y A A B B d Vasos Pavlika (1) (2)

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1 The alclaton of smmetrc ct Geometres for Incompressble otatonal Flow Usng a fferental Eqaton pproach a ondar Integral Formla based on Green s Theorem Vasos Pavlka bstract In ths paper a nmercal algorthm s descrbed for solvng the bondar vale problem assocated wth asmmetrc, nvscd, ncompressble, rotatonal rrotatonal flow n order to obtan dct wall shapes from prescrbed wall veloct dstrbtons The governng eqatons are formlated n terms of the stream fncton, the fncton, as ndependent varables where for rrotatonal flow, can be recognzed as the veloct potental fncton, for rotatonal flow, ceases beng the veloct potental fncton bt does reman orthogonal to the stream lnes nmercal method based on fnte dfferences solvng a Posson tpe eqaton on a nform mesh s emploed The technqe descrbed s capable of tacklng the so-called nverse problem where the veloct wall dstrbtons are prescrbed from whch the dct wall shape s calclated, as well as the drect problem where the veloct dstrbton on the dct walls are calclated from prescrbed dct wall shapes eslts for the case of prescrbng the rads e the so called rchlet bondar condtons are gven downstream condton s prescrbed sch that clndrcal flow, that s flow whch s ndependent of the aal coordnate, ests n alternatve formlaton s also derved based on sng Green s fncton for the Laplace eqaton on a rectangle Inde Terms Irrotatonal Incompressble flow, Upstream condtons ownstream lndrcal flow condton, dont eqaton, Green s fncton Integral Formla I INTOUTION esgners of dcts reqre nmercal technqes for calclatng wall shapes from a prescrbed veloct dstrbton The obectve of the prescrbed veloct s tpcall to avod bondar laer separaton t nlet a veloct s prescrbed to allow for a vortct to be present calclated from v where the denotes the sal cross prodct of vectors, s the vortct vector v the veloct vector respectvel Manscrpt receved gst 4, 3 VPavlka s wth SOS, Unverst of London, U, vp4@soasack The obectve of the present paper s to descrbe a nmercal algorthm for solvng the bondar vale problem that arses when the ndependent varables are where ma be dentfed as the veloct potental fncton for rrotatonal flow onl, for flow wth vortct ceases beng the veloct potental fncton bt does reman orthogonal to whch ma be dentfed as the stream fncton The dependent varable, s the radal coordnate the aal coordnate The nmercal technqe s based on the fnte dfference scheme on a nform mesh II THE ESIGN PLNE s shown n Pavlka [4] when the ndependent varables are,, where the,, have been prevosl defned t can be shown that the governng partal dfferental eqaton that the rads satsfes s gven b: + wth the speed calclated from q completon of the phscal coordnates provded from d d d where s the aal coordnate satsf ther own frst order qas-lnear hperbolc partal dfferental eqatons wth characterstcs parallel to the aes IMES 4

2 whch maps the phscal flow feld nto an nfnte strp n the, plane In fact the satsf: log log q q 3 4 egardng temporarl, q as known fnctons of the sstem 3 4 as prevosl mentoned s qas-lnear hperbolc wth characterstcs parallel to the aes whch maps the phscal flow feld nto an nfnte strp n the, plane earng n mnd the freedom avalable n the stream wse varaton of the cross stream varaton of, stable vales of can be prescrbed along one characterstc those of can be prescrbed along one characterstc III THE NUMEIL LGOITHM IN THE ESIGN PLNE ewrtng the partal dfferental eqaton that satsfes e eqaton as: where E h,,, N, g,, g, gn, N, E,,, N, ,,, N 7 + c 5 where, for problems posed n the desgn plane c=, the vale of wll var dependng on whether the flow feld s rrotatonal or swrl free etc Eqaton 5 wll be rewrtten as a Posson eqaton that s as: c log e where s the sal two dmensonal Laplacan operator so that g,,,, c where g s a fncton of the argments shown as defned b eqaton 6 Wrtng n fnte dfference form sng central dfferences gves: 6 On a nform mesh wth h IV IET SOLUTION OF THE IFFEENE EQUTIONS The matr-vector eqaton 7 s E Wth all of order NXN, colmn vectors E of order N To solve the vector recrrence relaton a speclaton s made that the - vector can be related lnearl to the vector as follows: where the the are at present nknown matrces colmn vectors respectvel Sbstttng 8 nto 7 gves W W W bt E W E E W Ths eqatng coeffcents mples E 8 IMES 4

