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CASE F I L E COPY NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS TECHNICAL NOTE 3169
ON THE DRAG AND SHEDDING FREQUENCY O F
TWO -DIMENSIONAL B L U F F BODIES By Anatol Roshko
California Institute of Technology
Washington July 1954
r
NATIONAL ADVISORY COlNLTTEE FOR AERONAUTICS e
TECHNICAL NOTE 3169
J
ON THE DRAG AND SHEDDING FREQUENCY OF TWO-DIMENSIONAL BLUFF BODIES By Anatol Roshko
SUMMARY A semiempirical study is made of the bluff-body problem.
Some experiments with interference elements in the wake close behind a cylinder demonstrate the need for considering that region in any complete theory.
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I
Dimensional analysis of a simple model of the region leads to a universal Stromal number S* which is then experimentally determined a s a function of wake Reynolds nuniber R*. This result, together with free-streamline theory, allows the drag to be calculated from a measurement of the shedding frequency and furnishes a useful correlation between different bluff cylinders. By allowing for some annihilation of the vorticity in the free shear l/ayys, it is shown how to corribine the free-streamline theory with Karman's theory of the vortex street to obtain a solution dependent on only one experimental measurement. INTRODUCTION
The problem of the drag of bluff bodies in incompressible flow is one of the oldest in fluid mechanics, but it remains one of the most important, for practical reasons as well as for its theoretical interest The two major contributions toward/ altheoretical understanding are the well-known ones of Kirchhoff and Karman. These attack two aspects of the problem that must be understood, namely, the potential flow in the vicinity of the cylinder and the wake farther downstream, but neither by itself can furnish a complete theory. Much of the work devoted to the problem through the years has been essentially the, elaboration and application of the theories of Kirchhoff and drman.
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It has become increasingly evident, however, that a complete solution will not be obtained until one is able to "join" the two parts of the problem and that this will require an understanding of the flow in the early stages of the wake, that is, the first few diameters downstream of the cylinder. On the experimental side, there has also been a continued activity. It is surprising that the influence on theoretical developments has been rather small, for considerable useful information has been compiled, some as long ago as 20 or 50 years. Much of this is referred to in chapters IX and XI11 of reference 1. Particular mention must be made of the work of Fage and his coworkers (e.g., refs. 2, 3 , and 4) who made very useflil investigations of the flow near the cylinder and in the early stages of the wake. The ideas of this report grew largely out of a study of their work. The interest at GALCIT in the flow past bluff bodies has been connected not so much with the problem of the drag as with that of turbulent wakes. Many of the turbulent flows that are used for experimental studies (e.g., behind grids) and almost all those that cause practical difficulties (e.g., buffeting) are produced in the wakes of bluff bodies. Much of the empiricism connected with these problems can be resolved only by a better understanding of how the wake is related to the body which produces it. This includes questions of wake scale, frequencies, energy, interference between wakes, and so forth. However, whatever the approach, one is led to consider the relation between the wake and the potential flow outside the wake and cylinder. A short review of the theory of flow past bluff bodies is presented and some experiments are described which demonstrate how critically the whole problem depends on that part of the wake immediately behind the cylinder. In the remainder of the report the free-streamline theory of reference 5 is combined with some experimental results to obtain a much needed correlation between bluff cylinders1 of different shapes. This furnishes at the same time some of the sought-after relations between wake and cylinder.
The experiments were performed in the 20- by 20-inch low-turbulence wind tunnel at GALCIT, under the sponsorship and with the financial assistance of the National Advisory Committee for Aeronautics. Standard %he term "cylinder" is used throughout to denote a body whose cross-sectional shape is the same at every section along the span. This is the so-called two-dimensional body, which was the only kind for which measurements were made here. The term is applied to all cross-sectional shapes, including the limiting case of a flat plate normal to the flow, in which case the cross-sectional shape is simply a line.
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hot-wire equipment, manometry, and so forth were employed. The work is part of a broader program of turbulence research directed by Dr. H. W. Liepmann; the author is indebted to him and to other members of the GALCIT staff for many discussions. SYMBOLS b'
distance between outer edges of free shear layers
CD
drag coefficient
cP
pressure coefficient
cPS
base-pressure coefficient
C
distance *om back of cylinder to trailing edge of interference element
d
cylinder diameter or breadth
d'
distance between free streamlines
h
width of vortex street
k
base-pressure parameter,
"r
base-pressure parameter, uncorrected for tunnel blockage
2
longitudinal vortex spacing
n
vortex shedding frequency
PS
base pressure
R
Reynolds number based on cylinder diameter, Ud/v
RT
Reynolds nurdber uncorrected for tunnel blockage
R*
Reynolds number based on wake parameters, Usd'/v
S
cylinder Strouhal number, nd/U,
ST
Strouhal number uncorrected for tunnel blockage
S*
wake Strouhal number, nd'/U,
U,/U,
or
\Il-cpi
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Fage's form of Stromal number, nb'/Um velocity along shear layer velocities at edges of shear layer free-stream velocity blockage correction to measured free-stream velocity velocity on free streamline at separation velocity of vortices relative to f'ree-stream velocity distance downstream angle measured from stagnation point of circular cylinder circulation per vortex fraction of shear-layer vorticity that goes into individual vortices vorticity
.
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coordinate normal to shear layer kinematic viscosity REVIEW OF THXORY OF FLOW PAST BLUFF BODIES
Although the main contributions to the theory of flow past bluff bodies are well-known, a review is useful to bring out the important features of the problem.
In the free-streamline theory developed by Kirchhoff, the free shear layers which are known to separate from bluff bodies a r e idealized by surfaces (streamlines) of velocity discontinuity. These f'ree streamlines divide the flow into a wake and an outer potential field. The possibility of treating the problem this way in two parts is important to note. Kirchhoff's theory, however, considerably underestimates the drag, and the failure is easily traced to the assumption which is made about the velocity on the free streamline. It is assumed that the velocity there is the free-stream velocity or, what amounts to the same thing, that the pressure in the wake and on the cylinder base is
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the free-stream pressure. It is known, however, that the pressure on the base, behind the separation points, is actually much lower, which corresponds to a velocity on the free streamline which is higher than the free-stream value. This lower base pressure accounts for the higher drag actually observed. Kirchhoff's theory has been applied, by a long line of successors, to many other cylinder shapes; and indeed it can be applied in principle to any shape whatever, the difficulty being only of computation. (In cases such as the circular cylinder, where there is not a well-defined, fixed separation point, an additional assumption must be made.) In all cases, however, the theory gives values of drag much lower than those observed, and always for the reason that the value assumed for the separation velocity is too low. /
Kardn, in his famous theory of the vortex street, attacked the problem by way of another characteristic feature of flow past bluff bodies, that is, the phenomenon of periodic vortex shedding. The theory is incomplete in that it cannot by itself relate the vortexstreet dimensions and velocities to the cylinder dimension and freestream velocity. Two additional relations are required, and these must come from elsewhere, usually from experiment.
d&
These two examples by Kirchhoff and are, however, representative of the two parts of the flow that will have to be considered in any complete theory. While each part the potential field and the wake may be considered separately, the complete solution will only be found by discovering how to join them. Indeed, Heisenberg (ref. 6) attempted to obtain such a closed solution by joining the Kirchhoff solution to K6rdn's vortex street. His solution gives a value for the drag in good agreement with that for a flat plate set normal to the flow, but it also gives the same value for any other cylinder shape, as pointed out by Ka'rdn in a footnote to the same paper. There is an inconsistency in the theory in that the Kirchhoff flow, which is taken as one element in the synthesis, predicts a drag coefficient which is different (lower) from the final result. In short, the Kirchhoff flow is not a suitable starting point for such solutions unless it is modified to allow more realistic base pressures.
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In reference 5 it was shown how such a modification may be made. Instead of restricting the separation velocity to the free-stream value U , it is allowed to assume an arbitrary value Us = kUm. The base-pressure coefficient is then Cps = 1 k2. For k = 1 this reduces to the Kirchhoff case, but for agreement with experiment k must be greater than 1. This modified Kirchhoff theory will be referred to as the notched-hodograph theory, after the hodograph upon which it is based.
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Another method by which t h e p o t e n t i a l p a r t of t h e flow might be "joined" t o t h e wake i s t o use t h e momentum-diffusion theory which has been a p p l i e d i n c a l c u l a t i o n s of base p r e s s u r e on supersonic p r o j e c t i l e s ( r e f s . 7 and 8) These c a l c u l a t i o n s a r e made on t h e b a s i s of a mixing theory, wherein t h e p r e s s u r e deficiency on t h e base i s assumed t o be "supported" e n t i r e l y by t h e d i f f u s i o n of momentum across t h e shear l a y e r s . I n f a c t , it appears i n e v i t a b l e t h a t some such c a l c u l a t i o n w i l l be necess a r y b e f o r e a complete t h e o r e t i c a l formulation of t h e problem can be obtained. The mechanism of t h e wake immediately downstream of t h e c y l i n d e r i s t h e main i t e m i n t h e coupling between wake and p o t e n t i a l flow, and it w i l l probably be e s s e n t i a l t o understand it. Unfortunately, t h e idea of simple momentum d i f f u s i o n across t h e shear l a y e r s , which appears t o be s u i t a b l e f o r t h e supersonic flows, i s not by i t s e l f s u f f i c i e n t f o r t h e incompressible flow p a s t a b l u f f c y l i n d e r . The "coupling region" immed i a t e l y downstream of t h e body i s a l s o that i n which t h e v o r t i c e s are formed, and t h e y are an e s s e n t i a l p a r t of the mechanism, as w i l l be shown i n t h e following s e c t i o n s .
.
EFFECT OF VORTEX FORMATION Figure 1 shows a p r e s s u r e t r a v e r s e made along t h e c e n t e r l i n e of t h e wake behind a f l a t p l a t e s e t normal t o t h e flow. The measurement i s not simple t o make because of t h e l a r g e t r a n s v e r s e v e l o c i t y f l u c t u a t i o n s a s s o c i a t e d w i t h t h e v o r t e x shedding. A s t a t i c probe o r i e n t e d along t h e c e n t e r l i n e of t h e wake experiences a nonstationary crossflow, f l u c t u a t i n g a t t h e shedding frequency. A s f a r as t h e mean flow i s concerned, t h e p r e s s u r e i s constant over t h e circumference of t h e c y l i n d r i c a l probe, s i n c e i t s dimensions a r e small compared w i t h t h e flow f i e l d . Because of t h e crossflow, however, t h e r e i s a p r e s s u r e v a r i a t i o n over t h e circumference, jhst as t h e r e i s for a cylinder placed normal t o a stream with p e r i o d i c v e l o c i t y . "herefore t h e p r e s s u r e measured v a r i e s with t h e p o s i t i o n of t h e o r i f i c e on t h e circumference of t h e probe. When t h e o r i f i c e i s a t one of t h e s t a g n a t i o n p o i n t s of t h e crossflow t h e p r e s s u r e i s t h e h i g h e s t , and when it i s a t 90° from t h e s t a g n a t i o n p o i n t t h e pressure i s t h e lowest. The pressures measured with t h e o r i f i c e a t t h e s e l i m i t i n g p o s i t i o n s are shown i n f i g u r e 1. The c o r r e c t p r e s s u r e should be somewhere between. Now on a c y l i n d e r i n steady crossflow t h e pressure c o e f f i c i e n t i s zero a t 30' i n p o t e n t i a l flow and a t about 35' i n r e a l flows. It w a s simply assumed t h a t t h e l a t t e r i s a l s o t h e c o r r e c t p o s i t i o n f o r t h e o s c i l l a t i n g crossflow. The intermediate curve i n f i g u r e 1 i s t h e r e s u l t with t h e o r i f i c e a t 3 5 O from t h e s t a g n a t i o n p o i n t ; it i s believed that t h i s gives very n e a r l y t h e c o r r e c t value f o r t h e mean s t a t i c p r e s s u r e . I n any case, what i s important i s t h e low-pressure region a t about 2 p l a t e widths downstream; t h a t i s , t h e p r e s s u r e on t h e base i s not t h e lowest p r e s s u r e i n t h e wake. This measurement had a l r e a d y been made by
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S c h i l l e r and Linke i n 1933 ( r e f . 9 ) , but does not seem t o have been noted s i n c e . I t s s i g n i f i c a n c e seems c l e a r - t h a t t h e low pressure i s a s s o c i a t e d with a low-pressure region a t t h e center of t h e vortex which i s being formed. This low pressure must f l u c t m t e , of course, as t h e v o r t i c e s form a l t e r n a t e l y , and it i s t h e mean e f f e c t which has been measured. It seems l i k e l y then t h a t a l a r g e p a r t of t h e low pressure on t h e base of t h e f l a t p l a t e i s a s s o c i a t e d with t h e v o r t e x pressure, that i s , t h a t t h e main mechanism f o r t h e base pressure i s t o be found not i n t h e d i f f u s i o n of momentum across t h e shear l a y e r s but i n t h e dynamics of t h e v o r t i c e s . I n f a c t , t h e moment-m-diffusion theory could p r e d i c t only a monotonically increasing pressure from t h e base. Now i f t h e vortex dynamics are indeed important, then i n t e r f e r e n c e with t h e i r f o r m t i o n should have a s t r o n g e f f e c t on t h e base pressure. This w a s e a s i l y i n v e s t i g a t e d by placing a " s p l i t t e r " p l a t e along t h e tent e r l i n e of t h e wake. Figure 2 shows pressure d i s t r i b u t i o n s with and without t h e s p l i t t e r p l a t e . (The r e s u l t s given i n t h i s f i g u r e a r e f o r a c i r c u l a r cylinder i n s t e a d of t h e f l a t p l a t e . ) With t h e s p l i t t e r p l a t e t h e p e r i o d i c vortex formation i s i n h i b i t e d and t h e base pressure increases considerably. It i s s t i l l below free-stream pressure, but whether t h i s r e s i d u a l underpressure can be accounted f o r by t h e momentum-diffusion theory i s not c e r t a i n . It i s p o s s i b l e that a kind of standing vortex i s formed on each s i d e , but t h i s p o i n t w a s not i n v e s t i g a t e d f u r t h e r . I n any case, t h e momentum-diffusion theory i s i n d i f f e r e n t t o t h e omission o r i n c l u s i o n of a p a r t i t i o n along t h e c e n t r a l streamline. It i s c l e a r t h a t without t h e s p l i t t e r p l a t e t h e p e r i o d i c formation of v o r t i c e s i s an e s s e n t i a l p a r t of t h e base-pressure mechanism. Figure 3 shows how t h e s p l i t t e r p l a t e a f f e c t s t h e pressure d i s t r i b u t i o n over t h e whole cylinder circumference. It shows t h a t i n t e r f e r e n c e i n t h e "coupling region" changes t h e outer p o t e n t i a l flow, as well as t h a t i n t h e wake. EFFECT OF INTERFERENCE ELEMENT ON RELATIONS BETWEEN BASE PRESSURE AND SHEDDING FREQUENCY I n t h e experiment of f i g u r e 2 t h e chord of t h e s p l i t t e r p l a t e w a s almost 5 diameters. One immediately asks how t h e i n t e r f e r e n c e changes with changing chord. Accordingly, some measurements were made with a s p l i t t e r p l a t e whose chord w a s about 1 diameter. This w a s found not t o i n h i b i t t h e vortex formation a t a l l , though it does change t h e shedding frequency s l i g h t l y . More i n t e r e s t i n g i s t h e e f f e c t of moving t h i s s h o r t p l a t e downstream, that i s , leaving a gap between it and t h e c y l i n d e r . The e f f e c t i s t o decrease t h e shedding frequency and t o increase t h e base pressure, as shown i n t h e lower p a r t of f i g u r e 4. The shedding frequency becomes a minimum, and t h e base pressure a maximum, when t h e t r a i l i n g edge of t h e i n t e r f e r e n c e element i s 3.85 diameters downstream of t h e cylinder
I
base. It i s c l e a r t h a t such a minimum must be reached, f o r when t h e element i s very f a r downstream of t h e cylinder i t s upstream influence should be n e g l i g i b l e . What i s remarkable i s t h e abrupt jump t h a t occurs t h e r e , p r a c t i c a l l y t o t h e o r i g i n a l value. Q u a l i t a t i v e l y , it appears t h a t c l o s e t o t h e cylinder t h e element has a streamlining e f f e c t ; t h a t i s , it extends the shear l a y e r s and f o r c e s t h e v o r t i c e s t o form downstream of i t s t r a i l i n g edge. When it i s f a r downstream, t h e v o r t i c e s form on i t s upstream s i d e , i n t h e normal p o s i t i o n c l o s e t o t h e cylinder, and t h e element has only a slight e f f e c t . There must then be some c r i t i c a l posit i o n where t h e flow must choose between one of t h e two configurations. I t i s observed, i n f a c t , t h a t when t h e i n t e r f e r e n c e element i s a t t h i s c r i t i c a l p o s i t i o n , t h e flow does jump, i n t e r m i t t e n t l y , from one configur a t i o n t o t h e o t h e r , as shown by t h e double values measured t h e r e .
