the effect of a velocity gradient on the aerodynamic forces of a circular cylinder with tangential...

The Effect of a Velocity Gradient on the Aerodynamic Forces of a Circular Cylinder with Tangential Blowing

Tsutomu Hayashi Fumio Yoshino Department of Mechanical Engineering, Tottori University, Tottori, Japan

l iThe lift and drag acting on a circular cylinder with tangential blowing immersed in a uniform shear flow are presented for a shear parameter K = 0.15 and Reynolds number Re = 6 x 104, and the effect of the velocity gradient of the uniform shear flow on characteristic values is discussed for coefficients of momentum of the blowing jet ranging from 0 to 0.4. The characteristic values include the angle and the pressure coefficient of the stagnation point, separation points, minimum pressure point, and base pressure. Consequently, the shear parameter in the two cases where the location of the blowing slot on the circular cylinder is on the cylinder surface on the faster side of the shear flow (K + ) and on the slower side (K - ) yields the difference of coefficients of lift and drag and different starting points for the forced reattachment phenomenon. A comparison of various features such as the stagnation point clarifies the contributions of each feature to the lift and drag on the cylinder.

Keywords: aerodynamic characteristics, uniform shear flow, circular cylinder, tangential blowing

I N T R O D U C T I O N

It is very important to understand the influence of the velocity gradient in a uniform shear flow on the aerodynamic characteristics of a circular cylinder with a tangential blowing jet because almost all the actual flow is shear flow with some velocity gradient.

The technique of using the tangential blowing jet on a circular cylinder is a method for controlling or preventing boundary layer separation, mainly to generate lift by increasing the circulation around the lifting cylinder. In the case of a circular cylinder, the facts that the model is easily made and the angular location of the slot is variable at will are advanta- geous feature. Therefore, a considerable number of investiga- tions have been carried out on cylinders with a tangential blowing jet and have produced basic data on industrial as- pects such as high-lift devices, film cooling, and improvement of the efficiency of diffusers [1-4]. However, there has been no investigation on a circular cylinder with tangential blowing immersed in shear flow.

In this paper, the fluid forces, such as lift and drag, are derived from the measured pressure distribution on the surface of the circular cylinder with a tangential blowing jet immersed in an artificial uniform shear flow. In a limited experimental case of uniform shear flow with a large shear parameter K = 0.15, an attempt was made to distinguish clearly the effect of the velocity gradient in comparison with the fluid force for the two cases where the slot of the blowing

jet was located in the faster velocity side ( K + ) and the slower velocity side ( K - ) of the uniform shear flow. A constant Reynolds number of 6 x 10 4, which was based on the velocity on the central axis ( X axis), was maintained throughout the experiment.

Consequently, the shear parameters in the two cases ( K + ) and ( K - ) yield the difference of the coefficients of lift and drag and yield different starting momentum coefficients of the blowing jet for the forced reattachment phenomenon. The contributions of various features such as the stagnation point to the lift and drag are discussed in detail and are evaluated for the individual components.

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

Experimental Apparatus

Experiments were conducted in an open-circuit low-speed wind tunnel with a working section 0.2 m high, 0.6 m wide, and 2 m long. Using an idea from Kotansky [5], a honeycomb was designed that gave various resistances across the working section in order to produce a uniform shear flow with a large constant-velocity gradient. This was installed at the entrance of the section. Instead of the ordinary honeycomb, however, a number of polypropylene straws of 6 mm outer diameter and 0.125 mm thickness were piled up to

Address correspondence to Dr. Tsutomu Hayashi, Department of Mechanical Engineering, Tottori University, Minami 4-101, Koyama, Tottori, 680 Japan.

Experimental Thermal and Fluid Science 1992; 5:317-324 © 1992 by Elsevier Science Publishing Co., Inc., 655 Avenue of the Americas, New York, NY 10010 0894-1777/92//$5.00

317

318 T. Hayashi and F. Yoshino

construct a tube-bundle conditioner. The shape of the conditioner was determined through trial and error by an iterative process of measuring the velocity distribution and cutting away the straws.

