**5.1 Flow around two circular cylinders in tandem configuration**

One of the main applications of this type of study is to obtain a better understanding of the flow around a set of risers, which is subject to shear flows by ocean currents. The flow interference over bluff bodies is responsible for changes in characteristics of the fluid load that acts on immersed bodies. Consequently, the study of cylinders pair even in twodimensional simulations has received considerable attention both from the standpoint of academic and industrial fields. In addition, flow over circular cylinders involve different fundamentals dynamic phenomena, such as boundary layer separation, shear layer development and vortex dynamic (Akbari & Price, 2005).

The configurations with a cylinders pair in tandem and side by side are the most extensively discussed in the literature (Sumner et al., 1999; Deng et al., 2006; Silva et al., 2009), although the form more general is the staggered configuration ( Sumner et al., 2008; Sumner et al., 2005; Silva et al., 2009). According to the literature, there are various interference regimes, which were based on flow visualization in experiments. The wake behavior of two cylinders can be classified into groups according to the pitch ratio between the cylinders centers by diameter (*P/D*) (Sumner et al., 2005).

Here, the two cylinders have equal diameters *d* and the distance center to center of the cylinders, is called *L*. The cylinder *A* is located upstream and cylinder *B* is located downstream of the inlet. In all simulated cases, the two cylinders are disposed such that the minimum distance from the surface of each cylinder to the end of the uniform grid region is 1.25*d* in the x direction and 2*d* in the *y* direction as shown in Fig. (7). The non-uniform grid

An Introduction of Central Difference Scheme Stability for High Reynolds Number 73

Figure (9a) shows the time evolution of the drag coefficient of the upstream (*A*) and downstream (B) cylinders and Fig. (9b) shows the time evolution of the lift coefficients. It is verified that the drag coefficient on the cylinder B is considerably smaller than the cylinder *A*, with mean close to zero. This can be understood by the fact that the cylinder B is inside of the upstream cylinder wake. The fluctuations of the lift coefficient of the two cylinders have zero mean, as shown in Fig. (9b). The amplitude obtained for the cylinder B is approximately seven times greater than the amplitude of cylinder *A*. The absence of vortices behind the upstream cylinder minimizes the lift fluctuations. Note also that the both fluctuations are in phase, Fig. (9b). This is consistent, once the vortices that are formed and

(a) (b)

The flow dynamics around a rotating cylinder is different from that observed for a stationary cylinder. The rotation of a cylinder in a uniform viscous flow modifies the vortices configuration and probably has an effect on flow-induced oscillations. As the cylinder rotates the flow is accelerated in one side and decelerated in the other side. This can be attributed to viscous effects injected by the cylinder on the flow. Therefore, the pressure on the accelerated side becomes smaller than the pressure at the decelerated side resulting on a lift force, transverse to the flow. In recent years more attention has been given to control the wake formed behind the cylinder, especially in order to suppress the vortices with the use of active or passive controls. The rotating motion of an immersed body can suppress partially or totally the vortex shedding process, so that the wake separation on one

Aiming to compare the present results with the literature, simulations were carried out at low Reynolds numbers, which are 60, 100 and 200. For this simulations, the grid is composed by 200x125 points, refined over the cylinder (twenty grids per diameter) to ensure good accuracy in the results. The rotating moviment is imposed clockwise around its axis

Fig. 9. Time evolution of the dynamic coefficients for Re=39,500 and *L/D=2*: a) drag

side of the body, be displaced from the axis of vertical symmetry.

transported induce forces on both cylinders simultaneously.

coefficients and b) lift coefficients.

**5.2.1 Comparison of results** 

**5.2 Flow around a rotating cylinder** 

region is composed by 600x300 points, the Reynolds number equal to 39,500 and the pitch ratio equal to *L/D=2*.

Fig. 7. Illustrative scheme of the distance from the cylinder surface to the uniform region boundaries.

#### **5.1.1 Instantaneous vorticity fields**

Figure 8 shows the flow visualization through the instantaneous vorticity field after the flow has reached steady state. It is noted that the shear layers originated from the surface of the upstream cylinder surrounding the downstream cylinder, forming a single wake behind the cylinder *B*. It is also noted, that the vortex wake oscillates around the symmetry line of the domain. The interaction between the two layers occurs only behind the downstream cylinder, which is within the wake of the upstream cylinder. For this case, the '2S' vortex shedding mode compose the wake. It is important to appreciate that for this pitch ratio and geometrical configuration, the two cylinders behave as a single body.

According to Naudascher & Rockwell (1994) no detectable vortex shedding behind the upstream cylinder occur, for *L/D<3.8*. Also according to these authors, as the spacing between the cylinders increases, vortex shedding occur in the upstream cylinder with a frequency that gradually approaches to the frequency for a stationary cylinder. Deng et al. (2006), in they work at low Reynolds number (Re=220), concluded that for two-dimensional simulations, each cylinder will produce a vortex wake only for *L/D* 4.0, with no vortex shedding between the cylinders for *L/D* 3.5. They also affirmed that even in threedimensional flows, for this configuration and *L/D* 3.5, the flow is equal to the twodimensional.

