7RD-A149 415 SECONDARY FLOW IN CSCDES(U) CINCINNATI

UNIV OH DEPT i/iOF AEROSPACE ENGINEERING AND APPLIED MECHANICS A HRMEDJUN 84 UC-ASE-AH-iMi RFOSR-TR-84-@9i2 AFOSR-80-0242

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SECONDARY FLOW IN CASCADES

A. Hained

University of CincinnatiCincinnati, Ohio 45221

.une 1994

Final Scientific RenortJuly 1, 1980 Through December 31, 1Ai83

Grant AFOSR-80-02'22

Preipared for

Directorate of Aerospacq SciencesAir Force Office of Scientific Research

Boiling Air Force Base* ~Washington, D.C. 20332

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secondary flow Rheoatinal FnLtr low, f iesnalytlov l

and experiuentally. .Ahuvork wasponsored by the Air Force*Oftioe of Scientific Research an~ts main objective wasn todevelop computational procedures £5r internal three-dimensional -rotattonel flow fields apd to obtain experimental measurements

SSW elm 01.O4.i"1yGo, T FA M IIIIIIII, Do

~ , b thr:ee velocity @@3pOn~nts of the r6tationalto a crvedy duct with shear inlet velocity %

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ZJ60W In curved passages for periodV, Ow 'e 31, 1983. 2 he research was

~ owinOf fice Of Scientific Research under

wwwre4 by ft T""me Wilson and Captain

~ftlIsnapr,# Directorate of Aerospace

W(2~0 a~o~ntiie kearch, United States

P allthe previous interim

'777, llthe t"ch6ica1 papers both publishedwhich waft written under this research

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A This technical1 raport hns been r6VIewed %"Id ill

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AoublOiabed Work. 0 . a . . . . . . . . . . . 0 0 * . 0 4

WL 0piaim . a 0 * * .

I -Zuiacid Solution for the Secondary Flow in Curved~~t5 9

J11404100fh 2 Ulliptic solution of the Secondary FlowProblea18

- 3 fte -Diamnsionaol Rotational Compressible FlowoUt JAi Variable. Area Channels 2

A~qwlix, 4 Internal Three-Dimensional Viscous Flow Solution 7Using the Streamlike Function37..

~eI~ 5 A* I aellitiSltonf 3-D Internal Viscous Flow .. *tbzpthStrealk.1nction 48

UNi 6-W asrstuoTre-Dimensional Flow ' --Development. in a Curved Rectangular Duct With InletShear Profile 66 C'

-4 V.9~ Th rsutsprvdeth nedd n

-wi h.th ine teoctyl pressurediswerein

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4

P rofessor

S.q JDAMlah - Graduate Research Assistant

c. X boufantomis Graduate Research Assistant-C. *~Ag - Graduaate Research Assistant .-

3. 3a1~ - Graduate Research Assistant

J. Copdto - Mechanic

ADVAUM DEGRESS ANARDSU

X.S. Degree jto M. Malakp Thesis Title:

Ohn Investigation Of The Trailing Edge Condition 1In ih. Finite glemmt Solution Of Inviscid Flow

In Cascade And Single Airfoil".

PhI.D. Degree to S. Abdallah,, Dissertation Title:

wAh Inviscid Solution For The Secondary Flow In-

Curved Passages"..: :

2.4

. P U .. ... V R THIS RESEARCH GRANT

Dmwt',AIJournal, Vol. 19., No. 8, 1981, ppy. 9

~ A~a,*.**The Elliptic Solution of the SecondarysJournal of Engineering for Power, Vol. 105,, 1983,

A.~~ ~ 4 i, hee mensional Rotational CoupressibU6 Flow~aziabe AeaChannels, AIAM Paper No.* 83-0259, and~U~eios in the AIM Journal.

SA- saG Aftalah, B., "Internal Threei-Dimensional ViscousO3bq the Streaulike Function," AZAA Paper No. 84-1633,

b-qhb mlh -9., "The Elliptic Solution of 3-D internalhewUs~ag ~heStreulk.Func tion," Recent Advances in

UboGJA ?luids Vol. III on Computational Methods inW~oVsPlnridg. Press,. 19 84.

a M3J, not "WOV Measurements of Three-Dimensionalti*b a s idva Motangular Duct With Inlet Shear

P~bt1. RiftPaper N0. 84-1601, June 1984.

**** ***

$9- 3

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U , _ OBJECTIVES

.e main objective of this research work was to investigate

the s e" flow jihnmsna analytically and experimentally.

s eapsimental work was conducted to obtain measurements for

the three velocity componen using Laser Doppler Velocimeter

(WV in a c=ved dUct with inlet shear velocity profile. The purpose

Of the analytical work was to develop a formulation and a

nminerical procedure for the solution of internal three

dimensonal. rotational flow fields.

ACCONLISNBD WORK

Since all six technical papers, which were generated under

AFOR sponsorship are attached in this report, only the

important contributions will be summarized here.

The first phase of the analytical work was aimed at develop-

ing a numerical computational procedure for inviscid incompressible-

rotational flow using a marching technique. The governing

eqation for the through flow velocity and vorticity components

and for the. streaulike functions in the cross-sectional planes

were derived from the conservation of mass and momentum. The

numerical results were obtained for the rotational flow field

in a curved rectangular duct with constant cross sectional

area and curvature, and compared with available experimental data.

We work was published in the AIAA Journal, Vol. 19, No. 8,

1981, pp. 993-999 (Appendix 1).

4.4...

4"

The next step in the analytical work was to develop an

elliptic numerical solution for internal inviscid rotational flows.

The equation of conservation of mass for three dimensional flows

was identically satisfied through the definition of three two-

dimensional streamlike functions on sets of orthogonal surfaces.

An iterative procedure was developed for the numerical solution

of the governing equations for the through flow vorticity, .

total pressure and streamlike functions. The results of the

numerical flow computations were compared with the experimental

data and with the results of other analytical studies. This work

was presented as ASME Paper No. 82-GT-242 at the 27th International

Gas Turbine Conference in London, England, April 1982, and later

published in the Journal of Engineering for Power, Vol. 105, 1983,

pp. 530-535 (Appendix 2). The analysis was then generalized to

compressible flows in curved ducts with variable cross-sectional .. :.

area, using orthogonal curvilinear coordinates. The results of

the numerical computations were compared with the experimental

measurements in Stanitz duct. This work was presented as AIAA.

raper Number 83-0259 at the AIAA 21st Aerospace Sciences Meeting

at Reno, Nevada, January 10-13, 1983, (Apendix 3), and is

accepted for publication in the AIAA Journal.

The final phase of the analytical study was to determine the

suitability of the streamlike function vorticity formulation for

obtaining elliptic solutions for three dimensional viscous flows.

The results of the computations were presented for the three'.. h .. . -

dimensional viscous flow in a square duct. ',.'

5art V % ' '

r

21ie comuted results were found in agreement with the experimentally

measured through flow velocity profiles. In addition the viscous ~

elliptic inflUence was reflected in the computed axial and cross

velocity components upstream of the duct inlet. Thiis work was

presented as AIMA Paper No. 84-1633, at the AIMA 17th Fluid Dynamics,

Plasma Dynamics and Lasers Conference at Snowmass, Colorado-

June 25-27, 1984 (Appendix 4). It will also appear in Recent

Advances in Numerical Methods in Fluids Volume III on Computational

Methods in Viscous Flows, Pineridge Press, 1984, (Appendix 5).

The experimental work was conducted to obtain measurements

of the three velocity components of the flow in a curved rectangular

duct, using Laser Doppler Velocimeter. A nearly linear inlet

shear velocity profile was produced using a grid of parallel

wires with variable spacing, resulting in secondary velocities

as high as 25% of the mean inlet velocity. This work was

presented as AIAA Paper No. 84-1601 at the AIAA 17th Fluid - -

Dynamics, Plasma Dynamics and Lasers Conference at Snowmass,

Colorado, June 25-27, 1984 (Appendix 6).

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SU NARY OF SIGNIFICANT RESULTS

LP 0to formulations were developed for odeling inviscid three---

dimensional rotational flow fields in curved passages. The first,

for a very efficient marching solution with hyperbolic equations

governing the development of through flow velocity and vorticity

profiles along the duct. In the second formulation, a new approach

was developed to satisfy the conservation of mass in three dimen- -

sional flows by defining two dimensional streamlike functions on

fixed orthogonal surfaces in the flow field The first formulation

, ieads to a faster numerical soluti6n in which the influence of

the pressure field on the three dimensional flow can be sensed

upstream only through the curvilinear coordinate system. The

second formulation on the other hand is more complex since it

models the elliptic influence of the three dimensional pressure

field. Computer time savings are realized through the two

dimensionality of the equations for the streamlike functions and

their Dirichlet boundary conditions. Agreement between the -

computed results and the experimental data was very good in both -

cases.

The streamlike function vorticity formulation was also

tested for its ability to model viscous flow effects and their

elliptic influence in a square duct.The viscous elliptic --

solution predicted the influence of the duct on the flow field '

upstream of the inlet. In addition, the computed results were

in very good agreement with the experimentally measured through

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of w th he vlct comvonents,

5~ WV 4y retn lar duct with an inlet shear1.kwas designed such that the

:77 be, Vas- to, validate both viscous and invis aid

tbee, 2iesional flow fields with strong

* e~daryflow velocities due to inlet~ distortions. A grid, of wires with variable

~~ *,4?~.anearly linear inlet velocity profile.

rMi On o usatufes up to 250 of the moan

we Oeamuted In the curved duct.S

iotal data for the three velocity components in

intis research work, provide detailed

A W the socondary flow pattern. This information cani M

~ Sedoflopoet of appropriate flow models for the

SO4i )Of internal flow. In addition, the presented

wkcan be used in both inviscid and viscous three

034o, a.utti to model the various secondary flow

A ib4~i555in turbomachines.

