Dynamics of a Partially Confined, Vertical Upward-Fluid-Conveying, Slender Cantilever Pipe with Reverse External Flow

Abstract

A linear theoretical model is established for the dynamics of a hanging vertical cantilevered pipe which is subjected concurrently to internal and reverse external axial flows. Such pipe systems may have instability by flutter (amplified oscillations) or static divergence (buckling). The pipe system under consideration is a slender flexible cantilevered pipe hanging concentrically within an inflexible external pipe of larger diameter. From the clamped end to the free end, fluid is injected through the annular passage between the external pipe and the cantilevered pipe. When exiting the annular passage, the fluid discharges in the counter direction along the cantilevered pipe. The inflexible external pipe has a variable length and it can cover a portion of the length of the cantilevered pipe. This pipe system has been applied in the solution mining and in the salt cavern underground energy storage industry. The planar motion equation of the system is solved by means of a Galerkin method, and Euler–Bernoulli beam eigenfunctions are used as comparison functions. Calculations are conducted to quantify the effects of different confinement conditions (i.e., the radial confinement degree of the annular passage and the confined-flow length) on the cantilevered pipe stability, for a long leaching-tubing-like system. For a long system, an increase in the radial confinement degree of the annular passage and the confined-flow length gives rise to a series of flutter and divergence. Additionally, the effect of the cantilevered pipe length is studied. Increasing the cantilevered pipe length results in an increase of the critical flow velocity while a decrease of the associated critical frequency. For a long enough system, the critical frequency almost disappears.

1. Introduction

This paper focuses on flow-induced vibration of a slender flexible cantilevered pipe which is hanging concentrically inside a shorter inflexible external pipe, thus constituting an annular passage between the cantilevered pipe and the shorter external pipe, as shown in Figure 1. The entire pipe system is submerged in incompressible fluid and is placed within a closed cavity, with the upper portion outside the cavity. As shown in Figure 1a, from the fixed end to the free end of the shorter inflexible external pipe, fluid is injected as external flow into the closed cavity through the annular passage, and flows out as a free jet into stagnant fluid. In consideration of conservation of mass, initially stagnant fluid in the closed cavity is then forced through the cantilevered pipe, as a bounded reverse internal flow, discharging from the closed cavity upwards along the cantilevered pipe. Comparing the pipe configuration shown in Figure 1a with that given in Figure 1b, it is clear that the fluid flowing directions are opposite from each other. Contrary to the direction of fluid circulation illustrated in Figure 1a, fluid [1] is injected as internal flow into the closed cavity through the cantilever pipe and discharges through the annular passage, as shown in Figure 1b.

The pipe configuration shown in Figure 1 has been applied in solution mining and in the salt cavern underground energy storage industry. Figure 2 shows the schematic view of pipe configurations applied in the salt cavern underground energy storage. Figure 2a,b correspond to Figure 1a,b, respectively. In fact, the pipe failure or excessive bending caused by flow-induced vibrations is a well-known issue in the salt cavern storage industry [2,3]. For example, excessive bending of the inner tubing in a Chinese salt cavern storage is shown in Figure 3. This paper focuses on the pipe model shown in Figure 1a, and the motivation is desired to potentially improve the design of slender cantilever pipes in order to avoid issues cause by flow-induced vibration.

For several decades, a few researchers have been done the work on conveying-fluid pipes subjected simultaneously to both internal and external axial flows. Most of the work done involves concurrent flows, i.e., internal and external axial flows in the same direction. In [4], Cesari and Curioni investigated the buckling instability of pipes subjected to internal and external axial fluid flow. In [5], Hannoyer et al. investigated the dynamics and stability of both clamped–clamped cylindrical tubular beams conveying fluid, which simultaneously are subjected to independent axial external flows. They also studied the dynamics of cantilevered pipes fitted with a tapered nozzle at the free end. They provided theoretical and experimental results to support their model. In [6], Païdoussis and Besancon studied various aspects of the dynamic characteristic of clusters of tubular pipes which were subjected to internal flows and concurrently surrounded by bounded outer axial flows. To establish general characteristics of free motions, they obtained the eigenfrequencies of the system and studied their evolution by increasing either external or internal flows. In [7], Wang and Bloom established a mathematical model to investigate the vibration and stability of a submerged and inclined concentric pipe system which subjected to internal and external flow, and obtained the resonant frequencies of the system. In [8], Païdoussis et al. developed a theoretical model to study the vibration of a hanging tubular cantilever which was centrally inside a cylindrical container, with fluid flowing within the cantilever, and discharging from the free end. The configuration thus resembles that of a drill-string with a floating fluid-powered drill-bit. Of particular interest is the study in [1]. Moditis and Païdoussis preliminarily discussed the dynamics of the pipe configuration shown in Figure 1b. Furthermore, they developed a theoretical model and carried out corresponding experiments.

Because the length of the cantilevered pipe system in practical applications is on the order of one kilometer, how the system behavior evolves is of interest as the length increases. Theoretical and experimental studies [9,10] regarding long hanging vertical pipes conveying fluid have shown that both the critical flow velocity and associated frequency for instability tend to be asymptotic toward different limiting values with increasing the length of the pipes. The literature related to aspirating cantilevered pipes was believed to be useful for analyzing the dynamics of a vertical hanging pipe shown in Figure 1a. For example, in [11,12], the authors theoretically and experimentally considered the dynamic stability of a submerged cantilever pipe aspirating fluid, which could be applied in deep ocean mining.

Several key dimensions of the system under consideration are defined in Figure 1a. The x-axis lies in the undeflected centerline of the internal cantilever pipe, while the z-axis lies along the lateral direction and is perpendicular to the x-axis. L is the length of the cantilever pipe, while L_s is the length of the shorter inflexible external pipe (i.e. the length of the annular passage). U_i is the internal flow velocity (area average flow velocity) inside the cantilevered pipe, and U_o is the external flow velocity (area average flow velocity) in the annular passage with reference to the cantilever pipe.

Firstly, the present paper studies the linear idealized system shown in Figure 1a. We present a derivation of a theoretical model as well as a method of solving it are discussed. Secondly, we give theoretical results for slender leaching-tubing-like systems which are associated with the industrial application of salt cavern storage. We also obtained some unexpected and interesting results by numerical simulation. Finally, we present general conclusions.

2. Problem Formulation

2.1. Derivation of the Motion Equation of Theoretical Model

The derivation of the linear theoretical model is carried out as follows. Take a small element of length δx of the cantilevered pipe into consideration, as shown in Figure 4a, under the action of fluid-related and structural forces and moments. Force balances in the x- and z-directions renders, respectively

$$\frac{\partial F_T}{\partial x}-\frac{\partial }{\partial x}\left(Q\frac{\partial w}{\partial x}\right)+M_{p}g-F_{it}+F_{et}-\left(F_{in}+F_{en}\right)\frac{\partial w}{\partial x}=0$$

(1)

$$\frac{\partial Q}{\partial x}+\frac{\partial }{\partial x}\left(F_T\frac{\partial w}{\partial x}\right)+F_{in}+F_{en}+\left(F_{et}-F_{it}\right)\frac{\partial w}{\partial x}-M_p\frac{{\partial }^{2}w}{\partial t^2}=0$$

(2)

where w is the lateral deflection; F_T is the axial tension; M_p is the mass per unit length of the cantilever pipe; Q is the transverse shear force in the cantilevered pipe; g is the gravitational acceleration; F_in and F_it are the normal and tangential hydrodynamic forces because of the internal flow U_i, respectively; F_en and F_et are the normal and tangential hydrodynamic forces because of the external flow, respectively.

