Blockage effects on drag of sharp-edged bodies

Journal of Industrial Aerodynamics, 1 (1975/76) 301--309 301 © Elsevier Scientific Publishing Company, Amsterdam -- Printed in The Netherlands

BLOCKAGE EFFECTS ON DRAG OF SHARP-EDGED BODIES

K.G. RANGA RAJU and VIJAYA SINGH Department of Civil Engineering, University of Roorkee, Roorkee (India) (Received July 15, 1975)

Summary This paper describes the results of an experimental study concerning blockage effects on the drag of sharp-edged bodies at Reynolds numbers high enough for viscous effects to be negligible. The authors' wind tunnel experiments were concerned with the drag of twodimensional rectangular prisms of various sizes with and without splitter plates. Based on these studies, relationships for the drag coefficient as a function of blockage and shape have been obtained. Further, the form of the proposed relationship is shown to be applicable to two-dimensional triangular prisms and also to circular discs. Interestingly, these relations appear suitable for correction foc blockage effects for bodies in the boundary layer also.

1. I n t r o d u c t i o n A s t r u c t u r e in a t m o s p h e r i c b o u n d a r y l a y e r f l o w is o f t e n m o d e l l e d in a w i n d tunnel with the object of determining the distribution of mean and maximum pressures a r o u n d t h e s t r u c t u r e , t h e t o t a l f o r c e o n t h e s t r u c t u r e , etc. R e q u i r e m e n t s o f g e o m e t r i c a n d d y n a m i c similarity, as well as t h e c o n s i d e r a t i o n t h a t the m o d e l be large e n o u g h to p e r m i t a c c u r a t e m e a s u r e m e n t s , govern t h e m o d e l d i m e n s i o n s . F r e q u e n t l y , t h e size o f such a m o d e l is large e n o u g h t o result in a p p r e c i a b l e " b l o c k a g e " ; in o t h e r words, t h e t u n n e l ceiling w o u l d d i s t o r t t h e f l o w p a t t e r n f r o m t h a t o b t a i n e d in t h e u n b o u n d e d a t m o s p h e r i c flow. O n e t h e n requires a m e t h o d o f c o r r e c t i n g t h e o b s e r v e d results f o r t h e b l o c k a g e e f f e c t b e f o r e t h e s e can b e c o n v e r t e d t o t h e p r o t o t y p e . R e c o g n i s i n g t h a t m o s t o f t h e s t r u c t u r e s in t h e a t m o s p h e r i c b o u n d a r y layer m o d e l l e d in t h e w i n d t u n n e l are fax f r o m s t r e a m l i n e d , o n e can see t h a t t h e m e t h o d s o f b l o c k a g e c o r r e c t i o n used b y a e r o n a u t i c a l engineers f o r airfoil sections [1,2] o b v i o u s l y w o u l d n o t a p p l y here. It is also p e r t i n e n t to p o i n t o u t t h a t a single t w o - d i m e n s i o n a l o b j e c t p l a c e d in a w i n d t u n n e l (say at t h e c e n t r e o f height) m a y b e r e v i e w e d as o n e o f a series o f identical o b j e c t s p l a c e d in a r o w p e r p e n d i c u l a r t o t h e f l o w d i r e c t i o n w i t h t h e spacing e q u a l t o t h e t u n n e l d e p t h , p r o v i d e d t h e t h i c k n e s s o f t h e b o u n d a r y l a y e r o n t h e f l o o r a n d t h e ceiling is small a n d m a y b e a s s u m e d t o i n t r o d u c e n o d i s t o r t i o n . E x p e r i m e n t s b y R a m a m u r t h y a n d Ng [3] i n d i c a t e t h a t such a p r e m i s e is justified; t h e drag o f an o b j e c t in a series was f o u n d t o