A t c/d = 1.13 t h e r e i s a p o i n t t h a t does not f a l l on t h e l i n e drawn f o r Cps. This w a s thought t o be an e r r o r , but f u r t h e r checks and readjustments indicated t h a t it i s real. This p o i n t corresponds t o t h e case where t h e s p l i t t e r p l a t e w a s touching t h e back of t h e cylinder; t h a t i s , t h e gap w a s completely closed. However, t h e j o i n t w a s by no means p r e s s u r e t i g h t . It seemed, r a t h e r , that some o t h e r kind of communication through t h e qap becomes e f f e c t i v e a t some f i n i t e gap width. A t p r e s e n t , t h i s has not been thoroughly i n v e s t i g a t e d . Experiments such as t h e s e may lead t o information u s e f u l f o r understanding t h e mechanics of t h e vortex formation. A n i n i t i a l s t e p i n t h i s d i r e c t i o n has been made by i n v e s t i g a t i n g t h e r e l a t i o n s h i p between t h e base pressure and t h e shedding frequency ( s e e t h e following s e c t i o n ) . BLUFF -BODY SIMILARITY Although a complete t h e o r e t i c a l s o l u t i o n seems remote a t present, t h e r e i s s t i l l t h e p o s s i b i l i t y of f i n d i n g a c o r r e l a t i o n between d i f f e r e n t b l u f f bodies, on a semiempirical b a s i s . Such a c o r r e l a t i o n m u s t be based on, and must account f o r , t h e following q u a l i t a t i v e f a c t s , which a r e indeed t h e main f e a t u r e s of t h e flow p a s t b l u f f cylinders: (1) The "bluffness" of a cylinder i s r e l a t e d t o t h e width of t h e wake, compared with t h e cylinder dimension. It i s almost i n t u i t i v e t h a t t h e b l u f f e r body tends t o diverge t h e flow more, t o c r e a t e a wider wake, and t o have l a r g e r drag. ( 2 ) The shedding frequency i s r e l a t e d t o t h e width of t h e wake, t h e r e l a t i o n being roughly inverse, so t h a t t h e b l u f f e r bodies have t h e lower Strouhal numbers. I
(3) For a given cylinder t h e shedding frequency i s r e l a t e d t o t h e base pressure. Generally, an increase i n base pressure i s accompanied
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by a decrease i n shedding frequency. This i s well i l l u s t r a t e d i n t h e case of f i g u r e 4 with t h e i n t e r f e r e n c e element (although t h e r e an anomalous behavior i s observed f o r C/d > 3.85). Thus, f o r a given cylinder a decrease i n wake width corresponds t o an increase i n drag, which seems a t variance with i n t u i t i o n . However, t h e decrease i n width i s a s s o c i a t e d w i t h an increase i n "wake v e l o c i t i e s , " t h e n e t e f f e c t being an increased wake energy corresponding t o t h e increased drag. Figure 5 shows measurements of Strouhal number made on t h r e e d i f f e r e n t cylinder shapes. These were a c i r c u l a r cylinder, a goo wedge, and a " f l a t p l a t e " normal t o t h e flow. The shedding frequencies were measured by a hot-wire anemometer placed i n t h e wake. The dimensions of t h e cylinders used a r e given i n t a b l e I. (For t h e c i r c u l a r cylinder, t h e curve has been taken from r e f . 1 0 ) . A t present, one i s concerned only w i t h t h e higher Reynolds numbers, f o r which t h e Strouhal number S i s d i s t i n c t l y d i f f e r e n t f o r t h e d i f f e r e n t cylinders, though approximately constant f o r each i n d i v i d u a l one. The f i g u r e shows that t h e c i r c u l a r cylinder, goo wedge, and f l a t p l a t e a r e increasingly b l u f f , i n that order. f o r t h e same Figure 6 i s a p l o t of base-pressure c o e f f i c i e n t Cps t h r e e cylinder shapes. The base pressure ps w a s measured a t s t a t i c o r i f i c e s on t h e backs of t h e c y l i n d e r s . I n t h e case of t h e c i r c u l a r cylinder t h e measurement w a s made a t about 130° from t h e s t a g n a t i o n p o i n t , which i s i n a region of constant pressure ( c f . f i g . 3) and i s believed t o be a more s i g n i f i c a n t p o i n t f o r t h e base pressure than t h e one a t 180°. Figure 6 shows t h a t t h e d i f f e r e n c e s i n base pressure f o r t h e d i f f e r e n t cylinders a r e not so marked as are t h e shedding frequencies and that they cannot be ordered i n terms of b l u f f n e s s . Included i n t h e f i g u r e are two cases with wake i n t e r f e r e n c e - a c i r c u l a r cylinder and a f l a t p l a t e for which t h e increase i n base pressure i s q u i t e marked. The i n t e r f e r e n c e elements which w e r e used a r e sketched i n t a b l e I.
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The r e s u l t s of f i g u r e s 5 and 6 have been corrected f o r t u n n e l blockage. The s c a t t e r i n f i g u r e 6 i s r a t h e r l a r g e , f o r t h e probable e r r o r i n t h e c a l c u l a t i o n of Cps i s i n h e r e n t l y l a r g e ; 1-percent accuracy i n m e a s urement of pressure and v e l o c i t y gives only about 5-percent accuracy i n Cps. Now it i s well-known that t h e wakes of d i f f e r e n t b l u f f bodies are s i m i l a r i n s t r u c t u r e . I n every case t h e flow separates on t h e two s i d e s of t h e cylinder, c r e a t i n g f r e e shear l a y e r s which continue f o r a s h o r t d i s t a n c e downstream and then "roll up" i n t o v o r t i c e s , a l t e r n a t e l y on e i t h e r s i d e . The region where t h i s occurs, which extends only a few diameters downstream, was r e f e r r e d t o i n t h e s e c t i o n "Review of Theory of Flow P a s t Bluff Bodies" as t h e coupling region. Because of t h e s i m i l a r i t y i n a l l b l u f f - c y l i n d e r wakes, one would expect t o f i n d a parameter t o compare t h e wakes of d i f f e r e n t cylinders. The clue i s i n t h e shedding frequency, as may be demonstrated by a simple dimensional a n a l y s i s .
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Consider two p a r a l l e l . shear l a y e r s ( s k e t c h 1) which may be imagined t o
1
US
d'
Sketch 1 have been formed by some b l u f f cylinder. The c i r c u l a r arrows i n d i c a t e t h e s i g n of t h e v o r t i . c i t y . The c h a r a c t e r i s t i c frequency associated with and one can d e f i n e a wake t h i s configuration i s p r o p o r t i o n a l t o U s / d ' Strouhal number S* = nd'/Us
where n i s t h e shed.ding frequency. Strouhal number S by
It i s r e l a t e d t o t h e usual cylinder
S d'
= E d The wake S t r o u h a l number S* i s expected t o be u n i v e r s a l , that i s , t o be t h e same f o r a l l b l u f f - c y l i n d e r wakes. O f course t h i s idea i s based on an i d e a l i z e d model i n . which t h e shear l a y e r s are surfaces of d i s c o n t i n u i t y , whereas a c t u a l l y t h e condition of t h e f r e e shear l a y e r s w i l l d i f f e r from one cylinder t o another, depending on t h e i r h i s t o r i e s up t o t h e p o i n t of vortex formation. This may, however, be a secondary e f f e c t , and it i s s u f f i c i e n t a t f i r s t t o assume t h a t S* depends only on a wake Reynolds number R* = Usd'/v = R k d'/d
NACA TN
L. b
51-69
11
The idea of such a s i m i l a r i t y i s e s s e n t i a l l y t h e same as that given by Fage and Johansen ( r e f . 3 ) , b u t they omitted t h e c h a r a c t e r i s t i c wake v e l o c i t y Us. Their parameter i s S ' = nb'/Um, where b ' i s a wake width measured between t h e o u t e r edges of t h e shear l a y e r s . They found a good c o r r e l a t i o n , S ' 0.26, f o r s e v e r a l d i f f e r e n t c y l i n d e r shapes, i n s p i t e of t h e omission of a wake v e l o c i t y . This seems t o be due t o t h e f a c t t h a t , f o r t h e cases of good c o r r e l a t i o n , t h e wake v e l o c i t i e s were very nearly t h e same. Also, they d i d not introduce a wake Reynolds number. I n w h a t follows, t h e wake s i m i l a r i t y i s s t u d i e d on t h e b a s i s of t h e parameter S*(R*). The method, which i s q u i t e d i f f e r e n t from that of Fage and Johansen, depends e s s e n t i a l l y on t h e r e s u l t s of t h e notchedhodograph theory ( r e f e r r e d t o i n t h e s e c t i o n "Review of Theory of Flow P a s t Bluff Bodies"). These r e s u l t s are b r i e f l y as follows: The v e l o c i t y a t s e p a r a t i o n and on t h e i n i t i a l p a r t of t h e f r e e streamline i s Us = kU, ( s k e t c h 2 ) . The base pressure i s t h e same as that a t s e p a r a t i o n , and t h e
T
d'
Sketch 2
-
base-pressure c o e f f i c i e n t i s t h e r e f o r e Cps = 1 k2. For a given value of k, t h e p o t e n t i a l flow outside t h e wake i s completely determined, and s o t h e drag c o e f f i c i e n t CD i s a function only of k . The important r e s u l t f o r t h e present consideration i s that a wake width d ' i s defined. This a l s o depends only on k . Figure 7, which has been computed from t h e r e s u l t s of reference 5 , shows how d ' / d v a r i e s with k f o r t h e t h r e e cylinder shapes being considered. It gives t h e "measure of b l u f f n e s s " alluded t o e a r l i e r ; that i s , t h e b l u f f e r cylinders have t h e wider wakes a t a given value of k, but f o r a given cylinder t h e wake width decreases with increasing k ( i . e . , increasing d r a g ) . The shedding frequencies and base pressures which had been measured f o r f i g u r e s 5 and 6 were used t o compute S*(R*) f o r t h e various cyli n d e r s . The computation i s straightforward: k = is calculated from t h e measured base-pressure c o e f f i c i e n t . The corresponding value of d ' / d i s found i n f i g u r e 7. With t h e s e and measured values of S and R t h e corresponding values of S* and R* a r e e a s i l y c a l c u l a t e d from
-4
a
NACA TN
12
equations (1)and ( 2 ) . i n f i g u r e 8.
