turbomachinery design and tonal acoustics computations€¦ · turbomachinery design and tonal...

MCAT Institute

Progress Report95-17

TurbomachineryComputations

Design and Tonal Acoustics

Aki I A. Rangwalla

January 1995 NCC2-767

_VICAT Institute

3933 Blue Gum Drive9-San Jose, CA : a127

(NASA-CR-197749)

D_SIGN AND TO_-_AL

COMPUTATIONS

1993 - Jan.

53 9

TURBOMAC _ INE RY N95- 26 777

ACCUST ICS

Final Report, Jun.

1995 (MCAT Inst. ) UncIas

G3/O? 0048502

https://ntrs.nasa.gov/search.jsp?R=19950020357 2020-04-16T20:51:53+00:00Z

Turbomachinery design and tonal acoustics computations

Akil A. Rangwalla

Objective

This report describes work performed under co-operative agreement NCC2-767 with

NASA Ames Research Center, during the period from June 1993 to January 1995. The

objective of this research was two-fold. The first objective was to complete the three-

dimensional unsteady calculations of the flow through a new transonic turbine and study

the effects of secondary flows due to the hub and casing, tip clearance vortices and the

inherent three-dimensionM mixing of the flow. It should be noted that this turbine was

and is still in the design phase and the results of the calculations have formed an inte-

gral part of the design process. The second objective of this proposal was to evaluate the

capability of rotor-stator interaction codes to calculate tonal acoustics.

Motivation

The ultimate motivation behind this proposed research is to be able to simulate a com-

plete propulsion system. In order to do this, certain key CFD methodologies such as

turbomachinery flow solvers have to evolve to a level of maturity so as to be used with

confidence. An important criterion for a numerical code to be a design tool for turbo-

machinery applications, is its ability to predict accurately, the secondary flow features,

losses and other more sensitive flow quantities such as tonal acoustics.

Background

Computational fluid dynamics (CFD) is playing an increasingly important role in the de-

sign of various propulsion components. Considerable progress has already been made in

using CFD in the design of turbopumps and impellers. Turbomachinery (rotor-stator)

flow solvers have been developed at NASA/ARC which incorporate some of the most

modern, high-order upwind-biased schemes for the solution of the thin-layer Navier-

Stokes equations (Ref. 1-2). The family of rotor-stator interaction codes (which includes

ROTOR-l, ROTOR-2-4 and STAGE-2 codes) are currently in use in several industrial

and government organizations. So far, the unsteady rotor codes have proven quite capa-

ble of predicting pressure variations on the surface of the airfoils. They have Mso demon-

strated their capability of predicting more "sensitive" flow variables such as the total

pressure losses in the flow field. In addition, two-dimensional versions have been used in

the actual design of turbomachinery (Ref. 3). However, the success of CFD in the de-

sign of turbomachinery has largely been in predicting the two-dimensional time-averaged

and unsteady pressure loads on airfoil surfaces. However, there are important three-

dimensional effects in the flow associatedwith turbomachines such as secondary flowsdue to the hub and casing, tip clearance vortices and the inherent three-dimensionalmixing of the flow which require additional detailed analysis. Additionally, the flow inthe tip clearance region is not well understood. There can be considerable flow turningand temperature variation in this region that can affect the overall performance of theturbomachine. An important aspect of the flow in a turbomachine is the impact on theoverall lossesdue to secondary flow features. Hence the timely completion of the three-dimensional unsteady flow in a new transonic turbine in the design stage would makerotor-stator interaction codesdevelopedat NASA/ARC a viable tool for turbomachinerydesign.

The secondaspect of this proposal was the calculation of tonal acoustics in turboma-chines. In Ref. 4-5, two-dimensional unsteady rotor-stator interaction calculations wereperformed to study the plurality of spinning modes that are present in such an inter-action. The propagation of these modes in the upstream and downstream regions wasanalyzed and compared with numerical results. It was found that the numerically calcu-lated tonal acoustics could be affected by the type of numerical boundary conditions em-ployed at the inlet and exit of the computational boundaries and the grid spacing in theupstream and downstream regions. Results in the form of surface pressure amplitudesand the spectra of turbine tones and their far field behavior were presented. The "mode-content" for different harmonics of blade-passagefrequency was shown to conform withthat predicted by a kinematical analysis. It was however assessedthat a similar three-dimensional calculation would require a highly accurate algorithm since relying on veryfine grids would be impractical. Also, three-dimensional non-reflective boundary condi-tions would have to be developed. It waswith the above in mind, that the developmentof a new high-order accurate multi-zone Navier-Stokes code was initiated. This code isbased largely on the ideas presented in Ref. 6. Figure I shows a comparision of the re-sults obtained by the new code with that obtained from ROTOR-2. These preliminaryresults look quite promising and have indicated a further study in the development ofthe new method.

Achievements

Flow predictions in an advanced transonic turbine was completed in a timely fash-ion. The task for the calculation was given at the same time as fabrication was initiated.The numerical results were obtained well in advanceof the first experimental runs andhave already played an itegral part in determining the placement of the probes and havealso facilitated in the understanding of the experimental results. This exercisehas madethree-dimensional rotor-stator interaction codesa viable tool in the designprocess.

A detailed study of the capability of the two-dimensional rotor-stator codesin com-puting tonal acoustics was completed. In anticipation of extending this capability forthree-dimensional predictions, development of a new high-order-accurate flow solver was

2

initiated. Preliminary results appear to be promising.

References

1. Rai, M. M., "Navier - Stokes Simulations of Rotor-Stator Interaction Using Patchedand Overlaid Grids," AIAA Journal of Propulsion and Power, Vol. 3, No. 5, pp. 387-396, Sep. 1987.

2. Rai, M. M., "Three-Dimensional Navier-Stokes Simulations of Turbine Rotor-StatorInteraction; Part I & 2," AIAA Journal of Propulsion and Power, Vol. 5, No. 3, pp.305-319, May-June 1989.

3. Rangwalla, A. A., Madavan, N. K. and Johnson, P. D., "Application of an Unsteady

Navier-Stokes Solver to Transonic Turbine Design", AIAA Journal of Propulsion and

Power, 8, 1079-1086, 1992.

4. Rangwalla, A. A. and Rai, M., M. "A kinematical/numerical analysis of rotor-stator

interaction noise", AIAA Paper No. 90-0281.

5. Rangwalla, A. A. and Rai, M., M. "A numerical analysis of tonal acoustics in rotor-

stator interactions", Journal of Fluids and Structures, 7, 611-637, 1993.

6. Rai, M. M. and Chakravarthy, S. "Conservative high-order-accurate finite-difference

methods for curvilinear grids", AIAA Paper No. 93-3380.

3

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The Ibll.wing aualysis is limiled I(_ Iwo-dJmCllSi(_nal II.w hut call I}l: exlemh_d Io thr,,e dhu_lls.h}ns in a straig, htforwar,I

fasldu. ('u.sid_.r a sin&l,:-_tage lulhim,, wh,'re S is Ihe nt=mh_.r _ff stalor airfl=ils a_d /I I1.' number of rotor airfoils. The

composite pil<:h is the transverse (lista==ce over which Ihe Ilow is periodic ,am1 is denoh:d by I The dislanre bel.wee, the

stntor airl_}ils is I/S a.d _l_e dista ce I e _',!ea rolor aidnils is I/IL The v_.lo¢il.y of Ih,, rotor is I_m)l _ I hv t'll The prPs-

sure at any axial plane in the I1_ v li_hl is n_sum,'d Io he p_rimlic i. lime wilh perk.I equal Io Ih_. miM_tun hme requir_'d

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Ihe period ia _=slalionary frame of reference is given hy I/l!/.llC This is also know== _s Ih,' hlado passing lime TIw pr,'s-

sure variatl(,, iu a=ty particular axial plane is givP. by

............. (T{,,,,_ - ,,/¢1¢';It)q &,,) (1)

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pa.ssing frequencies are absent. This Ihen yields

m = nR(s#n(Vl_)) _ kS 12)

where k = . .., - I, 0, 1,... is the spatial harlnonic of the disturhance produced hy the statnrs.

Propagatin_ Modes

Analytical solutions representing the unsteady flow in the far field can also he derived, as given in 181It is assulned

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_ R ,'_ln ,'1I_,

I,T _ + _ > z (3}

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A. = 1(1 - Mz 2)(,I},_,(,,/¢_t,_ + M_,,,)4 _,(v,}v,_- _ ,%,,,}_ _ (1- _£_},,,_

v,,here =n is given by IBq (2) For very low values of axial and transverse Math numhers, M., and /1/v respectively, tile

maxin)nm waveleugth A. can he apl,roximnted hy stzl}stil.ling 0 for _l t and zl/_, in E{I ('1} For low va _ es of ,_[H it {'allI)e see=t Ihal for the lower harmonics,

I(x.I..... = __

_R,Un (5)

For a 3-stalor/,l-rolor case thai is prPte.led here, [A_J,.,_r = I/(3fIM/_) since the fundameldal and the first ha_znonic do

llol propagate under Ihe presenl assumptions of ,_f., Mn << I V_'h_m a hm_ grid is used Io dissipate th,. i_r_q>al{ating nmd,.s

i. thP far eh r _gion, the grid _p:_ring_ ,mar Ihe Pxit at_d htlel hu,m{h_ri,.s are rlu_sen I,) h{. ahoul [.\.],,,._/2 h _houhl he

- 262 -

noted that there in a possibility of large _xial wave lengths when ,_l_Mn -_ V/(J-_-" _1_)-,, but this would occur only f_T

higher harmonics umler ihe prese_t _._sumplions of low Mach nunll_l.rs.

Derayin_ Modes

A fourier mode will decay with in{re_t_i.g axial dislance from Ih_. rotor.stah_r pai_ if Ihe inequ;dily sigu in I';t h Ill) is

reversed. Tht' amplitude of Ihe mode wmlhl vary _s

a,,,,, {x e ±'_-'_ 0})

where

,2_ 1 ,_r I ....v_l:'}'& = LTi--T_,]) _{ - Mh{,,,P-[,,_f,,, {_}

A decaying mode decays exponent.i_lly. It is nnderstoml that [or z <: (I the posJlive _igu in _q. (B} is nsed ;rod for Z ." [)

the negalive sign is used. The variation of the natural Ingardhn_ of Ihe amplitude (h_g(.m,,)) is liuear with slope :Ed..,,.

{_EOME'I'ItY AND GRII) SYSTIBM

The airfoil geometry used is that given in ['2]. TIw tw_.dilnensi,a_al computalicms ol tlHs st_ldy were perf(_rtoed ttsiug

the experimenl;d airfoil rro.%_-se{:LiOllS aL nlJdspan. "l'hesP rrt>_._-sertions, alollg with a ._,:hematie of th,. COmlaltalion,d grid

_re show_ in Figure I. A system of patdmd and ov*_rhdd grids is used to discretize thr' II_w r¢&kat of inl.erPsl. The ilu_el

grid_ ar,_ O-grid_ that were generaled u.i_g an elliplir grid gem.ralor. The outer II-grids were generah:d algehraically "1 I..

experilu_mt ct}nsisted of 2'2 stalor airfi}il._ aad 28 rtllol airfoils. Io mod_'l thor exp_rimet_lal setop the Ilow over "_=t le_t 25

airfoils (I I sin(or airfoils and 14 rolor atrfi_ils} wmdd have Io he caltulale,I. This wouhl require exressIve ¢ompulnllonal

resources It was therefore decided to solv,, a smaller prnhhmt I,y nsing resraling str_legit.s a._ shown ia [131 ia order Io r,:-

duc,_ tl_e airfoil c.unt. '['he IlUllll_er of slalor airfi,ils was rhal_g{'d frOlll 22 h_ 21 tlltd Ihe size of _'ach imlivid.al slalor airf.il

w._s enlarg_.d b} a lartor of '22/_1 This r_._r;din_ all_ws a :l-stalnr/I-roh,r ¢_,mputal.ion wherein period[cit} is imposed nn

the Ilow ovt_r 3 slator airfoils a_.l 4 rotor airh)ilt. (!hanging'the airfoil _ouut does (:hange Ihe i_alure of the tonal a¢oust i¢s

in the flow liehl herause Ihe mode {onll.ld. _f th,, propagaling nt_.le_ depcmls upon thP ;"rfi}il romd as indiral,'d hy IBq_.

(2-'/). llowPver, si_re IIw obje_'live c_f lhis pr_.hmi_arv ilB,,_ligati._n is Io evatoale Ilte c:qmhdil)' of _nt_r-stalgr i.t,:rac.

[inll codt.s :_1 rahu]atiltg I,}lla] ;iCOllS{ica. Ihe rc_, aled l_}lor,_l;t(,,r g_.ulllelry 'a';ts list,d, II Sll_llllt he iiiCllliOll_'d Ihal PiKtl[,t

I shows rmly a srh+.ntalir of the grid. 'l'h+r :+{lu:d numher (,( grid points is nnlrh latg,,r and Ih_: sl,n¢ing I.'twc_'. lhe grid

poinls is nul_h smaller, More d,,lails nlm.I Ihc grid syslem ,'all h. _ fi_utid in [1',11

NI:MEItI{'A I MITI I1(}11

'l'h<. iiIl_l,,adv. Ihin la._t.r, Navi,._-Slok,'_: eqllall,,ll_ ;irt: s_h','d tl';illt_. ,tt_ upw_nd-hi:t._',l Ihtit_:.dill_'_enre alt.t_rithm. The

kill_,lllalz_ _i:,,_,stt_ _;is ¢;d,-uhd,.,I o_itl K _qllihl,rlall,l'_ la_ _. al_d Ih, I_lzhlll_.ld .'d,I} xl_)_Jt '." wa_; rah.lat_d tl_ili& lilt

It;_ldwilt b_lll:tx m,,,h'l "1 hP II_ethod is Ihil(I-_|H,.i arC_llal,' ill spa, e alld _,_,_'l,lld _q_lPr.a£_ tlral_' ill 11111{% At _:afh lillll, sl_'p,

sc_urnl SPt_l,,ll ih i:tli.us :m" I.:r[t)rmPd. _,, Ih;d Ih,. hdly h.l,licil IhlHe ,liff_,r,.n{'_: ,',luali.ns ;*r," s.h'ed. Addili_nnt d_'lnil_

r_arding lh_ _{'h,'_.,' ,:._ I,_. l;.nBI i_ I1:11

lit }IINI)AIIY ('()N I}ITI{}NS

Th, I.,.r,da,y ,oiBlili{,n_ r,',l,,ire,I wl,,*. ,,si,,;_ In,,hil,l," _,}_,,'- , ;_,_ h,- I,r,};tdly ,l:_t,:ifi,_d inlo hvo lyp,.s. '1"1,{. flrsl lype

i',,l_ists ,,f Ih,' 7{,il:d ,'oildiliun ¢- whi, h all' iml_l¢lll,'.l,',l ;11 llle i_l, ,f:,, _:s .f Ih,. _'t,lllplll_tliollal &ri,l_ _l=,d Ih,. s_:,'l.ud I._pe

{4}l_Sisls of Ihl, ilatllr;tl I,olllldary ¢Olldilion_; inlp()t_'d _!11 lll_' stlrfal_. ;in_l IhP o111/,i I._tllld;llit.s of Ihe rlHillllll;iti{lllal &lid,

The [r,'almenl uf the zol_;d boumlarics cal_ I)e fi..iml ii_ [I I]. The uaturld I)mln(larv eomlitio_s ased ill Illis shldy are dis-

r =l_s_'d h,'hBv.