As approximated by the Euler–Bernoulli (E–B) beam theory, one obtains the following relationship

$$Q=-\frac{\partial }{\partial x}\left(EI\frac{{\partial }^{2}w}{\partial x^2}\right)$$

(3)

where EI is the flexural rigidity. Internal dissipation in the cantilevered pipe material is relatively small compared to dissipation by the surrounding fluid so it can be neglected [13]. Substituting Equation (3) into Equations (1) and (2), and neglecting second-order terms, one finds the following expressions in the x- and z-directions, respectively

$$\frac{\partial F_T}{\partial x}-\left(F_{it}+F_{in}\frac{\partial w}{\partial x}\right)+\left(F_{et}-F_{en}\frac{\partial w}{\partial x}\right)+M_{p}g=0$$

(4)

$$EI\frac{{\partial }^{4}w}{\partial x^4}-\frac{\partial }{\partial x}\left(F_T\frac{\partial w}{\partial x}\right)-\left(F_{in}-F_{it}\frac{\partial w}{\partial x}\right)-\left(F_{en}+F_{et}\frac{\partial w}{\partial x}\right)+M_p\frac{{\partial }^{2}w}{\partial t^2}=0$$

(5)

The F_in and F_it (hydrodynamic forces because of internal flow) are obtained by a force balance on an element δx of internal fluid (Figure 5b) in the manner of Païdoussis [14,15]. The F_in′ and F_it′ in Figure 5b are a pair of interaction forces with the F_in and F_it in Figure 4a, F_in′ = −F_in and F_it′ = −F_it. The resulting expressions are written as below in the x- and z-directions, respectively,

$$F_{it}+F_{in}\frac{\partial w}{\partial x}=-M_{f}g+A_f\frac{\partial p_i}{\partial x}$$

(6)

and

$$F_{in}-F_{it}\frac{\partial w}{\partial x}=-M_f\left(\frac{{\partial }^{2}w}{\partial t^2}-2U_i\frac{{\partial }^{2}w}{\partial x\partial t}+U_i^2\frac{{\partial }^{2}w}{\partial x^2}\right)-A_f\frac{\partial }{\partial x}\left(p_i\frac{\partial w}{\partial x}\right)$$

(7)

where M_f is the mass of internal fluid per unit length; p_i is the internal pressure in the cantilevered pipe; A_f is the cross-sectional area of the internal flow, and M_f = ρ_f A_f, in which ρ_f is the fluid density.

Substituting Equations (6) and (7) in (4) and (5) renders the following equations in the x- and z-directions, respectively

$$\frac{\partial F_T}{\partial x}+\left(M_{f}g-A_f\frac{\partial p_i}{\partial x}\right)+\left(F_{et}-F_{en}\frac{\partial w}{\partial x}\right)+M_{p}g=0$$

(8)

$$EI\frac{{\partial }^{4}w}{\partial x^4}-\frac{\partial }{\partial x}\left(F_T\frac{\partial w}{\partial x}\right)+\left[M_f\left(\frac{{\partial }^{2}w}{\partial t^2}-2U_i\frac{{\partial }^{2}w}{\partial x\partial t}+U_i^2\frac{{\partial }^{2}w}{\partial x^2}\right)+A_f\frac{\partial }{\partial x}\left(p_i\frac{\partial w}{\partial x}\right)\right]-\left(F_{en}+F_{et}\frac{\partial w}{\partial x}\right)+M_p\frac{{\partial }^{2}w}{\partial t^2}=0$$

(9)

The field of flow outside the cantilevered pipe can be simplified because the cantilever pipe displacements w are considered to be small according to E–B beam theory. More specifically, the external flow is modelled as the superposition of a perturbation potential because of the cantilever pipe vibrations in the mean axial flow [15,16]. Hence, the effects of fluid viscosity are considered to be different from this potential flow, and the viscosity-related forces are added separately to the system. When linear investigations of fluid-structure interaction issues are carried out, this is a common approach. There are more details in [5,15].

The resultant force of all external hydrodynamic forces acting on the cantilevered pipe (Figure 4a) can be broken down into F_en and F_et in the directions perpendicular and tangential to the cantilever pipe centerline, respectively. Projection of the external hydrodynamic force in the x- and z-directions yields the following force-balance equations

$$F_{et}-F_{en}\frac{\partial w}{\partial x}=F_L-F_{px},$$

(10)

and

$$F_{en}+F_{et}\frac{\partial w}{\partial x}=-\left(F_A+F_N\right)+F_{pz}+F_L\frac{\partial w}{\partial x}.$$

(11)

In Equations (10) and (11), F_A is the lateral inviscid hydrodynamic force; F_px and F_pz are the resultant forces due to external pressure in the x and z directions, respectively; F_N and F_L are the fluid frictional viscous forces perpendicular and tangential to the cantilever pipe centerline, respectively; and high-order terms have been omitted.

The added (hydrodynamic) mass M_a is the mass per unit length for the annular (external) fluid associated with motions of the cantilevered pipe. Based on the perturbation potential, for a thin boundary layer, the added (hydrodynamic) mass M_a = χρ_fA_o [17,18], where A_o = πD_o²/4 is the external cross-sectional area of the cantilevered pipe. The parameter χ quantifies the effect of radial confinement degree (or narrowness) of the annular flow passage as

$$\chi =\frac{{\left(D_{ch}/D_o\right)}^2+1}{{\left(D_{ch}/D_o\right)}^2-1},\hspace{1em}\underset{D_{ch}\to \infty }{\mathit{lim}}\left(\chi \right)=\underset{D_{ch}\to \infty }{\mathit{lim}}\frac{{\left(D_{ch}/D_o\right)}^2+1}{{\left(D_{ch}/D_o\right)}^2-1}=1.$$

(12)

In Equation (12), D_ch is the inner diameter of the passage (the inflexible external pipe) and D_o is the outer diameter of the cantilever pipe as shown in Figure 1. Regarding an unconfined pipe, D_ch → ∞ and as a result M_a is equal to ρ_fA_o, i.e., the displaced mass of the fluid per unit length. Hence, considering the variation of confinement along the cantilevered pipe, the added (hydrodynamic) mass may be written as

$$M_a=\left[\chi +\left(1-\chi \right)H\left(x-L_s\right)\right]{\rho }_f{A}_o$$

(13)

where χ pertains to the confined portion of the cantilevered pipe, 0 ≤ x ≤ L_s, and H(x − L_s) is the Heaviside step function.