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inhuwindly mespmee 1w esdylavuid.I. strhmlin curvature eho 'to a thee-variable prablem. In01111" 6tw. LbW hsey MY, oue MWMO thir goiih two imilcal Values wre calculated for tho

dabi uwrINl 1wt Nhtbu I smu N revisfom, -am through, flow velocity by Inuteatn two sits of firs-11the=ln d,4 sd the ma-a 3Mm cknuder a orde equeiiaf5w the tNenaine curvatures. The diffanc

uaubery ~ ~ ~ ~ ~ ~~ d beud ed.Fb Hwhre te ews the two values was Introduced as a momrc term in two~~~~u~~~~~mi~~~~ damn g10lldvrut qal.t0aHpi~ciut equationts, which were used. in turn. to*d Wifn- that 'an no is Hawthoones eur~o.A. 0. calculate thi two seconidary veocity cmponents. The '~OMW trived an apeslo o h mm vurtls I bouim conitions were not comptibl wit thi sceeinuft AmN wil La~len AM n Hhek P"w a which a duplicate variable is used for the forwar iduck,.a r undlty -ulon I a VWM foreeIle statflei, and the continuity equation Is integrated twice. The abr

Sod vahs. w. The ppio~e f Iwilurnesaiequeo rectified this situation by deriving two Imtae boudary .*.ft mUMmw vanly aeukowe mobise prior conldtin for the duplicate forward velocities from the twoImem b O'f tho ,,de fI sI aiU Comequsesly, sow coinlyauti after dropping the sorce tern. Alan"

apprimesum unlyhav be.Invled teapplication an the other head used Wu's techdnie'3 an two families of .-41 Hf ~ of 3hmesoqmio. 1wt tr umws worticity mu r aces On these surfaces the invhisi flw govetnn

qul.are combined to obtain Pso type equaimn. Thesaem surfae In this -m are sktewed and warped becaus ofthe streamwe vonlality. This analysis was used successfully .. *.In case of sma disturbances, but Fagan reported difficulties

7~4MM - "ape $&I 1 at 0he 1"~3IA~Mn Jeis In highl rotational flow, because of the warpage of the-WAf'BN 1 Comi F. H rd. Cone., lJes W0July 2. 19W. stream surfaces which was more than 90 deg. Corkscrew ***NNWily IS, "ft dsi Jan. 29 1"1. Copyghtle coordinates that rotate at a specified rate were introduced in

AS M111110 o MMNINa Avvennaes, Inc.. IM. AN Ref. I I to resolve this problem.ACMM aw Aminse. OW. of Afop aief n This perpresents a new analytical formulation and an

-11' 1 ' i ll" b AA efficient numerical technique tor the solution of the ,*.?P~bm. Deto Aeresm bgasmae amd AppWe rotational flow in curved ducts. Through cross dif.

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&.AVALIAMANDA. RAD AMA JOURNAL

mob. ON the weihciy comnor of ftI. remain practically%wdiwith me swicentg rotation. It can be noticd from

thwes rq.u. ditI 0- and 94 rs etos h

ofmdsIt lb I u the o ex center in Prp. 4, 6.andL ThdW - pmasbeubmeto tsoure tmIn __

bao fow vemut opom in Pehcton.U." 1UompUmi veloely contomrs atlk 30.W, 0and Wdeg

.0 mmmdmm~ wit ibm apu~bmwaldata oftRe. 1S 3ad doe kvbWi l1aw -melsIt of We. I Iin Pip. 96il for

ampe ~Ite.. b ese matus fgr ht h rsaM**ei powat te rowom of the velocit conour -amm at J& aqnd ngmss end in tih b qwily roo is t "-ftung angle. The aumen1eu the me*& and the expilmental data Is veyftlee In 48 hg weeVdy reIos, where the vismo=M ena mmet *doian. In addition. te reent analysisrI IN the e~apmuma measurd low velocity reom at

dinerwavia nasbe Owat the iner cormr in F 9 andII W.* " eetm I ~eb of the Crems sec in afg. 10. The tramlantionM66slb bee a Of fti low velocit reon fromt t corne at ib*~eg crams

Pe 0110 U * ob mo sua *Ath commain at t 6O0deg crosction isno

4 ~ ~ 1 he PO uamee OmeW In the solution of the Irsn studymilL I * -b . le ms than the mpute dme In Raft. 10, 11,

en 14. FPn" using the total, of sn wstrem surfame withno0 modal points Per surfac, rFported a CPU dome of 1500-2100 au the dwM 370/155. Stumr and Hmiehrngton'I

* the pruem analyis, fth CM im was 20 s on AMAIL 470V/E41 usin an I Ix I I x 4S gri In thenumeorlcal soluton.

Mie prI n analysis of Interval rotational flow lwed to aMrY effii numerica schune for pIctn t secondary

__ B~fow pheoxomn. The analysis Is applied to t rotationalflow in a N-fts bond wit rectangular crom section. Rooam tompariso Of t computed relts wi t emprisental

daa It as be concluded that the91 pIca of the secondaryflow ~ w prlva el fera tmii h analysis. The prsntI

___ ~~forumulton as be adapted, wt som madlfication"..o.r.variable aducts and turbomacisry poassnm ,.

* a mh sote 'p m-fo ffinern enteat ft. (Ij with repc to g and Eq. (3) with~em enbs~ ~ in repec tor and subtracting, -m obtains

ib W a~ vu nd te carrespasift seaftd) .. .....~Sith b~~u SM~Sin30 ofis s

nw t e 6aee In y vtlly reMhn a6111019 ft am mamim Ia is hportmnt to *ii8e ~ lowZo

Own to sAu uw, d he -m radius e 4..b to e do w the *2"4 - ales. ireS. k .q(l

00 o WNOW! 6s o Wll; iam ~Ntin. Thsrgo the divergnc of the vorticity vector is identically equal to~' ~ zeo. This as be expressed In terms of v. v, and w and f as

___de so I* slOfgin dW 7b, and 8b. It as be seen *, ~ ~ **W- *0 #bm . s-Midvd dacrns we W- t+ tN !W 'L' *.49 h amiftaft ft do W ad ftheWe duct crom S rrr N 8

~~bmemammynat 9 deg to0aTduc~th :%~= (Zdo Was wm ibl s apoelowlary ver +m +

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- *, i L ' A al'o toy. . I. 'hs In tioaf ml S o i It o"w"

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(AA)51 3972. O 9,Pa 1 ue 93 V 7

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01111161 %dw8% (u, . . "il Some So olmaine fr Hyaoalc uationa .of

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~b~hmSE&~iaimelEq. (~in fllows JHigLer Use e M.S. Tein Mu~pwdmackilfta o7M~tedOlogy, o5l. 7,Ot VS p N-06 -

"Rescs. J. H.. "The unusrnof SolPtionyofixoeN&vL.e-SwkedUft ft ft ~ iiia o~sibmwft F - v sui~aquati1ons orV fti A " ThesD msn sso Bla de Ro. ", NASipSPdW1O ftoff (71 saeo ntmo h 30I.AWPlAit DiferAcw MetoWds. Jwwof Englasdmrlq.

