Harmonic oscillations of a circular cylinder moving with constant velocity in a quiescent fluid

The flow around an oscillating circular cylinder which moves with constant velocity in a quiescent Newtonian fluid with constant properties is analyzed. The influences of the frequency and amplitude oscillation on the aerodynamic loads and on the Strouhal number are presented. For the numerical simulation, a cloud of discrete Lamb vortices are utilized. For each time step of the simulation, a number of discrete vortices are placed close to the body surface; the intensity of theirs is determined such as to satisfy the no-slip boundary condition.


Introduction
Understanding and being able to analyze the flow around an oscillating body which moves with constant velocity in a quiescent fluid with constant properties is of great fundamental and practical importance in aero and hydrodynamics analysis.Oscillatory motions of small amplitude are important in the analysis of immerse vibrating bodies and special care should be taken in the lock-in condition.Large amplitude motions, on the other hand, are of relevance in the analysis of bodies located in waves and currents such as the ones found in the offshore structures (WILLIAMSON; ROSHKO, 1988).
The oscillatory motion of small amplitude mainly modifies the near field changing the boundary layer flow and, as a consequence, having an important effect on the aerodynamic forces and the pressure distribution.If the amplitude of the oscillatory motion is large one observes, additionally, substantial changes in the far field wake which can be of importance in the presence of other bodies or near by surfaces.This paper deals with the analysis of a body oscillating around a fixed position which is located in an incoming uniform flow with constant velocity; to simplify matters the oscillatory motion is restricted to heave.In previous works, Silva (2004) analyzed the same situation with the restriction of small amplitude of oscillation and Mustto et al.
(1998) presented results for a rotating cylinder.
A simpler approach to the present problem would consider a fixed body located in an oscillating incoming flow; notice, however, that with this approach the whole fluid mass would oscillate with the same frequency and amplitude, which is not quite what, happens in real situations, mainly in the far field region.
Results for a circular cylinder fixed and heaving in a uniform flow are presented and compared with results found in the literature; these are experimental results as well as results obtained using numerical simulations.It is important to mention that although the Reynolds number used in the simulations is high, no attempt to include turbulence modeling (ALCÂNTARA PEREIRA et al., 2002) was made.
In the present simulations the integrated aerodynamic loads (such as lift and drag coeffi- In the body fixed coordinate system, the surface S b is defined by the function Thus, in the inertial frame of reference and, for a symmetrical body F b (x,y,t) = y b -y 0 (t) ± h(x) = 0 (3) .

Governing equations
For an incompressible fluid flow the continuity is written as where u ≡ (u, v) is the velocity vector.
If, in addition, the fluid is Newtonian with constant properties the momentum equation is represented by the Navier-Stokes equation as On the body surface the adherence condition has to be satisfied.This condition is better specified in terms of the normal and tangential components as on S b , the impenetrability condition on S b , the no-slip condition here n and t are unit normal and tangential vectors, u is the fluid particle velocity and v is the body surface velocity.Following Shintani and Akamatsu (1994), this function is then obtained using the following integral formulation where H = 1 in the fluid domain, H = 0.5 on the boundaries and G is a fundamental solution of the Laplace equation, Alcântara Pereira et al. (2002).

Convection and diffusion of vorticity
For the numerical implementation, the vorticity in the fluid domain is simulated by a cloud of Lamb vortices.
For each time step of the simulation, a number of discrete vortices are generated on the body surface; the intensity of these newly generated vortices is determined using the no-slip condition, see Eq. ( 7).
For the convection of the discrete vortices of the cloud, Eq. ( 10) is written in its Lagrangian form as being i = 1, N; N is the number of vortices in the cloud.
A second order solution to this equation is given by the Adams-Bashforth formula (FERZIGER, 1981) The diffusion of vorticity is taken care of using the random walk method (LEWIS, 1991).The Therefore the final displacement is written as 4 Numerical implementation The u (i) and v (i) components of the velocity induced at the location of the (i) vortex can be written as where: ui (i) ≡ [ui (i) , vi (i) ] is the incident flow velocity, uc (i) ≡ [uc (i) , vc (i) ] is the velocity induced by the cylinder at the location of vortex (i), uv (i) ≡ [uv (i) , vv (i) ] is the velocity induced at the vortex (i) by the other vortices of the cloud.
The ui (i) and uv (i) calculations present no problems and they follow the usual Vortex Method procedures; to the first approximation the same happens with the uc (i) when the body oscillation amplitude is small, see Silva (2004).
For large amplitude body oscillations, however, the body boundary conditions can not be -The velocity induced by the body, according to the panel method calculations, is indicated by [uc(X,Y), vc(X,Y)]; this is the velocity induced at the vortex (i), located at the point [x(t), y(t)]; thus uc (i) (x,y;t) = uc(X,Y;t) where the following relations remains The drag and lift coefficients can be expressed by (Ricci, 2002) where ∆S j is the panel length and θ j represents the panel angle.

Results
We start presenting the results for a stand still circular cylinder immersed in a uniform flow.
Table 1 shows the results for a circular cylinder, Re     Of course, the opposite is observed when the cylinder is moving from its lower position to the upper position; Figure 6 shows the near field wake pattern at t = 23, when the cylinder is in its uppermost position.The data from Table 2 also shows a reducing trend in the drag coefficient as the frequency of the cylinder oscillation increase.No solid explanation can be presented about this subject at this moment.

