Pressure measurements on flapped hydrofoils in cavity flows and wake flows

Pressure Measurements on Flopped Hydrofoils

in Cavity Flows and Wake Flows'

By M. C. Meijer^

The purpose o f the present experiments is to obtain a detailed information about the flow field, such as the pressure distribution, at the surface of a flapped hydrofoil in full cavity or wake flows. The model and the experimental procedure are described. The experimental results obtained have been used to compare with the theoretical predic-tions, to investigate the tunnel wall effect and to estimate the viscous effect at a sharp corner. An empirical method f o r correcting the tunnel wall efFect is developed here, the validity of which is supported b y tests with models o f three different sizes. An a p -preciable viscous efFect has been found near the hinge o f a deflected flap. Except for Ihis effect, the theory and experiments are found to be In good agreement.

Introduction

I n the classical theorj^ of free streamline flows and its application to cavity and wake phenomena, i t is well known t h a t even i n the limiting case of infinite cavities, the numerical calculation of the solution f o r arbitrary obstacles presents much difficulties, as has been ex-tensively discussed i n some recent survey literatures

' T h i s w o r k w a s c a r r i e d o u t u n d e r t h e B u r e a u o f S h i p s G e n e r a l H y d r o d y n a m i c s R e s e a r c h P r o g r a m , A d m i n i s t e r e d b y t h e D a v i d T a y l o r M o d e l B a s i n , O f f i c e o f N a v a l R e s e a r c h C o n t r a c t N o n r -2 -2 0 ( 5 -2 ) . , ^ S e n i o r R e s e a r c h E n g i n e e r , C a l i f o r n i a I n s t i t u t e o f T e c h n o l o g y , P a s a d e n a , C a l i f . , 1 9 6 2 - 1 9 6 4 . P r e s e n t a d d r e s s : S h i p b u i l d i n g L a b o r a t o r y o f t h e T e c h n o l o g i c a l U n i v e r s i t y o f D e l f t , N e t h e r l a n d s . i M a n u . s c r i p t r e c e i v e d a t S N A M E H e a d q u a r t e r s , F e b r u a r y 3 , 1 9 6 6 .

[1,2].5 For the general case of finite cavities, the prob-lem is further complicated by an additional parameter, namely the cavitation number. I n 1962 W u [3] intro-duced a simple wake model and developed a theory for plane wake and cavity flows past an inclined flat plate at an arbitrary cavitation number. This theory was subsequently extended by W u and Wang [4] to the general case of arbitrary body f o r m and arbitrary cavita-tion number. The original theory has been found i n good agreement w i t h the experimental results of Fage and Johansen [5] who measured the pressure distribution over a flat plate, inchned i n a separated flow i n a wind tunnel. The total force coefficients predicted b y this theory also compare satisfactorily w i t h several water

- N u m b e r s i n b r a c k e t s d e s i g n a t e R e f e r e n c e s a t e n d o f p a p e r . Nomenclature c = l e n g t h o f c h o r d Cp = p r e s s u r e c o e f f i c i e n t , g e n e r a l o r r e -s u l t i n g Cpo = p r e s s u r e c o e f f i c i e n t b a s e d o n t h e u p s t r e a m a v e r a g e p r e s s u r e C,,m = p r e s s u r e c o e f f i c i e n t b a s e d o n t h e r e l a t i v e m i n i m u m p r e s s u r e a t t h e w a l l , as o p p o s e d t o Cpa ƒ = flap l e n g t h /* = m a n o m e t e r r e a d i n g ( c o l u m n h e i g h t ) ho = m a n o m e t e r r e a d i n g f o r t h e u p -s t r e a m a v e r a g e p r e -s -s u r e h,n = m a n o m e t e r r e a d i n g f o r t h e r e l a t i v e m i n i m u m w a l l p r e s s u r e hr = m a n o m e t e r r e a d i n g f o r a n a r b i -t r a r y r e f e r e n c e p r e s s u r e hi; = m a n o m e t e r r e a d i n g f o r t h e c a v -i t y — o r w a k e p r e s s u r e H = t o t a l h e a d ( p r e s s u r e ) I = l e n g t h o f c a v i t y m e a s u r e d f r o m t h e l e a d i n g e d g e p = l o c a l p r e s s u r e Po = u p s t r e a m a v e r a g e s t a t i c p r e s s u r e Pm = r e l a t i v e m i n i m u m w a h p r e s s u r e Pr = a r b i t r a r y r e f e r e n c e p r e s s u r e Pj = c a v i t y — o r w a k e p r e s s u r e s = l e n g t h a l o n g c h o r d a n d flap, m e a s u r e d f r o m t h e l e a d i n g e d g e T = t u n n e l h e i g h t ( = 3 0 i n . ) V ^ r e f e r e n c e v e l o c i t y i n g e n e r a l Vo = u p s t r e a m a v e r a g e v e l o c i t y Vr = s o m e s p e c i f i e d r e f e r e n c e v e l o c i t y X = l e n g t h a l o n g t u n n e l w a l l , m e a s u r e d f r o m t h e c e n t e r o f t h e h y d r o f o i l m a i n b o d y , d o w n s t r e a m a = a n g l e o f i n c i d e n c e o f t h e p r e s s u r e s i d e o f t h e b a s i c h y d r o f o i l Ay = w e i g h t d e n s i t y o t m e r c u r y less w e i g h t d e n s i t y o f w a t e r e = f l a p d e f l e c t i o n a n g l e d i v i d e d b y t t ( f r o m W u ) p = m a s s d e n s i t y o f w a t e r c = c a v i t a t i o n n u m b e r , g e n e r a l o r r e -s u l t i n g CFo = c r i t i c a l ( o r c h o k i n g ) c a v i t a t i o n n u m b e r , b a s e d o n t u n n e l s t a t i c p r e s s u r e w i t h o u t m o d e l <r„ = c o n v e n t i o n a l c a v i t a t i o n n u m b e r b a s e d o n u p s t r e a m a v e r a g e c o n -d i t i o n s c r „ = c a v i t a t i o n n u m b e r b a s e d o n c o n d i -t i o n s a -t p o i n -t a l o n g w a l l w h e r e t h e p r e s s u r e i s a r e l a t i v e m i n i -m u -m 170 J O U R N A L O F SHIP R E S E A R C H