Airfoil .qu_face lloumlary

'1 h,. I,¢,,noh_L_ r,mditio._ eu Ih_: airli,ll sul[a('r s :*r,: the. "nr, slip" c..nditi(_ and mlial,alic wall c_._diti,ms. It should I,_.

n{_ted Ihat in Ihe _:ase of Ihe rotor airfoil, "no-slilP iml,lh:_ zero relative v[Io¢ity at the s.rfaee of Ihe airfoil. In additi(,n tothe "'m}.slip" condition, the derivative of pressure ill the direction _mrn_al to the wall surface is set to z_'ro.

Exit Iloundar)'

One refleclive aml two radiative houndary eonditi_ms were slu,li,'d, f'.r the refle{:ti_e boundary condition (st_ e.g.

[13]) the exit pressure w_ specified and three quanlilies _rl: ,.xlr_q,_dat_.d from the inlerior. The thre.: (luaatities are the

Iliemann variable Rz = u + 2c/(7 - 1), the entropy .S' = p/p_ aml _ the ira.swrrse velo¢ily. This I.vpe of houndary rondi-

don retie(Is Ihe pressare waves that reach the I)oul_tlary back i_do the s)'stera. Two lypcs of radiating I_ouadary ¢omlilious

'at:re al_o iinldemelll_'d. The lqrM was a nue-dintensio.al hotmd_ry ,'ondition forn_olaled hy Ilayli_s an,I "['urk,d ({I I] and

also [151) It is assumed that at th_ downstream boundary. [he Ilow is liuear. Two.dimensional boul.lary conditions as pre-

sented in [12] were also implemented. As in the pre_ious boundary condition, the flow at the exil is ,_sumed to have small

perturbations and hence linearizaBle about an umlerlying mean fl,}w, hnplementalion of this I}_}undary {onditio. reqaires

a k.o_h dge of the uaderlying exit flow variables, p., _,:_. t_, and p._. The first three quantities are lime li_gg,.d whereas

the exit pressure, p_, is kept ¢onstallt.

Inlet I|{_ut_dary

Ore: r,.tl,.cllv,, _1_1 two r._dialive houndar.', romlilions were als,_ slu,liod fi>r Ill,, iillct. 'l'h_ firs( w_S IhP refl_rtive

boumtary ¢on,litiou pro¢_.dl_re wherein three quaulhies have to h_. spe{die(I. The Ihree rhosen are the Riemann invariant

f& = ,, + '2c/{'t - I) = "t-_) 4. 2c_-_l/('t - I} the total pressure /',_,at = p(__)(l + (T - I/2}MI__: _ )_-_" aud the inlet

II_Bv angle, which in this ra-_e is cquivalenl to o.=_¢_ = 0 The fourth quantity .eeded to updale the points on thi_ boundary"

- 263 -

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TURBOCHARGER NOISE_ GENERATING MECHANISMS AND CONTROL

Sunil tl. Sahay

Product Engineering

Garrett Automotive Group

'['orrance, California 90505

U.S.A.

Denis Thouvenin

Engineering Department

Garrett, S.A.

Thaon-les-Vosges

France

ABSTRACT

Turbochargers, with maximum running speed in excess of 200,000 rpm,

are perhaps the fastest running turbomachJnery in today's Automotive

industry. Turbocharger noise problem came into prominence with thee

application of turbochargers on passenger cars in early L970. During

the early period of its application, a "pure tone" noise synchronous

with the tu,'bocharger speed, was the dominant objectionable, noise and it

was contt-ol led by reducing tile imbal_*nce level. Since thr_n, a(hlitlr)l*,_l

types of tu[bochargc:r noise, excited by mechani(:a] vibration ,_nd air

flow, have been encountered . The trend towards lighter and more compact

vehicles with reduced engine space has increased vehicle's response to

turbochazge_- generated vibration and press*ire pulsations that rust£1t in

vehicle noise. This presents an ever-increasing cha[].enge to the

turbocharger manufacturers [or producing quiet tur-bocharger_. "|hi';

paper presents a brief descript _on of the different types of

turbocharger noise, their generating mechani.';ms _nd control, in most.

cases, the noise reduction is achieved by design improvement and better

control of manufacturing process. This consists of chal,ging the rot()r-

bearing system design, improving the balance process to correct

extremely small amount of imbalance and tightening the casting and

machining tolerances for more symmetrical compressor and turbin(,- wheels.

INTRODUCTION

Turbochargers are very high speed turbomachinery, used primar[]y on

automotive vehicles to increase tile power output of a given size of"

engine. Till the end of last decade, turbocharger noise was addr_:s.u]

on a "fire fighting" basis. A prior assessment of potential noJ_e

problem was not done during the design and development phase which

resulted in high cost for noise control . This has now started to change

and potential noise problems are being addressed from the start of

design. This paper will describe the various noise genera* ing

mechanisms and the steps taken during the development of a new

turbocharger to control the noise generation. The development of *hi';

turbocharger is nearing completion and the information presets.ted is

based on development testing. Tile next step in the cost effective noise

control is for the turbochal'ger manufacturer to provide turbochargez"

related input to the vehicle manufacturer at the start of a new

engine/vehicle program and stay involved till the design is complete.

- 268 - - 269 -

i a ;;

Journal of Fluids and Structures (1993) 7, 611-637

- . ._ '': .).. -[ t .i" ¸ i

,.

!

A NUMERICAL ANALYSIS OF TONAL ACOUSTICS

IN ROTOR-STATOR INTERACTIONS

A. A. RANGWALLA

MCAT Institute, NASA Ames Research Center

AND

M. M. RAI

NASA Ames Research Center, Moffett Field, California 94035, U.S.A.

(Received 19 November 1991 and in revised form 21 January 1993)

In this study, the unsteady, thin-layer, Navier-Stokes equations are solved using a systemof patched grids for a rotor-stator configuration of an axial turbine. The study focuses onthe plurality of spinning modes that are present in such an interaction. The propagation ofthese modes in the upstream and downstream regions is analysed and compared withnumerical results. It was found that the numerically calculated tonal acoustics could beaffected by the type of numerical boundary conditions employed at the inlet and e:dt of thecomputational boundaries and the grid spacing in the upstream and downstream regions.Results in the form of surface pressure amplitudes and the spectra of turbine tones andtheir far field behavior are presented. Numerical results and experimental data arecompared wherever possible. The "mode-content" for different harmonics of blade-passage frequency is shown to conform with that predicted by a kinematical analysis.

1. INTRODUCTION

THE FLUID FLOW WITHIN A TURBOMACHINE is inherently unsteady. There are several

mechanisms that cause the unsteadiness, and some of these are, (a) the relative motion

between the rotors and stators (which is also called the inviscid or potential effect),

(b) the interaction of the downstream airfoils with the wakes generated by the

upstream airfoils, and (c) the shedding of vortices at blunt trailing edges. In general, asthe axial gap between the stator and rotor airfoils decreases, the magnitude of the

unsteady interactions increases. These interactions can even become strongly coupled.

Hence, to study unsteady processes involved within a turbomachine, it may be essential

to treat the rotor and stator airfoils as a single system.Pioneering work in predicting inviscid rotor-stator interaction was conducted by

Erdos et al. as far back as 1977. However, the subject has only very recently become

the focus of increasing attention due to the considerable increase in computational

resources. Lewis et al. (1987) solved the quasi three-dimensional inviscid equations, and

Jorgenson & Chima (1988, 1989) used the explicit Runge-Kutta method to solve the

quasMhree-dimensional Euter and thin-layer Navier-Stokes equations. Three-dimensional periodic calculations have also been presented by Koya & Kotake (1985).

Gibeling et al. (1986) presents results for the flow in a compressor stage obtained using

a shearing grid technique. Here, a single grid is used to discretize the flow domain and

is allowed to shear in order to allow relative motion between the rotor and statorairfoils. The data from the sheared grid are interpolated onto an undistorted initial grid

0889-9746/93/060611 + 27 $08.00 (_ 1993 Academic Press Limited

,+

612 A.A. RANGWALLA AND M. M. RAI

at regular time intervals so as to limit the cumulative distortion due to grid shear

The development of general zonal techniques and robust, accurate algorithms fornumerical solution of Euler and Navier-Stokes equations has contributed further to

development of rotor-stator interaction codes. Giles (1988a) has calculated t

dimensional rotor-stator interactions using the Euler equations. In this work, a ncconcept of a "time-inclined" computational plane is used in order to surmo

difficulties encountered when the stator-rotor pitch ratios is not a ratio of two sn

integers. Rai (1987) presented a two-dimensional calculation of rotor-stator interact

for an axial turbine. The airfoil geometry and flow conditions were those given in 13r

et al. (1982). More recently, Rai (1989) and Madavan et aL (1989) computed the fithree-dimensional flow fields for the same case. The hub, outer casin_ and the rotor

clearance were all included in the calculation. Rai (1987, 1989) and Madavan er

(1989) solved the thin-layer Navier-Stokes equations in a time-accurate manner us

an implicit, upwind-biased, third-order accurate method to compute the flow field.ability of their codes to predict near field flow quantities, such as the time-avera_

pressure distributions on airfoil surfaces and the pressure amplitudes and phase onsurface of the airfoils, was demonstrated. In addition, the two-dimensional codes we

used to predict accurately the total pressure defects in wakes. These compuprograms have also been recently used in the design process of turbomachir(Rangwalla et al. 1992). Rotor-stator interaction codes have also been used to calculi

"sensitive" flow quantities such as heat transfer (Rao er aL 1992a, b; Griffin

McConnaughey 1989) and three-dimensional unsteady hot streak migration (Dorney

aL 1990). Recently, the codes have also been extended for multiple stage calculatio

by Gundy-Burlet et aL (1991). More recently, Dorney (1992) performed a rigoro

validation of a modified rotor-stator algorithm through comparisons with analytical a:

linearized unsteady aerodynamic solutions. The present work is focussed on investig_ing the ability of rotor-stator codes to predict "correctly" the tonal acoustics in tiflow field due to rotor-stator interactions.

An axial flow turbine (or compressor) produces rotating pressure patterns callt

spinning modes that may propagate in a spiral path. In two dimensions, these spinnii

modes propagate at a non-zero angle to the axial direction. For any particulharmonic of blade passing frequency, the interaction field can produce an infini

number of spinning modes. Each of these modes rotates at a different speed. Some ,

these modes propagate, whereas others decay. The modes that are possible for son"multiple of blade passing frequency depends upon the number of stator and rot(

airfoils. This study focusses on an examination of the modes present in a rotor-stat(

interaction for both a single-stator/single-rotor and a 3-stator/4-rotor case usingNavier-Stokes solution procedure.

The numerical study of tonal acoustics involves the study of the flow field in tl_

upstream and downstream regions of the interacting rotor-stator airfoils. This raise

the issue of the numerical boundary conditions employed at the computational inl(

and exit boundaries and the grid spacing used in the upstream and downstream region:Traditionally, the boundary conditions employed have been reflective. Reflectiv

boundary conditions have been used because they provide greater control on turbin

operating conditions, such as mass flow and pressure ratio. However, reflectiv

boundary conditions are generally inadequate for the study of tonal acoustics since the

reflect the propagating modes back into the flow field. Rai & Madavan (1990performed a 1-stator/1-rotor calculation with non-reflective boundary conditions base_on Riemann invariants. These non-reflective boundary conditions were the same a

those developed by Erdos et al. (1977). However it was found that the use of suct

• _ . . • . ; -.., • ...J

• ', . ..

.- .'•7 •• . . •.

L-

ROTOR-STATOR INTERACTION ACOUSTICS 613

boundary conditions resulted in a loss of control over the mass flow rate through the

turbine.To overcome thisproblem, two approaches were tried.The firstapproach was

to use reflectiveboundary conditionsbut with a sufficientlylong computational domain

upstream and downstream of the rotor-stator airfoils. The estimated maximum of thewavelengths of the propagating modes was then used to determine grid spacings at the

far upstream and downstream regions, so that the propagating modes are numerically

dissipated near the boundaries. An acoustic analysis based upon the methodology

developed by Tyler & Sofrin (1970), Goldstein (1974) and Verdon (1989) is applied todetermine the wavelengths of the propagating modes. This approach will be referred to

as the "long-grid" approach. The second approach was to use a short grid. but with

non-reflective boundary conditions, which provided some measure of control over the

turbine operating conditions. These boundary conditions are based on the linearbehavior of the flow in the far field. Two types of non-reflective boundary conditions

were tested. The first was a differential one-dimensional boundary condition derived

from the far field acoustical behavior of the flow and similar to that developed Bayliss

& Turkel (1980, 1982a, b). The second was an approximate two-dimensional unsteady

boundary condition developed by Giles (1988b). This second approach (using differentnon-reflective boundary conditions) will be denoted the "short-grid" approach. For the

purpose of comparison, short-grid calculations with reflective boundary conditions were

also performed.The computations that have been carried out so far, are for two different

configurations: a single-stator/single-rotor case and a three-stator/four-rotor case. Bothcases were computed using the long-grid approach as well as the short-grid approach

and the corresponding results are compared. Additionally, results in the form of

surface pressure amplitudes, the spectra of turbine tines and the axial variation of

amplitudes in the near and far field regions of the different modes are presented.

2. KINEMATICAL PREDICTION OF INTERACTION TONE NOISE

Reflective boundary conditions, such as a fixed exit static pressure condition, can be

made to behave essentially as a non-reflective boundary condition with the use of

appropriate grids in the far field region. When the grid cell sizes in the far-field regionare of the order of the wavelength of the mode to be attenuated, the energy associated

with that mode decays rapidly because of numerical dissipation. Hence, in order to

attenuate reflections at the computational boundaries when reflective boundary

conditions are used, an estimate of the maximum wavelength in the pressure field is

required. Therefore a kinematical analysis of the Fourier modes was carried out. This

analysis, when coupled with a linearized analysis of the flow gives importantinformation about the relative magnitudes (of the different Fourier modes) as a

function of distance. The method was first used by Tyler & Sofrin (1970) for the

analysis of a single stage and this study is generalized for multiple stages. The single

stage results are presented first followed by the results for multiple stages. The

assumption made here is that the tone generating mechanisms occur at multiples of

blade passing frequency. It should be noted that contributions from wake shedding andother aerodynamic noise sources are ignored. The computed results seem to indicate

that these secondary contributions are small for the geometry and flow conditions

chosen.The analysis that follows is limited to two-dimensional flow. However, extension

to three dimensions is straightforward. Consider a single-stage turbine as shown in

\

614A. A. RANGWALLA AND M. M. RAI

.y"

Stator Suctionsurface

__q,..surface N_Inlet flowdirection

Rotor

Pressure /

Suctionsurface

Direction of motionfor rotor

Ca)

(b)

II

',: _ : f! U_', /2 1 , (c)

--¢: f 14--"upstream - I I J -50 Chords

downstream

Figure 1. (a) Rotor-stator geometry; (b) short grid; (c) long grid.

