NEW YORKUNlVERSlTYW ANT lNS

‘lTU

Cou rant Inst itute of"3 “BMW

Mathemat ica l Sc iences

A EC Computi ng and A pp l ied Mathemat icsC en te r

Nume r ica l So lut ion ofNon l inear

BoundaryV alue ProblemsUsing Reflect ion

A . D . Sn ide r

A EC Research and Deve lopmen t Report

Mathemat icsand Comput i ngA p r i l 1 97 1

New Yo rkU n ive rsi ty

UNCLA SS IFIED NEW YORKUN‘lVBRSlTVCOUR ANT i NSTi TUTE - U BRARY

A E C Comp ut in g an d A pp lie d Mathemat ic s Cen te rCour an t In st i tute of Ma thematica l Sc ien ce s

New York Un ive rs i ty

Ma thema ti c s an d Comp utin g NY O- 1 48 0— l 67

NUME R ICA L S OLUT ION OF NONLINE A R B OUNDA RY

PROB LEMS U SING RE FLE CT ION

A . D . Sn ider

Con t r a ct No . A T (B O- l ) - 1 48 0

UNCLA SS IFIE D

NEW YORKUNIVERSITYCOUIR AMT INSTITUTE - LIBRA RY

TA B LE OF CONTE NTS

1 . Introduction

2 . R e f lection

Formulation of Free Surface P rob lems

P arametri z ation by a C on formal T rans formation

C alculation o f Conformal Maps

Re f lection

Impl i cations o f the R e flection Rules

E xi s tence , Uniquenes s , and C onvergence

Summary

3 . The V ena C ontracta

The Phy s ical Prob lem and Parameters

The Con forma l Map

A ppl ication o f the R e f lecti on Laws to the X— axi s

and the Wal l

The R e f lection Law at the Free S urface

The Separation Point

The L ine o f T runcation

The S ingularities at Infini ty

The Inte rior Points

The Fortran Program

4 . Re sults

Dependence on C onvergence Factor s

A ccuracy of the Solutions

R ate of C onvergence of the Iterati ons

E s timates of the C ontraction C oe ffic ients

- i i i

B ib l iography

F igures

A ppendix I L i s ting of the Fortran P rogram

A ppendix I I . Dimens iona l P erturbation

A BST RA CT : Re cently the so lution of s ome non linear free

boundary prob lems has been e f fected numerical ly by

incorporating a change o f independent variab le s th rough

a conformal trans fo r mation , thus s impl i fy i ng the domain of

the solution ( c f . E . B loch , On e then so lves the

trans formed di f ferentia l equation in the s impler domain ,

s imul taneous ly determining the trans formation its el f , and

the re sult i s regarded as a parametri zed form o f the de s ired

so lution . There i s s ome que s tion as to what i s the be s t

way to handle the boundary condi tions in s uch an approach .

The aforementioned report employed a s teepes t des cent

procedure to produce di fference equations at the boundary

which un iquely de termined the so lution,but led to a rather

s low iterative s cheme .

The pre sent paper di s cus s es the results o f us ing

the re f lection property of solutions of e l l ipti c parti al

di f ferential equations [ cf . 8 ] to de termine the boundary

conditions to the trans formed di f ference equations . The

bas i c idea i s analy zed theoreti cal ly,and we demons trate

i ts appl icabi li ty to a special c las s of two - dimens iona l

prob lems . The procedure i s then app lied to a s imple

_v_

plasma containment prob lem and to the two—d imens ional and

the three — dimens ional ax ia l ly symmetric vena contracta

mode l s . The results are compared with those of B loch ,

and i t i s s een that the re fle ction s cheme requi res about

one — t enth as many iterations fo r convergen ce as the method

of s teepes t des cent . New computations for the contraction

coe ff ici ent ar e pres ented .

_ V j__

P art 1 . INTRODUCT ION

The advent of the modern computer h as s t imul ated

the f inite di f ference technique of f inding approximations

to s o lutions o f partial di ffe renti al equations . None the les s

the enormous number o f cal culati ons required by th i s

te chnique strains the capab i l i tie s of even the bigges t

machines , nece s s i tating s tudies se ek ing more e f f i cient ways

of for mulating and so lving the equations . In the case o f

e ll ipti c boundary value prob l ems one i s usual ly confronted

wi th the task o f solving a large sys tem o f algeb rai c equa

tions expres s ing the fini te di f ference anal ogue s of the

partia l di f ferenti al equation and the boundary condi tions .

The s oluti on o f th i s sys tem i s ach ieved by some ite rative

procedure . Thus the e f f ectivene s s and fe as ibi l i ty o f

a f inite di f fe rence s cheme are i n d i c a ted , by f irs t , the

rate at whi ch the iterates of the tri a l s o lutions of the

di f ference equations converge to the actual s o lution ,

and s e cond , the degree to whi ch th i s s o lution of the

di f ference equations approximates the s o lution o f the

di f ferentia l equation . The latte r is a meas ure o f the

accuracy of the f inite di f ference approximati on .

The ac curacy and rate o f convergence o f many

l inear sys tems can be es timated theore ti cal ly

The appearance of nonl ineari ties , however , makes thes e

e s timates prohib i tively di f ficu lt . In fact , in the cas e

o f nonl inear boundary condi t ions , parti cularly those

involving free boun daries , i t may n ot be cl ear how to

formula t e the f inite di f feren ce approximation . In thi s

paper we des cribe a method for handl ing s ome o f thes e

nonl inear examples . A pply ing the te chnique to a free

bound ary prob lem , we are a lso able to es tab l i sh the

ac curacy and convergence o f our s cheme by expe rimenting

wi th the number of mesh points .

In the pas t , free boundary prob lems have been

attacked by iterative pro cedure s wh ich gues s the location

of the boundary , s olve the re s t o f the equations , and then

improve the gues s A more accurate and fas ter s cheme

was recently reported by Bloch He incorporates a change

of independent variab l es V i a a conformal trans formati on ,

s imp li fying the domain o f the so lution . Then he s olves the

trans formed di f ferenti al equation in the new ( known ) domain ,

S imul taneous ly so lving for the tran s formation its e l f . The

res ult i s regarded as a parametri zed form o f the des i red

solut ion . The boundary condi tions for the trans formation

are derived from a s t eepes t des cent argument , s upplemented

by equations s erving to es tab l i sh the free boundary

cons traint , whi ch in hi s cas e expre ss es continui ty of

pres s ure acros s a fluid - air interface .

In thi s paper we propos e an improvement o f thi s

procedure whereby the boundary condi tions of the trans forma

tion are obtained by analytic continuati on , re sul ting in

re f lection rules ( cf . replacing the s teepe s t des cent

and cons tant pre s sure equati ons of the above method . The

s cheme i s j us t as accura te as B loch ' s , but the i terations

converge almos t ten times fas ter . Furthermore , it i s

directly app l i cab le to any prob l em in wh i ch the s o lution

can be continued acros s th e boundary in s ome pre s c ribed

manner as des cribed by H . Lewy in for example . Thi s

report purports to demons trate that the ref lection te chnique

i s s uperior to the method of s teepes t des cent in handl in g

thes e prob lems .

In P art 2 o f thi s paper we des cribe the theory beh ind

the re f lecti on s cheme . It i s introduced in the context of a

general free s ur face prob l em of fl uid me chani cs , but the

actual pro cedure for obtaining the conforma l map i s pre sented

wi thout such motivation . Proo fs of certain res ul ts are

given when avai lab le , and evidence o f other expe cted

cons equence s i s o f fered s o that a heuri s ti c unders tanding

of the s cheme ' s impl i cati ons and i ts l imitati ons i s obtained .

In fact , the who le theory can be neatly caps uli zed , and

thi s i s done in a s ummary at the end o f the se cti on .

The appl i cation o f the pro cedure in s o lving the

vena contracta prob lem i s given in P art 3 . The phys i ca l

s i tuation i s the fol lowing : one has an in compres s ib l e

invi s cid fluid under h igh pres s ure conf i ned beh ind a plane

wal l , and the wal l has an aperture through whi ch the f luid

es cape s as a j et . In the two— dimens ional mode l th e

aper t ure i s an in fini t e ly l ong s li t ; in the th ree—dimens ional

prob lem wi th ax ial symmetry , the apert ure is a ci rcular ho le .

The boundary of the j e t i s o f cours e unknown beforehand ,

so we have to deal wi th a free boundary prob lem .

To determine the vena contracta one mus t modi fy

the general procedure of P art 2 somewhat , so additional s tudy

of the theoreti cal aspe cts i s neces sary . Furthe rmo re , thi s

problem is ful l o f comp li cations not re lated to re f lection ,

and each o f thes e mus t be de alt with . In parti cular , the

behavior o f the f low ne ar the edge of the aperture i s quite

s ingular , and we deve lop s ome spe ci al techniques for handl ing

th i s point . Knowing the l ocati on o f thi s edge , one can

evaluate the contraction coe f fi cient , whi ch i s the ratio

of the are a of the j e t at inf in i ty to the are a o f the

aperture . Thus we are ab le to pres ent a new , accurate me thod

for computing thi s numbe r . The de tai l s of a l l these

cons iderations are pres ented in P art 3 , whi ch concl ude s

wi th a de s cription of a Fortran program ( l is ted in A ppendix I I )

fo r the s olution o f the prob lem .

Part 4 des cribe s the res ul ts of running thi s program

on New York Univers i ty ' s CDC 6 6 00 computer . The accuracy

and e f fi ciency are reported , and i t i s shown that the method

converges ab out ten time s fas ter than the s teepes t des cent

pro cedure mentioned above . The new cal culations of the

contrac tion coe f f i cient produced by the re f lection program

are also pres ented in th is se ction . They agree extreme ly we l l

wi th thos e of Bloch [ l ] .

The evidence o f fered here in cle arly demons trate s

the superiori ty of th i s s cheme in handl ing the vena contra cta

mode l , and we are l ed to conclude that re f le ction i s the

appropri ate method for s o lving a wider cl as s o f free

boundary prob lems whi ch ari s e in f luid dynami cs and plasma

phys i cs . Furthermore , the re s ul ts tend to en courage

the use of re f lection rules wheneve r they are app li cab le

in so lving more gene ra l nonl ine ar e l l ipti c boundary value

prob lems in two independent vari ab le s .

A s a fina l note , we pres ent in A ppendix I a

r e - evaluation of a cal culation of the axi symmetri c con t r a c

tion coe f f i cient by a perturbational te chni que whi ch

motivated the pre sent s tudy .

A . Formulati on o f Free Surface Probl ems

On e of the prin cipal s our ces o f nonl inear boundary

va lue prob lems for whi ch the re f le cti on s cheme i s s uitab le

i s the free s ur face prob lem in fluid dynami cs , s o a s urvey

of the known re sul ts in thi s are a wi l l s erve as our

in troduc tion to the me thod .

On e mus t s olve a parti al di f fe renti a l equation for

th e s t ream function w in a regi on , cal led the flow region ,

o ccupied by the fluid . Thi s regi on i s bounded part ly by

f ixed wall s and obs tacles and partly by a cons tant— pres sure

medi um , s uch as the atmos phere . The lo cations of the walls

and ob s tacles are known data , but the l ocation of the free

s urface mus t be de t ermined as part o f the s olution . The

value of w i s given on al l boundaries , and s ome other

condi tion (usual ly expre s s ing the fact that the pres s ure

in the flui d mus t match the atmospheri c pre s s ure ) i s given

on the free s ur face . Thus one i s pres ented with a di f ferenti al

equat ion to s olve in an unknown region wi th , us ual ly , nonl inear

boundary condi tions .

Spe ci fi cally , in two—dimens ional poten ti al f low

problems , the components of ve loc ity in the (x , y ) p lane

are ob t ained from w through the equations

El liVx By 8x

The equation for w then expres s es the condi ti on that the

f l ow i s i rro tational

curl v Aw 0

The f low on the f ixed and free b oundaries i s paral le l

to the boundarie s , so

w cons tant

there . O n a free boundary , the pre s s ure p i s cons tant .