Figure 1 shows a velocity profile of the uniform shear flow formed in the working section. The velocity gradient in the central region, which was obtained by the least squares approximation, was du / dz = 91 (s - 1), and shear parameter

K = 0.15. The turbulent intensity (u2)l/2/Uc fluctuated across the section within 1-2% in the central field of the shear flow but showed no fluctuation for the two one-third- width regions on the sides, having constant velocity distributions and low turbulent intensity of less than 1%. Because the velocity distribution in the central region of the working section had a constant velocity gradient and unevenness of less than 2%, it was confirmed that the uniform shear flow obtained afforded an optimum flowfield for this experiment.

The structure of a circular cylinder with tangential blowing and the relative location of the blown circular cylinder in the uniform shear flow is illustrated in Fig. 2. The specimen circular cylinder was constructed of a hollow cylinder of 40 mm outer diameter and a small circular cylinder of 8 mm diameter, which was inscribed in the hollow cylinder and operated as one side surface of the nozzle. The full-span slot had a constant width of 0.2 mm across the whole span, and the outer surface of the cylinder was chromium-plated. The model cylinder spanned the working section vertically and passed through both the upper and lower walls of the working section. The cross-sectional shape of the walls in contact with the surface of the cylinder formed a structure with a single- edged section such that the tangential jet blew smoothly along the inside of both walls.

Compressed air was supplied from both ends of the cylinder via a float-type area flowmeter, a reducing valve, and an air reservoir and was blown out through the slot to flow along the surface of the smaller cylinder as a part of the nozzle.

In the central cross section of the model cylinder, 34 static pressure holes 0.3 mm in diameter were spaced at l0 ° intervals except in the vicinity of the slot. The cylinder could also be rotated about its axis to give a desired angular location of the slot Oj.

0

U i ? I _7

U

Z I

K + let >o Z !

Cd>O

K w

Figure 2. Circular cylinder.

300 , , ,

200 1 i o 100

-1oo !i -200 I I I -3000 1 2 3 4

,r~l Uc (*/,)

E E

o N

i i v i o o i

Q o o

15 20 25 30 U (m/s)

Figure 1. Velocity profile.

35

Exper imen ta l Condi t ions

A series of experiments were carried out on the condition of the Reynolds number Re = 6.0 × 10 4, which was based on the diameter D and the representative velocity U c of the uniform shear flow on the X axis. The influence of the velocity gradient of the uniform shear flow was shown to be strong after a comparison of the results of the jet blowing at the faster velocity side ( K + ) with that of the slower velocity side ( K - ) under a limiting case of only one shear parameter K = 0.15.

The circumferential pressure distributions on the cylinder surface at midspan were measured under the conditions of three slot locations at 0j = 30 °, 60", 100 ° measured from the geometrical leading point of the cylinder and with various momentum strengths of the blowing jet C~ ranging from 0 through 0.4. The value of C~, was obtained by using the jet velocity calculated by assuming isentropic expansion of the compressed air in the cylinder to the static pressure of the

wind tunnel, and by using the measured central velocity on the X axis in the working section. It was confirmed that the experimental uncertainty of the measured pressure distribution on the cylinder surface at midspan was less than 5 % [6].

It is evident that the flow over the cylinder surface is nearly two-dimensional in the central region along the spanwise direction from the measurement of the pressure distributions and the aerodynamic forces. The two-dimensionality of the blowing jet was also confirmed, in particular from the visualization of the separation lines by the oil-flow method.

