# Prediction of the VIV Responses Based on the Numerical Solutions of Controlled Motion

Authors: Duanmu Yu; Chen Jianping; Chen Yizhong
Journal: Shock and Vibration
 (2021)

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## Abstract

The study of forced and free vibration of a cylinder has long been isolated. The internal relationship between free vibration and forced vibration has rarely been investigated. In this paper, the relationship between the forced and free vibration of a cylinder was established. A series of numerical simulations of a cylinder undergoing forced oscillations at a wide range of vibration amplitudes and frequencies were carried out, with the flow solver viv-FOAM-SJTU developed based on the open-source platform OpenFOAM. Complex demodulation analysis was conducted to quantify the spatial-temporal phase relationship between the forces and the displacement of the cylinder. It was found that, at some particular oscillating amplitudes and frequencies, the phase angle switched between positive and negative values, which corresponds to a vortex mode transferring from the 2P mode to the 2PO mode. This distinct new mode “2PO” was closely related to the intermittent jumping between lower and upper branches of the amplitude responses of VIV. A prediction model was developed to obtain the VIV amplitude responses based on the numerical results of forced oscillation. The prediction results of three points located separately in the initial, upper, and lower branches of VIV agreed well with experimental measurements of an elastically mounted cylinder. This prediction model was thus expected to be suitable for predicting the response of VIV.


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## Body

## 1. Introduction

When a flow passes a cylinder, the oscillatory shedding of vortices into the wake would cause fluctuating lift and drag forces on the cylinder. If the cylinder was elastically mounted, the fluctuating forces could excite the vibrations of the cylinder. The motions of the cylinder would in turn change the vortex wake. Such a problem of fluid-structure interaction was called vortex-induced vibrations (VIV). VIV was a highly specialized subject that incorporates fluid mechanics, structural mechanics, vibrations, computational fluid dynamics (CFD), etc. and occurs in many engineering situations such as marine riser pipes, mooring cables, tethered structures, and spar hulls. VIV could cause large stresses and fatigue damage of structures. So far, numerous contributions to flow-induced oscillations in general and to VIVs in particular had guided the acquisition of design data through physical and numerical experiments, theoretical, and physical research work, of which many had been discussed in the compressive reviews of Williamson [1]; Williamson [2]; Williamson [3]; Sarpkaya [4]; Gabbai [5]; Bearman [6]; Wu [7]. Much progress had been made during the past decade, both numerically and experimentally, toward the understanding of the kinematics of VIV. Studies for VIV could be divided into two categories: free vibration of a cylinder and forced vibration of a cylinder.For free vibrations of a cylinder, the vibration force was provided through the fluid forces exerted on the cylinder. Mass ratiom∗ was an important parameter that influences the amplitude response of VIV. Feng’s [8] experiment of free vibration of an elastically mounted cylinder in air, which represents the typical model of high mass ratio, was the earliest and perhaps the most famous experiment. According to Feng [8], there were two branches for the amplitude response: the vortex mode changes from 2S of the initial branch to 2P of the lower branch. The situation for low mass ratio was quite different. A new branch, called the upper branch, appeared in the amplitude response of VIV for low mass ratio [9–12], besides the initial and lower branches, as was also shown for the high mass ratio. The maximum response amplitude reached about one diameter (1D) of the cylinder, which was greater than Feng’s results (0.6D). Jauvtis and Williamson [13] focused their attention on the study of vortex-induced vibration with two degrees of freedom. The amplitude response obtained from combined X and Y motion was larger than Y-only motion (X is the in-line direction, and Y is the cross-flow direction). A new vortex wake mode “2T,” which consisted of a triple of vortices, was observed for low mass ratios VIV experiments (m∗=2.6). Other researchers carried out numerical simulations of freely oscillating cylinder with CFD methods. Navrose and Sanjay [14] reported free vibrations of a cylinder of low mass ratio m∗=10 at Reynolds number Re = 1000. Three branches were obtained, and the maximum amplitude of cross-flow (y) oscillations was 0.7D. Al-Jamal and Dalton [15] performed a series of 2-D LES simulations of the VIV response of a circular cylinder at Re = 8000 with a wide range of damping ratios and natural frequencies. Lucor et al. [16] presented DNS results of uniform flow past a rigid cylinder with low mass damping at Re = 1000, 2000, and 3000. Wanderley et al. [17] correctly predicted the amplitudes of the VIV oscillations by solving the slightly compressible Reynolds average NS equations with the Roe-Sweby scheme.For forced vibrations, the cross-flow (y) movement of the cylinder was predefined by the sinusoidal trajectory relative to the fluid. Williamson [18] conducted experiments of controlled oscillation, over a broad range of amplitudes (A∗ up to 5.0) and wavelengths (λ∗ up to 15.0). The results showed principally three vortex wake patterns, named 2S, 2P, and P + S modes, which were relevant to the fundamental synchronization regime. These experiments had been used as benchmarks and have been verified by experiments or numerical simulations conducted by later researchers. Williamson and Roshko [18] studied the cross-flow (y) forced vibration of a circular column in uniform flow. Nishihara et al. [19] presented the characteristics of fluid forces and wake patterns of a circular cylinder oscillating in the streamwise direction in a cross-flow. Jeon and Gharib [20] performed the experiments of a cylinder undergoing one-and two-degree-of-freedom motions and found that the addition of streamwise motion will result in qualitative changes in the wake. Numerical simulations were carried out as an alternative method to study forced oscillations of a circular cylinder. Atluri et al. [21] investigated the influence of Reynolds number (ranging from 500 to 8000) on the wake structure of a forced oscillating cylinder at different vibration frequencies. Dong and Karniadakis [22] presented DNS results of the flow past an oscillating cylinder at Re = 10000 by employing a multilevel-type parallel algorithm. Zhao and Wan [23, 24] presented DES results of the flow past an oscillating cylinder. Fu et al. [25] presented VIVs of a flexible cylinder in an oscillatory flow. Deng et al. [26] studied VIVs of a flexible cylinder by three-dimensional strip model.As was shown in the foregoing studies, the study on the forced and free vibration had long been isolated. The internal relationship between free vibrations and forced vibrations had received few investigations. However, the ultimate objective of VIV research was to predict, to the extent possible, the kinematics and dynamics of self-excited vibrations from forced vibration (physical/numerical) experiments and, equally important, the dynamics of forced oscillation (say, e.g., drag lift, and inertia coefficients and the phase angle) from the physical/numerical experiments with self-excited oscillations [4]. Morse and Williamson [27] performed 5680 runs of forced oscillating experiments in a water flume and accurately predicted the response and wake modes for vortex-induced vibration response. A new distinct mode of vortex formation, termed as the 2POVERLAP mode (2PO), was identified. This mode also consisted of two pairs of vortices (like 2P), but one vortex in each pair was much weaker than the other. The extreme care of control required for forced vibration experiments and extensive measurements needed to create very high-resolution data make it arduous to accomplish such experimental studies [27]. Moreover, many factors such as running condition, installation error, and collection of data could influence the experimental results. Compared to experiments, numerical simulations can easily control the simulation parameters and obtain the detailed vortex structures. One objective of our study was to identify the new vortex mode 2PO by numerical simulations. The new vortex mode was closely related to intermittent jumping between lower and upper branches of the amplitude responses of VIV. We found that the wake could switch intermittently between the 2P and 2PO modes, even if the cylinder was vibrating with constant amplitude and frequency. In other words, the two different vortex modes could exist simultaneously at some specific oscillating amplitudes and frequencies, contrary to the conventional notion that only one clear vertex formation can exist. Complex demodulation analysis was used to obtain the time history of the phase difference between the lift coefficient and the displacement. The phase angle switched between positive and negative values, corresponding to the change of vortex mode transition between 2P mode and 2PO mode. Another objective of this study was to predict the amplitude response of free vibration cylinders from forced vibration. The prediction function was derived based on the concept of an “energy balance” to compute the amplitude response of a free oscillating cylinder at different mass damping ratios. Our work helped explain several important questions of free vibrations of a cylinder.This paper presented a number of numerical researches of the forced oscillating cylinder at Re = 4000. The simulations were carried out with a newly developed flow solver viv-FOAM-SJTU, which was integrated with six-degree-of-freedom motion module and coupled dynamic deformation mesh module, based on open-source platform OpenFOAM. All the simulations were performed by solving the Reynolds-averaged Navier–Stokes (RANS) equations with the SSTk−ω turbulence model. The finite volume method (FVM) for spatial discretization and implicit Euler scheme for temporal discretization were adopted. The pressure-velocity coupling was dealt with by the PIMPLE (merged PISO-SIMPLE) algorithm, a large time-step transient solver for incompressible flow.This paper is organized as follows. Section2 gives an introduction of numerical methods, which were composed by four parts. In section 2.1, the numerical fluid model is introduced firstly. All the nondimensionalized parameters used in the paper are presented in section 2.2. Next, the complex demodulation method used to obtain the phase angle between the lift force and displacement is introduced in section 2.3. The prediction model, which is used to obtain the VIV amplitude responses based on the numerical results of forced oscillation, is described in section 2.4. Section 3 presents and discusses the numerical results. First, an overview of the computational domain was presented, and four different mesh verifications were carried out to confirm the mesh convergence. Next, numerical simulations of forced oscillation with constant frequency λ∗=6.2 and different amplitude A∗ ranging from 0.4 to 0.9 were carried out. And discussions about the wake modes and vortex transition were carried out to verify the new 2PO mode, which would give the explanation for the intermittent switch between the lower branch and upper branch. The last part of section 3 was about the validation of the prediction model by using the free vibration responses of Govardhan and Williamson [28]. The VIV responses were successfully predicted based on the forced vibration data with three typical cases λ∗=5.0,6.0,6.2, which were located in the initial, upper, and lower branches separately. Finally, a summary of the paper is presented.