F i g . 1 S u c t i o n s i d e ( l e f t ) a n d i n t e r f a c e s i d e ( r i g h t ) o f h y d r o f o i l m a i n b o d y

F i g . 2 M o u n t e d h y d r o f o i l m o d e l w i t h c o m p l e t e s e t o f flaps. P r e s s u r e s i d e s a r e a l l f a c i n g r i g h t

tunnel experimental results. I n order to establish the theory f u l l y , i t is still important to check experimentally w i t h the detailed flow field, such as the pressure dis-tribution over the body surface, for some typical cases of arbitrarjr body form. Such comparisons may clarify the validity of the theoretical model and the simplifying assumptions of neglecting the effects of viscositj^ and gravity.

From the viewpoint of experimental activities, i t may be pointed out that i n most of the previous model tests of cavity flows i n water tunnels only the forces and

mo-ments were recorded, whereas detailed surveys of the pressure field over the body surface are indeed very scarce. Furthermore the present technique of using water tunnels for cavity fiow studies is also handicapped by a lack of a well established method for wall corrections so that the experimental errors can be precisely elim-inated. Such a method for the most general condition should be extremely valuable for the future experimental purposes.

I n view of the ever increasing scope of applications of cavity flow theory, such as supercavitating hydrofoil

SIMPLE WEDGE € 7 r = 0 ° t = 0 . 6 I f7T= 2 0 ° • 0 . 4 ~-0.2 €77-= 2 0 c = 0 . 2 ^ STT'tO ^ = 0 . 2 ^ e77= 6 0 EXPLANATION OF SYMBOLS c = 6 INCHES F i g . 3 O r i g i n a l m o d e l c o n f i g u r a t i o n p r o f i l e s HALF SIZE f / c = 0.2

eiT

watercraft, stalled wing performance i n V T O L opera-tions, supercavitating propellers and cascades, a thor-oughljr verified theory for ready use and a well defined experimental interpretation are both of vitmost impor-tance. Furthermore i n hydrofoil applications, adoption

F i g . 5 F i i l l y m o u n t e d h y d r o f o i l m o d e l w i t h flap, s e e n f r o m p r e s s u r e s i d e . T h e p r e s s u r e o p e n i n g s a r e n u m b e r e d a s s h o w n

of flaps or other load modulators as control devices is necessary and a good knowledge of supercavitating flaps is stifl much i n need. For these reasons i t has become the main purpose of this experimental program to choose the supercavitating hydrofoil w i t h flaps as a concrete case for investigating the validity of the theory and developing a wall correction method.

Preliminary reduction of the data obtained early i n the course of this investigation drew the attention of the author to the fact that the local pressure coefficients are more sensitive to the wah effect than the force coeffi-cients. Consequently an adequate method for correct-ing the wall effect was urgently needed for a meancorrect-ingful interpretation of the present data. B y using several geometrically similar models of three different sizes, a method is developed here to reduce the data consistently to a single set of results, nearly insensitive to the model size, thus providing a reliable correlation to the theo-retical case of unbounded fiows. I n other words, the present method is supported at least by this experiment for the different model sizes tested. A f t e r the data were so reduced, the theory is found to be i n good agreement w i t h the experiments except i n a small region near the flap hinge, where the influence of viscosity is revealed by these experiments.

The experiments were designed to include force and moment measurements but i n the course of the investiga-tion i t turned out that pressure and force measurements could not w i t h the available instruments be performed at the same time. Also because of the complication i n

0 . 8 O O ^ E X T R E M E L Y LOJ^G CAVITIES E X T R E M E L Y LONG CAVITIES ( / c = 0.2 s / c - 0.267 C / T STA. 0 . 2 0 8 0 . 1 5 6 0 . 1 0 4 THEORY L I N E S PASSING T H R O U G H T H E P 0 I N T 5 ( Cp^ = I, <r, = - n AND T H E T H E O R E T I C A L VALUE OF Cp FOR a-= 0