S ince Bernoul l i ' s law re late s pres s ure and vel oci ty

p %' V2

cons tant

the velo ci ty i s al s o cons tant the re . Thus on the free

boundarie s where the vel ocity i s given by the normal

deriva tive a w/ a n , we mus t have

cons tant

O f cours e the values of al l the se cons tan ts mus t be chosen

t o s ati s fy condi ti ons at the boundarie s , at in fini ty , e tc .

In three - dimens ional prob lems wi th axi a l s ymmetry ,

we let x denote di s tance a long the axi s o f symmetry and

y denote radi al di s tance from thi s axi s . The ve lo ci ty i s

derived from the s tre am function w according to the equati ons

x y By y 8x

I f the f low i s irro tational ,

boundarie s as be fore . This i s now expres s ed by

l 92 cons tanty a n

I t has been shown that the se prob lems have

unique s o lutions , an d furthermore that the free boundaries

are analyt i c . Thi s latte r fact wi l l be helpful in

implementing the re f le ction s cheme .

A n example of thi s k ind of problem, whi ch wi l l be

tre ated in ou r paper , i s the vena contracta . Water is

kept under pres sure in an infinite tank bounded by the

y- axi s , and al lowed to es cape through an Opening in the

wal l . The Opening i s a s l i t in the two—dimens ional case ,

and a hole in the three — dimens ional axi symmetri c case .

If we neg lect gravity , the water es capes through the

aperture in a s tream , which eventual ly looks l ike a tube

o f fluid in uni form motion (see F i g . l ) . The boundary

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c al culat ed i t to be . 59 l 3 5 . 00004 . We wi l l pres ent

another evaluation bel ow .

B . P arame tri zati on by a C onformal T rans formation

The so luti on Of free boundary prob lems can of ten

be aided by a parame tri z ation via a conformal map . On e

i s trying t o s olve an equation for w in te rms O f x and y

in an un known region . Now i f we regard the regi on as a

domain in the comp lex plane O f z x i y , and i f i t i s

s imply conne cted and not the who le plane , then the R i emann

mapping theorem te l ls us that thi s domain can be regarded

as the image , under an analyti c function 2 z (w) , O f

s omething s pe ci f i c , l ike a re ctangl e , in the w- p lane .

Furthermore we can speci fy the boundary correspondences

O f three poin ts . I f we have thi s map , we can parame tri ze

the equation for w (x , y ) to yie ld an equation for w as a

function O f u and v , where w u + i v ; th i s equation now

has to be so lved in a s imple , known domain .

The parametri z ation O f the w equation in the case

dis cus sed above i s aided by certain identi ties invo lving

the gradient Operator V and the Lapla cian A ; the se

identi ties are immedi ate cons equences O f the C auchy - R iemann

equat ions for the analy ti c function z (w) . If Vzdenote s

the gradient Operator wi th res pect to x and y whi le Vw

denotes thi s Operator wi th respe ct to u and v , then for

- 1 0_

any real twi ce - di f ferenti able functions f (x ,y ) and

we have

Vw

V

d[é s (x , y ) V

zg (x , y )

2

Aw 13—3 A

Zf (x ,y )

There fore , fo r two - dimens i onal potenti al f low

prob lems , the equation

I

toAZW( l )

be comes

I

I

0Aw

ll’ ( u l V )

The boundary value s O f w j us t be come trans ferred to the

images O f thos e boundarie s . T O faci l i tate s tudying the

cons tant - speed condit ion , let us introduce the symbo ls

n t nIZ,z and t

wto denote normal and tangentia l

W

dire ctions in the z and w pl ane s . Then by the chain rule ,

fl i t i’

fr i m i fua n a n a n a t an

Z W Z W Z

But aw/atw 0 on the pre - image o f the free boundary

be caus e w i s cons tant there . S O the cons tant s peed

condition becomes

cons tant .

— 1 1 _

S ince z (w) i s conformal and maps boundarie s to boundarie s ,

the normal to the boundary i s preserved and we can wri te

a n a n2

lanw

an2

provided the pos i tive dire ctions along the normal s are

chosen appropri ate ly ; thus we can rewrite our condition as

l cons tantw w

on the free boundary .

The ax i al ly symmetri c equati on for w i s parametri zed

j ust as e as i ly when it i s wri tten in the form

A wZ Y

I t t rans forms to

w wA wW Y

in the w— p lane . The boundary values O f ware carried ove r

as be fore , and the cons tan t speed condition becomes

l a w 8 2

y 35" l an l cons tant .

w w

S O now the prob lem for w i s s imp ler , given the

analy t i c map ; one has an equation wi th boundary condi tions

S imi lar in form to the ori gin al , but the se are to be

so lved in a known regi on .

The f ly in the Ointment here i s that one mus t a lso

find the conformal map . A n d when there i s an unknown boundary ,

_ 1 2 _

as in free s urface problems , the mapping prob lem i s

coup led wi th the prob lem for w , through the free boundary

condition ( as in lggl so on e cannot , in

general , so lve the equation and f ind the mapp ing separate ly

There fore we shal l examine the conf ormal mapping prob lem

in de tai l .

C . C al cula tion Of C onformal Maps

Let z (w) be analytic ; then , O f cours e , x and y

are harmoni c fun ctions O f u and v

Ax 0 Ay 0

Now revers ing ou r point of V iew , i f we wi sh to f ind a

conformal map , we can s tart by regarding these as

di f fe renti al equations to be s o lved for x and y ;

but then the boundary conditions mus t be de te rmined to

f ix the so lution .

F irs t Of al l , the image z x + i y of a poin t on

the boundary mus t l ie on the curve bounding the image

region . This i s a rather s ub tle requirement , however ,

be caus e it doe s not spe ci fy x and y individual ly ; at mos t

i t give s one in terms O f the othe r , and in the case Of

an unknown free boundary , it te l ls us nothing . More

condi tions are needed .

In B lo ch uti l i z ed the theory o f P l ateau ' s

prob lem to ge t the ne ce s s ary boundary conditions . A s

_ 1 3 _

de s cribed in i f one has continuous ly d i f ferentiab le

functions x ( u , v ) and y ( u , v ) de f ined on a rectangle in the

( u , v ) plane and mapping this re ctang le onto a domain bounded

by a c los ed curve , and i f on e minimi z es the D ir i chlet

Dougl as integral

( ( Vx )2

( Vy )2) du dv

sub j e ct to a three — point normal i z ation the images

O f three points on the boundary O f the re ctangle are fixed ) ,

then the re sulting function x+ i y o f the comp lex variab le

u + i v i s un ique and an alyti c . T O minimi ze the integral

under any auxi l i ary conditions , Bloch shows , one mus t

have Ax Ay 0 , and also on the boundary of the re ctangle

the normal derivative O f the ve ctor (x , y ) mus t be

perpendi cul ar to i ts tangenti al derivative . On e can expre s s

th i s condi tion as

xtxn

ytyn

0

where n and t denote normal and tan gential derivatives

respe ctive ly . The le ft—hand s ide i s the dot product O f

the ve ctors (xn

, yn) and In the cas e where the

re ctangle in the u , v- pl ane has s ides paral le l to the axe s ,

the orthogonal i ty condi ti on as serts that

x x yuyV

01I V

on al l s ides .

- l 4

To unders tand exactly what thi s l as t condi tion

s ays about the so lutions to the equati ons Ax Ay 0 ,

one must inve s tigate i t further . Fo l lowing Bloch , we

note that the functi on xuxv+y

uyV

i s harmoni c i f x and y

are , s o i f it van i shes on the boundary then i t vani she s

ins ide the re ctangle as we l l . I ts harmoni c con j ugate i s

l 2 2

5 (xv yv

xu

yul 7

H'

as dire ct cal cu lation shows , so thi s l atter functi on mus t

be cons tant in the re ctangle . Furthermore , i f th i s

cons tant i s z ero , then the map x+ i y i s con formal

be cause i t s at is f ie s the C auchy - R iemann equations .

In summary , i f the D ir i ch le t—Dougl as integral i s

minimi zed sub j e ct to a three - point cons traint , the map i s

analyti c and the cons tan t H i s zero ; and i f i t i s minimi z ed

sub j e ct to addi ti onal constraints but with H 0 , the

map i s s ti l l analyti c .

B loch actual ly s olve s the prob lem by en forcing

a four - point condition spe ci fying the images O f the co rners

of the re ctangle and ite rating h i s answer by a s teepe s t

des cent corre ction according to the fo l lowing

1 . moving (x , y ) along the boundary in such a way

as to drive xuxv+y

uyV

to z e ro ,

moving the fre e boundary its e l f in s uch a way

as to drive the pre s sure to i ts corre ct value ,

moving the (pres cribed ) pos i tion Of two O f the

_ 1 5_

corners in such a way as to drive H to zero .

Thi s s cheme i s suc ce s s ful , but i t i s ou r goa l here to

e s tab l i sh that re flec tion i s equal ly succe s s ful and much

more e f fic ient .

D . R e f lection

To de s cr ib e the us e of re fle ction in seeking a

con formal map numer ica l ly , let us cons ider a rectang le

in the w- pl ane . It i s to be mapped onto a reg ion in the

z - plane bounded by real ana lyti c curves (F i g . T O

find the map , we Obs erve that x and y are harmoni c functions

O f u and v,and we us e t hi s in a finite di f ference s cheme .

Let u s fi rs t set up a mesh in the rectang le (F i g .

If we identi fy the points by integer pairs enumerating

them in the u and v directions respectively,then for

harmonic functions we have the approximation

1 2xi , j I (x

i + l , j i - l) O l h

where h i s the me sh s i z e , as sumed uni form . O f cours e ,

a s imilar equation ho lds for y . Thi s i s the di f ference

equation approx imating the equation Ax 0 .

Th is finite di f ference analogue of the underlying

di f ferentia l equation can be written at every interior mesh

point Of ou r rectangula r grid . In s tandar d treatments o f

boundary value prob lems,e . g . the D ir i chlet p r ob l em,

' a n other

- 1 6

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those me sh po int s ad j acent to the s ide s , and x+ i y can be

de f ined at the se re flected po int s (at lea st , for small

enough h ) (F i g . Thus we can wr ite the numer ical analogue

for Laplace ' s equation at each boundary point , even at the

corner s ; each one ha s a le ft and right and an upper and

lower neighbor .

Thi s leave s u s with the problem of wr iting

equations fo r the value s O f x and y at the refl ected point s ,

that i s , the formula for the re flection rul e . TO derive thi s

rul e , we must cons ider more careful ly the exac t de scr iption

O f the analytic curve s bound ing the image region in the

z —plan e .

Let us say the real - analytic curve P i s one O f

the bound ing curve s in the z - plane , and its pre - image i s

the bottom O f the rectangl e , v 0 (F i g . The equation

for F i s O f the form

g (x , y ) 0

where g can be expanded local l y in a real power ser ie s .

Making the sub sti tutions

xz+z 2

— 52 2 1

(where a bar denote s compl ex con j ugation ) we get the

equation for F in the form

F ( z , z) 0

where F ( zl

' 22) i s complex — analytic in Z

1and 2

2becau se

F ha s a power serie s expans ion , namely , that der ived from g .

- 1 8

refl ected image O f w ref lected through the l ine v 0 , and

i f z z (w and z then we a s sert that

F ( z+

, z O

Thi s i s the ba s ic formu la which give s va lue s O f 2 at the

reflec ted points in terms O f the value s at inter ior po int s .

Before we prove thi s , let u s examine two spec ial

ca se s .

Ca se 1 . P i s a stra ight l ine , say x l .

Rewr iting thi s as x- l 0 , then as

we get

z +zz

F ( zl

, zz)

21

We know the rule for re fl ecting in thi s ca se ; it

i s ( z+

- l ) — ( z - l ) . Thi s agree s with F ( zl

, z2

Ca se 2 . P i s a c irc le , say x2

y2

1 .

Thi s i s wr itten as zz- l 0 and

z lF ( z2

Z1

The rule for ref lection i s

Zl

2

again agree ing with F ( z+

, z 0 .

_ 1 9 _

O f cours e , i f 0 i s to spe ci fy a curve ,

then Re F ( z ,z ) and Im F ( z ,

z) are not independent functions ;

i f they were,the ( complex ) condi tion F ( z ,

z) 0 would

de f ine ,as a locus

,the inters ection O f two di fferent

curves,i . e . , certain i solated points . In the two cases

cons idered above ,Im F ( z ,

z) wa s identi ca l ly zero, as i t

wi l l a lways be i f P i s derived from g in the manner

des c ribed .