## 2. Numerical Method

### 2.1. Numerical Fluid Model

The fluid motion is represented by the incompressible unsteady Reynolds-Averaged Navier–Stokes (URANS) equations. Details of the solution procedure implemented in OpenFOAM are described in Jasak [29] and Rusche [30]; and only a brief introduction was presented here.The Reynolds-averaged Navier–Stokes equations are as follows:(1)∂ui¯∂xi=0,(2)∂∂tρui¯+∂∂xjρui¯uj¯=−∂p¯∂xi+∂∂xj2μSij¯−ρuj′ui′¯,where Sij¯=1/2∂ui¯/∂xj+∂uj¯/∂xi is the mean rate of strain tensor. −ρuj′ui′¯ results from the fluctuating velocity field and is generally referred to as the Reynolds stress τij.A turbulent model is required to compute the Reynolds stresses for turbulence closure. The SSTk−ω turbulence model is used because of its well performance in reverse pressure gradient and separated flow. The Reynolds stress is modelled by a linear constitutive relationship using Boussinesq hypothesis with the mean flow straining field as(3)τij=−ρuj′ui′¯=2μtSij¯−23ρkδij,where μt is the turbulent viscosity, obtained from solving the SST k−ω model, and k=1/2ui′ui′¯ is the mean turbulent kinetic energy.Applying equations (3) to (2), one has(4)∂∂tρui¯+∂∂xjρui¯uj¯=−∂peff¯∂xi+∂∂xj2μeffSij¯,where peff=p+2/3ρk, μeff=μ+μtThe solution of the governing equations is achieved by using the PIMPLE, which is a combined PISO-SIMPLE algorithm and is adapted for large time-step computation of incompressible flow.


### 2.2. Basic Parameters

All the parameters are nondimensionalized, so that different systems can be compared with each other. The nondimensional parameters of the paper are shown in Table1.Table 1
Definition of dimensionless variables.
Reynolds numberReUD/νMass ratiom∗m/πρD/22LDamping ratioζc/2mkAmplitude ratioA∗A/DNormalized wavelengthλ∗λ/D=U/fexDLift force coefficientCL2FL/ρU2DLDrag force coefficientCD2FD/ρU2DLFrequency ratiof∗fex/fnIn Table1, U is the inlet velocity; fex is the forced oscillating frequency; fn is the natural frequency in water; D is the cylinder diameter; L is the cylinder length; ν is the fluid kinematic viscosity; ρ is the fluid density; m is the cylinder mass; c is the structural damping; k is the spring constant; FL is the lift force of the cylinder; FD is the drag force of the cylinder.


### 2.3. Complex Demodulation

The phase angleφ between the fluid lift force and cylinder displacement is determined by a complex demodulation of the lift coefficient and displacement. A description of the technique is given as follows:The lift coefficient signal is chosen as the reference signal, as(5)CLt=CL0cos2πfext+φt,where CL0 is the amplitude of the lift coefficient, φ is the angle between the cylinder displacement and fluid lift, and fex is the frequency of forced oscillation.(6)CLt=CL0cos2πfext+φt=12CL0expiωext+φt+exp−iωext+φt,where ωex=2πfexis the angular frequency.Demodulation by multiplying byexp−iωext,(7)Yt=CLtexp−iωext=12CL0expiφt+12CL0exp−i2ωext+φt.abLow-pass filter to remove frequencies at or above frequencyωex: this removes term (b), and smooth term (a). The result is(8)Y′t=12CL0′expiφ′t.Extractφ′t: expiφ′t=2Y′/CL0′.


