Effect of a fixed downstream cylinder on the flow-induced vibration of an elastically-supported primary cylinder

The FIV of an upstream elastically supported cylinder with a downstream fixed control cylinder in the low Reynolds number regime is of present interest; see the problem schematic in Fig. 1. The single-degree-of-freedom elastically supported circular cylinder is located upstream and can move freely along the $y$ direction. The fixed control circular cylinder is located downstream. The initial state of the two cylinders is on the same horizontal line. The nondimensional distance between the centers of the two cylinders, $L/D$, is systematically varied from $1.25$ to $3$, where $D$ is the diameter of the circular cylinder.

To facilitate the analysis in the later sections, Fig. 1 shows the front stagnation point of the cylinder marked as FSP and the separation point of the boundary layer marked as SP. The angle $\alpha$ is defined as the angle between the front stagnation point and the center horizontal line of the cylinder. The angle is positive when the front stagnation point is above the center horizontal line of the cylinder, and negative when it is below. The angle $\beta$ is the angle between the separation point of the boundary layer and the front stagnation point.

The viscous Newtonian flow is governed by the continuity and unsteady incompressible Navier-Stokes equations as given in the dimensionless form in Eqs. 1 and 2.

The structural system consists of a fixed DC and an elastically supported UC with a single degree of freedom (sdof) along the cross-flow direction. A typical mass-spring-damper dynamic model describes the motion characteristics of the elastically supported circular cylinder. The motion of the UC in the ‘y’ direction is governed by a second-order linear differential equation describing the dynamics of a sdof system as follows:

The two-way coupled fluid-structure interaction (FSI) simulations are carried out using the CFD software Ansys Fluent based on the fully–implicit algorithm, where the structural equations are incorporated through user-defined functions. To calculate the two-way coupled response of the UC, the N-S equations are coupled with the structural governing equations in a staggered manner. The second-order accurate spatial and temporal discretization schemes are used in this study with the absolute and relative tolerance criteria kept at $10^{-6}$.

The entire computational domain is rectangular with a size of $70D\times 2H$, where $D$ is the diameter of the circular cylinder and $H$ is half of the width of the flow field. In the flow direction, the distance between the center point of the elastically supported cylinder and the inlet is $20D$, while the distance is $50D$ from the center of the elastically supported cylinder to the outlet. The boundary conditions are shown in Fig. 2. The boundary settings of the computational domain include the velocity inlet with $u=U_{in}$ and $v=0$, the pressure outlet with $du/dx=0$ and $dv/dx=0$, and the symmetric upper and lower walls with $du/dy=0,v=0$.

To avoid the numerical instability caused by the excessive mesh deformation due to the large displacement of the upstream elastically supported cylinder, the overset meshing strategy is used in the present study. The computational mesh is shown in Fig. 3, which consists of a background mesh and two overset meshes. The partial mesh of the tandem cylinder is shown in Fig. 3(a), and Fig. 3(b) shows an enlarged view of the cylindrical boundary grid. The boundary of the overset mesh domain is $3D$, as seen in Fig. 3(c). To accurately capture the flow-field structures, the perimeter of the cylinder is discretized at equal intervals, and the thickness of the first layer grid around the cylinder structure and the mesh growth ratio in the radial direction are $0.001D$ and $1.02D$, respectively. Additionally, the same grid density is used for both overset grid domains.

The effect of domain size on the results is checked by varying the cross-flow height $H$ ($20D$, $25D$, and $28D$) at $L/D=3$ and $Re=100$. The same foreground mesh for the cylinder walls and the same grid size growth rate are used in these three cases. The time step in the domain-independence study is set to 0.008, meeting the requirement of the maximum Courant number to be less than unity to ensure the stability of the solution scheme. The simulation results are shown in Table 1, the maximum difference is less than $1\%$. To reduce the computational cost, the $H$ value of $20D$ is considered by comparing the results of $Y_{max}$, ${C_{D}}_{mean}$, and ${C_{L}}_{rms}$ for these three cases.

