1 Introduction

Fluid-structure interaction is one of the most important problems in fluid mechanics, such as flow past cylinders^[1-2] and vortex-induced vibrations^[3]. For flow past a stationary circular cylinder, as the Reynolds number (Re) is increased to a critical value around 47, the wake flow loses stability, and vortices shed in the wake comprise the von Karman vortex street^[1]. In the vortex shedding regime, the Strouhal number (St), representing the dimensionless vortex shedding frequency, has been proven to be dependent on the Reynolds number. Recently, a simple form St=1/(A + B/Re) with A and B as two constant parameters, describing the relationship between the Strouhal and Reynolds numbers, has been suggested by Roushan and Wu^[2].

When a cylinder undergoes the forced or free oscillation, the cylinder motion will have important effects on the wake vortex mode and the hydrodynamic force on the cylinder surface^[3-15]. After the pioneering work of Bishop and Hassan^[4], flow past a cylinder oscillating in both cross-flow^[4-12] and streamwise^[13-14] directions, have been studied comprehensively. A significant phenomenon found in flow past an oscillating cylinder is lock-in/synchronization. When the vortex formation frequency is close to the vibration frequency, the vortex shedding frequency will be locked into the cylinder oscillation frequency. Olinger and Sreenivasan^[5] demonstrated that the region of synchronization is mainly dependent on the vibration frequency and amplitude of cylinder, and the Arnold's tongue was observed in such a dynamical system. Williamson and Roshko^[6] experimentally investigated flow over an oscillating cylinder with the Reynolds number ranging from 300 to 1 000. Based on the oscillation amplitude and frequency, three vortex shedding regimes named 2S (a single vortex is formed in each half oscillation cycle, as shown in Figs. 1(a), 1(b), and 1(f)), P+S (a vortex pair and a single vortex are shed in a vibration cycle, as shown in Figs. 1(c), 1(d), and 1(g)) and 2P (two vortex pairs are shed per cycle, as shown in Fig. 1(e)) modes are observed. A map of vortex synchronization regions in the wavelength-amplitude plane has also been given.

To our knowledge, there is little literature dealing with such a problem in the subcritical region (Re < 47). Buffoni^[11] found that vortex shedding could be triggered under subcritical conditions by transverse small-amplitude oscillations of a circular cylinder. Vortex shedding has not been observed at low frequencies, and as the frequency is increased, the intensity of vortex shedding increases to its maximum level and then abates gradually till it disappears. A modified Landau's model with least-squares curve fitting is proposed by Chen et al.^[12] to describe the resonant phenomenon of flow around a transversely vibrating cylinder in a subcritical regime. Compared with the numerical results, it is concluded that the resonant frequencies can be determined accurately with model fitting, and the four-parameter model is suitable for flow in the subcritical region with a small oscillation amplitude. Mittal and Singh^[15] numerically researched two degrees of freedom VIV of an elastically mounted cylinder at subcritical Reynolds numbers. Results indicated that the vortex shedding and self-excited vibration of the cylinder are possible even though Re is as low as 20. In the subcritical region, lock-in exists extensively, unlike the phenomenon observed in the supercritical region (Re > 47), and no hysteresis occurs in the subcritical region.

The previous works for flow past an oscillating cylinder in subcritical conditions mainly concentrate on the region of low vibration amplitude, and there is little literature about the case of a high amplitude. In this study, we numerically investigate flow past a large-amplitude transversely oscillating cylinder over a wide range of vibration frequency at subcritical Reynolds numbers with the lattice Boltzmann method (LBM). The behaviors of the wake flow and hydrodynamic force on the oscillating cylinder at different Reynolds numbers, oscillation amplitudes, and frequencies are investigated, and special attention is paid to the vortex shedding phenomenon.

