In-Line and Cross-Flow Coupling Vibration Response Characteristics of a Marine Viscoelastic Riser Subjected to Two-Phase Internal Flow

This paper studies the in-line and cross-ﬂow coupling vibration response characteristics of a marine viscoelastic riser subjected to two-phase internal ﬂow and aﬀected by the combined eﬀects of several parameters including the volume fraction of gas phase, sea water ﬂow velocity, viscoelastic coeﬃcient of the marine riser, axial tension amplitude, and the in-line and cross-ﬂow coupling eﬀect taking into account both the geometric and hydrodynamic nonlinearities. On the base of extended Hamilton’s principle for open systems, the dynamic equations of the marine viscoelastic riser subjected to the axial tension and gas-liquid-structure interaction are established. Two distributed and coupled van der Pol wake oscillators are utilized to model the ﬂuctuating lift and drag coeﬃcients, respectively. The ﬁnite element method is adopted to directly solve the highly coupled nonlinear ﬂuid-structure interaction equations. Model validations are ﬁrstly performed through comparisons with the published experimental data and numerical simulation results, and the characteristic curves of the in-line and cross-ﬂow vibration pattern, the in-line and cross-ﬂow displacement trajectories, the in-line and cross-ﬂow space-time response of displacement, and the in-line and cross-ﬂow space-time response of stress versus diﬀerent parameters are obtained, respectively. The results show that the volume fraction of gas phase, sea water ﬂow velocity, viscoelastic coeﬃcient of marine riser, axial tension amplitude, the in-line and cross-ﬂow coupling eﬀect, and multiphase internal ﬂow velocity have signiﬁcant inﬂuences on the dynamic response characteristics of the marine viscoelastic riser. Furthermore, the maximum displacements and stresses of the marine viscoelastic riser can be increased or decreased depending on the internal ﬂow velocity, and the critical internal ﬂow velocities result in the increase of mode order for diﬀerent cross-ﬂow velocities. It is also demonstrated that appropriate viscoelastic coeﬃcients are very important to eﬀectively suppress the maximum displacements and stresses.


