**7. Acknowledgments**

78 Computational Simulations and Applications

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

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

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

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

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

corresponding to the harmonics are not considered here.

peak corresponding to *St*1 has low energy level.

frequencies match up again.

problems of interst both academic and industrial.

**6. Conclusion** 

level in comparison with Fig. (12d), inside the lock-in regime.

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 Uberlândia, Brazil.

**0**

**4**

**A Fourth-Order Compact Finite Difference**

Convection-diffusion equations are widely used for modeling and simulations of various complex phenomena in science and engineering (Hundsdorfer & Verwer, 2003; Morton, 1996). Since for most application problems it is impossible to solve convection-diffusion equations analytically, efficient numerical algorithms are becoming increasingly important to numerical

Recently a great deal of efforts have been devoted to developing high-order compact schemes, which utilize only the grid nodes directly adjacent to the central node. In (Noye & Tan, 1989), Noye and Tan derived a class of high-order implicit schemes for solving the one-dimensional unsteady convection-diffusion equations. This method is very stable and accurate (third-order in space and second-order in time). In (Gupta et al., 1984), a fourth-order finite difference scheme for a steady convection-diffusion equation with variable coefficients was proposed. The scheme is defined on a single square cell of size 2Δ*x* over a nine-point stencil. In (Rigal, 1994), Rigal provided an extensive analysis of the properties of a class of two- and three-level second-order difference schemes which have been proposed in (Rigal, 1989; 1990). In (Spotz & Carey, 2001), the two-dimensional HOC (High Order Compact) scheme proposed in (Gupta et al., 1984) was extended to solve unsteady one-dimensional convection-diffusion equations with variable coefficients and two-dimensional diffusion equations. This method was further extended by Kalita et al. in (Kalita et al., 2002) to a class of HOC schemes with weighted time discretization, and successfully used to solve unsteady two-dimensional convection-diffusion equations. In (Karaa & Zhang, 2004), Karaa and Zhang proposed a novel high-order alternating direction implicit method, based on the technique in (Zhang et al., 2002), for solving unsteady two-dimensional convection-diffusion problems. This new method is second-order in time and fourth-order in space, and is computationally efficient. In (Tian & Dai, 2007), Tian and Dai proposed a class of high-order compact exponential finite difference methods for solving one- and two-dimensional steady-state convection-diffusion problems. This method is nonoscillatory, fourth-order in space, and easy to implement. Some more recent high-order ADI methods for unsteady convection-diffusion equations can be

simulations involving convection-diffusion equations.

found in (Tian & Ge, 2007; You, 2006).

**1. Introduction**

**Scheme for Solving Unsteady**

Wenyuan Liao<sup>1</sup> and Jianping Zhu<sup>2</sup>

<sup>2</sup>*University of Texas at Arlington*

<sup>1</sup>*University of Calgary*

<sup>1</sup>*Canada* <sup>2</sup>*USA*

**Convection-Diffusion Equations**

### **8. References**

