Predictions of fluctuating lift on a transversely oscillating square-section cylinder

lournal of Fluids and Structures (1990) 4, 219-228

PREDICTIONS OF FLUCTUATING LIFT ON A TRANSVERSELY OSCILLATING. SQUARE-SECTION CYLINDER S. C. L u o t AND P. W. BEARMAN

Department of Aeronautics, Imperial College of Science, Technology and Medicine, London, U.K. (Received 20 March 1989 and in revised form 21 September 1989)

Two mathematical models are examined in an attempt to explain the behaviour of the fluctuating normal force induced by the transverse oscillation of a square-section cylinder in flow. When the cylinder is at rest, the approaching flow is normal to one of its faces. The models used are the quasi-steady theory of galloping due to Parkinson and unsteady aerofoil theory. The validity of both models is assessed by comparing predictions of the unsteady side force at the oscillation frequency and the phase angle between the side force and displacement with the authors' own experimental measurements. The quasi-steady theory is found to predict the experimental data well at high reduced velocity but, as expected, due to the influence of vortex shedding and fluid inertia forces, does not follow the data so well at intermediate to low reduced velocity. The unsteady aerofoil theory, on the other hand, by making some allowance for shed vorticity and fluid inertia forces, is found to model the flow over a larger range of the ratio of reduced velocity to dimensionless amplitude of oscillation, Ur/(A/D), with 26 being the approximate lower bound. Neither model is able to predict accurately conditions in the vortex lock-in regime. 1. I N T R O D U C T I O N WHEN A FLEXIBLYMOUNTEDOBJECT, either streamlined (e.g. an aerofoil) or bluff (e.g. a square-section cylinder) is exposed to a steady stream, fluid-structure interactions can take place at different (but sometimes overlapping) reduced velocities Ur, where Ur is defined as UJfND, with U~ being the free stream velocity, fly the body oscillation frequency, and D the body width. The interaction m a y result in different types of flow-induced oscillation. T h r e e well known categories of flow-induced oscillation are vortex resonance, galloping oscillation and flutter [see Parkinson (1971)]. Parkinson (1971) further classifies galloping oscillation into four main types. In the present p a p e r only plunging oscillation will be considered. The most widely used mathematical model for predicting the galloping response of a square-section cylinder (and other shapes that are susceptible to galloping oscillations) is that due to Parkinson & Brooks (1961) and Parkinson & Smith (1964); see also Parkinson (1972). In their quasi-steady theory Parkinson and coworkers assume that the transverse fluid force acting on a cylinder moving with a transverse velocity dy/dt in the presence of a free stream of velocity U~ can be taken to be the same as one that is acting on a stationary cylinder at a similar instantaneous angle of incidence cr in a free stream with velocity Ures, where o: and Ures c a n be expressed in terms of dy/dt and U~ #Present address: Department of Mechanical and Production Engineering, National University of Singapore, 10 Kent Ridge Crescent, Singapore 0511. 0889-9746/90/020219 + 10 $03.00

~ 1990 Academic Press Limited

220

S. C. L U O A N D P. W. B E A R M A N

as follows: oc = tan

_ll-dy/dtl [ - - - ~ J,

U r e s = ~ U 2 -~- (dy

/dt) 2.

(1) (2)

The fluid force is represented by 0-5p D l U~Cy(OC), where Cy(te) is the coefficient of the force acting normal to the side faces, measured on a stationary cylinder that is inclined to the free stream at an angle re. The side force will often be referred to in the paper as the lift force, p, D and l are the fluid density, and the width and the span of the square-section cylinder, respectively. The sign convention is such that when the cylinder is descending below its equilibrium position, both cr and y are positive. The side force is positive when acting downwards. Within the range - 1 6 ° < 0:< 16°, Parkison represents Cy(OC) = Cy([dy/dt]/Uo~) by an odd power polynomial as follows: Cy(plU~) = A O I U ~ ) - B(plU~) 3 + C(plU~o) 5 - D(plU~) 7.

