Local and global instability of fluid-conveying pipes

Journal of Fluids and Structures (2002) 16(1), 1}14 doi:10.1006/j#s.2001.0405, available online at http://www.idealibrary.com on LOCAL AND GLOBAL INSTABILITY OF FLUID-CONVEYING PIPES ON ELASTIC FOUNDATIONS O. DOARED AND E. DE LANGRE LadHyX, CNRS-E! cole Polytechnique, 91128 Palaiseau, France (Received 6 July 2000, and in "nal form 19 March 2001) We investigate the relationship between the local and global bending motions of #uidconveying pipes on an elastic foundation. The local approach refers to an in"nite pipe without taking into account its "nite ends, while in the global approach we consider a pipe of "nite length with a given set of boundary conditions. Several kinds of propagating disturbances are identi"ed from the dispersion relation, namely evanescent, neutral and unstable waves. As the length of the pipe is increased, the global criterion for instability is found to coincide with local neutrality, whereby a local harmonic forcing only generates neutral waves. For sets of boundary conditions that give rise only to static instabilities, the criterion for global instability of the long pipe is that static neutral waves exist. Conversely, for sets of boundary conditions that allow dynamic instabilities, the criterion for global instability of the long pipe corresponds to that for the existence of neutral waves of "nite nonzero frequency. These results are discussed in relation with the work of Kulikovskii and other similar approaches in hydrodynamic stability theory. ( 2002 Academic Press 1. INTRODUCTION IN THIS PAPER, the in#uence of the local properties of bending waves on the global linear stability of a long #uid-conveying pipe with various boundary conditions is analysed. A schematic view of a cantilevered pipe on an elastic foundation is given in Figure 1. In the literature, considerable attention has been given to the lateral vibrations of pipes containing or surrounded by a moving #uid. Many studies [see PamK doussis (1998)] have sought to establish the instability conditions for #uid-conveying pipes of "nite length in various geometrical and physical con"gurations. It has been demonstrated that these pipes can be destabilized by #utter or buckling, depending on the boundary conditions at the ends, as the #uid velocity is increased. Such instabilities will be referred to as global. It has also been shown (Roth 1964; Stein & Tobriner 1970) that an in"nite pipe modelled with the same local equations can experience instability. In that case the instabilities are referred to as local, since only the local equations are considered, independently of end conditions. Recently, the concepts of absolute and convective instability have been applied to this particular problem (Kulikovskii & Shikina 1988; Triantafyllou 1992; de Langre & Ouvrard 1999). These concepts, "rst introduced in plasma physics (Briggs 1964; Bers 1983) and fruitfully applied to hydrodynamics (Huerre & Rossi 1998) pertain to the long-time impulse response of a spatially homogeneous system of in"nite extent. The objective of the present paper is to determine the respective roles of wave propagation and wave re#ections at the pipe ends on the global instability as the length takes large values, and thus to establish a link between local and global analyses. Such an approach has already been applied to the case of the Ginzburg}Landau equation, a simple substitute for 0889}9746/02/010001#14 $35.00/0 ( 2002 Academic Press 2 O. DOARED AND E. DE LANGRE Figure 1. The cantilevered #uid-conveying pipe on elastic foundation. the Navier}Stokes equations, and the key role of absolute instability has been brought out (Chomaz & Couairon 1999). In the case of #uid-conveying pipes (Benjamin 1960; PamK doussis 1970), the energy exchange rate between the #uid and the structure over one period of oscillation has been expressed in terms of the motion of the pipe ends. Lucey (1998) and Wiplier & Ehrenstein (2000) explored numerically the e!ect of "nite length on the instability of a #exible panel under #ow. In Lee & Mote (1997), wave re#ections at the boundaries were also shown to result in an energy gain that could lead to global