Introduction
Marine risers are indispensable equipment in offshore oil and gas exploitation, which connects the production platform on the surface and the subsea wellhead and provides key transmission channels transporting the drilling fluid or oil and gas. e marine risers are in a complex ocean environment and subjected to the combined actions of multiphase internal flow, its own gravity, the top tension, and a variety of external currents, such as uniform flow and shear flow, which lead to the complex force and make the marine riser have a variety of dynamic problems [1,2]. erefore, the marine riser is the weakest and easily damaged component in the offshore drilling platform system. As an important form of fluid-structure interaction response, the vortex-induced vibration (VIV) of the marine riser not only affects the fatigue life of the structure but also, sometimes, directly causes the destruction of the structure due to the large-amplitude oscillation generated by its frequencylocked resonance, which leads to the occurrence of major accidents in offshore oil and gas exploitation [3][4][5]. At present, the fatigue damage caused by the VIV has become one of the most serious problems in the dynamics of the marine risers. Consequently, it is urgent to study the VIV responses characteristics of the marine riser systematically and comprehensively.
Over the past several decades, models that predict the response of marine risers have received considerable attention. e majority of research efforts related to the cylinder VIV modeling, simulation, and experiment in the past have been focused on the study of flexibly-mounted rigid cylinders [6][7][8] and long slender cylinders [9][10][11][12]. For rigid cylinders, a new two-degree-of-freedom wake oscillator model was proposed by considering the relative flow velocity around the cylinder [13,14]. Moreover, geometric nonlinearities are very important to the accurate prediction of response amplitudes and hydrodynamic properties [15,16]. For flexible long cylinders, many research studies focus on the VIV phenomenon of a flexible pipe by experiments, numerical simulations, and theories. Based on the Euler-Bernoulli beam theory, Srinil and Ricciardi [17,18] established a vibration model of a marine riser considering the initial state of static equilibrium in the plane and studied the dynamic response of the marine riser under the action of linear shear flow. Subsequently, by simplifying the threedimensional vibration of the marine riser into a plane model, Srinil [10] studied the dynamic response of the initial plane bending and the response mechanism of vertical pipe under linear shear flow in plane based on the nonlinear theory. Sanaati et al. [19] discussed the influence of simple harmonic tension and axial stiffness on vibration amplitude and suppression characteristic by experiment and analyzed the dynamic response under axial simple harmonic tension through experiments and numerical simulations. Chaplin et al. [20] investigated the VIVs of a vertical tension riser in a stepped current by experiment and revealed that in-line (IL) and cross-flow (CF) displacements had a strong dependence on the modal composition of the motion. Multimode responses and the asymmetry of the bare pipe response in uniform flow were observed by laboratory tests of Xu et al. [21]. Franzini et al. [22] discussed the dynamic responses of a flexible riser subjected to harmonic excitation at the top by experiment and found that the Mathieu instability may simultaneously occur in more than one mode. Considering the phase difference of support motions at two ends of the pipe and a wake oscillator model based on the van der Pol equation to describe the vortex-induced force, He et al. [23] investigated the nonlinear dynamics of a pipe subjected to vortex-induced vibrations and unsynchronized support motions. Employing the van der Pol wake oscillators to simulate the dynamical behavior of the vortex shedding in the wake, Jiang et al. [24] numerically studied the nonplanar vibrations and multimodal responses of pinned-pinned risers under the combined action of internal and shear cross external fluid flows and discussed the effects of the shear parameter on the dynamic responses of the riser. Based on the wake oscillator model, Dai et al. [25] established the response model of pipe conveying fluid under the excitation of simple harmonic transverse acceleration at the top and found the jump of response amplitude and aperiodic phenomena. e vibration response characteristics of slender structures can be affected by the material properties. In this aspect, extensive researches have been published in the papers. Based on a four-unknown refined integral plate theory and Galerkin's approach, Rahmani et al. [26] investigated the influence of different boundary conditions on the bending and free vibration behavior of functionally graded sandwich plates resting on a two-parameter elastic foundation. On the basis of a novel integral first-order shear deformation theory, Bousahla et al. [27] investigated the buckling and vibrational behavior of the composite beam armed with single-walled carbon nanotubes resting on Winkler-Pasternak elastic foundation by applying Hamilton's principle and the Navier solution and presented several parametric studies and their discussions. Considering a new type of quasi-3D hyperbolic shear deformation theory and defined material properties by rule of the mixture with an additional term of porosity in the through-thickness direction, Kaddari et al. [28] studied the statics and free vibration of functionally graded porous plates resting on elastic foundations and discussed the influences of the porosity parameter, power-law index, aspect ratio, thickness ratio, and the foundation parameters on bending and vibration of a porous FG plate. Based on a simple quasi-3D hyperbolic theory and four different patterns of porosity variations, Addou et al. [29] investigated the effect of the Winkler/Pasternak/Ken foundation and porosity on the dynamic behavior of FG plates and discussed the influences of gradient index, porosity parameter, stiffness of foundation parameters, mode numbers, and geometry on the natural frequencies of imperfect FG plates. Considering exponential and power-law distributions of the functionally graded beam, applying a hyperbolic shear deformation theory, Chaabane et al. [30] studied the static and dynamic behaviors of functionally graded beams resting on the elastic foundation. Employing a quasi-3D hyperbolic shear deformation model, Boulefrakh et al. [31] investigated bending and dynamic behavior of functionally graded plates resting on visco-Pasternak foundations and discussed the effects of material index, elastic foundation type, and damping coefficient of the foundation, on the bending and dynamic behavior of rectangular functionally graded plates. Considering a simple quasi-3D higher shear deformation theory and the stretching effect, Boukhlif et al. [32] presented a dynamic investigation of functionally graded plates resting on an elastic foundation. Reference [33] presented a finite element formulation for transient dynamic analysis of sandwich beams with embedded viscoelastic material, and the results illustrated that the model had a good agreement with experimental data. Reference [34] proposed and investigated a concept to suppress vibrations of steel catenary risers by using viscoelastic sandwich layers, and a great increase of damping was observed. However, insights into the fully coupled CF, IL and axial (AX) VIV of a marine riser considering material viscoelasticity are still lacking in the literature. erefore, the present research aims at overcoming such model limitation by considering three-dimensional dynamic responses of a marine viscoelastic riser. e effects of internal flow on the VIV dynamic response of the pipe have been investigated by several researchers. References [35][36][37] had dedicated to investigate the VIV dynamic responses of a flexible fluid-conveying riser. Furthermore, Reference [38] examined cross-flow (CF) VIV of a flexible fluid-conveying pipe which internal velocities are from subcritical to supercritical. A marine riser often carries a mixture of multiphase energy sources such as oil and gas; therefore, it is important to study the vibration characteristics and instability of the marine riser with multiphase internal flow. Although the dynamics of single-phase pipe conveying fluid had been well studied, the dynamic behaviors of the marine riser subjected to multiphase internal flow need further research. e multiphase flow may lead to differences in material properties, phase-change process, and the excessive turbulence. So far, there has been very little research conducted on the dynamic behavior of marine risers transporting multiphase internal flow. Pettigrew and Taylor [39] reviewed the mechanism of vibration caused by two-phase fluid and analyzed the influence of relevant dynamic parameters on vibration. An and Su [40] used generalized integral transformation (GITT) to analyze the dynamic characteristics of flexible risers transporting twophase internal flow. Based on the generalized integral transformation method, Ma et al. [41] adopted the Timoshenko beam theory to establish the transverse vibration model of a marine riser transporting two-phase internal flow and studied the influence of different size parameters of the riser and two-phase flow characteristics parameters on the response of the riser.
Although there have been various investigations on the dynamic behaviors of the marine riser in the literature, almost no attention has been paid to the in-line and cross-flow coupling vibration response characteristics of a marine viscoelastic riser subjected to two-phase internal flow and the combined effects of the volume fraction of gas phase, sea water flow velocity, viscoelastic coefficient of the marine riser, axial tension amplitude, and the in-line and cross-flow coupling effect taking into account both the geometric and hydrodynamic nonlinearities on vibration response characteristics. In particular, there is a lack of an effective analytical model for the in-line and cross-flow coupling vibration response characteristics of a marine viscoelastic riser under the consideration of the gas-liquid-structure interaction and hydrodynamic nonlinearities.
is paper is concerned with the in-line and cross-flow coupling vibration response characteristics of a marine viscoelastic riser subjected to two-phase internal flow and the combined effects of the volume fraction of gas phase, sea water flow velocity, viscoelastic coefficient of the marine riser, axial tension amplitude, and the in-line and cross-flow coupling effect taking into account both the geometric and hydrodynamic nonlinearities on the dynamic response characteristics of marine viscoelastic riser. First, based on extended Hamilton's principle for open systems, the dynamic equations of a marine viscoelastic riser, which contain the internal flow velocity coupling term, coupling term of two-phase flow, viscous damping force of internal flow, the axial tension, and the gravity of the marine viscoelastic riser, are established. Two distributed and coupled van der Pol wake oscillators are utilized to model the fluctuating lift and drag coefficients, respectively. en, employing the finite element method, the highly coupled nonlinear equations of the system are solved and the corresponding characteristic equation is obtained. Finally, some numerical results on the in-line and cross-flow dynamic behaviors and response characteristics of the marine viscoelastic riser varied with different parameters of the volume fraction of gas phase, sea water flow velocity, viscoelastic coefficient of the marine riser, and axial tension amplitude are displayed in detail.

