**5.3.1 Different vortex shedding modes**

74 Computational Simulations and Applications

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

*Cl*

Fig. 10. Fluctuations amplitude of the dynamic coefficients: a) drag and b) lift. Full symbols:

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

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

sin(2 ) *V RA tg <sup>c</sup>*

 (r m s)

0.0

0.2

0.4

0.6

0.8

<sup>0</sup> <sup>1</sup> <sup>2</sup> <sup>3</sup> -0.2

*f t R* , (12)

*Re Re Re Re Re*= 200

= 60 = 60 = 100 = 100

numerical results of Kang et al. (1999), for Re=60 and Re=100.

*Re Re Re Re Re* = 200

= 60 = 60 = 100 = 100

present work and empty symbols: Kang et al. (1999).

present results with those of Kang et al. (1999).

velocity on the cylinder is given by the expression:

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

(a) (b)

<sup>0</sup> <sup>1</sup> <sup>2</sup> <sup>3</sup> -0.2

*Cd*

(r m s)


0.0

0.1

0.2

0.3

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 ratios are considered.

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

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 a cone-shaped wake.

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

An Introduction of Central Difference Scheme Stability for High Reynolds Number 77

0.2

0.5

0.5 1.0 1.5 2.0 2.5

0.1 0.2 0.3 0.4 0.5

*Pow er Spect rum*

Fig. 12. Power Spectra for Re=1,000: a) stationary cylinder; b) and c) *A=1* and / 1.05 *c o f f* and / 2.5 *c o f f* respectively; d) and e) *A=2* and / 0.5 *c o f f* and / 2.5 *c o f f* respectively

and f) g) and h) *A=3* and / 0.5 *c o f f* , / 2.5 *c o f f* and / 4.0 *c o f f* respectively.

*Pow er Spect rum*

1.0

*Power Spectrum*

1.5

*Pow er Spect rum*

0.4

0.6

0 0.2 0.4 0.6 0.8 1.0 <sup>0</sup>

0 0.2 0.4 0.6 0.8 1.0 <sup>0</sup>

0 0.5 1.0 1.5 <sup>0</sup>

0 0.5 1.0 1.5 <sup>0</sup>

*St2*

*St*

*St1*

*St*

*lock-in*

*St*

*lock-in*

*St*

*lock-in*

<sup>0</sup> 0.1 0.2 0.3 0.4 0.5 <sup>0</sup>

*St*

*St2*

*St2*

0 0.2 0.4 0.6 0.8 1.0 <sup>0</sup>

*St1*

(c) (d)

*St*

0 0.2 0.4 0.6 0.8 1.0 <sup>0</sup>

*St1*

(e) (f)

(g) (h)

*St*

0 0.5 1.0 1.5 <sup>0</sup>

*lock-in*

*St*

(a) (b)

0.1 0.2 0.3 0.4 0.5

0.2

0.1 0.2 0.3 0.4 0.5

0.1 0.2 0.3 0.4 0.5

*Pow er Spect rum*

*Pow er Spect rum*

*Pow er Spect rum*

0.4

0.6

*Pow er Spect rum*

same sign near the cylinder, resulting in large scale vortices, whose frequencies have values close to the vortex shedding frequency of the stationary cylinder (0.23). The return to the '2S' mode is observed for high *fc o* / *f* in all simulated amplitudes.

Fig. 11. Instantaneous vorticity fields for Re=1,000: a) stationary cylinder; b) and c) *A=1* and / 1.05 *c o f f* and / 2.5 *c o f f* respectively; d) and e) *A=2* and / 0.5 *c o f f* and / 2.5 *c o f f* respectively and f), g) and h) *A=3* and / 0.5 *c o f f* , / 2.5 *c o f f* and / 6.0 *c o f f* respectively.

(g) (h)

#### **5.3.2 Vortex shedding frequency**

Figure (12) shows the power spectra ( *EC* ) obtained by Fast Fourier Transform (FFT) of the lift coefficients signals. The frequency peak more energized are called by *St*1 and *St*2 where *St*1 will be considered equal to the dimensionless frequency corresponding to the cylinder oscillation *St f D U c c* / . When the power spectrum contains only one promiment peak it is

same sign near the cylinder, resulting in large scale vortices, whose frequencies have values close to the vortex shedding frequency of the stationary cylinder (0.23). The return to the '2S'

(a) (b)

(c) (d)

(e) (f)

(g) (h) Fig. 11. Instantaneous vorticity fields for Re=1,000: a) stationary cylinder; b) and c) *A=1* and / 1.05 *c o f f* and / 2.5 *c o f f* respectively; d) and e) *A=2* and / 0.5 *c o f f* and / 2.5 *c o f f*

Figure (12) shows the power spectra ( *EC* ) obtained by Fast Fourier Transform (FFT) of the lift coefficients signals. The frequency peak more energized are called by *St*1 and *St*2 where *St*1 will be considered equal to the dimensionless frequency corresponding to the cylinder oscillation *St f D U c c* / . When the power spectrum contains only one promiment peak it is

respectively and f), g) and h) *A=3* and / 0.5 *c o f f* , / 2.5 *c o f f* and / 6.0 *c o f f*

respectively.

