Interference galloping of a circular cylinder near a plane boundary

Interference galloping of a circular cylinder near a plane boundary

Journal of Sound and Vibration (1995) 181(3), 431–445 INTERFERENCE GALLOPING OF A CIRCULAR CYLINDER NEAR A PLANE BOUNDARY† A. B‡ Brown & Root L...

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Journal of Sound and Vibration (1995) 181(3), 431–445

INTERFERENCE GALLOPING OF A CIRCULAR CYLINDER NEAR A PLANE BOUNDARY† A. B‡ Brown & Root Limited, 150 The Broadway, London SW19 1RX, England (Received 27 August 1993, and in final form 4 January 1994) The galloping motion of an elastic circular cylinder, with oscillations restricted to a plane normal to the incident uniform flow, when it is located in the close neighbourhood of a plane boundary is investigated theoretically by using a quasi-steady assumption; that the cylinder and the flow are both parallel to the boundary. Closed form expressions are presented to describe the cylinder behaviour: namely, the variation of the reduced (relative) amplitude with the reduced velocity, the relative mean position of the vibration with the dimensionless free stream dynamic pressure and the relative motion frequency with the reduced amplitude. The first variation and also that of the cylinder mean position with the flow velocity approach asymptotically constant values, the second one is a function only of the lift coefficients, while the last one depends on both the dimensionless free stream dynamic pressure and the lift coefficients. The mathematical model predicts that an increase in the cylinder mass or structural damping causes the galloping to occur at a higher flow velocity. Simple expressions are presented for the prediction of the reduced velocity and the cylinder motion frequency at the galloping onset. The theory indicates that the galloping instability occurs only when the gap between the cylinder and the boundary is very small.

1. INTRODUCTION

Owing to its practical importance, the subject of response of suspended pipeline spans exposed to waves or currents, or a combination of both, has received much attention [1–4]. Experience with unburied offshore pipelines laid in strong current areas has shown that such pipelines may develop unsupported spans due to the sea bottom being scoured out under the pipeline by current action. The physics behind vibrations of pipelines lying on the sea bed is quite different from that of an isolated circular cylinder. It is an acknowledged fact that an elastic cylinder in the vicinity of other bodies does not conform to the pattern of behaviour recorded when the cylinder is isolated [5, 6]. For example, laboratory observations of Bokaian and Geoola [7–10] indicated that when a flexible circular cylinder, with oscillations restricted to a plane normal to the incident flow, lies in the close vicinity of a parallel and similar but rigid body, the dynamic response of the elastic cylinder may become significantly more complex. This means, amongst other things, that the mean position of the moving cylinder and also both its amplitude and frequency may increase with increased flow velocity; the variation of the amplitude with the flow velocity approaches asymptotically a constant value. Increasing the gap between † The original version of this paper was presented at the 18th Annual Offshore Technology Conference, OTC 5217, Houston, Texas, 5–8 May, 1986. ‡ Present address: Northern Ocean Services (McDermott), Offshore House, Tees Offshore Base, South Bank, Middlesbrough, Cleveland TS6 6UZ, England

431 0022–460X/95/130431 + 15 $08.00/0

7 1995 Academic Press Limited

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. 

