Numerical investigations of two vibrating cylinders in uniform flow using overset mesh

2.1 Governing equations

For the two-dimensional laminar incompressible flow, the governing equations can be expressed by a set of continuity and momentum equations as follows:

where the velocity components along x and y directions are u and v, respectively; p is the pressure; ρ is the fluid density, µ is the dynamic viscosity; Re represents the Reynolds number Re = ρ UD μ , where D is the cylinder diameter and U is the free-stream velocity. In the present study, the numerical parameters are set to adopt fixed Reynolds number Re = 200 for all computations. The cylinders are excited to oscillate transversely to the incoming flow according to a simple harmonic motion as follows:

where Y(t) is the cylinder oscillation in a cross-flow direction, A is the vibration amplitude of the cylinders, f e is the excitation frequency derived from the frequency ratio f e / f s , where f s is the Strouhal shedding frequency of the single stationary cylinder.

A finite volume-based commercial software ANSYS FLUENT 19.0 is used to solve the governing equations for the flow field. The incompressible Navier–Stokes equations are discretized using the finite volume method , where the integral form of the conservation equations are solved numerically for control volumes which form a partition of the computational domain (for more details, see ref. [23]). The transverse vibration of the two cylinders is implemented with the user-defined function tool integrated with the flow solver. A COUPLED algorithm is implemented for the incompressible flow to solve the pressure–velocity coupled equations. The first-order implicit scheme is implemented for the unsteady terms in the momentum equation. The second-order upwind scheme is implemented for convective terms. Time step sizes are restricted to Δt = 0.0025 s in all numerical computations.

2.2 Computational domain and overset mesh

The schematic diagram of the computational domain adopted in the present study is illustrated in Figure 1. Two transversely vibrating cylinders of identical diameter are placed in a rectangular domain 40D × 20D, where D is the diameter of the cylinders. The upstream cylinder is placed such that the center is 10D away from the inlet boundary, while the downstream cylinder is located such that the spacing distance between the centers of cylinders is L/D = 1.5 and 4, respectively. The cylinders are applied to no-slip boundary conditions, while the lower and upper boundary conditions are of slip-wall type. The boundary of inflow is set to have free stream velocity. The outflow boundary is set to have pressure at a reference value of zero.

Figure 1

The computational domain.

The dynamic overset mesh technique in ANSYS FLUENT 19.0 is used to perform numerical simulation on the flow around the two vibrating circular cylinders. This technique allows to build multiple separate meshes, one for each region of the problem and combines them into one complete domain [24,25]. In the current study, three individual meshes are created: the rectangular grid consists of 27,787 cells, and another two cylindrical grids are the moving component meshes consisting of 7,214 cells for each. The rectangular grid represents the static background mesh, while the moving component meshes, with outer diameter 2D and inner diameter D, are assigned to attach to the two vibrating cylinders. Hence, the two cylinders oscillate without any mesh deformation along the fluid domain. The overset static background mesh and the two moving component meshes are illustrated in Figure 2.

Figure 2

Overset mesh for the flow past two tandem cylinders. (a) Static background mesh, (b) two moving component meshes, and (c) component meshes are embedded in the background mesh.

4.1 Single vibrating cylinder

A uniform flow around a single vibrating circular cylinder is simulated at Re = 200 to establish a baseline for comparison purposes. A numerical simulation is carried out to investigate the influence of different vibration frequencies with constant amplitude A/D = 0.25 on the lift and drag forces acting on a single cylinder. The study focuses on determining the lock-in state, in which the vortex shedding frequency is dominated by the forced excitation frequency f _e completely rather than controlled by the Strouhal shedding frequency of a corresponding fixed cylinder f _s. Computations are performed on a single cylinder forced to vibrate at varying frequency ratios f _e/f _s = 0.8, 1.0, and 1.2.

Figure 9a–d shows that the lock-in state occurs for two cases of f _e/f _s = 0.8 and 1.0. It can be observed that the lift coefficients’ time history is absolutely periodic and purely sinusoidal. The power spectrum diagram of the lift coefficients clearly depicts that the dominant frequency is the excitation frequency f _e (lock-in state), where only one dominant peak at f/f _e = 1.0 appears, indicating that the excitation frequency f _e controls the lift forces. Figure 9e and f shows the no lock-in state for the case of f _e/f _s = 1.2. As can be seen, the time history of lift coefficients is no longer purely sinusoidal but exhibits a beating response. The power spectrum of the lift coefficients depicts two peaks: the primary peak at f _e and the secondary peak at f _m (modulation frequency). In general, the present results are consistent with the observations of Meneghini and Bearman [4], confirming the present technique’s efficiency and accuracy for modeling the flow around a single vibrating cylinder at Re = 200.

