Effects of boundary conditions and unsteadiness on draft tube flow

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L U L E A • U N I V E R S I T Y . J k ^t

O F T E C H N O L O G Y

2003:11

DOCTORAL THESIS

Effects of Boundary Conditions and Unsteadiness on Draft Tube Flow

b y

MICHEL J. CERVANTES

Department ot Applied Physics and Mechanical Engineering Division o f Fluid Mechanics

2003:11 • ISSN: 1402 - 1544 • I S R N : L T U - D T - - 03/11 - - SE

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Effects of Boundary Conditions and Unsteadiness on Draft Tube Flow

Michel J . Cervantes April 7, 2003

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ABSTRACT

T h e present research focuses o n flow p r o p e r t i e s o f t h e e l b o w d r a f t t u b e . T h i s element has a m a j o r f u n c t i o n i n low head t u r b i n e s , since u p t o h a l f of t h e flow losses m a y arise t h e r e away f r o m the best efficiency.

T h e use of c o m p u t a t i o n a l fluid d y n a m i c ( C F D ) t o redesign a d r a f t t u b e necessitates d e t a i l e d knowledged of the b o u n d a r y c o n d i t i o n s . T h e y are generally n o t available a n d q u a l i f i e d guesses m u s t be m a d e . T h i s applies i n p a r t i c u l a r t o t h e r a d i a l v e l o c i t y at t h e i n l e t . A m e t h o d t o e s t i m a t e t h i s c o m p o n e n t i n s w i r l i n g flows f r o m e x p e r i m e n t a l values of the a x i a l a n d t a n g e n t i a l velocities is d e r i v e d . T h e m e t h o d uses a t w o dimensional non-viscous d e s c r i p t i o n o f t h e flow, t h e Squire-Long formulation. I t is tested against s w i r l i n g flow i n a d i f f u s e r a n d a p p l i e d t o t h e T u r b i n e - 9 9 d r a f t t u b e flow.

A s several o t h e r b o u n d a r y c o n d i t i o n s are d i f f i c u l t t o e s t i m a t e , e.g. t h e t u r b u - lence l e n g t h scale, a n d m a n y parameters are available t o p e r f o r m a s i m u l a t i o n , e.g. t u r b u l e n c e models a n d difference schemes, t h e use o f factorial design is p r o p o s e d as an a l t e r n a t i v e t o design s i m u l a t i o n s i n a s y s t e m a t i c , o b j e c t i v e a n d q u a n t i t a t i v e way. T h e m e t h o d allows t h e d e t e r m i n a t i o n o f t h e m a i n a n d j o i n t effects o f i n p u t p a r a m e t e r s o n t h e n u m e r i c a l s o l u t i o n . T h e i n p u t p a r a m e t e r s m a y be e x p e r i m e n t a l u n c e r t a i n t y on b o u n d a r y c o n d i t i o n s , u n k n o w n b o u n d a r y c o n d i t i o n s , g r i d a n d t u r b u l e n c e models. T h e m e t h o d is a p p l i e d t o the T u r b i n e - 99 test case, w h e r e t h e r a d i a l velocity, t h e surface roughness, the t u r b u l e n c e l e n g t h scale a n d t h e g r i d were the f a c t o r s i n v e s t i g a t e d . T h e i n l e t r a d i a l v e l o c i t y is f o u n d t o have a m a j o r effect o n the pressure recovery.

T h e flow i n w a t e r t u r b i n e s is h i g h l y u n s t e a d y due t o t h e r u n n e r blade r o - t a t i o n , guide vanes a n d stay vanes. U n s t e a d y pressure measurements on a K a p l a n p r o t o t y p e p o i n t o u t unsteadiness i n t h e h i g h a n d l o w pressure region of t h e t u r b i n e . Since m o d e l a n d p r o t o t y p e are n o t r u n n i n g i n d y n a m i c a l l y s i m - i l a r c o n d i t i o n s , t h e influence of unsteadiness o n t h e losses is o f interest. T h e d e r i v a t i o n of t h e variation of the mechanical energy f o r t h e mean, o s c i l l a t i n g a n d t u r b u l e n t f i e l d p o i n t o u t the c o n t r i b u t i o n o f unsteadiness t o the losses a n d t h e t u r b u l e n t p r o d u c t i o n . A p p l i c a t i o n t o t u r b u l e n t channel flow reveals t h a t t h e c o n t r i b u t i o n is a f u n c t i o n of the a m p l i t u d e o f t h e o s c i l l a t i o n , t h e f r e q u e n c y a n d t h e f r i c t i o n velocity.

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Turbulent pulsating flow i n a generic m o d e l o f t h e r e c t a n g u l a r diffuser f o u n d a t t h e end o f elbow d r a f t t u b e is s t u d i e d i n d e t a i l w i t h laser D o p p l e r a n e m o m - e t r y ( L D A ) . T h r e e frequencies, c o r r e s p o n d i n g t o t h e quasi-steady, r e l a x a t i o n or i n t e r m e d i a t e a n d q u a s i - l a m i n a r regime w i t h a n a m p l i t u d e of a b o u t 10% are i n v e s t i g a t e d beside t h e steady r e g i m e . T h e results i n d i c a t e no a l t e r a t i o n of t h e m e a n f l o w b y t h e e x c i t a t i o n of a single frequency. F u r t h e r m o r e , t h e existence of t h e d i f f e r e n t regimes, as f o u n d i n t u r b u l e n t p u l s a t i n g t u r b u l e n t p i p e and channel flows, is c o n f i r m e d .

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ACKNOWLEDGMENTS

I w o u l d like t o t h a n k m y supervisor, Professor H . Gustavsson, f o r g u i d i n g me i n m y three years o f research a n d give me the o p p o r t u n i t y t o be i n v o l v e d i n t h e e d u c a t i o n p r o g r a m H y d r o Power U n i v e r s i t y .

I a m g r a t e f u l t o F . E n g s t r ö m f o r i n t r o d u c i n g me t o laser D o p p l e r anemome- t r y ( L D A ) technique a n d the c o n s t r u c t i v e c o l l a b o r a t i o n f o r three o f t h e papers presented i n t h e thesis.

V a l u a b l e discussions have be made w i t h P h D students o f t h e " V a t t e n t u r b i n - t e k n i k " p r o g r a m ; U . Anderssson ( V a t t e n f a l l U t v e c k l i n g A B , Sweden), M . G r e k u l a ( C h a l m e r s U n i v e r s i t y o f Technology, Sweden) a n d S. V i d e h u l t ( G E H y d r o , N o r - w a y ) .

Special t h a n k s are t o m y lovely w i f e Sara a n d c h i l d r e n , E m i l a n d Elise, f o r t h e i r comprehension a n d s u p p o r t despite m y l o n g absence f r o m t h e house t o w o r k o n t h e present thesis. Since m y c h i l d r e n c a n n o t r e a d yet, I take t h e l i b e r t y t o p u t t h e i r p i c t u r e . I t h a n k also D r . C a r r , D r . v a n Devanter, J . B e r r e b i a n d S. Leduc f o r t h e i r f r i e n d s h i p .

T h e present w o r k was financed b y t h e Swedish E l e c t r i c a l U t i l i t i e s Research a n d D e v e l o p m e n t C o m p a n y ( E L F O R S K ) , the Swedish N a t i o n a l E n e r g y A d m i n - i s t r a t i o n , G E E n e r g y (Sweden) a n d W a p l a n s M e k a n i s k a V e r k s t a d s A B t h r o u g h t h e p r o g r a m " V a t t e n t u r b i n t e k n i k " .

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Part I

General discussion

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Chapter 1 INTRODUCTION

T h e first p r o d u c t i o n of e l e c t r i c i t y b y water t u r b i n e s dates back t o 1882 [1].

