Forces acting on circular cylinders placed in a turbulent plane mixing layer

Journal o f Industrial Aerodynamics, 5 (1979) 13--33 © Elsevier Scientific Publiching Company, Amsterdam -- Printed in The Netherlands

13

F O R C E S ACTING ON C I R C U L A R CYLINDERS PLACED IN A T U R B U L E N T PLANE MIXING L A Y E R

M. KIYA, M. ARIE and I4_.TAMURA Faculty o f Engineering, Hokkaido University, Sapporo, 060 (Japan) (Received December 18, 1978; accepted in revised form May 1, 1979)

Summary Time-averaged pressure distributions along the surface of a circular cylinder placed in a turbulent plane mixing layer were measured in order to clarify the time-averaged aerodynamic forces acting on the cylinders, together with the flow patterns around them. The Reynolds number based on the diameter of the cylinder d and the mainstream velocity U outside the mixing layer was in the range (2.16--4.06) × 104. Both the angular position of the stagnation point and the stagnation-pressure coefficient were well correlated with a parameter d]6, 6 being the width of the mixing layer, if the location of the cylinders was expressed in terms of the ratio uc]U, where u c is the velocity of the otherwise undisturbed mixing layer at the centre of the cylinders. The drag coefficient was found to be rather insensitive to the parameter d/8, whereas a slight dependence of the drag coefficient on the cylinder diameter was observed owing to the effect of turbulence in the mixing layer. The lift force was always directed from the high-velocity side to the lowvelocity side of the mixing layer, its magnitude being approximately proportional to the parameter d ]~ .

1. Introduction Aerodynamic behaviour of bluff bodies in the wakes of other bodies has attracted the attention of many investigators of the industrial aerodynamics, especially in connection with the forces acting on downstream bodies (Hori [1], Mair and Maull [2], Ishigai et al. [3], Bearman and Wadcock [4], Price [5], Zdravkovich and Pridden [6], Zdravkovich [7], Reinhold et al. [8], Quadflieg [9], among others). These investigators were usually concerned with t w o bluff bodies of comparable dimensions. The aerodynamic behaviour of bluff bodies in the wake of upstream bodies of much larger dimensions have n o t been clarified as yet. If upstream bodies are much larger than downstream ones, the latter will be situated in one of turbulent shear layers separated from the surface of the former. Some practical examples of bluff bodies in this category may be smokestacks, towers or poles in the leeward vicinity of a corner of rectangular buildings. Pedestrians turning a corner of such buildings will be exposed to highly turbulent and nonuniform air flow resulting from the shear layer separated from the corner concerned. It is suggested

14 by Gandemer [10] that a strong horizontal or vertical gradient of mean velocity markedly increases the discomfort of pedestrians, especially if they are moving. However, influences of the velocity gradients have not yet been integrated into any criterion for the estimation of pedestrians' discomfort. Studies on these influences have thus to be pursued in order to secure the safety and comfort of pedestrians in strong winds. With the above-mentioned applications in mind, the present paper describes an experimental investigation of time-averaged aerodynamic forces acting on circular cylinders placed in a turbulent plane mixing layer. Circular cylinders were chosen because they could be considered as a typical representative of the smooth bluff bodies concerned, such as smokestacks, poles or pedestrians. The turbulent plane mixing layer is assumed to be an approximation to separated shear layers. It is well established that a circular cylinder in the wake of another circular cylinder of approximately the same diameter experiences a lift force in the direction towards the centre of the wake of the upstream cylinder. Although a number of theories have been developed to interpret the lift force of this nature, it seems that none of these are able to predict both the direction and magnitude of the lift force accurately enough to give quantitative agreement with experimental measurements. Within the authors' knowledge, the wakedisplacement hypothesis originally proposed by Mair and Maull [2] is most plausible at least qualitatively, among the theories presented heretofore, in the interpretation of the physical mechanism responsible for the origin of the lift force in the direction of the centre of the wake, as long as the downstream cylinder is not deeply submerged in the wake. It is expected that the aerodynamic bel~aviour of circular cylinders in a plane mixing layer will throw some light on the understanding of the lift, generation mechanism in the wake. Some considerations of this problem will be given in the discussion of the result of the present investigation. In passing it may be noted that, to the authors' knowledge, the forces acting on circular cylinders in turbulent plane mixing layers have not, as yet, been clarified. 2. Parameters to be included The definition sketch of the flow configuration treated in this investigation is shown in Fig. 1. One considers a turbulent plane mixing layer originating from the mixing of two streams of the same fluid with different velocities. The higher velocity will be denoted by U while the lower velocity is taken as zero. The origin of the Cartesian coordinate system x , y , z , is located at the middle of the trailing edge of the splitter plate. The streamwise coordinate x is parallel to the surface of the splitter plate, the transverse coordinate y in the direction of decreasing velocity and the z-axis normal to both x- and y-axes. Downstream of the point of encounter (the origin of the coordinate system) the width of the mixing layer will increase as a linear function of the down-

15

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Fig. 1. F l o w c o n f i g u r a t i o n a n d d e f i n i t i o n o f s y m b o l s , z-axis is n o r m a l t o b o t h x- a n d y-axes.

