Numerical Investigation of Vortex-Induced Vibrations of a Rotating Cylinder near a Plane Wall

Abstract

Numerical simulations are carried out to investigate the vortex-induced vibrations of a two-degree-of-freedom (2DOF) near-wall rotating cylinder. Considering the effects of gap ratio, reduced velocity and rotational rate, the amplitude response, wake variations and fluid forces are analyzed, with the Reynolds number of 200 and the mass ratio set to 1.6. The correlative mechanism in the wake–hydrodynamics–vibration is revealed. The results show that the influence of the wall dominates the vortex-induced vibration of the cylinder. The effect of the wall on the vibration weakens as the gap ratio increases, and the effect of the wall on the vibration is negligible when H/D > 1.1. The forward rotation of the cylinder enhances the wall effect, while the backward rotation presents the reverse effect. The vortex-induced vibration of the cylinder is suppressed when 0 < α < 1, and the amplitude range is concentrated at V_r ∈ (3, 5). The wake mode can be divided into five modes, and the wake modes are clarified under different rotation rates and reduced velocities.

1. Introduction

Vortex-induced vibrations (VIVs) of a cylinder frequently occur in ocean engineering fields, such as ocean pipelines, risers, mooring lines and bridge piers. VIVs are accompanied by variation in the flow of the cylinder, and the transformation of the wake mode leads to changes in the fluid forces and further causes the lock-in phenomenon [1]. VIVs trigger large amplitude vibrations and cause fatigue damage. Therefore, studying the impact of VIVs is meaningful due to its critical implications. To understand the structure–fluid interaction and vibration characteristics, scholars have performed much research under different conditions in the past few decades [2,3,4,5].

In practice, due to the influence of terrain, the pipeline may be near the wall. The wall affects the vortex shedding and the vibration of the cylindrical structure. The vibration in the gap ratio has a significant impact on the wake for a stationary cylinder [6]. When a cylinder is near a wall, the shedding of the vortices is suppressed by the wall, resulting in a delay in the shedding cycle of the cylinder [7]. For a small gap ratio (e.g., H/D = 0.1), no shedding vortices were observed around the cylinder. However, vortices were generated around the cylinder at an intermediate gap ratio (e.g., H/D = 0.25) [8]. An offshore drilling platform is a common vibrating cylindrical structure. The wall proximity enlarges the mean lift force, and the drag coefficient increases with the ratio of the gap to the boundary layer thickness [9,10]. The size of the lock-in zone increases and the peak vibration amplitude of the cylinder decreases with the decrement of the gap ratio [11]. Li et al. [12] found that with the increase in the gap ratio, the lock-in range shifted toward the direction of the higher reduced velocity. The vibration effects on the cylinder can vary significantly with different gap ratios. At H/D < 0.3, vortex shedding was found to be restrained [13]. For H/D ∈ (0.5, 7.5), the effect of wall proximity on the frequency response tended to disappear [14].

A drilling pipe can be considered for the structure of a spring-mounted rotating cylinder. The rotation of the cylinder influences the flow characteristics, which will dramatically affect the VIV behaviors. Chang and Chern [15] found that the shedding vortex is greatly affected by the Reynolds number and cylinder speed. The vibration amplitude reaches 1.9 times the diameter under rotation, which is three times the maximum amplitude under non-rotation [16]. The wake behind a cylinder varies significantly at different rotation rates. The rotation rate of α = 2 is an important critical value, and the wake of a cylinder is similar to a Karman vortex street for α < 2 [17]. For α ≥ 2, the generation of the vortex stops and forms a closed flow line around the cylinder [18]. VIVs mainly occur in the cross-flow direction, which is generated at α < 2 [19]. VIVs are completely suppressed at α > 1.3 at Re = 300 [20]. With the increase in the rotation rate, the cylindrical trajectory varies from a narrow ellipse to a circle and finally to a flat ellipse [21].