2 Problem statement and governing equations

Figure 2 shows the schematic diagram of flow configuration with a transversely oscillating cylinder immersed into the viscous incompressible fluid flow. The fluid is of the density ρ and the kinematic viscosity ν. The cylinder is fixed in the streamwise direction and undergoes the forced harmonic oscillation in the cross-flow direction,

(1)

where A and f_v denote the amplitude and frequency of cylinder oscillation, respectively. The fluid motion is governed by the Navier-Stokes equations. Let the cylinder diameter D and the uniform incoming velocity U be the characteristic length and velocity, respectively. The dimensionless continuity and momentum equations are written as follows:

(2)

where u is the velocity vector, P is the fluid pressure, and Re=UD/ν is the Reynolds number. In Fig. 2, O₁ (1D, 1D) and O₂ (16D, 1D) are two detection points, where both the streamwise and transverse velocities of fluid flow are recorded.

The drag and lift coefficients can be obtained by

(3)

where F_x and F_y represent the horizontal and vertical hydrodynamic forces on the cylinder surface, respectively.

The vibration Strouhal number, describing the dimensionless oscillation frequency, is defined as

(4)

It has been proven that, when the Reynolds number is less than about 190, the flow structure for the flow past a stationary cylinder is two-dimensional^[1]. In the present study, the Reynolds number is always less than 40. Therefore, a two-dimensional flow is considered here with a safe range.

3 Numerical method and validation of algorithm

The LBM is developed from the cellular automaton (CA) and has been proven to be a promising numerical tool in computational fluid dynamics^[16].

The discrete Boltzmann equation based on the Bhatnagar-Gross-Krook (BGK) collision term can be written as^[17-19]

(5)

where f_i(x, t) and f_i^eq(x, t) are the density distribution and local equilibrium distribution functions in the ith direction of the phase space, i=0, 1, …, N, δt is the unit time step, and τ is the relaxation time.

A two-dimensional nine-speed lattice model (D2Q9) is chosen in this study^[19]. In such a model, as shown in Fig. 3, the velocity vector in the phase space is discretized into nine directions,

(6)

where is the lattice speed.

The equilibrium distribution functions can be written as

(7)

where the weight coefficients are

(8)

The mass density of macroscopic fluid ρ and the momentum density of macroscopic fluid ρu can be obtained by

(9)

while the fluid pressure and the kinetic viscosity are

(10)

where is the lattice sound speed.

Through the multi-scaling expansion, the Navier-Stokes equations can be recovered with the lattice Boltzmann equation^[20].

A no-slip boundary condition is applied to the velocity of cylinder surface. A Dirichlet boundary condition, i.e., u=U and v=0, is used for the inlet, upper, and lower boundaries of the computational domain, whereas a Neumann boundary condition, i.e., , is chosen at the outlet boundary.

In the present study, an extrapolation method, which has been proven to be of second-order accuracy, and has well behaved stability characteristics^[21], is adopted for the moving curved boundary of the oscillating cylinder and the boundaries of the computational domain.

As shown in Fig. 4, a physical boundary node b is intersected between the fluid node f and the solid node w. The fraction of intersected link in the fluid region is defined as

(11)

After collision, the distribution function f_i(x_f, t) at the node f is known. However, we need to know f_i(x_w, t) moving from w to f in the streaming step. The basic idea of the extrapolation method is to decompose the distribution at the solid node f_i(x_w, t) into the equilibrium and non-equilibrium parts, which can be expressed as

(12)

In Eq. (13), the equilibrium distribution f_i^eq(x_w, t) can be approximated by a fictitious one,

(13)

with the velocity at the solid node,

(14)

where q_c ∈ [0, 1] is the judgment condition, and q_c=0.75.

The non-equilibrium part f_i^neq(x_w, t) can be approximated by that of the neighboring fluid nodes f and ff along the link,

(15)

In our previous work, the algorithm has been validated by applying it to simulate flows past two tandem stationary cylinders^[22] and vortex-induced vibrations of an elastically mounted cylinder^[23]. The convergence of the algorithm has also been studied^[22-23], and the scheme with the inlet dimensionless free-stream velocity U/c=0.05 and the lattice density D/δx=48 is used to ensure enough accuracy. The length and width of the computational domain are 48D and 32D, respectively. At the initial time, the cylinder center is placed at 16D downstream of the inlet boundary to keep the cylinder sufficiently away from the boundaries.