As is shown in Figure 1a, both the external flow velocity U_o and the confinement degree vary along the length of the cantilevered pipe. For the sake of simplicity, it is assumed that U_o is zero along the unconfined part, L_s < x ≤ L, and immediately attains the value of U_o along the confined portion of the cantilevered pipe, 0 ≤ x < L_s. The above assumptions can also be found in the literature [1]. Thus, the lateral inviscid hydrodynamic force F_A can be written as

$$F_A=\left(\frac{\partial }{\partial t}+U_o\frac{\partial }{\partial x}-U_{o}H\left(x-L_s\right)\frac{\partial }{\partial x}\right)\left\{\left[\chi +\left(1-\chi \right)H\left(x-L_s\right)\right]{\rho }_f{A}_o\left(\frac{\partial w}{\partial t}+U_o\frac{\partial w}{\partial x}-U_{o}H\left(x-L_s\right)\frac{\partial w}{\partial x}\right)\right\},$$

(14)

where the initial expression [16] and later [17] have been altered to explicate the spatial variation of both the external flow velocity and the added (hydrodynamic) mass Equation (13). After various simplifications, the formula given in Equation (14) was arrived at

$$F_A=\left(1-\chi \right){\rho }_f{A}_{o}H\left(x-L_s\right)\frac{{\partial }^{2}w}{\partial t^2}+{\rho }_f{A}_o\chi \frac{{\partial }^{2}w}{\partial t^2}-2A_o{U}_o{\rho }_f\chi H\left(x-L_s\right)\frac{{\partial }^{2}w}{\partial x\partial t}+2A_o{U}_o{\rho }_f\chi \frac{{\partial }^{2}w}{\partial x\partial t}-A_o{U}_o^2{\rho }_f\chi H\left(x-L_s\right)\frac{{\partial }^{2}w}{\partial x^2}+A_o{U}_o^2{\rho }_f\chi \frac{{\partial }^{2}w}{\partial x^2}$$

(15)

The fluid frictional viscous force F_L tangential to the cantilevered pipe centerline is

$$F_L=\frac{1}{2}{C}_f{\rho }_f{D}_o{U}_o^2\left[1-H\left(x-L_s\right)\right],$$

(16)

where C_f is the frictional damping coefficient, with a value of 0.0125 to give an acceptable estimate of the fluid frictional viscous force F_L [8].

Equation (16) was modified from the literature [17] to take the spatial variation of the mean external flow velocity into account. Likewise, in terms of the literature [17], the fluid frictional viscous force F_N perpendicular to the cantilevered pipe centerline is

$$F_N=\frac{1}{2}{C}_f{\rho }_f{D}_o{U}_o\left[1-H\left(x-L_s\right)\right]\left\{\frac{\partial w}{\partial t}+\left[1-H\left(x-L_s\right)\right]U_o\frac{\partial w}{\partial x}\right\}+k\frac{\partial w}{\partial t},$$

(17)

where the variation of the external flow velocity U_o has been taken into account; k is a viscous damping coefficient which only relates to lateral motions of the cantilever pipe, without average external flow U_o. Based on two-dimensional flow analysis, expressions used for k are derived in [13,18]. For a thin boundary layer, the coefficient k is calculated via the following expression

$$k=\frac{2\sqrt{2}}{\sqrt{S}}\frac{1+{\overline{\gamma }}^3}{{\left(1-{\overline{\gamma }}^2\right)}^2}{\rho }_f{A}_o\Re \left\{\Omega \right\}$$

(18)

In (10), S = R{Ω}D_o²/(4ν) represents the oscillatory Reynolds number (i.e., Stokes number), where ν and R{Ω} are the kinematic viscosity of the fluid and the circular frequency of oscillation, respectively; $\overline{\gamma }$ = D_o/D_ch is a measure of the radial confinement degree of the annular passage. Naturally, based on Figure 1a, the value of k varies along the cantilevered pipe. This variation of k is explained by letting

$$F_N=\frac{1}{2}{C}_f{\rho }_f{D}_o{U}_o\left\{\left[1-H\left(x-L_s\right)\right]\frac{\partial w}{\partial t}+\left[1-H\left(x-L_s\right)\right]U_o\frac{\partial w}{\partial x}\right\}+k_u\left[\frac{1+{\overline{\gamma }}^3}{{\left(1-{\overline{\gamma }}^2\right)}^2}+H\left(x-L_s\right)\left(1-\frac{1+{\overline{\gamma }}^3}{{\left(1-{\overline{\gamma }}^2\right)}^2}\right)\right]\frac{\partial w}{\partial t}.$$

(19)

$$k_u=\frac{2\sqrt{2}}{\sqrt{S}}{\rho }_f{A}_o\Re \left\{\Omega \right\}$$

(20)

In (19) and (20), k_u represents the friction coefficient applicable to the cantilevered pipe without confined portion, L_s < x ≤ L.

Finally, the forces (F_px and F_pz) caused by mean tensioning (induced by gravity) and pressurization are considered, respectively. F_px and F_pz are cleverly derived in [15,17] and they are expressed as

$$F_{px}=-\frac{\partial }{\partial x}\left(A_o{p}_o\right)+A_o\frac{\partial p_o}{\partial x}$$

(21)

and

$$F_{pz}=A_o\frac{\partial }{\partial x}\left(p_o\frac{\partial w}{\partial x}\right)$$

(22)

in the x- and z-directions respectively, where p_o represents the fluid pressure outside the cantilevered pipe.

For the purpose of analyzing the model shown in Figure 1a, the outlet effects of the flow exiting from the annular passage have been reduced to a tinily thin region at x = L_s. For 0 ≤ x < L_s, there is a pressure loss due to the friction of the flowing fluid inside the annulus, while there is a pressure increasing due to the gravity. For L_s < x ≤ L, there is a purely hydrostatic pressure distribution along the unconfined portion of the cantilever pipe. For 0 ≤ x < L_s, an annular fluid element (i.e., the external flow of length δx) (Figure 5a) is considered, and a force balance per unit length is obtained as

$$-A_{ch}\frac{\partial p_o}{\partial x}-F_f+A_{ch}{\rho }_{f}g=0,$$

(23)

where A_ch = π(D_ch² − D_o²)/4 represents the cross-sectional flow area of the annular passage; F_f represents the total wall-friction force acting on the fluid element.

It is assumed that an equal wall shear stress acts on both annular passage surfaces, the total wall-friction force F_f can be expressed as

$$\frac{F_f}{S_{tot}}=\frac{F_L}{S_o}$$

(24)

in which S_tot = π(D_ch + D_o) is the total wetted area of the annular flow per unit length, and S_o is the outside wetted pipe perimeter.

Combining Equations (23) and (24), multiplied by (A_o/A_ch), the resultant equation is rearranged as

$$A_o\frac{\partial p_o}{\partial x}=-F_L\left(\frac{D_o}{D_h}\right)+A_o{\rho }_{f}g,$$

(25)

where D_h = 4A_ch/S_tot = (D_ch − D_o) is the hydraulic diameter of the external flow in the annular passage. Integration of Equation (25), where for the confined portion external fluid pressure at x = 0 is the reference pressure (p_o|_{x = 0} = 0), with respect to x gives

$$p_o\left(x\right)=\left[-\frac{F_L}{A_o}\left(\frac{D_o}{D_h}\right)+{\rho }_{f}g\right]x,$$

(26)

for 0 ≤ x < L_s.

For the unconfined part, i.e., L_s < x ≤ L, the external fluid pressure distribution is hydrostatic

$$\frac{\partial p_o}{\partial x}={\rho }_{f}g,$$

(27)

which upon integration results in

$$p_o\left(x\right)={\rho }_{f}gx+C_1,$$

(28)

in which C₁ is an integration constant determined in the following description.