3169
The r e s u l t s a r e l i s t e d i n t a b l e I and p l o t t e d
"4
It i s necessary t o draw a t t e n t i o n t o s e v e r a l points: (1) The c h a r a c t e r i s t i c wake v e l o c i t y , which i n dimensionless form i s simply k, i s not measured, but i s computed from t h e base pressure. This
4
may be regarded as t h e e s s e n t i a l s t e p i n t h e coupling between t h e wake and t h e o u t e r p o t e n t i a l flow. ( 2 ) The w a k e width i s not measured b u t i s obtained from t h e theory.
( 3 ) I n computing S*, the shedding frequency and base pressure should not be f i r s t corrected f o r t u n n e l blockage. T h a t is, t h e wake parameters that a r e used must correspond t o t h e wake t h a t i s a c t u a l l y observed. Blockage gives only a second-order e f f e c t i n t h i s computation. The uncorrected parameters a r e l i s t e d i n t a b l e I as ST, RT, and k ~ , respectively.
(4)The
errors that
he made i n measuring
and
even though small i n d i v i d u a l l y (about 1 p e r c e n t ) , c o n t r i b u t e t o a p o s s i b l e e r r o r of 4 percent i n S*.
(3) A t R* < 8,000 t h e r e i s a l a r g e discrepancy between t h e values f o r t h e wedge and f o r t h e c i r c u l a r c y l i n d e r . It i s not c l e a r whether t h i s i s r e a l o r due t o experimental d i f f i c u l t i e s . The s m a l l wedge used t o o b t a i n t h e s e p o i n t s d i d not have a proper base-pressure o r i f i c e . Instead, a small tube with an open mouth w a s cemented t o t h e back of t h e wedge, and it i s not c e r t a i n that t h i s measures t h e c o r r e c t base pressure. Otherwise, t h e s i m i l a r i t y parameter S* does give a good c o r r e l a t i o n f o r a f a i r l y wide range of Reynolds number. It i s probable that it can be extended t o higher Reynolds numbers, say another order of magnitude, up t o t h e c r i t i c a l Reynolds number of t h e c i r c u l a r cylinder. It w i l l be noted that t h e s i m i l a r i t y p l o t ( f i g . 8) includes t h e cases with wake i n t e r f e r e n c e . For t h e s e t h e base pressure and corresponding wake v e l o c i t y d i f f e r considerably from those without i n t e r f e r e n c e . T h a t they f i t f a i r l y w e l l i n t o t h e s i m i l a r i t y p l o t i s taken as evidence f o r Nevertheless, an examination t h e s u i t a b i l i t y of t h e parameter S*(R*) of an i n d i v i d u a l case shows t h a t t h e r e a c t u a l l y i s a systematic v a r i a t i o n of S*. I n t h e upper p a r t of f i g u r e 4 t h e data of t h e lower p a r t have been used t o c a l c u l a t e S*, which i s seen t o vary s y s t e m a t i c a l l y with t h e p o s i t i o n of t h e i n t e r f e r e n c e element. (The "bad point" r e f e r r e d t o i n t h e s e c t i o n "Relations Between Base Pressure and Shedding Frequency" does not f i t t h i s curve e i t h e r . )
.
.*
NACA 1
TN 3169
13
-
J O I N I N G TRE FREE STREAMLINES AND VORTEX S’I!REXT I C
To close t h e K&rdn theory of r e l a t i o n s a r e needed t o r e l a t e t h e s t r e e t t o t h e free-stream v e l o c i t y hodograph theory may f u r n i s h t h e s e r e a l i s t i c way can be found t o j o i n
t h e vortex s t r e e t , two a d d i t i o n a l v e l o c i t i e s and dimensions of t h e and cylinder dimension. The notcheda d d i t i o n a l two r e l a t i o n s , provided a t h e r e s u l t s of t h e two t h e o r i e s .
One r e l a t i o n i s obtained by considering how c i r c u l a t i o n i n t h e vort i c e s i s r e l a t e d t o t h e v o r t i c i t y i n t h e f r e e shear l a y e r s . The r a t e a t which c i r c u l a t i o n flows p a s t any plane s e c t i o n of a shear l a y e r i s
L2
CU dv =
u12
- u;! 2 2
where (I i s t h e v o r t i c i t y and U 1 and U2 a r e t h e v e l o c i t i e s a t t h e edges of t h e shear layer; f o r t h e free-streamline case t h e s e a r e Us = kU, and 0, r e s p e c t i v e l y , and t h e r a t e of flow of c i r c u l a t i o n i s k%,2/2. On t h e o t h e r hand, t h e r a t e a t which Circulation i s c a r r i e d downstream by t h e v o r t i c e s i s nr, where I? i s t h e c i r c u l a t i o n per vortex and n i s t h e shedding frequency. The experiments of Fage and Johansen ( r e f . 3) i n d i c a t e t h a t only a f r a c t i o n E of t h e v o r t i c i t y i n t h e shear l a y e r s is found i n t h e i n d i v i d u a l v o r t i c e s f a r t h e r downstream. They estimated E t o be about 1/2. It i s necessary t o allow f o r t h i s i n w r i t i n g t h e r e l a t i o n between t h e c i r c u l a t i o n produced a t t h e cylinder and that c a r r i e d downstream by t h e v o r t i c e s . Thus
or
where u i s t h e v e l o c i t y of t h e v o r t i c e s r e l a t i v e t o t h e f r e e stream and 2 i s t h e spacing along a row. F i n a l l y , i n dimensionless form,
14
NACA 'I"
3169
. (3)
2
4
This i s similar t o t h e r e l a t i o n obtained by Reisenberg, b u t he assumed b o t h E and k t o be equal t o u n i t y . Equation (3) can be w r i t t e n i n another form by introducing one of t h e &rm& parameters: r
=
2
p
(4)
UZ
Elimination of
I' from equations (3) and (4) gives
which may t h e n be used i n K$rm'n's
where h t h e form
drag formula
i s t h e width of t h e vortex s t r e e t .
CD ; d =
5.63
U -
2.25(-)
2
J
This is better written i n
= f(k,E)
U, On t h e o t h e r hand, from t h e notched-hodograph theory, one can c a l c u l a t e CD(d/d') as a f u n c t i o n of k, where d ' i s t h e d i s t a n c e between t h e f r e e s t r e a m l i n e s . It seems reasonable t o assume that h = d ' , that is.,
15
NACA TN 3169
t h a t t h e c e n t e r s of v o r t i c i t y i n t h e vortex s t r e e t a r e t h e same d i s t a n c e
a p a r t as i n t h e f r e e shear l a y e r s . simply
This then gives t h e second r e l a t i o n ,
CD d/h = CD d/d' To f i n L s o l u t i o n s , t h e l e f t - and right-hand s i d e s are p i o t t e d as functions of k. The i n t e r s e c t i o n s a r e t h e s o l u t i o n s . The left-hand s i d e CD(d/h) gives a family of curves with E as parameter. The right-hand s i d e gives another family i n which t h e c y l i n d e r shape i s t h e parameter. Figure 9 shows t h e r e s u l t . The following p o i n t s should be noted: (1) The family of curves f o r d i f f e r e n t cylinders i s a c t u a l l y a s i n g l e curve ( w i t h i n t h e width of t h e l i n e on f i g . 9) up t o about k = 1.5.
( 2 ) There are two p o s s i b l e i n t e r s e c t i o n s f o r each value of E. The upper i n t e r s e c t i o n s correspond t o u/Um > 0.5 and t h e lower ones, t o U/Um C 0 . 7 . The l a t t e r are t h e observed values. This e m p i r i c a l f a c t determines t h e choice of s i g n f o r t h e square r o o t i n t h e equation f o r u/Um above.
(3) There are no s o l u t i o n s f o r
E
g r e a t e r than about 1/2.
To choose t h e c o r r e c t value of E, some reference t o experiment i s necessary; b u t t h e circumstance that there i s only one curve f o r a l l t h e c y l i n d e r s reduces t h e e m p i r i c a l aspect t o a minimum, f o r it i s necessary, i n p r i n c i p l e , t o f i n d E experimentally f o r only one c y l i n d e r shape. Another way of expressing t h i s i s that t h e value of k i s t h e same f o r a l l c y l i n d e r s having t h e s a m e value of E . I n f a c t , it i s observed ( f i g . 6) that t h e values of k, o r of Cps, a r e approximately t h e same f o r t h e d i f f e r e n t c y l i n d e r s ( b u t not when t h e r e i s i n t e r f e r e n c e i n t h e wake). From that f i g u r e an average value of k f o r c y l i n d e r s without wake i n t e r f e r e n c e i s 1.4, which gives t h e s o l u t i o n CD(d/d') = 0.96, The corresponding values of t h e drag coefU/Um = 0.18, and E = 0.43. f i c i e n t are 1.10, 1.32 and 1.74, f o r t h e c i r c u l a r c y l i n d e r , 90° wedge, and f l a t plate, r e s p e c t i v e l y . These may be compared w i t h experimental values: For t h e c i r c u l a r c y l i n d e r CD v a r i e s from about 0.9 t o 1 . 2 i n t h i s range of Reynolds number, while f o r t h e goo wedge and f l a t p l a t e it i s about 1.3 and 1.8, r e s p e c t i v e l y , a l s o varying somewhat b u t less t h a n f o r t h e c i r c u l a r cylinder.
16
NACA TN
The shedding frequencies may a l s o be c a l c u l a t e d . terms,
3169
I n dimensionless
S = nd/Um
where again use has been made of t h e r e l a t i o n n = (U, - u)/Z and t h e assumption d ' = h. The r a t i o h/2 is t h e d r d n spacing r a t i o of 0.281. The u n i v e r s a l Strouhal number i s then
s* = 0.2 ( 81l -
k)
k
S* = 0.164, and f o r t h e cylinder Strouhal numbers t h e values 0.206, 0.167, and O.l27, r e s p e c t i v e l y . These may be compared with t h e experimental r e s u l t s of f i g u r e 5 .
Using t h e s o l u t i o n obtained above, t h i s gives
(I 4
DISCUSSION
,
The s i m i l a r i t y parameter S*(R*) together with t h e notched-hodograph theory, reduces t o one t h e number of parameters that must be found empiric a l l y i n order t o have a complete s o l u t i o n . E s s e n t i a l l y , it allows t h e drag t o be determined from a measurement of t h e shedding frequency.2 2To determine t h e drag f o r a given cylinder and Reynolds number R, t h e procedure i s as follows: On t h e S*(R*) p l o t a value of S* i s picked o f f a t a value of R* which must be guessed, a t f i r s t , b u t which may be expected t o be a l i t t l e higher than R. Then t h e cylinder Strouhal number S i s computed from t h e measured shedding frequency. The r a t i o S/S* = k d/d' (eq. (1))e s s e n t i a l l y gives k from t h e p l o t of k d/d' versus k (which has not been included here but may e a s i l y be obtained from f i g . 7). Once k has been determined, t h e value of R* = R k d ' / d may be computed. If it does not check t h e o r i g i n a l assumed value, then a new value of S* I s found, and t h e procedure i s repeated. No more than t h e one i t e r a t i o n i s necessary, f o r S* v a r i e s q u i t e slowly with R*. Once k i s determined, t h e drag i s found from t h e p l o t of CD(k) given i n reference 5 .
3v
h-
?
NACA TN
3169
17
The range of Reynolds numbers f o r which t h e s i m i l a r i t y i s v a l i d has not been e s t a b l i s h e d . A t t h e upper end s i m i l a r i t y w i l l probably e x i s t so long as t h e r e i s p e r i o d i c shedding, which f o r t h e c i r c u l a r c y l i n d e r i s up t o R = 105. A t t h e lower end it i s not c e r t a i n whether t h e d i s c r e p ancy of f i g u r e 8 i s due only t o t h e experimental d i f f i c u l t y mentioned i n t h e s e c t i o n "Bluff-Body S i m i l a r i t y . " More probably, it i s i n s u f f i c i e n t i n t h a t range t o lump a l l t h e Reynolds number e f f e c t s i n t o t h e one parame t e r R*. An example of t h e d e t a i l s that may have t o be considered i s t h e problem of t r a n s i t i o n i n t h e free shear l a y e r s . A t higher Reynolds numbers t h e l a y e r s a r e t u r b u l e n t almost from s e p a r a t i o n , b u t a t lower values they remain laminar f o r some d i s t a n c e downstream. The p o i n t of t r a n s i t i o n may be expected t o be d i f f e r e n t f o r d i f f e r e n t c y l i n d e r s even a t t h e same R*, s i n c e t h e shear l a y e r s w i l l have experienced d i f f e r e n t p r e s s u r e g r a d i e n t s , and so f o r t h . The p r e s e n t s o l u t i o n m y be a s u i t a b l e s t a r t i n g p o i n t f o r a more d e t a i l e d study of such e f f e c t s . The s o l u t i o n obtained by j o i n i n g t h e f r e e - s t r e a m l i n e flow t o t h e vortex s t r e e t might a l s o be regarded as a kind of s i m i l a r i t y s o l u t i o n , depending on only a single experimental measurement. The Reynolds number dependence does not appear e x p l i c i t l y . Instead, t h e r e i s a dependence on E, t h e f r a c t i o n of t h e shear-layer v o r t i c i t y t h a t appears i n i n d i v i d u a l v o r t i c e s . The i n t e r e s t i n g r e s u l t i s that f o r a given E t h e d i f f e r e n t c y l i n d e r s have t h e same value of k, t h a t i s , of base p r e s s u r e , t h e i r drag c o e f f i c i e n t s t h e n being simply p r o p o r t i o n a l t o t h e wake widths and i n v e r s e l y p r o p o r t i o n a l t o t h e shedding frequencies. I n both cases t h e need f o r an a d d i t i o n a l emp'irical r e l a t i o n appears
t o be connected w i t h t h e need f o r more understanding of t h e flow i n t h e region of v o r t e x formation. may prove t o be u s e f u l .
