1 Introduction

Fluid-conveying pipes play important roles in engineering applications, e.g., oil transport lines, hydraulic and pneumatic systems, thermal power plants, and heat transfer equipment. They have also been regarded as a new and dynamical problem^[1] for the study on the stability of structures. It is the simplest fluid-structure interaction system in both theoretical and experimental investigations. Systematic studies have been compiled by Païdoussis^[2-3].

For practical applications, the corrugated pipes supported at both ends will be studied in this paper. In general, a fluid-conveying pipe first loses its stability by divergence when it is supported at both ends. It is known that the fluid-conveying pipe supported at both ends is a gyroscopic conservative system when the dissipative forces are neglected. while the gyroscopic forces do no work^[2]. This system has been investigated extensively in the past half century, primarily focusing on the fluid-conveying pipes with constant cross-sections, e.g., uniformly straight pipes and curved pipes.

In the 1970s, Chen^[4] first derived the equations of motion for curved pipes conveying fluid, and found that such conservative systems were subject to the buckling-type instability when the flow velocity exceeded a certain value. Païdoussis and Issid^[5] found that the conservative systems were subject not only to the buckling at sufficiently high flow velocities but also to the oscillatory instability (coupled-mode flutter) at high flow velocities. Later, the nonlinear theory was introduced to the investigation of fluid-conveying pipes.

Considering the first-order nonlinearities, Holmes^[6] drew the conclusion that flutter motions were impossible. However, the applicability needed to be further assessed, e.g., in fluid-conveying pipes involving more nonlinear factors or in pipes of nonuniform shapes. Semler et al.^[7] derived accurate and inclusive nonlinear equations for fluid-conveying pipes supported at both ends. Zhang and Huang^[8] found that the dynamics of fluid-conveying pipes was very complicated if the effects of Poisson, junction, and friction couplings were considered. Modarres-Sadeghi and Païdoussis^[9] studied the post-divergence behaviors of an extensible fluid-conveying pipe supported at both ends. They found that the fluid-conveying pipe remained in its undeformed static equilibrium state at low flow velocities and then underwent a static pitchfork bifurcation at a critical flow velocity. Zhang and Chen^[10-11] and Chen et al.^[12] investigated the nonlinear dynamics of pipes conveying fluid, i.e., post-divergence behaviors. They found that external and internal resonances occurred in the supercritical regime. Furthermore, the 3:1 internal resonance was first found by Mao et al.^[13].

Recently, the fluid-conveying pipes with constant cross-sections have been focused on several fields, e.g., in-depth nonlinear dynamics^[14-19], vibration control^[20-22], microtubes or nanotubes in microfluidic devices^[23-26], and pipes using functionally graded materials^[27-29]. At the same time, there are increasing studies on fluid-conveying pipes with variable cross-sections and the pipes having periodic structures. Wang et al.^[30] investigated the dynamics of a conical fluid-conveying pipe, and found that the transference of flutter instability from one mode to another, or known as "mode exchange", might frequently occur when the taper angle was successively increased. Yu et al.^[31] investigated the stability of a periodic fluid-conveying cantilevered pipe with different types of binary periodicity. Through the analyses of a geometrically periodic pipe, it showed that the dimensionless critical flow velocity might be larger or smaller than that of a uniform one, depending on the mass ratio.

In fact, fluid-conveying corrugated pipes play important roles in both theoretical analyses and practical applications. Theoretically, the corrugated pipe symbolizes the simplest system among variable cross-section pipes and periodic pipes. Moreover, it possesses great smoothness and periodicity from the perspective of mathematical equations. Corrugated pipes are also used in plenty of industrial and domestic applications, e.g., flexible risers in the oil and gas industry, automotive cabin conditioning systems, liquid propulsion systems in rockets, and compact heat exchangers. An appealing characteristic of corrugated pipes is that they can offer global flexibility combined with local rigidity^[32]. In addition, in many production plants, e.g., food and chemical industry, where heat and mass transfer is crucial, corrugated pipes will improve the heat transfer to a great extent^[33]. Nevertheless, the fundamental stability and dynamics of fluid-conveying corrugated pipes are limited in the literature. To the authors' knowledge, for the in-plane vibrations of corrugated pipes, the effects of the corrugations and the internal fluid flow have not been concerned. This motivates us to investigate the possible dynamical behaviors of fluid-conveying corrugated pipes.

