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Vortex wake of a transversely oscillating square cylinder: A flow visualization analysis
Journal of Wind Engineering and Industrial Aerodynamics, 45 (1992) 97 119
97
Elsevier
Vortex wake of a transversely oscillating square cylinder: A flow visualization analysis S.C. Luo Department of Mechanical & Production Engineering, National University of Singapore. 10 Kent Ridge Crescent, Singapore 0511 (Received March 2, 1992; accepted in revised form August 3, 1992)
Summary The flow past a transversely oscillating square cylinder is visualized by the smoke wire technique and studied. In the present investigation, by varying the cylinder oscillating frequency f~ in a free stream with constant velocity U, the reduced velocity U r varies from 7.65 to 97.38. Under the above conditions and at a constant amplitude of oscillation of A/d=0.675, four different flow structures were identified. They are called lock-on type A, lock-on type B, triple lock-on and quasi-steady in the present paper. The way that vortices are formed and shed are different in these four flow structures. The above results in different wake structures with rather different magnitudes of longitudinal and lateral vortex spacing (a/d and b/d, respectively). It was also found that the vortex shedding mechanism proposed by Gerrard is applicable only to lock-on type B and quasi-steady, but not the other two. The vortex spacing ratio b/a of 0.281 predicted by Von Karman does not apply to the present flow situations. During lock-on, flow visualization reveals that when fN decreases in magnitude at a constant A/d, both aid and F (circulation associated with each vortex in the wake) increase in magnitude. Under certain flow conditions the unconventional thrust type of vortex street exists. The position of the cylinder within its oscillation cycle when vortices are shed was observed to be a factor that affects the magnitude of the lateral vortex spacing and whether the cylinder is susceptible to flow induced oscillation.
Notation A
amplitude of oscillation
a
longitudinal vortex spacing lateral vortex spacing lift c o e f f i c i e n t s t a n d a r d d e v i a t i o n o f lift c o e f f i c i e n t side l e n g t h of the s q u a r e c y l i n d e r
b CL C[ d
Correspondence to: Dr. S.C. Luo, Department of Mechanical and Production Engineering, National University of Singapore, 10 Kent Ridge Crescent, Singapore 0511.
0167-6105/92/$05.00 { 1992 Elsevier Science Publishers B.V. All rights reserved.
98
f~ Re Sr U
ur Uro Y F P P (p
S.C. Luo/Vortex wake of a transversely oscillating square cyclinder
cylinder forced oscillation frequency vortex shedding frequency Reynolds Number = f~Ud/p Strouhal Number = f~ U/d free stream velocity reduced velocity = U/( f N d ) resonance reduced velocity = 1/Sr cylinder transverse displacement circulation of each vortex dynamic viscosity of air density of air phase angle between the forcing frequency component of lift and y
1. I n t r o d u c t i o n
In the last few decades, due to improvement in construction techniques. typical civil engineering constructions which include buildings and bridges were designed to become lighter and more slender. While the above may be considered as an improvement in terms of cost effectiveness over older designs. other problems, notably that of flow induced vibration became more critical. Flutter, galloping, and vortex resonance are the three most commonly encountered types of flow induced vibration [1]. While the circular cylinder is the most frequently employed shape in vortex resonance studies because of its isolation from the other types of flow induced vibrations mentioned above, in research related to galloping the square section cylinder (from now on simply referred to as "the square cylinder") is the more commonly used shape. The two main reasons that favour the use of a square are its resemblance to the basic rectangular shape of most buildings and bridge decks, and the challenge it offers as it is subjected to both galloping and vortex resonance. The author had carried out a series of measurement on aerodynamic forces acting on a transversely oscillating square cylinder and the findings were reported in Ref. [2]. It was reported in the above paper that at very low reduced velocity ( U r = U/fNd, where U, fN, and d are the free stream velocity, forcing frequency, and the characteristic dimension (in the present case the side length) of the square cylinder, respectively) the magnitude of the fluctuating lift coefficient C'L (defined as the dimensionless standard deviation of the lift force) was very large because of vortex resonance. As the reduced velocity increased, C'L reduced and reached a minimum of about 0.75. As the reduced velocity increased further, C'L increased initially rapidly but later approached asymptotically the stationary square cylinder's value of about 1.4. Based on the C' t versus Ur variation just described, Bearman and Luo [2] suggested that the flow around a square section cylinder oscillating at constant amplitude can be divided into three main regimes: the "multiple lock-in" regime which
S.C. Luo/Vortex wake of a transversely oscillating square cyclinder
99
I
Urm
Urs
UF Fig. 1. Classification of flow regimes ([2], Fig. 10).
c o r r e s p o n d s to the r e d u c e d v e l o c i t y r a n g e from low reduced v e l o c i t y to the r e d u c e d v e l o c i t y at which C'L is a m i n i m u m , the lift r e c o v e r y r e g i m e which c o r r e s p o n d s to r e d u c e d velocity from C'L = m i n i m u m to C'L = 1.4, and the " q u a s i - s t e a d y " r a n g e w h i c h s t a r t s from r e d u c e d v e l o c i t y at w h i c h C'~ = 1.4. A s k e t c h of the t h r e e regimes is r e p r o d u c e d as Fig. 1. The objective of the p r e s e n t p a p e r is to g a i n m o r e i n f o r m a t i o n of the flow by v i s u a l i z i n g it. It is hoped t h a t by b e i n g able to see the flow, c e r t a i n m e a s u r e m e n t s r e p o r t e d in Ref. [2] can be u n d e r s t o o d better.
2. Experimental arrangement The flow v i s u a l i z a t i o n was c o n d u c t e d in a closed r e t u r n wind t u n n e l with a 0.91 x 0.91 x 5.49 m (length) w o r k i n g section. B e c a u s e of the 9 to ] r a t i o e m p l o y e d in the c o n t r a c t i o n of the wind tunnel, the t u r b u l e n c e level in the w o r k i n g section is at a low v a l u e of no m o r e t h a n 0.05%. All the flow visualiza t i o n was c a r r i e d out at a c o n s t a n t wind speed of 2.72 m/s which was t h o u g h t to be a good c o m p r o m i s e b e t w e e n the c o n t r a d i c t i n g c r i t e r i a of m i n i m u m b u o y a n c y effect and t h i c k s m o k e which f a v o u r high and low wind speed, respectively. The s q u a r e c y l i n d e r model has a side length (d) of 50.8 mm. It was m a d e of light-weight wood and was p a i n t e d in m a t t b l a c k to provide o p t i m u m c o n t r a s t with the w h i t e smoke. To e n s u r e t h a t the flow was as t w o - d i m e n s i o n a l as possible, two p e r s p e x 304.8 x 228.6 m m r e c t a n g u l a r end plates (longer side p a r a l l e l to the flow) were installed at the two ends of the s q u a r e cylinder at a d i s t a n c e of 17d apart. The cylinder was installed h o r i z o n t a l l y with b o t h ends p r o t r u d i n g out of the sides of the w o r k i n g section. The ends of the c y l i n d e r were c o n n e c t e d to two v e r t i c a l push rods w h i c h were in t u r n c o n n e c t e d to
100
S.C. Luo/ Vortex wake of a transversely osciUating square cyclinder
a Scotch-Yoke mechanism installed underneath the working section. The Scotch-Yoke mechanism oscillated the cylinder in an up-and-down simple harmonic motion inside the working section. The Scotch-Yoke mechanism was designed in such a way that both the amplitude (A) and frequency (f~) of the oscillation can be varied in a continuous manner. The smoke wire method was employed in the present investigation to visualize the flow. In the present investigation, a thin (Reynolds number based on wire diameter <20) but high resistance Ni/Cr/Fe alloy wire was stretched vertically in front of the cylinder. Oil was either coated manually onto the wire or made to drip down it. When an electric current was passed through the wire, because of its high resistance, it heated up and caused the oil to vaporise to form numerous fine droplets which is seen as white smoke. The smoke was carried downstream by the free stream, thus visualizing the flow past an oscillating square cylinder. The visualized flow field was then captured on both still and cine films for detail analysis.
3. Experimental results In smoke wire flow visualization, because the optimum wind speed range is narrow, different reduced velocity was obtained by varying fN- AS mentioned in the previous section, all the flow visualization was conducted at a constant free stream velocity of 2.72 m/s. Based on the side length of the square cylinder, the corresponding Reynolds number (Re = p Ud/p, where p and # are the density and the dynamic viscosity of air, respectively) is approximately 8.76 x 103. The dimensionless amplitude of oscillation ( A / d ) was also kept constant at 0.675. As fN employed in the present investigation varied from 0.55 Hz to 7 Hz, the corresponding reduced velocity varied from 97.38 to 7.65. At the present Reynolds number the Strouhal number (Sr = f s d / U , where fs is the frequency of the shedding of vortices by one shear layer) is about 0.13. The lowest reduced velocity achievable in the present experiment is, thus, very slightly lower t h a n the resonance reduced velocity (Uro, = 1/Sr) of 7.69. The locking on of vortex shedding frequency to the cylinder oscillating frequency is, thus, expected at the low reduced velocity end. At A / d = 0.675 and within the range of reduced velocity investigated, at least four flow structures were identified by flow visualization. They will be discussed separately below. However, before describing the flow structure in detail, it is necessary to first define a few terms which will occur quite frequently in the following. During one cycle of the vertical simple harmonic motion, the highest and lowest positions reached by the cylinder are referred to as the top dead centre and bottom dead centre, respectively. They will from now on be abbreviated as TDC and BDC, respectively. The cylinder always oscillates with two faces normal and two faces parallel to the free stream. Of the two faces that are always normal to the free stream, the upstream and downstream faces are referred to as face A and C, respectively. For the other two faces, the top one (the one t h a t is closer to the ceiling of the working section) will be referred to as face B and the lower one
101
S.C. Luo/Vortex wake of a transversely oscillating square cyclinder
will be face D. The s h e a r l a y e r t h a t s e p a r a t e s from the c o r n e r of faces A and B will be r e f e r r e d to as e i t h e r s h e a r l a y e r B or the top s h e a r layer, while the s h e a r l a y e r t h a t s e p a r a t e s f r o m the c o r n e r of faces A and D will be r e f e r r e d to as e i t h e r s h e a r l a y e r D or the b o t t o m s h e a r layer. 3.1. Lock-on type A At fN = 7 Hz, Ur = 7.65. At this r e d u c e d v e l o c i t y v o r t e x shedding locks on to the body f r e q u e n c y and, hence, one v o r t e x per cycle is shed from e a c h s h e a r layer. The v o r t e x shedding process is s h o w n in Figs. 2 and 3. It should be n o t e d h e r e t h a t the p r e s e n t e x p e r i m e n t a l set-up is such t h a t it is easier to t a k e a flow p h o t o g r a p h on the side of the w o r k i n g section w h e r e the flow is from r i g h t to left. To be c o n s i s t e n t with the flow p h o t o g r a p h s for e a s y cross reference, the flow in Fig. 2 and s u b s e q u e n t figures are also from r i g h t to left and is, therefore, opposite to the "flow from left to r i g h t " c o n v e n t i o n a d o p t e d in m o s t o t h e r p a p e r s and t e x t books. S t a r t i n g from the TDC, a v o r t e x grows n e a r the top face (face B) as the c y l i n d e r m o v e s d o w n w a r d s . T h e v o r t e x grows to n e a r its m a x i m u m size w h e n the cylinder r e a c h e s the BDC. W h e n the c y l i n d e r r e v e r s e s its d i r e c t i o n and m o v e s upwards, the v o r t e x m o v e s d o w n s t r e a m a n d p a s t the r e a r c o r n e r ( j u n c t i o n of faces B and C). W h e n the cylinder r e a c h e s the TDC again, m o m e n t a r y flow r e a t t a c h m e n t o c c u r s at face B, the v o r t e x is cut off from the feeding s h e a r l a y e r and is shed. The s a m e s e q u e n c e r e p e a t s itself in s u b s e q u e n t cycles. I d e n t i c a l v o r t e x shedding s e q u e n c e o c c u r s on the b o t t o m face D b u t it is 180 ° out of p h a s e to t h a t on face B. A s k e t c h of the w a k e of lock-on type A is s h o w n in Fig. 4. Since the top s h e a r l a y e r sheds a v o r t e x w h e n the cylinder is n e a r the TDC and vice versa, the v o r t i c e s r e m a i n on the s a m e
LJ
TOC []
80C I
1
TocC"'~ ! 1
l I
1
1
2
BOC
Fig. 2. Vortex shedding sequence of lock-on type A.
