P.Pieringer, E.Göttlich, J.Woisetschläger, W.Sanz, F.Heitmeir (2005) Numerical investigation of the unsteady flow through a transonic turbine stage,

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1 P.Pg, E.Gölh, J.Wohläg, W.az, F.Hm (5) Numal gao of h uady flo hough a ao ub ag, Po. 6h Euoa of o Tubomahy, Lll,. 9-5

2 NUMERIAL INETIGATION OF THE UNTEADY FLOW THROUGH A TRANONI TURBINE TAGE UING AN INNOATIE FLOW OLER P. Pg - E. Gölh - J. Wohläg - W. az - F. Hm Iu fo Thmal Tubomahy ad Mah Dyam, Gaz, Uy of Thology, Aua, aul.g@ugaz.a ABTRAT I h a a umal gao of h uady h-dmoal flo hough h ao ub ag a Gaz Uy of Thology d. Th umal od ald ol h Ryold-aagd Na-ok uao ug a m-a haa mhod dd by h f-olum aoah. Tubul modld by h o-uao modl of ala ad Allmaa. A od oluo obad by alyg ha laggd bouday odo. Wall bouday lay mulad by u gad all fuo. Th ff of h lakag flo a odd by mhg h laa b oo blad ad ag. Th ul a dud ad omad h xmal daa gad by m-old la dol lomy (LD). Boh alulao ad maum d good agm h ak hddg fom h alg dg of ao ad oo blad o b ha lokd o h blad ag fuy. Th alulao ho, ha og hok a odally fld by ag ad blad a obl fo h homo. NOMENLATURE oy loy o u d of oud dy ao of f ha ll-fa aa Nabla-oao & ha flux o τ ha o,η, ula ooda do gd dx ll olum u o omal o ll fa u o of ula ooda do, u o ll fa, omal o ah oh, dff oao galu haa alu ω ud laxao aam fo No-Raho Iao I u max,b oa fo h all-fuo dfd by aldg (96) u odmoal aalll-o-all loy y odmoal all da k off of hmal oduy R ga oa ν l, ν lama ad ubul kma oy T oo blad ag od u T oao o ao max

3 INTRODUTION Th dg of ff ubomahy ag u dald koldg of flo homa ad h ma o ffy. Noaday ub ag of fau h-dmoal gomal dg ad hgh u ao. o h flo ao ad u hok aff h ulg omlx uady flo uao. ady a FD flo alulao au h m aagd ma flo ll bu lak by dfo h do of uady ff. Th umal alulao of h h-dmoal uady flo hough a ub ag hold a lo of hallg. To au h uady ff h gd oluo mu b f h hol flo doma. Bouday lay ff dm h oday flo ad h lo, o h all bouday lay mu b mulad oly. Rolg h bouday lay by a f gd lad o mall ll, hh fo h omuaoal m- alu o b y mall. To ah a od oluo a lo of blad ag od ha o b alulad. If h h ao of ao ad oo o uy, addoal ffo mu b ald o du h alulao doma o o ao ad oo hal. To g aoa ul fo uh a oblm aoabl oal omuaoal m, a FD ol mu fau a fa algohm, ha abl o aualy ol h Na-ok uao, allfuo, ha oly mula h bouday lay o du oal ou of ll ad o abl a lag m- alu, ad a o mhod o du h flo oblm o o hal fo ao ad oo. Th ud Na-ok od ol h Ryold-aagd Na-ok uao haa fomulao ogh h h o uao ubul lou dlod by ala ad Allmaa (99). Th od u uud gd a mul-blok aagm. Th bouday odo a ald h haom ll. To ga a abl ad aua odu h od u a domoo of h d flux, ha a dd by h oa fomulao of h Na-ok uao, ladg o a of a-y uao (o dmoal oo uao). Ug h haa fom of h flo uao h omuaoal ffo aud by mull max maulao of oa flo ol a b aodd, hu ladg o duo omu m. Th ulg hm o oa fo ma, momum d gy, bu h falu glgbl. Dao m do by a od-od aoxmao of h m oluo, hh old a ml ay. To o h uual h ao, ha laggd bouday odo (.g. Edo al., 977 ad H, 99) a ald. To fuh du alulao ffo ad m, u-gad- all fuo ald. Wall-fuo dd by h la-of-h-all fomulao (aldg 96) ha b uggd by Fk (996), h h a a aoah ll b d. Th ud FD-od a dlod a h Iu fo Thmal Tubomahy Gaz by ham o h mhod khd by Pg al. (), o h majo fou ll b lad o h do of h umal hm. Th ao ub ag a Gaz Uy of Thology a ho fo h gao, a lo of xmal ad umal ah ha alady b odud fo ha a (.g. Gölh al., Pg al. ). Exmal ad umal ul ll b omad, ad h mham of ha-lokd ak-hddg bhd h ao alg dg ll b aalyzd. NUMERIAL METHOD Dao a Wh ug m aabl ludg h oy, h Na-ok uao a b a follo.