Model Description.
A marine riser is an important channel connecting the subsea wellhead and the floating platform (fixed or movable) near the surface in offshore oilgas exploration. A classic marine riser system subjected to oil-gas internal flow is shown schematically in Figure 1(a). For a suitable mathematical modeling of the vibration of a composite marine riser transporting oil-gas internal flow, the following assumptions are introduced to describe the motions of the riser and loading status: (1) e shear strain is neglected owing to the high slenderness, and the effect of temperature on material properties is ignored. (2) Owing to the effect of the mooring line constraints and dynamic positioning systems, the platform is only allowed to move in the horizontal direction. Hence, the reacting forces at the two ends are considered to be shear and moment and not torsional loads.
e upper end of the composite marine riser is connected with the sea surface floating production platform through a hydraulic tensioner, and the lower end is connected with the riser base through the flexible joint. In this paper, it is assumed that the fully-submerged flexible pipe with length L is perfectly straight at its vertical static equilibrium due to the effective weight. e fluid-conveying marine riser simply supported at both ends is placed within a uniform fluid flow with cross-flow velocity, U (z), aligned with the axial z-direction, as shown in Figure 1(b). e model of a flexible conveying fluid marine riser is modeled by extended Hamilton's principle. It should be noted that the marine riser will vibrate in-line and cross-flow about the section profile in in-line (IL) x, cross-flow (CF) y, and axial (AX) z-direction.

