I.O-CYLDO 30 LJ=2,LJM1IF (F(I.d).LT.O.5) GO TO 30IF (AC(I:J). LT.ACELH) GO TO 30XCC=XI(I)YCC=O.5-(Y(J)+Y(J-1) )UMPL=O.OVMPL=O.OIF (AR(I.u).GT.EM6) uMpL=I.oIF (AR(I-I, IJ).GT.EM6) UM?L=l.O/(1IF (AT(I,d).GT.EM6) vMpL=I.oIF (AT(I. d-1) .GT.EM6) vMpL=I.o/(1UPLT=UMPL=(U( :-I,U)+U( I,LI))VPLT=VMPL=(V(I.J-1 )+V(I,LJ))VELNEW=AMAX1 (VELNEW,ABS(’JPLT ),ABS

ORAWORAWDRAWORAWORAWORAWDRAWORAWORAW.

262728293031323334

DRAW.35DRAW.36DRAW.37DRAW.38ORAW.39DRAW.40DRAW.41DRAW.42DRAW.43

O+UMPL) DRAW.44ORAW.45

O+VMPL) DRAW.46ORAW.47DRAW.48

VPLT)) ORAW.49

97

30c +++

c +++c +++

cc +++c

40

50

6070

UVEC=LJPLT=VELMX1+XCCVVEC=VPLT=VELMXI+YCCCALL DRWVEC (XCC.YCC .UVEC.VVEC. i)CALL PLTPT (XCC,YCC.53B,1 )CONTINUEIF(NCYC.GT.5 )VELNEW=VELNEWIF”(IMOVY.EO.1) CALL COLOR (2.5)COLtiR CODE : 0: WHITE. 1. RED, 1.5 YELLOW, 2. GREEN, 2.5 CYAN.3. BLUE, 3.5 MAGENTAVELMXl=AMINl (DXMIN,DYMIN)/VELMX

ORAW FREE SURFACE

CONTINUEFPL=O.500 60 I=2,1M100 60 J=2.JM1IF (BETA(I .IJ).LT.O.0) GO TO 60FATR=0.25=(F( I.I.J)+F(I+I ,J)+F(I.J+l)+F(I+l ,J+l))FxTR=0.5*( F(I+I,J+1 )+F(I+l,J) -F(I.d+l)-F( I.LJ))/(XI( 1+1 )-XI(I) )FYTR=o.5*(F(I,J+I )+F(I+l, d+l)-F(I,J)-F( I+l,LI) )/(Yd(d+l)-YJ(LJ) )FTRS=FXTR**2+FYTR=*2IF (FTRS.EO.O.0) FTRS=EP1OXTR=0.5*(XI (I+1)+XI(I ))+(FPL-FATR)*FXTR/FTRSXTR=AMAX1(XTR,XI (I))XTR=AMIN1(XTR.XI (I+I ))YTR=0.5=(YU( LJ)+YU(J+ l))+(FPL-FATR)*FYTR/FTRSYTR=AMAX1(YTR,YJ(J))YTR=AMINl (YTR,Yd(U+l ))IF (F(I, d) .GT.O.5.AND.F(I+l ,U).GT.O.5) GO TO 50IF (FII.d) .LT.o.5.AND.F(I+t .d).LT.o.5) Go TO 50FABR=o.25~(F( 1,d)+F(1+1 ;J)+F(I, J-l)+F(I+l.d- 1))FX5R=0. 5=(F(I+1.LJ)+F (I+l,d-1)-F(IFYBR=O. 5=(F(I.J)+F( 1+1.J)-F(I.J-1FBRS=CXBR-=2+FYBR=*2IF (FBRS.EO.C.!3J FB2S=EP1OXBR=O.5-(XI( 1+1 )+AI(I))+(FPL-FABRXBR=I.MAX1 (XBR.Y.1( I J )XBR=AMINIIXBF?.XI( :+1 1~“YB2=0.5- (YLJ(LJJ+YLJ( J-1) )+(FPL-FAeRYBR=AMAXl (YBG.YJ(u- 1 ) 1YBR=AMINI (YBR.YU(J) )CALL DRWVEC (XBR, YBR,XTR,YTR, l )CONTINUEIF (F(I. J).G7.O.5.AND.F( 1.d+l).GT.O.5) GO TO 60

~)-F(I,j:l)jj(XI( 1+1)-XI(I))-F(I+I..J- l))/(YJ(d)-YJ( d-l))

=FXBR/FBRS

=FVi3R/FBR3

IF (F(I. LJ). LT.O.5.AND.F(I ,J+l).LT.O.5) GO TO 60FATL=0.25=( F(I,J)+F( I, IJ+l)+F( I-1,d)+F(I-1 .LJ+l))FxTL=o. 5=(F(I.d+l)+F( I,u)-F(I-l. J+lj-F( I-l, J))/(XI(I)-XI( I-l))FYTL=o.5=( F(I-l,J+l )+F(I.J+l)-F( I-I ,d)-F(I .d))/(Yd(d+I )-yJ(J))FTLS=FXTL==2+FYTL=*2IF (FTLS.EO.O.0) FTLS=EPIOXTL=O.59(XI(I-I )+XI(I) )+(FPL-FATL )*FXTL/FTLSXTL=AMAXI (XTL,XI(I-I ))XTL=AMIN1(XTL,XI (I) )YTL=O.5*(YJ(J)+YJ(LJ+ l) )+( FPL-FATL )*FYTL/FTLSYTL=AMAX1(YTL, YJ(J) )YTL=AMINl (YTL,YJ(d+ 1))cALL DRwvEc (xTL.yTL.xTR. YTR.1)CONTINUECONTINUEIF (IMovY. Eo. l.AND.NcYc.Eo. 1) CALL ADVIF (IMOVY.EO.1) CALL COLOR (O.)IF (IMOVY.EC.1) GO TO 160!XAPX=XI(IMAX~-XI ( 1 )DMPY=YJ(JMAxj-YJ( 1)CALL CONTRdB (XI. -IMAx. YIJ.-dMAx.F.IBAR2

2)

dBAR2,11,-1. ,-I. ,0.,ZC1 ,OMPX,DM~Y,O:lHF, l ,IHX, l; IHY,l)

CALL CONTRJB (XI,- IMAX, YJ.-JMAX.P.IBAR2 .dBAR2,11,-l. .-l .,O.,ZC1 ,OMPX,DMPY,O, lHP, I,lHX. I,IHY, 1)

CALL ADV(l)cC +++ MESH PLOTc

IF (T.GT.o.0) GO TO 100CALL AOV (1)CALL ORWOBSDO 80 I.J=l,JM1YCC=Y(J)CALL ORWVEC (XMIN, YCC,XMAX,YCC,O)

DRAW.50ORAW.51ORAW.52ORAW.53ORAW.54DRAW.55DRAW.56ORAW.57DRAW.58ORAW.59ORAW.60ORAW.61ORAW.62ORAW.63DRAW.64ORAW.65DRAW.66DRAW.67ORAW.68ORAW.69DRAW.70DRAW.71ORAW.72ORAW.73ORAW.74oRAk!.75DRAW.76ORAW.77ORAW.78ORAW.79ORAW.80DRAW.81DRAW.82DRAW.83ORAU.84ORAki.85ORAW.8EORAW.87oRAk.8eDRAK.8SER&w.90GRAW.91ORAW.92ORAW.93ORAW:94DRAW.95ORAW.96EIRAW.S7ORAW.98ORAW.99ORAW. 100ORAW. 101ORAW. 102ORAW. 103ORAW. 104ORAW. 105ORAW. 106ORAW. 107ORAW. 108ORAW. 109DRAW. 110DRAW. 111DRAW. 112DRAW. 113ORAW. 114DRAW. 115DRAW. 116ORAW. 117ORAW. 118ORAW. 119ERAW. 120ORAW. 121ORAW. 122ORAW. 123ORAW. 124ORAW. 125DRAW. 126DRAW. 127ORAW. 128

98

8@ CONTINUEDO 90 1=1.IM1XCC=X(I)CALL ORWVEC (XCC.YMIN,XCC,YMAX, 1)

90 CONTINUE100 CONTINUE

cCALL AOV(l)

150 CONTINUE160 CONTINUE

RETURNc #

170 FORMAT (IX,2HT=, IPE12.5,6H NCYC=,16,6H ITER=,15,6H OELT=,EII.4,6HlPLOTS)

180 FORMAT (4HT = ,F4.2)190 F9RMAT (9HL10 voL =,F6.2)200 FORMAT (lx,21iT= .lPEll.4, 2x,6HcYcLE= ,17.2X. IOA8)

ENO

SUBROUTINE DRWOBS=CALL,COMMON1cc +++c

10

20

3040

50

60

70

8!)

ORAW AROUND ALL OESTACLES

D3 :70 I=2,1M1,LTR=l.O-EM6A7L=1.C)-EM6IF(I. EQ.2.ANO.CYL.EQ . 1.0) ATL=-EM6A7C=1.O-EM6DO 170 J=2.dMlIF (AC(I,LJ).LT.EM6) GO TO 170AFR=l.OAFT=l.OAFL=I.OAFB=l.OIF (AR(I,J).LT.ATR) AFR=AR(I.LJ)/ATRIF (AT(I.J).LT.ATC) AFT=AT(I.J)/ATCIF (AR(I-I, J).LT.ATL) AFL=AR(I-l,J)/ATLIF (AT(I,d-l). LT.ATC) AFB=AT(I,d-lJ/ATCIF (AC(I,J).GE.ATC) GO TO 120IF (1.EO.2) AFL=AFR-EM6IF (1.EO.IMI) AFR=AFL-EM6IF (J.EC).2) AFB=AFT-EM6IF (d.EO.JMl) AFT=AFB-EM6IF ((AFT+AFB ).LT.EM6.0R .(AFL+AFR). LT.EM6) GO TO 170M=lAMN=AFB+AFRIF ((AFR+AFT).GT.AMN) GO TO 10M=2AMN=AFR+AFTIF ((AFT+AFL).GT.AMN) co TO 20M=3AMN=AFT+AFLIF (!AFL+AFB).GT.AMN) GO TO 30~=4”

GO TO (40,60.80.100). MX1=X(I- I)+AFT*DELX(I)YI=Y(J)IF (AFT.LT. 1.0) GO TO 50YI=Y1-AFR*OELY(J)X2=X(1-1)f2=Y(d)-AFL*OELy(J)IF (AFL.LT. 1.0) GO TO 160Y.2=X2+AFB=DELX(I )GO TO 160X1=X(1-I)YI=Y(d- l)+AFL*OELY(J )IF (AFL.LT. 1.0) GO ‘- ‘-Xl=~l+AFT=OELX(I)X2=X( I-I)+AFB*DELX(Y2=Y(d-1)IF (AFB.LT. I.0) GOY2=Y2+AFR*DELY(d)GO TO 160XI=X(I) -AFB*DELX(I’Yl=Y(d-1)IF (AFB.LT. 1.0) GOY1=YI+AFL*DELY(IJ)

Iu Iu

I)

TO 160

TO 90

ORAW. 129ORAW. 13GORAkl. 131ORAW. 132ORAW. 133ORAW. 134ORAW. 135ORAW.136ORAW. 137ORAW. 138ORAU. 139ORAW. 140ORAW. 141Ol?AW. 142ORAW. 143DRAW. 144ORAW. 145DRAW. 146

ORWOBS.2ORWOBS.3ORWOBS.4ORWOeS.5ORWOBS.6ORWOB5.7ORWCPS.8ORWOBS.9ORWOBS.19ORWOBS.1~ORk103S. 12ORWOBS. 13DRWOBS.14DRWOBS.15oRkloBs.16ORWOBS.17ORWOBS. 18ORWOBS.19ORWOBS.20ORWOBS.21ORWOBS.22ORWOBS.23ORkiOBS.24ORWOBS.25ORWOBS.26ORWOBS.27ORWOBS.28DRWOBS.29ORWOBS.30DRWOBS.31ORWOBS.32oRkloBs.33ORWOBS.34ORWOBS.35ORklOBS.36ORWOES.37DRWOBS.38ORWOBS.39ORWOBS.40ORWOBS.41ORWOBS.42DRWOBS.43oRkloBs.44ORWOBS.45ORWOBS.46OF?WOBS.47ORWOBS.48ORWDBS.49ORWOBS.50ORWOBS.51ORWOBS.52ORWOBS.53ORWOBS.54ORWOBS.55ORWDBS.56ORWOBS.57ORWOBS.58ORWOBS.59DRWOBS.60

99

90 X2=X(I)Y2=Y(d- l)+AFR=DiELY(J)IF (AFR.LT. I.0) GO TD 160X2=X2-AFT*OELX(I)GO TO 160

109 X1=X(I)Yi=Y(J) -AFR*DELY(J)IF (AFR.LT. I.0) GO TO I1OX1=X1-AFB*DELX(I)

110 X2=X( I)-AFT=OELX(I)Y2=Y(J)IF (AFT.LT. I.0) GO TO 160Y2=Y2-AFL*DELY(J)GO TO 160

120 IF (AFR.GT.EM6) GO TO 130X1=X(I)YI=Y(J-1)X2=X1Y2=Y(d)CALL ORWVEC (X1,Y1.X2.Y2,1)

130 IF (AFT.GT.EM6) GO TO 140X1=X(1-I)YI=Y(J)X2=X(I)Y2=}!CALL ORWVEC (X1,V1 ,X.2.Y2. 1!

140 IF (AFL.G7.EM6) GO TC 150Xl=Xi I-!lvl=v(~ )x~.,l,!Y2=Y(U-I;CALL DRWVEC (XI .V1,X2.Y2.1)

150 IF IAFB.GT.EM6) GO TO 170Al=X(I-l]Yl=Y(d-1)X2=X(I)Y2=Y1CALL ORWVEC (X1 .Y1.X2.Y2.1)

160 CALL DRWVEC (X1,Y1,X2,Y2,1)170 CONTINUE

RETURNENO

,

SUBROUTINE ORWVEC=CALL,COMt40Nlcc +++ DRAW’ A VECTORC +++ ~ROVIOES A SYSTEMc

Ic=cX1=XONEY1=YONEY.2=XTW0Y2=YTW0

(XONE.YONE.XTWO.YTWO,ISYM)

DEPENDANT CALL

IO xOI=(X1-XMIN)=SF+XSHFTYO1=(YI -YMIN)-SF+YSHFTX02=(X2-XMIN) =SF+XSHFTY02=(Y2-YMIN )=SF+YSHFTIX1=50.+920.0-XOIIX2=50.+920.0*X02IY~=50.+920.0=( 1.0-yol )IY2=50.+920.0=( 1.0-y02 )CALL ORV (IX1.1Yl,1x2,1y2)IF (ISYMPLT. EO.O.OR.ISyM. EO.0) Go To 20IC=IC+IIF (IC.GT.1) GO TO 20X1=-X1x2=-x2GO TO 10

20 RETURNEND

ORWOBS.61ORWOBS.62ORWOBS.63ORWOBS.64DRWOBS.65ORWOBS.66DRWOBS.67ORWOBS.68ORWOBS.69ORWOBS.70oRwoBs.7iORWOBS.72ORWOBS.73ORWOBS.74ORWOBS.75DRklOBS.76ORWOBS.77ORWOBS.78OI?WOBS.79ORWOOS.80ORWOBS.81ORWOBS.82oRwoBs.e3DRW09S.8LORWOES.85oRkioBs.86ORW03S.875Rw09s.88oEw3es.89oRwoas.soDRWOBS.91DRWOBS.92ORWOBS.93ORWOBS.94ORWOBS.95ORWOBS.96ORWOBS.97DRWOBS.98ORWOBS.99ORWOBS. 100ORWOBS. 101ORWOES.102

ORWVEC.2DRWVEC.3ORWVEC.4DRWVEC.5ORWVEC.6DRWVEC.7CRWVEC.8ORWVEC.9ORWVEC.1OORWVEC.11ORWVEC.12ORWVEC.13DRWVEC.14ORWVEC.15ORWVEC.16ORWVEC.17ORWVEC.18ORWVEC.19ORWVEC.20ORWVEC.21ORWVEC.22ORWVEC.23ORWVEC.24ORWVEC.25ORWVEC.26ORWVEC.27oRWVEC.28ORWVEC.29

100

SUBROUTINE OVCAL=CALL.COMMON1

ccccc

c

20

40

c +++c +++c ++’+

c ++4c +++c +++

c +++c +++c +++

10

DATA ISLTP / O :

ISLIP = G FREE-SL:?-LIKE FOR CURVED INTERIOR BOUNOA21ES

ISLIP = 1 s7ANOAQc FREE SLIF CONDITloN MosT cooEs

DUOR=(UN(I+l .IJ)-UN( I,JJ)=ROX(I+I )DJDL=(UN~I tJ)-UN(I- I,d) )-ROX(I)DUOT=(UN(I,.J+l )-UN(I. d) )-2.O/(DELY(J )+OELY(U+l))Duo5=(uN(I. d)-uN(I .d-1))=2.o/(DELY(J)+oELY (J-I))DVOR=(VN(I+I .d)-VN( I,J))=2.O/(OELX( I)+OELX(I+I))DVOL=(VN( I,J)-VN(I-1, d) )=2.o/(DELx(I )+oELx(I-1)1DvDT=(VN(I.J+l )-VN(I, d))*RDY(J+l )DVOB=(VN(I.J)-VN(I ,J-l) )-ROY(J)

IF(ISLIP.NE.0) GO TO 20IF(AR(I+I, .J). LT.EM6)OUOR=0.()IF(AR(I -1 .LJ). LT.EM6)OUDL=0.C)IF(AR(I, J+l). LT.EM6)OUOT=0.OIF(AR(I, LJ-1).LT.EM6 )OUOB=O.OIF(AT(I+ l. J) .LT.EM6)DVOR=0.OIF(AT(I -l, d) .LT.EM6)ovoL=o.oIF(AT(I .J+l). LT.EM6)DVOT=0. oIF(ATII, J-1 ).LT.EM6)OVOB=0.OGO TO 40IF(ISLIP.NE.1) GO TO 40IF(AI?( I,J+I).LT.EM6) OUOT=O.OIF(AR(I, d-1) .LT.EM6) OUOB=O.OIF(AT(I+ ;.J).LT.EM6) DVDR=O.OIF(AT(I- I.J).LT.EM6) OVOL=O.OCONTINUERETURNENO

suEiRouTINE EOUIB (y,2,NX1.BOND,CANGLE,CYL )DIMENSIDN Y(NX1). Z(NXI)DATA PI /3.141592654/, EPS /1.E-04/

USEFUL INTERMEDIATE CXJANTITIES

RANGLE=cANGLE=?I/ 18c!. .COSTST=CCS(RANGLE)COS:A=( I.-CYL)=COS(RANGLE )OMCYL=l.-CYiNX=NXI-1DR=l./FLOAT(NX)ITER=O -

SET INITIAL GUESS FOR Y(1)

Y(1 )=0xF”iAi%(BoND) .GT.o.) Y(l)=-coscA/(2. *AEis(BoNo )]

NUMERICAL INTEGRATION

2(1)=0.ITER=ITER+IZ(NX1+I)=O.WRITE (8,100) Y(1)DO 20 J=2.NXIRJM=OR*FLOAT(J-2 )*CYL+OMCYLRJ=OR*FLOAT(J- I)*CYL+OMCYLRdH=o.5=(RJM+RJ)Z(J)=Z(d- l)=RJM/Rd+OR*(RIJH/Rd)=(COSCA-BONO= (Y(J-l)+O. 5=oR=Z(d-l )

I /50RT(l. -z(L1-1)~*2)))IF IABS(Z(d)).GE. l .) GO TO 30sLoPE=z(J )/soRT(l. -z(d)**2)SLOPF=Z(LJ- 1)/SORT( 1.-Z(J-l )*=2)Z(NXI+LJ)=(SLOPE-SLOPF )/(OR=( l.+SLOPE**2)** 1.5)Y($J)=Y(J- l)+o.5*oR=(z(J)/sclRT( 1. -z(u)**2)+z(J- l)/soRT( 1.-Z(J-I)==2

1 ))20 CONTINUE

GO TO 40

OVCAL.2DVCAL.3OVCAL.4OVCAL.5DVCAL.6DVCAL.7DVCAL.8OVCt.L.9OVCAL.1ODVCAL. 11OVCAL. 12DVCAL.13OVCAL.14OVCAL.15OVCAL. 16DVCAL.17OVCAL.18DvCAL. 19

OVCAL.20OVCAL.21DVCAL.22OVCAL.23ovcAL.24

OVCAL.25DVCAL.26DVCAL.27OVCAL.28OVCAL.29DVCAL.30OVCAL.31OVCAL.32OVCAL.33OVCAL.34OVCAL.35OVCAL.36OVCAL.37OVCAL.3B

EC)UIB.2EOUIB.3EOU16.4EOU16.5EOUIB.6EOUIB.7EOUIB.BEoiJIE.9EOU13.10EOUIB.11EOUIB.12EC)U16.13EOUIB. 14EcluIB.15EOUIB.16EOUIB.17EOUIB.18EOUIB.19EOUIB.20EOUIB.21EOUIB.22EOUIB.23EOUIB.24EOUIB.25EOUIB.26EOU18.27EOUIB.28EOUIB.29EOUIB.30EOUIB.31EOUIB.32EOUIB.33EOUIB.34EOUIB.35EOUIB.36EOUIB.37EOUIB.38EQUIB.39EOUIB.40

30

.+++r+++L ++

40

50

c+++c+++,. ‘++

60

70

c8090

1(XI

CONTINUEWRITE (8,110) J.Z(d)Y(l)=Y(l)*I.05WRITE (8.120) Z(NXI)IF (ITER.GT.400) Go To 60GO TO 10

CHECK CONSTRAINT ANO CONVERGENCE

CONTINUEWRITE (8.130)YSUM=O~5=OR=Y(NXl)DO 50 J=2,NXYSUM=YSUM+(FLOAT(d-l )*OR*CYL+OMCYL)=DR*Y(J)CONTINUEWRITE (8.140) YSUM, Y(I), Z(NXI),COSTSTIF (ABS(2(NXI )-COS(RANGLE )] .LE. Eps.ANO.BONO.NE.0. ) GO TO 70IF (A5s(YsuM) .LT.Eps.AND.BONO .EO.O.) GO TO 70Y(l)=Y( l)-YsuM/(oMcYL+cYL*2. )IF (ITER.GT.400) GO TO 60GO TO 10

EXIT IF CONVERGENCE FAILS

CONTINUEWRITE (59,150)CALL EXITLOC (5HEQUIB)CONTINUEWRITE (.s,80) (I,Y(I) .Z(I),Z(NXI+I ),I=I,Nx1)WRITE (59,90)WRITE (8,90)RETURN

FORMAT (///3x, iH8X,X, lHy, llx, lHz.9x.3HKxy/( lx,13. lP3E12.4))FORMAT (15H FINISHEO EOUIB)FORMAT (1X, 17HSTEP B WITH Y(I)=. 1PE14.6)

liO FORMAT (lX.9HZ TOO BIG.15,1PE12.4)120 FORMAT (1X, 17HG0 TO STEP B Z=,1PE12.4)130 FORMAT (IX.23HPARTS B ANO C CONVERGEO)140 FORMAT (lx,5HYsuM=, IpEf3.5,4H Y1=.E13.5,4H 2N=E13.5,5H COS=.E13.5)150 FORMAT (IX,28H*** EOUIB FAILEO TO CONVERGE)