Now we prove the rule for re flection . De fine

z and w as be fore.

Theorem . I f z (w) i s analytic and i f it i s de fined on both

s ide s of the l ine v 0 , mapping thi s l ine onto the ana lyti c

curve F ( z , z) 0,then 0 .

P r oof : N oticing that w+and w are re lated by w

we de fine the function

f (w) F ( Z (w)

where z (w) denotes the con j ugate O f z (w) . When w w+

,

w w and f (w) i s j us t We know F i s ana lyti c

in i ts f i rs t and s econd vari ab les . S ince z (w) i s al so

analyti c,

z (w) i s analyti c and thus f i s analyti c in w ,

by compos i tion . When w lies on the bottom O f the re ctangle ,

w u w,then 2 lies on P

, so

f (w) 0

There fore f (w) , an analyti c function O f w,vani shes on

the line v 0 ; hence i t i s identi cal ly zero . Thus

F ( z z f (w+) 0 Q . E . D .

From the above proo f , we can readi ly see that the

s ame rule ho lds for ref lection through the other s i de s o f

rectangle al so , i . e . 0 i s the general rule for

analytic continuation .

Now we have a feas ib le s cheme for cal culating

con formal maps . We wri te the numerical equivalent Of

Laplace ' s equation at each interior and boundary point ,

and we wri te 0 for each re f lected point .

The next section di s cus ses the cons equence

procedure .

E . Impl ications O f the Re fle ction Rule s

Now we mus t reverse our point O f V iew in this sens e ;

we set up B ap l a ce' s di f ference equation a t ea ch_ i n ter i or

and boundary point O f the rectang le , and we wri te the

re f lection rule for e ach o f the ref lected points . What

can we say about the solution to thi s sy s tem?

For the moment let us as s ume that the solution

exi s ts and i s un ique for every suf f ic iently smal l mesh s i ze h ,

and that the solution con verges to a function z ( u , v ) x+ i y ,

where x and y are harmonic , as h goes to zero . These

as sumptions wi l l be di s cus sed in the next s e ction . What are

the propertie s O f 2 on the boundary?

- 2 1 _

Let us examine a parti cular po int w on the boundary0

O f the re ctang le , together with its inner neighbor w and

its re f lected neighbor w (see F i g . A s h goes to z ero ,

the three points z z0

z (wo) , z

_z (w mus t

al l converge to the s ame po int 21

. But 0 , so

F ( zl ,

zl) 0 and z

1l ies on F . A nother way O f s eeing thi s

i s to noti ce that 2+

and 2 always l ie on Oppos ite s i des

of P, 8 0 2 mus t l ie on F

. We conc lude that the l imiting1

func tion 2 maps the boun dary of the re ctangle onto the

boundary O f the region in the z—pl ane .

Now it i s ou r goal to prove that the normal and

tangential derivatives O f z are perpendi cular , i . e . ,

xnxt+y

nyt= 0 . T O thi s end we mus t s tudy the geometry of

the curve F.

F ir s t we as sume that the function F ( zl ,

z2) was

derived from a real function g (x ,y ) as des cribed earlier ,

so that F ( z , z) i s always re al . If z i s g iven as a function

Of a re al variab le 5 , z z (s) , then gg- mus t be real .

There fore

Flzs

FzzS

is real . S ince z (s) is arb itrary,

we conclude

Re F Re F Im F in other words1

Im F2 ' 1 2

’

F2( c, c) F l a w)

Now i f z ( t ) is a parametri z ation Of the curve P,

0 . Thus

- 2 2_

0 Re s tIm F

zyt

Now (xt ,yt

) i s tangent to the curve P, so ( Re F

2 , Im F2) i s

normal to the curve and - Im F2 , Re F

2) i s again tangen t t o P

If we as sociate , wi th each complex numbe r , a vector

whose x and y components a r e the real and imaginary parts

respe ctively , we can say that F2( C , f) i s normal to the

curve 0 .

We can get more information about F by applying

the ana lyti c form O f the impl ici t function theorem . If

F ( z1 ,

z2) i s analyti c in both variab le s wi th F ( a ,b ) 0

and F l ( a , b ) 0 , then F ( zl , zz

) 0 de fines 21

as an analytic

function of 22in a nei ghborhood of

2( zl—a ) c

0cl( z2—b ) c

z( zz- b )

1

We have 0 de fining z+in terms o f z

F

Furthermore , c0

0 and cP1

Let us suppos e that 2+

and 2 both approach c as h t 0

we j us t proved C was on the curve P ; so F ( C , C ) 0 .

The theorem then te l ls us that we can wr i te

F2

( E - Z) ez(E_

- E)2

F l a m( z+- c)

Us ing and adding c- z to bo th s ides ,

we ultimate ly have

- 2 3 _

- i a [F2( 2 - E\ F

2( z O ( | z_

1"1

A s the mesh s i ze h O f the rectangle becomes smal ler ,

C— z z — z

the quanti tie s —

Hand “

2H_ should g ive approximations

to the de rivative of z in the di re ction normal to the s ide

O f the re ctangle , denoted zn

. Thus i f we divide the above

equations by 2h and take the l imit , we get

F2

zn

[F22n+F2zn]

2 | F 2 |

S ince the coe f fi cient O f F2i s real , the dire ction O f the

ve ctor Zn

’i f i t i s non— zero , i s the s ame as that O f F

2 ,

i . e . normal to the curve F . ( A cas e where zni s zero

wi l l be mentioned in the next section . )

O f cours e , the derivat ive ztOf 2 a long the s ide

O f the rectangle wi l l be paral le l to the tangent to P,

so we wi l l have

xnxt y

nyt

0

for the functi on z (u , v ) along the boundary O f the rectangle .

R eason i ng as be fore for thes e harmon i c functions x and y ,

we know that the latter equation ho lds throughout the

re ctang le and that xi yi xi yi H i s cons tant there .

From what we have s aid s o far we cannot Conc lude

th at z i s ana lytic , i . e . , that H 0 . For example,the

mapping X 2u, y v s atis fi es al l the above requirements

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Diri chle t and Neumann prob lems , whose fini te di f ference

analogue s have been analy zed In the latter cas es ,

the so lutions o f the di f ference equations approach thos e

O f the di f ferential eq u a ton wi th_ a n error Of the order O (h2) ,

so we expect the s ame O f ou r s cheme al so . Numerical

experiments con firm that the convergence i s O (h2) ( see below ) .

Let us brie fly examine the three po int normali zation .

I f we choos e t o map a re ctangle onto a domain bounded by

three an alyti c curves , we would natura l ly map two ad j acent

s ides onto one O f the curves,and we would n ot be Speci fy ing

the location of the included co rner . In F igure 4 we have

labe led the corne r wc

, whi le i ts ne ighboring mesh po ints on

the top edge are denoted W1and w and on the vertical edge ,

3 :

w2and W

4. The ne ighbor ing inte rior point is c al led w and

i ts re fl ected images through the top and s ide are cal led

wil ) and wi

z),respe ctive ly . Ob serve that the two re flec tions

O f w are mapped into on e and the s ame point , by the

rule 0 , because the s ame re flection rule i s

appl ied to both s ide s o f the re ctangle . If we denote the

image of each subs cripted w by a 2 with the s ame sub s cript ,

we write Lapl ace ' s di ffe rence equations

z 1( 2 + 2 + 2 + 2l I c 3

l22 I

so 4 21- 23

4 22— 24

. There fore we can wri te

3 z - 4 z + 20 1 3

3 2c

— 4zz+ z4

The le ft and right hand s ide s are fini te di f ference

approximations O f order O (h2) to gé- and gé- a t the corne r ,

S O i f we can as su me that thes e converge to the derivative s ,

e have 2 2 there . Thus x x n dW u v u va y

uYv

'so

2 2 2 2H x

v+y

v—x

u

- ~

yu

i s ze ro at the corner,hence zero everywhere ,

and there fore z i s analyti c . We conclude that i f the

efl ect i on s cheme conve rges for the three po int normal i zed

mapping on a re ctang le , the l imi ting function i s analyti c .

Noti ce that nowhe re in the above proo fs do we as sume

that the image O f boundary mesh points l ie on the boundary O f

the image region . On ly in the limi t as h.

+ 0 can we be sure

O f thi s correspondence O f boundaries .

G . Summary

The re sults O f the preceding s e ction can be s tated

conci se ly as fol lows .

C ons ide r analytic boundary curves de s cribed

equations O f the form

F ( z , z) 0

We write Lap lace ' s di fference equation for x and y in a

region in the w- pl ane ; the di fference eq uations are wr i tten

for each interio r and boundary point . Then we wri te

fo r the re flected points . Thi s procedure cons titute s the

re fle ction s cheme fo r con formal mapping prob lems .

When the w- region is a rectangle and the image i s

bounded by four analy ti c curves,each corre sponding to a

s ide Of the re ctangle,our numeri cal experiments lead us

to con j e cture the fol lowing theorem .

Theorem . The so lutions x and yhO f the equations O f the

h

re flection s cheme converge to ha rmoni c functions x ( u , v )

and y ( u ,v ) as the mesh s i ze h is dimini shed . The rate

O f converg ence i s O (h2) .

Furthe rmore , wheneve r the s olutions and thei r

di f ference quotients converge , we have proved the fol lowing .

Theorem . Unde r the convergence as sumptions s tated , the

function z (w) maps the s ides O f the rectang le onto the

boundary curve s , and on these s ide s the derivatives O f z

in the normal and tangenti a l dire ctions are perpendi cul ar

to each other .

A s we mentioned , i t fol lows immediate ly that

q Vyuyv

0

in the rectangle , and there fore

2 2

u yu

cons tant2 2

xv

+ yV

— x

E xpe rimentation Shows that the value O f th is

cons tant depends on the locations Of the curves in the

Z- pl ane , and the cons tant can be made zero by choos ing

—2 8

the locations appropri ate ly . Thi s condition wi l l then

make z (w) ana lytic .

The above procedure seems to b e mos t suitab le for

appl ications of the method O f re flection when the curves

in the z- p lane are ana lytic and known . However , as we

proceed in Part 3 it wi l l be come clear that neither O f

thes e conditi ons i s e s senti al , and on e can Often adopt

the te chnique to les s res tri ctive case s .

P art 3 . THE V ENA CONT RA CT A

The re flection technique w il l now be employed to

so lve the two and three —dimens i onal vena contracta prob lems .

This se ction de s cribe s the de tai l s of the procedure .

A . The Phys i cal P rob lem and P arameters

A s des cribed earl ie r, fo r the phys i cal s ituation in

the vena contracta , on e has an i n fi n i te \KflJmE E O f incompres s ib le

f luid unde r pres sure behind an in finite plane wal l ; the wal l

has an aperture which i s e i ther a S l it ( two dimens ions ) or

a ci rcular ho le ( three dimens ions ) . A j et of fluid es cape s

from the aperture and i ts cros s —se ct i on contracts as it moves

away from the wall ; asymptotica lly,it become s a tube O f f luid

in uni form motion .

In e i ther cas e, on e needs only t o examine the f low

in an (x ,y ) - plane . For the infini te S li t, on e has t r an sl a

t i on a l symmetry in the z—dire ction ; for the ci rcular hole ,

one has rotational symmetry about the x- axi s .

The motion i s de s cribed by the s tream function

whi ch gives the x and y components o f ve loci ty by

8 mVx fi / y

8 mV

y a—fi/ y

where m 0 in two dimens ions and m l in three dimens ions .

The equation determining wi s

m a wA llY B?

3 0

Let the y- ax i s be the wal l , wi th the j e t i s s uing

to the right and the aperture extending from y YOto

y - Y Then the motion i s symmetri c about the x— axi s

and we can mere ly cons ider the prob lem in the uppe r hal f

pl ane .