the pair of cylinders generally causes the tendency towards this interference type of galloping to decrease: i.e., a higher flow velocity may be needed in order to excite the cylinder to instability, the motion amplitudes drop considerably, and the derivation of the vibration frequency from the cylinder’s natural frequency as well as the cylinder mean position from its undisturbed one become smaller. For the case when the flexible cylinder is upstream of the rigid one, the interference galloping occurs only at very small separations, the mean position of the moving cylinder hardly changes with the flow velocity, and only one pair of vortices is shed during one motion cycle. For the opposite case, the change in the cylinder mean position is appreciable not only during galloping, but even before the cylinder starts to oscillate. Moreover, the galloping instability may occur at larger separations, the variation of the galloping frequency with the flow velocity is more intense, and more than one pair of vortices may be shed during one galloping cycle; the vortex-shedding frequency varies roughly linearly with the flow velocity. For a flexible cylinder near a plane wall, only vibrations in the transverse direction are significant [11, 12]. The test results of Tsahalis and Jones [12] and Tsahalis [13] indicated that when the gap between the pipe and the wall is about one diameter, or larger, the pipeline suffers only from a vortex-resonance. Many engineering situations involve gaps smaller than one diameter. If the sea bed is erodible, for example, a typical scour depth below a pipe is in fact in the range of 0·2–0·6 diameters. The very small gaps experimental observations of Tsahalis and Jones [12] and the more detailed ones of Sumer et al. [14] reveal that in such situations the cylinder exhibits a vigorous build-up of motion amplitudes with increasing flow velocity, for all flow velocities greater than a critical threshold value; the variation of amplitude with the flow velocity takes roughly a S-shaped form, as opposed to a near-symmetric bell-shaped curve associated with a vortex lock-in. The limiting vibration amplitude is of the same order of magnitude as in the case when the cylinder was suffering from a vortex-resonance at larger spacings. A fascinating feature of the instability at very small gaps is that on increasing the flow velocity, the cylinder is continually repelled from the wall and the motion becomes faster; the variation of the cylinder mean position with the flow velocity approaches asymptotically a constant value. At tiny gaps, the wall proximity effects are dominant and the cylinder suffers from large lift forces; the vortex-shedding no longer constituting the driving mechanism [14]. When moving upward, the contraction of the streamlines above the cylinder induces an upward force, which lifts the cylinder away from the wall. When the cylinder is moved away from the wall, more flow passes below more easily, reducing the upward lift force, and hence the cylinder falls back to its equilibrium position, after which the event repeats itself. Although the analogy between this interference galloping and that associated with a pair of cylinders is not completely valid, many features of the two are similar. Sumer et al. [14] did not observe any hysteresis effect in the cylinder response and found that the change in the cylinder mean position began immediately after galloping began. There is no reported experimental observation of the vortex-shedding process behind the moving cylinder. When a particular pipeline span is likely to undergo hydroelastic oscillations, most engineers try to alter the pipeline design by offsetting its natural frequency from the vortex-shedding frequency without due consideration for galloping phenomenon at tiny gaps. Hence, a reliable model for hydroelastic instability analysis of pipelines at small gaps is highly desirable. This paper presents a mathematical model for prediction of the galloping behaviour of a circular cylinder, in both the amplitude and frequency domains, in terms of the governing

    

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flow and structural parameters and the relative position with respect to the wall. In some applications, such as design of submarine pipelines, the cylinder may be totally immersed in a thick boundary layer. In this paper, attention is concentrated only on thin boundary layers: i.e., boundary layers thinner than the diameter of the cylinder.

2. MATHEMATICAL MODEL

The first sign of interference between a stationary cylinder and a plane wall is the displacement of the cylinder front stagnation point towards the gap [15]. This point is gradually displaced towards the gap side as the cylinder is placed closer to the wall. The pressure distribution as a whole remains symmetric with respect to the axis formed by the stagnation point and the cylinder centre; the resulting force being along the axis [16]. This force can be split into a drag component parallel to the free stream flow and a lift component perpendicular to it. The lift force is always directed away from the wall and is proportional to the displacement of the stagnation point. As the cylinder is moved away from the wall, the lift force falls off very rapidly, and the pressure distribution becomes more symmetric about the stagnation point. When a circular cylinder is positioned progressively closer to the wall, the flow pattern changes from a strong vortex street to a single row of regular vortices which are erratically arranged. The vortex-shedding process is suppressed if the gap to cylinder diameter ratio is less than a critical value of 0·3–0·4 [15, 17, 18]. In this case, the pressure distribution beocmes markedly asymmetric, but the displacement of the stagnation point continues in the same direction and at the same pace [16]. For the case of a moving cylinder, the vortices are always shed even at tiny gaps. One may assume that, in this case, the symmetry of the pressure distribution with respect to the axis formed by the stagnation point and the cylinder centre also holds. In Figure 1 is depicted a two-dimensional uniform flow of velocity V streaming past a circular cylinder placed near to a plane horizontal wall; the cylinder and the flow both being parallel to the wall. In Figure 2(a) are displayed the sketches of the variations of the static lift CL and drag CD coefficients with the cylinder spacing ratio y0 = r/D, where D is the cylinder diameter and r is the distance of the cylinder centre from the wall, as shown in Figure 1 (a list of nomenclature is given in the Appendix). These variations may

Figure 1. Lift and drag coefficients under a flow parallel and at an angle to the wall.