Here, the code is further validated by applying it to flow past a forced transversely oscillating cylinder. As we know, when flows pass through a forced transversely oscillating cylinder, the 2S, P+S, and 2P vortex wake modes depending on the Reynolds number, the oscillation frequency and amplitude, have been observed^{[3, 6-7]}, as shown in Fig. 1. We first simulate flows past a stationary cylinder at Re=200, and results show that the Strouhal number is around 0.197. Then, flow past an oscillating cylinder with f_v/f=0.8 and A/D=0.6 and 0.65, to keep all the physical parameters the same with those of Ref.[7], are computed. The 2S vortex shedding mode for A/D=0.6 in Fig. 1(f), and the P+S mode for A/D=0.65 in Fig. 1(g), are obtained. The present results agree well with the previous experimental and numerical results^{[3, 7]}. The above discussion demonstrates that the current algorithm is able to simulate flow past an oscillating cylinder, and the present numerical results are reliable.

4 Results and discussion

This work is comprised of two parts. In the first part, the Reynolds number is fixed at Re=20, and simulations are performed at A/D=0.125, A/D=0.25, and A/D=0.5, respectively. In the second part, the oscillation amplitude is fixed at A/D=0.25, and simulations are performed at Re=5, Re=10, Re=20, and Re=40, respectively. The vibration frequency St_v ranges from 0.025 to 0.8. In the following sections, we will discuss the two series of numerical results separately and give an explanation for the mechanisms of the observed phenomena.

4.1 Effects of vibration amplitude and frequency on flow past oscillating cylinder

For flow past an oscillating cylinder, the vibration amplitude and frequency are two important parameters. With a fixed Reynolds number of Re=20, we first investigate the coupled effects of the oscillation amplitude A/D=[0.125, 0.5] and the frequency St_v=[0.025, 0.8] on the wake structure, the flow velocity, and the hydrodynamic force on the cylinder surface.

Figure 5 shows the phase diagram of vortex wake modes in the plane. Results show that, for each oscillation amplitude, the resonance phenomenon is observed, and vortex shedding is excited in a range of frequency. For each oscillation amplitude, when St_v is 0.025, the wake flow is nearly steady. As St_v is increased, the wake flow fluctuates stronger and stronger, and when St_v is increased to a number around 0.075, vortices are shed in the wake of the cylinder to comprise a von Karman vortex street. The strongest vortex shedding regime appears at St_v ≈0.125 for each vibration amplitude. As the frequency is increased continually, the vortex shedding weakens gradually till it disappears finally. The vibration Strouhal numbers where vortex shedding disappears, are within [0.25, 0.3], [0.3, 0.4], and [0.4, 0.5] for A/D=0.125, A/D=0.25, and A/D=0.5, respectively. It can be found that the response region of vibration frequency, becomes wider with increasing oscillation amplitude.

Figure 6 shows the computed mean drag coefficient (C_D) and the mean amplitude of lift coefficient (C_L) versus the oscillation frequency (St_v) at different vibration amplitudes. Results show that, at each oscillation frequency, both the drag and lift forces of a larger-amplitude oscillating cylinder are larger than those of a smaller-amplitude one. For each oscillation amplitude, as St_v increases, C_D first increases slowly and then increases rapidly when the vortex shedding is excited, as shown in Fig. 6(a). As St_v is increased beyond the strongest vortex shedding regime, there is a decrease of drag force due to the disappearance of vortex shedding, and then the drag force increases slowly again with increasing St_v. A peak is formed on the St_v-C_D curve around the strongest vortex shedding regime. As shown in Fig. 6(b), for each vibration amplitude, as St_v is increased, C_L increases more and more rapidly due to the cylinder oscillation.

To have a better understanding of the observed resonance phenomena, a detailed analysis for the flow behavior is given. Based on the above discussion, we know that for each oscillation amplitude, the flow behavior is similar. For example, Figs. 7-11 show the time series of the lift coefficient, the transverse displacement, streamwise and transverse velocities at the detection points and the instantaneous vorticity contours for flow past an oscillating cylinder at a variety of oscillation frequencies with Re=20 and A/D=0.25.