4 [ ] ( )[ ] R T & & ) ( ) ( ) ( ) ( τ τ τ ( ) Th ul of h d a (Eul a) of h Na-ok uao ) ( ( ) a alo b obad by h oluo of a of o-dmoal haa uao. Fo a ula ooda ym h haa uao o dx do ado h fom, ad a aally dzd fom I fa h d uao o dx do a b a follo ( ) ( ) ( ) ( )[ ] ( )[ ] ( ) o, h haa dff ad h oodg galu a ( ) ( ) 5 5 ( ) ad h da m of h m aabl a o b oud.

5 ( ) [ ] [ ] ζ ζ ζ 5 5 ( 5) To alua h aal dff of h haa ddg o h g of h oodg galu (ud-dg), a mhod dd fom h TD (oal aao dmhg) odu of hakaahy (988) ald, hh lad o follog do. ( ) ( ) ( ) ( ) Φ Φ Φ Φ Φ Φ Φ Φ Φ Φ I I I I g g g g ab, ab m ab, ab m ab, ab m ab, ab m ( 6) Ddg o h g of h oodg galu, h aal dff I, ad I ad h o alu g ad g, ddg o h g of h aal dff mu b. < > I I I I ( 7) ( ) ( ) [ ] ( ) ( [ I I g g g g, g g )] ( 8) Wh hoog Φ /, h algohm lad o a abl aoxmao of aal hd od auay. Th ou m oud a al-dffg ma, ad uod o h Eul ul. Dao m Th aabolal aoxmao of h m-da lad o h ml odo ( 9) hh old ug h No-Raho mhod, I ad ω ( ) dg fo o h alu. Iad of ag h No-Raho ao h fo, h ag alu dmd by a aoxmao o a oa gd. Af aog h flo oblm o a oa gd ( Q ), old f od aua m ad a (Q ), ad h ul ud fo h alzao o h f gd, Q - Q fo. Th oagao of lo fuy duba, ha a b dd y ll by a oa gd, ll aud by h mhod, ad a dud ou of No-Raho o h f gd uff fo og.

6 Pha-laggd bouday odo To ga od oluo m ad a fo abay ao ad oo blad umb, ha laggd bouday odo a gomal od bouda ad a h fa b ao ad oo doma ald. A d-o hu ud, h uda o h bouda do by a ghd aag of h ly aalabl flo-fld ad h od aabl. To k h dmad of omu mmoy lo, o ah bu a uff amou of m- alu od ad a la olao ud fo h ouo of h bouday alu. Pu-gad- all fuo Th ald all-fuo bad o h la-of-h-all fomulao by aldg (96) ad od h ha- dbuo dula o h all ufa, hh maly dmd by u gad (Pg, 5). Th all fuo aum ady a odo. Th la-of-h-all fomulao by aldg (hh a lghly modfd fo h all-fuo aoah) db h o-dmoal aalll o h all loy dbuo of a ubul bouday lay fo zo u gad flo. (Th do of y omal o h all.) ( ) ( ) ( ) u u u B u y u u ( ) 6 Th aumo, ha h h bouday lay h ha oa, lad o a uao fo h ao of ubul ad lama kma oy. ν ( ) ( ) u u B u u ( ) ν l 6 By kog h aalll o h all loy a a o h a dfd da o h all ad mag h lama oy ad h dy a h all, o a fgu ou h dbuo of h aalll loy ad h ao of ubul ad lama kma oy b ha o ad h all. By ug haom ll o mo h bouday odo, a ay ay o ak all ko alu of h f alulao ll lo o h all ad o alula h alu of h haom ll, hag a dfd da o h all. o h a laa b h f alulao ll ad h all. Wh u gad ou, h aumo, ha h h bouday lay h ha oa, o ald aymo. lo o a all h ha ll ah b a fuo of kfo, da o all ad u gad aalll o h all. I fa h ulbum of fo d u ( ν l ν ) τ xy ( y) τ xy () y ( ) y x By alg h aal dbuo of h ao of lama ad ubul oy, ha a b dd by E. () ad E. (), o h h loal aalabl alu a h oo of h f alulao ll, E. () df h aal da of h loy u. u zo a h all ad ha o ado h ko alu a h f alulao ll h loy ofl ad h kfo a b alulad. No h loy ofl ud o df h loy h haom ll ad h ald dbuo of h ao of lama ad ubul oy ud o df h ala Allmaa ubul aabl h (fo dald do Pg, 5). Fo h-dmoal oblm, h u gad o dula do aalll o h all a dd od o oly df h bouday lay. I h haom ll h loy dula o h all ufa o zo ad h u xaolad fom h alulao doma. Wh oulag adaba all odo, h ulbum of ha flo ad ok by ha fo, 5

