theory and tunnel tests of rotor blades for supersonic turbines by b.s. stratford and g.e. sansome

8/13/2019 Theory and Tunnel Tests of Rotor Blades for Supersonic Turbines by B.S. Stratford and G.E. Sansome

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M I N I S T R Y O F V I T I O N

R . M . N o . 3 27 5~

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A E R O N A U T I C A L R E S E A R C H C O U N C I LR E P O R T S A N D M E M O R A N D A

T h e o r y an d T u n n e l T e s t s o f R o to r la desfor Supersonic TurbinesBy B S STRATFORD and G E SANSOME

L O N D O N : H E R M A J E S T Y S S T A T I O N E R Y O F F I C EI962

SEVENTEEN SHILLINGS NET


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T h e o r y an d T u n n e l T e s t s o f R o t o r la d es fo rSuperson ic Turbines

By B S STRATFORDand G E SANSOME

COMMUNICATED BY THE DEPUTY CONTROLLER IRCR FT RESEARCH AND DEVELOPMENT),MINISTRY OF AVIATION

Reports and M emoranda No . 3275December z96o

Summary I n sp e c i a l c i r c u m s t a n c e s w h e r e a l a rg e w o r k o u t p u t i s r e q u i r e d f r o m a t u r b i n e i n a s in g l e s ta g ei t i s n e c e s sa r y to u s e h i g h p r e s su r e r a t io s a c r o s s th e n o z z l e b l a d e s , t h u s p r o d u c i n g su p e r so n i c v e l o c it i es a ti n l e t to t h e r o t o r . A s p a r t o f a n i n v e s t i g a ti o n i n t o su c h t u r b i n e s , s e v e r a l d e s i g n s f o r t h e i n t e r - b l a d e p a s sa g e s o ft h e r o t o r h a v e b e e n t e s t e d i n a t w o - d i m e n s i o n a l t u n n e l , a d e s i g n t h e o r y b e i n g d e v e l o p e d c o n c u r r e n t l y .

T h e f i r s t d e s i g n , f e a tu r i n g c o n s t a n t p a s sa g e w i d t h a n d c u r v a t u r e a s i n s t e a m - t u r b i n e p r a c t i c e , b u t h a v i n gt h i n l e a d i n g a n d t r a i li n g e d g e s, w a s f o u n d t o su f f e r f r o m f o c u s i n g o f t h e c o m p r e s s i o n w a v e s f r o m t h e c o n c a v esu r f a c e , w i t h c o n se q u e n t f l o w se p a r a t i o n f r o m t h e o p p o s i t e c o n v e x su r f a c e . I t g a v e a v e l o c i t y c o e f f i c i e n t o f0 " 9 2 9 - - b a s e d u p o n t h e a r e a - m e a n t o t a l p r e s s u r e as m e a s u r e d a t a n i n le t M a c h n u m b e r o f 1 . 9 0 a n d t u r n i n ga n g l e o f 1 4 0 d e g . T h e m e a su r e d v a l u e c o m p a r e s f a v o u r a b l y w i t h v a l u e s f r o m p r e v i o u s s t e a m t e s t s ; w h e r e t h er e su l t s h a v e b e e n i n t h e r a n g e f r o m 0 . 6 5 t o 0 . 9 2 .

F r o m t h e o r e t i c a l r e aso n i n g , a n d f r o m a d d i t i o n a l t e s t o b se r v a ti o n s , a su b se q u e n t p a s sa g e w a s d e s i g n e dh a v i n g a n i n l e t t r a n s i ti o n l e n g t h o f sm a l l c u r v a t u r e , l e a d i n g t o a f r e e - v o r t e x p a s sa g e o f d o u b l e t h e t r a n s it i o nc u r v a t u r e ; a sm a l l a m o u n t O f c o n t r a c t i o n w a s in c o r p o r a t e d . S c h l i e re n p h o t o g r a p h s sh o w e d t h e f l ow i n t h i sp a s sa g e t o b e a l m o s t sh o c k f r ee . A t h i n r e g i o n o f l o w - e n e r g y a i r e x i s te d c l o se t o t h e c o n v e x su r f a c e, b u t l i q u i d -i n j e c t i o n t e s t s l o c a t e d o n l y o n e sm a l l b u b b l e o f r e v e r se d f l o w . P r e s su r e t r a v e r se s a t e x i t i n d i c a t e d a v e l o c i t yc o e f fi c ie n t o f 0 . 9 5 2, b a se d o n t h e a r e a - m e a n t o t a l p r e s su r e . W h e n a lo w a n c e is m a d e f o r t u r n i n g a n g l e a n dR e y n o l d s n u m b e r t h i s r e su l t a p p e a r s to c o m p a r e q u i t e f a v o u r a b l y w i t h p r e v i o u s w o r k .

I t w o u l d s e e m t h a t th e o p t i m u m b l a d e p i t c h i n g i n a t u r b i n e w o u l d b e a b o u t 2 0 t o 3 0 p e r c e n t c l o ser t h a n i n at w o , d i m e n s i o n a l c a sc a d e . H o w e v e r , t h e r e su l t a n t p i t c h i n g t e n d s t o b e c o m e v e r y c l ose , e x c e p t a t v e r y l a rg et u r n i n g a n g l e s , w i t h t h e r e su l t t h a t i n so m e a p p l i c a t i o n s d i f f i c u l t i e s c o u l d a r i s e i n t h e p r a c t i c a l d e s i g n a n dm a n u f a c t u r e .

S e v e r a l u n c e rt a i n ti e s r e m a i n a n d t h e p r e se n t d e s i g n m u s t b e r e g a r d e d a s st il l e x p e r im e n t a l .

1 0 Introduct ion F o r c e r t a i n a p p l i c a t io n s , s u c h a s s t e a m - p l a n t a u x il i ar i e s, ' a s t e r n ' t u r b i n e s i nm a r i n e s t e a m p l a n t , o r th e d r i v e o f f u e l p u m p s f o r li q u i d - p r o p e l l a n t r o c k e ts , t u r b i n e s a r e r e q m r e dt o g i v e a v e r y la r g e w o r k o u t p u t i n a s i n g le s t a ge . I m p u l s e w h e e l s a r e u s u a l l y e m p l o y e d , s o t h a t t h ew h o l e o f t h e e x p a n s i o n o c c u r s a c r o s s t h e n o z z le s . T h e g a s f r o m t h e n o z z le s e n t e rs t h e r o t o r a t a

* P r e v io u s l y is s u e d as N . G . T . E . R e p o r t N o . 2 4 5 - - A . R . C . 2 2 ,5 3 7.


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supersonic relative velocity and a largeswirl angle, to be turned through an angle of the order of90 to 160 deg. Previous reports have described the design and testing of such a turbine at theNational Gas Turbine Establishment1 and the tunnel performances of the supersonic nozzles ~.

The present Report describes the tunnel investigation of the first row of rotor blades, wherein isthought to reside the major source of l o s s - - s e e for example, Refs. 1 to 4. The design requirementsdemanded a relative inlet Mach number to the rotor of 1.90 and a turning angle of 140 deg, theinlet and ou tlet swirl angles being equal at 70 deg (relative to the axia direction).

The main experimental work in the present Report is represented by Figs. 1, 5, 18 and 22, whilethe main theoretical arguments are represented by Figs. 10 and 13; the quali fications given in thetext are needed, however, when applying Figs. 10 and 13.

2.0. T h e C a s c a d e T u n n e l . The apparatus, shown in Fig. 1, consisted of a closed-jet two-dimensional wind tunnel of Mach number 1.90, constructed of wood and Perspex and containingin the cascade a single blade and two passages. A Holland exhauster powered the tunnel, to which airwas supplied by a dry-air plant at just above atmospheric pressure and at a nominal dew point of- 80 deg C. A 1 deg diverging section at the Cascade outlet, followed by a subsonic diffuser, allowedthe tunnel to be operated at an overall pressure ratio of 1.8--a lmost in silence. The Perspex formingthe side walls fulfilled the dual purpose of holding the blade and provid ing Schlieren windows.As such it gave an unobstructed field of vision and an inexpensive mode of construction. TheSchlieren views were readily examined on the rig so that, with the preceding advantages, it wasdecided to accept the photographic imperfections resulting from the susceptibility of Perspex toscratching.

Of the two passages in the cascade the test passage was the inner one. The flow at its entry consistsof a uniform mainstream, together with a boundar y layer which has grown in the nozzle and which isapproaching the convex surface of the passage. The uniform flow represents that in the turbinedownstrea m of the pattern of bow waves 2 (strictly it should therefore be at a slightly higher Machnumber than the flow upstream of the bow waves in the turbine). The boundary layer should wellrepresent that which would approach the convex portion of a blade, after growing on the flat regionof the blade immediately upstream. Since in supersonic flow there can be no interaction betweenneighbouring passages, other than by the wave patterns upstream and downstream, the flow in theinner passage of the tunnel should be typical of that in an infinite cascade. The upstream and down-stream wave patterns, and hence the incidence and outlet angle, may be considered almostindependently of the internal flow as, for example, in Refs. 1 and 2.

The second passage in the test cascade, the outer, acts as a by-pass for the boundary layer fromthe other profiled wall of the nozzle. The outer passage is not representative of a passage in aninfinite cascade because of this boundary layer and because it traps the bow waves from the singleblade. In an infinite two-dimensional cascade having, as here, a subsonic axial component ofvelocity, the bow waves all pass upstream from the rotor rather than being trapped within thepassages. In all the present tests in which supersonic flow was established the flow separated fromthe convex surface of the outer passage; consequen tly this passage has been deleted from most of theSchlieren photographs in order to avoid confusion.

All the tests were performed at zero incidence relative to the flat region of the blade upper surface.Such, it is felt, is the cond ition which holds locally in the t urbine, i . e . at the leading edges and atthe passage entry, downstream of the rotor bow wave1, ~


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The traversing instruments are sketched in Fig. 2. The static probe with the annular slot wasused for the earlier traverses, but was found to have a slight error resulting from a 0. 002 in. step(unintentional) at the slot. Consequently, as discussed in Ref. 2, the static probe with the fourholes was used for the later traverses, from which all the main results are quoted, and it is thought 2that the readings from this instrument were accurate.

The velocity coefficients quoted are based on the area-mean total pressure, the relationshipbetween pressure coefficient and velocity coefficient at M - - 1.9 being shown in Fig. 3. Bothcoefficients take values for isentropic flow as datum as discussed in Ref. 2. In typical flows the veloci tycoefficient based on the area-mean total pressure is close in value to the mome ntum- mea n velocitycoefficient, i .e . to a coefficient based on the mass mean of velocity.

The main tests were carried out without tip clearance, at a Reynolds number based on chordof 1 3 x 10 6.

3 . 0 . A B l a d e D e s ig n b a s e d o n S t e a m - Tu r b h ze P r a c t i c e . The first blade profile to be tested wastha t designed for the tu rbine 1, the design having been carried out mainly on the basis of practice insteam plant. The profile, shown in Fig. 4, provides a passage having a constant width, w and acentre-line of constant radius of curvature, r m ; w / r m = 0 33, while the pitch/chord ratio, sic --- O. 47.The upper surface of the blade at the leading and trailing edges is parallel to the nominal gas flowat inlet and outlet respectively, except for the 20 deg chamfer at the edges themselves, which arevery thin.

In the turbine the chord and maximum thickness of the blades are constant throughout theirheight, with the result that the passages between the blades diverge towards the tips. The sectiontested two-dimensionally in the tunnel and shown in Fig. 4 was a four times magnification of thatin the turbine at mid-blade height. The aspect ratio of the blade was approximately the same as forthe turbine, the span in the tunnel being 3.11 in. and the chord 4 in.