**5.3.2 Vortex shedding frequency** 

mode is observed for high *fc o* / *f* in all simulated amplitudes.

Fig. 12. Power Spectra for Re=1,000: a) stationary cylinder; b) and c) *A=1* and / 1.05 *c o f f* and / 2.5 *c o f f* respectively; d) and e) *A=2* and / 0.5 *c o f f* and / 2.5 *c o f f* respectively and f) g) and h) *A=3* and / 0.5 *c o f f* , / 2.5 *c o f f* and / 4.0 *c o f f* respectively.

An Introduction of Central Difference Scheme Stability for High Reynolds Number 79

Firstly, simulations were performed with stationary cylinder, considering different Reynolds numbers and time discretization schemes. Results such as vorticity, time histories and fluctuations amplitude of dynamic coefficients and the Strouhal number are obtained. The concomitant use of second order temporal schemes with the spatial centered scheme is crucial for the stability of the methodology. The Adams-Bashforth temporal scheme presented more stable than the second order Runge-Kutta scheme. As the Reynolds number is increased the methodology showed to be unstable for all second-order temporal discretization schemes. This result is expected once the centered scheme has no numerical diffusion. Thus, it is concluded that for high Reynolds number, the use of turbulence modeling for the energy transfer process between the largest and smallest scales of turbulence is needed. It is important to appreciate that without the modeling and numerical diffusion the kinetic energy of the physical instabilities accumulates on the cutoff frequency and the simulation diverges. The cutoff frequency is determined by the mesh discretization. It is known that the use of developed flow boundary condition at the outlet of the domain is common in the literature. However when there are physical instabilities, which must leave the domain, there may be problems of numerical stability, especially when using centered spatial schemes. This is due to the fact that the physical instabilities carry spurious information from the outside of the domain to inside. The result is also the divergence of the simulations. To solve this problem the use of a damping function is essential to ensure

Aiming to illustrate the applicability of the Immersed Boundary method used togheter the second order spatial centered scheme and second order temporal discretization scheme, simulations were carried out with a circular cylinder pairs, rotating cylinder and rotationally-oscillating cylinder. For the rotating cylinder case, the results showed good agreement with literature data. It was found that the rotation has greater influence on the amplitude of the drag coefficient than on the amplitude of the lift coefficient. It's worth noting that with increasing rotation the amplitude of the dynamic coefficients tends to null,

For simulations with rotationally oscillating cylinder is analyzed the influence of amplitude and frequency ratio in the vortex shedding modes, as well as in the vortex shedding frequency. It is observed different vortex shedding modes when fixed the oscillation amplitude and varies the frequency ratios. It is important to appreciate the 2C mode obtained in this study once this mode is not found in the literature for rotatinally-oscillating cylinder and it is worth mentioning that, according to Williamson & Jauvtis (2004) the 2C mode is obtained for pivoted cylinder. It is also obtained for the amplitude and frequency ratios considered the lock-in regime, whose range increases as the oscillation amplitude

The authors are deeply gratefully to the following organizations: Minas Gerais State Agency FAPEMIG for the continued support to their research work, especially through the postdoctorate scholarship granted to A.R. da Silva; Brazilian Research Council – CNPq for the nancial support to their research activities; CAPES Foundation from the Brazilian Ministry of Education and the School of Mechanical Engineering College of the Federal University of

stability for higher values of Reynolds number.

increases.

**7. Acknowledgments** 

Uberlândia, Brazil.

as expected, once the vortex shedding process decreases.

called ressonance phenomenon or lock-in, ie, the cylinder is oscillating with a frequency equal to the vortex shedding frequency. It is worth remembering that the energy peaks corresponding to the harmonics are not considered here.

For the stationary cylinder case, Fig. (12a), the power spectrum shows a single energy peak, corresponding to the Strouhal number equal to 0.23. For *A=1* and / 1.05 *c o f f* given in Fig. (12b), only one prominent peak is observed, corresponding to the lock-in regime. It is important to observe that the lower limit of the lock-in regime for this amplitude, starts for the studied cases, in / 0.6 *c o f f* . The ratio / 1.05 *c o f f* correspond to the upper limit of this regime. Due to the large amount of data regarding all amplitudes and frequency ratios studied, only few results are reported here. With the increase of the frequency ratio and keeping the oscillation amplitude, Fig. (12c), there is more than one frequency peak in the spectrum, which indicates that the lock-in regime no longer exists. Interesting to note, for / 2.5 *c o f f* ('P+S' mode, as Fig. (11c)), that for this vortex shedding mode, the frequency peak corresponding to *St*1 has low energy level.