7 T u k u ( ν l ν ) ( ) y y dd by h fa, ha gy-flux oad h all mu b ual zo, hold a h all ad alo h h bouday lay. Thfo E. () ud o alula h mau of h haom ll. h bouday lay aoxmao dd fom ady-a odo, om addoal o ll ou by alyg o m-old alulao. ALIDATION To alda h d umal hm h blad KI L 59 a ho fo omao o h TTM ol, a flo-ol ha bad o ooal ad ll ablhd hm. Th TTM ol ol h Ryold-aagd Na-ok uao by ma of a fullyml m-mahg f-olum mhod ad dbd dal by Gh (999). Th gog uao a ad oa fom, h o a a dzd ug a hd-od-aua TD-ud hm, bad o h MUL y of a ud hm. Th KI L 59 modld D h a.m. Th mh fo h TTM ol dgd fo olg h bouday lay a h all, ladg o a ll ou of abou.. Fo h d umal hm h mh dgd fo mulag h bouday lay a h all ug h d all fuo, ladg o a ll ou of abou.. Bd fom bouday lay aa boh gd do o dff fom h oh. A h l oal u. ba ad oal mau T 8. K ad a h oul o Mah umb M.7 a ald, ubul modld by ala Allmaa ubul lou. (a) TTM ol (b) d ol Fg. : KI L 59, gayal da Mah umb, l da oal u Rul a md-a ad oul (Fg.) ho, ha h xll agm Mah umb (Fg., gayal) ad ll agm oal u (Fg., l). Th a oly mo dao oal u lo o h ag (Fg., ao). Aa fom m-ag du o h dud amou of ay omuaoal gd-ll, h d ol dd half h m fo dg h Eul-a of h Na-ok uao a ah ll omad o h TTM ol, h fuh dud omuaoal m. 6

TET AE Th ao ub ag a Gaz Uy of Thology a old flo ub faly (Ehad ad Gh, ), hh a uu ombao of a max.

Th ub ag allo a fo ooal a ll a oal maum d. Th ub ag oa ao blad ad 6 oo blad, hh lad o a h ao of /. Wha h dam of h ag oa (D.

Th dg x agl of h ao 69.7 fom h ax. Th ga b h oo blad ad h ag.95 mm, hh.56% of h blad hgh a h ladg dg ad.6% of h blad hgh a h alg dg.

: Tao ub ag a Gaz Uy of Thology, gomy o omuaoal Gd A mul-blok gd og of blok ad a oal ou of abou.. ll ud fo h umal alulao.