For t h i s , t h e technique of wake i n t e f e r e n c e
The r e s u l t s obtained may be a p p l i e d t o c y l i n d e r shapes o t h e r t h a n t h e three t h a t were considered so long as t h e y are of comparable " b l u f f n e s s . " For i n s t a n c e , t h e y probably cannot be a p p l i e d d i r e c t l y t o bodies of small percentage t h i c k n e s s , such as a t h i n wedge, f o r which t h e h i s t o r y of t h e boundary-layer development i s q u i t e d i f f e r e n t from that on b l u f f e r c y l i n d e r s .
C a l i f o r n i a I n s t i t u t e of Technology, Pasadena, C a l i f . , August 13, 1953.
NACA TN
18
3169
REFERENCES ,4
1. F l u i d Motion Panel of t h e Aeronautical Research Committee and Others ( S . Goldstein, e d . ) : Modern Developments i n F l u i d Dynamics. Vol. 11.
1
The Clarendon P r e s s (Oxford), 1938. 2. Fage, A., and Johansen, F. C.: On t h e Flow of A i r Behind an I n c l i n e d F l a t P l a t e of I n f i n i t e Span. R . & M. No. 1104, B r i t i s h A.R.C., 1927; a l s o Proc. Roy. SOC. (London), s e r . A, vol. 116, no. 773, Sept. 1, 1927, pp. 170-197.
3. Fage, A., and Johansen, F. C . : The S t r u c t u r e of Vortex Sheets. R . & M. No. 1143, B r i t i s h A.R.C., 1927; a l s o P h i l . Mag., ser. 7, v o l . 5, no. 28, Feb. 1928, pp. 417-441.
4. Fage, A.:
The A i r Flow Around a C i r c u l a r Cylinder i n t h e Region Where t h e Boundary Layer Separates From t h e Surface. R . & M. No. 1179, B r i t i s h A.R.C., 1928.
5.
Roshko, Anatol: A New Hodograph f o r Free-Streamline Theory. TN 3168, 1954.
NACA
6. Heisenberg, Werner:
D i e absoluten Dimensionen d e r Karmnschen. Wirbelbewegung. Phys. Zs., Bd. 23, Sept. 15, 1922, pp. 363-366. (Available i n English t r a n s l a t i o n as NACA TN 126.)
7. Chapman, Dean R.:
An Analysis of Base Pressure a t Supersonic V e l o c i t i e s and Comparison With Experiment. NACA Rep. 1051, 1951. (Supersedes NACA TN 2137.)
8. Crocco, Luigi, and Lees, Lester:
A Mixing Theory f o r t h e I n t e r a c t i o n Between D i s s i p a t i v e Flows and Nearly I s e n t r o p i c Streams. J o u r . Aero. S c i . , v o l . 19, no. 10, Oct. 1952, pp. 649-676.
9. S c h i l l e r , L., and Linke, W.:
Druck- und Reibungswiderstand des Zylinders b e i Reynoldsschen Zahlen 5000 b i s 40000. Z.F.M., Jahrg. 24, Nr. 7, Apr. 13, 1933, pp. 193-198. (Available i n English t r a n s l a t i o n as NACA TM 715.)
10. Roshko, Anatol: On t h e Development of Turbulent Wakes From Vortex S t r e e t s . NACA TN 2913, 1953.
4
TABLE I
CALCULATED AND MEASURED FLOW PARAMETERS FOR TEST CYLINDERS
C
~
~
Cylinder and symbol (as plotted in pigs. 4, 5, 6, and 8)
A
0
B
8
ST
d' % k T d
- -- S*
R*
8,030 1.48: 1.633 ).159 19,50C 1.48t 1.625 .154 26,60c ~2,1301.46: 1.674 .162 29,70C ~3,6401.46( 1.680 .I6733,50C L5,950 1.451 1.682 .166 39,loC L7,900 1.44: 1.710 .17144,2OC 3,220 1.42; 1.742 -173 8703C 3,900 1.45( 1.700 -175 9,61C 4,190 1.42t 1.747 .174 10,44C 5,670 1.47~1.659 .162 13,800~ 6,320 1.45( L .700 .166 15,60c 6,610 1.42; 1.742 .174 16,50c 7,950 1.45t 1.682 .164 19,50c L1,OW
,143C 3,850 1.45t 1.700 .168 9,50C .1425 4,790 1.45: 1.690 .165 ii,80c .141e 5,860 1.46: ~.678 .163 14,40C .1407 7,060 1.44; 1.706 .166 17,40C .1401 7,900 1.45: 1.700 .164 19,50C .1401 8,860 1.46; L .675 .161 21,70c .141C 9,630 1.441 ~.721.I69 23,gOC .140t! -0,630 .140? -1,1101.44E L 705 .166 27,40C
I)
D
0
0 d=2.22c
.2060 5,850 1.334 .2065 7,150 1.36E .2067 9,920 1.385 .2041 .1,6501.397 .2010 .j,200 1.40C .2015 .4,2001.401 .2004 -7,1301.421 .2020 -5,2201.42C .2100
E UaD
.2110 .2097 -2097 .2124 2075 .2065 .2088
-
0
d =0.638
.2103
.2090 .2091 .2090 .2081 .2050 -
L.206 L .151
t .12a L .120 L .116 L. 114
L .093
t .096
9,400 .i74 11,250 .168 15,500 .164 18,200 .160 20,600 .160 22,200 .i54 26,700 .156 25,800 .1%
885 1.395 L .120 .169 1,380 956 1.3& t.148 .176 1,510 1,078 1.35~L .179 .18j 1,720 1,270 1.325 ..215 .i92 2,050 1,510 1.342 t .190 .188 2,420 1,700 1.322 i.230 -193 2,770 1,935 1.311 t.250 -197 3,180 2,145 1.322 t.230 -194 3,500 2,950 1.322 ~.230 .196 4,800 3,250 1* 334 t.205 .18a 5,250 3,675 1.33~t.243 .195 6,060 3,950 1 . W .I95 4,330 1.33E L .2O0 4,890 1.34C - .195
&No base-pressure measurements made.
L
-
R
k
cPS
- -- -
3.144C .1411 ,142f .1445 .1441 .1442 .141C .14gC .142C .144C .14K .142C .142C
S
135 7,53c 1-391 -0.94
).
.I33 -0, 32c 1.396 -.95 .I33 .1,40C 1.37; -.68 .136 2,EQC 1 37C -.88
-
135 .135 .I33 .140 .I33 *
135
.132 *
133
- 133
-4,536~ 1.36E -.87 .6,80c 1* 35f -.a4 3,02C 1 335 -.ea 3,6& 1.36~ - .85 3,93c 1.34C -.80 5,32c 1.3& -.go 5,93c 1.36~ -.85 6,20c 1.335 -.eo 7,45c 1.36t -.88
-
3,69c 1.392 4,59c 1* 391 5,610 1.40c 6,76c 1.38: 7,570 1* 39C 8,500 1.40C 9,22C 1.3& 0,200 135 0,60c 1.38:
.I37 .136 ,136 .135 .134 * 135 .I35 135
-.93 -.94 - .95 -.g1
-.93 -.96
-.go -e92
-
5,730 1 309 -.71 .203 7,000 1.34c -.80 .205 9,720 1.360 -.85 .200 1,400 1* 37c -.88 .202
197 .198 * 197 .198
2,900 1* 372 3,900 1.375 6,800 1.394 4,900 1 * 392
-.88 -.89 -.94 -.94
.209
880 L 390 950 1.375 1,070 L .345 1,265 1.324 1,500 L. 337 1,690 1 317 1,930 1.306 2,135 L .317 2,940 L. 317 3,240 L. 334 3,660 1.325 3,930 1.335 4,310 L. 333 4,870 1.335 -
-.93 -.89 -.81
.210
.209 .209 .212
.207 .206 .208 .210
.208 .20a
.208 .207 .204
-.75
-.79 -.74 -.71 -.74 -.74 -.78 - .76 -.78 -.78 -.78
I
20
NACA TN
3169
TABLE I.- Concluded CALCULATED AND MEASURED
now
Cylinder and symbol (as plotted in Ygs. 4, 5, 6, and 8
T'
%
d'
% a
S*
4,16C 1.473 1.31: 1.16( 5,05C 1.515 1.29: .16: 7,UC 1.497 1.30( 155 8,15c 1.482 1.31~ .16j 8,75c 1.497 l.3OC .16: .la81 10,hOC 1.492 1.30; .16L .la112,93C 1.482 1.31~ .16: .la83 15,27C 1.497 l.30C .161 .la82 16,92c 1.485 l.3Ol .16f -18% 19,llC 1.480 1.311 .I64 .16; .1905 7,18C 1.489 1.3& .1856 8,43C 1.480 1.311 .16: .1&9 9,40C 1.495 1.302 .161 .le46 11,04c 1.487 1.30: .16z -1855 1 3 , 6 6 ~1.499 1.30C .161 ~88015,320 1.497 1.30C .16; .la52 17,300 1.490 1.504 .16; .le48 19,210 1.478 1.312 .164 .1%6 20,60c 1.490 1.304 .16: .:t872 22,700 1.489 1.306 .164
1.1886 .19lO .18jl .1&5 ,1878
1
R*
S 1.05:
1.175
1.395 1.437 .174 1.419 1.404 17: .17f 1,419 .17E 1.414 * 1-75 ~2,3001.404 * 175 14,500 1.419 .17E ~6,0001.407 .17t ~8,1001.402 .181 6,800 1.411 .176 7,980 1.402 17: 8,900 1.417 17: -0,500 1.409 .176 -2,900 1.421 .17~-4,500 1.419 .176 -6,400 1.412 * 175 -8,200 1.400 177 -9,500 1.412 177 !1,500 1.411 .181
.
G
4
PARAMETERS FOR TEST CYLINDERS
3,940 4,790 6,750 7,730 8,300 9,850
-
-0.9: -1.0' -1.0:
-.9' -1.0:
-1.a -.g
-1.0:
-.9t
-.Y -.95 -.g' -1.0.
-.9! -1.0: -1.0:
-.9: -.% -.95 -.9!
A
#
uLaf -A
.430cm 4
.1&6 .I830 -1865 .1&5 .la32
.la24 .1820
38 Pressure '\tube
.la18
.1%6
1,210
1,380 1,570 2,030 2,240 2,510 2,750 3,030 3,400
1.461 1.322 1.457 1.326 L.480 1.310 t.476 1.312 ..495 1.301 t.483 1.310 t.488 1.306 ..488 1.306 ..492 1.303
-.
D
J 1
+
I-
I 10 K
0
.0998 5,600 311 2.115 .io08 6,430 -.3oi 2.162 .io02 6,750 ..323 2.063 .io30 7,650 ..jO3 2.155 .io06 6,950 ..j42 1.990 -.?TO 2.035 .350 1.963 -330 2.035 330 2.035 -330 2.035 .323 2.060 .260 .27O .258 .240 .244 .236 .246
aNa base-pressure measurements made.
1.370 1.351 1.385 1.445 1.430 1.461 1.425
.165 .167 .165 .164
.01c
.161 .160 .160 .161 .161
.162 1.55 .154 .161 .158 .162 .157 *
1,200 1.446 -1.05 1,370 1.442 -1.Of
1,550 1.465 1.461 1.480 2,490 1.468 2,722 1.4'73 3,000 1.473 3,370 1.477
2,010 .181 2,220
.160
.167 .156 .170 .149 159 .156 .162 .161 .160 .162
.181 .181 .185 .183
.181 .180 .le0
.183 .o6t
.094 095 .094 * 097 .094 .097
-1.1: -1.lt -1.17 -1.1; -1.lE
1.230 -.51 1.220 -.45 1.241 -.54 1.223 -.5C 1.260 -.55 1.248 -.5t 1.266 - . 6 ~ 1.248 -.56 1.248 -.56 1.248 -.56 1.241 -.54
* 099 .098 .098 .098
5,250 6,030 6,340 7,180 6,500 7,850 0,400 1,900 3,300 5,000 6,700
.147 ,143 137 .135 135 .134 * 135
5,850 1.242 -.54 7,220 1.245 -.55 0,100 1.232 -.52 1,200 1.216 -.48 2,600 1.221 -.49 5,800 1.211 -.47 5,300 1.220 -.49
*
.lo1
.020
-1.1: -1.14
NACA TN
t
3169
21
NACA TN
22
W
a W
I-
-JI-
a aI3
0 X
I-
30
3169
NACA TN 3169
23
-3
?
0
'.
0
t
Q Q.
0
0
24
NACA 'I"
51-69
.I 8
S* .I 6
.I 4
.22
.20
0
S
-.2
.I 8
CPS .I 6
-.4
.I4
-. 6
.I 2
-.8
.I 0
- 1.0 5
4
c/d -C
Figure 4.- Wake interference.
6
7
,4v
NACA TN 3169
26
*
a
7-71" 0
a0
0
cu
NACA TN 3169
2 .I
2.4
2.2
d '/C 2 .c
8
I .a
4
I .e
I .4
1.2
I .o
NACA TN
28
3169
m
0
w
(I
w
I
J
5v
NACA TN
3169
3.5
3.c
d
COX 2.5
d
CDji 2.0
8
I .s
I .o
.5
0
I,
k Figure 9.- Wake solutions.