Increasing the amplitude for *A=2* the lock-in regime range is greater, which is given by 0.5 / 1.05 *c o f f* . Figure (12d), / 0.5 *c o f f* , represents the lower limit of the lock-in regime for this amplitude. Note a considerable increase in energy level with the amplitude. For / 2.5 *c o f f* , Fig. (12e), out of lock-in regime, it is noted a great reduction in the energy level in comparison with Fig. (12d), inside the lock-in regime.

Considering *A=3* and / 0.5 *c o f f* in Fig. (12f), there is only one prominent peak, which indicates that this frequency ratio is within the lock-in regime. Comparing Figs. (12d) and (12f), corresponding to the same frequency ratio and different oscillation amplitude, there is an increase in energy level for a greater amplitude. It is noteworthy that the range of lock-in regime, for this amplitude is greater than for *A=2*, being 0.2 / 2.5 *c o f f* , as Fig. (12g) (upper limit of the regime). It is Interesting to note that, within the lock-in regime, the increase of the frequency ratio from / 0.5 *c o f f* to / 2.5 *c o f f* leads to a great reduction in energy level, as shown in Figs. (12f) and (12g). This reduction is associated with different vortex shedding modes, as shown in the Figs. (11f) and (11g). For / 4.0 *c o f f* and *A=3*, Fig. (12h), one observes two frequency peaks, which indicates that, the lock-in regime no longer exists. It is verified for all considered amplitudes that as the frequency ratio is increased, the frequency called *St*2 gradually recovers the frequency corresponding to the stationay cylinder, due to the fact that for high oscillation frequencies, there is no more synchronization between the oscillating cylinder and vortex shedding downstream of it. Thus, the vortices tend to reorient themselves to form a classical von Kárman Street and the frequencies match up again.

#### **6. Conclusion**

One of the goals that motivated the development of this work was to demonstrate through analysis of the important parameters such as dynamic coefficients, obtained through twodimensional simulations of incompressible flows, that the second order centered spatial schemes can perfectly provide accurate results when used toghether the second order time discretization scheme. Another motivation was to continue the development of the Immersed Boundary method with the Virtual Physical model for further application in problems of interst both academic and industrial.

Firstly, simulations were performed with stationary cylinder, considering different Reynolds numbers and time discretization schemes. Results such as vorticity, time histories and fluctuations amplitude of dynamic coefficients and the Strouhal number are obtained. The concomitant use of second order temporal schemes with the spatial centered scheme is crucial for the stability of the methodology. The Adams-Bashforth temporal scheme presented more stable than the second order Runge-Kutta scheme. As the Reynolds number is increased the methodology showed to be unstable for all second-order temporal discretization schemes. This result is expected once the centered scheme has no numerical diffusion. Thus, it is concluded that for high Reynolds number, the use of turbulence modeling for the energy transfer process between the largest and smallest scales of turbulence is needed. It is important to appreciate that without the modeling and numerical diffusion the kinetic energy of the physical instabilities accumulates on the cutoff frequency and the simulation diverges. The cutoff frequency is determined by the mesh discretization.

It is known that the use of developed flow boundary condition at the outlet of the domain is common in the literature. However when there are physical instabilities, which must leave the domain, there may be problems of numerical stability, especially when using centered spatial schemes. This is due to the fact that the physical instabilities carry spurious information from the outside of the domain to inside. The result is also the divergence of the simulations. To solve this problem the use of a damping function is essential to ensure stability for higher values of Reynolds number.

Aiming to illustrate the applicability of the Immersed Boundary method used togheter the second order spatial centered scheme and second order temporal discretization scheme, simulations were carried out with a circular cylinder pairs, rotating cylinder and rotationally-oscillating cylinder. For the rotating cylinder case, the results showed good agreement with literature data. It was found that the rotation has greater influence on the amplitude of the drag coefficient than on the amplitude of the lift coefficient. It's worth noting that with increasing rotation the amplitude of the dynamic coefficients tends to null, as expected, once the vortex shedding process decreases.

For simulations with rotationally oscillating cylinder is analyzed the influence of amplitude and frequency ratio in the vortex shedding modes, as well as in the vortex shedding frequency. It is observed different vortex shedding modes when fixed the oscillation amplitude and varies the frequency ratios. It is important to appreciate the 2C mode obtained in this study once this mode is not found in the literature for rotatinally-oscillating cylinder and it is worth mentioning that, according to Williamson & Jauvtis (2004) the 2C mode is obtained for pivoted cylinder. It is also obtained for the amplitude and frequency ratios considered the lock-in regime, whose range increases as the oscillation amplitude increases.