Drill pipes are under near-wall rotation conditions when they conduct underwater operations in ocean trenches. Previous research has focused on how gap ratios or wall surfaces affect cylindrical vibrations. This paper is a useful reference for rotating drill columns in trench operations. What variations will occur in the amplitude response of the pipe? What are the variations of the wake mode and at what conditions would the vibration of a cylinder be suppressed? To answer these questions, a spring-mounted rotating cylinder under near-wall conditions was constructed and investigated. The research objectives of this paper are as follows: (1) analyze the VIVs of a cylinder rotating near the wall and explore the vibration of the cylinder under the combined action of the rotation and near wall; (2) establish the relationship among the wake flow, fluid dynamics, and structural vibration, and clarify the interaction mechanism; and (3) explore the effect of the near-wall condition on the suppression of vortex-induced vibration under rotation.

The content of this paper is as follows. In Section 2, the physical model is described. In Section 3, numerical methods, boundary conditions, and the verification process are provided. In Section 4, the vibration of the cylinder is analyzed with different rotation ratios at H/D = 0.8. In Section 5, the impact of the rotational direction of a cylinder on the suppression of vortex-induced vibration, as well as the transition of the wake and the wake–hydrodynamics–vibration interaction are discussed. Finally, in Section 6, the main findings and conclusions of this paper are summarized.

2. Physical Model

The spring-mounted rotating cylinder model is shown in Figure 1, where x means in-line direction and y means cross-flow direction. The diameter of the rotating cylinder is D = 0.01 m, and the fluid density in the flow field equals ρ = 998.2 kg/m³. The distance between the cylinder lower surface and the bottom wall is determined by the initial gap distance H. The Reynolds number defined as Re = UD/υ is set to 200 in this paper. The rotating cylindrical elastic system is equipped with springs (k₁ (cross-flow), k₂ (downflow)) and dampers (C₁ (cross-flow), C₂ (downflow)) in the cross-flow and downflow directions. This creates a spring-damped system that enables two degrees of freedom (2DOF) of vibration. The length of the computational domain in this paper is 65D, the width is 40D, and the distances from the center of the cylinder to the inlet and outlet boundaries are 15D and 50D, respectively. The blockage rate in this paper is set to 0.025, and the effect of the width on the structural response is negligible [22].

3. Numerical Method

3.1. Governing Equations and Kinematic Equation

In this study, the two-dimensional incompressible non-constant Navier–Stokes equation was used to describe the incompressible fluid, as follows:

$$\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}=0$$

(1)

$$\frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}=-\frac{1}{\rho }\frac{\partial p}{\partial x}+\upsilon \left(\frac{{\partial }^{2}u}{\partial x^2}+\frac{{\partial }^{2}u}{\partial y^2}\right)$$

(2)

$$\frac{\partial v}{\partial t}+u\frac{\partial v}{\partial x}+v\frac{\partial v}{\partial y}=-\frac{1}{\rho }\frac{\partial p}{\partial y}+\upsilon \left(\frac{{\partial }^{2}v}{\partial x^2}+\frac{{\partial }^{2}v}{\partial y^2}\right)$$

(3)

where u and v are the velocities in the in-line and cross-flow directions, respectively, ρ is the fluid density, p is the pressure, and υ is the coefficient of kinematic viscosity of the fluid.

The motion in the in-line direction and cross-flow direction of the oscillator is governed by the following equations [23]:

$$\left(m+m_{ad}\right)\ddot{x}+c\dot{x}+kx=F_x$$

(4)

where m_ad is added mass m_{ad =} πD²ρ_fluid/4, and x and F_x are physical coordinates and force, respectively. In dimensionless form, using designations $x_c=x/D$, ${\dot{x}}_c=\dot{x}/U$, ${\ddot{x}}_c=\ddot{x}D/U^2$, $C_d=2F_x/{\rho }_{fluid}{U}^{2}D$, we obtain Equations (5) and (6).

$$\left(1+\frac{1}{m^{\ast }}\right){\ddot{x}}_c+\frac{4\pi \zeta }{V_r}{\dot{x}}_c+{\left(\frac{2\pi }{V_r}\right)}^2{x}_c=\frac{2C_d}{\pi m^{\ast }}$$

