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Two dimensional numerical simulations of flow around a rotationally oscillating circular cylinder were performed at Re = 1000. A wide range of forcing frequencies,
*f*
_{r}, and three values of oscillation amplitudes, A, are considered. Different vortex shedding modes are observed for a fixed A at several values of
*f*
_{r}, as well as for a fixed
*f*
_{r} at different values of A. The 2C mode of vortex shedding was obtained in the present study. It is important to point out that this mode has not been observed by other investigators for rotationally oscillating case. Also, it is verified that this mechanism has great influence on the drag coefficient for high frequency values. Furthermore, the lift and pressure coefficients and the power spectra density are also analyzed.

It is widely known that the vibrations induced by vortex shedding process may have negative effects in engineering systems, such as economic loss, damage of installations, and very frequently with environment-related consequences. Thus, although the fluid flow around circular cylinder is a classical problem in fluid dynamic due to its simple geometry, this is a reason for which in the last decades, a great deal of effort has been devoted to the development of numerical and computational procedures for dealing with the problem of vortex shedding phenomena. Comprehensive studies on this subject have been reported in books [

In applications in which a circular cylinder under rotational oscillations is involved, the flow dynamic is different from those observed for stationary cylinders and has fascinated researchers for a long time [

Among some studies over rotationally oscillating cylinders, the so-named Hybrid Vortex Method and the Discrete Vortex Method have been proposed [^{4}. Ray and Christofides [

The present study focuses on two-dimensional, incompressible viscous flow over a rotationally oscillating circular cylinder by using the Immersed Boundary Methodology (IBM) [

One approach, which is not very common to solve the Navier-Stokes equations, is the so called velocity-vorticity formulation [

where ^{3}] and v [m^{2}/s] are the specific mass and the kinematic viscosity, respectively; ^{2}] are, respectively, the i-th velocity component and the pressure; ^{3}] is the i-th component of the Eulerian force calculated as follows:

In Equation (3) ^{−2}] is the distribution function [

At this point, the mixed Eulerian-Lagrangian formulation is retained, in which the Eulerian fixed grid describes the flow and the Lagrangian grid (which can be fixed or not) describes the immersed body. These meshes are geometrically independent from each other, and this fact enables to study the flows around simple, complex, movable and deformable geometries, without any remeshing process. These two formulations are coupled by a force field obtained at the Lagrangian points and then distributed over the Eulerian nodes in the body neighborhood. By this strategy, one can use a simple Cartesian grid and it is not necessary to move the grids. According to the VPM [

particle placed on the Lagrangian points.

By considering a particle of fluid placed on the fluid-solid interface as illustrated in

where

A number of mesh-free methods have been developed in recent years [

where

and the velocity field is updated by solving the algebraic equation

In this section, numerical simulations are performed to investigate the effects of the oscillating amplitude and forcing frequency on the flow structure of a circular cylinder.

left to the right side of the domain and at the inlet, a uniform velocity profile

Rotational oscillations at a prescribed set of frequency ratios and amplitudes are then imposed on the cylinder, where the tangential velocity over each Lagrangian point k, as shown in

The simulations are performed at Reynolds number 1000, with a time step chosen arbitrarily of

the cylinder and decreases away from it. When

The patterns of vortex shedding from the cylinder in the near and far wakes are shown in

formation process is completely different from that of the previous cases, in which a new mode of vortex shedding, named as 2 C mode, appears. This mode indicates that two vortex pairs of the same signal are shed per cycle. It is important to mention that this mode was not observed by the authors cited in the references for the case of circular cylinders subjected to rotational oscillations. However, it is noted in the case of a pivoted cylinder as reported in Williamson and Jauvtis [

In the frequency range

It is known that the vortex shedding process causes fluctuations in the dynamic coefficients and affects the behavior of the flow structure.

When

shown in

When

of the peak, as shown in

For

Another relevant aspect to be investigated is the energy level of the power spectra, as shown in

It is desirable to compare the present results with some numerical results from previous studies reported by other investigators.

Another aspect to be pointed out is that the maximum mean drag coefficients were obtained for the 2S mode for all the analyzed amplitudes, in which the longitudinal and transversal spacing for the oscillating cylinder is greater than those of the stationary cylinder. This fact can be observed by analyzing the vorticity contours corresponding to the maximum

The mechanism of increase and reduction of

The pressure coefficient is obtained by the relation_{r} = 0.6) to _{r} = 1.5).

By comparing _{r} = 0.6) to _{r} = 1.5). Thus, the forcing frequency plays an important role on the pressure distribution, enabling to verify that for low frequency ratios, the flow downstream of the cylinder is very dissipative due to viscous effects. This effect contributes to reduce the pressure and consequently to increase the drag over the cylinder. However, at high frequency ratios, the wake dynamic downstream of the cylinder is not so strong when compared to the corresponding ones obtained for low frequencies, which contribute to increase the pressure and, consequently, to reduce the drag coefficient.

The numerical simulations of the flow over the rotationally-oscillating circular cylinder by using the Immersed

Boundary Method combined with the Virtual Physical Model have been addressed. Given the strong influence of the oscillation amplitude and frequency on the flow around the cylinder, the influence of these parameters on the lift and drag coefficients, on the pressure distribution, as well as on the vortex shedding frequency has been investigated.

The simulations shown that the flow structure in the near wake was strongly dependent on the oscillation frequency. Also, it observed different vortex shedding modes (“2C”, “2P”, “2S”, and “P + S”) for a fixed oscillation amplitude at different values of frequency ratios, and also for a fixed frequency ratio at different values of oscillation amplitudes, in addition to the conical wakes. Another important feature of the numerical methodology is its capability to identify the 2 C mode of the vortex shedding, which has not been observed by other investigators for the case of rotationally-oscillating cylinders.

The study reported herein enabled to observe a number of important features that should be mentioned:

・ The range of resonance increases as the oscillation amplitude increases;

・ For high frequency ratios, the wake structure configuration is similar to the classical Von Kármán Street, and the values of the vortex shedding frequency take the values corresponding to the stationary cylinder case;

・ The pressure distribution over the cylinder is influenced by forcing frequency, and consequently affects the drag over the cylinder. It implies that drag control can be done by the rotational oscillations mechanism;

・ The Immersed Boundary Method combined with the Virtual Physical Model can be easily employed in the case of moving bodies, being a very useful tool to simulate problems involving prescribed motion.

The author is grateful to the following organizations: Agency of the Ministry of Science, Technology and Innovation―CNPq for the continued support to their research work, and the Minas Gerais State Agency FAPEMIG.

Alice Rosa daSilva,Aristeu daSilveira-Neto,Antônio Marcos Gonçalves deLima, (2015) Rotational Oscillation Effect on Flow Characteristics of a Circular Cylinder at Low Reynolds Number. World Journal of Mechanics,05,195-209. doi: 10.4236/wjm.2015.510019