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Dynamics of two-phase flow in vertical pipes
Journal of Fluids and Structures 87 (2019) 150–173
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Dynamics of two-phase flow in vertical pipes ∗
Ali Ebrahimi-Mamaghani, Rahmat Sotudeh-Gharebagh , Reza Zarghami, Navid Mostoufi Multiphase Systems Research Lab., School of Chemical Engineering, College of Engineering, University of Tehran, P.O. Box 11155/4563, Tehran, Iran
graphical
article
abstract
info
Article history: Received 5 November 2018 Received in revised form 2 March 2019 Accepted 12 March 2019 Available online xxxx Keywords: Pipe Gas–liquid Two-phase flow Fluid–structure interaction Dynamic behaviors Complex frequency Stability maps
a b s t r a c t In this study, a novel mathematical model was proposed for the dynamic analysis of two-phase flow in vertical pipes considering different two-phase flow models by including dissipative forces. To model the corresponding two-phase flow, common slipratio factors were utilized. The Galerkin discretization method and eigenvalue analysis were applied to solve the model equations. A detailed parametric analysis was also performed in order to elucidate the influence of various parameters such as volumetric gas fraction, flow velocity, structural damping and gravity parameter on dynamics of the system, critical flutter velocities and frequencies. The model was validated with experimental data and simulation results reported in the literature It was concluded that the pipes conveying two-phase flow are prone to experience several dynamic phenomena. Stability of the pipe structure was also examined for different two-phase flow models and the results indicated that the instability boundaries are significantly affected by the choice of the model. Furthermore, it was shown that the dynamical response of the pipe is substantially dependent on the volumetric gas fraction. Hence, the gas volume fraction can be introduced as a fundamental parameter for the vibration control of the two-phase flow systems. The results of this study would be beneficial for engineers to optimally design suitable structures for two-phase flow systems. © 2019 Elsevier Ltd. All rights reserved.
∗ Corresponding author. E-mail addresses: [email protected] (A. Ebrahimi-Mamaghani), [email protected] (R. Sotudeh-Gharebagh), [email protected] (R. Zarghami), [email protected] (N. Mostoufi). https://doi.org/10.1016/j.jfluidstructs.2019.03.010 0889-9746/© 2019 Elsevier Ltd. All rights reserved.
A. Ebrahimi-Mamaghani, R. Sotudeh-Gharebagh, R. Zarghami et al. / Journal of Fluids and Structures 87 (2019) 150–173
Nomenclature AC AG AL C EI g L Ks K M mG mL mP q qj (t) QC QG QL t UL UG uL uG uL c uL cd uL cf uL ∗ uL opt x y
pipe internal cross-sectional area, m2 area occupied by the gas in the pipe internal cross-section, m2 area occupied by the liquid in the pipe internal cross-section, m2 dimensionless damping matrix of the system pipe flexural rigidity, N.m2 gravitational acceleration, m/s2 length of pipe, m slip ratio defined by K s =U G /U L dimensionless stiffness matrix dimensionless mass matrix mass of gas per unit length, kg/m mass of liquid per unit length, kg/m mass of pipe per unit length, kg/m vector generalized coordinate jth dimensionless time-dependent generalized coordinate total volumetric flow rate, m3 /s volumetric gas flow rate, m3 /s volumetric liquid flow rate, m3 /s time, s average liquid velocity, m/s average gas velocity, m/s dimensionless liquid velocity defined by uL =(M L /EI)0.5 U L L dimensionless gas velocity defined by uG =(M G /EI)0.5 U G L dimensionless critical liquid velocity dimensionless divergence velocity dimensionless coupled-mode flutter velocity dimensionless liquid velocity at mode-exchange dimensionless optimum liquid velocity longitudinal coordinate, m lateral deflection of the pipe, m
Greek symbols
αG βG βL δ ij εG εG ∗ ζ η λj µ ξ ρG ρL τ ϕ j (x) ω ωc
gas void fraction defined by α G =AG /AC dimensionless gas mass ratio defined by β G =mG /(mG +mL +mP ) dimensionless liquid mass ratio defined by β L =mL /(mG +mL +mP ) Kronecker delta function gas volume fraction defined by ε G =Q G /Q C gas volume fraction at the mode-exchange phenomenon damping ratio of the system defined by ζ =Image(ω)/Real(ω) dimensionless deflection of the pipe defined by η =Q G /Q C eigenvalue of cantilever for mode j hysteretic structural damping coefficient dimensionless longitudinal coordinate defined by ξ =x/L gas density, kg/m3 liquid density, kg/m3 dimensionless time defined by τ = t/ ((mp +mL +mG )L4 /EI)0.5 jth eigenfunction of the cantilevered beam dimensionless frequency dimensionless critical flutter frequency
Subscripts L G
liquid phase gas phase
151
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1. Introduction Structures like pipes that convey fluid are widely used in chemical plants, municipal water supply, marine structures, heat exchanger tubes, hydropower systems, boiling water reactors, etc. Hence, predicting the performance of these systems from different points of view, such as heat transfer, computational fluid dynamics (CFD) (Delnoij et al., 1999), pressure fluctuations (Matsui, 1984) and flow pattern signature (Shaikh and Al-Dahhan, 2007) is vital. The presence of air in water pipelines causes many drawbacks, such as loss of carrying capacity, change of physical properties of fluid, disruption of hydrodynamics and reduction of the system efficiency (Pozos et al., 2010a,b). Therefore, it is important to understand the gas–liquid two-phase flow characteristics. Thus, the characterization of the hydrodynamics of such structures has been the subject of numerous investigations during the last decades (Shoham, 2006; Miwa et al., 2015; Pettigrew and Taylor, 1994; Boure et al., 1973; Ruspini et al., 2014; Chalgeri and Jeong, 2019; Lips and Meyer, 2011). Despite the low density of gas in gas–liquid systems, dynamics of pipes conveying two-phase flow is of great interest. For instance, Nakamura et al. (1995) experimentally investigated the instability conditions for tube arrays in air–water flow and reported critical flow velocities in extreme conditions. Pettigrew et al. (2001) studied the effects of mass flux and void fraction on vibrating excited mechanisms of heat exchanger tube bundles of various geometries by conducting multiple tests. They revealed that the vibration amplitude of the tubes is roughly proportional to the mass flux. Sheikhi et al. (2013) measured vibration and pressure fluctuations simultaneously coupled with photograph and image analysis in order to study the hydrodynamics of bubble columns. They showed that by calculating skewness and kurtosis of pressure fluctuations and wall vibration, one can easily detect the regime transitions. Monette and Pettigrew (2004) carried out several experiments on hanging cantilevered pipes carrying two-phase flow for understanding the fluid-elastic instability. Moreover, they proposed a modified two-phase flow model to analyze the vibrational behavior of hanging tubes. Zhang and Xu (2010) conducted an experimental study on the influence of internal bubbly flow in a pipe. They indicated that vibration characteristics of the pipe conveying two-phase flow principally depend on variations in void fraction. Shiea et al. (2013) experimentally determined the flow regime transition onset in a bubble columns by employing a resistivity probe. Adhami et al. (2018) carried out diverse experiments to characterize the hydrodynamics of bubble columns containing non-newtonian fluids. They evaluated the effect of superficial gas velocity on the bubble size based on pressure fluctuation analysis. Experimental investigations on the dynamics of two-phase flow systems have been widely reported in the literature, however, a limited number of theoretical studies have been conducted on this topic. Hara (1977) assessed two-phase flowinduced vibrations of horizontal pipe structures theoretically and experimentally in which the frequency of void signals drastically affects the fundamental frequency of the piping systems. Zhang et al. (2010) probed bubbly flow induced pipe vibrations experimentally and theoretically and reported that the power spectral density of pipe vibrations and pressure fluctuations are considerably increased by an increase in the bubble diameter. Seyed and Patel (1992) calculated pressure and internal slug flow induced forces on straight and curved pipes via a mathematically rigorous method. They disclosed that high internal flow-rates can amplify the internal pressure effects. Adegoke and Oyediran (2017) surveyed the nonlinear dynamics of cantilevered pipes conveying two-phase flow subjected to thermal loads. They utilized multiple scales method to survey the axial and transverse vibrations and extracted natural frequencies and critical velocities of the system for a set of void fractions. They explored the effects of Poisson’s ratio and pressurization on the dynamical response of the pipe. Wang et al. (2018) presented a dynamical model for horizontal pipes in the slug flow. They solved the governing equations of motion by employing the finite element method (FEM) and verified their results with experimental data. They showed that the slugging transition velocity has a significant effect on the system properties such as damping and stiffness. Ortega et al. (2012) performed a numerical study on the interaction between unsteady two-phase flow and the dynamical response of flexible pipes. They calculated stress variations, pressure, and lateral deflections by the linear FEM. An and Su (2015) presented a parametric study on the dynamical behavior of two-phase flow in pipes using generalized integral transform technique (GITT) and also investigated the influences of several parameters, such as volumetric gas fraction and void fraction, on time response of the system numerically. Liu and Wang (2018) obtained the natural frequencies of slug flow in a pipe using FEM. They predicted the conditions in which the divergence instability in the system is probable. They also showed that the root-mean-square of the natural frequencies of the system initially increases and then decreases by increasing the superficial liquid velocity. In the piping systems, preventing instability and undesirable vibrations is a mandatory engineering requirement. Nevertheless, due to complex hydrodynamics and operational challenges, a limited number of analytical investigations have reported the stability maps and natural frequencies of such systems. In this study, a comprehensive investigation was carried out on fluid-elastic vibrations of pipes. In this regard, the most significant dynamic aspects of these systems, including the natural frequencies and instability conditions, are obtained. Herein, hanging and upward cantilevered pipes were studied considering the gravitational and hysteretic effects based on the Euler–Bernoulli theorem. The homogeneous model and also other two-phase flow models available in the literature (Monette and Pettigrew, 2004; Chisholm, 1983) were also utilized. Galerkin discretization method was employed to obtain the reduced-order model equations. Eigenvalue analysis was implemented to evaluate the stability of the structure. The influences of various parameters such as flow velocity, gas volume fraction and structural damping on various modes of the system, critical flutter, and divergence velocities were discussed. Besides, Argand diagrams and stability maps, as useful tools for designing of the considered structure, were used to survey the dynamical behavior of the system.
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Fig. 1. Main flow patterns in vertical pipes.