odoft~~A~mnu~ Vol.~a 12. Oct v93 pp. th 03-3Ia23ow M p II Sm le. .i e.ii8W ? Aa. . mD HWa eito.R."h Solutions ofNnl~ Hprof tEuTions

~~~~q- ~ ~ ~ ~ Vl Don F.-~-(9 ndTerNmrl Equaation." oNA SID 304, onW Parr andP. we]' 8 Apcwwvd~aihemeuia. ol. VI . p. 19. 197.13. -30

lip ~1 r Fw.J .'teNesodSboi utFoAnalysis,"~~~~~ ~ ~ ~ ~ ~ Deto DislAlsn Dv fGnrl oosCr.

MWI~~~d ha My IN a .Jwdo nketfrf

1- 10

Iftm3 33 VolW 10,jt hak ~f Iaeisuin qfo Fw .adkopw~i

LKMM The Ellpti Soleutimonmdpo of h reinI the foSeca n ondr suFlow Probvndtvble Anti9.muainaAtt~y nbiyc *pka he sk ote e an the vunlowi hswrrii coponente Vn wv"f equo Ine tm f ese dukw vjlowji re solvend oeach,

fbwlt- qufam ung th smwWfeunhetwdanfrukon. A owi meachns s01111K o 406 &W~somd o - t gf -twam eteen te soluions the lsov th fmly

awmm - N int ea uniqe souaddag Irn aion thceibunak donibtion forAr enwt *rnow tof hewtoswf Thewv~ depene aln In thcisfomtn and

*~ssb~Wsam ftea d sornpouut. M oa nussy.d rsulsobaedo the rib~ otatioonlen-Thejb inuk eaun ar uu a hre daeanden uopa ith ae soloe at- ac

adrwa~d im tauoarch e homton tbetain bidthe ie solutions. Thee samelsommomnuinmel ao The faddthe mto. a asIo9 ben additionNothekband ery o nds by

Z;= S~w p- n od, -'"pxinsemins to ah solution. The auptions renld obtaid o/se or the rttoanph -djw An aw leu ths ied sa' sure ds and stvprad wilah thicbessx

2bs..~oui doe uuIn bodda iw urbomacIn his dilstton In to obtand S1d-o-d solutins. TesVsAI "k hmsa ftsm. th faNta h method siar alo bee susbsed e sy 9 aondecbyo

W" to 661oul *iG~y two n tlf11'o with t arix m,e1thodoti. e~~nl n aet-ldMM M o vime dhsui *d apoanng oeaesolutionsem assmpound ppromutin a sedorth

SOW At &@ uWbean ifn oi.fel s Wu stream sufcsap nd she-ieninltresalw ilet r thinesMu111. *a e isfwe tu ow asaelo.I this(b&*4& adistributon Inthe S, and S2 soutface sin this preblod are

bw *d is dstm g fraud of uoelmil i to thsedsused pfratiousl iewen conectio.. a ak Ws sousems m o eothe fam d s ouwiths thoncarnin th to srae hp ndte

amenV m bond va wb1 teatigonRu thickness. Novavrinkevrl rblm and enoner~ oned Incm utin a yn6Mlfm - smisAwsislaofo bbf intigtrs theue threMomenhiksna aluatheflied from the d-oknotsW W* s thfeoey to tir d n theS, ore anlde solution not ostatsra and dosuwce.hstrem are*5a Twol N*Am M. Thu swet isolutdoioe rthed taow Ths eihage inoration etwe the iet

E@Olb do~ Mm dmahgad the tumn ot and, sthoscnrnn the slw etream asrae shpet n the reamnsm1i. uhe ebtwe tesluin on e filamen thicunss.d Nova andec ohean 19oinef u thmat

han ~e Wa teor toobainsoltin o th S orS~ lad slutonIsnotcontat ustraand downstream of .in ~ ~~ AM Tw. 41uh have h be .used ain s he eudes th ae srow.e etising contradiction ihchanes e ue t

bwaM II ad Umkris M7, use the matrix miethod to blades, where the net angular momenu hne utbgal hdM O te S, mad 32 surfacu Katsanis wo. Stuart and Hetherington (121 tried to use a technique *

11a 1 m1 o MW I Is far a mdiinlslton1 (33 similar to Wu's in their solution for rotational flow in a 90-ml*. ~ ~ ~ J* f-suda abaaobaufof revolutdon doe bend, by synthesizing the three-dimensional flow field

dduspapmnm Ito theblade height. In through the iterations between two-dimensional solutions.aveage S~Stram urfceThey were unable to reach convergence in the iterative *

gemmetede by etigthe numerical solution and had to abandon this approach. TheybiSt osabutthenclsofreotion. speculd that the information conveyed between the two

vmethod has bea used by so~~awa o sfiieto roe onvergn yo

=SUWON is, M.rw m.r4 due to the presen of the secondary velocities. These

mvus.US, ALY19 Traneacthof@ the ASME

x *andix

fte Elliptic Solution of the Secondary

Flow ProblemI ,. AD *

%-. 9*

, •-

tramsvets velocities exist due to the streamwise vorticity streumlike functions. Dwnstream, the derivatives of the flow "whh I generated by the turning of a nonuniform flow with a properties in the throughflow direction is set to zero. Avarty component in the curvature plane 113, 141. The Poisson type equation with Neumann boundary conditions is

m ltu s of inlet velocities in turbomachines are derived and solved for the static pressure at the inlet pla,-associated with the hub and the tip boundary layers. Stream which Is then used together with the inlet velocity profile tosurfa warpae pose additional mathematical difficulties In determine the inlet total pressure profile.the solution of the rotational flow. Pan (151 could not Numerical resuts are obtained for the case of rotational -obtain a solutio for highly rotational flow In curved ducts flow in a curved duct with rectangular cross sections. Theusini We's aprach. He resolved the problems encountered results are discussed and compared with the experimentalwhen the seam surfaces warpae approaches 90 des through dam.using a corkscrew coordinate system that operates at aspe 4 =fai zeed and Hamed deLped an - u Formulaton

fectihe method for the solution of thre-dimensional For simplicity and to be able to compare with exiingrotational flow in curved ducts, in which the throughlflow experimental results .in ducts (191, the cylindrical polarvelocity and voridty components were computed using a coordinates ae used in the foilowing presentation. The basicsearching technique In the main flow direction. This led to a equations for nonviscous incompressible flow are expressed invery efficient numerical solution whose results compared terms of :he three velocity components, the total pressure andfavorably with the experimental data for duct flows, the throuahflowvortitycomponensafollows:However, because of the marching technique used in com- Consevatim of massputing the through flow velocity, the influence of thedownstream conditions on the flow characteristics is not _ 0 9) (n + + -0 (.)modeled. This effect, although not significant in duct flow, r • r DOmay be quite important in turbomachinery applications. Coniation of momentum n r-direcdonBarber and Langston (171 contrasted the blade row and duct r V I am :iflow problems and discussed the importance of the elliptic V (2)solution to the flow in blade rows. La r r OJ r

This Investigation represents an elliptic solution for the Conservation of momentum in g-direction .internal nonviscous incompressible rotational flow in curved .9•1passages. The streamlike function method, which was ap ...(3)developed by the authors [18] for the efficient numerical r at z,solution of the continuity and rotationAlity equations is where P Is the total pressure divided by the density, andimplemented in the present problem formulation. The (ku,w) are the three velocity components in (r,9,.8-drections.dependent variables in this formulation are the three The throughflow vorticity component, C, can be written instreamlike functions, the total pressure, and the throughflow terms of the cross velocities, u and w, as followsvorticity component. The equations of motion are satisfied on au awthree arbitrary sets of orthogonal surfaces. On these surfaces, (4)two-dimenulonal Poisson equations are derived, for the xstreailke functions, with source tems representing the flow "three-dimensionallty. The source terms are dependent uponthe variation of the flow properties In the direction normal to Bernoulli's equation is used instead of the momentumthese surfaces. Because of the dependency of the solution on equation in the fedirection.each set of surfaces on the solutions for the remaining two sets ap V aP apof surfaces, an iterative process Is involved in the solution. u- +r - - +W_ -(The three flow velocity components are determined by the ar r30 asource terms and the three streaminke function derivatives. HlNbOit IZAEnd.... .The total pressure and through flow vorticity are computedfrom Bernoulli and Helmholtz equations, respectively. u - + _ V + w

The initial and boundary conditions for a closed system of "r r 89 .zequations are carefully chosen to insure the existence and

. uniqueness of the solution. The no-flux condition at the solid Dv , au +v I .'-.boundaries leads to Dirichlet boundary conditions for the r r 0 °Z r "

Nomenclature . . .....

A -duct cross-sectionalarea •r 1, - arbitrary integration

C - contour enclosingA reference points on the component in 9-dC - incremental distance r- and z-coordinates, direction

along C respectively ( ,A9,Az) space increments inn W outward normal to the SO ,Sa - blade-to-blade and (r,Oz) directions

contour, C meridional streamP M total pressure divided surfaces, respectively Subscriptsby the density (u,vw) - velocity components h = horizontal surfacesP W Inlet static pressure In (rDr) directions, I = inlet conditions

* divided by the density respectively v = vertical surfaces(r,Dg) - cylindrical polar X streamlike function c = cross-sectional sur-

coordinates - main flow vorticity faces

i Jo lofl fin.hu nrhg for Fwer 20 JULY 1983, Vol. 1051"31~~~~~~~~~~~~~...................-........ ..-.... .,.....,,. ._..... ,...,..... .... •........ ..... ,...... .. .'.'.'

*O Ib V . I.Xb VW + +;rip

[W1+ of.

+x~ + (109 *.**

I fax

- W o + 0 - -

OF r -7(1-4 m(34 am an tmhus f

1SZ 2 1 ul tsi m t hsskft aeu6- nrovide 1. k : ~h d ~ j ](2th~*~n~uis.NW~wwhft th e hpeow ic

lb jj6 how" ohu n fu Atl de0lopd by the The elliptic equations (10), (11), and (12) are sole on theaudm 11W buI 11 Ussps prM. for P--P Ih pupoe oriutd. vsrticanmd cs-gectiomal surfa=e for u.

.1 ~ ~ ~ ~ ~ ~ ~~~1 IR OW Xmsdin 1. hlna qsi n e rstvdy. The solution on one me of surfaces is~~aupminsme m d fuamm , xv Xe sd xe, we inflenc by tine solutions on d te r two SMt of surfaCes

-long0 1 Sow I. Fig. through the SamI'm terms. Consequently. an iterative IpocsJ1.1 kbft slddoprinip of cousietiom of 1U, betweon the three fammilies of surface$ Is involved. The three-

dimensional solut iIs obtained by adding the comtputed1U Ma e~ fI~. xI . it dHn 'am the horboati vlocit components in each of t solutions a folOW

m~en .. eNin~+U ~(13.).86 .+ ! f,!!i (74) VWAb+P. (13b)

r M r WmwW+w, (13c)

- (7b) ft can be easily shown that the velocty componmeqa uAr detertmined by equation (13). rePsn a uniqi tsblution.

%hm ,~ bt he radft oordhte of an artr lateratoni-uum I srfa I ad t sbcit hW en refer a. the NoofarI Cemdiuleas The IWOe conditions for f and P aresohim on the borimui ad vertical sufaces rspactvely. given by

lb daillb function, X,, Is Meind an the vertical 10(14) 0

weeP, Is the inlet static pressure divided by the density. Th- me . stati pressure distribution at the duct inlet cmos section is

computed from the numerical solution of the followingwham c bt the add corinat of en arbiutry InegaP s

Minineurae. alp, +3 .+e,(6Mwb 0 l etims ;&,Is deind on themes plane F a + Z

a blowswhere

+ ( pqjt* (9. ) + 2IF r

wAm do mubsul6 . refer a. the solutlom on the cross- With tile following boundary conditions

lbt pieg ~ s f 1111 or, don tra functions are*P_ rR ,(7)d1. b~ haingsqMONG (7cb) Int equaion (2), & ,

IN L W ULl 21. ruwouwi of ftl AIME

-'i frxv. Xe. mad x.- 1 Inmrtoarte-esrforftb haedi qI dIm - (7-9) we chass bere to b

W~~tihsaebshm ~~ -euno r .iebyr, - Rssi:. .m respetvely. ob

*#** .r#.C zen 0I ~~~X ofe~i~-i ~ (S

S at

~4~~du~smg~uI~inuhebe6Awd x~; w*(2b

sod j f~f p *hg~5 q tate the b-

-~+w~rn@ W r~R..--O (Z7b)

WE- t =Xin o, (19b) + .0R.~~~o r...

0 + -. A.@ tXW. (9) R

+W~ Ih' Serg ft V P7kW 2b

Aixb -0 :---:

j~g ~(29.)or le bIk ON ho .. I (6.nd M k 2

0 (30s)

- sh 0odm m (30b)

wiS(1) The governng equatons ).(6). (10) (11) and (12) with thepw~(21b) conditions given by equations (14). (15). and (26-30) form awoo (21) dlosed system which is solved for the variables X6, Xe. x.. f.

Mtr - Rj.R. NWP

Resst and DlamaonsThe results of dhe computations of the secondary flow in

winOcurved duct caused by total pressure inlet distortion arAgotpresented. The iterative solution procedure is based on the use -

of successive ova relaxation method for the solution of thethree, Ltoniefnto qain, a .x's (20) marchingscheme for the total pressure and through flow vorticity4%

11W equationa. The results are presented and compared with the(240) experimental measurements of Joy (191 for the flow in a

curved duct with constant curvature and rectangular cross 2\section. Joy (191 obtained nlow measurements in a curved

-was 24 rectangular duct of 0.125 x 0.25-rn (5 x 10-In.) cross section.0.375-rn (154in.) mean radius, and 90 dog tutniag angle.A

k e aiim O M osw emidom Is use to uniquey large velocity gradient was produced in the experiment usingoA nusi1. X.- ad X.: screens plce before the curved portion of the duct, which

1*. Xe i~.resulted In a nearly symmetrical velocity ipiom at Ine to th.+-+ _+ - (20 bend. The velocity contours In the lower half of the duct are

~7 ~ shown In Fig. 2. The computations we.e carried out In thei14 W NOW W bWwith WJA A lwhl fthe duct to take advantage of the symmetry. The

22 JULY1963 Vol. 105 1 In.

1~~A

looWbuSwoidin not Skad90datoanganle.A14,11. in FWei-5 we nrmalize

Ab~

wel P1.5 the immmat tbr nesigtos11.

OWW*"" som apalhantgtq mob . 4

*bsw bmft Iwseparation t-r 3*d atriutd'

of the iput ausaed byqu wth secy'dar flow

#9m1wd# vIeeltiue in Figs. 6. Th nruse

~~~the .mlift vefoity. Jo 11ddnto tian bcpessur&umormae do coul the compeed wusith tre prntrwini kwat tde p semeu coniurs dollat the oadf*.**-

S n tw angles. hferelc of tescnare owa devernMihe dw ua jt aN M h Is deosrd I these angres nnerthog hagn henme o trtinihk w i doe dap u ofu Othe =Meaow s [it , 151 dutslto f-

M*A. Is am Aebbar TmbmW s M N Mmiii." Am' *

I ffawdi T. "Ralml Pemem IP.m W m - nCdnkd Vsdi.mbeubaw m do Mb~pia Almd two kbb d= Amle..to". a Nk6.11w Tinbs w a be.w Iw : WV.s md."