instability. More recently, Inada & Hayama (2000) analysed the role of travelling waves in #utter instability caused by leakage #ow. In a more general case, it has been shown by Kulikovskii (1966) that, as the length of a system is increased, the criterion for global stability tends asymptotically to a form that is independent of the boundary conditions. The linearized equation of motion governing the lateral in-plane de#ection >(X, ¹) of a #uid-conveying pipe is (Bourrières 1939; PamK doussis 1998) EI L2> L2> L2> L4> #(oA;2) #(2oA;) #(m#oA) #S>"F(X, ¹), LX2 LXL¹ L¹2 LX4 (1) where EI is the #exural rigidity of the pipe, oA the #uid mass per unit length, ; the plug #ow velocity, S the elastic foundation modulus and F(X, ¹) is the external force per unit length. We only consider here the onset of instabilities, and nonlinear e!ects are therefore neglected in the dynamics of the pipe. If one considers a pipe of length ¸, appropriate nondimensional variables are, following Gregory & PamK doussis (1966), x"X/¸, u";¸(oA/EI)1@2, y">/¸, t"(EI/(oA#m))1@2¹/¸2, b"oA/(oA#m), s"S¸4/EI, f"F¸3/EI. (2) Equation (1) then reads L2y L2y L2y L4y #u2 #2Jbu # #sy"f (x, t). LxLt Lt2 Lx2 Lx4 (3) In the following sections, we consider various boundary conditions at the pipe ends, x"0 and x"1: y"Ly/Lx"0 (clamped end), y"L2y/Lx2"0 (pinned end) or L2y/Lx2" L3y/Lx3"0 (free end). In Section 2, we give the global conditions of stability for various boundary conditions. The stability conditions of the in"nite pipe and the properties of propagating waves are analysed in Section 3. In Section 4, a comparison between the local and global approaches is made regarding the predicted behaviour of the pipe as its length is increased. The results are discussed in Section 5. LOCAL/GLOBAL INSTABILITY OF PIPES 3 Figure 2. Global stability curves of the clamped}free pipe in the (b, u) plane for di!erent values of the elastic foundation sti!ness s. 2. GLOBAL STABILITY Let us "rst analyse the linear stability of a #uid-conveying pipe of "nite length on an elastic foundation. Three di!erent sets of boundary conditions are considered at the upstream and downstream ends: clamped}free (cantilevered pipe), pinned}pinned and clamped}clamped. Free motion of the pipe is assumed, so that f (x, t)"0 in equation (3). Following Lottati & Kornecki (1986), the dynamics of the cantilevered (clamped}free) pipe is analysed by calculating its eigenfrequencies u , j"1,2, N, via a standard Galerkin j method (Gregory & PamK doussis 1966), y(x, t)"+N / (x)e~*uj t, / (x) being the jth eigenj j/1 j mode of the pipe without #ow and elastic foundation (u"0, s"0). These frequencies will be referred to as global. The real part of u is the dimensionless oscillation frequency, while its imaginary part is the temporal growth rate. For u"0, the system described by equation (3) is a neutrally stable beam with real eigenfrequencies. By increasing u, some eigenfrequencies become complex and a positive imaginary part for one of them induces a #utter-type instability. The critical nondimensional velocity u for the onset of instability is plotted in c Figure 2 as a function of b for several values of the elastic foundation sti!ness s. Up to 100 modes have been used at the highest values of u and s. The value of u clearly depends on b, c and the elastic foundation modulus s has a stabilizing e!ect, as noted in Lottati & Kornecki (1986). The stability conditions of the pinned}pinned and clamped}clamped pipes on an elastic foundation have been obtained by Roth (1964) and can be derived from that of a column under compressive load (Timoshenko & Gere 1961). The critical velocity for the pinned} pinned pipe is given by 1@2 s u "Nn 1# , (4) c (Nn)4 A B where N is the smallest integer satisfying N2(N#1)25s/n4. The critical velocity at s"0 is readily found as u "n. Similarly, the clamped}clamped pipe undergoes an instability at c a critical velocity 3s 1@2 u "2n 1# (5) c 16n4 A B for s4(84/11) n4, and u "n c A B 1@2 N4#6N2#1 s # N2#1 n4(N2#1) (6) 4 O. DOARED AND E. DE LANGRE Figure 3. Global stability curves of the pinned}pinned pipe and the clamped}clamped pipe; - - -, clamped} clamped pipe; - ' -, pinned}pinned pipe. otherwise, where N is the smallest integer satisfying N4#2N3#3N2#2N#65s/n4. The pinned}pinned and clamped}clamped pipes are unstable due to buckling and the critical velocity does not depend on b. The critical velocities for static instability of the pinned}pinned and clamped}clamped pipes are plotted as functions of the foundation sti!ness s in Figure 3. 