### 2.4. Prediction Model

The equation, which represents the vortex-induced vibrations of a cylinder oscillating in the cross-flow (y) direction, is defined as follows:(9)my¨+cy˙+ky=FL.According to Table1, FL=1/2CLρU2DL, and then, the vibration equations can be expressed as(10)my¨t+cy˙t+kyt=12CLtρU2DL.The parameters in the equation are shown in Table1. The cylinder motion yt, which is given by the controlled vibration, is a precisely sinusoidal function. The lift force coefficient CLt, caused by uniform flow around a controlled vibrating cylinder, is also approximated by a sinusoidal function(11)yt=Asin2πfext,(12)CLt=CL0sin2πfext+φ,where A is the amplitude of the controlled motion, and CLt can be resolved into a component in phase with the cylinder velocity CLv, and a component in phase with the cylinder acceleration CLa(13)CLt=CLvcos2πfext+CLa−sin2πfext,where CLv=CL0sinφ, CLa=−CL0cosφ.The structural dampingcand spring constant kare defined as c=2ζmωnk=mωn2, where ωn=2πfn, and fnis the natural frequency of the circular cylinder in water. Substitutingc,k, Y=y/Dinto equation (11) combined with nondimensional variables, then equation (11) can be expressed as(14)Y¨+2ζY˙+Y=CLρU22ωn2m.Transforming equations (13) and (14) to the dimensionless form and substituting them into(15)AD=14CL0sinφπ3m∗ζUfexD2fexfn.The prediction equation is given as follows, by applying Table1:(16)CL0sinφ=4π3A∗m∗ζλ∗2f∗.Different from experimental studies, added-mass coefficient is not considered in numerical simulations. Instead, the contribution of pressure and viscous forces is solved by N-S equations directly. The influence of added mass is considered by the fluid force in the right-hand side of N-S equations.The prediction method is based on the balance of the energy. The termCL0sinφ is the force coefficient in phase with the velocity. This term is denoted as the “fluid excitation” and represents a normalized energy into the system. For a free-vibration system oscillating at steady state this energy must be balanced by the normalized energy out of the system. The phase angle φ is an important parameter that influences the transition of energy from the fluid flow the cylinder oscillation. If the phase angle φ is between 0∘ and 180∘, then it will yield positive excitation from fluid to cylinder. If the phase angle φ is between −180∘ and 0∘, then it will yield negative excitation from cylinder to fluid.


## 2.1. Numerical Fluid Model

The fluid motion is represented by the incompressible unsteady Reynolds-Averaged Navier–Stokes (URANS) equations. Details of the solution procedure implemented in OpenFOAM are described in Jasak [29] and Rusche [30]; and only a brief introduction was presented here.The Reynolds-averaged Navier–Stokes equations are as follows:(1)∂ui¯∂xi=0,(2)∂∂tρui¯+∂∂xjρui¯uj¯=−∂p¯∂xi+∂∂xj2μSij¯−ρuj′ui′¯,where Sij¯=1/2∂ui¯/∂xj+∂uj¯/∂xi is the mean rate of strain tensor. −ρuj′ui′¯ results from the fluctuating velocity field and is generally referred to as the Reynolds stress τij.A turbulent model is required to compute the Reynolds stresses for turbulence closure. The SSTk−ω turbulence model is used because of its well performance in reverse pressure gradient and separated flow. The Reynolds stress is modelled by a linear constitutive relationship using Boussinesq hypothesis with the mean flow straining field as(3)τij=−ρuj′ui′¯=2μtSij¯−23ρkδij,where μt is the turbulent viscosity, obtained from solving the SST k−ω model, and k=1/2ui′ui′¯ is the mean turbulent kinetic energy.Applying equations (3) to (2), one has(4)∂∂tρui¯+∂∂xjρui¯uj¯=−∂peff¯∂xi+∂∂xj2μeffSij¯,where peff=p+2/3ρk, μeff=μ+μtThe solution of the governing equations is achieved by using the PIMPLE, which is a combined PISO-SIMPLE algorithm and is adapted for large time-step computation of incompressible flow.


## 2.2. Basic Parameters

All the parameters are nondimensionalized, so that different systems can be compared with each other. The nondimensional parameters of the paper are shown in Table1.Table 1
Definition of dimensionless variables.
Reynolds numberReUD/νMass ratiom∗m/πρD/22LDamping ratioζc/2mkAmplitude ratioA∗A/DNormalized wavelengthλ∗λ/D=U/fexDLift force coefficientCL2FL/ρU2DLDrag force coefficientCD2FD/ρU2DLFrequency ratiof∗fex/fnIn Table1, U is the inlet velocity; fex is the forced oscillating frequency; fn is the natural frequency in water; D is the cylinder diameter; L is the cylinder length; ν is the fluid kinematic viscosity; ρ is the fluid density; m is the cylinder mass; c is the structural damping; k is the spring constant; FL is the lift force of the cylinder; FD is the drag force of the cylinder.


## 2.3. Complex Demodulation

The phase angleφ between the fluid lift force and cylinder displacement is determined by a complex demodulation of the lift coefficient and displacement. A description of the technique is given as follows:The lift coefficient signal is chosen as the reference signal, as(5)CLt=CL0cos2πfext+φt,where CL0 is the amplitude of the lift coefficient, φ is the angle between the cylinder displacement and fluid lift, and fex is the frequency of forced oscillation.(6)CLt=CL0cos2πfext+φt=12CL0expiωext+φt+exp−iωext+φt,where ωex=2πfexis the angular frequency.Demodulation by multiplying byexp−iωext,(7)Yt=CLtexp−iωext=12CL0expiφt+12CL0exp−i2ωext+φt.abLow-pass filter to remove frequencies at or above frequencyωex: this removes term (b), and smooth term (a). The result is(8)Y′t=12CL0′expiφ′t.Extractφ′t: expiφ′t=2Y′/CL0′.


## 2.4. Prediction Model

The equation, which represents the vortex-induced vibrations of a cylinder oscillating in the cross-flow (y) direction, is defined as follows:(9)my¨+cy˙+ky=FL.According to Table1, FL=1/2CLρU2DL, and then, the vibration equations can be expressed as(10)my¨t+cy˙t+kyt=12CLtρU2DL.The parameters in the equation are shown in Table1. The cylinder motion yt, which is given by the controlled vibration, is a precisely sinusoidal function. The lift force coefficient CLt, caused by uniform flow around a controlled vibrating cylinder, is also approximated by a sinusoidal function(11)yt=Asin2πfext,(12)CLt=CL0sin2πfext+φ,where A is the amplitude of the controlled motion, and CLt can be resolved into a component in phase with the cylinder velocity CLv, and a component in phase with the cylinder acceleration CLa(13)CLt=CLvcos2πfext+CLa−sin2πfext,where CLv=CL0sinφ, CLa=−CL0cosφ.The structural dampingcand spring constant kare defined as c=2ζmωnk=mωn2, where ωn=2πfn, and fnis the natural frequency of the circular cylinder in water. Substitutingc,k, Y=y/Dinto equation (11) combined with nondimensional variables, then equation (11) can be expressed as(14)Y¨+2ζY˙+Y=CLρU22ωn2m.Transforming equations (13) and (14) to the dimensionless form and substituting them into(15)AD=14CL0sinφπ3m∗ζUfexD2fexfn.The prediction equation is given as follows, by applying Table1:(16)CL0sinφ=4π3A∗m∗ζλ∗2f∗.Different from experimental studies, added-mass coefficient is not considered in numerical simulations. Instead, the contribution of pressure and viscous forces is solved by N-S equations directly. The influence of added mass is considered by the fluid force in the right-hand side of N-S equations.The prediction method is based on the balance of the energy. The termCL0sinφ is the force coefficient in phase with the velocity. This term is denoted as the “fluid excitation” and represents a normalized energy into the system. For a free-vibration system oscillating at steady state this energy must be balanced by the normalized energy out of the system. The phase angle φ is an important parameter that influences the transition of energy from the fluid flow the cylinder oscillation. If the phase angle φ is between 0∘ and 180∘, then it will yield positive excitation from fluid to cylinder. If the phase angle φ is between −180∘ and 0∘, then it will yield negative excitation from cylinder to fluid.