0.2 OA 0.6 F i g . S C o m p a r i s o n o f p r e s s u r e c o e f f i c i e n t s , o b t a i n e d f o r s a m e r e l a t i v e s t a t i o n w i t h t h r e e m o d e l s i z e s a n d f r o m t h e o r y . <t a n d Cp w e r e b a s e d o n m e a s u r e d m i n i m u m p r e s s u r e s a t t u n n e l b o t t o m 1.0 1.2 1.4 1.6 1.8 2.0 F i g . 7 T h e i n f l u e n c e o f s i z e o n p r e s s u r e c o e f f i c i e n t w h e n <t a n d Cp a r e b a s e d o n u p s t r e a m a v e r a g e p r e s s u r e f / c = 0 . 2 0 * T = 2 0 ' 0 lO" 2 0 ° 3 0 ° 6Cf 4 5 ° f / c = 0 . 4 0 \ ° ^ ^ > - ^ 5 ° « 6 0 ° 5 ° _lol 3 0 ^ - ^ 0 = 5 1^ 10° o 2 0 ° « 3 0 ° » 4 5 ° o 6 0 ° n = 14.85 2 9 . 8 5 ° •/ 4 9 . 8 5 ° d 6 9 8 5 ° o" — T H E O R Y F i g . 9 C u r v e s o f c a v i t y l e n g t h r a t i o s , u s e d f o r e s t i m a t e o f <r,„ i n t h o s e c a s e s w h e r e a d i n g s o f w a l l p r e s s u r e s w e r e a v a i l a b l e 1 7 4 J O U R N A L O F SHIP R E S E A R C H

' •10° c T COMPARED 1 f / c = 0 . 2 E T T = 4 0 ° a = 5 1 0 ° 2 0 ° 3 0 ° 4 5 ° 6 0 ° F i g . 1 0 I n f l u e n c e o f m o d e l s i z e o n u p s t r e a m a v e r a g e p r e s s u r e c o e f f i c i e n t , b a s e d o n m i n i m u m p r e s s u r e a t t u n n e l b o t t o m

volved i n the determination of the tare forces and the influence of the end-gap, i t was decided to omit the force measurement and to investigate only the pressure dis-tributions.

F r o m the overall result i t may be concluded that pri-mary success has been achieved for both of the two main purposes of this study.

I n order to facilitate a large number of models to be tested i n this experimental program as well as future studies of the optimum profile, a basic model support was designed and constructed for this general purpose. I t serves as the conmion main body to which a hydrofoil of different profile and flap deflection can be easily fastened, or interchanged, f o r testing. This model sup-port can be permanently mounted to span across the tunnel, w i t h its cylindrical base passing through a hole i n the tunnel wall. The ducts and tubes imbedded i n this model support, as shown i n Fig. 1, serve as passages leading f r o m the pressure holes i n the model face to the pressure tubes outside the tunnel.

The complete model as shown i n Fig. 2 consists of the main body of 6-in span and the removable parts, whose outer surface bears the required hydrofoil profile and whose inner surface was made entirely flush at the inter-face w i t h the main body. The pressure holes drilled across these removable parts lead directly to the passages i n the main body; the sealing of the pressure leads at the

F i g . 1 1 T h e i n f l u e n c e o f a,,, o n p r e s s u r e d i s t r i b u t i o n a l o n g t u n n e l b o t t o m 0.2 f/c = 0.2 677 = 4 0 ° a = ro° c / T RUN 0 0.20 827 O 0.15 70G • 0.10 6 5 0 1.2 -0.1 • 0 . 2 S T A T I O N F i g . 1 2 I n f l u e n c e o f m o d e l s i z e o n p r e s s u r e d i s t r i b u t i o n a l o n g t u n n e l b o t t o m

intersection was accomplished w i t h 0-rings, seated i n the main body (Fig. 1 at right).

The basic profile has a simple wedge outline w i t h a vertex angle of 9 deg and a length of the pressure side c = 6 i n which is also the chord length of the hydrofoil. The vertex angle of 9 deg was chosen for the material stifi'ness, though i t limited the smallest incidence for the the l i f t i n g flow to about 5 deg. I n the present program the removable parts have the simple configuration of flat plates, each covering 0.2 of the chord, except for the

0.2 0

(D

I ' 0 , 2 6 7 f-0.467 @ _1-0.500 ®

leading segment which has a length of 0.4 chord. This part tapers towards the leading edge.

Different configurations of the flap were achieved by replacing one or more of the rear plates by wedge shaped prisms whose pressure side has basically the same length as its base so that the flap length, denoted by ƒ, remains fixed at different flap deflections. The wedge angle of the prisms gives the desired flap angle er (see Figs. 3, 4). For measuring the pressure distribution along the pressure side, a row of }i2-m pressure holes were drilled along the center of the 6 i n span of the removable parts at a distance of }4o chord apart and w i t h two additional holes at j-io chord length f r o m the leading and trailing edges (Fig. 5). Furthermore additional pressure taps

were installed at the intersection of each consecutive pair of removable parts. This was done i n order to measure the pressure at the hingepoints and to find if the local pressure deviates f r o m the stagnation pressure

(/4pV'') as predicted by the inviscid flow theory. T o

prevent leakage through the interface, the two opposing surfaces were greased before they were mounted to-gether.

Because of the small tapering of the forward part of the model, the pressure holes i n this p"art could not just be drilled through the plate. Instead, tubes were laid in the plate, leading to a row of holes, distributed span-wise i n the main body, slightly to the rear of the leading

176

F i g . 'L4(a,b) E x a m p l e s o f p r e s s u r e d i s t r i b u t i o n c o r r e l a t i o n s

edge. The method of using tlie sealed breaks i n the pressure leads proved to be f u l l y satisfactoiy.

As only the f u l l y cavitating and f u l l y separated flows were dealt w i t h , the back side of the main body was kept simple. Fig. 1 shows the open recess and layout of the brass tubings and the openings for air supply (for forced ventilation when necessarj^ and for measuring the cavity pressure or the base underpressure i n the separated wake flow. This measurement of cavity pressure was per-formed by measuring the static pressure near the dis-charge end i n a 3/16-in wide tube, through which slow moving water was directed, keeping the probe wet. The discharge of water into the recess f r o m a relatively wide tube was supposed to eliminate interference f r o m capillaritJ^ B y using this sj^stem, the cavitj^ pressure could be treated i n exactly the same way as all the other pressures.