Figure l(a). Let S be the number of sator airfoils and R the number of rotor airfoils

The composite pitch is the transverse distance over which the flow is periodic and idenoted by L Hence, the distance between the stator airfoils is L/S and the distanc_

between rotor airfoils is l/R. The velocity of the rotor is denoted by VR. The pressur(

at any axial plane in the flow field, is assumed to be periodic in time, with period equato the minimum time required for the rotor-stator geometry to repeat. In the case of

single-stage configuration with identical equispaced rotor airfoils,, the period instationary frame of reference is given by I/IVRI R. This is also known as the blade

passing time. The pressure variation in the axial plane can be written as

p(y, t) 2 p.(y) cos[2rcnt [VR` R )= _ + ¢.(Y) (1'_,,=o / '

where p,,(y) is the amplitude of the nth Fourier component and d_n(y ) is the phase. It

should be noted that p,,(y) and _b,,(y) are periodic functions in y with period equal to

the composite pitch t. Using this periodicity in y and appropriate trigonometricidentities, ultimately results in

p(y, t)= 2 2 P ..... (2)t/=() m=--_

i

!

where

• .i::;iii:i. L.i.i i:i.

ROTOR-STATOR INTERACTION ACOUSTICS615

p,,,, = a,,, cos --/- (my - nR IVRI t) + 49,_, , (3)

where n is the harmonic of blade passing frequency and rn corresponds to the spatial

harmonic in y. The next assumption that is made is crucial to the analysis. Consideringthat all stator airfoil in the stator row are equally spaced, the pressure variation is

assumed to satisfy the shift condition, which states that

p(y, t) =p y --_, t- • (4)

It should be noted that both equations (1) and (4) are valid only when frequencies that

are non-commensurate with the blade passing frequencies are absent. Substituting

equation (4) in equation (2) then yields

m = nR sgn(Vn) + kS (5)

where k =..., -1, 0, 1, ... is the spatial harmonic of the disturbance produced by the

stators.

• •.:'... •.

. f - .- .

2.1 PROPAGATING MODE

Analytical solutions representing the unsteady flow in the far field can also be derived

(Verdon 1989). It is assumed that in the far upstream and downstream regions, the

unsteady flow is a linear perturbation of a steady uniform flow. (The underlying steadyuniform flow in the upstream region 'is 'different from that in the downstream region.)

A linearized solution of the pressure field (Verdon 1989) indicates that a Fourier mode

propagates, if

nR MR MyL+ > 1, (6)

where MR is [VR[/c, M_ is the axial Mach number of the underlying flow and My is the

transverse Mach number of the underlying flow. The axial wave-number of the

propagating mode is

1 2[Mx_±sgn(Vn ) _c__f a2], (7)k_ = _1

where

2re 2_rmfl =--i-(nR [VnI + cMym), o_ =---_- .

and c is the sonic velocity in the underlying steady flow. The axial wavelength of the

propagating mode is denoted by ,_x and is equal to 2rc/kx. Using equation (7) we get

l(1 - (8)hx = M_(nRMR + Mym) ± sgn(Vn)_/(nRMR + Myrn) z - (1 - M_)rn 2'

where rn is given by equation (5). For very low values of axial and transverse Mach

• ': - ::,

-\

/•

/,"

/

. ~ -

616 A. A. RANGWALLA AND M. M. RAI

numbers, M_ and My, respectively, the maximum wavelength, X_, can be approximateby substituting 0 for M_ and My in equation (8). For low values of MR, it can be seethat for the lower harmonics,

l=_, (![h_]m,× nRMR

In the case of a single-stator/single-rotor calculation, all harmonics would have spati;modes that would propagate as indicated by equation (6) and hence the maximmpossible wavelength at the lower harmonics is

l

[A_]max ?fiR' (10_

whereas, for a 3-stator/4-rotor case,

l

['_xlmax - 3RMR' (10k

since for this case the fundamental and the first harmonic do not propagate under th

present assumptions of Mx, MR<< 1. When a long grid is used to dissipate thpropagating modes in the far field region, the grid spacings near the exit and inkboundaries are chosen to be about [A._]m,×/2. It should be noted that there is

possibility of large axial wavelengths when nRMR _ 1/(1 - M_)m, but this would occufor higher harmonics under the present assumptions of low Mach numbers.

2.2. DECAYING MODE

A Fourier mode would decay with increasing axial distance from the rotor-stator pairthe inequality sign in equation (6) is reversed. The amplitude of the mode would varas

where

a,,,,, = e *-'aÈ_, (11

( 1 )2d_. = 2;1 --M_ {(1 -M_)(m) 2- [nRMR + rnMy]2}. (12

A decaying mode decays exponentially. It is understood that for x < 0 the positive sig]in equation (11) is used and for x >0 the negative sign is used. The variation of th_natural logarithm of the amplitude, log(a,,,), is linear with slope +d,,,.

2.3. MULTIPLE STAGES

The analysis is similar in the case of multiple stages. Let the number of rotor rows beand number of stator rows be s. Denote the number of airfoils in the rotor rows a

R_, R,_,..., R_ and the number of stator airfoils in stator rows as S_, $2 ..... S,. Th_

_° _ ._.i)_ : .' -i'

2


composite pitch is again denoted by l, so that the distance between the airfoils in rotorrow i is l/Ri, and the distance between the airfoils in the stator row i is l/Si. Define

and

R = Highest common factor (RI, R2,..., Rr)

S = Highest common factors (St, $2,..., S,).

The stator and rotor rows can be aligned arbitrarily; however, it is assumed that all the

rotors have the same velocity, VR. It is also assumed that in each individual row, the

airfoils are identical and equispaced. The minimum time for the rotor-stator geometry

to repeat is again equal to l/IVRI R. Thus, the pressure variation on any axial plane is

given by equation (1), and the shift condition is given by equation (4). R and S can now

be considered to be the number of rotor and stator airfoils on an "equivalent"

single-stage configuration. The rest of the analysis is similar.

3. GEOMETRY AND GRID SYSTEM

The airfoil geometry used is that given by Dring et al. (1982). The two-dimensional

computations of this study were performed using the experimental airfoil cross-sections

at midspan. These cross-sections are shown in Figure l(a). A system of patched and

overlaid grids is used to discretize the flow region of interest. Figure l(b) shows a

typical system of grids used in this study. The inner grids are O-grids and were

generated using an elliptic grid generator. The outer H-grids were generated

algebraically. The experiment consisted of 22 stator airfoils and 28 rotor airfoils. To

model the experimental set-up the flow over at least 25 airfoils (tl stator airfoils and 14

rotor airfoils) would have to be calculated. This would require excessive computational

resources. It was therefore decided to solve a smaller problem by using rescaling

strategies adopted by Rai (1987) and Rai & Madavan (1990) in order to reduce theairfoil count.

The tonal acoustics for two different rescalings were computed. In the first case, the

number of rotor airfoils was changed from 28 to 22. The size of each individual rotor

airfoil was enlarged by a factor of 28/22 such that the pitch-to-chord ratio of the rotor

was unchanged. This rescaling results in a turbine which has equal number of stator

and rotor airfoils, thus allowing a single-stator/single-rotor calculation. (Flow periodi-

city is imposed over one stator airfoil and one rotor airfoil.) In the second case, the

number of stator airfoils was changed from 22 to 21. The size of each individual stator

airfoil was enlarged by a factor 22/21. This rescaling allows a 3-stator/4-rotor

computation wherin periodicity is imposed on the flow over three stator airfoils and

four rotor airfoils. Changing the airfoil count does change the nature of the tonal

acoustics in the flow field because the mode content of the propagating modes depends

upon the airfoil count as indicated by equations (5-12). However, the objective of this

preliminary investigation is to evaluate the capability of rotor-stator interaction codesto calculate tonal acoustics. Hence a numerical solution of a rescaled rotor-stator

geometry can be used to establish numerical boundary condition and grid require-

ments. Figure l(b) shows a typical grid for the single-rotor/single-stator calculation.

However, if the grid spacing at the upstream and downstream boundaries is chosen to

attenuate reflections as described in the previous section, the grid would have to be

lengthened as shown in Figure 1(c). It should be noted that both the long and the short

grids are identical in the near field region.

• .:: •


3.1. Gmo DENsrrY

It should be mentioned that Figure 1 only shows a schematic of the grid. The actual

number of grid points is much larger and the spacing between the grid points is much

smaller. In all the calculations, each inner O-grid had 151 points along the airfoil

surfaces and 41 points in the wall normal direction for a total number of 6,19t grid

points in each O-grid. Each H-grid had 71 grid points in the y-direction and 90 to 141

grid points in the x-direction (90 points for the short grid case and 141 points for the

long grid case). This results in each short H-grid having 6,390 grid points and each long

H-grid having 10,011 grid points. The total number of grid points used for the

3-stator/4-rotor short grid computation was 7 × (6,191 + 6,390) = 88,067 whereas for

the long grid computation, the total number of grid points was 7× (6,191 + 10,011)=

133,414. Rai and Madavan (1990) give more details about the grid system.

4. NUMERICAL METHOD

The unsteady, thin-layer, Navier-Stokes equations are solved using an upwind-biased

finite-difference algorithm. The method is third-order-accurate in space and second-

order-accurate in time. At each time step, several Newton iterations are performed, so

that the fully implicit finite-difference equations are solved. Additional details

regarding the scheme can be found in Rai (1987).

• .. _. _&_... - :.. : ,

&,.

5. BOUNDARY CONDITIONS

The boundary conditions required when using multiple zones can be broadly classified

into two types. The first type consists of the zonal conditions which are implemented at

the interfaces of the computational meshes and the second type consists of the natural

boundary conditions imposed on the surface and the outer boundaries of thecomputational mesh. The treatment of the zonal boundaries can be found in Rai

(1986). The natural boundary conditions used in this study are discussed below. In

particular radiating boundary conditions for the inlet and exit boundaries arepresented.

5.1. AIRFOIL SURFACE BOUNDARY

The boundary conditions on the airfoil surfaces are the "no-slip" condition andadiabatic wall conditions. It should be noted that in the case of the rotor airfoil,

"no-slip" does not imply zero absolute velocity at the surface of the airfoil, but rather,

zero relative velocity. In addition to the "no-slip" condition, the derivative of pressurein the direction normal to the wall surface is set to zero.

5.2. EXIT BOUNDARY

Two types of boundary conditions were used at this boundary. The first was a reflective

boundary condition where the exit pressure was specified and three quantities are

\

• . ".%

..


extrapolated from the interior. The boundary conditions are

Pstatic "=- constant, (13a)

_R1 0, _S 0 (13b, c)8x 8x

andc]l)-- = 0, (13d)8x

where R1 = u + 2c/(y - 1) is the Riemann variable, S =p/pY is the entropy and v is thetransverse velocity. This type of boundary condition reflects the pressure waves thatreach the boundary back into the system. Two types of radiating boundary conditions

were also implemented. The first was a one-dimensional boundary condition and itsformulation (Bayliss & Turkel 1982) is shown below. It is assumed that, at thedown-stream boundary, the flow is linear, that is, the unsteadiness can be considered a

linear perturbation to a steady flow. This steady flow need not be axial. However, wecan always rotate the coordinate system such that the x-axis is aligned with the

underlying steady flow. In the rest of the analysis, it is assumed that the coordinates areso aligned. The inearized Euler ec uations far downstream are ;1yen by

p"l ue pe 0 0" p' "0 o o_ 0 p'l

• 1 u'u'l 0 u_ 0 -- 0 0 0 0 u'l

-- ., + _ , + .... O, (14)0t v'l 0 0 u_ 0 v' 0 0 0 1 0y-- U p I

Pe

p'l 0 p_c2 0 u_ p' 0 0 p_c_ 0 !P't• j , • • _ k. J

where u_, p_ and ce are the underlying steady state velocity, density and speed of soundat the exit, whereas, p', p', u' and v' are the perturbations in the pressure, the density,

the velocity in the direction of the mean flow, and the velocity normal to the meanflow, respectively. Using the linearized x-momentum, y-momentum and energy

equations, we can obtainp[, 2u_px_ - (C 2 -- 2 , 2 ,- ' u_)px_- c_p,, = o. (_5)

Introducing the change in variables

and

x

sc= V-f--S_M_ , _7= Y

= c,_t + M._,

equation (15) is transformed to!

P_ -P_¢ -Pn,7 = O. (16)

Equation (16) admits solutions of the form

p' =f('r - _:cos 0 + 77sin 0), (17)

where 0 is the angle between the underlying mean flow and the axial direction

(0 = tan-_(v=/u=); ue = _). These solutions are planar waves propagating in theaxial direction. Since in the present calculation the airfoils extend from r_= -oo to + ocwaves of this form are expected to exist. It should be noted that a boundary condition


based on equation (17) will be non-reflective for one-dimensional cases. Equation (1:

suggests a differential operator of the form Sg= (a/&r)+ cos 0(8/a_)-sin 0(0/a_t

which annihilates the functional form in equation (17). Hence the radiating boundm

condition used is _p = 0. which when transformed into the actual physical non-rotatecoordinates give, for M_ << 1,

/A Jp/= p=c=(Z + M:_=)(u[ + =ux). (1_

Implementation of equation (18) is done by first setting u_ equal to zero. This

consistent with the zeroth order extrapolation of the velocities at the boundary. Tb

velocities u and v and the entropy p/p_' are extrapolated from the interior. Th

pressure is updated on the exit boundary by using equation (18). In this equation, th

terms p=, u= and v= are obtained by circumferentially averaging p, u and v at the ex:

at the previous time step. The exit sonic speed, c=, is evaluated by using c= =

where the value of p= is fixed and is equal to the exit pressure value used in threflective boundary condition procedure.