stream distance measured from an appropriate virtual origin whose location is designated as x0 in Fig. 1. It is well established that the flow in the mixing layer becomes self-preserving and thus the mean velocity and turbulent properties have similar nondimensional profiles at different downstream distances. Accordingly the properties of flow in the mixing layer can be fully described once the mainstream velocity U and its width 5 are given, as long as the turbulence level in the mainstream is sufficiently low. Here the width 5 may be defined as the transverse distance between locations where the mean velocity becomes ( 1 / 4 ) U and (3/4)U, respectively. The geometrical shape of a circular cylinder of infinite span, which is the case to be simulated in this study, can be determined by its diameter d. The location of the cylinder in the mixing layer is, of course, described by the coordinate of its centre. However, since the development of the mixing layer depends on the nature of the boundary layer at the trailing edge of the plate and probably on the turbulence level in the main flow outside the mixing layer, it is n o t necessarily convenient to introduce explicitly the streamwise coordinate x in order to assign the location of the cylinder. Accordingly, one defines here the location of the cylinder in terms of the velocity ratio uc/U, uc being the velocity of the otherwise undisturbed mixing layer at the centre of cylinders, and the ratio d/$ which measures the diameter of cylinders relative to the width of the mixing layer. The property of fluid can be represented by the density p and the kinematic viscosity v because the effects of compressibility and heat transfer will not be considered in the present study. The Reynolds number of the flow around the cylinders can be defined as Re = Ud/v. The foregoing considerations lead to the conclusion that the aerodynamic behaviour of circular cylinders in the turbulent plane mixing layer is described by three nondimensional parameters Uc/U, d/5 and Re. As an example, the stagnation pressure Pmax on the surface of the cylinders may be written as (Pmax --P0)/( 1/~P U2) = f(uc/U, d/5, R e ) , where P0 is the reference pressure outside the mixing layer.

(1)

16 3. Experimental apparatus and procedure The mixing layer e m p l o y e d in the present experiment was the boundary of a free jet issuing into the ambient atmosphere from a rectangular d u c t 34-cm high, 28-cm wide and 100-cm long. The uniformity of the mean velocity profile at the exit of the d u c t was good (within an error of + 0.5% at most) except for turbulent b o u n d a r y layers along the duct walls. The longitudinal turbulence intensity of the mainstream was a b o u t 3% at the mean velocity of 16.0 m/s. The mean velocity profiles in the mixing layer were measured by means of a Pitot~yaw tube which consisted of t w o tubes for the detection of the flow direction and total- and static tubes for the determination of the velocity magnitude; the diameter of each tube being 0.12 cm. The mainstream velocity U was maintained constant during measurements by monitoring the dynamic pressure in the potential core of the jet issuing from the duct. The velocity fluctuations in the mixing layer were detected by linearized constant, temperature hot-wire anemometers (KANOMAX 2700) whose o u t p u t signals were processed to obtain the foot, mean-square values of the velocity fluctuations. Calculation of correlation coefficients was performed by the use of a digital correlator (SAICOR 42A) after the hot, wire signals were passed through a low-pass filter (1 KHz) to escape the aliasing error which had its origin in discrete sampling of continuous analogue signals by digitizers. The frequency of the vortex shedding from the cylinders was determined from the p o w e r spectrum which is a Fourier transform of the autocorrelation function of velocity fluctuations.The p o w e r spectrum was obtained by means of a Fourier transformer (SAICOR 472). Circular cylinders tested in this experiment had diameters 2.0 cm, 3.0 cm and 3.7 cm, respectively. In order to secure the two-dimensionality of the flow around the test cylinders, a pair of thin circular end plates of diameter 25.0 cm was installed with a distance apart of 11.0 cm. The cylinders were set in position b e t w e e n the end plates by means of screws. The effective span of the cylinders was thus 11.0 cm for all tests, the aspect ratio being between 3.0 and 5.5. The diameter of the end plates was chosen such that the base pressures of the test cylinders measured in the potential core of the issuing jet were equal to those of circular cylinders immersed in a uniform stream at the same Reynolds number, which are well d o c u m e n t e d in the literature. As will be shown later, the two-dimensionality of the mixing-layer flow between the end plates was satisfactory when measured with the absence of the test cylinders. The cylinders had a pressure tap of diameter 0.05 cm at the mid-span to measure static pressure distributions along the surface of the cylinders by rotating the cylinders a b o u t their axes. At the outset, the surface pressure distributions were measured in the potential core in order to determine the direction of the main flow on the protractor by means of the s y m m e t r y of the pressure-distribution curve. This direction was taken as the reference to measure the angular position 0 of the cylinder surface. It may be n o t e d that

17 the d i r e c t i o n o f the main flow t h u s o b t a i n e d was practically parallel t o the f l o o r o f the e x p e r i m e n t a l air duct. All the m e a s u r e m e n t s t o be d e s c r i b e d in w h a t follows were m a d e at the m a i n s t r e a m v e l o c i t y o f U = 16.0 m/s. 4. E x p e r i m e n t a l results a n d d i s c u s s i o n 4.1