(3)

The equation of motion of the system is given by my + 2[3mOOny + ma~Ey = ~pDlU~Cy(~/Uoo), 1 2 •

(4)

where m is the mass of the cylinder, fl the fraction of critical damping and ton the natural frequency in radians/s. Equation (4) can be solved and the magnitudes of Ucr (critical reduced velocity--the reduced velocity above which galloping oscillation will commence), the build up time [time that the cylinder takes, from rest, to reach 99% (arbitrary criterion) of the maximum amplitude of oscillation for a particular free stream velocity] and the relation between dimensionless amplitude of oscillation ( A / D ) and reduced velocity can be estimated. For a system with sufficiently high mass and damping, the critical reduced velocity, Uc, = 4~rfl/nA, [where n is equal to half the ratio of the displaced fluid mass to the mass of the cylinder and A is the slope of Cy(a0 versus ol at 0°] is much higher than the reduced velocity for vortex resonance U , o ( - I / S , where S is the stationary cylinder Strouhal number). Parkinson's quasi-steady theory can accurately predict the response of a square-section cylinder galloping transversely to the free stream, including the magnitude of Uc,, the variation of A / D with U, and the build-up time. However, for a system with only moderate or low mass and damping, U~ may be predicted to be only slightly higher then, or even lower than, U,o. For these systems, it has been observed experimentally that galloping oscillations always commence at Ur-~ U,o [see Wawzonek (1979), Wawzonek & Parkinson (1979) and Bearman et al. (1987)]. The objectives of the present paper are the following: (i) to examine the range of validity of Parkinson's quasi-steady theory by comparing predictions of C'L(fN), the rms value of the component of lift at the body frequency fs, and ~ the phase angle between CL(fN) (the component of lift at the body frequency) and cylinder displacement y, with experimental measurements; and (ii) to determine whether unsteady aerofoil theory can provide useful results for a transversely oscillating square-section cylinder; its usefulness will be assessed by comparing again predicted values of C'L(fN) and ~b with experimental measurements.

LIFT ON SQUARE-SECTION CYLINDER

221

2. E X P E R I M E N T A L SET-UP A square-section cylinder was installed horizontally in a wind tunnel with working section 0.91 m × 0.91 m. The ends of the cylinder were connected to sYnchronized Scotch-Yoke mechanisms, which set the cylinder into forced simple harmonic motion in a vertical plane. The neutral position of the cylinder displacement coincided with the midheight position of the wind tunnel working section. Two opposite faces of the square-section cylinder were tapped at midspan position. Six evenly spaced pressure tappings were installed on each face. By connecting the six pressure tapping outputs to a pressure transducer via a six-to-one pressure averager [see Surry & Stathoupoulos (1977) and Bearman & Luo (1988)], the average instantaneous pressure acting on one side of the square-section cylinder can be measured. By measuring the average pressures on the two opposite sides of the square cylinder that are normal to its motion, the side force acting on the cylinder can be calculated by finding the difference between them. The rms lift coefficient C~ can then be found. Using Fourier analysis, the components of the side force at various frequencies can also be calculated. The components that are of particular interest here are the components at the body frequency fN and its higher harmonics. They are denoted by CL(fN), CL(3fN), etc. By performing correlation measurements between the side force and the displacement of the cylinder, the phase angle between the body frequency component of the lift and the cylinder displacement, ~b, can also be calculated. Details of the experimental setup and data acquisition system can be found in Luo (1985) and Bearman & Luo (1988). 3. EXPERIMENTAL D A T A The variation of the normal force coefficient Cy with angle of incidence te forms the foundation of Parkinson's quasi-steady theory. In the present experiment Cy was measured within the range - 6 °-< 0c-< 20 ° at three Reynolds numbers Re: 2.26 x 104, 4"6 × 104 and 6.6 x 10 4. The measurements are plotted in Figure 1. Since in the quasisteady approach the flow is determined by only one parameter, the angle of incidence or, it was decided to plot all the experimental measurements for oscillating cylinders against UJ(A/D) rather than /Jr. It can be shown that this quantity is related to the maximum value of a~ attained during a cylinder of oscillation via the relation a~ma~= tan-l~2sr A 1 ] .