2
102
S.C. Luo/ Vortex wake of a transversely oscillating square cyclinder
Fig. 3. Vortex shedding process of lock-on type A: (a) the cylinder is moving from the TDC to the BDC, a vortex is growing near face B; (b) the cylinder has almost reached the BDC, the top vortex is almost fully grown; (c) the reverse (upwards) motion of the cylinder forces the top vortex to go downstream; (d) temporary flow r e a t t a c h m e n t on face B cuts top vortex off from its feeding shear layer.
S.C. Luo/Vortex wake of a transversely oscillating square cyclinder
Fig. 3. Continued.
103
104
S.C. Luo/Vortex wake of a transversely oscillating square cyclinder
U
Fig. 4. Wake structure of lock-on type A.
side of the wake as their feeding shear layers and two rows of vortices are seen. Such a vortex street is known as the drag type of vortex street and is the type t h a t is commonly seen. In the present flow st ruct ure vortices in the same row are relatively closely packed and appear to be relatively weak in strength, In the near wake, vortices from one row do not appear to i nt eract with their c o u n t e r p a r t from the ot her row. Viscous i nt eract i on of vortices appears to:take place only at some three longitudinal vortex spacing (a) downstream of face C. As a result, f u r t her than three longitudinal vortex spacing downstream of face C, no distinct vortex is seen because of the cancellation o f vorticity t h ro u g h viscous interaction. By projecting (and hence enlarging) many frames of the film of the visualized flow onto a screen, the longitudinal and lateral vortex spacing of the vortex street were estimated to be a / d ~ 2 . 7 4 and b/d ~ 2.39, respectively. The use of the " ~ " sign is to emphasise t hat the values are only approximate figures. The vortex spacing ratio b/a is thus 0.87. By using the inviscid flow result of F/( Ud) = [2a/d(1 - Sra/d)]/tanh(rcb/a),
(1)
the dimensionless circulation F / ( U d ) of a vortex in the wake is estimated to be 3.71. In the above Sr is the Strouhal number from the results published in Ref. [2]. Since a/d and b/d were estimated from flow v i s u a l i z a t i o n , t h e i r magnitude as well as the magnitude of b/a and F / ( U d ) are of course not very accurate. The numerical values quoted above are, therefore, not meant to be quantitatively ac c ur at e but are meant to be qualitative figures t h a t can be used for the comparison of different flow structures identified. 3.2. Lock-on type B At fN = 4 Hz, Ur = 13.39. At this reduced velocity f~ is still locked on to fN. One vortex is thus shed by each shear layer per cycle of oscillation, The vortex shedding sequence is shown in Figs. 5 and 6. When the cylinder moves from the TDC to the BDC, a vortex grows ne a r face B (the side face t h a t t h e cylinder is moving away from). When compared with lock-on type A, because of the lower oscillation frequency and, hence, lower shedding frequency and longer shedding period, the vortex has more time to grow and this results in: (l) Each
S.C. Luo/Vortex wake of a transversely oscillating square cyclinder
105
©
TOC []
L
80C IOC
8DC
TOC
80C
Fig. 5. Vortex shedding sequence of lock-on type B.
v o r t e x appears to be s t r o n g e r (larger) and contains more circulation. (2) At the end of the first h a l f of the shedding cycle, when the cylinder is n e a r the BDC~ instead of located v e r y n e a r face B as in the case of lock-on type A, the v o r t e x fed by the top s h e a r l a y e r is located n e a r the upper trailing c o r n e r of the cylinder (i.e., j u n c t i o n of faces B and C). In the second h a l f of the cycle, when the cylinder reverses its direction and moves from the BDC to the TDC, the reverse upwards m o t i o n of the cylinder, and possibly with the opposite vorticity from the opposite shear layer D, " d r a w " the v o r t e x fed by s h e a r l a y e r B across the wake and cause it to end up on the o t h e r side of the wake as its feeding s h e a r layer. When the above takes place, the v o r t e x and its feeding s h e a r layer (shear l a y e r B) also seem to cut the grown v o r t e x fed by s h e a r layer D in the previous shedding cycle off from shear layer D. At the end of the present shedding cycle and when the cylinder is n e a r the TDC again, s h e a r layer B is also seen to m o m e n t a r i l y r e a t t a c h on face B. The shedding of vortices by the o t h e r s h e a r layer (shear layer D), at least in the vicinity n e a r the base of the cylinder, occurs in a m a n n e r t h a t is similar to shear layer B but is 180 out of phase to the former. As in lock-on type A, the various v o r t e x street p a r a m e t e r s are estimated. They are a/d~6.84, b / d ~ l . 1 6 , b/a~O.17 and F / ( U d ) ~ 14.36. A sketch of the vortex street is shown in Fig. 7. It should be noted t h a t in the present wake s t r u c t u r e vortices end up on the o t h e r side of
106
S.C. Luo/Vortex wake of a transversely oscillating square cyclinder
Fig. 6. V o r t e x s h e d d i n g process of lock-on type B: (a) t h e c y l i n d e r is n e a r t h e TDC, t h e b o t t o m v o r t e x is a l m o s t fully g r o w n a n d is located n e a r t h e b o t t o m t r a i l i n g c o r n e r CD; (b) t h e c y l i n d e r h a s a l m o s t r e a c h e d t h e BDC, t h e top v o r t e x is m e a r t r a i l i n g c o r n e r BC; (c) t h e c y l i n d e r b e g i n s to r e v e r s e its d i r e c t i o n a n d moves upwards, a n d t h e top v o r t e x is d r a w n across t h e w a k e to j o i n t h e b o t t o m row of v o r t i c e s (a s e c o n d a r y v o r t e x c a n be seen in t h e middle of t h e top p a r t of t h e flow field); (d) flow r e a t t a c h m e n t occurs o n face B n e a r t r a i l i n g c o r n e r BC (a s e c o n d a r y v o r t e x c a n be seen n e a r t h e top left c o r n e r of t h e flow field).
S.C. Luo/ Vortex wake of a transversely oscillating square cyclinder
Fig. 6. Continued.
107
108
S.C. Luo/ Vortex wake of a transversely oscillating square cyclinder
U.
~Bl. 0
Fig. 7. Wake structure of lock-on type B.
the wake as their feeding shear layer. This is sometimes referred to as the " t h r u s t " type of vortex street and is not common.
3.3. Triple lock-on At fN = 2.5 Hz, Ur = 21.4. The present reduced velocity is approximately three times the resonance reduced velocity Uro. A triple lock-on situation is observed in t h a t 3fs is seen to lock-on to fN and, hence, each shear layer sheds exactly three vortices per cycle of oscillation, at reduced velocity near 3Urol The vortex shedding sequence of the present wake structure is shown in Figs. 8 and 9. During the first half of the cycle, the cylinder moves from the TDC to the BDC. Two consecutive vortices are shed by the top shear layer B. In the second half of the cycle when the cylinder moves from the BDC upwards, a third and much weaker looking vortex is shed by shear layer B. At the same time the first of the two consecutive vortices shed by shear layer B during the first (downwards) half of the cycle interacts and annihilates with the third vortex shed by
i
BOC
,,
TOC
15
Ill I
BIE
Fig. 8. Vortex shedding sequence of triple lock-on.
S.C. Luo/ Vortex wake of a transversely oscillating square cyclinder
109
Fig. 9. V o r t e x s h e d d i n g process of triple lock-on: (a) t h e cylinder moves from t h e TDC to the BDC, two c o n s e c u t i v e vortices fed by layer B b e g i n to grow n e a r face B; (b) the cylinder c o n t i n u e s to move d o w n s t r e a m a n d almost r e a c h e s t h e BDC, t h e two c o n s e c u t i v e vortices fed by s h e a r layer B are a l m o s t fully grown; (c) t h e c y l i n d e r is m o v i n g u p w a r d s from the BDC to the TDC, a t h i r d and w e a k v o r t e x shed by t h e layer B c a n barely be seen n e a r t h e c o r n e r BC, a n d two vortices are shed by s h e a r layer D; (d) t h e cylinder c o n t i n u e s to move u p w a r d s a n d almost r e a c h e s t h e TDC.
110
SC.
Fig. 9. Continued.
