Effects of Mass and Damping on Flow-Induced Vibration of a Cylinder Interacting with the Wake of Another Cylinder at High Reduced Velocities

: Flow-induced vibration is a canonical issue in various engineering fields, leading to fatigue or immediate damage to structures. This paper numerically investigates flow-induced vibrations of a cylinder interacting with the wake of another cylinder at a Reynolds number Re = 150. It sheds light on the effects of mass ratio m * , damping ratio, and mass-damping ratio m * ζ on vibration amplitude ratio A/D at different reduced velocities Ur and cylinder spacing ratios L/D = 1.5 and 3.0. A couple of interesting observations are made. The m * has a greater influence on A/D than ζ although both m * and ζ cause reductions in A/D . The m * effect on A/D is strong for m * = 2–16 but weak for m * > 16. As opposed to a single isolated cylinder case, the mass-damping m * ζ is not found to be a unique parameter for a cylinder oscillating in a wake. The vortices in the wake decay rapidly at small ζ . Alternate reattachment of the gap shear layers on the wake cylinder fuels the vibration of the wake cylinder for L/D = 1.5 while the impingement and switch of the gap vortices do the same for L/D = 3.0.


Introduction
Flow over cylindrical structures is ubiquitous in engineering fields such as naval engineering (submarines, ship propellers), offshore engineering (semisubmersibles, spar, gravity platforms, and jackets), renewable energy engineering (offshore wind and tidal turbines), nuclear engineering (reactors, cooling tower, chimneys), civil engineering (skyscrapers and cables of suspension bridges), electrical engineering (power lines), etc. These cylindrical structures undergo undesirable flow-induced vibrations because of fluctuating forces induced by flow separation and alternate vortex shedding. When more than one cylinder is in a group, the fluid force acting on a cylinder in the group is different from that on a single isolated cylinder. The difference arises from the mutual fluid-structure interactions between the cylinders. The flow over a cylinder placed in the wake of another cylinder is considered as the baseline to study fluid-structure interactions between the cylinders in a group. In this case, the wake cylinder (downstream cylinder) receives a strong interaction from the wake-generating cylinder (upstream cylinder).
Flow-induced vibrations commonly include instability-induced excitation (vortexinduced vibration) and movement-induced excitation (galloping). Vortex-induced vibration (VIV) usually occurs over a limited range of reduced velocity, hence, is self-limiting and is a kind of resonance, which occurs when the shedding frequency equals the oscillation frequency of the structure. On the other hand, galloping is a movement-induced excitation where the vibration amplitude grows with increasing reduced velocity. The galloping vibration is generated when the shedding frequency is greater or smaller than the oscillation frequency of the structure. Flow-induced vibration is a function of mass ratio m * , damping ratio ζ, natural frequency f n , Reynolds number Re, reduced velocity Ur frequency away from the natural frequency of the cylinder. Hysteresis was observed in the responses of both cylinders. The review of the literature suggests that numerical investigations at low Re are scarce for the case where the wake cylinder is free to oscillate but the wake-generating cylinder is fixed. There are a few key issues to be resolved. For example, what are the effects of m * and ζ on VIV responses and wake topology for the wake cylinder interacting with a wake generated by another cylinder? Are the effects dependent on L/D? Is mass-damping ratio m * ζ a unique parameter to characterize the vibration? The objective of this work to investigate the effects of m * (=2-32), ζ (=0-0.5), m * ζ (=0-8) and L/D (=1.5 and 3) on flow-induced vibration of a cylinder submerged in the wake of another. Numerical simulations are conducted at Re = 150 for the parametric ranges mentioned above. The Ur is varied from 2.5 to 30. A low-Re flow provides a better understanding of flow physics, and one can clearly observe the development of shear layers, formation of vortices, evolution of vortices, etc. Although major flow phenomena at both low-and high-Re flows are the same (e.g., vortex shedding, lock-in, vibration nature), quantitative magnitudes of forces, Strouhal number, lock-in range, maximum vibration amplitude, etc. differ between the low-and high-Re flows. The results from a low-Re flow can, thus, not be extrapolated to a high-Re flow.