The boundary condition described above is essentially a one-dimensional boundar

condition. Two-dimensional boundary conditions as developed by Giles (1988b) wet

also implemented. As in the previous boundary condition, the flow at the exit :

assumed to have small perturbations and hence be lineafizable about an underlyinmean flow. The exit boundary conditions in terms of one-dimensional characteristivariables are

{c}c3c3ca _ c2 = O, (19a--+{0 u_= 0 v_}._t ay

Ca

Ox c2 O, (19b

C3

where the transformation between the characteristic variables and the perturbatio:

variables is given by

E-i o0c2 0 p,:c.: 0 u' l

c3 p_c_: 0 1 u' [

ca 0 -p=c, 0 1 ,p' J

and

p I

P/A

13

P

1 1 1

c_ 2c_ 2c_

1 10 0

2p_c_ 2p_c_

10 0 0

p_c_

o o _

ct

c2

c3

C,,

(20b

• • ,;°

ROTOR-STATOR INTERACTION ACOUSTICS •621

Implementation of this boundary condition requires a knowledge of the underlying exit

flow variables, p::, u=, v_:, and p_:. The first three quantities are time lagged whereas

the exit pressure, p_:, is kept constant.

5..3. INLET BOUNDARY

. - • . -,,

.. -, . . .

Here again two types of boundary conditions were used. The first was the reflective

boundary condition procedure wherein three quantities have to be specified. The threechosen are the Riemann invariant

the total pressure

2c 2c_:,Rl = u + - u_,_ + -- (21a)

y-I y-l'

\ yl-y- 1Ptota, =P-= 1 4 y -- 1 M2_ @ , (21b)i

and the inlet flow angle, which in this case is equivalent to

vi.l_t = 0. (21c)

The fourth quantity needed to update the points on this boundary is also a Riemann

invariant but is extrapolated from the interior and is given by

OR2-- = 0, (21d)3x

where

• .q .

"-':.7: " " .{.7.

. .- .%,, -. . " .

2cR2=u ---.

3'-1

In the above equations the quantities u and v are the velocities in the x and y

directions, p is the pressure and c is the local speed of sound. Specifying the total

pressure at the inlet results in the boundary condition being reflective.A non-reflective or radiative one-dimensional boundary condition (Bayliss & Turkel

1982) was also implemented. As in the case of the exit boundary, it is assumed that at

the upstream boundary the flow is linear, that is. the unsteadiness can be considered a

linear perturbation to a steady flow. Using an analysis that is very similar to that used

in developing the radiating boundary condition for the exit boundary we obtain the

following condition at the inlet:

p; = p-,4u-:_ - c_=)(u; + u_,_u'd. (22)

At the upstream boundary, the Riemann invariant, R_, and the flow angle, v_nie, = 0, are

_: aa

622 A.A. RANGWALLA AND M. M. RAI

still used. However, Ptotal is replaced by the condition that at the inlet, the flow isisentropic, which gives

p/pY = constant = p_=/p_'=, (23)

and equation (21d) is replaced by equation (22).

The radiating boundary condition described above is basically one-dimensional in

nature. Two-dimensional inlet boundary conditions (Giles 1988b) were also imple-mented. The inlet boundary conditions in terms of the one-dimensional characteristicvariables are

fclti 0 0 0j{ci}0-_ c2 + v _(c+u) ½(c-u) 0 c2 0, (24a)1 _ C3c3 _(c - u) v 0

C4

_C4

-- = 0. (24b)%x

The transformation between the one-dimensional characteristic variables and the

perturbation flow quantities are given by equation (20), with the quantities ()=replaced by the quantities ( )_=.

5.4. UPPER AND LOWER BOUNDARIES

The computations reported in this study assume that the flow is spatially periodic in the

y-direction. The spatial interval of,periodicity depends upon the airfoil count. (For

example, in the 3-stator/4-rotor case, periodicity is imposed after every three stator

airfoils and four rotor airfoils.) Further details regarding this boundary condition canbe found in Rai (1987).

6. RESULTS

In this section, results obtained for the single-rotor/single-stator and four-rotor/three-

stator configurations are presented. In particular, a comparison between the long-gridand short-grid computation with reflective boundary conditions and non-reflective

boundary conditions will be made. In addition, the spectrum of the turbine tones and

the variation of the amplitudes of the different modes in the far field will be presentedfor different cases.

The dependent variables are non-dimensionalized with respect to the far upstreampressure (p_=) and density (p_=). The free-stream velocities are

u_= = M_='V'TT , v_= = 0,

where M_= = 0.07 is the inlet Mach number. The pressure ratio across the turbine

(Psta,ic,,./Pto,al,.,o,) is 0"963. The rotor velocity was obtained so as to match the

experimental flow coefficient (ratio of average inlet velocity to rotor speed) of 0.78 as

given in Dring et al. (1982). The Reynolds number is 100,000/in. The kinematic

viscosity was calculated using Sutherland's law and the turbulent eddy viscosity was

calculated using the Baldwin-Lomax model. The calculations were performed at a

\


constant time-step value of about 0.i6 (this translates into 500 time steps for the rotor

to move through a distance equal to the distance between two successive blades).

• 3. • - . t ' ". , .

. _.- .

6.1. AIRFOIL SURFACE PRESSURE AMPLITUDES

The first comparison is made between the long and the short grid computations for the

single-rotor/single-stator case, Reflective boundary conditions are used in both

computations. Figure 2 shows the pressure amplitudes on the stator for the two cases.

The symbols in this figure are the experimental data of Dring et al. (1982). The

pressure amplitude (Cp) is defined as

_p ---_ Pmaxl -- Pmin2

_Pinlet O)

where o_ is the rotor velocity and Pm_x and Pmi, are the maximum and minimum

pressures that occur over a cycle. A cycle corresponds to the rotor moving by a

distance equal to the distance between adjacent rotor or stator airfoils. The pressure

amplitudes obtained in the short-grid computation show most of the qualitative

features that are found in the experimental results. However, the numerical data show

a wider large-amplitude region than that found experimentally. In addition, the

predicted peak is to the left of the experimental peak, and the pressure amplitudeminimum on the suction side seen in the experimental results (x _-2.4) is absent in

the computed results. The long grid computation yields an amplitude distribution closer

to the experimental data. The position of the peak and the extent of the large

amplitude region agree well with the data.

The improvement obtained using the long grid is due to the large grid spacing in the

far field region of the outer grid, which attenuates propagating modes. However, one

penalty incurred in using this approacl-r is the excessive computer time needed to obtain

a periodic state. A typical short grid computation for a 3-stator/4-rotor case to

converge to a time periodic state (including convergence in the tonal acoustics) is about

20cpu hours on a single processor of a CRAY YMP supercomputer. For a

corresponding long grid computation, approximately five times as much computing

time is required. Additionally, the time for convergence varies linearly with the number

of airfoils, provided the extent of the upstream and downstream regions of the grid is

the same. It should be mentioned that the time for convergence for flow quantities such

2"5

2-0

1.5

l.O

O.S

%

(a)

Z..(3

I I P.-4 0 4

®

f- -4 0

(b)

Axial distance along stator surface

Figure 2. Pressure amplitude on stator surface for (a) short grid, (b) long grid for a 1-stator/1-rotor case:O, suction surface; ,11,,pressure surface.

2: ¸


3

,®

2

1 o 3

OI r I

0-8 --4 0 4


Figure 3. Pressure amplitude on stator surface for a 1-stator/1-rotor case (using nonreflective boundar

conditions): O, suction surface; 41., pressure surface.

as pressure amplitudes on the airfoil surfaces and near field acoustics was considerablless.

Figure 3 shows the computed surface pressure amplitude distribution obtained usinlthe short grid in conjunction with the one-dimensional non-reflective inlet and exi

boundary conditions [see equation s (18,22)]. The agreement with the experimentadata is slightly better than that obtained on the long grid with reflective boundar"

conditions. It was found that the level of repeatability (solution periodicity in time

with these boundary conditions was much better and the solution converged to a timc

periodic state faster. In addition, turbine operating conditions were maintained unlik_

in Rai (1990), where the use of non-reflective boundary conditions required an iteratiw

process in which the Riemann invariant, Rz, (specified at the exit) had to be variec

until the proper average exit pressure was obtained. The flow coefficient and th(

turbine pressure ratio differed from that obtained using the reflective boundar?

condition by less than 1%. (This will be shown later in Table 3.) Figure 4 shows the

3.0

,G_ 1.5

2.5

2-0 --

1-0-

0.5

O

0 _ I

(a)'I

m

(3I I r

2-5 10 -10

Axial distance along rotor surface

(b)

Figure 4. Pressure amplitude on rotor surface with a (a) reflective boundary conditions and (b)

non-reflective boundary conditions for a 1-stator/1-rotor case: O, suction surface; _, pressure surface.

mlO -5 0 -5 0 2.5 10

ROTOR-STATORINTERACTIONACOUSTICS 625

2.5

20 i1-5 .___

1.0(D

0.5

0 !-7.5 -5.0 -2.5 0 2.5 5-0 7.5 -7.5 -5-0 -2.5 0 2-5 5.0 7.5

Axail distance along stator surface

Figure 5. Pressure amplitude on stator surface for 3-stator/4-rotor case with (a) a short grid and (b) a long

grid: O, suction surface; ,I,, pressure surface.

pressure amplitudes on the rotor for a short grid with and without reflective boundary

conditions. It is seen that the amplitudes obtained using the non-reflective boundaryconditions are generally lower.

In contrast to the single-stator/single-rotor case, the pressure amplitudes for the

3-stator/4-rotor case did not differ much for the long or short grid or for the reflective

or non-reflective boundary conditions. For the single-stator/single-rotor case, an

acoustic analysis (Tyler & Sofrin 1970) shows that every harmonic could have

propagating modes whereas the 3-stator/4-rotor case does not have any propagating

modes for the first two blade passing harmonics. The first two harmonics do have

decaying modes. In the original experimental configuration, there are 22 stator airfoils

and 28 rotor airfoils. For this casd also, the acoustic analysis does not predict any

propagating modes for the first two blade passing harmonics. Since the higher

harmonics are usually much smaller in magnitude, the unsteady pressures that reach

the computational boundaries in the 3-stator/4-rotor case are much smaller than that

for the single-stator/single-rotor case. Hence, it is expected that the reflective

properties of the boundary conditions would play a smaller role in determining the

unsteady pressures on the airfoils for the 3-stator/4-rotor case. Figure 5 shows the

pressure amplitudes on the stator surface for the short and long grids with reflective

boundary conditions. Figure 6 shows the pressure amplitude for the short grid with

2

07.5 -2.5 2-5 7-5


Figure 6. Pressure amplitude on stator surface for 3-stator/4-rotor case (using non-reflective boundary.

conditions): O, suction surface; II,, pressure surface.

j;

/


non-reflective boundary conditions. Clearly there is an improvement over the resul

depicted in Figure 2(a) and a slight improvement over that depicted in Figures 2(b) ar3. The extent of the large amplitude region and the location of the pressure pe_matches the experimental data. The slight improvement is due to a closer similaritythe geometry with that of the experimental geometry. However, it should not

concluded that reflective properties of the computational boundaries are unimportmfor the 3-stator/4-rotor case; they can still significantly alter the tonal acoustics in ttlinear region of the flow. The pressure amplitudes on the rotor surface for tt3-stator/4-rotor case also did not depend on the type of grid or the boundmconditions. The amplitudes on the rotor surface were very similar to that reportepreviously by Rai & Madavan (1990).

Besides the pressure amplitudes, phase information can also be obtained. The phmof the low pressure peak on the stator suction surface [see Rai and Madavan (i990) f<

details] for the 3-stator/4-rotor case did not depend on the type of grid or the boundmconditions. The numerical results compared well with experimental data of Dring:e_ c(1982) and were similar to that reported by Rai & Madavan (1990).

6.2. FAR FIELD LINEAR BEHAVIOR

The spectrum of turbine tones obtained from the computations is presented in thsection. Recall that the Fourier modes predicted by the kinematical analysis a_

denoted by P,,n [equation (3)], where m and n are related as given by equation (5). Tkvalues of am,, can be obtained by performing a Fourier decomposition of the pressmvariation upstream and downstream of the turbine. The upstream results we_calculated at two chord-lengths upstream of the leading edge of the stator airfoils arthe downstream results were calculated at two chord-lengths downstream of the trailiredge of the rotor airfoils. Figure 7ia, b) shows the contribution of the axially aligne

planar waves (m = 0) for the single-stator/single-rotor case. The x-axis corresponds I

xl0-*

6

3

2

II0

xl0-S

12.

(a) l10

8-

6-

4-

2tAI

12 0

Blade passing harmonic n

(b)

h|AhL_,l

4 8 12

Figure 7, Spectrum of the m= 0 mode (a) upstream of the stator and (b) downstream of the rot

(1-stator/1-rotor).

\


g

xlO-5

12

10-

8-

6-

4-

2-

i0

(a)

.I.I I4 6 8 10

xlO-'635

30-

25-

20-

15-

10-

I

(b)

A . I.A A4 6 8 10 129 12 0


Figure 8. Spectrum of the (a) m = 1 and (b) m = -1 modes downstream of the rotor (1-stator/1-rotor).

.,:. ,"7- _" - "'" ": _ _ ?_-_- - " _" 7"

harmonics of blade passing frequency and the y-axis corresponds to the computed

coefficients, amn. It is seen that, in general, the amplitudes of the higher harmonics are

smaller than the amplitudes of the lower harmonics. Equation (6) predicts that all these

harmonics propagate without decay. The numerical results conform with this prediction

[see Rangwalla & Rai (1990)]. Note that the contribution due to the subharmonics of

blade passing frequency is very small (by two orders of magnitude) compared to the

harmonics of that frequency, thus leading to the conclusion that, for this mode, thekinematical interactions dominate. It 'was also found that the contribution of the

non-planar modes (m ¢0) upstream of the stator was less by at least an order of

magnitude when compared to the planar modes. Figure 8 shows the m = 1 and m = -1modes downstream of the rotor. These modes are about an order of magnitude smaller

than the planar mode. The subharmonic content is very small as in the rn = 0 case.The situation in the 3-stator/4-rotor case is different. For the planar case (m = 0),

equation (5) predicts the existence of only the n = 3, 6, 9 .... harmonics of blade

passing frequency. Figure 9(a) shows the contribution of these planar waves upstream

of the stator. Although the subharmonic content is more than in the single-stator/

single-rotor case, most of the energy is seen to lie in the n = 3 and n = 6 harmonics.

Figure 9(b) shows the contribution of the planar waves downstream of the rotor. It

should be noted that the pressure variations downstream of the rotor are measured in

the rotor frame of reference. Hence for the m = 0 modes, the kinematical analysis

predicts the existence of only the n = 4, 8, 12 .... harmonics of blade passing frequency.

Once again, we notice a low level of subharmonic noise, thus leading to the conclusionthat for this case the kinematical interactions dominate. It should be noted that, for the

rn--0 modes, the subharmonic noise of the singte-stator/single-rotor case is of the

same order of magnitude as the 3-stator/4-rotor case than that in the single-

stator/single-rotor case.