## 3. Computational Results and Discussion

### 3.1. Grid Convergence Discussion

In this part, grid convergence is verified to check the mesh of forced oscillation simulations. We present a study of flow dynamics associated with simulated two-dimensional flows past a circular cylinder that is in simple harmonic cross-flow (y) oscillation. The cylinder oscillation in the cross-flow (y) direction is expressed by(17)y=Acos2πfext.Results are examined for Re = 4000 and a fixed motion amplitude ofA∗=A/D=0.4, and a fixed vibration frequency of λ∗=U0/fexD=4.Computational geometry is two-dimensional, as shown in Figure1. The length of computational geometry is from −10D of the boundary inlet to 20D of the boundary outlet. The width is from −15D of the boundary top to 15D of the boundary bottom.Figure 1
The calculational geometry.Four different mesh verifications were carried out by changing the nodes around the cylinder and the distance of nearest grid to the cylinder boundary. Table2 summarizes the information of the four meshes. Figure 2 shows the whole gird and mesh information around the cylinder. All grids used in this paper are structured grids, which is easy for systematic refinement in all directions.Table 2
The details of the mesh for the computation.
MeshNumber of cellsNumber of nodes around the cylinderThe height of nearest grid to the cylinder boundaryMesh I219001200.004DMesh II293001200.003DMesh III465001600.003DMesh IV716002000.002DFigure 2
(a) Uniform mesh for the global domain; (b) boundary layers around cylinder; (c) details of the four meshes around the cylinder.
(a)(b)(c)Table3 summarizes the values of the time average of drag coefficient CD¯ and lift coefficient amplitude CLmax¯, the vortex shedding frequency fst, the Strouhal number St, and the phase angle φ at four grid resolutions from the simulation for the flow past an oscillating cylinder at Re = 4000. It can be seen that large differences in the grid size only slightly change the results. The Strouhal number is the same for all grids. It indicates that dominant frequency of vortex shedding shifts to the frequency of oscillating cylinder. At this condition, the lock-in of frequency will occur. The differences for the drag coefficient are smaller than 1.1%, and the differences for the lift coefficient are smaller than 0.8%. The phase angle is stable around 3.5∘. Then, following this grid sensitivity study, it is dependable to proceed with the study of vortex mode transitions and prediction the free vibration responses by employing controlled oscillations. According the mesh convergence study, the mesh III is chosen for the next numerical simulations.Table 3
Force coefficient and phase between lift and displacement.
MeshCD¯CLmax¯fstStφ∘Mesh I1.43062.44840.6910.25−3.6786Mesh II1.41482.45920.6910.25−3.4629Mesh III1.41452.46850.6910.25−3.4916Mesh IV1.41442.46890.6910.25−3.4957


### 3.2. Wake Modes and Transitions

In this section, a forced motion cylinder controlled by sinusoidal oscillation with different amplitudes and a constant frequency is simulated to study the wake modes and transitions.Williamson and Roshko [18] studied the vortex wake patterns of an oscillating cylinder controlled by a sine function. They researched different vortex modes and clarified them into three different principal types: 2S pattern (each period alternatively shedding single vortex), 2P pattern (each period alternatively shedding vortex pairs), and P + S pattern (each period alternatively shedding single vortex and vortex pairs), which are shown in Figure 3(a). It identified the clear boundaries of the three principal vortex patterns.Figure 3
(a) Boundaries of wake modes identified by Williamson and Roshko [18]. (b) Boundaries of wake modes identified by Morse and Williamson [27].
(a)(b)Morse and Williamson [27] did 5680 controlled oscillation experiments over a more extensive range of normalized amplitude and wavelength. The map of vortex shedding regimes is shown in Figure 3(b). For the boundaries described by Williamson and Roshko [18], the transitions of the boundary separating different vortex patterns were clear and abrupt. However, Morse and Williamson [27] found several regions where two distinct vortex-formation modes can exist. In these regions, a distinct new mode of vortex formation was named “2POVERLAP,” abbreviated as “2PO” mode. The 2PO mode exists in its own clearly defined region of the amplitude-wavelength plane, overlapping the boundary between the 2S and 2P regions.Many researchers have performed experiments or numerical simulations to verify the 2S, 2P, and P + S modes. However, no research work has been conducted to verify the new distinct mode pattern 2PO. Therefore, it is essential for us to perform numerical simulations to verify the 2PO mode. A series of simulations about a forced oscillating cylinder with constant frequencyλ∗=6.2 and different amplitudeA∗ ranging from 0.4 to 0.9 are carried out.First, phase angleφ between the fluid lift force and cylinder displacement is studied by the method of complex demodulation to understand the system and its mechanism for vortex mode transitions. Figure 4 displays the time history of the y-direction motion, lift coefficient, and the phase angle at λ∗=6.2, A∗=0.4. As shown in Figure 4, the phase angle φ maintains a stable value of 140.0∘. The vorticity contours of the obtained 2P mode are shown in Figure 5. A pair of vortices, where the two vortices do not separate at all, shed alternatively from each side of the cylinder. Vorticity contours in this paper are defined by Q criterion: Q=1/2Ω2−S2. Here, Ω=1/2∂ui/∂xj−∂uj/∂xiis the vorticity; S=1/2∂ui/∂xj+∂uj/∂xiis the strain invariant.Figure 4
The cylinder motions, lift coefficient, and phase angle in time-domain atA∗=0.4, λ∗=6.2.Figure 5
Vorticity contours of 2P mode in one vibration period at A∗=0.4, λ∗=6.2.When amplitudeA∗ of vibration is added from 0.5 to 0.9, while oscillating frequency holds constant as before, we further study the vortex mode transitions with the changes of phase angle. Time histories of the phase angle are shown in Figure 6. This figure shows that, for amplitude A∗=0.5, the phase angle keeps at 159.7∘, while, for the other four amplitudes increasing from 0.6 to 0.9, the phase angle cannot remain unchanged as above. Abrupt jumps from a negative angle to a positive angle are observed. After a short time, the angle instantly goes back to its original negative value. The intermittent switch of angle appears to be irregular and unstable.Figure 6
Vortex field of2PO mode at the positive part of phase angle.Accompanied with the jump of the phase angle, the vortex wake pattern is also changed. Take the dotted box in Figure7 as an example. As the phase angle jumps to the positive part, the vortex wake exhibits a different pattern, as shown in Figure 6. This new pattern, named 2PO, is first found by Morse and Williamson [27] by physical experiments. The new pattern is distinguished with the 2P mode in that although two pairs of vortices alternatively shed each period, the strengths of the two vortices are unequal. One vortex in a pair is much weaker and decays quickly than the other. Figure 8 shows the 2PO mode vortex field obtained by digital particle image velocity (DPIV) measurements of Morse and Williamson [27]. The numerical simulations of the vortex patterns are consistent with the experimental results, as displayed in Figures 9 and 10. The good details of vortex street and vortex transition are presented. The very weak secondary vortex can only be observed in a very near region behind the cylinder. The weak vortex diffuses quickly when propagating downstream, which explains why many experimental researchers did not identify the 2PO mode before; one would confuse the 2PO mode with the 2S mode, which is decided by how well one could solve the secondary weak vortex. However, the subtle structure of vortex could be identified relatively easily by numerical method, which could provide small enough contours.Figure 7
Phase angle at different amplitude ratios varying from 0.5 to 0.9.Figure 8
2PO mode vortex field by using DPIV, Morse and Williamson [27].Figure 9
Vortex field of2P mode at the negative part of phase angle.Figure 10
Vortex patterns of the three VIV response branches, from Govardhan and Williamson [28]. (a) Initial branch. (b)Upper branch. (c)Lower branch.
(a)(b)(c)When the phase angle jumps back to the negative part, the vortex wake exhibits a2P pattern, as shown in Figure 9. The two vortices in a vortex pair have almost equal strength. However, this 2Pmode is different from that in Figure 5 in that the vortices in a pair are separated from each other when two pairs of vortices shed alternatively behind the cylinder.In this section, the new distinct mode pattern2PO is verified by numerical simulations. Moreover, the amplitude A∗ and wavelength λ∗ where the 2P and 2PO modes are observed in the simulations are located in the corresponding 2P and 2PO regions of Morse and Williamson [27]. In the overlapping area, the vortex mode is not fixed, the vortex mode changes between the two modes 2P and 2PO, and even the cylinder is vibrating with a fixed frequency and amplitude.As mentioned above in the introduction, for the typical model of low mass ratio, the amplitude responses are comprised by three branches: initial branch, upper branch, and lower branch. The vortex pattern is gradually turning from the 2S mode in the initial branch into the 2P mode in the upper and lower branches, while the 2Pupper mode of upper branch is different from the 2Plower mode of lower branch, as shown in Figure 10. The newly found 2PO vortex-formation mode is equivalent to the “2Pupper” mode found by Govardhan and Williamson [28]. Khalak & Williamson [12] found that the intermittent switches between upper branch and lower branch are observed in the low-mass ratio studies, as shown in Figure 11. The overlap regions can be used to explain the intermittent switch, because, in the overlap regions, the vortex mode switches intermittently between two distinct modes 2P and 2PO, which is closely related to the intermittent switch existing between the upper and lower branches. When the phase angle φ is positive, the vortex mode appears 2PO, the energy transfers from fluid to cylinder, and it will yield positive excitation. And the amplitude response is located in the upper branch. By contrast, the negative phase angle φ means the negative excitation from cylinder to fluid. The amplitude response is located in the lower branch.Figure 11
Intermittent switch (I) between upper branch and lower branch.