As shown in Fig. 7, the frequency of lift force fluctuation is always equal to the cylinder oscillation frequency. For St_v=0.025, there is a significant phase difference between the transverse displacement and the lift coefficient, and as St_v is increased, the phase difference decreases gradually. When St_v is increased to a threshold around 0.125, the transverse displacement and lift coefficient always behave as an in-phase state.

As shown in Fig. 8, the streamwise and transverse flow velocities at the detection point O₁ always fluctuate at the same frequency, which is equal to the cylinder vibration frequency, and the streamwise flow velocity fluctuates stronger than the transverse velocity. The streamwise flow velocity always behaves as a single cycle in one oscillation period. For St_v=0.025 and 0.05, the transverse velocity also behaves as a single cycle in an oscillation period (see Fig. 8(a)). When St_v is larger than 0.1, two subcycles with one stronger and the other one weaker are formed in a period, as shown in Figs. 8(b)-8(d). As St_v is increased from 0.025 to 0.1, the fluctuations of both streamwise and transverse velocities strengthen slowly and then weaken gradually with St_v=[0.1, 0.5] and St_v=[0.1, 0.25], respectively. When St_v is increased continually, the fluctuations of flow velocities become strong again. For A/D=0.125 and A/D=0.5, similar behaviors of flow velocities have been observed, and all the transition points are almost the same as those of A/D=0.25.

As a comparison, Fig. 9 shows the time series of transverse cylinder displacement, streamwise and transverse velocities at the detection point O₂ with A/D=0.25, St_v=0.125, and St_v=0.4, representing the strongest vortex shedding regime and the steady regime, respectively. As shown in Fig. 9(a), in the far wake of the cylinder at St_v=0.125, both the streamwise and transverse flow velocities fluctuate periodically at a coupled frequency which is equal to the cylinder oscillation frequency. The fluctuation amplitudes for both the streamwise and transverse velocities at O₂ are smaller than those at O₁, and in an oscillation period, only one single cycle is formed for both the streamwise and transverse velocities, which is different from those observed at O₁. For St_v=0.4 in the far wake (see Fig. 9(b)), both the streamwise and transverse velocities are almost steady, while at O₁, the large-amplitude fluctuation of flow velocity is observed (see Fig. 8(d)). The mechanism for the distinct behaviors of flow velocities at detection points O₁ and O₂ is the interaction of the cylinder oscillation and the triggered fluid flow. For St_v=0.125, both the cylinder oscillation and the excited vortex shedding have significant effects on the near flow, which makes the large-amplitude fluctuations of velocities. While in the far wake, the effect of cylinder motion on the flow is weak, and the fluctuation of velocity is mainly dependent on the shedding vortices. For St_v=0.4, significant fluctuation of the flow velocity still can be induced in the near wake due to the fast vibration of the cylinder. While in the far wake, the effect of cylinder motion on the fluid flow is very weak, and in such a case, no vortex shedding is excited. Therefore, both the streamwise and transverse flow velocities are almost steady.

Figure 10 shows the instantaneous flow structures at the dimensionless time of tU/D=800. When St_v is low (see Fig. 10(a)), two nearly stationary reversed vortices are formed behind the cylinder, just like the wake of a stationary cylinder. The wake flow has a slight fluctuation due to the cylinder motion. As St_v is increased to 0.075, two alternately rotating vortices are shed in the wake per cycle to comprise a von Karman vortex street, and the strongest vortex shedding occurs at St_v=0.125, as shown in Fig. 10(b). For St_v=0.2 (see Fig. 10(c)), the vortex shedding disappears, however, in the near wake, the flow still fluctuates significantly. When St_v is larger than 0.4, the vortex shedding is suppressed absolutely, and the wake flow behaves as two nearly stationary reversed vortices again, as shown in Fig. 10(d). In the near wake, there is an obvious fluctuation of flow due to the influence of cylinder oscillation, while in the far wake, the fluid flow is almost steady. As St_v is less than 0.175 in Figs. 10(a) and 10(b), three separation points with one in the upstream side and the other two in the downstream side of the cylinder are observed, just like the case of a stationary cylinder. As St_v is between 0.2 and 0.8, as shown in Figs. 10(c) and 10(d), only two separation points with one on the upstream side and the other one on the downstream side of the cylinder are formed. Videos (not shown here) of the whole cylinder motion and fluid flow process show that the two separation points are almost always at two poles of the cylinder, and the link between two separation points rotates with the cylinder oscillation. The maximum rotation angle of the link between two separation points increases with increasing oscillation frequency. For example, Fig. 11 shows the flow patterns of St_v=0.8 in a period at t=T/4, T/2, 3T/4, T, respectively, where T is the oscillation period. It can be seen from Fig. 11 that, as the cylinder is vibrating, there are always two separation points at two poles of the cylinder surface. Both separation points move anticlockwise in the first half period and clockwise in the second half period. When the cylinder is at its maximum displacement, as shown in Figs. 11(a) and 11(c), the flow around the cylinder is complex. On the cylinder surface, there are two thin flow layers with one of positive vorticity and the other one of negative vorticity, and both flow layers are surrounded by thick flow layers with the opposite vorticity.