ENO

EOUIB.41EOUIB.42EOUIB.43EOUIB.44EOUIB.45EOUIB.46EOUIB.47EOUIB.48EOUIB.49EOUIB.50EOUIB.51EOUIB.52EOUIB.53EOUIB.54EOUIB.55EOUIB.56EOUIB.57EOUIB.58EOUIB.59EOUIB.60EC)UIB.61EOUIB.62EOUIB.63EOUIB.64EOUIB.65EOUIB.66EOUIB.67EOUIB.69EOUIB.69EQUIB.70EQUIB.71EOUIB.72EOUIB.73EQUIB.74EOUIB.75EOUIB.76EOUIB.77EOUIB.78EOUIB.79EOUIB.80EOUIB.81EOUIE.82

102

EXITLDC.2SUBROUTINE EXITLDC (IILoc)*CALL,COMMON1

CALL ADV (1)WRITE [59.10) NCYC. IILOCWRITE (12:10) NCYC:IILOCCALL EXIT.*

10 FORMAT ( lY.,20HTERMINATION ON CYCLE. 17.6H FROM .Ae)ENO

SUBROUTINE FRAME (xxL,xxR, yyT.yYi3)cc +++ DRAW A FRAME AROUND THE PLOTL

CALL DRWVEC (XXL. YYT,XXR,YYT.0)CALL ORWVEC (XXL, YYT,XXL,YYB.0)CALL DRWVEC [X-XL .YYB.XXR.YYB.0)CALL DRWVEC iXXR, YYE:XXR:YYT:O)RETURNENO

SUBROUTINE LAVORE=CALL.COMMON1cc +++c

10

20

ccc

c70

LABEL VOIO REGIONS - - VOID REGIONS ARE NF.EQ.6 AND ABOVE

NNR=6NVR=6CJ 2C d=2,dvlDo 30 1=2.IMIIF (NF(I.J).LT.6) GC TO 30INFP=NF(I.J-:)INFL=NF(I-l,J)IF (INFE. LT.6.ANo.INFL.LT .6) GO TO 20IF (INFB. LT.6.oR.INFL.LT .6) GO TO 10NF(I, J)=MINO(INFB,INFL )INRB=NR(INFB)INRL=NR(INFL)INRMN=MINO( INRB,INRL )INRMX=MAXO( INRB.INRL )NR(INRMX)=INRMNNR(INFB)=INRMNNR(INFL)=INRMNGO TO 30NF(I,J)=INFBIF (INFB.LT.6) NF(I.d)=INFLGO TO 30NF(I,J)=NVRNR(NVR)=NNRNVR=NVR+lNNR=NNR+IIF(NNR.LE.NvOR) GO TO 30

NVOR TOO SMALL

WRITE (59,70) I.d.NNR,NvoR,NcYcWRITE (12,70) I,U,NNR,NVOR,NCYCCALL EXITLOC (6HLAVORE)

FoRMAT (lX.5H====* .IX,25HNVOR IS TOO sMALL1 15. IX,6HNVOR =,15, 1X,7HCYCLE =,17)

30 C6NTINUE

32

NVRI=NVR-1NNRI=NNR-I

REOUCE NR TO LOWEST VALUE

00 38 K=6,NvRIKK=NVRI+6-KKN=KKIF(NR(KN).E().KN) GO TO 34KN=NR(KN)GO TO 32

EXITLDC.3EXITLOC.4EXITLOC.5EXITLOC.6EXITLDC.7EXITLDC.eEXITLGC.9E)iITLDC. 10

FRAME.2FRAME.3FRAME.4FRAME.5FRAM5.6FRAME.7FRAME.&Fi?AME.9FRAME. 10FRAME.11

LAVORE.2LAVORE.3LAVORE.~LAVDRE.5LAVORE.ELAVORE.7LAVORE.8LAVORE.9LAVGRE. 10LAVORE.11LAVORE.12LAVORE. 13LAVORE.14LAVORE.15LAVORE. 16LAVORE.17LAVORE.18LAVORE.19LAVORE.20LAVORE.21LAVORE.22LAVORE.23LAVORE.24LAVORE.25LAVORE.26LAVORE.27LAVORE.28LAVORE.29LAVORE.30LAVORE.31LAVORE.32LAVORE.33LAVORE.34LAVORE.35LAVORE.36LAVORE.37LAVORE.38LAVORE.39

I,d =,215, 1X,4HNF = LAVORE.40LAVORE.41LAVORE.42LAVORE.43LAvogE.44LAVORE.45LAVORE.46LAVORE.47LAVORE.48LAVORE.49LAVORE.50LAVORE.51LAVORE.52LAVORE.53

103

34 NR(KK)=KN38 CONTINUE

cC +++ REDEFINE REGION NUMBEl?s TO BE CONSECUTIVEc

KKN=6DO 50 KK=6,NNf?lKFLG=O00 40 K=KK.NVRIIF (NR(K).NE.KK) GO TO 40NR(K)=KKNKFLG=I

40 CONTINUEIF (KFLG.EO.1) KKN=KKN+I

50 CONTINUENREG=KKN-6

cc +++ REDEFINE voIo NuMBERs To BE CONSECUTIVE IF NREG.GT.1c CHECK FOR CORRESPONOENCE WITH OLD REGIONSc

00 60 d=2.LJMlDO 60 I=2,1MIINF=NF(I,d)IF (INF.LT.6) GO To 60.NF(I,J)=NR(INF)

60 CONTINUERETURNENO

SUBROUTINE MESHSET=CALL,COMMONIcc +++~

1020

MESH sETuP (GENERATION)

1=1d=lX( l)=XL( 1)Y( l)=YL( 1)00 40 K=!.NKXIF (NXL(K).EO.G) GO TO 20DXML=(XC(K)-XL[K) )/NXL(K )NT=NXL(K)TN=NTTN=AMAXI(TN. 1.0+EM6)DXMNI=AMIN1 (DXMN(K) ,OXML)CMC=(XC(K)-XL(K) -TN=OXMN1 )=TN/(TN-l.0)IF (NT.EQ.1) CMC=O.OBMc=xC(K )-xL(K)-cMc00 10 L=l,NT1=1+1RLN=(FLOAT(L )-TN)/TNX( I)=XC(K)+BMC=RLN-CMC*RLN-RLNIF (NXR(K).EO.0) GO TO 40OXMR=(XL(K+l )-XC(K) )/NXR(K)NT=NXR(K)TN=NTTN=AMAXI(TN. 1.0+EM6)DXMN1=AMINI( OXMN(K) ,OXMR)CMC=(XL(K+I )-XC(K)-TN*DXMNI )=TN/(TN-l -O)IF (NT.EO.1) CMC=O.OB!.K=XL(K+l )-XC(K)-CMCDO 30 L=l.NT1=1+1RLN=FLOAT(L)/TN

30 X( I)=XC(K)+BMC=RLN+CMC*RLN*RLN40 CONTINUE

IF [KR.NE.4) GO TO 501=1+1X(1)=X(1-1)+X(2)-X(1 )

50 CONT:NUEDO 90 K=4,NKYIF (NYL(K).EO.0) GO TO 70DyML=(YC(K)-YL(K) )/NYL(K)NT=NYL(K)TN=NTTN=AMAXI(TN, 1.0+EM6)

LAVORE.54LAVORE.55LAVORE.56LAVORE.57LAVORE.58LAVORE.59LAVORE.60LAVORE.61LAVORE.62LAVORE.63LAVORE.64LAVORE.65LAVORE.66LAVORE.67LAVORE.68LAVORE.69LAVORE.70LAVORE.71LAVORE.72LAVORE.73LAVORE.74LAVORE.75LAVORE.76LAVORE.77LAVORE.78LAVORE.79LAVORE.80LAVORE.81

MESHSET.2M5sHsET.3M:51+5ET.4MES!-ISE7.5p~s~5ET.6

M5slisET.9REs~sET.loMESHSET.11MESHSET.12MESHSET.13MESHSET.14MESHSET.15MESHSET.16MESHSET.17MESHSET.18MESHSET.19MESHSET.20MESHSET.21MESHSET.22MESHSET.23MESHSET.24MEsHSET.25MESHSET.26MESHSET.27MESHSET.28MESHSET.29MESHSET.30MESHSET.31MESHSET.32MESHSET.33MESHSET.34MESHSET.35MESHSET.36MESHSET.37MESHSET.38MESHSET.39MESHSET.40MESHSET.41MESHSET.42MESHSET.43MESHSET.44MESHSET.45MESHSET.46MESHSET.47MESHSET.48

●

104

6C70

8090

loa

cc +++c

110120

130

140

DYMN 1=AMIN 1 ( DYMN( K ) . DYML )CMC=(YC(K) -iL(K):TN=OYMNl )*TN/(TN-I.0)IF (NT.EQ.1) CMC=O.OBMC=YC(K) -YL(K)-CMCDO 60 L=I,NTd=d+lRLN=(FLOAT(L) -TN)/TNY(J)=YC(K)+BMCBIRLN-CMC*RLN*RLNIF (NYR(K).EO.0) GO TO 90DYMR=(YL(K+l )-YC(K))/NYR(K)NT=NYR(K). .. . .. .TN=NTTN=AMAX1(TN, l .0+EM6)DYMN1=AMIN1 (DYMN(K),DYMR)CMC=(YL(K+I )-YC(K)-TN*DYMNI )*TN/(TN-l.0)IF (NT.EO.1) CMC=O.OBMC=YL(K+I )-YC(K)-CMCDO SO L=I,NTJ=J+IRLN=FLOAT(L)/TNY(J)=YC(K)+BMC*RLN+CMC=RLN*RLNCONTINUEIF (KT.NE.4) GO TO 100d=d+lY(lJ)=Y(lJ- l)+Y(2)-Y( 1)CONTINUENUMX=INUMY=dNUMXMI=NUMX-1NUMYMI=NUMY-INUMXP1=NUMX+INUMYP1=NUMY+lIBAR=NUMX-1dBAR=NUMY-1IMAX=IBAR+2dMAx=dBAR+2IM:=IMAX-1JM1=IJMAX-I

NEEDEE FOR VARIABLE MESH

G@ TO 110

CALCULATE VALUES

DC :20 I=I,NUMXIF (X(I).EO.O.0)RX(I)=l.O/X(I )GO iD 120RX(I)=O.OCONTINUEDO 130 I=2,NUMXXI(I)=O.5=(X(I-1 )+X(I))DELX(I)=X(I)-X( I-1 )RXI(I)=l.O/XI(I)RDX(Ij=l .(j;OELX(I)DELX(I)=DELX(2)XI(l)=XI (2)-DELX(2)RXI(l)=l.O/XI( 1)ROX(l)=l.O/OELX(I)DELXA=DELX(NUMX)IF (KR.EQ.4) DELXA=DELX(3)DELX(NUMXPI)=DELXAXI(NUMXPI )=XI(NUMX)+DELXAX(NUMXPl )=XI(NUiXPl)+O. 5*DELX(NUMXPI)RXI(NUMXP1)=I .O/XI(NUMXPl)RDX(NUMXPl )=l.O/DELX(NUMXP 1)DO 140 1=2.NUMYYd(IJ=O. 5=tY(I-l)+Y( I))RYu(I)=l.O/}J(I)DELY(I)=Y( I)-Y(I-1)RDY(I)=l.0/DELY(I)CONTINUEDELY(I)=DELY(2)RDY(l)=l.O/DELY(l)YJ(I)=YJ(2)-DELY(2)RYJ(l)=l.O/YIJ(l )DELYA=DELY(NUMY)IF (KT.EO.4) DELYA=DELY(3)DELY(NUMYPI)=DELYAYd(NUMYPl)=Yd(NUMY )+DELYAY(NUMYPI )=YJ(NUMYPI )+0.5*DELY(NUMYPI)RYJ(NUMYP1)=I .O/YJ(NUMYPl)RDY(NUMYP1)=l .O/DELY(NUMYPI)

MESHSET.49MESHSET.50MESHSET.51MESHSET.52MESHSET.53MESHSET.54MESHSET.55MESHSET.56MESHSET.57MESHSET.5aMESHSET.59MESHSET.60MESHSET.61MESHSET.62MESHSET.63MESHSET.64MESHSET.65MESHSET.66MESHSET.67MESHSET.68MESHSET.69MESHSET.70MESHSET.71MESHSET.72MESHSET.73MESHSET.74MESHSET.75MESHSET.76MESHSET.77MESHSET.78MESHSET.79MESHSET.80!NESHSET.81MESHSET.a2MESHSET.83MESHSET.a4NESHSET.85MESH5ET.86MES3SET.E7h!EsHsET.8eMESHSET.89MESI-!SET.90MESHSET.91MES!-ISET.92MESHSET.93MESI-!SET.94MEsiisET.95MESHSET.96MESHSET.97MESHSET.98MESHSET.99MESHSET. 100MESHSET. 101MESHSET.102MESHSET. 103MESHSET. 104MESHSET. 105MESHSET. 106MESHSET. 107MESHSET. 108MESHSET. 109MESHSET. 110MESHSET. 111MESHSET. 112MESHSET. 113MESHSET. 114!4EsHsET. lt5MESHSET. 116MESHSET. 117MESHSET. 118MESHSET. 119MESHSET. 120MESHSET. 121MESHSET. 122MESHSET. 123MESHSET. 124MESHSET. 125MESHSET. 126MESHSET. 127MESHSET. 128

105

cc

145

150

160

170

180190

cc +++c

c

c200

c2 lG220

SET R AND RI ARRAY FOR PLANE OR CYLINDRICAL GEOMETRYDO 145 1=1.IMAXR(I)=X(I)Rl(l)=xr(z)IF(ICYL.EO.1) GO TO 145R(I)=l.ORI(I)=l.OCONTINUEWRITE (6,210)DO 150 I=I,NUMXP1WRITE (6,220) I,X(I), I.RX(I),I.DELX( 1), l.ROX(I), I,XI(I).I ,RXI(I)CONTINUEWRITE (6.210)DO 160 I=I,NUMYP1WRITE (6.230) I.Y(I). I.OELY(I).I,RDY( 1) .I,YIJ(I), I.RYJ(I)CONTINUE”IF (IMOVY.EO.1) GO TO 190WRITE (12,210)DO 170 1=1.NUMXPIWRITE (12.220) I,X(I),I,RXCONTINUEWRITE (12.210)DO 180 I=I,NUMYP1WRITE (12,230) I,Y(I).I,DECONTINUECONTINUE

TEST ARRAY SIZE

MESHSET. 129MESHSET. 130MESHSET. 131MESHSET.132MESHSET. 133MESHSET. 134MESHSET. 135MESHSET. 136MESHSETMESHSETMESHSETMESHSETMESHSETMESHSETMESHSETMESHSETMESHSET.

137138139140141142143144145

MESHSET. 146MESHSET. 147MESHSET.148

1),1.DELX(I).I,ROX(I) .I,XI(I).I,RXI(I) MESHSET.149MESHSET. 150MESHSET. 151MESHSET.152

Y(I), I.RDY(I).I,YJ(I). I.RYLJ(I) MESHSET. 153MESHSET. 154MESHSET. 155MESHSET.156MESHSET. 157MESHSET. 158

If (IMAx. LE. rf3AR2.ANO.dMAx. LE. dBAR2.ANo. lsoR.Eo. 1) Go TO 200 MESHSET.159IF (IMAX. EO. IBAR2.AND.dMAX. Eo. d5AR2.ANo. lsoR.Eo.o) Go TO 200 MESHSET. 160WRITE (6,240) MESHSET. 161

MESHSET. Y62

CALL EXITLDC (7HMEShSET)

CONTINUERETURN

M~sHsET.j63MESHSET. 164MESHSET. 165MESHSET. 166MESHSET. 167

rORMAT ( 11-!l) MEsl-15ET.16e

FORMAT ( 1X,2HX[ .:2.2H)=. lpE12.5.2%.3HRX( .12.21( . 12.2H)=,

H)=. lPE12.5,2x.5HDELx MESHSET. 169:PE12.5,1X.4HRDXI .12.2HI=, IDE12.5,zx.311XI( .12.2HI=, IPE12 MESWSET. 170

2 .5,2X,4HRXI( .12,2H)=, 1PE12.51230 FORMA-

MES-ISET. 171( IX.2HY( ,12,2H)=, IPE12.5, 3x.5H!)ELY( . 12,2H)=. 1PE12.5.3X,4HRD MESI-ISET. 172

IY( .12.2H)= .IPE12.5.3X, 3HYJ(, i2,2H)=. lpE12. 5,3X,4HRYLJ( ,12.2H)=, lpEl MESHSET. 172975) MESHSET. 174--.-,

240 FORMAT (45H MESH SIZE INCONSISTENT WITH ARRAY DIMENSIONS) MESHSET. 175

END MESHSET. 176

SUBROUTINE PETACAL PETACAL.2=CALL,COMMON1 PETACAL.3Lc +++c +4-*c +++c

c

ccc

c

:cc

1

4

DETERh!INE TiiE PRESSURE INTERPOLATION FACTOR PETADETERMINE THE SURFACE TENSION pRESSURE ANDWALL ADHESION EFFECTS IN SURFACE CELLS

OATA KTRAN ; 1 /

GO TD [1,4),KTRAN

RESET KTRAN: PICK UP GEOMtTRIC ANO TRIGONOMETRIC FACTORS

KTRAN=4FEWLIM=25.0/AMINl (X( IMI).Y(JM1))FNSLIM=FEWLIMCSANG=COS(CANGLE)SANG=SIN(CANGLE)CONTINUE

SET 00 LOOPS TD GIVE DEFAULTCELLS: CELLS ARE FLUID CELLSPETA = 1.0

VALUES OF NF. PS, AND PETA IN ALL,SURFACE TENSION PRESSURE IS ZERO ANO

PETACAL.4PETAcAL.5PETACAL.6PETACAL.7PE7ACAL.8pETAcAL.9PETACAL. :oPETACAL.11PETACAL.12PETACAL.13PETACAL.14PETACAL.15PETACAL.16PETACAL.17PETACAL.18PETACAL.19PETACAL.20PETACAL.21PETACAL.22PETACAL.23PETACAL.24PETACAL.25PETACAL.26PETACAL.27PETAcAL.28PETAcAL.29PETACAL.3C)

DO 10 1=1.IMAX00 10 d=l.dMAXNF(I,J)=O”PS(I,d)=O.O

10 PETA(I,U)=I.O

106

cccccc

cccc

ccc

cccc

ccccc

cccccc

ccc

190ccccccc

ccc

ccc~

cccccccc

ccc

DEFAULT VALUES ARESET MAIN 00 LOOPS

CELL: IF SC. LOOPS

00 750 1=2Do 750 J-2

IF CELL IS

IF (AC(I.J

IMIJM1

NOWFORSET

AVAILABLEFINOING WHETHER (1.J) CELLNF. CALCULATE PS AND PETA.

OBSTACLE CELL SKIP TO END OF LOOPS

.LT. EM6.OR.BETA(I,U). LT.O.0) GO TD 750

DECLARE EMPTY CELL TO BE A VOIO CELL

IF (F(I.J).LT.EMF) NF(I.J)=6

IF CELL IS EMPTY (OR FULL BUT PSAT = 0.0) SKIP TOCELLS WILL HAVE OEFAULT VALUES

IS A SURFACELOOPS ENO AT

ENO OF LOOPS;

PETACAL.31PETACAL.32PETACAL.33

750 PETACAL.34

IF (F(I, J).LT.EMF.CR. (F(I.d).GT.EMFl .ANO.PSAT.EQ.O.0) ) GO TO 750

FOUR TESTS TO SEE WHETHER ONE OF THE FOUR NEIGHBOR CELLS IS BOTHEMPTY ANO OPEN TO FLOW FROM (I,J) CELL; IF SO, ENTER MAIN 00LOOPS THRU 190

IF (F(I+l.J) .LT.EMF.ANO.AR( 1.J).GT.EM6) GO TO 190IF (F(I-I,J) .LT.EMF.ANO.AR( 1-l,d).GT.EM6) Go To 190IF (F(I,U+l ).LT.EMF.AND.AT( I,LJ).GT.EM6) GO TO 190IF (F(I, d-1 ).LT.EMF.ANO.AT( 1.u-1).GT.EM6) Go To 190

CELL IS NOT A SURFACE CELL, OBSTACLE CELL,EMPTY CELL OR PARTIC-ULAR TYPE OF FULL CELL. IF IT SATISFIES PRESSURE TEST. SET NF = 5FOR ISOLATEO CELL ANO SKIP TO ENO OF LOOPS: OTHERWISE CELL ISFLUIO CELL ANO WE SKIP TO ENO OF LOOPS WITH OEFAULT VALUES

IF (p(I. d).LE.psAT-EM6pI .ANO.PSAT.GT.O.0) NF(I.J)=5GO 70”750

WE NOW ENTER CALCULATIONAL PARTS OF MAIN

CONTINUE

● ** CALCULATE THE PARTIAL DERIVATIVES OF

DO LOOPS

F

OISTANCES FROM MIOPOINT OF CELL TO MIOPOINT OF NEIGHBOR CELLSOISTANCE TO RIGHT ANO LEFT NEIGHBORS

OXR=0.5*(OELX(I )+OELX( 1+1))OXL=0.5*(OELX( I)+DELX( I-l))

OISTANCE TO TOP AND EOTTOM NEIGHBORS

OYT=0.5=(OELY[ LJ)+GELY[ d+”) 1DYB=0.5=(DELY[ LJ)+DELY( J-1))

D5NDM1NATGRs FOR FINITE DIFFERENCE FORMULAS FOR PARTIALOERIVATIVSS Ih A AND v DIRECTIONS

RXOEN=l .O/(DXR*DXL- (DXR+DXL ))RYCEN=l .O/’(2YT=OYB= (DYT+OYB))

FOFM (ANO FOFP) INOICATE WHETHER CELLS WITH LESSER (GREATER)DICES CONTRIBUTE TO AVERAGE FLUIO HEIGHTS IN THREE CELL ARRAYFOFM=l.O WHEN CELL CONTRIBUTEs:=o.o OTHERWISEINOEX IS d FOR VERTICAL HEIGHTS: I FOR HORIZONTAL HEIGHTSOBSTACLE CELL OOES NOT CONTRIBUTE IF NO FLUIO IN NEIGHBOi? CELLTHREE CELL ARRAY

FOFM=l.OIF(6ETA(I+l.d-l ). LE.O.O.ANO. F(I,J-I ).LTFOFP=l .0IF(BETA(I+l.u+I) .LE.O.O.ANO .F(I,J+I ).LT

Y FLUIO HEIGHT IN CELLS TO RIGHT = AVFRY HEIGHTS MEASUREO FROM FLOOR OF ( d -

EMF) FOFM=O.O

EMF) FCIFP=O.O

) CELLS

PETACAL.35PETACAL.36PETACAL.37PETACAL.38PETACAL.39PETACAL.40PETACAL.41PETACAL.42PETACAL.43PETACAL.44PETACAL.45PETACAL.46PETACAL.47PETACAL.48PETACAL.49PETACAL.50PETACAL.51PETACAL.52PETACAL.53PETACAL.54PETACAL.55PETACAL.56PETACAL.57PETACAL.58PETACAL.59PETACAL.60PETACAL.61PETACAL.62PETACAL.63PETACAL.64PETACAL.65PETACAL.66PETACAL.67PETACAL.68PETACAL.69PETACAL.70PETAcAL.7tPETACAL.72PETACAL.73PETACAL.74PETACAL.75PETACAL.76PETACAL.77PETACAL.78PETACAL’.79PETACAL.80PETACAL.81PETACAL.82PETACAL.83pETAcAL.8APETACAL.85PETACAL.86PETACAL.E7p5-ACAL.~8PETACLL.eQPETACAL.90PETACAL.91PETAcAL.92PETACAL.93

IN- PETACAL.94PETACAL.95PETACAL.96PETACAL.97

OF. PETACAL.98PETACAL.99PETACAL. 100PETACAL. 101PETACAL. 102PETACAL. 103PETACAL. :04PETACAL. 105PETACAL. 106PETACAL. 107

107

c

ccc

ccc

c

cccc

ccc

ccc

~

ccccc

ccccc

ccc

cccc

cc

AVFR=(l .O+AC(I+l,J- 1 )=(F(I+l,d-l )-l .O))=FOFM*DELy(d-l )+1( I.o+AC( I+I,J)=(F(I+ l.J)-l.O) )*DELy(J)+2( 1.o+Ac(l+l,d+l )=( F(I+l.d+l )-l .O))*FOFP=DELy( d+l)

FoFMxl.oIF(BETA( I-I,LJ-1).LE .O.O.AND.F( I,J-1).LT .EMF) FOFM=O.OFOFP=I.OIF(BETA( I-l,d+l).LE. O.O.ANO .F(I,J+I).LT.EMF ) FOFP=O.O

Y FLUIO HEIGHT IN CELLS TO LEFT = AVFL

AVFL=(l.O+AC(I-1 ,J-l)-(F( I-I.J-l )-I .O))*FOFM*OELY(U- l)+1( l.O+AC(I-l.JI=(F( I-1 .LJ)-l.O) )*OELy(J)+2( 1.O+AC( I-l,J+l)*(F( I-1, d+l)-l.O) )=FOFP*DELv(L.J+l )

FOFM=l.OIF(BETA( I.IJ-1).LE.O. O.AND. F(I-I,LJ-I)+F(I+ l.d-1).LT.EMF) FOFM=O.OFCIFP=I.OIF(BETA(I,J+I).LE

Y FLUIO HEIGHT IN

LVFCX=(I.O+AC(I.U1( l.O+AC(I.LJ)=(F(I2( 1.o+Ac(I.d+l)=(F

FOFM=I.OIF(BETA(I-IFOFP=I.OIF(BETA(I+I

X FLUID WIO”X FLUIO WIO”

O.O.ANO

CENTRAL

l)=(F(IJ)-1.o)I,J+l)-

F(I-I, d+l)+F(I+i,d+l ).LT.EMF) FOFp=O.0

CELLS = AVFCY.