Today, h y d r o p o w e r p l a n t s are present i n most countries a n d s t a n d f o r a b o u t 20%

o f t h e e l e c t r i c i t y p r o d u c t i o n w o r l d w i d e . H y d r o p o w e r is t h e m a j o r renewable source of energy w i t h m a n y b e n e f i t s . I t produces e l e c t r i c i t y w i t h a m i n i m a l emission of greenhouse gases. H i g h l y effective, u p t o 96% efficiency f o r the large size o f Francis t u r b i n e . I t has the a b i l i t y t o respond r a p i d l y t o m a r k e t demands, a n i m p o r t a n t f e a t u r e f o r deregulated m a r k e t s . F u r t h e r m o r e , reservoirs represent a h i g h l y efficient w a y t o store energy. T h e m a i n drawbacks of t h i s technology concern i t s dependence t o p r e c i p i t a t i o n , w h i c h can be h i g h l y v a r i a b l e a n d i t s influence on fish m i g r a t i o n . A p r o b l e m w h i c h has t o be e f f e c t i v e l y solved before h y d r o p o w e r can be f u l l y accepted as a green source o f energy. Since less t h a n a t h i r d of the h y d r o p o w e r p o t e n t i a l is i n s t a l l e d over the w o r l d a n d the concern over t h e e n v i r o n m e n t increases, h y d r o p o w e r can p l a y a m a j o r role i n the n e x t decades as i t does already i n m a n y countries such as N o r w a y , C a n a d a a n d Sweden where 99%, 6 1 % a n d 54% respectively o f the e l e c t r i c i t y was p r o d u c e d b y t h i s technology i n year 2000, a c c o r d i n g t o t h e O r g a n i z a t i o n f o r E c o n o m i c C o o p e r a t i o n a n d D e v e l o p m e n t ( O E C D ) .

Besides h y d r o p o w e r , Swedish e l e c t r i c i t y is p r i n c i p a l l y p r o d u c e d b y nuclear p l a n t s (37.4%) a n d fossil f u e l ( 6 . 1 % ) , a c c o r d i n g t o t h e O E C D i n year 2000. T h e p a r k of h y d r o t u r b i n e s i n Sweden is w i d e a n d composed o f a b o u t 700 p l a n t s w i t h a capacity larger t h a n 1.5 MW of w h i c h 14 have a c a p a c i t y o f m o r e t h a n 200 MW a n d a b o u t 1200 o t h e r smaller u n i t s [2]. T h e m a i n p r o d u c t i o n is con- c e n t r a t e d along t h e L u l e å r i v e r ( 2 0 % ) , Å n g e r m a n r i v e r (17%, i n c l u d i n g Fax r i v e r ) and I n d a l s r i v e r ( 1 5 % ) . M o s t o f t h e t u r b i n e s were b u i l d between 1940 a n d 1970 [3], t h e r e f o r e a n i m p o r t a n t p e r i o d o f r e n o v a t i o n is a p p r o a c h i n g . T h e r e f u r b i s h m e n t has t o be done i n c o n j u n c t i o n w i t h state o f t h e a r t i n t u r b i n e t e c h n o l o g y t o o b t a i n best efficiency. However, r a t i o n a l i z a t i o n o f t h e m a r k e t has considerably reduced the n u m b e r o f t u r b i n e m a n u f a c t u r e r s a n d R<kD i n the i n - d u s t r y and at universities decreased s u b s t a n t i a l l y . A w a r e o f t h e p r o b l e m , t h e Swedish E l e c t r i c a l U t i l i t i e s Research a n d D e v e l o p m e n t C o m p a n y ( E L F O R S K ) ,

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F i g u r e 1 . 1 : Schematic of a hydropower plant [5].

t h e Swedish N a t i o n a l E n e r g y A d m i n i s t r a t i o n , G E E n e r g y (Sweden) a n d W a - plans Mekaniska Verkstads A B s t a r t e d a p r o g r a m i n 1997 t o i m p r o v e Swedish w a t e r power competence [4]. T h e present w o r k is p a r t o f t h e second phase, w h i c h s t a r t e d i n J a n u a r y 2000 a n d ended i n F e b r u a r y 2003, where 9 research s t u d e n t s were i n v o l v e d . T h e f o l l o w i n g w o r k has been f i n a n c e d b y t h i s p r o g r a m .

1.1 Kaplan t u r b i n e

H y d r o p o w e r p l a n t s are c o m p l e x systems, cf. figure 1.1, where competence i n c i v i l , mechanical a n d e l e c t r i c a l engineering are c o m b i n e d . T h e a i m is t o convert p o t e n t i a l energy c o n t a i n e d i n a n elevated b o d y of water i n t o r o t a t i o n a l mechanical energy i n order t o d r i v e a generator. H y d r o p o w e r p l a n t s are generally composed of a reservoir d e l i m i t e d p a r t i a l l y b y the n a t u r a l e n v i r o n m e n t a n d a d a m , where water is stored. A s t h e c o n t r o l gate is opened, t h e w a t e r flows f r o m t h e reservoir t h r o u g h t h e penstock t o t h e t u r b i n e , where i t t r a n s m i t s i t s energy.

T h e t u r b i n e is coupled t o a s h a f t w i t h a generator p r o d u c i n g e l e c t r i c i t y . T h e electrical power generated is a f u n c t i o n of t h e t o t a l efficiency o f t h e power p l a n t , t h e w a t e r density, t h e acceleration due t o g r a v i t y , t h e flow r a t e a n d t h e s t a t i c head.

T h e r e are t w o types of t u r b i n e s ; r e a c t i o n a n d i m p u l s e . R e a c t i o n t u r b i n e s have t h e i r r u n n e r covered b y w a t e r , w h i l e i m p u l s e t u r b i n e s have t h e i r r u n n e r i n a i r . I m p u l s e t u r b i n e s are used f o r large head. T h e t u r b i n e s i n Sweden are of the f i r s t k i n d due t o t h e l o w head. T h e y can be either h o r i z o n t a l a x i a l , a x i a l

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T u r b i n e

I m p u l s e ( P e l t o n ) R a d i a - a x i a l (Francis) A d j u s t a b l e blade m i x e d - f l o w

10-50 80-400 300-500 A d j u s t a b l e blade a x i a l - f l o w ( K a p l a n ) 450-1200

T a b l e 1.1: T u r b i n e t y p e as a f u n c t i o n o f t h e specific speed [6].

( K a p l a n ) , m i x e d - f l o w , r a d i a l - a x i a l (Francis)or b u l b .

T h e choice o f t h e t u r b i n e is p r i n c i p a l l y based o n t h e specific speed ris , see K r i v c h e n k o [6].

where n , Q, rj a n d H represents t h e r o t a t i o n a l speed o f t h e r u n n e r ( r p m ) u n d e r r a t e d c o n d i t i o n s , t h e flow rate, the efficiency a n d t h e head. T h e specific speed represents t h e r o t a t i o n a l speed o f a given r u n n e r t o develop 1.36 k W (1 horse p o w e r ) under 1 m head. T a b l e 1.1 represents t h e v a r i a t i o n o f the specific speed f o r d i f f e r e n t t y p e o f t u r b i n e s .

T h e present w o r k focuses o n K a p l a n e l b o w d r a f t t u b e flow. I n v e n t e d i n the early 20^th c e n t u r y b y t h e A u s t r i a n V i k t o r K a p l a n (1876-1934), t h i s t y p e o f t u r b i n e is used f o r l o w head r a n g i n g f r o m 4 t o 70 m, cf. figure 1.2. A s t h e w a t e r leaves t h e penstock, i t enters the s p i r a l , w h i c h is designed t o d i s t r i b u t e a s y m m e t r i c , steady a n d s w i r l i n g flow over the l e a d i n g edge o f t h e stay vanes w i t h m i n i m u m losses. S p i r a l casings are either made of m e t a l f o r heads r a n g i n g f r o m 40 t o over 200 m or of concrete f o r l o w and m e d i u m heads, i.e. u p t o 75 m [8].

