Vortex-Induced Vibrations of A Long Flexible Cylinder in Linear and Exponential Shear Flows

A numerical study based on a wake oscillator model was conducted to determine the response performance of vortex-induced vibration (VIV) on a long flexible cylinder with pinned-pinned boundary conditions subjected to linear and exponential shear flows. The coupling equations of a structural vibration model and wake oscillator model were solved using a standard central finite difference method of the second order. The VIV response characteristics including the structural displacement, structural frequency, structural wavenumber, standing wave behavior, travelling wave behavior, structural velocity, lift force coefficient and transferred energy from the fluid to the structure with different flow profiles were compared. The numerical results show that the VIV displacement is a combination of standing waves and travelling waves. For linear shear flow, standing waves and travelling waves dominate the VIV response within the low-velocity and high-velocity zones, respectively. The negative values of the transferred energy only occur within the low-velocity zone. However, for exponential shear flow, travelling waves dominate the VIV response and the negative energy occurs along the entire length of the cylinder.


Introduction
The problems caused by vortex-induced vibration (VIV) have been studied by many researchers in the past decades. To investigate the intrinsic mechanisms of the VIV problem, in early research, the majority of studies focused on rigid cylinders (Sarpkaya, 1979;Bearman, 1984;Govardhan and Williamson, 2000). However, recently with the rapid development of oil and gas production in deep waters, the VIV responses of long, flexible cylinders have received considerable attention in the practice of ocean engineering.
The studies of VIV on long flexible cylinders contain physical experiments and numerical computations. The research contents from experimental methods are very abundant and reliable (Guo and Lou, 2008;Vandiver et al., 2009;Gao et al., 2015Gao et al., , 2016Song et al., 2011Song et al., , 2016Song et al., , 2017Xu et al., 2017Xu et al., , 2018, however, experimental methods have some limitations, such as model scale limit, expensive cost, difficulty of flow profile generation and others. Compared with experimental methods, numerical methods can overcome some of these limitations. Of the available numerical methods, the computational fluid dynamics (CFD) method is widely used for predicting the VIV responses of cylindrical structures (Newman and Karniadakis, 1997;Evangelinos and Karniadakis, 1999;Kang et al., 2017;Gao et al., 2018b). However, due to the high computational cost, in contrast to rigid cylinders, CFD simulations of long flexible cylinders are relatively rare. As the length-to-diameter aspect ratio of a cylinder exceeds 10 3 , the CFD simulation becomes too long and too difficult to meet the actual needs of engineering practice. Thus, it is very necessary to establish a model that can quickly predict the VIV responses of cylinders with large aspect ratios.
In recent years, the wake oscillator model has been widely used to predict VIV of long structures. Bishop and Hassan (1964) first proposed a self-exciting and self-limiting Van del Pol equation to simulate the lift force acting on a structure in VIV. Based on the suggestion of Bishop and Hassan (1964), Hartlen and Currie (1970) proposed the first wake oscillator model, in which the structural vibration velocity was coupled with the Van der Pol equation. After that, the wake oscillator model has been used to predict the VIV responses of rigid cylinders by many researchers (Facchinetti et al., 2004;Farshidianfar and Zanganeh, 2010;Jin and Dong, 2016;Kang et al., 2018). According to the directions of vibrations, the studies can be divided into vibration only in the cross-flow direction and in the cross-flow and in-line directions simultaneously.