J-l )- l.o))*FoFM*DELy( d-l)+*OELY(J)+.O))*FoFp=oELy(d+l )

J+I). LE.O.O.ANO.F(I -I,LJ).LT.EMF) FOFM=O.O

J+l). LE.O.O.ANO.F(I+l ,d).LT.EMF) FOFP=O.O

H IN CELLS ABOVE = AVFTH MEASURED FROM FLOOR OF (I-1) CELLS

AVFT=(l .O+AC(I-l,J+ l)*(F(I-l.J+i)-l .O))*FOFM=OELx( I-l )+1( 1.O+AC(I,U+l)*(F(I .IJ+l)-l .Ol)*DELx(I)+2( 1.o+Ac(I+l,J+I)*(F (1+1, d+l)-l.O) )=FoFp*oELx( 1+1)

FOFM=I.OIF(BETA(I-l,J-1).LE .O.O.ANO.F(I-l,d).LT .EMF) FOFM=O.OFOFP=I.OIF(BETA( I+l,J-1).LE. O.O.AND .F(I+~,J)-LT-EMF ) FoFp=o-o

X FLUIO WIOTH IN CELLS BELOW = AVFB

AVFB=(l .O+AC(I-I.J-I )=(F(I- l.d-l)-l .o))*FoFM=oELx( I-1)+1( 1.O+AC(I, d-l)*(F(I .J-l)-l.O) )=OELX(I)+2( 1.o+Ac(I+l.J-I)*(F (1+1 .d-l)-l .O))=FOFp*oELx( 1+1)‘FoI=M=l.(j“

IF(i3ETA( I-l,d). LE.0.0.ANo. F(I-l,U- l)+F(I-l,U+l) .LT.EMF) FOFM=O.OFOFP=l.OIF(BETA( I+l.d). LE.O.o-ANo. F(I+l.J+l )+F(I+l,d-l) .LT.EMF) FOFP=O.O

X FLUIO WIOTH IN CENTRAL CELLS = AVFCY

AVFCY=( l.O+AC(I-l,O)=( F(I-l,J)- l.O))=FoFM*oELx( 1-1)+1( 1.O+AC(I.J)-(F(I. J)-I.O))*OELX(I ~+2( l.O+AC( I+l.LJ~=(F(I+l ,d)-l.o) )=FoFp=oELx( 1+1)

PETACAL. 108PETACAL. 109PETACAL. 110PETACAL. 111PiITACAL. 112PETACAL. 113PETACAL. 114PETACAL. 115PETACAL. 116PETACAL. 117PETACAL. 118PETACAL. 119PETACAL. 120PETACAL. 121PETACAL. 122PETACAL. 123PETACAL. 124PETACAL. 125PETACAL. 126PETACAL. 127PETACAL. 128PETACAL. 129PETACAL. 130PETACAL. 131PETACAL. 132PETACAL. 133PETACAL. 134PETACAL. 135PETACAL. 136PETACAL. 137PETACAL. 138PETACAL. 139PETACAL. 140PETACAL. 141PETACAL. 142PETACAL. 143PETACAL. 144PETACAL. 145PETACAL. 146PETACAL. 147PETACAL. 148PETACAL. 149PETACAL. 150PETACAL. 151PETACAL. 152PETACAL. 153PETACAL. 154PETACAL. 155pETAcAL.156PETACAL. 157PETACAL. 158p5TAcAL. 159

PETACAL. 160PETACAL. 161PETACAL. 162PETACAL. 163p~TAcAL. 154

AVFL SET EY CCNVENTIi2N IN FIRST COLUMN OF CELLSIk BOTHX ANO Y DIRECTIONS OESTACLES ARE pLACEO ON FLOOR FROM WHICH PETACAL. 166OISTANCES ARE MEASURES: FLUID IS THEN A60VE OESTACLES pETAcAL. :67

IF NF=2 OR 4 FLOOR 1S. AT TOP oF CELL: AT 1+1 OR d+l RESPECTIVELY pETAcAL. 168

i=(I.E0.2) AVFL=AVFCX

THREE POINT FINITE DIFFERENCE FORMULAS FOR SURFACE SLOPES WITHVARIAELE MESti SIZES: FORMULAS EXACT FOR QUADRATICSSLOPE OH/OX FOR ALMOST HORIZONTAL FLUIO = PFX

PFX=RXOEN=( (AVFR-AVFCX) ‘0XL*=2+(AvFcx-AvFL) *OxR*=2)

SLOPE OW/OY FOR ALMOST VERTICAL FLUIO = PFY

PFY=RYDEN*( (AVFT-AVFCY )=OYB*=2+(AVFCy-AVFB )*OYT=*2)

PF = SUM OF SQUARES OF TANGENTS; USEo AS FLAG To oIFFERENTIATESURFACE CELLS FROM ISOLATED CELLS (NF=5)

PF=PFX-*2+PFY**2

IF PF VERY SMALL. CELL IS OECLAREO IsOLATEO (INSTEAO OF suRFAcE)

PETACAL. 165

PETACAL.15SPETACAL. 170PETACAL. 171PETACAL. 172PETACAL.1?3PETACAL. 174PETACAL. 175PETACAL. 176PETACAL. 177PETACAL. 178PETACAL. 179PETACAL. 180PETACAL. 181PETACAL. 182PETACAL. 183PETACAL. 184PETACAL. 185PETACAL. 186PETACAL. 187

108

cc

ccccc

655

cccc

660ci.,.

cc~

ccccccc

ccccc

;cc

cccc

ccc

ccc

ccc

AND WE CONTINUE: OTHERWISE GO TO 660

IF (PF.GT.EMIO) GO TO 660

SET NF(I,J) ANO p(I,~) FOR ISOLATEO CELL: BYPASS DETERMINATIoNOF PRESSURE INTERPOLATION CELL,CALCULATION OF SURFACE PRESSUREANO CALCULATION OF PETA

NF(I,LI)=5P(I, LJ)=O.25=(P(I+I ,J)+P(I,LJ+l )+P(I-I,J)+PI

HAVING SET NF ANO P FOR THE ISOLATED CELLMAIN LOOPS

GO TO 750CONTINUE

I,d-1))

WE NOW SKIP TO ENO OF

FOR SURFACE CELLS WE PICK UP CALCULATIONS OF MAIN LOOPS: HAVINGOETERMINEO SLOPES ANO SOME AUXILIARY QUANTITIES, WE NOW GET NF,PS. AND PETA1~.ORD:R” ~0 SET FLAGS TO BE USEO LATER. WE SUM THE F’S IN COLS.TO RIGHT AND LEFT ANO IN ROWS ABOVE ANO 8ELOW THE (Id) CELL

SUM OF F’S IN (1-1) COL.SFIM =SFIP =SFJP =SFdM =SFIC =SFJC =

SFIM=FSFIC=FSFIP=FSFJP=FSFdC=FSFJM=F

SUM OF F’S IN (r+{j COL.SUM OF F’S IN (J+l) ROWSUM OF F’S IN (J-1) ROWSUM OF F’S IN I COL.SUM OF F’S IN J ROW

I-l, d+l)+F(I-l, d)+F(I- I,J-1)I.J+I)+F( I,d)+F(I,J-I )1+1, J+I)+F( I+I,J)+F(I+l ,LJ-l)1+1, J+I)+F(I.J+I)+F( I-l,d+l)1+1, u)+F(I,J)+F(I-I .J)1+1, J-l )+F(I,u-I)+F(I- I,J-l)

IF THERE IS LITTLE FLUID IN THREE BY THREE ARRAY OF C~LLS: SETCELL PRESSURE AS FOR AN ISOLATEO CELL ANO GO TO ENO OF MAIN LOOPFLAGS ARE INITIALLY SET = O

IF(SFIM+SFIC+SFIP+SFJM+SFIJC+SFJP .LT.O.1O) GO TO 655

IF ANY CELL FACE IS COMPLETELY CLOSEO TO FLOW: SKIP ROW ANOCOLUMN TESTS

IF(AR(I, J).LT.EM6.OR. AR(I-I.U) .LT.EM6.OR.AT( 1.lJ).LT.EM6.I OR.AT(I,J- 1).LT.EM6) GO TO 670IFLGX=OJFLGY=O

IF EITHER COL. TO LEFT OR COL. TO R;X=l

IF (SFIM.LT. EMF.OR.SFIM.GT. 3.O-EMF)IF (SFIP. LTIEMF.OR.SFIP.GT .3.o-EMF)

IF EITHER ROW AEOVE OR ROW 6ELOW IS

IF (5FJP. LT.EMF.OR.5FJp .GT.3.0-EM=)IF (sFJM, L7.EMF.GR.sFdM. GT.3.0-EMFl

GHT IS EMPTY OR IS FULL: IFLG-

IFLGX=IIFLGX=I

EMPTY OR IS FULL: dFLGY = 1

tJFLGY=ltJFLGY=l

IF BSTH FLAGS = 1: CONTINUE EXECUTION AT 670 WITHOUT INTERVENTION

IF (IFLGX. EO. l.ANO.dFLGY .EQ.1) GC TO 670

IF EXAC7LY ONE FLAG = 1: CHANGE THE CORRESPONDING SLOPE

IF (IFLGX.EO.1) PFX=I.OEIO*PFXIF (JFLGY.EO.1) PFY=I.OE1O=PFY

670 CONTINUEcc WE HAVE CONCLUOEO SLOPE INCREASESccc =** OETERMINE THE PRESSURE INTERPOLATION CELL NFcc ALGORITHM GUARANTEES THAT A NEIGHBORING FLUIO CELL (ONE OR MORE

ALWAYS EXIST) ALWAYS LIES AT FLOOR OF THE SURFACE CELL. THE FLUIDCELL AT THE FLOOR IS USEO AS THE INTERPOLATION NEIGHBOR CELL INPRESIT OR PRESCR

PETACAL. 188PETACAL. le9PETACAL. 190PETACAL. 191PETACAL. 192PETACAL. 193PETACAL. 194PETACAL. 195PETACAL. 196PETAcAL. 197PETACAL. 198PETACAL. 199PETACAL.200PETACAL.201PETACAL.202PETACAL.203PETACAL.204PETACAL.205PETACAL.206PETACAL.207PETACAL.208PETACAL.209PETACAL.210PETACAL.211PETACAL.212PETACAL.213PETACAL.214PETACAL.215PETACAL.216PETACAL.217PETACAL.218PETACAL.219PETACAL.220PETACAL.221PETACAL.222PET13CAL.223PETACAL.226PETACAL.225PETACAL.226PETACAL.227PETACAL.228PETACAL.229PETACAL.230PETACAL.231PETAcAL.232PETACAL.233PETACAL.234PETACAL.235PETACAL.236PETACAL.237PETACAL.238PETACAL.239PETACAL.240PETACAL.241PETACAL.242PETACAL.243PETACAL.24APETACAL.245PETACAL.246PETACLL.247pETAcAL.~~8PETAC&L.2.19pETAcAL.2~DPETACAL.251PETACAL.252PETACAL.253PETACAL.25APETACAL.255PETACAL.256PETACAL.257PETACAL.258PETACAL.259PETACAL.260PETACAL.261PETACAL.262PETACAL.263PETACAL.264PETACAL.265PETACAL.266PETACAL.267

109

cccc

cccc

cccc

ccc

cccc

ccccccc

cccc

680ccccc

~ccccc

690ccccc

I

;ET ABSOLUTE VALUE OF SLOPES: MINIMUM WILL OETERMINE WHETHERSURFACE HAS NEAR HORIZONTAL OR NEAR VERTICAL ORIENTATION

iBPFX=ABSIPFX)iBPFY=ABS(PFY)

SET DEFAULT VALUES OF THE INDICES OF THE INTERPOLATION NEIGHBORCELL (L,M): (L,M) = (I,d)

L=IU=J

IF SURFACE IS MORE NEARLY HORIZONTAL: GO TO 680. OTHERWISESURFACE IS MORE NEARLY VERTICAL

IF (ABPFY.GE.ABPFX) GO TO 680

SET MINIMUM SLOPE. L INOEX. ANO

PFMN=PFYNF(I,J)=2L=I+I

COMPUTE LENGTH INTERVALS NEEDEOINTERPOLATION (INTERvAL5 NEEDEO

ANO WE ASSIGN NF = 2 OR 1

NF CONSISTENTLY WITH NF = 2

FOR EVALUATING THE PRESSUREIN BUILOING THE PETA ARRAY)

OMX=OELX(I)OMIN=0.5=(OMX+OELX( I+I ))ONBR=OELX(I+I)

PETACAL.268PETACAL.269PETACAL.270PETACAL.271PETACAL.272PETACAL.273PETACAL.274PETACAL.275PETACAL.276PETACAL.277PETACAL.278PETACAL.279PETACAL.280PETACAL.281PETACAL.282PETACAL.283PETACAL.284PETACAL.285PE7ACAL.286PETACAL.287PETACAL.288PETACAL.289PETACAL.290PETACAL.291PETACAL.292PETACAL.293PETACAL.294PETACAL.295PETACAL.296PETACAL.297PETACAL.298

ALGEBRAIC SIGN OF LARGER ABSOLUTE SLOPE (PFX) DETERMINES NF VALUE. PETACAL.299IF PFX > 0.0 THEN NF = 2 ANO WE SKIP TO END OF NF ROUTINE AT 690

OTHERWISE NF = 1 ANO WE GO ON, REPEATING IMMEDIATELY PREVIOUSSTEPS APPROPRIATELY FOR NF = 1

IF (PFX.GT.O.0) GO TO 690NF(I.J)=lPFMN=-PFYL=I-~DMX=OELX(I)OMIN=O.5*(DMX+DELX(I- 1))ONBR=OELX(I-1)

HAVING COMPLETED CALCULATIONS FORROUTINE HOLOING NF = 1 OATA

GO TO 690CONTINUE

WE ARE NOW IN SURFACE MORENF = 4SET MINIMUM SLOPE. L INOEX

PFMN=-PFXNF( 1.J)=4V=dt:OMX=CELY(J)DMIN=O. 5=(OMX+OELY( J+! ))ONBR=OELY(d+l)

NEARLY

ANO NF

NF = 1. WE SKIP TO ENO OF NF

HORIZONTAL CASE. WE START WITH

CONSISTENTLY WITH NF = 4

ALGEBRAIC SIGN GF LARGER SLOPE (PFY) DETERMINES N; VLLLJE. IF PFY> 0.0 THEN NF = 4 ANO WE SKIP TO ENO OF NF ROUTINEOTHERWISE NF = 3 ANO WE GO ON. REPEATING IMMEDIATELY PREVIOUSSTEPS APPROPRIATELY FOR NF = 3

IF (PFY.GT.O.0) GO TO 690NF(I,LJ)=3PFMN=PFXMxJ-1DMx=OELY(d)oMIN=o. 5=(DMx+oELY(d- 1))ONBR=DELY(U-1)CONTINUE

WE HAVE COMPLETEO SETTING NF ANO (L,M) FOR THE PRESSURE INTERPO-LATION

PETACAL.300PETACAL.301PETACAL.302PETACAL.303PETACAL.304PETACAL.305PETACAL.306PETACAL.307PETACAL.308PETACAL.309PETACAL.310PETACAL.311PETACAL.312PETACAL.313PETACAL.314PETACAL.315PETACAL.316PETACAL.317PETACAL.318PETACAL.319PETACAL.320PETACAL.321PETACAL.322PETACAL.323PETACAL.324PETACAL.3Z5PETACAL.326PETACAL.227PETACAL.228PETACAL.329PETACAL.330PE7ACAL.331PETACAL.332PETACAL.333PETACAL.334PETACAL.335PETACAL.336PETACAL.337PETACAL.338PETACAL.339PETACAL.340PETACAL.341PETACAL.342PETACAL.343PETACAL.344PETACAL.345PETACAL.346PETACAL.347

110

cc

cc~

c

Gr

(..c

:c

:ccccc

ccccccccc

:ccccc

ccccccc

DETERMINE AN INTERPOLATION LENGTH FOR THE PETA ARRAY PETACAL.348PETACAL.349

IF(NF(I,d).Eo.1) SDFS=(I .O+AC(I-I,J)= (F(I-I.J) -I.o))=DELX( I-I)+ PETACAL.3501 (1 .o+AC(I, J)*(F(I,J)- I.O))=DELX(I) PETACAL.351

IF(NF(I,U).EQ.2) SDFS=(l .O+AC(I+l,d)*( F(I+I, J)-I.0))-DELX(I+I )+ PETACAL.3521 (1 .C)+AC( I.J)=(F(I.d)- I.O))=DELX(I) PETACAL.353

IF(NF(I,J):EO.3) SDFS=(I.O+AC(I,~-l )=(F(I, LJ-l)-l .O))=DELY(d- 1)+f (l .o+Ac(I .d)=(F(I.d)-l .O))*DELY(d)

PETACAL.354PETACAL.355

IF(NF(I,J).EO.4) SDFS=(l.O+AC(I, J+l )=(F(I, J+I)-1.O))*DELY(J+ 1)+ PETACAL.3561 (1 .o+AC(I, d)*(F(I,d)- 1.O))=OELY(LJ) PETACAL.357DFS=0.5*DMX+DNBR-SDFS PETACAL.358DFS=AMINI (DFS,O.5=DMIN) PETACAL.359

PETACAL.360IF NO SURFACE TENSION BYPASS PS ROUTINE AND CALCULATE PETA(I,IJ)

IF (IsuRFIO.LT.1) GO To 740

*=* OETERMINE THE SURFACE PRESSURE

WE DETERNINE ONLY THE SURFACE TENSION CONTRIBUTION TO PS: VOIDPRESSURE IS AODED IN PRESIT OR IN PRESCR

SET NF ANO NW INTO NON-INOEXEO VARIABLES FOR USE IN IF STATEMENTSSET OEFAULT VALUES OF FLGE. .. .,AFFLGU. .FLGU .INW,CSANG, AND SANGFLAGS GIVE CHOICE OF WALL AOHESION OR SURFACE TENSIDN ANO, IF WALLAOHESION. THE OIRECTIDN OF THE FORCE

NFF=NF(I,d)FLGE=I.OFLGW=I.OFLGN=I.OFLGS=I.ONWW=NW(I.J)CSANG=COSO(I,LJ)SANG=SINO(I,J)FLGU=l.OAFE=l.OAFW=I.OAFN=I.OAFS=l.OINw=O

NWW=O INDICATES A FLUIO CELL FULLY OPEN TO FLOW WHOSE FACESMAY BE COMPLETELY OPEN TO FLOW, FULLY CLOSEO. OR PARTIALLYOPEN TO FLOW (SIMULATING POROUS BAFFLES) - 5KIp TD sTANOAROSURFACE TENSION CALCULATION

IF(NWW.EO.0) GO TO 100

RESET INW: THIS FORECLOSES THE POROUS BAFFLE OPTION(cELL FACES partially OPEN TO FLOw, PART 5uRFAcE TENsION.PART AOHESION)RESET CSANG FOR NF=l OR 2; WE NOW HAVE FINAL VALUES FOR CSANGSO MAGNITUDE OF WALL AOHESION FORCE HAS NOW BEEN DETERMINEIF NWW=NFF GO TO STANOARO CALCULATION OF SURFACE TENSIONUSING PARAMETERS SET FOR CURVED BOUNDARY CASES

INW=IIF(NFF.LE.2) CSANG=SINO~I.LJlSANG=O.OIF~NwW.EC.NFF) GC TO 10’J

CURVED BOUNDARY WALL ADHESION

FOR SOME CASES OF Nk’ = 1.3,4.5,6,7, OR 8 SET WALL AOHESIOh R4THERTHAN SURFACE TENSION. CHOOSING THE WALL ANO THE FORCE DIRECTIONUSE NF AND NEIGHBORING F VALUES TO MAKE THE DETERMINATION

GO TO (20.100,30,40,50,60,70. 80).NWW

WE ARE IN NW = 1 BRANCH. WE HAVE ALREADY TREATED NF = 1. FOR NF =3 OR 4 : BRANCH AHEAO TO 23 OR 24FOR NF = 2: RESET FLGU: IF NO FLUID ABOVE, PUT WALL AOHESION DNTOP WALL: IF NO FLUIO BELOW. PUT WALL AOHESION ON BOTTOM WALLENTER NEXT STAGE OF CALCULATION

20 IFINFF.EO.3) GO TO 23IF(NFF.EO.4) GO TO 24FLGU=-1.OIF(F(I, J+l).LE.EMF) AFN=O.O

PETACAL.361PETACAL.362PETACAL.363PETACAL.364PETACAL.365PETACAL.366PETACAL.367PETACAL.368PETACAL.369PETACAL.370PETACAL.371PETACAL.372PETACAL.373PET&cAL.374PETACAL.375PETACAL.376PETACAL.377PETACAL.378PETACAL.379PETACAL.380PETACAL.381PETACAL.382PETAcAL.3e3PETACAL.384PETACAL.385PETACAL.366PETACAL.387PETACAL.388PETACAL.389PETACAL.390PETACAL.391PETACAL.392PETACAL.393PETACAL.394PETACAL.395PETACAL.396PETACAL.397PETACAL.398PETACAL.399PETACAL.400PETACAL.401PETACAL.402PETACAL.403PETACAL.404PETACAL.405PETACAL.496PETACAL.407PETACAL.40ePETACAL.409PETACAL.410PETACAL.LIIPETACAL,412PETACAL.413PETACAL.414PETACAL.415PETACAL.416PETACAL,417PETACAL,418PETACAL.419PETACAL.420PETACAL.421PETACAL.422PETACAL.423PETACAL.424PETACAL.425PETACAL.426PETACAL.427

111

ccccc

23

ccccccc

24

ccccccc

4043

ccccc

41

ccccc

42

ccccccc

3033

ccccc

31

ccccc

32

ccc

112

IF(F(I, J-1).LE.EMF) AFS=O.OGO TO 100

FOR NF = 3: IF NO FLUIO ABOVE. PUT WALL AOHESIoNNO FLUID BELOW, PUT WALL AOHESION ON WEST WALLENTER NEXT STAGE OF CALCULATION

IFIF(I.J+I ).LE.EMF) AFE=O.OIF(F(I,d- 1).LE.EMF) AFW=O.OGO TO 100

ON EAST WALL: IF

FOR NF = 4: IN THIS CASE WE ARE CALCULATING -Y COMPONENT OFSURFACE FORCES. THUS WALL ADHESION ENTERS WITH OPPOSITE SIGN. IFNo FLulo ABovE, PUT WALL AOHESION ON WEST WALL: IF NO FLUIDBELOW, PUT WALL AOHESION ON EAST WALLENTER NEXT STAGE OF CALCULATION

IF(F(I, J+I).LE.EMF ) AFW=O.OIF(F(I. d-1).LE.EMF) AFE=O.GO TO 100

WE ARE IN NW=4 BRANCH. WE HAVE ALREAOY TREATEO NF=lFOR NF=l OR 2: t3RANCH AHEAD TO 41 OR 42FOR NF=3: IF NO FLUID TO RIGHT, PUT AOHESION ON EAST WALL:IF NO FLUIO TO LEFT, PUT AOHESION ON WEST WALLENTER NEXT STAGE.. .