2 Theoretical model

The governing equation is established based on the Newtonian method in this section. Schematic corrugated pipes supported at both ends are shown in Fig. 1. This system comprises a corrugated pipe of the length L, the frequency of the corrugations ω_p, the internal cross-sectional area of a sinusoidal type A(x), the mass per unit length m, the Young modulus E, and the incompressible fluid of mass per unit length M. The pipe is assumed to be slender and to lie along the x-axis, oscillating in the xy-plane. The lateral motion w(s, t), where s is the curvilinear coordinate, is considered to be small compared with the diameter. Thus, the curvilinear coordinate s, which is along the centerline of the pipe, and the coordinate x will be used interchangeably. Due to the variable cross-section, the fluid in the pipe will harmonically flow in an axial space-dependent velocity U(x).

The pipe is vertically placed so as not to consider the effect of the gravity. The effect of fluid friction is also neglected. Then, the stress situations of the fluid element and the pipe element are analyzed. The fluid element, as shown in Fig. 2(a), is subject to (ⅰ) the pressure forces pA, where p = p(s, t) is the pressure from the surrounding fluid; (ⅱ) the reaction forces of the pipe on the fluid associated with the wall-shear stress q, i.e., the forces normal to the fluid element Fds and the forces tangential to the fluid qSds. Considering the small deflection approximation and applying Newton's second law in the x- and y-directions, respectively, we have

(1)

(2)

where a_fx and a_fy are the accelerations of the fluid element in the x- and y-directions, respectively. Similarly, for the pipe element in Fig. 2(b), we have

(3)

(4)

where T is the longitudinal tension force, and Q is the transverse shear force.

The sum of Eqs. (1) and (3) will provide

(5)

At the same time, combining Eqs. (2) and (4), we have

(6)

The terms of acceleration in Eqs. (5) and (6) will be expressed as follows:

(7)

(8)

(9)

(10)

Then, substituting Eqs. (7)–(10) into Eqs. (5) and (6), respectively, yields

(11)

(12)

It is noted that Eqs. (11) and (12) are combined to provide

(13)

Here, the diameter of the pipe is small in comparison with its length so that the Euler-Bernoulli beam approximation can be used as follows:

(14)

After the Q in Eq. (13) is replaced, the governing equation for the transverse motions of the fluid-conveying corrugated pipe will be established, i.e.,

(15)

With regard to Ref. [30] and the comparison of Eqs. (15) and (16), we obtain that, for a conical fluid-conveying pipe, when the taper angle α of the conical pipe is limited (α < 2.05°),

(16)

Then, the position-varying velocity U(x) is processed. It is known that, for pipes conveying pulsating fluid, the velocity is assumed as follows:

(17)

where U₀^* is the mean flow velocity, and ε and ω^* are the perturbed amplitude factor and the frequency of the flow-velocity fluctuation, respectively. The velocity of this form will excite parametric dynamics. Here, for the corrugated pipe, the velocity is assumed as an analogous form as follows:

(18)

where U₀ indicates the mean flow velocity, and ω_p indicates the frequency of the flow-velocity fluctuation. As mentioned above, ω_p is also the frequency of the corrugations. Equation (18) will bring convenience to the succeeding solution, and there will be a symmetry between Eq. (17), which changes over time, and Eq. (18), which changes over space.

We refer to the continuity equation for incompressible fluid as follows:

(19)

where A(x) is the cross-sectional area, and C is an integral constant related to the initial condition of the fluid. Applying the Taylor formula yields

(20)

(21)

where I is the moment of inertia, and δ is the thickness of the pipe.

Then, substituting Eqs. (18) and (20) into Eq. (15) and simplifying the equation, we have

(22)

To gain the dimensionless form of Eq. (22), the following dimensionless parameters are introduced:

(23)

Eventually, Eq. (22) will be expressed in a dimensionless form as follows:

(24)

where P_c = Lω_p is defined as the total number of the corrugations, and the primes and overdots denote the partial differentiation with respect to ξ and τ, respectively.

3 Solution methodology

The differential quadrature method (DQM) is used, which is a numerical method well documented in Ref. [34]. The DQM has been initially used by Wang and Ni^[35-36] to investigate the dynamical behaviors of fluid-conveying pipes and curved pipes subjected to motion-limiting constraints and harmonic excitation. It is also adopted by Ni et al.^[37] to deal with the in-plane and out-of-plane dynamics of curved pipes conveying pulsating fluid. To solve Eq. (24), the DQM has an attractive feature in the computational efficiency due to the rapid convergence compared with the finite element method (FEM). Also, it is more general and convenient than the analytical method and the Green function method (GFM)^[38]. And the programming process of the DQM is plainer in comparison with the Galerkin Method, which will involve a great number of complex integral calculations. More importantly, the classical beam modes are crucial when the Galerkin method is used for discretization, but for corrugated pipes, these modes have not been clarified. With the DQM, we have

(25)

(26)

where N is the total number of the grid points (N = 18 is adopted in this paper), and A_ij and j are the weighting coefficients of the first-order derivative related to ξ and the grid points, respectively. Then, substituting Eq. (25) into Eq. (24) yields

(27)

(28)

where B_ij, C_ij, and D_ij are the weighting coefficients of the second-, third-, and fourth-order derivatives, respectively.