4.2 Effects of Reynolds number and vibration frequency on flow past oscillating cylinder

Based on the above discussion, we know that, for each current oscillation amplitude, vortex shedding can be excited under subcritical conditions in a range of frequency, and the dynamic behavior of the oscillating cylinder flow for each vibration amplitude is similar to each other. In this section, we will continue to investigate the effects of the Reynolds number and the oscillation frequency on the flow at a fixed oscillation amplitude of A/D=0.25.

Figure 12 shows the phase diagram of vortex wake modes in the St_v-Re plane. Results show that for Re=5, as St_v is changed in the whole range, no vortex shedding state has been observed. As St_v is 0.025, the wake flow behaves as a steady state. As St_v is changed between 0.05 and 0.15, the wake flow behaves as a fluctuation state. When St_v is larger than 0.175, the wake flow translates to a steady state again. For Re=10, Re=20, and Re=40, as St_v is increased, the wake flow first fluctuates strong gradually. When St_v is increased to a threshold, vortices are shed in the wake of the cylinder, and the thresholds of Re=10, Re=20, and Re=40 are St_v=0.1, St_v=0.075, and St_v=0.075, respectively. The strongest vortex shedding states for Re=10, Re=20, and Re=40 appear at St_v=0.1, St_v=0.125, and St_v=0.15, respectively. As St_v is increased continually, the wake flow fluctuates more and more weakly. For Re=10 and Re=20, as St_v is more than a threshold of St_v=0.25 and St_v=0.4, respectively, the vortex shedding is restrained absolutely, and the wake flow always behaves as a steady state. For Re=40, after the strongest vortex shedding regime, the fluctuation of the wake flow first becomes weak gradually till to a steady state, similar to that of Re=10 and Re=20. However, as St_v is continually increased to a value around 0.6, the wake flow begins to lose stability again, and as St_v is larger than 0.7, vortex shedding can be reproduced.

Figure 13 shows the computed mean drag coefficient (C_D) and the mean amplitude of lift coefficient (C_L) versus the oscillation frequency (St_v) at different Reynolds numbers. Results show that, at each oscillation frequency, both drag and lift forces of the cylinder at a larger Re are smaller than those at a lower Re. For Re=5 and Re=10, C_D increases monotonously with increasing St_v. For both Re=20 and Re=40, C_D increases with increasing St_v for most cases, and for each Re, a peak is formed near the strongest response regime in the St_v-C_D curve due to the excitation of vortex shedding (see Fig. 13(a)). For each Re, C_L increases more and more rapidly as St_v is increased in the whole current range, as shown in Fig. 13(b).

As discussed above, for Re=40, vortex shedding is excited twice at St_v=[0.075, 0.2] and St_v=[0.7, 0.8], respectively. The resonance phenomenon for Re=40 on the branch of low frequency is similar to that of Re=20, where two alternately rotating vortices are shed in the wake per oscillation cycle. While on the branch of high frequency, the behavior of vortex shedding is distinct from that on the branch of low frequency, and only a pair of alternately rotating vortices are shed for several oscillation cycles. We now give a detailed investigation on the behaviors of flow at high vibration frequencies. Figures 14-17 show the time series of lift coefficient, transverse displacement, streamwise and transverse velocities at the detection points and the instantaneous vorticity contours at a variety of oscillation frequencies with Re=40 and A/D=0.25. As shown in Fig. 14, for St_v=0.15, St_v=0.4, and St_v=0.6, the lift coefficient fluctuates at a single frequency, while for St_v=0.8, besides the primary frequency, a secondary frequency is formed in the time series of lift coefficient due to the appearance of the second vortex shedding regime. For each frequency, the primary frequency of the fluctuation of lift coefficient is always equal to that of the cylinder oscillation, and the transverse displacement and the lift coefficient always behave as an in-phase state.