S ince the x- axi s i s a s treaml ine , we can set w 0

there arbi trari ly . The wa l l and the free surface al s o form

on e s treamline , so w wi l l be a cons tant , mo, there . The

fluid behind the wal l i s un de r a pres sur e p at infin ity ,

and atmospheri c pre s s ure in front o f the wall i s a cons tant ,

po. The free s treaml ine approache s a hori zontal line , y Y

asymptotical ly . The speed O f the flow on the sur face mus t

be a cons tan t , U0 , by Bernoul l i ' s law , so i f n denote s the

dire ction normal to the s tre amline , we have

3 11)275

- 00

in two d i mens ions,and

l a wUy FH

' _

0

in three dimens ions on the free s urface .

Thes e cons tants are re lated , o f cours e . The speed

U0depends , via Be rnoul li ' s l aw , on the di fference p m po

.

The value mo de termine s the rate a t whi ch fluid e s capes in

the j e t , so i t depends on the other pa r ameters al so .

T O Obtain a cons i s tent set Of value s , we Ob s erve the fo l lowing .

_3 l

Fi rs t O f al l , the behavio r O f mat infini ty in the

j et i s S imple . In (m+ 2 ) dimens ions ,

for cons tants d and B . Th is fol lows from the di f ferential

equation and the fact that w wi l l ul timate ly n ot depend on x .

A l s o , fo r laro e x

B

S ince w appro aches mo when y Y and x increas es , we have

There fore wocan be cal culated from Y and ( p m

-po) .

Se condly , i t i s cle ar on phy s ica l grounds that

Speci fying YO , p m ,

and p 0complete ly de te rmines the solution ;

the se a r e pre ci s ely the variab le s which the experimenter can

contro l . Furthe rmore , only the di fference p oo—p 0 can be

re levant ; an equal incre as e in pre s s ure on both S ide s of

the wall canno t change the flow O f an incompres s ib le f luid .

_ 3 2_

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B . The C onformal Map

A s de s cribed in P art 1 , the prob lem wi l l be attacked

us ing a conformal map . Theoreti cal ly , we would l ike to map an

in finite re ctang le onto the f low region . The part O f the

re ctang le extending to inf ini ty would correspond to the jet ,

the top O f the rectangle would repres ent the free boundary ,

and the bottom would repre sent the x—axi s . The remaining

S ide would be mapped onto the fixed wal l , and a s imple pole

at the lower corner would map thi s corner to in fini ty i n the

se cond quadrant . These corre spondence s are depicted in

Figure 5 , where A B E i s the in f in ite rectang le .

In practi ce , we wi l l approximate thi s s ituation

with a fin i te re ctangle , trun cating A B E wi th the vertical

s ide CD . The boun dary corre spondences wi l l now be de fined

pre ci se ly .

The corner A of the rectangle , at the orig i n O f the

w—pl ane , wi l l be mapped to in fini ty in the second quadrant .

Thi s requi re s a pole in the trans formation z (w) , S O near

w 0 we have

z (w) W R/w

wi th a negative res idue R whi ch mus t be cal culated . The

s ide A B wi l l be mapped onto the wal l from y m to y YO

.

The s ide B C i s mapped onto the free boun dary F . 8 0 the

angle at the corner B i s expanded from fl/Z to W ; thi s i s done

- 3 4

because ne ar the corner w has an expans ion in ( z

and the doub l ing of the ang le whi ch we wi l l ge t keeps

the truncation error down to O (h2) ; see [ 1 ] for detai l s .

The S ide A D i s mapped on to the x~ ax i s.

The image o f the l ine CD i s a curve C 'D ' whi ch

becomes asymptotical ly a s traight segment as the rectang le

i s lengthened . We truncate th i s behavior an d as sume C ' D'

i s a s traight l ine , but i ts location mus t s ti l l be determin ed

(F i g .

The treatment O f a l l the boundary con d i tions aroun d

the rectan g le i s a compl icated af fair , and we wi l l devote

the next few s e ctions to thi s matter .

C . A ppl ication Of the Re f le ction Laws to the x- a xis

and the Wa l l .

The s ides A D and A B are each to be mapped onto kn own

curve s , so the re fl ection te chn ique i s appropriate here .

A D i s mapped to the x—axis,whos e equation i s z - z 0 .

Indicatin g images of interior points ad j acent to A D by x an d

y_ , and the ir re fle ctions by x+and y+

(F i g . we derive

x+

x_

and y+—y

A B i s mapped into the y—axi s , and we have a simi l ar

s i tuation . The re f le ction rule become s

y+y and x

+

- 3 5

A rguing as in Part 1 ,we conc lude that these equations

wi l l demand that the boundaries do corres pond and that

Xuxv

yuyV

0

along A D and A B .

D . The Re f lection L aw at the Free Sur face

B C i s to be mapped onto the free boundary P, but the

s ituat i on i s more compl i cated than that treated in Part 1 .

The location O f the free boundary i s unknown , and we have

n o s imple eq uation l ike O to work with . However ,

we can de rive a re flection rule from the integral equation

for w in two dimens ions, and we wi l l modi fy i t to make i t

suitab le in three dimens i ons . Then we wi ll have to s tudy

the impl i cations O f s uch a procedure al l over again .

To ob tain the integral equation fo r m, we aga i n

employ the impl i ci t fu ncti on theorem as in Part 2 to rewri te

the equation F ( z , z) 0 in the form

2 g ( Z )

where g i s an analyti c function . Wi th thi s formul ation ,

Garabedian has es tab li s hed in [ 3 ] that the s tre am function

fo r gene r a i zed ax ia l ly symmetri c flow in (m+2 ) dimens ions

i s given by

llj (x r y ;m) Im

4 hI

Q" ? 1 _( Z - t l (5

d t ,

( E- t ) ( z

- 3 6

where 21i s any point on the free s ur face F

, and

the hypergeome tri c func tion

mm(“ r t

-

7

The integrand on the

of z, i s e s sentia lly

2

Z ' i ' z Z—Z

¢( X r Y ) W( —r

_

2I)

and th i s equation i s j us t the complex form O f

Aw “lg 0

9

file

l

l 0

which i s the usual equation for the s tre am function .

The integrand , cons idered as a fun ction of t ,i s analyti c

S O the choi ce O f the path O f integration i s immater i al .

F ina l ly , we add that the normali z ation for win thi s

equation di f fers from ours , where w 1 on P,but the

addi tiv e con s tan t wi l l drop ou t in our app l i cation .

The equation i s parti cularly us e ful in two dimens ions,

m 0 ; it be comes

w (x ,y ) Im 9

'

( t ) dt

In two dimens ions , w i s harmonic . Thus i t can be comb ined

_ 3 7 _

r fi - yz

right- hand S ide , con s idered as a function

the R i emann function O f the equation

m(a w a w 0

2 ( z - E)52

az

wi th i ts harmoni c con j ugate to give an analyti c

func tion g o i w . The above formula immediate ly

di splays th i s property,s ince the righ t—hand s ide i s

an alyti c , and

i w dt

Di f ferenti ating , we see that

g ( z ) ( c ' )2d z

for some 22

. S taying with the two—dimens i onal case ,

we employ ou r usua l notation : w wo,

and w+denote

interior , boun dary,and re flected me sh points i n the

rectangle , whi le z 20 '

and 2+

are thei r images (F i g .

The free boundary equation s ays

20 g ( zo)

0

and the re f le ction law 0 takes the form

z+ g ( z 0

Sub s ti tuting for g ( z ) and subtracting , we have the refl ection

rule for the two—dimens ional free boundary prob lem ,

2

z 20 g ( Z ) -

g ( zo)

I t i s n ow ou r tas k to approximate this rule so that i t can

be us ed in ou r finite di f ference s cheme .

— 3 8

We can integrate the right - hand S ide by emp loying

the power s eries deve lopment O f C in z,and that Of z in w

E xpanding c' ( z ) about z

0and us ing the notation c

‘( zo)

we have

'2 '

2 l u 2C C

02 C

0C0( z 2

0) O ( l z z

ol

By the chain rule ,

w ) ldw0 z w

z0=z (w

0)

l a¢ a w lr (as

lW )

z0

s i nce ¢vmu

0 on the top edge O f the rectang le ,

C0 w

V/ZO

S imi larly we find

c

"

L. {l ’mfi

l ‘l’vv

‘l’vzo

—r0

d z z0

Z0

z0 26

where we use the C auchy—R iemann equations to e l iminate

Noti ce that s ince w i s harmonic ,

lllvv

w

and the latt er i s, again , zero along top edge O f the

re ctang le . Now we use the expans ion z in terms of w

to de rive

_ 3 9_

z — z 2 h2

z

' d zl

0 0O (h

z)

0 an W0

1 h 2 1 5

in terms O f the mesh S i ze h . When al l of this i s sub s ti tuted

into the integra l,the appe arance O f 2

0cance ls and we get

h2w2+ i h

3wkv 3

z z0

O (h

0

From thi s approximate equation we formul ate our

re flect i on rule as

2 2 3—h w+ ih w w

z z v v uv

0Z 2

0

The W— derivatives can be approximated by fini te di f ferences

accurate ly S O that the trun cation error on the right hand

s ide i s O f order h2higher than the le ft- hand s ide itse l f ,

which i s, O f cour se , O (h ) . Heuri s t i ca l ly thi s seems des irab le

be caus e we are us in g a di f fe rence equation whi ch approa ches

Lapla ce ' s equati on t o O (hz) ,

and we should try to match thi s

accuracy in al l ou r othe r formulae .

Howeve r,to r e a l ly evaluate the e f fec t O f thi s rule ,

we mus t s tu dy i ts cons equences as we did for the exact rule s

in P art 2 . We as sume we have a s olution O f Laplace ' s di f fer

ence equation de fined on the interior and boundary mesh po ints .

A t the points z+whi ch are re fle cted through T

,we have

- bzw+ i h

3wkv

_‘ zo

N

<:

N

I f we multiply by ( 2 —z0) , divide by h

2,and take the re al

part , we get

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Thi s ve ri f ies the fir s t con sequence O f the rule .

S econd , the cons tan t speed condi tion mus t hold al l

a long the top edge O f the rectang le , so we can di f ferentiate

wi th re spect to u . E xpres s ing the condition as

22v v

llJV

we h ave

uvzv

zvzu v

zwkv

If z (w) i s analyti c, z

u

— i zV

and the le ft—hand s ide can

ultimate l y be written as 2 Im ( zvzvv) . Thus we have

Im zvzvv

lllvll’uv

which i s j us t the s econd cons equence o f the rule . We

might add that , had the re flection law no t kept the h3term

with wkv'we would be impos ing the condi tion

2 z 0vv v

Im

along P, whi ch would be wrong .

A lthough we have shown that thes e two cons equences

a r e cons i s tent with ou r s olution,they do not appear to be

as s trong as t he ones generated by appl i cation Of the exact

re f lection laws ; name ly, 2 maps boundary to boundary and

xuxv+ y

uyV

0 . In fact numeri ca l experiments on a very

S imple plasma- containment probl em indi cated that the

appl i cation O f the inexa ct rules doe s n ot de termine a

unique s olu t ion to the di f ference equations,as does the

_4 2_

exact formulation (with four point normal i z ation ) .

A l so some of the so lutions cal cul ated with the inexact

rule a llowed the derivative s zu

and 2V

to be come

non ~ orthogonal, S O that x

uxv+y

uyVwas not zero .

Further experimentation led to the s uspi cion that

the latter fault was the cruci al de fect O f the procedure .

The di f fi culty i s reso lved by modi fying the re flection rule

in s uch a way as to en force the orthogonal i ty Of thes e

derivatives . On e can accompl i sh thi s by adding an

appropriate function O f the derivatives . The new term

should have the fol lowing properti es

( 1 ) it should caus e z+

to be Shi fted in a di re ction

S O as to make zu

and 2Vmore nearly perpendi cular ,

and the amount of the shi ft should be greater

when the n on —orthogonal i ty, as me asured by

xuxV

yuyv

, i s greate r .

( 2 ) it should be zero when xuxv+ y

uyVi s zero , so

that unde r thi s condition the term has no e f fec t

and the two con sequences o f the re f lection rule

wi l l ho ld .

( 3 ) it mus t not caus e ins tab i l i ty , or e ls e the

ite rate s wi l l s t i l l diverge .