With the rapid development of long cylinders used in ocean engineering, the wake oscillator model is further developed to predict the VIV responses of long structures (Mathelin and de Langre, 2005;Violette et al., 2007;Xu et al., 2008;Ge et al., 2009;Doan and Nishi, 2015;Gao et al., 2018a). According to the simplified models, the long cylinder models can be divided into two types: infinite tensioned cable model and finite tensioned beam model. For an infinite tensioned cable model, Gao et al. (2018a) validated that the VIV characteristics are not sensitive to the selected aspect ratio, therefore, we can use a finite tensioned cable model with periodic boundary conditions to study the VIV characteristics of an infinite cable model. However, as the contribution of the bending stiffness to the VIV responses is neglected, an infinite tensioned cable model is just an idealized model. In order to consider the VIV problems of long cylinders more realistically, the bending stiffness should be considered and the flexible cylinder should be simplified to a finite tensioned beam model. For a practical ocean flow, the flow variation with respect to the flow location usually becomes larger as the flow location moves from the sea surface to the seabed. That is to say, the type of flow encountered in the actual ocean engineering applications is neither uniform flow nor linear shear flow, but rather is closer to an exponential shear flow (Mathelin and de Langre, 2005). In contrast to the large number of publications dedicated to the VIV problems of a tensioned beam model in uniform and linear shear flows, there have been few studies of the VIV problems in exponential shear flow. Lucor et al. (2006) used the direct numerical simulation (DNS) method to study the VIV displacements of a long tensioned beam model (L/D = 2028) in exponential shear flow. Bourguet et al. (2013aBourguet et al. ( , 2013b gave a very abundant research on a tensioned beam model (L/D = 200) in exponential shear flow by the DNS method and they presented many useful and interesting results. From the past publications, we can see that the VIV characteristics of a tensioned beam model with very large aspect ratio (on the same order of 10 3 ) are still not well understood and further research is still necessary. For example, in the physical tests, as the generation of the exponential shear flow profile in the laboratory is very difficult, we usually study the linear shear flow instead of the exponential shear flow. What are the main differences between the VIV responses of a long flexible cylinder subjected to linear shear flow, which are usually studied by experimental method, and the VIV responses of a cylinder subjected to exponential shear flow, which are the actual ocean flow?
To answer the above question, in this paper, we conducted a study of a long tensioned beam model with the lengthto-diameter ratio of 2000 in both linear and exponential shear flows using the wake oscillator method. Although the VIV responses of real long structures always take place in both cross-flow and in-line directions, for simplicity, here the study is restricted to vibrate only in the cross-flow direction. The paper is divided into five sections. Brief descriptions of the structural and wake oscillator models are given in Section 2. In Section 3, the numerical method is described in detail. In Section 4, the VIV responses of a long circular cylinder in both linear and exponential shear flows are studied. Finally, in Section 5, some conclusions based on the study results are presented. Fig. 1 shows a long vertical circular cylinder with length L and diameter D, constrained to oscillate transversely in response to a stationary shear flow, the boundary conditions of the cylinder are pinned-pinned ends. Here, we just take the linear shear flow as an example. The origin O of the coordinate system is taken as the bottom end of the cylinder, X is the flow direction, Z is the upward direction, and Y is the transverse vibration direction. The tension and the bending stiffness of the cylinder are Θ and EI, respectively. In this paper, the shear parameter β is used to model the different flow profiles.

Model description
As shown in Fig. 1, the shear parameter β is defined as β = (U max -U min )/U max . With considering the long circular cylinder shown in Fig. 1 as a long tensioned beam model, the dimensional cross-flow displacement Y of the cylinder is de- GAO Yun et al. China Ocean Eng., 2019, Vol. 33, No. 1, P. 44-56 45 scribed as: where, m is the mass per unit length of the vibration system, R is the damping coefficient, Θ is the tension, T is the dimensional time, and p(Z, T) is the lift force per unit length, which can be expressed as: in which, C L (Z, T) is the lift force coefficient, and can be calculated as C L (Z, T) = C L0 q(Z, T)/2, where q(Z, T) is the dimensionless wake variable (Facchinetti et al., 2004), C L0 is the reference lift coefficient, and ρ is the fluid density. In Eq.