Luol Vortex wake of a transversely oscillating square cyclinder
S.C. Luo/Vortex wake of a transversely oscillating square cyclinder
-3Lgd
111
I
Fig. 10. Wake structure of triple lock-on. the other shear layer (shear layer D) during the previous cycle. At the end of the cycle, near the TDC, shear layer B is very close to face B and momentary r e a t t a c h m e n t may have occurred near the trailing corner BC. As in previous wake structures, the vortex shedding sequence by both shear layers are identical but 180 out of phase. In the present type of vortex shedding, because three vortices are shed by each shear layer per cycle, due to the relatively little time available for each vortex to be shed, all the vortices appear to look small and weak. Diffusion and cancellation of vorticity take place a short distance downstream from the cylinder and typically no distinct vortex can be seen at any distance that is larger than 1 to 1.5 longitudinal vortex spacing from face C. The present shedding pattern is also less reproducible from cycle to cycle as compared with the previous two. Sometimes it is almost impossible to detect the very weak third vortex shed by a shear layer when the cylinder is moving towards it. The various vortex street parameters are estimated to be a i d ~ 3.49, b / d ~ 1.73, b/a ~ 0.50, and F / ( U d ) ~ 3.93. A sketch of the vortex street is shown in Fig. 10. 3.4. Q u a s i - s t e a d y
At low fN and hence high Ur, the shedding and oscillation frequencies become unrelated. The magnitude off~ in the quasi-steady flow region is such that the Strouhal number is approximately constant and equals to the stationary cylinder's Strouhal number. In the quasi-steady region the wake is very similar to that of a stationary cylinder and, hence, resembles a classic Von Karman vortex street (see Fig. 11). Estimation of vortex street parameters were carried out at U, = 53.56 and 97.38. Their values are given in Table 1.
U~
0.87d
7
- -
6.t6~
Fig. 11. Wake structure of quasi-steady.
112
S.C. Luo/Vortex wake of a transversely oscillating square cyclinder
3.5. Intermediate stages At reduced velocity that is in between the reduced velocity of the four flow structures described above, the flow looks like a combination of the two adjacent flow structures. For example, some flow visualization were conducted at Ur = 26.78 and 35.71. The flow appears to have some of the characteristics of the two adjacent flow structures (i.e., triple lock-on and quasi-steady). The flow structure of the former looks more like a triple lock-on flow, whereas the structure of the latter looks more like a quasi-steady flow. It can, therefore, be said th at the transition of the flow s t r uc t ur e from one type to anot her takes place in a gradual and continuous manner. 3.6. Overall discussion Gerrard [3] looked into the mechanics of the formation of vortices in a region behind a bluff body. In his paper, Gerrard suggested t hat when a growing vortex becomes strong enough to draw the other shear layer across the wake, the approach of oppositely signed vorticity in sufficient concent rat i on cuts off furtl~er supply of circulation to the growing vortex and the vortex can then be considered to have ceased growing and is shed. One of the important features in Gerrard's model of vortex shedding is, therefore, t hat for a vortex to be cut off from its feeding shear layer and shed, the interaction of the oppositely signed shear layer plays an important part. Before the present visualization work is carried out, the a u t h o r did not expect the vortex shedding process of the present case to be significantly different from Gerrard's proposal. It is, therefore, interesting to note t ha t in the case of a transversely oscillating square cylinder, vortices can be shed in ways that are r a t h e r different from the mechanism proposed by Gerrard. Of the four flow structures identified and described above, the vortex shedding process involved in lock-on type B and quasi-steady is quite similar to that suggested by Gerrard. However, for the other two flow structures there does not appear to be any significant interaction between the two separated shear layers in the near wake. In lock-on type A, the momentary r e a t t a c h m e n t of the separated shear layer onto the cylinder's side face appears to be the mean of disrupting the further supply of vorticity to a growing vortex by the feeding shear layer and causes the former to be shed. In triple lock-on, two consecutive vortices are shed by one shear layer during half a shedding cycle and a third vortex during the second half. It cannot be seen clearly from the flow visualization as how the three vortices are shed by the same shear layer during one cycle of oscillation, although it appears that there is little interaction between the two separated shear layers in the n ear wake region where the vortices were formed. Therefore, it appears t h a t the shedding of vortices, at least for the present case of vortices shed by a transversely oscillating square cylinder, d o e s not always take place in the m an n er proposed by Gerrard [3]. Bearman and Obasaju [4] also conducted smoke flow visualization of flow past a transversely oscillating square cylinder at A / d = 0 . 2 5 . They observed two different flow structures at Ur = 4 and 7.8. They reported th a t Gerrard's type of vortex shedding is not observed in the
7 4 2.5 2 1.5 1 0.55 0
Lock-on type A L o c k - o n type B Triple lock-on Intermediate
7.65 13.39 21.4 26.78 35.71 53.56 97.38 3c
U~ 2.76 1.36 0.83 0.85 1.12 1.4 1.46 1.45
C'I~ 0.12 0.0715 0.139 0.129 0.13 0.13 0.13 0.128
Sr u 2.74 6.84 3.49 5.57 5.43 6.18 6.36 6.38
a/d"
2.39 1.16 1.73 2.01 1.58 1.29 0.87 0.86
b/d ~
D a t a e s t i m a t e d from flow v i s u a l i z a t i o n . b D a t a o b t a i n e d f r o m e x p e r i m e n t a l m e a s u r e m e n t a n d were r e p o r t e d in B e a r m a n a n d L u o [2]. D a t a e s t i m a t e d from b o t h flow v i s u a l i z a t i o n r e s u l t s a n d e x p e r i m e n t a l m e a s u r e m e n t s .
Stationary
Quasi steady
fN (Hz)
Structures
E s t i m a t i o n of v o r t e x s t r e e t p a r a m e t e r s
Table 1
0.87 0.17 0.50 0.36 0.29 0.21 0.14 0.13
b/a ~
3.71 14.36 3.93 3.82 4.31 4.22 5.47 6.13
I/( U ~ d) ~
- 7 5 ' : to - 1 0 ~ -22.5 ° 52.5 ~" 65 • 67.5 ~ 70 88 ~' 90
~pu
~-
114
S.C. Luo/Vortex wake of a transversely oscillating square cyclinder
Ur = 4 case. In Gerrard's mechanism the interaction provided by the other shear layer is always needed to dissociate a growing vortex from the feeding shear layer and cause it to be shed. It now appears that there are other possible means of providing the " i n t e r a c t i o n " mentioned above. Mom ent ary contact between the separated shear layer and the corresponding side face of the cylinder appears to be one of them. Other means, like the way that vortices are shed in the triple lock-on flow structure are not well understood at the moment. A tentative conclusion that can be drawn from the present flow visualization is t h a t Gerrard's vortex shedding mechanism is applicable to steady flow past bluff bodies and certain unsteady flow situations only. The magnitude of the dimensionless longitudinal and lateral vortex spacing (a/d and b/d, respectively) had been estimated and reported earlier. Their ratio (b/a) is called the vortex spacing ratio. The magnitude of the above terms are tabulated in Table 1. From Table 1 it is observed that b/a varies from 0.13 fbr a stationary square cylinder to 0.87 for lock-on type A. Von Karman [5] had shown that for a stable vortex street, the spacing ratio should assume the value of 0.281. Some doubt over the correctness of the value 0.281 had been previously brought up in Ref. [6]. Although the accuracy of b/a estimated in the present work is not very good, the values reported in Table 1 leaves one with little doubt th at for the case of flow past a square cylinder (both stationary and transversely oscillating), the value of b/a can be r a t h e r different from the Von Karman's prediction of 0.281 [5]. In the present experiment, b/a of a stationary square cylinder is only about 0.13. Returning to a/d and b/d, some comparison can be made here between the present results and the findings of Griffin and Ramberg [7]. Griffin and Ramberg's results apply to a circular cylinder oscillating transversely to a flow with a Reynolds number range of 150 to 300. They found that for a transversely oscillating circular cylinder, when the amplitude of oscillation is increased, the lateral vortex spacing reduces but the longitudinal vortex spacing remains fairly constant. They also suggested that since the lateral spacing decreases with increasing amplitude of oscillation, at very large amplitude of oscillation of A / d ~ 1, the alternating sign vortices in the wake become almost collinear (i.e., b/d-*O). At even larger amplitude of oscillation, instead of changing from a drag type to a thrust type (i.e., vortices end up on the other side of the wake as the original shear layer) of vortex street, the vortex street becomes asymmetrical in that two counterclockwise vortices are shed by the bottom shear layer for every clockwise vortex shed by the top shear layer. The flow is from left to right in Griffin and Ramberg's experiment [7]. On the other hand, when the frequency of oscillation is increased within the regime where the body and wake frequencies are synchronized (i.e., the lock-on regime), Griffin and Ramberg reported that the longitudinal vortex spacing decreases whereas the lateral vortex spacing remains more or less unchanged. As most of the present flow visualization was conducted at a constant amplitude of A/d=0.675. the effect of amplitude of oscillation cannot be examined in the present investigation. However, for the two lock-on cases, when the frequency of oscillation
S.C. Luo/Vortex wake of a transversely oscillating square cyclinder
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decreases from type A to type B, a/d increases from 2.74 to 6.84. The cause of the above observation is not difficult to explain. In both cases the frequency of vortex shedding was locked-on to the cylinder oscillation frequency. The cylinder oscillates at a lower frequency in type B than in type A. This is the same as saying that the shedding period is longer in type B. Provided t hat the vortex convection velocity is not significantly different in the two cases, a shed vortex in lock-on type B will have more time to travel downstream before the next vortex is shed, thus resulting in a larger value of a/d in lock-on type B when compared with lock-on type A. In fact, whenever the shedding frequency is locked-on to the cylinder oscillating frequency, there should always be an approximately inverse relationship between the longitudinal spacing and the Strouhal number. The present observation on the effect of varying the frequency of oscillation is, therefore, in agreement with that of Griffin and Ramberg [7]. On the other hand, Griffin and Ramberg also suggested that the lateral vortex spacing remains constant when the frequency of oscillation is changed in the lock-on region. This, however, is not supported by the present observation as bid was estimated to be 2.39 and 1.16 for lock-on type A and B ( fN = 7 and 4 Hz), respectively. This point will be returned to later when the a u th o r discusses the relation between the position of the cylinder when vortices are shed and the phase angle between the forcing frequency component of the lift force and the cylinder displacement. Griffin and Ramberg also suggested that at large amplitude of oscillation, collinear vortices is the ultimate situation and the vortex street becomes asymmetrical to avoid the transition of the vortex street from the drag to the thrust configuration. That again is not supported by the present visualization. Flow visualization in lock-on type B (Fig. 6d) shows that it is possible to have a thrust type of vortex street where vortices end up on the opposite side of the wake as the original feeding shear layer, and at the same time the wake has become asymmetrical with one shear layer shedding two vortices (one stronger and one weaker) for every vortex shed by the other shear layer. At a few longitudinal spacing downstream, the wake became very diffused looking, suggesting that the opposite vorticity from the two shear layers had annihilated each other, and there was no time for the vortices to rearrange themselves back to the conventional (drag type) configuration. The difference between the author's and Griffin and Ramberg's observation [7] may be related to the difference in flow separation position. The flow separation positions are fixed at the two upstream corners for the square cylinder, whether it is stationary or undergoing transverse oscillation. On the other hand, the flow separation positions can change with the direction of the incoming resultant flow (vector sum of free stream velocity and the cylinder's transversely velocity) for the case of a transversely oscillating circular cylinder. By representing the wake vortex street by a double staggered row of vortices, it was shown that the circulation (F) associated with each vortex can be estimated from eq. (1). By substituting the magnitude of Sr measured (see Ref. [2]), and a/d and b/a estimated from flow visualization into eq. (1), F/(Ud)