Facility and Experimental Setup

The experiments were performed i n the H i g h Speed Water Tunnel of the California Institute of Technology, using the new two-dimensional working section [6]. This tunnel is equipped w i t h a three-component balance for stationary flows to which the hj^drofoil model was fastened.

For measurement of pressure distribution, a 6-ft high, 21-tube multinianonieter, filled w i t h mercury and water was used. The reference pressure was taken f r o m a point upstream of the nozzle (see Section " D a t a Eeduc-t i o n " following). Two Eeduc-tubes i n Eeduc-the mulEeduc-tinianonieEeduc-ter were used for measuring the average static pressure i n the working section, upstream of the model, and the cavity pressure. Readings of the multimanometer and the profile of the cavity were recorded by two coupled "Recordak" cameras. A n example of this pair of re-cordings is shown i n Fig. 6. Separate measurements of the velocity head and tunnel static pressure were also made w i t h separate mercury manometers for the con-venience of conducting the experiment.

The average static pressure upstream of the hydrofoil was taken f r o m a manifold connected through resistors w i t h 15 taps distributed along the height of the working section, approximately one chord upstream of the leading edge of the hydrofoil. Fig. 6(a) shows the layout of the pressure taps i n the working section.

Experimental Procedure

Prior to each run, the angle of attack of the wetted surface of the model was set at a specified value. The manometer tubes were checked for air bubbles and bled if

F i g . 14(c,d) E x a m p l e s o f p r e s s u r e d i s t r i b u t i o n c o r r e l a t i o n s

air was supplied to the model suction side. The de-ficiency of further pressure control marked the end of each set of runs. Reynolds numbers between 0.6 X 10« and 1.5 X 10^ were obtained during the runs.

The first series of experiments was conducted w i t h pressure measurements f r o m 18 holes along the wetted side of the model, together w i t h the measurement f r o m the cavity pressure tap and the manifold for the tunnel static pressure The choice of the stations along the pressure side was such that i n the regions of large pressure gradient successive stations were used; these regions include those near the leading edge, trailing edge, and on both sides of, as well as within, the hinge-point. I n the two remaining regions, some pressure holes were skipped and not used.

A f t e r the first set of data was reduced on the basis of the measured upstream velocity (F») and the static pressure ip„), the resulting Cj, values (C„„) were found to be i n rather poor agreement w i t h the theory. I n this comparison no correction was made for the wall efi^ect.

Tunnel W a l l Effect

The wall correction f o r experiments w i t h thin bodies i n w i n d tunnels is fairly well established [7]. For the cavitating flow past a model i n a water tunnel, however,

J O U R N A L O F SHIP R E S E A R C H

necessary. Values of the pertinent flap-chord ratio, flap angle, angle of attack and other specifications of the r u n were indicated on the multimanometer. Each set of runs was started at a static pressure, measured by the separate manometer, or approximately 10 psi gage and was generally foUowed b y measurements at 5 and 0 psi gage. The velocity was set to give a maximum ma-nometer reading of 1.5 f t of the pressure i n the cavity^ or somewhat less i n case the flow showed noticeable un-steadiness; the latter measure was taken to provide an adequate safety margin for the model and its support. AU the photographic records, automatically numbered, were then taken after the mercury colunm heights had reached a sufficiently constant state.

For the succeeding measurements, air was supplied to the suction side of the model and the pressure level i n the tunnel lowered to give different values of the cavitation number. I n this connection i t is stressed that the cavi-ties i n the experiments have been both vapor and air fiUed. A f t e r a total of approximately nine runs w i t h graduaUy decreasing cavitation number, the pressure control became deficient due to the generation of a large air pocket i n the tunnel diffuser. This difficulty could be overcome to an extent by releasing water f r o m the tunnel into an evacuated vessel at the same time when

o.oe 0 , 6 0 f/c = 0.4 C 77- = 2 0' RUN 3 3 4 3 3 2 0 . 2 0 . 4 0 . 6 o ! 8 \ s

\

\

\

the situation is further complicated by the presence of a free boundary as well as a sohd boundary and by the fact that the body-cavity system is often not t h i n com-pared w i t h the channel w i d t h . I n the simple case of choked cavity flow (with the cavity extremely long i n a water tunnel) past a flat plate set normal to the stream, i t has been shown by Birkhoff, Plesset, and Simmons [8 ] that the wall correction for the drag coefflcient C D is small if i t is based on the velocity at the cavity boundary, but may be very large i f based on the upstream velocity. However, for the more general case of l i f t i n g flows w i t h a finite cavity, no definite formula or rules have been established for correcting the wall effect. I n this series of experiments, i t has been found that the local pressure coefficient is more sensitive to the wall effect than the force coefficients and therefore a reliable wall correction is necessary to present a meaningful result. Fortunately an empirical rule has been established here for a wall correction, which is supported b y observations w i t h geometrically similar models of three different sizes.

I n an earlier experiment Parkin [9] was able to obtain some good agreement for the force coefficient w i t h nearly the same theory (Wu, 1955) without maldng any corrections. The ratio of the model chord to the tunnel height (c/T) i n the present case differed only slightly

f r o m that i n Parkin's experiments. Therefore i t could be expected that again the influence of the walls should be small. B u t now poor agreement between the C'^ values necessitated an investigation of the wall effect.