Next, the mesh size and time-step independence studies are carried out, keeping the computational domain fixed at $70D\times 40D$ at $L/D=3$ and $Re=100$. Case 1, Case 2, and Case 3 are used to check for the grid convergence; and Case 2 and Case 4, with Case 2 being the reference case, are used for non-dimensional time step study. The comparison of vibration response results is presented in Table 2. The number of mesh elements of the three groups is gradually increased. Comparing Case 2 and Case 1, the differences in the maximum dimensionless amplitude $Y_{max}$, the mean value of the drag coefficient ${C_{D}}_{mean}$, and the root mean square of the lift coefficient ${C_{L}}_{rms}$ are found to be 0.17%, 0.04%, and 0.45%, respectively. When the number of elements is further increased to achieve Case 3, the differences are reduced to 0%, 0.08%, and 0.11%, respectively. In the time-step independence results, the difference is less than 0.2%. Based on these results, the Case 2 mesh with a non-dimensional time step of 0.003 is selected for further simulations in this paper.

With the chosen simulation setup, the results of the VIV of a single cylinder obtained using the present fluid-structure interaction solver are compared with the reference results of Bao et al. (2012) in the similar $Re$ regime to check its efficacy. Using the same parameters as Bao et al. (2012), the comparison of dimensionless displacement of the cylinder is shown in Fig. 4. It can be seen that the present results are in good agreement with the reference results. Hence, the present solver can be used for the low $Re$ regime with confidence.

Different wake regimes are characterized by varying $Re$ and $L/D$ values, and the wake regime map is presented in Fig. 5. The wake can be broadly distinguished into two different regimes according to the development of the shear layer (Ali, Islam, and Janajreh, 2022): the steady flow regime (denoted by I, where there is no vortex shedding behind the DC); the alternating attachment regime (denoted by II). As the $L/D$ value increases from 1.25 to 3, the range of $Re$, where steady flow occurs gradually decreases from $Re\leq 80$ to $Re\leq 60$, and no vortex shedding occurs in this region. On the other hand, the alternating attachment regime is predominant for $Re\geq 70$. The wake patterns are further classified according to the vortex shedding pattern of the two cylinders. As shown in Fig. 5, the alternating attachment regime can be divided into four sub-regions, namely (i) ‘2S’ (two single isolated vortices are shed in an oscillation cycle of the cylinder), (ii) ‘2C’ (two counter-rotating vortex pairs are shed in an oscillation cycle of the cylinder), (iii) ‘2P’ (two co-rotating vortex pairs are shed in an oscillation cycle of the cylinder)(Stremler et al., 2011), and (iv) aperiodic wake regime (vortex shedding takes place completely irregularly with no periodicity), respectively. It can be seen from the flow regime classification diagram that the characteristics of the flow field are sensitive to $Re$ and $L/D$. A detailed analysis of the flow field interactions is presented in the next subsection.

The wake patterns for the steady flow regime and the ‘2S’ vortex shedding pattern in the alternate reattachment regime are shown in Fig. 6 and Fig. 7, respectively. The vorticity is non-dimensionalized, considering $U_{in}$ as the reference velocity scale and $D$ as the reference length scale, as $\omega_{z}^{*}=\omega_{z}D/U_{in}$. In the steady flow regime (at $L/D$ = 1.75 and $Re$ = 60), the shear layer, separating from the UC, directly crosses over the DC, and no vortex shedding takes place behind the DC; see Fig. 6. The displacement ($Y$) and lift coefficient ($C_{L}$) of the UC are practically zero in this regime. Fig. 7 shows the temporal evolution of the flow structures around the cylinders, the pressure coefficient ($C_{P}$) around the cylinders, and the corresponding time-histories of the $C_{L}$ and $Y$ (at $L/D$ = 1.75 and $Re$ = 80). Here, the clockwise vortices are marked in blue (negative), while the counter-clockwise vortices are marked in red (positive). The resulting wake pattern is characterized as a typical Kármán vortex street.

At 1# instance, the vortex P1 is seen to be separated from the shear layer of the DC, while the positive shear layer P2 generated by UC remains completely attached to the lower surface of DC. The negative vortex above the DC N1 is shed alternatively, and the negative shear layer N2 in the gap begins to roll up, adhering to the DC. As a result, the pressure on the upper surface of UC is lower than that on the lower surface, and the pressure difference of the DC is opposite to that of UC at this moment. At 2# instance, as the UC reaches the maximum positive displacement, the negative shear layer in the gap starts adhering to the upper surface of DC. The shear layer P2 rolls up and squeezes the vortex N1. The pressure on the upper surface of UC increases significantly. At 3# instance, the vortex N1 is separated, and the remaining shear layer merges with the negative shear layer N2, forming a new shear layer labelled N2^∗. The negative pressure areas of UC and DC are seen to get reversed compared to the 1# instance. The vortex shedding process at 3# to 5# instances is the same as that at 1# to 3# instances, but the vortex shedding direction is the opposite. In Fig. 7(a), the $C_{L}$ and the $Y$ time histories follow the same trend. Within the ‘2S’ pattern sub-region, there is a distinctive wake pattern characterized by two-layered vortex streets, as shown in Fig. 7(b). This phenomenon is more pronounced at lower $Re$ and $L/D\geq 2$.