k

DU~J

(5)

Measured values of C'L(fN), C~.(3fiv) and q~ are plotted against Ur/(A/D) and are presented in Figures 2 to 4, respectively. 4. PREDICTIONS F R O M QUASI-STEADY T H E O R Y When a cylinder oscillates sinusoidally in a direction transverse to a free stream, its angle of incidence varies continuously between +tr . . . . where O~m~ is related to A/D and Ur via the relation given by equation (5). Assuming that the quasi-steady consideration is valid, the side force coefficient for the cylinder therefore varies continuously between Cy(o[----0dmax) and Cy(od = 0t'max). With the knowledge of the variation of Cy with re (Figure 1), the cyclic variation of Cy with time can therefore be estimated. By using Fourier analysis, the magnitudes of the various components of Cy such as Cy(fN) and Cy(3fN), etc. can also be estimated. Predicted rms values of the components of the normal force coefficient at the forcing frequency and its third

222

S. C. L U O A N D P. W. B E A R M A N I

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224

S. C. L U O AND P. W. B E A R M A N

harmonic, Cy(fN) and Cy(3fN), are compared with measured values and are shown in Figures 2 and 3, respectively. The quasi-steady theory assumes that the phase angle between the component of the normal force at the body frequency and the cylinder displacement is at a constant value of ~r/2. This is included in Figure 4. 5. UNSTEADY A E R O F O I L THEORY In Figures 2 to 4 the applicability and the limitations of the quasi-steady theory were illustrated. Since it assumes that the inertia component of the lift force acting on the cylinder is negligible, the quasi-steady theory predicts the fluctuating component of the side force at the cylinder frequency best at high values of Ur/(A/D), where the free stream velocity U~ is much larger than the transverse velocity of the cylinder 3). If one wishes to improve the quasi-steady theory or to propose a new theory so that the magnitude of CL(fN) can be predicted reasonably accurately at intermediate or even low values of G/(A/D), the inertia (or acceleration) effect and the influence of shed vorticity must be taken into consideration. For an aerofoil oscillating transversely in an incompressible flow, Bisplinghoff et al. (1957) have shown that the lift force per unit span L induced by the plunging oscillating motion is given by ~D 2 3CL D L = -CmoO --~-Y - - ~ pUoo-~ C(k)p,

(6)

where Cmo is the potential flow inertia coeffÉcient, C(k) is Theodorsen's function, equal to H~Z)(k)/{H~2)(k)+ iH(oZ)(k)}, with U(02) and H~2) being Hankel functions of the second kind of order 0 and 1, respectively; k is equal to Jr/Ur, and 3CL/3Ol is the li•curve slope. In equation (6) the first term represents the conventional inertia force term for attached flow, while the physical significance of the second term may be explained as follows. For an aerofoil oscillating transversely to a free stream, the lift force acting on the aerofoil and hence the circulation around it varies continuously. In order to satisfy Kelvin's circulation theory (DF/Dt=O), vorticity must be shed by the aerofoil continuously, so that the total circulation around a loop enclosing the aerofoil and the shed vorticity F remains conserved. The vorticity shed can affect the flow around the aerofoil, and hence the lift force acting on it in both magnitude and phase; this effect is treated by the second term on the right-hand side of equation (6). We have taken the bold step of assuming that an equation which is derived for an aerofoil oscillating with small amplitude can be applied to an oscillating cylinder. In the present case, C,,o will be taken as 1-513, the potential flow inertia coefficient for an oscillating square-section cylinder, and 3CL/30I equated to the negative value of dCy/doL at ol = 0. It should be noted that CL and Cy have opposite signs since in equation (6) the lift is positive upwards. The measured value used for dCy/do~ is 3.86. Although it is impossible to make rigorous defence of the use of this approach for a square-section cylinder, it should be noted that equation (6) makes allowance for two important contributions to side force that are not included in Parkinson's quasisteady theory. They are an inertia force resulting from body acceleration and a modification of lift due to shed circulation. Together, these effects can have an important influence on the phase angle between side force and displacement. Now let

y = Im[Aei'°Nq

(7)