GO TO (41,42,43,43), NFFIF(F(I+l,J) .LE.EMF) AFE=O.OIF(F(I- l,IJ).LE.EMF) AFw=O.OGO TO 100

FOR NF=l: RESET FLGU: IF NO FLUIO TO RIGHT, PUT AOHESIONON SOUTH WALL; IF NO FLUIO TO LEFT, PUT AOHESION ON NORTH WALLENTER NEXT STAGE. . .

FLGU=-1.OIF(F(I* l,J).LE.EMF) AFS=O-OIF(F(I- I.J).LE.EMF) AFN=O.OGO TO 100

FOR NF=2: RESET FLGU: IF NO FLUIO TO RIGHT PUT AOHESION ONNORTH WALL; IF NO FLUIO TO LEFT, PUT AOHESION ON SOUTH WALLENTER NEXT STAGE. . .

IF(F(I- 1.d).LE.EMF) AFS=O.OIF(F(I+l, d).LE.EMF) AFN=O.OFLGU=-I.OGO TO 100

WE ARE NOW IN NW=3 BRANCH. WE HAVE ALREADY TREATEO NF=3FOR NF=I OR 2: BRANCH A!-IEAO TO 31 OR 32FOR NF=4: ‘RESET FLGU, IF NO FLUID TO RIGHT, PUT AOHESIONON EAST WALL; IF NO FLUID TO LEFT. PUT AOHESION ON WEST WALLENTER NEXT STAGE. . .

GO TO (31.32,33.33), NFFIF(F(I+I, JI.LE.EMF) AFE=O.OiF(FII-l, d).LE.EMF ) AFW=O.OFLGU=-I.OGO 70 100

IF NO FLUIO TO RIGHT. PUT Ai)HESICIN Oh’ NORTi-I WALL: IF NO FLU~OTO LEFT, PUT AOHESION ON SOUTH WALLENTER NEXT STAGE. . .

IF(F(I+l, LJ).LE.EMF) AFN=O-OIF(F(I- l,d).LE.EMF) AFS=O.OGO TO 100

IF NO FLUIO TO RIGHT, PUT AOHESION ON SOUTH WALL: IF NO FLUIOTO LEFT, PUT AOHESION ON NORTH WALLENTER NEXT STAGE. . .

IF(F(I- I,J).LE.EMF) AFN=O.OIF(F(I+I,U) .LE.EMF) AFS=O.OGO TO 100

WE ARE NOW IN NW=6 BRANCHRESET FLGU: IF NF=2 OR 4

PETACAL.428PETACAL.429PETACAL.430PETACAL.431PETACAL.432PETACAL.433PETACAL.434PETACAL.435PETACAL.436PETACAL.437PETACAL.438PETACAL.439PETACAL.440PETACAL.441PETACAL.442PETACAL.443PETACAL.444PETACAL.445PETACAL.446PETACAL.447PETACAL.448PETACAL.449PETACAL.450PETACAL.451PETACAL.452PETACAL.453PETACAL.454PETACAL.455PETACAL.456PETACAL.457PETACAL.458PETACAL.459PETACAL.460PETACAL.461PETACAL.462PETACAL.463PETACAL.464PETACAL.465PETACAL.46GPETACAL.467PETACAL.468PETACAL.469PETACAL.470PETACAL.471PETACAL.472PETACAL.473PETACAL.474PETACAL.475PETACAL.476PETACAL.477PETACAL.478PETACAL.479PETACAL.480PETACAL.481PETACAL.482PETACAL.483PETACAL.484PETACAL.485PETACAL.486PETACAL.487pETACAL.488PETACAL.489PETACAL.49CPETACAL.491PETACAL.492PETACAL.493p~~AcAL.494PETACAL.495PETACAL.496PETACAL.497PETACAL.498PETACAL.499PETACAL.500PETACAL.501PETACAL.502PETACAL.503PETACAL.504PETACAL.505PETACAL.506PETACAL.507

ccc

BRANCH AHEAD TD 64; IF NF=3 OR 4FOR NF=l OR 2: IF NO FLUIO ABOVE, PUT ADHESION ON NORTH WALL:IF NO FLUIO TO LEFT. PUT AOHESION ON SOUTH WALL

PETACAL.508PETACAL.509PETACAL.51OPETACAL.511PETACAL.512PETACAL.513PETACAL.514PETACAL.515PETACAL.516PETACAL.517PETACAL.518PETACAL.519PETAcAL.520PETACAL.521PETAcAL.522PETACAL.523PETACAL.524PETACAL.525PETACAL.526PETACAL.527PETACAL.528PETACAL.529PETACAL.530PETACAL.531PETACAL.532PETACAL.533PETACAL.534PETACAL.535PETACAL.536PETACAL.537PETACAL.538PETACAL.539PETACAL.540PETACAL.541

cc

ENTER NEXT STAGE. . .

60 IF(NFF. EO.2.OR.NFF.EQ .4) FLGU=-1.OIF(NFF.GT.2) GO TO 64IF(F(I, LJ+l).LT.EMF) AFN=O.OIF(F(I- l,J).LT.EMF) AFS=O.OGO TO 100

AOHESION ON EAST WALL:ON WEST WALL

ccccc

FOR NF=3 OR 4: IF NO FLUID A80VE, PUTIF NO FLUID TO THE LEFT, PUT AOHESIONENTER NEXT STAGE. . .

64 IF(F(I. d+l).LT.EMF) AFE=O.OIF(F(I- I,LJ).LT.EMF) AFW=O.OGO TO IM

c WE ARE NOW IN NW=5 BRANCHFOR NF=I OR 4: RESET FLGUFOR NF=3 OR 4: BRANCH AHEAO TO 54FOR NF=I OR 2: IF NO FLUIO BELOW, PUTIF NO FLUID TO THE LEFT, PUT AOHESIONENTER NEXT STAGE. . .

AOHESION ON SOUTH WALL;ON NORTH WALL

c50 IF(NFF. EQ.1.OR.NFF.E(). 4) FLGU=-I.O

IF(NFF.GT.2) GO TO 54IF(F(I, J-1).LT.EMF) AFS=O.OIF(F(I- 1.LJ).LT.EMF) AFN=O.OGO TO 100

ccccc

FOR NF=3 OR 4: IF NO FLUIO 8ELOW, PUTIF NO FLUID TO THE LEFT, PUT AOHESIONENTER NEXT STAGE. . .

AOHESION ON EAST WALL;ON WEST WALL

PETACAL.542PETACAL.543PETACAL.544PETACAL.545PETACAL.546

54 IF(F(I, J-1).LT.EMF) AFE=O.OIF(F(I- l,d).LT.EMF) AFW=O.OGO TO 100

ccc

PETACAL.547PETACAL.548PETACAL.549PETACAL.550PETACAL.551PETACAL.552PETACAL.553PETACAL.554

WE ARE IN NW*7 8RANCHFOR NF=I OR 3: USE SURFACE TENSION ONLYFOR NF=4: BRANCH AHEAD TO 74FOR NF=2: IF NO FLUIO ABOVE. PUT ADHESION ON NORTH WALLENTER NEXT STAGE. . .

Go To (lOOOYZ,ICX),T4),NFFIF(F(I, J+l).LT.EMF) AFN=O.OGO TO 100

FOR NF=4: RESET FLGU: IF NO FLUIO TO RIGHT, PUT ADHESION ON EASTWALLENTER NEXT STAGE. . .

IF(F(I+I, LJ).LT.EMF) AFE=O.O,CLGu=.1 .oGO TO 100

WE ARE IN Nw=8 BRANCHFOR NF=l OR 4: USE SURFACE TENSION ONLYFOR NF=3: ERANCH AHEAO TC 93FOR NF=2: RESET FLGU: IF hog FLUID i3ELa*’, PUT AD”HE310N ON’ SouTH

c7072

74

PETACAL.555PETACAL.556PETACAL.557PETACAL.558PETACAL.559PETACAL.560c

c PETACAL.561PETACAL.562PETACAL.563PETACAL.564PETACAL.565PETACAL.566PETACAL.567p~T~cAL.568PETACAL.569PETACAL.570pETAcAL.571

cccccccc

W.4LLENTER NEXT STAGE.. .

GO TO (100.82.83.:00), NFFIF(F(I. d-l).LT.EMi) AFS=O.OFLGU=-1.OGO TO 100

FOR NF=3: IF NO FLUID TO THE RIGHT, PUT

IF(F(I+I, LJ).LT.EMF) AFE=O.O

PETACAL.572PETACAL.573PETACAL.574PETACAL.575PETACAL.576PETACAL.577PETACAL.578PETACAL.579PETACAL.580PETACAL.581PETACAL.582PETACAL.583PETACAL.584PETACAL.585PETACAL.586PETACAL.587

8082

83

100

ccc

AOHESION ON EAST WALL

cccc

CALCULATION OF SURFACE TENSION FORCE IS OIFFERENT FOR VERTICALANO HORIZONTAL SURFACES: GO TO APPRCPIATE BRANCH TO BEGIN

GO TO (700,700.710,710). NFF

WE HAVE A NEAR VERTICAL SURFACEcc

ccc

700cccccc

701

cccc

:cc

ccc

cccc.

E

cccc

cc

:702

cccc

cccccc

cccc

114

CALCULATE A HORIZONTAL OISTANCE (ROEW) ANO A RECIPROCAL Y PETACAL.588INTERVAL (RONS) FOR LATER USE

RDEW=RXI(I)

IF(INW=O: GC TO CALCULATION FOR POROUS BAFFLE CAPABILITY: CALCULATLINEAR COMBINATION OF SURFACE TENSION AND ADHESION FORCESOTHERWISE. BYPASS POROUS BAFFLE ROUTINE AND CALCULATE EITHERSURFACE TENSION OR WALL ADHESION

IF(INW.EO.0) GD TO 701RDNS=ROY(J)IF(I.EO.2) GO TO 702IF(AR(I- l,J).ST.EM6) RDNS=RDY( d)/AR(I-l,J)GO TO 702FLGU=-1.ORDNS=RDY(d)

SET FRACTIONAL AREA OPEN TO WOW ON EAST, OR RIGHi, CELL WALLANO ON WEST, OR LEFT, WALL (I.E.. AFE AND AFW) BY CONVENTION

AFE=I.OAFW=I.O

SET FRACTIONAL AREA OPEN TO FLOW ON NORTH, OR TOP, WALL AND ONSOUTH, OR BOTTOM. WALL ACCOROING TO OBSTACLE GEOMETRY

AFN=AT(I,d)AFS=AT(I.d-1)

SET DEFAULT VALUE OF AFLOOR, FOR LATER USE IN WALL ADHESION FORCE

AFLOOR=O.O

SET AFLOOR FOR NF = 1: EXCEPT FOR FIRST COL. OF CELLS AFLOOR =FRACTIONAL AREA OPEN TO FLOW ON EAST CELL WALL

FOR NF = 2 AFLOOR = FRACTIONAL AREA OPEN TO FLOW ON WEST WALL

IF (X(I-1).GT.EM6) AFLOOR=AR(I-l.LJ)IF (NFF.EO.2) AFLOOR=AR(I.U)

IF TOP.DR BOTTOM CELL WALL IS OPEN TO FLOW BUT CELL IS EMPTY, SETCORRESPONDING FLAG TO AFLOOR VALUE

IF (AFN.GT. EM6.AND.F(I, d+l).LT.EMF) FLGN=AFLOORIF (AFS.GT. EM6.AN0.F(I,d- 1).LT.EMF) FLGS=AFLOOR

BEGIN CALCULATION OF. SURFACE TENSION FORCES. ESPECIALLY OF WETANGENTS THAT APPEAR ON THE VARIOUS WALLS

OHENE=O.OOHESE=O.OOHENW=O.OOHESW=O.OOHNNE=( AVFT-AVFCY)”2 .0/DHNNW=OHNNEOHNSE=( AVFCY-AVF6)”2 .o/DHNSW=OHNSE

IF PLANE GEoMETRY; sKIpTENSIoN FoRcEs pRccEEDS

IF(CYL.LT.O.5) GO TO 720

DELYIJI+DEL!’(J*I ))

DELY(J)ADELY(J- l))

TO 720. WHERE CALCULATION OF SIJRFACE

FOR CYLINDRICAL CASE WE CALCULATE EXTRA TERM OUE TC AZIMUTHALCURVATURE OF SURFACE: CALCULATE AN AVERAGE SLOPE. MULTIPLY BY SOMEGEOMETRIC FACTORS, SWITCH ALGEBRAIC SIGN WHEN NF = 2 (ASSURFACE CURVATURE CHANGES SIGN). ANO STORE RESULT IN FLGN ANO FLGS

OHNA=0.5=(OHNNE+DHNSE )FCYL=0.5=DELY( J)=OHNA=RXI (11IF(NFF.EO.2) FCYL=-FCYLFLGN=FLGN*( l.O+FCYL)FLGS=FLGS=( l.O-FCYL)

HAVING STOREO CYLINDRICAL INFORMATION: PROCEED TO FURTHERCALCULATION OF SURFACE TENSION FORCE AT 720

GO TO 720

PETACAL.5B9PETACAL.590PETACAL.591PETACAL.592PETACAL.593PETACAL.594PETACAL.595PETACAL.596PETACAL.597PETACAL.598PETACAL.599PETACAL.600PETACAL.601PETACAL.602PETACAL.603PETACAL.604PETACAL.605PETACAL.606PETACAL.607PETACAL.608PETACAL.609PETACAL.610PETACAL.611PETACAL.612PETACAL.613PETACAL.614PETACAL.615PETACAL.616PETACAL.617PETACAL.618PETACAL.619PETACAL.620PETACAL.621PETACAL.622PETACAL.623PETACAL.624PETACAL.625PETACAL.626PETACAL.627PETACAL.628PETACAL.629PETACAL.630PETACAL.631PETACAL.632PETACAL.633PETACAL.634PETACAL.635PETACAL.636PETACAL.637PETACAL.638PETACAL.639PETACAL.640PETACAL.641PETACAL.642PETACAL.643PETACAL.644PETACAL.645PETACAL.646PETACAL.647PETACAL.64ePETACAL.649PETACAL.65GPE7ACAL.651PETACAL.652PETACAL.653PETACAL.654PETACAL.655PETACAL.656PETACAL.657PETACAL.658PETACAL.659PETACAL.660PETACAL.661PETACAL.662PETACAL.663PETACAL.664PETACAL.665PETACAL.666PETACAL.667

cc WE HAVE A NEAR HORIZONTAL SURFACE: CALCULATE RDEW AND RDNSc APPROPRIATE TO THIS CASEc

710 RONS=RXI(I)Lccccc

713

11

c

\

ccc

cccc

712

(,

Ec

c

ccc:~

720

cccc~

ccc

:ccc

IF INW=O GO TO POROUS BAFFLE ROUTINE, GIVING LINEARCOMBINATION OF SURFACE TENSION AND ADHESION FORCESOTHERWISE ,BYPASS POROUS BAFFLE ROUTINE, CALCULATING EITHERSURFACE TENSION OR AOHESION, EITHER HERE OR AFTER 713

IF(INW.EO.0) GO TO 711ROEW=ROX(I)IF(NW(I, ti).EO.3.OR.NW( 1. LJ). EO.5.OR.NW(I,LJ). EQ.7) GO TO 713IF(AT(I, LJ).GT.EM6) ROEW=ROX( I)/AT(I,J)GO TO 712IF(AT(I, J-1).GT.EM6) ROEW=ROX(I)/AT(I ,J-1)GO TO 712FLGU=-I.OROEW=ROX(I)

SET FOUR FRACTIONAL AREAS OPEN TO FLOW APPROPRIATELY

AFE=AR(I,J)AFW=O.OIF (X(I-1).GT.EM6) AFW=AR(I-1.J)AFN=I.OAFS=l.O

SET AFLOOR FOR LATER USE: TOP OR BOTTOM AREA OPEN TO FLOW

AFLOOR=AT(I,d-1)IF (NFF.EC).4) AFLOOR=AT(I.J)

IF LEFT OR RIGHT CELL IS OPEN TO FLOW BUT EMPTY, SETCORRESPONDING FLAG TO AFLOOR VALUE

IF (AFE.GT. EM6.ANo.F(I+I ,J).LT.EMF) FLGE=AFLOORIF (AFW.GT. EM6.ANO.F(I- l,d).LT.EMF) FLGW=AFLOORIF (CYL.GT.O.5) FLGE=FLGE*X( I)*RXI(I)IF (CYL.GT.O.5) FLGW=FLGW*X( I-I)*RXI( I)

LOAD THE SURFACE TANGENTS IN THE SAME EIGHT VARIABLES:ACCUMULATING IN MANNER APPROPRIATE TO HORIZONTAL CASE

OHNNE=O.OOHNSE=O.OOHNNW=O.OOHNSW=O.OOHENE=(AVFR-AVFCX)-2 .O/(DELX( I)+OELX(I+I ))OHESE=OHENEOHENW=(AVFCX-AVFL)*2 .O/(OELX( I)+OELX(I-l ))DHESW=OHENW

WE HAVE COLLECTEO SURFACE TANGENTS AND GO ON TO FURTHEREvaluation oF suRFAcE TENsioN ANo wALL ADH5sIoN AT 7Qo:

ALL PATHS THRU THIS ROUTINE MEET AT 720 AND GO ON TOWE CONSOLIDATE VARIABLES FOR GIVING SURFACE TANGENTS

OHEE=O. 5-(OHENE+DHESE ]CI-INE=O .5-[OHNNE+OHNSE )OHEW=9. 5-(OHENW+OHESW )OhN’4=0. 5-(OHh!4W+DHNSW J

BEGiN TO ASSEMBLE TERMS THAT WILL PROVIOE DENOMINATORS INEXPRESSIONS FOR SIGN OF ANGLE SURFACE MAKES WITH APPROPRIATEAXIS; OIRECTLY NEEOED FOR SURFACE TENSION FORCE

TERM1=I .O+OHNE*OHNETERM2=1.0+OHNW*DHNW

NOW CONSTRUCT THE DENOMINATOR

RHOE=SORT(TERMI+DHEE=OHEE )RHOW=SORT(TERM2+OHEW=OHEW )

CALCULATE SURFACE FORCES ON EAST ANO WEST WALLSFORCE ’IS FY FOR NF=3 AND -FY FOR NF=4ONLY FOR INW=O IS FORCE REALLY A LINEAR COMBINATION OFSURFACE TENSION ANO WALL AOHESION FORCES. OTHERWISE. IT IS

PETACAL.668PETACAL.669PETACAL.670PETACAL.671PETACAL.672PETACAL.673PETACAL.674PETACAL.675PETACAL.676PETACAL.677PETACAL.678PETACAL.679PETACAL.680PETACAL.681PETACAL.682PETACAL.683PETACAL.684PETACAL.685PETACAL.686PETACAL.687PETACAL.688PETACAL.689PETAcAL.690PETACAL.691PETACAL.692PETACAL.693PETACAL.694PETACAL.695PETACAL.696PETACAL.697PETACAL.698PETACAL.699PETACAL.70CIPETACAL.701PETACAL.702PETACAL.703PETACAL.704PETACAL.705PETACAL.706PETACAL.707PETACAL.708PETACAL.709PETACAL.710PETACAL.711PETACAL.712PETACAL.713PETACAL.714PETACAL.715PETACAL.716PETACAL.717PETACAL.716PETACAL.719PETAcAL.720PETACAL.721PETACAL.722PETACAL.723PETACAL.724FETACAL.725PE7ACAL.726PETACAL.727PE7ACAL.728PETACAL.725PETACAL.730OETACAL.731PETAcAL.732PETACAL.723PETACAL.734PETACAL.735PETACAL.736PETACAL.737PETACAL.738PETACAL.739PETACAL.740PETACAL.741PETACAL.742PETACAL.743PE7ACAL.744PETACAL.745PETACAL.746PETACAL.747

c EITHER SURFACE TENSION OR WALL AOHESION: ACCOROING TO RULESc GIVEN EARLIERL

FEW=FLGE=(AFE=DHEE/RHOE+FLGU= (l .O-AFE)-CSANG)-i FLGW=~AFW=DHEW/RHDW+FLGU= (1 .o-AFw)*csANG)

ccccc

cccc

730

ccccccc

ccccc

cccc

ccccc

c740

c.

c

750c

116

——..IF (CY~lLT.O.5.OR.NFF.GT .2) GO TO 730

IN CYLINDRICAL CASE AOO A TERM TO FEW WHEN NF = 1 OR NF = 2(HELpS BUlLD AZlMUTHAL cuRvATuRE FoRcE, so ALLow FoR cHANGE

OF SIGN OF CURVATURE BETWEEN NF = 1 ANO NF = 2)

FEw=-0.5=(TERMl/RHOE+TERM2/RHDW )IF (NFF.EO.2) FEW=-FEW

ASSEMBLE TERMS TO CONSTRUCT SINES OF ANGLES SURFACE MAKES WITHNORTH ANO SOUTH CELL WALLS

DHEN=O.5=(OHENE+DHENW)OHNN=O.5-(EIHNNE+OHNNW)DHES=O. 5*(OHESE+DHESW)DHNS=0.5* (DHNSE+DHNSW)TERM1=I .O+DHEN-DHENTERM2=1.0+OHES*OHESRHON=SORT(TERMI+OHNN=OHNN)RHDS=SORT(TERM2+DHNS*DHNS )

CALCULATE SURFACE FORCES ON NORTH ANO SOUTH WALLSFORCE IS FX FOR NF=l AND -FX FOR NF=2ONLY FOR INW=O IS FORCE REALLY A LINEAR COMBINATIONOF SURFACETENSION AN!I WALL AOHESION FORCES. OTHERWISE, IT IS EITHERSURFACE TENSION OR WALL AOHESION: ACCOROING TO RULES GIVEN EARLIER

FNS=FLGN*(AFN=OHNN/RHON+FLGU= (I .O-AFN)*CSANG)-FLGS-(AFS=OHNS/RHOS+FLGU*( I.O-AFS)*CSANG)

FOR THE PDROUS BAFFLE OPTION (INw=O): coMpuTE AooITIoNAL suRFAcEFORCES. ALLOWING FOR SOME CASES OF WALL AOHESION WHICH THEPRECEOING ALGORITHMS MISS WHEN INW=O

FEW=FEW-(2.0-FLGE-FLGW) *SANGFNS=FNS-(2.0-FLGN-FLGS) *SANGAFEW=ABS(FEW)

