On collapse and flutter phenomena in thin tubes conveying fluid

Journal of Sound and Vibration (1977) 50(l), 117-132

ON COLLAPSE AND FLUTTER PHENOMENA IN THIN TUBES CONVEYING FLUID D. S. WEAVER Department of Mechanical Engineering, McMaster University, Hamilton, Ontario, CaMda AND

M. P.

PAIDOUSSIS

Deportment of Mechanical Engineering, McGill University, Montreal, P.Q., Conaria (Received 24 June 1976, and in revised form 9 September 1976)

The stability hehaviour of thin-walled tubes conveying fluid is examined with an emphasis on the effects of tube flattening. TWQ simplified theoretical models are developed which represent a flattened tube as two paralIe1flat plates. The first model is in terms of standing waves on finite length plates whereas the second model is in terms of travelling waves on in6niteIy long, kite width plates. The fluid forces are determined by using potential flow theory. Experiments were conducted tith three different tubes, one of which was initially fiat. The experimental observations agree with the predictions of the 6rst theoretical model, at least qualitatively.

1. INTRODUCTION

The dynamics and stability of pipes conveying fluid has been studied extensively as indicated in the historical review given by Paidoussis and Issid [I]. Applications vary from above ground oil pipelines, hydraulic penstocks and rocket fuel lines to human lung airways. Most analyses have been concerned with beam-like modes, with a linearized theory and plug flow being used. More recently, shell modes have been studied, with use of potential flow theory to model the fluid forces [24]. Through all this work, a clear picture of tube behaviour has evolved, at least for flow velocities up to the critical divergence velocity. When both ends of the pipe are fixed (either clamped or pinned) but are free to move axially, the behaviour is as shown schematically in Figure 1. As the flow velocity increases, the natural frequencies of the pipe gradually reduce from that at zero flow until buckling in the lowest axial mode occurs. This instability is referred to as “static divergence” and the critical velocity is denoted by V, in the figure. Upon further increase in flow velocity, the lowest mode frequency reappears and increases until it “coalesces” with the second mode to produce “coupled mode flutter”. The buckled mode shape always involves the first axial mode but may be associated with any number of circumferential waves depending on tube thickness and length; the shorter and thinner the shell, the higher the number of circumferential waves. Long, relatively thick tubes buckle in the first circumferential mode: i.e., like beams. The behaviour of cantilevered tubes is fundamentally similar except that the critical velocity is associated with flutter-no buckling occurs. Experimental corroboration for the frequency dependence on velocity and the critical divergence velocity has generally been quite good, at least for pipes without axial constraint. Several recent papers [5, 61 have indicated the importance of axial restraint in preventing 117

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D. S. WEAVERAND M. P. PAIDDUSSIS

Coupled mode flutter 1st mode 6

a \r

Diverge”ce Flow

--II

1

I.

velocity

Figure 1. General stability behaviour of fluid conveying tubes.

buckling. However, the non-linear analysis of Liu and Mote [5] failed to improve significantly the agreement between their theory and experiments. The discrepancy is apparently associated with stiff pipes for which extremely high flow velocities are required to produce buckling (in excess of 150 ft/s (46 m/s)). The authors have found no such difficulties in previous experiments with thin, low modulus tubes (see, for example, reference [4]) and certainly Dodds and Runyan [7] found excellent agreement for an axially unconstrained aluminum pipe which buckled at 127 ft/s (38.7 m/s). On the other hand, flutter predictions for fixed-ended tubes have not been very satisfactory, whether axially constrained or not. In experiments with very thin tubes, shell flutter usually appears immediately after divergence, well below the flow velocities expected. Furthermore, the flutter mode shape appears to be more like the flapping of two adjacent parallel plates as shown in Figure 2(a) than the classical shell mode predicted by the theory (Figure 2(b)). This is especially true when the tubes used in experimentation have an initial curvature or are ovalled in cross-section due to coil storage. In the latter case, the critical buckling velocity is also apparently reduced. These observations have led to speculation [4] that the “flapping” behaviour is an entirely different phenomenon than that predicted by previous theories. While the authors’ experiments on thin-walled low modulus elastic tubes leave no doubt that flutter occurs, it should be noted that the possibility of post-divergence flutter has been the subject of considerable controversy (see, for example, the paper by Kornecki [8] and the references in that paper).