302 be the same as that o£ a single object at identical blockage values. In other words, the blockage correction developed from experiments with a single object in the midstream would permit evaluation of the effect of interference when a number of these objects are located in series. Practical examples of such a series of objects are to be found in the cases of bridge piers, heat exchanger tubes, cooling towers, etc. A few studies concerning the problem of blockage have been carried out in the past [3,4,5]. Maskell [4] proposed that the blockage correction for bluff bodies may be looked at as a velocity increment. However, the base pressure coefficient under zero blockage (required for making blockage corrections by his method) is not known for many body shapes. Raju and Garde [ 5] evolved experimentally a blockage correction procedure for two-dimensional normal plates of negligible thickness provided with a symmetrical tail plate. R a m a m u r t h y and Ng [3] provided experimental data on the drag of two-dimensional cylinders and triangular prisms at various blockage ratios. The present experimental study was devoted to the determination of drag of two-dimensional rectangular prisms held in midstream with one face normal to the flow. Prisms with and without symmetrical tailplates were tested. As pointed out by Arie and Rouse [6], flow past an object in midstream with a symmetrical tailplate may be looked at as that past the same object placed on the boundary, but with zero thickness of the approach boundary layer. The analysis of the experimental data has led to simple blockage correction relations for these bodies. 2. Collection of experimental data The experimental data were collected in an open circuit wind tunnel, 32.4 cm X 32.4 cm in section and 4.9 m in length, located at the Hydraulics Laboratory of the University of Roorkee. Details of the tunnel characteristics are available elsewhere [7]. Figures l(a) and l(b) show the placement of the test prism in midstream with and without the rear splitter plate respectively. The length of the splitter plate provided was considered adequate in view of the results of Arie and Rouse [6] concerning the length of standing eddy behind a normal wall. The test prisms were hollow and were made of wood. Pressure points were provided on both faces of the prism along the centre line of width. The pressures at any desired velocity were read with the help of an inclined manometer using methyl alcohol as the indicating liquid. Tests were carried out at three different velocities for each prism. Prisms of different heights and thicknesses were used to cover a range of h/D from 0.075 to 0.24 and t/h from 0 to 2.0. (Here h is the height of the prism, t the thickness of the prism and D the tunnel depth.) The data of Arie and Rouse [6], Fage and Johansen [4] and Raju and Garde [5] were also used in the analysis.

303 (a)

l

EST PRISM

SPLITTER PLATE

[ ,(b)

10 h

"I

TUNNEL CEILING

TEST PRISM

~ltk-

TUNNEL

FLOOR

Fig. 1. Diagram showing placement of test prism. 3. Analysis o f data 3.1 Pressure distribution The pressure distribution around sharp-edged bodies is know n to be indep e n d e n t of the Reynolds n u m b e r if it exceeds 103 . This was confirmed in the present study wherein the Reynolds n u m b e r for all runs was in excess of 103 . As such, th e Reynolds n u m b e r was n o t considered in the analysis of pressures. The average values of the local pressure coefficient and drag coefficient for the different velocities differed insignificantly from the individual values and, hence, the average values were used. Figure 2 shows the pressure distribution around plates of zero thickness but different heights. In this figure, y is the distance from the centre line, p is the mass density of the fluid, U is the ambient velocity and Ap = p -- P0, P being the pressure at any point and P0 the ambient pressure. The base pressure coefficient remains sensibly constant over the height of the plate, as one might expect, b u t it changes considerably with changes in blockage ratio. On the o th er hand, the pressure distribution on the upstream side of the plate is practically in d ep e nde nt of blockage. This is clear p r o o f t hat blockage correction c a n n o t be looked at as a velocity increment, since, if this were the case, the pressure distribution on b o t h faces of the plate ought to have been affected by blockage. The upstream pressure distribution in plates with various thicknesses was also f ound to be i ndependent of blockage. F u r t h e r m o r e , it was interesting to n o t e t ha t the average pressure coefficient for the upstream face was i n d e p e n d e n t of the thickness of the plate and presence (or absence) of

304 the splitter plate also. This is revealed by Fig. 3 wherein all available data on rectangular prisms are plotted. Here, CD is the drag coefficient defined as:

Co

F =

-

-

(1)

hp U2/2

where F is the f o r m drag per unit length. z~p + 10

0

r

~U~

-10

I

-20 I

10

/®

~

1

----~

'

o D/h -- 4.0'5 • D/h = 5 . 4 0

I

•

~1

I

/

I

D/h

= 8.10

0.5

~

1' + FLOW .

.

.

.

.

.

I

.

--

,0.

o

,i,I

I ~'SPLITTER I I

TEST PLATE

I

ID

I

I

I

I

I

I

t

PLATE

q

I

~.