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S.C. Luo/ Vortex wake of a transversely oscillating square cyclinder
is estimated to be 3.71, 14.36, 3.93, 5.47, and 6.13 for lock-on type A, B, triple lock-on, quasi-steady, and stationary cylinder flow structure, respectively. The magnitude of F / ( U d ) estimated by eq. (1) is in fairly good agreement with the circulation observed from flow visualization. In a visualized flow, the amount of circulation associated with each vortex can be approximately estimated from the size of the vortex and the amount of smoke within the vortex. During lock-on when the vortex shedding frequency is related to the cylinder oscillating frequency and if the free stream velocity is kept constant, a decrease in the shedding frequency (i.e., Strouhal number) will not only lead to an increase in longitudinal spacing as explained earlier, but also to an increase in the strength of each vortex shed. The later is also not difficult to understand because the longer shedding period associated with the lower shedding frequency means that more time is allowed for the shear layer to feed each growing vortex with vorticity. This is thought to be so, even if it is quite welt known that only a fraction of the shed vorticity ends up in the vortices in the wake downstream. The magnitudes of Sr, a/d, and F~ Ud for lock-on type A and B, and triple lock-on tabulated in Table 1 support the above deduction, at least qualitatively. The magnitude of the phase angle between the forcing frequency component of the lift force (CL(fN)) and the cylinder displacement (y), is designated as ~0- ~0 has been measured and reported in Ref. [2]. The value of ~ that corresponds to the flow visualization flow conditions are also included in Table 1. Also, the present definition of (p is such that if ~0 is between 0' to 180, galloping type of flow induced oscillation will take place and vice versa. In lock-on type A and B, when the cylinder is moving downwards from the TDC to the BDC, a vortex is growing adjacent to the top side face (face B, the side face that is downstream to the cylinder's transverse motion) and vice versa. Because of fluid entrainment, a growing vortex is normally considered to be associated with a region of fairly high negative pressure (i.e., suction). Since in lock-on type A and B the suction is always downstream to the cylinder's transverse motion, from flow visualization one expects the lift force generated by the cylinder motion to oppose the cylinder motion and is, hence, a stabilizing one. This means galloping oscillation will not occur naturally and the angle ~ should have a negative value. This deduction is confirmed by experimental measurement reported in Ref. [2]. As shown in Table l, (p~ -42.5 ~' and -22.5 ° for lock-on Type A and B, respectively. Flow visualization thus reveals that the stability of a cylinder to transverse galloping oscillation is dependent on the position of the cylinder within an oscillation cycle when vortices grow and get shed. For triple lock-on, vortices appear to be shed by both shear layers during both the downwards and upwards motion of the cylinder. It is, therefore, not so easy to deduce the sign of ~ from flow visualization observation alone. For quasi-steady, f~ and fs are uncoupled. Therefore, (p also cannot be deduced from flow visualization. The author wishes to add here that for the uncommon thrust type of vortex wake that is associated with lock-on type B, because the associated (p is negative, it does not occur
S.C. Luo/Vortex wake of a transversely oscillating square cyclinder
117
naturally. On the other hand, triple-lock on type of wake structure where vortices are NOT shed in accordance to Gerrard's vortex shedding mechanism [3] will occur naturally because the associated (p is positive. The magnitude of b/d is also related to (p because b/d is affected by the position of the cylinder when a vortex is shed. In lock-on type A, vortices fed by the top and bottom shear layers are shed when the cylinder is near the TDC and BDC, respectively. Thus, bid is expected to be relatively large (b/d~2.39 in lock-on type A). On the other hand, in lock-on type B, when the cylinder in near the BDC, the vortex fed by the top shear layer has already moved downstream of the top rear corner (junction between faces B and C). When the cylinder begins its upward motion from the BDC to the TDC, that vortex is left behind and eventually ends up on the other side of the wake as the feeding shear layer. This, therefore, illustrates that during lock-on, even if A/d is kept constant, the frequency of oscillation affects when a vortex is shed which in turn affects the magnitude of bid. Therefore, b/d is not necessarily a constant when A/d is constant. This is in contradiction to the earlier mentioned observation of Griffin and Ramberg [7]. The aut hor is not sure whether this difference is caused by the difference in the flow separation position between an oscillating square and an oscillating circular cylinder mentioned earlier. In triple lock-on, not one but three vortices are shed by one shear layer during one cycle of oscillation, the vortices are, therefore, shed when the cylinder was at different position within its oscillation cycle. Flow visualization reveals that the vortices shed have r a t he r little interaction with vortices shed by the other shear layer. They do not cross the wake centre line and the magnitude of b/d is a moderate 1.73. Similar situation applies to quasi-steady (i.e., there is insufficient interaction to cause the vortices to cross the wake centre line), b/d was estimated to be 1.29 and 0.87 at UF =53.56 and 97.38, respectively. 4. C o n c l u s i o n s
The smoke wire method is used to visualize the flow in the present work. At
A/d = 0.675, and in the reduced velocity range of 7.65 to 97.38, four different flow structures were identified. They are called lock-on type A, lock-on type B, triple lock-on and quasi-steady in the present paper. As suggested by their names, in lock-on type A and B, the vortex shedding frequency is synchronized with (or locked-on to) the cylinder oscillation frequency. The resulting flow structures are, however, r a t he r different and had been described in the text of the present paper. In triple lock-on, the vortex shedding frequency is locked-on to three times the cylinder oscillation frequency. Three vortices are thus shed per cycle of oscillation. In the last flow structure called quasi-steady, the cylinder oscillates so slowly that the resulting flow structure is almost indistinguishable from that of a stationary cylinder. Other main findings of the present flow visualization investigation can be summed up in the following:
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S.C. Luo/Vortex wake of a transversely oscillating square cyclinder
(1) The way that vortices are shed by a transversely oscillating square cylinder does not always take place in the manner that is suggested by Gerrard [3]. Of the four flow structures identified in the present investigation, only in lock-on type B and in the quasi-steady flow structures that vortices are shed in a similar manner as Gerrard's suggestion. It appears that the interaction from an oppositely signed shear layer is not the only way that a growing vortex can be cut off from its feeding shear layer and be shed. Momentary reattachment of the shear layer onto the cylinder's downstream corner appears to be one of the other possible means that causes a vortex to be shed. It appears that Gerrard's vortex shedding mechanism is applicable to steady flow past bluff bodies and certain types of unsteady flow past bluff bodies only. (2) For both stationary and oscillating square cylinder, the vortex spacing ratio can assume values that are rather different from the Von Karman's prediction of 0.281 [5]. (3) When the oscillation frequency of the cylinder is decreased within the lock-on range, the longitudinal vortex spacing increases. This observation is in agreement with that of Griffin and Ramberg [7]. On the other hand, Griffin and Ramberg suggested that under such circumstance the lateral vortex spacing remains unchanged. This is not so in the present experiment. Also, Griffin and Ramberg's suggestion of a collinear row of alternatively signed vortices being the limiting situation for very large amplitude of oscillation is also not supported by the present flow visualization observation. A thrust type of vortex wake is observed in lock-on type B. It is suggested that the difference between the author's and Griffin and Ramberg's observation may be attributed to the fundamental difference between flow past an oscillating square and circular cylinder. That is, the flow separation positions are fixed in the former's case but not so for the latter. (4) During lock-on, if the Strouhal number increases, both the longitudinal vortex spacing and the circulation associated with each vortex decrease~ (5) The position of the cylinder within an oscillation cycle when vortices grow and get shed affects both the magnitude and sign of (p and the magnitude of b/d. The sign of q~ determines whether the cylinder is susceptible to the galloping type of flow induced oscillation. 5. P o s t s c r i p t n o t e
Since all the results reported in the present paper were obtained at a constant A/d of 0.675, the reduced velocity Ur was the only flow parameter used to describe the different flow structures observed. However, any attempt to directly compare the results reported in the present paper with those from other investigations by merely matching the reduced velocity Ur should be avoided. This is so because A/d is also an important flow parameter. Comparison of results should be made only when the parameter U J ( A / d ) is matched. It can be shown quite easily that the parameter Ur/(A/d) is related to the maximum flow
S.C. Luo/Vortex wake of a transversely oscillating square cyclinder
119
incidence angle experienced by the square cylinder during a cycle of oscillat i o n , ~ .... via the following relationship: ffmax = t a n
1 2~
~
.
Acknowledgement The work reported in the present paper was carried out by the author under t h e s u p e r v i s i o n o f Prof. P . W . B e a r m a n w h e n t h e a u t h o r w a s a p o s t g r a d u a t e student at the Department of Aeronautics, Imperial College of Science, Techn o l o g y a n d M e d i c i n e . T h e a u t h o r is v e r y g r a t e f u l to Prof. B e a r m a n for a l l t h e a d v i c e s h e h a d so k i n d l y o f f e r e d d u r i n g t h e c o u r s e o f r e s e a r c h . References [1] G.V. Parkinson, Mathematical models of flow-induced vibrations of bluff bodies, in: Naudascher (Ed), Flow-Induced Structural Vibrations (1974) pp. 81 127. [2] P.W. Bearman and S.C. Luo, Investigation of the aerodynamic instability of a squaresection cylinder by forced oscillation, J. Fluids Struc. 2 (1988) 161 176. [3] J.H. Gerrard, The mechanics of the formation region of vortices behind bluff bodies. J. Fluid Mech. 25 (1966) 401 413. [4] P.W. Bearman and E.D. Obasaju, An experimental study of pressure fluctuations on fixed and oscillating square-section cylinders, J. Fluid Mech. 119 (1982) 297 321. [5] L.M. Milne-Thomson, Theoretical Hydrodynamics, Macmillan, 5th edn., pp. 377 380. [6] P.W. Bearman, On vortex street wakes, J. Fluid Mech. 28 (1967) 625 641. [7] O.M. Griffin and S.E. Ramberg, The vortex-street wakes of vibrating cylinders, J. Fluid Mech. 66 (1974) 553 576.