(1), m includes the structural mass m s and the added mass m f . The mass per unit length is calculated as m = m s +m f = m s +C M ρD 2 π/4, in which C M is the added-mass coefficient (C M = 1.0 for a circular cylinder) (Mathelin and de Langre, 2005). Furthermore, R includes the structural damping coefficient R s and fluid damping coefficient where Ω f is the local vortex shedding angular frequency (Ω f = 2πStU(Z)/D, in which St is the Strouhal number). γ is the stall parameter and γ = C D /(4πSt), in which C D is the mean drag coefficient. The structural damping coefficient R s is set equal to zero (R s = 0) to allow the maximum vibration amplitude to develop. A classical van der Pol equation was used to model the wake dynamics, expressed as (Nayfeh, 1993): where A and ε are two empirical parameters, and other variables are previously described. To transform Eqs.
(1) and (3) to a dimensionless form, the following definitions are used.
where t is the dimensionless time, and y and z are the dimensionless vibration response and the dimensionless space location, respectively.  (1) and (3), the dimensionless equations of the structural and wake models can be obtained, as follows: in which, ω f (z) reflects the flow profile and is expressed as As introduced before, the maximum velocity is chosen as the reference velocity U ref ; thus, ω f (z) at the top end of the cylinder (z = L/D) has a constant value of 1, ω f (z) at a random location (z) of the cylinder for linear and exponential shear flows can be expressed, respectively, as follows: (Facchinetti et al., 2004); c and b are the dimensionless tension and dimensionless bending stiffness, respectively, expressed as follows: 3 Numerical method y m n q m n Eq. (5) was discretized using a standard central finite difference method of the second order in both the time and space domains. Supposing that the total calculated time t total can be divided into N segments, and space length L/D can be divided into M segments, each time step can be determined as Δt = t total /N and each space step as Δz = L/(D×M). The discrete space locations were expressed as z = z i with i = 0, 1, 2, …, M, and the discrete times were expressed as t = t j with j = 0, 1, 2, …, N. The dimensionless parameters y and q at the location z m with time t n are denoted as and , respectively. The related partial derivatives of the second order with respect to y and q in Eq. (5) can be expressed as: As initial conditions, the displacement y and the velocity ∂y/∂t along the span are all zero. A one-wavenumber excitation with the amplitude of order O (10 -3 ) is applied to the fluid variable q, and the first time derivative ∂q/∂t is also set to zero. Whether the VIV responses are sensitive to the selected initial values of the fluid variable q or not will be discussed in detail in Section 4. Thus, the values of and when 0≤m≤M can be directly determined from the initial conditions. However, when determining the values of and when 0≤m≤M, the initial conditions ∂y/∂t = 0 and ∂q/∂t = 0 should be used. From ∂y/∂t = 0 and ∂q/∂t = 0, we have y m 1 q m 1 Substituting Eq. (10) into Eq. (9), the values of and can be written as: y m n q m n So far, we have obtained the values of and when n = 0 and n = 1. However, for n ≥ 2, the boundary conditions need to be used. Boundary conditions are posed as pinned at both ends, meaning that the displacement and the bending moment on the boundaries are always zero, expressed as: The first expression of Eq. (9) shows that can be calculated directly when 2 ≤ m ≤ (M -2). However, when m = 0 and m = M, the zero displacement boundary condition needs to be used and can be written as: when m = 1 and M -1, the zero bending moment of the boundary condition needs to be used. From the zero bending moment when m = 1 and m = M -1, we have Substituting Eq. (14) into the first expression of Eq. (9), the y m n+1 expression for when m = 1 and m = M -1 can be developed as: After the values of are obtained, the values of can also be evaluated based on the second expression of Eq. (9). The numerical method used in the present work was identical to the method used by Violette et al. (2007), thus, no additional validation was performed.

Results and discussion
All parameters used in this section had the same values as those used by Lucor et al. (2006) The values of dimensionless tension c and dimensionless bending stiffness b are selected to cause high-wavenumber vibrations, similar to those usually encountered in the long flexible cylinders used in ocean engineering. The results calculated using the wake oscillator model are highly dependent on the values of selected empirical parameters (A and ε) (Xu et al., 2008(Xu et al., , 2010. Here for simplicity, A = 12 and ε = 0.3, which were selected to match the values in Facchinetti et al. (2004). Furthermore, Δz = 1 and Δt = 0.001 were applied to discretize the space and time domains, respectively.