Kinetic Energy of Internal Flow.
e velocity vector of an arbitrary point on the center line of the marine riser can be written as follows: where the displacement vector r, which actually means the deformed marine riser position vector at any point, is Shock and Vibration where i, j, and k denote the unit vectors along the x-, yand z-axis. u, v, and w, respectively, denote in-line, crossflow, and axial displacements, and z is the initial axial coordinate value of the marine riser. Substituting equations (2) to (1), one has According to the schematic diagram of the relative velocity between internal flow and the marine riser, taking the motion of the riser into account, the absolute fluid velocity of the internal fluid element at the same point in the Cartesian coordinate system can be expressed as follows: Due to the deformation of the marine riser, the relative velocity vector v r of the fluid at the center point in the riser cross section is given as en, substituting equations (3) and (5) into equation (4), the absolute internal fluid velocity V f can also be rewritten as follows: It is assumed that the rotary inertia effect and the secondary flow effect are neglected; then, the corresponding kinetic energy T f of the internal flow can be described as follows: where m i is the mass of the fluid per unit length (m i � ρ i πd 2 /4 with ρ i being the inner fluid density) and V is the internal flow velocity. e over dot (prime) denotes differentiation with respect to time t (axial coordinate z), respectively. e variation result of kinetic energy of the internal flow is obtained as follows:  Figure 1: A schematic 3D model of a marine riser system subjected to two-phase internal flow and its simplified model.

Material Viscoelastic
Characteristics. In the current work, the pipe material is assumed to be viscoelastic according to the Kelvin-Voigt formulation. erefore, the relationship between the stress and strain may be expressed as follows: where σ * and ε * are the stress and strain, respectively, E * is the coefficient of internal dissipation, and t is the time.
Let a � E * /E, and equation (9) can be rewritten as follows:

Multiphase Internal Flow Characteristics.
It has been observed that the fluid transported in the marine riser always occurs in multiphase flow conditions, and there are several flow regimes in multiphase flow. In this work, we studied a marine riser conveying gas-oil mixtures. Considering different physical properties and flow velocities between different phases, based on Monette et al.'s work [42], m f , m f U f , and m f U 2 f of the multiphase flow can be written as follows: In order to facilitate the study, it is necessary to define the relevant parameters of two-phase flow in the composite marine riser. ε g is the gas volume fraction, K is the slip coefficient, and U is the apparent velocity; ε g , K, and U are, respectively, expressed as where Q g is the volumetric flow rate of the gas, Q l is the volumetric flow rate of oil, and the corresponding flow velocity are U g and U l , respectively. In addition, the slip coefficient can be expressed as the function of the gas volume fraction, which can be written as