NACA-Langley
-
1-1-54
-
1000
cr, CD I-!
m
z b E
CASE F I L E COPY NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS TECHNICAL NOTE 3169
ON THE DRAG AND SHEDDING FREQUENCY O F
TWO -DIMENSIONAL B L U F F BODIES By Anatol Roshko
California Institute of Technology
Washington July 1954
r
NATIONAL ADVISORY COlNLTTEE FOR AERONAUTICS e
TECHNICAL NOTE 3169
J
ON THE DRAG AND SHEDDING FREQUENCY OF TWO-DIMENSIONAL BLUFF BODIES By Anatol Roshko
SUMMARY A semiempirical study is made of the bluff-body problem.
Some experiments with interference elements in the wake close behind a cylinder demonstrate the need for considering that region in any complete theory.
v
I
Dimensional analysis of a simple model of the region leads to a universal Stromal number S* which is then experimentally determined a s a function of wake Reynolds nuniber R*. This result, together with free-streamline theory, allows the drag to be calculated from a measurement of the shedding frequency and furnishes a useful correlation between different bluff cylinders. By allowing for some annihilation of the vorticity in the free shear l/ayys, it is shown how to corribine the free-streamline theory with Karman's theory of the vortex street to obtain a solution dependent on only one experimental measurement. INTRODUCTION
The problem of the drag of bluff bodies in incompressible flow is one of the oldest in fluid mechanics, but it remains one of the most important, for practical reasons as well as for its theoretical interest The two major contributions toward/ altheoretical understanding are the well-known ones of Kirchhoff and Karman. These attack two aspects of the problem that must be understood, namely, the potential flow in the vicinity of the cylinder and the wake farther downstream, but neither by itself can furnish a complete theory. Much of the work devoted to the problem through the years has been essentially the, elaboration and application of the theories of Kirchhoff and drman.
2
NACA TN 51-69
It has become increasingly evident, however, that a complete solution will not be obtained until one is able to "join" the two parts of the problem and that this will require an understanding of the flow in the early stages of the wake, that is, the first few diameters downstream of the cylinder. On the experimental side, there has also been a continued activity. It is surprising that the influence on theoretical developments has been rather small, for considerable useful information has been compiled, some as long ago as 20 or 50 years. Much of this is referred to in chapters IX and XI11 of reference 1. Particular mention must be made of the work of Fage and his coworkers (e.g., refs. 2, 3 , and 4) who made very useflil investigations of the flow near the cylinder and in the early stages of the wake. The ideas of this report grew largely out of a study of their work. The interest at GALCIT in the flow past bluff bodies has been connected not so much with the problem of the drag as with that of turbulent wakes. Many of the turbulent flows that are used for experimental studies (e.g., behind grids) and almost all those that cause practical difficulties (e.g., buffeting) are produced in the wakes of bluff bodies. Much of the empiricism connected with these problems can be resolved only by a better understanding of how the wake is related to the body which produces it. This includes questions of wake scale, frequencies, energy, interference between wakes, and so forth. However, whatever the approach, one is led to consider the relation between the wake and the potential flow outside the wake and cylinder. A short review of the theory of flow past bluff bodies is presented and some experiments are described which demonstrate how critically the whole problem depends on that part of the wake immediately behind the cylinder. In the remainder of the report the free-streamline theory of reference 5 is combined with some experimental results to obtain a much needed correlation between bluff cylinders1 of different shapes. This furnishes at the same time some of the sought-after relations between wake and cylinder.
The experiments were performed in the 20- by 20-inch low-turbulence wind tunnel at GALCIT, under the sponsorship and with the financial assistance of the National Advisory Committee for Aeronautics. Standard %he term "cylinder" is used throughout to denote a body whose cross-sectional shape is the same at every section along the span. This is the so-called two-dimensional body, which was the only kind for which measurements were made here. The term is applied to all cross-sectional shapes, including the limiting case of a flat plate normal to the flow, in which case the cross-sectional shape is simply a line.
I
4 b
NACA TN
3169
3
hot-wire equipment, manometry, and so forth were employed. The work is part of a broader program of turbulence research directed by Dr. H. W. Liepmann; the author is indebted to him and to other members of the GALCIT staff for many discussions. SYMBOLS b'
distance between outer edges of free shear layers
CD
drag coefficient
cP
pressure coefficient
cPS
base-pressure coefficient
C
distance *om back of cylinder to trailing edge of interference element
d
cylinder diameter or breadth
d'
distance between free streamlines
h
width of vortex street
k
base-pressure parameter,
"r
base-pressure parameter, uncorrected for tunnel blockage
2
longitudinal vortex spacing
n
vortex shedding frequency
PS
base pressure
R
Reynolds number based on cylinder diameter, Ud/v
RT
Reynolds nurdber uncorrected for tunnel blockage
R*
Reynolds number based on wake parameters, Usd'/v
S
cylinder Strouhal number, nd/U,
ST
Strouhal number uncorrected for tunnel blockage
S*
wake Strouhal number, nd'/U,
U,/U,
or
\Il-cpi
NACA TN
3169
Fage's form of Stromal number, nb'/Um velocity along shear layer velocities at edges of shear layer free-stream velocity blockage correction to measured free-stream velocity velocity on free streamline at separation velocity of vortices relative to f'ree-stream velocity distance downstream angle measured from stagnation point of circular cylinder circulation per vortex fraction of shear-layer vorticity that goes into individual vortices vorticity
.
3
coordinate normal to shear layer kinematic viscosity REVIEW OF THXORY OF FLOW PAST BLUFF BODIES
Although the main contributions to the theory of flow past bluff bodies are well-known, a review is useful to bring out the important features of the problem.
In the free-streamline theory developed by Kirchhoff, the free shear layers which are known to separate from bluff bodies a r e idealized by surfaces (streamlines) of velocity discontinuity. These f'ree streamlines divide the flow into a wake and an outer potential field. The possibility of treating the problem this way in two parts is important to note. Kirchhoff's theory, however, considerably underestimates the drag, and the failure is easily traced to the assumption which is made about the velocity on the free streamline. It is assumed that the velocity there is the free-stream velocity or, what amounts to the same thing, that the pressure in the wake and on the cylinder base is
*
5
NACA TN 3169
c.
the free-stream pressure. It is known, however, that the pressure on the base, behind the separation points, is actually much lower, which corresponds to a velocity on the free streamline which is higher than the free-stream value. This lower base pressure accounts for the higher drag actually observed. Kirchhoff's theory has been applied, by a long line of successors, to many other cylinder shapes; and indeed it can be applied in principle to any shape whatever, the difficulty being only of computation. (In cases such as the circular cylinder, where there is not a well-defined, fixed separation point, an additional assumption must be made.) In all cases, however, the theory gives values of drag much lower than those observed, and always for the reason that the value assumed for the separation velocity is too low. /
Kardn, in his famous theory of the vortex street, attacked the problem by way of another characteristic feature of flow past bluff bodies, that is, the phenomenon of periodic vortex shedding. The theory is incomplete in that it cannot by itself relate the vortexstreet dimensions and velocities to the cylinder dimension and freestream velocity. Two additional relations are required, and these must come from elsewhere, usually from experiment.
d&
These two examples by Kirchhoff and are, however, representative of the two parts of the flow that will have to be considered in any complete theory. While each part the potential field and the wake may be considered separately, the complete solution will only be found by discovering how to join them. Indeed, Heisenberg (ref. 6) attempted to obtain such a closed solution by joining the Kirchhoff solution to K6rdn's vortex street. His solution gives a value for the drag in good agreement with that for a flat plate set normal to the flow, but it also gives the same value for any other cylinder shape, as pointed out by Ka'rdn in a footnote to the same paper. There is an inconsistency in the theory in that the Kirchhoff flow, which is taken as one element in the synthesis, predicts a drag coefficient which is different (lower) from the final result. In short, the Kirchhoff flow is not a suitable starting point for such solutions unless it is modified to allow more realistic base pressures.
-
In reference 5 it was shown how such a modification may be made. Instead of restricting the separation velocity to the free-stream value U , it is allowed to assume an arbitrary value Us = kUm. The base-pressure coefficient is then Cps = 1 k2. For k = 1 this reduces to the Kirchhoff case, but for agreement with experiment k must be greater than 1. This modified Kirchhoff theory will be referred to as the notched-hodograph theory, after the hodograph upon which it is based.
-
-
6
NACA TN
3169
Another method by which t h e p o t e n t i a l p a r t of t h e flow might be "joined" t o t h e wake i s t o use t h e momentum-diffusion theory which has been a p p l i e d i n c a l c u l a t i o n s of base p r e s s u r e on supersonic p r o j e c t i l e s ( r e f s . 7 and 8) These c a l c u l a t i o n s a r e made on t h e b a s i s of a mixing theory, wherein t h e p r e s s u r e deficiency on t h e base i s assumed t o be "supported" e n t i r e l y by t h e d i f f u s i o n of momentum across t h e shear l a y e r s . I n f a c t , it appears i n e v i t a b l e t h a t some such c a l c u l a t i o n w i l l be necess a r y b e f o r e a complete t h e o r e t i c a l formulation of t h e problem can be obtained. The mechanism of t h e wake immediately downstream of t h e c y l i n d e r i s t h e main i t e m i n t h e coupling between wake and p o t e n t i a l flow, and it w i l l probably be e s s e n t i a l t o understand it. Unfortunately, t h e idea of simple momentum d i f f u s i o n across t h e shear l a y e r s , which appears t o be s u i t a b l e f o r t h e supersonic flows, i s not by i t s e l f s u f f i c i e n t f o r t h e incompressible flow p a s t a b l u f f c y l i n d e r . The "coupling region" immed i a t e l y downstream of t h e body i s a l s o that i n which t h e v o r t i c e s are formed, and t h e y are an e s s e n t i a l p a r t of the mechanism, as w i l l be shown i n t h e following s e c t i o n s .
.
EFFECT OF VORTEX FORMATION Figure 1 shows a p r e s s u r e t r a v e r s e made along t h e c e n t e r l i n e of t h e wake behind a f l a t p l a t e s e t normal t o t h e flow. The measurement i s not simple t o make because of t h e l a r g e t r a n s v e r s e v e l o c i t y f l u c t u a t i o n s a s s o c i a t e d w i t h t h e v o r t e x shedding. A s t a t i c probe o r i e n t e d along t h e c e n t e r l i n e of t h e wake experiences a nonstationary crossflow, f l u c t u a t i n g a t t h e shedding frequency. A s f a r as t h e mean flow i s concerned, t h e p r e s s u r e i s constant over t h e circumference of t h e c y l i n d r i c a l probe, s i n c e i t s dimensions a r e small compared w i t h t h e flow f i e l d . Because of t h e crossflow, however, t h e r e i s a p r e s s u r e v a r i a t i o n over t h e circumference, jhst as t h e r e i s for a cylinder placed normal t o a stream with p e r i o d i c v e l o c i t y . "herefore t h e p r e s s u r e measured v a r i e s with t h e p o s i t i o n of t h e o r i f i c e on t h e circumference of t h e probe. When t h e o r i f i c e i s a t one of t h e s t a g n a t i o n p o i n t s of t h e crossflow t h e p r e s s u r e i s t h e h i g h e s t , and when it i s a t 90° from t h e s t a g n a t i o n p o i n t t h e pressure i s t h e lowest. The pressures measured with t h e o r i f i c e a t t h e s e l i m i t i n g p o s i t i o n s are shown i n f i g u r e 1. The c o r r e c t p r e s s u r e should be somewhere between. Now on a c y l i n d e r i n steady crossflow t h e pressure c o e f f i c i e n t i s zero a t 30' i n p o t e n t i a l flow and a t about 35' i n r e a l flows. It w a s simply assumed t h a t t h e l a t t e r i s a l s o t h e c o r r e c t p o s i t i o n f o r t h e o s c i l l a t i n g crossflow. The intermediate curve i n f i g u r e 1 i s t h e r e s u l t with t h e o r i f i c e a t 3 5 O from t h e s t a g n a t i o n p o i n t ; it i s believed that t h i s gives very n e a r l y t h e c o r r e c t value f o r t h e mean s t a t i c p r e s s u r e . I n any case, what i s important i s t h e low-pressure region a t about 2 p l a t e widths downstream; t h a t i s , t h e p r e s s u r e on t h e base i s not t h e lowest p r e s s u r e i n t h e wake. This measurement had a l r e a d y been made by
NACA TN
J
b
3169
7
S c h i l l e r and Linke i n 1933 ( r e f . 9 ) , but does not seem t o have been noted s i n c e . I t s s i g n i f i c a n c e seems c l e a r - t h a t t h e low pressure i s a s s o c i a t e d with a low-pressure region a t t h e center of t h e vortex which i s being formed. This low pressure must f l u c t m t e , of course, as t h e v o r t i c e s form a l t e r n a t e l y , and it i s t h e mean e f f e c t which has been measured. It seems l i k e l y then t h a t a l a r g e p a r t of t h e low pressure on t h e base of t h e f l a t p l a t e i s a s s o c i a t e d with t h e v o r t e x pressure, that i s , t h a t t h e main mechanism f o r t h e base pressure i s t o be found not i n t h e d i f f u s i o n of momentum across t h e shear l a y e r s but i n t h e dynamics of t h e v o r t i c e s . I n f a c t , t h e moment-m-diffusion theory could p r e d i c t only a monotonically increasing pressure from t h e base. Now i f t h e vortex dynamics are indeed important, then i n t e r f e r e n c e with t h e i r f o r m t i o n should have a s t r o n g e f f e c t on t h e base pressure. This w a s e a s i l y i n v e s t i g a t e d by placing a " s p l i t t e r " p l a t e along t h e tent e r l i n e of t h e wake. Figure 2 shows pressure d i s t r i b u t i o n s with and without t h e s p l i t t e r p l a t e . (The r e s u l t s given i n t h i s f i g u r e a r e f o r a c i r c u l a r cylinder i n s t e a d of t h e f l a t p l a t e . ) With t h e s p l i t t e r p l a t e t h e p e r i o d i c vortex formation i s i n h i b i t e d and t h e base pressure increases considerably. It i s s t i l l below free-stream pressure, but whether t h i s r e s i d u a l underpressure can be accounted f o r by t h e momentum-diffusion theory i s not c e r t a i n . It i s p o s s i b l e that a kind of standing vortex i s formed on each s i d e , but t h i s p o i n t w a s not i n v e s t i g a t e d f u r t h e r . I n any case, t h e momentum-diffusion theory i s i n d i f f e r e n t t o t h e omission o r i n c l u s i o n of a p a r t i t i o n along t h e c e n t r a l streamline. It i s c l e a r t h a t without t h e s p l i t t e r p l a t e t h e p e r i o d i c formation of v o r t i c e s i s an e s s e n t i a l p a r t of t h e base-pressure mechanism. Figure 3 shows how t h e s p l i t t e r p l a t e a f f e c t s t h e pressure d i s t r i b u t i o n over t h e whole cylinder circumference. It shows t h a t i n t e r f e r e n c e i n t h e "coupling region" changes t h e outer p o t e n t i a l flow, as well as t h a t i n t h e wake. EFFECT OF INTERFERENCE ELEMENT ON RELATIONS BETWEEN BASE PRESSURE AND SHEDDING FREQUENCY I n t h e experiment of f i g u r e 2 t h e chord of t h e s p l i t t e r p l a t e w a s almost 5 diameters. One immediately asks how t h e i n t e r f e r e n c e changes with changing chord. Accordingly, some measurements were made with a s p l i t t e r p l a t e whose chord w a s about 1 diameter. This w a s found not t o i n h i b i t t h e vortex formation a t a l l , though it does change t h e shedding frequency s l i g h t l y . More i n t e r e s t i n g i s t h e e f f e c t of moving t h i s s h o r t p l a t e downstream, that i s , leaving a gap between it and t h e c y l i n d e r . The e f f e c t i s t o decrease t h e shedding frequency and t o increase t h e base pressure, as shown i n t h e lower p a r t of f i g u r e 4. The shedding frequency becomes a minimum, and t h e base pressure a maximum, when t h e t r a i l i n g edge of t h e i n t e r f e r e n c e element i s 3.85 diameters downstream of t h e cylinder
I
base. It i s c l e a r t h a t such a minimum must be reached, f o r when t h e element i s very f a r downstream of t h e cylinder i t s upstream influence should be n e g l i g i b l e . What i s remarkable i s t h e abrupt jump t h a t occurs t h e r e , p r a c t i c a l l y t o t h e o r i g i n a l value. Q u a l i t a t i v e l y , it appears t h a t c l o s e t o t h e cylinder t h e element has a streamlining e f f e c t ; t h a t i s , it extends the shear l a y e r s and f o r c e s t h e v o r t i c e s t o form downstream of i t s t r a i l i n g edge. When it i s f a r downstream, t h e v o r t i c e s form on i t s upstream s i d e , i n t h e normal p o s i t i o n c l o s e t o t h e cylinder, and t h e element has only a slight e f f e c t . There must then be some c r i t i c a l posit i o n where t h e flow must choose between one of t h e two configurations. I t i s observed, i n f a c t , t h a t when t h e i n t e r f e r e n c e element i s a t t h i s c r i t i c a l p o s i t i o n , t h e flow does jump, i n t e r m i t t e n t l y , from one configur a t i o n t o t h e o t h e r , as shown by t h e double values measured t h e r e .