Computational Domain
The flow is given in a rectangular computational domain where two circular cylinders are arranged in tandem at the horizontal centerline of the domain (Figure 1). The wakegenerating cylinder is fixed, and the wake cylinder is spring-mounted. The latter cylinder is allowed to oscillate in the transverse direction only. The total spring stiffness is k. The Cartesian coordinate system (x-y) has its origin at the center of the wake-generating cylinder. The inlet and outlet boundaries of the computational domain are placed at x = −15 D and L + 45 D, respectively, where L is the distance between the centers of the cylinders, and D is the diameter. The upper and lower boundaries are symmetrically separated by 30 D from each other [17,18]. The corresponding cylinder blockage ratio is 3.3% [19].
affected the upstream cylinder vibration response. Compared to a single cylinder, the upstream cylinder had lock-in at smaller Ur, with the vortex shedding frequency away from the natural frequency of the cylinder. Hysteresis was observed in the responses of both cylinders. The review of the literature suggests that numerical investigations at low Re are scarce for the case where the wake cylinder is free to oscillate but the wake-generating cylinder is fixed. There are a few key issues to be resolved. For example, what are the effects of m * and ζ on VIV responses and wake topology for the wake cylinder interacting with a wake generated by another cylinder? Are the effects dependent on L/D? Is massdamping ratio m * ζ a unique parameter to characterize the vibration? The objective of this work to investigate the effects of m * (=2-32), ζ (=0-0.5), m * ζ (=0-8) and L/D (=1.5 and 3) on flow-induced vibration of a cylinder submerged in the wake of another. Numerical simulations are conducted at Re = 150 for the parametric ranges mentioned above. The Ur is varied from 2.5 to 30. A low-Re flow provides a better understanding of flow physics, and one can clearly observe the development of shear layers, formation of vortices, evolution of vortices, etc. Although major flow phenomena at both low-and high-Re flows are the same (e.g., vortex shedding, lock-in, vibration nature), quantitative magnitudes of forces, Strouhal number, lock-in range, maximum vibration amplitude, etc. differ between the low-and high-Re flows. The results from a low-Re flow can, thus, not be extrapolated to a high-Re flow.

Computational Domain
The flow is given in a rectangular computational domain where two circular cylinders are arranged in tandem at the horizontal centerline of the domain ( Figure 1). The wake-generating cylinder is fixed, and the wake cylinder is spring-mounted. The latter cylinder is allowed to oscillate in the transverse direction only. The total spring stiffness is k. The Cartesian coordinate system (x-y) has its origin at the center of the wake-generating cylinder. The inlet and outlet boundaries of the computational domain are placed at x = −15 D and L + 45 D, respectively, where L is the distance between the centers of the cylinders, and D is the diameter. The upper and lower boundaries are symmetrically separated by 30 D from each other [17,18]. The corresponding cylinder blockage ratio is 3.3% [19].
Wake-generating cylinder Wake cylinder

Governing Equations and Numerical Technique
The continuity and Navier-Stokes equations are the governing equations that can be written as: The parameters in the equations are non-dimensional and have the following expressions.
Here, u and ν are the streamwise and transverse components of the flow velocity, respectively, p is the static pressure, µ is the fluid viscosity, U is the freestream velocity, and t is the time.
The cylinders are surrounded by an O-xy grid system while a rectangular grid system is provided away from the cylinders, as shown in Figure 2. The meshes around the cylinders are given a greater density. We set the first grid level 0.009D away from the cylinder surface, with a mesh expansion ratio of less than 1.1. The simulation uses a dynamic mesh scheme where the ANSYS-Fluent 15 solver is used to move boundaries and/or objects and to adjust the mesh accordingly. The grid box around the cylinder moves with the cylinder displacement. The user-defined function is feed into the solver to estimate the cylinder displacement. At each time step, the domain deformation is handled by the dynamic meshing tool in ANSYS-Fluent 15, and the mesh is updated using the Laplace smoothing method.