Conclusions
The    The sub-grid turbulence modeling is of significant importance for the numerical simulation.The results of this analysis, taking into account the subgrid turbulence modeling, are also being generated and will be presented in due time, elsewhere.
Finally, despite the differences presented in this preliminary investigation, the results are promising, that encourages performing additional tests in order to explore the phenomena in more details.
cient), the pressure distribution and the Strouhal number agree quite well with the experimental results when the cylinder is kept without oscillation.Due to the alternate vortex shedding the lift coefficient oscillates, around zero, during the numerical simulation; the amplitude of the lift coefficient oscillation is increased with the cylinder oscillation keeping, however, the mean value almost identically to zero.It is also possible to identify three different types of flow regime as the cylinder oscillation frequency increases.The first type -Type I -is observed for low frequency range of the cylinder oscillation; in this situation the Strouhal number remains almost constant.Type I is followed by an intermediate range of frequency -Type II, the transition regime -where apparently the shedding frequency does not correlate to the frequency of the cylinder oscillation.Finally, in Type III -high frequency of cylinder oscillation -the vortex shedding frequency is locked-in with the cylinder oscillation frequency.around a moving body in a large two-dimensional domain.The body moves to the left with constant velocity; an oscillatory motion with finite amplitude A and constant angular velocity w is added to body motion.This is represented, in Fig. 1, by a heaving cylinder immersed in a uniform incoming flow with velocity U.In this figure the (x, o, y) is the inertial frame of reference and the (X, O, Y) is the coordinate system fixed to the cylinder; this coordinate system oscillates around the x-axis as y o = Acos (wt).The boundary S of the fluid domain is S = Sb ∪ S ∞ ; being S ∞ the far away boundary, which can be viewed as r = → ∞, and S b the body surface.
transferred from the actual position to the mean position.As the body surface is simulated by NP straight line panels on which singularities are distributed (Panels Method) it is convenient to calculate the body induced velocity in the moving coordinate system.For that one has to observe the following -The fluid velocity on the body surface of the j component of the right hand side of the fluid velocity (in the above expression) one gets an additional singularities distribution on the body surface.Of course, the induced velocity due to this additional singularities distribution fades away from the body.

= 10 5 .Figure 2
Figure2shows the pressure distribution on the cylinder surface.As can be noted the obtained values follows closely the ones obtained experimentally; there are some small discrepancies from 60 o to 80 o which probably could be reduced with a proper turbulence modeling.
To better understand what is happening, let us start analyzing the flow behind a stand still cylinder.From each side of the symmetry line (x-axis passing through the cylinder center) large structures formed by clusters of point vortices are shed alternately forming the Karmann vortex street.For low frequency of the body oscillation, the behavior is almost the same although the positions of the cluster shedding move according to the oscillation amplitude; this is shown in Figure 4.For a heaving cylinder in the transition regime, the wake structure becomes intermittent and the vortex clusters are shed irregularly from the cylinder.For the lock-in regime a high strength cluster of vortices just behind the cylinder is observed when it passes through the symmetry line; this situation is illustrated in Figure7, when the cylinder is moving from top to bottom.This cluster -referred as the actual cluster -has a slow rotation in the anti-clockwise direction which decelerates the flow in the upper part of the cylinder and accelerates the flow in the lower part; thus vorticity is fed into a new cluster, which has started to develop in the upper part, enhancing its strength.This newly developed cluster pushes the actual cluster and when the cylinder reaches its lower position it separates into the wake, see Figure5.This figure shows the near field wake pattern at t = 15, when the cylinder is its lowest position.

Figure 8
Figure 8 is assembled with the data from the same simulation used in the previous figure; it is presented to illustrate the lift coefficient behavior during the numerical simulation.As can be observed, the lift coefficient oscillates with the same frequency of the body oscillation and its amplitude can reach values as high as 1.5 to 2.0.In this figure, the black line is the cylinder motion and the blue one is the lift coefficient.

Figure 3 :
Figure 3: Strouhal number behavior as a function of the body oscillation frequency, Re = 10 5 Source: The authors.

Figure 4 :
Figure 4: Wake pattern when the body oscillates with small amplitude and low frequency, Re = 10 5 Source: The authors.
three-dimensional effects present in the experiments are very important for the Reynolds number used in the simulations.Therefore a purely two-dimensional computation of the flow must produce differences in the comparison of the numerical results with the experimental results.The differences encountered in the comparison of the computed values with the experimental results for the distribution of the mean pressure coefficient along the cylinder surface as shown in Figure 2 are attributed mainly to the inherent three-dimensionality of the real flow for such a value of the Reynolds number, which is not modeled in the simulation.The results for the pressure distribution indicated that there was a lack of resolution near the stagnation point and the position of the separation point.The position of the separation point is predicted to occur at about 78°, whereas the experimental value(Blevins, 1984) is about 82°.This seems to indicate that a higher value of M would improve the resolution and probably produce a better simulation with respect to the pressure distribution.More investigations are needed and one can imagine that with the use of more panels (and therefore more free vortices in the cloud) the results tend to be in closer agreement with the experiments.Some discrepancies observed in the determination of the aerodynamics loads may be also attributed to errors in the treatment of vortex element moving away from a solid surface.Because every vortex element has different strength of vorticity, it will diffuse to different location in the flow field.It seems impossible that every vortex element will move to same ε-layer normal to the

Figure 7 :
Figure 7: Center of the cylinder passing through the symmetry line in the downward motion (t = 30, when w = 1.5 and A = 0.5) Source: The authors.

Figure 8 :
Figure 8: Lift coefficient during the numerical simulation Source: The authors.

Table 1 :
Lift and drag coefficients and Strouhal number for a circular cylinder Source: The authors.

Table 2
Source: The authors.

Table 2 :
Circular cylinder: results for an oscillatory motion, Re = 10 5 Source: The authors.