Fig. 8. Instantaneous vorticity field for *L/D=2* and Re=39,500.

region is composed by 600x300 points, the Reynolds number equal to 39,500 and the pitch

cylinder B

2*d*

30*d*

Fig. 7. Illustrative scheme of the distance from the cylinder surface to the uniform region

50*d*

Figure 8 shows the flow visualization through the instantaneous vorticity field after the flow has reached steady state. It is noted that the shear layers originated from the surface of the upstream cylinder surrounding the downstream cylinder, forming a single wake behind the cylinder *B*. It is also noted, that the vortex wake oscillates around the symmetry line of the domain. The interaction between the two layers occurs only behind the downstream cylinder, which is within the wake of the upstream cylinder. For this case, the '2S' vortex shedding mode compose the wake. It is important to appreciate that for this pitch ratio and

According to Naudascher & Rockwell (1994) no detectable vortex shedding behind the upstream cylinder occur, for *L/D<3.8*. Also according to these authors, as the spacing between the cylinders increases, vortex shedding occur in the upstream cylinder with a frequency that gradually approaches to the frequency for a stationary cylinder. Deng et al. (2006), in they work at low Reynolds number (Re=220), concluded that for two-dimensional simulations, each cylinder will produce a vortex wake only for *L/D* 4.0, with no vortex shedding between the cylinders for *L/D* 3.5. They also affirmed that even in threedimensional flows, for this configuration and *L/D* 3.5, the flow is equal to the two-

geometrical configuration, the two cylinders behave as a single body.

Fig. 8. Instantaneous vorticity field for *L/D=2* and Re=39,500.

ratio equal to *L/D=2*.

boundaries.

dimensional.

**5.1.1 Instantaneous vorticity fields** 

1.25*d*

cylinder A

Figure (9a) shows the time evolution of the drag coefficient of the upstream (*A*) and downstream (B) cylinders and Fig. (9b) shows the time evolution of the lift coefficients. It is verified that the drag coefficient on the cylinder B is considerably smaller than the cylinder *A*, with mean close to zero. This can be understood by the fact that the cylinder B is inside of the upstream cylinder wake. The fluctuations of the lift coefficient of the two cylinders have zero mean, as shown in Fig. (9b). The amplitude obtained for the cylinder B is approximately seven times greater than the amplitude of cylinder *A*. The absence of vortices behind the upstream cylinder minimizes the lift fluctuations. Note also that the both fluctuations are in phase, Fig. (9b). This is consistent, once the vortices that are formed and transported induce forces on both cylinders simultaneously.

Fig. 9. Time evolution of the dynamic coefficients for Re=39,500 and *L/D=2*: a) drag coefficients and b) lift coefficients.

#### **5.2 Flow around a rotating cylinder**

The flow dynamics around a rotating cylinder is different from that observed for a stationary cylinder. The rotation of a cylinder in a uniform viscous flow modifies the vortices configuration and probably has an effect on flow-induced oscillations. As the cylinder rotates the flow is accelerated in one side and decelerated in the other side. This can be attributed to viscous effects injected by the cylinder on the flow. Therefore, the pressure on the accelerated side becomes smaller than the pressure at the decelerated side resulting on a lift force, transverse to the flow. In recent years more attention has been given to control the wake formed behind the cylinder, especially in order to suppress the vortices with the use of active or passive controls. The rotating motion of an immersed body can suppress partially or totally the vortex shedding process, so that the wake separation on one side of the body, be displaced from the axis of vertical symmetry.

#### **5.2.1 Comparison of results**

Aiming to compare the present results with the literature, simulations were carried out at low Reynolds numbers, which are 60, 100 and 200. For this simulations, the grid is composed by 200x125 points, refined over the cylinder (twenty grids per diameter) to ensure good accuracy in the results. The rotating moviment is imposed clockwise around its axis

An Introduction of Central Difference Scheme Stability for High Reynolds Number 75

the oscillation or imposed frequency and *t* is the physical time. The simulations were performed for Reynolds number equal to 1,000, the non-uniform grid is composed by 400x125 points and the turbulence model and damping function in the outlet of the domain

In Fig. (11) the flow visualizations are presented, through the instantaneous vorticity fields for the dimensionless time equal to 200, at different amplitude values and frequency ratios. Figure (11a) corresponds to the stationary cylinder. Figures (11b) and (11c) correspond to *A=1*, for / 1.05 *c o f f* and / 2.5 *c o f f* respectively. Figures (11d) and (11e) correspond to *A=2*, for / 0.5 *c o f f* and / 2.5 *c o f f* respectively. Figures (11f), (11g) and (11h) correspond to *A=3*, for / 0.5 *c o f f* , / 2.5 *c o f f* and / 6.0 *c o f f* respectively. It is observed that there are different vortex shedding modes, when the same amplitude and different frequency