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. 

Figure 2. A sketch of the variation of CL and CD with y0 , u with y0 and CL and CD with u.

be approximated as CL = sIi =+11 A(2i − 1) y0(2i − 1) and CD = sIi= 0 B(2i) y0(2i), respectively, where i is a counter, I is an integer and A(2i − 1) and B(2i) are constant coefficients. If S0 represents the stagnation point, the angle of OS0 with the wall may be approximated as tan u = CL /CD , as indicated in Figure 1. The stagnation angle u decreases with increased spacing y0 ; thus dCL /du q 0 while dCD /du Q 0, as shown in Figures 2(b) and 2(c). The variation of dCL /du may similarly be approximated as dCL /du = sIi= 0 C(2i) y0(2i), where C(2i) are constant coefficients. The above description of the force coefficients is for the case in which the flow direction is parallel to the wall. When the incident flow is at an angle g with respect to the wall, as also depicted in Figure 1, the effective gap between the cylinder and the wall increases; thus the cylinder suffers from a smaller lift but a larger drag force. It is assumed that the effect of this flow is to reduce the stagnation angle u by g: i.e., to move the stagnation point from S0 to Sg , as shown in Figure 1. The corresponding changes in the lift and the drag coefficients are DCL = −(dCL /du)g and DCD = −(dCD /du)g respectively. In most cases the variation of u with y0 for small values of y0 where the galloping instability can occur may be approximated linearly. Thus dCL /du simply becomes directly proportional to dCL /dy0 . In Figure 3 the situation is illustrated in which the cylinder is excited by the flow, the cylinder motion being constrained at right angles to the flow direction. In this figure, O indicates the cylinder centre in its undisturbed position (is still fluid). The structural differential equation of motion of the cylinder may be written in the usual form as my¨ + cy˙ + ky = Fy ,

(1)

where y is the downward displacement of the cylinder from its undisturbed position, an overdot represents differentiation with respect to time, c is the cylinder structural damping

    

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coefficient, k is its stiffness, Fy is the transverse component of the flow-induced force on the cylinder and m is the vibrating mass of the cylinder. For a cylinder in water, such as a submarine pipeline, this mass is equal to the cylinder mass as measured in air minus the cylinder buoyancy plus the hydrodynamic added mass [2]. In motion, as depicted in Figure 3, the relative velocity between the oncoming flow and the oscillating body is Vrel = zV 2 + y˙ 2 , where y˙ is the cylinder velocity, the angle of flow attack being a = tan−1 (y˙ /V). This relative velocity will give rise to a force having two components of drag FDa in the direction of Vrel and life FLa in an orthogonal direction. The projection of these components on the y-direction yields the relationship 2 my¨ + cy˙ + ky = Fy = −(rDLVrel /2)(CDa sin a + CLa cos a),

(2)

2 ) and where r is the fluid density, L indicates the cylinder length, and CDa = FDa /( 12 rDLVrel 1 2 CLa = FLa /( 2 rDLVrel ) denote the cylinder dynamic drag and lift coefficients when the cylinder is at position O' and the flow is at an angle a with respect to the wall. Now consider the case of incipient (small) cylinder motion: that is, the condition in the vicinity of y˙ = 0, wherein sin a 1 tan a 1 a 1 y˙ /V. Based on a quasi-steady assumption, the dynamic coefficients CDa and CLa may be related to the static coefficients CD and CL by the expressions

CDa = CD − (dCD /du)a,

CLa = CL − (dCL /du)a.