For a practical ocean flow, the flow velocity at the bottom end of the cylinder is usually much smaller than that at the top end. Here a high value of the shear parameter (β = 0.95) was selected for study. The VIV characteristics of a long circular cylinder subjected to linear and exponential shear flows were quantified. An overview of the main performances of the structural response is given in Section 4.1, and the spectral analysis of the structural response is described in Section 4.2. Finally, in Section 4.3, the fluid forces and fluid-structure energy transfer are investigated. The selected time intervals in the analyses are all the stable vibration time, i.e., the time after the transient vibration has dissipated out. Fig. 2 shows the variation of the structural displacements versus time and span location for linear and exponential shear flows, respectively. The VIV displacements are the combination of standing waves and travelling waves for both linear and exponential shear flows. As indicated by the RMS values of the structural displacements, there exist several wave troughs and wave peaks along the span. The locations corresponding to the wave troughs and wake peaks are defined as nodes and anti-nodes, respectively. It is notable that we are using the term "node" loosely to represent a minimum value of the displacement envelope; strictly speaking, it is not a node where the displacement is zero. The nodes and anti-nodes distribution along all the spans indicates that the VIV displacement has standing wave behavior along the entire length of the cylinder. However, the RMS values of the VIV displacements at all nodes are larger than zero, implying that the VIV displacement also has the travelling wave behavior along the entire cylinder length. As shown in Fig. 2a for linear shear flow, within the low-velocity zone (z ≤ 200), the RMS values of the VIV displacements at the nodes are obviously different from the RMS values of the anti-nodes, indicating that the VIV response is dominated by the standing waves. However, as z reaches a higher velocity zone (z > 200), the RMS values of the VIV displacements at the nodes become close to the RMS values at the anti-nodes, implying that the VIV response is dominated by the travelling waves. As shown in Fig. 2b for exponential shear flow, along the entire span of the cylinder, the RMS values of the nodes are close to the RMS values of the anti-nodes, meaning that travelling waves dominate the VIV response along the entire span.

Space-time evolutions of VIV displacement
The travelling waves can be divided into two groups according to their directions. The directions in one group are from the high-velocity zone to low-velocity zone, i.e., the negative z direction; and the directions in the other group are from the low-velocity zone to high-velocity zone, i.e., the positive z direction. For linear shear flow, the directions of the travelling waves are preferentially oriented from the high-velocity zone to the low-velocity zone, i.e., the negative z direction. However, for exponential shear flow, there exists no tendency from one direction to the other direction; in other words, the travelling waves propagate from the high-velocity zone to the low-velocity zone, and the waves propagate from the low-velocity zone to the high-velocity zone exist simultaneously. The maximum RMS values of the VIV displacements along the span of the cylinder are 0.189 and 0.086 for linear and exponential shear flows, respectively. The maximum values both appear at the high-velocity zones. Comparison of the RMS values of the linear and exponential shear flows indicates that the maximum value of the linear shear flow is obviously larger than that of the exponential shear flow. The location of the maximum value for the exponential shear flow is nearer to the top end of the cylinder than the location of the maximum value for the linear shear flow.

Spectral analysis of the structural response
The fast Fourier transform (FFT) analysis was performed on a selected stable time series (t = 2700-3000) of the VIV displacements. Fig. 3 shows the vibration frequency characteristics along the span for different flow profiles. Fig. 3 shows that the vibration displacements have several peak frequencies, displaying obvious multi-frequency characteristics. As expressed by the white dashed lines in Fig. 3, there are 6 and 8 vibration frequencies for linear and exponential shear flows, respectively, which distribute along the cylinder span and compete with each other. For linear shear flow, the vibration frequency ranges from 0.75 to 0.96; for exponential shear flow, the frequency ranges from 0.19 to 0.88. It can be seen that the width of the distributed frequency for the exponential shear flow is obviously larger than that for the linear shear flow. However, the maximum peak frequency of the exponential shear flow is a little smaller than that of the linear shear flow.