97
Elsevier
Vortex wake of a transversely oscillating square cylinder: A flow visualization analysis S.C. Luo Department of Mechanical & Production Engineering, National University of Singapore. 10 Kent Ridge Crescent, Singapore 0511 (Received March 2, 1992; accepted in revised form August 3, 1992)
Summary The flow past a transversely oscillating square cylinder is visualized by the smoke wire technique and studied. In the present investigation, by varying the cylinder oscillating frequency f~ in a free stream with constant velocity U, the reduced velocity U r varies from 7.65 to 97.38. Under the above conditions and at a constant amplitude of oscillation of A/d=0.675, four different flow structures were identified. They are called lock-on type A, lock-on type B, triple lock-on and quasi-steady in the present paper. The way that vortices are formed and shed are different in these four flow structures. The above results in different wake structures with rather different magnitudes of longitudinal and lateral vortex spacing (a/d and b/d, respectively). It was also found that the vortex shedding mechanism proposed by Gerrard is applicable only to lock-on type B and quasi-steady, but not the other two. The vortex spacing ratio b/a of 0.281 predicted by Von Karman does not apply to the present flow situations. During lock-on, flow visualization reveals that when fN decreases in magnitude at a constant A/d, both aid and F (circulation associated with each vortex in the wake) increase in magnitude. Under certain flow conditions the unconventional thrust type of vortex street exists. The position of the cylinder within its oscillation cycle when vortices are shed was observed to be a factor that affects the magnitude of the lateral vortex spacing and whether the cylinder is susceptible to flow induced oscillation.
Notation A
amplitude of oscillation
a
longitudinal vortex spacing lateral vortex spacing lift c o e f f i c i e n t s t a n d a r d d e v i a t i o n o f lift c o e f f i c i e n t side l e n g t h of the s q u a r e c y l i n d e r
b CL C[ d
Correspondence to: Dr. S.C. Luo, Department of Mechanical and Production Engineering, National University of Singapore, 10 Kent Ridge Crescent, Singapore 0511.
0167-6105/92/$05.00 { 1992 Elsevier Science Publishers B.V. All rights reserved.
98
f~ Re Sr U
ur Uro Y F P P (p
S.C. Luo/Vortex wake of a transversely oscillating square cyclinder
cylinder forced oscillation frequency vortex shedding frequency Reynolds Number = f~Ud/p Strouhal Number = f~ U/d free stream velocity reduced velocity = U/( f N d ) resonance reduced velocity = 1/Sr cylinder transverse displacement circulation of each vortex dynamic viscosity of air density of air phase angle between the forcing frequency component of lift and y
1. I n t r o d u c t i o n
In the last few decades, due to improvement in construction techniques. typical civil engineering constructions which include buildings and bridges were designed to become lighter and more slender. While the above may be considered as an improvement in terms of cost effectiveness over older designs. other problems, notably that of flow induced vibration became more critical. Flutter, galloping, and vortex resonance are the three most commonly encountered types of flow induced vibration [1]. While the circular cylinder is the most frequently employed shape in vortex resonance studies because of its isolation from the other types of flow induced vibrations mentioned above, in research related to galloping the square section cylinder (from now on simply referred to as "the square cylinder") is the more commonly used shape. The two main reasons that favour the use of a square are its resemblance to the basic rectangular shape of most buildings and bridge decks, and the challenge it offers as it is subjected to both galloping and vortex resonance. The author had carried out a series of measurement on aerodynamic forces acting on a transversely oscillating square cylinder and the findings were reported in Ref. [2]. It was reported in the above paper that at very low reduced velocity ( U r = U/fNd, where U, fN, and d are the free stream velocity, forcing frequency, and the characteristic dimension (in the present case the side length) of the square cylinder, respectively) the magnitude of the fluctuating lift coefficient C'L (defined as the dimensionless standard deviation of the lift force) was very large because of vortex resonance. As the reduced velocity increased, C'L reduced and reached a minimum of about 0.75. As the reduced velocity increased further, C'L increased initially rapidly but later approached asymptotically the stationary square cylinder's value of about 1.4. Based on the C' t versus Ur variation just described, Bearman and Luo [2] suggested that the flow around a square section cylinder oscillating at constant amplitude can be divided into three main regimes: the "multiple lock-in" regime which
S.C. Luo/Vortex wake of a transversely oscillating square cyclinder
99
I
Urm
Urs
UF Fig. 1. Classification of flow regimes ([2], Fig. 10).
c o r r e s p o n d s to the r e d u c e d v e l o c i t y r a n g e from low reduced v e l o c i t y to the r e d u c e d v e l o c i t y at which C'L is a m i n i m u m , the lift r e c o v e r y r e g i m e which c o r r e s p o n d s to r e d u c e d velocity from C'L = m i n i m u m to C'L = 1.4, and the " q u a s i - s t e a d y " r a n g e w h i c h s t a r t s from r e d u c e d v e l o c i t y at w h i c h C'~ = 1.4. A s k e t c h of the t h r e e regimes is r e p r o d u c e d as Fig. 1. The objective of the p r e s e n t p a p e r is to g a i n m o r e i n f o r m a t i o n of the flow by v i s u a l i z i n g it. It is hoped t h a t by b e i n g able to see the flow, c e r t a i n m e a s u r e m e n t s r e p o r t e d in Ref. [2] can be u n d e r s t o o d better.
2. Experimental arrangement The flow v i s u a l i z a t i o n was c o n d u c t e d in a closed r e t u r n wind t u n n e l with a 0.91 x 0.91 x 5.49 m (length) w o r k i n g section. B e c a u s e of the 9 to ] r a t i o e m p l o y e d in the c o n t r a c t i o n of the wind tunnel, the t u r b u l e n c e level in the w o r k i n g section is at a low v a l u e of no m o r e t h a n 0.05%. All the flow visualiza t i o n was c a r r i e d out at a c o n s t a n t wind speed of 2.72 m/s which was t h o u g h t to be a good c o m p r o m i s e b e t w e e n the c o n t r a d i c t i n g c r i t e r i a of m i n i m u m b u o y a n c y effect and t h i c k s m o k e which f a v o u r high and low wind speed, respectively. The s q u a r e c y l i n d e r model has a side length (d) of 50.8 mm. It was m a d e of light-weight wood and was p a i n t e d in m a t t b l a c k to provide o p t i m u m c o n t r a s t with the w h i t e smoke. To e n s u r e t h a t the flow was as t w o - d i m e n s i o n a l as possible, two p e r s p e x 304.8 x 228.6 m m r e c t a n g u l a r end plates (longer side p a r a l l e l to the flow) were installed at the two ends of the s q u a r e cylinder at a d i s t a n c e of 17d apart. The cylinder was installed h o r i z o n t a l l y with b o t h ends p r o t r u d i n g out of the sides of the w o r k i n g section. The ends of the c y l i n d e r were c o n n e c t e d to two v e r t i c a l push rods w h i c h were in t u r n c o n n e c t e d to
100
S.C. Luo/ Vortex wake of a transversely osciUating square cyclinder
a Scotch-Yoke mechanism installed underneath the working section. The Scotch-Yoke mechanism oscillated the cylinder in an up-and-down simple harmonic motion inside the working section. The Scotch-Yoke mechanism was designed in such a way that both the amplitude (A) and frequency (f~) of the oscillation can be varied in a continuous manner. The smoke wire method was employed in the present investigation to visualize the flow. In the present investigation, a thin (Reynolds number based on wire diameter <20) but high resistance Ni/Cr/Fe alloy wire was stretched vertically in front of the cylinder. Oil was either coated manually onto the wire or made to drip down it. When an electric current was passed through the wire, because of its high resistance, it heated up and caused the oil to vaporise to form numerous fine droplets which is seen as white smoke. The smoke was carried downstream by the free stream, thus visualizing the flow past an oscillating square cylinder. The visualized flow field was then captured on both still and cine films for detail analysis.
3. Experimental results In smoke wire flow visualization, because the optimum wind speed range is narrow, different reduced velocity was obtained by varying fN- AS mentioned in the previous section, all the flow visualization was conducted at a constant free stream velocity of 2.72 m/s. Based on the side length of the square cylinder, the corresponding Reynolds number (Re = p Ud/p, where p and # are the density and the dynamic viscosity of air, respectively) is approximately 8.76 x 103. The dimensionless amplitude of oscillation ( A / d ) was also kept constant at 0.675. As fN employed in the present investigation varied from 0.55 Hz to 7 Hz, the corresponding reduced velocity varied from 97.38 to 7.65. At the present Reynolds number the Strouhal number (Sr = f s d / U , where fs is the frequency of the shedding of vortices by one shear layer) is about 0.13. The lowest reduced velocity achievable in the present experiment is, thus, very slightly lower t h a n the resonance reduced velocity (Uro, = 1/Sr) of 7.69. The locking on of vortex shedding frequency to the cylinder oscillating frequency is, thus, expected at the low reduced velocity end. At A / d = 0.675 and within the range of reduced velocity investigated, at least four flow structures were identified by flow visualization. They will be discussed separately below. However, before describing the flow structure in detail, it is necessary to first define a few terms which will occur quite frequently in the following. During one cycle of the vertical simple harmonic motion, the highest and lowest positions reached by the cylinder are referred to as the top dead centre and bottom dead centre, respectively. They will from now on be abbreviated as TDC and BDC, respectively. The cylinder always oscillates with two faces normal and two faces parallel to the free stream. Of the two faces that are always normal to the free stream, the upstream and downstream faces are referred to as face A and C, respectively. For the other two faces, the top one (the one t h a t is closer to the ceiling of the working section) will be referred to as face B and the lower one
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S.C. Luo/Vortex wake of a transversely oscillating square cyclinder
will be face D. The s h e a r l a y e r t h a t s e p a r a t e s from the c o r n e r of faces A and B will be r e f e r r e d to as e i t h e r s h e a r l a y e r B or the top s h e a r layer, while the s h e a r l a y e r t h a t s e p a r a t e s f r o m the c o r n e r of faces A and D will be r e f e r r e d to as e i t h e r s h e a r l a y e r D or the b o t t o m s h e a r layer. 3.1. Lock-on type A At fN = 7 Hz, Ur = 7.65. At this r e d u c e d v e l o c i t y v o r t e x shedding locks on to the body f r e q u e n c y and, hence, one v o r t e x per cycle is shed from e a c h s h e a r layer. The v o r t e x shedding process is s h o w n in Figs. 2 and 3. It should be n o t e d h e r e t h a t the p r e s e n t e x p e r i m e n t a l set-up is such t h a t it is easier to t a k e a flow p h o t o g r a p h on the side of the w o r k i n g section w h e r e the flow is from r i g h t to left. To be c o n s i s t e n t with the flow p h o t o g r a p h s for e a s y cross reference, the flow in Fig. 2 and s u b s e q u e n t figures are also from r i g h t to left and is, therefore, opposite to the "flow from left to r i g h t " c o n v e n t i o n a d o p t e d in m o s t o t h e r p a p e r s and t e x t books. S t a r t i n g from the TDC, a v o r t e x grows n e a r the top face (face B) as the c y l i n d e r m o v e s d o w n w a r d s . T h e v o r t e x grows to n e a r its m a x i m u m size w h e n the cylinder r e a c h e s the BDC. W h e n the c y l i n d e r r e v e r s e s its d i r e c t i o n and m o v e s upwards, the v o r t e x m o v e s d o w n s t r e a m a n d p a s t the r e a r c o r n e r ( j u n c t i o n of faces B and C). W h e n the cylinder r e a c h e s the TDC again, m o m e n t a r y flow r e a t t a c h m e n t o c c u r s at face B, the v o r t e x is cut off from the feeding s h e a r l a y e r and is shed. The s a m e s e q u e n c e r e p e a t s itself in s u b s e q u e n t cycles. I d e n t i c a l v o r t e x shedding s e q u e n c e o c c u r s on the b o t t o m face D b u t it is 180 ° out of p h a s e to t h a t on face B. A s k e t c h of the w a k e of lock-on type A is s h o w n in Fig. 4. Since the top s h e a r l a y e r sheds a v o r t e x w h e n the cylinder is n e a r the TDC and vice versa, the v o r t i c e s r e m a i n on the s a m e
LJ
TOC []
80C I
1
TocC"'~ ! 1
l I
1
1
2
BOC
Fig. 2. Vortex shedding sequence of lock-on type A.