2. The examined system 2.1. Background The flow pattern is a qualitative description of the phase mixture in the system, which strongly depends on the flow direction, flow rate, physical properties of fluids and geometrical properties of the pipe. Different mass transfer mechanisms in two-phase flow systems can be identified by flow patterns. Various flow patterns have been reported in the literature (Rouhani and Sohal, 1983). Among them, four main patterns may co-exist in vertical pipes shown in Fig. 1, as described below (McQuillan and Whalley, 1985): (i) Bubbly flow: the liquid flow rate is high enough to disperse gas into discrete distorted bubbles, randomly distributed through the continuum liquid phase (Fig. 1a). (ii) Slug flow: at higher gas flow rates, bubble coalescence occurs and large bubbles are formed that occupy most of the cross section of the column (Fig. 1b). The slugs of gas move with a higher velocity than the average liquid phase velocity. (iii) Churn flow: as the gas flow is further increased, gas slugs become unstable and then collapse, resulting in a turbulent chaotic flow (Fig. 1c). The main characteristic of this pattern is the chaotic oscillatory motion of the liquid phase. (iv) Annular flow: this flow pattern occurs at relatively high gas and low liquid flow rates. In this case, the gas becomes the continuous phase. The liquid flows as a film along the pipe wall while high-velocity gas moves through the center, carrying dispersed liquid droplets (Fig. 1d). The principle parameters for defining a two-phase flow are gas volume fraction (εG ), the ratio of gas to total flow rates (QG /QC ), the void fraction (αG ), the ratio of the area occupied by the gas to the total area (AG /AC ) and the slip ratio (Ks = UG /UL ). Defining UL = QL /AL and UG = QG /AG , the relation between the aforementioned parameters can be presented by Chisholm (1983): 1 − εG
εG
( =
1 − αG
αG
)
1 Ks
(1)
The most basic model giving a relation between gas and liquid phases is the homogeneous model in which both phases have the same velocity (Ks = 1, αG = εG ). Due to neglecting many effects, such as the buoyancy effect, this model may not be realistic, especially at high values of void fraction (Feenstra et al., 2002). To overcome this problem, several experimental investigations have been carried out to quantify the slip velocity (Woldesemayat and Ghajar, 2007). In this study, the slip ratio in vertical tubes was obtained from:
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Fig. 2. Slip ratios (Ks ) estimated by various available models against the volumetric gas fraction (εG ) for vertical pipes.
For hanging pipes (Monette and Pettigrew, 2004):
( Ks =
1 − εG
)0.5 (2)
εG
For upward pipes (Chisholm, 1983): 1 Ks = [ ( 1 − εG 1 −
ρG ρL
)]0.5
(3)
It is noteworthy that in the model presented by Chisholm (1983), the maximum value of the slip ratio is (Ks )max = (ρL /ρG )0.25 . The slip-ratios evaluated by different models are demonstrated in Fig. 2 with respect to the gas volume fraction for the air–water flow. 2.2. Problem formulation Fig. 3 illustrates upward and downward cantilevered pipes that carry two-phase flows. The pipes are of length L, mass per unit length mp , flexural rigidity EI and hysteretic structural damping coefficient µ. The transverse displacements of the pipes are demonstrated by y(x, t). The flow consists of discrete gaseous bubbles in the continuous liquid phase or discrete liquid droplets in the continuous gas phase medium. The parameters mL and mG indicate the mass of liquid and gas per unit length of the pipes, respectively, and their corresponding velocities are shown by UL and UG The dynamical equation of motion of the aforementioned system can be written as (Monette and Pettigrew, 2004; An and Su, 2015; Paidoussis, 1970):
) 2 ∂ 4y ( ∂ 2y 2 2 ∂ y + m U + m U + 2 m U + m U ( ) L G L L G G L G ∂ x4 ∂ x2 ∂ x∂ t ) ( ∂ 2y ∂ 2y ∂ y + (mP + mL + mG ) 2 + (mP + mL + mG ) g (x − L) 2 + =0 ∂t ∂x ∂x
EI (1 + µi)
(4)
The terms of Eq. (4) represent, respectively: elastic restoring force with structural damping effects, centrifugal, Coriolis, √ inertial and gravitational forces (i = −1). The equation can be applied to bubbly flow and annular flow regimes. In order
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155
Fig. 3. A schematic view of (a) upward and (b) hanging cantilevered pipes conveying two-phase flow.
to obtain the dimensionless form of the equation of motion, the following variables are introduced: x
y
t
L
L
T
ξ = ,η = ,τ =
(5)
where T is defined as: T =L
2
(
mP + mL + mG
)0.5 (6)
EI
By substituting Eq. (5) into Eq. (4), the dimensionless equation of motion is acquired as: (1 + µi)
) ∂ 2η ( ) ∂ 2η ∂ 4η ( 2 ∂ 2η + uL + u2G + 2 βL0.5 uL + βG0.5 uG + 2 4 2 ∂ξ ∂ξ ∂ξ ∂τ ∂τ ( ) ∂ 2 η ∂η =0 + γ (ξ − 1) 2 + ∂ξ ∂ξ
(7)
The dimensionless parameters appeared in Eq. (7) are defined as: mG
βG =
mP + mL + mG ( m )0.5 L uL = UL L , EI
,
βL =
mL
,
mP + mL + mG ( m )0.5 mP + mL + mG G uG = U G L , γ = ±gL3 EI EI
(8)
It is worth mentioning that in the gravity parameter (γ ), positive and negative signs refer to hanging and upward pipes, respectively. By utilizing Eqs. (1) and (8), one may write:
βG =
mP AC
ρG αG ρ G εG β L = ρL Ks (1 − εG ) + ρL (1 − αG ) + ρG αG
(9)
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[ uG = uL K s
ρG εG ρL (1 − εG )
]0.5 (10)
where ρG and ρL are gas and liquid density, respectively. Substitution of Eqs. (9) and (10) into Eq. (7) yields:
( ) 2 ∂ 4η ρ G εG ∂ η 2 (1 + µi) 4 + uL 1 + Ks ∂ξ ρL (1 − εG ) ∂ξ 2 ( ) 2 ρ G εG ∂ η ∂ 2η + 2βL0.5 uL 1 + + 2 ρL (1 − εG ) ∂ξ ∂τ ∂τ ( ) 2 ∂ η ∂η + γ (ξ − 1) 2 + =0 ∂ξ ∂ξ
(11)
Third and fourth terms in this equation present Coriolis and centrifugal forces in upward pipes conveying two-phase flow, respectively, equivalent to what occurs in the pipes conveying single liquid-phase flow. 3. Solution procedure 3.1. Galerkin approach The reduced-order model of the equation of motion, which is a simplification of a high accuracy dynamical model that preserves dominant behavior and necessary effects of the system, can be acquired from the partial differential equation (Eq. (11)) using the Galerkin method. Hence, the transverse displacement of the pipe can be approximated as (Mamaghani et al., 2016; Hosseini et al., 2017; Mamaghani et al., 2018):
η (ξ , τ ) =
n ∑
ϕj (ξ ) qj (τ )
(12)
j=1
where n is the number of modes to be considered, qj (τ ) is the jth time-dependent generalized coordinate and ϕj (ξ ) is ∫1 the jth eigenfunction of the cantilevered beam. Note that ϕj (ξ ) is normalized such that 0 ϕj (ξ )ϕk (ξ )dξ = δjk . These eigenfunctions are defined in the following equation (Dehrouyeh-Semnani et al., 2016; Gregory and Paidoussis, 1966):
( ) ( ) ) ( ) ( ) sinh λj − sin λj ( ( ) sinh(λj ξ ) − sin(λj ξ ) ( ) ϕj (ξ ) = cosh λj ξ − cos λj ξ − cosh λj + cos λj
(13)
In this equation, values of λj can be evaluated from the frequency equation cosh(λj )cos(λj ) + 1 = 0. By substituting Eq. (13) into Eq. (11) and multiplying by ϕk and then integrating over the domain [0, 1], the partial differential equations is transformed into an ordinary equation as:
¨ (τ ) + Cq( ˙ τ ) + Kq(τ ) = 0 Mq
(14)
in which dots indicate temporal derivation. Eq. (14) represents the discrete form of the equation of motion in the matrix form, where q is the vector generalized coordinate, M is the dimensionless mass matrix, C is the dimensionless damping matrix and K is dimensionless stiffness matrix. These matrices are functions of geometrical and physical properties of the system and are defined as follows: q = [q1 (τ ) , q2 (τ ) , . . . , qn (τ )]T
∫
(15)
1
Mjk =
φj (ξ )φk (ξ )dξ = δjk
0 (
Cjk = 2 βL0.5 uL (1 +
Kjk = (1 + µi)
∫
ρ G εG ) ρL (1 − εG )
1
φj (ξ )
0
1
φj (ξ ) 0
ρG εG ρL (1 − εG )
1
∫
(ξ − 1) φj (ξ ) ( 0
∂φk (ξ ) dξ ∂ξ
(17)
∂ 4 φk (ξ ) dξ ∂ξ 4
( ( + u2L 1 + Ks +γ
(16)
)∫
1
)) ∫
φj (ξ ) 0
∂ 2 φk (ξ ) dξ ∂ξ 2
∂φk (ξ ) ∂ 2 φk (ξ ) + )dξ ∂ξ ∂ξ 2
(18)
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3.2. Stability methodology The standard second-order matrix form of Eq. (14) can be reduced to a first-order differential form by the following procedure (Paidoussis, 1998): BZ˙ (τ ) + EZ (τ ) = 0
(19)
where:
[ B=
0 M
M C
]
,
[ E=
−M 0
0 K
]
,
Z(τ ) =
[
q(τ ) ˙ τ) q(
] (20)
Assuming the solution to have the formZ(τ ) = Aeiωτ , the reduced-order Eq. (19) yields the following eigenvalue problem: YA − iωI = 0
(21)
in which I shows the identity matrix and Y = −B−1 E. Finally, the complex frequencies (ω) of the system can be computed numerically as a function of the flow velocity, volumetric gas fraction, structural damping and gravitational parameter. 4. Results and discussion To investigate the dynamical behavior of the system, Argand diagrams of pipes conveying air–water two-phase flow with respect to dimensionless fluid velocity are plotted for typical values of the physical parameters (ρG = 1.2 kg/m3, ρL = 1000 kg/m3 ). Horizontal and vertical axes in these diagrams are real and imaginary parts of the dimensionless complex natural frequencies of the system and show the resonant frequency of oscillations and damping of the system, respectively. Damping ratio of the system can be defined by ζ =Image(ω)/Real(ω). As one of the branches of the frequency diagram crosses the horizontal axis (i.e., Real(ω)̸ =0, Image(ω) =0), Hopf bifurcation occurs in the system leading to flutter instability (point H). The minimum velocity at which the flutter phenomenon happens is known as the critical flutter velocity (ucL ) and its corresponding frequency is known as the critical flutter frequency of the system (ωc ). Buckling or divergence instability happens when one of the branches crosses the origin from positive to negative on the [Image(ω)]axis. In addition, Paidoussis coupled-mode flutter bifurcation (points Pi ) takes place when two loci of branches whether leave or coalesce on the [Image(ω)]-axis (Paidoussis, 1998). Furthermore, velocities shown in Argand diagrams are related to the liquid phase, while one can calculate gas phase velocity from Eqs. (8)–(10). Paidoussis (1970) investigated the effect of number of terms in Eq. (12) in the estimation of dynamical characteristics of the system and found that a 10-term approximation yields reliable results. 4.1. Hanging pipes In order to evaluate the numerical results, complex frequencies for the lower modes of a hanging pipe conveying single liquid-phase flow are plotted in Fig. 4a) for γ = 10, µ = 0 and βL = 0.3. It can be seen in this figure that in the single-phase case, the Argand diagram is in close agreement with results reported by Paidoussis and Issid (1974). Then, by increasing the gas volume fraction, while keeping other parameters constant, the behavior of the loci was drawn based on Monette’s model (Eq. (2)) and its dynamical behavior was analyzed (Fig. 4a, b). Neglecting damping effects, if one considers the flow velocity equal to zero, all frequencies located on the real axis for both single and two-phase flows indicate natural frequencies of a corresponding pipe without flow. At lower velocities, since the difference between the momentums of single and two-phase flows is insignificant, both systems behave almost the same. However, the more the flow velocity, the more the corresponding branches separate from each other. Also, increasing the gas volume fraction results in a prominent deviation from the single liquid-phase case and subsequently displaces bifurcation points more and more. For instance, Fig. 4 demonstrates that in the single-phase case, Paidoussis coupled-mode flutter bifurcations are located at ucf L ≈6.5 and 8.4. However, when two-phase flow exists in the pipe (εG = 0.4), these bifurcations occur at ucf ≈ 6.8 and 9. In other words, overdamped responses were acquired at higher speeds when considering the two-phase L flow and in this situation, the pipe does not vibrate for the first mode. In addition, for all vibration modes, except the first mode, values of imaginary part of the frequencies have decreased significantly. Consequently, as compared with the single liquid-phase flow, the two-phase flow encounters a higher damping ratio for the first mode and smaller for higher-order modes. Since transforming the flow regime from single liquid-phase to two-phase decreases the critical flow velocity (single liquid-phase flow: ucL ≈8.7 and two-phase flow (εG = 0.4): ucL ≈6.5), one can conclude that in the two-phase flow, the stability of hanging pipes decreases. Also, in comparison with the single-phase case, increasing the gas volume fraction results in instability of the fourth mode at uL ≈14. In Fig. 5, the Argand diagram of two-phase flow is shown in comparison with the single liquid-phase case for γ = 100, µ = 0 and βL = 0.2 (Paidoussis, 1970) and various gas volume fractions. It can be seen in this figure that the third mode is unstable in the single liquid-phase case (at ucL ≈10.5). By increasing the volumetric gas fraction to the critical value of εG∗ = 0.54, the imaginary part of the second mode decreases to the proximity point where instability occurs (at
158
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Fig. 4. Dimensionless complex frequency of hanging cantilevered pipe (ω) as a function of dimensionless fluid velocity (uL ) for γ = 10, µ = 0, βL = 0.3.