To determine the relationship between excited frequencies (ω) and spatial wavenumbers (k), a spatiotemporal spectral analysis was performed based on a two-dimensional FFT of the structural response. Fig. 4 shows how the displacement power spectral density (PSD) varied with the temporal frequency (ω) and spatial wavenumber (k) for linear and exponential shear flows, respectively. Here, the PSD was normalized by the maximum values observed in both (time and space) domains. The white vertical dashed lines and yellow horizontal dashed lines indicate the excited frequencies and the corresponding wavenumbers, respectively. It is worth noticing that the excited mode number (n) could be expressed using the wavenumber as n = 2k (L/D). Based on the values of the wavenumbers shown in Fig. 4, it can be ∈ ∈ seen that the excited wavenumbers correspond to the excited modes n {18, 19, 20, 21, 23, 24} for linear shear flow and n {5, 6, 7, 8, 12, 14, 16, 22} for exponential shear flow. These results indicate that for linear shear flow, only high modes are excited; however, for exponential shear flow, both the high modes and low modes are excited. The frequency bandwidth of the structural response in the exponential shear flow is obviously much larger than that in the linear shear flow.
It can be seen from Fig. 3 that at a specific span location, the VIV response always has multi-frequency characteristics. However, the variation in vibration frequencies with time could not be observed directly by the FFT analysis. For a better understanding of how the frequency characteristics shifted with time, the wavelet transform analysis was performed for the selected three span locations (z = 500, 1000, and 1500) in linear and exponential shear flows, respectively. Figs. 5 and 6 show the time-frequency analysis of VIV displacements at three different locations. For each subgraph, the upper part, the left part and the lower part denote the displacement variation with the time, the displacement variation with the frequency and the frequency  It can be seen from Fig. 5 that the selected time interval (t = 2700-3000) could be divided into two "periods" T 1 and T 2 . During period T 1 , the response frequency fluctuated from one peak frequency to another. For a specified instantaneous time, the response preferentially presents a narrowband distribution, implying that the response displacement exhibits mono-frequency characteristics. However, for the whole selected time interval, the response displays a broadband distribution, meaning that the response indicates multifrequency characteristics, as shown in the left part of each subgraph in Fig. 5. The vibration characteristics during period T 1 are very similar to those during period T 2 , as shown in the top and lower parts of each subgraph in Fig. 5. For exponential shear flow, the response always presents a broadband distribution, showing that the response always indicates multi-frequency characteristics.
As shown in Figs. 5 and 6, for a specified flow profile, the response frequency displays different characteristics at different span locations, implying that the response frequency does not occur simultaneously along the span. In other words, the peak frequencies compete with each other along the span of a long circular cylinder in both linear and exponential shear flows.

Lift force coefficient and fluid-structure energy transfer
In this section we describe the lift force coefficient that acted on the structure and how the energy transferred from the fluid to the structure. From Section 2, it can be seen that the instantaneous lift force coefficient C L (z, t) exerted on the cylinder can be expressed as C L (z, t) = C L0 q(z, t)/2. Fig. 7 presents the variation of the lift force coefficient versus time and span location and the RMS values of the lift force coefficients along the cylinder span. For linear shear flow, the lift force coefficient is comprised of a mixture of standing wave patterns and travelling wave patterns. Within the lowvelocity zone, the lift force coefficient indicates obvious standing wave characteristics. However, as the span location reaches a high-velocity zone, the lift force coefficient displays obvious travelling wave characteristics. For exponential shear flow, within the very-high-velocity zone (z > 1600), the lift force coefficient shows slight travelling wave behaviors, and the direction of the travelling waves is from the high-velocity zone to low-velocity zone, i.e., negative z direction. However, within the left zone of the cylinder (z ≤ 1600), the lift force coefficient displays very chaotic behaviors with time and span location.