3. LOCAL STABILITY Consider now an in"nite pipe modelled with the same local equation (1) and introduce dimensionless variables that do not refer to the pipe length ¸ but to the local scale g"(EI/S)1@4, associated with the ratio between bending sti!ness and foundation modulus. The length g may be interpreted as the wavelength that appears in the static response of the pipe without #ow A B y(x)"e~x@g sin 2n x g (7) under transverse unit point loading. Appropriate nondimensional variables are now x"X/g, y">/g, v";(oA)1@2/(SEI)1@4, t"(S/(oA#m))1@2¹, b"oA/(oA#m), f"F/(S3EI)1@4. (8) These will be used throughout the remainder of the present paper. Equation (1) now reads L4y L2y L2y L2y #v2 #2 Jbv # #y"f (x, t). Lx4 LxLt Lt2 Lx2 (9) If the pipe displacement is sought in the form y(x, t)"y e*(kx~ut), 0 the linear dispersion relation is readily obtained as D(k, u, v, b) y(u, k)"[k4!v2k2#2Jbvku!u2#1] y(u, k)"/(u, k), (10) (11) where /(u, k) is the Fourier transform in x and t of the forcing function f (x, t). Local properties of bending waves propagating along the pipe are now analysed in terms of the wavenumber k and frequency u through the examination of this dispersion relation. 5 LOCAL/GLOBAL INSTABILITY OF PIPES The system is stable if, for any sinusoidal wave of in"nite extent in the x-direction and associated to a real wavenumber k, the corresponding frequencies given by equation (11) are such that the displacement remains "nite in time. Stability is therefore ensured if for any real wave number k, Im[u(k)]40. This approach is said to be temporal, since it examines the evolution of waves in time. According to Roth (1964) and Stein & Tobriner (1970), the pipe is locally unstable when S 2 . v'v " i 1!b (12) Conversely, the spatial approach refers to the development in space of waves generated by a localized time-harmonic forcing. It is more appropriate for the analysis of the global instability of "nite-length systems, since no assumption of in"nite extent is made on the perturbation. Let us "rst consider the impulse response G(x, t) of the system to an impulsive loading, f (x, t)"d(x)d(t). (13) If v(v , the system is stable and the impulse response is an evanescent wavepacket. In the i case of instability, we may di!erentiate between two cases. The absolute or convective nature of the instability is characterized by the long-time impulse response G(0, t) at the impulse point (Bers 1983). If lim G(0, t)"0, the instability is said to be convective. t?= The ampli"ed wavepacket created by the impulse forcing is convected downstream of the point of excitation. If lim G(0, R)"R the instability is said to be absolute. The t?= wavepacket grows near x"0 and is dominated by the absolute frequency u such that 0 LD(k, u) "0. (14) Im(u )'0, and D(k, u )" 0 0 Lk u0 This saddle point (u , k ) of the dispersion relation must also be associated with a pinching 0 0 in the complex k plane between two branches that correspond to waves found on either side of the impulse perturbation. This gives a criterion of transition between convective and absolute instability (Bers 1983). Let us also analyse the properties of waves generated by forcing at a real frequency u . f This is done by considering the harmonic forcing K f (x, t)"d(x) H(t) e*uf t, (15) where H(t) is the Heaviside unit step function. In the case of absolute instability the transient associated with the switching on of the harmonic forcing at t"0, modelled by H(t), contains all the frequencies, including the absolute frequency u and the system 0 evolution will be dominated by the absolute instability. In the case of convective instability, the switch on transient is advected downstream, and such a forcing generates four waves corresponding to four wavenumber roots of the dispersion relation at u"u . Two f wavenumbers correspond to waves that propagate downstream of the excitation