--\ 2% I\

‘\

\

\

\

’

:

‘.__

(a)

(b)

Figure 2. Unstable flutter mode shapes. (a) Flapping mode; (b) classical shell mode (n = 2). This paper is concerned exclusively with collapse due to static divergence and subsequent flutter phenomena. Two simplified mathematical models have been developed to predict the behaviour of flattened thin-walled tubes conveying fluid, at least qualitatively. In addition, new experiments have been conducted with thin elastic tubes, with the expressed purpose of examining more closely the behaviour at flow velocities equal to or exceeding the critical.

FLUTTER

PHENOMENA

119

IN THIN TUBES

Beyond its intrinsic interest, this work has applications in the field of bio-mechanics. Apparently, human breathing capacity is limited by airway collapse and flutter. Furthermore, the upper airways may be more oval than round, especially in disease, thereby increasing an individual’s tendency towards breathlessness or pulmonary insufficiency [9]. 2. THEORETICAL

2.1.

SIMPLIFIED MODEL-PARALLEL

ANALYSIS

PLATES ON PERIODIC

SUPPORTS

Experimental observations, as noted above, indicate that the mode of flutter immediately following divergence resembles the flapping of two parallel plates as shown schematically in Figure 2(a). This behaviour is quite distinct from that of the classical shell modes (Figure 2(b)), as predicted by previous theories [2, 31. These observations suggest the use of a simple model based on the small deformations of two parallel flat plates on periodic supports as shown in Figure 3. Such a simplification may be further rationalized by observing that the mathematical models for plates and shells in a subsonic flow are fundamentally similar and that their behaviour is qualitatively the same [lo]. This mathematical class of problems has been treated more generally by Huseyin and Plaut

Ull.

Figure

3. Parallel plate model for flattened tube.

There is no doubt that such a simple model cannot represent some of the important characteristics of the actual tube-notably the large deformation, non-linear elastic response. However, it is felt that the principal features of the observed phenomena should be predicted, at least qualitatively. The analysis below follows the formulation given by Flax [12]. That paper dealt with a single two-dimensional infinitely long panel on periodic supports exposed to a flowing fluid of infinite extent but few details were given. The equation of motion for the lower plate in Figure 3 is given by (a list of symbols is given in Appendix B) 9v4w+p,ha2w~at2+p,haw~at+Pd=o,

(1)

with boundary conditions W(0) = W(Z) = 0,

a2 w(oya2

= a2 w(lya9

= 0.

(2)

The dynamic pressure on the plate, Pd, is determined from the unsteady Bernoulli equation : (3)

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D. S. WEAVER

AND M. P. PAIDOUSSIS

where C#J is the perturbation velocity potential. If the fluid is assumed to be inviscid, incompressible and irrotational, the velocity potential must satisfy Laplace’s equation, v24=0,

(4)

subject to the appropriate boundary conditions. This boundary value problem may be solved by using separation of variables or by integrating known solutions as outlined below. The cylindrical motion of a surface in the plane z = q may be represented by a source solution of Laplace’s equation of strength H(& q, t) integrated along the surface :

(5) It can be shown (see reference [ 131)that the source strength is equal to twice the perturbation velocity, w (c, q, t), in the z-direction. This, in turn, may be expressed in terms of the displacement of the surface since the latter is a material surface through which no fluid may pass. With the upper plate denoted by a subscript u and the lower plate by a subscript I, the source strengths are given by

(6) (7) Hence, the perturbation velocity potential for the flow between the two flexible plates in terms of the plate’s displacements is given by

This expression indicates that there are two cases of relative plate motion which are of interest and quite easy to handle : (a) when the plates are moving 180”out-of-phase with each other, W y = - W,, and (b), when the plates are moving in phase with each other, W, = W,. Intuitively, one would guess that these should be the significant modes as well. Upon assuming, then, that W, = + W, and using this, together with equations (8) and (3), the dynamic pressure on the plate in terms of plate motion is

(9) where the plus or minus sign refers to case (a) or (b), respectively. Assuming a solution of the form llZX

W=

2 A,sin-j--e““’ n-1

(10)

FLUTTER PHENOMENA IN THIN TUBES

121

and substituting into equation (9) leads to integrals of the type:

1 x-r x-r+(X-5)2+b2

d5.