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

Influence of Veloci ty G r a d i e n t on Aerodynamic Charac te r i s t i c s

Pressure Distribution Some examples of a pressure distribution on the surface of a circular cylinder with a tangential blowing jet are reproduced in Fig. 3. The features of the results for both K + and K - at C# = 0 are an asymmetric pressure distribution with a shifted stagnation point, a rearward-shifted laminar separation point on the faster velocity side of the uniform shear flow, and a forward-shifted laminar separation point on the slower side, leading to a nonzero and opposite direction C / a t C a = 0. Increasing C a results in the appearance of a large and wide negative pressure region on the upper surface of the cylinder (0" < 0 < 180°), in the further rearward movement of the upper separation point, and also in the raising of the base pressure.

In comparison with results of K + and K - at the same C , the effect of the shear parameter appears as a difference otepressure level on the whole circumferential surface; that is, the pressure level in the case of K + is always lower than

1

0

-1

-2

-3

-¢

-5

-6

-7

-B

-9

90"

C/z K Oj

• 0 -

o 0.15 + 100"

* 0 . 1 5 -

o 0.35 +

. 0 35 -

,' 0 +}60" • 0 -

I I I

Figure 3. Pressure distribution.

Effect of a Velocity Gradient 319

that for K - . As C a is further increased, therefore, the location of 0st, the stagnation point on the surface, is further shifted in a downward direction. Also, the difference between the K 4- and K - values of the stagnation pressure Cpst, the minimum pressure Cpm, and the base pressure Cpb are all increased.

Lift Coefficient C t The experimental pressure distributions are integrated to give the lift and drag coefficients, C t and Ca, respectively.

The relation of C a and C t is shown in Fig. 4 to institute a comparison between K + and K - . One effect of the velocity gradient of the uniform shear flow is a difference of the lift coefficient C t acting on the cylinder with tangential blowing at the same C a for 0 = 30", 60", and 1000, and the different jumping points for the forced reattachment phenomenon in the case of Oj = 1000. It is especially worthy of note that the lift coefficient C/ in the case of K - is always higher than that for K + .

Although it is confirmed that C t in K = 0 has an approxi- mate intermediate value between the C t values for K - and K + at the same C a, these were not plotted in the figure in order to avoid unnecessary confusion.

The tendency of the lift to increase with increases in the momentum coefficient of the blowing jet can be

Lx

I ' I I

~ ~ K+ K-

100 ° o • f i0 ° A •

30" [] •

0- .1 0.2

II

0.3

Figure 4. Lift coefficient.

0.4


approximately divided into the following three categories, whatever the value of Oj.

I. A region of gradual increase. Keeping the differences of the lift that appear at C u = 0 due to alternating K + and K - , C I gradually increases with C~ (approximately 0 < C~, < 0.05).

II. A region of steep increase or jumping. The effect of the jet blowing increases abruptly and encourages the difference of C z due to K + and K - (approximately 0 . 0 5 < C <0 .1 ) .

III. A high li~ generating region. The cylinder with tangential blowing performs as predicted by the Coanda effect (approximately 0.1 < C~,).

It is already known that with simple circular cylinders without any facility such as the blowing jet in a uniform shear flow the velocity gradient of the shear flow acts to shift the front stagnation point on the surface toward the side having a greater velocity than the center velocity, and the stagnant streamline is also emitted originally from the upstream side of even greater velocity [7].

In category I, nonzero C z in opposite direction are produced by an asymmetric pressure distribution that rises from the stagnant streamline that is emitted originally from the side of uniform shear flow of greater velocity and reaches the stagnation point existing in the greater velocity side on the surface. In the case of K + , for instance, because the pressure level in the range of 0 ° < 0 < 70 ° on the upper surface of the cylinder is higher than that of the lower surface, the lift coefficient C l < 0 acts toward the slower side of the uniform shear flow from the faster side. Therefore, the lift coefficients C t for K + and K - are absolutely equal in value but in opposite directions to each other. As C~, is further increased in this category, C 1 varies only slowly because the effect of the blowing jet is not so active. There exists an exceptional case of Oj = 60 ° in which the slot groove acts as a tripping facility so that C / is also positive for both K + and K - at C~ ,=0 .