It is assumed that x₁ = L_s⁻ is the axial location just inside the annular passage, and x₂ = L_s⁺ is the location just outside the annular passage. Based on Bernoulli’s equation, an energy balance of the fluid is obtained from x₁ to x₂

$$p_o|{}_{x_2}=p_o|{}_{x_1}+\frac{1}{2}{\rho }_f{U}_o^2-{\rho }_{f}g{h}_e,h_e=\frac{K_1{U}_o^2}{2}$$

(29)

where the quantity h_e, with K₁ = 1, is the head-loss associated with sudden enlargement of the external flow in the annular passage into the surrounding fluid at x = L_s⁺ [19]. Combination Equations (28) and (29) yields

$$C_1=\left[-\frac{\Vert F_L\Vert }{A_o}\left(\frac{D_o}{D_h}\right)L_s\right]+\frac{1}{2}{\rho }_f{U}_o^2-{\rho }_{f}g{h}_e,$$

(30)

in which ||F_L|| = 0.5ρ_fC_fD_oU_o².

Through the above analysis, the resulting fluid pressure gradient and fluid pressure distribution over the whole cantilevered pipe, for 0 ≤ x ≤ L, are, respectively

$$\frac{\partial p_o}{\partial x}=-\frac{\Vert F_L\Vert }{A_o}\left(\frac{D_o}{D_h}\right)\left[1-H\left(x-L_s\right)\right]+{\rho }_{f}g+\left(\frac{1}{2}{\rho }_f{U}_o^2-{\rho }_{f}g{h}_e\right){\delta }_D\left(x-L_s\right),$$

(31)

and

$$p_o=-\frac{\Vert F_L\Vert }{A_o}\left(\frac{D_o}{D_h}\right)x+\frac{\Vert F_L\Vert }{A_o}\left(\frac{D_o}{D_h}\right)\left(x-L_s\right)H\left(x-L_s\right)+{\rho }_{f}gx+\left(\frac{1}{2}{\rho }_f{U}_o^2-{\rho }_{f}g{h}_e\right)H\left(x-L_s\right),$$

(32)

where δ_D(x − L_s) is the Dirac delta function.

Substituting Equations (16) and (21) in (10), and subsequently substituting (10) into (8), the force balance on the cantilevered pipe is obtained in the x-direction, namely

$$\frac{\partial }{\partial x}\left(F_T-A_f{p}_i+A_o{p}_o\right)+M_{p}g+{\rho }_f{A}_{f}g-A_o\frac{\partial p_o}{\partial x}+\frac{1}{2}{C}_f{\rho }_f{D}_o{U}_o^2\left[1-H\left(x-L_s\right)\right]=0$$

(33)

Substituting Equations (15), (16), (19), and (22) in (11), and subsequently substituting the result in (1), the motion equation is obtained in the z-direction,

$$\begin{array}{l}EI\frac{{\partial }^{4}w}{\partial x^4}-\frac{\partial }{\partial x}\left[\left(F_T-A_f{p}_i+A_o{p}_o\right)\frac{\partial w}{\partial x}\right]+M_p\frac{{\partial }^{2}w}{\partial t^2}+\left[M_f\left(\frac{{\partial }^{2}w}{\partial t^2}-2U_i\frac{{\partial }^{2}w}{\partial x\partial t}+U_i^2\frac{{\partial }^{2}w}{\partial x^2}\right)\right]+\left(1-\chi \right){\rho }_f{A}_{o}H\left(x-L_s\right)\frac{{\partial }^{2}w}{\partial t^2}+{\rho }_f{A}_o\chi \frac{{\partial }^{2}w}{\partial t^2}\\ -2A_o{U}_o{\rho }_f\chi H\left(x-L_s\right)\frac{{\partial }^{2}w}{\partial x\partial t}+2A_o{U}_o{\rho }_f\chi \frac{{\partial }^{2}w}{\partial x\partial t}-A_o{U}_o^2{\rho }_f\chi H\left(x-L_s\right)\frac{{\partial }^{2}w}{\partial x^2}+A_o{U}_o^2{\rho }_f\chi \frac{{\partial }^{2}w}{\partial x^2}+\frac{1}{2}{C}_f{\rho }_f{D}_o{U}_o\left[1-H\left(x-L_s\right)\right]\frac{\partial w}{\partial t}\\ +k_u\left[\frac{1+{\overline{\gamma }}^3}{{\left(1-{\overline{\gamma }}^2\right)}^2}+H\left(x-L_s\right)\left(1-\frac{1+{\overline{\gamma }}^3}{{\left(1-{\overline{\gamma }}^2\right)}^2}\right)\right]\frac{\partial w}{\partial t}=0.\end{array}$$

(34)

In Equation (34), the tensioning and pressurization term (F_T − p_iA_f + A_op_o) is determined by integrating Equation (33) from x to L, which yields

$$\begin{array}{c}\left(F_T-A_f{p}_i+A_o{p}_o\right)=\frac{1}{2}{C}_f{\rho }_f{D}_o{U}_o^2\left(\frac{D_o}{D_h}+1\right)\left(L_s-x\right)\left[1-H\left(x-L_s\right)\right]-A_o\left(\frac{1}{2}{\rho }_f{U}_o^2-{\rho }_{f}g{h}_e\right)\left[1-H\left(x-L_s\right)\right]\\ +\left(M_{p}g+{\rho }_f{A}_{f}g-A_o{\rho }_{f}g\right)\left(L-x\right)+\left(F_T-A_f{p}_i+A_o{p}_o\right)|{}_L.\end{array}$$

(35)

In Equation (35) |_L represents an evaluation of the term in parentheses at x = L.

Hence, the final form of the motion equation is obtained in the z-direction by substituting Equation (35) in (34), as follows

$$\begin{array}{l}EI\frac{{\partial }^{4}w}{\partial x^4}+\left\{\left(M_p+{\rho }_f{A}_f-{\rho }_f{A}_o\right)g+\frac{1}{2}{C}_f{\rho }_f{D}_o{U}_o^2\left(\frac{D_o}{D_h}+1\right)\left[1-H\left(x-L_s\right)\right]-A_o\left(\frac{1}{2}{\rho }_f{U}_o^2-{\rho }_{f}g{h}_e\right){\delta }_D\left(x-L_s\right)\right\}\frac{\partial w}{\partial x}\\ +\left\{\left(-M_p-{\rho }_f{A}_f+{\rho }_f{A}_o\right)g\left(L-x\right)-\frac{1}{2}{C}_f{\rho }_f{D}_o{U}_o^2\left(\frac{D_o}{D_h}+1\right)\left(L_s-x\right)\left[1-H\left(x-L_s\right)\right]+A_o\left(\frac{1}{2}{\rho }_f{U}_o^2-{\rho }_{f}g{h}_e\right)\left[1-H\left(x-L_s\right)\right]-{\left(T-A_f{p}_i+A_o{p}_o\right)|}_L\right\}\frac{{\partial }^{2}w}{\partial x^2}\\ +M_p\frac{{\partial }^{2}w}{\partial t^2}+{\rho }_f{A}_f\frac{{\partial }^{2}w}{\partial t^2}-2U_i{\rho }_f{A}_f\frac{{\partial }^{2}w}{\partial x\partial t}+{\rho }_f{A}_f{U}_i^2\frac{{\partial }^{2}w}{\partial x^2}+{\rho }_f{A}_o\chi U_o^2\left[1-H\left(x-L_s\right)\right]\frac{{\partial }^{2}w}{\partial x^2}+2A_o{U}_o{\rho }_f\chi \left[1-H\left(x-L_s\right)\right]\frac{{\partial }^{2}w}{\partial x\partial t}+\left(1-\chi \right){\rho }_f{A}_{o}H\left(x-L_s\right)\frac{{\partial }^{2}w}{\partial t^2}\\ +{\rho }_f{A}_o\chi \frac{{\partial }^{2}w}{\partial t^2}+\frac{1}{2}{C}_f{\rho }_f{D}_o{U}_o\left[1-H\left(x-L_s\right)\right]\frac{\partial w}{\partial t}+k_u\left\{1+\left[1-H\left(x-L_s\right)\right]\left(\frac{1+{\overline{\gamma }}^3}{{\left(1-{\overline{\gamma }}^2\right)}^2}-1\right)\right\}\frac{\partial w}{\partial t}=0\end{array}$$