(12)

These integrals may be evaluated in closed form (see Appendix A). Substituting the results into the equation of motion (1) and applying the method of Galerkin (multiplying by sinqrrx/l and integrating over the length of the plate between supports) then results in the equations

where a,, is the Kronecker delta. This becomes, upon defining the reference frequency o. = (n/Z)’(g/p,,, h)l12 and the following dimensionless variables : mass ratio, p = p. l/p,,,h ; complex frequency, C = o/oo; flow velocity, V = U/w, I; damping coefficient, g = E/o~; and separation ratio, jI = b/l,

The form of these equations is the same as that found by Flax [12] except for the 1 + e-@ term which accounts for the proximity of the other plate. Since it is known that this system loses stability first by static divergence and that this form of instability is not affected by damping, the damping coefficient in equation (14) may be set equal to zero. Upon taking a two term approximation and setting the determinant of the coefficients, A,, equal to zero, the following equation is obtained : (1%

where a1 = 1 - np V2(1 + eex4), a2 = 16 - 211~V2(1 f e-t=b), #I1= 1 + (p/n) (1 f e+),

/I2 = 1 + b/n) (1 + ev2nB),

y1 = (8c(V/3~) (1 + emZX6), yz = (16/3) (pV/rr)(l + e-*#). By setting the complex frequency, C, equal to zero, the divergence stability boundary is seen to occur when a1 = 0 and in the mode for which the plates are 180” out of phase (+ve sign): Vcr = [l/ad1 + e-‘8)]1’2.

(16) Several significant conclusions may be drawn from this equation. When the separation ratio is large, /I --f co, the critical divergence velocity is that given by Flax [12]. This result has been corroborated by Weaver and Unny [14] and Ellen [15] using different solutions. As the separation between the plates is reduced to below about one axial wavelength, the interaction between the plates reduces the critical flow velocity. Furthermore, there appears to be a limit to the reduction in divergence velocity to about (l/d) V,,_ or about 29 “/d.It is also interesting to note that if the in-phase plate mode is considered, the critical divergence

122

D. S. WEAVER

AND M. P. PAIDOUSSIS

Figure 4. Effect of separation on critical flow velocity-first theoretical model. Mass ratio. p = 10.

velocity increases without limit as the plates come very close together: i.e., thein-phasemodeis stable. These conclusions are independent of mass ratio and are illustrated in Figure 4. In terms of the tube being modelled then, it is expected that the tube will collapse with adjacent sides of the tube moving towards one another. This suggests a possible explanation for the non-classical “flapping mode” flutter observed in the experiments. When the tube collapses, there will be a sudden reduction in discharge due to an increase in pressure drop across the constriction. The disappearance of the fluid forces associated with the high velocity flow coupled with a possible surge pressure allows the constriction to open up and the flow to be re-established. The cycle then repeats itself. Note that this flutter behaviour would not be predicted theoretically because viscosity effects have been neglected. Note also that any initial constriction or like imperfection in the tube has the effect of lowering the critical flow velocity. If equation (15) is solved for the complex frequency, C, one sees that flutter may occur if the discriminant of C2 is imaginary. After some rearrangement this requirement may be shown to be (rr ~2)’+ (ai 82 - a2 A)’ + 2~~ 72b 19~ + a2 PI) < 0. (17) This can only happen for flow velocities exceeding V,, : i.e., a1 < 0. Since the /I’s and y’s are positive and both a’s become negative for sufficiently high flow velocities, coupled mode flutter is a clear possibility. Solution of equation (15) for complex frequency as a function of flow velocity yields results similar to those reported by Weaver and Unny [14] and further details here are unnecessary. Suffice it to say that coupled mode flutter is predicted at some flow velocity considerably in excess of the critical divergence velocity. This is also in agreement with previous analyses for thin tubes conveying fluid [2,3]. Another interesting feature may be observed by setting the flow velocity in equation (15) equal to zero. This give6 the zero-flow natural frequencies of the plate: C, ={l/[l +@/a)(1 +e-nB)n1’2, C, = 4{1/[1 + @/a)(1 + e-2n8)n1’2.

(18) (1%

123

FLU’lTER PHENOMENA IN THIN TUBES

It appears that when the plates are moving 180” out-of-phase, the close proximity of the plates increases the added mass effect which reduces the natural frequencies. On the other hand, when the plates are in-phase, the natural frequencies of the plates increase as the mass of fluid between them reduces and the frequencies approach their in vucuo values (C, -+ 1, c, --f 4). 3. TRAVELLING

WAVE SOLUTION-FINITE

WIDTH PARALLEL PLATES

Another very simple but perhaps more realistic model for the collapsing tube consists of two infinitely long parallel plates of finite width with constrained edges as shown in Figure 5. Dowel1 [16] considered such a single panel exposed to a flowing fluid of infinite extent above the surface of the panel. He found that when damping was present, static divergence occurred first, followed immediately by travelling wave flutter. While there is some question about the applicability of the results for an infinitely long model to a long but finite plate, such an approach offers several potential advantages in the problem considered here. Firstly, the constrained edges parallel to the flow more reasonably represent the tube being modelled. Secondly, the short wavelength compared with the tube length and the spatial wave variation more closely approximates the experimental observations of tube instability.