Governing Equations and Numerical Technique
The continuity and Navier-Stokes equations are the governing equations that can be written as: The parameters in the equations are non-dimensional and have the following expressions.
Here, u and ν are the streamwise and transverse components of the flow velocity, respectively, p is the static pressure, μ is the fluid viscosity, U is the freestream velocity, and t is the time.
The cylinders are surrounded by an O-xy grid system while a rectangular grid system is provided away from the cylinders, as shown in Figure 2. The meshes around the cylinders are given a greater density. We set the first grid level 0.009D away from the cylinder surface, with a mesh expansion ratio of less than 1.1. The simulation uses a dynamic mesh scheme where the ANSYS-Fluent 15 solver is used to move boundaries and/or objects and to adjust the mesh accordingly. The grid box around the cylinder moves with the cylinder displacement. The user-defined function is feed into the solver to estimate the cylinder displacement. At each time step, the domain deformation is handled by the dynamic meshing tool in ANSYS-Fluent 15, and the mesh is updated using the Laplace smoothing method. The boundary conditions are given as u * = 0 and v * = 0 at the surfaces of the upstream and downstream cylinders, u * = 1 and v * = 0 at the inlet, ∂u * /∂y * = 0 and v * = 0 at the lateral sides, and ∂u * /∂x * = 0 and ∂v * /∂x * = 0 at the outlet. In the governing Equations (1)-(3), p * , u * and v * are the unknown parameters, solved by coupling the governing equations. They are solved for the unsteady and incompressible flow with constant fluid properties. The computations are conducted using the finite volume method. The convective components are discretized using the second-order up- The boundary conditions are given as u * = 0 and v * = 0 at the surfaces of the upstream and downstream cylinders, u * = 1 and v * = 0 at the inlet, ∂u * / ∂y * = 0 and v * = 0 at the lateral sides, and ∂u * / ∂x * = 0 and ∂v * / ∂x * = 0 at the outlet. In the governing Equations (1)-(3), p * , u * and v * are the unknown parameters, solved by coupling the governing equations. They are solved for the unsteady and incompressible flow with constant fluid properties. The computations are conducted using the finite volume method. The convective components are discretized using the second-order upwind scheme. The coupling between the velocity and pressure fields is done by the pressure correction-based iterative algorithm SIMPLE (Semi-implicit method for pressure linked equations) proposed by Patankar [20]. We used a first-order implicit formulation for the time discretization. The governing equation of the cylinder motion in the dimensionless form can be written as: ..
where Y is the cylinder displacement measured from y = 0, . Y is the cylinder velocity, and ..
Y is the cylinder acceleration. The C Li is the instantaneous lift coefficient of the cylinder.
The F n = f n D/U, where f n is the natural frequency of the cylinder. The m * (=4 m/ρπD 2 ) is the cylinder mass ratio, where m stands for the mass of the cylinder. We solved Equation (4) using the fourth-order Runge-Kutta method for every time step. The Re = 150 for all simulations including validation. The flow around a single cylinder transits from two-to three-dimensional at Re ≈ 190 [21]. When two cylinders are placed in tandem, the transition Re (Re cr2 ) from two-dimensional to three-dimensional flows is delayed for L/D ≤ 3.0, see Figure 3 reproduced from Rastan and Alam [22]. The flow around two tandem cylinders at L/D = 1.5 and 3.0 examined is thus assumed to be two-dimensional.
wind scheme. The coupling between the velocity and pressure fields is done by the pressure correction-based iterative algorithm SIMPLE (Semi-implicit method for pressure linked equations) proposed by Patankar [20]. We used a first-order implicit formulation for the time discretization. The governing equation of the cylinder motion in the dimensionless form can be written as: where Y is the cylinder displacement measured from y = 0, is the cylinder velocity, and is the cylinder acceleration. The CLi is the instantaneous lift coefficient of the cylinder. The Fn = fnD/U, where fn is the natural frequency of the cylinder. The m * (=4 m/ρπD 2 ) is the cylinder mass ratio, where m stands for the mass of the cylinder. We solved Equation (4) using the fourth-order Runge-Kutta method for every time step. The Re = 150 for all simulations including validation. The flow around a single cylinder transits from two-to three-dimensional at Re ≈ 190 [21]. When two cylinders are placed in tandem, the transition Re (Recr2) from two-dimensional to three-dimensional flows is delayed for L/D ≤ 3.0, see Figure 3 reproduced from Rastan and Alam [22]. The flow around two tandem cylinders at L/D = 1.5 and 3.0 examined is thus assumed to be two-dimensional.  [22]. For the definitions of the legends, please refer to Rastan and Alam [22]. Recr1 represents the flow transition from steady to two-dimensional unsteady while Recr2 indicates the flow transition from two-dimensional unsteady to three-dimensional unsteady.