2
102
S.C. Luo/ Vortex wake of a transversely oscillating square cyclinder
Fig. 3. Vortex shedding process of lock-on type A: (a) the cylinder is moving from the TDC to the BDC, a vortex is growing near face B; (b) the cylinder has almost reached the BDC, the top vortex is almost fully grown; (c) the reverse (upwards) motion of the cylinder forces the top vortex to go downstream; (d) temporary flow r e a t t a c h m e n t on face B cuts top vortex off from its feeding shear layer.
S.C. Luo/Vortex wake of a transversely oscillating square cyclinder
Fig. 3. Continued.
103
104
S.C. Luo/Vortex wake of a transversely oscillating square cyclinder
U
Fig. 4. Wake structure of lock-on type A.
side of the wake as their feeding shear layers and two rows of vortices are seen. Such a vortex street is known as the drag type of vortex street and is the type t h a t is commonly seen. In the present flow st ruct ure vortices in the same row are relatively closely packed and appear to be relatively weak in strength, In the near wake, vortices from one row do not appear to i nt eract with their c o u n t e r p a r t from the ot her row. Viscous i nt eract i on of vortices appears to:take place only at some three longitudinal vortex spacing (a) downstream of face C. As a result, f u r t her than three longitudinal vortex spacing downstream of face C, no distinct vortex is seen because of the cancellation o f vorticity t h ro u g h viscous interaction. By projecting (and hence enlarging) many frames of the film of the visualized flow onto a screen, the longitudinal and lateral vortex spacing of the vortex street were estimated to be a / d ~ 2 . 7 4 and b/d ~ 2.39, respectively. The use of the " ~ " sign is to emphasise t hat the values are only approximate figures. The vortex spacing ratio b/a is thus 0.87. By using the inviscid flow result of F/( Ud) = [2a/d(1 - Sra/d)]/tanh(rcb/a),
(1)
the dimensionless circulation F / ( U d ) of a vortex in the wake is estimated to be 3.71. In the above Sr is the Strouhal number from the results published in Ref. [2]. Since a/d and b/d were estimated from flow v i s u a l i z a t i o n , t h e i r magnitude as well as the magnitude of b/a and F / ( U d ) are of course not very accurate. The numerical values quoted above are, therefore, not meant to be quantitatively ac c ur at e but are meant to be qualitative figures t h a t can be used for the comparison of different flow structures identified. 3.2. Lock-on type B At fN = 4 Hz, Ur = 13.39. At this reduced velocity f~ is still locked on to fN. One vortex is thus shed by each shear layer per cycle of oscillation, The vortex shedding sequence is shown in Figs. 5 and 6. When the cylinder moves from the TDC to the BDC, a vortex grows ne a r face B (the side face t h a t t h e cylinder is moving away from). When compared with lock-on type A, because of the lower oscillation frequency and, hence, lower shedding frequency and longer shedding period, the vortex has more time to grow and this results in: (l) Each
S.C. Luo/Vortex wake of a transversely oscillating square cyclinder
105
©
TOC []
L
80C IOC
8DC
TOC
80C
Fig. 5. Vortex shedding sequence of lock-on type B.
v o r t e x appears to be s t r o n g e r (larger) and contains more circulation. (2) At the end of the first h a l f of the shedding cycle, when the cylinder is n e a r the BDC~ instead of located v e r y n e a r face B as in the case of lock-on type A, the v o r t e x fed by the top s h e a r l a y e r is located n e a r the upper trailing c o r n e r of the cylinder (i.e., j u n c t i o n of faces B and C). In the second h a l f of the cycle, when the cylinder reverses its direction and moves from the BDC to the TDC, the reverse upwards m o t i o n of the cylinder, and possibly with the opposite vorticity from the opposite shear layer D, " d r a w " the v o r t e x fed by s h e a r l a y e r B across the wake and cause it to end up on the o t h e r side of the wake as its feeding s h e a r layer. When the above takes place, the v o r t e x and its feeding s h e a r layer (shear l a y e r B) also seem to cut the grown v o r t e x fed by s h e a r layer D in the previous shedding cycle off from shear layer D. At the end of the present shedding cycle and when the cylinder is n e a r the TDC again, s h e a r layer B is also seen to m o m e n t a r i l y r e a t t a c h on face B. The shedding of vortices by the o t h e r s h e a r layer (shear layer D), at least in the vicinity n e a r the base of the cylinder, occurs in a m a n n e r t h a t is similar to shear layer B but is 180 out of phase to the former. As in lock-on type A, the various v o r t e x street p a r a m e t e r s are estimated. They are a/d~6.84, b / d ~ l . 1 6 , b/a~O.17 and F / ( U d ) ~ 14.36. A sketch of the vortex street is shown in Fig. 7. It should be noted t h a t in the present wake s t r u c t u r e vortices end up on the o t h e r side of
106
S.C. Luo/Vortex wake of a transversely oscillating square cyclinder
Fig. 6. V o r t e x s h e d d i n g process of lock-on type B: (a) t h e c y l i n d e r is n e a r t h e TDC, t h e b o t t o m v o r t e x is a l m o s t fully g r o w n a n d is located n e a r t h e b o t t o m t r a i l i n g c o r n e r CD; (b) t h e c y l i n d e r h a s a l m o s t r e a c h e d t h e BDC, t h e top v o r t e x is m e a r t r a i l i n g c o r n e r BC; (c) t h e c y l i n d e r b e g i n s to r e v e r s e its d i r e c t i o n a n d moves upwards, a n d t h e top v o r t e x is d r a w n across t h e w a k e to j o i n t h e b o t t o m row of v o r t i c e s (a s e c o n d a r y v o r t e x c a n be seen in t h e middle of t h e top p a r t of t h e flow field); (d) flow r e a t t a c h m e n t occurs o n face B n e a r t r a i l i n g c o r n e r BC (a s e c o n d a r y v o r t e x c a n be seen n e a r t h e top left c o r n e r of t h e flow field).
S.C. Luo/ Vortex wake of a transversely oscillating square cyclinder
Fig. 6. Continued.
107
108
S.C. Luo/ Vortex wake of a transversely oscillating square cyclinder
U.
~Bl. 0
Fig. 7. Wake structure of lock-on type B.
the wake as their feeding shear layer. This is sometimes referred to as the " t h r u s t " type of vortex street and is not common.
3.3. Triple lock-on At fN = 2.5 Hz, Ur = 21.4. The present reduced velocity is approximately three times the resonance reduced velocity Uro. A triple lock-on situation is observed in t h a t 3fs is seen to lock-on to fN and, hence, each shear layer sheds exactly three vortices per cycle of oscillation, at reduced velocity near 3Urol The vortex shedding sequence of the present wake structure is shown in Figs. 8 and 9. During the first half of the cycle, the cylinder moves from the TDC to the BDC. Two consecutive vortices are shed by the top shear layer B. In the second half of the cycle when the cylinder moves from the BDC upwards, a third and much weaker looking vortex is shed by shear layer B. At the same time the first of the two consecutive vortices shed by shear layer B during the first (downwards) half of the cycle interacts and annihilates with the third vortex shed by
i
BOC
,,
TOC
15
Ill I
BIE
Fig. 8. Vortex shedding sequence of triple lock-on.
S.C. Luo/ Vortex wake of a transversely oscillating square cyclinder
109
Fig. 9. V o r t e x s h e d d i n g process of triple lock-on: (a) t h e cylinder moves from t h e TDC to the BDC, two c o n s e c u t i v e vortices fed by layer B b e g i n to grow n e a r face B; (b) the cylinder c o n t i n u e s to move d o w n s t r e a m a n d almost r e a c h e s t h e BDC, t h e two c o n s e c u t i v e vortices fed by s h e a r layer B are a l m o s t fully grown; (c) t h e c y l i n d e r is m o v i n g u p w a r d s from the BDC to the TDC, a t h i r d and w e a k v o r t e x shed by t h e layer B c a n barely be seen n e a r t h e c o r n e r BC, a n d two vortices are shed by s h e a r layer D; (d) t h e cylinder c o n t i n u e s to move u p w a r d s a n d almost r e a c h e s t h e TDC.
110
SC.
Fig. 9. Continued.