The maximum RMS values of the lift force coefficients along the span of the cylinder are 0.415 and 0.318 for linear and exponential shear flows, respectively, showing that the maximum value in linear shear flow is slightly larger than that of the exponential shear flow. Both the maximum values for linear and exponential shear flows occur at the high-  GAO Yun et al. China Ocean Eng., 2019, Vol. 33, No. 1, P. 44-56 51 velocity zone. Compared with the locations where the maximum displacements occur, discussed in Section 4.1, the location where the maximum lift force coefficient occurred is almost coincident with the location where the maximum displacement occurs. The instantaneous energy W L (z, t) developed by the lift force coefficient C L (z, t) acting on the structure which moves at a velocity v(z, t) can be expressed as (Newman and Karniadakis, 1997): Fig. 8 shows the evolution of the energy transferred from the fluid to the structure versus time and span location for linear and exponential shear flows. The transferred energy having a positive value implies that the fluid supplies energy to the structure and excites the cable vibration; in contrast, a negative value indicates that the energy is transferred from the structure to the fluid and damps the structural vibration. From the color bars shown in Fig. 8, the maximum transferred energy in linear shear flow is larger than that in exponential shear flow. For linear shear flow, at the high-velocity zone, the energy is always not smaller than zero, meaning that the energy always transfers from the fluid to the cylinder and excites the structural vibration. However, at the low-velocity zone, there exist negative values for the instantaneous energy, implying that the energy sometimes also transfers from the structure to the fluid and damps the cylinder vibration. For exponential shear flow, there always exist positive and negative energy transfer values along the entire span, indicating that the fluid sometimes excites the structural vibration and sometimes damps the structural vibration.
To further confirm the intrinsic reason for the change of the energy transferred from a positive value to a negative value, we studied the lift force coefficient, structural displacement, structural velocity and transferred energy from the fluid to the structure, simultaneously. The time interval (t = 2020-2060) was selected to conduct the study. Figs. 9 and 10 show the variations in the instantaneous lift force coefficient C L (t), structural displacement y(t), structural velocity v(t) and transferred energy W L (t) with time at four different span locations (z = 250, 500, 1000 and 1500). As shown in Fig. 9d for the high-velocity location (z = 1500), when the lift force coefficient is at its maximum value (Position a), the vibration displacement is near zero and the structural velocity has its maximum value. However, when the lift force coefficient has its minimum value (Position c), the vibration displacement is also near zero and the structural velocity is at its minimum value. These responses indicate that the structural displacement always lags the lift force coefficient by π/2. But the structural velocity is always inphase with the lift force coefficient, such that the transferred energy multiplied by the lift force coefficient and the structural velocity is always not smaller than zero. As the location along the cylinder changes from the high-velocity zone to the low-velocity zone, the lagged phase between the lift force coefficient and the structural displacement decreases, further resulting in the increasing phase shift between the lift force coefficient and the structural velocity. As shown in Fig. 9a for the low-velocity location (z = 250), at Position a, both the lift force coefficient and the structural velocity have their moderate negative values; however, the two moderate negative values lead to the maximum value of the transferred energy. When the time moves from Position a to Position b, the lift force coefficient still has its moderate negative value, but the structural   GAO Yun et al. China Ocean Eng., 2019, Vol. 33, No. 1, P. 44-56 53 velocity has its moderate positive value; thus, one moderate negative value and one moderate positive value cause the minimum value of the transferred energy. When the time has further moved to Position c, both the lift force coefficient and the structural velocity have their moderate positive values resulting in the maximum value of the energy again. Comparing Fig. 9 and Fig. 10 shows that for linear shear flow, the vibration characteristics of the four parameters (lift force coefficient, structural displacement, structural velocity and transferred energy) are regular; however, for exponential shear flow, the vibration characteristics of all four parameters are very chaotic. For both the high-velocity location (z = 1500) and low-velocity location (z = 250), there always exist a phase shift between the lift force coefficient and the structural velocity, such that the transferred energy always has positive and negative values. Zhao (2015) also observed a similar phenomenon (negative transferred energy) for the VIV of a rigid cylinder. Here we found that for exponential shear flow, the negative transferred energy from the fluid to the structure occurs along the entire cylinder span. However, for linear shear flow, the negative value occurs just within the low velocity zone. The change of the transferred energy from a positive value to negative value is induced by the phase shift between the lift force coefficient and the structural velocity.