Besides being tested in the 'turbine configuration' shown in Fig. 4 the design was also testedreversed, so that the 10 deg cut-back shown at the leading edge was transferred to the trailing edge,while the tangent to the concave surface at the leading edge became parallel to the oncoming flow'.This second arrangement is referred to as the 'cascade configuration'.

3 . 1 R e s u l t s f r o m t h e F i r s t D e s i gn . The main results are shown in Figs. 5 to 8.The Schlieren photographs show a focusing of the compression waves from the forward region

of the concave surface and suggest that, resulting from the focusing, there is a strong separation fr omthe convex surface. Subsequent tests injecting small quantities of methylated spirits confirmed thepresence of separation and reverse flow for about a n. length Of the convex surface. In the regionnear mid-chord, where the photographs indicate the vortex sh~eet to have returned close to thesurface, the flow at the wall was found to be a mixture of forward and reversed flow.

Near the convex wall at the passage exit the measured total and static pressures are very nearlyidentical for most of the span, indicating a region of almost stagnant air. This region accounts formost of the deficiency in velocity coefficient. In the turbine configuration the region for which thedynamic head is less than a half tha t of the mainst ream ranges from approximately 20 per cent of thepassage width at 2 per cent span, .rising to a maximum of 35 per cent at 20 per cent span anddecreasing to 30 per cent of the passage width at mid-span. There is good agreement between thewid th of these areas of large total-pressure loss, as seen in the traverses, and the distance between thewall and the apparen t vortex sheet as seen in the photographs of Fig. 5, measu rement of this distance

384943) A 2


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i n t h e p h o t o g r a p h s i n d i c a ti n g t h a t i t ex t e n d s o v e r 3 0 t o 3 5 p e r c e n t o f t h e p a s s a g e w i d t h . T h e m e a nv e lo c i t y c o e ff i c i en t f o r t h e w h o le s p a n , f o r t h e b l a d e i n t h e t u r b in e c o n f ig u r a t i o n , i s g iv e n b y F ig . 8t o b e a p p r o x i m a t e l y 0 . 9 3 .

T h e r e s u l ts f o r t h e c a s c a d e c o n f i g u r a t io n w e r e n o t s o c o n s i s t e n t a s fo r t h e t u r b i n e c o n f i g u r a ti o n .T h e y g a v e a m e a n v e l o c i t y c o e ff ic i e nt o f a p p r o x i m a t e l y 0 . 9 0 , t h e g r e a t e r l o ss m e a s u r e d i n c o m p a r i s o nw i t h t h e t u r b i n e c o n f i g u r a t io n b e i n g t h e r e s u l t o f a p p a r e n t l y w i d e r r e g io n s o f h i g h - l o s s a ir a d j a c e n tt o t h e c o n v e x w a l l i n th e t r a v e rs e s . H o w e v e r , t h e S c h l i e r en p h o t o g r a p h s s u g g e s t t h a t th e w i d t h o ft h e h i g h - l o s s r e g i o n s h o u l d b e a b o u t t h e s a m e f o r b o t h c o n f i g u r a t i o n s . S i n c e t h e p h o t o g r a p h s a r ec o n s i s t e n t w i t h t h e t r a v e r s e s f o r t h e t u r b i n e c o n f i g u r a t i o n a n d s i n c e t h e t r a v e r s e s f o r t h e c a s c a d ec o n f i g u r a t i o n w e r e p e r f o r m e d v e r y e a r l y i n t h e e x p e r i m e n t a l p r o g r a m m e , w h e n t h e s p e c i a l c a r en e e d e d w a s n o t f u l l y a p p r e c i a t e d see S e c t i o n 8 . 0 ) , t h e r e s u l t s f o r t h e t u r b in e c o n f ig u r a t i o n a r ec o n s id e r e d r e a s o n a b ly r e l ia b l e , w h i l e t h b s e fo r t h e c a s c a d e c o n f ig u r a t i o n a r e r e g a r d e d w i th r e s e r v a t i o n .A c tu a l l y , f o r t h e t u r b in e c o n f ig u r a t i o n , a p a r t i a l r e p e a t p i t o t t r a v e r s e u s in g t h e s t a t i c t r a v e r s e f r o mth e m a in t e s t o f t h i s c o n f ig t~ ra t io n f o r c o n v e r t i n g t h e p i t o t p r e s s u r e s t o t o t a l p r e s s u r e s , g a v e a v e lo c i t yc o e f fi c ie n t a b o u t 0 - 0 1 5 h i g h e r t h a n i n t h e m a i n t e s t, b u t t h e d i s c r e p a n c y w a s n o t a p p r e c i a t e d e a rl ye n o u g h t o b e i n v e s t i g a t e d f u r t h e r .

F r o m F i g s . 5 a n d 6 t h e f o c u s i n g i s p e r h a p s m o r e s e v e r e f o r t h e c a s c ad e c o n f i g u r a t i o n t h a n f o rt h e t u r b i n e c o n f i g u r a t i o n .

4.0. T h e o r e t i c a l A n a l y s i s a n d I n t e r m e d i a t e T e s ti n g . I n t h e p r e s e n t s e c t i o n a n a t t e m p t w i l l b em a d e t o a n a l y s e a p p r o x i m a t e l y , b u t i n g e n e r a l t e r m s , t h e s u p e r s o n i c f l o w i n c u r v e d p a s s a g e s . T h ef l o w a t t h e e n t r y t o t h e p a s s a g es i s a s s u m e d t o b e s t ra i g h t, a n d o f u n i f o r m M a c h n u m b e r M 0.C o n s i d e r a t i o n i s a t f ir s t l i m i t e d t o p a s s a g es o f v e r y s m a l l c u r v a t u r e s .

I n a p a ss a g e o f c o n s t a n t c u r v a t u r e t h e p r e s s u r e d i s t r i b u t i o n o n e i t h e r s u r fa c e h a s t h e z i g - z a gf o r m o f F ig . 9 , as m a y b e s e e n f r o m c o n s i d e r a t i o n o f th e c h a r a c t er i s ti c s p a t t e r n . T h u s , i n w h a tm a y b e t e r m e d t h e e n t r y r eg i o n , i . e . , A B C D o f F i g . 9 , t h e p r e s s u r e w i l l d e c r e a s e r o u g h l y l i n e a rl yt h r o u g h t h e e x p a n s i o n w a v e s a l o n g A B u n t i l , a t B , t h e c o m p r e s s i o n w a v e s s t a r t b e c o m i n g i n c i d e n tf r o m C D . A l o n g B E t h e e x p a n s i o n w a v e s w h i c h w o u l d b e c a u se d b y t h e c u r v a t u r e o f B E j u s t c a n c elt h e r e f l e ct i o n s o f t h e c o m p r e s s i o n w a v e s f r o m C D b u t t h e c o m p r e s s i o n w a v e s t h e m s e l v e s r e m a i n .T h u s t h e p r e s s u r e r i s e s a l o n g B E u n t i l , a t E , i t h a s r e tu r n e d t o i t s e n t r y v a lu e , i . e . , t h a t a t A . T h es a m e p a t t e r n i s r e p e a t e d a l o n g t h e n e x t s e c t i o n o f s u r fa c e , E G I , w h i l e t h e c o r r e s p o n d i n g ( e q u a l a n do p p o s i t e ) p r e s s u r e c h a n g e s o c c u r a l o n g t h e c o n c a v e s u r fa c e . T h e r e s u l ti n g z i g - z a g p r e s s u r ed i s t r i b u t i o n s u b j e c t s t h e b o u n d a r y l a y er s to r e p e a t e d s u c t i o n p e a k s a n d a d v e r s e p r e s s u r e g r a d i e n t s- - p o s s i b l y c a u s i n g f l o w s e p a r a t i o n - - s o t h a t a p a s s a g e o f c o n s t a n t c u r v a t u r e i s n o t l i k el y t o b ee f fi c ie n t . B e f o r e c o n s id e r in g t h e d e s i gn mo d i f i c a t i o n r e q u i r e d , h o w e v e r , t h e f l o w o f F ig . 9 ma y b ee x a m i n e d f u r t h e r .

I n a c u r v e d f l o wr

O p l a y = p l r =

w h e r e y i s t h e d i s t a n c e m e a s u r e d n o r m a l t o t h e s t r e am l i n e s a n d K i s t h e c u r v a t u r e o f t h e f l o w .T h u s t h e p r e s s u r e d i f f er e n ce , b e t w e e n a p o i n t o n o n e s u r f a c e o f th e p a s s a g e , s a y B , a n d t h e p o i n to p p o s i t e o n t h e o t h e r s u r f a c e , i . e . , D , i s p r o p o r t i o n a l t o t h e c u r v a t u r e o f t h e f l o w b e t w e e n t h e t w op o in t s ( t h e l o c a l f l o w c u r v a tu r e i s n o t n e c e s s a r i l y t h e s a me a s t h e l o c a l p a s s a g e c u r v a tu r e ) .


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Therefore, the zig-zag pressure distribution shown in Fig. 9 also represents the flow-curvaturedistribution, so that across each of AC, EF and IJ there is a zero pressure difference and zerocurvature of the flow, whilst the peak flow curvatures which occur across BD and GH are twicethe magnitude of the mean flow curvature. But the mean flow curvature must be equal to thepassage curvature, which is const.ant in Fig. 9, Consequently across BD and GH the flow curvatureis double the passage curvature. The 2/1 ratio would be exact, of course, only for infinitesimalcurvatures, where the waves become exact diagonals of rectangles.Suppose that the passage curvature of BEDF in Fig. 9 had been double that of the inlet regionABCD. Considering first the characteristics pattern, the expansion waves from BE would nullifythe effects of both the compression waves from CD and the reflections of those compression waves,so that the pressure along BE would remain constant. The new design with its improved pressuredistribution is illustrated in Fig. 10, B KD L being an arbitrary length of passage whose curvature isdouble that of ABCD. N ow in the previous paragraph it was shown that the flow curvature acrossBD was double the passage curvature ABCD; the flow curvature, therefore, matches the curvatureof the new region BKD L and it becomes clear how the pressure di stribution and flow curvature areable to remain constant in the new passage.

The preceding arguments indicate that, for small curvatures, a passage entry length of a certaincurvature acts as a trans ition region for a passage of double the curvature, the en try length convertingthe straigh t flow to the ~tppropriate curved flow.

A similar transition region at exit will return the free-vortex flow to straight flow with a smoothchange of pressure.

For the very small curvatures so far considered, ABCD is approximately a rectangle, with ADand CB straight diagonals at an angle/x0 to AB, CD. Hence the length required for t he transitionregion is then

AB CD w cot ~o. 1)In order to determine the sharpest curvature which may be utilized for the central portion of thepassage three aspects may be considered, the behaviour of the boundar y layer, the geometry of the

characteristics pattern, and the avoidance of diffusion through sonic speed.Were the flow in the passage strictly two-dimensional, the boundary layer on the convex surface

would be able to withst and a considerable pressure rise, as would t end to occur between mid-chordand the trai ling edge if a high curvature were employed at mid-chord. With a large total turningangle for the passage, however, secondary flow would be expected from the end walls on to possiblythe whole span of the convex surface, the resulting boundary layer containing cross flows and beinga potential cause of loss in any significant pressure rise. Consequently a fairly mild curvature isprobably required.