Dynamic
Equations. Based on Zanganeh and Srinil's work [43], accounting for the effect of multiphase internal flow, and combined with equations (8), (10), and (11) previously derived, it can be shown that the nonlinear partial-differential equations of 3D coupled cross-flow and in-line motions of a marine viscoelastic riser subjected to two-phase internal flow have the forms of Shock and Vibration 5 where F x and F y are the associated hydrodynamic forces in three directions, respectively, m is the mass of the marine riser per length, m i is the internal fluid mass per unit length, and m a is the additional fluid mass per unit length (m a � C A ρ o πD 2 /4, with C A being the added mass coefficient and ρ o being the outer fluid density); it should be noted that the still water added mass coefficient C A taken as unity for circular cylinder. e parameters related with the marine riser with constant Young's modulus (E), damping coefficient (c), outer diameter (D), inter diameter (d), moment of inertia (I), cross-sectional area (A r ), bending stiffness (EI), and axial stiffness (EA r ). e static effective tension T without internal flow effect can be spatially varied by accounting for the buoyancy effect; then, the tension of the fluid-conveying pipe can by expressed as follows: where T t is the top pretension and g is the gravity. e boundary conditions of the riser are assumed as a simply supported pipe and given by

Hydrodynamic Forces
According to the theory of Zanganeh and Srinil [43], the projected three-dimensional hydrodynamic force components can be expressed as where C d0 and C l0 are the associated drag and lift coefficients of a stationary cylinder (assumed as C d0 � 0.2, C l0 � 0.3, and C d � 1.2). e total velocity relative to the pipe can be written as follows: In this paper, the van der Pol nonlinear vibration equations are used to describe the shedding characteristics of vortex, according to the theory of Facchinetti et al. [44], and the variations of p and q can be described as Herein, Ω f � 2πStU/D is the vortex-shedding angular frequency, and the right side of the equations are the excitation terms simulating the effect of pipe motion on the near wake. ε u , ε v , Λ u , and Λ v are the wake and coupling empirical coefficients adopted equally as ε u � 0.3, Λ u � Λ v � 12, and ε v calibrated by Zanganeh and Srinil [43].

Numerical Results and Discussion
Numerical results are performed in this section to analyze the effects of sea water flow velocity, viscoelastic coefficient of marine riser, axial tension amplitude, multiphase internal flow property, and gas-phase volume fraction on vibration responses characteristics of the marine viscoelastic riser. A long flexible straight riser model experimentally tested by Song et al. [45] can be used for numerical simulations. Geometry parameters and physical property parameters are listed in Tables 1  and 2. e highly nonlinear partial-differential equations (13a) and (13b) in conjunction with equations (18) and (19) are solved by the FEM in COMSOL program. Initial conditions are specified at the static equilibrium for the marine riser e Lagrange quadratic finite elements are used for the discrete of the riser, and Δt � 0.001s and Δz � 0.1 m have been chosen through a series of convergence yielding stable simulations for all considered flow velocities. In this study case, validation tests of numerical simulations with the experimental model and finite difference approach of Zanganeh and Srinil [43] have been performed. 6 Shock and Vibration