General discussion
As shown in Fig. 7b, the lift force coefficient for exponential shear flow displays very chaotic characteristics. Is this phenomenon caused by the instability of the numerical method used in this paper? To answer this question, we conducted two independency tests to study it. First, a space-step and time-step sizes independency test was conducted for exponential shear flow when β = 0.95. Four different step sizes were employed in the independence test and the calculated RMS values of the VIV displacements are given in Fig. 11. Fig. 11 shows that when the step size is refined from the first case (Δt = 0.01, Δz = 10) to the second case (Δt = 0.005, Δz = 5) and from the second case to the third case (Δt = 0.002, Δz = 2), the RMS values of the dimension-less displacements are obviously changed. However, when the step size is further refined from the third case to the fourth case (Δt = 0.001, Δz = 1), the RMS values remain stable, such that the selection of the fourth case in this paper can assure the precision of the numerical model.
The independency test of the initial excitation conditions for exponential shear flow when β = 0.95 was also conducted here. Four different initial values of q, including two sinusoidal functions with different space wavelengths (q 0 = 10 -3 sin2πz/(L/D) and 10 -3 sin4πz/(L/D)) and two constants (q 0 = 10 -3 and 2), were selected to study whether the VIV responses are sensitive to the initial excitation values or not. Fig. 12 shows the RMS values of VIV displacements along the span for different initial values. The comparison of the numerical results demonstrates that the VIV displacement is not sensitive to the selected initial values of q during the stable vibration period. The above two independency tests have confirmed the stability of our numerical method and the chaotic characteristics for the exponential shear flow are not caused by the present numerical method.

Conclusions
The VIV responses of a long circular cylinder having a large aspect ratio (L/D = 2000) with pinned-pinned boundary conditions subjected to both linear shear flow and exponential shear flow were studied numerically. A standard central finite difference method of the second order was used to solve the coupling equations of a structural model and wake oscillator model. The VIV response parameters (including VIV displacement, VIV frequency, VIV wavenumber, standing wave behavior, travelling wave behavior, lift force coefficient, structural velocity and transferred energy from the fluid to the structure) were determined and evaluated systematically. Based upon these results, the following conclusions are justified.
(1) The VIV displacements caused by both linear shear flow and exponential shear flow are characterized by a mixture of standing waves and travelling waves. For linear shear flow, standing waves dominate the VIV response within the low-velocity zone, while travelling waves domin- Fig. 11. Comparison of the RMS values of VIV displacements using different space-step and time-step sizes. ate the VIV response within the high-velocity zone. However, for exponential shear flow, the VIV response is dominated by travelling waves along the entire length of the cylinder span.
(2) The VIV responses display obvious multi-frequency characteristics in both linear and exponential shear flows. For linear shear flow, only high vibration modes are excited. However, for exponential shear flow, both the high vibration modes and low vibration modes are excited. Compared with linear shear flow, the bandwidth of the excited frequencies is much larger for exponential shear flow.
(3) Compared with linear shear flow, the vibration characteristics of the lift force coefficient, structural displacement, structural velocity and the transferred energy for exponential shear flow are all very chaotic. For linear shear flow, the negative values of the energy occur only within the low velocity zone. However, for exponential shear flow, the negative values occur along the entire span of the cylinder.