point and will be referred to as k 1 and k 2 . The two other wavenumbers correspond to waves that d d propagate upstream of the excitation point and will be referred to as k 1 and k 2 . u u In the convectively unstable regime, the direction of each of the waves emerging from the point of excitation may be found by calculating the four roots k(u) at a complex frequency u with Im(u) larger than the maximum growth rate q (Bers 1983) de"ned by .!9 q "max Im[u(k)]. .!9 k real (16) 6 O. DOARED AND E. DE LANGRE Complex wavenumbers with a positive imaginary part de"ne waves propagating downstream of the point of excitation, while wavenumbers with a negative imaginary part de"ne waves propagating upstream. By reducing Im(u) to zero and following the evolution of k-branches, we may identify upstream- and downstream-going waves for a real value of u. A downstream wavenumber with a positive imaginary part corresponds to a spatially evanescent wave and with a negative imaginary part to a spatially ampli"ed wave. A wavenumber that is real refers to a spatially neutral wave. If the system is locally stable, the maximum growth rate as de"ned in equation (16) equals zero and the direction of propagation for a real forcing frequency may be found directly by considering the imaginary part of k or the group velocity if k and u are both real. Depending on the parameter values and the forcing frequency, four possible combinations of waves are found by solving equation (11) for a given real value of u, as illustrated in Figure 4: Case 1, four evanescent waves; Case 2, two evanescent waves and two neutral waves, one of each downstream, one of each upstream; Case 3, four neutral waves; Case 4, one ampli"ed and one decaying wave travelling downstream, two neutral waves travelling upstream. Figure 5(a) and 5(b) show the evolution of Im(k) with u for two typical sets of parameters. When b"0)1, v"1 [Figure 5(a)], only Cases 1 and 2 are observed. Conversely, with b"0)94, v"7 [Figure 5(b)], Cases 2}4 are found. For the latter values of b and v, the unforced pipe is convectively unstable. For a given set of physical parameters b and v, the waves generated by forcing are clearly dependent on the forcing frequency. Assuming that u explores all real values, we may now analyse the wave-bearing capabilities of the medium in the (b, v) plane. In the domain of instability, the limit between convective and absolute instability is related to the existence of a triple root of the dispersion relation (Crighton & Oswell 1991; Triantafyllou 1992; de Langre & Ouvrard 1999), which arises at A B 12b 1@4 . v " ac 8/9!b (17) When v(v , the instability is convective and spatially ampli"ed waves exist in a range of ac frequencies. In the domain of stability, v(v , we may also di!erentiate between several subdomains i in the (b, v) plane. As shown in Figure 6, a set of four neutral waves may appear in two distinct frequency ranges. The "rst range, further referred to as &&static range'', is bounded by u"0 and u , while the second, further referred to as &&dynamic range'' is bounded by 1 u and u . The frequencies u , u and u are associated with wavenumbers that are 2 3 1 2 3 double roots of the dispersion relation. Therefore, the emergence of the static range [0, u ] 1 may be related to the existence of a double wavenumber root at u"0 for a critical value v of the reduced velocity. We then have s LD(k, u, b, v ) s D(k, 0, b, v )" s Lk K "0, u/0 (18) which yields v "J2. s (19) Thus, if v'v there exists a range [0, u ] in which all the waves are neutral. At the s 1 emergence of the dynamic range, the two double roots at u and u are identical (see 2 3 Figure 6), forming a triple root of the dispersion relation. This second threshold for the appearance of a dynamic range is therefore also given by equation (17), but extended in the LOCAL/GLOBAL INSTABILITY OF PIPES 7 Figure 4. Schematic view of waves generated by the harmonic forcing: Case 1, evanescent waves; Case 2, neutral and evanescent waves; Case 3, neutral waves; Case 4, neutral, evanescent and ampli"ed waves. Figure 5. Imaginary part of the wavenumbers as functions of the real frequency: - - -, Downstream wavenumbers k 1 and k 2 ; ' ' ' ' ', upstream