Validation
A mesh independence test was conducted based on the mesh system adopted in Zafar and Alam [23] and Abdelhamid et al. [24]. Three different meshes (M1, M2, and M3 consisting of 76,350, 80,050, and 9400 elements, respectively) are tested for a single fixed cylinder at Re = 150. The results of global parameters including time-mean drag coefficient  Table 1 and compared with those in the literature. The results for the three mesh systems are essentially converged as these mesh systems were decided based on the mesh systems in [23][24][25][26]. The maximum deviation in D C is found to be less than 2.5%. The present results overall show quite good agreement with those from the literature. Mesh M2 system, with the same grid distributions around the two cylinders, was,   [22]. For the definitions of the legends, please refer to Rastan and Alam [22]. Re cr1 represents the flow transition from steady to two-dimensional unsteady while Re cr2 indicates the flow transition from two-dimensional unsteady to three-dimensional unsteady.

Validation
A mesh independence test was conducted based on the mesh system adopted in Zafar and Alam [23] and Abdelhamid et al. [24]. Three different meshes (M1, M2, and M3 consisting of 76,350, 80,050, and 9400 elements, respectively) are tested for a single fixed cylinder at Re = 150. The results of global parameters including time-mean drag coefficient C D , fluctuating lift coefficient C L , and Strouhal number St for three different meshes are presented in Table 1 and compared with those in the literature. The results for the three mesh systems are essentially converged as these mesh systems were decided based on the mesh systems in [23][24][25][26]. The maximum deviation in C D is found to be less than 2.5%. The present results overall show quite good agreement with those from the literature. Mesh M2 system, with the same grid distributions around the two cylinders, was, however, used for the computations of two tandem cylinders when the wake cylinder is free to oscillate, and the results are validated again in Figure 4. To compare our results with the numerical results of Carmo et al. [2], we first simulated vibration responses for m * = 2, ζ = 0.003, and L/D = 3 which are the same geometrical and physical conditions used by Carmo et al. [2]. A comparison between the  Figure 4 where the vibration amplitude ratio A/D is presented against Ur. Here, A represents the cylinder vibration amplitude obtained from displacement signal Y. First, root-mean-square (rms) value of Y (Y rms ) is obtained, and then A is calculated as A = Y rms × √ 2. The present results agree well with those by Carmo et al. [2], with a maximum deviation of 5% occurring at Ur = 10. however, used for the computations of two tandem cylinders when the wake cylinder is free to oscillate, and the results are validated again in Figure 4. To compare our results with the numerical results of Carmo et al. [2], we first simulated vibration responses for m * = 2, ζ = 0.003, and L/D = 3 which are the same geometrical and physical conditions used by Carmo et al. [2]. A comparison between the present and Carmo et al.'s results is made in Figure 4 where the vibration amplitude ratio A/D is presented against Ur. Here, A represents the cylinder vibration amplitude obtained from displacement signal Y. First, rootmean-square (rms) value of Y (Yrms) is obtained, and then A is calculated as 2 rms A Y =.
The present results agree well with those by Carmo et al. [2], with a maximum deviation of 5% occurring at Ur = 10.   Figure 5 shows the dependence of vibration amplitude ratio A/D on Ur for m * = 2, ζ = 0.005 and L/D = 3.0. The A/D firstly increases with increasing Ur, reaching a peak at Ur = 7 before declining with a further increase in Ur. The decline is rapid for Ur = 7-15 but mild  Figure 5 shows the dependence of vibration amplitude ratio A/D on Ur for m * = 2, ζ = 0.005 and L/D = 3.0. The A/D firstly increases with increasing Ur, reaching a peak at Ur = 7 before declining with a further increase in Ur. The decline is rapid for Ur = 7-15 but mild for Ur > 15. The vorticity structures shown in Figure 5b,c illustrate that each of the wakes at Ur = 7 and 15 is characterized by two rows of opposite sign vortices. Yet, the two wakes differ in different aspects. Firstly, the lateral separation between the two vortex rows is wider for Ur = 7 than for Ur = 15. Secondly, the streamwise distance between two consecutive vortices is large for Ur = 15, compared to that for Ur = 7. As the vibration amplitude gently decreases with Ur for Ur > 15 (Figure 5a), there is no galloping. In a water tunnel test, Bokaian and Geoola [1] observed galloping vibration at 1.09 ≤ L/D ≤ 3 in the range of Re = 600-6000 with m * ζ = 0.109. When both cylinders are allowed to Energies 2021, 14, 5148 7 of 13 vibrate, Kim et al. [12] in a wind tunnel test found the occurrence of galloping vibrations for 1.2 ≤ L/D ≤ 1.6 at Re = 4365-74,200 and m * ζ = 0.64. It suggests that the occurrence of galloping is highly sensitive to L/D, Re, and m * ζ. Interestingly, as seen in Figure 5, the A/D value remains high (A/D > 0.4) even at high Ur (>15). It is worth investigating whether galloping occurs at other m * and ζ values. We will therefore focus on the vibration response at large Ur (>10) only.