LIMIT SIZE OF FEW ANO FNS WHEN ALGORITHMS GIVE UNPHYSICAL RESULTSAND MULTIPLY BY GEOMETRIC FACTORS NEEOEO FOR PS

IF(NFF.LE.2) FEW=AMIN1(ROEW=AFEW, RX( 2), FEWLIM)=SIGN( l.0.FEw)IF(NFF.GT.2) FEW=AMINl (ROEW=AFEW.ROEw, FEWLIM)=SIGN( l.O.FEw)AFNS=ABS(FNS)FNS=AMINI (RDNS*AFNS.RONS eFNSLIM)*SIGN( l.O,FNS)

COMPUTE SURFACE TENSION PRESSURE ANO CALL IT THE SURFACE PRESSURE(PS). ANY VOIO PRESSURE CONTRIBUTION TO PS IS AOOED IN PRESSUREtiPOATE ROUTINE

PS(I. J)=-SIGMA*(FEW+FNS )

CCNTINUE

=.= CALCULATE PETA

IFIFIFIFPE.IFIFIFIFIFIFIF

(F(I, d+I). LT.EMF.OR.ATl I,d[F(I-l.IJ ).LT.EMF.OR .AR(I-1(F(I. J-1 ).LT.EMF.OR.AT( I,d(NFSB.EO. 151 PS(I,J)=O.OA(I.d)=l.O/( l.O-DFS/OMIN)

----- -IF \F[i+l, J).LT.EMF.OR .AR(I.J .LT.EM6) NFSB=NFS6+1

LT.EM6) NFSB=NFSB+2J).LT.EM6) NFs0=NFs0+41).LT.EM6) NFSB=NFSB+8

(L.EO:l.Oli~L.EO. IMAX) PETA(I,U)=l.O(M. EO.l.OR.M. EO.JMAX) PETA(I,J)=l.O(NF(L.M).NE.0) PETA(I.J)=l.O(NF(I.LJ).EO.1 .AND.AR( I-l,JJ.GT.EM6) GO TO 750(NF(I. J) .Eo.2.ANO.AR(I. J).GT.EM6) GO TO 750(NF(I, U) .EO.3.AND.AT( 1.d-1).GT.EM6) GO TO 750INF(I.U) .EO.4.ANO.AT( 1.J).GT.EM6) GO TO 750

PETAII~J)=l.OCONTINUE

CALL LAVORE

PETACAL.748PETACAL.749PETACAL.750PETACAL.751PETACAL.752PETACAL.753PETACAL.754PETACAL.755PETACAL.756PETACAL.757PETACAL.758PETACAL.759PETACAL.760PETACAL.761PETACAL.762PETACAL.763PETACAL.764PETACAL.765PETACAL.766PETACAL.767PETACAL.768PETACAL.769PETACAL.770PETACAL.771PETACAL.772PETACAL.773PETACAL.774PETACAL.775PETACAL.776PETACAL.777PETACAL.778PETACAL.779PETACAL.7B0PETACAL.781PETACAL.782PETACAL.7B3PETACAL.784PETACAL.785PETACAL.786PETACAL.787PETACAL.788PETACAL.789PETACAL.790PETACAL.791PETACAL.792PETACAL.793PETACAL.794PETACAL.795PETACAL.796PETACAL.797PETACAL.798PETACAL.799PETACAL.SOOPETACAL.801PETACAL.802PETACAL.803PETACAL.804pETAcAL.805PETACAL.806?ETAC&L.8G7PETACAL.B08PETACAL.809PETACAL.810PETACAL.811PETACAL.812PETACAL.B13PETACAL.814PETACAL.815PETACAL.816PETACAL.817PETACAL.818PETACAL.819PETACAL.820PETACAL.B21PETACAL.822PETACAL.823PETACAL.824PETACAL.825PETACAL.826

c

cc

:c

760

770

780

790

800

810

CALL CAVOVO

IF NEcESsARy. DETERMINE PRESSURES PR FOR VOID REGIONS NF

=== SET PETA IN ADUACENT FULL CELL

DO 830 LJ=i,dMAX00 820 1=1, IMAXNFF=NF(I.d)IF (NFF. EO.O.OR.BETA(I,d) .LT.O.0) GO TO 820IF (NFF.LE.5) GO TO 760P(I,J)=PR(NFF)GO TO 820L=IM=dGCI TO (770,780,790,800,820) , NFFL=I-ltiMx=OELX(L)OMIN=O. 5*(DMX+DELX(I) )AMN=AR(L.IJ)*R(L)GO TO 810L=I+lDMX=OELX(L)OMIN=O.5=(DMX+DELX(I) )AMN=AR(I.J)*R(I)GO TO 810M=d-1OMX=OELY(M)oMIN=o.5=(DMx+DELY(d) )AMN=AT(I,M)*RI(I)GO TO 810M=d+lOMX=OELY(M)OMIN=O.5*(DMX+OELY(J) )AMN=AT(I,u)*RI(I)CONTINUEIF (NF(L.M) -GT.o.oR.AMN.LT. EM6) GCI TO 820BPO=l.O/PETA( L.M)-BETA( L,M)*(l.O-PETA(I ,d))*AMN/

1 (AC(L, M)*RI(I ))=DELT/(DMIN=OMX)/RHOFPETA(L,M)=AMINI( 1.0/6PD. l.98/OMG)

320 CONTINUE830 CONTINUE

RETURNEND

subroutine PLTPT (xONE, YONE,ICHAR.ISYM )*CALL.COMMON1Lc +++ pLoT (oRAW) A POINTC ‘*+ PROVIDES A SYSTEM DEPENOANT CALLL

IC=OXI=XONEv1=YONE

10 XOl=(X1-XMIN) -SF+xSHFTYOI=[YI -YMIN)=SF+YSHFTiX1.50.+920.o-XOlIY1=50.+920.0-( 1.O-YO1 )CALL PLT (IXI, IYI,42)IF (ABS(XI).LE.EM6) GO TO 20IF (IsYMPLT .EO.O.OR. ISYM. Eo.o) Go To 20IC=IC+IIF (IC.GT.1) GO TO 20X1=-X1GO TD 10

20 RETURNENO

PETACAL.827PETACAL.828PETACAL.829PETACAL.830PETACAL.831PETACAL.832PETACAL.833PETACAL.834PETACAL.835PETACAL.83EPETACAL.837PETACAL.838PETAcAL.839PETACAL.840PETACAL.841PETACAL.842PETACAL.843PETACAL.844PETACAL.845PETACAL.846PETACAL.847PETACAL.848PETACAL.849PETACAL.85CIPETACAL.851PETACAL.852PETACAL.853PETACAL.854PETACAL.855PETACAL.856PETACAL.857PETACAL.858PETACAL.859PETACAL.860PETACAL.867PETACAL.862PETACAL.863PETACAL.864PETACAL.865PETACAL.866PETACAL.867PETACAL.868PETACAL.869PETACAI..87OPETACAL.871

PLTPT.2PLTPT.3PLTPT.4PLT?T.5PLTPT.6PLTPT.7PLTPT.8PLTPT.9PLTPT. 10PLTPT. :1PLTPT. 12PLTDT. 13PLTPT. 14PLTPT.15PLTPT.16PLTPT. 17PLTPT. 18PLTPT. 19PLTPT.20PLTPT.2tPLTPT.22PLTPT.23

117

SUBROUTINE PRESCR*CALL,coMMoNlc

CIMENSION FF(IBAR2, JEAR21.AF(16AR2 ,JBAR2),VLM( IBAR2.J6AR2 ),I AO(IBAR2. JBAR2).00( IBAR2. JBAR2),SSM(IBAR2 .JBAR21.2 RXR(16AR2. $JBAR2),RY.L( IBAR2,dBAR2).o( IBA~2.JBAR2).3 Ry7(IBAx2,~3AR2).RYB( 13t.R2,dBAR2) .0P(IBAR2.JE3AR2)

L

cccc

1c

c

723

724

725

726727

;28

DA7A ADEFM,8DEFM / 100.0, 0.70 /OATA ITMAX. ITMIN.ITMOST / 500, 5, 90 /OATA KTRAN / 1 /

THIS SUBROUTINE MUST BE MOOIFIEO FOR BOUNOARY CONDITIONSOTHER THAN NORMAL VELOCITY EQUAL TO ZERO

cio To (f,2),KTRANKTRAN=2

EPSIP=2.O=EPSIEPSIM=O.5*EPSI

IF(NCYC.GT.1) GO TO 200 750 J=I.JMAX00 750 1=1. IMAXIF(AC(I, J):LT.EMF.OR .BETA(I,d). LE.O.0) GO TO 740NFF=NF(I,LJ)IF(NFF.EO.0) GO TO 750IF(NFF.GT.5) GO TO 730L=IM=dGo ?o (723,724,725,726,728) , NFF

L=I-lGO TO 727L=I+IGO TO 727M=d-1GO TO 727M-d+lNFEL=NF(I-I,J)NFER=N#(I+l,J)NFEB=NF(I.J-1)NFET=NF(I.J+l)NFE=MAXO(NFEL,NFER.NFEB ,NFET)PSURF=PS( I.J)+PR(NFE )PLM=P(L.M)IF (NF(L,M) .NE.O.ANO.BE7A( I.J).GT.O.0) pLM=pSURFP(I.d)=( i.O-PETA( I.d))*PLM+pETA( I,J)*PSURFG9 TO 745IF(PSAT.LE.O.0) GO TO 736IF(F(I.d).LT.EMFl ) GO TO 733PMPSXO;OIF(PN(I,d) .LT.PSAT) PMPS=P(I,J)-PSATDIU=ROX( I)*(AR(I.J)=R( I)=U(I.J)-AR( I-l,J)*R( I-l)=U(I-l.J))+

1 RDY(d)=(AT( I,J)=RI(I)*V( I,d)-AT( I.J-l)*RI(I)=v( I.J-l))OId=OId/(AC( I,J)=RI( I))-PMPS”*2/DELTP(I,J)=PN( I,J)-DIJ*BETA( 1. J)/(1.O-2-0”pMpS*5ETA( I.U)/oELT)GO TO 745

73o P(I,J)=PR(NFF)GO TO 745

733 P(I.J)=PSATGO TO 745

736 P(I.J)=PN(I.J)GO TO 745

740 P(I,J)=O.O745 PN(I.J)=P(I,U)750 CONTINUE

L2 CONTINUE

cOELTCR=OELT/RHOFCl=l.oOD=l.OLVEC=IMAX*dMAXDO 150 J=l.LJMAXDO 150 I=I:IMAXIF(NF(I, J).GT.O.OR.AC( I,d). LT.EMF.OR.BETA(I. LJ).LE-o.o) Go To 140OId=ROX( I)=(AR(I.J)=R( I)=U(I,J)-AR(I-I ,J)*R(I-l)*u(I-l .J))+

1 ROY(J) *(AT(I. J)*RI(I)*V( I. IJ)-AT( I,J-l)*RI(I)*V( I.U-l))VLM(I,d)=AC( I.d)*TPI=RI (I )*OELX(I)*OELY(d)D(I,LJ)=OIJ/(AC( I,LJ)*RI (I))=VLM(I.d)

PRESCR.2PRESCR.3PRESCR.4PRESCU.5PRESCR.6PRESCR.7PRESCR.8PREiCR.5PRESCR.1OPRESCR.11PRESCR.12PRESCR.13PRESCR.14PRESCR.15PRESCR. 16PRESCR.17PRESCR.18PRESCR.19PRESCR.20PRESCR.21PRESCR.22PRESCR.23PRESCR.24PRESCl?.25PRESCR.26PRESCR.27PRESCR.28PRESCR.29PRESCR.30PRESCR.31PRESCR.32PRESCR.33PRESCR.34PRE5CR.35PRESCR.36PRESCR.37PRESCR.38PRESCR.39PRESCR.40PREscR.4~PRESCR.42PRESCR.43PRESCR.44PRESCR.45PRESCR.46PRESCR.47PRESCR.48PRE!jcR.49PRESCR.50PRESCR.51PRESCR.52PRESCR.53PRESCR.54PRESCR.55PREsCR.56PRESCR.57PRESCR.58PRESCR.59PRESCR.60PRESCR.61PRESCR.62PRESCR.63PRESCR.64PRESCR.65PRESCR.66PRESCR.67PRESCR.68PRESCR.69PRESCR.70PREscR.7~PRESCR.72PRESCR.73PRESCR.74PRESCR.75PRESCR.76PRESCR.77PRESCR.78PRESCR.79PRESCR.80PRESCR.81

118

IF(IDEFM. EQ.O.OR.F(I ,J).GE.EMF1) GG TO 5

D(Z. J)=D(I,J)+AMIN1 (AOEFM=EPSI .BDEFM={ l.O-F(I.J))/CELTl

CONTINUESshl::,dl=o.oRXR[ i.d\=C:.C2XRt=0.@kXRE=G.0IF(AR(I, J).LT.EM6) G3 TO 10RXRII. J)=AR(I.J)=TPi -R(I)*DELY( J)_2.O/(DELX( I )+DELX[I*l])SSM(I, J)=SSM( I.J)+DELTCR=RXR( Id)IF(NF(I+l. d).EQ.o) GO TO 10IF(NF(I+I. J).GT.5) GO TO 9IF(NF(I+l. J).NE.1) GO TO 8RXRA=I 1.O-PETA(I+l,U) )-RXR(I.J)RXRB=(( l.O-PETA(I+I ,J))=PN(I, J)+PETA( 1+1, JI=(PS(I+l, J)+PSAT )-

1 PN(I+l, J))*RXR(I.J)GO TO 9

8 RXRB=(P( I+l,d)-PN(I+l ,J))=RXR(I,J)9 RXR(I,LJ)=O.O

10 RXL(I,J)=O.ORXLA=O.ORXLB=O.OIF(AR(I- I.J).LT.EM6) GO TO 20RXL(I, J)=AR(I-I.J)*TPI -R(I-l) *OELY(d )=2.0/(OELX( I)+OELX(I-1))SSM(I.LJ)=SSM(I .LI)+DELTCR=RXL(I ,d)IF(NF(I- 1.LJ).EO.0) GO TO 20IF(NF(I-I. J).GT.5) GO TO 19IF(NF(I-I. J).NE.2) GO TO 18RXLA=(l.O-PETA( I-l ,J))=RXL(I.U)RxLB=(( ~.O-PETA(I-l ,LJ))=PN( I.U1+PETA(I-1 ,U)=(PS( I-I,U)+PSAT)-

1 PN(I-!. J))=RXL(I.J)GO TO 19

18 RXLE=(O( I-I,LJ)-PN(I- l,d))=RXL(I,d)19 RXL(I.J)=O.O20 RYT(I:J)=O.O

RYTA=O.ORYT8=0.OIF(AT(I,d).LTRYT(I,Ji=AT(ISSM(I,J)=SSM(IF(NF(I,J+i).IF(NF(I,d+4).IF(NF(I.J+l)-

‘.EM6) GOT030,u)*TPI*RI(I)-OEI.J)+DELTCR=RYT(EQ.0) GO TO 30GT.5) GO TO 29NE.3) GO TO 28

::xjj)”2.0/(oELY(d )+oELY( d+l))

RYTA=(l ;O-PkTA(I,ti+l ))*RyT(I,d)RYTE=((I.O-PETA(I ,IJ+l))-PN( I,J)+PETA(I ,J+l)=(PS(I,d+l )+PSAT)-

1 PN(I.u+I ))*RYT(I,LJ)GO TO 29

28 2YTB=(p( !,J+l)-PN(I.LJ+l ))*RyT(I.IJ)29 RYT(I,J)=o.o30 RYE(I.d)=O,O

l?Y5A=0.ORYBB=O.OIF(AT(I,J- 1).LT.EM6) GO TO 40RYB(I,J)=AT(I ,IJ-I)*TPI-RI(I )*DELX(:SSM(Io J)=SSM( I,J)+DELTCR=RyB( I.J)IF(NF(I,d-l). EO.0) GO TO 40IF(NF(I,J-1).GT.5) GO TO 39IF(NF(I.J-1).NE.4) GO TO 38RYBA=(I .O-PETA(I.J- I))*RYB(I,J)RY6B=(( I.O-PETA(I,J- l))=PN(I,U)+PE”

1 PN(I, d-l ))*RYB(I,LI)GO TO 39

38 RYBB=(P(I.d-I )-PN(I,U- I))*RYB(I$d)39 RYB(I,J)=O.O40 SSM(I, J)=SSM( I,J)-DELTCR* (RXRA+RXLA+RYTA+RyBA )

D(I.J)=(D( I.IJ)-OELTCR=( RXRB+RXLB+RyT6+RyBB) )/SSM( Id)

140

FF(I:~)=O;O”AF(I,J)=O.O00(1.d)=O.OAD(I.LJ)=O.OOP(I,U)=O.OGO TO 150VLM(I,LJ)=l.OD(I,J)=O.OFF(I,J)=O.OAF(I,J)=O.OAD(I,U)=O.O

)*2.o/(DELY(LJ)+DELY(J-l ))

A(I,J-l)*(PS(I, d-1 )+PSAT)-

PREsCR.82PRESCR.83PP.ESCR.84PRESCR.85PRESCR.86PR5SCR.67PREscR.8epR5s~R.g9PREscR.9@PR5SCR.91PRESCR.92PRESCR.93PRESCR.94PRESCR.95PRESCR.96pp5scp.97PRESCR.98PRESCR.99PRESCR.1OOPRESCR. 101PRESCR. 102PRESCR. 103PRESCR. 104PRESCR. 105PRESCR. 106PRESCR. 107PRESCR. 108PRESCR. 109PRESCR. I1OPRESCR. 111PRESCR. 112PRESCR. 113PRESCR. 114PRESCR. 115PRESCR. 116PRESCR.117PRESCR. 118PREscR. f19PRESCR. 120PRESCR. 121PRESCR. 122PRESCR. 123PRESCR. 124PRESCR. 125PRESCR. 126PRESCR. 127pREscR. 128PRESCR. 129PRESCR. 130PRESCR. 131PRESCR. 132PRESCR. 133PREscR. f34PRESCR. 135PRESCR.136PRESCR. 137PRESCR. 138PRESCR. 139PRESCR. 140PRESCR. 141PRESCR. 142PRESCR. 143PRESCR. 144PRESCR. 145PRESCR. 146PRESCR. 147PRESCR. 148PRESCR. 149PRESCR. 150PRESCR. 151PRESCR. 152PRESCR. 153PRESCR. 154PRESCR. 155PRESCR. 156PRESCR. 157PRESCR. 158PRESCR. 159PRESCR. 160PRESCR. 161

150c200

350

245250

450

>Jo

Lcc +++c

510cc +++c

523

524

525

526527

OO(I,d)=O.OSSM(I.J)=l.Ol?XR(I.J)=O.ORXL(I,J)=O.ORYT(I.J)=O.ORYB(I.d)=O.ODP(I,d)=O.OCONTINUE

A=CII’ODCALL SAXPY(LVEC,A,FF.CALL SAXPY(LVEC.-A,00DMAX=O.ODO 350 d=l,dMAXDO 350 1=1.IMAXIF(NF(I,J).GT.ODIV=O(I,J)=SSM(:OMAX=AMAX1(OMAXCONTINUEIF(DMAX.LE.EPSIDO 250 d=t,JMAX00 250 I=l,IMAXIF(NF(I.J).GT.OAD(I,~)=OELTCR=

1 RYT(I,J)*D(I,UGO TO 250

llR.AC,J)/v

,DP,l)I,D.1)

I. J).LT. EMF.OR.BETA( I.J).LE.O.0) D(I,J)=O.OM(I.U)=AC(I, LJ)-TPI=RI (I)

ABS(OIV))

) GO TO 500

OR.AC(I .J).LT.EMF.OR .BETA(I. J).LE.O.0) GO TO 245RXR(I,J)*D( I+l.IJ)+RXL( I,LJ)*O(I-l,J)+I)+RYB( I,J)=D(I,d-1 ))-SSM(I,U)*D(I,d)

AO(I.LJ)=O.OCONTINUEC2=SOOT(LVEC.D, l ,AO,l)B=C2/ClDD=O.OCALL SSSAL(LVEC,B,FF, l)CALL SAXPY(LVEC. I. ,D,’CALL SSCAL(LVEC.B.AF.CALL SAXPY(LVEC, l. ,AD00 450 J=I,JMAXDO 450 I=I,IMAXOQ(I, J)=AF(I.J)/SSM(ICONTINUEDD=SD9T(LVEC.00. 1.AF.CI=C2ITER=ITER+IF(ITER.GTGO TO 200IF(ITER.GTIF(ITER.LTEPSI=AMINIEPSI=AMAXI

,FF,I))l,AF,I)

J)

)

ITMAX) GO TO 230

ITMOST) EPSI=I.05=EPSIITMIN) EPSI=0.95-EPSIEPSI,EPSIP)EPSI,EPSIM)

UNFOLD PRESSURE FOR INTERIOR FLUID CELLS

00 510 d=l.dMAX00 510 I=I;IMAXIF(NF(I, d).GT.O.OR.AC( I,d).’LT. EMF.OR.BETA(I ,d).LE.O.0) GO TO 510P(I. d)=PN(I,J)+OP(I, J)CONTINUE

CALCULATE PRESSURE CHANGES FOR SURFACE CELLS

00 550 d=l.tJMAX00 550 1=1.IMAXIF(AC(I,LJ) :LT.EMF.OR.BETA( I.IJ).LE.O.0) GO TO 540NFF=NF(I,J)IF(NFF.EQ.0) GO TO 550IF(NFF.GT.5) GO TO 530L=IM=JGO TO (523.524.525,526,528)L-I-IGO TO 527L=I+IGO TO 527Mx~-1GO TO 527MxJ+lNFEL=NF(I-I.LI)

NFF

PRESCR. 162PRESCR. 163PREscR. 164pRFscR. 165PRESCR. 166DREscR. le?pREscR. j6epR5scR.169PRESCR. 170PRESCR. 171PRESCR. 172PRESCR. 173PRESCR. 174PRESCR. 175PRESCR. 176PRESCR. 177PRESCR. 178PRESCR.179PRESCR. 180PRESCR. 181PRESCR.1B2PRESCR. 183pR5.5cR.j34PRESCR. ;85PRESCR.186PRESCR. 187PRESCR. 188PRESCR. 189PRESCR. 190PRESCR. 191PRESCR. 192PRESCR. 193PRESCR. 194PRESCR. 195PRESCR.196PRESCR. 197PRESCR. 198PRESCR. 199PRESCR.200PRESCR.201PRESCR.202PRESCR.203PRESCR.204PRESCR.205PRESCR.206pREscR.207PRESCR.208Pi?ESCR.209PRESCR.210PRESCR.211PRESCR.212PRESCR.213PRESCR.214PRESCR.215PRESCR.216PRESCR.217PREscR.218PRESCR.219PRESCR.220PRESCR.221PRESCR.222PRESCR.223PRESCR.224PRESCR.225PRESCR.226PRESCR.227PRESCR.228PRESCR.229PRESCR,230PRESCR.231PRESCR.232PRESCR.233PRESCR.234PRESCR.235PRESCR.236PRESCR.237PRESCR.238PRESCR.239PRESCR.240PRESCR.241

NFER=NF(I+l.d)NFEB=NF(I,J-1)NFET=NF(I,d+l)

120

528

530

533

536

540550

cccc

551

552

553

555

556

557

558560cc +++c

570c

c

230

c240

NFE=MAXO(NFEL.NFER.NFEE .NFET)PSURF=PS( I,LJ)+PR(NFE )PLM=P(L,MIIF (NF(L. M) .NE.0.&NO.EETA( I,d].GT.0.0) PLM=PSURFDP(I.LJ~=( 1.O-PETA(I,J) )*PLM-PETA( I,J)-PSURF -PNII.JIGD TO 550I=(PSAT.LE.O.0) GG T@ 536IC(F(l$J).LT.~MFl ) Go TO 533PMPS=C.GIF(PN(I, J).LT.PSAT) PMPS=P(I,J)-PSATOILI=ROX(I)=(AR(I .Ll)=RII)*u~I ,LI)-AR( I-!,LJ)=R(I-I )=U(I-1.J))+