Now the mainstream flow in the region BKDL of Fig. 10 may be considered as part of a freevortex, for which the characteristics network is given in Ref. 5 and also in Fig. 11 of the presentReport. Fu rther properties are given in Fig. 12. The only limitation ideally from the shape of thecharacteristics in the free-vortex fl ow is tha t the flow does not exist inside a certain radius. However,it might be judged that in practice a compression wave from the concave surface must reach not tooremote a point on ,the convex surface in order tha t some unprescribed) factor shall not causebreakdown of the flow. Thi s requirement wou ld be met by the criterion that a straight linetangential to the Mach line at mid-passage shall intersect or be tangential to the convex surface.


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B y t h e s i m p l e t r i g o n o m e t r i c c o n s t r u c t i o n o f F i g . 1 3 a th e c u r v a t u r e a t m i d - c h o r d i s t h e n s e e n tos a t i s fy

r o o ~ ~ o = / r m ~ . ~ n > 1 cos ff~, , (2)w h e r e / x m = s i n - l l / M m , M m b e in g t h e M a c h n u m b e r a t r = r o i . e . o n t h e p a s s a g e c e n t r e - l i n e .A t h i g h M a c h n u m b e r s E q u a t i o n ( 2 ) c a n b e e x p r e s se d

I l M ? 114M 3 )W h e n M , ~ = 1 9 0 E q u a t i o n ( 2) g iv e s w / r ~ = 0 3 0 , w h i l e t h e f i r s t t e r m o f E q u a t i o n (3 ) w o u l dg ive 0 . 277 .

A f u r t h e r c o n d i t i o n w h i c h c o u l d b e s p e c i f i e d f o r t h e p a s s a g e f l o w is t h a t t h e v e l o c i t y o n t h ec o n c a v e s u r f a c e s h o u l d r e m a i n s u p e r s o n i c , a s o t h e r w i s e d i f f i c u l ty m i g h t b e e x p e c t e d i n d i f f u s i n gt h e r e l e v a n t p a r t o f t h e f l o w t h r o u g h s o n ic s p e ed . T h i s l i m i t a ti o n w o u l d o n l y b e c o m e e f f e c ti v e w h e nt h e M a c h n u m b e r a t e n t r y i s i n th e l o w s u p e r s o n i c r a ng e .

I t i s h e l p f u l n o w t o r e t u r n t o t h e t e s t s o f t h e f i r s t b l a d e d e s i g n i n o r d e r t o c o n s i d e r h o w t h ef l o w m a y b e c o n t r o l l e d i n t h e e n t r y r e g i o n o f t h e p a s s a g e .

E x a m i n a t i o n o f F i g s . 5 a n d 6 s u g g e s t s t h a t , i f t h e c o n v e x w a l l w e r e b r o u g h t n e a r e r t h e c o n c a v e ,t h e ' f a n ' o f c o n v e rg i n g c o m p r e s s io n w a v e s e m a n a t i n g f r o m t h e c o n c a v e w a ll w o u l d r e a c h t h ec o n v e x s u r f a c e b e f o r e f o c u s i n g . M o r e o v e r , i f t h e r e w e r e n o s e p a r a t i o n a t t h e c o n v e x s u r f a c e t h ee x p a n s i o n w a v e s w h i c h w o u l d r a d i a te f r o m i t w o u l d t e n d t o p r e v e n t t h e f o c u s i n g o f t h e f a n o fc o m p r e s s i o n w a v e s a f t e r r e f l e c t io n . C o n s e q u e n t l y , t e s t s w e r e p e r f o r m e d o n a s e c o n d b l a d e d e s i g nh a v i n g a m o r e g e n t l e p a s s a g e c u r v a t u r e , a n d F i g . 1 4 s h o w s t h a t a c o n t r o l l e d e n t r y f l o w i s t h u so b t a i n e d . ( I n F i g . 1 4 t h e i m p o r t a n t c o m p r e s s i o n w a v e s a r e t h e s e c o n d a n d t h i r d l i n e s s p r i n g i n gf r o m t h e c o n c a v e s u r f a c e a s , w i t h t h i s p a r t i c u l a r b l a d e , t h e c u r v a t u r e d i d n o t s t a r t u n t i l a p o i n to p p o s it e t h e s h o u l d e r o f t h e l e a d i n g - e d g e c h a m f e r , i . e . a b o u t a h a lf pa s sa g e w i d t h d o w n s t r e a mo f t h e e n t r y . )

T h e d e c r e a s e i n p a s s a g e c u r v a t u r e s h o w n i n F i g ' . 1 4 h a s , a s i t w e r e , b r o k e n a ' v i c i o u s c i r c l e 'e x i s t i n g i n F i g s . 5 a n d 6 . F o r t h e f l o w to b e ' c o n t r o l l e d ' t h e l e a d i n g - e d g e w a v e f r o m t h e c o n c a v es u r f a c e m u s t r e a c h t h e c o n v e x s u rf a c e . H o w e v e r , w i t h t h e g e o m e t r y o f F i g s . 5 a n d 6 , t h e l e a d i n g - e d g ew a v e w o u l d c u r v e a r o u n d t o r e a c h t h e c o n v e x su r f a c e o n l y i f t h e f l o w a t t h e c o n v e x s u r f ac e u p s t r e a mr e m a i n e d a t t a c h e d a n d t h e r e b y e m i t t e d e x p a n s i o n w a v e s. I n f a c t, h o w e v e r , F i g s . 5 a n d 6 s h o w t h a tt h e f l o w u p s t r e a m s e p a r a t e s , t h e l e a d i n g - e d g e w a v e r e m a i n s s t r a i g h t i n s t e a d o f c u r v i n g , a n d s od o e s n o t r e a c h t h e s u r f a c e ; f o c u s i n g o c c u rs , a n d t h e r e s u l t a n t s h o c k w a v e c a u s e s t h e u p s t r e a ms e p a ra t io n . T h e c i rc l e o f e v e n ts h a s b e e n b r o k e n w i t h t h e s m a l l e r c u r v a t u r e b y e n s u r i n g t h a t t h el e a d i n g - e d g e w a v e c a n r e a c h t h e c o n v e x s u r f a c e w h e t h e r o r n o t i t i s c u r v e d b y e x p a n s i o n w a v e sf r o m u p s t r e a m . H e n c e f o r a th e o r e t ic a l g u i d e t o t h e c u r v a t u r e a l lo w a b l e a t e n t r y o n e m a y s p e c if y ,a s a fi r s t r o u g h a p p r o x i m a t i o n , t h a t t h e s t r a i g h t - li n e c o n t i n u a t i o n o f th e w a v e f r o m t h e l e a d in g e d g eo f t h e c o n c a v e s u r f a c e m u s t a t l e a s t t o u c h t h e c o n v e x s u r f a c e , a s i n F i g . 1 3 b . A s d r a w n i n F i g . 1 3 bt h e l e a d i n g - e d g e w e d g e a n g l e o f t h e c o n c a v e s u r f a c e i s z e r o a n d s o t h e w a v e i s j u s t a M a c h l i ne .T h e f i rs t c r it e r io n o n e n t r y c u r v a t u r e i s t h e n

r e o ~ - c o ~ : / r o o n e ~ ~ ~ > >. cos fro , (4)w h e r e # 0 i s t h e M a c h a n g l e , i . e . s i n - l l / M 0 , M 0 b e i n g th e M a e h n u m b e r a t t h e p a s sa g e e n t r ya f t e r p a s s i ng t h r o u g h a n y u p s t r e a m b o w - w a v e s y s t e m ~. W h e n M o = 1 . 9 0 E q u a t i o n ' (4 ) g i v e s


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w /r m


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T h e p r e c e d i n g d i s c u s s i o n m a y b e s u m m a r i s e d i n t h e f o l l o w i n g d e s i g n p r o c e d u r e . T h e b a s i cp a s s a g e d e si g n w o u l d f o l l o w t h e s c h e m e o f F i g . 1 0 . T h e l e n g t h r e q u i r e d f o r e a c h t r an s i t i o n r e g i o n ,i .e. , a t e n t r y a n d e x i t , w o u l d b e a p p r o x i m a t e l y w c o t / x 0 , a s g i v e n b y E q u a t i o n ( 1 ) . T h e c u r v a t u r eo f t h e t r a n s i t io n r e g i o n s w o u l d i n i t ia l ly b e s e t a t 2 0 t o 3 0 p e r c e n t l e ss th a n g i v e n b y E q u a t i o n s ( 4 )a n d ( 5 ), s o t h a t, w i t h t h e m a i n c u r v a t u r e d o u b l e th e t r a n s it i o n c u r v a t u r e , t h e m a i n c u r v a t u r e s h o u l db e c o m e a b o u t 2 0 t o 3 0 p e r c e n t l es s t h a n g i v e n b y E q u a t i o n s ( 2) a n d ( 3 ). I t w o u l d t h e n b e n e c e s s a r yt o c h e c k t h a t t h e l e a d i n g - e d g e w a v e r e a c h e d t h e c o n v e x s u r f a c e a t a p o s i t i o n h a v i n g a n a c c e p t a b l et u r n i n g a n g l e - - s a y 1 0 t o 1 5 d e g , p r e fe r a b l y t o w a r d s t h e l o w e r e n d o f t h e r a n g e f o r a t u r b i n e - - a n dt h i s c h e c k C o u ld b e c a rr i e d o u t e i t h e r e x p e r i m e n t a l l y o r b y c o n s t r u c t i n g a r u d i m e n t a r y c h a r a c te r i s ti c sp a t t e r n i n t h e e n t r y r e g i o n .

S h o u l d i t b e r e q u i r e d t o c a l c u l at e t h e r a d ia l d i s t r i b u t i o n o f M a c h n u m b e r i n th e c e n t r a l r e g i o no f t h e p a s s a g e , w h e r e t h e f l o w s h o u l d a p p r o x i m a t e t o a fr e e v o r t e x , F i g . 1 2 c o u l d c o n v e n i e n t l y b eu s e d . A k n o w l e d g e w o u l d f i r s t b e n e e d e d o f th e M a c h n u m b e r , M m , a t th e m e a n r a d i u s ; f o r a f i r sta p p r o x i m a t i o n e i t h e r t h i s i s t a k e n a s t h e e n t r y M a c h n u m b e r - - i f t h e p a s s a g e a r e a i s c o n s t a n t a n dt h e b o u n d a r y - l a y e r g r o w t h n e g l e c t e d - - o r i t i s c a l c u l a t e d f r o m t h e e n t r y M a c h n u m b e r , a l l o w i n g f o rp a s s a g e c o n t r a c ti o n a n d g r o w t h o f t h e b o u n d a r y - l a y e r d i s p l a c e m e n t t h i c k n e s s, a c c o r d i n g t o t h eu s u a l o n e - d i m e n s i o n a l r e la t io n s f o r i s e n tr o p i c f l o w . G i v e n Mm , F ig . 1 2 d e t e r min e s rm/rl , t h u s f i x in gt h e p o s i t i o n o f t h e p a s s a g e c e n t r e - li n e r e l at i v e t o th e n o n - d i m e n s i o n a l c u r v e . T h e M a c h n u m b e rd i s t r i b u t i o n m a y t h e n b e re a d f r o m t h e c u r v e . T h e p v g r a p h o f F ig . 1 2 a l lo w s a s imp le s e c o n da p p r o x i m a t i o n t o M , ~. T h e d e p a r t u r e o f t h e p v g r a p h f r o m a s t r a ig h t l i n e i n t h e r a d iu s r a n g ec o n c e r n e d , i .e. , t h e d i f f e r e n c e b e t w e e n i t s me a n l e v e l a n d i t s v a lu e at t h e me a n r a d iu s , g iv e s t h el o s s o f f l o w c a p a c i t y d u e t o t h e c u r v a t u r e o f t h e p a s s a g e ; t h e o n e - d i m e n s i o n a l e s t i m a t e o f M , ~ c a nt h e n b e a d j u s t e d . T h e a d j u s t m e n t i s u s u a ll y v e r y s l ig h t .