In category II, two kinds of rapid increase, continuous steep and jumping phenomena, are recognized for both K + and K - . The slope of the C I - C ~ curve is steep and continuously increases in the cases of Oj = 30 ° and 60 ° where the slot is located upstream of the original separation point. However, the C t curve jumps suddenly at some C~ value in the case of Oj = 100 ° where the slot is located downstream of the original separation point. It was determined that this jumping phenomenon arises from the forced reattachment of the separated shear layer. The jumping phenomenon of C t includes two significant processes, a different starting C~, of the forced reattachment for both shear parameters K + and K - and a historical cycle that has different jumping points for the two directions of increase and decrease of C~. The starting C~, of the forced reattachment was determined to be C~, = 0.0879 in the case of K + and Cu = 0.0681 for K - on the basis of an experiment on the gradual increase of the blowing momentum coefficient, while the reverse jump C~, was determined to be C~, = 0.0555 for K + and C~, = 0.0520 for K - on the basis of an experiment on gradual decreasing of the blowing.

In category III, it was recognized that the further to the rear the location of the slot, the larger was the yielded lift, such that a final maximum value C / = 7.2 was reached in this experiment. The difference between the lift coefficient values

for K + and K - , which reached the maximum AC l = 0.5-0.6 immediately after the jumping process, decreased gradually with Cu.

Drag Coefficient C a A difference also appeared in the drag coefficient C d due to the velocity gradient of the uniform shear flow, as shown in Fig. 5. In category III, C d for K + was lower than that for K - a t Oj = 30*; a t Oj = 60 ° the difference between K + and K - C a values was maximum near C = 0.1 and disappeared with increasing Cu, and at Oj= 100 g C a for K + was higher than that for K - This contrary decrease of C a in K - originated from the extreme increase in the base pressure Cpb. Except for the difference of C a due to K + and K - , it seems that a tendency for C a to increase with C~, for Oj = 100 ° arose from the large induced drag in the range of the larger C/ because the aspect ratio of the cylinder was comparatively small at 5.175.

In category I, the influence of the shear parameter on C a disappeared, but C a had a larger value owing to the laminar separation. The highest value C a = 1.2 at Cu = 0 for 0j = 60* arose from the considerable peak suction near the upper separation point due to the tripping effect of the slot groove.

Inf luence of Velocity Gradien t on Characterist ic Variables

Various characteristics were determined from the pressure distribution curve. These include the angular location and pressure coefficient of the stagnation point, Ost and Cp. st, respectively; the angular location of the upper and lower separation points, Ou and Or, respectively; the base pressure coefficient Cpb; and the angular location and pressure coefficient of the minimum pressure point, 0 m and Cpm, respectively. The variations in these characteristics are discussed as a function of C~, or C r However, this detailed explanation is restricted to results for the case of Oj = 100 ° because this case includes a jumping phenomenon.

Angular Location of Front Stagnation 0st If it is assumed that the maximum pressure point agrees with the stagnation point, it can be stated that the factors closely related to the

1.5 I , , . , . , . , e , -

• " - - - o

1.0 ~ ~ - = 3 0 °

0.5 - ~---~--~. ~

0 x I , I ~ I ~ I 0.1 0.2 0.3 0.4

C/Z

Figure 5. Drag coefficient. Symbols as in Fig. 4.


effect of the shear parameter are 0st and Cp~ t of the front stagnation point. Figure 6 shows the relation of 0st to C~. The stagnation point was originally shifted toward the faster side of the shear flow even at C , = 0 and exists at 0st nearly equal to +6* and for K +_ , which agrees with the angle on the simple circular cylinder without any facility where the phenomenon is already known as the displacement effect [7].

In category I of C l, 0s, is shifted only 1-2 ° toward the lower side on the front surface of the cylinder with increasing c..