As shown in Fig. 15, for each oscillation frequency, there is only one single frequency in the fluctuation of streamwise velocity at the detection point O₁, and two subcycles with one stronger and the other one weaker are observed in a period for the fluctuation of transverse velocity. The fluctuation of streamwise velocity is always stronger than that of transverse velocity, and the primary frequencies of the streamwise and transverse flow velocities are always the same. For St_v=0.15, St_v=0.4, and St_v=0.6, both the streamwise and transverse flow velocities fluctuate periodically. For St_v=0.8, the flow velocities fluctuate quasi-periodically, and a secondary frequency is formed in the evolution of both streamwise and transverse velocities.

Figure 16 shows the time series and frequency spectrums of transverse cylinder displacement, streamwise and transverse velocities at the detection point O₂ with St_v=0.15 and St_v=0.8, representing the two distinct vortex shedding regimes. As shown in Fig. 16(a), in the far wake of the cylinder at St_v=0.15, both the streamwise and transverse flow velocities fluctuate periodically at the same frequency which is equal to the cylinder oscillation frequency, similar to that observed at Re=20 and St_v=0.125 (see Fig. 9(a)). Only one single peak is formed in the frequency spectrums of streamwise and transverse velocities, and the primary frequency is equal to that of transverse cylinder displacement. For St_v=0.8 in the far wake, as shown in Fig. 16(b), both the streamwise and transverse flow velocities fluctuate quasi-periodically, and the primary frequency is much lower than the cylinder oscillation frequency. Besides the primary frequency, a secondary frequency equal to the cylinder vibration frequency is also observed. In a whole period, several tiny fluctuations are formed in the evolution of both the streamwise and transverse velocities. The main fluctuation of flow velocity is excited by the shedding vortices, while the tiny fluctuation is induced by the cylinder oscillation. In the near wake, the effect of the fast cylinder oscillation on the flow plays a leading role. Therefore, the flow velocity fluctuates strong with a primary frequency equal to the cylinder vibration frequency. Moreover, the low-frequency vortex shedding also has an effect on the fluid flow, which makes the formation of the secondary frequency observed in the time series of hydrodynamic force and flow velocity (see Figs. 14(d) and 15(d)). A further data analysis demonstrates that the vortex shedding frequencies for St_v=0.7 and St_v=0.8 are both around 0.109. The mechanism for the two distinct types of vortex shedding regimes will be further discussed in the next section.

Figure 17 shows the instantaneous flow structures at tU/D=800. For St_v=0.15 representing the strongest vortex shedding regime, two alternately rotating vortices shed per oscillation cycle comprise a von Karman vortex street, as shown in Fig. 17(a). As St_v is increased from 0.15, vortex shedding weakens gradually till it disappears. For St_v=0.4 representing the steady regime (see Fig. 17(b)), two nearly stationary reversed vortices are formed in the wake flow with no vortex shedding. For St_v=0.6 (see Fig. 17(c)), the far wake becomes unsteady again, and for St_v=0.8 (see Fig. 17(d)), vortex shedding is reproduced in the wake of cylinder.

4.3 Mechanism for excitation of vortex shedding

The above discussion demonstrates that two distinct types of vortex shedding are excited in the subcritical region. The first type of vortex shedding regime (VSR Ⅰ) occurs on the branch of the low vibration frequency with the Reynolds number larger than 10, while the other type of vortex shedding regime (VSR Ⅱ) is excited on the branch of high frequency with the Reynolds number close to the critical value. In this section, the mechanisms for excitations of the two types of vortex shedding will be discussed.