wavenumbers k 1 and k 2 . (a) b"0)1, v"1, stability, evanescent and neutral d u u d waves only; (b) b"0)94, v"7, convective instability, evanescent, neutral and ampli"ed waves. domain of stable parameters. Thus, if v'v there exists a dynamic range [u , u ] in which ac 2 3 all the waves are neutral, provided that v(v (stability). i Let us summarize the response of the in"nite pipe to localized time-harmonic forcing in terms of the parameters b and v. (a) When A v(J2 and v( B 12b 1@4 , 8/9!b 8 O. DOARED AND E. DE LANGRE Figure 6. The four roots of the dispersion relation as a function of u at b"0)15, v"1)5; (a) real part of k, (b) imaginary part of k; d, location of a second-order root of the dispersion relation. The static and dynamic ranges refer to the values of u at which there are four neutral waves. there exist evanescent waves at all forcing frequencies. This domain is further referred to as that of &&evanescence'' (E). (b) When S v' A B 12b 1@4 2 and v' , 8/9!b 1!b the response to any forcing is dominated by the complex absolute frequency u and no 0 wave direction may be de"ned. This is the domain of &&absolute instability'' (AI). (c) When S v' 2 1!b A and v( B 12b 1@4 , 8/9!b there exists a range of forcing frequencies that generates ampli"ed waves. This is the domain of &&convective instability'' (CI). (d) In the remaining domain of the (b, u) plane, there exist ranges of forcing frequency where four neutral waves are generated, and no ampli"ed waves exist at other frequencies. If parameters are such that neutral waves are generated by frequencies in the static range as de"ned above (see Figure 6), the medium is said to be &&statically neutral'' (SN). In the same manner, a &&dynamically neutral'' (DN) domain refers to the existence of the dynamic range de"ned above. The various domains of the (b, v) plane associated with (a)}(d) are shown in Figure 7. 4. LOCAL NEUTRALITY AS A CRITERION FOR GLOBAL INSTABILITY We consider again a pipe of "nite length ¸. In nondimensional variables pertaining to wave propagation, as de"ned in equation (8), the nondimensional length of the pipe is l"¸ A B S 1@4 . EI (20) In this section we increase the value of l, which is interpreted as increasing the length of the pipe. This is strictly equivalent to increasing the elastic foundation modulus in equation (3) (DoareH & de Langre 2000). 9 LOCAL/GLOBAL INSTABILITY OF PIPES Figure 7. Properties of waves in an in"nite pipe: 0, local stability criterion, equation (12); - - - -, criterion for absolute/convective instability transition in the unstable region and for the existence of four neutral waves in the stable region, equation (17); ' - ' -, criterion for the existence of four neutral waves at u"0, equation (19). E, evanescent; DN, dynamic neutrality; SN, static neutrality; CI, convective instability; AI, absolute instability; grey region, local stability. Let us "rst analyse the behaviour of the pinned}pinned and clamped}clamped pipes. The critical velocities of equations (4)}(6) are "rst rewritten in the local dimensionless variables of equation (8). For the pinned}pinned pipe, they are C A BD Nn l 4 1@2 v" 1# , c l Nn (21) where N is the smallest integer satisfying N2(N#1)25(l/n)4, and for the clamped}clamped pipe C A BD C D 2n n N4#6N2#1#(l/n)4 1@2 3 l 4 1@2 v" or v " 1# c c l l 16 n N2#1 (22) for l44(84/11)n4 or l45(84/11)n4, respectively, using N4#2N3#3N2#2N#65 (l/n)4. In Figure 8(a), the evolution of the critical velocities is plotted in the (b, v) plane as l is varied. When l is increased to in"nity, critical velocities (21) and (22) tend to the same limit, namely (23) v "J2. = As noted in the analysis of buckling under compressive load (Timoshenko & Gere 1961), the critical velocity for global static instability becomes independent of the boundary conditions. The above limit is exactly the lower boundary for static neutrality as sketched in Figure 8(b). Thus, when the pipe length tends to in"nity, the global static instability arises as soon as neutral waves exist at zero frequency (static range) which may be combined into a static global mode of in"nite extent. Indeed, it was already noted by PamK doussis (1998) that for "nite-length pipes