(36)

According to the Bernoulli equation, the pressures P_o|_L and P_i|_L are related to the energy balance of the fluid at x = L

$$p_i|{}_L=p_o|{}_L-\frac{1}{2}{\rho }_f{U}_i^2-{\rho }_{f}g{h}_a,h_a=\frac{K_2{U}_i^2}{2}.$$

(37)

In Equation (37), the quantity h_a is the headloss due to the stagnant fluid entering the cantilevered pipe at x = L and acquiring internal flow velocity U_i, calculated with 0.8 ≤ K₂ ≤ 0.9, independent of flow velocity [19]. Consequently, evaluation of Equation (32) at x = L yields P_o|_L

$$p_o|{}_L=\frac{1}{2A_o}{C}_f{\rho }_f{D}_o{U}_o^2\left(\frac{D_o}{D_h}\right)L^{\prime }+{\rho }_{f}gL+\frac{1}{2}{\rho }_f{U}_o^2-{\rho }_{f}g{h}_a.$$

(38)

Finally, the internal and external flow velocities are related through conservation of mass, i.e., U_iA_f = U_oA_ch, which after some transformation is expressed as

$$U_o=U_i\frac{D_i^2}{D_{ch}^2-D_o^2}.$$

(39)

2.2. Boundary Conditions

The motion Equation (36) is subjected to the boundary conditions

$$w\left(0,t\right)={\frac{\partial w}{\partial x}|}_{x=0}=0,{\frac{{\partial }^{2}w}{\partial x^2}|}_{x=L}={\frac{{\partial }^{3}w}{\partial x^3}|}_{x=L}=0.$$

(40)

2.3. Dimensionless Motion Equation and Boundary Conditions

The motion Equation (36) is non-dimensionalized using the following dimensionless quantities

$$\begin{array}{l}\xi =\frac{x}{L},\tau ={\left[\frac{EI}{M_p+{\rho }_f{A}_f+{\rho }_f{A}_o}\right]}^{\frac{1}{2}}\frac{t}{L^2},\eta =\frac{w}{L},\\ u_i={\left(\frac{{\rho }_f{A}_f}{EI}\right)}^{\frac{1}{2}}L{U}_i,u_o={\left(\frac{{\rho }_f{A}_o}{EI}\right)}^{\frac{1}{2}}L{U}_o,{\beta }_o=\frac{{\rho }_f{A}_o}{M_p+{\rho }_f{A}_f+{\rho }_f{A}_o},\\ {\beta }_i=\frac{{\rho }_f{A}_f}{M_p+{\rho }_f{A}_f+{\rho }_f{A}_o},\gamma =\frac{\left(M_p+{\rho }_f{A}_f-{\rho }_f{A}_o\right)g{L}^3}{EI},\Gamma =\frac{{T|}_L{L}^2}{EI},\\ {\Pi }_{\mathit{iL}}=\frac{{p_i|}_L{A}_f{L}^2}{EI},{\Pi }_{\mathit{oL}}=\frac{{p_o|}_L{A}_o{L}^2}{EI},c_f=\frac{4C_f}{\pi },\\ {\kappa }_u=\frac{k_u{L}^2}{{\left[EI\left(M_p+{\rho }_f{A}_f+{\rho }_f{A}_o\right)\right]}^{\frac{1}{2}}},\epsilon =\frac{L}{D_o},h=\frac{D_o}{D_h},\\ \alpha =\frac{D_i}{D_o},{\alpha }_{ch}=\frac{D_{ch}}{D_o},r_{ann}=\frac{L_s}{L}.\end{array}$$

(41)

Thus, the resulting dimensionless form of Equation (36) is

$$\begin{array}{l}\frac{{\partial }^4\eta }{\partial {\xi }^4}+\left\{\gamma +\frac{1}{2}{c}_f\epsilon u_o^2\left(h+1\right)\left[1-H\left(\xi -r_{ann}\right)\right]-\frac{1}{2}{u}_o^2\left(1-K_1\right){\delta }_D\left(\xi -r_{ann}\right)\right\}\frac{\partial \eta }{\partial \xi }\\ -\left\{\gamma \left(1-\xi \right)+\frac{1}{2}{c}_f\epsilon u_o^2\left(h+1\right)\left(r_{ann}-\xi \right)\left[1-H\left(\xi -r_{ann}\right)\right]-\frac{1}{2}{u}_o^2\left(1-K_1\right)\left[1-H\left(\xi -r_{ann}\right)\right]+\left(\Gamma -{\Pi }_{\mathrm{iL}}+{\Pi }_{\mathrm{oL}}\right)\right\}\frac{{\partial }^2\eta }{\partial {\xi }^2}\\ +\left\{u_i^2+\chi u_o^2\left[1-H\left(\xi -r_{ann}\right)\right]\right\}\frac{{\partial }^2\eta }{\partial {\xi }^2}+\left\{1+\left(\chi -1\right){\beta }_o\left[1-H\left(\xi -r_{ann}\right)\right]\right\}\frac{{\partial }^2\eta }{\partial {\tau }^2}+2\left(\chi u_o{\beta }_o^{\frac{1}{2}}\left[1-H\left(\xi -r_{ann}\right)\right]-u_i{\beta }_i^{\frac{1}{2}}\right)\frac{{\partial }^{2}w}{\partial \xi \partial \tau }\\ +\frac{1}{2}{c}_f\epsilon u_o{\beta }_o^{\frac{1}{2}}\left[1-H\left(\xi -r_{ann}\right)\right]\frac{\partial \eta }{\partial \tau }+{\kappa }_u\left\{1+\left[1-H\left(\xi -r_{ann}\right)\right]\left(\frac{1+{\alpha }_{ch}{}^{-3}}{{\left(1-{\alpha }_{ch}{}^{-2}\right)}^2}-1\right)\right\}\frac{\partial \eta }{\partial \tau }=0,\end{array}$$

(42)

with corresponding dimensionless boundary conditions

$$\eta \left(0,\tau \right)={\frac{\partial \eta }{\partial \xi }|}_{\xi =0}=0,{\frac{{\partial }^2\eta }{\partial {\xi }^2}|}_{\xi =1}={\frac{{\partial }^3\eta }{\partial {\xi }^3}|}_{\xi =1}=0.$$