Flattened

tube

Figure 5. Fixed edge model for flattened tube.

The equation of motion is the same as considered in the previous model and is given by equation (1). In this case, the boundary conditions are ~(*,~,*)=~(~,-~,I)=o. ~(x,~,r)=~(x,-~,r)=o,

(20)

A solution is assumed in the form of a travelling wave with a transverse plate deflection which satisfies the boundary conditions (20). As above, the deflection of the upper plate is designated with a subscript u, while I is used for the lower one, and the in-phase and 180” out-of-phase solutions will be examined : w,(x,y,

t) = *W,(X, y, t) = A cos

nY

-fz*(er-r)‘~,

d

d

2

2

--dy<-.

124

D. S. WEAVER

AND M. P. PAIDOUSSIS

The dynamic fluid pressure on the lower plate is found as in the previous model by solving Laplace’s equation (4) subject to the appropriate boundary conditions (6) and (7) and then employing the unsteady Bernoulli equation (3) at the surface of the lower plate. Weaver and Unny [ 171studied the effect of a liquid free surface on the stability of a panel similar to that considered here. For an approximate solution presented in that paper a free surface wave was assumed of the same shape as that of the panel but multiplied by an amplitude scale factor, x, which was determined by using hydrodynamic theory. If this factor is set equal to upity, the solution will be identical to that required for the present model. Thus, the presence of the second plate does not alter the essential character of the solution and only the final results will be presented below. The interested reader is referred to reference [ 171 for the details. Fourier integral theory was used together with a Galerkin solution in the spanwise mode : i.e., multiply through by cos ny/d and integrate over the width of the panel. Note that some of the variables have been changed in order to avoid confusion and facilitate comparison with the results of the first model developed. The characteristic equation thus derived is (1 +jiF)CZ+{(igl/2d)-2jiP~C+~~*F-L/4=0,

(22) where the reference wavespeed is co = 2n@/p,hd *) I/*, the mass ratio is ii = pod/p,h, the .wave speed is c = c/c,, the flow velocity is P = U/co, the damping coefficient is g = .sd/nc, and L = (2d/1)* + 2 + (1/2d)*. The function F is given by the sum of two integrals which were evaluated numerically : F= Fl rfrF2,

(23)

where

s 03

Fl =fi

cos2 k

2d o tanh{[(kf/d)* + n*]l’* (26/1)}[(n/2)* - k*]* [(kZ/d)* + x*]~/~ m

F,?

cos* k

dk,

dk.

2d os sinh {[(kl/d)* + x*]~‘~@b/l)} [(n/2)* - k*]* [(kZ/d)* + TC*]~/*

The plus or minus sign in equation (23) corresponds to the 180” out-of-phase and in-phase modes, respectively. Clearly, it is integrals F1 and F2 which account for the close proximity of the parallel plate. If the plates are far apart, tanh{ } -+ 1, sinh{ } --f co and the solution reduces to that found by Dowel1 [16]. The characteristic equation (22) is quadratic in the wavespeed c and is easily solved. The interested reader is referred to Dowell’s paper for details of finding the square root of a complex function. Examination of the solution indicates that stability is lost by static divergence (c = 0) and that the stability boundary is given by VD= $(L/jiF)‘/‘. (24) As expected with static instabilities, the result is independent of the damping factor. In contrast to the previous model, travelling wave flutter is predicted to occur immediately following divergence. However, Dowel1 [16] has shown that the rate of increase in wave amplitude is very small until the flow velocity reaches the flutter boundary predicted for zero damping (g = 0): i.e., when VP> $[L( 1 + jiF)/jiF]“‘. (25) Comparison of equation (25) with equation (24) shows that the flutter boundary occurs for flow velocities somewhat in excess of the divergence boundary. Thus, the behaviour of the two models presented is qualitatively the samestatic divergence followed by flutter at some higher flow velocity.