As we have known, for flows past a transversely vibrating cylinder in the supercritical Reynolds number region, when the natural vortex shedding frequency is close to the oscillation frequency, the phenomenon of synchronization has been observed, and in the lock-in region, the fluid flow will acquire energy from the cylinder oscillation, which will strengthen the vortex shedding in the wake of the cylinder. The natural vortex shedding frequency has been proven to be a function of Reynolds number, and a simple form, St=1/(A+B/Re), was suggested by Roushan and Wu^[2] to describe the relationship between the Strouhal and Reynolds numbers.

In this work, the range of oscillation frequency in the VSR Ⅰ is always between 0.075 and 0.2, and becomes wider with increasing vibration amplitude and Reynolds number. Using the relationship St=1/(A+B/Re) with the constant parameters A and B of the values suggested by Roushan and Wu (see Table I in Ref.[2]), the intrinsic Strouhal numbers for Re=10, 20, and 40 are St ≈ [0.03, 0.05], St ≈ [0.05, 0.07], and St ≈ [0.08, 0.11], respectively. The intrinsic vortex shedding frequencies obtained from Roushan and Wu's formula are found to be close to the present oscillation frequencies in the VSR Ⅰ. Therefore, in such a region, the fluid flow around the cylinder can acquire energy from the cylinder oscillation, and vortex shedding is induced.

In the VSR Ⅱ, the efficient transverse length of the cylinder is increased because of the high-frequency oscillation (we can assume that if the oscillation frequency was infinite, the efficient transverse length of the cylinder would be D + A), which corresponds to an increase of Reynolds number. In the VSR Ⅱ, the Reynolds number (Re=40) is close to the critical value (Re_c=47), however, the efficient Re of the system is increased due to the fast cylinder vibration. Therefore, the translation point will move forward, and vortex shedding with its frequency close to that at Re_c, is excited. Results show that the vortex shedding frequency in the RSV II is around 0.109, which is close to the Strouhal number of 0.12 for flows past a stationary circular cylinder at the critical Re_c (see Refs.[1]-[2]).

5 Conclusions

In summary, the LBM is used to simulate laminar flows past an oscillating cylinder in a subcritical region. The effects of Reynolds number, vibration amplitude and frequency on the resonance phenomenon, the hydrodynamic force on the cylinder surface, flow velocity, and wake structures are systemically investigated. Four Reynolds numbers, Re=5, 10, 20, and 40, and three vibration amplitudes, A/D=0.125, 0.25, and 0.5 with the oscillation frequency St_v ranging from 0.025 to 0.8 are considered. The main conclusions are summarized as follows:

(ⅰ) Vortex shedding can be excited extensively in the wake for flows past an oscillating cylinder in the subcritical region. For Re=5, as St_v is changed in the whole current range, no vortex shedding is observed, and as Re is larger than 10, the vortex shedding occurs, and the response range of vibration frequency broadens with increasing Reynolds number and vibration amplitude.

(ⅱ) Two distinct types of vortex shedding excitation are observed. The VSR Ⅰ, in which the vortex shedding frequency is locked into the oscillation frequency, is excited in the branch of low vibration frequency that is close to its intrinsic frequency, and the VSR Ⅱ, in which a pair of alternately rotating vortices are shed for several oscillation cycles, and the vortex shedding frequency is close to that at the critical conditions, occurs on the branch of high oscillation frequency. The mechanism for the excitation of VSR Ⅰ is that, when the oscillation frequency is close to the intrinsic frequency of the system, the cylinder oscillation will enhance the flow fluctuation of the fluid around the cylinder, and vortex shedding is excited. For the VSR Ⅱ, the fast oscillation of the cylinder makes the efficient transverse length large, which corresponds to a larger efficient Reynolds number, and the critical Reynolds number moves forward.

(ⅲ) For most cases, as the vibration frequency is increased, the drag and lift coefficients of the cylinder increase monotonously, and when the VSR Ⅰ occurs, there is a sudden change of the drag coefficient to form a tiny peak in the curve. Both the drag and lift coefficients increase with increasing oscillation amplitude and decreasing Reynolds number.