(5)

$$\left(1+\frac{1}{m^{\ast }}\right){\ddot{y}}_c+\frac{4\pi \zeta }{V_r}{\dot{y}}_c+{\left(\frac{2\pi }{V_r}\right)}^2{y}_c=\frac{2C_l}{\pi m^{\ast }}.$$

(6)

where m* is the mass ratio of the spring-mounted cylinder, ζ ($\zeta =c/2\sqrt{km}$) is the damping ratio, c represents the damping of the system, and C_l and C_d are the lift coefficient and drag coefficient, respectively.

$$y(n+1)=y(n)+\frac{\mathrm{\Delta }t}{6}(K_1+K_2+K_3+K_4)$$

(7)

$$\dot{y}(n+1)=\dot{y}(n)+\frac{\mathrm{\Delta }t}{6}(L_1+L_2+L_3+L_4)$$

(8)

where K₁–K₄ and L₁–L₄ represent the fourth-order Runge–Kutta functions, Δt is the time step, and subscript n is the number of time steps.

Numerical simulations were carried out with ANSYS/Fluent software. To solve the equations of motion for the cylinder, a user-defined program based on the C programming language was introduced via user-defined functions (UDFs). The flow chart is shown in Figure 2. The entire solution process is as follows: The software solves the flow field and inputs the solved cylindrical surface load into the UDF, which calculates the lift and drag forces of the structure. Then, the cylindrical equations of motion based on the fourth-order Runge–Kutta functions are solved to obtain the motion parameters of the cylindrical structure. The UDF feeds the solution parameters back into the software, which completes the motion of the cylindrical structure and updates the mesh in time, finally outputting the cylindrical displacements, fluid forces, and flow field parameters.

The inlet boundary conditions are u = U and v = 0, and the outlet boundary conditions are ∂u/∂x = 0 and ∂v/∂x = 0. The lower and upper boundaries are slip wall surfaces (u = U, v = 0), and the cylindrical surface is a no-slip wall surface. u and v denote the velocity components of fluid flow in the x and y directions, respectively. The constrained velocity is defined as V_r = U/(f_n × D), where f_n represents the intrinsic frequency of the cylinder, and $f_n=\left(1/2\pi \right)\times \sqrt{k/m}$, where k represents the structural stiffness of the spring, and m represents the mass of the cylinder. The cylinder is forced to rotate counterclockwise at a rotational velocity ratio α (α = DΩ/2U, Ω is the angular velocity of the cylinder rotation). f* (f/f_n) is the frequency ratio in this paper, used to describe the frequency response of the rotating cylinder, and f represents the vibration frequency of the cylindrical structure.

3.2. Mesh and Validation

Figure 3 shows the mesh division of the whole computational domain, which is divided into three parts: A is the rotating cylindrical vibration region, B is the wake region, and C is the buffer region. To simulate the vibration and vortex shedding of the cylinder more accurately, the mesh density of area A is the highest, the density of area B is lower than that of area A, and the mesh density of area C is the lowest. Table 1 shows the numerical results for different parameters compared with previous studies (∆x is the grid size near the cylinder). To ensure the accuracy of the numerical simulation results, a mesh density of 0.5 mm was chosen.

A comparison of the amplitudes under near-wall conditions and under rotating conditions is shown in Figure 4. Cylindrical vibration in the cross-flow direction was compared with the data from previous studies [24,25]. Under the near-wall and rotating conditions, the cross-flow direction amplitude ranges were V_r ∈ (2, 10) and V_r ∈ (3, 10), respectively. The results match well with the results of Gao et al. [24] and Zhao et al. [25], which indicates an acceptable accuracy of the current numerical method.

Table 1. Parameters and comparison of flow around a cylinder with Re = 200 [26,27,28].