8 TET AE Th ao ub ag a Gaz Uy of Thology a old flo ub faly (Ehad ad Gh, ), hh a uu ombao of a max..8 MW axal ub ad a dly ould omo ha ga mo ha half of h omd a dd o d h ub. To o h lo addoal a odd by a aa, lally d omo ao. Th ub ag allo a fo ooal a ll a oal maum d. Th ub ag oa ao blad ad 6 oo blad, hh lad o a h ao of /. Wha h dam of h ag oa (D. m) h ou ag ho a ogg-dgg oou. Th mmum of h flo hal hgh a h af o of h ao (D m.5 m), h h ou oou a o h a og agl of (D oul.55 m). Th dg x agl of h ao 69.7 fom h ax. Th ga b h oo blad ad h ag.95 mm, hh.56% of h blad hgh a h ladg dg ad.6% of h blad hgh a h alg dg. l ag fa laa oul. [m] fll ao blad oo blad fll.5. hub la B la x[m].5 Fg. : Tao ub ag a Gaz Uy of Thology, gomy o omuaoal Gd A mul-blok gd og of blok ad a oal ou of abou.. ll ud fo h umal alulao. ag blad h blok fll Fg. : Gd dal, lf: -laa of h oo-blad, gh: mhg of h fll Th laa b oo blad ad ou ag mhd by (oly) 6 ll, hough h u of all-fuo ll allo a good aal oluo h (Fg., lf). Th fll a h ao hub ad ad a h oo hub a modld a ll. To gula h, a h blok o h all doma a h blad (Fg., gh). Th hal mhd by 7 ll fom hub o ou ag. 7

9 Bouday odo Il Bouday odo: oal u:. ba oal mau:. K l agl: (hal) ala-allmaa ubul aabl: ~ν /ν l. Th ala-allmaa ubul modl a dgd fo fully ubul flo h lo fam ubul. A h ubul y a h l a o ko, h alu fo ~ν /ν l a o., hh a uggd alu by ala ad Allmaa (99) fo lo f-am ubul. Oul Bouday odo: a u:.95 ba umb of : 55.5 m. alulao Paam A ady-a oluo a ahd by ug a mxg la hu b ao ad oo. To-dmoal o-flg bouday odo ug h umfally aagd a u a h oo l ald a h ao oul. Th flo agl, h oal u ad h oal mau xad fom h umfally aagd alu a h oul of h ao gd dmd h l of h oo gd. ag fom h ady a ul. m h a alu alulad o oba a od uady oluo. Th maxmum FL umb a abou 7. O ao od la 9 m ad o oo od la 5 m, o mo ha ao od ad oo od omud. Th aoxmao o h oa gd (y od gd-o) fo ah m a alulad f-od aua m ad a. Afad o h f gd ah m a old by 6 No-Raho h a ud-laxao alu of ω.6. O h f gd h odmoal all da y a blo 8 a ah ll lo o a all. Fo h ha-laggd bouday odo y 6 h m a od, gg a oluo of 5 od alu- fo h ao ad 5 fo h oo. Th ghg fao fo h uda a o.7. Th alulao m a abou fou k o a mod.5 GHz. 6-b omu. Th alulao aam d by a ua D gao of h d a. Blla agm fo mul-aag (o ao hal, h oo hal) ad ha-laggd (o hal ah) ul a ahd, og h ald alao of h ha-laggd bouday odo m of mhod ad aam fo ha a. MEAUREMENT ETUP Th oal loy maum of h flo fomd ug a o-dmoal LDym (Gölh al. ). Oal a a alzd hough a mall gla do, o ga h axal ad umfal loy. Th daa odd f axal la by o- do alog adal l, umfal ag a ahd by ug h ao-ag. To allo a oo-ha-old aaly of h maud lo, a f gal odd by h moog ym of h ub a ud o gg h daa amlg. Th daa of o la ud o oma h h umal ul, la B load lo bhd h alg dg of h ao ad la load lo bhd h oo (Fg. ad Fg. ). 8

ad alulad daa fomd by aluag h m-h lo ad h m-a lo of axal ad

Pla B Th m-h lo la B a 5% a o ou a hok-a ha ogly al h umfal

Th maum ad h FD ul dag oo, bu boh ho h hok-oo o b almo m dd.

Th ak hog ha lokd ak hddg ha mu b odally ggd.

Th ak lghly dfld h- do a abou /T.5 fo FD ad /T. fo maum.

5, mak ) ad a gfa og ox h ak a b dd af ha h maum ad FD ul

Th FD ul au boh homa ll, bu hy a m-dlayd omad o h maum.