4 Validation

First, the effects of the variable cross-sectional area and the varying flow velocity are analyzed in detail in the process of deriving the governing equation. In addition, despite the different approaches of the derivation process, Eq. (15) is equivalent to Eq. (16) for a conical pipe and a pipe with a variable cross-section.

Second, the correctness of the solution is verified. Referring to the dimensionless equation (24), if ε = 0, the corrugated pipe will degrade to a uniform pipe whose stability has been determined thoroughly. For the clamped-clamped pipe, when ε = 0 and β = 0, the evolutions of frequencies are shown in Fig. 3(a). Perfect agreement is observed with the plot of Païdoussis^[2]. The critical flow velocity of divergence is u_cr¹ = 2π. For the pinned-pinned pipe, when ε = 0 and β = 0, the evolutions of frequencies are plotted in Fig. 3(b). Again, they are in good agreement with the curves obtained by Dai et al.^[39]. Here, the critical flow velocity is u_cr¹ = π. Thus, the validations of both the equation and the solution are demonstrated.

5 Results and discussion

In this section, the effects of the parameters, e.g., the amplitude of the corrugations ε, the total number of the corrugations P_c, and the dimensionless mean flow velocity u, are investigated. The in-plane stability and dynamics of the fluid-conveying corrugated pipes are demonstrated in detail. Some results are found distinctly different from the fluid-conveying pipes with constant cross-sections, i.e., uniform pipes. In general, a uniform fluid-conveying pipe loses stability by divergence rather than by flutter when it is supported at both ends, as mentioned above. However, for the corrugated pipes, the flutter instability is observed even if it is supported at both ends. Furthermore, the stiffness of the corrugated pipes will be reduced to a great extent with the increase in the total number of the corrugations, and then such pipes will lose their stability by flutter at a micro flow velocity, since the stiffness becomes insufficient to maintain the stability of the pipes. Before moving on, it is recalled that ω is the eigenfrequency and is a complex number. Re(ω) is the dimensionless oscillation frequency, while Im(ω) is linked to damping. For small damping, Re(ω) is approximate to the natural frequency. In addition, Re(ω) ≅ Im(λ) and Im(ω) ≅ Re(λ), where λ represents the eigenvalues of Eq. (27).

5.1 Dynamics and stability of the clamped-clamped corrugated pipe 5.1.1 Dynamical behaviors

The dynamical behaviors of the clamped-clamped pipe vary with the increase in the dimensionless mean flow velocity (see Fig. 4). The evolutions of the eigenfrequencies of the lowest four modes are plotted for P_c = 0, 5, 8, 16, respectively. The amplitude of the corrugations is selected as ε = 0.1.

First, in the case of P_c = 0, i.e., the corrugated pipe degrades to a uniform one. As illustrated in Fig. 4(a), the eigenfrequencies are real numbers when the dimensionless flow velocity u is zero. With the increase in u, the real parts of the eigenfrequencies decrease while the imaginary parts remain zero. When u reaches the critical flow velocity u_cr¹ = 2π, the eigenfrequency of the first mode vanishes. Thereafter, it will be a pure imaginary number before the eigenfrequency of the second-mode diminishes to zero. Therefore, this system becomes unstable in terms of the first-mode divergence when 2π < u < 9.02. The divergence instability occurs since the destabilizing centrifugal force becomes strong enough to overcome the restoring flexural force. Then, with the increase in the flow velocity u, the Païdoussis flutter will be observed. When the flow velocity u increases further, the fourth mode will lose stability by divergence at u_cr³ = 12.58. The "mode exchange" phenomenon occurs. Here, the third mode turns to the first-mode, thus the coupled-mode flutter is switched from the coupled-mode flutter of the first- and second-modes to a coupled-mode flutter of the second- and third-modes.

Next, the effect of the corrugations is analyzed with the increase in P_c. As shown in Fig. 4(b), the pipe first loses stability by divergence. It is found that the critical velocity of the first-mode divergence is smaller than that of a uniform pipe. Actually, it is more complicated. Since the critical flow velocity for the divergence instability is what we are interested in, it will be specially discussed in the subsequent section. Back to Fig. 4(b), when the flow velocity reaches u = 8.52, the imaginary part of the first-mode eigenfrequency crosses the coordinate axis, and becomes negative, whereas the second-mode stays stable. Thereafter, the corrugated pipe loses stability via a single-mode flutter (Hopf bifurcation) instead of the coupled-mode flutter (Païdoussis flutter). This phenomenon is also observed when P_c increases. Besides, the frequency of the first-mode stays close to the frequency of the second-mode with the further increase in the flow velocity u, which indicates the similarity between the corrugated pipe and the uniform one.