with "xed ends, all wavenumbers are real at static instability. Following a similar approach for the cantilevered pipe, the evolutions of the critical velocity, calculated in Section 2, are now rescaled in terms of local dimensionless variables and plotted in the (b, v) plane as l is varied [Figure 9(a)]. The critical velocity appears to tend to a limit that also falls into the neutral domains of Section 3 (Figure 7) but it di!ers 10 O. DOARED AND E. DE LANGRE Figure 8. (a) Critical velocities of the pinned}pinned (- - - -) and the clamped}clamped (' ' ' ' ') pipe for increasing pipe length; the (0) refers to the asymptotic criterion for static instability given by equation (23). (b) Schematic view of the local properties of travelling waves; in the grey region, there exist evanescent waves at u"0. from the limit of the preceding static case. Up to b"2/3, this limit appears to coincide with that of dynamic neutrality in Figure 9(b). (Above this limit, a more complex behaviour seems to arise, and much higher values of l should be explored numerically.) Thus, when the length of the pipe tends to in"nity, the global dynamic instability arises as soon as neutral waves exist at nonzero frequencies (dynamic range) that may be combined into a dynamic global mode of in"nite extent. The fact that only the dynamic range is relevant in this latter criterion is further supported by plotting the critical #utter frequency u at the critical c velocity in comparison with the boundaries u and u of the dynamic range (Figure 10). 2 3 Clearly, the global instability sets in the dynamic range of frequencies. These results may now be analysed following the approach of Kulikovskii, when l tends to in"nity. In this framework, all global eigenfrequencies u of the pipe asymptotically j satisfy the equation (24) Im[k 1 (u )!k 1 (u )]"0, u j d j where k 1 and k 1 are, respectively, the downstream wavenumber with the smallest imagid u nary part and the upstream wavenumber with the greatest imaginary part. Depending on the imaginary part of the u , conclusions regarding asymptotic global stability may be j directly drawn from equation (24). In our case, three di!erent situations arise in the (b, v) plane. In the domain of &&evanescence'' (Figure 7), it may be proven by standard branch analysis in the complex u-plane that no frequency with a positive imaginary part may satisfy equation (24). Hence, stability is ensured. Conversely, in the domains of &&instability (convective or absolute)'', equation (24) implicitly de"nes a curve in the complex u-plane, with a positive imaginary part. Global asymptotic instability arises. Finally, in the domains of &&neutrality'', one may not identify most ampli"ed or less evanescent wavenumbers k 1 , d k 1 for all real frequencies, as some range of forcing frequencies generates four neutral u waves. Although equation (24) may have some meaning in speci"c regions of the u-plane, no general conclusions as to asymptotic global stability may be drawn. Figure 11 summarizes these three cases in the (b, v) plane. Clearly, our results are consistent with the approach of Kulikovskii, as our limit stability cases (Figures 8 and 9), fall into the intermediate domain between the domain of instability and the domain of evanescence of Figure 11. It should be noted that Kulikovskii's approach implies that boundary conditions have a negligible role at in"nite length. It cannot therefore di!erentiate between our two limit curves (cantilevered and "xed ends). LOCAL/GLOBAL INSTABILITY OF PIPES 11 Figure 9. (a) Critical velocities of the cantilevered pipe for increasing pipe lengths (thin lines); the (- - -) line refers to the asymptotic criterion of dynamic instability, equation (23). (b) Schematic view of the local properties of travelling waves; in the grey region, the dynamic range of neutral waves does not exist. Figure 10. Flutter frequency versus b: 0, critical #utter frequency at l"20; shaded domain, dynamic range. Figure 11. Asymptotic behaviour of the #uid-conveying pipe of in"nite length. 