### 3.3. Prediction Results and Discussion

In this section, we try to predict free-vibration response of Govardhan and Williamson [28], as shown in Figure 12, by the above prediction method. In order to predict the response by controlled-vibration data, the inlet velocity U remains the same, and forced oscillating frequency fex is changed. Thus, we can get different amplitude responses at differentλ∗=U/fexD.Figure 12
Amplitude response plots at a constant Reynolds numberRe≈4000 at different mass damping values m∗+CAζ=α=0.000;0.059;0.187;0.252, taken from Govardhan and Williamson [28]. ▲,α=0.000; ○, α=0.059; ●, α=0.187; □, α=0.252.The availability of the force coefficient in phase with velocityCL0sinφ enables us to predict the response of a free-vibration cylinder. We can simply solve numerically for amplitude (A∗) and frequency (f∗) and build up a response plot at a given m∗ζ. Continuing §3.2 wake modes and transitions study above, we try to predict the amplitude response at λ∗=6.2. Table 4 gives the amplitude of the lift coefficient CL0, the phase angle φ, and the force coefficient in phase with velocity CL0sinφ at different amplitude ratio A∗ and constant wavelength λ∗=6.2 of the forced oscillation. It is particularly worth mentioning that the phase angle listed in Table 4 is the stable angle. In the overlap region, there exist two phase angles. The average negative angle is the stable angle, and the jumping positive angle is the unstable angle.Table 4
Forced oscillating response at different amplitude ratioA∗ at λ∗=6.2.
A∗λ∗CL0φCL0sinφ0.46.20.4478107.61560.42680.56.20.5083159.66840.17660.66.20.6030−159.6776−0.20940.76.20.8782−126.7985−0.70320.86.21.2734−112.6470−1.17520.96.21.6000−110.9883−1.4939After we have got the value of the force coefficient in phase with the velocity, we can readily use our controlled-vibration data to predict the response of a freely vibrating cylinder at different mass dampingm∗ζ=0.0,0.059,0.187,0.252, in the example of Figure 13. The red circles in Figure 13 are the values of the term (CL0sinφ) at different forced amplitude ratio A∗. The blue asterisks denote the intersections of two lines corresponding to the right and left terms in prediction equation (16). We find close agreement with the measured free vibration amplitude response (taken from [28], from the error analysis shown in Table 5).Figure 13
Prediction of amplitude responseA∗ at different mass damping when λ∗=6.2.Table 5
Comparison ofA∗ at λ∗=6.2 with experiment results.
m∗ζA∗_predictA∗_experiment of Govardhan and Williamson [28]Error (%)0.00.41500.39584.90.0590.44640.43801.90.1870.51640.50921.40.2520.54570.53561.9For the low-mass-dampingm∗ζ case, there are three-branch responses, named initial branch, upper branch, and lower branch. In the free vibration of Figure 13, there exists an intermittent switch between the upper and lower branches at λ∗=6.2. The point λ∗=6.2 is located in the initial part of lower branch. According to the introduction of the part §3.2, the phase angle φ is unstable at λ∗=6.2, which is accompanied by the abrupt jump from negative angle to positive angle. When the phase angle φ is negative, the response amplitude A∗ is located in lower branch. When the phase angle φ jumps to the positive, extra energy is transported from fluid to cylinder. Then, it will cause a larger amplitude response, which is located in the upper branch. But the state is very unstable, and the response will come down to the lower branch again.Analysis of wake modes and transitions in part §3.2 shows that if the vortex mode is 2P, then there will be one stable equilibrium at the lower branch amplitude. However, this equilibrium will only persist if the vortex-formation mode continues to be 2P. If the wake switches to the 2PO mode, the fluid excitation will raise. This will cause the energy into the system to be more than the energy dissipated by damping, and therefore, the amplitude will increase until a new stable equilibrium for the 2PO mode is reached, corresponding to an upper branch amplitude. Later in time, the vortex-formation mode could possibly switch back to 2P, causing the amplitude to decrease back toward the lower branch amplitude. In this way, the amplitude could switch intermittently between the upper and lower branches.For VIV problems, the maximum displacement of cylinder is our focus. From Figure12, we find that the maximum displacement is around U∗=6.0. So, we carry out several numerical simulations of controlled oscillating cylinder undergoing a constant vibration frequency λ∗=6.0, and different amplitude A∗ varied from 0.4 to 0.9 in this paper. And the forced oscillating results are listed in Table 6.Table 6
Forced oscillating response at different amplitude ratioA∗ at λ∗=6.0.
A∗λ∗CL0φCL0sinφ0.46.00.4480107.50060.42730.56.00.5798111.20180.51840.66.00.6748140.43020.42980.76.00.8774168.82710.17000.86.01.2751−164.1542−0.34820.96.01.6286−156.4862−0.6498The prediction ofA∗ at λ∗=6.0 is shown in Figure 14 and Table 7. Although the prediction results offer less accuracy than that at λ∗=6.2, we believe that the error would be reduced if the distribution of the forced vibration amplitude ratio A∗ is refined, e.g., with a resolution of 0.01.Figure 14
Prediction of amplitude responseA∗ at different mass damping when λ∗=6.0.Table 7
Comparison ofA∗ at λ∗=6.0 with experiment results.
m∗ζA∗_predictA∗_experiment of Govardhan and Williamson [28]Error (%)0.00.73280.870715.80.0590.70300.783610.30.1870.60370.691212.60.2520.53500.633215.5The above two cases have verified the amplitude responses in the upper and lower branches. In order to verify the good versatility of the prediction method, we try to predict the amplitude responses in the initial branches. We carry out several numerical simulations of controlled oscillating cylinder undergoing a constant vibration frequencyλ∗=6.0, because the amplitude in the initial branch is small. Different amplitudes A∗ are chosen varying from 0.1 to 0.6, and the forced oscillating results are listed in Table 8.Table 8
Forced oscillating response at different amplitude ratioA∗ at λ∗=5.0.
A∗λ∗CL0φCL0sinφ0.15.00.5767101.50210.56510.25.00.6198112.13540.57410.35.00.7451−176.3427−0.04750.45.00.8267−161.7941−0.25830.55.00.9762−153.2176−0.43990.65.01.1301−145.5132−0.6399The prediction ofA∗ at λ∗=5.0 is shown in Figure 15. The prediction results and comparison with the experimental results at λ∗=5.0 are shown in Table 9. The experimental results are very close to each other. Especially for m∗ζ=0.059,0.187,0.252, all the amplitude responses are the same A∗=0.22. This is because the experiment accuracy is limited. It is difficult to distinguish the amplitude difference when the amplitude response is small. However, the predicted amplitudes by numerical method are discrete distribution. The smaller the mass damping is, the larger the amplitude response is. The upper and lower bounds of the predicted amplitudes are close to the experimental results, but the intermediate values are quite different, which may be the error caused by the experiment accuracy.Figure 15
Prediction of amplitude responseA∗ at different mass damping when λ∗=5.0.Table 9
Comparison ofA∗ at λ∗=5.0 with experiment results.
m∗ζA∗_predictA∗_experiment of Govardhan and Williamson [28]Error (%)0.00.2920.284.20.0590.2700.2222.70.1870.2510.2214.10.2520.2390.228.6