As shown in Fig. 5, the ‘2P’ and ‘2C’ vortex-shedding patterns occurs in a wide range of $Re$ and $L/D$. Figs. 8 and 9 show the evolution process of ‘2P’ and ‘2C’ vortex-shedding patterns, the $C_{P}$ around the cylinders for different $Re$ and $L/D$ values, respectively. In Fig. 8, the cylinders are situated at the middle position at 1# instance. The remaining shear layer from the negative shedding vortex of the previous cycle is marked as N1, while the negative shear layer within the gap is labeled as N2. Below the tandem cylinder, the positive shear layer from the UC has merged completely with that of the DC and is marked as P2. Both UC and DC have small pressure on the upper surface and high pressure on the lower surface. At 2# instance, the UC reaches the positive maximum displacement. The downward-curling vortex N1 squeezes the shear layer P2, separating it from DC. On the upper surface of the DC, the shear layer N2 is fully attached to the DC, while the shear layers N1 and N2 remain separate. Conversely, the shear layer N2 promotes the shear layer N1 to separate from DC. At 3# instance, the UC returns to the middle position. The shear layer P2 detaches from the DC due to the downward-curling shear layer N1. Due to the pushing and truncation effect of the shear layer N2 and P3, the shear layer N1 also sheds from DC. During the half cycle (from 3# to 5# instances) with negative motion displacement, the vortices N2 and P3 shed from the DC in a similar manner. In the downstream far-field of the DC, the vortices with the same direction between the two periods merge. The interference effect of DC results in a noticeable harmonic component of the frequency spectra of $C_{L}$, with the two peaks being almost equal.

Figure 9 illustrates the evolution process of the ‘2C’ vortex shedding pattern. This vortex shedding pattern is observed between the ‘2S’ and the ‘2P’ or ‘aperiodic’ wake regimes (when $L/D\leq 2$) and on the right side of the ‘2S’ pattern (when $L/D>2$). At 1# instance, the positive shear layers P1+F1 below DC are connected, and there are two shear layers, named E1 and N1, above DC and in the gap. At 2# instance, the shear layer N1 attaches to the upper surface of DC and combines with E1 to form the new vortex named N1+E1. At time 3#, vortex P1+F1 shedding from the DC. Compared with shear layer P3 at 3# instance in Fig. 8, the truncation effect of the remaining shear layer F2 is weakened, and the combination of N1+E1 is not destroyed, but shedding together from the DC at 5# instance in the negative half cycle. In the downstream far field of the DC, two vortices with the same direction merge completely into a vortex, the same as the typical Kármán vortex street. The evolution process of the ‘2S’, ‘2C’ and ‘2P’ patterns reveals that with increasing $Re$, the vortex shedding pattern gradually transitions from the ‘2S’ pattern to the ‘2P’ pattern, while the downstream far-field will evolve to the typical Kármán vortex street.

At small $L/D$ and high $Re$, the wake behind the tandem cylinders is characterized by an aperiodic vortex shedding pattern (‘aperiodic’ pattern), represented by a five-pointed star in the sub-region of Fig. 5. The corresponding time-histories of $C_{L}$ and $Y$ of the UC, the $C_{P}$ around the cylinders, and the evolution process of vortex shedding are shown in Fig. 10. It can be noted that the shedding pattern around the UC and the displacement of the UC are mostly periodic. However, the interference of the gap flow, and in turn, the wake downstream becomes aperiodic. At 1# instance, vortices N1 and P1 are formed behind the DC. At 2# instance, as the UC reaches its maximum positive displacement, the negative shear layer N2 is attached to the upper surface of DC. The positive shear layer below UC is truncated by DC, forming parts P2 and P3. Simultaneously, vortices N1 and P1 shed from the DC. At 3# instance, shear layer P3 cuts vortex N2 into two parts, forming the shear layers N3 and N4 at the 4# instance. It can be seen that at 4# instance, the shear layer P3 has attached to the lower surface of DC, and the vortex P2 begins to separate from DC. At 5# instance, the vortex P3 is divided into two parts by the shear layer N4. A new vortex labelled N3^∗ emerges due to the merge of N2 and N3. At 6# instance, the shear layer N4 attaches to the upper surface of DC, and N3^∗ separates from the DC. Below the DC, the vortices P2, P3, and P4 form a long, narrow vortex strip. The vortex P3 merges with part of P4 to form a new vortex, P3^∗, at the 7# instance. The vortex N4 is divided into two parts by the shear layer P5 at 7# instance and evolves into vortices N4, N5 and N6 at 8# instance. Finally, the vortex N4 and N3^∗ also merge to form N4^∗ at 9# instance, and the same is for vortex P4 and P3^∗. The time-history, frequency spectra, wavelet spectra of $C_{L}$ for the DC, and the phase portraits of $C_{L}$ and $C_{D}$ of the DC at $L/D=1.5$ and $Re=160$ are presented in Figs. 11(a)-11(d), respectively. It can be seen that the time history of $C_{L}$ is irregular with broadband frequency spectra, indicative of the aperiodic nature of the flow field. It is evident that there is no distinct periodic attractor in the phase space. In comparison to the other three sub-regions in Fig. 5, this sub-region exhibits a complex process of wake vortices shedding and merging, lacking regularity in the vortex shedding. The instability of the shear layer separated from the UC can be attributed as the primary agency for triggering this aperiodicity over the DC. Hence, we classify this flow regime as an ‘aperiodic’ wake pattern.