4.1. Cont rac t ion . T h e b l a d e p a s s a g e i s r e f e r r e d to a s c o n t a i n i n g c o n t r a ct i o n w h e n i t s w i d t h a tmid - c h o r d i s l e s s t h a n a t e n t r y o r e x i t , i .e. , w h e n t h e p a s s a ge is c o n v e r g e n t - d i v e r g e n t . S i n c e t h ef l o w is s u p e r s o n i c t h e c o n t r a c t i o n d i f f u s es t h e i n l e t f lo w t o g i v e a r e d u c e d m e a n M a c h n u m b e r a tm i d - c h o r d a n d a s u b s e q u e n t a c c e l e r at i o n b e f o r e t h e e x i t. T h e b o u n d a r y l a y e r s a t e n t r y a r e t h i n a n ds o t h e d i f f u s io n s h o u ld c a u s e l i t t l e d i f f i c u l t y , w h e r e a s a t e x i t t h e b o u n d a r y l a y e r s a r e t h i c k e r a n dc o n t a i n s e c o n d a r y f l o w , s o t h a t t h e a c c e le r a ti o n w o u l d b e e x p e c t e d t o b e b e n e f i c ia l . T h u s a n e t g a ins h o u l d r e s u l t f r o m c o n t r a c t i o n . M o r e o v e r w i t h c o n t r a c t i o n t h e v e l o c i t i e s a t m i d - c h o r d a r e l o w e r ,t h u s r e d u c i n g t h e s k i n - f ri c t io n l o ss e s ; f u r t h e r m o r e t h e p a s s a g e i s n a r r o w e r , s o th a t f o r a g i v e nv a l u e o f t h e n o n - d i m e n s i o n a l c u r v a t u r e , w/rm, t h e a b s o l u t e c u r v a t u r e , 1 / rm , i s s h a r p e r , s o t h a t t h ep a s s a g e i s s h o r t e r a n d a g a in t h e s k in f r i c t i o n i s r e d u c e d . L a s t l y , a s t h e Ma c h n u mb e r i s l o w e r ,E q u a t i o n ( 2 ) a l l o w s a s t e e p e r c u r v a tu r e , w / rm , a n d h e n c e a f u r t h e r r e d u c t i o n o f s k in f r i c ti o n .

T h e r e i s a p o s s ib i l i ty o f c h o k in g , h o w e v e r , i f m u c h c o n t r a c t i o n i s u s e d . T h e c o n t r a c t i o n f o r m s as e c o n d t h r o a t a n d , w h e n f i rs t tr y i n g t o e s t a b li s h s u p e r s o n i c f l o w i n t h e p a s s a g e, t h e l o s s o f t o t a lp r e s s u r e t h r o u g h t h e n o r m a l s h o c k , w h i c h i s s t il l u p s t r e a m d u r i n g t h e s t a r t i n g p r o c e s s , re q u i r e st h a t t h e s e c o n d t h r o a t b e c o n s i d e r a b l y l a r g e r t h a n t h e f i r s t t h r o a t i n o r d e r t o p a s s t h e s a m e m a s sf l o w . T h e c o n t r a c t i o n w h i c h h a s b e e n f o u n d p o s s i b l e i n s t r a i g h t w i n d t u n n e l s d u r i n g s t a r t i n g i sc o n s i s t e n t w i t h t h a t c a l c u l a t e d a s s u m i n g o n e - d i m e n s i o n a l f l o w a n d j u s t a s im p l e n o r m a l s h o c k i nt h e w o r k i n g s e c t i o n s F o r e x a m p l e , a s s h o w n b y t h e g r a p h o f F ig . 1 5 , t h e a l l o w a b l e c o n t r a c t i o n f o ro n e - d i m e n s i o n a l f l o w is 1 6 p e r c e n t W h e n t h e u p s t r e a m M a c h n u m b e r i s 1 . 9 . F o r c u r v e d p a s s a g e st h e t e s t s o f t h e p r e s e n t R e p o r t s u g g e s t t h a t t h e l i m i t at i o n i s s li g h t ly m o r e s e v e r e, t h e a l l o w a b l e


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c o n t r a c t i o n f o u n d i n S e c t i o n 7 . 0 b e i n g a b o u t 1 2 p e r c e n t . T h u s , a l l o w i n g a m a r g i n f o r m a n u f a c t u r i n ga r t d o t h e r t o l e r a n c e s , t h e d e s i g n c o n t r a c t i o n i n a c u r v e d p a s s a g e p r o b a b l y s h o u l d n o t e x c e e d s a y8 p e r c e n t fo r a n e n t r y M a c h n u m b e r o f 1 . 9 - - u n l e s s s o m e d e v i c e i s i n c o r p o r a te d f o r p r e v e n t i n gc hok ing .

I n g e n e r a l, c o n t r a c t i o n w o u l d b e e x p e c t e d t o b e b e n e f i ci a l p r o v i d e d t h e s e c o n d t h r o a t l i m i t a ti o n i ss a t is f i ed ; t h e r e c o m m e n d a t i o n f o r t h e p r e s e n t l e v el o f M a c h n u m b e r i s t h e r e f o r e t o i n c o r p o r a t e ah a l f o f t h e c o n t r a c t io n i n d i c a t e d b y F i g . 1 5 , u s i n g a s th e M a c h n u m b e r f o r t h a t f i g u r e t h e v a l u ea t t h e p a s s a g e e n tr y . B e l o w a M a c h n u m b e r o f 1 . 3 n o c o n t r a c t io n s h o u l d b e i n c o r p o r a t e d .

T h e p r e c e d i n g r e c o m m e n d a t i o n i s f o r a t u n n e l c a s c a d e i n w h i c h t h e w a l l s a t t h e e n d o f t h e s p a na r e p a r a l l e l . F o r t u r b i n e s h a v i n g a f l a r e d a n n u l u s t h e a l l o w a b l e c o n t r a c t i o n t w o - d i m e n s i o n a l l ym i g h t b e g r e a t e r t h a n j u s t s u g g e s t e d , b y a p r o p o r t i o n o f t h e e x p a n s i o n i n a re a w h i c h i s p r o v i d e da t m i d - c h o r d b y t h e f l ar e .

C o n t r a c t i o n m a y r e a d i l y b e o b t a i n e d e x p e r i m e n t a l l y o n a g i v e n s e t o f b l a d e s b y s l i g h t l y c l o s i n gt h e i r p i t c h . F o r e x a m p l e , a t th e p r e s e n t i n l e t a n g l e o f 7 0 d e g , e a c h d e c r e a s e in p i t c h o f 1 p e r c e n td e c r e a s e s t h e p a s s a g e w i d t h a t i n l e t b y a p p r o x i m a t e l y 1 p e r c e n t , w h i l e t h e p a s s ag e w i d t h a t m i d - c h o r di s d e c r e a s e d b y 3 p e r c e n t , s o t h a t t h e r e s u l t i n g c o n t r a c t i o n w o u l d b e a b o u t 2 p e r c e n t .

4 .2 . Reac t ion . R e a c t i o n , i.e., a d r o p o f s t a ti c p r e s s u r e t h r o u g h t h e p a s s a g e, w o u l d b e e x p e c t e dt o s t a b il i s e t h e f l o w , i n t h e s e n s e o f r e d u c i n g t h e l i k e l ih o o d o f b o u n d a r y - l a y e r s e p a r a t i o n . T h eu n f l a r e d f l o w s o f t h e p r e d e n t R e p o r t h a v e n e g a t i v e r e a c t i o n b e c a u s e o f t h e l o ss e s , t h e g e o m e t r i ce x i t a n d e n t r y a r e a s b e i n g e q u a l : i n m o s t o f t h e t e s t s t h e s t a ti c p r e s s u r e a t e x it is 4 0 to 6 0 p e r c e n tg r e a t e r t h a n a t i n le t . R e a c t i o n h a s t o b e c o n s i d e r e d i n c o n j u n c t i o n w i t h t h e t u r b i n e d e s i g n a s a w h o l e ,a s t he ve lo c i t y t r i a ng l e s a r e a f f e c te d .

4.3. T he T h i r d B lade D es ign . T h e t e st s w h i c h h a d b e e n p e r f o r m e d o n t h e se c o n d b l ad e d e si g n( F i g . 1 4 ) h a d c o n f i r m e d t h a t a r e d u c t i o n i n t h e c u r v a t u r e r a t i o w / r ~ a t t h e p a s s a g e e n t r y c o u l d c u r et h e f o c u s in g o f t h e c o m p r e s s i o n w a v e s f r o m t h e c o n c a v e s u rf a c e a n d t h e s e p a ra t io n f r o m t h e c o n v e x .R e l i a b l e n u m e r i c a l v a l u e s w e r e n o t o b t a i n e d f o r t h e c u r v a t u r e a s a s s e m b l y i n s p e c t i o n f i g u r e s w e r en o t t a k e n a t t h e t i m e o f th a t t e s t a n d i t w a s f o u n d l a t er t h a t s m a l l d i s c r e p a n c ie s i n a s s e m b l y c o u l ds ign i f i c a n t ly a l t e r t he e f f e c t i ve c ur v a tu r e r a t io .

T h e t h i r d b l a d e w a s d e s i g n e d t o t e s t an d t o t a k e a d v a n t a g e o f t h e a r g u m e n t o f S e c t i o n 4 . 0 th a tt h e c u r v a t u r e c o u l d b e d o u b l e d i m m e d i a t el y d o w n s t r e a m o f t h e e n t r y r e g io n . A t t h e t im e o f t h ed e s i g n , h o w e v e r , t h e f u l l s t r e n g t h o f t h e a r g u m e n t s l i m i t i n g t h e c u r v a t u r e a t m i d - c h o r d w a s n o ta p p r e c i a t e d a n d s o , in s t e a d o f f o l lo w i n g F i g . 1 0, a s e c o n d i n c r e a s e o f c u r v a t u r e w a s i n c o r p o r a t e d .T h e V a l ue s u s e d f o r t h e v a r i o u s r e g i o n s o f t h e p a s s a g e w e r e :

Pa ssa g e t u r n ing a ng l e r e l a t i ve t o i n l e t , 0 t oPa ssa ge t u r n ing a ng l e . r e l a t i veP a s s a g e t u r n i n g a n g l e r e la t iv e

t h e b l a d e b e i n g s y m m e t r i c a l a b o u t m i d - c h o r d .

15 deg, w/r m = 0 . 1 7to inle t , 15 to 40 deg, w/~ = 0 . 3 4to i n l e t , 40 t o 70 de g , w / r m = 0 . 4 2T h e s e v a l u e s w e r e f o r t h e n o m i n a l d e s i g n p o s i t io n ,

f o r w h i c h t h e p a s sa g e w a s o f co n s t a n t w i d t h ; b y p i t c h w i s e m o v e m e n t o f th e b l a d e i n t h e t u n n e lt h e a b s o l u t e v a l u e o f t h e p a s s a g e w i d t h c o u l d b e c h a n g e d a n d a c o n t r a c t i o n o f t h e p a s s a g e o b t a i n e db e t w e e n e n t r y a n d m i d - c h o r d , a s d e s c r i b e d i n S e c t i o n 4 . 1 .