Case	∆x/mm	∆t/s	Cd-Mean	St
A1		0.005	1.34	0.197
A2	0.2	0.01	1.33	0.190
A3		0.02	1.29	0.185
A4		0.005	1.38	0.195
A5	0.5	0.01	1.36	0.193
A6		0.02	1.33	0.189
A7		0.005	1.31	0.196
A8	1	0.01	1.30	0.193
A9		0.02	1.28	0.190
Braza et al. [26]			1.38	0.20
Liang et al. [27]			1.46	0.21
Mendes et al. [28]			1.39	0.20

Table 1. Parameters and comparison of flow around a cylinder with Re = 200 [26,27,28].

Case	∆x/mm	∆t/s	Cd-Mean	St
A1		0.005	1.34	0.197
A2	0.2	0.01	1.33	0.190
A3		0.02	1.29	0.185
A4		0.005	1.38	0.195
A5	0.5	0.01	1.36	0.193
A6		0.02	1.33	0.189
A7		0.005	1.31	0.196
A8	1	0.01	1.30	0.193
A9		0.02	1.28	0.190
Braza et al. [26]			1.38	0.20
Liang et al. [27]			1.46	0.21
Mendes et al. [28]			1.39	0.20

3.3. Effect of the Gap Ratios

To further investigate the vortex-induced vibration of a rotating cylinder under near-wall conditions, the gap ratio effect is discussed, and a range of H/D ∈ (0.2, 1.4) with an increment of 0.3 was selected. Figure 5 shows the vortex distributions at different H/D and V_r. When H/D = 0.2, at V_r ∈ (6, 10), a collision occurs between the cylinder and the wall. The shedding of vortices only occurs at V_r ∈ (11, 12). For H/D = 0.5 and V_r ∈ (5, 9), the wall suppresses the shedding of the positive vortex, and only the shedding of a negative vortex remains. At H/D = 0.8, the wall effect weakens. The presence of wall vortices causes the positive vortex to be smaller than the negative vortex, and the movement speed is lower than that of the negative vortex. At H/D ≥ 1.1, the characteristic of the wake is the Karman vortex street, indicating that the wall effect on the cylinder can be neglected.

For different gap ratios (H/D) and reduced velocities (V_r), the amplitudes of a spring-mounted cylinder near a wall are shown in Figure 6. The cylinder is in contact with the wall at a certain reduced velocity for H/D = 0.2. This gap ratio dramatically affects the amplitude of the cylinder. When H/D > 0.5, the peak is mainly concentrated at V_r = 5, but for H/D = 0.8, the peak appears at V_r = 6, and these peak values are between 0.5 and 0.7 in the cross-flow direction. When H/D = 0.5 and V_r = 8, the peak value reaches 0.7. The mean time-averaged displacement in the cross-flow direction of H/D = 0.5 and 0.8 is substantially different from that of other gap ratios. The wall affects the wake of the cylinder. At H/D = 0.5 and 0.8, the wake modes are significantly different from H/D ≥ 1.1. The transition of the vortex shedding mode is synchronized with the variation in the vibration amplitude. This is why the amplitudes at H/D = 0.5 and 0.8 are different from those of other gap ratios.

Previous studies have shown that the wall has little effect on the cylinder when H/D > 1 [29,30,31,32]. The motion trajectory of the cylinder is shown in Figure 6c. At H/D = 0.2, there is a collision between the cylinder and the wall. The minimum distance between the cylinder and the wall is 0.03 when H/D = 0.5, and if the cylinder is rotating, a collision may occur. For H/D = 0.8, the minimum distance is 0.28, and the wall effect needs to be considered. To further investigate the flow characteristics and vibration response of a spring-mounted rotating cylinder near the wall, the gap ratio of H/D = 0.8 was chosen as follows.