Characteristics o f m i x i n g layer

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constant-velocity lines are found to be nearly parallel to the z-axis in the higher-velocity side of the mixing layer, thus showing good two-dimensionality, whereas they have a little warping in the region izl 1> 2.0 cm of the lowervelocity side. The deviation from the two-dimensionality is 0.05U at most in that region. All the measurements for the mixing layer were thus performed in the xy-plane for which z = 0. Figure 4 shows the development of the mixing layer in terms of y ~ , y~, y~ (see Fig. 1) and Y0.9, which is the y-coordinate where the mean velocity is equal to 0.9U, plotted against the downstream distance x. This figure permitted

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to determine the virtual origin of the mixing layer to be located at x0 = --1.2 cm. The width of the mixing layer 6 can conveniently be defined as the distance between Yv, and Y3~, i.e. ~ = y,/, - - Yv,. The development shown in Fig. 4 yields an empirical formula 5 = 0.11(x--x0).

(3)

The existence of the end plates has negligible influence on the mean-velocity profile in the central region, the thickness of the mixing layer thus being described als0 by eqn.(3). Figure 2 also includes the distribution of the longitudinal turbulence intensity (u'2) v"/U at the same downstream stations as the mean-velocity measurements. For the _purpose of comparison the measurements of Wygnanski and Fiedler [11] and Liepman and Laufer [12] are also shown to indicate that the present data fall between these two results except for a higher-velocity region V < --0.1, where the present results give higher turbulence intensity. This feature was brought a b o u t by a higher level of mainstream turbulence in the present experiment (about 3%) than that in the experiments of the above investigators. The turbulence intensity is found to have reasonably good similarity at the stations where the measurements were performed. The cross-correlation coefficient of longitudinal velocity fluctuations R l l ( A x ) was measured at various downstream stations, where Ax was the distance along the line y = Yv, ( x ) from a fixed hot-wire probe to a downstream hot-wire probe whose position was systematically changed. The t w o hot-wire signals were processed by a SAICOR 42A digital correlator to obtain the cross-correlation coefficient. The results are shown in Fig. 5 in the form of R ll plotted against Ax Ix, where x implies the streamwise coordinate where the fixed hot-wire probe is located. A reasonable similarity of the correlation curves is obtained at different stations, together with a rather good agreement with the measurement of Wygnanski and Fiedler [11]. The integral

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20

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(4)

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The foregoing results indicate that the mixing layer realized in the present experiment has acquired a self-preserving state at least in the region 20 cm ~< x ~< 60 cm, where the test cylinders are to be installed.

4.2 Characteristics o f circular cylinders in the mixing layer The experiments were performed under the following conditions:

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Since the Reynolds number relevant to full-scale flows is much higher than the above values (probably more than 10s), it is natural to suspect whether or n o t the present results can be applied to actual situations. The authors are of the opinion that some of the fundamental nature of aerodynamic behaviour of circular cylinders in the mixing layer can be obtained from the experiments in subcritical R e y n o l d ~ n u m b e r range.

21

Pressure distributions along the surface of the cylinders with diameter 3.0 cm are shown in Fig. 7 for various values of the parameters d/5 and uc/U in the form o f the pressure coefficient Cp versus 0. Here the pressure coefficient Cp has been defined by

Cp = (p --Po)/(%P U : ) ,

(5)

where p is the pressure on the cylinder surface. The pressure distributions shown in Fig. 7 are typical in the sense that qualitatively similar pressure distributions were also obtained for another two cylinders tested. The nature of the pressure distributions for each cylinder will be described later in some detail. For the purpose of simplicity in the discussion of the results, a part of the cylinders corresponding to 0 ~< 0 ~< ~ will be designated by the lowvelocity side and another part corresponding to n ~< 0 ~< 2~ by the highvelocity side because they are exposed to the lower- and higher-velocity sides of the mixing layer, respectively. Figure 7 discloses a few characteristic features of the pressure distributions