(3)

With this and by replacing CDa and CLa in equation (2) by their equivalents from equation (3) and the linearization of the resulting equation with respect to y˙ , the following expression is obtained: Y + nvn U(CD − dCL /du + 2b/nU)Y + vn2 (Y + nU 2CL ) = 0.

Figure 3. The galloping of a circular cylinder near a plane boundary.

(4)

. 

436

This relationship is written in terms of the independent variables of the cylinder natural circular frequency vn = zk/m, its reduced (relative) displacement Y = y/D, its reduced damping b = c/(2mvn ), its mass parameter n = rD 2L/(2m) and the reduced velocity U = V/(vn D). The coefficients CD , CL and dCL /du should be calculated for (y0 − Y), the cylinder spacing ratio at position O'. In equation (4), the terms (Y + nU 2CL ) denote the variation of the cylinder motion frequency with the reduced velocity U. The self-excitation begins with CD − dCL /du + 2b/nU Q 0. As Zdravkovich [19] has shown, the static lift forces are significant only when the gap between the cylinder and the wall is less than about one diameter. In Figures 2(b) and 2(c) it is indicated that dCL /du is large only when the spacing ratio y0 is small. Thus the galloping instability can occur only at very small gaps. Replacement of CD , CL and dCL /du in equation (4) by their equivalent polynomial approximations results in the expression Y + vn2 Y = −nvn U{Y {(B0 − C0 ) + (B2 − C2 )(y0 − Y)2 + (B4 − C4 )(y0 − Y)4 + · · · + [B(2I − 2) − C(2I − 2) ](y0 − Y)(2I − 2) + [B(2I) − C(2I) ](y0 − Y)(2I) + 2b/nU} + vn U[A1 (y0 − Y) + A3 (y0 − Y)3 + A5 (y0 − Y)5 + · · · + A(2I + 1) (y0 − Y)(2I + 1)]}.

(5)

Upon the substitution Y = Z + p, the following relationship is found: p + nU 2[A1 (y0 − p) + A3 (y0 − p)3 + · · · + A(2I + 1) (y0 − p)(2I + 1)] = 0.

(6)

This expression gives the variation of the relative mean position of the galloping cylinder, p, in terms of the dimensionless free stream dynamic pressure nU 2. As expected, this variation depends only on the lift coefficients. It can easily be shown that the variation of p with the reduced velocity U asymptotically approaches a constant value. The differential equation of motion of the cylinder in terms of the new variable Z takes the form Z + vn2 Z = −nvn Uf(Z, Z ) = −nvn U{Z [h0 + h1 Z + h2 Z 2 + · · · + h(2I) Z (2I) + 2b/nU] + vn U[z1 Z + z2 Z 2 + z3 Z 3 + · · · + z(2I − 1) Z (2I − 1) + z(2I) Z (2I) + z(2I + 1) Z (2I + 1)]}, (7) where the coefficients h0 , h1 , . . . and z0 , z1 , . . . may be calculated from the two following upper triangular matrices, in which for two positive integers n1 and n2 the symbol

01

n1 = n1 !/[n2 !(n1 − n2 )!]: n2

437

0

01

2I

01

0 1

01

01

(B0 − C0 ) K h0 L K 0 G H G G h1 H G 0 G H G G H G 2 (B2 − C2 ) G h2 H G 2 G H G G h3 H G 0 G H G G H G 4 (B4 − C4 ) G h4 H G 4 G H G G h5 H= G 0 G H G G··· H G ··· G··· H G ··· G H G Gh(2I − 2)H G [B(2I−2)−C(2I−2) ] 2I−2 2I−2 G H G G H G 0 Gh(2I − 1)H G G H G k h(2I) l k [B(2I) − C(2I)] 2I

01

4 3

01

2 1

01

0

[C(2I) − B(2I)]

1

2I − 1 2I

2I 2I − 2

1

[B(2I) − C(2I) ]

0

[B(2I − 2) − C(2I − 2) ]

1

2I 4

01

2I − 2 2

0

..................