The wake characteristics behind the tandem cylinder system are analyzed in this subsection. When the UC is in the equilibrium position just before the start of the upward movement, the flow field contour is considered to calculate the vortex width at a position of $x/D$ = 10 downstream. For the streamwise distance, the center-to-center distance between the two adjacent co-rotating vortex cores is calculated at $x/D$ = 10 downstream. All the cases are represented with the same contour levels. It is worth noting that, for the aperiodic cases, the average and standard deviation of the wake width and streamwise distance of the 10 consecutive oscillation cycles are calculated; the results are plotted with the corresponding error bars.

Figure 12(a) illustrates the variation of the near-wake vortex width ($L_{w}/D$) for different $U_{r}$ and $Re$ values. The error bars in $L/D$ = 1.25, 1.5, and 1.75 represent the statistical results of the varying $L_{w}/D$ in the ‘aperiodic’ pattern region. Two representative cases are presented in Fig. 12(b). At $L/D\leq 2$, except for the ‘aperiodic’ regime, the wake width gradually increases with the increase of reduced velocity, however, with a decreasing rate of increment. In the ‘aperiodic’ regime, a variety of shedding patterns appear due to complex vortex shedding and merging; there is a different wake width in the wake flow. Comparing the vortex shedding pattern classification in Fig.5, the $L_{w}/D$ value increases rapidly in the ‘2S’ and ‘2C’ regimes, while only small changes are observed in the ‘2P’ region. For $L/D$ values between 2.5 to 3 and $Re<110$, the vortex shedding exhibits a ‘2S’ pattern with a double-layered vortex street, and the varies of $L_{w}/D$ is small. Subsequently, as the vortex shedding transformed into ‘2S’ pattern with the single-layer vortex street, the $L_{w}/D$ gradually increases.

Figure 13(a) presents streamwise distance ($L_{s}/D$) for the two adjacent co-rotating near-wake vortices as a function of $U_{r}$ and $Re$. Two representative cases are presented in Fig. 13(b). Different $L_{s}/D$ values are observed in the consecutive vortex shedding periods in the ‘aperiodic’ wake regime, which can be attributed to the complex vortex shedding behind the DC. When $L/D$ ranges from 2.5 to 3, $L_{s}/D$ increases with increasing $U_{r}$, indicating a decrease in the vortex shedding frequency. Based on the vortex shedding pattern classification in Fig. 5, the $L_{s}/D$ gradually increases as the vortex shedding pattern evolves from ‘2S’ to ‘2C’ and ‘2P’. When located in the two-layered vortex street, the change of $L_{s}/D$ is small.