4 .T. cmwPoo ft Cdodoft Vii~ il

ft JI.3 . 1, "4upW OdMim of We's Euwdom fw Cow ~1ws.Iuni~d~mb.7.LsL p43-4 3914.

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3 6 9 12 15 16 AkkbsIW 9mbmwi"AUWmkAwwwAmmVsL. 3ftwott" Ift.It FeM~ J. I. @Ia bnm fthblelim g w Ambew

u~~sas Dw4*hDIed A~am%=Dblsgt mwWtfmuftp.. 743.3912If AdddU.**m. "pb~~fw* iPo

is "Imft~bIWWW f *g38 Numi, A,. ad Abftk8S, -"b.llk fodm A Mm. Como. imUMIIIms.~ b 937 flt InI of ~Flow pt~m ftodoW "medAmw. Vu4 If. NM& I2. Oft 19.

MI o ljorom39 Jay. W.. -Uawii bi , f bSow Im le ssw

31 [a& P. D. "Wok Smielm of Nfevlkn Nihuibs qAmdum mi .p .

1W. amb un ipsuas by the U.S. Air Fate Office Nuowhii aL . m,' -of& Md samoosaft" 1nNI.INO.10242 23 N.M.. H.. " Ad EaWOMEM h a~.dii Flw Low.*~

le 3.6 oftb 3oae Cue Smash." CIUDA-TwWMl "r

22 Dow, F. W., "Thu abs. Soloa o doi Dbom Pulmm m10W4 C. 11. -A Omd ThMYo aThm Dbmmhed Mme Im Suba.kupi. N.4N^-SAshw. V.1.32.MNo2. Mms. 96.pp. MS-Ul

ead ~loeuf TubsmoaMi st AuULNo mmii Nbi 4Pbw Tlpu 22 lSumi. L A.. "A Cycs Rlii. Aibmmm for l.1gUidw~c~mum..Is..Twd~od Spimw of Aelbup Dkeado." SUAm ms~ -wAb

2 N IL U.. A BW CMMwt~ F fs w do iilkb Plow KAM Ab VOL 14.NHa 4.3mpg. 1977. pp.755-72L.0

244

JULY gn Vl. 10 IO

252

Mold Im

e7.'

t.

op"M AOW OM ofress t ndblea ima 0 MMOwkNY1N

22U8-DZIUZOVRL RTATONAL COMPRZOSIBL FLOW BOLUTIOW ..

ZN VARIABLE ARIA CHANIMLI . .. '

A nmed and C.* LiuOf Aerospace of gineering and AppliedMehnc

Unversity of CincinnatiCincinnati, Ohio 45221

Introduction ::Thia p Par psnts the analytical ?imoet developments of cosputational *$

AR .... A -A moweaLsoLutice of comn- Methods for three dimenional fliows inreeta~alflowin a tuzhmachine blade passages and curved ducts

= ="abe are m o lb. " three dimsrni*Aal in'Olkude both viscous and invisfi~ flowfLOW flw Is e06t124sd PIes the stream- Models. The parabolic methods-* wereUbhe towetios siblutiM an three sets of first developed for the solution of internalowtheysIna surfaes In the passage. This viscus flows. These methods are based on--too"s to a Very esom0mical elliptic solu- the assiption of the small influence ofti tht Is aII -able to tmubmchiner the downstream pressure field on the up-21 itimen sius It, sAinlates the down- stream flow conditions, which leads toAeOO-WWWO edtlm and AW cha 9e araVeia- parabolic governing equations. The use oftins. Ibe compted results for osinpressi- marching techniques leads to a very effi- .Mle zatatiomel fCUw with sheer inlet velo- cteut solution but it also limits its *.city In w~rlable ae curved ducts are application to flow in ducts with mildpresented to determine the influence of ourvatUge , Later partially parabolic ,-doe Ie variation ad the effects of methods 3 "q vere developed in which the

~res~biity ow the L'~~-ional diffusion of mass, momentum and energy Lafla fild.the stre nowis* direction were still neglected

but the elliptic influence was transmittedupstream through the pressure field.

Numerical methods have also beenh total enthelPy evlg for the solution of inviscid

~ ~ ~ fows~LS.The approaches used La the -* l-h2V ses coetricalt solution are considerably dif-*Ptota"peea ferest depending on the flow model and the

problem formulation Ealier quasi-threep sttic rssre iamnsional methodsl consisted of an

0 entropy interactive p1ocedure between two dimes- %-T total teoerature aoal flwu oltmn on blade to blade andhub to tip stream surfaces.* These efficient

* V1V2 1V3 velocity ceponeatetwo dimensional solvers allowed idab- ~ .xi~2I3 rtognal cuvilinear coor- trarily superimposed loss models ,L but

211,12113 for soft of the solutions, the blade toblade stream pqfface wes taken as a surface

0density of revolution''. The atteopts to extend -a sourcee" methos to rotational flow were not-;-a sorce erm9UCCSaflJ6J beausethestream surfaces

X stressliket function beoetwisted and warped. it was foundthat the iterative procedure between the.*-

al*'n2 13 verticity components two solutions does not converge in thisSub"Llpts case through the exchanged informationduring iterations La the form of stream

aGossectional surface shape and strem filament thickness.o crossTim dependent techniques have been devel-

h horisontal oped for the sollin of three dimensional *aX inlet rotational flows~i, and were used in f low 7,field calculations14. Aside from the ir**'--a maxi"mp dependent technique, two other methods..A U %

r reference point were developed for the solution of internalV vextical

4 This research wee supported by the Air Force Office of Scientific Research, underam" No. I0-0242.

* Profeseor, Deprtment of Aerospace Engineering and Applied Mechanics, University of aCiAmInmatit Cincinnati, Ohio. Associate Fellow AZAA.

**Visiting Scientist. Senior Engineer, The Shenyang Aeroengine Research institute :

27 *

- v ~ ~ __ ~ .Y ~ a-

se~@~Lrlcs* Lessor and coordinates, respectively, p Is the flow4wvinda. Very slamle model for density and hit h2 h3 are the metric coef -

~f ~irap~cflow Zields by.14t ~ igA o ficients of the orthogonal curvilinear

mialoa arts. Ibyused codntsS Si) iON"On to describe the rotation- Teeuto fcnevto fmsIfewWA h " We1ia class of P90 i he le qucllati f coe athroug th mo

*A4%tberothalpy and entropy i dnial aife hog ha" Orthgoal~ to the Verticity definition of three streamlike functions 1 7

'XoX ad)C.Testreamlike function Xh %~ musiA' reeened ais defined snob that its derivatives on the

two eloit~com otslh Vh as follows: .

JOL *IuhSM*seid froM tmm dlmensioa (on -hhhti 09o three stremalk functions on Ad31 ~

tbe o fotontsurfaces. Aside 1so additional 4J ~ (~ 2 Y,)d 2 (awvioedthat might limit (hr lhP~)d2(a

toa speial class of pro- rbo. ft* present work represents an ikndds""l of that amlysis to compressible'tew fields with generalized channel

Mse ee not"n orthogonal curvilinear T'R- (h3Xh) - 2h3P Vlh2bbe*g fitted coordinate. Yb. probleb for-2)AMom leads to a very efficient solution Tealoigdfnto ftescn

Thee geverig defuations ff ore theone~e~ike fnctions ons fo to's streaslike function Xv is given in terms

agoatlems with Dirichiet boundary condi- of its derivatives on the surface x2 -con-tiow iile the governing equations for astants1Watg" prsre, total enthalpy and stream-

tu wrtiit ae hyperbolic. The con- ax, (hxv hwI eoef si three-dimensional solution S~ lh 2 PV3 v *

is av .taand does not suffer from I1mcootere withMe txadi-(hh d 3a4in ot-thire "~Imensional methods In I j(l 3PV2h) x13 a

psmeo teoievorticity. 1 3rand .

Analysis a (hXv h hhPi (3b) *,*ftg* emia eas" s ed IS the

three dimmsional tawisoid rotational the third streamilike function xcis similar-_internal flow at* derived fro th bai ly defined as follows --equations of conservation of moss momentumand energy in orthogonal curvilinear body h- hx)*-hhD~fitted coordinates * In the analysis,* hyper-, 30~L31ia equations are derived for the athrough flowv vorticitye the total enthalpy,+ 2a (hhO dx 4)while elliptic equations are derived for - h1h~ ~2 3va2three sets of etreamlike functions which 12r T3are defined on fixed orthogonal surfaces. andThe details of the analysis for inoompres- ~*sible flow can be found in reference 16. a-**to the following derivations, the same l~r 1hx0 h 2PV3c **appoashl is applied to compressible flow rx2 ~ .**

sigorthogonal curvilinear coordinates. *3

Th0 Stremlike function Formulation rX (h2 3Vh x3(b

fte eqution of conservation of massfor ccpesbefo ncriierThe Integrals on the right hand sides ofcoordinates Is given by equations (2) throutqh (4) represent source

terms which are dependent upon the flow3 C (hY I) + a- (b hcrossing tethree sots offixed orthogonal

Sr + I (h 3PV2) surfaces. The superposition of the velo-1 2 city components in the above streamlike+ (- h 1 function definitions gives the three asI (h 2PV3) -0velocity components Vie V2 and V3 which

wher V~ V2andV 3 re te vlocty am- identically satisfy the conservation of2 3 mas equation (1)ot in the directions of the x.0 x2, 13

28

(k) In the above equations Ole a~ and a3 ae

vatives from the definition of the vorticity

' The elliptic governi g equations tattheatemIefunctions and x callbe otaied zm he ubstutions of

equations (2) through (5) into the x2 and

pis- oaponents of equation (t) and the xowponent of equation (8) reepeotiveiji.