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22

along the surface o f circular cylinders in the mixing layer. Firstly, the point of m a x i m u m pressure is not located at 0 = 0 but shifted to the high-velocity side. Secondly, the m a x i m u m pressure coefficient increases with an increase of the parameter d/~. The point of m a x i m u m pressure could be interpreted as the stagnation point of flow and thus the m a x i m u m pressure as the stagnation pressure. These two facts indicate that the stagnation streamline is shifted from the line 0 = 0 towards the high-velocity side. Thirdly, the minimum pressures at the high- and low-velocity sides are not equal, the minimum pressure at the high-velocity side being generally larger than that at the low-velocity side. This feature, together with the fact that the stagnation point is located in the high-velocity side of the cylinder, suggests that the lift force (which is the c o m p o n e n t of force in the positive y-direction) is positive, i.e. it is directed from the high-velocity side to the low-velocity side. This is interesting in view o f the fact that the lift force experienced by a circular cylinder in the wake o f another cylinder, the diameter o f each cylinder being almost the same, is in the direction towards the centre o f the wake, as shown experimentally by many investigators (see a review by Zdravkovich [7] ). Therefore it may be safe to say that the mechanism of flow to produce the lift force has m u c h in c o m m o n in both cases, i.e. in the mixing layer and in the wake. Kiya [13] and Snyder [14] considered the aerodynamic characteristics of circular and elliptical cylinders immersed in a uniform shear flow which had a linearly-varying velocity profile. They found that the minimum pressure on the high-velocity side is lower than that on the low-velocity side, the lift force thus being in the direction from the low-velocity side to the high-velocity side. Since the direction of the lift force is contrary to that in the mixing layer or in the wake, the uniform shear-flow model is ineffective in the interpretation of the lift force acting on cylinders placed in these turbulent shear flows. However, if such cylinders have diameters much smaller than the width of the wake or mixing layer in which they are immersed, the cylinders will experience a lift force in the direction suggested by the uniform shear-flow model, as was actually observed by Price [5]. Figure 8 shows the angular position 0 st of the stagnation point on the cylinder surface as a function of the parameter d/~ for each value of uc/U. The magnitude of shift of the stagnation point towards the high-velocity side becomes larger and larger as the parameter uc/U decreases. The stagnation angle 0st is found to tend to a constant value for each uc/U when d/~ becomes larger than about 0.6. The case of Uc/U = 0.5 deserves special attention, in which the centre of the cylinders coincides with the inflexion point of the mean velocity profile in the mixing layer. Thus the undisturbed velocity profile will vary almost linearly with the y-coordinate in the neighbourhood of the cylinders. If the mixing layer is tentatively replaced by a uniform shear flow whose velocity gradient is denoted by G, the shear parameter Gd/u c becomes equivalent to d/~. The mean velocity profile shown in Fig. 2 seems to justify replacing the

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24

figuration shown in Fig. 1. As is also evident in the photographs of uniform flow past a circularcylinder with circulation(see Batchelor [16], plate 12), the stagnation point is significantlyrotated towards the direction opposite to that of circulation,i.e.to the high-velocity side in the case of Fig. 1. The same mechanism may be responsible for the rotation of the stagnation point on the surface of circularcylinders placed in the mixing layer. Although Rawlin's estimation (see Price [5] ) of the liftforce based on this concept predicted a correct tendency of change of the liftforce, the predicted magnitude of the lift was smaller by 20--30% than the experimental result of Price [5]. The authors are not aware of the magnitude of the stagnation-point shift predicted by Rawlin's method. It seems that a correct theory remains to be constructed in order to predict the aerodynamic behaviour of circular cylinders placed in the wake or in the mixing layer. The variation of the stagnation-pressure coefficient Cpmax on the cyhnder surface with respect to the parameter d/~ is shown in Fig. 9 for three values of uc/U. The asymptotic state d/5 -~ 0 corresponds to the case of a circular i-0

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Fig. 9. S t a g n a t i o n - p r e s s u r e c o e f f i c i e n t o n t h e s u r f a c e o f circular c y l i n d e r s as a f u n c t i o n o f r a t i o d/6. D, d = 2.0 c m ; o, d = 3.0 c m ; zx, d = 3.7 c m ; e , Cpmax f o r u n i f o r m i n c i d e n t f l o w o f v e l o c i t y u c. Solid lines are t e n t a t i v e l y d r a w n t h r o u g h d a t a p o i n t s . A r r o w s % i m p l y t h a t t w o d a t a p o i n t s o a n d z~ coincide. (a) uc/U = 0.75, (b) uc/U= 0.5, (c) uc/U ffi 0.25.

the

25 cylinder immersed in an u n b o u n d e d uniform stream of velocity u c. Thus the stagnation-pressure coefficient at d/5 = 0 will b e c o m e (uc/U) 2, which is also shown in Fig. 9 by closed circles. It is observed that Cpmax is generally larger than (uc/U) 2 except that Cpmax's somewhat smaller than (uc/U) 2 appear when u c / U = 0.25. Since the measured pressure level is rather low in this particular case, it is probable that the pressure coefficients smaller than (u c/U) 2 may be attributed to experimental uncertainty. The fact that Cpmax is generally larger than (u c/U) 2 is another aspect of the rotation of the l"ncident local-flow vector towards the low-velocity side of the mixing layer. The stagnation-pressure coefficient increases monotonically with an increase of the parameter d/5, the possible maximum value of Cpmax being unity. At first sight it may seem strange that, despite rather constant values of 0st in the range d/5 > 0.6, the stagnation-pressure coefficient continues to show a substantial increase. This situation can be interpreted as follows. For simplicity consider the case where 6 is constant. The y-coordinate of the stagnation point, i.e. Yst = (d/2)sin•st, becomes larger and larger as the diameter d is increased even if the stagnation angle 0 st remains constant. Since larger value of Yst corresponds to a shift of the stagnation streamline towards the side of higher velocity o f the mixing layer, the stagnation pressure can increase with an increase o f the diameter d. Accordingly the result is that an increase of the parameter d/5 is accompanied by an increase of the stagnationpressure coefficient. It deserves mention that 0 st and Cpmaxcollapse almost onto single curves respectively for each value of u c / U when they are plotted against the ratio d/5. Attention will n o w be turned to the aerodynamic forces acting on the circular cylinders located in the mixing layer. Since, as mentioned at the beginning of this section, the Reynolds number is of the order of 104 in the present experiment, the forces experienced by the cylinders are almost entirely due to the nonuniform pressure distribution along the surface of the cylinders, the contribution from the viscous shear stress being negligibly small. The drag force D is defined as the c o m p o n e n t of force in the x-direction The lift force L is the c o m p o n e n t of force in the y
(6)