10 REULT h (a) la B h Fg. : Maum gd (b) la omao of Maud ad alulad Daa Th omao of maud ad alulad daa fomd by aluag h m-h lo ad h m-a lo of axal ad umfal (aga h- do) loy. Pla B Th m-h lo la B a 5% a o ou a hok-a ha ogly al h umfal loy (Fg.5, mak A). Th maum ad h FD ul dag oo, bu boh ho h hok-oo o b almo m dd. I h ak (Fg.5, mak B) ox uu a b dd. Th ak hog ha lokd ak hddg ha mu b odally ggd. Th maum ho o a od, h FD ul x. Th ak lghly dfld h- do a abou /T.5 fo FD ad /T. fo maum. Podally a hok-a o h aluao doma (Fg.5, mak ) ad a gfa og ox h ak a b dd af ha h maum ad FD ul (Fg.5, mak D). Th FD ul au boh homa ll, bu hy a m-dlayd omad o h maum. Th axal loy ho h m-a lo of a adal l ood a h. (h h ak) da yldal ox hddg FD ul (Fg. 6(a)), h maum d h baku of ha yldal bhao by a addoal ox blo a % (Fg. 6(b), ao). Th axal loy da (Fg. 6, mak A), ha ll dd FD ul bu m dlayd omad o h maum, da ha h ggg mham of h ak hddg dff a oo. Fuh gao ha o b ad ou o d h ao fo h m dlay of h umal ul. 9

11 A A B B /T (a) umfal loy, FD /T (b) umfal loy, LD D D /T /T () axal loy, FD (d) axal loy, LD Fgu 5: Pla B, a 5% a 8% 8% a A a A 5% /T 5% /T (a) FD (b) LD Fgu 6: Pla B, a. h, axal loy

Pla Th abolu-fam o of au all ff, ha f o h

Th m-h lo (h h ll h ao h) la a 5% a ho a ll

Th om od a of h axal loy omo h h ak good

7, makd h a h l), bu o og hok a a b dd, o

B A A B /T (a) umfal loy, FD /T (b) umfal

LD Fgu 7: Pla, a 5% a Th umfal loy lo ho

12 Pla Th abolu-fam o of au all ff, ha f o h mog oo of h oo blad, o b ld. Th m-h lo (h h ll h ao h) la a 5% a ho a ll ha-lokd ak hddg (Fg.7, mak A) bhd h oo alg dg. Boh, maum ad FD ul ho o o ao blad ag od. Th om od a of h axal loy omo h h ak good agm b FD ul ad maum (Fg.7, makd h a h l), bu o og hok a a b dd, o au h ha lokg of h ox hddg. B A A B /T (a) umfal loy, FD /T (b) umfal loy, LD /T () axal loy, FD /T (d) axal loy, LD Fgu 7: Pla, a 5% a Th umfal loy lo ho fuhmo h oo of h uo d alg dg hok (Fg.7, mak B), ad aoh o (Fg.7, mak ), ha aumd o b h flo of h u d alg dg hok of h ghbog blad, bg fld a h af o of h uo d. I good agm of FD ul ad maum h oo of h hok a y muh m-dd fg o h mog oo blad.

Wak hddg Tggg Mham a h ao Talg Edg Th uo d hok xd ao h fa

8, dahd l ad Fg.5, mak ) ad mg o h alg dg (Fg.8, /T.8).

Fom h o h ak hddg udubd, ul h x fld hok a gg a ox aga. h /T.

8 /T.5 9 Fgu 8: ao ak a 5% a, l da oba, gayal da oy omao of

ma ady a.5.5 od Fgu 9: Po of o blad Th m old o of o blad (Fg.

13 Wak hddg Tggg Mham a h ao Talg Edg Th uo d hok xd ao h fa oad h oo blad (Fg.8, fa l a /T., ad Fg. 5, mak A), bg odally fld. Th fld hok a mo bak aga oad h uo d of h ao blad (Fg.8, dahd l ad Fg.5, mak ) ad mg o h alg dg (Fg.8, /T.8). Th fo h ak o a ox-aao (Fg. 8, ao). Fom h o h ak hddg udubd, ul h x fld hok a gg a ox aga. h /T. /T. /T.8 /T.5 /T. /T.9 Fgu 8: ao ak a 5% a, l da oba, gayal da oy omao of Tm Rold ad ady a alulao o of o blad [kw] m old m ma ady a.5.5 od Fgu 9: Po of o blad Th m old o of o blad (Fg. 9) ho o ak a od ad % ak-o-ak alu fg o h m ma alu. Th ady a alulao oma h o by.% omad o h m ma alu. Th oy dbuo bhd h ao alg dg ho a d ak fo h ady a alulao (Fg., lf, dahd l), ha h m ma of h uady alulao ho a