(43)

Additionally, the dimensionless (complex) frequency of vibration, ω, is related to the dimensional one, Ω, by

$$\omega ={\left[\frac{M_p+{\rho }_f{A}_f+{\rho }_f{A}_o}{EI}\right]}^{\frac{1}{2}}{L}^2\Omega .$$

(44)

It is noted that the external and internal fluid pressures in dimensionless form at the free end have the relationship

$${\Pi }_{\mathit{iL}}={\alpha }^2{\Pi }_{\mathit{oL}}-\frac{1}{2}{u}_i^2-A_f{\rho }_{f}g{h}_a\left(\frac{L^2}{EI}\right),$$

(45)

where

$${\Pi }_{\mathit{oL}}=\frac{1}{2}{c}_{f}h{r}_{ann}\epsilon u_o^2+\frac{1}{2}{u}_o^2\left(1-K_1\right)+\frac{A_o{\rho }_{f}g{L}^3}{EI},$$

(46)

with h_a given in Equation (37), and K₁ as shown in Equation (29).

Finally, the dimensionless flow velocities are expressed as

$$u_o=\frac{\alpha }{{\alpha }_{ch}^2-1}{u}_i,$$

(47)

where α and α_ch are defined in (41).

3. Solution of Equations by a Galerkin Method

The governing Equation (42) of lateral vibration of the cantilever pipe contains the Heaviside step function, which is a discontinuous function, and a conventional Galerkin method is used here to discretize this system into an ordinary differential equation. The cantilever pipe is divided into N units. Consequently, solution of Equation (42) was effected as follows. Let an approximate solution be

$$\eta \left(\xi ,\tau \right)\approx \tilde{\eta }\left(\xi ,\tau \right)={\displaystyle \sum _{j=1}^N{\Phi }_j\left(\xi \right)q_j\left(\tau \right)}.$$

(48)

In (48), q_j(τ) is the generalized coordinate of the system and written as

$$q_j\left(\tau \right)=s_j{e}^{i{\omega }_j}{}^{\tau }=\frac{s_j}{e^{\mathrm{Im}\left({\omega }_j\right)\tau }}{e}^{i\mathrm{Re}\left({\omega }_j\right)}{}^{\tau }.$$

(49)

The Φ_j(ξ) are the appropriate comparison functions, in this case the normalized beam eigenfunctions for the fixed-free E–B beam, and satisfy the geometric and natural boundary conditions of the problem. The q_j(τ) are the generalized coordinates of the system. In Equation (49), ω_j are the eigenfrequencies; s_j is a complex amplitude; and Im(ω_j) and Re(ω_j) are the imaginary and real part of ω_j, respectively. It is assumed that the solution of Equation (42) is separable in terms of the dimensionless spatial variable ξ and the dimensionless time τ.

Equation (48) denotes a truncated series, in which N is finite positive integer and it represents the number of the appropriate comparison functions. For the sake of compactness of the notation, the following integral expressions are defined as

$$\begin{array}{l}{a}_{i{j}_{\left(m,n\right)}}\equiv {\displaystyle {\int }_m^n{\Phi }_i{\Phi }_{j}d\xi },b_{i{j}_{\left(m,n\right)}}\equiv {\displaystyle {\int }_m^n{\Phi }_i\left(\frac{\partial {\Phi }_j}{\partial \xi }\right)d\xi },\\ c_{i{j}_{\left(m,n\right)}}\equiv {\displaystyle {\int }_m^n{\Phi }_i\left(\frac{{\partial }^2{\Phi }_j}{\partial {\xi }^2}\right)d\xi },d_{i{j}_{\left(m,n\right)}}\equiv {\displaystyle {\int }_m^n\xi {\Phi }_i\left(\frac{{\partial }^2{\Phi }_j}{\partial {\xi }^2}\right)d\xi },\end{array}$$

(50)

where i, j are indices corresponding to the relevant quantity to be integrated; m and n of (m, n) are the lower/upper bounds of integration.

Substituting Equations (48) and (50) into (42), simultaneous equations of the form are obtained by pre-multiplying by Φ_i and integrating from ξ = 0 to ξ = 1

$$\mathbf{M}\frac{{\mathrm{d}}^2}{\mathrm{d}{\tau }^2}\left(\left\{\begin{array}{c}{q}_1\\ q_2\\ ⋮\\ q_3\end{array}\right\}\right)+\mathbf{C}\frac{\mathrm{d}}{\mathrm{d}\tau }\left(\left\{\begin{array}{c}{q}_1\\ q_2\\ ⋮\\ q_3\end{array}\right\}\right)+\mathbf{K}\left(\left\{\begin{array}{c}{q}_1\\ q_2\\ ⋮\\ q_3\end{array}\right\}\right)=0.$$

(51)

In Equation (51), M, C, and K matrices are mass matrix, damping matrix and stiffness matrix, respectively. The components of the M, C, and K matrices are

$${\mathbf{M}}_{ij}=a_{i{j}_{\left(0,1\right)}}-{\beta }_o\left(1-\chi \right)a_{i{j}_{\left(0,r_{ann}\right)}},$$

(52)

$${\mathbf{C}}_{ij}=-2u_i{\beta }_i^{\frac{1}{2}}{b}_{i{j}_{\left(0,1\right)}}+2\chi u_o{\beta }_o^{\frac{1}{2}}{b}_{i{j}_{\left(0,r_{ann}\right)}}+\frac{1}{2}{c}_f\epsilon u_o{\beta }_o^{\frac{1}{2}}{a}_{i{j}_{\left(0,r_{ann}\right)}}+{\kappa }_u{a}_{i{j}_{\left(0,1\right)}}+{\kappa }_u\left(\frac{1+{\alpha }_{ch}{}^{-3}}{{\left(1-{\alpha }_{ch}{}^{-2}\right)}^2}-1\right)a_{i{j}_{\left(0,r_{ann}\right)}},$$

(53)

$$\begin{array}{c}{\mathbf{K}}_{ij}={\lambda }_j^4{a}_{i{j}_{\left(0,1\right)}}+\gamma b_{i{j}_{\left(0,1\right)}}-\frac{1}{2}{c}_f\epsilon u_o^2\left(h-1\right)b_{i{j}_{\left(0,r_{ann}\right)}}-\frac{1}{2}{u}_o^2\left(1-K_1\right)\left({{\Phi }_i|}_{\xi =r_{ann}}{\frac{\partial {\Phi }_j}{\partial \xi }|}_{\xi =r_{ann}}\right)-\left(\Gamma -{\Pi }_{\mathrm{iL}}+{\Pi }_{\mathrm{oL}}\right)c_{i{j}_{\left(0,1\right)}}-\gamma \left(c_{i{j}_{\left(0,1\right)}}-d_{i{j}_{\left(0,1\right)}}\right)\\ +\frac{1}{2}{c}_f\epsilon u_o^2\left(h-1\right)\left(r_{ann}{c}_{i{j}_{\left(0,r_{ann}\right)}}-d_{i{j}_{\left(0,r_{ann}\right)}}\right)+\frac{1}{2}{u}_o^2\left(1-K_1\right)c_{i{j}_{\left(0,1\right)}}+u_i^2{c}_{i{j}_{\left(0,1\right)}}+\chi u_o^2{c}_{i{j}_{\left(0,r_{ann}\right)}};\end{array}$$

(54)

where λ_j is the jth dimensionless eigenvalue of the fixed-free E–B beam.