In category II, a jumping angle of 0st for K + is larger than that for K - . A difference in C~, at the jumping point of the forced reattachment due to K + and K - , as shown in Fig. 7, was determined from the requirement that the circulation must be able to reverse the curvature of the stagnation streamline. At an instant of the jumping process for K + , the stagnation point jumps from 0st = 4.5 ° on the upper side of the front surface to 0st = - 7 ° on the lower side. There- fore, the stagnation streamline, which is emitted from the faster side upstream of the uniform shear flow and collides with the upper side of the front surface, must jump for a moment to the streamline that is emitted from the slower side and collides with the lower side of the front surface, as illustrated in Fig. 7. At this moment, C~, should be large enough that the circulation around the cylinder due to blowing causes the curvature of the stagnation streamline to reverse. In the case of K - , however, C~, does not need to be so large because the stagnation streamline exists on the same side both before and after the jumping process and the curvature of the streamline is not reversed.

The flow around the cylinder can be rotated in its entirety

~-5 (D

STAGNATION STAGNATION

C~ = 0 C~ = 0

C~.=00879 LIFT F_,p.:O0681 LIFT

~G ~OR~G (o) K+ (b) K-

Figure 7. Starting C~ of the reattachment phenomenon.

during this process so that 0, , 0 t, and 0 m have a large shifted angle that has a strong connection with 0st.

In category III, although the stagnation point must move further downward on the front surface, the difference between the angle 0st in K + and K - becomes more and more narrow with increasing C~.

Pressure Coefficient of Front Stagnation Point Cps t In category I in Fig. 8, it was found from the results for all Oj that a situation Cps t > 0 is always satisfied so that the stagnation streamline is emitted from the faster side upstream of the uniform shear flow. The stagnation point on the cylinder surface has already shifted toward the faster side even at C u = 0; that is, the stagnation streamline is emitted from the faster location z = D[(Cpst) 1/2 - 1] /K = 6.59 mm and collides on the surface at the location z = ( D / 2 ) s i n 0~t = 2.09 mm.

The relation between the Reynolds number Re z, which is based on the stagnant velocity U(z) , and Re, which is based on the central velocity Uc, is Re z = Re × Cps t. It was found that the difference between the starting Cu values for K + and K - at the jumping process is not caused by the difference of Reynolds numbers because Cps t just before the process has the same value and therefore the same Re z.

In category III, the greater the value of Cu, the larger the displacement of Cp, st from 1.0; and the larger the values of C t or O j, the stronger is this tendency.

-10

-15

-20

-25

-30

15 t L

1.1! . ~ • ~. •

10 . .u 0.9 '~ .

08

T , I , I L I , I o.1 02 0.3 o 4

Figure 6. Angular location of the stagnation point. Figure 8. Stagnation pressure coefficient. Symbols as in Fig. 4.


1 8 0 o F ' I ' I ' I ' I _

F cB160 F

140 t '~

10080 t~l-_~ '~ 3060: a[] Ko•• -

BoJ-- T I , I , I , I 0 0.1 0.2 0.3 0.4

C4 Figure 9. Upper separation angle.

Location of Upper Separation Point 0 u The relation of 0 u to C~ is shown in Fig. 9. The separation point is defined as the intersection of the extended line of the base pressure curve, which shows an almost constant value, and the extended line of the recovery pressure distribution shown in Fig. 3. It was also confirmed that these separation points are located on the separation line that is obtained by using the visualization of the oil-flow method. 0 u is the most shifted and most important variable for the lifted cylinder because the width of the large negative pressure region on the cylinder surface is determined by this variable. The upper separation point at C~ = 0 is located at 0 u = 82* for K + and at 0 u = 74" for K - , which corresponds to the laminar separation on the cylinder without any facility such as the blowing jet. When C I is larger than 3.0, the angle of the upper separation point in Oj = 100", which is retarded by the blowing jet, is linearly related to the square root of the lift coefficient and agrees with the estimated result obtained from calculations based on the assumption of the transition to the turbulent boundary layer at 0 = 90*.