uL ≈10). Nevertheless, at u∗L ≈10.2, the loci of second and third modes are extremely close to each other and the system undergoes mode-exchange phenomenon which eventually results in instability on the third mode while the second mode
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Fig. 5. Dimensionless complex frequency of hanging cantilevered pipe (ω) as a function of dimensionless fluid velocity (uL ) for γ = 100, µ = 0, βL = 0.2 and various εG .
Fig. 6. Dimensionless complex frequency of hysterically damped hanging cantilevered pipe (ω) as a function of dimensionless fluid velocity (uL ) for γ = 10, µ = 0.1, βL = 0.65.
remains stable. The mode-exchange phenomenon is an exclusive characteristic of non-conservative systems and has been investigated by Ryu et al. (2002) for single liquid-phase flows. In this case, for a specific velocity, the eigenvalues of different modes approach each other and the probability of transference branches increases in the system. Further increase in the gas volume fraction (e.g., εG = 0.8) causes the second mode to become unstable while the third mode becomes stable. Therefore, the volumetric gas fraction may be interpreted as a fundamental parameter which controls transversal vibrations of elastic pipes conveying two-phase flow.
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Argand diagrams are depicted in Fig. 6 for both single liquid-phase and two-phase flows, considering the hysteretic effect, where γ = 10, µ = 0.1 and βL = 0.65. As expected, at zero flow velocity, due to the existence of structural damping, the loci of frequencies of the system separate from the real axis. Also, the Argand diagram loses its symmetry with respect to the imaginary axis and negative modes (the modes starting with the negative real part) have a tendency to cross the imaginary axis and move towards higher +Real(ω). The first negative mode (1− ) has caused the system to become unstable for both single liquid-phase and two-phase cases at ucL ≈10.6 and ucL ≈10.1, respectively. In addition, for values of εG∗ = 0.23 and u∗L = 6.87, mode-exchange occurs between 1− and 2+ modes in the two-phase case. Generally, the mode-exchange phenomenon occurs for sequential positive modes in pipes conveying single-phase flow. However, in the case of two-phase flow regarding the hysteretic effect, this is not valid. Another feature of the two-phase flow in Fig. 6 is that at higher velocities, meanders are observed in the loci of the fourth mode (uL ≈16). An important feature in Fig. 6 is that in the two-phase flow, the imaginary part of the first negative mode (the unstable mode) reaches its maximum value at uL ≈6.87 and this velocity can be introduced as the optimum velocity of the pipe opt (uL ). At this specific velocity, the system reaches the maximum damping value (decaying rate of the amplitude) for the first negative mode which eventually destabilizes the system. At this point, the system is maximally stabilized. Sugiyama et al. (1996) confirmed that the Coriolis force has a stabilizing effect on the pipe conveying single-phase flow, hence, can be utilized as the oscillation absorber. It is worth mentioning that for the single liquid-phase flow, the optimum velocity opt of the pipe is uL ≈6.4. To clarify this issue, the following time responses of the system for various values of the fluid velocity are demonstrated in Fig. 7. Effects of flow velocity on dynamical behavior of the pipe and the transition of the system to flutter instability are shown in Fig. 7 for γ = 10, µ = 0.1, βL = 0.65 and εG = 0.23. The dimensionless initial condition is set as q1 (0) = 10 and q2 (0) = · · · = qn (0) = q˙ 1 (0) = · · · = q˙ n (0) = 0. First, considering the flow velocity to be zero, the system behaves like a hanged cantilevered beam (Fig. 7a). Increasing the flow velocity results in two mutual effects: the Coriolis forces try to stabilize the system while centrifugal forces trying to destabilize it. Because of the linear relationship with the flow velocity, Coriolis forces dominate the system behavior in the lower flow velocities. Therefore, the system initially tends opt to dampen the oscillations (Fig. 7b). These dissipative effects continue up to the optimum velocity uL = 6.87. Increasing the flow velocity increases the centrifugal forces which have a second order relationship with the flow velocity. This leads to an increase in the oscillation amplitude (Fig. 7c). At the critical flutter velocity, the amplitude of system oscillations becomes constant (Fig. 7d). Finally, any enhancement in the flow velocity causes the induced vibration to amplify and the amplitude of oscillations to grow exponentially, hence the system becomes unstable (Fig. 7e). Generally, at lower flow velocities, the system loses energy while at velocities higher than the critical flutter velocity, the system gains energy. Comparison between various models available for two-phase hanging pipes is made in Fig. 8, with µ = 0, γ = 100 and βL = 0.65. This figure demonstrates a significant difference between corresponding critical flutter velocities and frequencies and a distinct trend can be observed for dynamical behavior of both cases, especially for lower modes. As shown in Fig. 2, for two-phase homogeneous and Monette’s models, the difference of slip ratios is much more prominent in high and low gas fractions (εG ) and, obviously, its influence increases for higher values of flow velocity. In Fig. 8, in comparison with the single liquid-phase case, both two-phase models indicate that the fourth mode becomes unstable. It can be concluded from Fig. 8 that determining the equivalent fluid velocity (or slip ratio), especially at higher gas volume fractions, has a considerable effect on the dynamical response of pipes carrying two-phase flow. Undoubtedly, two-phase flows cause fluctuations in average velocity and mass flux of the system. It is noteworthy to state that variations in the loci of frequencies are significant in the vicinity of the imaginary axis, or when a jump or mode exchange occurs. Hence, to precisely draw the plots, one must utilize a higher number of modes and simultaneously make the velocity intervals smaller. In order to investigate the stability of the two-phase flow system as compared with single liquid-phase flows, the stability map is plotted in Fig. 9 for two different values of the gas volume fraction in ucL -γ and ωc -γ planes. This figure portrays flutter instability boundaries. Within the stable region, any arbitrary initial conditions lead to the elimination of the pipe vibration. On the other hand, for velocities above each curve, any dynamical perturbation grows until a failure occurs in the system. Moreover, in the stability maps, sequential jumps are observed at various γ values. In the case of jumps, the system is stable to the right side of the curve and is unstable otherwise (Fig. 9a). By scrutinizing the stability map, it can be concluded that increasing γ leads to an increase in critical flutter velocities and frequencies. This trend can be justified by the fact that any increase in γ results in obtaining a stiffer system, hence, the stability region becomes wider. Therefore, one can conclude that the greater γ , the more stable the system becomes. It can be seen in the stability map shown in Fig. 9, especially for larger γ values, that two-phase flow regimes can significantly influence flutter boundaries. As expected, in the two-phase case in comparison with the single liquid-phase flow, smaller critical flutter velocities and frequencies are acquired which are indicative of shrinkage of the stability region. In other words, in the case of two-phase flow, the pipe becomes more flexible. On the other hand, the stability boundaries of two-phase flow are above those of single gas-phase flow. In the case of single liquid-phase flow in the pipe, four of the so-called S-shaped segments can be observed, each of which demonstrates three unique dynamical characteristics related to instability-restabilization-instability sequences (Paidoussis, 1998). Before the formation of each S-shaped segment, the number of specific mode destabilizing the system changes and the mode exchange phenomenon occurs. Another aspect observed in Fig. 9 is that smaller jumps
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Fig. 7. Variation of dimensionless transverse tip deflection of a hysterically damped hanging cantilevered pipe conveying air–water two-phase flow against dimensionless time for γ = 10, µ = 0.1, βL = 0.65, εG = 0.23.
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Fig. 7. (continued).