The m = 1 and m = -1 modes upstream of the stator are shown in Figure 10(a, b).

Substituting R = 4 and S = 3 in equation (5), the positive integer values that are

possible for n when m = -1 are n = 1, 4, 7 .... and when m = 1, the positive integervalues that n can take are n = 2, 5, 8 .... We observe that the dominant frequencies

Y

i'"


x10-6

20-

i5-

10-

(a)

5

0 2 4 6 8 lO 12

25

20

35-

30-

I5

10

5

(b)

m

20 4 6 8 10 12'


Figure 9. Spectrum of the (a) m = 0 mode upstream of the stator and (b) the m = 0 mode downstream othe rotor for the 3-stator/4-rotor case.

conform with the kinematical analysis. The same is true downstream of the rotor. Sinc_

the pressure variations downstream of the rotor are measured in the rotor frame o

reference, R and S in equation (5) should be interchanged. Hence, for the m = 1 mode

downstream of the rotor-stator pair, equation (5) predicts the existence o:

n = 3, 7, 11 .... blade passing harmonics. Similarly, for the m = -1 mode. equation (51

predicts that the blade passing, harmonics that can be present are given b?n = 1, 5, 9 ..... The m = 1 and m ='-1 modes downstream of the rotor are shown ir

Figure 10(c, d). Once again we observe that the dominant frequencies conform with the

kinematical analysis.

The computations can also be used to study the propagation or decay of the variou_

modes. To do this, either a long grid has to be used and the amplitudes of each mode

calculated in the region of the grid where the solution has not suffered from numerica

dissipation (due to grid coarseness) or a short grid with non-reflective boundar)

conditions should be used. A study of the numerical propagation or decay of the

different modes has the advantage of determining the grid spacing required in the fm

field to maintain a propagating mode or to capture accurately the decay rate of

decaying mode. Additionally, the effect of boundary conditions on the different mode_can be assessed.

Figure ll(a, b) shows the effect of grid coarsening on propagating waves for the

3-stator/4-rotor case. Figure 11(a) shows the amplitudes of three propagating modes

[ao.3, ao.6 and ao.9 as given by equation (8)] upstream of the rotor-stator pair. As

propagating modes, these amplitudes should remain constant. However. the amplitudes

do decay as the grid spacing in the axial direction increases: (the symbols indicate the

axial location of the grid points). As expected, the higher harmonics decay faste_

because they have smaller wavelengths. In the present case, the wavelength of the

(0, 3) mode is approximately 21-11 in. (536-2 ram), the wavelength of the (0, 6) mode is

approximately 10-55 in. and the wavelength of the (0, 9) mode is about 7-04 in. From

Figure ll(a) we see that numerical dissipation sets in when there are five or fewer mesh

points within a wavelength. Figure 11(b) shows the amplitudes of the propagating

x10--5

5

4 -

3-

2-

£-

ROTOR-STATOR INTERACTION ACOUSTICS

x10--540,

(a)351--

3ol-

25 b-

20-

15-

10-

.J- A

(b)

629

x 10-5

5

4--

3-

2-

1-

2

(c)

xl0-66

5 -

4-

3-

(d)

i

0

I I8 i0 12 2 4 6 8 I0 12

Blade passing frequency n

Figure 10, Spectrum of the (a) m = 1 and (b) m = -1 modes upstream of the stator and of the (c) m = 1and (d) m = -1 modes downstream of the rotor (3-stator/4-rotor).

modes (ao.4 and ao.8) downstream of the rotor-stator pair. Unlike the upstream results,these amplitudes exhibit rapid variations near the rotor-stator pair. These oscillationseventually subside and the amplitudes remain constant till they monotonically decaybecause of grid coarsening. The rapid axial variation of the amplitudes immediatelydownstream of the rotor-stator pair is due to the nonuniformity of the mean flow. Thisnonuniformity is largely due to the velocity defects in the wakes of the rotor airfoils.

The axial range over which the effect of this nonuniformity is felt depends upon themode. It will be seen later that the rate of decay of the decaying modes and the axialwavelength of propagating modes can be significantly affected by the nonuniformity ofthe underlying mean flow. The effect of numerical dissipation due to grid coarsenessdownstream of the rotor-stator pair is similar to that observed in the upstream region.In the present case, the wavelengths of the (0, 4) and (0, 8) modes are approximately

.J


xlO-63O

25

20

15

(a) .. -- ao.3

l°L5

,_e_ ao.6

/ ....o" ._ a0.9a/ ....... _.._'= el._.4r- I t--50" "-40 -30 -20 -10

xlO-5

7

6

5

4

3

I

2

Iw

10

(b)

)

. ao,8

• -_--J o-o o_-_..o..al._.20 30 40 50

Axial distance upstream of turbine

Figure 11. Axial variation of the amplitudes.of some propagating modes (a) upstream of the stator and (b)

downstream of the rotor (3-stator/4-rotor); @, axial location of grid points.

20-96 and 10.48 in., respectively. In Figure 11(b) it is seen that numerical dissipation

affects the propagating modes when the number of mesh points within a wavelengthare five or less.

Figure 12(a, b) shows the instantaneous variation in the axial direction of the (0, 3)and (0, 4) modes. In the figure, the effect of grid coarsening on the waveform can beseen. Grid coarsening can affect the amplitude as well as the wavelength of the mode.

The present numerical results seem to indicate that the wavelength variation due togrid coarseness is slight, as long as there are more than five grid points per wavelength.

xl0-5 xt0-5

.}

2

_" I

g 0

g-t

-2

-3

--4

(b)

q t I I-50 -40 -30 -20 -10 0 0 10 20 30 40 50

Axial distance upstream of turbine Axial distance downstream or turbine

Figure 12. Instantaneous axial variation of the (a) (0, 3) mode upstream of the stator and (b) (0, 4) mode

downstream of the rotor (3-stator/4-rotor); O, axial location of grid points.

_. " .7:


TABLE 1

Axial wavelengths of some propagating modes (upstream);(Theory in parentheses)

3-stator/4-rotor case.

n--1n=2n=3n=4n=5n=6n=7n=8n=9n = 10

-2 -1 0 1 2

decaying decayingdecaying

decaying

10"57(10-82)

20-81(21-63)

9-46(9.81)

6.35(6.58)

20-54(21-11)

10-49(10-55)

7.03(7-04)

decaying

14.96(15.09)

8.11(8.41)

decaying

14-15(14-48)

7"16(7"55)

However the amplitude can be significantly affected by grid coarseness and decays to

about half its value when there are six grid points per wavelength.

Comparisons between the theoretical axial wavelengths and that obtained numeri-

cally are shown in the Tables 1 and 2 for the 3-stator/4-rotor case.

Table 1 shows comparisons upstream of the rotor-stator pair. In general, the results

are good. The differences are less than the maximum grid spacing in the region where

the wavelengths were measured. Table 2 shows a similar comparison downstream of

the rotor-stator pair. The numerical results are generally in fair agreement with the

theoretical predictions for all (m, n) modes where m-<0. The differences are of the

same order as the maximum grid spacing in the region where the wavelengths were

measured. However, the calculated wavelengths of the (m, n) modes where m > 0 do

not agree well with theoretical predictions. It should be recalled that the theoretical

prediction of axial wavelengths [equation (8)] was obtained under the assumption of a

uniform mean flow. However, downstream of the rotor-stator pair, the underlyingmean flow is not uniform due to the wakes of the airfoils. It is believed that this

nonuniformity in the mean flow is the main reason for the discrepancy between the

numerical results and theoretical predictions.

One objective of the present study is to see if non-reflective boundary, conditions can

be used along with a short grid to predict the tonal acoustics present in rotor-stator

TABLE 2.

Axial wavelengths of some propagating modes (downstream); 3-stator/4-rotor case.(Theory in parentheses)

-2 -1 0 1 2

n=l

n = 2 decayingn=3n=4n=5

n = 6 decayingn=7n=8n=9n = 10 11-89(13-46)

decaying

23.78(26-91)

10-81(10-47)

23-51(20-96)

11"65(10"48)

decaying

18.38(13.39)

decaying

decaying

20.81(11.12)


xl0--s

8

B

6-

4

<

2

0--60

ao,3

/r F"

-40 -20

(a)

aO,4

\

\\

\

1I

xl0-_

_5

5-

4.-

3- al.2

,.)_

%0

a! .3

(b)

0 20 40 60 0 20 40 60

Axial distance from turbine

Figure 13. Comparison between a long grid solution and a short grid solution with reflective boundmconditions for (a) the (0, 3) mode upstream and (0, 4) mode downstream of the rotor-stator pair, and (b) tP

(1,2) mode upstream and (1,3) mode downstream of the rotor-stator pair. . Long grid; ...., short grid.

interactions. Three different boundary conditions were compared. The first boundar

condition was a reflective boundary condition as given in Rai & Madavan (1990). Thes

boundary conditions have been widely used in the numerical simulations of rotorstator interactions. However, they are not adequate in the study of tonal acoustic

because of their reflective proper/ies. Figure 13(a, b) shows the comparison of sho_

and long grid calculations with reflective boundary conditions for the 3-stator/4-rotc

case. In Figure 13(a) the axial variation of the amplitudes of two propagating mode

are shown. The effect of using reflective boundary conditions can be seen in this figur,

At the exit boundary of the short grid, the amplitudes become zero whereas at the inl_

boundary, the amplitudes are very small.

The short and long grid solutions also do not compare well within the flow domaiJ

The short grid solutions do not show an axial region where the amplitudes remaJconstant. This is because a reflective boundary condition reflects any propagating moc

back into the flow domain. As expected the long grid solution does exhibit an axi;

region over which the amplitudes remain constant. The reason for this is that tl:

coarseness in the grids near the inlet and exit boundaries essentially dissipates tt

propagating waves, thus minimizing the effects of reflection. In contrast to tl:

propagating modes, the effect of the reflective boundary conditions on the decayir

modes is only significant near the boundaries. Figure 13(b) shows the axial variation

the (1, 2) mode upstream of the stage and the (1, 3) mode downstream of the stag

The results show that the amplitudes remain unaffected in the near field region of ttairfoils. It should be recalled that these results are for the 3-stator/4-rotor case. For th

case the boundary conditions did not significantly affect the pressure amplitudes on tt

airfoil surfaces. The amplitudes on the airfoil surfaces are mainly composed of ttlower harmonics. For the 3-stator/4-rotor case, the lower harmonics decay wi_

increasing axial distance from the airfoils. Reflective boundary conditions do retiethese modes but the effect of the reflection is confined to the region near tt

boundaries.

.a¢

_.. 4--

-<

2

ao.3

ROTOR-STATOR INTERACTION ACOUSTICs 633

XI0-5

0 r I 1 l--60 -40 -20 0 20

0-0.4

(a) (b)

6

w

",, a0. 3 ,,,

, ...',... \2 \

) l 0 l t I )

40 60 ---60 --40 -20 0 20 40 60


Figure I4. Comparison between a long grid solution and a short grid solution for the (0, 3) mode upstreamand the (0,4) mode downstream of the rotor-stator pair using the (a) one-dimensional non-reflective

boundary condition and the (b) two-dimensional non-reflective boundary condition. Long grid;..... , .... , short grid.

Figures 14 and 15 show the axial variation of the amplitudes of the same modes [as in

Figure 13(a, b)] obtained on the short and long grids with non-reflective boundaryconditions. Figure 14(a, b) shows the upstream axial variation of the amplitudes of the(0,3) mode and the downstream variation of the (0,4) mode with one-dimensional

boundary conditions [equations (18,22)], and two-dimensional boundary conditions

[equations (19, 24)], respectively. The variation of the amplitudes of the "propagating

xl0-_

6

5

4-

"& 3=_

<

2

!

0-60

at.2

I

-40 -20

at.3

l\

I Ix'...`_ .l

0 20 40

×10--4

6

" Et

i0

60 --60 --40


(b)

al.2• s;

/-20

al.3

, I I_. I0 20 40 6O

Figure 15. Comparison between a long grid solution and a short grid solution for the (1, 2) mode upstream

and the (1, 3) mode downstream using (a) the one-dimensional non-reflective boundary condition and (b) the

two-dimensional non-reflective boundary condition. , Long grid; ,, short grid.

/

634A. A. RANGWALLA AND M. M. RAI

Pstati%xit

P totalinlet

Velocity_,_¢,

L Ptotallnlet

TABLE 3.

Converged operating conditions

Experimental Reflective B.C. 1-D B.C.

0.963

0-0831"0034

i

0-963

0-0841"0034

0-9636

O.O831-0031

2-D B.C.

0-963

0.0841.0034

modes for the short grid case are similar to that for the long grid. However, the overall

levels are a bit different. This difference may be due to the difference in the converged

operating conditions as shown in Table 3. The differences between the short and long

grid solutions with one-dimensional boundary conditions are larger than that obtainedwith the two-dimensional boundary conditions. The results also show some reflectivity

at the boundaries, as evidenced by the oscillations in the amplitudes near the upstream

boundaries. Figure 15(a, b) shows the upstream axial variation of the (1, 2) mode and

the downstream axial variation of the (1, 3) mode. In contrast to the propagating

modes, the amplitudes of these modes remain unaffected in the near-field region of the

airfoils. At the inlet and exit computational boundaries of the short grid, there are

differences between the, results obtained between the long and short grid solutions.

However, the solutions obtained by the one-dimensional and two-dimensional non-

reflective boundary conditions are slight. Examination of other decaying modes show

the same overall behavior, i.e., the amplitudes of the modes remain nearly unaffected

in the near-field regions of the airfoils.The amplitudes of the decaying modes vary exponentially in the upstream and

downstream directions [equation (11}]. The rate of decay depends upon the underlying

mean flow and the temporal and spatial frequencies of the mode as given in equation

(12). lit should be recalled that equation (12) is derived under the assumption of linear

perturbations to a steady uniform mean flow]. The axial and transverse Mach numbersof the underlying mean flow upstream and downstream of the turbine is obtained from

the numerical solutions. The mean flow quantities in nondimensional units are given in

Table 4. The nondimensional velocity of the rotor airfoils (VR) is 0-1051282. The

quantities M_, My and MR in equation (9) can be evaluated from U_, U>,, VR and the

sonic velocity of the underlying mean flow.•Figure 16(a. b) shows the axial variation of the amplitudes of some decaying modes

upstream and downstream of the rotor-stator pair. It should be noted that equations

(11) and (12) yield only the rate of decay and not the amplitude level. Hence in Figure

16(a. b), only the slopes of the curves are of interest. The behavior of these modes

upstream of the rotor-stator pair is in excellent agreement with the linear theory in the

region where the grid is sufficiently fine. In the far upstream region, the numerical

TABLE 4.