RDY(J)*( AT(I,d)-RI( I)=V(i.J)-AT(I. J-l )*RI(I)=V(I.J-1))DId=Dxd/(Ac( I.J)=RI (1 ))-pMps==2/OELTDP(I, J)=-OIJ*BETA(I,J )/( 1.O-2.O=PMPS=BETA(I ,J)/DELT)GO TO 550OP(I. J)=PR(NFF)-PN(I .J)GO TO 550DP(I. J)=PSAT-PN(I,J)GO TO 550OP(I,U)=O.OGO TO 550DP(I,J)=O.OCONTINUE

CALC NEW VELOCITIES CONSISTENT WITH SURFACE PRESSURE ASSUMPTIONSNOTE: YIELOS 0=0.0 FOR INTERIOR CELLS ADdACENT TO SURFACE CELLS

00 560 tJ=2,dMl00 560 1=2.IMIIF(AC(I, J):LT.EMF.OR .BETA(I, d).LE.O.0) GO TO 560IF(AR(I. J).LT.EM6) GO TO 555IF(NF(I, J).EO.O.ANO.NF [1+1.J).Eo.0) GO TO 551IF(NF(I,d) .EQ.O.ANi3. NF(I+l;J).NE. 1) GO TO 552IF(NF(I .LJ).NE.2.ANO. NF(I+I,U).E().0) GO TO 553U(I, J)=U(I,J)+DELTCR*(OP( I,J)-DP( I+l,J))*2.o/(oELx( I)+DELX(I+l))GO TO 555u(I, LJ)=u(I, J)+DELTcR*DP( I,J)*2.0/(DELX( I)+DELX( 1+1))GO TO 555U(I, J)=U(I.d)+OELTCR= (-DP(I+l, IJ))*2./(DELx( I)+OELX(I+I))G9 TO 555IF(AT(I, dJ.LT.EM6) GO TO 560IF(NC(I, J).EO.O.ANO. NF(I.U+I).EQ.0) GOIF(NF(I, J).EQ.O.ANO. NF(I.d+l).NE.3) GOIF(NF(I, LJ).NE.4.ANO.NF( I.J+l).EQ.0) GOV(I, J)=V(I, d)+OELTCR*( DP(I,J)-DP(I,U+IGO TO 560

TO 556TO 557TO 558)*2.O/(DELY(U)+DELY(d+ 1))

v(I. J)=V(I,J)+DELTCR*DP( I, J)*2.O/(OELY(LJ)+DELY( LI+l))GO TO 560V(I,j)=V( I,J)+OELTCR*( -DP(I,U+1))*2CONTINUE

PUT IN PRESSURES FOR SURFACE CELLS,

DO 570 J=l,dMAXDO 570 1=1.IMAX

/(DELY( d)+13ELY(d+t))

ISOLATED CELLS, VOIDS

IF(AC(I,d) :LT.EMF.OR .BETA(I, d).LE.O.0) GO TO 570IF(NF(I,J).EO.0) GO TO 570P(I, d)=PN(I, d)+DP(I,J)CONTINUE

CALL BC

RETURNCONTINUEWRITE (59,240) NCYCwRITE (12,240) NCYCWRITE (6,240) NCYCCALL EXITLDC (6HP 17ER)

FoRMAT (IX,29HT00 MANY CR ITERATIONS. cYcLE,17)ENO

PRE’5CR.242PRESCR.243PRESCR.244PRESCR.245PRESCR.246PRESC,?.2L7PRESCR.249PRESCR.249pREscR.~50PRESCR.254PRESCR.252PRESCR.253PRESCR.254PRESCR.255PRESCR.256PRESCR.257PRESCR.25ePRESCR.259PRESCR.260PRESCR.261PRESCR.262PRESCR.263PRESCR.264PRESCR.265PRESCR.266PRESCR.267PRESCR.268PRESCR.269PREsCR.270PRESCR.271PRESCR.272PRESCR.273PRESCR.274PRESCR.275PRESCR.276PRESCR.277PRESCR.278PRESCR.279PRESCR.280PRESCR.281PRESCR.282PRESCR.283PRESCR.284PRESCR.285PRESC2.286PRESCR.287PRESCR.288PRESCR.289PREsCR.290PRESCR.291PRESCR.292PREsCR.293PRESCR.294PRESCR.295PRESCR.296PRESCR.297PRESCR.298PRESCR.299PRE’5CR.300PRESCR.301pR5scR-3~~PRESCR.303PRESCR.3Q4PRESCR.305PRESCR.306PRESCR.307PRESCR.308PRESCR.309PRE!5CR.310PRESCR.311PREsCR.312

SUBROUTINE PRESIT=CALL,COMMON1

cc ++*cc +4.+

P.

10

20cc +++c

cc +++c

30

40

50

60

70

%

100

c

c

c5

cc +++c

110

DATA ITMAX /1000/

PRESSURE ITERATIOF;

TEST ‘OR “CONVERGENCE

IC (FLG.Ec.o. ) GO TO 160ITER=ITER~lIF (ITER.LT.ITMAXI GO TO 20FNOC=I.ONOCON=NOCON+IWRITE (12,170) NCYC,T.NOCONWRITE (59.1701 NCYC.T,NOCONGO TO 160FLG=o.O

COMPUTE UPDATEO CELL PRESSURE ANO

00 150 J=JPB,JPT00 150 I=IPL.IPRIF (BETAII.U). LT.O.0) GO TO 150IF (NF(I,d).GT.5) GO TO 150IF (NF(I.J).EO.0) GO TO 100

VELOCITIES

CALCULATE PRESSURE FOR SURFACE CELLS

NFF=NF(I,J)L=I .M=IJGO TO (50,60,70,80.30), NFFIF (PSAT.LE.O.0) Ga T5 150IF (F(I,J).LT.EMFI) GO TO 40PMPS=O.OIF (P(I,d).LT.pSAT) PMPS=P(I,d)-PSATOIIJ=ROX( I)-(AR(I,J)=R( I)*U(I. IJ)-AR(I-I,IJ)=R( I-I)*U(I-l,J))+

1 ROY(J)* (AT(I,IJ)=RI(I)*V( I.J)-AT( I,LJ-l)=RI(I)=V( I,J-l))DILl=DIJ/(AC( I,d)=RI (1 ))-PMPS*-2/OELTDELp=-oId-BETA( I,U)/( f-o-2.0*pMp5xBETA( I.d)/DELT )GO TO 110OELP=PSAT-P(I,J)GO TO 110L=I-lGO TO 90L=I+IGO TO 90M=J-lGO TD 90M=LJ+ICONTINUENFEL=NF(I-I,J)NFER=NF(I+l,d)NFEB=NF(I.J-I )NFET=NF(I,J+I)NFE=MAXO(NFEL,NFER,NFE13 .NFET)PSURF=PS( I,d)+PR(NFE)PLM=P(L.M)IF (NF(L.M) .NE.o.ANO.BETA( I.u).GT.o.0) PLM=PSURFDELP=(I .O-PETA(I.LJ) )*PLM+PETA( I.d)=PSURF-P( I.J)GO TO 110CONTINUEDIu=RDX( I)=(AR(I.J)*R( I)=U(I. J)-AR(I-l,J)=R( I-l )~U(I-l,d))+

1 RDY(J)=(AT( I,J)=RI(I)*V( I,d)-AT( I.J-l)*RI(I)*v( I,d-l))OIJ=DId/(AC( I,IJ)*RI(I ))

IF(IDEFM.EO.O.OR. F(I, tJ).GE.EMFl) GO TO 5

DIJ=OIU+AMINI( AOEFM*EPSI ,BOEFM*( l.O-F(I.d))/OELT )

CONTINUE

SET FLAG INDICATING CONVERGENCE

IF (ABS(OId).GE. EPSI) FLG=I.ODELP=-EETA(I ,J)*OIJ*PETA(I,J)CONTINUEP(I, d)=P(I.J)+OELPOPDT=2.O=DELT=OELPIF (AR(I,J).LT-EM6) GO TO t20

PRESIT.2PRESIT.3pR~~IT.4PRESIT.5pR~slT.6PRESIT.7PRESI?.EPQESIT.9PRESIT.10PRESIT.11PRESIT.12PRESIT.13PRESIT.14PRESIT.15PRESIT. 16PRESIT.17PRESIT.18PRESIT.19PRESIT.20PRESIT.21PRESIT.22PRESIT.23PRESIT.24PRESIT.25PRESIT.26PRESIT.27PRESIT.28PRESIT.29PRESIT.30PRESIT.31PRESIT.32PRESIT.33PRESIT.34PRESIT.35PRESIT.36PRESIT.37PRESIT.38PRESIT.39PRESIT.40PRESIT,41PRESIT.42PRESIT.43PRESIT.44PRESIT.45PRESIT.46PRESIT.47PRESIT.48PRESIT.49PRESIT.50PRESIT.51PRESIT.52PRESIT.53PRESIT.54PRESIT.55PRESIT.56PRESIT.57PRESIT.58PRESIT.59PRESIT.60PRESIT.61PRESIT.62PRESIT.63PRESIT.64PRESIT.65PRESIT.66PRESIT.67PRESIT.68PRESIT.69PRESIT.70PRESIT.71PRESIT.72PRESIT.73PRESIT.74PRESIT.75PRESIT.76PRESIT.77PRESIT.78PRESIT.79PRESIT.80PRESIT.81

122

RHOXR=RHOF*(DELX( I+I ~+DELX(I))U(I. J)=U(I, Jl+DPOT/RHOXRIF (AR(I-I. J).LT.EM6J GO TO 130RHCXL=RHOF=(OELXI I I+DELx(I-l 1)U(I-1.U)=U( I- 1.Jl-DPCT/Ri+OY.LIF (LT(I.d\.LT.EM6) GO TC 140RHOYT=RH9F- IOELYILJ- 1 I+2ELV[J))V( I .~lJ=V{I.JJ+DpCT lRHO}’TIF (AT(I.J-11.LT.EM6 ! GO 70 :50RH9YB=RHCF- (OELY(J)+OELY( d-l ))V(x,u-1 )=v(I,d- 11-oPDT/R!’ioYeCONTINUECALL ECGO TO 10CONTINUERETURN

PRESIT.82pR~sI~.83PRESIT.84PRESiT.E5PRESIT.86PRESIT.87P!ZESIT.8ePRESiT.89PRESIT.50PRESIT.91PRESIT.92pREsiT.93PRESXT.94PRE51T.95PRESIT.96PRESIT.97PRESIT.98PRESIT.99PRESIT. 100

120

130

14(j

150

160

c170 FORMAT (IX,35HPRESIT CONVERGENCE

1 .7,1x, 12,7HTH TIME)FAILURE ON CYCLE, 17,1X,2HT=,1PE 15

PRESIT.1OIEND

SUBROUTINE”PRTPLT (N]=CALL,COMMON1

FVOL=O.YCFL=l.E759DO 10 d=2.LJMl

PRTPLT.2PRTPLT.3PRTPLT.4PRTPLT.5PRTPLT.6

~F (F(2.di.LE. 1.-EM6.ANO.F(2,J) .GE.EM6) YCFL=Y(LJ- l)+F(2.d)r (Y(d)-v4 i,l-f)}

PRTPLT.~pR)p~l,8

PRTPL?.9PRTPLT. 10PRT~LT. 11PRTPL7.12PRTPLT.13PRTPLT. 14PRTPLT.15PRTPLT.16PRTPLT. 17PRTPLT. 1“8PRTPLT. 19PRTPLT.20PRTPLT.21

1~-(y~~L.GT. 1. E+9e. AND.F(2.Jj.LE .EM6! YCCL=Y(d-l)DO 10 1=2.IM1FVOL=FVOL*F( I.J)=AC( I.U)-TPI *RI(I)*(R(I )-R(I-i ))-[Y(LJ)-V(d- :))CONTINUE

PRINT ANO PLOTPROVIOES FDRMATTED WRITES TO PAPER AND FILM

GO TO (20,120,140,200) , N

PRTPLT (1) WRITE OUT INITIAL OATA ANO MESH OATA

WRITE (6,290)

10cc +++c +++c

cc +++c

20WRITE (6.300) NAMEWRITE (6,340) IBAR, dBAR.DELT,XNU, ICYL ,EPSI,GX,GY .Ul,VI.VELMx.TWFIN

1 .PRTOT, PLTOT.OMG.ALPHA .KL,KR.KT,KB, IMOVY.AUTOT ,FLHT, PSAT, ISYMPLT

PRTPLT.22PRTPLT.23PRTPLT.24PRTPLT.25PRTPLT.26PRTPLT.27PRTPLT.28PRTPLT.29PRTPLT.30PRTPLT.31PRTPLT.32PRTPLT.33PRTPLT.34PRTPLT.35PRTPLT.36PRTPLT.37PRTPLT.38PRTPLT.39PRTPLT.40PRTPLT.41PRTPLT.42PRTPLT.43PRTPLT.44PRTPLT.45PRTPLT.46PRTPLT.47PRTPLT.48PRTPLT.49PRTPLT.50PRTPLT.51PRTPLT.52PRTPLT.53PRTPLT.54PRTPLT.55PRTPLT.56PRTPLT.57PRTPLT.58PRTPLT.59

2 ,SIGMA, ISURFIO,CANGLE ,RHOFIF (IMOVY.GT.0) GO TO 70WRITE (12,290)”WRITE (12.300) NAMEWRITE (12,340) IBAR,J6AR,0ELT,XNU, ICYL, EPSI,GX,GY,UI.VI .VELMX

1 .TWFIN.PRTDT, PLTDT,OMG, ALPHA,KL,KR,KT,K6 .IMOVY ,AUTOT,FLHT,PSAT2 ,ISYMPLT, SIGMA,ISURFIO. CANGLE,RHOF

cc +++c

30

40

50GO70

cc +++c

80

WRITE ON FILM VARIABLE MESH INPUT DATA

WRITE (12.370)00 30 I=I,NKXWRITE (12,380)CDNTINUEWRITE (12,400)DO 40 I=l,NKYWRITE (12,390)CONTINUEWRITE (12,440)

NKX

I,XL(I:

NKY

I,YL(I’

,XC(I),XL( I+I),NXL(I) ,NXR

,YC(I).YL( I+I).NYL(I ).NYR

1).DXMN(I)

1).DYMN(I)

NOBSGO TO 60IF (NOBS.LE.0)

00 50 I=l,NOBSWRITE (12,450)CONTINUECOrJTINUECONTINUE

PRINT VARIABLE

WRITE (6,370)00 80 1=1.NKXWRITE (6,380)CONTINUEWRITE (6.400)00 90 1=1.NKYWRITE (6,390)

I),OCI(I),IOF!(I)1.0A2(I ),0AI(I).082(1 ).0BI(I),0C2

MESH INPUT DATA

UKXI

I

I,XL(I),XC(I) ,XL(I+I ).NXL(I).NXR(I) .DXMN(I)

UKY

I,YL(I), YC(I),YL(I+I ),NYL(I )._NYR(I),oyMN(I )

123

90

100110

rc +++c

120

130

cc +++c

140

150

CONTINUEWRIT,E (6,440) NOESIF (NOBS.LE.0) GO To 11000 100 I=l,NOBSWRITE (6,450) I,0A2(I ),0AI(I)OOB2(I) ,0BI(I),0C2( I),0CI(I), IOH(I)CONTINUECONTINUEGO TO 240

PRTPLT (2) WRITE TIME STEP, CYCLE INFORMATION

CONTINUEFORCE=O.OWRITE (6,330) ITER.T,OELT,NCYC,VCHGT ,FORCEIF (IMOVY.EO.1) GO TO 130IF (T.GT.o. ) GO TO 130WRITE (12.330) ITER,T,OELT,NCYC,VCHGT ,FORCECONTINUEGO TO 240

PRTPLT (3) WRITE FIELD VARIABLES TO FILM

IF (IMOVY.EO.1) GO TO 190CALL AOV (1)WRITE (12,360) NAMEWRITE (12.S30) ITER,T,OELT.NCYC. VCHGT ,FORCEWRITE (12.350)WRITE (12,410) NREGWRITE (,2.420)KNR=NREG+5DO 150 K=6.KNRWR:TE (:2.430) K,VOL(K).PR(K)CONTINJEWRITE ~12,260) FVOL.NCYC,YCFLwRITE (l2,3IO)00 170 I=l,IMAX

.

DO 170 d=l.JMAXDId=O.IF (d. EO. l.OR.I.EO. l.OR .d. EO. JMAX.OR. I.EO.IMAX ) GO TO 160DIU=RDXII)=(AR(I. d )=R(I)=U( I.J)-AR(I- I.LJ)=R(I-I )-U(I-I,J))+

PRTPLT.6CIPRTPLT.61PRTPLT.62PRTPLT.63PRTPLT.64PRTPLT.65PRTPLT.66PRTPLT.67PR-rPLT.68PRTPLT.69PRTPLT.70PRTPLT.71PRTPLT.72PRTPLT.73PRTPLT.74PRTPLT.75PRTPLT.76PRTPLT.77PRTPLT.78PRTPLT.79PRTPLT.80PRTPLT.81PRTPLT.E2PRTPLT.83PRTPLT.84PRTPLT.85PRTPLT.86PR7PLr.87PRTPLT.E8PRTPLT.8EPRTPLT.90PRTPLT.91PRTPLT.92PRTPLT.93PRTPLT.94PRTPLT.95PRTPLT.96PRTPLT.97PRTPLT.98PRTPLT.99PRTPLT. 100PRTPLT. 101

170

180

1B5190

cc +++c

200

210

124

1 ROY(d)=( AT(I,J)*RI(I )=V(I, J)-AT(I,J-I)=RI (I)=VII,J-1)) I

IF (Ac(I.d).GT.o-o) DIJ=OIJ/(AC( I,d)*RI(I )) I160 CONTINUE PRTPLT. 102

WRITE (12,320) I,J,U(I,J).V(I,d),P(I, J),OId,PS(I,J),F(I,d),NF (I,J) PRTPLT.1031 .PETA(I,J) PRTPLT. 104CONTINUE PRTPLT. 105WRITE (12,270) NCYC,T,OELT, (I.1=2, IMAX,2) PRTPLT. 106

PRTPLT. 107PRTPLT. 108

00 180 ddd=l,JMAXJ=JMAX+l-JJJWRITE (12.280) d,CONTINUEWRITE (12,275) NC’00 185 ddd=l,dMAXU=JMAX+I-JJUWRITE (12.2S0) d,CONTINUECONTINUEGO TO 240

PRTPLT (4) WRITE

WRITE (6.290)

NF(I,J)

C.T,DEL-

NW(I,d)

1=1.IMAX) PRTPLT. IG9PRTPLT. I1O

.(I.1=2.IMAX.2) PRTPLT. 111PRTPLT. 112PRTPLT. 113

1=1. IMAX) PRTPLT. 114PRTPLT. 115PRTPLT. 116PRTPLT. 117

FIELO VARIABLES TO PAPER

WRITE (6:360) NAMEWRITE (6,330) ITER,T,OELT,NCYC,VCHGTWRITE (6.350)WRITE (6.410) NREGWRITE (6,420)KNR=NREG+500 210 K=6,KNRWRITE (6,430) K.VCL(KI,PR(K)CONTINUEWRITE (6,260) FVOL,NCYC.YCFLWRITE (6,350)kRITE (6,310)00 230 I=I,IMAXDO 230 d=l,LJMAXDId=O.IF (J.EO. I.OR.I.EO. I .OR. J.EO.JMAX.OR.I .EO.IMAX) GO TO 220OIU=RDX( I)=(AR(I.J)*R( I)*U(I.LJ)-AR(I-I ,J)*R(I- l)*U(I-l,IJ) )+

PRTPLT. 118PRTPLT. 119PRTPLT. 120PRTPLT. 121PRTPLT. 122PRTPLT. 123PRTPLT. 124PRTPLT. 125PRTPLT. 126PRTPLT. 127PRTPLT. 128PRTPLT. 129PRTPLT. 130PRTPLT. 131PRTPLT. 132PRTPLT. 133PRTPLT. 134PRTPLT. 135PRTPLT. 136PRTPLT. 137PRTPLT. 138

1 RDY(J)=IF (AC(I

220 CONTINUEWRITE (6

,PETA(I2301CONTINUE240 RETURN

c

AT(I,U)*RI(I)=V(I,J) -AT(I,J-l)*RI(I)*v(I, J-1))J).GT.O.0) OId=DId/(AC( I,d)*RI(I ))

320) I,U, U(I, J),V(I,IJ),P(I, J),OIJ,PS( I,J),F(I,d) ,NF(I,LJ)J)

250 FORMAT (IX,5HT(F)=, IPE14.6.2X.8HON CYCLE,16)260 FORMAT (ix, 14HFLu10 vOLuME =, IPE14.6,9H ON CYCLE, 16,2X, 15HYCENT OF

1 FLuID=,E14.6)270 FORMAT (IHI, IX,43HNF FIELo (INcL. FICTITIOUS CELLS) FOR CYCLE.16,2

1 X.2HT=, IPE14.7,2X,5HOELT= .E12.5//5X,32I4)275 FORMAT (IH1,1X,43HNW FIELO (INCL. FICTITIOUS CELLS) FOR CYCLE.16.2

1 X,2HT=, IPE14.7,2X,5HDELT= .E12.5//5X,32I4)280 FORMAT [IX, 13.1X,6312/(5X06312 ))290 FORMAT (lH1)300 FORMAT (1OA8)310 FoRMAT (4XOIHI,5X, IHU,9X, IHU ,14x, 1HV,15X,1HP,15X, IHO.