Generally, the s p i r a l i n m e t a l have a r o u n d cross-section a n d t h e concrete spirals have a t r a p e z o i d a l cross-section t o d i m i n i s h t h e overall dimensions a n d s i m p l i f y t h e c o n s t r u c t i o n . P o t e n t i a l flow t h e o r y was first used t o design s p i r a l casing.

Today, C o m p u t a t i o n a l F l u i d D y n a m i c ( C F D ) is used t o h a n d l e t h e c o m p l e x i t y of t h e flow as done e.g. b y O h n i s h i [9].

T h e stay vanes are present t o s t r e n g t h t h e s t r u c t u r e . T h e guide vanes are used t o regulate t h e flow r a t e a n d a d j u s t the f l o w d i r e c t i o n t o m a t c h t h e r u n n e r blades. T h e i r n u m b e r ranges f r o m 20 t o 32 [6]. B o t h types o f vanes are s t r e a m - l i n e d t o reduce h y d r a u l i c losses. T h e r u n n e r consists of a h u b o n w h i c h the r u n n e r blades are m o n t e d . T h e blades are a d j u s t a b l e a l l o w i n g h i g h e f f i c i e n c y at d i f f e r e n t loads. R e l a t i v e l y t h i n , t h e n u m b e r o f r u n n e r blades m a y range f r o m 4 t o 8 as the head increases.

T h e d r a f t t u b e f o l l o w s t h e r u n n e r . T h e role o f t h e d r a f t t u b e is t o convert k i n e t i c energy i n t o pressure energy w i t h a m i n i m u m o f losses. For l o w h e a d t u r b i n e s , t h e d r a f t t u b e c a n represent a s u b s t a n t i a l p a r t of t h e h y d r a u l i c losses, u p t o 50% a t p a r t l o a d . M a n y t y p e s o f d r a f t t u b e have been developed over t h e

Hi (1.1)

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(i

F i g u r e 1.2: Kaplan turbine [7].

years [8]; s t r a i g h t conical d r a f t t u b e , b e n d conical d r a f t t u b e , M o o d y spread- i n g d r a f t t u b e a n d elbow d r a f t t u b e . T h e elbow d r a f t t u b e , cf. f i g u r e 1.3, is e x t e n s i v e l y used i n r e a c t i o n t u r b i n e since i t has a h i g h pressure recovery a n d t h e h e i g h t o f i t s u n d e r g r o u n d c o n s t r u c t i o n is l o w , t h u s l i m i t i n g excavation. A n e l b o w d r a f t t u b e is composed of a s t r a i g h t cone d i f f u s e r , f o l l o w e d b y an elbow a n d a d i f f u s e r . T h e m a i n p a r a m e t e r d e s c r i b i n g such a d r a f t t u b e are the d i f f u - sion r a t e d e f i n e d as the r a t i o of t h e i n l e t area t o t h e o u t l e t area ( 1 / 4 ~ 1/3), t h e r e l a t i v e l e n g t h , represented t h e l e n g t h of t h e d r a f t t u b e c e n t r a l line d i v i d e d b y t h e i n l e t d i a m e t e r (4 ~ 6) [8].

T h e f l o w leaving the runner is h i g h l y u n s t e a d y due t o t h e p e r i o d i c wake o f t h e r u n n e r blades. F u r t h e r m o r e , t h e c o m b i n a t i o n o f l o w mechanical energy a n d h i g h v e l o c i t y induce a low pressure, w h i c h m a y give rise t o c a v i t a t i o n . M o s t o f t h e pressure recovery ( ~ 80%) occurs i m m e d i a t e l y a f t e r t h e r u n n e r i n t h e s t r a i g h t cone d i f f u s e r . T h e conical angle is q u i t e large; i t ranges f r o m 1 4 ° t o a b o u t 1 8 ° . T h e p e r f o r m a n c e of t h e d r a f t t u b e increases w i t h t h e l e n g t h o f

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F i g u r e 1.3: Hölleforsen d r a f t tube model.

t h e cone d i f f u s e r , since the losses i n t h e elbow a n d t h e s t r a i g h t d i f f u s e r are m i n i m i z e d w i t h a lower k i n e t i c energy. T h e distance f r o m t h e b o t t o m o f t h e g u i d e vanes t o t h e d r a f t t u b e b o t t o m is f o u n d between 1.9 t o a b o u t 2.4 r u n n e r d i a m e t e r s . T o get g o o d p e r f o r m a n c e , t h e f l o w has t o be a t t a c h e d o n t h e cone.

T h e f l o w e n t e r i n g the elbow is s u b j e c t t o a c e n t r i f u g a l force, w h i c h gives rise t o a r a d i a l as w e l l as l o n g i t u d i n a l pressure g r a d i e n t [10]. T h e l o n g i t u d i n a l pressure g r a d i e n t is m o r e i m p o r t a n t o n t h e i n n e r r a d i u s at t h e i n l e t a n d t h e o u t e r r a d i u s a t t h e o u t l e t . Since t h e d i f f u s e r f o l l o w s a f t e r t h e elbow, t h e f l u i d is s u b j e c t t o an a d d i t i o n a l l o n g i t u d i n a l pressure g r a d i e n t , w h i c h c o m b i n e d w i t h t h e precedent m a y induce separation. A c o n t r a c t i o n m a y be present f r o m t h e m i d d l e t o the e n d o f t h e e l b o w t o a v o i d such p h e n o m e n o n . T h e u p w a r d angle o f t h e d i f f u s e r r o o f varies generally f r o m 10° t o a b o u t 1 3 ° .

1.2 M o d e l testing and C F D

H y d r o p o w e r t u r b i n e s have a life cycle of a p p r o x i m a t e l y 50 years. A f t e r t h i s p e r i o d , t h e y are r e f u r b i s h e d a n d i t is m a n d a t o r y t h a t t h i s process is done w i t h s t a t e of t h e a r t i n t u r b i n e technology. A s t h e e l e c t r i c i t y p r o d u c e r s w a n t t o m i n i m i z e t h e cost o f such an o p e r a t i o n , t h e y f a v o r solutions w h i c h c a n be i m - p l e m e n t e d w i t h o u t i m p o r t a n t c o n s t r u c t i o n , i.e. solutions f i t t i n g i n t h e e x i s t i n g i n s t a l l a t i o n s . T h e m o d i f i c a t i o n o f t h e H ö l l e f o r s e n d r a f t t u b e sharp heel done b y D a h l b ä c k [11] is one example. Such m o d i f i c a t i o n s m a y be done by m o d e l t e s t i n g as done b y D a h l b ä c k or w i t h C o m p u t a t i o n a l F l u i d D y n a m i c s ( C F D ) as done b y L i n d g r e n [12].

M o d e l t e s t i n g has been used since decades t o design a n d o p t i m i z e h y d r o p o w e r t u r b i n e s . T h e y a l l o w efficiency measurement w i t h a n accuracy below 0.5 % [13].

F r o m such measurements, scale-up f o r m u l a are used t o e s t i m a t e t h e p r o t o t y p e

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efficiency since m o d e l a n d p r o t o t y p e are n o t o p e r a t i n g i n d y n a m i c a l l y s i m i l a r c o n d i t i o n s . V a l i d a t i o n of scale-up f o r m u l a are d i f f i c u l t , since accurate measurements o n a p r o t o t y p e offer larger challenges c o m p a r e d t o measurements o n a m o d e l i n a l a b o r a t o r y , the m a i n d i f f i c u l t y concerns t h e d e t e r m i n a t i o n of the flow rate. Several f o r m u l a s have been proposed since t h e b e g i n n i n g of t h e 20th

century. A s u m m a r y o f t h e d i f f e r e n t c o n t r i b u t i o n s have been made b y A n t o n [14]. T h e c o m p l e x i t y o f t h e flow i n h y d r o p o w e r t u r b i n e s does n o t a l l o w a n exact a n a l y t i c a l d e s c r i p t i o n . T h e r e f o r e , scale-up f o r m u l a are e m p i r i c a l f o r m u l a , where t h e losses are separated i n t o k i n e t i c a n d viscous losses, an idea o r i g i n a t i n g f r o m A c k e r e t (1931). T h e viscous losses are due t o f r i c t i o n o n c o n d u i t s walls a n d are scalable, w h i l e t h e k i n e t i c losses are concerned w i t h energy d i s s i p a t i o n i n t h e t u r b u l e n t flow, e.g. i n the wake of the r u n n e r blades [15] a n d are n o t scalable.