Mean flow quantities

Upstream

Axial velocity, U_ 0-0842535Transverse velocity, U>. 0.00Sonic velocity, c 1-1829303

Downstream

0-08640040.07016661.1774635

\i

-6

,_ -i0

- -14


(a)

I P I I

-5

-10

d -i5u

e-20

-18 -_9_5

(c)

I I I I

• • [• . "[•_

, . .., . ,"

2

-5

-10

-15

-20-50 0

(b)f

B 1/

1/ 1

///1

" ) I I I-40 -30 -20 -10

-7-5

-10

-12.5d

-15

-17.5

-200

(d)

I I I )I0 20 30 40 50

Axial distance downstream or turbine

Figure 16. Comparison between numerical and theoretical decay rates upstream for (a) mode (1,2) and

(b) mode (-1, 1) upstream of the stator and for (c) mode (-1, 1) and (d) mode (I, 3) downstream of the

rotor: --, numerical; ---, theoretical.

solution deviates from the theoretical exponential decay because of the very. coarse gridin this region.

Downstream of the rotor-stator pair, a difference between the numerical results andthe linear theory is seen [Figure 16(c, d)]. The axial variation of the natural logarithmof amplitude of the (-1, 1) mode (log a-1.1) is shown in Figure 16(c). Even though theoverall decay matches that of the theoretical exponential decay, the numerical variationof the amplitude is not a "pure" exponential. Rather, the axial variation of thecomputed amplitude is exponential as well as oscillatory in nature. This behavior wasobserved for all (l, k) modes, where l < 0. Figure 16(d) shows the axial variation of the

natural logarithm of the amplitude of the (1, 3) mode. The decay of this mode isexponential, however, the decay rate does not match the theoretical prediction. Thisdifference between the numerical and theoretical decay rates was observed for all (/, k)modes where l > 0. It should be recalled that one of the assumptions underlying thetheoretical results is a uniform mean flow. However, downstream of the rotor-stator

pair, the underlying mean flow does deviate from a uniform flow because of the wakes

of the rotor airfoils. This deviation is much more than the deviation upstream of therotor-stator pair due to potential effects. It is believed that the nonuniformity in themean flow is the main reason for the discrepancy between the numerical results andtheory.

7. SUMMARY

This study focuses on the numerical computation of tones in rotor-stator interactions.

the numerical predictions are obtained by solving the thin-layer Navier-Stokesequations on a system of patched grids. The mode-cotent of interaction tone noise is


obtained for two different airfoil counts and is shown to conform with a kinematical

analysis of the flow. In addition, the propagation characteristics of different modes are

compared with the predictions of linear theory. Numerically computed pressure

amplitudes on the surface of the airfoils are compared with experimental data.The effects of both reflective and non-reflective boundary conditions on the

calculated flow field were assessed. It was found that for the short-grid calculations

non-reflective boundary conditions had to be used in order to predict the tonaacoustics in the flow field. Use of non-reflective boundary conditions becomes more

important for those cases where there is a high energy content in the propagatin_

modes. (The singte-stator/single-rotor case for example, had propagating towmharmonics. These harmonics had high energy content. Reflection of these harmonic:

from the computational boundaries resulted in a degradation of the computed pressure

amplitudes on the airfoil surfaces.) It was also shown that reflective boundar,Conditions can be made to behave essentially as non-reflective boundary condition'.

with the use of appropriately long grids in the upstream and downstream regions. Th_

long grid approach in conjunction with an increasing grid cell size in the far upstrean

and downstream regions results in numerical dissipation of propagating modes, thu:

avoiding the problems due to reflections at the computational boundaries.

ACKNOWLEDGEMENT

This study was partially supported by NASA Marshall Space Flight Center. Some o

the computing resources were provided by the NASA OAST NAS programme. Theauthors would like to thank Dr R. P. Dring of United Technologies Research Cente

for providing the turbine airfoil geometries used in this investigation.

REFERENCES

BAYLISS, A. & TUrUCEL, E. 1980 Radiation boundary, conditions for wave-like equationsCommunications on Pure and Applied Mathematics 33, 707-725.

BAYLISS, A. & TURKEL, E. 1982a Far field boundary conditions for compressible flows. Journaof Computational Physics 48, 182-t99.

BAYLmS, A. & TURI<EL, E. 1982b Outflow boundary conditions for fluid dynamics. SIAAJournal on Scientific and Stastical Computing 3, 250-259.

DORNEY, D. J., DAVIS, R. L., EDWARDS, D. E. & MADAVAN, N. K. 1990 Unsteady analysis ohot streak migration in a turbine stage. AIAA Paper No. 90-2354.

DORNE¥, D. J. 1992 Numerical simulations of unsteady flows in turbomachines. PhD. ThesisThe Pennsylvania State University, Department of Aerospace Engineering, University ParkPennsylvania, U.S.A.

DRING, R. P., JOSLYN, H. D., HARDIN, L. W. & WA_NER, J. H. 1982 Turbine rotor-statointeraction. ASME Journal of Engineering for Power 104, 729-742.

ERDos, J. I., ALZNER, E. & McNALLY, W. 1977 Numerical solution of periodic transonic flo_through a fan stage. AIAA Journal 15, 1559-1568.

GruELING, H. J., WEINaEaG, B. C., SHAMRO'rH, S. J. & McDoNALD, H. 1986 Flow throughcompressor stage. Report R86-910004-F, Scientific Research Associates. Glastonbury, C'IU.S.A.

GiLzS, M. B. 1988a Stator/rotor interaction in transonic turbine. AIAA Paper No. 88-3093.GILES, M. B. 1988b Non-reflecting boundary conditions for the euler equations. Repox

CFDL-TR-88-1, MIT Computational Fluid Dynamics Laboratory.GOLDS'rEIN, M. E. 1974 Aeroacoustics. NASA Report SP-346.GUNDY-BURLET, K. L., RAI, M. M., STAUTER, R. C. & DmNG, R. P. 1991 Temporarily an,

spatially resolved flow in a two stage axial compressor: Part 2-Computational assessmenASME Journal of Turbomachinery 113, 227-232.

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GRIFFIN, L. & McCoNNAU_HZY, H. 1989 Prediction of the aerodynamic environment and heattransfer for rotor/stator configurations. ASME Paper No. 89-GT-89.

JORGENSON, p. C. E. & CmMA, R. V. 1988 An explicit Runge-Kutta method for unsteadyrotor/stator interaction. AIAA Paper No. 88-0049.

JORCZNS0N, p. C. E. & CmMA, R. V. 1989 An unconditionally stable Runge Kutta method forunsteady flows. AIAA Paper No. 89-0205.

KoYA, M. & KOTAKE, S. 1985 Numerical analysis of fullv three-dimensional periodic flows

through a turbine stage. ASME Journal of Engineering for Gas Turbines and Power 107,945-952.

Lzwm, J. p., DELAtqE'C, R. A. & HALL, E. J. 1987 Numerical prediction of turbine vane-bladeinteraction. AIAA Paper No. 87-2149.

MADAVAN, N. K., RAI, M. M. & GAVAI_I, S. 1989 Grid refinement studies of turbinerotor-stator interaction. AIAA Paper No. 89-0325.

RA_, M. M. 1986 An implicit conservative zonal boundary scheme for euler equationcalculations. Computers and Fluids 14, 99-131.

RA_, M. M. 1987 Navier-Stokes simulations of rotor-stator interaction using patched andoverlaid grids. AIAA Journal of Propulsion and Power 3, 387-396.

RAI, M. M. 1989 Three-dimensional Navier-Stokes simulations of turbine rotor-stator interac-tion; Parts 1 & 2. AIAA Journal of Propulsion and Power 5, 305-319.

RAI, M. M. & MADAVAN, N. K. 1990 Multi-airfoil Navier-Stokes simulations of turbinerotor-stator interaction. ASME Journal of Turbomachinery 112, 377-384.

RAN_WALt, A, A. A., MAt)AVAN, N. K. & JOHNSON, P. D. 1991 Application of an unsteady

Navier-Stokes solver to transonic turbine design. AIAA Journal of Propulsion and Power 8,1079-1086.

RANCWALLA. A. A. & RAZ, M. M. 1990 A kinematical/numerical analysis of rotor-statorinteraction noise. AIAA Paper No. 90-0281.

RAO, K. V., DELANEY, R. A. _ DUNN, M. G. 1992a Vane-blade interaction in a transonicturbine. Part 1-Aerodynamics. AIAA Paper No. 92-3323.

RAO, K. V., DELANEY, 1_. A. & DUNN, M. G. 1992b Vane-blade interaction in a transonicturbine. Part 1-Heat transfer. AIAA Paper No. 92-3324.

TYLER, J. M. &: SOFRIN, T. G. 1970 Axial flow compressor noise studies. SAE Transactions 70,309-332.

VERDON, J. M. 1989 The unsteady flow in tia'e far field of an isolated blade row. Journal of Fluidsand Structures 3, 123-149.

/"

f"/

. "/-

- 12:,. . .

AIAA-94-2835

Unsteady Navier-Stokes Computations forAdvanced Transonic Turbine Design

Akii A. RangwallaMCAT InstituteNASA Ames Research CenterMoffett Field, CA 94035-1000

30th AIAA/ASME/SAE/ASEE JointPropulsion Conference

June 27-29, 1994 /Indianapolis, IN

For permission to copy or republish, contact the American Institute of Aeronautics and Astronautics370 L'Enfant Promenade, $.W., Washington, D.C. 20024

UNSTEADY NAVLER-STOKES COMPUTATIONS FORADVANCED TRANSONIC TURBINE DESIGN

Akil A. Rangwallat

MCAT Institute, Mountain View, CA

Abstract

This paper deals with the application of a three-

dimensional, time-accurate Navier-Stokes code for pre-

dicting the unsteady flow in an advanced transonic

turbine. For such advanced designs, prior work in

two dimensions has indicated that unsteady interac-

tions can play a significant role in turbine performance.

These interactions affect not only the stage efficiency

but can substantially alter the time-averaged featuresof the flow. This work is a natural extension of the

work done in two dimensions and addresses some of

the issues raised therein. These computations are be-

ing performed as an integral part of an actual de-

sign process and demonstrate the value of unsteady

rotor-stator interaction calculations in the design of

turbomachines. Results in the form of time-averaged

pressures and pressure amplitudes on the airfoil sur-

faces are shown. In addition, instantaneous, contours

of pressure and Math number are presented in order to

provide a greater understanding of the inviscid as well

as the viscous aspects of the flowfield. Relevant sec-

ondary flow features such as cross-plane contours of to-

tal pressure and span-wise variation of mass-averaged

quantities are also shown.

Introduction

The traditional design of new turbines has re-

lied upon empirical correlations, extensive experimen-

tal data, and a technology data base comprising pre-

vious design s . This design process has proven veryret.iable for new designs that do not deviate very much

from those in the existing data base. However, a more

general predictive capability is needed when the oper-

ating conditions of a new design demand radical devi-ations from the data base.

Considerable progress has been made in using

computational fluid dynamics (CFD) to predict flowswithin turbomachines. Much of the early work has fo-

cused on predicting the flow in airfoil cascades. An

extensive body of experimental and numerical resultsin the literature deals with a wide variety of two-and

three-dimensional cascade geometries. While such meth-

ods of analysis of flows in isolated airfoil rows have

helped improve our understanding of flow phenomena

t Research Scientist. Senior Member, AIAA

Copyright (_)1994 by the American Institute of Aero-nautics and Astronautics, Inc. All rights reserved.

in turbomachinery and have gained widespread accep-tance in the industrial community as a design tool,

they do not yield any information regarding the un-

steady effects arising out of rotor-stator aerodynamic

interaction. However, it is becoming increasingly im-

portant to consider interaction effects in the design of

new generation turbines. This has come about due tothe constraints of low weight, small size, high specific

work per stage, high efficiency, and durability, which

results in very high turning angles and unconventional

airfoil shapes, potentially giving rise to nonlinear un-

steady interactions.

In the past few years, advanced transonic tur-

bines have been designed by Pratt and Whitney in

support of the Consortium for CFD Application in

Propulsion Technology sponsored by NASA MarshallResearch Center. These turbines are characterized by

very high flow turning angles (160 ° per stage) and rel-

atively high loading coefficients. The current status

of turbine design technology is shown in Fig. 1 (pri-

vate communication, L. Griffin, NASA Marshall Space

Flight 'Center). The figure shows the turning angles

and turbine loading coefficients of some existing tur-bine designs. The figure shows two designs (the G3T

and the G2OT) that have turning angles of 160 ° which

is 20 ° higher than the traditional design limit of 140 °.

Numerical methods that simulate the unsteady

flow associated with rotor-stator configurations have

been developed in recent years. References 1-3 present

a zonal approach for solving the unsteady, thin-layer,

Navier-Stokes equations for rotor-stator configurationsin a time-accurate manner, both in two and three di-

mensions. The present work is an application of thethree-dimensional rotor-stator code described in Ref.

3, to evaluate the design of the advanced Gas Genera-

tor Oxidizer turbine designed by Pratt and Whitney,and will henceforth be referred to as the G2OT. It

should be mentioned that the first application of an un-

steady two-dimensional Navier-Stokes solver for design

purposes was carried out for the GST (Ref. 4). The nu-

merical predictions were obtained at a constant radius

corresponding to the midspan of the rotor airfoils. The

primary issue was the effect of unsteady interactions

on boundary layer separation. The results from the

unsteady two-dimensional analysis led to design mod-

ifications (Ref. 4) and provided the designers with abetter understanding of the physics of the flow. The

results also validated the concel_t of using high turn-

ing angles and high specific work per stage. One out-

comewasthepossibilityof usingsingle-stageturbinesfor certainapplications,thusconsiderablysimplifyingthe designprocess.This directlyinfluencedthe de-signoftheG202 _. The unsteady two-dimensional code

was again used to aid in the design of the G2OT and

was reported in Ref. 5. The flow was predicted at

a constant radius which was equal to the midspan of

the rotor airfoil. Results were obtained for two power

settings (100% and 70%) and it was found that theturbine loads were within the tolerances specified by

design requirements and were acceptable. The two-dimensional analysis used in l_efi 4-5 contained quasi-three-dimensional source terms to account for stream

tube contraction effects (Ref. 6). The numerical algo-rithm was an extension of that previously reported in

P_ef. 7.

One drawback of the two-dimensional analysis

is that it is not complete. Since the flow is three-

dimensional, the issue of secondary flow influencing

flow features such as strength and position of the shockshas to be addressed. Other issues such as the effect of

unsteady interactions on the end-wall boundary layershave to be assessed. Hence a three-dimensional inter-

action study was initiated.