## 3.1. Grid Convergence Discussion

In this part, grid convergence is verified to check the mesh of forced oscillation simulations. We present a study of flow dynamics associated with simulated two-dimensional flows past a circular cylinder that is in simple harmonic cross-flow (y) oscillation. The cylinder oscillation in the cross-flow (y) direction is expressed by(17)y=Acos2πfext.Results are examined for Re = 4000 and a fixed motion amplitude ofA∗=A/D=0.4, and a fixed vibration frequency of λ∗=U0/fexD=4.Computational geometry is two-dimensional, as shown in Figure1. The length of computational geometry is from −10D of the boundary inlet to 20D of the boundary outlet. The width is from −15D of the boundary top to 15D of the boundary bottom.Figure 1
The calculational geometry.Four different mesh verifications were carried out by changing the nodes around the cylinder and the distance of nearest grid to the cylinder boundary. Table2 summarizes the information of the four meshes. Figure 2 shows the whole gird and mesh information around the cylinder. All grids used in this paper are structured grids, which is easy for systematic refinement in all directions.Table 2
The details of the mesh for the computation.
MeshNumber of cellsNumber of nodes around the cylinderThe height of nearest grid to the cylinder boundaryMesh I219001200.004DMesh II293001200.003DMesh III465001600.003DMesh IV716002000.002DFigure 2
(a) Uniform mesh for the global domain; (b) boundary layers around cylinder; (c) details of the four meshes around the cylinder.
(a)(b)(c)Table3 summarizes the values of the time average of drag coefficient CD¯ and lift coefficient amplitude CLmax¯, the vortex shedding frequency fst, the Strouhal number St, and the phase angle φ at four grid resolutions from the simulation for the flow past an oscillating cylinder at Re = 4000. It can be seen that large differences in the grid size only slightly change the results. The Strouhal number is the same for all grids. It indicates that dominant frequency of vortex shedding shifts to the frequency of oscillating cylinder. At this condition, the lock-in of frequency will occur. The differences for the drag coefficient are smaller than 1.1%, and the differences for the lift coefficient are smaller than 0.8%. The phase angle is stable around 3.5∘. Then, following this grid sensitivity study, it is dependable to proceed with the study of vortex mode transitions and prediction the free vibration responses by employing controlled oscillations. According the mesh convergence study, the mesh III is chosen for the next numerical simulations.Table 3
Force coefficient and phase between lift and displacement.
MeshCD¯CLmax¯fstStφ∘Mesh I1.43062.44840.6910.25−3.6786Mesh II1.41482.45920.6910.25−3.4629Mesh III1.41452.46850.6910.25−3.4916Mesh IV1.41442.46890.6910.25−3.4957