FLUTER

PHENOMENA IN THIN TUBES

125

The divergence velocity predicted by equation (24) is dependent on the wavelength, 1, and has a minimum value when the wavelength is between two and two and one half times the plate width, 1 x 2d. This “critical” divergence velocity is plotted as a function of plate separation, /I = b/Z, in Figure 6. It is seen that these results confirm the conclusions drawn above regarding the effect of the adjacent plate. There is no effect until the separation is less than a wavelength and, for separation ratios less than I, the critical velocity is reduced for the 180 out-of-phase mode. The in-phase mode is stabilized.

Figure 6. Effect of separation on critical flow velocity-travelling

wave model. Mass ratio, p = 4.0.

On the other hand, equation (24) shows an important difference with its counterpart, equation (16), of the first model developed. The travelling wave model indicates that, as the plates come very close together, the critical velocity approaches zero. There is no limit to the percentage reduction in critical divergence velocity as found in the previous case. The reason for this significant difference is not clear. The divergence velocities given by equations (16) and (24) for the two different models afford an interesting comparison when cast in terms of dimensional variables : equation (16): UC,= {x&/(1

+ e-nB)1/2}(B/p, 13)1’2,

(26)

equation (24): UC,= nm(9/p,d3)1’2.

(27) Upon setting 1= 2d and assuming large separation, the travelling wave, finite width plate solution predicts critical velocities about 6-3 times those for the infinitely wide standing wave solution. This difference is no doubt in large part due to the additional stiffness provided by the edge constraints of the former configuration. However, it is questionable whether this can account for all of the discrepancy. 4. EXPERIMENTS

The purpose of the experimental programme was two-fold: (i) to observe carefully the stability behaviour of thin, not necessarily circular cylindrical, tubes conveying fluid; (ii) to

126

D. S. WEAVER

AND M. P. PAIDOUSSIS

ascertain whether the results obtained by the theory are, at least qualitatively, in agreement with experimental observations. Thus, the experiments should bridge the gap between the theoretical models presented here, which involve essentially parallel flat plates, and the available experimental observations which relate exclusively to circular cylindrical tubes. To this end, experiments were conducted both with round (i.e., circular cylindrical) tubes and with flattened (ovalled) tubes. The latter, when there is no internal flow, are almost completely collapsed; with flow, they take various quasi-elliptical cross-sectional shapes depending on internal pressure. Clearly, the flattened tubes are relatively close to the analytical models presented here, at least at low flow, when internal pressure is quite low. Two types of round tubes were used: (i) colostomy (“Penrose”) tubes of 0.67 cm diameter, D, and wallthickness, h = O-026cm, and (ii) specially cast silicon rubber (“silastic E”) tubes of D = 1.29 cm and h = O-046 cm. The flattened tubes were colostomy tubes with equivalent diameter D = I.25 cm and h = 0.028 cm. The tubes were clamped at both ends, the free length being typically 15 cm. First described is the dynamical behaviour of a flattened tube with increasing flow-as shown in Figure 7. At low flow rates the system is quite stable, the only effect of increasing flow being in gradually enlarging the cross-sectional area. However, at a certain flow velocity self-excited flutter of the tube occurs in the so-called “flapping” mode as depicted in Figure 2(a). This phenomenon may be precipitated at slightly lower flow if the tube is pinched. As the flow rate is increased further, flapping persists up to a certain point, at which there is a well-defined, sharp transition to classical flutter (Figure 2(b)), with an attendant increase in frequency; this is accompanied by the emission of a clear sound. The amplitude of this oscillation is much larger than that of flapping. Upon decreasing flow, classical flutter gives way to flapping with little or no hysteresis. Similar observations were made with cantilevered flattened tubes.

-1 =OzooClassical

flutter

50I

o

Stable 4x10-4

r' I 6~10.~ *110-4 Flaw rate (m’/s)

I( 10-a

Figure 7. Unstable modes and frequency as a function of flow rate ; I = 152 mm, ID = 12 mm, h = 0.28 mm, E = 2.59MPa, p. = 1.206 kg/m3, p,,,= 1121 kg/m3.

It is noted that, at the maximum flow rate at which tipping (rather than classical flutter) may be sustained, the tube is considerably less flat than at the minimum flow at which flapping becomes possible (aide Figure 7); in terms of the theoretical models discussed here, this corresponds broadly to saying that as the two “flat” sides of the tube come closer together, the critical flow for flapping is reduced-which agrees with theory.