12 O. DOARED AND E. DE LANGRE 5. DISCUSSION In the analysis of the preceding section, the respective roles of the ends and bulk of the pipe have been found to strongly depend on the ranges of #ow velocity v and mass ratio b. A "rst type of behaviour has been identi"ed, which is fully consistent with the approach of Kulikovskii: in the domains of local evanescence and local instability, when the pipe length tends to in"nity, the asymptotic behaviour of the pipe is independent of the boundary conditions at the pipe extremities. This may be understood by considering that evanescent (respectively ampli"ed) waves play an increasingly relative role in the energy balance as the pipe length is increased. Any energy input or output associated with end re#ections is ultimately overwhelmed by the stabilizing (respectively destabilizing) e!ect in the bulk of the pipe. The latter therefore controls global stability. Conversely, in the domain of neutrality there may exist sets of neutral waves that convey energy upstream and downstream without interfering. Even with increasing length, boundary conditions e!ectively control global stability. In this second type of behaviour, boundary conditions also play a crucial role through the selection of ranges of frequencies. Some end conditions allow #utter to develop [see PamK doussis (1970) and Lee & Mote (1997)], so that energy transfer between #ow and pipe may occur in the course of pipe motion. In our approach, for a #utter global mode to actually develop in the domain of dynamic neutrality it is necessary that two conditions be satis"ed: (a) the frequency be such that only neutral waves are generated and (b) the corresponding wavenumbers be such that the boundary conditions are satis"ed. While the latter condition selects discrete eigenfrequencies, the former requires that one of them falls into the &&dynamic range'', as de"ned in Section 4. For in"nite pipe length the spectrum of eigenfrequencies becomes continuous and #utter therefore arises as soon as the #ow velocity enters the domain of local &&dynamic neutrality''. The pipe of "nite length may yet be stable in this domain, as it may happen that none of its discrete eigenfrequencies fall into the existing &&dynamic range'' that would allow instability to set in. This is observed in Figure 9, where the stability curve for cantilevered "nite pipes is seen to lie above that for the in"nite pipe. S-shaped irregularities in the stability curves of "nite-length pipes are known to be associated with changes in modal contributions and frequency of instability (Gregory & PamK doussis 1966; PamK doussis 1998). They may also be interpreted as the consequence of discrete frequencies entering or leaving the dynamic range. The same approach may be used when boundary conditions allow only static instability: as the length increases, the spectrum of eigenfrequencies densi"es and the occurrence of one of them being equal to zero increases. In Figure 8, the instability threshold is seen to decrease with increasing pipe length. It should be noted that static neutrality may also be de"ned as marginal local divergence instability (Carpenter & Garrad 1986). The extension of the present approach to other hydroelastic systems modelled by dispersion relations of the same form (Brazier-Smith & Scott 1984; Crighton 1991; Peake 1997; de Langre & Ouvrard 1999; de Langre 2000) may also be considered. It raises the question of the existence of static or dynamic neutrality in parameter space. In our case, we have shown that dynamic neutrality takes place when the criterion for the existence of a third-order root, equation (17), is satis"ed in the domain of local stability. It is an extension into the domain of stability of the criterion for transition between absolute and convective instability (Crighton 1991). Systems such as pipes without foundation (Kulikovskii & Shikina 1988) and plates bounded by uniform #ow (Crighton 1991) are locally unstable at the onset of #ow. Hence, domains of static or dynamic neutrality may not exist, and it is therefore expected that global instability for long systems arises at zero velocity. In the case of membranes with bounded or unbounded #ow (Kelbert & Sazonov 1996; de Langre 2000) where local instability arises at "nite #ow velocity, the criterion of LOCAL/GLOBAL INSTABILITY OF PIPES 13 equation (17) is also found