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Fig. 7. (continued). Table 1 Characterization of the experimental setup used by Monette and Pettigrew (2004). EI (N.m2 )
L (m)
mp (kg/m)
D (m)
µ
0.003
0.547
65.7 × 10−3
9.25 × 10−3
0.1
are seen when the flow is in two-phase (εG = 0.4 and 0.7), which shifts the stability boundaries toward larger γ values. In other words, for two-phase flows, these S-shaped segments may occur in wider ranges of gravity parameter, as can be clearly observed in Fig. 9. Hence, the number of S-shaped segment decreases in the case of two-phase flow. This means that two-phase flow delays the formation of S-shaped segments. In single gas-phase flow as compared with liquid-phase, the average mass flux of the internal flow in the system decreases substantially. As shown in Ref. (Paidoussis, 1998), S-shaped segments and jumps are vanished at small mass ratios for various gravity parameters. Furthermore, Monette and Pettigrew (2004) have experimentally examined stability boundaries for single liquid and gas-phase flows and they did not observe S-shaped segments and jumps in the stability maps. The effect of damping on the stability of the structure is shown in Fig. 10. Generally, including the damping effect makes the jumps to become smaller. It is also important to note that for some specific ranges of γ (for instance, εG = 0.7 and γ >700), critical flutter velocities and frequencies become larger than the corresponding ones when neglecting dissipative effects. This phenomenon was discovered for the first time by Ziegler for non-conservative gyroscopic systems (Ziegler, 1952) and then was observed for horizontal cantilevered pipes conveying single liquid-phase flow by Semler et al. (1998). According to Fig. 10, one can state that the damping influence on the single-phase flow is more prominent than in two-phase flows. Fig. 11 shows the stability boundaries of hanging cantilevered pipes considering homogeneous and Monette’s twophase models. According to this figure, instability boundaries, especially jumps phenomena, are considerably affected by two-phase flow models. As mentioned before, by choosing a gas fraction close to either zero or one, the difference between critical flutter velocities and frequencies evaluated from the two proposed models increases. In Fig. 11, for εG >0.5, the critical flutter velocities and frequencies evaluated by the homogeneous model are less than those proposed by the Monette’s model, owing to the fact that in the Monette’s model, the slip ratio is predicted to be greater than in the homogeneous model, which causes greater gas fraction (εG ). In other words, when εG >0.5, the Monette’s model predicts higher values of flexural rigidity of the system than the homogeneous model. In conclusion, employing a precise model capable of explaining the accurate relation between the gas and liquid phases is a must at lower or higher gas volume fractions.
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Fig. 8. Real and imaginary components of the dimensionless complex frequency of hanging cantilevered pipe (ω) as a function of dimensionless fluid velocity (uL ) for µ = 0 and γ = 100, βL = 0.65 (a) single liquid-phase flow (εG = 0) (b) Monette’s model (εG = 0.8) (c) homogeneous model (εG = 0.8).
In order to examine results of the two-phase flow case, variations of calculated dimensionless critical liquid and gas phase velocities versus gas void fraction are compared in Fig. 12 with the experimental data reported by Monette and Pettigrew (2004) for the conditions of Table 1. As illustrated in this figure, the critical liquid phase velocity decreases by an increase in the gas void fraction, leading to increase in the critical gas phase velocity. The performance of the model was assessed quantitatively by extracting the average relative error and correlation coefficient. The average relative error of the dimensionless critical liquid and gas phase velocities are found to be 15.65% and 7.87%, respectively. Furthermore, the correlation coefficients of the considered parameters are 0.99 and 0.97, respectively. As discussed earlier, a notable phenomenon in the pipe structures is the occurrence of jump as a result of variation of the gas volume fraction. The structure stability boundaries are plotted against the gas volume fraction variation in Fig. 13. Obviously, εG = 0 corresponds to the single liquid-phase flow in the pipe. It can be seen in these figures that a larger gravity parameter leads to a more stable structure. Hence, the results shown in Fig. 13 demonstrate that employing a larger gravity parameter stabilizes the pipe in general, yielding higher critical flutter velocities and frequencies. Fig. 13 shows that in contrast to the stability maps, the ucL -εG and ωc -εG curves are overall descending with increasing the gas volume fraction. Moreover, it can be seen in this figure that when gravity parameter is small (e.g., γ = 10), due to a reduction in the average mass flux of the two-phase mixture, critical flutter velocity and frequency of the system monotonically decrease with increasing the gas volume fraction. At a relatively large γ (e.g., γ = 50 or 100), however, the critical flutter velocities and frequencies do not monotonously decrease with increasing the gas volume fraction and the ucr -εG curves exhibit Z-shaped segments. Extensive calculations for larger γ s (e.g., γ = 200 or 300) show more clear Z-shaped segments. Argand diagrams are shown in Fig. 14(a, b) demonstrate that the occurrence of each Z-shaped segment is associated with the instability-restabilization-instability sequence when the dimensionless flow velocity is monotonically increased. Consequently, at small γ values, the flutter instability may occur in a certain mode, while at large γ values, exchange of unstable modes appears and various modes experience instability. To better understand the role of gas volume fraction in vibration characterization of pipe structures, Argand diagrams are illustrated in Fig. 14(a, b) for γ = 100, µ = 0 and βL = 0.65. According to these figures, two jumps occur at εG ≈0.1 and 0.83. In the single liquid-phase flow, instability occurs in the first mode of the system. However, at εG ≈0.1, the preferred mode of flutter instability is the fourth one. Increasing the gas volume fraction leads to instability in the third mode. Since the third mode is approximately tangent to the +Real(ω) axis, the critical flutter velocity and frequency of the system undergo dramatic changes as a result of minor variations in the gas volume fraction.
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Fig. 8. (continued).
165
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Fig. 9. Stability map of hanging cantilevered pipe as a function of dimensionless gravity parameter (γ ) for βL = 0.65, µ = 0 and (a) dimensionless critical flutter velocity (ucL ) and (b) dimensionless critical flutter frequency (ωc ).
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Fig. 10. Effect of internal damping (µ) on critical flutter velocity (ucL ) of hanging cantilevered pipes for βL = 0.65.
Fig. 11. Stability map of hanging cantilevered pipe as a function of dimensionless gravity parameter (γ ) by considering various two-phase flow models for βL = 0.65 and µ = 0.
4.2. Upward pipes Argand diagrams of upward pipes conveying single liquid-phase flow are portrayed for various γ s and are shown in Figs. 15 and 16, where µ = 0, βL = 0.2 and γ =−10 and −50, respectively. These diagrams are in good agreement with
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Fig. 12. Comparison between simulation and experimental data of dimensionless critical liquid (ucL ) and gas (ucL ) phase velocities of hanging cantilevered pipe carrying two-phase flow as a function of gas void fraction (αG ) for conditions of Table 1.
Fig. 13. Dimensionless critical flutter velocity (ucL ) of hanging cantilevered pipe conveying two-phase flow as a function of volumetric gas fraction (εG ) for µ = 0 and βL = 0.65.
those reported by Paidoussis (1970). By increasing the gas volume fraction, while keeping other parameters constant, the behavior of complex natural frequencies of the system are drawn based on Chisholm’s model (Eq. (3)). Similar to hanging
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Fig. 14. Dimensionless complex frequency of hanging cantilevered pipe (ω) as a function of the dimensionless fluid velocity (uL ) for γ = 100, µ = 0, βL = 0.65.
pipes, increasing the gas volume fraction results in decreasing critical flutter velocities and frequencies of the system. According to Fig. 16, for small γ values, however, changes in gas volume fractions does not significantly influence the critical flutter velocities and frequencies of the system. In Fig. 15, the optimum fluid velocities for single liquid-phase and opt
two-phase flows are uL
= 3.25 and uopt = 3.15, respectively. L
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Fig. 15. Dimensionless complex frequency of upward cantilevered pipe (ω) as a function of the dimensionless fluid velocity (uL ) for γ =−10, βL = 0.2 and µ = 0.
Fig. 16. Dimensionless complex frequency of upward cantilevered pipe (ω) as a function of the dimensionless fluid velocity (uL ) for γ =−50, βL = 0.2 and µ = 0.
It is observed in Figs. 15 and 16 that the frequencies of upward pipes become pure imaginary for the first mode. As the flow velocity is increased, these frequencies move toward the origin. As mentioned before, the velocity at which the
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Fig. 17. Stability map of upward cantilevered pipe as a function of the dimensionless gravity parameter (γ ) for βL = 0.65 and various εG .
frequency of the system becomes absolutely zero is known as the thresholds of divergence for upward pipes under their own weight (i.e., Real(ω)̸ =0, Image(ω)=0). The critical flutter velocity at which this phenomenon is observed is known as the critical divergence flow velocity (udL ). This phenomenon is also probable to occur in pipes conveying single liquid-phase flow with both ends supported (Paidoussis and Issid, 1974). The calculations of the previous section show that in hanging pipes conveying two-phase flow, similar to the single liquid-phase case, the system does not undergo the divergence instability for any value of flow velocity. However, in the upward pipes, this phenomenon happens for a specific range of γ . It is worth mentioning that in the upward pipes, the structure first undergoes divergence instability and then, by increasing flow velocity as the frequency branch crosses the origin, the pipe stabilizes. Continuing to increase the flow velocity causes the frequency branches to cross the real axis, thus, flutter instability initiates. For a better illustration of the vibrational behavior of upward pipes, stability map of the system is shown in Fig. 17 for upward pipes conveying single and two-phase flows for βL = 0.65 and µ = 0. For an upward pipe, divergence instability regions are appeared on the stability map and depending on the values of uL and γ , the occurrence of either flutter, buckling or both is probable. Since compressive centrifugal force is linearly proportional to the flow velocity (Eq. (11)), and also the average velocity of the system is higher in the two-phase flow, divergence occurs at lower values of dimensionless velocities in comparison with the single liquid-phase flow. Hence, the strength of the pipe against buckling increases and the divergence regions become smaller gradually with increasing the gas volume fraction. On the other hand, since the average mass flux decreases, the flutter velocity in the system reduces and consequently, stability region is condensed with the enlargement of the flutter region. In other words, two-phase flows have a decreasing effect on divergence and flutter velocities. Generally, unstable regions are wider in two-phase flows than single liquid-phase cases. Physically, this implies that in comparison with the two-phase case, containing a single liquid-phase flow makes the pipe stiffer. The operational c range for flow velocity is considered to be ucd L
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Journal of Fluids and Structures journal homepage: www.elsevier.com/locate/jfs
Dynamics of two-phase flow in vertical pipes ∗
Ali Ebrahimi-Mamaghani, Rahmat Sotudeh-Gharebagh , Reza Zarghami, Navid Mostoufi Multiphase Systems Research Lab., School of Chemical Engineering, College of Engineering, University of Tehran, P.O. Box 11155/4563, Tehran, Iran
graphical
article
abstract
info
Article history: Received 5 November 2018 Received in revised form 2 March 2019 Accepted 12 March 2019 Available online xxxx Keywords: Pipe Gas–liquid Two-phase flow Fluid–structure interaction Dynamic behaviors Complex frequency Stability maps
a b s t r a c t In this study, a novel mathematical model was proposed for the dynamic analysis of two-phase flow in vertical pipes considering different two-phase flow models by including dissipative forces. To model the corresponding two-phase flow, common slipratio factors were utilized. The Galerkin discretization method and eigenvalue analysis were applied to solve the model equations. A detailed parametric analysis was also performed in order to elucidate the influence of various parameters such as volumetric gas fraction, flow velocity, structural damping and gravity parameter on dynamics of the system, critical flutter velocities and frequencies. The model was validated with experimental data and simulation results reported in the literature It was concluded that the pipes conveying two-phase flow are prone to experience several dynamic phenomena. Stability of the pipe structure was also examined for different two-phase flow models and the results indicated that the instability boundaries are significantly affected by the choice of the model. Furthermore, it was shown that the dynamical response of the pipe is substantially dependent on the volumetric gas fraction. Hence, the gas volume fraction can be introduced as a fundamental parameter for the vibration control of the two-phase flow systems. The results of this study would be beneficial for engineers to optimally design suitable structures for two-phase flow systems. © 2019 Elsevier Ltd. All rights reserved.