0

[B(2I) − C(2I) ]

···

···

0 ···

0

01

6 (B6 − C6 ) 4

0

4 2

01

(B4 − C4 )

0

2 0

01

(B2 − C2 )

···

6 (C6 − B6 ) 5

0

(C4 − B4 )

0

(C2 − B2 )

0

[C(2I) − B(2I) ]

0

1

2I 3

01

2I − 2 1

0

0

6 2

···

[B(2I) − C(2I) ]

0 2I 2

01

1

1

[C(2I) − B(2I) ]

2I 1

01

···

2I − 2 5

0

0

[C(2I − 2) − B(2I − 2)]

2I − 2 0

0

...........

0

6 0

01

(B6 − C6 )

·················

················

0

6 1

01

(C6 − B6 )

0

[B(2I − 2) − C(2I − 2) ]

01

(B6 − C6 )

0

4 0

01

(B4 − C4 )

[C(2I − 2) − B(2I − 2) ]

2I 5

01

[C(2I) − B(2I) ]

0

0

6 3

01

(C6 − B6 )

0

4 1

01

(C4 − B4)

0

[B(2I) − C(2I) ]

0

2I 0

1

1 2I − 2 3

L H H HG (y0 − p) H HG H HG H 2 HG (y0 − p) H HG H HG (y0 − p)3 H HG H HG H,  (8) 4 HG (y0 − p) H HG H HG (y0 − p)5 H HG H HG . . . H HG . . . H HG H HG(y0 − p)(2I − 2)H HG H HG (2I − 1)H HG(y0 − p) H HG H lk (y0 − p)(2I) l

LK

0 1 HG HG

1 2I − 2 4

0

0 [C(2I − 2) − B(2I − 2) ]

·················

[B(2I − 2) − C(2I − 2) ]

················

0

K G G G G G G G G G G G G G G G G G G G G G G G G G G k

z(2I + 1)

z(2I)

z(2I − 1)

z(2I − 2)

z(2I − 3)

z(2I − 4)

... ...

z5

z4

z3

z2

z1

0

0

0

01

01

01

1 L K −A1 1 H G H G 0 H G H G 3 −A3 H G 3 H G H G 0 H G H G 5 −A5 H G 5 H G ... H =G ... H G H G 0 H G H G−A 2I − 3 H G (2I − 3) 2I − 3 H G 0 H G H G H G−A(2I − 1) 2I − 1 2I − 1 H G H G 0 H G H G H G−A(2I + 1) 2I + 1 2I + 1 l k

1

1

1

3 2

5 4

A(2I + 1)

A(2I − 1)

1

A(2I − 1)

A(2I − 3)

0

1

1 −A(2I + 1)

1

0

2I − 1 4

0

0

−A(2I + 1)

0

1

2I + 1 5

1

2I − 1 3

0

0

−A(2I − 1)

0

2I − 3 2

A(2I + 1)

1

0

A7

0

A5

0

7 4

01

5 2

01

A(2I + 1)

A(2I − 1)

0

1 2I + 1 4

0

0

1

9 5

01

−A(2I + I)

0

0

−A(2I − I)

1 1 2I + 1 3

A(2I + 1)

1

A(2I + 1)

0

1

1

−A(2I + 1)

2I + 1 2I − 4

0

2I − 1 5

0

...........

...........

...........

...........

− A(2I + 1)

2I + 1 2

0

0

...........

2I − 1 1

1

0

2I + 1 2I − 3

0

0

... ...

−A9

7 3

01

0

−A7

5 1

01

0

−A5

−A(2I + 1)

2I − 1 2

0

1

1

0

2I + 1 2I − 2

0

2I − 1 2I − 4

0

... ... A(2I − 1)

2I − 3 1

0

1

0

0 2I + 1 2I − 1

1

−A(2I − 3)

1

0

0

−A(2I − 1)

2I − 1 2I − 3

0

2I + 1 2I

0

0 2I − 1 2I − 2

0

2I − 3 2I − 4 0

A(2I − 3)

7 5

01

0

−A7

5 3

01

0

−A5

0

3 1

01

... ...