Luol Vortex wake of a transversely oscillating square cyclinder
S.C. Luo/Vortex wake of a transversely oscillating square cyclinder
-3Lgd
111
I
Fig. 10. Wake structure of triple lock-on. the other shear layer (shear layer D) during the previous cycle. At the end of the cycle, near the TDC, shear layer B is very close to face B and momentary r e a t t a c h m e n t may have occurred near the trailing corner BC. As in previous wake structures, the vortex shedding sequence by both shear layers are identical but 180 out of phase. In the present type of vortex shedding, because three vortices are shed by each shear layer per cycle, due to the relatively little time available for each vortex to be shed, all the vortices appear to look small and weak. Diffusion and cancellation of vorticity take place a short distance downstream from the cylinder and typically no distinct vortex can be seen at any distance that is larger than 1 to 1.5 longitudinal vortex spacing from face C. The present shedding pattern is also less reproducible from cycle to cycle as compared with the previous two. Sometimes it is almost impossible to detect the very weak third vortex shed by a shear layer when the cylinder is moving towards it. The various vortex street parameters are estimated to be a i d ~ 3.49, b / d ~ 1.73, b/a ~ 0.50, and F / ( U d ) ~ 3.93. A sketch of the vortex street is shown in Fig. 10. 3.4. Q u a s i - s t e a d y
At low fN and hence high Ur, the shedding and oscillation frequencies become unrelated. The magnitude off~ in the quasi-steady flow region is such that the Strouhal number is approximately constant and equals to the stationary cylinder's Strouhal number. In the quasi-steady region the wake is very similar to that of a stationary cylinder and, hence, resembles a classic Von Karman vortex street (see Fig. 11). Estimation of vortex street parameters were carried out at U, = 53.56 and 97.38. Their values are given in Table 1.
U~
0.87d
7
- -
6.t6~
Fig. 11. Wake structure of quasi-steady.
112
S.C. Luo/Vortex wake of a transversely oscillating square cyclinder
3.5. Intermediate stages At reduced velocity that is in between the reduced velocity of the four flow structures described above, the flow looks like a combination of the two adjacent flow structures. For example, some flow visualization were conducted at Ur = 26.78 and 35.71. The flow appears to have some of the characteristics of the two adjacent flow structures (i.e., triple lock-on and quasi-steady). The flow structure of the former looks more like a triple lock-on flow, whereas the structure of the latter looks more like a quasi-steady flow. It can, therefore, be said th at the transition of the flow s t r uc t ur e from one type to anot her takes place in a gradual and continuous manner. 3.6. Overall discussion Gerrard [3] looked into the mechanics of the formation of vortices in a region behind a bluff body. In his paper, Gerrard suggested t hat when a growing vortex becomes strong enough to draw the other shear layer across the wake, the approach of oppositely signed vorticity in sufficient concent rat i on cuts off furtl~er supply of circulation to the growing vortex and the vortex can then be considered to have ceased growing and is shed. One of the important features in Gerrard's model of vortex shedding is, therefore, t hat for a vortex to be cut off from its feeding shear layer and shed, the interaction of the oppositely signed shear layer plays an important part. Before the present visualization work is carried out, the a u t h o r did not expect the vortex shedding process of the present case to be significantly different from Gerrard's proposal. It is, therefore, interesting to note t ha t in the case of a transversely oscillating square cylinder, vortices can be shed in ways that are r a t h e r different from the mechanism proposed by Gerrard. Of the four flow structures identified and described above, the vortex shedding process involved in lock-on type B and quasi-steady is quite similar to that suggested by Gerrard. However, for the other two flow structures there does not appear to be any significant interaction between the two separated shear layers in the near wake. In lock-on type A, the momentary r e a t t a c h m e n t of the separated shear layer onto the cylinder's side face appears to be the mean of disrupting the further supply of vorticity to a growing vortex by the feeding shear layer and causes the former to be shed. In triple lock-on, two consecutive vortices are shed by one shear layer during half a shedding cycle and a third vortex during the second half. It cannot be seen clearly from the flow visualization as how the three vortices are shed by the same shear layer during one cycle of oscillation, although it appears that there is little interaction between the two separated shear layers in the n ear wake region where the vortices were formed. Therefore, it appears t h a t the shedding of vortices, at least for the present case of vortices shed by a transversely oscillating square cylinder, d o e s not always take place in the m an n er proposed by Gerrard [3]. Bearman and Obasaju [4] also conducted smoke flow visualization of flow past a transversely oscillating square cylinder at A / d = 0 . 2 5 . They observed two different flow structures at Ur = 4 and 7.8. They reported th a t Gerrard's type of vortex shedding is not observed in the
7 4 2.5 2 1.5 1 0.55 0
Lock-on type A L o c k - o n type B Triple lock-on Intermediate
7.65 13.39 21.4 26.78 35.71 53.56 97.38 3c
U~ 2.76 1.36 0.83 0.85 1.12 1.4 1.46 1.45
C'I~ 0.12 0.0715 0.139 0.129 0.13 0.13 0.13 0.128
Sr u 2.74 6.84 3.49 5.57 5.43 6.18 6.36 6.38
a/d"
2.39 1.16 1.73 2.01 1.58 1.29 0.87 0.86
b/d ~
D a t a e s t i m a t e d from flow v i s u a l i z a t i o n . b D a t a o b t a i n e d f r o m e x p e r i m e n t a l m e a s u r e m e n t a n d were r e p o r t e d in B e a r m a n a n d L u o [2]. D a t a e s t i m a t e d from b o t h flow v i s u a l i z a t i o n r e s u l t s a n d e x p e r i m e n t a l m e a s u r e m e n t s .
Stationary
Quasi steady
fN (Hz)
Structures
E s t i m a t i o n of v o r t e x s t r e e t p a r a m e t e r s
Table 1
0.87 0.17 0.50 0.36 0.29 0.21 0.14 0.13
b/a ~
3.71 14.36 3.93 3.82 4.31 4.22 5.47 6.13
I/( U ~ d) ~
- 7 5 ' : to - 1 0 ~ -22.5 ° 52.5 ~" 65 • 67.5 ~ 70 88 ~' 90
~pu
~-
114
S.C. Luo/Vortex wake of a transversely oscillating square cyclinder
Ur = 4 case. In Gerrard's mechanism the interaction provided by the other shear layer is always needed to dissociate a growing vortex from the feeding shear layer and cause it to be shed. It now appears that there are other possible means of providing the " i n t e r a c t i o n " mentioned above. Mom ent ary contact between the separated shear layer and the corresponding side face of the cylinder appears to be one of them. Other means, like the way that vortices are shed in the triple lock-on flow structure are not well understood at the moment. A tentative conclusion that can be drawn from the present flow visualization is t h a t Gerrard's vortex shedding mechanism is applicable to steady flow past bluff bodies and certain unsteady flow situations only. The magnitude of the dimensionless longitudinal and lateral vortex spacing (a/d and b/d, respectively) had been estimated and reported earlier. Their ratio (b/a) is called the vortex spacing ratio. The magnitude of the above terms are tabulated in Table 1. From Table 1 it is observed that b/a varies from 0.13 fbr a stationary square cylinder to 0.87 for lock-on type A. Von Karman [5] had shown that for a stable vortex street, the spacing ratio should assume the value of 0.281. Some doubt over the correctness of the value 0.281 had been previously brought up in Ref. [6]. Although the accuracy of b/a estimated in the present work is not very good, the values reported in Table 1 leaves one with little doubt th at for the case of flow past a square cylinder (both stationary and transversely oscillating), the value of b/a can be r a t h e r different from the Von Karman's prediction of 0.281 [5]. In the present experiment, b/a of a stationary square cylinder is only about 0.13. Returning to a/d and b/d, some comparison can be made here between the present results and the findings of Griffin and Ramberg [7]. Griffin and Ramberg's results apply to a circular cylinder oscillating transversely to a flow with a Reynolds number range of 150 to 300. They found that for a transversely oscillating circular cylinder, when the amplitude of oscillation is increased, the lateral vortex spacing reduces but the longitudinal vortex spacing remains fairly constant. They also suggested that since the lateral spacing decreases with increasing amplitude of oscillation, at very large amplitude of oscillation of A / d ~ 1, the alternating sign vortices in the wake become almost collinear (i.e., b/d-*O). At even larger amplitude of oscillation, instead of changing from a drag type to a thrust type (i.e., vortices end up on the other side of the wake as the original shear layer) of vortex street, the vortex street becomes asymmetrical in that two counterclockwise vortices are shed by the bottom shear layer for every clockwise vortex shed by the top shear layer. The flow is from left to right in Griffin and Ramberg's experiment [7]. On the other hand, when the frequency of oscillation is increased within the regime where the body and wake frequencies are synchronized (i.e., the lock-on regime), Griffin and Ramberg reported that the longitudinal vortex spacing decreases whereas the lateral vortex spacing remains more or less unchanged. As most of the present flow visualization was conducted at a constant amplitude of A/d=0.675. the effect of amplitude of oscillation cannot be examined in the present investigation. However, for the two lock-on cases, when the frequency of oscillation
S.C. Luo/Vortex wake of a transversely oscillating square cyclinder
115
decreases from type A to type B, a/d increases from 2.74 to 6.84. The cause of the above observation is not difficult to explain. In both cases the frequency of vortex shedding was locked-on to the cylinder oscillation frequency. The cylinder oscillates at a lower frequency in type B than in type A. This is the same as saying that the shedding period is longer in type B. Provided t hat the vortex convection velocity is not significantly different in the two cases, a shed vortex in lock-on type B will have more time to travel downstream before the next vortex is shed, thus resulting in a larger value of a/d in lock-on type B when compared with lock-on type A. In fact, whenever the shedding frequency is locked-on to the cylinder oscillating frequency, there should always be an approximately inverse relationship between the longitudinal spacing and the Strouhal number. The present observation on the effect of varying the frequency of oscillation is, therefore, in agreement with that of Griffin and Ramberg [7]. On the other hand, Griffin and Ramberg also suggested that the lateral vortex spacing remains constant when the frequency of oscillation is changed in the lock-on region. This, however, is not supported by the present observation as bid was estimated to be 2.39 and 1.16 for lock-on type A and B ( fN = 7 and 4 Hz), respectively. This point will be returned to later when the a u th o r discusses the relation between the position of the cylinder when vortices are shed and the phase angle between the forcing frequency component of the lift force and the cylinder displacement. Griffin and Ramberg also suggested that at large amplitude of oscillation, collinear vortices is the ultimate situation and the vortex street becomes asymmetrical to avoid the transition of the vortex street from the drag to the thrust configuration. That again is not supported by the present visualization. Flow visualization in lock-on type B (Fig. 6d) shows that it is possible to have a thrust type of vortex street where vortices end up on the opposite side of the wake as the original feeding shear layer, and at the same time the wake has become asymmetrical with one shear layer shedding two vortices (one stronger and one weaker) for every vortex shed by the other shear layer. At a few longitudinal spacing downstream, the wake became very diffused looking, suggesting that the opposite vorticity from the two shear layers had annihilated each other, and there was no time for the vortices to rearrange themselves back to the conventional (drag type) configuration. The difference between the author's and Griffin and Ramberg's observation [7] may be related to the difference in flow separation position. The flow separation positions are fixed at the two upstream corners for the square cylinder, whether it is stationary or undergoing transverse oscillation. On the other hand, the flow separation positions can change with the direction of the incoming resultant flow (vector sum of free stream velocity and the cylinder's transversely velocity) for the case of a transversely oscillating circular cylinder. By representing the wake vortex street by a double staggered row of vortices, it was shown that the circulation (F) associated with each vortex can be estimated from eq. (1). By substituting the magnitude of Sr measured (see Ref. [2]), and a/d and b/a estimated from flow visualization into eq. (1), F/(Ud)