Validity of the Method and Convergence Verification.
Owing to the lack of VIV experiment concerned with internal flow, model validations are firstly performed through comparisons with the published experiment and numerical simulation results. Previous prediction results concluded by Zanganeh and Srinil [43] have highlighted the importance of considering the axial dynamic coupling, the amplification of IL mean displacements, and the geometric and hydrodynamic nonlinearities. erefore, variations of maximum IL and CF displacement amplitudes of the flexible riser with respect to cross-flow velocity are plotted in Figure 2 for different analysis methods. It is found that the IL displacement amplitudes between experiment and numerical simulations agree very well at lower cross-flow velocity. Outside this region of cross-flow velocity, the IL displacement amplitudes obtained by experiment keep almost larger than numerical simulations. As for the CF displacement amplitude, it is noted that an increase in the cross-flow velocity results in an increase in the CF displacement amplitude and the peak value of CF displacement amplitude occurs at the cross-flow velocity of around 0.3 m/s for FDM and FEM results. For the maximum displacement, the simulation results are larger than the experimental results. is is because the damping errors of the numerical calculation model in this paper and the experimental model in the references lead to different results, which are related to imperfect physical models and numerical calculation methods. In addition, the boundary conditions are also different, and the upper boundary in the experiment is a spring support, while the upper boundary in the numerical simulation in this paper is a simply supported support. In general, our model can effectively predict the VIV dynamic responses of a flexible riser. However, the even-order modes of the marine riser are not found in the analysis. e reason for this phenomenon is that the characteristic parameters of the marine riser determine that the locking interval of the even-order mode response of the riser under condition that the outflow excitation is small, so it is difficult to capture under several outflow velocities selected in the paper.  Figure 4(c). It can be see that, at a large displacement, the horizontal "8" type appears in the cross-flow phase diagram. At a small displacement, the cross-flow phase diagram shows a circular shape, but the velocity is zero, and there is a large change at the position with the largest displacement. As shown in the  local magnification, the cross-flow phase trajectory is a single closed curve without crossing, so the riser shows a singleperiod vibration in the cross-flow direction. In Figure 5 In addition, the stress changes in Figure 5(c) along the in-line are more complex, and the stress shows a trend of periodic changes along the axis direction of marine riser. e relationships curves in Figure 5(b) represent the cross-flow stress change. It can be seen that the crossflow stresses show a periodic change with time, and the vibration appears as a standing wave form. In addition, the maximum stress and minimum stress alternation happened in three locations. In Figure 5(d), the axial distribution of the maximum stress and minimum stress along the marine riser is more. In addition, the change that stress varies with time is also more complex and characterized by shear wave and standing wave superposition state. e reasons causing the abovementioned phenomenon is that the marine riser vibrates in multiple modes at this velocity.          respectively, while Figures 9(c) and 9(d) are the stress responses of the in-line and cross-flow under the condition that a � 0.11, respectively. It can be viewed that the absolute value of the three-direction maximum stress decreases with the increase of the viscoelastic coefficient and the three-direction maximum stress is symmetrically distributed along the axial direction of the marine riser. e spatial-temporal distribution of stress in the riser varies with the change of the viscoelastic coefficient. e stress distribution in both the in-line and the cross-flow directions shows that the spatial-temporal region of maximum stress increases with the increase of viscoelastic coefficients. e results show that the maximum stress and stress amplitude of the marine riser can be effectively reduced with the increase of viscoelasticity of the riser at low sea water flow velocity.

Effects of Axial Tension Amplitude on the Vibration Response Characteristics.
e influences of axial tension amplitude on the vibration response characteristics of marine riser considering the in-line and cross-flow coupling effect are discussed in this section. e cross-flow displacement response curve under different axial tension excitation amplitudes and different time intervals is displayed in Figure 10, where the axial tension excitation amplitudes in Figure 10(a) are A/D � 1.2, while that in Figure 10(b) are A/ D � 3.0. It can be indicated that the cross-flow displacement response of the riser is dominated by the outflow excitation at low amplitude excitation; however, the displacement response of the riser is dominated by the axial tension excitation when the axial tension amplitude excitation increases to a certain degree. e space-time response characteristic curves of displacement of the marine riser under different axial tension excitation amplitudes are plotted in Figure 11 Figure 11 show that the cross-flow displacement at the top of the riser oscillates periodically according to the given boundary conditions. Due to the influence of the multimode response, the displacement response of the riser in the cross-flow is relatively complex and it is presented as the second-order modal response and a certain periodicity in a large time range under the condition that A/D � 1.2. However, the displacement response along the direction of the riser shows a lag. e displacement response of the riser is relatively regular, which is greatly affected by the axial tension excitation under the condition that A/D � 3.0. e results of axial displacement show that the axial displacement of the riser presents strong asymmetry at low amplitude and tends to be symmetric at high amplitude. e space-time response characteristic curves of stress under different axial tension excitation amplitudes are plotted in Figure 12 erefore, when the axial tension excitation amplitude is large enough, the three-direction stress will increase a lot.
In Figure 13, the phase diagram of the in-line and crossflow displacement trajectories of the axial center of the marine riser considering the coupling effect is displayed, where the curves in Figures 13(a)