0

01

0

01

−A3

... ...

A5

A3

0

1 2I + 1 1

LK 1 GH GH GH (y0 − p) GH GH (y0 − p)2 GH GH GH (y0 − p)3 GH GH (y0 − p)4 GH ... GH ... GH GH (y0 − p)(2I − 5) GH GH GH (y0 − p)(2I − 4) GH GH (y0 − p)(2I − 3) GH GH (2I − 2) GH (y0 − p) GH GH (y0 − p)(2I − 1) GH GH (2I) lk (y0 − p)

L H H H H H H H H H H H H H. H H H H H H H H H H H H H l

(9)

438 . 

    

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Most cases of flow-excited instabilities involve a very small value of the reduced damping. In some cases the mass ratio is also very small. For n = 0 the solution of equation (7) is Z = a cos (vn t + f), where a is the reduced amplitude, defined as the vibration amplitude divided by the cylinder diameter, f is the phase and t is the time. For n W 1, it may be assumed that both f and a are slowly varying functions of time. Within a first approximation [20] nU da = K0 (a) = 2p dt = −nvn U

+

$

g

2p

f(a cos c, −avn sin c) sin c dc

0

01

01

b 1 1 0 1 1 2 a+ × 1 h a+ × 3 h a3 + · · · nU 1 2 0 0 2 2 1 2

01

%

2I 1 1 h a (2I + 1) , × I + 1 2(2I + 1) I (2I)

(10)

where the variable c = vn t + f. The steady state solution corresponds to da/dt = 0, which results in the equation

01

01

01

1 1 2 1 1 2I 2b 1 1 0 h + × h a2 + · · · + × h a (2I) + = 0. × I + 1 2(2I) I (2I) nU 1 20 0 0 2 2 2 1 2

(11)

The polynomial (11) contains the coefficients h with even number subscripts only. The circular motion frequency of the cylinder, vc , may similarly be calculated as vc2 = vn2 +

nUvn pa

g

2p

f(a cos c, −avn sin c) cos c dc.

(12)

0

The steady state vibration frequency is obtained from

$ 01

V 2 = 1 + nU 2

01

0

1

%

1 2 2I + 2 1 4 1 z + z a 2 + · · · + (2I + 1) z a (2I) , 2 1 1 1 23 2 3 I + 1 (2I + 1) 2

(13)

where V = vc /vn is the reduced motion frequency. Equation (13) contains the coefficients z with odd number subscripts only. It should be noted that in the even powered polynomials (11) and (13) the sets of coefficients h0 , h2 , . . . and z1 , z3 , . . . themselves vary with n and U; the latter depend only on the lift coefficients, while the former depend on both the drag and the lift coefficients (see equations (8) and (9)). The preceding relationships clearly indicate that for a given reduced damping, mass parameter and a set of static force coefficients, multiple solutions are possible. A motion with amplitude a = a0 is stable if [dK0 (a)/da]a = a 0 Q 0, in which K0 (a) is defined by equation (10). In the opposite case, the motion is unstable. The condition for the self-excitation of the cylinder from its undisturbed position is [dK0 (a)/da]a = 0 q 0. The reduced velocity at the galloping onset U0 is obtained by putting a = 0 in equation (11) and can be written as U0 = −2b/(nh0 ) in which, as described earlier, the coefficient h0 itself generally varies with U0 . It is seen that an exact value of U0 cannot easily be determined. An approximate solution for U0 , however, can be obtained by making use of the fact that at the galloping onset, the deviation of the cylinder mean position of motion from its undisturbed one is negligible: that is, p 1 0 at U = U0 . With this, the coefficient h0 is simplified to h0 1 (B0 − C0 ) + (B2 − C2 )y02 + · · · + [B(2I) − C(2I) ]y0(2I),

(14)

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. 