, 1.O L

(

/

I

Fig. 2. Diagram showing effect of blockage on pressure distribution. I

I

I

I

I

I

I

I

Eq. ( 3 )

13 U 2

•

.,~

•

•

•

I O

I

I

I

I

I

oee

"

SYMBOL

o

• e.

1

I.

-1.O

SOURC E

RE MAR KS

ARIE-ROUSE

WITH SPLITTER PLATE, ZERO THICKNESS PRISM

RAJU-GARDE

WITH SPLITTER PLATE, ZERO THICKNESS PRISM

RAJU- SINGH

WITH AND WITHOUT SPLITTER PLATES, VARYING THICKNESSES AND WIDTHS OF" PRISM

I

f -20

I

l

I -3.0

Cb

Fig. 3. Relation between base pressure coefficient and drag coefficient for two-dimensional prisms.

305

Cb

-

Pb --Po pU2/2

(2)

Pb being the base pressure. The data on Fig. 3 are well represented by the equation Co = 0.8 -- Cb

(3)

But, by definition: (4)

C D = C u -- C b

in which PU - - P 0

Cu-

pU2/2

Here/Su is the average pressure on the upstream face over the height of the prism. Combining eqns. (3) and (4), one obtains Cu = 0.8. While Cb is strongly affected by blockage, Cu remains practically independent of it illustrating the inapplicability of the concept of velocity increment for blockage correction.

3.2 Blockage correction for rectangular prisms Ranga Raju and Garde [5] suggested an equation for blockage correction of the form: CDo = CD(1 -- h/D) n

(5)

in which CDo is the drag coefficient corrected for blockage. For normal plates of negligible thickness provided with a splitter plate t h e y gave CDo = 1.38 and n = 2.85. The form of eqn. (5) appears well suited for plates w i t h o u t a splitter plate as well as for prisms of finite thickness as may be seen from Fig. 4. The

0

0.02

(704

0.06 ~g(1-

0.0a h/D)

0.10

0.12

0.14

Fig. 4. Typical plots showing variation of CD with (1 -- h/D) for rectangular prisms without splitter plate.

306

2.3

I

I

I

I

I

22

I

I

\ \

21

PLATE

\

\ \ \

2.0

\ \ \ \

1.9

\ \ \

(o18'

1.7

1.6

1.5

~

_

1.4

1..~

0

I

0.25

I

i

0.50

0.75

I

1.0

t/h

I

1.25

I

1.50

1

1.75

20

Fig. 5. Drag coefficient of two-dimensional rectangular prisms for no blockage.

LITTER PLATE 2.4,-

-WlTHOUT SPLITTER PLATE 1

.

2

~

0.8

0.4 0

"~ " ~

I 0.25

I 0.50

I 0.75

I :,1 I / 1.00 125 150 1.75 2,0 t/h Fig. 6. Value o f n in eqn. ( 5 ) for t w o - d i m e n s i o n a l rectangular prisms.

307 values of CDo and n were determined for all cases from figures similar to Fig. 4 and, as might be expected, CDo and n are functions of t/h and the nature of the standing eddy as decided by the splitter plate. The variations of CDo and n are shown in Figs. 5 and 6. It is to be noted t h a t CDo for a normal plate of zero thickness (without a splitter plate) comes out to be 1.80 instead of the usually listed value of 1.90 in fluid mechanics text books. The highervalue of 1.90 is obviously based on the data of Fage and Johansen [4], which are n o t also free from blockage effects as may be seen from Fig. 4. The considerable effect of the introduction of a splitter plate on the drag is well known and is well confirmed by Fig. 5. Use of Figs. 5 and 6 along with eqn. (5) obviously enables one to calculate the drag of a series of prisms placed in a row perpendicularto the flow.