∗ Corresponding author. E-mail addresses: [email protected] (A. Ebrahimi-Mamaghani), [email protected] (R. Sotudeh-Gharebagh), [email protected] (R. Zarghami), [email protected] (N. Mostoufi). https://doi.org/10.1016/j.jfluidstructs.2019.03.010 0889-9746/© 2019 Elsevier Ltd. All rights reserved.
A. Ebrahimi-Mamaghani, R. Sotudeh-Gharebagh, R. Zarghami et al. / Journal of Fluids and Structures 87 (2019) 150–173
Nomenclature AC AG AL C EI g L Ks K M mG mL mP q qj (t) QC QG QL t UL UG uL uG uL c uL cd uL cf uL ∗ uL opt x y
pipe internal cross-sectional area, m2 area occupied by the gas in the pipe internal cross-section, m2 area occupied by the liquid in the pipe internal cross-section, m2 dimensionless damping matrix of the system pipe flexural rigidity, N.m2 gravitational acceleration, m/s2 length of pipe, m slip ratio defined by K s =U G /U L dimensionless stiffness matrix dimensionless mass matrix mass of gas per unit length, kg/m mass of liquid per unit length, kg/m mass of pipe per unit length, kg/m vector generalized coordinate jth dimensionless time-dependent generalized coordinate total volumetric flow rate, m3 /s volumetric gas flow rate, m3 /s volumetric liquid flow rate, m3 /s time, s average liquid velocity, m/s average gas velocity, m/s dimensionless liquid velocity defined by uL =(M L /EI)0.5 U L L dimensionless gas velocity defined by uG =(M G /EI)0.5 U G L dimensionless critical liquid velocity dimensionless divergence velocity dimensionless coupled-mode flutter velocity dimensionless liquid velocity at mode-exchange dimensionless optimum liquid velocity longitudinal coordinate, m lateral deflection of the pipe, m
Greek symbols
αG βG βL δ ij εG εG ∗ ζ η λj µ ξ ρG ρL τ ϕ j (x) ω ωc
gas void fraction defined by α G =AG /AC dimensionless gas mass ratio defined by β G =mG /(mG +mL +mP ) dimensionless liquid mass ratio defined by β L =mL /(mG +mL +mP ) Kronecker delta function gas volume fraction defined by ε G =Q G /Q C gas volume fraction at the mode-exchange phenomenon damping ratio of the system defined by ζ =Image(ω)/Real(ω) dimensionless deflection of the pipe defined by η =Q G /Q C eigenvalue of cantilever for mode j hysteretic structural damping coefficient dimensionless longitudinal coordinate defined by ξ =x/L gas density, kg/m3 liquid density, kg/m3 dimensionless time defined by τ = t/ ((mp +mL +mG )L4 /EI)0.5 jth eigenfunction of the cantilevered beam dimensionless frequency dimensionless critical flutter frequency
Subscripts L G
liquid phase gas phase
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1. Introduction Structures like pipes that convey fluid are widely used in chemical plants, municipal water supply, marine structures, heat exchanger tubes, hydropower systems, boiling water reactors, etc. Hence, predicting the performance of these systems from different points of view, such as heat transfer, computational fluid dynamics (CFD) (Delnoij et al., 1999), pressure fluctuations (Matsui, 1984) and flow pattern signature (Shaikh and Al-Dahhan, 2007) is vital. The presence of air in water pipelines causes many drawbacks, such as loss of carrying capacity, change of physical properties of fluid, disruption of hydrodynamics and reduction of the system efficiency (Pozos et al., 2010a,b). Therefore, it is important to understand the gas–liquid two-phase flow characteristics. Thus, the characterization of the hydrodynamics of such structures has been the subject of numerous investigations during the last decades (Shoham, 2006; Miwa et al., 2015; Pettigrew and Taylor, 1994; Boure et al., 1973; Ruspini et al., 2014; Chalgeri and Jeong, 2019; Lips and Meyer, 2011). Despite the low density of gas in gas–liquid systems, dynamics of pipes conveying two-phase flow is of great interest. For instance, Nakamura et al. (1995) experimentally investigated the instability conditions for tube arrays in air–water flow and reported critical flow velocities in extreme conditions. Pettigrew et al. (2001) studied the effects of mass flux and void fraction on vibrating excited mechanisms of heat exchanger tube bundles of various geometries by conducting multiple tests. They revealed that the vibration amplitude of the tubes is roughly proportional to the mass flux. Sheikhi et al. (2013) measured vibration and pressure fluctuations simultaneously coupled with photograph and image analysis in order to study the hydrodynamics of bubble columns. They showed that by calculating skewness and kurtosis of pressure fluctuations and wall vibration, one can easily detect the regime transitions. Monette and Pettigrew (2004) carried out several experiments on hanging cantilevered pipes carrying two-phase flow for understanding the fluid-elastic instability. Moreover, they proposed a modified two-phase flow model to analyze the vibrational behavior of hanging tubes. Zhang and Xu (2010) conducted an experimental study on the influence of internal bubbly flow in a pipe. They indicated that vibration characteristics of the pipe conveying two-phase flow principally depend on variations in void fraction. Shiea et al. (2013) experimentally determined the flow regime transition onset in a bubble columns by employing a resistivity probe. Adhami et al. (2018) carried out diverse experiments to characterize the hydrodynamics of bubble columns containing non-newtonian fluids. They evaluated the effect of superficial gas velocity on the bubble size based on pressure fluctuation analysis. Experimental investigations on the dynamics of two-phase flow systems have been widely reported in the literature, however, a limited number of theoretical studies have been conducted on this topic. Hara (1977) assessed two-phase flowinduced vibrations of horizontal pipe structures theoretically and experimentally in which the frequency of void signals drastically affects the fundamental frequency of the piping systems. Zhang et al. (2010) probed bubbly flow induced pipe vibrations experimentally and theoretically and reported that the power spectral density of pipe vibrations and pressure fluctuations are considerably increased by an increase in the bubble diameter. Seyed and Patel (1992) calculated pressure and internal slug flow induced forces on straight and curved pipes via a mathematically rigorous method. They disclosed that high internal flow-rates can amplify the internal pressure effects. Adegoke and Oyediran (2017) surveyed the nonlinear dynamics of cantilevered pipes conveying two-phase flow subjected to thermal loads. They utilized multiple scales method to survey the axial and transverse vibrations and extracted natural frequencies and critical velocities of the system for a set of void fractions. They explored the effects of Poisson’s ratio and pressurization on the dynamical response of the pipe. Wang et al. (2018) presented a dynamical model for horizontal pipes in the slug flow. They solved the governing equations of motion by employing the finite element method (FEM) and verified their results with experimental data. They showed that the slugging transition velocity has a significant effect on the system properties such as damping and stiffness. Ortega et al. (2012) performed a numerical study on the interaction between unsteady two-phase flow and the dynamical response of flexible pipes. They calculated stress variations, pressure, and lateral deflections by the linear FEM. An and Su (2015) presented a parametric study on the dynamical behavior of two-phase flow in pipes using generalized integral transform technique (GITT) and also investigated the influences of several parameters, such as volumetric gas fraction and void fraction, on time response of the system numerically. Liu and Wang (2018) obtained the natural frequencies of slug flow in a pipe using FEM. They predicted the conditions in which the divergence instability in the system is probable. They also showed that the root-mean-square of the natural frequencies of the system initially increases and then decreases by increasing the superficial liquid velocity. In the piping systems, preventing instability and undesirable vibrations is a mandatory engineering requirement. Nevertheless, due to complex hydrodynamics and operational challenges, a limited number of analytical investigations have reported the stability maps and natural frequencies of such systems. In this study, a comprehensive investigation was carried out on fluid-elastic vibrations of pipes. In this regard, the most significant dynamic aspects of these systems, including the natural frequencies and instability conditions, are obtained. Herein, hanging and upward cantilevered pipes were studied considering the gravitational and hysteretic effects based on the Euler–Bernoulli theorem. The homogeneous model and also other two-phase flow models available in the literature (Monette and Pettigrew, 2004; Chisholm, 1983) were also utilized. Galerkin discretization method was employed to obtain the reduced-order model equations. Eigenvalue analysis was implemented to evaluate the stability of the structure. The influences of various parameters such as flow velocity, gas volume fraction and structural damping on various modes of the system, critical flutter, and divergence velocities were discussed. Besides, Argand diagrams and stability maps, as useful tools for designing of the considered structure, were used to survey the dynamical behavior of the system.
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Fig. 1. Main flow patterns in vertical pipes.
2. The examined system 2.1. Background The flow pattern is a qualitative description of the phase mixture in the system, which strongly depends on the flow direction, flow rate, physical properties of fluids and geometrical properties of the pipe. Different mass transfer mechanisms in two-phase flow systems can be identified by flow patterns. Various flow patterns have been reported in the literature (Rouhani and Sohal, 1983). Among them, four main patterns may co-exist in vertical pipes shown in Fig. 1, as described below (McQuillan and Whalley, 1985): (i) Bubbly flow: the liquid flow rate is high enough to disperse gas into discrete distorted bubbles, randomly distributed through the continuum liquid phase (Fig. 1a). (ii) Slug flow: at higher gas flow rates, bubble coalescence occurs and large bubbles are formed that occupy most of the cross section of the column (Fig. 1b). The slugs of gas move with a higher velocity than the average liquid phase velocity. (iii) Churn flow: as the gas flow is further increased, gas slugs become unstable and then collapse, resulting in a turbulent chaotic flow (Fig. 1c). The main characteristic of this pattern is the chaotic oscillatory motion of the liquid phase. (iv) Annular flow: this flow pattern occurs at relatively high gas and low liquid flow rates. In this case, the gas becomes the continuous phase. The liquid flows as a film along the pipe wall while high-velocity gas moves through the center, carrying dispersed liquid droplets (Fig. 1d). The principle parameters for defining a two-phase flow are gas volume fraction (εG ), the ratio of gas to total flow rates (QG /QC ), the void fraction (αG ), the ratio of the area occupied by the gas to the total area (AG /AC ) and the slip ratio (Ks = UG /UL ). Defining UL = QL /AL and UG = QG /AG , the relation between the aforementioned parameters can be presented by Chisholm (1983): 1 − εG
εG
( =
1 − αG
αG
)
1 Ks
(1)
The most basic model giving a relation between gas and liquid phases is the homogeneous model in which both phases have the same velocity (Ks = 1, αG = εG ). Due to neglecting many effects, such as the buoyancy effect, this model may not be realistic, especially at high values of void fraction (Feenstra et al., 2002). To overcome this problem, several experimental investigations have been carried out to quantify the slip velocity (Woldesemayat and Ghajar, 2007). In this study, the slip ratio in vertical tubes was obtained from:
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Fig. 2. Slip ratios (Ks ) estimated by various available models against the volumetric gas fraction (εG ) for vertical pipes.