T h e t w o m a i n s t a n d a r d s , JSME S008 a n d t h e IEC 60193 consider t h e k i n e t i c losses equal o n t h e m o d e l a n d the p r o t o t y p e . C o n s i d e r i n g t h e d o m i n a n t r u n - ner losses, I d a [16] demonstrates t h a t t h e y are n o t equal, g i v i n g rise t o a new scale-up f o r m u l a . T h e IEC 60193 s t a n d a r d is considered less accurate, since t h e r a t i o o f t h e viscous losses t o the t o t a l losses is considered constant f o r each t y p e o f machine, w h i l e i t is a f u n c t i o n o f t h e specific speed f o r the JSME S008 s t a n d a r d .

T h e d i f f e r e n t scale-up f o r m u l a consider the flow steady. However, t h i s is n o t t h e case i n r e a c t i o n t u r b i n e s due t h e g e o m e t r y (stay a n d guide vanes, r u n n e r r o t a t i o n ) a n d flow p h e n o m e n a such as v o r t e x rope a n d s t a l l . I n Stokes second p r o b l e m [17], i.e. a n o s c i l l a t i n g p l a t e i n a fluid at rest, t h e losses increases w i t h the square r o o t of t h e frequency. T h e effects of unsteadiness i n t u r b i n e flow are m o r e c o m p l e x , since t h e flow is also three d i m e n s i o n a l a n d t u r b u l e n t . T h e unsteadiness arises especially f r o m p e r i o d i c wakes, w h i c h d i s t u r b t h e b o u n d a r y layer, see F a r h a t , A v e l l a n & Siedel [18] a n d C i o c a n & A v e l l a n [19]. T h e w o r k o f H o l l a n d & Evans [20] o n t h e effect of p e r i o d i c wake s t r u c t u r e s o n t u r b u l e n t b o u n d a r y layers i n t u r b o machines indicates an increase of t h e b o u n d a r y layer thickness a n d s k i n f r i c t i o n a n d therefore, an increase o f t h e viscous losses. T h i s m a y i n d i c a t e , t h a t unsteadiness influences t h e viscous losses. I f i m p o r t a n t , t h e y s h o u l d be i n c l u d e d i n t h e step-up f o r m u l a t o o b t a i n p r o t o t y p e efficiency, i f scaled d i f f e r e n t l y t h a n t h e steady viscous losses. W h i l e t u r b u l e n t p u l s a t i n g flows i n p i p e a n d channel have received p a r t i c u l a r a t t e n t i o n d u r i n g t h e last t w o decades [21], f e w w o r k s have been r e p o r t e d f o r t u r b u l e n t p u l s a t i n g flow i n diffusers. W o r k i n t h i s area m a y help t o f u r t h e r u n d e r s t a n d flow i n d r a f t t u b e . A l s o , t h e l a b o r a t o r y a l l o w i n g m o d e l t e s t i n g are large f a c i l i t i e s , w h e r e reservoirs, p u m p s a n d m a n y measuring devices such as t o r q u e , flow r a t e , r o t a t i o n a l speed a n d pressure measurements are necessary. F u r t h e r m o r e , m o d e l b u i l d - i n g , m o d i f i c a t i o n s a n d d e t a i l e d i n v e s t i g a t i o n of t h e f l o w w i t h e.g. Laser D o p p l e r A n e m o m e t r y ( L D V ) or P a r t i c l e Image Velocimeter ( P I V ) are expensive a c t i v i - ties. A n a l t e r n a t i v e t o m o d e l t e s t i n g is c o m p u t a t i o n a l fluid d y n a m i c s ( C F D ) .

C F D is a p o w e r f u l technology, t h a t has emerged over t h e last 10 years f r o m t h e academic research t o t h e i n d u s t r y . T h e m a j o r i t y o f t h e c o n t r i b u t i o n s pre-

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sented t o the last I A H R s y m p o s i u m used t h i s t o o l . T h e m a j o r b e n e f i t s o f C F D are i t s cost, t h e d e t a i l e d i n f o r m a t i o n generated a n d i t s a b i l i t y t o test r a p i d l y n e w m o d i f i c a t i o n s , c o m p a r e d t o m o d e l t e s t i n g .

A c c u r a t e c a l c u l a t i o n of u n s t e a d y t u r b u l e n t f l o w is possible w i t h D i r e c t N u - m e r i c a l S i m u l a t i o n s ( D N S ) . However, several decades are necessary t o get a r e s u l t w i t h the present c o m p u t e r s due t o the s m a l l l e n g t h scales i n v o l v e d i n h y - d r o p o w e r flows, w h i c h d e m a n d s a huge g r i d . A m o r e generally used a l t e r n a t i v e is t h e Reynolds A v e r a g e Navier-Stokes ( R A N S ) e q u a t i o n . I t s i m p l e m e n t a t i o n i n v o l v e s t u r b u l e n c e m o d e l l i n g a n d several models have been developed since t h e 70^th such as algebraic or zero e q u a t i o n m o d e l , one e q u a t i o n m o d e l , t w o equa- t i o n m o d e l and R e y n o l d s stress m o d e l , see e.g. W a l l i n [22]. T h e s t a n d a r d k—e m o d e l is w i d e l y used i n the i n d u s t r y due t o its robustness. H y d r o p o w e r flows are h i g h l y c h a l l e n g i n g f o r t u r b u l e n c e m o d e l l i n g , since several p h e n o m e n a ap- p e a r s i m u l t a n e o u s l y such as 3-dimensionality, unsteadiness, s e p a r a t i o n , s w i r l i n g flow i n diffuser a n d e l b o w a n d t u r b u l e n c e . T h e space d i s c r e t i s a t i o n , g r i d size a n d topology, is a n o t h e r issue a n d g r i d independent solutions are d i f f i c u l t t o o b t a i n w i t h t h e present c o m p u t e r s , m a k i n g accurate s i m u l a t i o n of t h e e n t i r e flow, i n l e t t o o u t l e t o f t h e s t a t i o n , p r a c t i c a l l y unfeasible t o d a y . T h e b o u n d a r y c o n d i t i o n s o n t h e d o m a i n , i.e. i n l e t velocities, surface roughness, o u t l e t pressure m u s t be k n o w n i n d e t a i l . T h i s is generally d i f f i c u l t f o r d r a f t t u b e flows, o b l i g i n g t o q u a l i f y guesses.

T h e assumptions o n b o u n d a r y c o n d i t i o n s a n d t h e m u l t i t u d e o f possible i n p u t p a r a m e t e r s make i t d i f f i c u l t t o o b t a i n reliable results. T h e T u r b i n e - 9 9 W o r k s h o p [23], a i m e d t o d e t e r m i n e t h e state o f t h e a r t i n C F D s i m u l a t i o n s o f d r a f t t u b e flows, reflects t h i s d i f f i c u l t y . E x p e r i m e n t a l d a t a , m e a n a x i a l a n d t a n g e n t i a l v e l o c i t y components measured along a r a d i a l line, a t t h e i n l e t of t h e H ö l l e f o r s e n d r a f t t u b e m o d e l , cf. f i g u r e 1.3, were p r o v i d e d . T h e p e r i o d i c fluctuations a r i s i n g f r o m t h e blade passages a n d t h e t u r b u l e n t fluctuations i n t h e measurements were b o t h chosen t o c o n t r i b u t e t o t h e steady t u r b u l e n t q u a n t i t i e s . Some d a t a c o u l d n o t be measured such as t h e r a d i a l velocity, some o f t h e R e y n o l d s stresses a n d t h e t u r b u l e n c e l e n g t h scale a t t h e i n l e t of t h e d r a f t t u b e . A s these c o m p o n e n t s need t o be k n o w n t o p e r f o r m a n u m e r i c a l s i m u l a t i o n , various q u a l i f i e d guesses w e r e a t t e m p t e d r e s u l t i n g i n d i f f e r e n t results. T h e pressure recovery o b t a i n e d b y t h e d i f f e r e n t p a r t i c i p a n t s presents a scatter of ± 4 5 % .