1 IHF,11X,2HNF,9X,4HPETA )320 FOQMAT (2x. 13.3x. 13,6(3x, IPE f2.5),3x, 13,3xsE12.5)330 FORMAT (6x,6HITER= ,15,5x,6HTIME= ,1PE12. 5,5x.6HDELT=

1 7HCYCLE= ,16,2X,7HVCHGT= .IPE12.5,2X,4HFY= .E12.5)

2X,2HPS,13X,

.IPE12.5,5X,

340 FORMAT (IH ,5X,6HIBAR= ,13/6X,6HLJBAR= .13/6X.6HDELT= ,1PE12.518X,41 HNU= ,E12.5/6x.6HIcyL= . 12/6X,6HEPSI= .E12.518X.4HGX= .E12.518X,42 HGY= El~.5/8~,4~lJI= ,E:2.5/SX,4HVI= ,E12.5/5X.7HVELMX= ,E12.5/5X~ ,7HTW;IN= E12.5/5X.7HPRTDT= ,E12.515X,7HPLTDT= ,E12.5;7X.5H3MG=4 .~;2.5/5X, +HALpHA= .E12.5/8X.4HKL= ,1~/8X,4HKR= ,1~~8X.4HKT= 12/5 8.L,4HKB= .12/5X,7HIMOVY= .12/5X,7hAUTOT= ,E:2.5!C$X.6iiFLHT= .Ei2.56 ‘EA,6H?SAT= E12.5/3X, 9HISYMPLT= ;2/5X,7HSIGMA= .E12.5/3X.9HISUR~c~~= . 12/4X.81iCANGLE= .E12.5;7X.6HRH3F= ,E12.5/’)

350 FCRMA? (lHO)360 F02MAT (IH , 18x, IOAS. IX.A10,2( 1X, A8))370 FORMAT (2X.5HNKX= .14)380 FORMAT (2X,8HMESH-X= ,14.3X.4HXL= ,IPE12.5,3X,4FIXC= ,E12.5,3X,4HXR

,E12.5,3X,5HNXL= 14,3X.5HNXR= ,14,3X.6HOXMN= .E12.5)3901~ORMAT (2X,8HMESH-Y~ ,14.3X,4HYL= ,1PE12.5.3X,4HYC= ,E12.5,3X,4HYR

1= ,E12.5.3X,5HNYL= ,14,3X.5HNYR= ,14,3X,6HOYMN= .E12.5)400 FORMAT (2X,5HNKY=41O FORMAT (2X,6HNREG=’1%420 FORMAT (15X, 1HK,6X.6HVOL(K) .9X .5HPR(K))430 FORMAT (13X, 13,2X. IPE12.5,3X, E12.5)440 FORMAT (2X,6HNOBS= 12)450 FORMAT (2X.2HI=, 12,~X.4HOA2= , IPE12.5, 2X,4HOAI=,E 12.5,2X,5HOB2=

,E12.5,2X,5HOBI= ,E12.5.2X,5HOC2= .E12.5,2X,5HOCI= .E12.5.2X.5HI0~H= ,12)

ENO

PRTPL1 .139PRTPLT. 140PRTPLT. 141PRT?LT; 142PRTPLT. 143PRTPLT. !44PRTPLT. 145PRTPLT. 146PRTPLT. 147PRTPLT. 148PRTPLT. 149PRTPLT. 150PRTPLT. 151PRTPLT. 152PRTPLT. 153PRTPLT. 154PRTPLT. 155PRTPLTPRTPLTPRTPLTPRTPLTPRTPLTPRTPLT

~56157158159160J61

PRTPLT. 162PRTPLT. 163PRTPLT. 164PRT?LT. 165PRTPLT. 166PRTPLT. 167PRTDLT. 168PRTPLT .169PRTPLT.17QPRTPLT. 171PR7PLT. 172PRTPLT. 173PRTPLT. 174PRTPLT. 175PRTPLT. 176PRTPLT. 177PRTPLT. 178DRTPLT. 179PRTPLT. 180PRTPLT. 181PRTPLT. 182PRTPLT. 183PRTPLT. 184PRTPLT. 185

125

.

SUBROUTINE SETUP=CALL.COMMONI

NAMELIST /XPUT/’ DELT. XNU,ICYL.EDSI .GX.GY.UI .VI,VELMX.TW=IN. PRTDT,PLTDT.OMG, ALPHA,KL .KR,KT.K3, 1MOVY.AUTOT .FLHT. DSAT, ISYM?LT ,SIGMA

: ,ISURFIO.CANGLE, RHOF .iEOIC3 .NAME,NOUMP, OVOL.lSOR .IOEFM,NPACK. CON.OTCRMX .IOI\’

NAMELIST /MSHSET/ NKX.XL. XC.NXL.NXR,OXVN .NKY. YL.YC.NYL.NY.?. 9YMNNAMELIST lASETiN I’ luOES,0A2.OL:

c +++.022.oFl.ac2.oZl. ICk

c +++ INITIALIZE SELECTEEI VARIABLESc +++

OATA T !0./, ITER /0/. FLGC /0./. FNOC /0./. NCYC /0/. NFLGc /0/.1 NOCON /0/. TWPLT /0./. TWPRT /0./. VCHGT /0./, NEOUIB /501/

DATA ISOR’. IOEFM.NPACK;NOUMP .NOBS.OVOL / O. 0. 0. 0. 0. 0.0 /

c +++c +++c +++

30

c +++c +++c +++c +++

40

cc +++.-.c ++.?c

50

cc +++c

126

.

DATA CON,6TCRMX;IOIV j 0.30: 0.00i, 1 )

REAO INITIAL INPUT OATA

READ (5,XPUT)WRITE( 12,330) NAMEWRITE( 12,360) NOUMP, OVOLWRITE( 12,320) ISOR,IOEFMREAO (5,MSHSET)00 30 K=l,NOBOOA2(K)=0.OAI(K)=O.OB2(K)=0.OB1(K)=O.OC2(K)=0.OC1(K)=O.CONTINUEREAD (5.ASETIN)

IF THE DUMP SEOUENCE NUMBER IS GREATER THAN ZERO, REAO A RESTARTOUMP ANO SKIP THE REST OF THE SETUP ROUTINE.

IF (NOUMp.LE.0) Go To 40CALL TAPINGO TO 290CONTINUECALL MESHSET

COMPUTE CONSTANT TERMS ANO INITIALIZE NECESSARY VARIABLES

SET PARAMETER STATEMENT VALUE INTO CONSTANT

NOUM?=lIF(ICYL.Eo.0) TPI=I.OCYL=FLOAT(ICYL)EMFI=I.O-EMFIF (CANGLE.EO.90.0) CANGLE=CANGLE-EM6OANGLE=CANGLECANGLE=CANGLE=RPDIF (CON=l.3. LE.FCVLIM) GO TO 50WRITE (12.380) CON,FCVLIMWRITE (59,380) CON,FCVLIMFCVLIM=l.3*CONCONTINUEBONO=1.E+1OOIF (SIGMA.GT.O. ) BONO=RHOF~GY*X( IMl)*=2/SIGMAWRITE (59.390) BONDWRITE (12,390) BONDIPL=2IF (KL.EO.5) 1PL=3IPR=IMIIF (KR.EO.5) IPR=IBARdPB=2IF (KB.EO.5) JPB=3dPT=dMlIF (KT.EO.5) JPT=d8AR

SET CONSTANT TERMS FOR PLOTTING

XMIN=X(I)XMAX=X(IM1)IF (IsYMpLT.GT.0) xMIN=-xMAXYMIN=Y(l)YMAX=Y(dMl)DI=XMAX-XMIND2=YMAX-YMIN

SETUP.2SETUP.3SETUP.4SE7U?.5SETUP.6SETU?.7SETUP.8SETU?.9SETU?.10sETup.11SETUP.12SETLJP.13SETUP.14SETUP.15SETUP. 16SETUP.17SETUPSETUPSETUPSETUPSETUPSETUPSETUPSETUPSETUP.26SETUP.27

1819

;?22232425

SETUP.28SETUP.29SETUP.30SETUF’.3ISETUP.32SETUP.33SETUP.34SETUP.35SETUP.36SETUP.37SETUP.38SETUP.39SETUP.40SETUP.41SETUP.42SETUP.43SETUP.44SETUP.45SETUP.46SETUP-47SETUP.48SETUP.49SETUP.50SETUP.51SETUP.52SETUP.53SETUP.54SETUP.55SETUP.56SETUP-57SETUP.58SETUP.59SETUP.60SETUP-61SETUP.62SETUP-63SETUP.64SETUP-65s~Tup.66SETUP.67SETUP.68SETUP-69SETUP.70SETUP.7;SETUP.72SETUP.73SETUP-74SETUP.75SETUP.76SETUP.77SETUP.78SETUP.79SETUP.80SETUP.81

63

7(3

80c +++c +++c +++

cc +++cc +++

90

100

110

120

cc +++cc +++c

130

cc +++c

D3=AMAx1(DI,D2)sc.q.o,l~3XSHFT=O.5-( l.@-Dl-sc)~’sH!=T=o.5-( 1 .0-!22.s;)c,Y#,IN=EplcDC 60 1=2.IM:CX~IN=AMINl(3ELX( I],DXMIN)DJMIN=E=1O50 70 1=2.JM:DYVIN=AMINI(DELY( 1).DYMIN)VELMX1=AMIN1 [DXMIN, DYMIN)/VELMX00 80 1=1. IMAXDO 80 LJ=l.IJMAXBETA(I,d)=O.OAC(I,d)=l.OAR(I .LJ)=l.OA7(~l~)=l”CDNTINUE

SET SPEC

CALL ASET

DETERMINE

o

A+ A VALUES FOR

SLOPED BOUNOARY

OBSTACLES AND 8.C.

LOCATION

COMPUTE INITIAL VOLUME FRAC” ION FUNCTION F IN CELLSIF(IEOIC.LE.0) GO TO 90IF (NEOUIB+2. LT.IBAR2=JBARZWRITE (59,4oO) NEOU18,1BAR2WRITE (12,400) NEOUIB, IBAR2NEOU19=IBAR2=LJBAR2-2

GO TO 90JBAR2dBAR2

WRITE (59,410)-NEIjUIBWRITE ( 12,410) NEOUIBCON-INUEIF (IEOIC.GT.0) CALL EOUIB (U,V,NEOUIB,BONO,DANGLE ,CYL)SFLHT=FLHTDO 120 1=1, IMAX00 110 J=2,JMAXF(I,J)=l.OIF (IEOIC.LE.@) GO TO 100LocK=FLoAT(NEou18-I )*xI(I)/x( IMI)+I.000DoILOCK=MINO(NEOUIB. LOCK )LOCK=MAXO(I,LOCK)FLHT=SFLHT+U( LOCK, I)=X(IMl)CONTINUEIF (FLHT.GT.Y(LJ-1 ).ANO. FI%T.LT.Y(U)) F(I.J)=ROY(U)=(FLHT-Y( d-l))IF (Y(J-1).GE.FLHT) F(i,J)=O.OCONTINUEF(I, l)=F(:.2)CONTINUE”FLHT=SFLHT

GENERATE SPECIAL f-FuNcTION (FLuIO) CONFIGURATION

CALCULATE :TVIS ANO OTSFT

OS=I.OE+IOOTVIS=l.OE+IOOTSFT=I.OE+IODST=l.OEIO00 130 I=2,1M100 130 d=2.dMlOXSO=OELX(I )--2oYsa=oELY(d)*=2ROSO=OXSO=OYSO/(OXSO+DYSO)ROSO=ROSQ/( 3.O*XNU+l .oE-60)OTVIS=AMIN1 (OTVIS.RDSO)DS=AMIN1(OELX( I),OELY (d).OS)OSTX=OELX( I)=OELX(I)/(3 .O*OELY(d)=RDX( I)+CYL )DSTY=OELY( IJ)=DELY(J)/( 3.0*(DELX( I)*RDY(J))**Z+CYL)DST=AMINl (DST.DsTX*OELX( 1).OSTY=DELX(I))CONTINUESIGX=SIGMAIF (SIGX.EO.O.0) SIGX=EM1ODTM=soRT( RHoF*osT/(sIGx*4 .0))DTSFT=AMIN1 (OTSFT,OTM )

CALCULATE BETA(I,J) FOR MESH

00 140 I=2,1M1

SETIJP.82SETUO.E3SETUG.84SETL~.85SETU?.86s~Tu?.E~SE7UD.86SETU~.g9 .SETUD.90SETUP.91SETUP.92SETUP.93SETUP.94SETUP.95SETUP.96SETUP.97SETUP.98SETUP.99SETUP. 100SETUP. 101SETUP. 102SETUP. 103SETUP. 104SETUP. 105SETUP. 106SETUP. 107SETUP. 108SETUP. 109SETUP.11OSETUP. 111SETUP. 112SETUP. 113SETUP. 114sETup. 115SETUP. 116SETUP. 117SETUP. 118SETUP. 119SETUP. 120SETUP. 121sETup. 122SETUP. 123SETUP. 124SETUP. 125SETUP. 126SETUP. 127SETUP. 128SETUP. 129

.SETUP. 130SETUP. 131SETUP. 132SETUP. 133SETUP. 134SETUP. 135SETUP. 136SETUP. 137SETUP.13BSETUP. 139SETUP. 140SETUP. 141SETUP. 142SETUP. 143SETUP. 144SETUP. 145SETUP. 146SETUP. 147SETUP. 148SETUP. 149SETUP. 150SETUP. 151SETUP. 152SETUP. 153SETUP. 154SETUP. 155SETUP. 156SETUP. 157SETUP. 158SETUP. 159SETUP. 160SETUP. 161

127

140cc +++cc~ +++c

DO 140 J=2,dMlIF (AC(I.UI.LT.EM6) GO TO 1~~RHXR=QHOF-(OELX( 1+1 )+DELXII 1 1RHXL=RHOF-(9ELX( I)+3ELX(i-1))QHYT=RHOF- [OELY(C*l )-OELYIJ) IRHY6=Rr10F-(OELY(J )+OEL’i(J-l i 1X<=OELT-2.0=(.?OX[ I I-[AR( 1-1 .J1-R(I-1 J.’2+XL+AR~I .J}-l?!i 1 ‘2HXRl-

1 R2Y(u I=(A7(I.J)=RI! i);RHyT4A~( I.J-l )-RIii )/RHfEI 1XX=XX, (AC(I,d)-Gi(I ~ 16ETA(I.J)=OMG/’XXCONTINUE

SET BETA(I,u)= -1.0 IN OBSTACLE CELLSMUST BE OONE 9Y hANO IN GENERAL

PRINT BETA(I,d) ON FILM ANO PAPER

IF (Ih40vY.E0. t) J30 To 160WRITE (12,420)WRITE (12,430)00 1

Td=l.dMl

00 1 0 1=1.IMIWRITE (12.440) I.d,BETA( I,d).AC(I,d). AR(I.J),AT(I.J) .SINO(I.J)

1 COSO(I,J)150 CONTINdE160 CONTINUE

WRITE (6.420)WRITE (6.430)00 170 d=l.dMl00 170 1=1.IM1WR:TE (6.440) I. J,BETA( I.J).AC(I.J) ,AR(I. J) .AT(I.J).SINO( I.LJ).

1 COSO(I.J)170 CONTINUE

CALCULATE HYDROSTATIC PRESSU2Ecc +++&

00 180 I=2,1MIP(I.dMAX)=O.ODO 18C LJ=2.dMl

180c“c +++c

260cc +++c

270cc +++c

280290

c300310320330340350360

128

dd=dM~-d+2”RHOYA=(AMINI(F(I .dJ+l),O. 5)=DELY(JJ+ 1)+AMAX:(O.O,F

1 (I, UJ)-0.5)=DELY(JJ ))=RHOFP(I,dJ)=P(I ,Ud+l)-GY*RHOYACCINTINUE

oo260J=2~M, ‘PRESSURE

SET INITIAL SURFAC

00 260 I=2:IM1PS(I.J)=O.OCONTINUE

SET INITIAL VELOCITY FIELO INTO U ANO V ARRAYS

00 270 1=1.IMAX00 270 d=l.JMAXu(I.J)=UIv(I.d)=vIIF (A7(I.d).LT.EM6) v(I.J)=O.0IF (AR(I,d).LT.EM6) u(I.u)=o.0IF (F(I.J).GT.EMF) GO Tc 270U(I.U)=O.OV(I,LI)=O.OCONTINUE

SET INITIAL VOIO REGION QUANTITIES

00 280 K=I.NVORNR(K)=OPR(K)=PSATVOL(K)=O.ORETURN

FORMAT (lX,2HK=, lPE12.4,2X,3HxI =,E12.4.2X.4HPER= oE12 .4)FORMAT (lX. llH-==-* ISOR=.110.21H MUST BE O OR 1 *-*==)FORMAT (4x.6HIs0R =015.4x,7H10EFM =,I5)FORMAT (1OA8)FORMAT (1OX,I1O)FORMAT (IOX.E20.6)FORMAT (3x,7HN0uMP =.Ilo/4x.6Hw0L R.1PE20.6)

SSTlj~. lSSSETUP. !?9SETUP. ~71SETUP. 172SETUP. 173SETUP. 1-4SETUP. 175SETUP. 176SETU?. 177SETUP. 178SETUP. 179SETUP. 180SETUP. 181SETUP. 182SETUP. I83SETUP. 184SETUP. 185SETUP. 186SETUP. 187SETUP. 188SETUP-1895ETUP. 190SETUP. 191SETUP. 192sETu~. 193SETUP. 194SETUP. 195SETUP. 196SETU13. 197SETUP. 198SETUP. 199SETUP.200SETUP.201SETUP.202SETUP.203SETUP.204SETUP.205SETUP.206SETUP.207SETUP.208SETUP.209SETUP.21OSETUP.211SETUP.212SETUP.213SETUP.214SETUP-215SETUP.216SETUP.217SETUP.216SETUP.219SETUP.220SETUP-221SETUP-222SETUP.223SETUP.224SETUP.225SETUP.226SETUP.227SETUP.22eSETUP.229SETUP.230SETUP.231SETUP.232SETUP-233SETUP.234SETUP-235SETUP.236SETUP.237SETUP.23eSETUP-239SETUP.240SETUP.241

370 FORMAT (IOX,7110)380 FORMAT (IX.AHCON. .IPEIO.3. lx, IIHANO FCVLIM= .EIO.3,1X,AIHARE INCOMP

ILTIBLE. SETTING FCVLIM=l.3=CON. )390 FoRMAT (lx. 13H50No NuMBER =,$pEIp,4]400 FORMAT (/lX.8HNEOUIB =,15. IY.,29HIS TOO EIG FOR THE OIMENSIONS,215)410 FORM&T (IX, 17HCUTTING NEoUIB T0,15,15H ANO CONTINUING, )420 FORMAT (liil)43C FGRK!AT (4X. lt-!i ,5X. ltid.8X,4HBETA, IIX,2HAC,13X.2HAR , i3X.2H.L7, 12Y..

! 41-!S?N0 .11X,4HCOSJJ)440 FORMAT (2x, 13.3x,13,6[3x. 1>E12.5))

ENO

SUBROUTINE TAPIN=CALL.COMMONIc +++c +++c +++

10

REAO RESTART DUMP FROM TAPE7

2EA0 (7) MOUMPIF (MDUMP.EO.NOUMP) GO TO IGh’RITE (6.20) MOUMP,NDUMDWRITE (12,20) MOUMP.NDUMPWRITE (59,20) MDuMP,NDuMPCALL EXITLOC (etiTAPiN jCONTINUENOUMP=NOUMP+lREAO (7) UN,VN.PN, FN,U,V,P.F,PETA ,BETA. NF,PS.AR,AT.ACREAO (7) X,XI, RXI,OELX.ROX,RX, Y, YJ, RYJ,OELY,ROY,XL ,XC,OXMN,NXL.NXR

1 ,YL, YC,OYMN,NYL,NYR ,ZCREAO (7) NR,PR.VOL. IBAR,JBAR. IMAX, LJMAX,IMl,dMl,NKX ,NKY,NCYC,OELT,T

1 .TWPRT,TWPLT. RHOF.NREG. VCHGT2 ,CANGLE, ICYL,CYL,ITER ,FLG; FLGC.FNOC,NOCON, NFLGC. ISYMPLT3 ,IMOVY,VELMX. VELMX1 ,XSHFT. YSHFT.XMIN,XMAX .YMIN,YMAX,SF4 ,IPL, IPR,JPB,JPT,DTVIS .OTSFT.OXMIN,OYMIN, PSAT5 ,LITER,EMF1

WRITE (6.30) NCYC,MOUMP.TWRITE (12.30) NCYC.MOUMP,TWRITE (59.30) NCYC,MOUMP,TRETURN

c20 FORMAT (Ix,22HTAPIN ERRoR - GOT OUMP.14.16H, EXPECTING OUMP,14)30 FORMAT (IX, 19HRESTARTING ON CYCLE,I7,1OH FROM OUMP,13,4!-I T =,IPE13

1 .6)ENO

SETUP.242SETUP.243SETUP.244SETLIP.245SETdP.246SETUP.247SETUP.248SE7UP.245SETUP.250SETUP.251SETUP.252

TAPIN.2TAPIN.3TAPIN.4TAPIN.5-rAPIN.6TAPIN.7TAPIN.9TAPIN.9TAPIN. 10TAPIN. 11TAPIN. 12TACIN. 13TAPIN. 14TAPIN.15TAPIN.16TAPIN.17TAPIN. IeTAPIN. 19TAPIN.20TAPIN.21TAPIN.22TAPIN.23TAPIN.24T~PIN.25TAPIN.26TAPIN.27TAPIN.2eTAPIN.29TAPIN.30TAPIN.31TAPIN.32

129

SUBROUTINE TAPOUT (NKKP)=CALL,COMMON1c +-++c +++c +++c +-+c +++c +-+-+

1020

WRITE RESTART DUMP 0!’4 TAPE7NKKP = -1. EOLIILIBRIUM MESH DUMp

= O, FINAL DUMP ON A RUN. TERMINATE=1. PERIODIC BACK-UP OUM? OURINS A RUN

IF (NKKP.GE.G) GO TD 20CALL AOV (1)WRITE (6.30) NCYCWRITE (12.30) NCYCWRITE (59.30) NCYC00 10 1=1.IMAX00 10 J=l,dMAXu(I.IJ)=O.V(I,J)=O.CONTINUECONTINUELCMS=7HBACK UPIF (NKKp.Eo.0) LCMS=7HRES-WRITE (6,40) LCMS,NCYC,TIF (IMOVY.EO.0) WRITE (12WRITE (59.40) LCMS.NCYC,TREWINO 7WRITE (7) NOUMP

ART

,40) LCMS.NCYC.T

WRITE (7) UN,VN, pN. FN,U,V,poF.pETA ,BETAONF, PS,AR,AT.ACWRITE (7) X.XI, RXI.OELX.ROX,RX, Y, YJ, RYJ,OELY, RDY,XL.XC ,OXMN,NXL

1 ,NXR,YL. YC,OYMN,NYL.NYR .ZCWRITE (7) NR,PR,VOL, IBAR.LJBAR,IMAX ,dMAX, IMl,JMl,NKX.NKY ,NCYC,OELT

1 ,T, TWPRT.TWPLT,RHOF ,NREG,VCHGT2 ,CANGLE, ICvL.CYL,ITER ,FLG. FLGC. FNOC.NOCON,NFLGC3 ,ISYMPLT. IMOVY.VELMX, VELMX1.XSHFT, YSHFT,XMIN, XMAX.YMIN,YMAX,SF4. I?L, IpR.dpB,dpT.DTvIs ,QTSFT,DXMIN,OYMIN

5 .psAT.L~TER.EMFlIF [NKKP.LE.0) CALL EXITLDC (6HTAPOUT)RETURN

c30 FORMAT (IX. 39HEOUILIBRIUM MESH OUMP TO TAPE7 ON CYCLE.17)40 FDRMAT (1X,A7,1X,13HOUMP ON CYCLE,17,2X,3HT =,1PE13.5)

ENO

TAPOUT.2TAPOUT.2TAPOUT.4TAPOUT.5TAPOUT.67APOIJT.7TAPilUT.bTAP3UT.9TAPOUT.10TAFIOUT. 11TAPOUT. 12TAPOUT. 13TAPOUT.14TAPOUT.15TAPOUT.16TAPOUT.17TAPOUT. IBTAPOUT.19TAPOUT.20TAPOUT.21TAPOUT.22TAPOUT.23TAPOUT.24TAPOUT.25TAPDUT.26TAPOUT.27TAPOUT.28TAPOUT.29TAPOUT.30TAPOUT.31TAPOUT.32TAPOUT.33TAPOUT.34TAPOUT.35TAPOUT.36TAPOUT.37TAPOUT.38TAPOUT.39TAPOUT.40TAPOUT.41TAPOUT.42

130

SUBROUTINE TILDE=CALL.COMMON1cc +++ CoMDuTE TEMPORARY U AND V EXPLICITLYL

VISX=O.OvIsy=o.c.00 20 d=2,dMlGO 20 1=2.IM:CALL OVCALU(I.LJ)=O.OROELX=l .O/(OELX(I)+OELX( 1+1))ROELY=l .0/(12ELY( IJ)+OELY( u+I))IF (NF(I.d)+NF(I+l .d).GE. 12) GO TO IOIF (AR(I.d).LT.EM6) GO TO 10SGU=SIGN(l.O,UN(I ,J))ROXA=DELX(I)+DELX( I+I)+ALPHA*SGU*(OELX( I+I)-OELX(I)ROXA=l.O/RDXAFUX=ROXA*UN(I ,d)=(OELX( I)=OUDR+DELX( I+l)*OUDL+ALPHA SGU=(DELX(I+l)