127

FLUTTER PHENOMENA IN THIN TUBES

The same experimental procedure was then followed with round tubes. If the tube was initially perfectly round, then the only obvious form of flutter observed was that in the classical mode. In this case, however, flutter could be induced at much lower flow velocities by pinching the tube. It is clear that pinching the tube amounts to momentarily flattening part of the tube. In that sense, the occurrence of induced flutter in the round tubes may be viewed as a form of flapping: i.e., 180” out-of-phase motion of adjacent sides of the tube. It became evident, therefore, that the flapping type of flutter deserves more careful investigation. In the experiments described above with flattened tubes, the cross-sectional shape of the tube is itself a function of flow, so that the separation between the two “flat” sides of the tube could not be controlled independently. To remedy this situation the tube was constricted at a point, by means of two movable flat plates. Constriction was normally at a point l/3 or l/4 of the tube length from the upstream end, but the location of this point apparently has no appreciable effect on the tube’s behaviour. The experimental procedure was then as follows : at successive openings of the flat plates, the flow was increased from zero, and the flow rate was noted where flapping was precipitated. Experiments in which this procedure was used were conducted with both round and normally flattened tubes. It was found that the smaller the opening at the constriction, the lower is the critical flow, as in the previous experiments. Flapping seemed to be generated at the constriction point and travel downstream. The results are shown in Figure 8, where b is the internal gap between the two flattened surfaces of the tube at the constriction, and D is the tube diameter. The critical velocity for flapping, U,was based on an equivalent round crosssection in all cases and non-dimensionalized with respect to a parameter similar to that derived theoretically as seen in equations (26) and (27), u = U/(9/p0 D3)“‘. Note that this implies that the instability is initiated by static divergence. The choice of flow velocity based on the round tube cross-sectional area was a matter of convenience. Clearly, the local flow velocity at the constriction will be higher than that so calculated, especially for small b/D,

1

0

0.1

0.2

0.4

0.5

Dimensionless

gap, b/D

0.3

O-6

0.7

I

Q6

Figure 8. Experimental results for critical flow velocity as a function of dimensionless gap and comparison with first theory. ?? , 1a23 cm, flattened tube; o, 1a27 cm, round tube; a, 0.66 cm, round tube.

128

D. S. WEAVER AND M. P. PAIDOUSSIS

since the cross-sectional area reduces as the sides of the tube come closer together. The actual shape formed by the tube makes calculation of the cross-sectional area quite difficult. While use of this round tube flow velocity exaggerates the effect of constricting somewhat, it does not alter the essential features of the results-constricting the tube significantly reduces the critical flow velocity and the instability observed is a flapping mode of flutter. It is seen that the results for the flattened tube (full line) are qualitatively similar to those predicted by the first theory; as the gap widens the critical flow for flapping reaches an asymptotic value, in this case YN 2. The actual reduction% flow velocity based on crosssectional area is difficult to determine but it is less than the 50% indicated. At yet higher velocities, when the opening in the constricting plates becomes wide enough that the sides of the tube are actually free (dashed line), the critical velocity increases rather sharply as the tube becomes progressively rounder. The initially round tubes behave similarly to the flattened one at small b/D; in fact, the agreement is quite remarkable. However, as the separation ratio, b/D, increases the curves for the round tubes diverge from that of the flattened tube, presumably because a flat plate model is quite inappropriate for anything but a very flat tube. If one assumes that the wavelength is approximately twice the plate width as suggested by the second theory, 1 N 2d, and that for a very flattened tube the plate width approaches one half the tube circumference, d N nD/2, the results of the first theory may be compared quantitatively with the experimental data. Equation (16) may thus be written in terms of the critical flapping velocity : u = U,,/(9/po D3)lIz = {l/(1 + e-blD)}1’2. (28) For large separation, u --f 1.0, while as b/D -+ 0, v -+ O-71as shown in Figure 8. Of course, the flat plate model of the flattened tube is only reasonable for small side separation, and in this region, the quantitative agreement is about as good as one could expect. The results of the second theory are not shown in Figure 8 as the agreement is poor and the curve would only serve to confuse the graph. It predicts zero critical flow velocity as b/D --f 0 and this tendency was not observed experimentally. Furthermore, the predicted critical flapping velocity increases rapidly above the experimental data as the gap increases and hasavalueofv=2+1 atb/D=O.l.