## 3.2. Wake Modes and Transitions

In this section, a forced motion cylinder controlled by sinusoidal oscillation with different amplitudes and a constant frequency is simulated to study the wake modes and transitions.Williamson and Roshko [18] studied the vortex wake patterns of an oscillating cylinder controlled by a sine function. They researched different vortex modes and clarified them into three different principal types: 2S pattern (each period alternatively shedding single vortex), 2P pattern (each period alternatively shedding vortex pairs), and P + S pattern (each period alternatively shedding single vortex and vortex pairs), which are shown in Figure 3(a). It identified the clear boundaries of the three principal vortex patterns.Figure 3
(a) Boundaries of wake modes identified by Williamson and Roshko [18]. (b) Boundaries of wake modes identified by Morse and Williamson [27].
(a)(b)Morse and Williamson [27] did 5680 controlled oscillation experiments over a more extensive range of normalized amplitude and wavelength. The map of vortex shedding regimes is shown in Figure 3(b). For the boundaries described by Williamson and Roshko [18], the transitions of the boundary separating different vortex patterns were clear and abrupt. However, Morse and Williamson [27] found several regions where two distinct vortex-formation modes can exist. In these regions, a distinct new mode of vortex formation was named “2POVERLAP,” abbreviated as “2PO” mode. The 2PO mode exists in its own clearly defined region of the amplitude-wavelength plane, overlapping the boundary between the 2S and 2P regions.Many researchers have performed experiments or numerical simulations to verify the 2S, 2P, and P + S modes. However, no research work has been conducted to verify the new distinct mode pattern 2PO. Therefore, it is essential for us to perform numerical simulations to verify the 2PO mode. A series of simulations about a forced oscillating cylinder with constant frequencyλ∗=6.2 and different amplitudeA∗ ranging from 0.4 to 0.9 are carried out.First, phase angleφ between the fluid lift force and cylinder displacement is studied by the method of complex demodulation to understand the system and its mechanism for vortex mode transitions. Figure 4 displays the time history of the y-direction motion, lift coefficient, and the phase angle at λ∗=6.2, A∗=0.4. As shown in Figure 4, the phase angle φ maintains a stable value of 140.0∘. The vorticity contours of the obtained 2P mode are shown in Figure 5. A pair of vortices, where the two vortices do not separate at all, shed alternatively from each side of the cylinder. Vorticity contours in this paper are defined by Q criterion: Q=1/2Ω2−S2. Here, Ω=1/2∂ui/∂xj−∂uj/∂xiis the vorticity; S=1/2∂ui/∂xj+∂uj/∂xiis the strain invariant.Figure 4
The cylinder motions, lift coefficient, and phase angle in time-domain atA∗=0.4, λ∗=6.2.Figure 5
Vorticity contours of 2P mode in one vibration period at A∗=0.4, λ∗=6.2.When amplitudeA∗ of vibration is added from 0.5 to 0.9, while oscillating frequency holds constant as before, we further study the vortex mode transitions with the changes of phase angle. Time histories of the phase angle are shown in Figure 6. This figure shows that, for amplitude A∗=0.5, the phase angle keeps at 159.7∘, while, for the other four amplitudes increasing from 0.6 to 0.9, the phase angle cannot remain unchanged as above. Abrupt jumps from a negative angle to a positive angle are observed. After a short time, the angle instantly goes back to its original negative value. The intermittent switch of angle appears to be irregular and unstable.Figure 6
Vortex field of2PO mode at the positive part of phase angle.Accompanied with the jump of the phase angle, the vortex wake pattern is also changed. Take the dotted box in Figure7 as an example. As the phase angle jumps to the positive part, the vortex wake exhibits a different pattern, as shown in Figure 6. This new pattern, named 2PO, is first found by Morse and Williamson [27] by physical experiments. The new pattern is distinguished with the 2P mode in that although two pairs of vortices alternatively shed each period, the strengths of the two vortices are unequal. One vortex in a pair is much weaker and decays quickly than the other. Figure 8 shows the 2PO mode vortex field obtained by digital particle image velocity (DPIV) measurements of Morse and Williamson [27]. The numerical simulations of the vortex patterns are consistent with the experimental results, as displayed in Figures 9 and 10. The good details of vortex street and vortex transition are presented. The very weak secondary vortex can only be observed in a very near region behind the cylinder. The weak vortex diffuses quickly when propagating downstream, which explains why many experimental researchers did not identify the 2PO mode before; one would confuse the 2PO mode with the 2S mode, which is decided by how well one could solve the secondary weak vortex. However, the subtle structure of vortex could be identified relatively easily by numerical method, which could provide small enough contours.Figure 7
Phase angle at different amplitude ratios varying from 0.5 to 0.9.Figure 8
2PO mode vortex field by using DPIV, Morse and Williamson [27].Figure 9
Vortex field of2P mode at the negative part of phase angle.Figure 10
Vortex patterns of the three VIV response branches, from Govardhan and Williamson [28]. (a) Initial branch. (b)Upper branch. (c)Lower branch.
(a)(b)(c)When the phase angle jumps back to the negative part, the vortex wake exhibits a2P pattern, as shown in Figure 9. The two vortices in a vortex pair have almost equal strength. However, this 2Pmode is different from that in Figure 5 in that the vortices in a pair are separated from each other when two pairs of vortices shed alternatively behind the cylinder.In this section, the new distinct mode pattern2PO is verified by numerical simulations. Moreover, the amplitude A∗ and wavelength λ∗ where the 2P and 2PO modes are observed in the simulations are located in the corresponding 2P and 2PO regions of Morse and Williamson [27]. In the overlapping area, the vortex mode is not fixed, the vortex mode changes between the two modes 2P and 2PO, and even the cylinder is vibrating with a fixed frequency and amplitude.As mentioned above in the introduction, for the typical model of low mass ratio, the amplitude responses are comprised by three branches: initial branch, upper branch, and lower branch. The vortex pattern is gradually turning from the 2S mode in the initial branch into the 2P mode in the upper and lower branches, while the 2Pupper mode of upper branch is different from the 2Plower mode of lower branch, as shown in Figure 10. The newly found 2PO vortex-formation mode is equivalent to the “2Pupper” mode found by Govardhan and Williamson [28]. Khalak & Williamson [12] found that the intermittent switches between upper branch and lower branch are observed in the low-mass ratio studies, as shown in Figure 11. The overlap regions can be used to explain the intermittent switch, because, in the overlap regions, the vortex mode switches intermittently between two distinct modes 2P and 2PO, which is closely related to the intermittent switch existing between the upper and lower branches. When the phase angle φ is positive, the vortex mode appears 2PO, the energy transfers from fluid to cylinder, and it will yield positive excitation. And the amplitude response is located in the upper branch. By contrast, the negative phase angle φ means the negative excitation from cylinder to fluid. The amplitude response is located in the lower branch.Figure 11
Intermittent switch (I) between upper branch and lower branch.