For hanging pipes (Monette and Pettigrew, 2004):
( Ks =
1 − εG
)0.5 (2)
εG
For upward pipes (Chisholm, 1983): 1 Ks = [ ( 1 − εG 1 −
ρG ρL
)]0.5
(3)
It is noteworthy that in the model presented by Chisholm (1983), the maximum value of the slip ratio is (Ks )max = (ρL /ρG )0.25 . The slip-ratios evaluated by different models are demonstrated in Fig. 2 with respect to the gas volume fraction for the air–water flow. 2.2. Problem formulation Fig. 3 illustrates upward and downward cantilevered pipes that carry two-phase flows. The pipes are of length L, mass per unit length mp , flexural rigidity EI and hysteretic structural damping coefficient µ. The transverse displacements of the pipes are demonstrated by y(x, t). The flow consists of discrete gaseous bubbles in the continuous liquid phase or discrete liquid droplets in the continuous gas phase medium. The parameters mL and mG indicate the mass of liquid and gas per unit length of the pipes, respectively, and their corresponding velocities are shown by UL and UG The dynamical equation of motion of the aforementioned system can be written as (Monette and Pettigrew, 2004; An and Su, 2015; Paidoussis, 1970):
) 2 ∂ 4y ( ∂ 2y 2 2 ∂ y + m U + m U + 2 m U + m U ( ) L G L L G G L G ∂ x4 ∂ x2 ∂ x∂ t ) ( ∂ 2y ∂ 2y ∂ y + (mP + mL + mG ) 2 + (mP + mL + mG ) g (x − L) 2 + =0 ∂t ∂x ∂x
EI (1 + µi)
(4)
The terms of Eq. (4) represent, respectively: elastic restoring force with structural damping effects, centrifugal, Coriolis, √ inertial and gravitational forces (i = −1). The equation can be applied to bubbly flow and annular flow regimes. In order
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Fig. 3. A schematic view of (a) upward and (b) hanging cantilevered pipes conveying two-phase flow.
to obtain the dimensionless form of the equation of motion, the following variables are introduced: x
y
t
L
L
T
ξ = ,η = ,τ =
(5)
where T is defined as: T =L
2
(
mP + mL + mG
)0.5 (6)
EI
By substituting Eq. (5) into Eq. (4), the dimensionless equation of motion is acquired as: (1 + µi)
) ∂ 2η ( ) ∂ 2η ∂ 4η ( 2 ∂ 2η + uL + u2G + 2 βL0.5 uL + βG0.5 uG + 2 4 2 ∂ξ ∂ξ ∂ξ ∂τ ∂τ ( ) ∂ 2 η ∂η =0 + γ (ξ − 1) 2 + ∂ξ ∂ξ
(7)
The dimensionless parameters appeared in Eq. (7) are defined as: mG
βG =
mP + mL + mG ( m )0.5 L uL = UL L , EI
,
βL =
mL
,
mP + mL + mG ( m )0.5 mP + mL + mG G uG = U G L , γ = ±gL3 EI EI
(8)
It is worth mentioning that in the gravity parameter (γ ), positive and negative signs refer to hanging and upward pipes, respectively. By utilizing Eqs. (1) and (8), one may write:
βG =
mP AC
ρG αG ρ G εG β L = ρL Ks (1 − εG ) + ρL (1 − αG ) + ρG αG
(9)
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[ uG = uL K s
ρG εG ρL (1 − εG )
]0.5 (10)
where ρG and ρL are gas and liquid density, respectively. Substitution of Eqs. (9) and (10) into Eq. (7) yields:
( ) 2 ∂ 4η ρ G εG ∂ η 2 (1 + µi) 4 + uL 1 + Ks ∂ξ ρL (1 − εG ) ∂ξ 2 ( ) 2 ρ G εG ∂ η ∂ 2η + 2βL0.5 uL 1 + + 2 ρL (1 − εG ) ∂ξ ∂τ ∂τ ( ) 2 ∂ η ∂η + γ (ξ − 1) 2 + =0 ∂ξ ∂ξ
(11)
Third and fourth terms in this equation present Coriolis and centrifugal forces in upward pipes conveying two-phase flow, respectively, equivalent to what occurs in the pipes conveying single liquid-phase flow. 3. Solution procedure 3.1. Galerkin approach The reduced-order model of the equation of motion, which is a simplification of a high accuracy dynamical model that preserves dominant behavior and necessary effects of the system, can be acquired from the partial differential equation (Eq. (11)) using the Galerkin method. Hence, the transverse displacement of the pipe can be approximated as (Mamaghani et al., 2016; Hosseini et al., 2017; Mamaghani et al., 2018):
η (ξ , τ ) =
n ∑
ϕj (ξ ) qj (τ )
(12)
j=1
where n is the number of modes to be considered, qj (τ ) is the jth time-dependent generalized coordinate and ϕj (ξ ) is ∫1 the jth eigenfunction of the cantilevered beam. Note that ϕj (ξ ) is normalized such that 0 ϕj (ξ )ϕk (ξ )dξ = δjk . These eigenfunctions are defined in the following equation (Dehrouyeh-Semnani et al., 2016; Gregory and Paidoussis, 1966):
( ) ( ) ) ( ) ( ) sinh λj − sin λj ( ( ) sinh(λj ξ ) − sin(λj ξ ) ( ) ϕj (ξ ) = cosh λj ξ − cos λj ξ − cosh λj + cos λj
(13)
In this equation, values of λj can be evaluated from the frequency equation cosh(λj )cos(λj ) + 1 = 0. By substituting Eq. (13) into Eq. (11) and multiplying by ϕk and then integrating over the domain [0, 1], the partial differential equations is transformed into an ordinary equation as:
¨ (τ ) + Cq( ˙ τ ) + Kq(τ ) = 0 Mq
(14)
in which dots indicate temporal derivation. Eq. (14) represents the discrete form of the equation of motion in the matrix form, where q is the vector generalized coordinate, M is the dimensionless mass matrix, C is the dimensionless damping matrix and K is dimensionless stiffness matrix. These matrices are functions of geometrical and physical properties of the system and are defined as follows: q = [q1 (τ ) , q2 (τ ) , . . . , qn (τ )]T
∫
(15)
1
Mjk =
φj (ξ )φk (ξ )dξ = δjk
0 (
Cjk = 2 βL0.5 uL (1 +
Kjk = (1 + µi)
∫
ρ G εG ) ρL (1 − εG )
1
φj (ξ )
0
1
φj (ξ ) 0
ρG εG ρL (1 − εG )
1
∫
(ξ − 1) φj (ξ ) ( 0
∂φk (ξ ) dξ ∂ξ
(17)
∂ 4 φk (ξ ) dξ ∂ξ 4
( ( + u2L 1 + Ks +γ
(16)
)∫
1
)) ∫
φj (ξ ) 0
∂ 2 φk (ξ ) dξ ∂ξ 2
∂φk (ξ ) ∂ 2 φk (ξ ) + )dξ ∂ξ ∂ξ 2
(18)
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3.2. Stability methodology The standard second-order matrix form of Eq. (14) can be reduced to a first-order differential form by the following procedure (Paidoussis, 1998): BZ˙ (τ ) + EZ (τ ) = 0
(19)
where:
[ B=
0 M
M C
]
,
[ E=
−M 0
0 K
]
,
Z(τ ) =
[
q(τ ) ˙ τ) q(
] (20)
Assuming the solution to have the formZ(τ ) = Aeiωτ , the reduced-order Eq. (19) yields the following eigenvalue problem: YA − iωI = 0
(21)
in which I shows the identity matrix and Y = −B−1 E. Finally, the complex frequencies (ω) of the system can be computed numerically as a function of the flow velocity, volumetric gas fraction, structural damping and gravitational parameter. 4. Results and discussion To investigate the dynamical behavior of the system, Argand diagrams of pipes conveying air–water two-phase flow with respect to dimensionless fluid velocity are plotted for typical values of the physical parameters (ρG = 1.2 kg/m3, ρL = 1000 kg/m3 ). Horizontal and vertical axes in these diagrams are real and imaginary parts of the dimensionless complex natural frequencies of the system and show the resonant frequency of oscillations and damping of the system, respectively. Damping ratio of the system can be defined by ζ =Image(ω)/Real(ω). As one of the branches of the frequency diagram crosses the horizontal axis (i.e., Real(ω)̸ =0, Image(ω) =0), Hopf bifurcation occurs in the system leading to flutter instability (point H). The minimum velocity at which the flutter phenomenon happens is known as the critical flutter velocity (ucL ) and its corresponding frequency is known as the critical flutter frequency of the system (ωc ). Buckling or divergence instability happens when one of the branches crosses the origin from positive to negative on the [Image(ω)]axis. In addition, Paidoussis coupled-mode flutter bifurcation (points Pi ) takes place when two loci of branches whether leave or coalesce on the [Image(ω)]-axis (Paidoussis, 1998). Furthermore, velocities shown in Argand diagrams are related to the liquid phase, while one can calculate gas phase velocity from Eqs. (8)–(10). Paidoussis (1970) investigated the effect of number of terms in Eq. (12) in the estimation of dynamical characteristics of the system and found that a 10-term approximation yields reliable results. 4.1. Hanging pipes In order to evaluate the numerical results, complex frequencies for the lower modes of a hanging pipe conveying single liquid-phase flow are plotted in Fig. 4a) for γ = 10, µ = 0 and βL = 0.3. It can be seen in this figure that in the single-phase case, the Argand diagram is in close agreement with results reported by Paidoussis and Issid (1974). Then, by increasing the gas volume fraction, while keeping other parameters constant, the behavior of the loci was drawn based on Monette’s model (Eq. (2)) and its dynamical behavior was analyzed (Fig. 4a, b). Neglecting damping effects, if one considers the flow velocity equal to zero, all frequencies located on the real axis for both single and two-phase flows indicate natural frequencies of a corresponding pipe without flow. At lower velocities, since the difference between the momentums of single and two-phase flows is insignificant, both systems behave almost the same. However, the more the flow velocity, the more the corresponding branches separate from each other. Also, increasing the gas volume fraction results in a prominent deviation from the single liquid-phase case and subsequently displaces bifurcation points more and more. For instance, Fig. 4 demonstrates that in the single-phase case, Paidoussis coupled-mode flutter bifurcations are located at ucf L ≈6.5 and 8.4. However, when two-phase flow exists in the pipe (εG = 0.4), these bifurcations occur at ucf ≈ 6.8 and 9. In other words, overdamped responses were acquired at higher speeds when considering the two-phase L flow and in this situation, the pipe does not vibrate for the first mode. In addition, for all vibration modes, except the first mode, values of imaginary part of the frequencies have decreased significantly. Consequently, as compared with the single liquid-phase flow, the two-phase flow encounters a higher damping ratio for the first mode and smaller for higher-order modes. Since transforming the flow regime from single liquid-phase to two-phase decreases the critical flow velocity (single liquid-phase flow: ucL ≈8.7 and two-phase flow (εG = 0.4): ucL ≈6.5), one can conclude that in the two-phase flow, the stability of hanging pipes decreases. Also, in comparison with the single-phase case, increasing the gas volume fraction results in instability of the fourth mode at uL ≈14. In Fig. 5, the Argand diagram of two-phase flow is shown in comparison with the single liquid-phase case for γ = 100, µ = 0 and βL = 0.2 (Paidoussis, 1970) and various gas volume fractions. It can be seen in this figure that the third mode is unstable in the single liquid-phase case (at ucL ≈10.5). By increasing the volumetric gas fraction to the critical value of εG∗ = 0.54, the imaginary part of the second mode decreases to the proximity point where instability occurs (at
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Fig. 4. Dimensionless complex frequency of hanging cantilevered pipe (ω) as a function of dimensionless fluid velocity (uL ) for γ = 10, µ = 0, βL = 0.3.