4. Numerical Results

4.1. Vibration Responses

To investigate the vortex-induced vibration of a spring-mounted rotating cylinder under near-wall conditions, numerical simulations were conducted for H/D = 0.8 and α ∈ (−1.5, 1.5). The amplitude response is shown in Figure 7. The amplitude first increases rapidly and then decreases as V_r increases when α < 1. Due to the vibration of the wake and wall vortices, one peak occurs in the cross-flow and in-line directions. The increase and decrease in the amplitude decrease with an increasing rotation rate. The peak maximum values are 0.6 and 0.3, and the minimum values are 0.3 and 0.2, respectively. The peak range decreases from V_r ∈ (3, 12) to V_r ∈ (3, 5). At α = 1, the amplitude increases and then decreases rapidly at V_r = 4, increases rapidly at V_r = 5 and then decreases rapidly again. The two peak values are concentrated at V_r = 4 and V_r = 8. The amplitude values are 0.4 and 0.5 in the cross-flow direction and 0.6 and 0.3 in the in-line direction. The wake modes shift, multiply, and lead to a double peak. When α = 1.25, the cylinder amplitude is too large and collides with the wall at V_r ∈ (7, 8). The same reason applies to V_r ∈ (6, 7) for α = 1.5. However, when α > 1, the amplitude range is V_r ∈ (3, 9) and does not change with t, an increasing rotation ratio.

For reversed rotation of the cylinder, the amplitude of the cylinder first increases rapidly and then decreases rapidly with an increasing reduced velocity in the cross-flow direction. The peak value first decreases and then increases, and the peak value also arrives earlier and is mainly concentrated at V_r = 4 and V_r = 5 with an increasing rotational ratio. The peak range is concentrated around V_r ∈ (4, 8), with the maximum peak equal to 0.7 and the minimum equal to 0.5. In the in-line direction, the amplitude first increases at V_r = 4 and then decreases at V_r = 7 rapidly when the reduced velocity increases. However, amplitude and the peak value increase with an increasing rotation ratio at α ∈ (−1.5, −0.75). However, the peak value reaches 0.4 and no longer decreases for α ∈ (−0.75, −0.25) at α = −0.75. The peak value range is concentrated between V_r ∈ (4, 8). When V_r > 8, the effect of the reduced velocity and rotation ratio on the amplitude can be neglected. It is seen that inversion reduces the influence of the wall, making the wake mode easier to distinguish and the amplitude of the response simpler. The weakening of the wall effects leads to the wake pattern primarily being concentrated in the 2S mode, with the amplitude response showing consistent changes across all rotation ratios.

The time-averaged displacement is shown in Figure 8. The mean value of the amplitude increases as the reduced velocity increases, and the maximum mean value is concentrated at (0.3, 0.6) and (0.3, 1.0) in the cross-flow and in-line directions, respectively. At α < 1, there is almost no peak, and it steadily increases stably with an increasing reduced velocity. However, in the in-line direction, the variation in amplitude is not obvious with an increasing rotation ratio. For α = 1, the peak occurs at V_r = 8 in the cross-flow and in-line directions, and the values are −0.2 and 0.6, respectively. At α > 1, the amplitude increase starts earlier than that at α = 1, and the amplitude values vary distinctly from α < 1 with an increasing rotation ratio.

For positive rotation, the amplitude increases with an increasing reduced velocity. The rate of increase in the average displacement starts to increase when V_r = 6. As the rotation rate increases, the displacement growth rate varies with different rotation ratios, and the growth rate increases as the rotation ratio increases in the cross-flow direction. The displacement value is concentrated between 0.25 and 2.25. In the in-line direction, the amplitude first increases, then decreases at V_r = 8, and increases again as the reduced velocity increases. The displacement variation is slight for different rotation ratios, and the values are mainly concentrated between 0.5 and 0.8. The cylinder inversion weakens the influence of the wall, and there is no significant difference in the average displacement change at different rotation ratios.