,u~yuy 2 a ah h1 3 h7)

193 T'xl 2 23 r2L 40

~ ue(7) b hlh2 @ (13) 7

low" veos

*060016 bh 994101 in m m fe Veocity

a 2 3M a a V f

*w&I" dmnloeutos "Nam aqae o deive fx2 12 2 2 x 3 'x~tsI mewe t the Ihog low a -l

9.yp 'V (10 a7jUi 1

me I~z~11c equation can alobe derived (hhIme4the §Wbrf. Pfl m ow t de lown 3 2, 13

22 and a

am bseetta the ove eationh alot 12 a 1ofth elocitetor i the13 33 1 k

I~ I ,h (hu-(1)".(10) F1 Tx(2 b)

3h 2r

(a% V -0 V S~~S

Ph~hB~- ti- ph3 V,,3-Results and Discussion3x ?Vh ThO results of the numerical compu-

tations are presented in an accelerating3 retagua elbow and compared, ith exiot-

awwa deindbySai2 using ivsiAIncmpressible two dimensional Analysis to

Sh, X2 avoid boundary layer separation. This -Ig~- I b-(ph hV) dj exrimental data was also used for cow-

'13 39V 3 parison with results of the numef~ganalysis by other inveistigators,1C1

a. -Sn The experimental measurements were obtained -3t 33:, h"272 1k. ~ for different inlet total pressure profiles

- to investigate the effects of secondaryPnodur flow. The inlet total pressure profiles

were generated using perforated plates of1%0 4asaytical formulation results in; :different heights as spoilers. figure 2

~ ~ S~eehoeiPoisson' a equations for shbows the duct geometry and the comiputation-t"~Wdh h1hetbefuntions. Tese equations. &I grid in the x. and X2 directions. DueaWe Obtaimd from satisfying the equations to symetry, the flow computations weregof tlo on three "ets of orthogonal performed in the lower half of the ductmuftsee crmarseted by cons1tant values using a (9 x 13 x 55) grid in the

of he@O~~llats ~, ad 3 as shi :, and xl directions, respectively.I114. 1. "he source term in the result-: Figure 3 ahows the inlet velocity profilesAm, equtiom ar dependent upon the which were used in the numerical calculations ;

Jateeo h flow properties and an viAth no variation In the x2 direction. !4thO flux la the direction ormal to these The experimental measurements toe 2.5 Inchsurfces.15 A iterative procedure is used ;npoillor which were obtained half way be-la the smrioal solution because of the tween the pressure and suction surfaces are

deenenyof the source term in each set shown in the same figure. The results ofOF setreemlike function equations on the i the computations are presen ted in non-solutioms Obtained for the remaining two dimensional form in ]Pigs. 4 through 9. The

sets.flow velocities are normalized with respect

The Iterative procedure for the Rumor- to the maximum Inlet velocity, V1ax whileIMaL Cmputations copyists of the use of the pressures are normalized with respecta marching techniquLb in the solution of to the critical pressure, which corresponds

equtles (), 10)and(11) along the to a tank gauge pressure of 20 inchesthrogh fow irec io anda sccesive water in the experimental measurements.

over relaxation method W the solution ofThorognlcrinerbdthe two diensional elliptic equationsThorognlcvinerbd

13.(14) and (15) for the streamlike fitted coordinates for the flow computationsfftAmtioms. The flow density, which is were generated nuallioally using the codeallowed to lag one iteration in the numeri- developed by Davis" which is based on the --081 computations is determined from the :Schwrtz-KZistoffel transformatiod. The- ~local total pressure, total stagnation values of the orthogonal cordinate in ,.enthalpy, and frmtefow velocityl the transformed plane were between 0 and 1in the x direction and 0 and 6.791 in the .. *.

- - V2 (1/y-1) a1 direciion. The computational grid is -. '0 P(17) uniform in the x direction where half theduct height is cijual to 1.875 times theduct exit width. The convergence of thenumerical solution was fast with CPU time

Ihe boundary conditions for the streaulike' of 3.S minutes for 25 outer iterations .functions are carefully chosen t.9 insure on AMDAHL 370. The inner iterations did .. **.*~-the uniqueness of the solution."6 Dirichlet not exceed 25 in all the iterative solu-boundary conditions are determined for the tosfrXexad sltoso listreedmk functions from the requirawent rin o h vad .sltoso lOf Sero flux at the duct boundaries i'' The 77 surfaces.

xderivatives of the flow velocities areThreutarpesndfitfose Iqa to zero at the duct exit, while the computed static pressure coefficient

thet Diiblet boundary conditions for the distribution over the duct curved wallsstreemilike functions at x - 0, are and are compared with the experimentalinpeessed in terms of intigrals of the measurements. Complete spanwise staticilet flux. The total pressure, total pressure distributions over the pressure4Nratre and through flow vorticity and suction surfaces of thq elbow were

pefle are required at the duct Inlet reported by Stanitz et al L9, for the flow7to start the marching solution. more with nominal exit Mach number of 0.26. ..-details about the numerical procedure can 'be Sound In reference 16. Figures 4 and 5 show the computed

static pressure coefficient distributionover the duct curved boundaries at two

30

Ka -- A %.*% *

~ at~iS W55W0 The analysis is general and applicable to*5 a the di "am*e flow fields with both total pressuare and .

and the total temeraturt gradients. The resultsNOA84by the of the omputations are presented for the

- - flo in an acceler rectangular elbowwihsea n e .oca'y profile and to- *

tugung anle. The analysis predicts the

somata~~~-rr ar-nt ihte xei

ts n heregonewov

~4 S2~s hlfOf 1. Patanksr, S.V., 1paldingr 0.3., *Amm Pmathese figures, Calculation Procedure for Nfeat, Mass

Jr~ n h and Meaent=m Transfer in 3-D Parabolicaid 00eSq mental Flows. Int. .7. Neat adMs rnfr

9 the auction Vo1. iS. p. 1.157, 12 1.7r aeTase

WOW ugte 0In the comuted 2. Briley# V.R., umenrical Method forAW16 wereo at atet Of Predicting 3-D Steady Viscous Flow in

IS a Uw atribued o vLs-Ducts," .. Cam. Phys., Vol. 14, P. 8,

0, 1bu flaw begins to voll or half 3. Pratap, V.5., Spalding, 0.5., "Fluid"W' efo as at 2.559th towards the Flow and Beat Transfer in 3-D Duct11 IC1 PS@.1 Flows,- int. j. meat and Mass Transfer,

Vol. 1.9, pp. 1133-lrf1, 17rM.W~ases the oputed total

Wo -we, - ttecos e oa 4. Dodge, PR, -A Numerical Method for*al eeq -- im to N a 1, 23, 31 2-0 and 3-D Viscous Flows," AIM *--4).exa ere s emios correspond Paper no. 76-423, 1976.

as.So. 3.0 and 4.5 In Stanits"a~ui - fellow the development S. ft, C.R., -A General Theory of Three

saemnit u thecsg the rotation Dimensional Flow in subsonic -a super-injoes Wer sonic Turhasachines of Axial, Radial

ad 1bus athe 4,1 and Mixed-Flow Types,w MACA Ts-2604, *.19320 .

ft"We ohm dbos the tveise vorti- 6. Smith, 0.J.L., *Computer Solutions Of :at diftvaet me"5 sectional wals Equations for Compressible Flow

Fluid Mechanics Acoustics and Desi

W w" th espct o o Tubomchiel-IbParaw, pp . 43!r4-

reslt, rodetanImtena dcrease in the 7. Poaper C . 6Z-FE-23,v 197.k..

no og-i et cosnns the ta- "QuasiThree-Dimensional Flowicansurfahecet engtowarsithe Soluia ofs Turbin N .7.o En ineerin

met Welueteut. The reionofno" hoo LWeotct ow the is Sn er, Apri 197e Calculation of thW l"iteem te Thiaro s i gs-D~ Cuaing Trainsonc Veloc ien ass" f m3 sinc ponsue anfaee froad Fig. 3xa at-s d Strbiea Suface f a*svtha thro flow ccuplaoty hal ubmchnNrem the * du th nlet.aPrgrm o

100 . Xatsanis, T., "FridFota rgaIskitLE tor tea Tiisntcalculating Velsnc oci s and Sra-

lines on the Bub-Shroud MidchannelA -ehe is presented for the analysis Stream Surface of an Axial-, Radial- . ...

0 a9esmel theee dimensional --- presible or Mixed-Flow Turbomachine or Annular- £*5514 =,latiamal floow. The stresulike Duct. Zsz User's Manual," NASA TNO-

ff5m umets leeds to a ,'ery' 6430, July 1977.esemalelelliptic solution. that is

isulW for schima* applications.

2.gfl..m J63. MAn Ntheril~too, It, 18. Lakabminarayana, 5. , wEffects of ZnJletsu eSsile of Ube Three Variable Tmiperature. Gradients on Turbomachinery

Sms 2ev dst~a.' SP 5-304, Performance," J. of zncine~rzi forIai OmsCh e aess and Design Power Vol. 97, No. I, pp. 54-74,

6* - -, Part It 1974," 19. Stanits, 3.0., Osborn, LxM., )Iizisin,

w ~~ONM 13.. e.s *.. A "e Moeia thod J., *ass ZxperimentaJ. Investigation offts *an three Die410eal Seds- Seoay Flow in an Accoleratinq,teaU ftej Flamee* AM M 3775, 1975. ecagua lbow with 900 of Turning,"

13. esbai N ~ m.a~* *hs- ACA flI 3015, October 1953.910snesm ZesiesMA aepweeeib" 20. Sta&its, J.0., 'Design of 2-0 ChannelsSotfttt l 1).u.w- r esl moets with Prescribed velocity Distributions *..eM 1SM 10116 SAA161 Sobs- Alon" the Channel walls, X-Ralaxation

ftwt AM &U & tty mm- otions,- MACA TY 2593, Jan. 1952.

AME Smeeoii. pp. 141-149, 1979. 21. Iftore, %7., vMore# J.G. , OA CalculationProcedure for Three Dimesisonal Viscous.14. Dfer, 1. aT5mesm 75. atre Co~cosible Duct Flow," (Parts I

DisisS004 inelliq of Cases& Fumow and 11), 7. of Fluids !enqinerinc,hAL per 79N7 1979. V0l. 101. pp. 4L5-4ZO, 1979.

23. lawes. C. and Elsesh, Ch., Ofstational 22. Davis, 5.?., NGrid Generation for11w Inelalm i s ee Dxmosicsal Finite Difference Schemes," VRI*&ASs pasae, AM lever: no. Lectur* Series (1981-5). on Computational824-ft316. H92. Fluid Dynamics.

16. Ahiallah, S. amd Sia" A. onheM~lptiA Solution of Uhe Secondary

* Plow ProblM.' LI Paper NO.712-40-242, 1962.

17. inind, A. and Ahdallah, S., 'Stream-like Functions A New Concept In FlowProblems Formlation.' Journal of

Airrat#Vol. i6, no. 12, wim-erj.,,,w. 01-802.

x on MCONTANN x3CtCNSTAAN

SURFACES

x0

IUIT

Fig. 1. Schematic of the Curved Ducts Showincg The Three .* ....Sets of Orthogonal Surfaces and the CorrespondingStramlike Functions.

32

I J-023.

COO 1m.Go 24.0 S..0'b.0 1a 0 M

1.0.

~0.

U0.

-3. 3

0.1.

rig. . 3.e Mty aelcit C otaiol. i

33 .. ,

All.

A SUTION UUWACZ :

30

A* A

"Lao~~~~~~~~~r~, 1.1 -0 .0 f . 0M O 10 -- 0 -.

gymi~~ M~3 .(Xi) mnagqzsz M3 CO (i

Wal (3 3). i-p'(3 1)

34.

% ~ .*S.