Vibration Response at Small m* and ζ
for Ur > 15. The vorticity structures shown in Figure 5b,c illustrate that each of the wakes at Ur = 7 and 15 is characterized by two rows of opposite sign vortices. Yet, the two wakes differ in different aspects. Firstly, the lateral separation between the two vortex rows is wider for Ur = 7 than for Ur = 15. Secondly, the streamwise distance between two consecutive vortices is large for Ur = 15, compared to that for Ur = 7. As the vibration amplitude gently decreases with Ur for Ur > 15 (Figure 5a), there is no galloping. In a water tunnel test, Bokaian and Geoola [1] observed galloping vibration at 1.09 ≤ L/D ≤ 3 in the range of Re = 600-6000 with m * ζ = 0.109. When both cylinders are allowed to vibrate, Kim et al. [12] in a wind tunnel test found the occurrence of galloping vibrations for 1.2 ≤ L/D ≤ 1.6 at Re = 4365-74,200 and m * ζ = 0.64. It suggests that the occurrence of galloping is highly sensitive to L/D, Re, and m * ζ. Interestingly, as seen in Figure 5, the A/D value remains high (A/D > 0.4) even at high Ur (>15). It is worth investigating whether galloping occurs at other m * and ζ values. We will therefore focus on the vibration response at large Ur (>10) only.

Effect of Damping Ratio on Vibration Amplitude
The vibration response is investigated for Ur > 10 when ζ is increased from 0 to 0.5.

Effect of Damping Ratio on Vibration Amplitude
The vibration response is investigated for Ur > 10 when ζ is increased from 0 to 0.5.

Effect of Mass Ratio on Vibration Amplitude
More detail of the effect of m * on A/D is presented in Figure 8 for L/D = 3.0 and 1.5 with ζ = 0. For both L/D values, A/D decreases with increasing m * , the decrease being large at small Ur. Considering the steps of the increase in m * , the decrease in A/D is largest between m * = 2 and 4. When m * is large enough (i.e., m * = 32 and 16 for L/D = 3.0 and 1.5, respectively), A/D becomes very small and almost independent of Ur. There is a large drop in A/D for m * = 4 from Ur = 10 to 15. As seen, A/D is highly sensitive to m * for m * = 2-8. At m * = 4, it seems that A/D at Ur = 10 and Ur ≥ 15 has similar characteristics to that at m * = 2 and 8, respectively. There might be a transition from high-to low-amplitude vibration between Ur = 10 and 15, which requires further investigation.
between Ur = 10 and 15, which requires further investigation. Figure 9 shows the dependence of A/D on m * , Ur and ζ for L/D = 1.5. At Ur = 10 ( Figure  9a), with increasing m * , the A/D exponentially drops between m * = 2 and 16 and gently for m * > 16. The same figure (Figure 9a) further proves that there is an effect of ζ on A/D, the effect being small between ζ = 0 and 0.05 but strong between ζ = 0.05 and 0.5. At Ur = 15 (Figure 9b), again the effect of m * on A/D is very strong for m * = 2-16 and weak for m * > 16.
Here, the effect of ζ on A/D is very small. Figure 9c further describes the relationship between A/D, m * and Ur. For all Ur values, A/D exponentially declines with increasing m * up to 16. In addition, A/D drops significantly between Ur = 10 and 15 and mildly between Ur = 15 and 30. It can be generalized that A/D is inversely linked to m * , ζ and Ur.