In this paper, the three-dimensional as well assome two-dimensional results for the G20T will be

presented. The two-dimensional results were obtained

for two power settings (100% and 70%) whereas thethree-dimensional results were obtained for only the

100% power setting. Comparisions will be made wher-

ever possible. In particular, it was found that therewere similarities as well as differences between the two-

dimensional and three-dimensional results. The over-

all loading on the airfoils obtained from the three-

dimensional analysis at midspan compared fairly with

the two-dimensional predictions. However, some of the

details such as strength and positions of the shocks dif-fered. This also resulted in weaker unsteady interac-

tion predictions by the three-dimensional calculations.

Two grid systems (one with twice as many points

in the radial direction than th e other) were used for a

limited grid independance study. Each grid system

contains multiple patched and overlaid grids as de-

scribed in Ref. 3. These grids can move relative toone another to allow for the relative motion of the ro-

tor airfoils with respect to the stator airfoils.

Numerical Method

The numerical method solves the unsteady, three-

dimensional, thin-layer Navier-Stokes equations. The

Navier-Stokes equations in three dimensions are nondi-mensionalized and transformed to a curvilinear time-

dependent coordinate system, and a thin-layer approx-

imation is then made. The unsteady, thin-layer, Navier-

Stokes equations are solved using an upwind-biased

finite-difference algorithm. The method is third-order-

accurate in space and second-order-accurate in time.

Several iterations are performed at each time step, so

that the fully implicit finite-difference equations aresolved to ensure a time-accurate solution. Further de-

tails of the method can be found in Ref. 3.

Boundary Conditions

The boundary conditions required when using

multiple zones can be broadly classified into two types.The first are the zonal conditions which are imple-

mented at the interfaces of the computational meshes,

and the second are the natural boundary conditions

imposed on the surface and the outer boundaries ofthe computational mesh. The treatment of the zonal

boundary conditions can be found in Ref. 2. The nat-

ural boundary conditions used in this study are dis-

cussed below.

Airfoil Surface Boundary

The boundary conditions on the airfoil surfaces

are the "no-slip" condition and adiabatic wall condi-

tions. It should be noted that in the case of the rotor

airfoil, "no-slip" does not imply zero absolute velocityat the surface of the airfoil, but rather, zero relative

velocity. In addition, the derivative of pressure in the

direction normal to the wall surface is set to zero.

Exit Boundary

The flow in the axial direction is subsonic at the

exit boundary and hence only one flow quantity has to

be specified. The flow quantity chosen in this study

is the exit static pressure as a function of radius. To

completely specify the flow variables at the boundary,

four other flow quantities are extrapolated from theinterior. The four chosen are the Reimann invariant,

2cRl =u+_

7-1

the entropy,

p7

and the velocities in the transverse directions. One dis-

advantage of this type of boundary condition is that

the pressure waves that reach the boundary are re-flected back into the flow domain. However, this bound-

ary condition was chosen since, in general, it provides

greater control on the turbine operating conditionsand results in the correct pressure drop and mass flow

throughtheturbine.

Inlet Boundary

Theflowat the inlet boundaryissubsonic.Fourquantitiesneedto bespecifiedat thisboundary.Thefourchosenwerethe l%eimanninvariant,

2c/Z_1 =U-t----

7--1

the total pressure as a function of radius,

Pto,al= Pinle_ (1 + ---_lY, inl_t/

and the inlet flow angles,

v_"l_--------2-_= Can(O)Uinlet

and

winz.t __ tan(C)Uinle_

The fifth quantity needed to update the points on this

boundary is also a Reimann invariant that is extrapo-

lated from the interior and is given by

2cR2=u-- --

7--I

In the above equations, the quantities u and v and w

are the velocities in the axial (x) tangential (0) and

the radial (r) directions, p is the pressure and c is

the local speed of sound. Specifying the total pressure

at the inlet results in a reflective boundary condition,

but together with the specification of the exit staticpressure, has the advantage of determining uniquely

the turbine operating conditions.

Periodic Boundaries

Turbomachines are designed with unequal airfoilcounts in the stator and rotor rows in order to mini-

mize vibration and noise. A complete viscous simula-

tion including all of the airfoils in the stator and rotor

rows is yet impractical in a design environment. The

approach used here is to assume that the ratio of thenumber of stator to rotor airfoils is a ratio of two small

integers. This is achieved by scaling the stator or the

rotor geometries such that the blockage remains the

same. Periodicity conditions are then imposed over

the composite pitch. For the case of the G20T tur-

bine, the number ofstator airfoils is 20 and the number

of rotor airfoils is 42. By changing the number of sta-

for airfoils to 21 and rescaling the stator airfoils by a

factor of 20/21, a stator to rotor airfoil count of 1 to2 is achieved. The calculation assumes that the flow

exhibits spatial periodicity over one stator airfoil andtwo rotor airfoils. Note that the pitch of one rescaied

stator airfoil is equal to the composite pitch of tworotor airfoils.

Geometry and Grid System

A schematic diagram of the G20T is shown in

Fig. 2. This is a single stage turbine that is designed

to operate in the transonic regime. It is characterized

by very high turning angles and high specific work.

Figures 3a-b show the system of overlaid gridsused to discretize the flow domain. The figure shows

the fine grid with 51 grid points in the spanwise direc-

tion. Figure 3a shows the grid at the midspan. Eachairfoil has two zones associated with it; an inner zone

and an outer zone. The inner zone contains an O-grid

that is generated using an elliptic grid generator. This

grid is clustered near the airfoil surface in order to re-solve the viscous effects. The outer zone is discretized

with an H-grid and is generated algebraically. The in-

ner and outer grids overlap one another. This position-

ing of the inner and outer grids facilitates informationtransfer between the two zones. The outer H-grids of

the stator airfoils and rotor airfoils overlap and slip

past each other as the rotor airfoils move relative tothe stator airfoils. Figure 3b shows the surface grid

(minus the casing). Here, the grid in the tip clearance

region can also be seen. This grid was also generated

by means of an elliptic grid generator and maintains

metric continuity with the inner O-grid. The fine _id

contains approximately 940000 grid points whereas the

coarse grid has half as many.

Results

It should be mentioned that the numerical method

has been validated both in two and three-dimensional

applications. In particular, the ability to predict the

time-averaged pressures and pressure amplitudes on

airfoil surfaces and total pressure losses in airfoil wakes

have already been demonstrated for turbines as well as

for compressors (see Refs. 1-4, 7, 8).

G20T Two-Dimensional Computations

A brief description of the two-dimensionai results

(Ref. 5) will first be presented for the purpose of

comparison with the three-dimensionai results. Two-

dimensional predictions were obtained for two power

settings. The first setting is at 100% power and the

second is at 70% power. The operating conditions for

thetwopowersettingsareshownin TableI.

100%Power 70%PowerInlet MachNo. 0.46 0.54Inlet ReynoldsNo. 2.6× loS�inch 1.1× lOS�inch

RPM 7880 6232

Inlet Pt,,tal 542. 77 psia 313.82psia

Exit P_tatic 200.OOpsia 144.5psia

Inlet Ttot,_l 1307.02°/_ 1080.88°R

Table I. Turbine oPerating conditions

Static Pressure Variation on Airfoils Fig-

ures 4 and 5 show the time-averaged and unsteady en-velope of static pressure on the airfoil surfaces for the

two different power settings, respectively. The pres-sure coefficient on the surface of the stator airfoils in

this case is defined as

Cp.__ P,tatic

P_otallnltt

where Pstatic is either the time-averaged static pres-

sure on the surface of the airfoil (to obtain the time-

averaged pressure distribution) or the maximum or

minimum pressure over a cycle (which results in the

pressure envelope). The time averaging is performedover a cycle which corresponds to the rotor airfoils

moving through two airfoil pitches. The pressure co-efficient on the surface of the rotor airfoils is defined

as

Cp__ Pstatic

Ptotat(relative)_o,°.,=_.,

Here, the pressure is normalized with respect to the

time-averaged relative inlet total pressure to the ro-

tor rows. The figures show that the results of the

two power settings are qualitatively similar. The pre-

dicted pressure amplitudes are slightly smaller for the

70% power setting than for the design setting (100%power). The pressure distribution indicates a weak

(nearly stationary) shock on the suction surface of the

stator airfoil that impinges on the rotor suction surface

near the leading edge (Ref. 5). It is this shock that

accounts for the moderately high pressure amplitudes

near the leading edge of the rotor airfoils. The pressure

distributions also indicate a shock near the trailing

edge of the rotor airfoils. This second shock is nearlystationary with respect to the moving rotor airfoils. Itshould be noted that these are two-dimensional results

at constant radius. In the three-dimensional case, theinteraction effects are found to be less severe due to

the relaxation effects of the spanwise direction.

G_OT Three-dimensional results

The results for the three-dimensional computa-

tions of the G20T for the 100% power setting are pre-sented in this section. These results were obtained

by integrating the governing equations and boundaryconditions described earlier. A modified version of the

Baldwin-Lomax turbulence model (Refs. 9-11) wasused to determine the eddy viscosity. The modifica-

tion involves the use of a blending function that varies

the eddy viscosity distribution smoothly between theblade and endwall surfaces. Further details can be

found in Refs. 10-11. The kinematic viscosity was

calculated using Sutherland's law.

Static Pressure Variation on Airfoils Fig-

ures 6-8 show the time-averaged and unsteady enve-

lope of static pressure on the stator and rotor airfoils

at three spanwise locations. Figures 6a-8a show the

pressure variations on the stator airfoil at the hub, the

midspan and at the casing, whereas Figs. 6b-Sb show

the variations at the hub, the midspan and at the tip

of the rotor airfoils. These results were obtained by

the fine grid calculations and do not show any signif-icant differences when compared with those obtained

from the coarser grid. The level of unsteadiness on

the stator airfoils is small compared to that on the

rotor'airfoils. The amplitudes also are smaller at the

casing than at the hub. The figures seem to indicate

the existence of a weak shock (made clearer by contour

plots) on the suction surface of the stator near the hub.

The pressure amplitudes on the rotor airfoil are larger.

The rotor airfoils are unloaded considerably at the tip.

However, it was found that this is very localized near

the tip region and is not very critical. The predicted

pressure amplitudes of the three-dimensional results at

midspan, are smaller than the two-dimensional results.

This is mainly due to the difference in the strength of

the predicted axial gap shock.

Figures 9a-b show the comparisions of the time-

averaged pressures between the three-dimensional and

the two-dimensional results. On the stator airfoil, the

two-dimensional calculations predict a lower unload-

ing at the airfoil nose than that shown by the three-dimensional calculations. It should be noted that the

two-dimensional calculations were performed on a sur-

face of constant radius. Quasi-three-dimensional sourceterms associated with stream-tube contraction were

included in the calculation, but the terms associated

with radius variation were not. To properly account

for these terms, the two-dimensional calculations would

have to be performed on a cylindrical surface with an

axially varying radius. The overall loading on the ro-tor airfoils compares better. However, the details are

different. In particular, the position and strength of

thetrailingedgeshockon thesuctionsurfaceisdiffer-ent. Also,otherdetailsthat werepresentin thetwo-dimensionalcalculations,suchas a largelocalvari-ation on the suctionsurface,is absentin the three-dimensionalresults.

InstantaneousMach Number ContoursFig-ures10a-cshowinstantaneousMachnumbercontoursat 20%,50%(midspan),and80%of spanrespectively.Theseresultswereobtainedfor thefinegrid system.At 20%of span,the shocknearthe trailingedgeoftherotor airfoilcanbeseen.At this location,anaxialgapshockwasalsoseen. However,unlike the two-dimensionalcalculationsthethree-dimensionalcalcu-lationspredictan intermittentaxialgapshock. Atmidspan(Fig. 10b),theshockin theaxialgapregionismuchweaker,whereasthereisnoshockatthedown-streamlocation.In fact, it wasfoundthat theradialextentof the axialgapshockvariedwith timewith amaximumextentof about50%.Figure10cshowstheMachcontoursat 80%.Thecontoursseemto indicatethat theremightbeunsteadyseparationonthesuctionsurfaceof therotorairfoils.Recallthattheturningan-glesin this turbineareveryhigh,andthereis a con-cernaboutmassiveboundarylayerseparationunderthe influenceof unsteadyinteractions.Thenumeri-calcalculationsdonotpredictmassiveboundarylayerseparation,asindicatedbytheMachnumbercontours,thusincreasingtheconfidencein thedesign.

InstantaneousStatic PressureContoursFig-ureslla-c showthe instantaneouspressurecontoursat 20%50%(midspan)and80%of spanrespectively.These contours basically highlight the inviscid features

of the flow. As expected, the rotor shock near the hub

can be seen.

Mass-Averaged Quantities versus Span Fig-

ure 12 shows the mass-averaged meridonial angle ver-

sus normalized span at four different axial stations.

The axial stations correspond to the inlet of the tur-

bine, the midgap, half a chord downstream of the rotor

airfoil, and about one and a half chord lengths down-stream of the rotor airfoil. The figure does show that,

to a large extent, the flow turns about 160 ° through

the stage, however, it also shows a region of under-turning at the midspan. Figure 13 shows the mass-

averaged radial pitch angle. Recall from the schematic

of the G20T (Fig. 2) that the casing angle is -30 ° at

the inlet, and is positive aft of the rotor (approximately

11.75°). This is reflected in the mass averaged pitch.

Figures 14-17 show the variation of the mass-averagedMach number, the relative Mach number (relative with

respect to the rotating rotor airfoils), the absolute to-

tM pressure and the relative rotational total pressure.

One surprising aspect of the results is the local increase

in total pressure losses at the midspan.

Time-Averaged Contours Figures 18a-b show

the time-averaged contours of the relative rotational

total pressure at midgap and half a chord length down-stream of the rotor airfoils. The circumferential extent

of Fig. 18a equals the circumferential pitch betweentwo successive stator airfoils whereas that of Fig. 18b

equals the pitch between two rotor airfoils. Also, itshould be noted that the time averaging is done in two

different frames of reference. At the midgap (Fig. 18a)

the frame of reference is stationary, whereas, down-

stream of the rotor airfoils (Fig. 18b) it is rotating.

The contours at the midgap do show the expected

(nearly uniform in span) stator wake along with thehub and casing secondary flows. However, aft of the

rotor blades, at the midspan, a region of slightly higherlosses exists. This was also observed in the mass-

averaged numerical data (Figs. 16-17). Figures 19a-b

show the time:averaged contours of Mach number rel-ative to the rotor airfoils at the same axial location.

The relative Mach number of the flow is subsonic at

the midgap, but downstream of the rotor it becomes

supersonic and eventually shocks.

$ummary

detailed numerical calculation of the three-

dimensional unsteady flow in an advanced gas gener-

ator turbine is presented. The computational results

are obtained by solving the three-dimensional, thin-

layer, Navier-Stokes equations on a system of overlaid

grids. The numerical results do capture many aspectsof the flow that could aid in the understanding of the

flow. In addition, the results do not indicate any sig-

nificant boundary layer separation, (an object of con-

cern). The unsteady loadings were found to be within

acceptable limits.