225

LIFT ON SQUARE-SECTION CYLINDER

and

C(k) = F(k) + iG(k), where

F(k)

and

G(k)

(8)

are real. Substituting equations (7) and (8) into equation (6),

[ p~D 2 ~CL D L=[Cmo -~ Wu+-~ pU~-~G(k)](AWNsinwNt)

DF(k)J(AwNq

[~C L

COS O)Nt)

(9)

= Lin + Lout,

where

[- pc~D D G(k)](AO)Nj Lin=[Cmo -4 2-WN + ~CL 3a" pU~o~

sin tONt),

~CL D Lout= - [ - ~ PU~~ F(k)](AWNCOSWNt), respectively the part which is in phase and that which is 90 ° out of phase with the displacement of the cylinder. Lin and Lout are made dimensionless by dividing them by -(¼pe~D2)y and 0.5pDU~, respectively; thus, Lin

Cmt- _ ~perD2y 1 DC: G(k) = C~o -t ~k (10)

= Cmo "[- Cma.

The term (1/zk)(aCL/3O:)G(k) modifies the potential flow inertia coefficient and is therefore called the "added inertia coefficient," Cma, in the present paper. The interesting feature here is that since both 3CL/3OI and G(k) are negative, the added inertia coefficient has the effect of increasing the inertia force above its potential flow value. For Lout, Lout Clo -- 1

~pDU~2

= -Cto cos wNt, where (~to is the amplitude of

C1oand

(11)

is given by

C,o - 8CLF(k) 2¢c~ = 8C: F(k) tan

(12)

¢~'max;

a~maxhas been defined in equation (5). Now by substituting equations (10) and (11) back into equation (9) and nondimensionalizing by 0.5pDU 2, it can be shown that C L =

Cml~7~2tan

1 O~ma x - -

sin

Vr

¢Olvt

OC L

F(k) tan

The component of the lift force at the body frequency,

o ~ x cos ogNt.

CL(fN), can

(1.3)

also be expressed

226

S. C. LUO AND P. W. BEARMAN

as

CL(fN) = CL(fN) sin(fONt + q)) = V~ C'L(fN) sin((oNt + q0(14) Comparing equations (13) and (14) and equating the sin wNt and cos (Out coefficients separately, it can be shown that V ~ CL(fN ) COS (p =

Cml.TT,2 tan

1

ITmax =

(Jr

(15)

and

V~ C[(fN) sin q~= -- -OCz. - F(k) tan 3te

t~max.

(16)

By combining equations (15) and (16),

C~(fu) = N/~ [L (\ Cm'~eUtan r O{max)2 + ( ~

F(k) tan tXmax)2]

(17)

/.

(18)

and q) = tan-1 /

L

~

J Equations (17) and (18) are the expressions of C'L(fN ) and q) as predicted by the Cml'Tr'

unsteady aerofoil theory. The theoretical predictions are compared with experimental measurements and are shown plotted in Figures 2 and 4.