Effects of Gas-Phase Volume Fraction on the Vibration Response Characteristics.
e influences of gas-phase volume fraction on the vibration response characteristics of the marine riser considering the in-line and cross-flow coupling effect are discussed in this section.
For some prescribed values of different gas-phase volume fractions, the response characteristic curves of the cross-flow displacement response of the marine riser versus different gasphase volume fractions with U � 0.3 m/s and ε g � 0, 0.7, and 0.8 are plotted in Figure 14, where different color curves in the figure represent the cross-flow displacement response of the riser with the same time interval. It can be viewed that the cross-flow response mode of the riser is four orders under the condition that ε g � 0, while the cross-flow response mode of the riser is a transition from the 4th-order mode to the 3rd-order mode under the condition that ε g � 0.7, and the cross-flow response mode of the riser is three orders under the condition that ε g � 0.8. e space-time response characteristic curves of displacement of the marine riser under different gas-phase volume fractions are plotted in Figure 15, where Figures 15(a), 15(c), and 15(e) are the in-line displacement responses under the condition that ε g � 0, ε g � 0.7, and ε g � 0.8, respectively, while Figures 15(b), 15(d), and 15(f ) are the cross-flow displacement responses under the condition that ε g � 0, ε g � 0.7, and ε g � 0.8, respectively. It can be viewed that the response of the two-dimensional displacement of the riser is similar under the condition that ε g � 0 and ε g � 0.8, and the space-time response of the displacement of the marine riser is mainly in the form of a standing wave. However, the cross-flow displacement of the riser responds in mode 4 under the condition that ε g � 0, while the cross-flow displacement of the riser responds in mode 3 under the condition that ε g � 0.8. e space-time response of the two-dimensional displacement of the riser is shown as the superposition state of standing wave and traveling wave under the condition that ε g � 0.7 and the cross-flow displacement of the marine riser responds in mode of 4 order. e space-time response characteristic curves of the inline and cross-flow stress of the marine riser under different gas-phase volume fractions are plotted in Figure 16, where Figures 16(a), 16(c), and 16(e) are the in-line stress responses of marine riser under the condition that ε g � 0, ε g � 0.7, and ε g � 0.8, respectively, while Figures 16(b), 16(d), and 16(f ) are the cross-flow stress responses of the marine riser under the condition that ε g � 0, ε g � 0.7, and ε g � 0.8, respectively. It can be viewed that the in-line and cross-flow stress of the riser vary regularly with time under the condition that ε g � 0 and ε g � 0.8. In addition, the stress distribution along the direction of the riser is symmetrical and the stresses are mainly in the form of standing wave. While the stress distribution along the direction of the riser is asymmetric, the space-time response of the in-line and cross-flow stress of the riser is shown as the superposition state of standing waves and traveling waves.
In Figure 17, the characteristic relationship curves of the displacement trajectories along different axial positions of the marine riser with different gas-phase volume fractions are displayed, where the curves in Figures 17(a), 17(c), and 17(e) are the displacement trajectories of the marine riser with ε g � 0, while the ones in Figures 17(b), 17(d), and 17(f ) are the displacement trajectories of the marine riser with ε g � 0.8. It is obvious that the displacement trajectory of the riser is close to a single closed curve, but the displacement trajectory along the axial symmetric position of the riser is no longer symmetric. In addition, the displacement trajectories of the marine riser in the x-y and y-z planes are mainly in the shape of "8," while the displacement trajectories in the x-z plane are mainly in the shape of a ring. e phase trajectories in Figures 17(e) and 17(f ) are a mixture of "8" shape and "crescent" shape, and the phase trajectories near the center of the riser are "crescent" shaped and near both ends of the riser are "8" shaped. e displacement-time response characteristic curves of the axial center of the marine riser under different gas-phase volume fractions are plotted in Figures 18, where Figures 18(a), 18(c), and 18(e) are the in-line displacementtime responses of the marine riser under the condition that ε g � 0, ε g � 0.8, and ε g � 0.8, respectively, while Figures 18(b), 18(d), and 18(f ) are the cross-flow displacement-time responses of the marine riser under the condition that ε g � 0, ε g � 0.8, and ε g � 0.8, respectively. It can be viewed that the motion is close to periodic motion under the condition that ε g � 0 and ε g � 0.8, while the in-line and cross-flow displacement response of the marine riser show a strong aperiodic property under the condition that ε g � 0.7. In addition, there is a large "jump" in the cross-flow displacement response; the cross-flow displacement response shows the superposition state of multiple harmonics, and the axial displacement response shows a "locking" interval. Moreover, the nonlinear response of the marine riser is         Figure 17: e displacement trajectories of the marine riser at different gas-phase volume fractions in the x-y plane with (a) ε g � 0 and (b) ε g � 0.8, in the x-z plane with (c) ε g � 0 and (d) ε g � 0.8, and in the y-z plane with (e) ε g � 0 and (f ) ε g � 0.8.