I t w i l l b e n o t i c e d t h a t , i n t h e n o m i n a l d e s i g n p o s i t i o n , t h e c u r v a t u r e i n t h e s e c o n d r e g i o n , i.e.,b e t w e e n 1 5 a n d 4 0 d e g , is s l i g h t ly g r e a t e r t h a n E q u a t i o n ( 2 ) w o u l d a l lo w , w h i l e b e t w e e n 4 0 a n d


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7 0 d e g i t i s m u c h g r e a te r . H e n c e , i f E q u a t i o n ( 2) r e p r es e n t s a t r u e l i m i t a ti o n , f l o w s e p a r a t i o n w o u l db e e x p e c t e d . A s c o n fi r m a t i o n , t h e u p p e r p h o t o g r a p h s o f F ig . 1 6 s h o w a s e p a ra t i o n o f t h e f l o wo c c u r r i n g n e a r t h e b e g i n n i n g o f t h e s e c o n d r e g i o n , th e s u b s e q u e r / t M a c h l in e s b e i n g s u c h t h a t t h e i rc o n t i n u a t i o n w o u l d m i s s th e c o n v e x s ur f ac e .

F o r t h e p h o t o g r a p h o f F ig . 1 6 c t h e p i t c h h a s b e e n r e d u c e d 5 p e r c e n t a n d t h e f o r w a r d s e p a r a t io nh a s b e e n c u r e d ( u n f o r t u n a t e l y t h i s i s n o t c l e a r i n t h e p h o t o g r a p h , w h i c h i s a n e a r l y o n e ) . T h er e d u c t i o n b o t h i n w/r~ a n d i n M a c h n u m b e r r e s u l t i n g f r o m t h e c o n t r a c t i o n i n t h e p a s s a g e a l l o w st h e c r i te r i o n o f E q u a t i o n ( 2) t o b e s a ti s fi e d, a n d t h e f o r w a r d M a c h l in e s a re s e e n t o r e a ch t h e c o n v e xs u r f a c e . T h u s E q u a t i o n ( 2 ) a p p e a r s t o b e a n e c e s s a r y c r i t e r i o n . H o w e v e r , s e p a r a t i o n i s s t i l l s e e n t oo c c u r n e a r m i d - c h o r d e v e n t h o u g h a t t h i s c o n t ra c t i o n t h e M a c h l i n e c o n t i n u a t i o n w o u l d i n t e rs e c tt h e s u r f a c e a n d h e n c e w/r~ s a t is f y E q u a t i o n ( 2) . T h e c r i te r i o n o f E q u a t i o n ( 2) is th e r e f o r e n o t a l w a y ss u f fi c ie n t . I t w o u l d a p p e a r t h a t t h e l i m i t a t io n h e r e i s t h a t t h e b o u n d a r y l a y e r is n o t a b l e t o s u s t a i nt h e p r e s s u r e r i s e r e q u i r e d b e t w e e n m i d - c h o r d a n d t h e p a s s a g e e x i t .

5.0. The Fourth or Shock-Free Design. F o r t h e f o u r t h d e s i g n t h e p a s s ag e c o n t a i n e d o n l y as i n g le i n c re a s e o f c u r v a t u r e , t h u s f o l l o w i n g th e s c h e m e o f F ig . 1 0. T h e n o m i n a l d e s i g n w i t h z e r oc o n t r a c t i o n i s s h o w n i n F i g . 1 7 . A t t h e b l a d e p o s i t i o n f in a l ly a d o p t e d t h e p i t c h w a s 4 p e r c e n t b e l o wth e n o min a l , g iv in g a n 8 p e r c e n t c o n t r a c t i o n i n t h e p a s s a g e . I t w i l l b e n o t i c e d t h a t t h e c h a m f e r a n g l ea t th e l e a d i n g e d g e h a s b e e n r e d u c e d t o 1 0 d e g , a m o d i f i c a t i o n i n t r o d u c e d w i t h t h e s e c o n d d e s i g ni n o r d e r t o b e w e l l b e l o w t h e a n g l e f o r s h o c k d e t a c h m e n t .

T e s t r e s u l t s f o r t h e p a s s a g e w i th t h e b l a d e i n i t s f i n a l d e s ig n p o s i t i o n a r e s h o w n in F ig s . 1 8 t o 2 1 .T h e r e s u lt s s h o w t h a t a p as s a g e f lo w a h n o s t f r e e o f s h o c k s a n d s e p a r a t io n h a s b e e n o b t a i n e d w i t h

t h e f o u r t h b l a d e d e s i g n a t 8 p e r c e n t c o n t r a c t i o n ; t h e m e a s u r e d v e l o c i ty c o e f f c i e n t w a s 0 . 9 5 2 .T h e t r a n s i t i o n f r o m s t r a i g h t t o c u r v e d f l o w a p p e a r s t o h a v e t a k e n p l a c e s a t i s f a c t o r i l y a n d t h ec o m p r e s s i o n w a v e f o c u s i n g f r o m t h e b l a d e le a d i n g e d g e h a s b e e n a v o id e d . T h e r e i s a sl ig h t b u b b l eo f s ep a r a t i o n w h i c h o c c u r s w h e r e t h e l e a d i n g - e d g e s h o c k i s r e fl e c t e d f ro m t h e c o n v e x s u rf a c e, b u t ,f r o m e x t e n s iv e t e s ts i n j e c ti n g l i q u i d ( m e t h y l a t e d s p i r i ts d y e d w i t h e n g i n e e r s b l u e ) t h e f l o w a p p e a r st o r e a tt a c h w i t h i n a v e r y s h o r t d i s t a n c e d o w n s t r e a m . T h e s t e a d i n e s s in th e g r o w t h o f t h e b o u n d a r yl a y e r s u g g e s t s t h a t t h e p r e s s u r e d i s t r i b u t i o n i s s m o o t h , a n d t h e f l o w a p p e a r s t o c o n f i r m t h e d e s i g np r in c ip l e s d e v e lo p e d i n S e c t i o n 4 . 0 .

A s b e f o r e th e m a i n l o s se s ar e i n t h e l o w - e n e r g y r e g i o n w h i c h l ie s b e t w e e n t h e c o n v e x s u r f ac e a n dt h e v o r t e x s h e e t . T h e s e l o s se s ap p e a r t o r e s u l t f r o m t h e b o u n d a r y l a y e r o n t h e c o n v e x s u r f a c e, f r o mt h e s e c o n d a r y f l o w i n t o t h e b o u n d a r y l a y e r f r o m t h e e n d w a l l s o f t h e p a s s ag e , a n d f r o m t h e a d v e r s ep r e s s u r e g r a d i e n t w h i c h a c t s o n t h i s s e c o n d a r y f l o w a n d b o u n d a r y l a y e r n e a r t h e p a s s a g e e x i t .T h e s e c o n d a r y f l o w t e n d s t o r e m o v e t h e l o w - e n e r g y a i r f r o m n e a r t h e e n d w a l l s , s w e e p i n g i t i nt o w a r d s m i d - s p a n , a s i n d i c a t e d b y F i g s . 1 9 a n d 2 0 . T h e t h i c k n e s s o f t h e r e g i o n o f l o w - e n e r g y f l o wa d j a c e n t t o t h e c o n v e x s u r f a c e i s r a t h e r o v e r 2 0 p e r c e n t o f t h e p a s s a g e w i d t h a c c o r d i n g t o t h et r a v er s e s, a n d r a t h e r u n d e r 2 0 p e r c e n t a c c o r d i n g t o t h e S c h l i e re n p h o t o g r a p h s . ( C a r e f u lc o m p a r i s o n w i t h t h e l i m i t e d p h o t o g r a p h s f o r t h e t h i r d d e s i g n s h o w e d th a t a n a p p r e c i a b l e i m p r o v e -m e n t h a d b e e n o b t a i n e d . )

T e s t s o n a n e a r l ie r b u i l d o f t h e f o u r t h d e s i g n h a d g i v e n s l i g h tl y h i g h e r l o s s e s t h a n i n t h e m a i nt e s t , a s w i l l b e d i s c u s s e d i n S e c t i o n 8 . 0 , b u t t h e r e s u l t s f r o m t h e m a i n t e s t w e r e c o n f i r m e d t ow i th in 0 - 0 0 2 o n v e lo c i t y c o e f fi c i e n t b y l a t e r r e p e a t t r a v e r s e s m a d e a t t h e 5 0 p e r c e n t a n d 2 0 p e r c e n ts p a n p o s i t i o n s .

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S u b s e q u e n t t o th e m a i n t es t a n a t te m p t w a s m a d e t o e n s u r e t h a t t h e b o u n d a r y l a y e r a p p r o a c h i n gt h e s e p a r a t i o n b u b b l e w a s t u r b u l e n t . A p ie c e o f t a p e a p p r o x i m a t e l y 0 . 0 0 3 i n . t h i c k a n d 0 . 5 i n. w i d ec h o r d w i s e w a s s t u c k s p a n w i s e a c r o s s t h e i n n e r b l o c k w i t h i ts t ra i li n g e d g e 0 . 0 5 i n . u p s t r e a m o ft h e s e p a r a t i o n p o s i t i o n . E x c e p t f o r a fa i n t M a c h l in e f r o m t h e l e a d in g e d g e o f t h e t a p e t h e f l o w i n t h et u n n e l r e m a i n e d e ss e n t ia l l y t h e s a m e , t h e f l o w re v e r s a l n o w e x t e n d i n g u p t o t h e t a p e , b u t n o t u po n t o i t. T h e e x p e r i m e n t w a s r e p e a t e d u s i n g a p i e c e o f t a p e 0 . 0 1 0 i n . th i c k , l e s t t h e p r e v i o u s t a p eh a d b e e n o f i n s u f fi c ie n t th i c k n e s s t o e n s u r e t r a n s it i o n , a n d t h e f l o w i n t h e p a s s a g e w a s s t i ll f o u n d t ob e e s s e n ti a l ly u n a l t e r e d . T h e r e t h e n r e m a i n e d t h e p o s s i b i l i t y t h a t t h e r e v e r s e d f l o w w i t h t h e t a p ei n p o s i t i o n m i g h t b e c a u s e d b y t h e s t e p - d o w n a t t h e tr a i li n g e d g e o f t h e t a p e . T o e l im i n a t e t h i sp o s s i b i l i t y t h e t r a i l in g - e d g e s e c t i o n o f t h e 0 . 0 1 0 i n. t a p e w a s c a r e fu l l y fa i r e d - in t o t h e c o n v e xsur f a c e w i th P l a s t i c ine . Sc h l i e r e n p i c tu r e s e.g., F i g . 2 1 g ) t h e n s h o w e d t h e m a i n s t r e a m f l o w in th eb l a d e p a s s a g e t o b e a l m o s t u n a l t e r e d , b u t l i q u i d - i n j e c t i o n t e s t s i n d i c a t e d t h a t t h e s m a l l p o c k e t o fr e v e r s e fl o w n e a r t h e l e a d i n g - e d g e s h o c k r e f l e c ti o n h a d d i s a p p e a r e d a n d t h a t a m u c h l a rg e r p o c k e th a d a p p e a r e d a t 6 0 p e r c e n t c h o r d . T h u s a v e r y s m a l l c h a n g e i n p a s s a g e g e o m e t r y h a d c a u s e d ac h a n g e i n t h e d i s t r i b u t i o n o f r e v e r s e d f l o w w i t h i n t h e l o s s r e g io n , b u t n o c h a n g e i n t h e m a i n s t r e a mf l o w c o u l d b e d e t e c t e d f r o m t h e S c h l i e r e n p i c tu r e s . N o f u r t h e r t e s t s w e r e m a d e t o i n v e s ti g a t e t h i sp h e n o m e n o n .