3 W E 9 W For = ths gves E W To determne the, f the frst terate then t t t Frthermore p t p p The matr vector seqences are now defned b eqatons 9 for = to M The vectors are M now calclated startng from rght to left as s known sng M M M M V XISMMETI FLOW IN THE SENE OF O FOES Here nmercal soltons to nvscd asmmetrc flow wth constant vortct a swrl veloct wll be derved The aal veloct component at nlet wll be chosen to be of the form =+, where are constants chosen sch that = = where represents the rads the oter rads at nlet The swrl veloct, wll be of the form k l where the k l are constants wth k representng sold bod rotaton l/ the so-called free vorte term respectvel VI THE FLOW EQUTIONS IN THE PHSIL PLNE,,X doptng clndrcal polar coordnates wth beng the radal coordnate, the crcmferental the aal coordnate, defnng veloct components, wth correspondng vortct components,,, n the drecton of ncreasng, respectvel, then the eqaton of moton wth nt denst becomes: t p Where s the materal dervatve Eqaton can be t wrtten sng well known vector denttes as: can be wrtten once agan sng an approprate vector dentt as p t q Ths for stead flow rocco s form of the eqaton of moton s obtaned, e H 3 where H s the total head defned b H = p q alclatng the cross prodct on the left h sde of eqaton 3, gves H H In addton for asmmetrc flow the vortct vector becomes = + The eqaton of contnt becomes VII THE ESIGN PLNE OUNTEPTS In order to compte nmercal soltons n the desgn plane, epressons are reqred for the terms,, ths IMES 4

4 q log s or q log, bt q log q ths = f, that s n The arbtrar f fncton f represents the freedom n the cross stream dstrbton of choosng f to be nt everwhere f can be dentfed as the sal Stokes stream fncton gven b ; Eqaton, crcmferental component gves eferrng to the merdan plane fgre, t ma be dedced that: q ; q s s s ds where q = In terms of the vortct vector dt epresson 5 becomes = + + = + +, b defnton n epresson for s reqred as ths appears n the epresson for, so sng the radal component of eqaton 4 gves sng the Stokes stream fncton ths becomes d dh d d whch s the reqred epresson to be sed n calclaton of accordng to defnton 4 If far pstream the flow s assmed to be clndrcal so that all qanttes are ndependent of, then wth nt denst the eqaton of moton the Stokes Stream fncton gve p p ; ; ; ; gvng d d d d Wth prevosl defned Once dh d l k as has been calclated pstream t takes ths vale throghot the, snce as s self evdent the epresson s ndependent of Ths last epresson for s reqred n the calclaton of nmercal coplng wth eqaton gves the nmercal solton n the desgn plane VIII OWNSTEM ONITIONS ownstream a clndrcal flow condton as dscssed below wll be prescrbed efnng the pressre fncton as H the fncton p H for clndrcal flow radal eqlbrm from eqaton radal component gves dp d Integratng gves d p p d 3 Whch gves H as H IMES 4

5 p H Now Therefore 3 d d 3 d/ d d d where p, p,,, d,, wth, n ths case gven b 7 IX LULTION POEUE 8 The calclaton of the downstream rad follow from eqaton 8 wth, gven b eqaton 7, whch can be wrtten as p H d d d, g, where g, d d d 9 Sppose,,,,, where the sbscrpt denotes pstream condtons, then,,,, are reqred as fnctons of, where the sbscrpt smlarl denotng downstream condtons, so that p, H,, d d d d d,, 6 Frthermore,,, eqaton 6 now gves p, p,,,,, d or,, d 7 In order to calclate the n+ th terate t s known that: bt, oter = d g 3, n d n n, oter, oter, oter n n n from whch as can be seen from eqaton the n mst be calclated teratvel wth =, Once the n+ has been calclated t s ntrodced nto eqaton 9, gvng n rse to a new, whch n trn gves a new n, from eqaton 8 the process repeated ntl some convergence crtera s satsfed X PESIPTION OF WLL GEOMETIES In ths paper the rchlet bondar condtons wll be prescrbed on the wall bondares so that t s the rad vales, that are gven as a fncton of on the bondares The fncton chosen to gve a dstrbton s based on the hperbolc tangent, choosng =tanha+b+k where, a, b k are constants, applng the condtons that = at = = d at = takng a+b=3arbtrar b=- 3, so that tanha+b tanhb -, then t follows that IMES 4