3.3 Blockage correction for other body shapes R a m a m u r t h y and Ng [ 3] measured the form drag of two-dimensional equilateral triangular prisms in a wind tunnel. Their data on these prisms held symmetrically in the flow can also be fitted by eqn. (5) enabling extrapolation to the case of zero blockage. For a prism with one face facing the flow normally Coo and n have the value~ 2.05 and 2.32 respectively. When this prism is held with one face facing the downstream side, the values of CDo and n are 1.21 and 2.07 respectively. These values of CDo for triangular prisms as well as for rectangular prisms without a splitter plate (viz. Fig. 5) compare excellently with the values given by Hoerner [8] for apparently zero blockage condition; such agreement is deemed to indicate the correctness of the form of eqn. (5). Equation (5} seems well suited for three-dimensional objects if one replaces h/D by the ratio of the projected area of the b o d y to the tunnel cross-section. It was found that the data of Mckeon and Melbourne [9] on circular discs of area a held normal to the flow in a tunnel of area A can be described by the relation

Cm

=

1.17(1 -- a/A)

-3.45

It is thus strongly felt that eqn. (5) provides a simple and satisfactory means of estimating blockage effects in case of sharp-edged bodies.

3.4 Blockage correction for bodies in the boundary layer Quite often the model to be tested is submerged partly or fully in the tunnel boundary layer; it would be of interest to know whether the foregoing procedure evolved from data on bodies in midstream can be used to correct for blockage effects in this case. Fortunately, the limited data on hand appears to show that it is possible to do so. For a two-dimensional normal plate of negligible thickness placed in a boundary layer, one can write:

(6)

308

c~ = f

Uh,D 8 ) -

v

(7)

h'h

w h e r e ~ is the n o m i n a l thickness o f t h e b o u n d a r y layer at t h e section w h e r e the plate is placed prior t o its p l a c e m e n t . Obviously, if ~/h and Uh/u are held c o n s t a n t ( b u t D / h c h a n g e d ) in a pair o f runs and if Coo c o m p u t e d in accord a n c e with eqn. (5) for these runs remains the same, t h e applicability o f the b l o c k a g e c o r r e c t i o n p r o c e d u r e for b o d i e s in the b o u n d a r y layer is p r o v e d as a little t h o u g h t c o n c e r n i n g eqn. (7) will show. S u c h a pair o f runs was fort u n a t e l y p r o v i d e d b y an earlier s t u d y o f R a n g a R a j u [7] and the results o f the c o m p u t a t i o n s are s h o w n in Table 1. TABLE 1 Drag coefficients of plates in a boundary layer

h

,5/h

D/h

Uh/v

CD

CDo

2.0 4.0

0.84 0,85

16.2 8.1

1.64 X 104 1.73 × 104

1.12 1.36

0.93 0.93

The fact that CD in the above table is strongly dependent on D/h, whereas CDois not, proves the applicability of eqn. (5) for evaluating blockage effects in the case of bodies in the boundary layer, In fact, Ranga Raju and Garde [ 5] corrected their boundary layer data for blockage effects using eqn. (5). 4. Conclusion

A rational and simple method for estimating blockage effects in the case of two-dimensional rectangular prisms has been presented. It has also been shown that the form of this relation is well suited for other body shapes. The proposed relation may also be used to evaluate blockage effects in the case of bodies submerged in the boundary layer. References I A. Pope, Wind Tunnel Testing, Wiley, New York, 1954, p. 291. 2 R.C. Pankhurst and D.W. Holder, Wind Tunnel Technique, Pitman, London, 1948, pp. 327--349. 3 A.S. Ramamurthy and C.P. Ng, Effect of blockage on steady force coefficients, J. Eng. Mech. Div., Proc. A.S.C.E., 99 (1973) 755--772. 4 E.C. Maskell, A Theory for the Blockage Effects on Bluff Bodies and Stalled Wings in a Closed Wind Tunnel, R and M 3400, A.R.C. (1963). 5 K.G. Ranga Raju and R.J. Garde, Resistance of an inclined plate placed on a plane boundary in two-dimensional flow, Trans. ASME, J. Basic Eng., 92 (1970) 21--31. 6 M. Arie and H. Rouse, Experiments on two-dimensional flow over a normal wall, J. Fluid Mech., 1 (1956) 129--141. 7 K.G. RangaRaju, Mechanism of resistance to flow over artificial roughness elements, Ph.D. thesis, Univ. of Roorkee, 1967.

309

8 S.F. Hoerner, Fluid Dynamic Drag, published by the author, 1958, pp. 3--18. 9 R.J. Mckeon and W.H. Melbourne, Wind tunnel blockage effects and drag on bluff bodies in a rough wall boundary layer, Proc. Int: Conf. on Wind Effects on Buildings and Structures, 1971 Tokyo, Section 11-9.