Figure 9

Lift coefficients’ time history and power spectrum diagram of the single vibrating cylinder at Re = 200. (a) and (b) for f _e/f _s = 0.8, (c) and (d) for f _e/f _s = 1.0, and (e) and (f) for f _e/f _s = 1.2.

The vorticity fields over the vibrating single cylinder are depicted in Figure 10. For the lock-in state, vortices in the wake at f _e/f _s = 0.8 (Figure 10a) are arranged regularly and display 2S mode where two single vortices of opposite circulation are shed per cycle. In contrast, as the frequency ratio increases to f _e/f _s = 1.0, the near wake displays 2S mode, while the positive and negative vortices in the far wake form two parallel row 2S* mode, as shown in Figure 10b. For no lock-in state at f _e/f _s = 1.2 (Figure 10c), the vortices in the wake are 2S mode with irregular arrangement due to the multiple frequency contents in the lift force time history.

Figure 10

Instantaneous vorticity fields over the single vibrating cylinder. (a) f _e/f _s = 0.8, (b) f _e/f _s = 1.0, and (c) f _e/f _s = 1.2.

4.2 Two vibrating cylinders

Computations are performed on the flow around two vibrating circular cylinders in a tandem arrangement at Re = 200. Both cylinders are forced to vibrate transversely in phase at frequency ratios f _e/f _s = 0.8, 1.0, and 1.2, while their vibration amplitude is kept at A/D = 0.25. As illustrated previously, two different spacing between the cylinders are considered in this study (L/D = 1.5 and 4).

For the case of small spacing L/D = 1.5, the lock-in state occurs at all frequency ratios f _e/f _s = 0.8, 1.0, and 1.2. As can be noticed from Figure 11, the time histories of lift coefficients are purely sinusoidal at different frequency ratios for the downstream cylinder. Only one dominant peak can be observed in the power spectrum corresponding to the cylinder excitation frequency f _e (dominant peak at f/f _e = 1.0). It can be concluded that the lock-in state for the case of spacing L/D = 1.5 occurs within a broader frequency range compared to the case of the single vibrating cylinder.

Figure 11

Lift coefficients’ time history and power spectrum diagram of the downstream cylinder, L/D = 1.5 and Re = 200. (a) and (b) for f _e/f _s = 0.8, (c) and (d) for f _e/f _s = 1.0, and (e) and (f) for f _e/f _s = 1.2.

The vorticity fields over the two cylinders for the case L/D = 1.5 are shown in Figure 12. A clear 2 S mode vortex shedding around the downstream cylinder is observed at f _e/f _s = 0.8, as shown in Figure 12a. However, at f _e/f _s = 1.0 and 1.2, the near-downstream cylinder wake exhibits 2S* mode while the far wake turns into 2S mode, as depicted in Figure 12b and c.

Figure 12

Instantaneous vorticity fields over the two vibrating cylinders, L/D = 1.5. (a) f _e/f _s = 0.8, (b) f _e/f _s = 1.0, and (c) f _e/f _s = 1.2.

On the other hand, for the case of larger spacing L/D = 4, the lock-in state of two vibrating cylinders is only observed at frequency ratio f _e/f _s = 0.8. Figure 13a and b shows that at f _e/f _s = 0.8, the lift coefficients’ time history for the downstream cylinder is purely sinusoidal, and only the excitation frequency is dominated in the power spectrum diagram. However, no lock-in is observed at frequency ratios f _e/f _s = 1.0 and 1.2, as the beating behavior appears in the time histories of lift coefficients for the downstream cylinder (Figure 13c and e). Two peaks (primary peak at f _e and secondary peak at f _m) are observed in the frequency spectrum diagram for each cylinder (Figure 13d and f). It can be concluded that the lock-in state for spacing L/D = 4 occurs in a narrower frequency range than the single vibrating cylinder.

Figure 13

Lift coefficients’ time history and power spectrum diagram of the downstream cylinder, L/D = 4 and Re = 200. (a) and (b) for f _e/f _s = 0.8, (c) and (d) for f _e/f _s = 1.0, and (e) and (f) for f _e/f _s = 1.2.

The vorticity fields for the case L/D = 4 are shown in Figure 14. For the lock-in state at f _e/f _s = 0.8 (Figure 14a), a pure 2S* mode vortex shedding over the downstream cylinder is seen. For no lock-in state at f _e/f _s = 1.0 (Figure 14b), the vortex shedding over the downstream cylinder evolves from the 2S mode in the near wake to the 2S* mode in the far wake. While for no lock-in state at f _e/f _s = 1.2 (Figure 14c), the vorticity fields display a clear 2S mode vortex shedding around the downstream cylinder.

Figure 14

Instantaneous vorticity fields over the two vibrating cylinders, L/D = 4. (a) f _e/f _s = 0.8, (b) f _e/f _s = 1.0, and (c) f _e/f _s = 1.2.