T h e r e f o r e , i t is obvious t h a t m u c h e f f o r t s h o u l d be d e d i c a t e d t o i m p r o v e t r u s t & q u a l i t y i n C F D besides t h e development of t h e t u r b u l e n c e m o d e l . T h e Q N E T - C F D , a t h e m a t i c n e t w o r k o n q u a l i t y and t r u s t f o r t h e i n d u s t r i a l a p p l i - c a t i o n s of C F D , w o r k s i n t h i s d i r e c t i o n [24]. I t is set t o "assemble, s t r u c t u r e a n d collate e x i s t i n g knowledge encapsulating the p e r f o r m a n c e o f C F D m o d e l s " . E R C O F T A C special interest g r o u p i n Q u a l i t y a n d T r u s t f o r t h e I n d u s t r i a l A p - p l i c a t i o n o f C F D [25] w o r k s also i n t h i s d i r e c t i o n a n d r e c e n t l y p u b l i s h e d t h e Best Practice Guidelines f o r i n d u s t r i a l CFD-users [26].

T h e above r e m a r k s indicates, t h a t C F D is s t i l l u n d e r d e v e l o p m e n t a n d n o t m a t u r e t o c o m p l e t e l y take over m o d e l t e s t i n g . I t s h o u l d r a t h e r be seen as a p o w e r f u l c o m p l e m e n t t o m o d e l t e s t i n g today, w h i c h can m a k e t h i s m o r e e f f i c i e n t ,

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b u t i t w i l l have increased i m p o r t a n c e i n t h e f u t u r e .

1.3 Thesis aim and list of publications

T h e present thesis focuses o n t h e f l o w i n a n e l b o w d r a f t t u b e a n d t r y t o h e l p answer t w o p r o b l e m s , w h i c h are r e l a t e d t o unsteadiness a n d C F D :

• H o w can t h e a s s u m p t i o n s o n b o u n d a r y c o n d i t i o n s t o s i m u l a t e d r a f t t u b e f l o w be h a n d l e d i n a s y s t e m a t i c , o b j e c t i v e a n d q u a n t i t a t i v e w a y i n o r d e r t o increase t r u s t a n d q u a l i t y i n C F D ?

• H o w does unsteadiness i n f l u e n c e s t h e viscous losses?

P a p e r s A t o C are r e l a t e d t o t h e first q u e s t i o n , where t h e use of f a c t o r i a l design is proposed as a n a l t e r n a t i v e t o h a n d l e t h e m u l t i t u d e o f i n p u t a n d as- s u m p t i o n s . F u r t h e r m o r e , special a t t e n t i o n is g i v e n t o t h e i n l e t r a d i a l v e l o c i t y a n d pressure o f t h e d r a f t t u b e , a n issue o f t h e T u r b i n e - 9 9 w o r k s h o p .

P a p e r s D t o F are r e l a t e d t o t h e second q u e s t i o n , where t h e v a r i a t i o n o f t h e m e c h a n i c a l energy is f i r s t s t u d i e d i n d e t a i l f o r u n s t e a d y t u r b u l e n t f l o w . F r o m t h i s analysis, unsteady f l o w m e a s u r e m e n t s i n a s t r a i g h t a s y m m e t r i c d i f f u s e r a n d u n s t e a d y pressure measurements o n a K a p l a n t u r b i n e were p e r f o r m e d .

T h e d i f f e r e n t papers, t h e i r p u b l i c a t i o n s t a t u s a n d the c o n t r i b u t i o n o f t h e c o - a u t h o r are:

• P a p e r A - M . J . Cervantes & L . H . Gustavsson, Estimation of the Radial Velocity from the Squire-Long Equation and Experimental Data, s u b m i t t e d t o t h e J o u r n a l o f H y d r a u l i c Research ( c o n d i t i o n a l l y accepted f o r p u b l i c a - t i o n ) .

Professor Gustavsson h a d t h e o r i g i n a l idea o f u s i n g t h e S q u i r e - L o n g equa- t i o n t o s t u d y t h e i n l e t r a d i a l v a r i a t i o n o f t h e pressure.

• P a p e r B - M . J . Cervantes & T . F . E n g s t r ö m , Influence of Boundary Con- ditions Using Factorial Design, t o be p u b l i s h e d i n t h e P r o c e e d i n g o f t h e T u r b i n e - 9 9 W o r k s h o p o n D r a f t T u b e F l o w , Ä l v k a r l e b y , Sweden, 2 0 0 1 . T h e c o m p u t a t i o n a l g r i d s were s u p p l i e d b y D r . J. B e r g s t r ö m ( c u r r e n t l y a t V o l v o Cars, Sweden) a n d D r . Y . G . L a i ( c u r r e n t l y at I I H R - H y d r o - science a n d E n g i n e e r i n g , I o w a C i t y , U S A ) . D r . J . B e r g s t r ö m s u p p l i e d also t h e o r i g i n a l F O R T R A N r o u t i n e s . T h e i m p l e m e n t a t i o n o f t h e b o u n d a r y c o n d i t i o n s a n d m o d i f i c a t i o n o f F O R T R A N r o u t i n e s were done b y F . E n - g s t r ö m . T h e p r e p a r a t i o n o f t h e e x p e r i m e n t a l b o u n d a r y c o n d i t i o n s , t h e v i s u a l i z a t i o n s a n t h e i n l e t pressure analysis were done b y M . Cervantes.

T h e s i m u l a t i o n s a n d e v a l u a t i o n o f d a t a were p e r f o r m e d b y b o t h a u t h o r s .

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• P a p e r C - M . J . Cervantes & T . F . E n g s t r ö m , Factorial Design Applied to CFD, s u b m i t t e d t o t h e J o u r n a l o f F l u i d E n g i n e e r i n g ( c o n d i t i o n a l l y accepted f o r p u b l i c a t i o n ) .

T h e w o r k was o r g a n i z e d as i n Paper B

• P a p e r D - M . J . Cervantes & S. V i d e h u l t , Unsteady Pressure Measurements at Porjus U9, p u b l i s h e d i n t h e P r o c e e d i n g o f t h e X X I^{s t} I A H R S y m p o s i u m o n H y d r a u l i c M a c h i n e r y a n d Systems at Lausanne, S w i t z e r l a n d , Septem- ber, 2002.

S. V i d e h u l t p a r t i c i p a t e d t o t h e P i t o t t u b e measurements.

• P a p e r E - M . J . Cervantes & L . H . Gustavsson, Unsteadiness and Viscous Losses in Hydraulic Turbines, t o be s u b m i t t e d t o t h e J o u r n a l o f H y d r a u l i c Research.

• P a p e r F - T . F . E n g s t r ö m & M . J . Cervantes, Pulsating Turbulent Flow in a Straight Asymmetric Diffuser, t o be s u b m i t t e d t o t h e J o u r n a l o f F l u i d M e c h a n i c s .

T h e v e l o c i t y measurements a n d t h e paper were done j o i n t l y . E r r o r analysis a n d pressure measurements were p e r f o r m e d b y T . F . E n g s t r ö m , w h i l e t h e v e l o c i t y analysis a n d c o m p i l a t i o n were done b y M . J . Cervantes.