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is estimated to be 3.71, 14.36, 3.93, 5.47, and 6.13 for lock-on type A, B, triple lock-on, quasi-steady, and stationary cylinder flow structure, respectively. The magnitude of F / ( U d ) estimated by eq. (1) is in fairly good agreement with the circulation observed from flow visualization. In a visualized flow, the amount of circulation associated with each vortex can be approximately estimated from the size of the vortex and the amount of smoke within the vortex. During lock-on when the vortex shedding frequency is related to the cylinder oscillating frequency and if the free stream velocity is kept constant, a decrease in the shedding frequency (i.e., Strouhal number) will not only lead to an increase in longitudinal spacing as explained earlier, but also to an increase in the strength of each vortex shed. The later is also not difficult to understand because the longer shedding period associated with the lower shedding frequency means that more time is allowed for the shear layer to feed each growing vortex with vorticity. This is thought to be so, even if it is quite welt known that only a fraction of the shed vorticity ends up in the vortices in the wake downstream. The magnitudes of Sr, a/d, and F~ Ud for lock-on type A and B, and triple lock-on tabulated in Table 1 support the above deduction, at least qualitatively. The magnitude of the phase angle between the forcing frequency component of the lift force (CL(fN)) and the cylinder displacement (y), is designated as ~0- ~0 has been measured and reported in Ref. [2]. The value of ~ that corresponds to the flow visualization flow conditions are also included in Table 1. Also, the present definition of (p is such that if ~0 is between 0' to 180, galloping type of flow induced oscillation will take place and vice versa. In lock-on type A and B, when the cylinder is moving downwards from the TDC to the BDC, a vortex is growing adjacent to the top side face (face B, the side face that is downstream to the cylinder's transverse motion) and vice versa. Because of fluid entrainment, a growing vortex is normally considered to be associated with a region of fairly high negative pressure (i.e., suction). Since in lock-on type A and B the suction is always downstream to the cylinder's transverse motion, from flow visualization one expects the lift force generated by the cylinder motion to oppose the cylinder motion and is, hence, a stabilizing one. This means galloping oscillation will not occur naturally and the angle ~ should have a negative value. This deduction is confirmed by experimental measurement reported in Ref. [2]. As shown in Table l, (p~ -42.5 ~' and -22.5 ° for lock-on Type A and B, respectively. Flow visualization thus reveals that the stability of a cylinder to transverse galloping oscillation is dependent on the position of the cylinder within an oscillation cycle when vortices grow and get shed. For triple lock-on, vortices appear to be shed by both shear layers during both the downwards and upwards motion of the cylinder. It is, therefore, not so easy to deduce the sign of ~ from flow visualization observation alone. For quasi-steady, f~ and fs are uncoupled. Therefore, (p also cannot be deduced from flow visualization. The author wishes to add here that for the uncommon thrust type of vortex wake that is associated with lock-on type B, because the associated (p is negative, it does not occur
S.C. Luo/Vortex wake of a transversely oscillating square cyclinder
117
naturally. On the other hand, triple-lock on type of wake structure where vortices are NOT shed in accordance to Gerrard's vortex shedding mechanism [3] will occur naturally because the associated (p is positive. The magnitude of b/d is also related to (p because b/d is affected by the position of the cylinder when a vortex is shed. In lock-on type A, vortices fed by the top and bottom shear layers are shed when the cylinder is near the TDC and BDC, respectively. Thus, bid is expected to be relatively large (b/d~2.39 in lock-on type A). On the other hand, in lock-on type B, when the cylinder in near the BDC, the vortex fed by the top shear layer has already moved downstream of the top rear corner (junction between faces B and C). When the cylinder begins its upward motion from the BDC to the TDC, that vortex is left behind and eventually ends up on the other side of the wake as the feeding shear layer. This, therefore, illustrates that during lock-on, even if A/d is kept constant, the frequency of oscillation affects when a vortex is shed which in turn affects the magnitude of bid. Therefore, b/d is not necessarily a constant when A/d is constant. This is in contradiction to the earlier mentioned observation of Griffin and Ramberg [7]. The aut hor is not sure whether this difference is caused by the difference in the flow separation position between an oscillating square and an oscillating circular cylinder mentioned earlier. In triple lock-on, not one but three vortices are shed by one shear layer during one cycle of oscillation, the vortices are, therefore, shed when the cylinder was at different position within its oscillation cycle. Flow visualization reveals that the vortices shed have r a t he r little interaction with vortices shed by the other shear layer. They do not cross the wake centre line and the magnitude of b/d is a moderate 1.73. Similar situation applies to quasi-steady (i.e., there is insufficient interaction to cause the vortices to cross the wake centre line), b/d was estimated to be 1.29 and 0.87 at UF =53.56 and 97.38, respectively. 4. C o n c l u s i o n s
The smoke wire method is used to visualize the flow in the present work. At
A/d = 0.675, and in the reduced velocity range of 7.65 to 97.38, four different flow structures were identified. They are called lock-on type A, lock-on type B, triple lock-on and quasi-steady in the present paper. As suggested by their names, in lock-on type A and B, the vortex shedding frequency is synchronized with (or locked-on to) the cylinder oscillation frequency. The resulting flow structures are, however, r a t he r different and had been described in the text of the present paper. In triple lock-on, the vortex shedding frequency is locked-on to three times the cylinder oscillation frequency. Three vortices are thus shed per cycle of oscillation. In the last flow structure called quasi-steady, the cylinder oscillates so slowly that the resulting flow structure is almost indistinguishable from that of a stationary cylinder. Other main findings of the present flow visualization investigation can be summed up in the following:
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S.C. Luo/Vortex wake of a transversely oscillating square cyclinder
(1) The way that vortices are shed by a transversely oscillating square cylinder does not always take place in the manner that is suggested by Gerrard [3]. Of the four flow structures identified in the present investigation, only in lock-on type B and in the quasi-steady flow structures that vortices are shed in a similar manner as Gerrard's suggestion. It appears that the interaction from an oppositely signed shear layer is not the only way that a growing vortex can be cut off from its feeding shear layer and be shed. Momentary reattachment of the shear layer onto the cylinder's downstream corner appears to be one of the other possible means that causes a vortex to be shed. It appears that Gerrard's vortex shedding mechanism is applicable to steady flow past bluff bodies and certain types of unsteady flow past bluff bodies only. (2) For both stationary and oscillating square cylinder, the vortex spacing ratio can assume values that are rather different from the Von Karman's prediction of 0.281 [5]. (3) When the oscillation frequency of the cylinder is decreased within the lock-on range, the longitudinal vortex spacing increases. This observation is in agreement with that of Griffin and Ramberg [7]. On the other hand, Griffin and Ramberg suggested that under such circumstance the lateral vortex spacing remains unchanged. This is not so in the present experiment. Also, Griffin and Ramberg's suggestion of a collinear row of alternatively signed vortices being the limiting situation for very large amplitude of oscillation is also not supported by the present flow visualization observation. A thrust type of vortex wake is observed in lock-on type B. It is suggested that the difference between the author's and Griffin and Ramberg's observation may be attributed to the fundamental difference between flow past an oscillating square and circular cylinder. That is, the flow separation positions are fixed in the former's case but not so for the latter. (4) During lock-on, if the Strouhal number increases, both the longitudinal vortex spacing and the circulation associated with each vortex decrease~ (5) The position of the cylinder within an oscillation cycle when vortices grow and get shed affects both the magnitude and sign of (p and the magnitude of b/d. The sign of q~ determines whether the cylinder is susceptible to the galloping type of flow induced oscillation. 5. P o s t s c r i p t n o t e
Since all the results reported in the present paper were obtained at a constant A/d of 0.675, the reduced velocity Ur was the only flow parameter used to describe the different flow structures observed. However, any attempt to directly compare the results reported in the present paper with those from other investigations by merely matching the reduced velocity Ur should be avoided. This is so because A/d is also an important flow parameter. Comparison of results should be made only when the parameter U J ( A / d ) is matched. It can be shown quite easily that the parameter Ur/(A/d) is related to the maximum flow
S.C. Luo/Vortex wake of a transversely oscillating square cyclinder
119
incidence angle experienced by the square cylinder during a cycle of oscillat i o n , ~ .... via the following relationship: ffmax = t a n
1 2~
~
.
Acknowledgement The work reported in the present paper was carried out by the author under t h e s u p e r v i s i o n o f Prof. P . W . B e a r m a n w h e n t h e a u t h o r w a s a p o s t g r a d u a t e student at the Department of Aeronautics, Imperial College of Science, Techn o l o g y a n d M e d i c i n e . T h e a u t h o r is v e r y g r a t e f u l to Prof. B e a r m a n for a l l t h e a d v i c e s h e h a d so k i n d l y o f f e r e d d u r i n g t h e c o u r s e o f r e s e a r c h . References [1] G.V. Parkinson, Mathematical models of flow-induced vibrations of bluff bodies, in: Naudascher (Ed), Flow-Induced Structural Vibrations (1974) pp. 81 127. [2] P.W. Bearman and S.C. Luo, Investigation of the aerodynamic instability of a squaresection cylinder by forced oscillation, J. Fluids Struc. 2 (1988) 161 176. [3] J.H. Gerrard, The mechanics of the formation region of vortices behind bluff bodies. J. Fluid Mech. 25 (1966) 401 413. [4] P.W. Bearman and E.D. Obasaju, An experimental study of pressure fluctuations on fixed and oscillating square-section cylinders, J. Fluid Mech. 119 (1982) 297 321. [5] L.M. Milne-Thomson, Theoretical Hydrodynamics, Macmillan, 5th edn., pp. 377 380. [6] P.W. Bearman, On vortex street wakes, J. Fluid Mech. 28 (1967) 625 641. [7] O.M. Griffin and S.E. Ramberg, The vortex-street wakes of vibrating cylinders, J. Fluid Mech. 66 (1974) 553 576.