T u r b i n e t e s t s 1 o f t h e f o u r t h d e s i g n s h o w e d n o r a d i c a l i m p r o v e m e n t o v e r t h o s e o f t h e f i r s t d e s ig n .I t is n o w t h o u g h t t h a t t h r e e -d i m e n s i o n a l s e p a ra t io n s m a y h a v e d o m i n a t e d t h e f l o w s o b t a i n e d i nt h e t u r b i n e ; t h e f u t u r e a p p l i c a ti o n o f th e d e s i g n i s d i s c u s s e d f r o m t h i s v i e w p o i n t i n t h e n e x t s e c t io n ,t o g e t h e r w i t h p o s s i b l e d e v e l o p m e n t s o f t h e d e s i g n .

5.1. Deve lo pmen t a n d Fu tu re Ap p l i ca t io n o f t h e , Fo z l rth Des ig n . F o r t h e f l o w i n t h e f i n a l d e s i g np o s i t i o n t h e p r e s s u r e g r a d i e n t s a p p e a r t o b e c r i t ic a l o v e r a w i d e a r ea , a s s h o w n b y t h e s m a l l s e p a r a ti o nb u b b l e , b y t h e r a p i d g r o w t h i n th i c k n e ss o f t h e b o u n d a r y l a y e r o n t h e r e a r h a l f o f t h e c o n v e x w a ll ,a n d b y t h e e x t r e m e s e n s it iv i ty o f t h e b o u n d a r y - l a y e r f l o w t o t h e s m a l l c h an g e i n g e o m e t r y p r o d u c e di n t h e t e s t w i t h t h e f a i r e d - i n t a p e . B e c a u s e t h e p r e s s u r e g r a d i e n t s a r e c ri ti c al t h e c u r v a t u r e r a t iow/r,~, a n d h e n c e t h e p i t c h / c h o r d r a ti o a n d t h e b l a d e l o a d in g , a r e p r o b a b l y c lo s e t o o p t i m u m f o r t h i st y p e o f d e s ig n . R e d u c t i o n o f t h e c u r v a t u r e r a t i o m i g h t p e r h a p s i m p r o v e t h e e f f i c ie n c y b y r e d u c i n gt h e t e n d e n c y t o s e p a r a ti o n . I n t h e s e t e s t s t h e b l a d e p a s s a g e w a s o p e r a t i n g a t a s l ig h t l y n e g a t iv e r e a c t i o na s a r e s u l t o f t h e g r o w t h o f t h e b o u n d a r y l a y er . T h e i n t r o d u c t i o n o f a n e x p a n s i o n , o r p o s i ti v e re a c t io n ,i n t o t h e b l a d e p a s s a g e w o u l d s u p e r p o s e a f a v o u r a b l e p r e s s u r e g r a d i e n t , a s w o u l d , t o s o m e e x t e n t,t h e f l a re in a t u r b i n e . T h i s c h a n g e i n p r e s s u r e g r a d i e n t m i g h t b e e x p e c t e d t o r e d u c e t h e r a t e o fb o u n d a r y - l a y e r g r o w t h o n t h e r e ar h a l f o f t h e c o n v e x s u r f ac e a n d i t m i g h t h e l p t o p r e v e n t t h es e p a r a t i o n b u b b l e . B o t h o f t h e s e e f f ec t s w o u l d c o n t r i b u t e t o i m p r o v i n g t h e e f f ic ie n c y . F o r o b t a i n i n gs ti ll h i g h e r e f fi c ie n c ie s i t m i g h t b e p o s s i b l e to u s e a m o r e c o m p l e x d e s i g n i n w h i c h t h e c o n t r a c t i o ns t a rt e d u p s t r e a m o f t h e e n t r y t o t h e p a s sa g e p r o p e r, a t a p o i n t j u s t i n s i d e t h e u p s t r e a m M a c h c o n ef r o m t h e l e a d i n g e d g e o f t h e c o n c a v e s u r f ac e .

A c ri t ic a l e x a m i n a t i o n o f F ig . 1 8 r e v e a ls a s m a l l d i s c r e p a n c y f r o m t h e d e s i g n a s s u m p t i o n s o fS e c t i o n 4 .0 . I n F i g . 1 8 t h e d e s i g n a t t h i s b l a d e p i t c h i n g j u s t s a t i s fi e s E q u a t i o n ( 4) in t h e e n t r yr e g io n a n d y e t t h e c o m p r e s s i o n w a v e i s in c i d e n t o n t h e c o n v e x s u r f a c e at a n e a r ly p o s i t i o n ~ i n s t e a do f b e i n g t a n g e n t i a l t o i t a s i n F i g . 1 3 b . T h e e x p l a n a t i o n i s t h a t t h e w a v e f ro m t h e l e a d i n g e d g e h a sb e e n d i s t o rt e d v e r y l o c al ly j u s t a t t h e l e a d i n g e d g e , p e r h a p s b y g r o w t h o f t h e b o u n d a r y l a y er o nt h e c o n c a v e s u r fa c e , th e l e a d i n g e d g e b e i n g s h a r p i n t h e s e n s e o f f e e li n g s h a rp t o t h e t o u c h ; a ls o ,

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in the inner half of the pass age the wave has been curved under the influence of the expansionwaves from the convex surface. As a result the mean angle of the wave is some 4 to 5 deg larger thanthe Mach angle. The change is sufficient almost to halve the distance from the passage entry atwhich the wave is incident on the convex surface; the passage--satisfying Equation (4 )--then has asatisfactory entry flow. However, t he distortion of the bow wave in the vicinity of the leading edgedoes not seem to be a reliable phenomenon--for instance, it does not occur in Fig. 14. Thus, evenfor tunnel testing, where conditions are known fairly accurately, it might be preferable to use asomewhat smaller curvature than given by Equation (4) such that a bow wave would be incidentat a suitable position on the convex surface as in Fig. 18, even when not distorted in the vicinity ofthe leading edge.

In a turbine the annulus boundary layer might be thicker than in the present cascade--dependentupon the chord of the nozzles--while the Mach number would not be known to such a high degreeof accuracy and would vary along the blade span, as would the degree of reaction. Moreover,the secondary flow would be increased by the three-dimensional pressure fields. In order to allowfor these more difficult conditions, and to allow for the effect discussed in the preceding paragraph,it would be advisable to reduce the pitching of the blades and the values of the curvature, w/rmprobably by about 20 to 30 per cent, provided manufacturing considerations would so permit.This would give the Mach lines a stronger intersection with the convex surface and would reducethe adverse pressure gradient on the rear of that surface. Thus for a turbine rotor operating at arelative inlet Mach number of 1.9, more practical values of w/r~ ~would probably be about 0. 125for the inlet and outlet transition curves, and 0.25 for the central section. The degree of turningaccomplished in the transition curves would be reduced accordingly. With the passage curvaturesreduced 30 per cent the velocity coefficient in the t unnel wou ld still be expected to be 0.937 orhigher, the value 0.937 being obtained by scaling the losses of the present tests according to theincreased wetted area. If the reduced curvatures ensured a controlled flow in the turbine, and if avelocity coefficient in the region of 0- 93 could thereby be achieved, the resulting performance woul dbe very attractive.

As indicated in the previous paragraph the annulus boundary layer may be thicker in a turbinethan in the tunnel. Not only does this boundary layer represent a direct loss in the nozzles, it causessecondary flow in the rotor passages and tends to prevent the pressure rise required on t he rearwardconvex surface. Thus it would be desirable to use the maximum possible number of nozzle blades,consistent with strength and other design requirements, in order to reduce the chord of the nozzles,and hence the thickness of the annulus boundary layer, to a mini mum. T he larger number of bladesmight also reduce the nozzle secondary flow--as discussed in Ref. 1.

It is to be not ed th at the small passage curvatures recommended for the turbine rotor lead to veryclosely pitched blades and hence, probably, to a difficult manufacturing problem, except where theturning angle is very large.

A comparison of the four blade shapes tested in the present experiment, together with t he passageflow pattern attributed to each, is shown in Fig. 22.

6.0. Comparison with Previous Work. The early work on impulse blading for steam turbines, asreported for example by Stodola3 and Kearton~, gave velocity coefficients based on momentummeasurements ranging between 0.65 and 0-92. The lower coefficients may be attribut ed to thevery crude shape of some of the blades--such as those made from sheet metal--while the higher

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va lue s c or r e l a t e sa t i s f a c to r i l y w i th t he r e su l t s f o r t he f i r s t b l a de d e s ign o f t he p r e se n t se r i e s 1, t ha td e s i g n h a v i n g b e e n d e r i v e d f r o m w h a t a p p e a r e d t o b e t h e b e s t s t e a m p r a c t i c e . T h e r e s u l t s f o r t h a td e s ig n w h e n r e c al c u la t e d a s a m o m e n t u m m e a n , i . e . a m a s s m e a n o f v e lo c i t y , g a v e v e l o c i t yc o e f fi c ie n t s o f 0 . 9 4 a n d 0 . 9 2 f o r t h e t u r b i n e a n d c a s c a d e c o n f i g u r a ti o n s r e s p e c ti v e l y . I n t h ec a l c u l a ti o n o f t h e m o m e n t u m m e a n , t h e t r u e m a s s f l o w is t a k e n i n a n y s tr e a m t u b e , b u t t h e v e l o c i tya d o p t e d i s t h a t w h i c h w o u l d r e s u l t f r o m t h e m e a s u r e d t o t a l p r e s s u r e a n d t h e i s e n t r o p i c s t a t ic p r e s s u r e .