The frequency ratios of the forward rotation and reverse rotation of the cylinder are shown in Figure 9. When the cylinder is rotating forward, the frequency ratio becomes larger with an increasing reduced velocity. The frequency ratio changes slowly, close to the natural frequency of the cylinder, and reaches the locking range when V_r ∈ (4, 8). The frequency ratio reaches locking at approximately 0.8 as the rotation rate becomes larger. Conversely, the frequency ratio at locking slowly decreases as the rotation rate increases. There is no obvious locking phenomenon when α < 1. When α > 1, a clear locking phenomenon occurs, and the locking range is concentrated between V_r ∈ (3, 8). For cylinder inversion, the locking range is concentrated on V_r ∈ (5, 8). The value of the rate decreases as the rotation ratio increases, and the frequency ratio is concentrated between 0.7 and 0.9. When locking occurs, the wake pattern changes, the amplitude response exhibits a peak, and the locking range is distributed similarly to the peak range.

4.2. Wake Structures

In near-wall conditions, there are two types of vortex shedding: cylindrical wake shedding (cylindrical vortex) and shedding from wall vortex layers (wall vortex). The distribution of the different vortex shedding modes at different rotation rates and reduced velocities is shown in Figure 10. At α > 0, the vortex shedding can be divided into five modes: S, 2S, U, US, and FS. At α < 0, vortex shedding appears in two modes: 2S and U. (1) The S mode is shown with only one vortex shedding in one vibration period. (2) The 2S mode has one positive vortex shedding and one negative vortex shedding in one vibration period. (3) The U mode has no vortex shedding but has positive and negative wakes [33]. (4) The US mode has a single negative vortex shedding on top of the U mode, as shown in Figure 11a. (5) The FS mode exhibits all the wakes coalescing, and vortex shedding occurs, as shown in Figure 11b.

The vortex mode is US in Figure 11a. During one vibration period, the negative vortex produces vortex shedding based on U. The vortex shedding merges with the wall wake to form a larger vortex. Figure 11b shows that the FS vortex shedding is mainly manifested by the cylindrical wake and completely wraps the rotating cylinder. Vortex shedding occurs after the wake moves for a certain period.

By observing the distinct distribution of the vortex mode, we further understand the impact of the wall on the wake. When the cylinder rotates forward and becomes closer to the wall, the effect of the wall on the wake is strengthened. Therefore, when α > 0, there are more wake modes because the fluid force changes more frequently, and the amplitude response of the cylinder varies more.

The wake modes of a spring-mounted rotating cylinder in the positive direction are shown in Figure 12. At 0 < α < 1 (shown in Figure 12a,d,g), the mode of the shedding vortices is that a pair of counterrotating vortices are shed within one vibration period, where the positive vortices are very little influenced by the rotation and wall, the positive vortices disappear quickly, and the wall layer vortices become more significant and merge with the negative vortex more strongly as the rotation rate increases. In Figure 12b,c,e,f,h,i, the influence of the rotation and wall increases, and the positive vortices are suppressed and cannot be shed. The suppression of the positive vortices is strengthened as the rotation rate increases, and the mergence between the wall vortex layer and the upper vortex layer becomes stronger.

At 1 < α < 1.5 (shown in Figure 12j,m,p), the positive vortex is generated on the cylinder but does not escape, and the vortex that escapes at this time is the vortex layer of the upper and lower negative vortex. As the rotation ratio increases, the positive vortex layer decreases, and the wall vortex layer curls upward and tends to merge with the negative vortex layer. In Figure 12k,n,q, a small amount of positive vortex is generated in the interval where the cylinder is close to the wall, and the vortex falls off into a negative vortex in one vibration period. Two periods of shedding vortex tend to merge. As the rotation rate increases, the upward of the wall vortex layer increases, and the mergence with the negative vortex also increases. In Figure 12l,o,r, the cylinder is close to the wall, and the positive vortex is attached to the surface of the cylinder, which induces smaller vorticity. The negative vortex is a stable one-sided wake formed by the merging of the boundary layer on the left side of the cylinder and the negative vortex.