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Chapter 2 TRUST AND QUALITY I N C F D

U n t i l n o w , m o s t academic w o r k has been d i r e c t e d towards t h e v a l i d a t i o n o f p a r t i c u l a r codes a n d t u r b u l e n c e models i n s i m p l i f i e d geometries [24], w h e r e t h e b o u n d a r y c o n d i t i o n s were w e l l d e f i n e d a n d g r i d independent s o l u t i o n s was n o t an issue. A p p l i e d t o i n d u s t r i a l flows, C F D meets new problems; accurate b o u n d a r y c o n d i t i o n s are d i f f i c u l t t o o b t a i n e d , t h e large geometries make g r i d i n d e p e n d e n t s o l u t i o n s d i f f i c u l t t o o b t a i n a n d t h e the t u r b u l e n c e models have d i f f i c u l t i e s t o c a p t u r e t h e c o m p l e x i t y o f c e r t a i n flows.

T h e T u r b i n e - 9 9 W o r k s h o p [23] i l l u s t r a t e s such problems. T h e scatter i n t h e r e s u l t s m a y have several e x p l a n a t i o n s . T h e p a r t i c i p a n t s used d i f f e r e n t t u r b u - lence models (k-e, k-u>, R S M ) a n d t h e grids ranged f r o m 41000 t o 728000 cells, w e l l b e l o w t h e e s t i m a t i o n of B e r g s t r ö m [23]; a g r i d w i t h 3.9 • 1 06 t o 222 • 1 06

cells was necessary t o lower t h e g r i d e r r o r t o 1 % f o r t h e pressure recovery u s i n g a R e y n o l d s stress m o d e l . A n o t h e r reason f o r the scatter i n t h e results m a y be a t t r i b u t e d t o t h e i n l e t r a d i a l v e l o c i t y w h i c h was n o t prescribed b y t h e organizers. Page & G i r o u x [23] p e r f o r m e d calculations b o t h w i t h zero a n d non-zero r a d i a l v e l o c i t y , u s i n g t h e results of a separate K a p l a n r u n n e r s i m u l a t i o n . T h e pressure recovery increased b y 15% w i t h the s i m u l a t e d r a d i a l v e l o c i t y . T h e r e - f o r e , t h i s c o m p o n e n t is o f great i m p o r t a n c e , i f d r a f t t u b e flow w i l l be s i m u l a t e d accurately.

2.1 Boundary C o n d i t i o n s (Summary of paper A ) 2 . 1 . 1 R a d i a l v e l o c i t y a t t h e i n l e t o f t h e d r a f t t u b e

T h e p a r t i c i p a n t s of the T u r b i n e - 9 9 W o r k s h o p faced a p r o b l e m w i t h t h e u n k n o w n i n l e t r a d i a l v e l o c i t y , a p r o b l e m c o m m o n t o m a n y engineers s i m u l a t i n g i n d u s t r i a l flows. Several a l t e r n a t i v e s were used b y the p a r t i c i p a n t s t o h a n d l e the p r o b l e m . T h e f i r s t a l t e r n a t i v e was t o set t h i s b o u n d a r y c o n d i t i o n equal t o zero. However, s i m p l e g e o m e t r i c a l considerations show t h a t t h e v o l u m e of fluid t r a n s p o r t e d t o w a r d t h e cone is m o r e t h a n t w i c e the v o l u m e t r a n s p o r t e d t o w a r d t h e w a l l .

1 3

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1 4

4I , , , , , , 1 l _

0.08 0.1 0 . 1 2 0 . 1 4 0.16 0 . 1 8 0.2 0 . 2 2 0 . 2 4 Radius (m)

F i g u r e 2 . 1 : Radial velocity profile obtained w i t h the Squire-Long equation and proposed by the organizers of the second Turbine-99 Workshop.

S e p a r a t i o n occurs i f such a v o l u m e is n o t t r a n s p o r t e d t o w a r d t h e cone a n d t h e w a l l , decreasing s u b s t a n t i a l l y t h e p e r f o r m a n c e of t h e d r a f t t u b e . F u r t h e r m o r e , m a n y p a r a m e t e r s w o r k against t h e transfer o f m o m e n t u m t o w a r d t h e cone.

T h e angle of t h e cone is i m p o r t a n t at the i n l e t a n d t h e t a n g e n t i a l v e l o c i t y creates a c e n t r i f u g a l force, w h i c h p u l l s the fluid t o w a r d s t h e w a l l . T h e r e f o r e , a r a d i a l v e l o c i t y is necessary t o a v o i d a p r e m a t u r e s e p a r a t i o n o n t h e cone. T h e second a l t e r n a t i v e was t o consider t h e flow a t t a c h e d t o t h e w a l l s a n d the t h i r d a l t e r n a t i v e t o use t h e p r o f i l e o f a separate K a p l a n r u n n e r s i m u l a t i o n , since t h e r u n n e r g e o m e t r y was n o t available; Page & G i r o u x [23].

T h e Best Practice Guidelines [26] recommends a s e n s i t i v i t y analysis f o r u n - k n o w n b o u n d a r y c o n d i t i o n s i n w h i c h the b o u n d a r y is s y s t e m a t i c a l l y changed w i t h i n c e r t a i n l i m i t s t o see t h e v a r i a t i o n i n the results. Since t h e a x i a l a n d t a n - g e n t i a l velocities were available i m m e d i a t e l y a f t e r t h e cone, c o m p a r i s o n of t h e c o m p u t a t i o n a l results w o u l d have g i v e n a n a p p r o x i m a t i o n o n t h e r a d i a l velocity.

A fifth a l t e r n a t i v e w o u l d have been t o use t h e e x p e r i m e n t a l values of t h e m e a n a x i a l a n d t a n g e n t i a l velocities a n d a t w o d i m e n s i o n a l non-viscous descrip- t i o n o f t h e s w i r l i n g flow g i v e n b y t h e S q u i r e - L o n g e q u a t i o n (2.1) f o r t h e s t r e a m f u n c t i o n ip as described i n p a p e r A .

T h e S q u i r e - L o n g e q u a t i o n has t h e p e c u l i a r i t y t o d e p e n d o n t h e b o u n d a r y c o n d i t i o n s , since t h e B e r n o u l l i f u n c t i o n H a n d the c i r c u l a t i o n C are constant o n streamlines. T h e m e t h o d takes advantage of t h i s d e p e n d e n c y b y c o m p u t i n g t h e e q u a t i o n r e l a t e d t o the d o m a i n w i t h a n i t e r a t i v e process. T h e r a d i a l v e l o c i t y

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15

F i g u r e 2.2: Variation of the pressure at the inlet of the d r a f t tube i n the radial di- rection for the T mode, Turbine-99 Work- shop test case. H u b on the left and shroud on the right.

F i g u r e 2.3: Variation of the radial pres- sure gradient (~^r) and the different terms i n the radial component of the Euler equation. (^§7)2 represents the pressure gradient i n equation 2.2.

p r o f i l e is t h e n o b t a i n e d . T h e p r o f i l e o b t a i n e d w i t h t h e present m e t h o d is represented i n figure 2.1 w i t h t h e p r o f i l e proposed b y t h e organizers of t h e second T u r b i n e - 9 9 W o r k s h o p , w h i c h is s i m i l a r t o t h e p r o f i l e r e c e n t l y o b t a i n e d b y N i l s - son [27] w i t h a n u m e r i c a l s i m u l a t i o n of the H ö l l e f o r s e n r u n n e r . T h e discrepancy between b o t h profiles is i m p o r t a n t a n d a t t r i b u t e d t o the lack of i n f o r m a t i o n b e n e a t h the cone. T h i s lack o f d a t a can be c o m p e n s a t e d b y an i t e r a t i v e process between t h e viscous a n d non-viscous region u s i n g e.g. t h e a p p r o x i m a t e m e t h o d due t o v o n K a r m a n a n d Pohlhausen f o r t w o d i m e n s i o n a l f l o w [17].