Figure 10 shows the drag coefficient Co plotted against the parameter d/5 for each value of the velocity ratio uc/U. The drag coefficient is found to be rather insensitive to the parameter d/6 while some systematic change of C D with respect to the cylinder diameter is observed. At first sight, one may be inclined to attribute this dependence of C D on the cylinder diameter to the Reynolds number effect. In the case of u c / U = 0.75, for instance, the Reynolds n u m b e r defined by ucd/v is 1.6 × 104, 2.4 × 104 and 3.0 × 104 for the cylinders of diameter 2.0 cm, 3.0 cm and 3.7 cm, respectively. Since it is well established that the drag coefficient of a circular cylinder in this range

26 0°6

I

n

(o)

0.5

I

I

I

o

t]

o O

0.4

I

0 o

0

0

•

0

•

drnnl

a o •

0-3

uu

(b)

0-2

20 30 37

OII

o

o 0

•

O

o •

AO

2

• °

•

0.1

(c) 0

0

o" ° oaa• op I (~2

I 0.4

I 0-6

d/$

I 0.8

I 1.0

1.2

Fig. 1 0 . D r a g c o e f f i c i e n t o f c i r c u l a r c y l i n d e r s as a f u n c t i o n o f t h e r a t i o ( b ) uc/U = 0 . 5 , ( c ) ue/U -- 0 . 2 5 .

d/6. ( a ) uc/U=

0.75,

of the Reynolds number is almost constant in the case of a uniform incident stream o f small turbulence intensity, the above-mentioned dependence of the drag coefficient on the cylinder diameter is t o o large to be explained by the Reynolds-number effect especially in the range of smaller d/~ (see Fig. 10(a)). It is observed that a lower value of C D is obtained for larger diameter o f the cylinders except for the rightmost data points for u c / U = 0.5 and 0.25, which may be attributed to an experimental error. The authors are of the opinion that this p h e n o m e n o n can be explained by the effect of turbulence originally included in the surrounding mixing layer. In order to examine the effect of turbulence intensity and scale, the drag coefficient of circular cylinders placed in a uniform flow which contains the grid-generated turbulence is plotted against a parameter T u = [(u'2) ~/Uo. ] ( d / L x ) , where U~ is the free-stream velocity, (u'2) ~ the root-mean-square value of the longitudinal turbulent velocity fluctuation and L x denotes the integral length scale defined exactly in the same way as eqn. (4). The results are summarized in Fig. 11, which shows that C D generally decreases with an increase of the parameter Tu. On the other hand, as will be observed in Fig. 2, a value o f u c / U is accompanied by a particular value of the nondimensional turbulence intensity (u-if) ~ / U . Since the integral length scale of turbulence L x increases in linear proportion to 5, an increase in d results in an increase of the ratio d / L x if the width is maintained to be constant. Accordingly, provided that/~ and u c / U remain unchanged, the value of the parameter Tu increases with an increase o f the cylinder diameter d. The result is that, if

27

0.35 i

°°&

o

1.0

o

A

0.30 o

o

0.8

A

$

o

0.25

o

0.(

'#,o!

0.4

r

0.20

•

A Solid

symbols for

St

~.15

02 I 0

0

011 012Tu 013 o',

~10

Fig. 11. Drag coefficient and Strouhal number of circular cylinders placed in grid-generated turbulent flows as functions of the parameter Tu. D, • Surry [21] ; o, a, o, • measurements in authors' laboratory.

a n d u c / U are constant respectively, a cylinder of larger diameter will have

a smaller drag coefficient. The same tendency is clearly observed in Fig. 10 especially in the case of u c / U = 0.75. At this stage of the discussion it will be appropriate to mention a few parameters similar in nature to Tu, which were proposed by previous investigators. Taylor [17] derived a parameter, n o w called Taylor number Ty = [(u'2) ~ / U = ] ( d / L x )1/5, in his theoretical study of the effect of free-stream turbulence on the critical Reynolds n u m b e r for boundary layers along the surface of circular cylinders or spheres. Taylor showed that the critical Reynolds number plotted against T y collapsed approximately onto a single curve. When the drag coefficient C D shown in Fig. 11 was replotted against the Taylor number Ty, the drag coefficient was again found to decrease accompanied by an increase of Ty with a somewhat broader band of scatter of data than that shown in Fig. 11. Bearman [18] derived a parameter, say TB, for square plates by considering an enhanced entrainment rate in the separated shear layers owing to the freestream turbulence. His parameter may be reduced to T B = [(u'2)~,/U~ ] ( L x / d ) for two-dimensional bodies. Although Bearman obtained a very good correlation between the drag coefficient or base pressure coefficient and the parameter T B for the square plates, the same correlation was found to be rather poor for circular cylinders placed in the grid-generated turbulent flows. This situation may suggest that the free-stream turbulence plays different roles in both cases, i.e. square plates and circular cylinders, in the determination of the base-pressure level. It is well established that the free-stream turbulence decreases the base pressure of thin normal plates, thus producing larger drag force, whereas the pressure at the rear of circular cylinders is raised by the free-stream turbulence, thus leading to a smaller drag coefficient. Accordingly, an appropriate parameter to represent the effect of the free-stream turbulence