6. DISCUSSION OF RESULTS The variation of the steady side force with the angle of incidence, shown in Figure 1, has been reported and discussed by many researchers. A fairly detailed discussion can be found in Luo's (1985) work. It will therefore only be pointed out in the present paper that the variation of Cy with ol is weakly Reynolds-number dependent and that the change in the sign of ~Cy/~O{a t cr = 13-5 ° is caused by the reattachment of flow on one side of the square cylinder. From Figure 2 it is seen that when plotted against Ur/(A/D), the rms values of the component of the side force at the body frequency C'L(fN) measured at different values of A/D show a good collapse at high values of Ur/(A/D). However, scatter increases rapidly at low values of Ur/(A/D). The prediction of C'L(fN) by the quasi-steady theory [equations (3) and (4)] agrees well with experimental measurements only at high values of Ur/(A/D). This, is likely due to the exclusion by the quasi-steady theory of the inertia component from the total fluid force acting on the cylinder and the influence of vortex shedding. In Figure 3 the quasi-steady theory is only capable of predicting the trend of the experimentally measured C~(3fN). The theory under-estimates the magnitude of the experimental data throughout the range of Ur/(A/D) investigated. The quasi-steady theory also assumes that the body frequency component of the side force always leads the cylinder displacement by av/2. A horizontal line representing (I) = 90° is therefore included in Figure 4 for comparison with experimental measurements. Again, it can be seen that the agreement between predictions and measurements exists only at high values of Ur/(A/D).

LIFT ON SQUARE-SECTION CYLINDER

227

Both C'm(fu) and 4, predicted by the unsteady aerofoil theory, are also included in Figures 2 and 4, respectively, for comparison with experimental data and with the predictions of the quasi-steady theory. From Figures 2 and 4 two important observations can be made. First, at high values of Ur/(A/D) predictions by both theories agree well with each other and with the experimental measurements. Second, for intermediate values of Ur/(A/D), between 25 to 75 approximately, the unsteady aerofoil theory predicts experimental measurements more accurately. Both observations support the earlier suggestion that the inertia component of the total force is not negligible at low and intermediate values of UJ(A/D). From measurements conducted on a stationary inclined cylinder (Figure 1), it is known that one of the two separated shear layers reattaches onto the trailing corner of the cylinder at a~-~ 13.5 ° and causes a sudden change in the flow structure. For a cylinder oscillating transversely to the free stream, if O(max is equated to 13-5° then this angle corresponds to Ur/(A/D) equal to approximately 26. This suggests, therefore, that for UJ(A/D) <--26, the separated shear layers may reattach onto the sides of the oscillating square-section cylinder during part of the cycle, and the flow field may be altered significantly. The unsteady aerofoil theory, which does not cater for the effect of shear layer reattachment at large trmax and only makes use of the value of 3CL/~O:at tr = 0°, is therefore not expected to be able to predict experimental measurements for U~/(A/D)<-26. This is confirmed in Figures 2 and 4. The theories discussed above cannot address the problem of what happens when the frequency of oscillation is close to the cylinder natural vortex shedding frequency. In this case the shedding frequency can lock on to the body frequency and, for a square section cylinder undergoing small amplitude oscillations, this will occur when U~ is in the range 7 to 8; the lock-on range increases with increasing A/D, as discussed by Bearman and Obasaju (1982). Caution has to be exercised over the use of the parameter UJ(A/D) in this regime because small amplitudes of oscillation can lead to large values of Ufl(A/D), suggesting, incorrectly, that it is permissible to use the quasi-steady or unsteady aerofoil theories. In practice it is found that there is a large component of the side force at the body frequency but that the phase angle is small. Finally, some consideration is given as to how the results presented here, derived from forced oscillation experiments, might apply to freely a vibrating cylinder. A flexibly-mounted cylinder can oscillate when the phase angle q~ is in the range 0° to 180°. It can be seen from Figure 4 that constant amplitude oscillations will only occur when UJ(A/D) is greater than about 20. The amplutide level at a particular value of U~ will depend on the mass and damping of the cylinder. The quantity Lin given in equation (9) acts as an inertia force and will add to the inertia term on the left-hand side of equation (4). For a cylinder in air it is unlikely that the additional inertia force will affect the motion of the cylinder. However, in denser fluids such as water the additional inertia component derived from unsteady aerofoil theory could influence the oscillatory behaviour. 7. CONCLUSIONS The effectiveness of quasi-steady theory in predicting the relationship between Ur and