The Lower Separation Locations 0 t 0 t always has a tendency to move in the same direction as 0st with increase C/z. The relation of 0/ to C t is plotted in Fig. 10, which shows that 0 t is irrelevant to Oj and is determined only by K. In both K + and K - , 0 x has an asymptotic tendency to a limited value 0 t nearly equal to 270", which the separation point never exceeds. This situation means that the stronger the circulation around the cylinder, the weaker is the effect of the velocity gradient on 0 t, and the separated width Osb = 0 z - 0,, is finally restricted to the range of one-fourth of the surface of the back and lower sides of the cylinder.

The Base Pressure Coefficient Cpb Cp~ contributes to the lift and drag directly with strong relationships to the width and location of the separated regions. The relation of Cpb to

290*u , f , , , , ,

c~ 28 5a k

27(

T I I L I I I I 0 1 2 3 4 5 6 7 C~ Figure 1O. Lower separation angle. Symbols as in Fig. 9.

the width of the separated region Osb is shown in Fig. 11, in which it is shown clearly that the smaller 0sb is, the greater the difference of Cpb due to increases in K, and Cpb for K - is finally raised to an extreme value.

The Minimum Pressure Location 0 m 0 m and Cpm a r e

important and unique characteristics in representing the large negative pressure region, which includes the basic mecha- nism of the lift acting on the cylinder with tangential blowing. Because 0st is ,shifted in the negative direction and 0 u in the positive direction, the large negative pressure region is stretched with C~, and therefore 0 m is positioned midway

0.8

0.6 N_

0.4

0.2

0

-0.2

-0.4

-0.6

-0.8

-1.0

I I

60*

i ~ ° 30°

"~\~. 150

I I I I

Figure 11. Base pressure coefficient.

K+ K-

o •

/x •

[] •

200 ° J I

~sb

&@ o%°~ o l l

& A


E

90

8O

70,

60

I ' I ' I

e j ~ K+ K- 100 = o •

60" A At 30 ° [] •

"[ t I ~ I i I L I 0 0.1 0.2 0.3 0.4

F i g u r e 12. Angle of minimum pressure.

between them. In category III in Fig. 12, 0 m for K + is slightly more retarded than that for K - .

T h e M i n i m u m P r e s s u r e C o e f f i c i e n t Cpm Cpm is a characteristic feature that decreases remarkably with C~ and is clearly influenced by K. In the range of the stronger circulation in Fig. 13, it is shown that Cpm for K -'l- is lower than that for K - . This finding agrees with the results of the theoretical pressure distribution previously presented using the potential theory for cylinders with a circulating flow, applying perturbation theory to uniform shear flow [8].

c~ 0 0.1 0.2 0.3 0.4

I [ I 1 ]

-41 o ~-5

-6

-7 1 | E) j \ K+ K-

-8 I 100" o • o

60 ~ •

_g 30" o •

F i g u r e 13. Minimum pressure coefficient.

S I G N I F I C A N C E / U S E F U L N E S S OF R E S U L T S

The technique of using a tangential blowing jet on a circular cylinder is a practical method for controlling or preventing boundary layer separation, mainly for generating lift by increasing circulation around the lifting cylinder.

An effect of the velocity gradient of the uniform shear flow yields a difference in the lift coefficient C t acting on the circular cylinder with tangential blowing at the same momentum coefficient C , of the blowing jet in all cases of angular location Oj of the slot. This effect also shows different values for the momentum coefficient at the jumping points for the forced reattachment phenomenon of the separated boundary layer in the case of Oj = 100".

Knowledge of the influence of the velocity gradient of the flow on the aerodynamic forces of a circular cylinder with a tangential blowing jet, which was distinguished clearly in the present paper, should be useful in various industrial applica- tions such as high-lift devices, film cooling, and improvement of the efficiency of diffusers because almost all actual flows are shear flows with some velocity gradient.