24
Shock and Vibration intensified when the gas-phase volume fraction makes the riser near the critical point of "jumping" in the cross-flow mode.
In Figure 19, the phase diagram of the in-line and crossflow displacement trajectories of the axial center of the marine riser considering the coupling effect is displayed, where the curves in Figures 19(a), 19(c), and 19(e) are the inline displacement trajectories of the marine riser under the condition that ε g � 0, ε g � 0.8, and ε g � 0.8, respectively, while the curves in Figures 19(b), 19(d), and 19(f ) are the cross-flow displacement trajectories of the marine riser under the condition that ε g � 0, ε g � 0.8, and ε g � 0.8, respectively. It is obvious that the phase trajectory is close to a single closed curve under the condition that ε g � 0 and ε g � 0.8, while the phase trajectory is more complex and shows aperiodic motion under the condition that ε g � 0.7. e results show that the orbital motions showed the IL-CF and IL-CF trajectories dominate figure-of-eight shapes, and the IL-AX trajectories depict a majority of "O" shapes. e phase diagram of the IL and CF directions depicted that the dynamic responses behaved as periodic and aperiodic responses depending on internal flow velocity. Furthermore, the maximum displacements and stresses of the marine viscoelastic riser can be increased or decreased depending on the internal flow velocity, and the discontinuous jumping phenomenon of the in-line response modal is discovered.

Data Availability
Data used to support the findings of this study can be obtained from the corresponding author on request.

Additional Points
Summary: owing to the lack of VIV experiment concerned with internal flow, model validations are firstly performed through comparisons with the published experiment and numerical simulation results. Previous prediction results concluded by Zanganeh and Srinil [38] have highlighted the importance of considering the axial dynamic coupling, the amplification of IL mean displacements, and the geometric and hydrodynamic nonlinearities. erefore, variations of maximum IL and CF displacement amplitudes of the flexible riser with respect to cross-flow velocity are plotted in Figure 2 for different analysis methods. It is found that the IL displacement amplitudes between experiment and numerical simulations agree very well at lower cross-flow velocity. Outside this region of cross-flow velocity, the IL displacement amplitudes obtained by experiment keep almost larger than numerical simulations. It indicates that the present solution is very close to the experimental result, which validates the present solution.

Conflicts of Interest
e authors declare that they have no conflicts of interest.