2.1.2 R a d i a l p r e s s u r e a t t h e i n l e t o f t h e d r a f t t u b e

A n o t h e r issue o f t h e first w o r k s h o p was t h e pressure d r o p t o w a r d the o u t e r w a l l at t h e i n l e t o f t h e d r a f t t u b e f o u n d b y some p a r t i c i p a n t s , a result i n c o n t r a d i c t i o n w i t h s w i r l c a l c u l a t i o n s i n a d i f f u s e r . F i g u r e 2.2 represents t h e v a r i a t i o n of t h e pressure a t the i n l e t o f t h e d r a f t t u b e o b t a i n e d i n paper B f o r t h e 2^{n d} Turbine—99 W o r k s h o p w i t h t h e e x p e r i m e n t a l values of A n d e r s s o n presented at t h e second w o r k s h o p . T h i s figure w i l l be c o m m e n t e d i n section 2.2.2.

T h e pressure d r o p t o w a r d t h e o u t e r w a l l m a y be e x p l a i n e d using i n v i s c i d flow t h e o r y , due t o the h i g h R e y n o l d s n u m b e r at t h e i n l e t o f t h e d r a f t t u b e . For such a flow, t h e r a d i a l v a r i a t i o n o f t h e pressure is c o u p l e d t o t h e a x i a l , r a d i a l a n d t a n g e n t i a l v e l o c i t y c o m p o n e n t s (U, V, W) t h r o u g h t h e r a d i a l c o m p o n e n t o f t h e E u l e r e q u a t i o n .

IdP W 2 TTdV ÖV , .

- — = ( 7 — - V — (2.2) p or r oz or

T h e sign o f t h e r a d i a l d e r i v a t i v e o f t h e pressure depends o n t h e m a g n i t u d e o f the d i f f e r e n t t e r m s a n d t h e i r v a r i a t i o n is shown i n figure 2.3. T h e first t e r m is always p o s i t i v e . T h e second t e r m is m a i n l y p o s i t i v e except close t o t h e cone

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1 6

a n d t h e w a l l . T h e t h i r d t e r m is m a i n l y p o s i t i v e , except close t o the cone. T h e second t e r m dominates due t o t h e i m p o r t a n t v a r i a t i o n o f the r a d i a l v e l o c i t y w i t h z, except close t o the cone. T h u s , t h e large angle o f the cone i m p l i e s a negative r a d i a l v e l o c i t y at the i n l e t o f t h e d r a f t t u b e t o avoid separation. T h e r a d i a l v e l o c i t y decreases r a p i d l y i n m a g n i t u d e , w h i c h gives a p o s i t i v e second t e r m i n t h e above r e l a t i o n . T h e r e f o r e , t h e pressure decreases t o w a r d t h e w a l l .

T h e s u m o f the d i f i e r e n t t e r m s ((dP/dr)² represented b y a dashed l i n e ) is presented i n figure 2.3 w i t h t h e r a d i a l pressure g r a d i e n t t a k i n g i n t o account t h e viscous effect. I f g o o d agreement is o b t a i n e d i n t h e m i d d l e o f the f l o w , i n d i c a t i n g a n e a r l y i n v i s c i d fluid, s t r o n g discrepancies appear close t o t h e cone a n d t h e s h r o u d , where viscous effects a n d t u r b u l e n c e cannot be neglected.

2.2 Simulation of the d r a f t tube (Summary of paper B ) 2 . 2 . 1 T h e k — e t u r b u l e n c e m o d e l

I n paper B , t h e s t a n d a r d k-e m o d e l is used t o s i m u l a t e the m o d e l d r a f t t u b e flow o f t h e T u r b i n e - 9 9 W o r k s h o p on t h e t o p ( T m o d e ) a n d the right ( R m o d e ) o f t h e p r o p e l l e r curve. T h i s t w o e q u a t i o n m o d e l uses the eddy viscosity h y p o t h e s i s f o r t h e R e y n o l d s stresses, w h i c h relates t h e m l i n e a r l y t o t h e m e a n v e l o c i t y g r a d i e n t :

2

- UiUj = 2 v^TS^{l j} - -kSij, (2.3)

where, i>i is t h e eddy viscosity a n d is t h e m e a n s t r a i n r a t e tensor [28].

T h e t u r b u l e n t k i n e t i c energy (k) a n d i t s d i s s i p a t i o n r a t e (e) are used t o get t h e v e l o c i t y a n d l e n g t h scales f o r t h e e d d y viscosity g i v e n as:

^ T = C ^M - (2.4)

e T h e steady equations f o r fe a n d e are:

u.

dk dxj de

.9Ui

1 dx. e +

dxj (u + v^T/ a^k) dk

r t dUi

'^Câ kÛⁱ Ûⁱd x ~

c J - + —

€ k dx.

(2.5)

(2.6)

where t h e s u m m a t i o n c o n v e n t i o n is assumed a n d t h e m o d e l constants are:

0.09, Cêl = 1.44, C â = 1-92, a^k = 1 a n d crê = 1.3.

A s y —> 0, e q u a t i o n (2.5) has t h e l i m i t i n g b e h a v i o r [29]:

e^w = v dy2

(2.7)

Since t h e t u r b u l e n t k i n e t i c energy is e q u a l t o zero a n d t h e d i s s i p a t i o n is non-zero o n t h e w a l l , t h e s t a n d a r d k-e fails t o p r e d i c t the flow near t h e w a l l . Some o f t h e

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1 7

a l t e r n a t i v e s developed t o handle the p r o b l e m are d a m p i n g t h e viscosity w i t h a f u n c t i o n , w a l l f u n c t i o n s a n d t w o layer models. T h e second a l t e r n a t i v e was used i n t h e s i m u l a t i o n s . I t decreases s u b s t a n t i a l l y t h e c o m p u t a t i o n a l t i m e , since no fine g r i d is needed close t o t h e w a l l . T h e m e t h o d abandons t h e k-e equations i n regions n e x t t o t h e walls a n d imposes b o u n d a r y c o n d i t i o n s a t t h e t o p o f t h a t zone. I n t h i s region, the flow is assumed t o f o l l o w t h e l a w of t h e w a l l :

U+ = -log(y⁺)+C, (2.8)

where, U⁺ = U/u*, y⁺ = yu*/v a n d it* = ß(dU/dy)^w. A s s u m i n g p r o d u c t i o n a n d d i s s i p a t i o n n e a r l y equal, the t u r b u l e n t k i n e t i c energy a n d t u r b u l e n t dissi- p a t i o n m a y be given as a f u n c t i o n o f t h e f r i c t i o n v e l o c i t y f o r 40 < y⁺ < 0.25gg:

k =

= ]å ny'

(2.9)

(2.10)

T h e w a l l f u n c t i o n m e t h o d m a y f a i l f o r c o m p l e x flows, where t h e flow does n o t f o l l o w t h e log-law. O f interest f o r d r a f t t u b e s i m u l a t i o n s , flows i n a conical d i f f u s e r a n d a s t r a i g h t a s y m m e t r i c d i f f u s e r do n o t f o l l o w t h e s t a n d a r d log-law;

see A z a d [30] a n d paper E .