to fall into the domain of instability. Following the present approach, one deduces that no domain of dynamic neutrality may exist. The asymptotic criterion for global instability is probably that of local instability or that of a transition from convective to absolute instability. We may now draw some comparison with the results of similar approaches in the "eld of hydrodynamic stability. The Ginzburg}Landau equation is known to be a simple yet fruitful model of a wide class of open #ows (Huerre & Monkewitz 1990). It has been shown that for such a model the global criterion for instability tends asymptotically to the local criterion for absolute instability when the length of the medium is increased (Chomaz & Couairon 1999). This conclusion, which di!ers from that of the present paper, may be understood by taking into consideration the fact that for the Ginzburg}Landau equation no neutral domain exists. Moreover, when convective instability arises, upstream-going waves are strongly evanescent. Any balance involving the ampli"ed downstream-travelling and damped upstream-travelling waves yields a total decay of the energy. The absence of a domain of neutrality implies that boundary conditions play a relatively minor role. The strong evanescence of upstream waves precludes global instability even in the range of local convective instability. These results are fully consistent with those derived from Kulikovskii's approach. The preceding comparison may be used to shed some light on the very classic problem of destabilization by damping in "nite or in"nite systems (PamK doussis 1970, 1998; Carpenter & Garrad 1986). If damping terms were to be added in our model of the #uid-conveying pipe the domain of neutrality would vanish. Depending on the value of parameters it would be transfered into the stable or unstable domains. When the length of the pipe is increased (or alternatively the damping coe$cient), the criterion for global stability would come closer to the criterion for absolute instability instead of neutrality. This is being further investigated. 6. CONCLUSION The investigation of the behaviour of the #uid-conveying pipe on an elastic foundation has been carried out as its length is increased. This behaviour has been interpreted in terms of the local properties of the waves in the pipe. In that sense we have de"ned distinct local con"gurations pertaining to the properties of the waves generated by a localized harmonic forcing. The range of #ow velocity and mass ratio where spatially evanescent waves exist at all real forcing frequencies is said to be that of &&evanescence''. In the domain of &&static neutrality'', there exist only neutral waves in a range of forcing frequencies containing the zero frequency. In the domain of &&dynamic neutrality'', there exist only neutral waves in a range of nonzero frequencies. In the domain of &&convective instability'', there exist ampli"ed waves at some forcing frequencies. In the domain of &&absolute instability'', no wave direction can be found at any real frequency and the instability is dominated by the absolute frequency; the response is ampli"ed at the source and gradually contaminates the entire medium. We have thus found that the criterion for global instability as the length is increased becomes closely related to the local properties of the waves in the pipe, but also depends on the boundary conditions. For sets of boundary conditions that allow only static instability, such as pinned}pinned or clamped}clamped ends, we have determined that the asymptotic criterion for global instability is that of static neutrality. Conversely for the cantilevered pipe, which is known to be destabilized by #utter, the asymptotic criterion for global instability is that of dynamic neutrality. By di!erentiating the particular e!ects of the boundary conditions, we have given here a set of results that may not be derived from more classical criteria such as that of Kulikovskii. 14 O. DOARED AND E. DE LANGRE REFERENCES BENJAMIN, T. B. 1960 Dynamics of a system of articulated pipes conveying #uid*I. Theory. Proceedings of the Royal Society (London) 261, 457}486. 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