R e c e n t i n v e st ig a t io n s h a v e b e e n m a d e o n i m p u l s e b l a d in g a t th e N a t i o n a l A d v i s o r y C o m m i t te ef o r A e r o n a u t i c s 5 ,7 ,s , w h i l e s o m e a d d i t i o n a l u n p u b l i s h e d w o r k i n G r e a t B r i t a in h a s b e e n m a d ea v a i la b l e t o t h e a u t h o r s . I t a p p e a r s , a l s o , t h a t a c o n s i d e r a b le v o l u m e o f w o r k h a s b e e n c a r r i e d o u t i nR u s s i a 9, b u t f e w d e t a il s a r e a v a i la b l e. O f t h e A m e r i c a n a n d B r i t i sh t e s t s m o s t s e e m t o i n d i c a t e f a i rl yh i g h l o s s e s . T h o s e b l a d e d e s i g n s w h i c h g a v e re l a t iv e l y lo w l o s s e s a p p e a r t o h a v e b e e n t e s t e d a t a b o u tt h e m a x i m u m p o s s i b le c o n t ra c t io n c o n s i st e n t w i t h t h e e s t a b l i sh m e n t o f s u p e r s o n ic f l o w , a n d w o u l dt h e r e f o r e h a v e n o m a r g i n i n h a n d o n c o n t r a c t i o n t o c o v e r m a n u f a c t u r i n g a n d d e s i g n t o l e r a n c e s . A ne x a m p l e o f a l o w - l o s s b l a d e i s L i c c in i s t h i r d d e s i g n 7 ,s t e s t e d a t M a c h n u m b e r s o f 1 . 7 1 a n d 1 . 8 1 .L i c c i n i s r e s u l t s a r e q u o t e d a s a m a s s m e a n o f t o t a l p r e s s u r e , b u t w h e n r e c a l c u l a t e d g i v e v e l o c i t yc o e f fi c ie n t s b a s e d o n t h e a r e a - m e a n t o t a l p r e s s u r e s o f 0 . 9 7 6 , 0 . 9 83 a n d 0 . 9 7 0 , t h e s e t h r e e v a l u e s b e i n gr e s p e c t i v e l y f o r th e t r a v e r s e s u p s t r e a m o f t h e t r a il i n g e d g e in R e f s . 7 a n d 8 , a n d d o w n s t r e a m o f t h et r a il i n g e d g e i n R e f . 8; t h e m e a n v a l u e is t h u s 0 . 9 7 6 . N o w d i r e c t c o m p a r i s o n i s n o t p o s s i b l e w i t h t h ep r e s e n t t e s t a s s e v e r a l c o n d i t i o n s a r e d i f f e r e n t. F i r s tl y , t h e t u r n i n g a n g l e is 9 0 d e g : i f it w e r e a s s u m e da r b i t ra r i ly t h a t l o s s es w e r e d i r e c t l y p r o p o r t i o n a l t o t u r n i n g a n g l e a c o e f f ic i e n t o f 0 . 9 7 6 a t 9 0 d e gt u r n i n g w o u l d b e e q u i v a le n t to 0 . 9 6 3 a t 14 d e g tu r n in g . S e c o n d l y, t h e R e y n o l d s n u m b e r s i nL i c c i n i s t e s t s w e r e b e t w e e n a b o u t f o u r a n d s e v e n t i m e g h i g h e r t h a n i n t h e p r e s e n t t e s ts . I f t h e l o s se sw e r e p r o p o r t i o n a l t o R -115 t h e v a l u e 0 - 9 63 w o u l d b e e q u i v a l e n t t o 0 - 9 5 0 a t t h e R e y n o l d s n u m b e r o f-t h e p r e s e n t t e s t s . T h i r d l y , L i c c i n i s b l a d e s i n c o r p o r a t e s o m e p o s i t iv e re a c t io n , t h e r e b e i n g a 2 0 p e rc e n t a r e a e x p a n s i o n f r o m e n t r y t o e x i t a n d , c o r r e s p o n d i n g l y , a f a v o u r a b l e s t a t i c - p r e s s u r e r a t i o o fa b o u t 2 - 0 / 1 b e t w e e n t h e e n t r y a n d t h e t r a v e r s e s t a ti o n i n t h e f i r s t t e s t , a n d 1 . 3 / 1 i n t h e s e c o n d t e s t .I n t h e p r e s e n t t e s t s t h e r e w a s a n o v e r a ll a d v e r s e p r e s s u r e g r a d i e n t , o r n e g a t iv e r e a c ti o n , r e p r e s e n t e db y a s t a t i c - p r e s s u r e r a ti o b e t w e e n t h e t ra v e r s e s t a ti o n a n d t h e e n t r y o f 1 . 4 5 / 1 . F o u r t h l y , a s m e n t i o n e da b o ve , L i c c i n i s d e s i g n a p p e a r s t o h a v e b e e n t e s t e d a t t h e m a x i m u m p o s s i b l e c o n t r a c ti d n , w h i l e i no r d e r t o b e s u i t a b le f o r u se i n a t u r b i n e a m a r g in w o u l d b e n e e d e d t o a l l o w f o r m a n u f a c tu r i n g a n dd e s i g n t o l e r a n c e s . C o n s e q u e n t l y a s m a l l e r c o n t r a c t i o n w o u l d h a v e t o b e u s e d i n a t u r b i n e a n d t h i sw o u l d b e e x p e c t e d t o i n c r e as e t h e l o ss e s . T h u s , w h e n a c c o u n t is t a k e n o n l y o f t h e p r e c e d i n g f a c t o r st h e p r e s e n t d e s i g n c o m p a r e s f a v o u r a b l y w i t h L i c c i n i s . T h i s c o n c l u s io n , h o w e v e r , n e e d s q u a l i f ic a t io n ,a s i n t h e p r e s e n t c a s c a d e t h e t u n n e l n o z z l e h a s b e e n m a d e v e r y s h o r t i n o r d e r t o re d u c e t h e b o u n d a r y -l a ye r t h i c kne sse s a t t he c a sc a de e n t r y , w he r e a s L ic c in i s noz z l e s a r e t ong r e l a t i ve t o h i s b l a de spa n , a ndt h e r e s u lt in g t h i c k b o u n d a r y l a y er s o n t h e e n d w a l ls w o u l d b e e x p e c t e d t o c a u s e s e v er e s e co n d a r y f l o wi n th e r o t o r b l a d e s u n d e r t e s t s , t h u s m a k i n g t h e m a p p e a r a t a d i s a d v a n t a g e . T h e m o s t th a t c a n b e s a i df r o m c o m p a r i s o n o f t h e e x p e r i m e n t a l r e s u lt s i s t h a t t h e t w o t y p e s o f d es i g n w o u l d b e e x p e c t e d t op r o d u c e t h e s a m e o r d e r o f e ff ic i e n cy u n d e r s i m i la r c o n d i t i o n s .

T h e d e s i g n m e t h o d s o f t h e N . A . C . A . u t i l iz e t h e t h e o r y o f ch a r a c te r i st i cs , t h u s o b t a i n i n g t h e e x a c ti n v i s c i d - f l o w p a t t e r n f o r a n y p r e s c r i b e d c o n d i t i o n s . H o w e v e r , s e v e r a l d i f f i c u l t i e s a r i s e , i n c l u d i n gt h a t o f d e c i d i n g t h e b o u n d a r y - l a y e r b e h a v i o u r , e s p e c ia l ly i n t h e p r e s e n c e o f s e c o n d a r y f l o w , a n d t h a to f k n o w i n g w h a t c o n d i t i o n s o f p r e s s u r e d i s t r i b u t i o n e tc . i ni ti a ll y to p r e s c r i b e . M o r e o v e r t h es i g n i fi c a n t f e a t u r e s o f t h e d e s i g n a r e l e ss o b v i o u s , a n d t h e r e f o r e r a t h e r m o r e d i ff ic u l t t o d i s c u s s

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when considering, for example, the modifications which would be appropriate in appl ying a tunneldesign to an engine. In the present investigation the approach has been to regard the wind t unnel andSchlieren photographs rather as a computing machine, both for the characteristics network and forthe boundary layer. The choice of design submitted to this computing machine has been decidedupon on general principles-including the broad theory of the characteristics pattern--and theemphasis in the theoretical design work has been on developing these general principles.

The present passage consists of a circular arc approached by a straigh t line and a transit ion circulararc, the outlet nominally being symmetrical with the inlet. The simplicity of this geometry wouldfacilitate the methodical testing of design modifications either for other Mach numbers or forvarying amounts of reaction, and makes it unlikely to suffer as much as would a more complexgeometry in the course of being transformed into a practical blade. It has been an aim in the designto prevent any of the fans of compression waves from even coming close to focusing, because,although it is possible, ideally, to 'cancel' compression waves at a surface by correspondingexpansions, the cancellation of a sharp compression fan would finally rely on the characteristicsnetwork being accurately reproduced in the turbine. In the present design, the increase of pressurethrough the fans is kept fairly gradual, in order that the design may not be over-sensitive tovariations of manufacture or inlet Mach number.

Both the design method of Ref. 5 and that for the present blade utilize a transition curve leadingto a free-vortex flow, Ref. 5 using a transition shape obta ined in detail from a characteristics networkwhile the present method uses a transition circular arc. Both methods could lead to the same designfinally if the engine designer.chose to approximate the detailed transition shape by a transitioncircular arc- -as has in fact been carried out in some of the unpublished British work.

7.0. Contraction on the First and Fourth Designs Several tests were carried out with the firstblade design in the cascade configurat ion applying contraction to the inner blade passage as describedin Section 4.1.

Tests with 5 per cent and 10 per cent contraction showed that the cascade remained supersonicwith no major alteration in either the shock pattern or flow separation. At 15 per cent contractionhowever, the t unnel had ceased to be fully supersonic; an almost normal shock wave could not bedrawn beyond a position slightly upstream of the bt ade leading edge. At 12} per cent contraction,an intermediate type of flow pattern was observed, the flow becoming almost fully supersonic.Thus it is deduced that the critical contraction is between 10 per cent and 12 per cent. Pressuretraverses carried out at 5 per cent and 10 per cent cont raction at 50 per cent span gave an insignificantdifference in velocity coefficient from the value for zero contraction. It is therefore concluded thatcontraction is not necessarily an important parameter as regards losses in a passage where the flowremains separated.

Tests with contraction in the outer blade passage showed that the flow was subsonic with 6 percent contraction and supersonic with 2 per cent contraction. In the top Passage there is a 'bu ilt-in'contraction of approximately 7 per cent due to the chamfer at the blade leading edge, so that ineffect the passage was choking with between 9 per cent and 13 per cent contraction, which isconsistent with the results for the inner passage. It is to be noticed that in the tunnel the thickness ofthe leading edge causes a contraction, whereas in the turbine it causes positive incidence and anexpansion1, 9. Thus, provided incidence is catered for in the design, a turbine should not be quite assusceptible to choking as the outer passage of the present tunnel.

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For the final design, Fig. 23 shows a graph of velocity coefficient against percentage contractionfor the inner blade passage, together with an indication of whether fully expanded supersonic flowcould be established; the percentage contractions shown are values taken from inspection figures.Most of the traverses were carried out on the earlier build mentioned in Section 5.0 and the velocitycoefficients may be about 0.01 low.

With 11.8 per cent contraction it appeared from the Schlieren photographs, with confirmationfrom the traverses, that the flow separated, but rejoined the convex surface near the passage exit.On contracting the passage a further 0.7 per cent, i . e . to 12.5 per cent contraction, the flowfluctuated between subsonic and transonic. Thus the maximum permissible contraction to avoidstarting and choking difficulties for this design at an inlet Mach number of 1.9 appears to beapproximately 12 per cent, which is consistent with the values obtained from the first design, andwhich is the value quoted in Section 4.1. Because of the long bubble of separation at the 11.8 percent position and because that was very close to the critical contraction, the design position adoptedwas at 8 per cent contraction, even though the measured losses were lower at the 11.8 per centposition. The nearest test position to the nominal design of zero contraction was at 1 per centexpansion, when the flow separated from the convex surface near the foot of the leading-edge shock.

Expanding the inner passage either 16 per cent or 10.4 per cent caused choking, which mayhere be attributed to the contraction in the outer passage. Traverses at 50 per cent and 20 per centspan at 16 per cent expansion yielded high velocity coefficients, probably due to the loss through thenormal shock at entry be ng more than compensated by the reduced skin friction of the transonic orsubsonic flow. Such a flow is not of interest in a turbine because the shock at entry would reduce thenozzle outlet velocity.

8.0. E x a m p l e s o f F l o w S e n s i t iv i t y . In most of the designs the flows were found to be highlysensitive to small changes in the test conditions. In particular it was so sensitive to small changes ingeometry that it became necessary to specify very closely several dimensions on which the buildshould be based and to take inspection figures for each build. These precautions appearedto eliminate the small geometrical discrepancies which would ordinarily be expected whenthe structural materials are wood and Perspex. It was also found necessary to seal all thejoints very carefully, as well as the bolts and locating pins, in order to prevent leakage into thetunnel.