1 =OUOL-OELX( I)*OUOR))VBT=(DELX( I)=VN(I+I ,J)+OELX( I+I)=VN(I,J) )*RDELxVBB=(OELX( I)*VN(I+I .d-l)+OELX( I+l)=VN(I .LJ-l))=RDELXVAV=O.5=(VBT+VBB)oYT=o.5-(oELY(d)+oELY (d+l))OYB=O.5-(OELY(J-I )+OELY(LJ))SGV=SIGN(I.O.VAV)OYA=OYT+DYB+ALPHA=SGV=(OYT-OYB )FUY=(VAV/DYA)*(DYB*DUDT+OYT=DUDB+ALPHA=SGV*(DYT*DUDB-Oy3*DuDT ))IF(XNU.EO.O.O)GO TO 5OUDXSO=(OUDR-DUDL )/(XI (1+1)-XI(I))OUOYSO=(OUOT-OUOB )/(YJ( d+l)-YJ(J-l))RXDUDX=RX( I)=(i3ELX(I+ l)=OUOL+OELX( I)*DUOR)/(DELX( I)+OELX(I+l))RXSOU=UN(I,U)*RX(I )=*2VISX=XNU*(DUOXSO+OUOYSO+CYL=RXOUDX-CYL*RXSOU )

5 CONTINUERHOX=RHOF*(OELX( I+I)+DELX( I))U(I. d)=UN(I.d)+OELT*( GX-FUX-FUY+VISX)+DELT= (P(I,J)-P( l+I.J))*2.0

1 jRHox10 CONTINUE

V(l.d)=o.oIF (NF(I,Ji+NF(I.d+I ).GE. I2) GO TO 20IF (AT(I,LJ).LT.EM6) GO TO 20UBR=(DELY(J+I )*uN(I.J)+OELy(d)*uN( I,J+UBL=[DELY(J+I )-uN(I- I,d)+OELy(J)*uN( I-UAV=O.5*(UBR+UBL)OXR=0.5=(DELX( I)+DELX( 1+1))DXL=0.5-(OELX( I~+DELX( I-I))SGU=SIGN(l.O,UAV)OXA=DXR*OXL+ALPHA*SGU=(OXR-DXL)FVX=(UAV/DXA)*(OXL*OVOR+OXR=OVDL+ALPHA>SGV=SIGN( I.O.VN(I.J) )

))=ROELY,d+l))*RoELY

SGU*(DXR*OVOL -OXL=OVDR))

DYA=DELY(d+l )+OELY(ii)+ALPHA*SGV*(DELY (J+l)-OELY(J))FVY=(VN( I,J)/OYA)*(OELY( d)*DVOT+OELY(d+l )*OVDB+ALPHA*SGV*(OELY( d+l

1 )=DVDB-OELY(U)*DVDT))IF(XNU.EQ.O.O)GO TO 15OVDXSO=(DVDR-DVDL)/(XI (l+l)-X1(l-l))OVOYSO=(OVOT-DVDB)/(Yd(J+l )-Yd(IJ))OVDXRX=O. 5=(OVOR+DVDL )=RxI(I)vISY=XNU*(DVOXSO+DVOYSQ+CYL*OVOXRX)

15 CONTINUERHOY=RHOF*(DELY(J+l )+DELY( d))v(r,d)=vN( I.J)+DELT= (Gy-FvX-FVy+VISy )+OELT=( p(l. J)-p(1,d+1))*2.o

1 /RHOY20 CONTINUE

RETURNEND

TILOE.2TILOE.3TILOE.4TILDE.5TILOE.6TILOE.?TILEIE.8TILO:.9TILDE.1OTILoE.~1TILDE.12TILDE.13TILDE.14TILOE.15TILDE.16TILDE.17TILDE. 18TILOE.19TILOE.20TILDE.21TILDE.22TILOE.23TILDE.24TILOE.25TILDE.26TILOETILOETILDETILDETILOE

.27

.28

.29

.30

.31TILOE.32TILOE.33T:LOE.34TILOE.35TILDE.36TILOE.37TILOE.38TILDE.39TILOE.40TILDE.47TILOE.42TILDE.43TILOE.44TILDE.45TILDE.46TILOE.47TILDE.48TILDE.4STILDE.50TILDE.51TILDE.52TILOE.53TILOE.54TILOE.55TILDE.56TILDE.57TILOE.58TILOE.59TILDE.60TILDE.61TILOE.62TILOE.63TILDE.64TILDE.65TILOE.66TILDE.67

131

-CALL

cc +++c

10

20

30

40

SUBROUTINE VFCONVCOMM’JN1DATA FCVLIh! /.5/

SONVECT THE VLILIJME OF CLUID FUNCTICN F

7= (NC?C.L7. 1) GO ?0 ICK3FLGC=O.OCCl 50 d=l.JMlDO 50 1=1.IM1IF(BETA(I,~).i-~.o.o) Gc To 50VX=U(I.IJ)=DELTVY=V(I,J)=DELTABVX=ABS(VX)ABVY=ABS(VY)IF (ASVX. LE.FCVLIM=DELX( 1) .AND.ABVY .LE.FCVLIM=OELY (d)) GO TO 10~LGC=l .WRITE (59.150) NCYC,T,OELT .;.J.FCVLIM ,ABVX.OELX( I),ABVY,OEL} (d)WRITE (12,150) NCYC, T, DELT,I,J.FCVLIM, ABVX,OELX( I).ABVY,OELY( J)CONTINUEIF (AR(I,J).LT.EM~) GO TO 30IA=I+l10=1IDM=MAXO(I-1,1 )ARDM=AR(IOM,J)RB=AR(I.J)*R(I)RA=AC(I+l, d)*RI(I+l )RO=AC(I.J)*RI(I)IF (VX.GE.O.0) GO TO 20IA=lID=I+lIDM=MINO(I+2,1MAX)AROM=AR(IDtU-l,d)RA=AC(I,J)~RI(I)RO=AC(I+l, d)*RI(I+l )CONTINUEIAO=IAIF (NF(ID.d) .EO.3.OR-NF( Io,d).Eo.4) IAD=IOIF (FN(IA, J).LT.EMC.OR. FN(IOM,J).LT.EMF) IAO=IAFDM=AMAXI (FN(IOM.J). FN(ID,J),O.1O)IF(ARDM.LT.EM6) FDM=I.OFXt=FN( IAO,J)=ABS(VX)+AMAX l((FOM-FN( IAO.d))=ABS(VX )-(FOM-FN(ID.IJ))

1 -OELX(ID).O.0)FX=AMINI (FXI,FN(IO,J) ●OELX(ID)=RD/RB)

4F(IL?, J)=F(iO, d)-FX-RDX( IO)=(RB RD)F(IA. IJ)=F(IA,d )+FX=RDX( IA)=(RBjRA)IF (AT(I,J).LT.EM3) GD TO 50JA=J+lLJD=ddOM=MAXO(d-1.1)ATOM=AT(I.dOM)RBxAT(I.J)RA=AC(I,d+l)Ro=Ac(I,J)IF (VY.GE.O.0) GO TO 40dA=dJDxd+lJoM=MINo(.J+2.dMAx)ATDM=AT(I.JDM-1)RA=Ac(IoJ)RD=Ac(l,d+l)CONTINUEuJAO=dAIF (NF(I.UD) .EO.1.OR.NF( I,UD).EO.2) JAO=dDIF (FN(I,JA ).LT.EMF.OR.FN( I,UOM).LT.EMF) UAO=JAFDM=AMAX1( FN(I,JDM),FN( I.JD).O.1O)IF(ATDM.LT.EM6) FOM=l.OFYI=FN( I, JAO)=ABS(VY)+AMAXI (( FDM-FN( I.dAD))*ABS(VV )-(FOM-FN(I,LJD))

1 *DELY(JD),O.0)FY=AMINl (FY1.FN(I,dO) ●DELY(dD)=RO/RB)F(I, JO)=F(I.JD)-FY=RDY (dD)=(RB/RO)F(I,JA)=F( I,JA)+FY=RDY(JA )=(RB/RA)

50 CONTINUEc +++ DEFDAM IT

IF (NPACK.EO.0) GO TO 7000 60 1=2.IMIDO 60 J=2,dBARIF (AT(I,LJ).LT.EMF) GO TO 60IF (NF(I.J).NE.0) GO TO 60IF (F(I,J).GE.EMF1) GOTO 60

“132

VFCONV.2VFCONV.3VFCONV.4VFCONi”.5vFcaNv.6vCcoNv.7VFCONV.8VFCONV.9VFCGNV. 10VFCONV. 11VFCONV. 12VFCONV. 13VFCONV. 14vFcoNv. f5VFCONV.16VFCONV.17VFCONV. 18VFCONV. 19VFCONV.20VFCONV.21VFCONV.22VFCONV.23VFCONV.24VFCONV.25VFCONV.26VFCONV.27VFCONV.28VFCONV.29VFCONV.30VFCONV.31vFCONV.32VFCONV.33VFCONV.34VFCONV.35VFCONV.36VFCONV.37VFCONV.38VFCONV.39VFCONV.40VFCONV.41VFCONV.42VFCONV.43VFCONV.44VFCONV.45VFCONV.46VFCONV.47VFCONV.48VFCONV.49VFCONV.50VFCONV.51VFCONV.52VFCONV.53VFCONV.54VFCONV.55VFCONV.56VFCONV.57VFCONV.58VFCONV.59VFCONV.60VFCONV.61VFCONV.62VFCONV.63VFCONV.64VFCONV.65VFCONV.66VFCONV.67VFCONV.68VFCONV.69VFCONV.70VFCONV.71VFCONV.72VFCONV.73VFCONV.74VFCONV.75VFCONV.76VFCONV.77VFCONV.78VFCONV.79VFCONV.80VFCONV.81

c

cc-..

c

FADD=AMIN1(l. -F(I,LJ),F(I.d+l ))F(I. Jj=F(I,LJ)+FADDF(I.J+I)=F(I.J+I )-FADD

60 CONTINUE70 CONTINUE

+++*++ Divergence Correction TERMIFIIDIV.EO.0) GC TO 90IF(IDEFM.EO. 1) GO TC 90DO 8G d=2,JMlDO 80 I=2,1M:IF (NC(I, IJI.NE.O.OR.AC( I,J).LE.O. ) GO TO 80DIJ=(ROX( I)-(AR(I,J )=R(I)=U( I,J)-AR( I-1 ,J)=R(I -l)*U(I-I.JI )+

1 RDY(U)*( AT(I.IJ)*RI(I )*V(I, UJ)-AT(I,J-IJ*RI (I )=V(I.d-: )))/2 (AC(I.tJ)-RI(I))

F(I, LJ)=F(I, LJ)+OELT=FN( I,d)=DId80 CONTINUE90 CONTINUE

100 CONTINUEDD 140 J=2.LJM100 140 I=2,1MIIF (BETA(I.J). LT.O.0) GO TO 140VCHG=O.OIF (F(I, d) .GT.EMF.ANO.F( I,U).LT.EMFI) GO TO 120IF (F(I.IJ).GE.EMFI) GO TO 110vCHG=F(I,d)F(I,J)=O.OGO TO 120

110 CONTINUEVCHG=-( 1.O-F(I,IJ))F(I.J)=I.O

120 CONTINUEVCHG~=VCHGT+VCHG*OELX( :)*DELY(d)*ACI I,d)*RI (I)*TPIIF (F(I,u).LT.EMF1) GO TO 1401= (F(I+l,U) .LT.EMF.AND.AR( I,J)..GT.EM6) GO To 130IF (F(I-l,J ).LT.EMF.ANO.AR( I-1.J).GT.EM6) GO TO 130IF (F(I.d+l ).LT.EMF.ANO. AT(l,u).GT.EM6) GO TO 130IF (F(I, J-1 ).LT.EMF.ANO. AT(l.d-1).GT.EM6) GO TO 130

130

140

‘r++

150

GO TO 140F(I,J)=F(I.J)-I. I*EMFVCHG=I. I=EMFVCHGT=VCHGT+VCHG*OELX( I)=DELY(dCONTINUE

SPECIAL BOUNDARY CONDITIONS FOR

RETURN

-AC(I.J)-RI(I)=TF’I

F

VFCONV.82VFCONV.83VFCONV.84~:coNv.85VFCONV.86VFCDNV.87V;CONV.88VFCONV.89VFCONV.90VFCONV.91VFCONV.92VFCONV.93VFCONV.94VFCONV.95VFCDNV.96VFCONV.97VFCONV.98VFCONV.99VFCONV. 100VFCONV. 101VFCONV. 102VFCONV. 103VFCONV. 104VFCONV. 105VFCONV. 106VFCONV. 107VFCONV. 108VFCONV. 109VFCONV. 110VFCONV. 111VFCONV. 112VFCONV. 113VFCONV. 114VFCONV. 115VFCONV. 116VFCONV. 117VFCDNV. 118VFCONV. 119VFCONV. 120VFCONV. 121VFCONV. 122VFCONV. 123VFCONV. 124VFCONV. t25VFCONV. 126VFCONV. 127VFCONV. 128

FORMAT (1x, 12HVFCONV ERROR/lX,5HNCYC=,17, 1X. 2HT=, IPE14.6,1X,5HOELT VFCONV. 1291=.E12.4, IX.4HI.J=,214, lX,7HFCVLIM=, Etl .3/3X .5HABVX=,E12.4. IX.5HOEL VFCONV. 1302X=, E12.4.1X.5HABVY= ,E12.4, 1X,5HOELY=.E12 .4) VFCONV. 131

ENO VFCONV. 132

133

: +..~ +++c +.-C +..+.

Et-.c.c

:cccccc:c:cccccccccccccccccccccc

ic~:cccccccccccccccccccc

:ccccccccc

134

SUBROUTINE EPI

EPiLOGLIE - ON-LINE DOCUMENTATION

VARIABLES LISTEC IN COMMON (EXCLUOING INPUT PARAMETERS)

LOEFUEc~FK4c.:OTSCT

DTV:S

DUDEDUDLOUJTDUOROVORDVOLOVDTOVDBOXMINOYMINEMF

EME1EM6EM6PIEM1OEP1OFCVLIM

FLG

FLGC

FNOC

IBAR

IBAR2

IMAX

IM1

18AR

IPL

IPR

ITERd8AR

d8AR2

dMAX

dMl

dBAR

dPB

dPT

LITERNCYC

NFLGC

OE?OAM OPTION PARAMETER DEFINED IN PQESCR SUBROUTINE~EFaAR o?TIo& pARAMETERMESU GEoffETRY IN31CATOR (= ICYL)MAXIMUM DELT VALUE ALLOWED 8} THE SURFACE TENSION FORCESSTA91L17y CRITERION (OELT IS AUTOMATICALLY ADJUSTED)MAXIMUM DELT VALUS ALLOWED BY THE VISCOUS FORCESSTABILITY CRITERION IOELT IS AUTOMATICALLY AOIJUSTEO)DERIVATIVE OF U VELOCITY RESPECT TO Y AT (1+1/2.J-l/2)DERIVATIVE OF U VELOCITY RESPECT TO X AT (1.J)DERIVATIVE OF U VELOCITY RESPECT TO Y AT (1+1/2.d+l/2)DERIVATIVE OF U VELOCITY RESPECT TO X AT (1+1 .LJIDERIVATIVE OF V VELOCITY RESPECT TO X AT (141/2,d+l/2)DERIVATIVE OF V VELOCITY RESPECT TO X AT (1-1/2,J+t/2)OERIW!TIVE OF V VELOCITY RESPECT TO } AT (1.d+l)DERIVATIVE OF V VELOCITY RESPECT TO Y AT (Id)SMALLEST OELX(I) IN MEsHSMALLEST OELY(UJ) IN MESHSMALL VALUE. TYPICALLY 1.OE-06, USEO TO NEGATE ROUNO-OFFERROR EFFECTS WHEN TESTING F=I.O OR F=O.O=1.O-EMF=1 .OE-06

=1.O-EM6=1.OE-10=1 .OE+IOF CONVECTION LIMIT ABS(UOT/OX) < FCVLIM

A6S(VOT/OY) < FCVLIMPRESSURE ITERATION CONVERGENCE TEST INOICATOR (~0.O WHENTHE CONVERGENCE TEST IS SATISFIEO. =1.0 WHEN THECONVERGENCE TEST ?S NOT SATISFIED)VOLUME OF FLUIO FUNCTION CONVECTION LIMIT INOICATOR(DELT REoUcEO ANO CYCLE STARTEO OVER IF LIMITIS EXCEEOED)PRESSURE CONVERGENCE FAILURE INOICATOR (=1.0,CONVERGENCE FA:LED ANO OELT IS REOUCEO. =0.0 OTHERWISE)NUMBER OF REAL CELLS IN X-DIRECTION (EXCLUOES FICTITIOUSCELLS)=IBAR+2. SPECIFIEO IN PARAMETER STATEMENT(=IBAR+3. IF PERIOOIC IN X-DIRECTION)TOTAL NuMBER oF M H cELLS IN x-OIREcTION (=18AR+2)(=I13AR+3 IF PERI % XC IN X-OIRECTION)VALUE OF”THE INOEX I AT THE LAST REAL CELL IN THEX-OIRECTION (=IMAX-1)VALUE OF THE INOEX I AT THE NEXT TO THE LAST REAL CELLIN THE X-OIRECTION (=IMAIN THE x-2)LEFTMOST PRESSURE ITERATION INDEX IN X-OIRECTION(=3 FOR CONSTANT PRESSURE BOUNDARY CONOITION, =2 FORALL OTHER CASES)RIGHTMOST PRESSURE ITERATION INDEX IN X-OIRECTION

(=IBAR FoR cONSTANT pREssuRE 6ouNoARY cONOITION. =It41 FORALL OTHER CASES)PRESSURE ITERATION COUNTERNUMBER OF REAL cELLs IN Y-OIRECTION (ExCLUOES FIcTIcIOUsCELLS)=JBAR+2, SPECIFIEO IN PARAMETER STATEMENT(=dBAR+3. IF PERIOOIC IN Y-OIRECTION)TOTAL NUMBER OF MESH CELLS IN Y-OIRECTION (=LJBAR+2)(=t16AR+3.IF PERIOOIC IN Y-DIRECTION)VALUE OF THE INOEX d AT THE LAST REAL CELL IN TFIEY-OIRECTION (=dMAX-1)

VALUE OF THE INDEX J AT THE NEXT TO THE LAST REAL CELLIN THE Y-DIREcTION (=JMAx-2)BDTTOM PRESSURE ITERATION INOEX IN Y-OIRECTION(=3 FoR coNSTANT pREssuRE BouNDARY CONDITION, =2 FORALL OTHER CASES)TOP PRESSURE ITERATION INDEX IN Y-OIRECTION

(=d3AR FoR cONsTANT PRESSURE 50uN0ARY CONOITION. =JM1 FORALL OTHER CASES)NUMBER OF ITERATIONS ON THE PREVIOUS CYCLECALCULATIONAL TIME CYCLE NUMBERTAPE7 DUMP SEOUENCE NUMBER. SET TD ZERD TOSKIP TAPE RESTART. OTHERWISE. IT MUST EQUAL THESEOUENCE NUMBER ON TAPE7 TO SUCCESSFULLY RESTART.NUMBER OF CYCLES THE VOLUME OF FLUID FUNCTION CONVECTIONLIMIT (FLGC) IS EXCEEDEO

EDi.2EP:.3EP!.4EPI.5Ep:.6E?:.7EP:.8EPI.95?1. :0EPI.l!EPi.12EPI.13EPI.14EPI.15EPI.16EPI.17EPI.18EPi.19EPI.20EPI.21EPI.22EPI.23EPI.24EPI.25EPI.26EPI.27EPI.26EPI.29EPI.30EPI.31EPI.32EPI.33EPI.34EPI.3’5EPI.36EPI.37EPI.38EPI,39EPI.40EPI.41EPI.42EPI.43EPI.44EPI.45EPI.46EPI.47EPI.4BEPI.49EPI.50EPI.51EPI.52EPI.53EPI.54EPI.55EPI.56EPI.57EPI.58EPI.59EPI.60EPI.61EPI.62EPI.63EPI.64EPI.65EPI.66EPI.67EPI.68EPI.69EPI.70EPI.71EPI.72EPI.73EPI.74EPI.75EPI.76EPI.77EPI.78EPI.79EPI.80EPI.81.

cc~

N08D NUMBER OF OBSTACLE OEFINING FUNCTIONS ALLOWECsPEcIFIEc IN PARAMETER STATEMENT

NUM35R OF CYCLES PRESSUQE CCJNVEGENCE HAS FAiL~~(USEO TO SET &RRAY SIZE - h!JST BE > o]NUh%EG OF VCIC REGIONS GENERAT50 IN CALCU~LTIOKMAXIMUM NUMSER OF VOID 2EGIQNS ALLOWE2. SDECIF:EO INPARAMETER STATEMENTNUMBER CF SUSMES!-I REGIONS IN >-OIRECTION, S~ECIFIE5IN PARAMETER STATEMENTNUMBER 0= SUBMESH REGIONS IN Y-OIRECTION, SPECIFIEDIN PARAM5TE2 STATEMENT=3.141592654DEGREES TO RADIANS CONVERSION FACTORPLOT SCALING FACTORPROBLEM TIME2.O=PIACCUMULATED FLUIO V9LUME CHANGEVELMX NORMALIZED TO MINIMUM MESH CELL OIMENSIONLOCATION OF RIGHT-HANO SIOE OF MESHLOCATION OF LEFT-HANO SIOE OF MESHCOMPUTEO SHIFT ALONG THE PLOTTING A8SCISSA TO CENTERTHE PLOT FRAME ON FILMLOCATION OF THE TOP OF THE MESHLOCATION OF THE BOTTOM OF THE MESHCOMPUTEO SHIFT ALONG THE PLOTTING OROINATE TO CENTER THEPLOT FRAME ON FILM

NOCONc

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PRESSURE ITERATION RELAXATION FACTOR IN CELL (1.J)FRACTIONAL CELLFRACTIONAL CELLFRACTIONAL CELLCOSINE OF ANGLEMESH SPACING OFMESH SPACING OFVOLUME OF FLUIOLEVEL N+lVOLUME OF FLUIOLEVEL N

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FN(I.J) PER UNIT VOLUME OF CELL (I,u) AT TIME EPI.121EPI. 122EPI.123EPI. 124

PROSLEM IDENTIFICATION LINEFLAG OF SURFACE CELL (1.J) INDICATING THE LOCATIONOF ITS NEIGHBORING PRESSURE INTERPOLATION CELLLABEL OF VOID REGION, K > 5FLAG INO?CATING TYPE OF CELL IN COMPUTING MESHPRESSURE IN CELL (I,d) AT TIME LEVEL N+lPREsSURE INTERPOLATION FACTOR FOR CELL (I,J)PRESSURE IN CELL (Id) AT TIME LEVEL N

EPI.125EPI. 126EPI.127EPI.128EPI. 129EPI.130EPI. t31EPI.132EPI. 133EPI. 134EPI.135EPI.136EPI.137EPI. 138EPI.139EPI.140EPI. 141EPI.142EPI. 143EPI.144EPI.145EPI. 146EPI.147EPI.148EPI. 149EPI.150EPI. !51EPI. 152

NR(K)NW(I.d)P(I.J)PETA(I,J)P$i(:.d)PI?(K) PRESSURE IN Vo:o REGION NR(K)Ps(I;Ll) SURFACE PRESSURE IN CELL (Id) COMPUTEO FROM SURFACE

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EPI.153EPI.154EPI. 155EPI.156EPI.157EPI.15SEPI.159EPI.160EPI.161

135

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