Figure 4. Two possible profiles of the variation of the reduced amplitude with the reduced velocity. ——, Stable motion; – – – –, unstable motion.

which is a constant value and is a function of both the lift and the drag coefficients. The corresponding reduced frequency is V0 = (1 + z1 nU02 )1/2 , where the coefficient z1 is z1 1 −1A1 − 3A3 y02 − 5A5 y04 · · · − (2I − 1)A(2I − 1) y0(2I − 2) − (2I + 1)A(2I + 1) y0(2I),

(15)

which is also a constant value, being a function only of the lift coefficients. The galloping instability begins when U q U0 = −2b/(nh0 ), h0 Q 0 and the expression [dK0 (a)/da]a = 0 1 −(nvn U/2){(2b/nU) + (B0 − C0 ) + (B2 − C2 )y02 + · · · + [B(2I) − C(2I) ]y0(2l)}

(16)

is positive. In this situation, the equilibrium position becomes unstable and strong motion builds up. When h0 q 0 this position remains stable for all flow velocities. It should be noted that the closed form expressions (10)–(16) are valid only when the cylinder mass ratio is very small. Otherwise, equation (7) must be solved numerically [21]. As mentioned earlier, for large values of the reduced velocity U, the cylinder mean position p takes a constant value. Thus the variations of the coefficients h0 , h1 , . . . , h(2I) and also the reduced amplitude a with U asymptotically approach constant values. Two possible profiles of the variation of a with U are depicted in Figure 4. In case (a), a stable motion grows smoothly from the cylinder undisturbed position as the flow velocity increases beyond U0 . In case (b), in the reduced velocity range U1 Q U Q U0 , an initial displacement of the cylinder greater than the unstable amplitude will trigger the galloping instability; the required displacement decreases with increased flow velocity. For U e U0 , the galloping begins spontaneously. Both profiles have been observed experimentally: for the former see references [14, 12], and for the latter reference [22]. More complex profiles are possible. 3. APPLICATION OF THE MODEL

Shown in Figure 5(a) is the variation of the measured lift coefficient on a smooth circular cylinder in a shear free flow as a function of the spacing ratio y0 . In drawing these data the positive direction of the lift force was assumed away from the wall. These are taken from the work of Fredsoe and Hansen [23]. The free stream Reynolds number of the data is 2 × 104 Q Re Q 3 × 104 and the cylinder has an aspect ratio of 22·4. Unfortunately, these researchers did not measure the corresponding drag coefficient. Thus the measurements of

    

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Go¨kten [24], which are plotted in Figure 5(b), and have a Reynolds number of 90 000, but a smaller aspect ratio of 6·7 is employed. This undermines the prediction, as ideally, in order to obtain the best result, both the static lift and drag coefficients have to be measured in situ: i.e., in the very same facility in which the cylinder motion is measured [25]. The lift and the drag coefficients were approximated by polynomials with degree 19 and 2, respectively. The angle u was calculated by using the measured lift coefficient and the fitted curve of the drag coefficient. The variation of u with y0 is approximated by a straight line, as depicted in Figure 5(c). In the curve fitting procedure, in order to improve the predictions, the polynomials were fitted to the data with small spacings where the galloping actually occurred. The constant coefficients of the fitted curves are presented in Table 1. The heavy lines in Figure 5 represent the fitted polynomials. Indicated in Figure 6 is a comparison between the galloping observations of Sumer et al. [14] and the predicted values. The data in the amplitude domain are shown in Figure 6(a), with the reduced amplitude, a, versus the reduced velocity U. The corresponding data in

Figure 5. The variation of static force data with spacing. (a) Lift coefficient; (b) drag coefficient; (c) stagnation angle coefficient; u° = −242·86y0 + 172·43.

. 