KID______ $PAN _____ MID $PAN

1.6g U

L S1.885 ___ __

7''

1.89 % -4 S.-%-

.88-

M WALEND WALL *~-.

Ma ffl, 1 (d)X 4

rig. ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ -6.MrilzdTtlPesr otusi h rs etoa ufcs

35..

_M_& I__ _ _R_ _I__ _ _ _ _ -W SP A N

0.2.

.0

(b) N3-'3

0.4 '

0.8.

0.4II* 0.

1.awI8 SEND KA EN WL

, 31 (d I4

t~.7 omlxdTboq lo atciyCnousi heCosE*- nl Sracs

10 6U

36.

'N. V~? 4'",

A"..'' L'L'- *&~*

~ft-',&-ft

ft. ..

ftj%4~=.icftft*ft. '9. .-

~~-.i' '9,

"IF,

I

*~~9~ft%.

ft..-..'A

1'

ft.--Appendix 4* ft. *.ft.-.ftfts*C C%

I .'..,.ft'ft

Iatznal lliree-Diusnsioual Viscous Flow

Solution Using the Stremlike FunctionB.

B.* %ftF,

.ft 'ft -ft.

0* ft. - . .

ft 'F- *. 'F..

* ~*ftft~ftft.ft 'F.

~ft

~ -ft

*~S~ ""

..-. ft*.

37ft

- ~~%j%%'%''F.%%%%%.. ..*.*ft ~'

Iviscous a

Univesit of Cincinnati,,OtI an S.Abdallah,

colWe PA.

.. -N

AIAA 17th Fluid Dynamics,Plasma Dynamics, and

UsersConfrenc

June 25-27, 1984/Snowmass, Coloradoa

103 buwan". Wlw YOM, NY 1001.38

z~~Offin sinnza ST=MW ow nuw'o

A. named8004mmot cc Aerospace NagIner sd Applied mechanics

Uiversity of CiM!antiCismmti. shio 45221

and

4- Abdllab~e searcb laboratory

Pennsylvania state UniversitySt11e Couege Pennsylvania 16001

ae dfor the numerical solution of the Para- .'a now methodvfor-Stlkesdequatioge in primitive

.U ti olution Of %vari 1 e fonr of_$ ibj cr.Pressi-S 0based tuflng'7 ca flows. 2b o ndyY~gU~ityelliptic Influence which is accounted for

vat eveOped by the In the parabolized solution is that dueho ~ ~ o StfOth ee155t to the potential pressure field. Partially

Oa tto""-s vorioty parabolised solution prooedures-7- wereflowt n W~e ducts. thi suseuetl developed for the numerical

96 soluion wof viscous flow fields in which'I dW~5lpr~esure Is the dominant transmitter i. . . .. no deft Lgluecesin the upstream direction.~i~i ireested fthese flows a"e still characterized by the~uq~rimma masurmnt for absence of recirculation in the primary~pa w-amaa.viscous flum is a flew direction and by high roynolds number,

so that the viscous diffusion and thermalconduction are significant only in thelateral direction. The solution p.oceure

lemingunhanedfor the main and lateralbbl01k ometim equations, but the pressure

correction equation at each marching stop1 dust2atgthcontains term that link the pressure

Correction In a given lateral plane toUSnube 31d uptream and downstream pressure correc-

t thus tosin both 11t 1s the velocityI ,Yad componentse otie rmtemmntn..-U~vv wosi~~ ~ee ts s x y equations * and the continuity equation is

* t*Uonly satisfied indirectly by the pressure9velocity Wbotor field. this Indirect approach to satisfying-

YB momaliing vlociy at slet the continuity equation is common'to all -V&y senlsn veoiyape arabolisedt- and partially parabolized5 -7

XV'X2.X3 etroe"Like functionus .The full Xavier-Stokes equations areV kinematic viscosity required to model flows with significant

vortolt vetor in , yseparatinor shear layers not aligned

,11(C. vortifity components ixyad the parabolixed and partially parabolizeda direcin methods are not suitable for obtainingsolutions to flow fields in which viscous

phenomna significantly affect pressuredistribution. Several methods have been ~ 4developed for the solution of the tim

A large n P of i cial methods depnd~ni form of the governing equa-he been developed over the years f'or th tions. -J These methods can be very"Ofiut of Internal Viggo" flow fields. expensive, when used in the solution o0So eellwt olulem J wre btanedfor viscous three dimensional flow fields,"th arbesle Nuim-Sto weuotindfo but have been demonstrated to predict

-2ea ueesn h it irt iv solu- complex three dimensional phenomena suchties ~eim ee' have sinebe developed as the horseshoe vortex in turbine bladepassages. The numerical solution to the

full steady state Wavier-Stokes equations5 k~esin Asociae PllowAIMin primitive variables was reported in~eu ~ ~ gpp.reference 11 for natural convection and .>reference 12 for laminar flow in curved

39J

IRV t1 1% - forme for incompressible viscous flowtIII Ws WakShown to' be depndent

emthe difference sed s fewo ecniutrYYD-(.) 149sate especialyin the prediction of )

ONO14 01URfluttered cc the LasIde and -. 4Oe w o su affered a Mofmmand

full WIL nOVerge. andt author 0*- (2)that mer attentAin to the aun-

)MMaItiSS Of the flow My possibly4114dato the last two problem Ote n the above equations a -veo/v whenin~td developed for the elliptic solution the velocities are normalized by V*, the

Of hestedystate full Eavier-Utokes space dimensions by D and the vrticity by .. .egutles rebae" an the extension Of

0"e wall-hem. 2-0 strew untUnt&&--I*1 o di A--.Tbe solution to the three dimensional

Ivm@ ~ ~ deie etrptn viscous flow is obtained in term of thetua t Mootooly saisfythe quthree* vorticity components and three stream-of 1 ftorvaldnof ao*In hre diensonlikeM funticnS1 S Which are defined, to

amo fiel". So Vector potential ro - Identically satisfy the equation of cn-441W formelation hes been used to solve servation of moos for general three-Mthe sislm of I&lamna natural cone- dimensional rotational flow fields. Unlikeat":Xyxr In this formlationth the traditional stream function solutions"Oetm lovermimgsqeuatiose consist of which must be obtained an stream surfaces,bfto 3-D oiseon equations fo th eco the following streamlike function velocity

Potential and trevrciytasot relations permit the definition of thes"emteS. ase Main advwantage of Whi two dimens~ional functions X1 (xoy), X2 Cy,:),fesmlatonin two dimensions , Snely the N(, a) on fizxed non-stream surfaces In

smaer aso of the governing eqain thX lo3ied$a actually reversed in three almeonst. thflwied

eI~e three- Iaseal. differential equs- *rei~k -mbosVlct eaintite have to be solved for M RMhe three components of thme vector poten hmte-iefucinomlto

tial.~lkefncin frultowas developed by the authors to model

One oam conclude from the preceding Internal three dimensional flow fields. 1 -iS~ueioe tht eistii sace-llitic aredetails and general definition* in

selves of the 3-0 Nwaver-Stokes equations curvilinear coordinates of the sralkam 7Very costly In term of CPU tie function can be found in reference 17.store ge requirmeents. in -addition, goe Poe the sake of simplicity, the equationsof thee methods which were develpd for will be preseted here for incompressible . .-simple convection problems have not been flow using Cartesian coordinates.Very successful in through flow calcula-time. The presented work represents a Definition of Xam formulation for the 3-0 wavier-Stokes7

u + 2 dx(3a)solution, the formulation is based gn the T- I rSe of the 2-0 strealie functions1 , to aOidentically satisfy the equation of canser- -

vaiolf mass for 3-0 rotational xflws1 "'The governing equations con- v(3b)flat of the Yorticit.y transport equation 'a"i 2-0 Poisson equations for the stream-like functions. The present method is Def inition of Xvey general, *in that invisci4. flow solu-tions can be obtained In the limit whenSe Im n fact, numerical solutions have 3X2 eabeen obtained for izviscid rotational ~ y 2 -ImoompreasibleA9 and compressiblejL7 flows z0.4Ima cuve ducts and It was demonstratedthat the method can predict significant -seesadary flow and straeiso vorticity aX2 (bdevelopment due to inlet vorticity. The vT n2 (bfollowing work represets the generalize- Do~.tie of this formulation to internal three- Dfinition of Xdimeional viscous flow problems. 3 3 x3

Analysis IX0 (athe governing equations consist ofx uthe vortiaity transport equation and the 1X x(b

equation of conservation of mess, which z--w a *(bawe written In the following dimensionless a

40.

WV~

,'SO V

A,~* NO M 'I

~G w' 2 2t~~hmm+~s -k IX

fie Isme.~-so

Wtis St +' me S-X ;!iL (1

1

twily(17)

* 0Ai)

a--- go eig eutions (7)-C2) -aa m amm da.O the (M1-15) are solved for the vorticity -

~ the S~OS~its , C and the streamlike ., %*be ~~ei~lb ~umion X1,x 2 nd x. rsp.ctively. *..~~~ the boundary conditions ased for the sli- ~

g lm ties of these equations are given for thevisceous flow In a square duct. because ofsymer E4SI one ter ~ of the square

~~ YR ~dust is ousi= allwigderivations. Yb coordinate y along the .;,

* ,.*,straight 4uct io mesured from the duct* k ~ (6) entrance, while x and a represet theoordinates 1A the cross sectional planes

eeaured fram the duct centerl ine an shown '"5ia rig. 1.

w.A teilet station which extends far *2UmeS~~el.~ ifeumtal psteamof the duct entrace, the flow

-- qmmlibe fe ies are veoiyis taken to be uniform (v - i1 00dt a e u, w 0) *leading to the following

:4N-IlibW ;bIMAA1~ Oftlaity bo ayconditiones ''.

- 1) At the duct boundaries, the no slip condi-IUHIJ(12) tion is used to obtain the boundary condi- .,.

tions for the vorticity components, while*90"fles A the "no""ta t"o di=Mm- the mero flux condition is used to obtain4.,. *.

~S 48eew~tie ~the boundary conditions for the streamlikefunctions.

- ~ ~ I Mw S 2 h streamlike function boundary;;&. 4C 4 j dx (13) conditions are simplified through the

so appropriate choice of the reference coor-dinates x 0 so in the lover limit of theintegrals in equations M3-0S). Thefollowing boundary conditions result when .

6 ds (14)soY ~'.