Effect of Mass-Damping Ratio on Vibration Amplitude
In the previous section, both m * and ζ causes a reduction in A/D. For a single isolated cylinder, mass-damping ratio (m * ζ) has been proven to be another characteristic parameter that has a connection with A/D [36]. The A/D is found to monotonically decrease with increasing m * ζ for a single isolated cylinder [37]. To see the relationship between A/D and m * ζ for the cylinder interacting with the wake, A/D is plotted against m * ζ in Figure 10 for Ur = 10 and 15. For these data, ζ varies from 0 to 0.5 for both m * = 2 and 16 (see the legends in the figure). Interestingly, although A/D decreases with increasing m * ζ for a given m * , the decrease rate is contingent on m * , high at small m * , and vice versa. The A/D data fail to collapse onto a single line against m * ζ, which suggests that m * ζ is not a unique parameter to characterize the vibration for a cylinder interacting with the wake of another cylinder.

Effect of Mass-Damping Ratio on Vibration Amplitude
In the previous section, both m * and ζ causes a reduction in A/D. For a single isolated cylinder, mass-damping ratio (m * ζ) has been proven to be another characteristic parameter that has a connection with A/D [36]. The A/D is found to monotonically decrease with increasing m * ζ for a single isolated cylinder [37]. To see the relationship between A/D and m * ζ for the cylinder interacting with the wake, A/D is plotted against m * ζ in Figure 10 for Ur = 10 and 15. For these data, ζ varies from 0 to 0.5 for both m * = 2 and 16 (see the legends in the figure). Interestingly, although A/D decreases with increasing m * ζ for a given m * , the decrease rate is contingent on m * , high at small m * , and vice versa. The A/D data fail to collapse onto a single line against m * ζ, which suggests that m * ζ is not a unique parameter to characterize the vibration for a cylinder interacting with the wake of another cylinder. It is worth investigating how the wake structure is modified when L/D, m * , and ζ are changed. Figure 11 shows vorticity structures for L/D = 3.0 and 1.5 for different values of m * and ζ. The first-and second-column snapshots correspond to Y ≈ 0 (i.e., mean position)