As has been mentioned i n the Introduction, the cause of the difference i n the results is obvious if one considers that Cp is a function of the position along the chord where i t is measured and so is the error of Cp. A t the leading and trailing edges of the model and i n the forward stagnation point the error is zero as a consequence of the method of data reduction used. The force coefficient is equal to the line integral of the pressure coefficient along the profile boundarj^, which is exact i n the above mentioned points. The error of the force-coefficient is therefore always less than the maximum error occurring in the pressure coefficients. This maximum error is definitive for the comparison. The difference between the pertinent errors is increased w i t h increased cavitation number.

Although the value of a should reduce to zero as the cavity becomes very long, i t is well known t h a t i n a tunnel this is never reached i f velocity and pressure are related to the working section pressure i n undisturbed fiow. A t a certain critical CTC. > 0, the cavitation index cannot be reduced further; this is known as "choking"

O I/c . 0.2 €n = 20° a . 2 0 ' 0 0 3 0 . 0 2 8 0 2 1 0 9 7 0 8 4 0 . 7 8 7 9 6 1 0 9 0 F i g . I4(g,h) E x a m p l e s o f p r e s s u r e d i s t r i b u t i o n c o r r e l a t i o n s

or "blockage" of the tunnel. Contrary to what is sug-gested by these terms, i t is still possible to increase the average tunnel velocity, but tr remains unchanged be-cause of a simultaneous increase of the average tunnel static pressure. From experience and theorj^ (Cohen

[10]) i t is known that the larger the ratio c/T, the higher a-c is.

I n cases when an extremely long cavity was recorded, an attempt was made to improve the correlation in data reduction by expressing the pressure coefficient based on the pressure and velocity adjacent to the cavity. To this end Po and Vo were corrected theoretically by using Bernoulli's eciuation and the overall continuity condi-tion, the latter depending on the working section cross-sectional area reduced by the cross-sectional area of the cavity at the downstream end of the tunnel window, which was measured f r o m the photographic recording. The result of this attempt gave a nearly perfect correla-tion between experiment and theory w i t h the value of a equal to zero. This type of wall correction i n which both

tr and Cp were corrected is i n agreement w i t h Birkhoff,

Plesset, and Simmons [8]. I n the case of the infinite cavity w i t h i n tunnel walls i t is known that the flow far downstream is parallel and constant. The pressure at the cavity boundary therefore must be equal to that i n

the main stream according to the present method of wall correction and hence i t corresponds to the case o- = 0 i n an unbounded flow.

Such a simple consideration for long cavities must however be modified for the general case of moderate and short cavity lengths. For this reason, consideration was given to the measurement of the relative minimum pressure on the upper or lower tunnel wall, occurring near the trailing edge of the model, i n which region the maxi-mum width of the cavity occurs. I t appears that this pressure can represent the static pressure i n infinite flow better than any other specified pressure available in. the tunnel. This minimum pressure should occur at the tunnel wall adjacent to the cavity i f the viscous eft'ect is neglected and i t should occur at a point where the stream-lines are horizontal. The pressure gradient i n the flow direction at this point is zero and the transverse gradient can be shown to be zero on account of the small curvature of the streamlines near this point.

To investigate this possible solution, i t was decided to make use of the versatility of the model and have new flaps made which could be combined w i t h the available face plates to form half size and three-quarter size hydro-f o i l models. The combinations ohydro-f the model parts are sketched i n Fig. 4. I t should be noted that only those

0.4 O 0 . 0 3 • 0 . 7 5 0 . 1 0 l.OO 1.00 R U N 2 5 B 2 4 5 F i g . l 4 ( / , y ) E x a m p l e s o f p r e s s u r e d i s t r i b u t i o n c o r r e l a t i o n s

flow conditions would be valid where the rear part of the main body would be f u l l y enveloped by a wake or cavity.

Furthermore, pressure taps Avere made i n the lower window of the working section which were connected to the multimanometer, thus reducing the number of re-cording stations on the model to 9. W i t h this setup measurements were made, starting w i t h the smallest model size and the data were reduced on the basis of the relative minimum pressure on the tunnel bottom (see Figs. 11, 12).

Fig. 8 gives an example of the results. I t shows that w i t h this method a very reasonable agreement is ob-tained between the experimental Cp values (here referred to as C.pm) for the models of different sizes and the theo-retical results [11]. As a comparison Fig. 7 shows another set of data which are now reduced on the basis of the upstream average pressure po, again without other corrections. I t is shown that the m i n i m u m value of ao depends on the model size and on the flow conditions. Furthermore i t is found that the relationship between

C po and (To for very large cavity length can be expressed

hy the formula of a straight line which intersects the 0- = 0 point of the theoretical curve and the point

( — 1 , 1); hence

(1 - C p ) / ( 1 + <r) = const. (1)

The constant is determined by the condition that Cp assumes the theoretical value at cr = 0. This means that the points related to the choked flow condition obtained experimentally w i t h models of diff'erent sizes can be made to coincide w i t h the theoretical result of the infinitely long cavity by shifting the experimental points along the line of equation (1). This shift of experimental data can be achieved eff'ectively by taking the m i n i m u m pressure p„j on the wafi to replace po and the flow velocity at the same point to be the reference velocity i n the definition of Cp and a. The results of this rule for wall correction is so effective that the dependence of model size is seen to be practically eliminated.

W i t h the measurement of pressures along the tunnel bottom included, the tests were repeated w i t h all model configurations, i n order to obtain a f u l l set of data as a result of the method based on the m i n i m u m bottom pressures under various conditions. The example of Fig. 6 was taken f r o m these tests.