5. DISCUSSION The experiments demonstrated two distinct modes of flutter. The so-called “flapping” mode is associated with tubes which are either flat initially or flattened by constricting. The classical mode flutter had two circumferential nodes, a higher frequency and occurred for tubes which were either initially round or rounded out by internal pressure at relatively high flow velocities. The thinness of the tubes used in these experiments permitted dilation due to internal pressure and probably accounts for the fact that no flapping was observed in the absence of flattening. A very thin tube would undoubtedly behave more like a membrane and become unstable in the dilation mode (n = 0). The thicker tubes used in the experiments described in reference [43buckled at divergence. Of course, a tube flattens or ovals considerably as it bends and hence flapping mode flutter was observed at the onset of buckling. It appears that the first theory best describes the flattened tube’s behaviour. Divergence instability with the shell sides moving 180”out-of-phase leads to limit cycle oscillations which cannot be predicted by the theory as suggested above. The critical divergence velocity is reduced a finite amount as the sides of the tube come close together. Although constricting the tubes led to a form of travelling wave as predicted by the second theory, there is no evidence

FLUTTERPHENOMENAIN THIN TUBES

129

that the critical flow velocity approaches zero as the separation between the sides of the tube becomes very small. The inadequacy of the second model must be associated with the assumption of infinite length as, intuitively, it is certainly the most appealing model. The reasonable quantitative agreement between the first theory and experimental observations for critical flow velocity at small separation ratios seems remarkable in view of the crudeness of the model. The fact that u II 1 at small separation ratios for the three quite different tubes tested seems more than fortuitous and suggest the use of equation (28) as a lower bound on the critical flow velocity for flattened tubes. More importantly, it tends to validate the assertion that the flapping instability is initiated by static divergence. In any event, there is no doubt that the principal features of the stability behaviour are predicted by the theory. 6. CONCLUSIONS

Two simple theories have been developed to model the stability behaviour of thin-walled tubes conveying fluid. Both theories are based on the representation of a flattened tube by parallel flat plates. The finite length, standing wave model best describes the behaviour observed in the experiments. It is found that two distinct modes of flutter are possible : a flapping mode flutter resulting from the symmetric divergence of adjacent sides of the tube, and classical shell mode flutter. The flapping mode will normally be the critical one for thin-walled tubes and any constriction of the tube reduces the critical flow velocity. The results are significant in that they bring to light phenomena which apparently have not been previously reported. Furthermore, experimentally observed classical mode flutter is reported for the first time. While the latter is predicted by the present theory, its possibility has been questioned. REFERENCES and N. T. Issm 1974 Journal of Sound and Vibration 33, 267-294. Dynamic stability of pipes conveying fluid. 2. M. P. PAIDOUSSIS and J.-P. DENISE1972 Journal of Soundand Vibration 20,9-26. Plutter of thin cylindrical shells conveying fluid.. 3. D. S. WEAVERand T. E. UNNY 1973 American Society of Mechanical Engineers Journal of 1. M. P. P~USSIS

Applied Mechanics 40,48-52. Dynamic stability of fluid conveying pipes. 4. D. S. WEAVERand B. MYKLATUN 1973 Journal of Sound and Vibration 31, 399410. On the stability of thin pipes with an internal flow. 5. H. LIU and C. D. MOTE 1974 American Society of Mechanical Engineers Journal of Engineering for Industry %, 591-596. Dynamic response of pipes transporting fluids. 6. J. L. HILL and C. G. DAVIS 1974 American Society of Mechanical Engineers Journal of Applied Mechanics 41, Series E, 355-359. The effect of initial forces on the hydroelastic vibration and stability of planar curved tubes. 7. H. L. DODDSand H. L. RUNYAN1965 NASA TN D-2870. Effect of high velocity fluid flow on the bending vibrations and static divergence of a simply supported pipe. 8. A. KORNECKI1974Journal of Sound and Vibration 32, 251-263. Static and dynamic instability of panels and cylindrical shells in subsonic potential flow. 9. E. G. VAIL 1973 Aerospace Medicine 44, 649-665. Pulmonary insticiency with airway flutter,

closure and collapse. 10. D. S. WEAVER1974 Journal of Sound and Vibration 36,435-437. On the non-conservative nature of gyroscopic conservative systems. 11. K. HUSE~N and R. H. PLAUT 1974 Journal of Structural Mechanics 3, 163-178. Transverse

vibrations and stability of systems with gyroscopic forces. 12. A. H. FLAX 1960 Aero a.ndHydroelasticity, Proceedings of the First Symposium on Naval Structural Mechanics. New York : Pergamon Press.