6 d d tanh a b replacng, b n eqaton gves a dstrbton The rads s prescrbed to be eqal to nt n ths paper arbtrar The geometres prodced are shown n fgres, 3 4 respectvel XI LTENTIVE SOLUTION USING N INTEGL FOMUL SE ON GEEN S THEOEM Here a second method of solton s derved sng an ntegral formla ommencng wth the generalzed form of Green s theorem for the self adont ellptc operator Et n normal form gven b: ve t te v dd v t t v ds n n where t =, E E where E s the adont of E v s the fndamental solton to the adont eqaton In ths case the adont eqaton s gven b Et = g as defned b eqaton 6 The contor bondng the srface s traversed n the conter clockwse sense For a dobl connected regon ntrodcng a snglart at the pont, nsde or on the contor assmng v, F, log r e so that the dstance r s r gven b: / wth F, analtc, then t can be shown that m t, F, wth v t t v ds n n t L vg, dd, m, L log e t Now m = f, s wthn, s on the m= case can be shown sng m= f the approprate Plemel formlae or b ndentng the contor at, For the rchlet case of bondar condton of t, the reqrement s that v, on n addton to v, beng harmonc for the Nemann condtons on t, the reqrement s that v = v, once agan satsfng Laplace s n eqaton Mch lteratre s avalable for the Green s fncton for the Laplace eqaton see Wllams [6] need not be mentoned here Hence for the rchlet problem wthot loss of generalt settng F for nteror ponts:,, t, v t ds n t L v g, dd e, the Green s fncton v satsfes Laplace s eqaton on where s defned b:, M N vanshes on For the Nemann problem t, t n v ds N v N t L g, dd Whch gve ntegral formlae for the sqare of the rads t, from whch the rads can be determned, above Green s fncton v N satsfes the Laplace eqaton on wth v N t vanshng on nowledge of the dervatves n t are also reqred for the determnaton of the speed q gven b eqaton hence dfferentatng nder the ntegral sgn above wth respect to t t gves ntegral formlae for both, sch that: t, v smlarl for t v n g v g dd t, v t t v n g v g dd v t ds n v t ds n IMES 4

7 XII ITETIVE SOLUTION To convert formla to a sstem of lnear algebrac eqatons the pont t, nsde s related to ts bondar vales on To obtan the frst terates t,, g s set eqal to zero, so that NM4 v n t t s Usng the trapezodal rle v t s s t NM4 4 n NM4 t v, s t, v s = s s Where, v 4 n Usng ths method there s a smple self-consstenc check e the t are known pstream downstream for =,,,,N+ =N+M+3,N+M+4, N+M+3, hence the frst teraton ma be wrtten as: N N 3 NM4 tn t N N3 N4 NM4 N3 t N NM4 NM4 t t t where the smmatons on the rght h sde are performed over =,, N+ =M+N+3, M+N+4,,,M+N+3 Once the frst terate t has been calclated the feld ntegral contanng g s then compted, where the central dfference appromaton to the second dervatve s sed, ths s then ntrodced nto the rght h sde of eqaton 3 compte the second terate t The procedre s repeated ntl some convergence crtera s satsfed eg k k t t p, where s a constant the p denotes the p-norm p=, or XIII ONLUSIONS s shown, geometres have been prodced sbect to gven pstream downstream condtons wth prescrbed rchlet bondar condtons In ths case vortct at nlet has been specfed b defnng the aal veloct to be of the form =+, the swrl veloct of the form l k, where the k l are constants, defnng the so-called free forced vorte whrl respectvel The downstream condtons where sch that: clndrcal flow was present, rchlet bondar condtons were prescrbed, however the case wth Nemann condtons can be accommodated sng the algorthm, n addton so can the case wth obn bondar condton Frther eamples of the algorthm wth a combnaton of bondar condton s gven n Pavlka [5] It was fond that at most eght teratons were reqred to acheve an acceptable level of convergence, wth the technqe accelerated sng tken s Method EFEENES [] osns JM, Specal omptatonal problems assocated wth asmmetrc flow n Trbomachnes Ph Thess, N 976 [] rle N, aves HJ, Modern Fld namcs, van Nostr ehhold ompan, 97, chapter [3] ler M, erodnamc esgn of nnlar cts Ph thess 99, chapter [4] Pavlka V, Vector Feld Methods the Hdrodnamc esgn of nnlar cts, chapter VI, Unverst of North London 995 [5] Pavlka V, Vector Feld Methods the Hdrodnamc esgn of nnlar cts, chapter VIII, Unverst of North London 995 Fg The merdan plane so that t b 3 IMES 4

8 Fg The geometr speed dstrbton along the top bondar prodced gven a Swrl veloct 5 a veloct at nlet gven b Fg 4The geometr speed dstrbton along the top 6 bondar prodced gven a Swrl veloct an aal veloct at nlet gven b Fg 3 The geometr speed dstrbton along the top bondar prodced gven a Swrl veloct 6 3 an aal veloct at nlet gven b IMES 4