I n order to be able to use the original data obtained w i t h 18 stations along the pressure side of the model, an adequate method had to be found for reducing those data. The relationship between o-„j and do could not be used since the Variation of the required value cr,a is very large compared w i t h the variation of the available

F i g . I 4 ( k , l ) E x a m p l e s o f p r e s s u r e d i s t r i b u t i o n c o r r e l a t i o n s

t i t y based on the upstream average pressure p». The only reasonable method seemed to be to estimate the minimum bottom pressures by comparing the length of the cavities f r o m the photographic recordings i n com-parable flow conditions (Fig. 9). Due to dynamic effects i n the flow and the presence of much air i n the wake be-hind the cavities, the estimate was not precise, but was nevertheless adequate.

To show the relationship between the upstream average pressure and a-,,,, C^,,, values for the upstream cross section of the tunnel are plotted i n Fig. 10, i n which the i n -fluence of the chord to tunnel height ratio can also be noticed. I n Figs. 11 and 12 an example is given of plots of tunnel bottom pressure coefflcients, to show the i n -fluence of o-„, and model size, respectively.

The method of ehminating the wall effect on the pres-sure coefficients along the hydrofoil model does not eliminate the influence of the walls on the length of the cavity. More about this matter is given i n the chapter on cavity lengths.

Data Reduction

The data obtained f r o m the multimanometer and the cavity lengths, both by photographic recording, were used for reduction. (Fig. 6). The heights of the mercury

columns (/i) were read i n feet and immediately corrected for paraflax ( - 0 . 0 0 2 through + 0 . 0 0 1 f t ) .

The pressure upstream of the nozzle was supposed to be equal to the t o t a l head (H) of the flow. The error made b y thus neglecting the finite velocity i n the settling chamber was small (0.4 percent of the stagnation pres-sure i n the working section without a cavity and 0.3 per-cent w i t h a 6-in wide cavity at o-,„ = 0) due to the large contraction ratio of the nozzle (15.7:1; Kiceniuk [6]).

Because this total head was used for reference i n the multimanometer, all columns measured the difference between the total head and the local pressure:

H - p = (Ay){h).

The local pressure coefficients were derived by simple division of the pressure readings by the reading of the reference pressure. The procedure is based on the application of Bernoulli's equation, considering that the reference pressure (p^) is the static pressure in a point where the velocity equals the reference velocity (Vr) of the flow

P + = H

and

182

0.8 O 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 i . 0 . 2 1 = 2 0 ° J = 0.8 (HINGE) C/T STA O 0.20 INDIRECT 24 V 0 2 0 DIRECT 24 O 0.15 DIRECT 18 FULL MARKS = F U L L WETTED

O Q2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8

K g . 15(a) H i n g e p o i n t p r e s s u r e c o e f f i c i e n t s

ilo

f r o m which i t follows . r r w i t h the relative m i m m u m wall pressure lor reference

i p j n = H - p = ( A T ) ( / I ) (/i^ = h,n) i t becomes

and 1 - Cp,„ = 7^ 'lm

= H - Pr = ( A T ) ( / 0

The cavitation number based on the actual cavity pres-Now sure is i n fact the pressure coefhcient of the cavity w i t h

the opposite sign

p - p , - ( H - p ) + i H - p r ) _ ( A 7 ) ( f e ) , i ^ _ ^

' yVr' " - P. ( A 7 ) ( M <r = = - C , ( c a v i t y )

so

1 - = f 1 + , = • I n the various data plots, the values of 1 - Cp were now

simply plotted downwards f r o m the value = 1. o- = (r„ if K = K I f the upstream average condition (tunnel without

model) is taken for reference Qi, = ha) and

O- = O-,,, if hr = /(.,„

The ordinate i u Fig. 10 which reads

C.p,n of the upstream average tunnel pressure

_ 1 1 + cr,„ 1 + <ro

is i n fact

C„„(at 0) = 1 - = 1 _

'hn Ilk llm

As has been pointed out before, the data were reduced w i t h the relative minimum pressure at the bottom w i n -dow as reference pressure. For practical reasons only one window was available for this purpose. The bottom was chosen because the relative minimum was expected to be more pronounced behind the overpressure region below the model than along the part of an underpressure region above. Future experiments have to show whether the top window would have given the same results.

184

Correlation with Theory; Errors

Theoretically and experimentally obtained pressure coefficients for several stations have been compared as a function of the cavitation number. This has been done for all configurations of the model, w i t h almost the same results i n all cases. A n example is shown i n Fig. 13. The agreement is reasonable to very good.

I n judging the discrepancies which do occur, the f o l -lowing possible experimental errors should be taken into account.

1 The multimanometer zero level may have been approximately 0.002 f t low, which is due to a lack of rigidity of the floor and the thickness of the zero line on the manometer scale. The exact level was not clearly discernible and the error was therefore neglected.

2 The meniscus of the mercury columns reflected the light i n such a way that i t was difficult to perceive pre-cisely. Readings may have been 0.003 f t low for this reason.

f z -C/T STA 0.2 O 0 2 0 INDIRECT 2 4 6 0 ° V 0.20 DIRECT 2 4 6 0 ° 0 0.15 DIRECT 18 0.8 (HJNGE) • 0.10 DIRECT 12

FULL MARKS • F U L L WETTED F i g . 1 5 ( c ) H i n g e p o i n t p r e s s u r e c o e f f i c i e n t s

3 The mercury cokmms in the multimanometer were interconnected b y a manifold, which i n t u r n was con-nected to the well. This system gave rise to swinging of the mercury w i t h a tendency for the high columns on one side to be overestimated and the low columns on the other side to be underestimated. This means that model face pressure readings may have been up to 0.035 f t low (in few extreme cases 0.035 f t was read near the stagnation point); tunnel bottom pressure readings may have been the same amount high (which lead to 1

-1-<r™ < 1).