28

may possibly be different for thin normal plates and circular cylinders, respectively. Moreover, it may be possible to assume that, in the case of circular cylinders, the part after the separation point will influence the behaviour of the separated shear layers so as to reduce the suction pressure at the rear of the cylinders. The parameter Tu was obtained by trial and error, the base-pressure coefficient and the spanwise correlation length of the shed vortices in addition to the drag coefficient having been incorporated during the search for the parameter. A detailed report on this subject will be published in the near future elsewhere. At present the authors are of the opinion that the parameter Tu yields the best general collapse of the aforementioned aerodynamic properties of circular cylinders placed in the grid-generated turbulent flows. The parameter Tu was first introduced by Batchelor and Proudman [19] in their rapid distortion theory of turbulence in which the parameter Tu was interpreted as the ratio of time scales relevant to the distortion of mean flow d/Uoo and relevant to the distortion of turbulent eddies Lx/(U'2) ~ The ratio actually yields

Tu = (d/U.. )/[Lx / ( ~ - ) ~ l = [ (u'~)~ /U,~ 1 (d/Lx ) , which is assumed to be much smaller than unity in the rapid distortion theory. In passing it may be noted that the theory of Batchelor and Proudman [ 19] was extended by Hunt [20] to the turbulent flow past two-dimensional bluff bodies. However, Hunt's theory can tell little about the influence of the freestream turbulence on the behaviour of the separated shear layers. The physical reasoning is not clear to the authors why, of the three parameters mentioned above, the parameter Tu is most effective in collapsing the drag and base pressure coefficients and the spanwise correlation length of the shed vortices. A plausible explanation remains to be investigated in the future. The value of the drag coefficient at the limit d/~ ~ 0 corresponds to that of circular cylinders placed in a uniform flow of infinite extent with zero or nonzero free-stream turbulence. Since, as clarified above, the drag coefficient depends upon the intensity and scale of the free-stream turbulence even in the case of uniform incident flow, it is not possible to assign a unique value of Cv at d/6 = 0. However, the drag coefficient for each u c / U is smaller than that of circular cylinders placed in the uniform stream of velocity u c with negligibly small turbulence intensity. The lift coefficient C L of circular cylinders is plotted in Fig. 12 as a function of d/~ for three values of Uc/U. In the range of the present experiment, the lift force is positive, viz. it directs from the high-velocity side to the lowvelocity side of the mixing layer. The lift coefficient increases with an increase of the parameter d/~. Although a somewhat large scatter of data exists, any systematic change of the data points with regard to the cylinder diameter cannot be observed. Since the lift coefficient seems to increase approximately in linear proportion to the parameter d/~, the least-squares-method has been applied to obtain the best-fit straight lines, which are also shown in

29

Fig. 12. Although the best-fit straight lines were not forced to pass through the origin, the intersections of the resulting lines coincided with the origin with very good accuracy. 0.30 0.4O

I

~

,

I

I

i

lb/

I

6

¢a)

•

0.25

/

0~35 0.20 0"3( 0.2',

?°

?0

0.15

O-05

/

0-10

o o o= oa

o

/

/

o

0.05 /

0

o/°°

O~

0I'

dram

016 018 d~

1.10

1-2

o 20 o 30 37

~

(c)

'

' oo

0

a

o

0-20

0.10

0.15

I L 0-6 d/~ 0-8

1!0

1-2

0~ -0

'

'

, A

° ° ° °t2r'~=-~-°~ },-o 02 0-4 O.6a 0.B d,~

I

J

o

1.0

J 1.2

FiE. 12. L i f t c o e f f i c i e n t o f c i r c u l a r c y l i n d e r s as a f u n c t i o n o f t h e r a t i o d / 6 . o, d = 2.0 c m ; o, d = 3.0 cm; z~, d = 3.7 cm. S o l i d l i n e s are o b t a i n e d b y t h e l e a s t - s q u a . r e s - m e t h o d . (a) uc/U= 0.75, (b) uc/U= 0.5, (c) uc/U= 0.25.