A t c/d = 1.13 t h e r e i s a p o i n t t h a t does not f a l l on t h e l i n e drawn f o r Cps. This w a s thought t o be an e r r o r , but f u r t h e r checks and readjustments indicated t h a t it i s real. This p o i n t corresponds t o t h e case where t h e s p l i t t e r p l a t e w a s touching t h e back of t h e cylinder; t h a t i s , t h e gap w a s completely closed. However, t h e j o i n t w a s by no means p r e s s u r e t i g h t . It seemed, r a t h e r , that some o t h e r kind of communication through t h e qap becomes e f f e c t i v e a t some f i n i t e gap width. A t p r e s e n t , t h i s has not been thoroughly i n v e s t i g a t e d . Experiments such as t h e s e may lead t o information u s e f u l f o r understanding t h e mechanics of t h e vortex formation. A n i n i t i a l s t e p i n t h i s d i r e c t i o n has been made by i n v e s t i g a t i n g t h e r e l a t i o n s h i p between t h e base pressure and t h e shedding frequency ( s e e t h e following s e c t i o n ) . BLUFF -BODY SIMILARITY Although a complete t h e o r e t i c a l s o l u t i o n seems remote a t present, t h e r e i s s t i l l t h e p o s s i b i l i t y of f i n d i n g a c o r r e l a t i o n between d i f f e r e n t b l u f f bodies, on a semiempirical b a s i s . Such a c o r r e l a t i o n m u s t be based on, and must account f o r , t h e following q u a l i t a t i v e f a c t s , which a r e indeed t h e main f e a t u r e s of t h e flow p a s t b l u f f cylinders: (1) The "bluffness" of a cylinder i s r e l a t e d t o t h e width of t h e wake, compared with t h e cylinder dimension. It i s almost i n t u i t i v e t h a t t h e b l u f f e r body tends t o diverge t h e flow more, t o c r e a t e a wider wake, and t o have l a r g e r drag. ( 2 ) The shedding frequency i s r e l a t e d t o t h e width of t h e wake, t h e r e l a t i o n being roughly inverse, so t h a t t h e b l u f f e r bodies have t h e lower Strouhal numbers. I
(3) For a given cylinder t h e shedding frequency i s r e l a t e d t o t h e base pressure. Generally, an increase i n base pressure i s accompanied
c
NACA TN
3169
9
by a decrease i n shedding frequency. This i s well i l l u s t r a t e d i n t h e case of f i g u r e 4 with t h e i n t e r f e r e n c e element (although t h e r e an anomalous behavior i s observed f o r C/d > 3.85). Thus, f o r a given cylinder a decrease i n wake width corresponds t o an increase i n drag, which seems a t variance with i n t u i t i o n . However, t h e decrease i n width i s a s s o c i a t e d w i t h an increase i n "wake v e l o c i t i e s , " t h e n e t e f f e c t being an increased wake energy corresponding t o t h e increased drag. Figure 5 shows measurements of Strouhal number made on t h r e e d i f f e r e n t cylinder shapes. These were a c i r c u l a r cylinder, a goo wedge, and a " f l a t p l a t e " normal t o t h e flow. The shedding frequencies were measured by a hot-wire anemometer placed i n t h e wake. The dimensions of t h e cylinders used a r e given i n t a b l e I. (For t h e c i r c u l a r cylinder, t h e curve has been taken from r e f . 1 0 ) . A t present, one i s concerned only w i t h t h e higher Reynolds numbers, f o r which t h e Strouhal number S i s d i s t i n c t l y d i f f e r e n t f o r t h e d i f f e r e n t cylinders, though approximately constant f o r each i n d i v i d u a l one. The f i g u r e shows that t h e c i r c u l a r cylinder, goo wedge, and f l a t p l a t e a r e increasingly b l u f f , i n that order. f o r t h e same Figure 6 i s a p l o t of base-pressure c o e f f i c i e n t Cps t h r e e cylinder shapes. The base pressure ps w a s measured a t s t a t i c o r i f i c e s on t h e backs of t h e c y l i n d e r s . I n t h e case of t h e c i r c u l a r cylinder t h e measurement w a s made a t about 130° from t h e s t a g n a t i o n p o i n t , which i s i n a region of constant pressure ( c f . f i g . 3) and i s believed t o be a more s i g n i f i c a n t p o i n t f o r t h e base pressure than t h e one a t 180°. Figure 6 shows t h a t t h e d i f f e r e n c e s i n base pressure f o r t h e d i f f e r e n t cylinders a r e not so marked as are t h e shedding frequencies and that they cannot be ordered i n terms of b l u f f n e s s . Included i n t h e f i g u r e are two cases with wake i n t e r f e r e n c e - a c i r c u l a r cylinder and a f l a t p l a t e for which t h e increase i n base pressure i s q u i t e marked. The i n t e r f e r e n c e elements which w e r e used a r e sketched i n t a b l e I.
-
The r e s u l t s of f i g u r e s 5 and 6 have been corrected f o r t u n n e l blockage. The s c a t t e r i n f i g u r e 6 i s r a t h e r l a r g e , f o r t h e probable e r r o r i n t h e c a l c u l a t i o n of Cps i s i n h e r e n t l y l a r g e ; 1-percent accuracy i n m e a s urement of pressure and v e l o c i t y gives only about 5-percent accuracy i n Cps. Now it i s well-known that t h e wakes of d i f f e r e n t b l u f f bodies are s i m i l a r i n s t r u c t u r e . I n every case t h e flow separates on t h e two s i d e s of t h e cylinder, c r e a t i n g f r e e shear l a y e r s which continue f o r a s h o r t d i s t a n c e downstream and then "roll up" i n t o v o r t i c e s , a l t e r n a t e l y on e i t h e r s i d e . The region where t h i s occurs, which extends only a few diameters downstream, was r e f e r r e d t o i n t h e s e c t i o n "Review of Theory of Flow P a s t Bluff Bodies" as t h e coupling region. Because of t h e s i m i l a r i t y i n a l l b l u f f - c y l i n d e r wakes, one would expect t o f i n d a parameter t o compare t h e wakes of d i f f e r e n t cylinders. The clue i s i n t h e shedding frequency, as may be demonstrated by a simple dimensional a n a l y s i s .
NACA TN
10
3169
Consider two p a r a l l e l . shear l a y e r s ( s k e t c h 1) which may be imagined t o
1
US
d'
Sketch 1 have been formed by some b l u f f cylinder. The c i r c u l a r arrows i n d i c a t e t h e s i g n of t h e v o r t i . c i t y . The c h a r a c t e r i s t i c frequency associated with and one can d e f i n e a wake t h i s configuration i s p r o p o r t i o n a l t o U s / d ' Strouhal number S* = nd'/Us
where n i s t h e shed.ding frequency. Strouhal number S by
It i s r e l a t e d t o t h e usual cylinder
S d'
= E d The wake S t r o u h a l number S* i s expected t o be u n i v e r s a l , that i s , t o be t h e same f o r a l l b l u f f - c y l i n d e r wakes. O f course t h i s idea i s based on an i d e a l i z e d model i n . which t h e shear l a y e r s are surfaces of d i s c o n t i n u i t y , whereas a c t u a l l y t h e condition of t h e f r e e shear l a y e r s w i l l d i f f e r from one cylinder t o another, depending on t h e i r h i s t o r i e s up t o t h e p o i n t of vortex formation. This may, however, be a secondary e f f e c t , and it i s s u f f i c i e n t a t f i r s t t o assume t h a t S* depends only on a wake Reynolds number R* = Usd'/v = R k d'/d
NACA TN
L. b
51-69
11
The idea of such a s i m i l a r i t y i s e s s e n t i a l l y t h e same as that given by Fage and Johansen ( r e f . 3 ) , b u t they omitted t h e c h a r a c t e r i s t i c wake v e l o c i t y Us. Their parameter i s S ' = nb'/Um, where b ' i s a wake width measured between t h e o u t e r edges of t h e shear l a y e r s . They found a good c o r r e l a t i o n , S ' 0.26, f o r s e v e r a l d i f f e r e n t c y l i n d e r shapes, i n s p i t e of t h e omission of a wake v e l o c i t y . This seems t o be due t o t h e f a c t t h a t , f o r t h e cases of good c o r r e l a t i o n , t h e wake v e l o c i t i e s were very nearly t h e same. Also, they d i d not introduce a wake Reynolds number. I n w h a t follows, t h e wake s i m i l a r i t y i s s t u d i e d on t h e b a s i s of t h e parameter S*(R*). The method, which i s q u i t e d i f f e r e n t from that of Fage and Johansen, depends e s s e n t i a l l y on t h e r e s u l t s of t h e notchedhodograph theory ( r e f e r r e d t o i n t h e s e c t i o n "Review of Theory of Flow P a s t Bluff Bodies"). These r e s u l t s are b r i e f l y as follows: The v e l o c i t y a t s e p a r a t i o n and on t h e i n i t i a l p a r t of t h e f r e e streamline i s Us = kU, ( s k e t c h 2 ) . The base pressure i s t h e same as that a t s e p a r a t i o n , and t h e
T
d'
Sketch 2
-
base-pressure c o e f f i c i e n t i s t h e r e f o r e Cps = 1 k2. For a given value of k, t h e p o t e n t i a l flow outside t h e wake i s completely determined, and s o t h e drag c o e f f i c i e n t CD i s a function only of k . The important r e s u l t f o r t h e present consideration i s that a wake width d ' i s defined. This a l s o depends only on k . Figure 7, which has been computed from t h e r e s u l t s of reference 5 , shows how d ' / d v a r i e s with k f o r t h e t h r e e cylinder shapes being considered. It gives t h e "measure of b l u f f n e s s " alluded t o e a r l i e r ; that i s , t h e b l u f f e r cylinders have t h e wider wakes a t a given value of k, but f o r a given cylinder t h e wake width decreases with increasing k ( i . e . , increasing d r a g ) . The shedding frequencies and base pressures which had been measured f o r f i g u r e s 5 and 6 were used t o compute S*(R*) f o r t h e various cyli n d e r s . The computation i s straightforward: k = is calculated from t h e measured base-pressure c o e f f i c i e n t . The corresponding value of d ' / d i s found i n f i g u r e 7. With t h e s e and measured values of S and R t h e corresponding values of S* and R* a r e e a s i l y c a l c u l a t e d from
-4
a
NACA TN
12
equations (1)and ( 2 ) . i n f i g u r e 8.