,~ ~ ~ Dsultand Discussion

I, The elliptic system of equations aresolved uoing an iterative proeue At .*,eah gl Ia Iteration* the linear equations .- *were solved by successive relaxationmothods. Numerical comptations in ac-sragh duct with WO/De = 0.1 were pr

fomdusing a uniform grid with . %a. /OMe a As/Oft - 0.001 and Ly/DE. - 0.0033.. reN

- Vz eZ Mue to ajmeetryp the computations were only1 Carried out in one quarter of the duct for

X1 VC and 4 since C(x,y#,s) -q(s,yox)and XIL(xey)--X (sy). Relaxation para-meters Of l.d for 1.9lo for X3 and 0.4

M* ffor C and Iwere used In the inner itera-tiome with a convergence criteria of "

It's I x 0, C~ 1 x 10~ dI *X I XO10

*~ -. :umx 3.06 acrding to the following.-

and T"e numerical solutions required 50N3 ~~global iterations and a CPUtieo2_X3 mminutes and 13 seconds an AMML 370

00ps of the boundary. conditions are uig-1 1x1 nfr rdavoty at the planes x a0 and The overall numer of Iterations was 351

~ 4for the vorticity equations and 179 for theOtaw" equtin an. nhe numeri-

At a- cal solution domain extended 1.67 diameters* . upstream of the duct Inlet, where the flow.

U to one. The results of the numericalcomputatin are presented at y/026 - 0.*0,

0.01 and 0.10. The through flow contours0- - at the duct inlet are presented in rig. 2.*

a"d IM flow development f rom a uniform* * throg velocity to the profile of Fig. 2

at the duct inlet is acmpanied by lateral'flow displacements due to the secondary

At * velocities. The secondary velocity con-At 2 -4tours at the duct inlet are shown inrig. 3 for the vertical velocity component

0-- we The ellipticity of the numerical solu-tion is demonstrated in the velocity con-0tours at the duct inet, and in the velo-city fields up to 0.83 diameters upstream *of the duct inet. Figure 4 shows the

and contours for the secondary velocity com-and ponent, w, at y/Dft - 0.01. A comparisonX2 -3 0 of Figs. 3 and 4 reveals the change in

both the magnitude and the location of themaximum secondary velocities along the

Polly developed flow conditions are duct. The development of the secondaryq Ied at the duct exit. velocity component, w, along the plane of

syme try, x - 0p is presented in Fig. S.0 ~one can see that the maximum secondary

Yvelocities are found near the solid.'boundaries at the duct inlet. As the flowr-

0* proceeds towards fully developed conditions,the secondary velocities decrease and thelocation of the maximum values moves

IX I toward the center of the duct. The re-e sults of the numerical computations at the

-~ duct exit are shown in Fig. 6 for the

4 2

%

"M"Ir4t me stasak function 1. Patakr 3.V. and Spalding, D.3.,so MISsho In wife Vt through 11. OA Calculation Procedure for Beat,Wohraoa Mh. vwueity cmupcmet G Maso and Momentum Transfer in Three-

44 V/s 0.0 and 0.1 In Fige. Dimensional Parabolic loem t16 _!"VIO Wg e toi 10w the co- oura" Of neat and uss Trankfl-7i" 1 . w S U famw . X3 6t VOL*. 1.5, 1j#3 PP. 1757.I h/o p *X 0 &U mi t.". lstwfigue show 2. Caeto L.. COI R.K. and Spalding,LU ~~~ ~.28 0710 0e1h um- Numerical methods for

*Smms e t stzemlik function X at 29~rMtosi f.e ehncM am1m I '. *t O*A 7j p 1

W ~S3. Uriley, 3.3., g3umerica1L Method for-9~a4. La s a* new the Vale. ftedi ng Three-D.enional Steady .4

r ~ ~ flow vit~Viscous ?low in Ducts's ourna ofpm63. s ang n plo of symnsry, x Of 2 .SilamovI~l-i

S 764With the MnEtiiNta*LL mesueash memes161 Wg. 2. 04.~ Sugg, B.C. t McDonald* a. # reskovsky,

~ ~3 9 ~ J.P. and Levy, a., -Computation ofmeaure Three-Disnsional Viscous Supersonic

salq mt aoes he 5 case lie arFlow In Inlets,' AIMK Paper no.

W4 EooS10 ad remit 1 a the gieSa . Pratap, V.6. and Spaldings, D.B., 'Fluid

M~sima dateas - bw In Psigs* 12 an Flow and Meat Transfer in Three Dime-W . sional, Duct Flows," 1st. Journal ofto To y eMtistbmho An VIAw Of the Neat and Mae Transfer, Val.* 19, 1!4bm~dam inrs rid used In the m sriosalpp13-15

6. Roberts, D.3. and Forester, C.K., *AParabolic Computational Procedure for . .0Three-Dismnsional Flown in Ducts with

thiapape prssut a se mthodforArbitrary Cross-Sections,' AIMA PaperOh ~e-v9mW a elliptic solu Mton fof go. 79-143, 1976. A A alu

the amir-ce euntione*G is basedy 7.* Moore, J. and Moore, J.Q, ' Cacuan th trealike-inctim voricitylation Procedure for Three-DimenIsional.... ~

wkoiltcm. h Vicous reltnsqor e Viscous, Compressible Duct Flow.thre-dlsmsimalvisous lo In sqarePart IX ; Staynation Pressure Losses

5.0 s preseted and compared with na stguar Ebw ora f~4eaedatal mesrnt.The remults Vldurt that the streanlike function Vl ~,

com swmweftUly model viscous effects in p.434fthe three disiual fl1ow field campu-

~ii5. ine te fomultin hs I. Briley , W. R. and McDonald,, H.. 'Selu-taesuseful uIc t sem formuion roas tion of the MUlti-Dimensional Compres- *Basea- flostal o nLvsi oe sible Wavier-Stokes Equations by atina, loto model secondary f low Ceneralised Method,* Journal of Cmu

evelomt due to inlet shear velocityttoVlif77~WOWt the effect of curvature, mne an ainl hsis 01" -comtude that the present method can be e32effeftively developed to obtain efficient ~ en .. adWrig .. Anumerical elliptic solutions of internalImlctFtreShmefrteC-viscos flow In curveid passages. pressible Wavier-Stokes Equations,**. *

AIMA Journal, Vol. 16, 1976, p. 393.

10. nab, C., *A Wavier-Stokes Analysis ofThree-Dimensional Turbulent Flows

This work was sponsered by U.S. Air Inside Turbine Blade Rows at Designforce Office of Scientific Research under and off-Design Conditions," ASE Paper .Contract Do. 63-0242 No. 83-MT-40, 1983.

IMe authors wish to express their 11. Williams, G.P., *Numerical Integrationaprsation to Mr. M. Nansour for his help of the Three-Dimensional Wavier-Stokes ~h efrming the numerial computations. Equations for Incompressible Flow," '

Journal of Fluid Mechanics, Vol. 37,Part 4, 196, pp. 227-35.4..

%P

43

ot foloSolution 16. AbdaUak, So* and Namned, A., 0"m"~~*4minmimlLoonar rum, In Flow Probleam, Journal of aiern~rpFinmqPlite ifferemo. for lovere VOL. Lo, No. lr 76

inA v Me ZTW OctberZW~7~JS~33. 17. Nined, A. a"d Liu, C., -Three Mlen-

@ionaJl Rotational Compressible FlowIS' S ABU 6ad 341am., J.D., *hmsorioal Solution Lu Variable Area Channels,w

3e2*ios of the Yhree-olamaeiomal AXAA Paper no. 83-0259, 1963.~stlamof 11ftom for LOWmnr

Pmtga~vetimI~g g~osof 14. Goldstein, 3.7. and Ereid, Dze,10 t &"It pp. 'Measurement of lomminar Flow Develop-

U~I. Vl.sent In a Square Duct Using a laser-Doppler Flow Metor,* jora fMled.

't4 4" Dvist . Devab, Noais, Decmer L967 7 1113117

15. 3 a. md A ewel S., lteati-

&an. lNoms &am evcomoept Lu flow

MinmZ -0. 7I 6 3.i-wne

0.1.

1.0.

-j 100

doe.

Fig. 1. oordinate Systeim in the Duot. Fig. 2. Through Flow Velocity Contours at - . 'the Duct Inlet (Y/DRe -0-00).

VI

44

'--V.

40 4

-.006 0.0 -01

.02v

Planet x 0.0.

0.2S

0

.0

no. 4. Secondary velocity Contours at Fig. 6. Through Flow Velocity Contours/De-0.01. at y/DR* 0.1.

45I

0.00L

-fty. ~ ~ @.6iliy otusa h ut ri.9 tesiorntoxCnor

l~p. 7. Vbzt~tcty, ~ Cant~uraat the Duct Ii.9 temieFnlt, x3DR 0.0t).

-0.5

-4..0

3.0M 0 0.1

Y/Dft 0.1.at V/DRo 0.01.

46

4.~ ~ . 3PMNWZEAL DATA (RtF. (161)PRSTRESULTS

0.5

0. 4

0*30

*0.2 - *0.0075 0.020 0.10 .0-s. 3L

1 2n1 2

Vg1.Stzesmlike Funation, X, Contours at s 0.

Maet Inlet

0.54-0 5

0.4X

0.3

454

0.14

0.2.

-100.0 1.0 2.0 3.0 4.0 5.0 -Y/D

rig. 12. Devmlopusnt of the Through Flow velocity Profile at the Central-Plane xz-0.

2. ZPEMETAL DATA (REF. 181)

2.2.

2.0

1.6 Fig. 13. Through Flow Velocity Develop-ment Along the Duct Centerline.

1.2

0.00 0.05 0.10y/Dfe -

U 47

~Lvt~.1~,6:'

1;..'.. *. .*.

b. -~'I

L-~

* *- *.-' -**-*.,*

~ *- *,. ~.

% .'*~* *-

4-

'9

4,. "-4

'4...,.,

Appendix 5

1!he Ulliptic Solution of 3-D Internal Viacoum

Flow Uming the Streamlike Function

4Q ;i'-~S.-.

*.~.... 4' '

. .4.*-4.

*~ S.-.

*.T %1'~4.,S.,.

~. ~

%Jq.

.4.' 4.4'

484.

* *4.* 4

11 q; Agr

As M5 Ledeas bga.i piA M~rch Laboratoryso a Mlee UshmdmleinLvana State University

~4M~m~ t C .mtistate College, PennsylvaniaM-4MOt-R Ode U.S.A. U.S.A.

"a -froiltilas of the onale 3-0 amo field in turbo-iMIMn blad """a Continue to be te subJect. of manyftim.. ad - aVIagi flow studies. Maetly developed 3-D*Awrisaid methods 11-31 awe coable of rdcig3D0 flow

inasmistice sacb as secomary £1... (31. While theemabay flm is eaused byr vartioitr which is produced by

I a In - thane InYLsedd methods cannt predict the,,ism flow M is t a " in laedes passage. internal vis-

4110go ouinmtoshave been deve2.oped using pare- - 9.behead ftv- -am equtions. While the"e methods are veryfea't. their IMUCatio is lUmited due to their inabilitiesto. Sioux#"e We M 'blocago and strong curvature effects. .betially pabolised methods maintain the advantages of the

iwOU~isd MeUM& In that the streewise ditffusion of mass,m~it nd M eergy awe sti4 neglected but the elliptic -

issa.. is traCNIMitted opetrom through the pressure field."NOW well developed methods are not discussed here since thereader em reter to the eZtezuive review of Davis and Rubin (41ad ftin (51 . Our following discussion will be limited tothe fully elliptic methods for the solution of the 3-Dftvift-6tIkee equations for internal flows.