D. S. WEAVER AND M. P. PAIDOUSSIS

130

13. R. L. BISPLINGHOFF,H. ASHLEY and R. L. HALFMAN 1955 Aeroelasticity.

Cambridge, Mass.:

Addison-Wesley. 14. D. S. WEAVER and T. E. UNNY 1970

15. 16. 17.

18.

19.

American Society of Mechanical Engineers Journal of Applied Mechanics 37, 823-827. The hydroelastic stability of a flat plate. C. H. ELLEN 1973 American Society of Mechanical Engineers Journal of Applied Mechanics 40,68-72. The stability of simply supported rectangular surfaces in uniform subsonic flow. E. H. DOWELL 1966 American Institute of Aeronautics and Astronautics Journal 4, 1370-1377. Flutter of infinitely long plates and shells, Part 1: Plate. D. S. WEAVER and T. E. UNNY 1972 American Society of Mechanical Engineers Journal of Appfied Mechanics 39, 53-58. The influence of a free surface on the hydroelastic stability of a flat panel. V. A. DITKIN and A. P. PRUDNIKOV1965 Integral Transforms andOperational Calculus. London: Pergamon Press Ltd. F. B. HILDEBRAND1964 Advanced Calculus for Applications. Englewood Cliffs, N.J. : PrenticeHall. APPENDIX

A : INTEGRAL

EVALUATION

The integrals (11) and (12) may be evaluated by considering the first and second terms in brackets separately. Consider the first term in integral (11) as a function of the parameter x :

Taking gives

the derivative

of F(x) with respect to x and considering

the Cauchy

m elnnC/l pd{. F’(x) = 2 $ x-t -m

principal

value

(A2)

This may be evaluated by using contour integration or by recognizing the integral (A2) as the Hilbert transform of e’nnC’l. Upon using the tables of Ditkin and Prudnikov [ 181, equation (A2) becomes m el”“ell de = -2ni einnxl[. F’(x) = 2 (A3) $ x--r -m This is the solution for the first term of integral (12) and, integrating the solution to the first term of integral (13) :

with respect to x yields

(A4) The integrals breaking each The second differentiating

required in the analysis are determined from equations (A3) and (A4) by side up into real and imaginary parts. term of integral (12) may be derived from the second term of integral (11) by with respect to x as above : G’(x) = 2 f einxC/’(x *,‘i -rO

Upon making

the substitution

b2 de.

x - 5 = -c1, equation (A5) &comes ma einxolr G’(x) = -2@“*/l

-da. s a2 + b2 -m

645)

FLUTTER

PHENOMENA

IN THIN TUBES

131

With a considered as a complex variable, the integrand of equation (A6) has simple poles at u = Gb. This may be integrated by using a contour around the upper half-plane and the Cauchy residue theorem (see, for example, reference [19]) : m s --(o

gda

+ /sda

= 2ni Res(bi).

(A7)

CR

The second integral on the left-hand side of equation (A7) is evaluated around the curved part of the contour of infinite radius and is equal to zero since lim a/(a’ + b2) = 0. (I-OD

(A@

The residue at the pole a = bi inside the contour may be evaluated as Res (bi) = lil& (a - ib)

(A9)

Hence,

This is the second term of integral (12) and the second term of integral (11) is determined by integration of expression (AlO) : G(x) = f einrEI1 In [(T)’

+ (!)2]dr

= _~eJ~.xlle-“W~

= e-“rbl[F(x).

(Al 1)

--9

As above, the integrals required in the analysis may be determined by breaking expressions (AlO) and (Al 1) into their real and imaginary parts.

APPENDIX B: NOMENCLATURE b

distance between plates wave speed co 2749/p,,,hd2)“2 = reference wave speed C o/w0 = dimensionless frequency c c/c0 = dimensionless wave/speed plate width d D tube diameter 9 Eh3/12(1 - Vz)= plate stiffness C

E g

modulus of elasticity E/W~= dimensionless damping coefficient &? &/SC = dimensionless damping coefficient h plate thickness 1 wavelength Pd dynamic pressure time t u flow velocity U/(S/po D3)l12 = dimensionless flow velV ocity for flapping V U/w01 = dimensionless flow velocity r V/c, = dimensionless flow velocity plate deflection b/I = separation ratio

132

D. S. WEAVER

damping coefficient p. I/p, h = mass ratio pod/pmh = mass ratio Poisson’s ratio plate density fluid density perturbation velocity potential frequency (x/l)* (g/pm h)“* = reference frequency

AND M. P. PAIDDUSSIS