The f i rs t two cons ide ration s immediate ly s ugge s t t hat

the term be proportiona l to xuxv+ y whi ch , O f cours e ,

uyv

depends on the cos ine O f the angle between zuand s

v

T O dete rmine the direction in whi ch z+

Should be

sh i fted , we re fer to F i g . 6 , where zL

, zo,

and zRdenote

the image s Of three cons e cutive mesh points on the boundary ,

2 and 2+denote interior and re fle cted points as us ual .

zR

— zL

z+- z

Then zuand z

Vcorrespond to —

§H- and —

7Hf- re spective ly .

When the ang le between zuand 2

Vis les s than n/2 , as

Shown ,z+

should be shi fted in a di re ction 9 0 ° counter

clockwi se from the di recti on O f z+- z

_ ,that i s, in the

di re ction Of i ( z+

When the enclosed angle i s greate r

than n/2 , on e Sh i fts z in the Oppos i te di re ction ; the

s ign change i s exactly corre lated with that of xuxv+ y

uyv

.

U l timate ly on e s ee s that the s imp les t way of

introducing thi s sh i ft into the re f lection rule given above

i s through a term O f the form

1 (Xuxv+y

uyv)

Z — 20

Here we have approximated the di recti on O f z - z_by that of

zO- z

_because we don ' t want 2 to appear in the right—hand

s ide O f the re flection rule ; als o we put this in the

denominator to match the other te rms in the formula .

Furthermore ,we have taken account Of the comp lex con j ugation

Of z+

in the law .

Geometrical ly it i s c lear that the shi f t in 2+

should be of higher order than z+

— zo;

otherwi s e we would

expe ct that this orthogonal i ty- correcting procedure would

be un s tab le . In addition , the other te rm in the re flection

rule i s O f order O (h ) O ( | z+ and i f the shi f t i s

n ot O f higher orde r,it may become the dominant term and

Ob s cure the free -boundary condition . Re cal l that th i s

te rm al so appears to lowest orde r in the s teepe s t des cent

equations , and that ou r goal i s to produce a fas ter

convergence proces s . Final ly , the appearance O f a tan gential

derivative ( Zn) in the formulation should make us wary of

ins tab i l ity unles s the term i s O f hi gher order . Thes e

cons iderations lead us to the convi ction that the correction ,

as formulated , should be mul tipl ied by h3, and the modi fied

re flection ru le n ow re ads

2 2 3-h llI

v

' l l h‘ lllvllju v 3

z z Ah0 z

_-z0

z_- z0

where l i s a cons tant to be cho s en experimental ly .

The value O f the cons tant A whi ch gives good res ults

for al l the prob lems cons idered and al l mesh s i zes s eems to

be about 50 . However ,experimentation has revealed that thi s

type O f re f lection law , coupled with an Over r e l axa t i on

te chnique to s olve the equations, i s not very s ens itive to

the particu lar value O f A used . E vident ly the correction

Succes s ful ly achieve s ou r goal , i . e . ,the enforcement O f

orthogona li ty O f zuand Z

v' so tha t the solution i s

independent of A .

Ga r abed i an 's b as i c integra l formula quoted at the

beginning o f thi s s ection from re ference [ 3 ] also provide s

a re flection rule for the axisymmetri c cas e , in theory ,

but i t i s much more di f fi cult to apply the formula in a

fini te —di f ference s cheme be cause w i s n ot harmoni c and

the integral i s n ot analyti c . However,an Obvious modi fi ca

tion o f the two- dimens ional rule sugges ts i ts e l f .

Ins te ad Of$2

vFr?

we want

W2

1 v 11

along T ; di f ferentiating thi s wi th res pe ct to u , we al s o wan t

Wl 3 v 2

These wil l be consequences O f the fo l lowing modi fi cation

ou r two— dimens ional re fle ction rule

3 wv 2

2

h

llJV

+3

_

2'

T i l—

J y

H

0

where we again write in a se l f—a nnihi lating term whi ch

urges orthogonali ty . Reas oning in an exactly analogous

manne r , we see that thi s rule impl i es the condi tions

s tated above .

We point ou t here that we have j us t arrived at a

re flection rule by way of a heuri s ti c reas oning proces s

- 4 6

quite di fferent from the procedure outl ined in P art 2 .

We do not know the location o f T, i . e . we do n ot know

and we have not invoked the ana l ytici ty o f the

curve,ye t we are ab le to us e re f lection to expre s s the

boundary conditions . Thi s demons trate s the pos s ib i l i ty

O f apply ing the method to a very general c las s O f

prob lems .

We conc lude that us ing the above approximations

re f lection rule s in the vena contracta prob lem

insure the fo l lowing three conditions

2 2TV 1 w

v

$2

1 3 2 l 3 v( 2 ) - Im ( z

vvzv)

7'

Ffi

l (wv) or — Im ( z

vvzv)

;7.

( 3 ) xuxV

yuyv

0

C ondi tion ( 3 ) i s es s enti a l for un iquene s s for the

di f ference equations and for analyticity ; as shown in

Part 1 , i f it ho lds on the boundary it holds everywhere ,

2 2 2

V

- xu

—yu) /2 mus tand there fore i ts harmonic con j ugate (x$+y

be a constant , which wi l l be z ero when z (w) i s analyti c .

E quation ( 1 ) expres ses the cons tant pres sure condi tion

when z (w) i s analytic , and ( 2 ) i s j us t a cons equence O f

( l ) and analytic ity .

C lear ly , us ing a fini te di f ference approximation

to the re f l ection rule requires more care than the exact

formula tion . F i rs t ,we can only conclude that ( 3 ) holds

i f our solution i s insens itive to the value O f A ; thi s

mus t be te s ted . Second ,we have n o direct proo f that 2

wil l map boundary to boundary ; our argument for this i s

b as ed on the Ob servation that the s o lution to the

di f ference equati on s seems to b e unique , as indicated by

the fact that the ove r r e l axed iterative method O f so lution

does converge , whi le we would expect i t to os c i l late i f

there were more than on e solution . A n d s ince ou r equations

are certainly cons i stent , our solution mus t be the right

on e , wi th the right boundary correspondence s . Third , we

have the prob lem , inherent in any re f lection s cheme us ing

2 2 2—x -

y need n ot be zero ;v u ufour point normal i z ation , tha t x3+ythi s wi l l be taken care of in s ection F .

T o summari z e , Ga r abed i an ' s integral formula

m/2Im ( z - t ) ( z

( E- t ) ( z

provide s a re f le cti on law along the free boundary in both

the pl ane and axisymmetric cas es . In two dimens ions , the

law can be wr itten s imply as

z 20 g ( z g ( zo)

but in three dimens ions it is much more compl icated .

- 4 8

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and w2and W

4re flect w

1and w

3through B C . The z - image s

O f thes e po ints,carrying corres ponding sub s cripts , are

also shown . The separation point i s z z l ies above 200

'1

on the Y - axi s,

26i s below 2

0on the free boundary P

,

and Z3lies ins ide the flow region .

The mapping z (w) doub le s the angl es at the corner

wo

, so points which l ie 1 80 ° apart in the w- pl ane become

3 60 ° apart, o r nearly coincident

,in the z—p lane , For

example , w and i ts re fl ection w2lie on Oppos i te s ides of W

01

on the v - axi s,so 2 and z l ie on the s ame s ide of 2

0on

1 2

the y- ax i s . S imi larly the points 2

4and 2

5 ,and 2

6and z

7

coale s ce .

The rule for re flecting through the y- axi s

i s exact and can be appl ied accurate ly at th is corner ,

y ie lding 25and Z

7from 2

3and Z

6. But ou r approximation

for the rule for re flecting through I becomes us eles s ;

wv

and z

'

,which appeared in the numerator and denominator ,

respect ively , are both zero here,be caus e w E 1 on A B and

z (w) has a branch point at B . A more accurate rule mus t be

derived for re f lec ting 2 through 2l 0

‘

T here fore we go back to the integral repre s entation

in two dimens ions

2 z2

A s we indi cated above, C has an expans ion in powers O f

- 50

More expl ici tly,

l /2£ 0

Al( z— z

0) A

z( z— z

o) A

3( z - z

o)

where Ali s zero ( cf . Thi s g ive s

c' ( z ) c

'( zo) O ( l z - z

oll /Z

)

so we can write

222

z0

C0( 21

zo) O ( l z 1 Z

ol

O O 0 O

S ince z i s the s eparation po int , z 0 .

0 0

zl- z0i s O (h

z) and the error here i s O (h

3) , whi l e

le ft- hand s ide i s O (hz) . A lso we have

C0 dz ¢

xl lll

x

but analyticity requi re s

¢xwy

0

on A B, S O

QO

- l ll’x

and the integra l repres entation yi elds

2 z ( zl-

zo) m i O (h3)

2 0

In order to use this as a re f le ction rule we

find a suff i cient ly accurate approximation for wx

at

We expand waround z0to ge t w (x 3 , y 3 ) $ 3 ,

namely

ll'3 l! o

S ince w 1 on the y— axi s , m

yvani shes and the above

e s timates yield

3$ 3 wo ll»l

X( X3- XO) O (h

giving the approximation

W —wx x

3- x

0O (h

3 0

Us ing thi s in the approximate equation fo r the

re f lected point 22

z (w2) ,

we Obtain

ll“3“l’o 2 3

z z ( zl— zo) (

x3_xo

) O (h

There fore we take as ou r re flection law at the corner B ,

ll’ 3“l’o 2

22

20

( zo

‘ zil

Let u s examine the consequence s O f th is rule .

S ince ( zz— zo) /h and ( z

O- zl) /h are approximations fo r az/av

at wo, i f we as sume convergence of the s cheme and take

l imi t s as h 0 we get

2z zv v ll’

x

can conclude e i ther

Im z 0 and w

S O xv

0 and e ither conc lus ion leads to zv=0 .

_52

With thi s in mind we divide by h2/2 and take the

l imit . We get z Now we can conc lude Re 2 0vv

(whi ch i s no s urpr i s e ) and e ither Im z 0 al so or mi 1 .

In the former case , we have 2W

0 and we divide by h3/3 ! ,

e tc . , etc . , unti l we eventua l ly get a non— zero derivative

at which point we wi l l final ly conclude ti 1 .

There fore we see that the cons equences o f this

re f le ct i on rule are

The firs t condition i s clear ly cons istent .

S ince wxi s the norma l derivative O f w in the z- p lane at

the separation point , the s econd equation i s the cons tant

Speed condi tion , and thus i t i s cons i s tent too .

Summari z ing , we have shown that in the two—dimen s ional

cas e the re flection rule for Obtaining 22

z (wz) from 2

1

should n ot be the same rule as was us ed a long B C ; ins tead

i t Should be cal culated from

ll’ 3“l’o

x3

‘Xo

2

2z0

( zO— zl)

In three dimens ions , thi s i s changed to

1lJ —‘lJ2.

20

( 20- 21) —

x

3

- x

°>2 £

23 0 y

E ntire ly analogous calculations Show that this l aw imp l ies

l 2Illx

Y

wh ich i s the cons t ant speed condi tion a t separation .

l

- 53

In clos ing thi s secti on we add a note about the

implementati on of thi s theory . On e de s ired e ffect o f thi s

re flecti on rul e i s to caus e the coe f fi c ient of ( zo

— zl) ,

whi ch approx imate s the speed of the f low at ZO '

to approach

unity as h goes to z ero . However , S ince the Speed i s

already forced to as s ume this value along the f r ee boundary

by the other re flection rule , on e might as sume tha t this

coe f fi cient may we ll be replaced by l in the formul a ,

merely al lowing continuity to en force the constan t Speed

condi tion . The re sults O f numer ical experiments con firm

this reas on ing , and we found that comparab le accuracy could

be achieved by the s imple rule

in both the p lanar and axi symmetric cas es .

F . The Line O f T runcation

The values Of x , y and w on the vertical l ine C D

truncating the infinite rectang le i n the w—plane wi l l approach

the ir l imiting values at infinity as the length of the

re ctangle , A D, hereafter ca l led U i s increas ed ( s ee F i g .max '

A s an approx imat i on we set

y V x xl

cons tant,

_ 54_

on CD , and solve the resul ting set O f di f ference equations

for several value s O f Um

then extrapo late ou r data forax '

UmaxT

. T O do thi s we mus t dec ide how to s e lect xl

.