Solutions of Equation (51) constitute approximate solutions of Equation (42) with boundary conditions in Equation (43). In order to obtain a non-trivial solution of Equation (51), it is required that the determinant of coefficient matrix is equal to zero. This corresponds to a generalized eigenvalue problem, and the stability of the system can be determined by the generalized eigenvalue (i.e., eigenfrequency) ω of matrix E which consists of M, C, and K, as

$$\mathbf{E}=\left[\begin{array}{cc}-{\mathbf{M}}^{-1}\mathbf{C}& -{\mathbf{M}}^{-1}\mathbf{K}\\ \mathbf{I}& 0\end{array}\right].$$

(55)

For a given internal dimensionless flow velocity u_i, the stability of the cantilever pipe is determined by Im(ω). When Im(ω) < 0, the q_j(τ) (shown in Equation (49)) grows exponentially in time, so Im(ω) < 0 indicates instability. Consequently, for Im(ω) < 0 the cantilevered pipe becomes unstable, and for Im(ω) = 0, divergence or buckling of the cantilever pipe happens. Divergence represents a static loss of stability, vulgarly known as buckling and, in the nonlinear dynamics milieu, as a static pitchfork bifurcation. In the following passages, any discussion of the system stability involves only the stability of the internal cantilevered pipe.

4. Theoretical Analysis for Slender, Leaching-Tubing-Like Systems

The geometry shown in Figure 1 has been applied in the salt cavern underground energy storage industry, called leaching-tubing systems. Salt caverns serving as underground energy storage are generally located at a depth between 500 and 2000 m [3], and storage volumes of salt caverns range from 5 × 10⁴ m³ to 2 × 10⁵ m³ [20,21]. The typical depths of a salt cavern height and the salt cavern ceiling are about 400 m and 600 m, respectively, learned from the industry survey described in [2]. Hence common lengths of the leaching tubing are on the order of one kilometer.

Based on information from industrial applications, theoretical analyses for slender leaching-tubing-like systems are carried out in this section. Sample dimensional and associated dimensionless parameters for the leaching tubing are given in

Table 1. Sample properties of real-life leaching tubings and associated dimensionless parameters [1].

Dimensional parameters	Di (m)	Do (m)	Dch (m)	L (m)	Ls (m)	EI (N·m2)	Mp (kg/m)
	0.159	0.1778	0.298	1283	1085	3.47 × 106	38.7
Dimensionless parameters	α	αch	ε	βi	βo	h	γ	rann
	0.897	1.676	7216	0.239	0.297	1.479	2.019 × 105	0.85

. In the salt cavern storage industry, the geometric parameters α_ch, r_ann and L shown in Figure 1 can be varied by operators, as required by the engineering practice. Therefore, it is physically meaningful to discuss the effects of the geometric parameters α_ch, r_ann, and L on the pipe system. The effects of varying α_ch, r_ann and L are discussed from Section 4.1 to Section 4.3.

4.1. Effect of the Radial Confinement α_ch

Here, it is interesting to see the effect of different radial confinement values α_ch on the leaching-tubing-like system behavior. According to Equation (47), decreasing α_ch corresponds to an increase in radial confinement and to a related increase in the dimensionless annular flow velocity u_o.

The leaching-tubing-like systems behavior with the variation of α_ch from 1.10 to 20 is summarized in

Table 2. Summary of the leaching-tubing-like systems behavior with the vibration of αch from 1.10 to 20.

Behavior a
	D1		F2	F3	F1		F2
αch	1.10	1.20	1.27	1.32	1.35	…	4.46	…	20	…

F and D represent two kinds of unstable modes, respectively, i.e., flutter instability and divergence. As an example, D1 denotes first-mode divergence and F1 denotes first-mode flutter. ^a [r_ann = 0.50, ε = 1124.9 (L = 200 m)].

, based on r_ann = 0.50, and ε = 1124.9 (L = 200 m) and the remaining parameters shown in Table 1. Theoretical results of both u_cr and ω_cr are shown in Figure 6. u_cr is a dimensionless critical instability flow velocity and indicates the onset of instability of the internal pipe. For example, Figure 7 shows the vibration and stability of the system with α_ch = 1.20, for the first three modes. From Figure 7, we can see that when u_cr = 4.34 the first predicted instability of the system occurs, and the associated frequency ω_cr = 0, which indicates the onset of divergence. Naturally, these results presented here are related to this particular parameters selection, and only indicate what the leaching-tubing-like systems behavior could be.

As shown in Table 2, when α_ch ≈ 20, i.e., the leaching-tubing-like system is effectively unconfined, flutter instability occurs in the second mode. For 1.35 ≤ α_ch < 20, the decrease in α_ch results in flutter instability in which the modes are sequentially lowered. However, for 1.32 ≤ α_ch < 1.35, flutter instability occurs in the third mode. A further decrease of α_ch below α_ch < 1.32 can still result in flutter instability in which the modes are sequentially lowered, until first-mode divergence (i.e., a static instability) arises for 1.10 ≤ α_ch < 1.27. The vanishing of Re(ω) indicates the beginning of divergence, as clearly seen in Figure 6.

With regard to the dimensionless critical flow velocity u_cr, and with the very confined case (α_ch = 1.10) as a reference, the increase of α_ch leads initially to the rapid increase of u_cr. This is followed by a steep and significant decrease of u_cr as α_ch is increased continuously. Moreover, when α_ch > 1.48, u_cr is less than 1.00. However, with a further increase of α_ch above α_ch > 2.00 approximately, u_cr remains almost unaffected. As mentioned above, when 1.10 ≤ α_ch < 1.27, a first-mode divergence arises, so ω_cr is equal to 0. An increase of α_ch to 1.27 ≤ α_ch < 1.35 causes a significant increase of ω_cr. However, when α_ch > 1.35, decreased confinement causes a very steep reduction in ω_cr. When 1.35 ≤ α_ch < 4.46, ω_cr remains almost unaffected with the increase of α_ch. When 4.46 ≤ α_ch < 20, the system instability state changes from first-order flutter to second-order flutter, ω_cr steeply increases and remains almost unaffected with the increase of α_ch.

4.2. Effect of the Confinement Length r_ann

The effect of the confinement length r_ann (i.e., confined-flow length) on the slender leaching-tubing-like system is studied next. A larger value of the confinement length r_ann corresponds to a larger part of the cantilevered pipe being subjected to annular flow. In the schematic view of the theoretical model shown in Figure 1, the inflexible external pipes cannot be longer than the internal cantilevered pipe. Consequently, the studied values of r_ann are less than 1 and the studied maximum value of r_ann is 0.9.

The leaching-tubing-like systems behavior with the variation of r_ann from 0 to 0.90 is summarized in

Table 3. Summary of the leaching-tubing-like systems behavior with the vibration of r_ann from 0 to 0.90.