14 xo of h flu doma by h ak o abou / of h hol hal, du o h ox hddg. Th oo of h alg dg hok, dad by a ha oy, y h am fo h ady a alulao ad h m ma of h uady alulao (Fg., ao). ady a m ma of h uady alulao Fgu : ao x flo, gayal ad l da oy ady a m ma of h uady alulao Fgu : Flo a adu.5 m (abou md-a), gayal da la Mah umb, l da u Fgu ho h flo fld of h ady a ad h m ma of h uady alulao a abou md-a. Th y ll agm m of Mah umb ad u fo ao ad oo doma, aa fom h ao ak (Fg., h ao), ha moohd by h

15 uady alulao, ad h ao alg dg hok (Fg., blak ao) ha lghly og dd by h ady a alulao. Nhl h ady a alulao y aabl of dg h m ma flo. ONLUION A umal udy of h h-dmoal uady flo hough a ao ub ag a ad ou ug a oa flo ol. Th od bad o h haa flo uao o aod m oumg max maulao ay fo oa ol, hu ladg o a fa algohm. To fuh du omu m a ol aoah fo a bouday-lay mulag all-fuo a odud, ha od h flu of h u gad o h ha- dbuo dula o h all ufa. To ga od oluo m ad a, ha laggd bouday odo a gomal od bouda ad a h fa b ao ad oo doma ald. By omao h maum daa h umal mhod od o d h uady h-dmoal ff ll, hough om dagm ould by dd. Th homo of ha lokd ak hddg ould b fd by FD ul ad mham bhd h ao alg dg ould by dbd. Th uo d alg dg hok a of h ao o o b obl fo ha homo. Th hok a odally h h oo uo d, ad h flo mo bak o h ao alg dg, ggg h ox hddg. REFERENE hakaahy.r. (988) Hgh Roluo Ud Fomulao fo h Na-ok Euao, KI Lu 988-5, omuaoal Flud Dyam Edo J, Alz E, MNally W, (977) Numal oluo of Pod Tao Flo Though a Fa ag, AIAA Joual, ol. 5, o., Ehad J, Gh A () Dg ad ouo of a Tao T-Tub Faly, AME Pa -GT-8, Iaoal Ga Tub ad Aog og Fk N, (996) Am of a Uuud-Gd Mhod fo Pdg -D Tubul ou Flo, AIAA Pa 96-9 Gölh E, Numay F, Wohläg J, az W, Hm F (), Igao of ao- Roo Iao a Tao Tub ag Ug La Dol lomy ad Puma Pob, AME Joual of Tubomahy, ol. 6, 97-5 H L (99), A Eul oluo fo Uady Flo Aoud Ollag Blad, AME Joual of Tubomahy, ol., 7-7 Pg P, Gh A, az W (), alulao of ao-roo Iao of a Tao Tub ag Ug a Ioa Uady Flo ol, Podg of h h Euoa of o Tubomahy, Fz, 9- Pg P, az W, Numay F, Gh A, Hm F (), Numal Igao of h Th-Dmoal Flo Though a Tao Tub ag, Podg of h 5 h Euoa of o Tubomahy, Pag, 5- Pg P, az W (5), A Pu-Gad- Wall Fuo fo h Pdo of Tubul Flo Thmal Tubomahy, GT5-687, ad a h AME Tubo Exo 5 Gh A. (999), Eklg D-Na-ok od zu umh Bhug d Tubomahömug, Dooal Th, Gaz, Uy of Thology ala P.R., Allmaa.R., (99), A O Euao Tubul Modl fo Aodyam Flo, La Rhh Aoaal, No., 5- aldg D.B. (96), A gl Fomula fo h La of h Wall, Ta AME A, J Al Mh ol8, o, -58

Numerical Solution of Transient Thermal Stresses in a Functionally Graded Cylinder

Numerical Solution of Transient Thermal Stresses in a Functionally Graded Cylinder La d gg Mha gg Glgy al l f a hal a Fally Gadd yld IQ H KHOLO I-LMH aal gg a Jda y f ad hlgy P.O x Ibd JO al: daabh@.d. ba: - h a d h a hal a la yld ad f a fally gadd aal FGM. h yld aal dd b gadd alg h