3169
The r e s u l t s a r e l i s t e d i n t a b l e I and p l o t t e d
"4
It i s necessary t o draw a t t e n t i o n t o s e v e r a l points: (1) The c h a r a c t e r i s t i c wake v e l o c i t y , which i n dimensionless form i s simply k, i s not measured, but i s computed from t h e base pressure. This
4
may be regarded as t h e e s s e n t i a l s t e p i n t h e coupling between t h e wake and t h e o u t e r p o t e n t i a l flow. ( 2 ) The w a k e width i s not measured b u t i s obtained from t h e theory.
( 3 ) I n computing S*, the shedding frequency and base pressure should not be f i r s t corrected f o r t u n n e l blockage. T h a t is, t h e wake parameters that a r e used must correspond t o t h e wake t h a t i s a c t u a l l y observed. Blockage gives only a second-order e f f e c t i n t h i s computation. The uncorrected parameters a r e l i s t e d i n t a b l e I as ST, RT, and k ~ , respectively.
(4)The
errors that
he made i n measuring
and
even though small i n d i v i d u a l l y (about 1 p e r c e n t ) , c o n t r i b u t e t o a p o s s i b l e e r r o r of 4 percent i n S*.
(3) A t R* < 8,000 t h e r e i s a l a r g e discrepancy between t h e values f o r t h e wedge and f o r t h e c i r c u l a r c y l i n d e r . It i s not c l e a r whether t h i s i s r e a l o r due t o experimental d i f f i c u l t i e s . The s m a l l wedge used t o o b t a i n t h e s e p o i n t s d i d not have a proper base-pressure o r i f i c e . Instead, a small tube with an open mouth w a s cemented t o t h e back of t h e wedge, and it i s not c e r t a i n that t h i s measures t h e c o r r e c t base pressure. Otherwise, t h e s i m i l a r i t y parameter S* does give a good c o r r e l a t i o n f o r a f a i r l y wide range of Reynolds number. It i s probable that it can be extended t o higher Reynolds numbers, say another order of magnitude, up t o t h e c r i t i c a l Reynolds number of t h e c i r c u l a r cylinder. It w i l l be noted that t h e s i m i l a r i t y p l o t ( f i g . 8) includes t h e cases with wake i n t e r f e r e n c e . For t h e s e t h e base pressure and corresponding wake v e l o c i t y d i f f e r considerably from those without i n t e r f e r e n c e . T h a t they f i t f a i r l y w e l l i n t o t h e s i m i l a r i t y p l o t i s taken as evidence f o r Nevertheless, an examination t h e s u i t a b i l i t y of t h e parameter S*(R*) of an i n d i v i d u a l case shows t h a t t h e r e a c t u a l l y i s a systematic v a r i a t i o n of S*. I n t h e upper p a r t of f i g u r e 4 t h e data of t h e lower p a r t have been used t o c a l c u l a t e S*, which i s seen t o vary s y s t e m a t i c a l l y with t h e p o s i t i o n of t h e i n t e r f e r e n c e element. (The "bad point" r e f e r r e d t o i n t h e s e c t i o n "Relations Between Base Pressure and Shedding Frequency" does not f i t t h i s curve e i t h e r . )
.
.*
NACA 1
TN 3169
13
-
J O I N I N G TRE FREE STREAMLINES AND VORTEX S’I!REXT I C
To close t h e K&rdn theory of r e l a t i o n s a r e needed t o r e l a t e t h e s t r e e t t o t h e free-stream v e l o c i t y hodograph theory may f u r n i s h t h e s e r e a l i s t i c way can be found t o j o i n
t h e vortex s t r e e t , two a d d i t i o n a l v e l o c i t i e s and dimensions of t h e and cylinder dimension. The notcheda d d i t i o n a l two r e l a t i o n s , provided a t h e r e s u l t s of t h e two t h e o r i e s .
One r e l a t i o n i s obtained by considering how c i r c u l a t i o n i n t h e vort i c e s i s r e l a t e d t o t h e v o r t i c i t y i n t h e f r e e shear l a y e r s . The r a t e a t which c i r c u l a t i o n flows p a s t any plane s e c t i o n of a shear l a y e r i s
L2
CU dv =
u12
- u;! 2 2
where (I i s t h e v o r t i c i t y and U 1 and U2 a r e t h e v e l o c i t i e s a t t h e edges of t h e shear layer; f o r t h e free-streamline case t h e s e a r e Us = kU, and 0, r e s p e c t i v e l y , and t h e r a t e of flow of c i r c u l a t i o n i s k%,2/2. On t h e o t h e r hand, t h e r a t e a t which Circulation i s c a r r i e d downstream by t h e v o r t i c e s i s nr, where I? i s t h e c i r c u l a t i o n per vortex and n i s t h e shedding frequency. The experiments of Fage and Johansen ( r e f . 3) i n d i c a t e t h a t only a f r a c t i o n E of t h e v o r t i c i t y i n t h e shear l a y e r s is found i n t h e i n d i v i d u a l v o r t i c e s f a r t h e r downstream. They estimated E t o be about 1/2. It i s necessary t o allow f o r t h i s i n w r i t i n g t h e r e l a t i o n between t h e c i r c u l a t i o n produced a t t h e cylinder and that c a r r i e d downstream by t h e v o r t i c e s . Thus
or
where u i s t h e v e l o c i t y of t h e v o r t i c e s r e l a t i v e t o t h e f r e e stream and 2 i s t h e spacing along a row. F i n a l l y , i n dimensionless form,
14
NACA 'I"
3169
. (3)
2
4
This i s similar t o t h e r e l a t i o n obtained by Reisenberg, b u t he assumed b o t h E and k t o be equal t o u n i t y . Equation (3) can be w r i t t e n i n another form by introducing one of t h e &rm& parameters: r
=
2
p
(4)
UZ
Elimination of
I' from equations (3) and (4) gives
which may t h e n be used i n K$rm'n's
where h t h e form
drag formula
i s t h e width of t h e vortex s t r e e t .
CD ; d =
5.63
U -
2.25(-)
2
J
This is better written i n
= f(k,E)
U, On t h e o t h e r hand, from t h e notched-hodograph theory, one can c a l c u l a t e CD(d/d') as a f u n c t i o n of k, where d ' i s t h e d i s t a n c e between t h e f r e e s t r e a m l i n e s . It seems reasonable t o assume that h = d ' , that is.,
15
NACA TN 3169
t h a t t h e c e n t e r s of v o r t i c i t y i n t h e vortex s t r e e t a r e t h e same d i s t a n c e
a p a r t as i n t h e f r e e shear l a y e r s . simply
This then gives t h e second r e l a t i o n ,
CD d/h = CD d/d' To f i n L s o l u t i o n s , t h e l e f t - and right-hand s i d e s are p i o t t e d as functions of k. The i n t e r s e c t i o n s a r e t h e s o l u t i o n s . The left-hand s i d e CD(d/h) gives a family of curves with E as parameter. The right-hand s i d e gives another family i n which t h e c y l i n d e r shape i s t h e parameter. Figure 9 shows t h e r e s u l t . The following p o i n t s should be noted: (1) The family of curves f o r d i f f e r e n t cylinders i s a c t u a l l y a s i n g l e curve ( w i t h i n t h e width of t h e l i n e on f i g . 9) up t o about k = 1.5.
( 2 ) There are two p o s s i b l e i n t e r s e c t i o n s f o r each value of E. The upper i n t e r s e c t i o n s correspond t o u/Um > 0.5 and t h e lower ones, t o U/Um C 0 . 7 . The l a t t e r are t h e observed values. This e m p i r i c a l f a c t determines t h e choice of s i g n f o r t h e square r o o t i n t h e equation f o r u/Um above.
(3) There are no s o l u t i o n s f o r
E
g r e a t e r than about 1/2.
To choose t h e c o r r e c t value of E, some reference t o experiment i s necessary; b u t t h e circumstance that there i s only one curve f o r a l l t h e c y l i n d e r s reduces t h e e m p i r i c a l aspect t o a minimum, f o r it i s necessary, i n p r i n c i p l e , t o f i n d E experimentally f o r only one c y l i n d e r shape. Another way of expressing t h i s i s that t h e value of k i s t h e same f o r a l l c y l i n d e r s having t h e s a m e value of E . I n f a c t , it i s observed ( f i g . 6) that t h e values of k, o r of Cps, a r e approximately t h e same f o r t h e d i f f e r e n t c y l i n d e r s ( b u t not when t h e r e i s i n t e r f e r e n c e i n t h e wake). From that f i g u r e an average value of k f o r c y l i n d e r s without wake i n t e r f e r e n c e i s 1.4, which gives t h e s o l u t i o n CD(d/d') = 0.96, The corresponding values of t h e drag coefU/Um = 0.18, and E = 0.43. f i c i e n t are 1.10, 1.32 and 1.74, f o r t h e c i r c u l a r c y l i n d e r , 90° wedge, and f l a t plate, r e s p e c t i v e l y . These may be compared w i t h experimental values: For t h e c i r c u l a r c y l i n d e r CD v a r i e s from about 0.9 t o 1 . 2 i n t h i s range of Reynolds number, while f o r t h e goo wedge and f l a t p l a t e it i s about 1.3 and 1.8, r e s p e c t i v e l y , a l s o varying somewhat b u t less t h a n f o r t h e c i r c u l a r cylinder.
16
NACA TN
The shedding frequencies may a l s o be c a l c u l a t e d . terms,
3169
I n dimensionless
S = nd/Um
where again use has been made of t h e r e l a t i o n n = (U, - u)/Z and t h e assumption d ' = h. The r a t i o h/2 is t h e d r d n spacing r a t i o of 0.281. The u n i v e r s a l Strouhal number i s then
s* = 0.2 ( 81l -
k)
k
S* = 0.164, and f o r t h e cylinder Strouhal numbers t h e values 0.206, 0.167, and O.l27, r e s p e c t i v e l y . These may be compared with t h e experimental r e s u l t s of f i g u r e 5 .
Using t h e s o l u t i o n obtained above, t h i s gives
(I 4
DISCUSSION
,
The s i m i l a r i t y parameter S*(R*) together with t h e notched-hodograph theory, reduces t o one t h e number of parameters that must be found empiric a l l y i n order t o have a complete s o l u t i o n . E s s e n t i a l l y , it allows t h e drag t o be determined from a measurement of t h e shedding frequency.2 2To determine t h e drag f o r a given cylinder and Reynolds number R, t h e procedure i s as follows: On t h e S*(R*) p l o t a value of S* i s picked o f f a t a value of R* which must be guessed, a t f i r s t , b u t which may be expected t o be a l i t t l e higher than R. Then t h e cylinder Strouhal number S i s computed from t h e measured shedding frequency. The r a t i o S/S* = k d/d' (eq. (1))e s s e n t i a l l y gives k from t h e p l o t of k d/d' versus k (which has not been included here but may e a s i l y be obtained from f i g . 7). Once k has been determined, t h e value of R* = R k d ' / d may be computed. If it does not check t h e o r i g i n a l assumed value, then a new value of S* I s found, and t h e procedure i s repeated. No more than t h e one i t e r a t i o n i s necessary, f o r S* v a r i e s q u i t e slowly with R*. Once k i s determined, t h e drag i s found from t h e p l o t of CD(k) given i n reference 5 .
3v
h-
?
NACA TN
3169
17
The range of Reynolds numbers f o r which t h e s i m i l a r i t y i s v a l i d has not been e s t a b l i s h e d . A t t h e upper end s i m i l a r i t y w i l l probably e x i s t so long as t h e r e i s p e r i o d i c shedding, which f o r t h e c i r c u l a r c y l i n d e r i s up t o R = 105. A t t h e lower end it i s not c e r t a i n whether t h e d i s c r e p ancy of f i g u r e 8 i s due only t o t h e experimental d i f f i c u l t y mentioned i n t h e s e c t i o n "Bluff-Body S i m i l a r i t y . " More probably, it i s i n s u f f i c i e n t i n t h a t range t o lump a l l t h e Reynolds number e f f e c t s i n t o t h e one parame t e r R*. An example of t h e d e t a i l s that may have t o be considered i s t h e problem of t r a n s i t i o n i n t h e free shear l a y e r s . A t higher Reynolds numbers t h e l a y e r s a r e t u r b u l e n t almost from s e p a r a t i o n , b u t a t lower values they remain laminar f o r some d i s t a n c e downstream. The p o i n t of t r a n s i t i o n may be expected t o be d i f f e r e n t f o r d i f f e r e n t c y l i n d e r s even a t t h e same R*, s i n c e t h e shear l a y e r s w i l l have experienced d i f f e r e n t p r e s s u r e g r a d i e n t s , and so f o r t h . The p r e s e n t s o l u t i o n m y be a s u i t a b l e s t a r t i n g p o i n t f o r a more d e t a i l e d study of such e f f e c t s . The s o l u t i o n obtained by j o i n i n g t h e f r e e - s t r e a m l i n e flow t o t h e vortex s t r e e t might a l s o be regarded as a kind of s i m i l a r i t y s o l u t i o n , depending on only a single experimental measurement. The Reynolds number dependence does not appear e x p l i c i t l y . Instead, t h e r e i s a dependence on E, t h e f r a c t i o n of t h e shear-layer v o r t i c i t y t h a t appears i n i n d i v i d u a l v o r t i c e s . The i n t e r e s t i n g r e s u l t i s that f o r a given E t h e d i f f e r e n t c y l i n d e r s have t h e same value of k, t h a t i s , of base p r e s s u r e , t h e i r drag c o e f f i c i e n t s t h e n being simply p r o p o r t i o n a l t o t h e wake widths and i n v e r s e l y p r o p o r t i o n a l t o t h e shedding frequencies. I n both cases t h e need f o r an a d d i t i o n a l emp'irical r e l a t i o n appears
t o be connected w i t h t h e need f o r more understanding of t h e flow i n t h e region of v o r t e x formation. may prove t o be u s e f u l .