The wake modes of the reversing cylinder are shown in Figure 13. The positive vortex in V_r ∈ (1, 4) is enhanced by counterrotation but still does not dislodge at −0.75 < α < 0. A single negative vortex dislodges within one vibration cycle of the cylinder. The shedding period gradually increases as the rotation ratio increases. The negative vortices gradually connect, and the wall vortex layer and negative vortex mergence are enhanced. At V_r ∈ (5, 7), the vortex shedding mode is similar for all rotation rates, and all are alternating. However, due to the difference in the velocity of the flow field caused by the rotation of the cylinder, the velocity of the negative vortex is higher than that of the positive vortex. The mergence of the wall vortex layer and the negative vortex occurs earlier as the rotation rate increases, while the wall vortex layer in front of the cylinder tends to merge with the laminar flow on the cylinder. For −1.5 < α < −0.75, the wall vortex layer merges with the laminar flow in V_r ∈ (1, 4). At V_r ∈ (8, 12), a pair of counterrotating vortices sheds with an increasing rotation rate. With an increasing rotation rate, the equilibrium position of the cylinder moves up, and the influence of the wall vortex decreases. The wall vortex layer is more stable, and the influence of the vortex decreases when α = −1.5. The asymmetry of the flow field is mainly due to the formation of a shear layer near the wall, where the flow profile in front of the cylinder is no longer symmetrical.

4.3. Lift and Drag Coefficients

The lift and drag coefficients of the cylinder are shown in Figure 14. The lift coefficient increases with an increasing rotation ratio. At α < 1, the lift coefficient peak occurs at V_r = 4. The drag coefficient first increases and then decreases as the reduced velocity increases, and the peak value decreases and then increases as the rotation rate increases. At α = 1, the peak of the fluid force coefficient occurs at V_r = 4 and V_r = 8. At α > 1, the lift coefficient increases rapidly as the reduced velocity increases, and the growth rate of the drag coefficient exceeds α < 1. The changes in the lift and drag coefficients are consistent with the reduced velocity for each rotation ratio because the wake mode has been changed. This is also the reason for the appearance of peak values in the amplitude curve.

The lift and drag coefficients of the reversing cylinder are further shown in Figure 15. At α ≤ −0.25, the lift coefficient increases and then decreases with an increasing reduced velocity, and the peak value increases and then moves forward. For α > −0.25, the lift coefficient first decreases and then increases as the reduced velocity increases, and all peaks occur at V_r = 5. The lift coefficient increases with an increasing rotation rate in general. For the drag coefficient, the drag coefficient first increases and then decreases, with a peak value at V_r = 5 when the reduced velocity increases. The wake mode is relatively simple, and the variations in the lift and drag coefficients at different rotation ratios are also quite similar.

5. Discussion

As shown in Figure 7a,c, the vibration of the cylinder is mainly affected by forced excitation when α < 0. In contrast, when α > 0, the effect of the wall is more significant and has a greater impact on the vibration of the cylinder. At 0 < α < 1, the vortex-induced vibration of the cylinder is mainly concentrated in V_r ∈ (3, 9), and the vibration range further narrows to V_r ∈ (3, 5) with an increasing rotation ratio. For α > 1, the range is concentrated in V_r ∈ (3, 9), but the amplitude value is greater than α < 1 and does not vary with the rotation ratio. The presence of the wall reduces the peak amplitude of vibration and causes the first jump of amplitude to occur even earlier. The forced rotation of the cylinder mainly alters the degree to which the wall affects the vibration of the cylinder. When a cylinder is in a near-wall rotational environment, the effect of the wall dominates the vibration of the cylinder, and rotation can enhance or weaken the effect of the wall on the cylinder.

As shown in Figure 10, as the rotation rate and the reduced velocity increase, the main concentration of the vortex shedding pattern of the cylinder is at α > 0, and the shift of the vortex shedding mode is advanced (α = 0.25, V_r = 9 (2S to US); α = 0.75, V_r = 5 (2S to PS)). At α = 1 and V_r < 6, the vortex shedding mode mainly changes to U, expect at α = 1 and V_r = 5 (2S). S is mainly concentrated in α > 1 and V_r ∈ (4, 8). At α < 0, the vortex shedding mode shows regionalization, and U is mainly concentrated at α = −1 and V_r = 4.