Since the mixing layer includes a high level of turbulence, the periodic vortex shedding will become undetected when the circular cylinders are deeply immersed into the mixing layer. Table 1 shows the vortex-shedding frequency in the form o f the Strouhal number St( = fd/U, f being the frequency) versus u c / U for two cases of d/5. When smaller u c / U is approached, a spectrum peak which corresponds to the vortex-shedding frequency becomes lower and lower, together with the broadening of the spectrum, until the spectrum peak is hidden within the background turbulence spectrum. The value of u c /U beyond which the vortex shedding is undetected is found to become smaller as the ratio d/5 increases. Although the Strouhal number St generally decreases as Uc/U decreases, another Strouhal number defined by St c = fd/u c nevertheless increases slightly with an increase of uc/U. This feature may be interpreted as the effect of the mixing-layer turbulence in the following way. The Strouhal number St = fd/U= for a circular cylinder in the grid-generated turbulent flows is a slightly increasing function of the parameter Tu as already shown in F__~. 11. On the other hand, the nondimensional turbulence intensity (u'2) ~ / U increases with a decrease of u e / U in the range uc/U > 0.5 while it may be assumed that d / L x remains approx-

3O TABLE 1 Strouhal number of vortex shedding from d (cm)

d/6

uc/U

2.0

0.30 0.97

St c

0.17 0.17 0.16 0.16 0.15 0.14 0.15 0.14 0.16 0.13 0.15 0.12

0.18 0.18 0.18 0.18 0.19 0.17 0.20 0.19 0.22 0.18 0.22 0.18 --

0.68

h l h l h l h l h l h

0.60

l h

--

l

--

h l h l h

0.16 0.16 0.15 0.13 0.17

0.87 0.81 0.75 0.73

3.7

St

0.55 0.97 0.81 0.73

circular cylinders

0.17 0.17 0.19 0.16 0.23

h; high-velocity side, l; low-velocity side.

i m a t e l y c o n s t a n t t h r o u g h o u t t h e m i x i n g l a y e r f o r a p a r t i c u l a r value o f t h e ratio d/5. T h u s a decrease o f uc/U is a c c o m p a n i e d b y an increase in Tu and t h e result is a slight increase in t h e S t r o u h a l n u m b e r St c. T h e r e l a t i o n b e t w e e n uc/U and St c s h o w n in T a b l e 1 m a y be f o u n d t o validate the above discussion. T a b l e 1 also shows t h a t t h e v o r t e x - s h e d d i n g f r e q u e n c y is a little d i f f e r e n t o n b o t h sides o f t h e cylinders, St c o n the high-velocity side being larger t h a n t h a t o n the low-velocity side. A l t h o u g h the physical m e c h a n i s m responsible f o r this p h e n o m e n o n is n o t clear t o t h e authors, it m a y deserve m e n t i o n as an interesting f e a t u r e o f t h e v o r t e x shedding in highly t u r b u l e n t shear flows. 5. C o n c l u s i o n In t h e p r e s e n t paper, m e a n pressure distributions along the surface o f circular c y l i n d e r s i m m e r s e d in a t u r b u l e n t p l a n e free-mixing l a y e r have b e e n d e s c r i b e d in o r d e r t o clarify t h e m e a n a e r o d y n a m i c forces e x p e r i e n c e d by t h e cylinders, t o g e t h e r with t h e f l o w p a t t e r n s a r o u n d t h e m . T h e R e y n o l d s n u m b e r based o n t h e c y l i n d e r d i a m e t e r and the free-stream v e l o c i t y outside the m i x i n g l a y e r was in t h e range ( 2 . 1 6 - - 4 . 0 6 ) × 104. T h e main results o f the p r e s e n t s t u d y m a y be s u m m a r i z e d as follows:

31

(1) The stagnation point on the cylinder surface shifts towards the highvelocity side of the mixing layer. This can be interpreted as the rotation of the incident velocity vector to the low-velocity side. The shift of the angular position of the stagnation point becomes larger and larger as the velocity ratio u c/U decreases, but it is almost fixed when the parameter d/6 is more than about 0.6, the asymptotic values being larger for smaller uc/U. (2) The stagnation pressure on the cylinder surface increases monotonically with an increase of d/a. It is much larger than the dynamic pressure of the otherwise undisturbed mixing layer at the centre of the cylinders. The stagnation streamline shifts to the high-velocity side of the mixing layer. (3) The angular position of the stagnation point and the stagnation-pressure coefficient can be well correlated with the parameter d/a for each value of uc/U. (4) The drag coefficient of circular cylinders in the mixing layer is almost independent of the parameter d/a, although some systematic effect of the cylinder diameter is observed owing to the high turbulence level in the mixing layer. (5) The lift coefficient is in the direction from the high velocity side to the low-velocity side of the mixing layer, its magnitude being approximately in linear proportion to the parameter d/6. (6) The periodic vortex shedding from the cylinders becomes undetectable when the velocity ratio Uc/U is sufficiently small.