As an example of an early experience of the sensitivity it was found that on retesting the seconddesign in a new build the first compression wave, i . e . the second line from the concave surface inFig. 14, see Section 4.0, no longer intersected the convex surface and the flow separated rather asin the first design. It appeared that the blade pitch was very slightly greater tha n in the first build.

As a further example of the flow sensitivity it was found that in some instances, where thegeometry was very close to the critical contraction, a change from dry air to atmospheric damp airwas sufficient to cause the flow in the passages to change from supersonic to a type of flow referredto as 'transonic'. In this state the shock wave emanating from the blade leading edge appeared tobecome transonic, i . e . its angle became much greater than usual and the flow downstream--asgiven by shock-wave charts--became subsonic. The phenomenon is attributed to choking fartherdownstream, this in turn resulting from the reduction in the nozzle Mach number wh en using dampair. Depending on the degree of the throttling effect from downstream th6 transonic shock wassometimes detached from the leading edge of the blade.

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Figs. 21c and d provide a comparison of the flows obtained with wet and dry inlet air for thefourt h blade in its adopted design position, the wet air being taken direct from the test cubicle on aday when the atmospheric humidity was 0.55 x 10 3 lb/lb. From the two photographs it can beseen that the Mach angle of the leading-edge shock wave in t he i nner passage is slightly larger whenusing wet air, so that the Mach number is slightly lower. As a result the wave is incident on theconvex surface rather earlier than with dry air, and the region of low-energy flow using wet air alsoappears slightly thinner than with the dry.

A photograph is shown in Fig. 21e of the flow when the tunnel control valve in the ductingdownstream was partially closed, so that insufficient pressure ratio was being applied to the cascade.Ordinarily, on increasing the pressure ratio across the tunnel the flow pattern returned to thatseen in Figs. 18 and'21d. However, du ring the tests on the fourth blade at 8 per cent contraction asecond type of flow was obtained having apparently a more severe separation than in the standardflow of Fig. 18, as shown in Fig. 21f. In this f low the region between the convex wall and the vortexsheet in the Schlieren photographs occupied, approximately 33 per cent of the passage widt h atmid-chord, but decreased in width to approximately 23 per cent at the plane of the trailing edge.Even when the pressure ratio across the tunnel was increased above the usual value there was nodiscernible improvement in the flow pattern within the passage. No change had been made in thepassage geometry since the previous tests when the usual flow (i.e., as shown in Fig. 18) had beenobtained; also, the flow in the outer passage appeared to be the same as usual. However, on the dayof this particular test the second type of flow would sometimes be obtained on first opening thecontrol valve, while, on waiting, the flow would sometimes change to the usual flow patte rn-- for noapparent reason. Attempts to obtain the second type of flow on subsequent days were unsuccessful.Thu s a second type of flow could exist for the final blade design as tested in the tunnel, but it wasunpredictable and unstable.

As mentioned in Section 5.0, tests on an earlier build of the fourth design indicated slightly higherlosses than for the main tests, the velocity coefficient being 0.94. The discrepancy between t he twosets of tests might be associated with leakage into the tunnel in the earlier tests, with a slightdimensional change, or with an effect such as just discussed in connection with Fig. 21f.

9.0. L a p a n d T i p C l e a r a rw e . L a p is the term applied to a sudden increase in annulus depthsometimes inserted between two blade rows of an impulse turbine, the blades of the downstreamrow extending into the expansion as in Fig. 24. Examples of a design with lap are shown in Ref. 3and tests are reported in Ref. 1.

The use of lap can facilitate construction. Alternatively it may be defended, for a high-pressure-ratio turb ine stage, on the basis that upersonic flow prefers to travel in straigh t lines, rather thanalong the helix prescribed by a closed annulus' . T h e latter argument is equivalent to saying thatlap could el iminate the difficulties associated with radial pressure fields in supersonic flow. A fur the rpossible reason for using lap--again in a turbine containing transonic or supersonic flow--is to tryto ensure that the static pressure downstream of one blade row is transmitted through the deadair in the lap region to the next blade row upstream. This precaution would be aimed both atensuring that the upstream blade row operates at its design pressure ratio and at making thedownstream blade row less susceptible to choking.

Lap produces the feature which aerodynamicists usually endeavour to avoid, namely separationof the flow from the boundary surface, with subsequent mixing between the mainstream and thedead air. Lap would therefore be expected to reduce efficiency.

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W i t h i n t h e l i m i ta t i o n s o f t h e r ig , l a p w a s m o s t r e a d i ly i n c o r p o r a t e d b y i n s e rt i n g P e r s p e x b l o c k si n th e t u n n e l n o z z l e a t o n e o r b o t h e n d s o f t h e s p a n . I n o r d e r n o t t o i n t e rf e r e w i t h t h e n o z z l e f l o wt h e l i n e o f t h e l a p i n m o s t o f t h e t e s t s w a s m a d e p e r p e n d i c u l a r t o t h e c e n t r e - l i n e o f t h e n o z z l e,i . e . p e r p e n d i c u l a r t o t h e s u p e r s o n i c f l o w , a s s h o w n i n F i g . 2 5 a . I n a t u r b i n e , h o w e v e r , t h e l i n e o ft h e l a p w o u l d b e p e r i p h e r a l, a s i n F i g . 2 5 b , t h e c o m p o n e n t o f v e l o c i t y p e r p e n d i c u l a r t o t h is e d g ep r o b a b l y b e i n g s u b s o n i c . A l s o , i n t h e t u n n e l , t h e f l o w i s c o m p l e t e l y c o n t a i n e d b y t h e s i d e w a ll s ,t h e s e h a v i n g n o e q u i v a l e n t in t h e t u r b i n e . T h u s t h e r e s u l t s o f t h e p r e s e n t t e s t s m a y b e a p p l i e d o n l yt e n t a t i v e l y t o t u r b i n e s . I n o n e g r o u p o f t e s t s t h e o u t e r b l a d e p a s s a g e c o n t a in e d p e r i p h e r a l l a p, a sse e n i n F ig . 25c .

F o r t h e l a p te s t s t h e a n n u l a r - r i n g t y p e p r o b e w i t h t h e 0 . 0 0 2 in . s t e p a t th e a n n u l a r s lo t , d i s c u s s e di n R e f . 2 , w a s u s e d f o r t h e s t a t i c -p r e s s u r e t r a v e r s e s . T h e e r r o r r e s u l t i n g f r o m t h e u s e o f t h isi n s t r u m e n t s h o u l d b e s m a ll , h o w e v e r , i n t h e s e te s t s , a s t h e e x i t M a c h n u m b e r i s l o w , d u e t o h i g hl o s s e s , a n d t h e r e f o r e t h e c o r r e c t i o n f r o m p i t o t t o t o t a l p r e s s u r e i s n o t s e n s i t i v e t o t h e a c c u r a c y o ft h e s t a ti c p r e s s u r e .

T h e f i r st a n d f o u r t h b l a d e d e s i g n s w e r e t e s t e d w i t h v a r i o u s a m o u n t s o f l a p, t h e l o s se s b e i n gm e a s u r e d i n b o t h p a s s a g e s o f t h e t u n n e l a t a l i m i t e d n u m b e r o f s p a n w i s e p o s i t io n s , f o r v a r i o u sa m o u n t s o f c o n t r a c t io n . R e s u l t s a r e g iv e n in T a b l e 1 a n d F i g s . 2 6 t o 2 8 , c o m p a r i s o n b e i n g s h o w nb e t w e e n t h e t e s ts w i t h a n d w i t h o u t l a p .

F r o m T a b l e 1 i t w i l l b e s e e n t h a t t h e c r i t ic a l c o n t r a c ti o n , i . e . t h e c o n t r a c ti o n a t w h i c h t h e t u n n e lWi ll j us t s t a r t , i s i nc r e a se d by r o ugh ly 100 pe r c e n t o f t he t o t a l a r e a o f t he a pp l i e d l a p . T h e l os se s ,h o w e v e r , a r e a l so m u c h g r e a t e r w i t h l a p. I t c a n b e s e e n f r o m F i g . 2 6 t h a t, a s t h e l a p i s i n c r e a s e d ,p r o g r e s s i v e l y m o r e o f t h e l o w - e n e r g y a ir is s w e p t i n t o m i d - s p a n . S c h l i e r e n p h o t o g r a p h s s h o w e d t h a tt h e s m a l l es t a m o u n t o f l a p t e st e d , i . e . 1 0 p e r c e n t , w a s s u f f i c i e n t t o c a u s e a b r e a k d o w n i n t h e f l o wp a t t e r n f o r t h e f o u r t h d e s i g n a t 8 p e r c e n t c o n t r a c ti o n , t h e f l o w s e p a r a ti n g f r o m t h e c o n v e x s u r f a c em i d w a y b e t w e e n t h e p a s sa g e e n t ry a n d m i d - c h o r d .

S i n c e t h e f l o w c o n t a i n s l a rg e d e a d - a i r r e g i o n s i n t h e s e t e s ts t h e v e l o c i t y co e f fi c ie n t s s h o w n , w h i c ha r e b a s e d o n t h e a r e a - m e a n t o t a l p r e s s u r e , a r e p r o b a b l y p e s s im i s t ic . T h i s w o u l d e s p e c i a ll y b e s o f o ra p p l i c a t i o n t o p e r f o r m a n c e c a l c u l a t io n s f o r a s i n g l e -s t a g e t u r b i n e , i n w h i c h t h e r e w o u l d b e n od o w n s t r e a m b l a d e r o w w h e r e t h e m a l d i s t r ib u t i o n s c o u l d c a u s e l os s. H o w e v e r , t h e t r e n d s i n d i c a te db y t h e a r e a - m e a n c o e f fi c ie n t s w o u l d b e e x p e c t e d t o b e c o r r e c t, a s p a r t o f t h e l o s s e s c a u s e d b y t h em i x i n g w o u l d h a v e o c c u r r e d w i t h i n t h e t e s t p a s s a g e i t s e l f .

T h e r e s u lt s a n d c o n c l u si o n s c o n c e r n in g l a p m a y b e s u m m a r i s e d a s f o ll o w s :( i) L a p r e d u c e s t h e l i k e l i h o o d o f c h o k i n g . T h e i n c r e a se i n t h e a m o u n t o f a r e a c o n t r a c t i o n w h i c h

m a y b e i n c o r p o r a t e d i n t h e p a s s a g e b e f o r e c h o k i n g o c c u r s is a b o u t e q u a l t o t h e t o t a l a r e a o f l ap .( ii) L a p c a u s e s a n i n c r e a se o f l o ss o f b e t w e e n 0 a n d 1 0 0 p e r c e n t o f t h e l o s s w i t h o u t l ap .

( iii) T h e l o s s i n c r e m e n t s h o w s n o r e g u l a r b e h a v i o u r b u t , v e r y g e n er a ll y , i t i n c r e a se s w i t h i n c r e a seo f l ap . T h e i n c r e m e n t i s g r e a t er f o r a f l o w w h i c h d o e s n o t d i s p l a y s e p a r a t i o n w i t h o u t l a p ,a n d i n s u c h a fl o w e v e n l 0 p e r c e n t l a p c a n c a u s e a b r e a k d o w n o f th e f l o w p a t te r n .

( iv ) T h e m a i f l s tr e a m f l o w p a s s e s in t o t h e l a p r e g io n , w h i l e t h e d e a d a ir i s s w e p t a s s e c o n d a r yf l o w t o m i d - s p a n .

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theory and tunnel tests of rotor blades for supersonic turbines by b.s. stratford and g.e. sansome

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