· NEW YORKUNIVERSITY COUIRAMT INSTITUTE- LIBRA RY TA B LE OF CONTE NTS 1 . Introduction 2 . R...
Transcript of · NEW YORKUNIVERSITY COUIRAMT INSTITUTE- LIBRA RY TA B LE OF CONTE NTS 1 . Introduction 2 . R...
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
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
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
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_
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
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
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
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
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
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)
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_
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_
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
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
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
1 480 - 1 67 Sn iderA U T H O R
Numerica l so lution of n on
B O R R O W E R'
S N A ME
N.Y .U . Courant Institute of
Mathematica l Sciences251 Mercer St.
New York, N . Y . 10012