Figure 14 shows the variation of the dimensionless amplitude ($A/D$) and the frequency ratio ($f_{osc}/f_{n}$) of the flow-induced vibration response of the UC as a function of $U_{r}$, in comparison to the results of a single cylinder and large spacing ratio $L/D=15$. Here, the amplitude $A$ is defined as $0.5(y_{max}-y_{min})$, where $y_{max}$ and $y_{min}$ are the maximum and minimum response amplitude, respectively. Figure 15 presents the dynamical classification of the vibration response, in which I represent the steady flow regime, where the displacement of the UC is close to 0; II is the alternating attachment regime. Two vibration types of the UC are distinguished by colour blocks, extended VIV and interference galloping (IG). As can be seen in Fig. 14(a), the dimensionless amplitude of the UC increases gradually with increasing $U_{r}$ and has a similar trend when $L/D$ varies between 1.25 and 1.75. In addition, the amplitude is the largest when the $L/D=1.5$. When $L/D\geq 2.5$ and $U_{r}\geq 5.4$, the dimensionless amplitude shows a decreasing trend with increasing $L/D$; the amplitude does not change significantly for $L/D=2$. It can be seen from Fig. 14(b) that the dimensionless frequency of the UC gradually increases with increasing $U_{r}$ after the UC starts vibrating. Besides, the dimensionless frequency also increases with increasing $L/D$.

When $L/D$ is 1.25, 1.5, and 1.75, the dimensionless frequencies are locked-in at 0.62, 0.77, and 0.87, respectively, and the vibration response of the UC can be characterized as the interference galloping (Dielen and Ruscheweyh, 1995). The interference effect of the DC on the vibration of the UC gradually weakens with the increasing $L/D$. When $L/D$ is greater than a critical threshold, the response of the UC transitions to pure VIV (Chunning et al., 2014). As can be seen from Figs. 14(a) and 14(b), when $L/D=15$, the dimensionless amplitude and the frequency ratio of the UC response are completely consistent with that of a single circular cylinder. In Fig. 14(a), when $L/D$ is between 2.5 and 3, the dimensionless amplitude reaches its maximum at $U_{r}$ = 4.8, and then gradually decreases with the increasing $U_{r}$. It is worth noting that, when $U_{r}\geq 7.8$, the dimensionless displacement of the UC is larger than that of a single cylinder. The dimensionless frequency of the UC increases from 0.68 to 1.37 when $L/D$ is between 2.5 and 3. When $L/D$ = 2, the maximum frequency ratio is 1.02 at $U_{r}$ = 10.2, which is smaller than that observed for the $L/D$ between 2.5 to 3. This indicates that due to the interference effect of the DC, there is a wider lock-in region when $L/D$= 2. According to Chen et al. (2022), when $L/D$ falls between 2 and 3, it can be classified as extended VIV. The range of $L/D$ also aligns with Bokaian and Geoola (1984).

According to the classification results of the wake regime presented in Fig. 15, there is no vortex shedding behind the cylinder in the steady flow regime, and the displacement of the UC is close to zero in this regime. As $U_{r}$ increases, the wake changes from the steady flow to the alternate attachment regime, and the UC begins to vibrate with a large amplitude. Comparing with the classification in Fig. 5, the wake shifts from steady flow to the ‘2S’, and then evolves into ‘2C’, ‘2P’, and ‘aperiodic’ pattern with increasing $U_{r}$ at $L/D$ = 1.5, is different from the evolution process from ‘2C’ to ‘aperiodic’ pattern at $L/D$ = 1.25, resulting a small starting velocity and larger vibration amplitude of the former. At $L/D$ = 1.75, the larger transition range of the ‘2P’ pattern and smaller distribution of the ‘aperiodic’ pattern results in smaller vibration amplitudes compared with $L/D$ = 1.5.

Using the Hilbert transform, the instantaneous phase of the lift coefficient and the dimensionless amplitude is calculated, and then the instantaneous phase difference between the two is obtained. Finally, the root mean square value of the instantaneous phase difference ($\phi_{rms}$) is estimated to evaluate the phase characteristics of the lift coefficient and the dimensionless amplitude; see Fig. 16. When $L/D\leq 1.75$, the $\phi_{rms}$ gradually increases with the increasing $U_{r}$. The $\phi_{rms}$ increases to $51^{\circ}$ when $U_{r}=10.2$ and $L/D=1.75$, and increases to $110^{\circ}$ when $L/D$ increases to 2. Consequently, the increase in the dimensionless displacement decreases with increasing $U_{r}$. At $L/D$ = 2, the $\phi_{rms}$ increases significantly, and the enhancing effect of lift on the cylinder vibration weakens, resulting in a small change in the amplitude of the UC. The $\phi_{rms}$ of the UC are similar to that of the single circular cylinder for $L/D\geq 2.5$. When $U_{r}>7.8$ and $L/D\geq 2.5$, the $\phi_{rms}$ reaches $180^{\circ}$ (the lift coefficient and the dimensionless displacement are in anti-phase condition), resulting in continuous decreases of the response amplitude of the UC with increasing $U_{r}$.