Behavior a
	F1					F2
rann	0	0.15	0.30	0.45	0.60	0.72	0.75	0.80	0.85	0.90
Behavior b
	F2		F1			F2
rann	0	0.15	0.19	0.45	0.60	0.69	0.75	0.80	0.85	0.90

F represents a kind of unstable mode, i.e., flutter instability. As an example, F1 denotes first-mode flutter. ^a [α_ch = 1.676, ε = 562.5 (L = 100 m)]. ^b [α_ch = 1.676, ε = 1124.9 (L = 200 m)].

, respectively based on α_ch = 1.676, ε = 562.5 (L = 100 m) and α_ch = 1.676, ε = 1124.9 (L = 200 m) and the remaining parameters shown in Table 1. Theoretical results of both u_cr and ω_cr are shown in Figure 8. The value of α_ch = 1.676 is used, due to the typical degree of confinement in the real application. Here, two different leaching-tubing lengths (i.e., L = 100 m and L = 200 m) are used to show the qualitative difference in system behavior.

The slender leaching-tubing-like system behavior is discussed by considering r_ann values increased from r_ann = 0. For the shorter, (L = 100 m) leaching-tubing-like system initially loses stability by the first-mode flutter. A further increase of r_ann results in the second-mode flutter. For the longer, (L = 200 m) leaching-tubing-like system, an increasing r_ann leads initially to loss of stability by the second-mode flutter. A further increase of r_ann leads to first-mode flutter and second-mode flutter.

For the shorter leaching-tubing length considered, in terms of the dimensionless critical flow velocity u_cr, an increase of r_ann from 0 to 0.50 results in a slight stabilization. u_cr increases nonlinearly with a further increase of r_ann from 0.50 to 0.72. When r_ann increases from 0.72 to 0.90, u_cr begins to decrease and reaches a minimum at r_ann = 0.78, after which u_cr continues to increase. As shown in Figure 8a, ω_cr is almost unaffected by r_ann during the first-mode flutter phase and the second-mode flutter phase. However, when the unstable state changes from the first-mode flutter phase to the second-mode flutter phase, ω_cr increases steeply. For the longer leaching-tubing length considered, the dimensionless critical flow velocity u_cr remains almost unaffected with the increase of r_ann from 0 to 0.30. Then, u_cr increases nonlinearly with a further increase of r_ann from 0.30 to 0.69. When r_ann increases from 0.69 to 0.90, u_cr begins to decrease and reaches a minimum at r_ann = 0.81, after which u_cr continues to increase. As shown in Figure 8b, ω_cr is almost unaffected by r_ann during the first-mode flutter phase and the second-mode flutter phase. However, when the unstable state changes from the second-mode flutter phase to the first-mode flutter phase, ω_cr steeply decreases. When the unstable state returns from the first-mode flutter back to the second-mode flutter, ω_cr steeply increases. Hence, for both leaching-tubing lengths considered, the curve of u_cr has a U shape, and the curve of ω_cr is stepped.

4.3. Effect of the Cantilevered Pipe Length L

Dimensional length L of the cantilevered pipe has been used in the nondimensionalization of both the flow velocity u_i and the frequency ω. Consequently, the dimensionless parameters given in Equation (41) are unsuitable for studying the dependence of the dynamics on the system length. Another set of dimensionless parameters have been defined in [9] to study the dynamic characteristics of slender hanging pipes conveying fluid and in [22] to study the dynamic characteristics of cylinders in axial flow. In the present paper, no new dimensionless quantities have been defined, but rather the dimensional critical flow velocity U_cr and associated frequency Ω_cr have been used to carried out a brief analysis. Notwithstanding the loss of generality, this approach facilitates illustrating the variation of the U_cr and Ω_cr with increasing the pipe system length L. Based on diameters, mass, and stiffness typical of a leaching-tubing, as shown in Table 1, numerical results are obtained to predict dynamical behaviors of the leaching-tubing-like systems with increasing lengths L for various values of r_ann.

As shown in

Table 4. Summary of the leaching-tubing-like systems behavior with increasing L from 5 to 500 m.

Behavior a, b, c
	F1
L (m)	5		150		300		450		500
Behavior d	F2
L (m)	5		150		300		450		500

F represents a kind of unstable mode, i.e., flutter instability. As an example, F1 denotes first-mode flutter. ^a [α_ch = 1.676, r_ann = 0.125], ^b [α_ch = 1.676, r_ann = 0.25]. ^c [α_ch = 1.676, r_ann = 0.50], ^d [α_ch = 1.676, r_ann = 0.85].

, the leaching-tubing-like systems behavior with the variation of L from 5 m to 500 m is summarized, based on r_ann = 0.125, 0.25, 0.50, and 0.85. When 5 m ≤ L ≤ 500 m, with increasing of the length, the leaching-tubing loses stability by first-mode flutter, for r_ann = 0.125, 0.25, and 0.50, while the leaching-tubing loses stability by second-mode flutter, for r_ann = 0.85. Theoretical results of both U_cr and Ω_cr are shown in Figure 9 and Figure 10. When the length L is below 40 m, an increasing of L leads initially to a nearly negligible stabilization. After that, U_cr increases significantly with a further increase of L from 40 m to 500 m. U_cr increases approximately linearly, for r_ann = 0.125, 0.25, and 0.50, while U_cr increases approximately nonlinearly, for r_ann = 0.85. For all studied values of r_ann, with increasing the pipe system length L, the critical frequency Ω_cr decrease. Especially when L < 50 m, Ω_cr decreases sharply. Moreover, for a long enough system, the frequency Ω_cr almost vanishes, which indicates a divergence is about to begin.

5. Conclusions

A linear theoretical model has been formulated for a hanging vertical cantilevered pipe which is subjected concurrently to two dependent axial flows. It should be pointed out that after the external flow exits from the outside annular passage, fluid is conveyed upwards as the internal flow in the cantilever pipe. The motion equation of the system is solved by means of a Galerkin method, and eigenfunctions of Euler–Bernoulli beam are used as comparison functions. Theoretical predictions were obtained for a long leaching-tubing-like system with parameters related to the salt cavern energy storage.

A theoretical study into the effect of radial confinement (i.e., the degree of radial confinement of the annular passage) has shown that, the system loses stability in flutter and even in divergence regardless of length, as the radial confinement degree and consequently the annular flow velocity is increased. When α_ch is greater than a certain value, u_cr decreases as α_ch increases, even to a small value. Considering that the flow velocity in industrial applications is relatively large, the value of α_ch should not be too large. Additionally, only when values of α_ch exceed a certain parameter-dependent threshold does the radial confinement has no appreciable effect on the pipe system.

The confinement length r_ann (i.e., confined-flow length), was shown to have a destabilizing or stabilizing effect on the pipe system, depending on value ranges of r_ann. The effect is a clear and significant stabilization, especially when the annular flow extends over most portion of the cantilevered pipe.

Furthermore, investigations have been conducted in dimensional terms on the effect of the pipe system length L, for the leaching-tubing-like system. The pipe system length L was shown to have a stabilizing effect on the pipe system. Increasing the pipe system length L results in an increase of dimensional critical flow velocity U_cr, while it results in a decrease of the associated dimensional frequency Ω_cr. For a long enough system, the frequency Ω_cr almost vanishes, which indicates a divergence is about to begin.

The next step is to conduct experiments to check the validity of the linear theoretical model, based on the device described in [23], and further research results are deferred to another paper.