The variation in the vortex shedding mode causes shifts in the fluid force, and the change in the fluid force further causes vibrations in the cylinder. As shown in Figure 16, at α = 1, the wake mode of a spring-mounted cylinder changes from U to 2S, and the fluid force on the cylinder always jumps with the change in amplitude. At V_r = 4, it changes again to U, and the lift and amplitude are similar to those at V_r < 4. The lift amplitude jumps again when the wake mode shifts from U to S. When it shifts from S to FS, the lift and amplitude also change again. At α = −1, the variation in the vortex shedding mode also causes a jump in the lift and amplitude, and the change may be caused by the shedding of the wall vortices. At V_r > 4, the wake mode is 2S, but at V_r ∈ (4, 8), there is shedding of the wall vortices. At V_r ∈ (9, 12), there is no shedding of the wall vortices, which also causes a change in the lift and amplitude.

Figure 7a,c show that the cylindrical vibration is suppressed at 0 < α < 1 under the near-wall rotating environment. Figure 17 shows the range of vibrations at 0 < α < 1. The size of the vibration area varies significantly at different rotation ratios, and it decreases with an increasing rotation ratio. The size of the amplitude range decreases from 4.5 to 0.66 as the rotation ratio increases, and the cylindrical vibration is more significantly suppressed at 0 < α < 1. The reduction in the area of the vibrations is caused by the shift of the wake. The cylindrical wake mode shifts from 2S mode to FS mode earlier with an increasing rotation ratio. The longer the FS mode range length, the smaller the vortex vibration area.

6. Conclusions

A numerical study of the 2-DOF vibration of a rotating cylinder under near-wall conditions was carried out. H/D ∈ (0.2, 1.4), α ∈ (−1.5, 1.5) and V_r ∈ (1, 12) were chosen to obtain the wake of the cylinder. We analyzed the fluid forces on the surface of the cylinder and the vibration response of the cylindrical system. The main conclusions are as follows:

(1)

In a near-wall rotating environment, the influence of the wall on the cylinder dominates over the effect of rotation, with rotation mainly affecting the position of the cylinder. At α > 0, there are five wake modes (S, 2S, U, US, FS) for the cylinder, and the amplitude of the cylinder varies considerably at different rotation ratios. VIVs are mainly concentrated in V_r ∈ (3, 9), and they are suppressed as the rotation ratio increases at 0 < α < 1 but are enhanced when α > 1. Positive rotation brings the cylinder closer to the wall, resulting in a stronger influence of the wall on the cylinder. For α < 0, the cylinder moves away from the wall, and there are only two wake modes (2S, U) for the cylinder, with similar amplitudes at each rotation ratio.

(2)

The critical point of the wake transition and the changes in the vibration and fluid forces during the wake transition are discussed. The U-mode is distributed over V_r ∈ (1, 4) at V_r ∈ (−1.5, −1) for α < 0. For α > 0, the vortex shedding mode transition is advanced as α and V_r increase, and the wake mode shifts advance from V_r = 8 to V_r = 4 at 0 < α < 1. However, the wake mode enters another mode completely at α > 1. The 2S wake mode disappears, and the S and U modes appear simultaneously. At the same time, the connection between the wake shedding and fluid force and amplitude is established, and the mechanism of the wake flow–fluid force–amplitude interaction is clarified.

(3)

α = 1 is a dividing line for the vibration of the cylinder. The vibration is suppressed at $0<\alpha <1$ but enhanced for α > 1. As the rotation ratio increases, the value of the amplitude decreases from 0.6 to 0.3, and the amplitude interval narrows from V_r ∈ (3, 12) to V_r ∈ (3, 5) when 0 < α < 1. The shift in the wake mode is consistently advanced, from V_r = 8 at α = 0.25 to V_r = 4 at α = 0.75. The suppression of the cylindrical vibrations becomes more pronounced as the rotation ratio increases, and vibration suppression is most pronounced at α = 0.75. However, an increase in the rotation ratio enhances the cylinder vibration and results in a larger amplitude range from V_r = 3 to V_r = 9 when α > 1.