uL ≈10). Nevertheless, at u∗L ≈10.2, the loci of second and third modes are extremely close to each other and the system undergoes mode-exchange phenomenon which eventually results in instability on the third mode while the second mode
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Fig. 5. Dimensionless complex frequency of hanging cantilevered pipe (ω) as a function of dimensionless fluid velocity (uL ) for γ = 100, µ = 0, βL = 0.2 and various εG .
Fig. 6. Dimensionless complex frequency of hysterically damped hanging cantilevered pipe (ω) as a function of dimensionless fluid velocity (uL ) for γ = 10, µ = 0.1, βL = 0.65.
remains stable. The mode-exchange phenomenon is an exclusive characteristic of non-conservative systems and has been investigated by Ryu et al. (2002) for single liquid-phase flows. In this case, for a specific velocity, the eigenvalues of different modes approach each other and the probability of transference branches increases in the system. Further increase in the gas volume fraction (e.g., εG = 0.8) causes the second mode to become unstable while the third mode becomes stable. Therefore, the volumetric gas fraction may be interpreted as a fundamental parameter which controls transversal vibrations of elastic pipes conveying two-phase flow.
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Argand diagrams are depicted in Fig. 6 for both single liquid-phase and two-phase flows, considering the hysteretic effect, where γ = 10, µ = 0.1 and βL = 0.65. As expected, at zero flow velocity, due to the existence of structural damping, the loci of frequencies of the system separate from the real axis. Also, the Argand diagram loses its symmetry with respect to the imaginary axis and negative modes (the modes starting with the negative real part) have a tendency to cross the imaginary axis and move towards higher +Real(ω). The first negative mode (1− ) has caused the system to become unstable for both single liquid-phase and two-phase cases at ucL ≈10.6 and ucL ≈10.1, respectively. In addition, for values of εG∗ = 0.23 and u∗L = 6.87, mode-exchange occurs between 1− and 2+ modes in the two-phase case. Generally, the mode-exchange phenomenon occurs for sequential positive modes in pipes conveying single-phase flow. However, in the case of two-phase flow regarding the hysteretic effect, this is not valid. Another feature of the two-phase flow in Fig. 6 is that at higher velocities, meanders are observed in the loci of the fourth mode (uL ≈16). An important feature in Fig. 6 is that in the two-phase flow, the imaginary part of the first negative mode (the unstable mode) reaches its maximum value at uL ≈6.87 and this velocity can be introduced as the optimum velocity of the pipe opt (uL ). At this specific velocity, the system reaches the maximum damping value (decaying rate of the amplitude) for the first negative mode which eventually destabilizes the system. At this point, the system is maximally stabilized. Sugiyama et al. (1996) confirmed that the Coriolis force has a stabilizing effect on the pipe conveying single-phase flow, hence, can be utilized as the oscillation absorber. It is worth mentioning that for the single liquid-phase flow, the optimum velocity opt of the pipe is uL ≈6.4. To clarify this issue, the following time responses of the system for various values of the fluid velocity are demonstrated in Fig. 7. Effects of flow velocity on dynamical behavior of the pipe and the transition of the system to flutter instability are shown in Fig. 7 for γ = 10, µ = 0.1, βL = 0.65 and εG = 0.23. The dimensionless initial condition is set as q1 (0) = 10 and q2 (0) = · · · = qn (0) = q˙ 1 (0) = · · · = q˙ n (0) = 0. First, considering the flow velocity to be zero, the system behaves like a hanged cantilevered beam (Fig. 7a). Increasing the flow velocity results in two mutual effects: the Coriolis forces try to stabilize the system while centrifugal forces trying to destabilize it. Because of the linear relationship with the flow velocity, Coriolis forces dominate the system behavior in the lower flow velocities. Therefore, the system initially tends opt to dampen the oscillations (Fig. 7b). These dissipative effects continue up to the optimum velocity uL = 6.87. Increasing the flow velocity increases the centrifugal forces which have a second order relationship with the flow velocity. This leads to an increase in the oscillation amplitude (Fig. 7c). At the critical flutter velocity, the amplitude of system oscillations becomes constant (Fig. 7d). Finally, any enhancement in the flow velocity causes the induced vibration to amplify and the amplitude of oscillations to grow exponentially, hence the system becomes unstable (Fig. 7e). Generally, at lower flow velocities, the system loses energy while at velocities higher than the critical flutter velocity, the system gains energy. Comparison between various models available for two-phase hanging pipes is made in Fig. 8, with µ = 0, γ = 100 and βL = 0.65. This figure demonstrates a significant difference between corresponding critical flutter velocities and frequencies and a distinct trend can be observed for dynamical behavior of both cases, especially for lower modes. As shown in Fig. 2, for two-phase homogeneous and Monette’s models, the difference of slip ratios is much more prominent in high and low gas fractions (εG ) and, obviously, its influence increases for higher values of flow velocity. In Fig. 8, in comparison with the single liquid-phase case, both two-phase models indicate that the fourth mode becomes unstable. It can be concluded from Fig. 8 that determining the equivalent fluid velocity (or slip ratio), especially at higher gas volume fractions, has a considerable effect on the dynamical response of pipes carrying two-phase flow. Undoubtedly, two-phase flows cause fluctuations in average velocity and mass flux of the system. It is noteworthy to state that variations in the loci of frequencies are significant in the vicinity of the imaginary axis, or when a jump or mode exchange occurs. Hence, to precisely draw the plots, one must utilize a higher number of modes and simultaneously make the velocity intervals smaller. In order to investigate the stability of the two-phase flow system as compared with single liquid-phase flows, the stability map is plotted in Fig. 9 for two different values of the gas volume fraction in ucL -γ and ωc -γ planes. This figure portrays flutter instability boundaries. Within the stable region, any arbitrary initial conditions lead to the elimination of the pipe vibration. On the other hand, for velocities above each curve, any dynamical perturbation grows until a failure occurs in the system. Moreover, in the stability maps, sequential jumps are observed at various γ values. In the case of jumps, the system is stable to the right side of the curve and is unstable otherwise (Fig. 9a). By scrutinizing the stability map, it can be concluded that increasing γ leads to an increase in critical flutter velocities and frequencies. This trend can be justified by the fact that any increase in γ results in obtaining a stiffer system, hence, the stability region becomes wider. Therefore, one can conclude that the greater γ , the more stable the system becomes. It can be seen in the stability map shown in Fig. 9, especially for larger γ values, that two-phase flow regimes can significantly influence flutter boundaries. As expected, in the two-phase case in comparison with the single liquid-phase flow, smaller critical flutter velocities and frequencies are acquired which are indicative of shrinkage of the stability region. In other words, in the case of two-phase flow, the pipe becomes more flexible. On the other hand, the stability boundaries of two-phase flow are above those of single gas-phase flow. In the case of single liquid-phase flow in the pipe, four of the so-called S-shaped segments can be observed, each of which demonstrates three unique dynamical characteristics related to instability-restabilization-instability sequences (Paidoussis, 1998). Before the formation of each S-shaped segment, the number of specific mode destabilizing the system changes and the mode exchange phenomenon occurs. Another aspect observed in Fig. 9 is that smaller jumps
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Fig. 7. Variation of dimensionless transverse tip deflection of a hysterically damped hanging cantilevered pipe conveying air–water two-phase flow against dimensionless time for γ = 10, µ = 0.1, βL = 0.65, εG = 0.23.
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Fig. 7. (continued).
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Fig. 7. (continued). Table 1 Characterization of the experimental setup used by Monette and Pettigrew (2004). EI (N.m2 )
L (m)
mp (kg/m)
D (m)
µ
0.003
0.547
65.7 × 10−3
9.25 × 10−3
0.1
are seen when the flow is in two-phase (εG = 0.4 and 0.7), which shifts the stability boundaries toward larger γ values. In other words, for two-phase flows, these S-shaped segments may occur in wider ranges of gravity parameter, as can be clearly observed in Fig. 9. Hence, the number of S-shaped segment decreases in the case of two-phase flow. This means that two-phase flow delays the formation of S-shaped segments. In single gas-phase flow as compared with liquid-phase, the average mass flux of the internal flow in the system decreases substantially. As shown in Ref. (Paidoussis, 1998), S-shaped segments and jumps are vanished at small mass ratios for various gravity parameters. Furthermore, Monette and Pettigrew (2004) have experimentally examined stability boundaries for single liquid and gas-phase flows and they did not observe S-shaped segments and jumps in the stability maps. The effect of damping on the stability of the structure is shown in Fig. 10. Generally, including the damping effect makes the jumps to become smaller. It is also important to note that for some specific ranges of γ (for instance, εG = 0.7 and γ >700), critical flutter velocities and frequencies become larger than the corresponding ones when neglecting dissipative effects. This phenomenon was discovered for the first time by Ziegler for non-conservative gyroscopic systems (Ziegler, 1952) and then was observed for horizontal cantilevered pipes conveying single liquid-phase flow by Semler et al. (1998). According to Fig. 10, one can state that the damping influence on the single-phase flow is more prominent than in two-phase flows. Fig. 11 shows the stability boundaries of hanging cantilevered pipes considering homogeneous and Monette’s twophase models. According to this figure, instability boundaries, especially jumps phenomena, are considerably affected by two-phase flow models. As mentioned before, by choosing a gas fraction close to either zero or one, the difference between critical flutter velocities and frequencies evaluated from the two proposed models increases. In Fig. 11, for εG >0.5, the critical flutter velocities and frequencies evaluated by the homogeneous model are less than those proposed by the Monette’s model, owing to the fact that in the Monette’s model, the slip ratio is predicted to be greater than in the homogeneous model, which causes greater gas fraction (εG ). In other words, when εG >0.5, the Monette’s model predicts higher values of flexural rigidity of the system than the homogeneous model. In conclusion, employing a precise model capable of explaining the accurate relation between the gas and liquid phases is a must at lower or higher gas volume fractions.