442

T 1 Coefficients of the fitted polynomials (I = 9). (a) Lift coefficients; (b) drag coefficients; (c) dCL /dt coefficients (a)

(b)

(c)

A1 = 14·21058 A3 = −92·16156 A5 = 251·46527 A7 = −373·31030 A9 = 332·66186 A11 = −185·54611 A13 = 65·19551 A15 = −14·00227 A17 = 1·67729 A19 = −0.08579

B0 = 0·67 B2 = 0·76 B4 = 0·00 B6 = 0·00 B8 = 0·00 B10 = 0·00 B12 = 0·00 B14 = 0·00 B16 = 0·00 B18 = 0·00

C0 = −3·35257 C2 = 65·22855 C4 = −296·62972 C6 = 616·50223 C8 = −706·33733 C10 = 481·51651 C12 = −199·95289 C14 = 49·55145 C16 = −6·72703 C18 = 0·38455

frequency domain are plotted in Figure 6(b) with the reduced frequency V against U. The variation of the mean position of the galloping cylinder p with U is shown in Figure 6(c). Superimposed on these graphs are the corresponding theoretical curves which were calculated by using equation (6) and solving equation (5) numerically, and were found to be all stable. It is seen that the onset velocity for galloping is reasonably well predicted. 4. CONCLUDING REMARKS

The quasi-steady linearized theory successfully predicts many features of the interference galloping of a circular cylinder in the neighbourhood of a plane boundary. Closed form

Figure 6. A comparison of theory and experiments (n = 0·27, b = 0·06) [15].

    

443

expressions have been presented for the variation of the reduced amplitude with the reduced velocity, the reduced frequency with the reduced amplitude, and the relative mean position of the cylinder with the free stram dynamic pressure. The variation of this position with the flow velocity and also the first variation both approach asymptotically constant values. The second and the third variations are functions only of the lift forces. The theory clearly shows that the onset flow velocity for galloping increases with increased cylinder mass or structural damping, and a simple formula is presented for this. The theory clearly shows that the galloping instability only occurs when the gap between the cylinder and the wall is tiny.

ACKNOWLEDGMENT

The opinions and ideas presented in this paper are those of the author, and do not necessarily represent the views of any company or organization.

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APPENDIX: NOMENCLATURE A1 , A3 , . . . , A(2I + 1) a a0 a˙ B0 , B2 , . . . , B2I CDa CD CL CLa C0 , C2 , . . . , C2I c D Fy f(Z, Z ) I i K0 (a) k L m n O O' n1 n2 p Re r S0 Sg t

constant lift coefficients reduced (relative) amplitude defined as cross flow vibration amplitude divided by the cylinder diameter a constant =da/dt, derivative of reduced amplitude with respect to time constant drag coefficients drag coefficient at angle of flow attack a drag coefficient in the dirction of free stream flow lift coefficient normal to free stream flow direction lift coefficient at angle of flow attack a constant coefficients of polynomial approximation to dCL /du viscous damping coefficient of cylinder cylinder diameter transverse component of flow-induced force on cylinder a function of Z and Z an integer a counter a function of variable a stiffness coefficient of cylinder cylinder length total vibrating mass of cylinder =rD 2L/(2m), cylinder mass parameter position of cylinder centre when it is stationary position of cylinder centre when it is oscillating positive integer positive integer relative mean position of galloping cylinder Reynolds number based on free stream flow velocity and the cylinder diameter transverse distance between the cylinder centre and the plane boundary cylinder front stagnation point when free stream flow velocity is parallel to wall cylinder front stagnation point when free stream flow velocity is at angle g to wall. time

     U U0 V Vrel Y Y Y y y0 y˙ y¨ Z a b z1 , z2 , . . . , z(2I + 1) h0 , h1 , . . . , h(2I) p r

=V/(vn D), reduced velocity reduced velocity at galloping onset time-mean free stream flow velocity relative flow velocity =y/D, reduced (relative) amplitude, =y˙ /D, reduced velocity =y¨ /D, reduced acceleration cylinder displacement measured from its undisturbed position =r/D, cylinder spacing ratio cylinder vibration velocity cylinder vibration acceleration a variable as defined in text angle of attack of relative flow to cylinder =c/(2mvn ), cylinder reduced (relative) structural damping coefficients as defined in text coefficients as defined in text 3·1415926 fluid density

s

summation sign

f c V vc vn u g

vibration phase angle a variable as defined in text =vc /vn , cylinder reduced (relative) vibration frequency circular motion frequency of cylinder circular natural frequency of cylinder stagnation angle angle of incident flow with respect to wall

445