Recal l that n o provi s ion has ye t been made for

2 2 2 2driving the constan t H xv+y

V

-xu

-

yu

to zero . S ince xl

i s

the only undetermined parameter le ft in the prob lem , it i s

here that we en force thi s con dition . E xperimen tation on

several mode l s has indicated that the so lution to the

di f fe rence equations i s un ique for any fixed xl , and further

that H i s a function Of x There fore it i s through our1

'

choice of xlthat we make H 0 .

If xli s too large , the s tretching in the u - di r ection

o f the map z (w) i s greater than in the v- d irection , so H

wi l l be negative ; on the other hand i f xli s too smal l H wil l

be pos i t i ve . We can use this knowledge to calculate xlwhi le

we are i teratively so lving the di f ference equations. We could

calculate H at each i te ration,then shift x

lby an amount

where u i s a positive cons tant . Thi s i s es s en tial ly the

s cheme in [ l ] .

The s cheme which we shal l use i s a s l ight variation

O f thi s whi ch i s fas ter and more s tab le . More speci f ical ly ,

H i s cal culated by finite di f ferences at every po i nt in the

w- re ctang le to the right O f u 1 , and an average i s formed .

A l though H should be cons tant once the so lution to the

di fference equations has been ach ieved , i t need not be

cons tant during the iterations , and th is averag ing makes

the procedure more s tab le . S ince we shal l exclude from

ou r average the part of the rectang le contain ing the po le

and the branch po int , the proce s s i s s tab i l i z ed even more .

We calculate 6x by the above equati on ; however ,1

we not only shi ft x by this amount , but al l the x—coordinatesl

o f the mesh po ints to the right of u l are shi fted by a

proportional amount . Thi s not on ly s tab i l i zes the s cheme ,

but als o speeds up convergence .

F inal ly , rather than apply thi s procedure at each

i teration , we wa i t an appropri ate number of i terations after

shi fting x for the e f fect to s ettle,so that the next

1

cal culation of H is more meaningful and real i s tic . Thi s

al so increas es s tab i l i ty . E xperimentation indi cates that

the optimal number of ite ration s performed between shi fts

o f X1should be about equal to the numb er o f me sh po ints

along the u —axis , or in other words Umax/

h ’

We can es timate the optimal value of the cons tant u

by the fol low ing argument . If a 4 X l rectang le i s mapped onto

a ( 4a ) X I rectangl e us ing our re flection laws , the mapping

i s given by

Thi s give s

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be handled on the computer . The me thod we use i s s imi l ar

to that in In the triangul ar reg ion o f the rectangl e

de termined by the bas e A D , the s ide A B , and the l ine

running from w i /2 to w l /2 ( F i g . we s ubtract o f f

the pole and s olve for the function

C (w) z (w)£

|

w

Th is function i s s ti l l analyti c ; thus it too can be

approximated as a s olution to Lapl ace ' s di f ference equation .

Furthermore g (w) has no s ingulari ty in the triangul ar

region . There fore the trun cation error i s bounded and

a machine s olution i s feas ib le .

C lear ly , knowing z (w) and the re s idue R i s equivalent

to knowing z (w) . I t remains fo r us to speci fy the boundar y

condi ti ons to go with the di ffe rence equation for z(w) , and

to cal culate R . We wi l l dis cus s the boundary cond i tions first .

There i s n o probl em in spec ify i ng the values of z(w)

a long the hypotenus e of the tri angle , because we have va lue s

of z (w) at the se po ints ; we s imply sub tract the va lues o f R/w.

The detai l s of the interfacing are quite s traightfor ward ,

and we omit them here ( c f .

The boundary condi tions fo r C along the two s ides

of the triang le enclos ing the corner A are derived from the

re f le ct ion laws for 2 . S ince A B maps into the y—axis

,

the law for z i s

2+

z z (w+) z (w 0

- 58

where

w+

h i v w h i V

w ith h, as us ual , denoting the me sh s i ze and v denoting

the ordinate of the mesh point .

Sub s tituting c R/w for z and no ting that R i s

rea l,we see that the re f lection rule for C i s given by

C+

i . e . ,it i s identical with the rule fo r 2 .

S imi lar cons iderations show that the law for

re f lecting C along the bottom of the triang le i s

c+

— c_= 0

The procedure for de te rmining g (w) i s now complete ly

speci fied except for the calculation o f the res idue R .

We turn our attention to thi s prob lem .

Firs t we point ou t that i f the function z (w) i s

to be analyti c , the res idue R i s de termin ed . We have

completely s peci fied the f low reg ion by s etting YOO

1

( c f . S ection A ) , and we have f ixed the images o f three

boundary points on the semi - in finite rectang le in the w- plane

( the po int at in fini ty and the two corners ) , so there i s

no more freedom le ft in the choi ce o f the conformal map .

Numer ica l experiments were performed , so lving the

Sys tem of equations de s cribed above with fixed preas s igned

value s for R to see the ef fect o f thi s parameter . It was

found that the i terate s conve rged to solutions where in the

_59

free boundary detached from the fixed wal l at s ome n on — zero

angle,rathe r than detaching smoothly as the theory predicts .

Furthermore ,in the ne ighborhood of the s eparation point

the C auchy— R iemann equatio n s were i vo l a ted . However , we

noticed that thi s angle at detachment was a monotoni c function

of the parame ter R, so that any de s ired angle could be

achieved by the proper choi ce o f the value of R .

Thi s immediately s ugges ts an iterative procedure

for cal culating the re s idue ; one s tarts wi th an initi al gues s

and then adds to th is value a correction depending on the

angle formed by the free surface and the wa ll at detachment,

unti l this angle i s ze ro . The i teration of the value of R

can proceed s imul taneous ly with the iteration o f the

di f ference equations . There are many way s of implementing

th i s s cheme . We wil l report the de tai l s of the method

whi ch seems most accurate , as j udged by i ts succe s s in

the two- dimens ional cas e .

We re fer again t o F igure 7 ,where the mesh points

along the top edge of the rectangle are called wO , W

6 , w8 ,

and W9and their z - image s are s imi larly de s ignated . The

smooth detachmen t property at the s ep aration point i s

demons trated analytical ly in the power s eries o f w along

the t O p edge , expanded at w w Denoting the ab s cis s aO

.

by u ( the ordinate i s i ) ,we have

- 6 0

a x 82x u

24

x ( u+ i ) u —

7O ( u

Bu 07

i ! 32Y u

2

y ( u+i ) ua“

o auz02

Recal l ing that x0

0 , al l firs t derivative s are zero ,

x x

we see that

x O ( u3)

2

at separation . Hence

X

Y-YO

O ( u )

so that the tangent to the free boundary i s verti cal at 20

and we have smooth detachment .

C learly the cruci al fact here i s that the fir s t

three coe f fic ients in the expans ion of x are zero . Ou r

di f fe rence s cheme drive s x0to zero through the re flection

law on A B, and

q i s driven to zero through Lapl ace‘s

di f ference equation by equating i t with - xvv

_whi ch , again ,

i s zero be cause of the re f lection law on A B . Let us then

introduce a mechani sm for null i fy ing the coe f f ici ent of u

in the expans ion . This coe f ficient wi l l contro l ou r

ite rations on the res idue R .

We there fore pos tul ate that the value o f the res idue

shal l be changed by an amount propo r tional to xuevaluated

- 6 1

at s eparation

GR n

The con s tant n i s cho sen experimental ly to achieve optimal

convergence,but once the sys tem has been solved

,the

so lution should be ins ens itive to changes in n , s ince xu

must b e zero .

The bes t formula for cal culating xu

in terms of

di f ferences involving ne ighboring mesh points i s

3 l 3( 3 x

6 7- x8 j

x9) /h O (h

Thi s i s the exact formula for the derivative of a cub i c

polynomial , s o i ts use i s mos t appropri ate here , where

we seek the dependence in the form

3x N u

near s eparation .

We can summari ze our procedure for calcul at i ng the

re s idue R as fol lows . I t was ob s erved that the cal culated

angle between the free boundary and the f ixed wal l i s a

monotonic fun ction of the value us ed for R . S ince the

theory s tate s that this angle mus t be zero,thi s property

i s used to find the correct value of R . A n accurate measure

of the deviation of thi s angle from zero i s provided by

the derivative xu

as approximated through the above

di fference formula , and thi s value i s us ed in evaluating

the corre ctive term

- 62

GR n 55

in an iterative s cheme for ca lculatin g R .

Thi s iterative s cheme proceeds along with the

i teration o f the dif ference equations for z and w,

but rather than correcting R at each cycle , we a llow

the e f fect of a correction to s ettle down be fore correcting

again . In fact , the i teration o f R i s done s imu l taneously

wi th the sh i fting of x as de s cribed i n Se ction C .

1

We add that the above proces s works s atis factori ly

for the vena contracta , and that when the i terations have

converged the C auchy- Riemann equa tions ho ld at wo

, The

cubi c approximation for xugives s igni ficantly higher

accuracy at every mesh si ze . The optimal value for n

seems to be about 3 0 .

We conc lude this s ection w i th a brie f di s cus s ion

o f the equation for w near the orig in . S ince

w 1

on A B and

w 0

on A D, this dis con t inui ty wi l l af fect the accuracy o f the

di f ference equation un les s i t i s properly handl ed .

The treatment here proceeds exactly as in

In the triangular region we sub tract off a so lution wl '

whi ch incorporate s the behavior near the origin , so that

the remainde r has continuous boundary values . In two

dimens ions,

2$1 F

arccos

and in three dimens ions ,

In three dimens ions the truncation error in the

di f ference equation for w becomes unbounded at the or i g i n ,

and thi s, too ,

i s hand led by a modi fi cation ins ide the

triangular region . The variab le s 2 and w are trans formed

to z' and w' via the Ke lvin trans formation

2' l /z

w' w/ l z l

Then the equation for has the s ame form as

that for and the truncation error i s again bounded .

Fo r detai ls see

H . The Interior Po ints

A s we s aid be fore,the equation

Ax 0

i s approximated by Laplace ' s di f ference equation

x .

1(x . x . x . x .

I l — l , j—1 i + l , j i

, j + l

wri tten at every interior mesh point . A s imi l ar equation

i s written for

Ay= 0

— 6 4

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I . THE FORT RA N P ROGRAM

In A ppendix II we l i s t the Fortran program us ed

to calculate the vena contract as des cribed above . A typi cal

run involving 9 00 ite rations on a grid of mesh points

required 4 0 minutes o f machine time . However , due to the

operating sy s tem on the CDC 6 6 00 , the same re su lt cou ld

have been achieved in about a third of th is time i f the

unknowns x X , y Y , and w P had been s tored in s ingl e

ins tead of doub le arrays . We al s o mention that when a

s imi lar large r un was tri ed on the IBM at the

Univers ity of South F lorida it fai led because o f the smal ler

word s i ze .

C are ful s tudy o f the code wi l l enab le the re ade r

to identi fy the computations de s cribed in the text , but

i t wi l l be he lp ful i f we de fine the input para meters .

M,N are the number of mesh points in the v and u

di rections respective ly

NOHA LF i s the number o f times the mesh is to be halved

IPOLE i s the location , on the v axi s, of the interface

de fining the triangle at the origin ( S ection G )

ITPS IO is the number of i terations performed on x and y,

per iterat ion on w

ITE RMA X i s the maximum number of i terations al lowed

E PS i s m , where m+2 i s the number of dimens ions

CON i s A of S ection A , P art 4

RE LA XZ,

GONVE RG

RE LA XS I are the parameters c in the re laxation

factors rI_%EH' in the computation of z and w

re spective ly

i s the number with which the res idual s are compared

to de termine con ve r gence

i s n of Secti on A ,Part 4

i s u of S e ction A , Part 4 .

Part 4 . RE SULTS

The program de s cribed above was run on New Y ork

Unive rs ity ' s C DC 6 600 computer for many tes t cas es .

Here we shal l de s cribe the results wi th regard to the

parameters,accuracy

,and convergence . We conclude with

ou r new es timates of the contraction coe fficient .