S U M M A R Y

1. The lift coefficient in the slower side blowing ( K - ) is always higher than the lift coefficient in the faster side blowing (K + ) at the same C~.

2. The tendency of C I to increase can be readily distinguished in the following three categories with C/z: I: a gradually increasing region of C t II: a steep increase or jumping region of C t III: a high lift generation region

3. The starting C~ of the jumping process is different for K + and K - ; that is, the jumping for K - begins at a smaller C~ than that for K + .

4. In category III, 0st , Cpst, Cpb , Or, Ore, and C~ are characteristic variables that actively influence the . . ~ t and drag due to the existence of a velocity gradient in the main flow.

NOMENCLATURE

C t section lift coeffcient [= ( - 1/2)/2~Cpsin 0 dO], dimensionless

C a section drag coefficient [= (1/2)/o2~CpCOSO dO], dimensionless

Cp pressure coefficient [= ( P - Po)/(pUc2 /2)], dimensionless

Cub base pressure coefficient, dimensionless Cpm minimum pressure coefficient, dimensionless Cps t stagnation pressure coefficient, dimensionless

C~ momentum coefficient of the blowing jet [= (momentum of jet per unit span)/((1/2)pU2D)], dimensionless

D diameter of circular cylinder, m K shear parameter [= (D/Uc) dU/dZl ,

dimensionless K + the slot of the blowing jet is located in the faster

velocity side of the uniform shear flow K - the slot of the blowing jet is located in the slower

velocity side of the uniform shear flow


P

Re U v~

x , y , z

Oj

Om Osb 0st

0u, 0t

pressure, Pa Reynolds number ( = UcD / u), dimensionless velocity of the uniform shear flow, m / s velocity on the center axis, m / s coordinates of the center and midspan of the circular cylinder

Greek Symbols 0 angle measured clockwise from the geometrical

leading edge of the cylinder, deg angular location of the slot, deg angular location of the minimum pressure, deg

angular width of the dead water region, deg angular location of the stagnation point, deg angular location of the separation point on the upper and lower surface of the cylinder, respectively, deg

v kinematic viscosity of air, m2/s p density of air, K g / m 3

REFERENCES

1. Lockwood, V. E., Lift Generation on a Circular Cylinder by Tangen- tial Blowing from Surface Slots, NASA TN D-244, 1-38, 1960.

2. Cheeseman, I. C., Circulation Control and Its Application to Stopped Rotor Aircraft, Aeronaut. J. Aeronaut. Soc., 72, 635-646, 1968.

3. Dunham, J., Experiments Towards a Circulation-Controlled Lifting Rotor. Part 1. Wind Tunnel Tests, Aeronaut. J. Aeronaut. Soc., 74, 91-103, 1970.

4. Waka, R., Yoshino, F., Hayashi, T., and lwasa, T., The Aerody- namic Characteristics at the Mid-span of a Circular Cylinder with Tangential Blowing, Bull. JSME, 26(215), 755-762, 1983.

5. Kotansky, D. R., The Use of Honeycomb for Shear Flow Generation, AIAA J., 4(8), 1490-1491, 1966.

6. Moffat, R. J., Describing the Uncertainties in Experimental Results, Exp. Fluid Thermal Sci., 1, 3-17, 1988.

7. Hayashi, T., and Yoshino, F., An Analytical Evaluation of the Aerodynamic Forces Acting on a Circular Cylinder in a Uniform Shear Flow, First ASME-JSME Fluid Engineering Conference, Port- land, Ore., 112, 83-88, June 1991.

8. Yoshino, F., Hayashi, T., Waka, R., and Yoshida, T., A Theory of Circulation Control by Tangential Blowing, Applied to a Circular Cylinder in Uniform Shear Flow, Rep. Fac. Eng. Tottori Univ., 14(1), 29-37, 1983.

Received January 1, 1991; received October 1, 1991

the effect of a velocity gradient on the aerodynamic forces of a circular cylinder with tangential...

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