The extreme total error is estimated at 0.040 f t low reading for the model and 0.035 f t high reading for the minimum pressure at the tunnel b o t t o m and p^- This leads to experimental Cp values which may have been 0.071 high (for instance i n one case when e7r = 0 deg,

a = 69.85 deg, a,„ = 0.305, at station 29 which is near

to the traihng edge). Those extreme errors occurred w i t h the large « values as a result of dynamic effects i n the flow.

Discrepancies were found, ranging f r o m —0.090 through +0.105, i f the region near the hingepoint is neglected. The extreme negative deviation occurred only i n regions w i t h a very large pressure gradient and at very low <r values, when vapor lock i n the manometer leads may have caused a large error. Some low points were plotted f o r the smallest a values at higher cavita-t i o n numbers.- These should be negleccavita-ted as i cavita-t is doubcavita-t- doubt-f u l that doubt-f u l l separation occurred i n these cases.

When the possible errors are taken into account, i t must be concluded that the probability is large that the theory is correct i n all cases considered, except for the region near the hingepoint.

Hingepoint Separation

A t the hingepoint of the flap the experimental Cp„, values were found to deviate considerably f r o m the theoretical value, which i n this case is one.

I n Fig. 14, a selection of theoretical and experimental pressure distribution curves are compared. The

§ » 0 . 8 (HINGE)

7 0.20 DIRECT 24

0 0.20 INDIRECT 24

FULL MARKS - F U L L V/ETTED

F i g . 1 5 ( ( / ) H i n g e p o i n t p r e s s u r e c o e f f i c i e n t s

retical curves have beeu calculated for a cavitation num-ber value for which experimental data were available, obtained w i t h I S pressure holes. Other data, obtained under comparable conditions w i t h nearly eciual a, Avere added i f they were available. The curves show that just upstream of the hingepoints a large positive pressure gradient is predicted by the theory. I t is therefore speculated that the pressure deficiency i n the hingepoints is caused by a local separation of the flow. Another possibility is that the finite boundary layer thickness is responsible for the phenomenon. A n attempt to trap air i n the expected separated region Avas not successful. I n Fig. 1 5 complete plots are given of all the pressure coefficients obtained i n the hingepoints, including those obtained Avith the smaller scale model configurations.

Correction for Theoretical Forces and Moments

Of major interest to most designers of hydrofoils are the forces and moments Avhich can be anticipated i n the various conditions. T o obtain these directly f r o m the

experiments is illogical, as the integration of the experi-mental pressure data cannot be achieved by the same accuracy as can be obtained f r o m the theory. For t h a t reason no force and moment coefficients are given here. A comprehensive set of curves of l i f t coefficients, leading edge moment coefficients, hinge-moment coefficients and lift-drag ratios, can be found i n a separate paper by Harrison and Wang [ 1 1 ] .

I n order to be able to account f o r the inffiience o f hingepoint separation, corrections Avere estimated f r o m average experimental pressure coefficients i n this region. The results are given i n Fig. 16. The corrections should alAA'ays be subtracted f r o m the theoretical A^alues. Be-cause the viscous effect is small, the corrections Avere estimated only for the Avorst conditions, which are de-fined by a small cavitation number and a small angle of incidence. Low a values mean IOAV C„ values and conse-quently low values for the force and moment coefficients and a large rise toAvards the theoretical stagnation

186

i = 0 . 6 ^ STA O 0.20 INDIRECT 24 ^ . 0 . 8 ( H I N G E I FULL MARKS • F U L L WETTED

F i g . 1 5 ( e ) H i n g e p o i n t p r e s s u r e c o e f f i c i e n t s

pressure, A low cavitation number is connected w i t h wide peaks and therefore a wide separated region.

As a basis for the estimate i t was considered that at the point of separation and at the point of re-attachment of the flow, discontinuities i n the pressure curves n i a j ' exist and that probablj' the region of separation can be treated as a constant pressure region as is done i n a wake. A rather coarse approximation could be allowed, f o r the related error would be of the second order small, w i t h the correction itself being only up to 3 percent of the force and moment coefhcients. The approximation

consists of the replacement of the theoretical dis-t r i b u dis-t i o n cur^'es on bodis-th sides of dis-the hingepoindis-ts, by straight lines. I n this way the integration of the triangle between the theoretical and experimental distrit)u-tions w i l l probabl.y be somewhat overestimated, which may be counterbalanced b y the fact that the experi-mental Cp values w i l l probably be a little on the high side, as has been discussed before.

C a v i t y LengtFis

As has been discussed earlier, the cavity length data

have been used as a basis for the achievement of values of the m i n i m u m pressure at the bottom of the working section, i n those cases when no direct measurement of this quantity was performed. A l l these cases were related to the largest model size. I n one case also the cavity lengths have been measured for the half size model configuration. The plotting of both sets of data i n one diagram (Fig. 17) has disclosed the interesting fact that the length of long cavities is much influenced b y the proximity of the tunnel walls. I t was found that i n comparable conditions the smaller model has the longer cavity relative to its chord length. Fig. 17 also shows that the longer the cavity, the larger the relative differ-ence is. The author's explanation for this phenomenon is that the longer cavity is influenced b y the pressure recovery i n the diffuser of the tunnel. Although no cfirect comparison is possible, this finding seems to be i n accordance w i t h the results obtained theoretically b y H i r s h Cohen and DiPrima [10] for a 15-deg wedge i n sj'mnietrical flow.