Asia And Uellum (6) developed the vector potential-voctiolty formulation, and used it to solve the problem ofIsmiater natural. convection.. This approach is an extension ofthe well in 2-4D strem function vorticity formulation to3-0 YCOblam. the equation of conservation of mass. is iden- A~tically satisfied through the definition of the vectoraptental. Zn this formulation, the resulting governingequations consist of three 3-0 Poisson equations for theveckt potenal& and throw vorticity transport equations.Wliams (71 used a time marching method for the solution ofthe- haMmerW inanrssble flow field due to thermal convection

49

~ ~iaq p LA ive VaRIAbesS. In WdisU414A Im Q0M~W am th. soltio of a

~tioawith Wewesn boundary onditions. BothOAWMe wewLze, "- solution of these para-~bs~u. "h first method requires in

~ut*of the 3-0 koiss.. equations with~ seditum Ablea two boanndaries and Sos

"a" the third bUffdary, Vwie

elittnuma 3-0 Poisson equation for34 1)diseowsed me relativ* mrits of these

of .1@utimse *ompater strae require- '~"Latea O POLOftequation.

~~eiUWU4outio fg Viscou flow Lu a*1 ~sotw dla~t (10] fun. te governing

Lu et1* vavr.ebles ing a new finite dif1ference*01OO ware eiMntered In the application Of

a ~ Sle oelulatoumI the tendency of theS1W Aw lows of ass beftwen the diact inlet

4b~~'111111u1a," ANOe XeCert1y Dodge [1111:'-*Ut rcedare IS WhiQh the Velocity

1isou saOM d potntia components, and thedoteaimefbtr the potential velocity

~US~ he WSUY eqtion for te Viscou, velocityVSW m~ ~Ia s aba bar the inent= equation with

pfil 11 61msed Lu m to f f1the derivatives411FIu11111CI11W veoorcaan -to While the govern-

'ti qgts.hrt Velocity potential vector componentis ~~hiauI bee mtty. seya" this formulation,

00ft*** ~WOICI soltion pocedre is partially paraboliceiiihe malsted the stremdise diffusion of mambotii in

ges~~AA soml~pelution for the viscous velocity carn-psss~.ash analysis itself, in terms of the type of the

pouhet te velocity pressure formulation since the govern-

equatOn. Dodge did not discuss the boundary conditions forthe vlesi0tY potetiAtl. Be only mentioned that it can be

=4u that he used mr. potential gradient normal tothe wall In his nwerical. solution. Other formulations thatam lead to elliptic- solution, were described in references(51 end (63o however they will not be discussed here sincethey have not yet been applied to 3-0 flow computations.

Zn sommul existing elliptic solvers for the Navier-StOkes equations require the solution of one or three 3-0laese.m equstions in addition to the three ininentan or theIke VOrticity transport equations. In the came of theV*20mity pressure famlation, the velocity vector is evalu-Sak Of Lier tentwe eqation and the continuity equationis satisfied indirectly in the pressure equation. te con-

vegeceof th itraiv nVa roceni Aol be very

50

'.-' r

,

-m 0 a" is" Amr ab. su o .* theso""= boom""y ooaditims for the

Sthe Other hade. the rntnuity .quation -

,~thmse 3-0 btiasmequations for~jmnti~a.Omaeweasof the iterative

~~ LaW Wx ows ae, sIwc irichietof the bomrnie ht ft"Puter

j~ ~t~y Abresed bg two additiasal

~k ~sm a a m £ Muuan~ for the~~A tat 2.eb to a MYeomim.~ Us~a is base e the use Of '~

bto MUM satisfy thelww a4 90"twiesluS 1.sii addition to

ULrIchlet heundary con-~- ~ WU~U~ £~stics. me, pwsmtud mthod

tb* l~ iS.4d flo seltiess am be oh-

J141 timasla c09,11 ducts and it

, 6 1aft Stlaty "ae waity ot theOW. 9 flow WSAuk are indirecty

!LW fIpwe;6y hew. maridrn-........ 1.3 the M flwigu~mk pe

"MIA qem t the Ia tatis Are rsented forY#~S Slu 301.Sof a =az entrance flowe Ln

2*5.1 a to imlop* a method forta~edmena flow tieA, and the losses

~~~~u ita So ieldsvwit larg surface cutur andUiornt i mem effect an In the case of turbine blade

me "oVegai" eqUtIoms ae the vfticity transport9AMUm d Uhe aqutic. of aosmsevtAn of mass, which are

givn elo L anaia~e~t~Y vctor form for Incomprssible

A,. Y9*0 (2)

51

W~ uel bs M,. (R- mqears in the equations, as

-w.t D 4W WISh wrftICty Oy V/D* Where V is the

~ US~~s~ ~S)are wittem in Cartesian coiatsas

3. 2

(6)

16 volt wre the owuata of the vortict vector, Lin~ 3. 7 5 ~EW e U iieY utvwv are the

f tin eloityVector V.

~~diOf Cistaonof ae

U gicty Velocity Relations -

(9)

(10)

.2 4%.pnlk linitioa FOICUlation .

"me sremlike functions 1121 xh(xi), xv(yz), x C(xx) f*

we defIN to identimally satisfy the eqtution of conserva-Mien ed es In the case qf internal rotational flown (3.3,141. Us velity f ield is determined from the steaelike %!

fesc~e..accordif, to the following relationst

52

Ohl (12)

(13)~ ISa4

#1iw

(16)14)

Ww lbin*aD be V MIA0 24hff to InoUtiiAM On the* I~L Yu~*EoleM .s eatoi plas spective ly

U~h*S1 v o(17)

~tis(21-M7 IntIsty Ow egmtn Of meVtio Of- (7) *.iint~smlyt

~tI~~a~.qstlsms (1) -(17)Infto equtrAs (s)- (10)

2 a Sda.aw(19) %

3; a

x aaxIx In Sao

(20)

53

(22)

* (23)

INM~ "119i (4)-(G) iu Cl8)-(20) are solved- ------ 11*4 OM the stremhtke

- :~ X, e xer wsspsobilvlys the boundaXy coaft-.hM OUi CC m t ese uations are given for -

il omI - a6, u m Injiavl flow is assud

Amm *OWSMM 604"m. apply over the extended

tdot suao. mueo symmwetry.low to osidered in the -9..p."VO WIIN Otthhem 1

JM J @.Ift= he aft enterline r yffWIN- ms k bm In rig. 1.

- CASCADE ILEST

______ 1 u~~~~1 .. L5.

IL -- Le ~

Fig 1.A Schemtic of the Duct with Cascade Entrance. ~

54 ~

;Sao,

of 00 boSrcm-t

........ 0'q . ~ b U y a n ii. ~

~t 5g~Urn setlotuttoa of the n*Of$$ ft -4CLUM ~htefa

~~ ~st~jo w C7 b ~y~±VL&

~~~ ..~ .f .h .e ....

th bL.lgsajto

axv4.

0 0tWjz

01 )(VX,.O#X,']e 2 ~

"If 3 w ~ ondyc*to frtes0mUfU ____________of_____________________

at th 4k fsyt ~tm

5s -

hI ....... ....

-roo-

rL ~

NOtA3ase th iatCndto

W x,J-1 w* ........

* IvA0

tw A -1MI teUdfo odtosaeasod edn

00

I ay .

00

2'~ - -- - - - . I -i . *

45..I 15satisfie'd tbrough the vortiotty.0 qnty osutioas wre used along

dpSsftW2 *ptga& plame to deteguine the~ o@UMtle.shm La ftq. 4.

40

I Is

ax.

I a,

40

* ** .x

tt0 S.~-

G&WGOVA lOW OOL~iOU AV*WSM~, ledin

tt'0*

4i#

rngm

1-57

MOPMO.

,..0.

, z

If

I IL .

I gure

3. IN~SnMDSM

ft easo t wrcl cmuain r rsneIn a 9wter if *"w dut. n i ae nytooth 0atet eqatca ;..p (4 n*Sladtoo h

ftmol~~ik afluaain e.(9an(21aesvd

at 3.e duct enrac an eit -h iflenefthcscd

watrigare lts tic solutionl codmtonsae inesontedroig. Guart.o qaeds.Z hscs nytoo

slt wasotain for e 0dinyavecty cithponet , 0.aw ~De sna - 0.0an sin75 501is.7 and a 1 1b 4)gi.

.4..

Lbftr ad the lob shew the through s o eocary vlcitursate the dutetrwen t. he influencen of the tho lwvcascad

Iwo &latws the nof scomeary #elocty i componen.1a

ares~mat /D~ -00 ad 00058 nFg.7 n bDim ~ ~ ~ ~ ~~~A!;1 thes fi e necn e alrg hag i ot h

1.0

114. e. s ro n ow 1. Velocity COmtours at the Duct.

0.5

1.0

x0

Fife $be llwovh Flow Velocity Contours at the Duct Exit S

59

*A'*.,.& -.-

nowe

8600040ty~~~~'. Veoiycotusat.w .

2*V 9b OOdZ OI-tyCnor tYD& 007

608

%A %

0 -th Ing"la

z* .. '1447

et xefexam 1163. One can am that the computedU~ gmed agreemnt with the exiperiuental measure- .**

,* yWM 0.0075 and 0#02. but that the computed1 veloities at y/DO - 0.1 did not reach the

W~a3Jy MOSewe ftuy developed profile.

.4.1

0. LL(e. 16)

0. 3. 2-4

0 1@2

046

atteCeta.lae10.% Z

'0via 14 ohm thDevelopment of the secrougr vlw elocityl

-*ooot w~e v la the duct plane of my see try, z - 0.~ emerieatl masuromets are available for comparison .

With hec atdsecondary velocities. Figure 9 shows that -*the sexism secondary flow is initially located near thewlJA bouidrims then moves towards the center of the ductand decreases as the flow pr ods towards fully developedcondtloft. the computed through flow velocity development 0.

- %7 alIga the dhot centerline are compared with the eperimentalmeumeats of reference (161 in Fig. 10. One can see in

this figure that the elliptic solution predicts an increaseinl the centerline through flow velocity in the cascade entryre~on Sceceding the actual duct entry. Figure 10 shows thatthe omUtations slightly underestimate the centerlinevelocity. Considering the coarse grid used in the numerical .

c~ttiono C((li x 34) grid points in the duct],* theagressatof the coputed results with the experimental data -'**.

as shouwn in Figures 6 and 10 is very satisfacto