When P_c = 8 (see Fig. 4(c)), this system first loses stability by flutter in the fourth mode, but not distinct. This fourth mode will regain stability when its imaginary parts turn to positive with the increase in the flow velocity. However, the flutter in the third mode occurs apparently before its divergence. After that, the changing from the divergence to the single-mode flutter of the third-mode will take place. By means of the analyses on higher flow velocities, it is inferred that the evolution of the third-mode is similar to that of the first-mode while the evolution of the fourth-mode is similar to that of the second-mode, as shown in Fig. 4(c). It is partly because, after the "mode exchange", the third-mode turns to the first-mode, and the fourth-mode becomes the new second-mode. In addition, it is observed that the even order modes are more stable than the odd order modes. A comprehensive comparison shows that the second-mode is more stable than any other modes, indicating a stable manifold of this system. For instance, in Figs. 4(b), 4(c), and 4(d) the imaginary parts of the eigenfrequencies of the second mode constantly stay positive.

In the case of P_c = 16, the corrugated pipe loses its stability from the beginning, as shown in Fig. 4(d). It is attributed to the decrease in the stiffness of the system. Referring to Eq. (27), the stiffness term may turn non-positive when the total number of the corrugations increases. Thus, the initial position changes to an unstable position for the system. Even a micro flow velocity will lead the pipe to instability. In addition, it is a coupled-mode flutter of the second- and third-modes since the real parts of the eigenfrequencies of the second- and third-modes are the same while the imaginary parts are opposite. Subsequently, this coupled-mode flutter will turn to a single-mode flutter when the eigenfrequencies of the real parts of the second- and third-modes gradually separate.

The above mentioned shows that the fluid-conveying corrugated clamped-clamped pipe can lose stability by flutter. At P_c = 16, this system will lose stability from the beginning. These results indicate that the total number of the corrugations can be used to tune or modulate the stability of such pipes and to realize their parameter optimization and control in practical applications.

5.1.2 Critical flow velocity

It is recognized that the divergence in the first mode occurs when the mean flow velocity is beyond a certain critical value. The effects of the total number of the corrugations P_c and the amplitude of the corrugations ε are investigated. The value of P_c will be limited in [0, 9] so that the pipes cannot lose stability from the beginning.

The relations between the critical flow velocity u_cr¹ and the total number of the corrugations P_c are shown in Fig. 5. When P_c < e, where e is the Euler Number, the larger P_c is, the smaller the critical flow velocity will be. But the contrary behavior occurs when e < P_c < 7.7. When 7.7 < P_c < 9, the critical flow velocity basically remains unaltered. An extreme point is observed at P_c = e, where this system is the most prone to buckling. This extreme point appears at P_c = e, regardless of the value of ε. When ε increases, the critical velocity u_cr¹ decreases. This phenomenon is illustrated clearly in Fig. 6. The critical flow velocity monotonically decreases with the increase in ε. The lowest bound is determined by the curve, which denotes P_c = e. The value, i.e., P_c = e, is actually related to the geometric parameters of the corrugated pipe since the harmonically varying cross-section area is adopted. In the engineering applications, this value, which most easily leads the corrugated pipe to instability, is to be avoided. Finally, compared with the traditional uniform fluid-conveying pipe represented by the red line in Fig. 6, the existence of the corrugations reduces the critical flow velocity, and hence speeds up the divergence instability. Thus, although corrugated pipes have advantages over the uniform pipes in engineering applications, the parameter of corrugated pipes should be designed for the purpose of avoiding the extreme point, therefore avoiding the instability of the corrugated pipes.

5.2 Dynamics and stability of the pinned-pinned corrugated pipe 5.2.1 Dynamical behaviors

For the pinned-pinned pipe, the evolution of eigenfrequencies of the lowest four modes versus the dimensionless mean flow velocity is shown in Fig. 7. The eigenfrequencies when P_c = 0, 5, 8, 10 are plotted, and the amplitude of the corrugations is selected as ε = 0.1.