2.2.2 A p p l i c a t i o n t o t h e T u r b i n e - 9 9 b e n c h m a r k

T h e results o f t h e s i m u l a t i o n s p e r f o r m e d f o r t h e T u r b i n e - 9 9 b e n c h m a r k a t t h e T a n d R o p e r a t i n g c o n d i t i o n s are presented i n t a b l e 2 . 1 . T h e T m o d e is t h e p o i n t o n t h e t o p of t h e p r o p e l l e r curve, i.e. a t best efficiency. For t h e R m o d e , t h e guide vanes are m o r e o p e n a n d the e f f i c i e n c y is lower t h a n f o r t h e T m o d e . T h e m a i n difference between t h e modes resides i n t h e i n l e t t a n g e n t i a l v e l o c i t y , w h i c h is negative as t h e r a d i u s varies f r o m 0.1 m t o 0.16 m f o r t h e R m o d e . T h i s difference induces d i f f e r e n t d i r e c t i o n s of r o t a t i o n of t h e v o r t e x rope, w h i c h emanates f r o m t h e cone; i t c o n t r a - r o t a t e s i n t h e R m o d e a n d co-rotates i n t h e T m o d e . T h u s , d e p e n d i n g o n t h e m o d e , t h e v o r t e x r o p e is o r i e n t e d d i f f e r e n t l y a f t e r t h e elbow due t o the gyroscopic effect, cf. figures 2.4 a n d 2.5.

T a b l e 2 . 1 : Computational and experimental results of the T and R modes.

E n g . q u a n t i t i e s _CPwall _Cpmean ^axial III ßm Q {mA/s)

T m o d e ( C F D ) 1.288 0.971 1.223 1.224 0.522

T m o d e (exp.) 1.120 - 1.090 1.020 0.522

R m o d e ( C F D ) 1.283 0.945 1.188 1.149 0.542

R m o d e (exp.) 1.100 - - - ^0.542

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1 8

F i g u r e 2.4: Path of the vortex rope through the d r a f t tube, T mode. Scale represents velocity magnitude.

* 0 . 0

F i g u r e 2.5: Path of the vortex rope through the d r a f t tube, R mode. Scale represents velocity magnitude.

T h e w a l l pressure recovery Cp^wau based o n t h e m e a n value o f t h e pressure o n t h e w a l l a t t h e i n l e t a n d o u t l e t of the d r a f t t u b e is o v e r e s t i m a t e d f o r b o t h s i m u l a t i o n s . I t is of interest t o compare t h e i n l e t r a d i a l pressure o b t a i n e d w i t h C F D a n d t h e e x p e r i m e n t a l values of A n d e r s s o n¹, cf. figure 2.2. T h e e x p e r i m e n - t a l value o f t h e pressure increases s i g n i f i c a n t l y close t o t h e w a l l , a n expected characteristic since t h e s h r o u d acts as the w a l l o f a n o r d i n a r y d i f f u s e r , where the pressure increases t o w a r d t h e w a l l . However, t h e s i m u l a t i o n does n o t reproduce t h i s t r e n d . T h e r e f o r e , a higher value of Cpwau is o b t a i n e d w i t h the s i m u l a t i o n s . T h e m e a n pressure recovery presented i n t a b l e 2.1 is t h e difference between the m e a n pressure at t h e o u t l e t a n d i n l e t o f t h e d r a f t t u b e n o r m a l i z e d w i t h t h e m e a n k i n e t i c energy based o n t h e m e a n flow r a t e . I t is lower f o r the R m o d e t h a n f o r t h e t h e T m o d e . However, the same pressure difference n o r m a l i z e d w i t h t h e i n l e t k i n e t i c energy is s i m i l a r f o r b o t h cases; 0.921 f o r t h e T m o d e a n d 0.923 f o r the R m o d e . T h i s indicates t h a t t h e d r a f t t u b e has n e a r l y t h e same efficiency f o r b o t h modes.

T h e pressure recovery along the upper a n d lower walls is s h o w n i n figures 2.6 a n d 2.7, as a f u n c t i o n o f t h e center line c o o r d i n a t e . T h e e x p e r i m e n t a l values m a y be f o u n d i n A n d e r s s o n [31]. T h e pressure recovery occurs a l o n g several elements represented i n t h e figures b y dashed lines; t h e r u n n e r cone (0 < L < 0.042), t h e s t r a i g h t cone d i f f u s e r (0 < L < 0.094), the e l b o w (0.094 < L < 0.33), t h e elbow corner (L = 0.21), t h e o u t l e t d i f f u s e r p a r t I (0.33 < L < 0.68) a n d t h e o u t l e t d i f f u s e r p a r t I I (0.68 < L < 1). T h e pressure recovery is i n g o o d agreement along t h e s t r a i g h t cone d i f f u s e r . A s the elbow s t a r t s , t h e pressure o n t h e lower a n d u p p e r walls are u n d e r e s t i m a t e d . Since i m p i n g e m e n t occur a t the b o t t o m o f t h e elbow, one e x p l a n a t i o n m a y be t h e p a r t i c u l a r b e h a v i o r of t h e s t a n d a r d k — e m o d e l near a s t a g n a t i o n p o i n t [29], w h i c h gives rise t o a n excessive level o f t h e t u r b u l e n t k i n e t i c energy k a n d the eddy viscosity v?- I n such a case, t h e

XI thank Urban Andersson for allowing me the use of his precious experimental data. The following comparison is also available in the proceeding of the 2^{n d} Turbine-99 Workshop.

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19

O 0.2 0.4 0.6 0.8 1 Center line L

F i g u r e 2.6: Experimental and simulated values of the pressure recovery along the upper wall.

- Q - Experiments

- t - CFD

f

0 0.05 0.1 0.15 0.2 0.25 0.3 Radius (m)

F i g u r e 2.8: Experimental and simulated values of the axial velocity at section l b for the T mode.

0 0.2 0.4 0.6 0.8 1 Center line L

F i g u r e 2.7: Experimental and simulated values of the pressure recovery along the lower wall.

_ 1 .⁶I , ; 1 . 1 1

0 0.05 0.1 0.15 0.2 0.25 0.3 Radius (m)

F i g u r e 2.9: Experimental and simulated values of the tangential velocity at section l b for the T mode.

w a l l shear stress is o v e r e s t i m a t e d a c c o r d i n g t o r e l a t i o n ( 2 . 9 ) , w h i c h gives h i g h e r viscous losses a n d t h u s , a lower pressure recovery. A f t e r t h e elbow corner, t h e pressure recovery is w e l l represented f o r t h e lower w a l l a n d o v e r e s t i m a t e d f o r t h e u p p e r w a l l .

T h e e x p e r i m e n t a l a n d s i m u l a t e d values o f the a x i a l a n d t a n g e n t i a l velocities a t section l b are represented i n figures 2.8 a n d 2.9, where t h e dashed l i n e represents t h e diameter a t t h e end o f t h e cone. Close agreement is o b t a i n e d f o r the a x i a l v e l o c i t y f o r 0.07 < r < 0.2, where t h e flow can be supposed i n v i s c i d . Near t h e b o u n d a r y , t h e results are n o t as g o o d ; t h e a x i a l v e l o c i t y is o v e r e s t i m a t e d close t o the s h r o u d a n d u n d e r e s t i m a t e d close t o t h e center o f t h e d r a f t t u b e . T h e t a n g e n t i a l v e l o c i t y presents a s y s t e m a t i c bias w i t h t h e e x p e r i m e n t a l p r o f i l e f o r 0.07 < r < 0.24, however t h e t r e n d is w e l l c a p t u r e d . Once again, there is a n i m p o r t a n t difference near t h e cone a n d t h e s h r o u d , especially t h e p o s i t i o n a n d a m p l i t u d e of the m i n i m u m t a n g e n t i a l v e l o c i t y are n o t c a p t u r e d w i t h t h e present s i m u l a t i o n . F r o m these comparisons, one m a y suggest, t h a t t h e w a l l f u n c t i o n is

Effects of boundary conditions and unsteadiness on draft tube flow

DOCTORAL THESIS

Effects of Boundary Conditions and Unsteadiness on Draft Tube Flow

MICHEL J. CERVANTES

ABSTRACT

ACKNOWLEDGMENTS

Contents

Part I

General discussion

Chapter 1

INTRODUCTION

Chapter 2

TRUST AND QUALITY I N C F D