In Fig. (11a), corresponding to the stationary cylinder, as already mentioned, there is the classical Von Kárman Street, represented by the classical '2S' vortex shedding mode. This mode indicates the generation of a positive vortex in one side of the cylinder and a negative vortex on the other side, at each oscillation cycle, forming a single vortex wake with displaced vortices around the symmetry line of the flow. In Fig. (11b), / 1.05 *c o f f* , the vortex wake is similar to pattern wake ('2S' mode), however, the vortices are presented more rounded and with smaller longitudinal and transversal spacing between them when compared with Fig. (11a). Increasing the frequency ratio to / 2.5 *c o f f* and keeping the amplitude *A* 1 , Fig. (11c), there is a new vortex shedding mode called 'P+S'. This mode corresponds to a pair of vortices and single vortex composing the wake. Pairs of vortices having opposite signs are located at the inferior side of the central line of the flow, while the

For / 0.5 *c o f f* and *A=2* it is also observed a new vortex shedding mode called '2P ', which corresponds to pairs of vortices of opposite signs along the wake. Keeping the same oscillation amplitude and increasing the frequency ratio to / 2.5 *c o f f* , Fig. (11e), it is noted the same vortex shedding mode of the previous case, Fig (11d). Interesting to note, in this case, that the pairs are disposed symmetrically about the centerline of the flow forming

Increasing the amplitude to *A=3*, and taking / 0.5 *c o f f* again, a new vortex shedding mode is obtained, called '2C ', as quoted in Williamson & Jauvtis (2004). It is noteworthy that the '2C' mode is not taken by other authors for the case of circular cylinder in rotationallyoscillating moviment. According to Williamson & Jauvtis (2004) this mode was obtained for pivoted cylinders. For / 2.5 *c o f f* , Fig. (11g), there is a new standard of vortex emission, in which the double vortex wake near the cylinder, composed by vortices of the same sign in each row, after a given distance away from the cylinder are coupled to form a single wake. The double wake length decreases with increasing the frequency ratio. In Fig. (11h), corresponding to / 6.0 *c o f f* the instability caused by the cylinder oscillation is limited to a region near the cylinder, while far from the immersed body, the vortices reorient themselves to form the stable Von Kármán Street. Occurs, therefore, a vortex-vortex interaction of the

single vortices are released at the superior side of the cylinder.

is the angular velocity, *A* is the oscillation amplitude, *R* is the cylinder radius, *fc* is

where 

were applied.

ratios are considered.

a cone-shaped wake.

**5.3.1 Different vortex shedding modes** 

and is achieved by the imposition of the velocity components in each Lagrangean point. Figure 10 shows the amplitude of the drag and lift coefficients in function of the specific rotation (the ratio of the tangential velocity and free-stream velocity) compared with the numerical results of Kang et al. (1999), for Re=60 and Re=100.

Fig. 10. Fluctuations amplitude of the dynamic coefficients: a) drag and b) lift. Full symbols: present work and empty symbols: Kang et al. (1999).

Note that the drag coefficient amplitude, Fig. (10a) increases until given and then decreases, reaching a near-zero amplitude. Note also that the amplitudes increase with the Reynolds number and the rotation in which the amplitude decreases is different for each Reynolds number. For Re=60, the amplitude of the drag is reduced for >1.0 and for Re=100 and Re=200, the reduction occur for >1.5. On the other hand, the amplitude values of the lift coefficient, Fig. (10b), shows small variations for 1.0, for all Reynolds numbers and then decreases, tending to zero. As observed, there was good agreement between the present results with those of Kang et al. (1999).

#### **5.3 Flow over a rotationally-oscillating circular cylinder**

For the stationary cylinder at low Reynolds numbers, it is known that the vortex wake is aligned and symmetrical about the central axis of the flow. The behavior is not verified when the cylinder is subjected to rotationally-oscillating moviment around its own axis. The mutual interaction between cylinder moviment and the adjacent fluid modifies the pattern wake of the flow through the acceleration and deceleration of the flow around the cylinder. Thus, there is a transition between different vortex shedding modes as the relationship between oscillation frequency and the vortex shedding frequency for the stationary cylinder varies for the same amplitude *A* . Commonly, some authors present two different flow regimes, being the no lock-in regime and the lock-in regime (Cheng et al. 2001a, 2001b). According to Löhner & Tuszynski (1998), the flow around a rotationally-oscillating cylinder is a forced oscillator form, or a nonlinear system, that in some cases, can become chaotic. Here, the rotationally-oscillating cylinder is started impulsively from rest and the tangential velocity on the cylinder is given by the expression:

$$V\_{\circ \circ} = oR = A \sin(2\pi f\_c t)R \, , \tag{12}$$

where is the angular velocity, *A* is the oscillation amplitude, *R* is the cylinder radius, *fc* is the oscillation or imposed frequency and *t* is the physical time. The simulations were performed for Reynolds number equal to 1,000, the non-uniform grid is composed by 400x125 points and the turbulence model and damping function in the outlet of the domain were applied.