## 3.3. Prediction Results and Discussion

In this section, we try to predict free-vibration response of Govardhan and Williamson [28], as shown in Figure 12, by the above prediction method. In order to predict the response by controlled-vibration data, the inlet velocity U remains the same, and forced oscillating frequency fex is changed. Thus, we can get different amplitude responses at differentλ∗=U/fexD.Figure 12
Amplitude response plots at a constant Reynolds numberRe≈4000 at different mass damping values m∗+CAζ=α=0.000;0.059;0.187;0.252, taken from Govardhan and Williamson [28]. ▲,α=0.000; ○, α=0.059; ●, α=0.187; □, α=0.252.The availability of the force coefficient in phase with velocityCL0sinφ enables us to predict the response of a free-vibration cylinder. We can simply solve numerically for amplitude (A∗) and frequency (f∗) and build up a response plot at a given m∗ζ. Continuing §3.2 wake modes and transitions study above, we try to predict the amplitude response at λ∗=6.2. Table 4 gives the amplitude of the lift coefficient CL0, the phase angle φ, and the force coefficient in phase with velocity CL0sinφ at different amplitude ratio A∗ and constant wavelength λ∗=6.2 of the forced oscillation. It is particularly worth mentioning that the phase angle listed in Table 4 is the stable angle. In the overlap region, there exist two phase angles. The average negative angle is the stable angle, and the jumping positive angle is the unstable angle.Table 4
Forced oscillating response at different amplitude ratioA∗ at λ∗=6.2.
A∗λ∗CL0φCL0sinφ0.46.20.4478107.61560.42680.56.20.5083159.66840.17660.66.20.6030−159.6776−0.20940.76.20.8782−126.7985−0.70320.86.21.2734−112.6470−1.17520.96.21.6000−110.9883−1.4939After we have got the value of the force coefficient in phase with the velocity, we can readily use our controlled-vibration data to predict the response of a freely vibrating cylinder at different mass dampingm∗ζ=0.0,0.059,0.187,0.252, in the example of Figure 13. The red circles in Figure 13 are the values of the term (CL0sinφ) at different forced amplitude ratio A∗. The blue asterisks denote the intersections of two lines corresponding to the right and left terms in prediction equation (16). We find close agreement with the measured free vibration amplitude response (taken from [28], from the error analysis shown in Table 5).Figure 13
Prediction of amplitude responseA∗ at different mass damping when λ∗=6.2.Table 5
Comparison ofA∗ at λ∗=6.2 with experiment results.
m∗ζA∗_predictA∗_experiment of Govardhan and Williamson [28]Error (%)0.00.41500.39584.90.0590.44640.43801.90.1870.51640.50921.40.2520.54570.53561.9For the low-mass-dampingm∗ζ case, there are three-branch responses, named initial branch, upper branch, and lower branch. In the free vibration of Figure 13, there exists an intermittent switch between the upper and lower branches at λ∗=6.2. The point λ∗=6.2 is located in the initial part of lower branch. According to the introduction of the part §3.2, the phase angle φ is unstable at λ∗=6.2, which is accompanied by the abrupt jump from negative angle to positive angle. When the phase angle φ is negative, the response amplitude A∗ is located in lower branch. When the phase angle φ jumps to the positive, extra energy is transported from fluid to cylinder. Then, it will cause a larger amplitude response, which is located in the upper branch. But the state is very unstable, and the response will come down to the lower branch again.Analysis of wake modes and transitions in part §3.2 shows that if the vortex mode is 2P, then there will be one stable equilibrium at the lower branch amplitude. However, this equilibrium will only persist if the vortex-formation mode continues to be 2P. If the wake switches to the 2PO mode, the fluid excitation will raise. This will cause the energy into the system to be more than the energy dissipated by damping, and therefore, the amplitude will increase until a new stable equilibrium for the 2PO mode is reached, corresponding to an upper branch amplitude. Later in time, the vortex-formation mode could possibly switch back to 2P, causing the amplitude to decrease back toward the lower branch amplitude. In this way, the amplitude could switch intermittently between the upper and lower branches.For VIV problems, the maximum displacement of cylinder is our focus. From Figure12, we find that the maximum displacement is around U∗=6.0. So, we carry out several numerical simulations of controlled oscillating cylinder undergoing a constant vibration frequency λ∗=6.0, and different amplitude A∗ varied from 0.4 to 0.9 in this paper. And the forced oscillating results are listed in Table 6.Table 6
Forced oscillating response at different amplitude ratioA∗ at λ∗=6.0.
A∗λ∗CL0φCL0sinφ0.46.00.4480107.50060.42730.56.00.5798111.20180.51840.66.00.6748140.43020.42980.76.00.8774168.82710.17000.86.01.2751−164.1542−0.34820.96.01.6286−156.4862−0.6498The prediction ofA∗ at λ∗=6.0 is shown in Figure 14 and Table 7. Although the prediction results offer less accuracy than that at λ∗=6.2, we believe that the error would be reduced if the distribution of the forced vibration amplitude ratio A∗ is refined, e.g., with a resolution of 0.01.Figure 14
Prediction of amplitude responseA∗ at different mass damping when λ∗=6.0.Table 7
Comparison ofA∗ at λ∗=6.0 with experiment results.
m∗ζA∗_predictA∗_experiment of Govardhan and Williamson [28]Error (%)0.00.73280.870715.80.0590.70300.783610.30.1870.60370.691212.60.2520.53500.633215.5The above two cases have verified the amplitude responses in the upper and lower branches. In order to verify the good versatility of the prediction method, we try to predict the amplitude responses in the initial branches. We carry out several numerical simulations of controlled oscillating cylinder undergoing a constant vibration frequencyλ∗=6.0, because the amplitude in the initial branch is small. Different amplitudes A∗ are chosen varying from 0.1 to 0.6, and the forced oscillating results are listed in Table 8.Table 8
Forced oscillating response at different amplitude ratioA∗ at λ∗=5.0.
A∗λ∗CL0φCL0sinφ0.15.00.5767101.50210.56510.25.00.6198112.13540.57410.35.00.7451−176.3427−0.04750.45.00.8267−161.7941−0.25830.55.00.9762−153.2176−0.43990.65.01.1301−145.5132−0.6399The prediction ofA∗ at λ∗=5.0 is shown in Figure 15. The prediction results and comparison with the experimental results at λ∗=5.0 are shown in Table 9. The experimental results are very close to each other. Especially for m∗ζ=0.059,0.187,0.252, all the amplitude responses are the same A∗=0.22. This is because the experiment accuracy is limited. It is difficult to distinguish the amplitude difference when the amplitude response is small. However, the predicted amplitudes by numerical method are discrete distribution. The smaller the mass damping is, the larger the amplitude response is. The upper and lower bounds of the predicted amplitudes are close to the experimental results, but the intermediate values are quite different, which may be the error caused by the experiment accuracy.Figure 15
Prediction of amplitude responseA∗ at different mass damping when λ∗=5.0.Table 9
Comparison ofA∗ at λ∗=5.0 with experiment results.
m∗ζA∗_predictA∗_experiment of Govardhan and Williamson [28]Error (%)0.00.2920.284.20.0590.2700.2222.70.1870.2510.2214.10.2520.2390.228.6


## 4. Conclusion

In this paper, numerical simulations of a forced oscillating cylinder at a wide range of oscillating amplitudes and frequencies at Re = 4000 were performed by a newly developed solver viv-FOAM-SJTU. First, grid convergence study was conducted to further validate the computational results. Four grids with different nodes around the cylinder and grid’s height of first layer were applied. The convergence study was performed for a forced oscillating cylinder, and good convergence was achieved. Complex demodulation was chosen to analyze the spatial-temporal phase relationship between the forces and cylinder displacement. The time history of the phase angle was presented to understand the mechanism of vortex mode transitions. Abrupt jump from a negative value to a positive value was observed for the phase angle. Accompanied by the jumping of the phase angle, the vortex wake pattern also changed. By numerical simulations, we identified a new distinct mode of vortex formation. This new vortex model was comprised of two pairs of vortices formed per cycle, where the secondary vortex in each pair was much weaker than the primary vortex. This vortex mode was defined as2PO, which was closely related to the intermittent switch existing between the upper and lower branches. The good details of vortex after cylinder and the vortex propagating downstream were presented. The simulation region calculated in this part was in well agreement with the “OVERLAP” region of Morse and Williamson [27] experiments. Prediction function of a free-vibration cylinder based on vibration equation of one-degree-of-freedom was derived. Controlled-vibration data was used to predict the response of a freely vibrating cylinder at different mass damping m∗ζ. Take the free vibration of Govardhan and Williamson [28] as an example, we had successfully predicted the VIV response by forced vibration data at three typical casesλ∗=5.0,6.0,6.2 located in the initial, upper, and lower branches separately. The method predicted the amplitude response with acceptable accuracy. Future work included building up an entire response plot for all reduced velocity and investigating whether the prediction approach is held for all Reynolds number.



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