Another comparative in v e stig a tio n o f the D iscrete Methods as a group was carried out in 1966, th is time by Finlay son and/ f-, \
Scriven' « A study o f the lo c a lis e d , or gen eralised , forms o f th ese
(52)
methods was made by Taylorv /v.who was p r in c ip a lly concerned w ith the concept o f lo c a lis a tio n and i t s ap p lication to the g en era lisa tio n o f th e rem aining D iscrete Methods, which had been la r g e ly ignored. This en ligh ten ed trend i s r e fle c te d by C roll*s use o f the G eneralised C ollocation (or Hypar F in ite D ifference) method^58^(59)^ 93^3 method could be described as the C ollocation F in ite Element method and i s the technique to be adopted in th is study.
4 .3 The D iscrete Methods and F luid Flows
Since th e ir introduction to the f ie ld o f flu id dynamics, there has been a sca rcity o f so lu tio n s o f the R avier-Stokes
equations, mainly because o f th eir degree o f high order and th e ir n o n - l i n e a r i t y ^ S o m e so lu tio n s o f the lim itin g ca ses o f zero and in f in it e ly large Reynolds numbers were obtained, but i t took the advent o f the e lectro n ic d ig ita l computer to bring about
(25)
Favier—Stokes equations required such an immense ammount o f computation as to he im possible for p r a c tic a l purposes.
Probably the f i r s t attempt to in teg r a te the K avier-Stokes (61}
equations -was made hy Thom' \ As early as 1933, he managed to obtain a numerical solu tion for the wake behind a cylin d er in a uniform stream. The exhaustive computation, which he performed by hand, proved to be th e forerunner o f a number o f la te r stu d ies conducted a3ong the same lin e s . These were made by such authors as
( 62)
Kawaguti and Allen and Southwell' • They did not appear u n til the m id -n in e te e n ~ fiftie s, when the F in ite D ifference method coupled with the u se o f the elecro n ic d ig ita l computer were becoming esta b lish ed .
The U n ite D ifference method has been the most w idely used numerical technique employed for the treatm ent o f the N avier-Stokes
equations sin ce i t s introduction in to the f ie ld o f f lu id m echanics^^^ A wide v a riety o f flow con figu ration s have been tack led by a number o f d iffe r e n t authors, prominent amongst th ese being Kawaguti (1961), Sinuni (1964) and Burggraf (1966) ^ .
The o b sta cle o f numerical divergence, which had p reviou sely prevented the so lu tio n o f high Reynolds number flow s, was overcome by Runchal and Wolf ste in in 1966^ ^ \ Since that time a number o f authors asso cia ted w ith Imperial C ollege have contributed towards the development o f t h is approach regarding a number o f a sso cia ted problems. The development o f a group o f solu tion schemes, based on
the Patankar-Spalding F in ite D ifference rou tin e enabled the
(6 5) in v e stig a tio n o f a number o f flow con figu ration s' .
The replacement o f the F in ite D ifference method by the 7-Ll
F in ite Element as the most w idely used numerical technique has not y e t occurred in the f ie ld o f f lu id mechanics, as i t did in the
f ie ld o f stru ctu ral mechanics. A contributory factor to th is
situ a tio n i s that the method o f adaptation o f a technique i p r in c ip a lly developed for the so lu tio n o f problems in the f ie ld o f
stru ctu ral a n a ly sis (e s s e n tia lly a Langrangian method) to the solu tion o f problems in flu id mechanics (n atu rally described by a Eulerian approach) i s not im m ediately ob vious^ *^ . More im portant,
however, i s the fa c t that for the Navi er-Stokes equations for v isco u s flow there i s no simple v a r ia tio n a l p r in c ip le . The R itz F in ite Element method was employed in the solu tion o f problems for
for which there wore p o te n tia l fu nction s a v a ila b le ^ie; the
v a r ia tio n a l form ulation was p o ssib le ) n otably by Z ienkiew icz^ "^, Martin and dp V ries and N orrie^^^e
. E ecen tly, w ith the continuing development o f a lte r n a tiv e d isc r e te methods, a.vgrowing number o f problems in flu id mechanics have been trea ted w ith F in ite Element type methods* A Least Squares U n it e Element technique was employed by H u t t o n i n 1974*
( 66)
Galerkin F in ite Element techniques were employed by Oden'1 ' .