Throughout P art 4 we sha ll compare our re sults

with thos e of B loch in re ference becaus e the meas ure

of our succes s i s the deg ree to whi ch the re flection s cheme

improve s on the e f fi ciency of the method o f s teepes t

de s cent . In Section C on rate s of convergence the

superior ity o f the re flection te chnique i s clearly demons trated

by computer runs requiring about on e— tenth the number of

i terations needed fo r analogous cas es employ ing s teepes t

des cent .

A . Dependence on E xperimental Parameters

There are five parameters in our vena contracta

mode l which a r e chosen exper imental ly to achieve certain

goal s .

( 1 ) A ,

( 2 ) H r

( 3 ) n !

( 4 )

( 5) Umax '

These are

the coe f ficient o f xuxv+y

uyV

in the free

boundary re flection rule , which en forces orthogonali ty .

the coe f ficient of x3+y3—xi —yi in the equat i on

for shi fting x1 ,

the truncated end of the j et .

The goal here i s to nul l i fy x3+y$—x§eyi .

the coe f fi cient o f gg- a t s eparation , whi ch o ccurs

in the equation fo r corre cting the re s idue R in

order to achieve smooth de tachment .

the location of the hypotenus e of the triang le

bounding the region containing the origin o f the

w- plane ; in thi s region s ingular terms are

subtracted off from z (w) and w(w) . In the text

we choos e thi s l i ne to run from w i /2 to w l /2 ,

but oth er choi ce s are permis s ible .

the length of the rectang le in the w—plane .

Thi s rectangle mus t be long enough to produce

accurate approximations to the theore ti cal res ults

o f the in finite j et .

It i s clear that the rate o f convergence of the

iterate s o f the over r e l axed sys tem of equat i on s wil l be

af fe cted by the choice o f A, u ,

and r1 ,but the final so lution

mus t be insens i tive to them i f they are to be e f fective

in achieving thei r g oals ; i . e . ,the terms which they

multiply mus t go to zero . To get fas tes t convergence ,

we begin the iterations with the value s A 50 , u

n = 40 .

Once convergence i s achieved as indi cated by al l

res idual s being les s than s ome f ixed number , usual ly 1 06,

we doub le an d triple each of the numbers A , u and n to

te s t the insens itivity of the solution to them . In the

cases te s ted,th i s perturbation requi red only a few more

i te rations ( le s s than 1 0% of the number of i te rations

needed for ini tial convergence ) to reproduce the convergence

cri teria , and the value s o f the contraction coe f fic ient were

changed by no more than one un it in the fourth decimal place .

The locati on o f the inter face between the triangular

sub region at the orig in and the res t of the rectang l e should

have very l ittle e ffect on the data fo r smal l mesh si z e ,

s ince the term be ing subtracted i s an exact so lution o f the

appropriate di fferential equa tion . T e s ting thi s w i th

di f ferent cho i ces of the i nterfacing l ine produced n o change

in the fourth decima l of the contraction coe f fi ci ent nor

in the rate of convergence of th e i terate s .

The e f fect of trun cating the rectangle at uUmax

can be es timated by pe rforming the computation s for s everal

value s of Um

and extrapo lating for U m. In f act ,ax max

B loch [ 1 ] has argued that the extrapolated contraction

coe f fi cient , Cc' di f fers from the approximate va lue

Cc(Umax)

according to the asymptot ic formula

“A UmaxCc

Cc(Umax

) A e

where A an d A are cons tants . We shall exam i ne th i s

phenomenon care ful ly in Section D , but f irs t we turn

ou r attention to the accuracy and convergen ce of the method

on a re ctangle of fixed dimensions.

B . A ccuracy o f the So lutions

In th is section we shal l ve r i fy that as the mesh s i ze

h i s decreas ed , the contraction coe fficien t , which i s

repres entative of the value s o f the solutions on the mesh

points , approaches a l imit with an error varying as the

square of the mesh s ize .

We anti cipate th is kind o f behavior becaus e

throughout the program we use approximate formulas who s e

error are no bigger than O (hz) . For example

,our re fl ect ion

rule s are a l l at least th is accurate,and we write centered

di fferences o r high orde r extrapol ated di f ferences fo r

derivatives . The data pres ented herein supports ou r

contention that the accu racy i s O (hz) .

We tabu late the value s of the contraction coe f fici ent

cal cu l ated at the variou s mesh s i ze s in two dimens ions , with

U 4max

_7 1 _

C omputations show that th is fitted exce l lently

by the function

Cc(h ) . 6 1 005 . 2 1 h

2

so we conclude that the convergence i s O (hz) and the

limit of Cci s . 6 1 005 . O f cours e , th i s mus t s ti l l b e

corrected forUmax

m. For compari son , we quote B loch ' s

value for Um

4 as . 6 1 2 6 5 . Thus both values di f ferax

from the exact value . 6 1 1 01 5 by about . 001 , and the accuracy

i s comparab le .

In the axi symme tric case the fol lowing data were

computed wi th U 4

Thes e numbers are des cribed by the function

Cc(h) . 59 1 3 3 . 70 h

2

and we again confi rm that the accuracy is O (hz) .

uncorre cted l imi t here i s there for e . 59 1 3 3 .

7 2

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were reduced to 1 0- 6

. The number of i terations required

at each mesh s i z e i s pres ented in the fo llowing tab le

We mention that as h was de creas ed , the amount of over

re l axation us ed was increas ed much more rapidly than i s

pres cribed in re ference E vidently the non linearity

inherent in thi s prob lem makes an analys i s o f the optimal

re laxation factor extreme ly del i cate .

For compari son , B loch quote s i terations

required to reduce res iduals to 1 06in a cas e with

mesh points .

We conclude from thi s da ta that the re flection s cheme

requires ab out one— tenth as many i tera tions as s teepes t

de scent for the s ame degree of converg ence .

_ 7 4_

D . E s t imation of the C ontraction Coe f fi c i ents

A s mentioned i n Section A , the calculation fo r

the contraction coe f ficient should be corre cted for the

truncation of the rectang le by a formula ( cf . [ l ] )

— AUmax

Cc

Cc( Umax

) A e

Thi s rul e should be val id at any mesh s i ze , with A and A

independent o f h , at leas t when h i s sma ll enough .

We te s ted this behavior by computing Ccfor di f ferent

value s of Um

The re sults for the mesh s i ze hax

are as fol lows

where , as usual , m 0 in the planar model and m l in the

axi symmetric cas e . Thes e ob servation s fit the formula with

the fo l lowing v alue s of A and A :

_75

The va lues fo r A di f fer on ly ins igni f icantly from

those o f as we should expect . The coe f fic ient A i s o f

the oppos i te s ign , however . Th is is probab ly due to the

fact that B loch use s di fferent boundar y conditions for y

and w on the trun cating l ine CD .

To e s timate the ' con t r a ct i on coe f fi cients , we s tart

with ou r most accurate value s , computed for

U 4 h and re s idual s les s than 1 0max

We fi rs t extrapo late thes e va lue s for h 0 by the h2

formula of s ection B , then extrapo late forUmax

the above . Thi s leads to the final e s timates

I

I 0C . 6 1 1 06 mc

1Cc

. 59 1 42 m

where at leas t the fourth decimal i s s igni f i cant .

For compari son , B loch [ 1 ] quotes

CO

. 6 1 1 00 . 00002

CO

. 59 1 3 5 . 00004

and our computations con fi rm these values .

In clos ing , we add the fol lowing ob servation .

The coe f fi cient in the pl anar vena contracta i s cal culated

from the exact solution to be

so ou r es timate i s high by . 000045 . There fore a prudent

gues s for a five place value of Ccin the axi symmet ric cas e

is obtained by sub tracting thi s s ame corre ction,yie lding

Cc . 59 1 3 7 5 .

BIBLIOGRA PHY

1 . B loch , E . ,A Finite Di f feren ce Method for the Solution

Mathemati cs Center , Courant Ins t . Math . S c i . , New York

Un iv . ,NYO — l 4 80- 1 1 6 , 1 9 6 9 .

Wi ley,New York ,

1 9 6 4 .

Je ts and C avi tie s ,Paci fi c J. Math . 6

, 1 956 .

Garabedian , P . R . ,E s timation of the Re laxation Factor

for Smal l Me sh S i ze , Math . T ab le s A ids Comp . 1 0 , 1 9 56 .

Garabedi an ,P . R . and Spencer , D . C . ,

E xtremal Methods

in C av i t at i on a l F low ,J. Ra t . Mech . A n al . 1 , 1 952 .

Giese , J. H . ,On the T runcation E rror in a Numerical

J. Math . Phy s . 3 7 , 1 9 58 .

7 . G i l ba r g ,D . Jets and C avitie s

,E ncyc l . o f Phys ics 9

,

Springer—V er lag , 1 9 6 0 .

8 . Lewy ,H . , On the Re fle ction Laws o f Se cond orde r Di f fer

Bul l . A mer . Math . Soc . 6 5 , 1 9 59 .

Nebari ,Z . , C onformal Mapping , McG r aw- H i l l , New York , 1 9 52 .

S outhwel l , R . and V a i sey , G . , F luid Motions Cha racteri zed

by"Free "

S treaml ines,Phi l . T rans . Roy . Soc . London ,

Se r . A 2 40, 1 9 4 8 .

_7 7_

FR E E SU R FA C Ev

FIGU R E 1

(Plotted from computed data for the

axisymmetr ic case)

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7 0

PROGR AM REFLC TR ( INPUT . OUTPU T )INTEGER FPS

DA TA

COMMON

COMPLE X 2

RE A DCCONVERG. R ESCON . SENS

RE A D 1 . L .JUMPCA LL SECOND ( T )

PR INT 1 4 1 a T

PR INT 7 0.L

FORMA T I1 M 0 I1 0 . t L 0 )

-1 )

1 1 =N¢ 1

I2 8 M¢ 2UM A X-Nfi H

1 FORMA TCALL IN I T

2 0 PR INT

CRESCON. UMA X ; SENS . NOHALF

2 FO RMA T ( t o N IPOLE ITPS IO ITERMA XC EPS t / i H . 6 1 1 0/ t 0 CONT ROL REL A X Z RE L A XGI* . t C ONVERSE RESCCN t / i H

CO O UM A X SENS NOHA LF 0 /

C i H . Q E 1 5 . 4 . I1 0 )

IHA VE=4 0

ITER I O

IPRNT 8 500 $JPRNT0 1 SERRHAxt l ofi . SSHIFT i . 1 1 1 1 1 1 1 1

TEST 8 1 0 .

ITPS IS O

C A LL PRNTR ( NOHA LF . SHIFT )

0A B Y4=. 25¢ SR

SR 1 .-SR

SRP: 1 .-PA 8Y4

IF P A 8 Y4 3 . 25t PA 8Y4

4 ITER =IT ER¢ 1C ALL REFLEKC A LL INTRR

ISsN- i

D O 80 J' soH

80

IF ( ( ITER GO T O 7

IF GO TO 6 0

— 86

6 1

9 9

1 1

40

8 1

1 3

1 4

1 4 1

50

1 2

P A GE

CA L L PRNTRINOHALF . SMIFT )PR I N T 6 1

F ORM A T ( t ERROR HA S GROWN . t )

CA L L E X I TERRMAX T EST

IPRNT : IPRNT*500

CALL PRNTR IO. SHIFT )

IF 00 T O 9 9

CA L L PRNTR IO. SMIFT )CAL L EX I T

IF GO TO 1 1

IF GO T O 4

CA L L COMPMODISHIFT . RESCON. IHA VE oLI

00 T O 4

IF ( ( ITER GO TO 9

JPRNT 8 JPRNT t JUMP

CA L L PRNTR I1 . SHIFT )

00 T O 9

I F GO T O 4 0

CA L L COMPMODISHIFT n RESCON. IHA VE oL )

00 TO 4

CA L L PRNTR¢NOHA LF . SHIFT )

PR I N T 3 1 . SH I FTFORM A T ( ao L AST SH I F T HAS t oE i UJS )

CA L L SECONDIT )

PRIN T 1 41 0 7

IF ( NOHALF ) 1 2 . 1 2 . 1 3

CA L L HA L F ERNOHALF=NDHALF~ 1

PR IN T 1 4

FORM A T ( 0 0 MESH S IZE HAS B EEN HA LVED . 0 )

CALL SECOND( T )

PR I N T 1 4 1 oT

FORM A T ( 0 T IME IS t oF1 0 . 3 )

REA D 50

F ORM A TLI Lt L

GO TO 2 0

CALL EX ITEND

suaaour xue PRNTR tJAX.SHIFT )

ATA

IN T EGER EPS

- 87

P A GE

C OMMON

PR INT 1 . IT ER .