Acknowledgements The authors express their sincere thanks to Mr T. Yamazaki and Mr T. Sampo, Technical Officials of the Faculty of Engin¢ering, Hokkaldo University, for their assistance in the construction of the experimental apparatus. The present study was supported by the Grant-In-Aid for Scientific Research from the Ministry of Education, Science and Culture (Japan). Nomenclature

d

f P P0 Pmax u u ~

uc x,y,z Xo

Yv, Yv,

diameter of cylinder frequency of vortex shedding from circular cylinder pressure on the surface of cylinder reference pressure outside mixing layer stagnation pressure on the surface of cylinder local mean longitudinal velocity c o m p o n e n t longitudinal turbulent velocity fluctuation mean velocity at centre of cylinder Cartesian coordinate system (see Fig. 1) virtual origin of mixing layer y-coordinate where u = U/2 y-coordinate where u = U/4

32 Y~A

Yo.9 Ca CL

~¢~pms:x

D L

Rll St St c

%

Tu

Ty U

u® ~7 0

Ost v

p

y-coordinate where u = 3U/4 y-coordinate where u = 0.9U drag coefficient = D~ [ (l/~)p U~d ] lift coefficient = L~ [(l~)p U~d] pressure coefficient = (p -- po)/[ (1/~)p U~ ] stagnation-pressure coefficient drag force lift force longitudinal length scale o f turbulence correlation coefficient Strouhal number of vortex shedding = f d / U Strouhal number of vortex shedding = f d / u c nondimensional parameter = [(u_~_)~/U~ ] (L x / d ) nondimensional parameter [ (u'2)~ /U~ ] (d /Lx ) Taylor number = [(u-~)~/U® ] (d/Lx)l/5 mainstream velocity outside mixing layer velocity o f uniform flow past circular cylinders width of mixing layer = Yl/, -- Y3/, nondimensional coordinate = (y -- y~ ) / ( x - - Xo) angle (see Fig. 1) angular position of stagnation point kinematic viscosity of fluid density of fluid

References 1 E. Horl, Experiments on flow around a pair of parallel circular cylinders, Proc. 9th Japan Nat. Cong. Appl. Mech., 1959, pp. 231--234. 2 W.A. Mair and D.J. Maull, Aerodynamic behaviour o f bodies in the wakes of other bodies, Philos. Trans. R. Soc. London, A260(1971) 425---437. 3 IC Ishigai, E. Nishikawa, Y. Nishimura and K. Cho, Experimental study on structure of gas flow in tube banks with tube axes normal to flow (Part 1. Karman vortex flow around two tubes at various spacing), Bull. JSME, 15(1972) 949--956. 4 P.W. Bearman and A.J. Wadcock, The interaction between circular cylinders normal to a stream, J. Fluid Mec.h., 61(1973) 499--511. 5 S.J. Price, The origin and nature o f the lift force on two bluff bodies, Aeronaut. Q., XXVI(1976) 1 5 4 - 1 6 8 . 6 M.M. Zdravkovich and D . L Pridden, Interaction between two-circular cylinders: series of unexpected discontinuities, J. Ind. Aerodyn., 2(1977) 618--633. 7 M.IVLZdravkovich, Review o f flow interference between two circular cylinders in various arrangements, Trans. ASME, Set. I, J. Fluids Eng., 99(1977) 618--633. 8 T.A. Reinhold, H.W. Tieleman and F.J. Maher, Interaction of square prisms in two flow fields, J. Ind. Aerodyn., 2(1977) 223--241. 9 H. Quadflieg, Vortex induced load on the cylinder pair at high Reynolds numbers (in German), Forsch. Ingenieurwes., 43(1977) 9--18. 10 I~ Gandemer, Aerodynamic studies of built-up areas made by C.S.T.B. at Nantes, France, J. Ind. Aerodyn., 3(1978) 227--240. 11 I. Wygnanski and H.E. Fiedler, The two-dimensional mixing region, J. Fluid Mech., 41(1970) 327--361.

33 12 H.W. Liepman and J. Laufer, Investigation of free turbulent mixing layer, N.A.C.A. Wash., Tech. Note No. 1257(1947). 13 M. Kiya, Study on turbulent shear flow past a circular cylinder, Bull. Fac. Eng., Hokkaido University, 50(1968) 1--101. 14 M..A. Snyder, Testing of circular cylinders in shear flow, J. Aircr., 8(1971) 593--596. 15 M. Arie, Characteristics of two-dimensional flow behind a normal plate in contact with a boundary o n half plane, Mem. Fac. Eng., Hokkaido University, 10(1956) 211-310. 16 G.I~ Batchelor, An Introduction to Fluid Dynamics, Camb. Univ. Press, 1967. 17 G.I. Taylor, Statistical theory of turbulence Part V. Effect of turbulence on boundary layer. The theoretical discussion of the relationship between scale of turbulence and critical resistance of spheres, Proc. R. Soc. London, A, CLVI(1936) 307--317. 18 P.W. Bearman, An investigation of the forces on flat plates normal to a turbulent flow, J. Fluid Mech., 46(1971) 177--198. 19 G.K. Batchelor and L Proudman, The effect of rapid distortion of a fluid in turbulent motion, Q. J. Mech. Appl. Math., 7(1954) 83. 20 J.C.R. Hunt, A theory of turbulent flow round two-dimensional bluff bodies, J. Fluid Mech., 61 (1973) 625--706. 21 D. Surry, Some effects of intense turbulence on the aerodynamics of a circular cylinder at subcritical Reynolds number, J. Fluid Mech., 52(1972)543--563.