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Fig. 8. Real and imaginary components of the dimensionless complex frequency of hanging cantilevered pipe (ω) as a function of dimensionless fluid velocity (uL ) for µ = 0 and γ = 100, βL = 0.65 (a) single liquid-phase flow (εG = 0) (b) Monette’s model (εG = 0.8) (c) homogeneous model (εG = 0.8).
In order to examine results of the two-phase flow case, variations of calculated dimensionless critical liquid and gas phase velocities versus gas void fraction are compared in Fig. 12 with the experimental data reported by Monette and Pettigrew (2004) for the conditions of Table 1. As illustrated in this figure, the critical liquid phase velocity decreases by an increase in the gas void fraction, leading to increase in the critical gas phase velocity. The performance of the model was assessed quantitatively by extracting the average relative error and correlation coefficient. The average relative error of the dimensionless critical liquid and gas phase velocities are found to be 15.65% and 7.87%, respectively. Furthermore, the correlation coefficients of the considered parameters are 0.99 and 0.97, respectively. As discussed earlier, a notable phenomenon in the pipe structures is the occurrence of jump as a result of variation of the gas volume fraction. The structure stability boundaries are plotted against the gas volume fraction variation in Fig. 13. Obviously, εG = 0 corresponds to the single liquid-phase flow in the pipe. It can be seen in these figures that a larger gravity parameter leads to a more stable structure. Hence, the results shown in Fig. 13 demonstrate that employing a larger gravity parameter stabilizes the pipe in general, yielding higher critical flutter velocities and frequencies. Fig. 13 shows that in contrast to the stability maps, the ucL -εG and ωc -εG curves are overall descending with increasing the gas volume fraction. Moreover, it can be seen in this figure that when gravity parameter is small (e.g., γ = 10), due to a reduction in the average mass flux of the two-phase mixture, critical flutter velocity and frequency of the system monotonically decrease with increasing the gas volume fraction. At a relatively large γ (e.g., γ = 50 or 100), however, the critical flutter velocities and frequencies do not monotonously decrease with increasing the gas volume fraction and the ucr -εG curves exhibit Z-shaped segments. Extensive calculations for larger γ s (e.g., γ = 200 or 300) show more clear Z-shaped segments. Argand diagrams are shown in Fig. 14(a, b) demonstrate that the occurrence of each Z-shaped segment is associated with the instability-restabilization-instability sequence when the dimensionless flow velocity is monotonically increased. Consequently, at small γ values, the flutter instability may occur in a certain mode, while at large γ values, exchange of unstable modes appears and various modes experience instability. To better understand the role of gas volume fraction in vibration characterization of pipe structures, Argand diagrams are illustrated in Fig. 14(a, b) for γ = 100, µ = 0 and βL = 0.65. According to these figures, two jumps occur at εG ≈0.1 and 0.83. In the single liquid-phase flow, instability occurs in the first mode of the system. However, at εG ≈0.1, the preferred mode of flutter instability is the fourth one. Increasing the gas volume fraction leads to instability in the third mode. Since the third mode is approximately tangent to the +Real(ω) axis, the critical flutter velocity and frequency of the system undergo dramatic changes as a result of minor variations in the gas volume fraction.
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Fig. 8. (continued).
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Fig. 9. Stability map of hanging cantilevered pipe as a function of dimensionless gravity parameter (γ ) for βL = 0.65, µ = 0 and (a) dimensionless critical flutter velocity (ucL ) and (b) dimensionless critical flutter frequency (ωc ).
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Fig. 10. Effect of internal damping (µ) on critical flutter velocity (ucL ) of hanging cantilevered pipes for βL = 0.65.
Fig. 11. Stability map of hanging cantilevered pipe as a function of dimensionless gravity parameter (γ ) by considering various two-phase flow models for βL = 0.65 and µ = 0.
4.2. Upward pipes Argand diagrams of upward pipes conveying single liquid-phase flow are portrayed for various γ s and are shown in Figs. 15 and 16, where µ = 0, βL = 0.2 and γ =−10 and −50, respectively. These diagrams are in good agreement with
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Fig. 12. Comparison between simulation and experimental data of dimensionless critical liquid (ucL ) and gas (ucL ) phase velocities of hanging cantilevered pipe carrying two-phase flow as a function of gas void fraction (αG ) for conditions of Table 1.
Fig. 13. Dimensionless critical flutter velocity (ucL ) of hanging cantilevered pipe conveying two-phase flow as a function of volumetric gas fraction (εG ) for µ = 0 and βL = 0.65.
those reported by Paidoussis (1970). By increasing the gas volume fraction, while keeping other parameters constant, the behavior of complex natural frequencies of the system are drawn based on Chisholm’s model (Eq. (3)). Similar to hanging
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Fig. 14. Dimensionless complex frequency of hanging cantilevered pipe (ω) as a function of the dimensionless fluid velocity (uL ) for γ = 100, µ = 0, βL = 0.65.
pipes, increasing the gas volume fraction results in decreasing critical flutter velocities and frequencies of the system. According to Fig. 16, for small γ values, however, changes in gas volume fractions does not significantly influence the critical flutter velocities and frequencies of the system. In Fig. 15, the optimum fluid velocities for single liquid-phase and opt
two-phase flows are uL
= 3.25 and uopt = 3.15, respectively. L
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Fig. 15. Dimensionless complex frequency of upward cantilevered pipe (ω) as a function of the dimensionless fluid velocity (uL ) for γ =−10, βL = 0.2 and µ = 0.
Fig. 16. Dimensionless complex frequency of upward cantilevered pipe (ω) as a function of the dimensionless fluid velocity (uL ) for γ =−50, βL = 0.2 and µ = 0.
It is observed in Figs. 15 and 16 that the frequencies of upward pipes become pure imaginary for the first mode. As the flow velocity is increased, these frequencies move toward the origin. As mentioned before, the velocity at which the
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Fig. 17. Stability map of upward cantilevered pipe as a function of the dimensionless gravity parameter (γ ) for βL = 0.65 and various εG .
frequency of the system becomes absolutely zero is known as the thresholds of divergence for upward pipes under their own weight (i.e., Real(ω)̸ =0, Image(ω)=0). The critical flutter velocity at which this phenomenon is observed is known as the critical divergence flow velocity (udL ). This phenomenon is also probable to occur in pipes conveying single liquid-phase flow with both ends supported (Paidoussis and Issid, 1974). The calculations of the previous section show that in hanging pipes conveying two-phase flow, similar to the single liquid-phase case, the system does not undergo the divergence instability for any value of flow velocity. However, in the upward pipes, this phenomenon happens for a specific range of γ . It is worth mentioning that in the upward pipes, the structure first undergoes divergence instability and then, by increasing flow velocity as the frequency branch crosses the origin, the pipe stabilizes. Continuing to increase the flow velocity causes the frequency branches to cross the real axis, thus, flutter instability initiates. For a better illustration of the vibrational behavior of upward pipes, stability map of the system is shown in Fig. 17 for upward pipes conveying single and two-phase flows for βL = 0.65 and µ = 0. For an upward pipe, divergence instability regions are appeared on the stability map and depending on the values of uL and γ , the occurrence of either flutter, buckling or both is probable. Since compressive centrifugal force is linearly proportional to the flow velocity (Eq. (11)), and also the average velocity of the system is higher in the two-phase flow, divergence occurs at lower values of dimensionless velocities in comparison with the single liquid-phase flow. Hence, the strength of the pipe against buckling increases and the divergence regions become smaller gradually with increasing the gas volume fraction. On the other hand, since the average mass flux decreases, the flutter velocity in the system reduces and consequently, stability region is condensed with the enlargement of the flutter region. In other words, two-phase flows have a decreasing effect on divergence and flutter velocities. Generally, unstable regions are wider in two-phase flows than single liquid-phase cases. Physically, this implies that in comparison with the two-phase case, containing a single liquid-phase flow makes the pipe stiffer. The operational c range for flow velocity is considered to be ucd L
- 0.85), no jump is seen in oscillatory instability boundaries of the upward pipe. Therefore, it can be stated that the two-phase flows have a significant role in instability boundaries of the pipes conveying upward flow. Comparing the results obtained by both homogeneous and Chilsom’s models, it can be concluded that the homogeneous model shows an appropriate agreement with Chisolm’s experimental data.
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5. Conclusions Dynamics of hanging and upward cantilevered pipes was investigated by different two-phase flow models including the dissipative force. Argand diagrams and stability maps were obtained numerically. The model validation was performed with the simulation results and experimental data reported in literature and a close agreement was found. For the twophase flow, it was observed that the stability of the system is affected by various parameters. For instance, flow velocity has a substantial effect on the dynamical response of the pipe conveying two-phase flows, i.e., selecting an optimum flow velocity can minimize its unwanted vibrations. It is also deduced that different two-phase models predict the stability regions larger or smaller for higher or lower gas volume fractions. However, for a moderate gas volume fraction, choosing different two-phase models has not a remarkable effect on unstable boundaries. It was found that for the two-phase flow the damping ratio of the fundamental mode in hanging pipes is higher compared to single liquid-phase flow, unlike to higher modes. In addition, although in the upward two-phase pipes the unstable divergence regions shrink as the gas volume fraction increases, in the hanging pipes, divergence instability never occurs. Moreover, S-shaped and Z-shaped segments emerge in intersections of stable and unstable regions of the stability maps and the number of these segments profoundly depend on the chosen values of εG and γ . The results showed that the two-phase flow is more unstable compared to the single liquid-phase case, highlighting the dramatic effect of hysteretic damping. Moreover, the gas volume fraction has a fundamental role in the determination of vibration characteristics of the pipe (i.e., modeexchange, divergence, and flutter instabilities). Furthermore, increasing the gas volume fraction results in enlargement of the unstable regions. 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