The present results indicate that a proper un-derstanding of the unsteady interaction effects could

play an important role in the design of advanced gas

generator turbines.

ACKNOWLEDGEMENT

This study was partially supported by NASA

Marshall Space Flight Center. Computing resources

were partially provided by the NAS program.

I%EFER.ENCES

1. Rai, M. M., "Navier- Stokes Simulations of

Rotor-Stator Interaction Using Patched and Overlaid

Grids," AIAA Journal of Propulsion and Power, Vol.

3, No. 5, pp. 387-396, Sep. 1987.

S

2. Ral, M. M., '_rhree-Dimensional Navier-Stokes

Simulations of Turbine P_otor-Stator Interaction; Part

[ & 2," AIAA Journal of Propulsion and Power, Vol.5, No. 3, pp. 305-319, May-June 1989.

3. Madavan, N. K., Rai, M. M. and Gavali, S.,"A Multi-Passage Three-Dimensional Navier - Stokes

Simulation of Turbine Rotor-Stator Interaction," Jour-

nal of Propulsion and Power, Vol. 9, pp. 389-396,May-June 1993.

4. Rangwalla, A. A., Madavan, N. K. and John-

son, P. D., "Application of an Unsteady Navier-Stokes

solver to Transonic Turbine Design," AIAA Journal of

Propulsion and Power, Vol. 8, No. 5, pp. 1079-1086,September-October 1992.

5. R.angwalla, A. A., "Unsteady Flow Calcula-tion in a Single Stage of an Advanced Gas Genera-

tor Turbine." Invited paper at the Fifth Conference

on Advanced Earth-to-Orbit Propulsion Technology,

at NASA Marshall Space Flight Center, Huntsville,Alabama, May 19-21, 1992.

6. Chima, 1_. V., "Explicit Multigrid Algorithmfor Quasi-Three-Dimensional Viscous Flows in Turbo-

machinery," Journal of Propulsion and Power, Vol. 3,No. 5, Sept-Oct. 1987, pp. 397-405.

7. Ral, M. M., and Madavan, N. K., "Multi-Airfoil Navier-Stokes Simulations of Turbine Rotor-

Stator Interaction ," Journal of Turbomachinery, Vol.112, pp. 377-384, July 1990.

8. Gundy-Burlet, K. L., Ral, M. M., Stauter, 1_.

C., and Dring, 1_. P., '_remporarily and Spatially l_e-

solved Flow in a Two Stage Axial Compressor: Part

2-Computational Assessment," ASME Journal of TUr-

bomachinery, vol 113, pp. 227-232, April t991.

9. Baldwin, B. S., and Lomax, H., '_rhin Layer

Approximation and Algebraic Model for Separated Tur-

bulent Flow," AIAA Paper No. 78-257, 1978.

10. Dorney, D. J., and Davis, R. L., "Navier-Stokes Analysis of Turbine Blade Heat Transfer and

Performance," Journal of Turbomachinery, Vol. 114,pp. 795-806, 1992.

11. Vatsa, V. N., and Wedan, B. W., "Navier-

Stokes Solutions for Transonic Flow over a Wing Mountedin a Wind Tunnel," AIAA Paper No. 88-0102.

• , ._ "..,..., ...

180

160

140

120

100 -

80 -

- E)G3T GGOTQ

ATD LOX STME fuel ...E) J2L.O.X ........ C)..... . C_ J2f.ueJ..(i)RL10 Traditional

E) SSME LOX Design limitE) F1

E) ATD fuel

O SMME fuel

I I I I I I I I60 0 2 4 6 8 10 12 14 16

Turbine loading coefficient gJ&h'STGu_

aOVO_

_M_ cun_

..... If] -: ....

Ul

Fig. 1 Current status of turbine design technology.

Fig. 3a Rotor-stator airfoil grid systems for the G2OT,

at mid-span

(1.40)-1

3o,_ (_i]°)

(-0.75 li.072F,

i1.437R

V

(20)

(2.80)

r(2.9o),(1.9o) I _I

ti .087R

B

(42)

"I (× = 01. 10) (!}00)(1.60) (2.70)

8.973R

Fig. 2 Schematic of G20T flow path.

Fig. 3b Rotor-stator airfoil grid systems for the G2OT,

surface grid.

c_

.l=._

u m.

O

I..+

=m m.0tl oO_

tfl

.m.

I

a

I + I r r + t r f

.3 -t.2 -t.!

_r)

<z)

(.o

0

Lr')

-p

r_

......... - .................................... ..

'......

I|i "

b

,.J l r f t _ ]

-i.O -0.9 -0.8 -0.7 -0.8 -0.5 -0.4 -0.3 0.2 0.4 0.6 0.8 1.0 1.2 1.4

Axial Distance (x) Axial Distance (x)

Fig. 4 Static pressure variationon (a) stator and (b) rotor airfoilsurfaces for the G20T (100% power).

el_

Oo

U •

o

O_

m 6m<3,3

o

o

o

r_ _

a _.I 1 I I 1 I l 1 1 I ¢=

-,.:-t.2-t._-_.0-0._-0.8-0.7-o.6-0.s-o._-o._

Axial Distance (x)

". "+...

.... . ,....,.+..°..

i " ....... .

bI I T I T ,1

0.2 0._ o.s 0.8 t.o _.2 ,.4

Axial Distance (x)

Fig. 5 Static pressure variation on (a) stator and (b) rotor airfoil surfaces for the G20T (70% power).

- . ................. :.............. - ................................. ................,

Pressure Variation on Stator

-t.3-1.2 -1.1 -1.0 -0.9-0.8 -0.7-0.6 -0.5 -0.4-0.3

Axial Distance (x)

1211.1¢,j

ff-_ 1.0-

C9

"-_ 0.9-

0.8-

._ o.7-tO

_C0.6"

O_J

0.5"

0.4-cD

0.3-

0.2

Pressure Variation on Rotor

.i i i i ! _ i i:_,_ i i i -_",°d I-i-'_-----T-T, =:.... I.....

.......--T-i..............................._......T_........

........_.....!....!....!........!.........!.......!........_....!-!.........0.30.4 0.5 0.6 0.7 0.8 0.9 1.0 l.i 1.2 1.3 1.41.5

Axial Distance (x)

Fig. 6 Static pressure variation on (a) stator and (b) rotor airfoil surfaces for the G20T (Hub).


1-05 / i i

_ 1.ooI....,.-/T-_........._........i....- i

_ o._,l......-/-_i ! i ! _............. i0.85q.... --

_" 0.80 ....

0.75

0.70--_.. ---T--_.,---A., -7-

) 0.60 .....

= oi,--1_ I-._---_ ....'-'o4o........i.........i.........i.......i......i....7.......i......T-T-o.35 ; ; ; ; ' ' ' ' '

-1.3-1.2 -1.1 -1.0 -0.9 -0.8 -0.7 -0.6 -0.5 ,.0.4-0.3

Axial Distance (x)


"e.9

o

d:

:-I i i_ i i 'i ii2,_..o:,, i_"I--'_-_----h.--_ _-,ol__EL_T.?_._-_-__-_......0.9---! .._' :: : : i-'=F.q_.... ....

i..............................0.7-' . :.-i--':._-i.... i.....r-i .... i i--i-_--i..... !ii-

0.30.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.41.5

Axial Distance (x)

Fig. 7 Static pressure variation on (a) stator and (b) rotor airfoil surfaces for the G20T (Mid-span).

.-N

¢D

o_26DO

r..)


1.05 i

0.95-

°°iiii0.85-

0.80-

0.75-

0.70-

0.65-

0.60- . ....../

0.551..........._............_............t..........._..........r-_....-........i.........-......|

°4571-":_ I i................_-# ..... i _---i..... ....

0.407.........._............i...........i............i............i ....... "....... -"--" ........ ".....o35" " _, _ [

-[.3-t.2 -1.1 -1.0 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4-0.

Axial Distance (x)


i ..,"}.....i.....L.... ! ! i _ & _ i I__-_,o-...........i__:__.......'-_::?--i--_:_-_' i.........

_ o._-i_:,L"!i"-::i-i_ _!.i i i___i_!ii__i!!1_ 0.8 ..................

._ 0.7"---- :.......

_ °" iii__' -i-,_ o.5....... t......Yi .... T--'t-:r ........i_i i-!?;.:r,i .........

0.4"

o._-i i ! ___,_,,.0.2 ' ' , ' , , ,

0.30.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.41.5

Axial Distance (x)

Fig. 8 Static pressure variation on (a) stator (casing) and (b) rotor airfoil (tip) for the G_OT.

"U


1°5/ i1.ool.........}:_--_:......_...........:.,........._.........r---_---;----...............T---............................................o,oT--,__--r--

0.80 .......

07,............_-_-0.70........_: ....b

o.oo...........__A_..-:70........_i:i

-- :ND_m_u*_u_l C.al_lati*_ J0.45" .... :-Dm_*_m_ C._,_U,i_

0.40 ........ i ...............................................0.3.* .....

-1.3-1.2 -1.1 -I.0 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4-0.3

Axial Distance (x)

1.2

1.1"

£ _.O"e_

"---- 0.9"

_" 0.8"

0.7""U

0.6"

00.5"

0.4

_'* 0.3"

0.2


"-rj---.'----r---'T----V'---......_.....-"---'q--7---T

-4-i----.,-''--:4'.---,&=:{:.--4' .... 4----i .... g---÷......._........

.... _. .'..i';..,.____.._-_._ ..... _..... _...... ._--_-.: ..+ ......

0.30.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.41.5

Axial Distance (x)

Fig. 9 Comparision between two- and three-dimensional predictions of time-averaged static pressures (a)

stator and (b) rotor airfoil surfaces (Mid-span).

"_ds %0_ (_) P"_ _d_ %Og (q) '_d_%Og (_) _ _n_ad _o _no_uo_ _no_u_u_ I II "_._I

"u'eds%0_ (_) pu_ u_d_ %0_ (q) 'u'eds %Og

(_) _ _qmnu q_ _o g_no_uo_ sno_u_u_su I OI "S!_

<

100

80

60-

40-

20-

0

_ _,,_i, _,... 4- ,;., -;- _7 .7 z _ ....... : ........ :: ....................

1 Scauom ................et tO St._OOr

.............. 1='=N,_'_ ...................

-20....................................................i ...........................................

-40......................... _..........................-6o........................ !...................-so _....... _ ----.:_=='__.-_-C_

-1000.0 0.2 0.4 0.6 0.8 1.0

Radial span

Z

e.D

1.1

1.0-

0.9"

0.8-

0.7

0.6"

0.5

0.4-

0"3I-"

0.2 .....

0.1 ..........

0.00.0

........ ÷.......... _ ................... -_-'_,'i'-"_'

i _ ! ".'_ '

[ Stations

t tO sr_tor i

0.2 014 0.6 018 1.0

Radial span

Fig. 12 Mass-averaged meridonial angle as a functionof span.

FIE." 14 Mass-averaged absolute mach number as afunction of span.

20

10-

_

-10-

-20-

-3O

,_.,_. • _, _•A. _,.q*_ ! :

et tO StaU3f : _ "

0.0 0.2 0.4 0.6 0.8 1.0

Radial span

1211.1 "5_'_.--_-.UL

• 1 h'4"k¢'_-:--:'_'"_z_'_'_'_---t-----_"_'-*"=: ........

! .... +-'w.2_ .... _ .... _.... _-. .......ft..", *.,,_ • _ _ \_o.____-_'---.=-__- .=--.... - .... ..._.__._.._.,

u, ! _ " ""_-- ! _ ..d_,/Tt

0.7_'_"..... "r-.... T .... -Z-.... "_;.'_-=_- _

o.6_----+---_ :_,.,,= i---+ ..... ",_i ---': d-'°'=" ____]t;

_. 0.5 = ' i i:0.4_--___---- I..... _ .... _ ...... ;

0.2 _ i i '

0.00.0 0.2 0.4 0.6 0.8 1.0

Radial span.

Fig. 13 Mass-averaged pitching angle as a function of

span.

Fig. 15 Mass-averaged relative maeh number as a func-

tion of span.

0

1.2 t I

.o-if: ...............- ...............i............. ::.................._.................:_I

0.9 .................. _......................-:.................._..................._................

0.8 ..........................................................................................................

_.....°._._÷...--....._....... _'_....... ...._.-_.

o.6]_................i..............s_,o.." i..............i..........._]

o.4.1 ................i....................i..................!.................i....................-t0.3

0.0 0.2 0.4 0.6 0.8 1.0

Radial span

•............ _ '., - ,:. ¢ f_:_,- --....

, " :" ". =, ".. 'xN\\\\X.. "\ ,_._ :" ..: . :, :.. '- _ \ . \ _ ._'t .. i , ,l:[li ; :: ."

- ': - - " _tlil_ :\ llilil.7 1: " , :. \ t t i • / f ] _ I:l:!ll ;"

:' \ " ". t : 1 l!! I , t _i i:#;:'ti:tii::_i: :: :. \ <,_.{ii!i i

::::::::::::::::........

Fig. 16 Mass-averaged absolute total pressure as a

function of span.

Fig. ,18a Time-averaged relative rotational total pres-

sure at mid-gap.

O°_

>

VCl

1.2

1.1 ................................................................

1.0 .....................................................................

0.9- "-'_....... '--.......... -.......... -......... _--.,...... :"_

0.8-'_"7 ......... _"-_ ............ -_-.----.-t'i # i *',,_

i;. ......................................................... "..................... ?' :0.7-it _ "_

0.6...................................i.....................................................Sl_tkx_

0.5- __ _,.,,,

0.4 ................. _ ............... " ............. r ...........................

0.30.0 0.2 0.4 0.6 0.8 1.0

Radial span

Fig. 17 Mass-averaged relative rotational total pres-

sure as a function of span.

Fig. 18b Time-averaged relative rotational total pres-sure aft of the rotor airfoil.

-'_ " X /"

,,,'_ .\ ,, ,,, ... ,.............-------_.__.__._,'i//_ \ y.... ".... \ "X I "" .."'----- ------ttl. .'f t_ \ \ \

•.\.,,, \.-,.,,, ///,,----.---. _-// ",.,,,

•. ., \ <,\ / ]i ./" /_ _ .." _ \ _.

"\\'\",,\ /// / ..... -,. // , \v X \

\\\,x X...///li.... --..Jl_---/" \ _..J" \ \

X\ "X\.'_-J//!/,., _'---&?' ..... 2" \\_ i \. "x ..... ". '_--/...'I// ",-,.,--W' .-.....,\ _ _ \ ',

',,._\ "------" .",."'/ ,-.---.,. . ,/" X,' • _ , ; /_", \

_::::: :::::

Fig. 19a Time-averaged relative mach number at mid-

gap.

Fig. 19b Time-averaged relative mach number aft of

the rotor airfoil.

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