For t h i s , t h e technique of wake i n t e f e r e n c e
The r e s u l t s obtained may be a p p l i e d t o c y l i n d e r shapes o t h e r t h a n t h e three t h a t were considered so long as t h e y are of comparable " b l u f f n e s s . " For i n s t a n c e , t h e y probably cannot be a p p l i e d d i r e c t l y t o bodies of small percentage t h i c k n e s s , such as a t h i n wedge, f o r which t h e h i s t o r y of t h e boundary-layer development i s q u i t e d i f f e r e n t from that on b l u f f e r c y l i n d e r s .
C a l i f o r n i a I n s t i t u t e of Technology, Pasadena, C a l i f . , August 13, 1953.
NACA TN
18
3169
REFERENCES ,4
1. F l u i d Motion Panel of t h e Aeronautical Research Committee and Others ( S . Goldstein, e d . ) : Modern Developments i n F l u i d Dynamics. Vol. 11.
1
The Clarendon P r e s s (Oxford), 1938. 2. Fage, A., and Johansen, F. C.: On t h e Flow of A i r Behind an I n c l i n e d F l a t P l a t e of I n f i n i t e Span. R . & M. No. 1104, B r i t i s h A.R.C., 1927; a l s o Proc. Roy. SOC. (London), s e r . A, vol. 116, no. 773, Sept. 1, 1927, pp. 170-197.
3. Fage, A., and Johansen, F. C . : The S t r u c t u r e of Vortex Sheets. R . & M. No. 1143, B r i t i s h A.R.C., 1927; a l s o P h i l . Mag., ser. 7, v o l . 5, no. 28, Feb. 1928, pp. 417-441.
4. Fage, A.:
The A i r Flow Around a C i r c u l a r Cylinder i n t h e Region Where t h e Boundary Layer Separates From t h e Surface. R . & M. No. 1179, B r i t i s h A.R.C., 1928.
5.
Roshko, Anatol: A New Hodograph f o r Free-Streamline Theory. TN 3168, 1954.
NACA
6. Heisenberg, Werner:
D i e absoluten Dimensionen d e r Karmnschen. Wirbelbewegung. Phys. Zs., Bd. 23, Sept. 15, 1922, pp. 363-366. (Available i n English t r a n s l a t i o n as NACA TN 126.)
7. Chapman, Dean R.:
An Analysis of Base Pressure a t Supersonic V e l o c i t i e s and Comparison With Experiment. NACA Rep. 1051, 1951. (Supersedes NACA TN 2137.)
8. Crocco, Luigi, and Lees, Lester:
A Mixing Theory f o r t h e I n t e r a c t i o n Between D i s s i p a t i v e Flows and Nearly I s e n t r o p i c Streams. J o u r . Aero. S c i . , v o l . 19, no. 10, Oct. 1952, pp. 649-676.
9. S c h i l l e r , L., and Linke, W.:
Druck- und Reibungswiderstand des Zylinders b e i Reynoldsschen Zahlen 5000 b i s 40000. Z.F.M., Jahrg. 24, Nr. 7, Apr. 13, 1933, pp. 193-198. (Available i n English t r a n s l a t i o n as NACA TM 715.)
10. Roshko, Anatol: On t h e Development of Turbulent Wakes From Vortex S t r e e t s . NACA TN 2913, 1953.
4
TABLE I
CALCULATED AND MEASURED FLOW PARAMETERS FOR TEST CYLINDERS
C
~
~
Cylinder and symbol (as plotted in pigs. 4, 5, 6, and 8)
A
0
B
8
ST
d' % k T d
- -- S*
R*
8,030 1.48: 1.633 ).159 19,50C 1.48t 1.625 .154 26,60c ~2,1301.46: 1.674 .162 29,70C ~3,6401.46( 1.680 .I6733,50C L5,950 1.451 1.682 .166 39,loC L7,900 1.44: 1.710 .17144,2OC 3,220 1.42; 1.742 -173 8703C 3,900 1.45( 1.700 -175 9,61C 4,190 1.42t 1.747 .174 10,44C 5,670 1.47~1.659 .162 13,800~ 6,320 1.45( L .700 .166 15,60c 6,610 1.42; 1.742 .174 16,50c 7,950 1.45t 1.682 .164 19,50c L1,OW
,143C 3,850 1.45t 1.700 .168 9,50C .1425 4,790 1.45: 1.690 .165 ii,80c .141e 5,860 1.46: ~.678 .163 14,40C .1407 7,060 1.44; 1.706 .166 17,40C .1401 7,900 1.45: 1.700 .164 19,50C .1401 8,860 1.46; L .675 .161 21,70c .141C 9,630 1.441 ~.721.I69 23,gOC .140t! -0,630 .140? -1,1101.44E L 705 .166 27,40C
I)
D
0
0 d=2.22c
.2060 5,850 1.334 .2065 7,150 1.36E .2067 9,920 1.385 .2041 .1,6501.397 .2010 .j,200 1.40C .2015 .4,2001.401 .2004 -7,1301.421 .2020 -5,2201.42C .2100
E UaD
.2110 .2097 -2097 .2124 2075 .2065 .2088
-
0
d =0.638
.2103
.2090 .2091 .2090 .2081 .2050 -
L.206 L .151
t .12a L .120 L .116 L. 114
L .093
t .096
9,400 .i74 11,250 .168 15,500 .164 18,200 .160 20,600 .160 22,200 .i54 26,700 .156 25,800 .1%
885 1.395 L .120 .169 1,380 956 1.3& t.148 .176 1,510 1,078 1.35~L .179 .18j 1,720 1,270 1.325 ..215 .i92 2,050 1,510 1.342 t .190 .188 2,420 1,700 1.322 i.230 -193 2,770 1,935 1.311 t.250 -197 3,180 2,145 1.322 t.230 -194 3,500 2,950 1.322 ~.230 .196 4,800 3,250 1* 334 t.205 .18a 5,250 3,675 1.33~t.243 .195 6,060 3,950 1 . W .I95 4,330 1.33E L .2O0 4,890 1.34C - .195
&No base-pressure measurements made.
L
-
R
k
cPS
- -- -
3.144C .1411 ,142f .1445 .1441 .1442 .141C .14gC .142C .144C .14K .142C .142C
S
135 7,53c 1-391 -0.94
).
.I33 -0, 32c 1.396 -.95 .I33 .1,40C 1.37; -.68 .136 2,EQC 1 37C -.88
-
135 .135 .I33 .140 .I33 *
135
.132 *
133
- 133
-4,536~ 1.36E -.87 .6,80c 1* 35f -.a4 3,02C 1 335 -.ea 3,6& 1.36~ - .85 3,93c 1.34C -.80 5,32c 1.3& -.go 5,93c 1.36~ -.85 6,20c 1.335 -.eo 7,45c 1.36t -.88
-
3,69c 1.392 4,59c 1* 391 5,610 1.40c 6,76c 1.38: 7,570 1* 39C 8,500 1.40C 9,22C 1.3& 0,200 135 0,60c 1.38:
.I37 .136 ,136 .135 .134 * 135 .I35 135
-.93 -.94 - .95 -.g1
-.93 -.96
-.go -e92
-
5,730 1 309 -.71 .203 7,000 1.34c -.80 .205 9,720 1.360 -.85 .200 1,400 1* 37c -.88 .202
197 .198 * 197 .198
2,900 1* 372 3,900 1.375 6,800 1.394 4,900 1 * 392
-.88 -.89 -.94 -.94
.209
880 L 390 950 1.375 1,070 L .345 1,265 1.324 1,500 L. 337 1,690 1 317 1,930 1.306 2,135 L .317 2,940 L. 317 3,240 L. 334 3,660 1.325 3,930 1.335 4,310 L. 333 4,870 1.335 -
-.93 -.89 -.81
.210
.209 .209 .212
.207 .206 .208 .210
.208 .20a
.208 .207 .204
-.75
-.79 -.74 -.71 -.74 -.74 -.78 - .76 -.78 -.78 -.78
I
20
NACA TN
3169
TABLE I.- Concluded CALCULATED AND MEASURED
now
Cylinder and symbol (as plotted in Ygs. 4, 5, 6, and 8
T'
%
d'
% a
S*
4,16C 1.473 1.31: 1.16( 5,05C 1.515 1.29: .16: 7,UC 1.497 1.30( 155 8,15c 1.482 1.31~ .16j 8,75c 1.497 l.3OC .16: .la81 10,hOC 1.492 1.30; .16L .la112,93C 1.482 1.31~ .16: .la83 15,27C 1.497 l.30C .161 .la82 16,92c 1.485 l.3Ol .16f -18% 19,llC 1.480 1.311 .I64 .16; .1905 7,18C 1.489 1.3& .1856 8,43C 1.480 1.311 .16: .1&9 9,40C 1.495 1.302 .161 .le46 11,04c 1.487 1.30: .16z -1855 1 3 , 6 6 ~1.499 1.30C .161 ~88015,320 1.497 1.30C .16; .la52 17,300 1.490 1.504 .16; .le48 19,210 1.478 1.312 .164 .1%6 20,60c 1.490 1.304 .16: .:t872 22,700 1.489 1.306 .164
1.1886 .19lO .18jl .1&5 ,1878
1
R*
S 1.05:
1.175
1.395 1.437 .174 1.419 1.404 17: .17f 1,419 .17E 1.414 * 1-75 ~2,3001.404 * 175 14,500 1.419 .17E ~6,0001.407 .17t ~8,1001.402 .181 6,800 1.411 .176 7,980 1.402 17: 8,900 1.417 17: -0,500 1.409 .176 -2,900 1.421 .17~-4,500 1.419 .176 -6,400 1.412 * 175 -8,200 1.400 177 -9,500 1.412 177 !1,500 1.411 .181
.
G
4
PARAMETERS FOR TEST CYLINDERS
3,940 4,790 6,750 7,730 8,300 9,850
-
-0.9: -1.0' -1.0:
-.9' -1.0:
-1.a -.g
-1.0:
-.9t
-.Y -.95 -.g' -1.0.
-.9! -1.0: -1.0:
-.9: -.% -.95 -.9!
A
#
uLaf -A
.430cm 4
.1&6 .I830 -1865 .1&5 .la32
.la24 .1820
38 Pressure '\tube
.la18
.1%6
1,210
1,380 1,570 2,030 2,240 2,510 2,750 3,030 3,400
1.461 1.322 1.457 1.326 L.480 1.310 t.476 1.312 ..495 1.301 t.483 1.310 t.488 1.306 ..488 1.306 ..492 1.303
-.
D
J 1
+
I-
I 10 K
0
.0998 5,600 311 2.115 .io08 6,430 -.3oi 2.162 .io02 6,750 ..323 2.063 .io30 7,650 ..jO3 2.155 .io06 6,950 ..j42 1.990 -.?TO 2.035 .350 1.963 -330 2.035 330 2.035 -330 2.035 .323 2.060 .260 .27O .258 .240 .244 .236 .246
aNa base-pressure measurements made.
1.370 1.351 1.385 1.445 1.430 1.461 1.425
.165 .167 .165 .164
.01c
.161 .160 .160 .161 .161
.162 1.55 .154 .161 .158 .162 .157 *
1,200 1.446 -1.05 1,370 1.442 -1.Of
1,550 1.465 1.461 1.480 2,490 1.468 2,722 1.4'73 3,000 1.473 3,370 1.477
2,010 .181 2,220
.160
.167 .156 .170 .149 159 .156 .162 .161 .160 .162
.181 .181 .185 .183
.181 .180 .le0
.183 .o6t
.094 095 .094 * 097 .094 .097
-1.1: -1.lt -1.17 -1.1; -1.lE
1.230 -.51 1.220 -.45 1.241 -.54 1.223 -.5C 1.260 -.55 1.248 -.5t 1.266 - . 6 ~ 1.248 -.56 1.248 -.56 1.248 -.56 1.241 -.54
* 099 .098 .098 .098
5,250 6,030 6,340 7,180 6,500 7,850 0,400 1,900 3,300 5,000 6,700
.147 ,143 137 .135 135 .134 * 135
5,850 1.242 -.54 7,220 1.245 -.55 0,100 1.232 -.52 1,200 1.216 -.48 2,600 1.221 -.49 5,800 1.211 -.47 5,300 1.220 -.49
*
.lo1
.020
-1.1: -1.14
NACA TN
t
3169
21
NACA TN
22
W
a W
I-
-JI-
a aI3
0 X
I-
30
3169
NACA TN 3169
23
-3
?
0
'.
0
t
Q Q.
0
0
24
NACA 'I"
51-69
.I 8
S* .I 6
.I 4
.22
.20
0
S
-.2
.I 8
CPS .I 6
-.4
.I4
-. 6
.I 2
-.8
.I 0
- 1.0 5
4
c/d -C
Figure 4.- Wake interference.
6
7
,4v
NACA TN 3169
26
*
a
7-71" 0
a0
0
cu
NACA TN 3169
2 .I
2.4
2.2
d '/C 2 .c
8
I .a
4
I .e
I .4
1.2
I .o
NACA TN
28
3169
m
0
w
(I
w
I
J
5v
NACA TN
3169
3.5
3.c
d
COX 2.5
d
CDji 2.0
8
I .s
I .o
.5
0
I,
k Figure 9.- Wake solutions.
NACA-Langley
-
1-1-54
-
1000
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