The dynamical behaviors demonstrated in Fig. 7 are analogous with that of the clamped-clamped pipe in Fig. 4. The pinned-pinned pipes are also able to lose stability by flutter. As shown in Fig. 7(d), the pinned-pinned corrugated pipe loses its stability from the beginning, though it is a coupled-mode flutter of the third- and fourth-modes unlike the clamped-clamped pipe, where the coupled-mode flutter is of second- and third-modes. The results in Subsection 5.1.1, such as the single-mode flutter, the "mode exchange", and the stable second mode, are observed again. Considering that clamped-clamped pipes and pinned-pinned pipes are both conservative systems, similar behaviors displayed above are actually inherent in such systems. Moreover, the analyses of the pinned-pinned pipe can be regarded as a typical verification of this paper, illustrating and reproducing the dynamics and stability characteristics of fluid-conveying pipes from another respect.

5.2.2 Critical flow velocity

In this subsection, the effects of P_c on the dimensionless critical flow velocity of the first-mode u_cr¹ are investigated. As shown in Fig. 8, the relation between u_cr¹ and P_c is quasi-periodic. When P_c < e, the larger P_c is, the smaller the critical flow velocity is. When e < P_c < 7.7, however, the contrary behaviors occur. With the increase in the total number of the corrugations, the circle will repeat itself. Once more, the extreme point appears at P_c = e. Referring to the previous clamped-clamped one, it is concluded that P_c = e is an invariant of the fluid-conveying corrugated pipes supported at both ends due to its physical geometric parameters.

5.2.3 Validation via the Runge-Kutta method

To verify the phenomenon derived above, especially that fluid-conveying corrugated pipes will first lose its stability by flutter rather than divergence, we obtain the time history plots and phase diagrams with the Runge-Kutta method. According to Fig. 8, u = 2 ensures that the divergence in the first mode will not occur.

As shown in Fig. 9, the results are in good agreement with the dynamical stability derived from the above. When P_c = 0 and P_c = 5, the fluid-conveying corrugated pipes stay stable in its initial position. Referring to Figs. 9(a) and 9(b), the motions of the pipe will gradually diminish. However, when P_c = 8 and P_c = 10, the amplitude of the vibration gradually increases, suggesting that the corrugated pipes are subjected to flutter. In Figs. 10(c) and 10(d), the pipe midpoints are moving away from their equilibrium positions. The energy from the fluid feeds into such systems. Although the flutter instability is not distinct as shown in Fig. 6(c) and Fig. 7(c), it is confirmed here. In addition, the flutter, which occurs at the beginning, is verified. As shown in Fig. 9(d), the pipe loses its stability by flutter at a micro velocity u = 0.2.

6 Conclusions

In this paper, the in-plane dynamics of corrugated pipes is investigated under the clamped-clamped and pinned-pinned constraints. The governing equation for the fluid-conveying corrugated pipe is constructed based on the Newtonian method. Compared with the previous studies, the flow velocity of this system harmonically varies along the pipe rather than with time. Subsequently, the DQM is utilized to solve the governing equation. The results are reliable since they correspond precisely to the previously reported data and to the results calculated via the Runge-Kutta method.

The existence of the corrugations has a significant effect on the in-plane dynamics of the pipes in comparison with the traditional uniform fluid-conveying pipes. It is suggested that the total number of the corrugations P_c can be used to tune or modulate the stability of the corrugated pipes.

(ⅰ) The analyses on both clamped-clamped and pinned-pinned corrugated pipes show that such systems can first lose stability by flutter rather than by divergence. When P_c is sufficiently large, a noticeable flutter is observed at a micro flow velocity.

(ⅱ) The area of the Païdoussis flutter in uniform pipes is replaced by an area of single-mode flutter in the first-mode, while the second-mode mainly stays stable, indicating the difference and similarity between the corrugated pipe and the uniform one.

(ⅲ) The second-mode is the most stable one compared with any other order mode. Meanwhile, even order modes are more stable than odd order modes.

In addition, the effects of two parameters of the corrugated pipes, i.e., the amplitude of the corrugations ε and the total number of the corrugations P_c, on the critical flow velocity u_cr¹ are investigated. Compared with the traditional uniform pipes, the existence of the corrugations will reduce the critical flow velocity, and hence accelerates the divergence. With the increase in ε, u_cr¹ decreases monotonically. However, the relation between u_cr¹ and P_c is quasi-periodic. It is noted that there is an extreme point, which will most easily lead the corrugated pipe to buckling. More surprisingly, the extreme point is constantly taken at P_c = e, regardless it is clamped-clamped or pinned-pinned, which is related to the sinusoidal structure of the corrugated pipe and the corresponding geometric parameters.

An extension of the present work is to design an experimental research program to confirm some of the interesting findings of this paper. Furthermore, we plan to investigate the out-of-plane behaviors of the corrugated pipe.