Effects of L/D, m * , and ζ on Wake Structure
It is worth investigating how the wake structure is modified when L/D, m * , and ζ are changed. Figure 11 shows vorticity structures for L/D = 3.0 and 1.5 for different values of m * and ζ. The first-and second-column snapshots correspond to Y ≈ 0 (i.e., mean position) and Y ≈ Y max (i.e., maximum displacement), respectively. In the first-and second-row wake structures (Figure 11a,b), the effect of increasing m * from 2 to 16 on the wake structure is displayed. For m * = 2 and ζ = 0 with L/D = 3.0 (Figure 11a), the wake behind the wake cylinder is characterized by 2S mode (i.e., two vortices shed in one oscillation cycle). The shear layers emanating from the wake-generating cylinders roll up into the gap between the cylinders. When the vibrating cylinder is close to the mean position (Y ≈ 0), the vortices from the wake-generating cylinder impinge on the vibrating cylinder. On the other hand, they both pass over the same side of the vibrating cylinder when the vibrating cylinder is at its maximum displacement (Y ≈ Y max ). When the m * is increased to 16 (Figure 11b), the shear layers from the wake-generating cylinder do not roll up in the gap between the cylinders but alternately reattach on the vibrating cylinder. Another remarkable difference in the wake structures between m * = 2 and 16 (Figure 11a,b) is that a greater number of vortices appear in the same downstream distance (x * = 4-26) for m * = 2 than for m * = 16. Given the 2S mode for both cases, it can be argued that the cylinder oscillation frequency reduces when m * is increased from 2 to 16, which is consistent with our intuition.  A comparison of snapshots between the first and third rows gives out the effect of ζ (=0 and 0.5) on the wake structure. The arrangement and structure of vortices in the wake do not differ much between ζ = 0 and 0.5. The vortices in the wake, however, decay more rapidly for ζ = 0 than for ζ = 0.5. The effect of L/D can be understood by comparing the vortex shedding between L/D = 3.0 and 1.5 (Figure 11a,d), both for the same m * and ζ. Although the wake structure is the same for both L/D values, the flow structure around the cylinders is completely different in the two cases. While vortex shedding occurs from both cylinders for L/D = 3.0 (Figure 11a), only the wake cylinder sheds vortices for L/D = 1.5 (Figure 11d). Vortex shedding does not take place in the gap between the cylinders for L/D = 1.5. The alternate reattachment of the shear layer from the wake-generating cylinder A comparison of snapshots between the first and third rows gives out the effect of ζ (=0 and 0.5) on the wake structure. The arrangement and structure of vortices in the wake do not differ much between ζ = 0 and 0.5. The vortices in the wake, however, decay more rapidly for ζ = 0 than for ζ = 0.5. The effect of L/D can be understood by comparing the vortex shedding between L/D = 3.0 and 1.5 (Figure 11a,d), both for the same m * and ζ. Although the wake structure is the same for both L/D values, the flow structure around the cylinders is completely different in the two cases. While vortex shedding occurs from both cylinders for L/D = 3.0 (Figure 11a), only the wake cylinder sheds vortices for L/D = 1.5 (Figure 11d). Vortex shedding does not take place in the gap between the cylinders for L/D = 1.5. The alternate reattachment of the shear layer from the wake-generating cylinder sustains the wake cylinder vibration for L/D = 1.5 while alternate impingement and switch of the gap vortices do the same for L/D = 3.0.

Conclusions
Flow-induced vibrations of a cylinder interacting with the wake of another cylinder is investigated at Re = 150. The focus is given on the effect of m * (=2-32), ζ (=0-0.5), and m * ζ (=0-8) on A/D at different Ur and L/D values. While investigations at high Re found galloping vibration for the wake cylinder (e.g., [1]), no galloping occurs at this low Re. Following VIV peak, A/D remains high (A/D > 0.4) even at high Ur (>15).
The A/D is more sensitive to m * than to ζ. Both m * and ζ reduce A/D. The effect of ζ on A/D is larger at smaller m * . On the other hand, the m * has a considerable effect on A/D for all ζ values. The effect is, however, strong for m * = 2-16 and relatively weak for m * > 16. Generally, the A/D is inversely linked to m * and ζ.
Although m * ζ is believed to be a unique parameter determining the vibration amplitude for a single isolated cylinder, this is not the case for the cylinder interacting with the wake generated by another. Although A/D decreases with m * ζ, the decrease rate is high at small m * . As such, the A/D data fail to collapse onto a single line against m * ζ, which signifies that m * ζ is not a parameter that uniquely characterizes interaction of a cylinder with the wake of another cylinder.
The shear layers of the wake-generating cylinder do roll up in the gap between the cylinders at m * = 2 for ζ = 0, Ur = 10, and L/D = 3. With the same initial conditions (ζ = 0, Ur = 10, and L/D = 3), when m * is increased to 16, the shear layers do not roll up in the gap but alternately reattach on the vibrating cylinder. Vortices in the wake appear more for m * = 2 than for m * = 16, decaying more rapidly at small ζ. While the wake-generatingcylinder shear layers reattaching alternately on the wake cylinder fuels the vibration of the wake cylinder for L/D = 1.5, the gap-vortex impingement and switch do the same for L/D = 3.0. The results are useful to riser designers to understand the role of m* and ζ in flow-induced vibrations.
Funding: This research was funded by the Khalifa University of Science and Technology through Grants CIRA-2020-057.

Data Availability Statement:
The data that support the findings of this study are available within the article.