AID and the magnitude of Ucr has been well demonstrated by Parkinson and his coworkers. In the present paper the quasi-steady theory is used to estimate the magnitude of C[(fN), and the prediction is then compared with experimental data measured by the authors. It is found that the quasi-steady theory can predict the magnitude of C[(fN) fairly accurately for moderate and high values of Ur/(A/D)

228

s . c . LUO AND P. W. BEARMAN

(>75). The quasi-steady theory assumes that ~ - - 9 0 °, and experimental data shows that this value is approached only for Ur/(A/D) > 100. T h e discrepancies between the experimental m e a s u r e m e n t s and the predictions of the quasi-steady theory are thought to be mainly due to the influence of fluid inertia and the presence of shed vorticity, which are not taken into consideration. A n o t h e r mathematical model examined in the present paper, unsteady aerofoil theory, was devised originally for a thin aerofoil oscillating with small amplitude transversely to a free stream. Unsteady aerofoil theory is used here because it takes both the inertia and the effect of shed vorticity on the lift force into consideration. It should be noted, of course, that this theory was not derived to treat bluff bodies. However, it is found that unsteady aerofoil theory is capable of predicting the experimental m e a s u r e m e n t s as accurately at the quasi-steady theory at high Ur/(A/D); at intermediate values of Ur/(A/D) of between 26 to 75, it is m o r e accurate than the latter. UJ(A/D)--~26 approximately corresponds to the value of U J ( A / D ) below which flow reattachment begins to take place during part of the oscillation cycle. This appears to set a lower bound to the range of U J ( A / D ) where unsteady aerofoil theory is applicable. REFERENCES BEARMAN, P. W., GARTSHORE, I. S., MAULL, D. J. & PARKINSON, G. V. 1987

Experimentson

flow-induced vibration of a square-section cylinder. Journal of Fluids and Structures 1, 19-34. BEARMAN, P. W. & LUO, S. C. 1988 Investigation of the aerodynamic instability of a square-section cylinder by forced oscillation. Journal of Fluids and Structures 2, 161-176. BEARMAN, P. W. & OBASAJU, E. D. 1982 An experimental study of pressure fluctuations on fixed and oscillating square section cylinders. Journal of Fluid Mechanics 119, 297-321. BISPLINGHOFF, R. L., ASHLEY, H. & HALEMAN, R. L. 1957 Aeroelasticity. Reading, Mass.: Addison-Wesley. Luo, S. C. 1985 A forced vibration study of the aerodynamic instability of a square-section cylinder. Ph.D. Thesis; Imperial College, University of London. PARmNSON, G. V. 1971 Wind induced instability of structures. Philosophical Transactions of the Royal Society of London, A 269, 395-409. PARKINSON, G. V. 1972 Mathematical models of flow-induced vibrations of bluff bodies. In Flow-Induced Structural Vibrations (ed. E. Naudascher), pp. 81-127. Berlin: SpringerVerlag. PARKINSON, G. V. & BROOKS, N. P. H. 1961 On the aeroelastic instability of bluff cylinders. Journal of Applied Mechanics 28, 225-258. PARmNSON, G. V. ~ SMITH, J. D. 1964 The square prism as an aeroelastic non-linear oscillator. Quarterly Journal of Mechanics and Applied Mathematics 17, 225-239. SURRY, D. & STATHOPOULOS, T. 1977 An experimental approach to the economical measurement of spatially-averaged wind load. Journal of Industrial Aerodynamics 2, 385-397. WAWZONEK, M. A. 1979 Aeroelastic behaviour of square prisms in uniform flow. M.A.Sc thesis, Department of Mechanical Engineering, University of British Columbia, Vancouver, B.C., Canada. WAWZONEK, M. A. & PARKINSON, G. W. 1979 Combined effects of galloping instability and vortex resonance. In Proceedings Fifth International Conference on Wind Effects on Buildings and Structures, VI-2-1-VI-2-12.