CSHIFT a RES

1 FO RMA Tpsx-x

fl

SHIFTt / i H -nEso a

PR INT 3 0 . CC

3 0 FORMA T ( 0 0 CONTR A CT I ON COEFF I C I ENT 0 . F1 1 . 7 I

PR INT 3 00 .

3 00 FO RMA T ( o XIEND ) l fi oF1 0 o 6 )

RETU RNEND

SU B ROU T I NE REFLEK

D A TACOMMON

I NTEGER EPS

COMPLE X Zo RHS

IPMIN2= IPOL E 2

DO 1 I' IPMIN2 . N

J-Mo iDO 2 I8 IPMINE . J

Y Ii o IIO Y I3 o I)

IPMINZ' IPMIN2 ~ 1

Jl IPMIN2-1DO 3 IP J. IPMIN?2 8

X ( I, 1 ) S XIIo S ) - RE ALIZ )

¢ A IMAGIZI2 :

XIl v I ) a ' X‘S o I ) t REALIZI- A IMA G( Z )

Jl J-i

D0 4 1 : 2 0J

_ 88_

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SU B ROU T I NE I N ITD A T AC OMMON

INT EGER EPS

COMPLE X H . ZETA . E . E2 . 8 . L . 2

M1 uM¢ 1DO 1 I' Zo N

DO 1 J’ 2 0 "1

u : CMPLXIFLOA T IIIF ( IO J‘l ) 2 9 2 0 3

2

Y‘2 0 2 ) ° 0

P ( 2 . 2 ) 8 0

GO T O 1

3 2 ETA ' 1 . 57 07 96 3 3 ' H

ZET A ' . SO ICEXP IZET A ) OE XP ( -2ETA ) )

ZETA I . 6 3 6 6 1 9 9 8 O CLOGIZETA I

P II.J) = A IMAGIZET A )

E 8 -1 . 57 07 9 6 3 3 0 2ET A

E U CEXPIE )

52-E0 50 1 .

S O ' CSORT IEP I

Gz A IMAGIS )IF S I CONJGIS )

LS CLOGI1 . ¢S )

Gt A IMAGIL )IF L l CONJGIL )

Z=ZETA ¢ . 6 3 6 6 1 9 9 8-( E-S¢L)XIIO J) 3 ' RE‘LIZI

1 CO NT INUEN1 : N. 1

YIN1 3 2 ) 3 0 o

DO 4 J=3 .M1

Y ( N1 .J) ( J-2 ) * H

IF ( E PS )

5 P ( N1 .J) I YIN1 . J)

DA

P A GE

00 T O 4

P INi oJ)‘ Y IN1 .J) flv2

CO N T IN UEPIN1 . 2 ) 8 0 .

ID' IPOLE-4

R58 1

I F GO TO 8

D O 1 0 I0 2 0 I1

D O 1 0 J' 2 0M1

CA L L FLIPZI' 1 . II )

CA L L FLIPSII-i o II )

RET URNEND

SUBRO U T I NE FLIPS IIK. ID )

TA

CO M M O NCNYI40 . 4 0I0 22 I4 0 .‘0 I0C ITPS IoSR .GA BY4 oSRP . PA9Y4 0 CONo IoJo ITPQ IGo SE NS

I N TEGER EPS

D O 3 I 3 2 . I DJc ID- 1 0 2

. ( J' E IP O E II

I F ( EPS )

P IIaJ) fi p IIoJ) 0. 50KC I1 .

' FLOAT II-2 ) INI

00 T O 3

PIIA JI=PII0JI¢ -E I/NII

CONT IN UERET U R NEND

S U BROU T I NE FLIPZIK. ID )

D A TACO M M O N

I N T EGER EPS

COMPLE X 2 . N

_9 1 _

DO 1 I=2 . ID

Jt ID- Io zw8 CMPLX<( I -2 I4 H )

a Kt RES/ U

V II.J) = A IMA G ( 7 )RETURNEND

SU BROUT I NECOMPLE X 2

C OMMON XI405 .

CHY ( 4 0 . 22 t4 0 . an ) .

C ITPS I. SR . GA BY4 . SPP . PA BY4 . SRNQ

IHA VE=IHA VE ¢ N

SOD l 0 .

DO 2 0 I’ L O N

DO 2 0 J8 3 .M-X ( I- 1 .J)

- 1 )

JI-Y II 1 .J)YV ' YII.J¢ 1 ) -V II.J 1 )

-XV )

SOD-SOD/ I4 .

-2 ) t ( N

M1=M0 1SMIFT a -SENS ' SODIF SH I F T

XI N0 1 .2 )

DO 1 00 II L. I1

DO 1 00 J' Zo fi i

Jt MiSLOPE z -FLOA T IMDl -RESCONwSLOPE

IF

RES-RES-DDsD/ R

DO 3 0 I! 2 . N

INTFC=MA XO ( 2 . IPOLE - I-1 )

DO 3 0 J=INTFC .M1

-RE A L ( Z )- A IMA G ( Z)

_ 9 2_

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P A GE

p ( 3 0 2 ) =0 p

IPOLE 3 2 0 IPOLE-6II=IPOLE ~ 4

DO 3 3 III-S o II

C ALL FL IPZI ' 1 . III )

CALL FLIPS II' 1 . III )

I1 8 N¢ 1

I2=M4 2I1 08 IPOLE ' 6

DO 7 8 9 I' 2 3 1 1 0 3 2

INTFC=IPOLE ' I- 3

DO 7 8 9 JI 3 . INTFC . 2

-1 )

7 8 9 CONT INU EI1 0I IPOLE-5

DO 9 8 7 Il 3 . I1 0 . 2

INTFC=IPOLE ' I- 3

DO 9 87 JI 2 . INTFC

C ONT INU ERETURNEND

SU B ROU T INE INTRR

D A TAC OMMON

INTEGER E PS

COMPLE X N . Z

M1 =M¢ 1E RX=0 . sERvu o. SLOC I 1

_9 4_

N2 R N¢N

ERYsA B S ID-Y I2 .M1 I)

XI2 .M1 ) 8 0 .

INTFC=IPOLE -2

D O 1 1 1 J-INTFC .M00 TO 1 000

1 01 CO N T IN UE1 1 1 CONT I N UE

LOO-2D O 1 It 3 . N

INTFCcMAXO( 2 . IPOLE

D O 1 J=INTFC .M100 To 1 000

1 1 C ONT INUE1 C ONT IN UEIF ( ITPS I)

2 IF ( EPS )

22 ERS I =0 .

00 22 1 I: 3 . N

INTFCsMAXOI3 . IPOLE-I )

D O 22 1 J-INTFC .M00 T O 2000

22 1 1 CONT IN UE2 2 1 CONT IN UE

00 TO 3

2 1 ERS I-O .

00 2 1 1 I-S . N

INTFC :MAXO ( 3 . IPOLE -I )

D O 2 1 1 J-INTFC .MNHY' Y II i JI

EgA BS ( D

IF GO TO 3 001

ERS IgEIS I-IJS IsJ

3 D01 p IIoJ) =D2 1 1 C ONT INUE3 C ALL FLIPZI- i . IPOLE-2 )

C ALL FLIPZI- i . IPOLE-3 )

LOC z 3

MA RKa IPOLE-SDO 4 I=2 .MA RK

INTFC =2 oMA RK- IDO 4 J: 2 0 1NTFC

IF 4 0 4 3 1 000

CONT INUEC ONT INU EIF ( ITPS I )

CALL FL IPZ ( 1 . Ip O l E -E )

C ALL FL IRZ ( 1 . IPOLE -3 )

ITPS I=ITPS I' 1

RETU RNC A LL FL IPS I( ' 1 . IPOLE -2 )

C ALL FL IPS I ( ' 1 . IPOLE- 3 ) $MA RKO IPOL 3 -4

ITPS I=ITPS IO

IF ( EPS )

DO 6 2 1 I8 3 .MA RKINTFC 3 3 +MA RK- IDO 6 2 1 JP 3 . INTFC

GO TO 2 000

CONT INUEC ONT INUEGO TO 7

MA RK IPOLE-2DO 6 1 1 I=2 . NA RK

INTEG S IPOLE-I

DO 6 1 1 J' 2 . INTFC

IF

RES/W

C ONT INUEMA RK =IPOL E ' 3

DO 6 1 2 Il 3 .MA RKINTFC B IPOLE- I

DO 6 1 2 JI 3 . INTFC

F ( IA J) = 2 2 IIa J) * p IIoJ)MA RK=IPOLE -4DO 6 1 3 1-3 . M A RKINTFC 3 MA RK- It 3DO 6 1 3 J’ 3 ’ INTFCNHY B NY ( I.J)

P A GE 1 1

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A PPE NDIX II . DIME NS IONA L PE RTU RBA T ION

In thi s appendix we report a correction to a

calculation des cribed in re ference

Garabedian des cribe s therein an entirely di f ferent

me thod for e stimat ing certain parameters in axi al ly symmetric

cav i t a t i on a l flows and je ts . In parti cular,the vena

contracta in three d imens ions i s treated as a perturbation

of the two—dimens ional mode l with m as the pe rturb ing

parameter ,where m+2 denotes number o f spacial dimension s ,

as usua l . Hence the te chniq ue i s know as dimensiona l

perturbation .

The results lead to the con j ecture that the ratio

o f the radius o f the jet at inf inity to the radius o f the

ape rture, Y

w/Y O , may be expre s s ed in m+2 dimens ions as a

power s er i es expanded around m 0 . Thus

Ym (m) Y

oo( 0 )

Regarding m as a parameter in the eq uations for the vena

contracta , Garabedi an i s ab le to ca lculate

veg— 1 )

and , o f cours e,

— 9 8

Furthermore ,the derivative BY

O/Bm, evaluated

at m 0 for cons t ant Y0°

l , i s expre s s ed as fol lows

BYO

2 l

m i3+a tana a

l tan- l 2 a

b da5

4n a b +yb

where

Y 2 ( 1 - a2) e

flb/Zae-flb/a

]

Thi s integral was eva luated numeri cal ly in [ 3 ] and the value

i s q uoted as

BY

m0

. 6 5054 40)

Inte rpo lating the above da ta wi th a cubi c polynomial in

6 lea ds to the exp ression

K:

. 6 1 1 0 . 4 857 <3 . 1 1 1 0 52

. 01 4 3 63

0

as an approximation to the power series. For three dimensions,

l<

6 i s l /3 and the resul t i s

Yoga )

leading to the es timate o f the contraction coe f f i cient

Ym ( 1 )

12

. 57 9 3

. 1 1 1 0 52

. 01 2 3

. 01 4 3 63

. 0005

and Garabedian arrive s at an error es timate o f l / l o per cent .

U s ing our larger CDC 6 6 00 computer we r e- evaluated

the integral for 8Y0/8m and found that a more accurate va lue

i s

. 707 2 . 0003

leading to the cub i c polynomia l

. 6 1 1 0 . 52 7 9 a . 1 1 1 0 52

. 02 7 9 53

whi ch fits the data at m l,0

,and w as be fore .

The revi s ed computation o f Ccfor m l i s then

Ym ( 1 ) 2 2

CC

[Y—

OTI

-

r] . 7 7 3 7 . 59 85 “I“ . 0002

Thi s , o f course , i s larger than the value in [ 3 ]

and s l ightly c lo s er to our evaluation,although i t s t il l

di f fers from the latter by . 007 . A glance at the magnitudes

o f the terms in the po lynomi al n ow reveal s that the error

e s timate mus t also be revi s ed upwards ,fo r

- 1 00

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1 480 - 1 67Snide r

Nume ri ca l so lut ion of nonl i nea r bounda ry va lue p r ob . m

NYO ‘ C . 2

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Numerica l so lution of n on

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