in 1974 and Cooke and B la n c h a r d ^ °\ Murphy and I u c h i^ 2 \ i n 1977*
The.method to be adopted in th is study i s the G eneralised C ollocation method, which may be regarde as a Eypar F in ite D ifferen ce method in which certa in F in ite Element type concepts are included*
4*4 The G eneralised C ollocation Method
The D iscrete Methods provide a means o f obta5.ning
approximate so lu tio n s to the p a r tia l d iffe r e n tia l equations which govern a given p h ysical problem* Any approximate d isc r e te technique for th e solu tion o f p a r tia l equations c o n s is ts o f the determ ination o f the unknown fu n ction s a t a f in it e number o f d isc r e te p o in ts
d istrib u ted throughout the domain o f the problem* The v a lu es o f the fu n ction s at th ese p o in ts are obtained from the so lu tio n o f a system o f algeb raic equations determined from a d iscre te rep resen ta tio n o f . t h e governing d iffe r e n tia l equations*
The manner in which th is i s accomplished by the G eneralised • C ollocation method may be broadly o u tlin ed in the fo llo w in g sta g es
F ir s t, a to p o lo g ica l id e a lisa tio n o f the domain o f the problem i s . obtained by d efin in g a number o f d isc r e te p o in ts, or nodes, in a glob al mesh* Each node has a se t o f gen eralised co -o rd in a tes
a sso cia ted with i t r e la tin g to the unknown parameters o f th e problem* Second, the nodal mesh i s considered to c o n sist o f s e ts o f lo c a lly r e la te d nodes* Each set has an independent approximate lo c a l fu n c tio n , or fu n ctio n s, appertaining to the elementary nodal se t and the
a sso cia ted gen eralised co-ordinates* These s e ts and th e ir immediate environs are termed f in it e elem ents,the elem ents being in te r
connected as required by ..the p a rticu la r problem d efin itio n * Third, the approximating fu n ctio n s, in terms o f the lo c a l g en era lise d •-co-ord in ates, are subjected to the C ollocation method c r ite r io n ,
which incorporates the governing f ie ld equations and the d isc r e te p r in c ip le , to produce an algeb raic lo c a l f ie ld analogue or analogues* F in a lly ; the se t o f glob al algeb raic equations i s obtained by the . appropriate co-ord inate transform ation and the a p p lica tio n o f
boundary conditions,, •
The mathematics o f the process may be d elin ea ted by con sid erin g the governing f ie ld equationj or equations* to be o f the form ;
Iiu f c *«<•«. (4® 1)
togeth er with the kinem atic boundary con d ition s $
3) u = 0 o ( 4 .2 )
and the s tr e s s boundary conditions* which may or may not e x is t •
3>nU ” o o . o c o o ( 4« 3)
where *L* end. *£* are d iffe r e n tia l operators and , f^* and , ^ i f are p rescrib ed sp a tia l fu n ction s w ith hi* and * e ! as se t su b scrip ts.
An approximate d isc r e te function* in the form o f a truncated series* may be p o stu la ted for the gen era lised co -o rd in a tes ;
- _ r M
u - >i-k k
y . . . ( 4 .4 ;f, ,\
j=0 J
w ith resp ect to two-dim ensional problems, where * a . f are *aH-l* k k
unknown c o - e f f ic ie n t s , *xJ "y * i s one o f the 'nrf-11 lin e a r ly independant terms in *x! and *y* v?ith k ^ j* and im* f in it e . : , • I n i t i a l l y , *x* and *y! can be taken as v a lid through the e n tir e
f ie ld (ie ; o v era ll fu n ctio n s).
The transformed version o f equation (4*4) may be w ritten $
Z
m-flu . f . (x ,y ) . . . ( 4. 5)
i= l
1j3i
the corresponding 1 shape fu n ction *v''"‘V* The ob ject o f .the. D iscrete Methods i s to approximate the so lu tio n o f 'u* by means o f the
su b stitu tio n o f equation ( 4*5) in to the r e sp e c tiv e method c r ite r io n $ the la tt e r in v o lv in g the c la s s ic a l d e fin itio n and the d isc r e te
p r in c ip le . The method c r ite r io n adopted by C ollocation i s th at the inherent error i s se t to zero at a f in it e number o f p o in ts throughout the domain, as opposed to being zero over the whole domain as in the c la s s ic a l approach. That i s $
h i \ ± ~ f (fo r u^) . . . ( 4*6)
Jbuj.. — g^ (fo r u..) . . . ( 4»7)
where *i* and *3 * ty p ify in ter n a l and boundary nodes r e s p e c tiv e ly . G en eralisation , or lo c a lis a tio n , takes the form %
JjuJ. = f (fo r u^) . . . (4*8)
^ (fo r (4«9)
whore se t n o ta tio n a l su b scrip ts imply th a t *i* and * jT are in sub-regions ’k 1 and *h* r e sp e c tiv e ly . The terms and *y* which appear in the a sso cia ted exp ression s (represented by equations —(4 , 4 ) -and (4 *5)) now r e la t e s to the appropriate element s u b -fie ld .
■Global d e fin itio n i s now achieved by the co-ord inate transform ation o f the lo c a l elem ents with resp ect to the domain o f the solution*
Although the p oin tw ise manner o f d ealin g w ith, the in h eren t error i s somewhat crude, the method has the dual advantago o f
r e la t iv e ly stra ig h t forward form ulation and the ob viation o f non-no dal
( 5 2 ;
p iecew ise in ter-elem en t co n tin u ity requirem ents as suchv .The technique may be further refin ed by incorporating Hermitian, as opposed to
(58)
Lagrangian, fu n ction s w ithin the G eneralised C ollocation method .
4 .5 Summary
The proposed numerical treatment o f the f ie ld equations and appropriate boundary con d ition s have been described. I t may be noted that expressions o f the form (4. 2) are gen erally c la s s ic a lly
defined when th is approach i s adopted.
Having described the numerical technique to be employed and d elin eated the p h ysical problem to be trea ted , consider now the ap p lication o f the G eneralised C ollocation Method to the problem o f the turbulent plane j e t . ,
CHAPTER 5
THE DISCRETE MODELLING OP THE TORBULEHT PLAITE JET