5 10 2 0 3 0

O 5 10 2 0 3 0

F i g . 1 6 H i n g e p o i n t s e p a r a t i o n c o r r e c t i o n s f o r f o r c e a n d m o m e n t c o e f f i c i e n t s

Dynamic Effects

A t all the angles of attack, but more so at larger angles, cb'namic effects could be observed i n the flow. I t seems possible that these effects have the same cause as the K a r m a n vortex street behind a cylinder for instance. I t must be noted, however, that a rather elastic support has been used which was the force balance, and which caused severe oscillations of the angle of attack to occur i n extreme cases.

Conclusions

From the results of the present experiments i t can be concluded that the free streamline theory by W u gives extremely good results for f u l l y separated wake and f u l l cavity flows around flapped flat plate hydrofoils. The

theory fails i n a small region near the hingepoint of the flap, where the sharp rise of the pressure causes separa-tion of the flow. The influence of this separasepara-tion on the generated forces and moments is very small and can be neglected at larger angles of attack. W i t h small a values and small flap length ratios, however, the influence runs into several percent.

I t was found that the tunnel wall effect can not be neglected w i t h f u l l cavity flow experiments. Measure-ments w i t h different scale models and theory were found to be i n good agreement w i t h each other, i f the unbounded flow conditions at infinitj^ were supposed to exist i n a point at the pressure side tunnel wall where the pressure had its minimum value.

Additional conclusions may be drawn here. The con-ventional concept of "tunnel blockage" is misleading

188

because i t actually conforms w i t h the condition of zero cavitation number.

The length of long cavities is influenced bj^ the presence of tunnel walls. This infiuence seems to have little effect on the flow i n the v i c i n i t j ' of the model itself.

A c k n o w l e d g m e n t

The author is indebted to those working i n the H y d r o -dynamics Laboratory of the California Institute of Technology for their assistance, cooperation and interest i n his work during his academic visit i n Pasadena, while on leave of absence f r o m the Technical University of D e l f t . He is also indebted to the Technical University of D e l f t and The Netherlands Organization for Pure Research " Z . W. 0 . " for their support of his visit at the California Institute of Technology.

References

1 G. Birkhoff and E. H . Zarantonello, Jets, Wakes

and Cavities, Academic Press Incorporated, New Y o r k ,

N . Y . , 1957.

2 D . Gilbarg, "Jets and Cavities," Handbuch der

Physik, Springer Verlag, Berlin, Germany, vol. 9, 1960,

pp. 311-445.

3 T. Y . Wu, " A Wake ]\Iodel for Free Streamline Flow Theory, Part I : F u l l y and Partially Developed Wake Flows Past an Oblique Flat Plate," Journal of

Fluid Mechanics, v o l . 13, 1962, pp. 161-181.

4 T. Y . W u and D . P. Wang, " A Wake Model for Free Streamline Theory, Part I I : Cavity Flows Past Obstacles of A r b i t r a r y Profile," Journal of Fluid

Me-chanics, vol. 18, 1964, pp. 65-93.

6 A. Fage and F. C. Johansen, " O n the Flow of A i r Behind an Inclined Flat Plate of Infinite Span,"

Pro-ceeding of the Royal Society, London, England, vol. 116,

1927, pp. 170-197.

6 T. Kiceniuk, • " A Two-Dimensional W o r k i n g Section for the H i g h Speed Water Tunnel at the Cali-fornia Institute of Technology," A S M E , Cav. Res. Fac. and Tech., 1964.

7 A. Pope, Wind-Tunnel Testing, John Wiley and Sons, Inc., New Y o r k , N . Y . , and Chapmann and Hall Limited, London, England.

8 G. Birkhoff, M . Plesset, and N . Simmon, "WaU Effects i n Cavity Flow," Quarterly of Applied

Mathe-1 0 0 8 0 6 _ l l + o"n. 0.4j 0.2 2 0 ' 5 ^ / f f / / /'' r/'l ^ / / A / f o ' / A S ' B 6 0 ' -i/c •- 0. 2 a --5 * €TT • 4 0 ' 1 0 * O c / T = 0.1 and FULL MARKS 2 0 ° a e/T ' 0 . 2 ond OPEN MARKS 3 0 *

4 5 ° 6 0 °

o

c

F i g . 1 7 W a l l e f f e c t o n c a v i t y l e n g t h

matics, v o l . 8, 1950, pp. 151-168; ibid.. Part I I , vol. 9,

1952, pp. 413-421.

9 B . R. Parkin, 1956, "Experiments on Circular Arc and Flat Plate H y d r o f o ü s i n Noncavitating and F u l l C a v i t y Flows," J O U R N A L OF S H I P R E S E A R C H , vol. 1, 1958, pp. 34-56.

10 Cohen, Hirsh, and R. C. Diprima, " W a l l Effects, in Cavitating Flows," Second Symposium on Nav. Hydro., 1958 (ONR).

11 Z. Harrison and D . P. Wang, "Evaluation of Pressure D i s t r i b u t i o n on a Cavitating H y d r o f o i l W i t h Flap," California Institute of Technology Report No. E-133.1, 1965.