Métodos basados en características

In programming the C simulation, a scheme was adapted based on the model of

Sarpkaya [50]. He used a potential-flow model of 2D vortex shedding, using image

vortices and a Joukowski transformation to model the unsteady flow past a stationary plate. The vortices present in the flow are convected using a complex potential. At each timestep two nascent point vortices are generated to account for the effect of separation. For a flat plate in a transverse flow, the separation is located at the two edges of the flat plate, which will be referred to as the separation points. The strengths of the nascent vortices shed at the separation points are computed through an approximation of the shear layer velocity at these points. The position of the nascent vortices is then deter- mined by enforcing the Kutta equation at the separation points on the body surface. At this point, it is important to distinguish the separation points which are defined as

the points on the body surface where the flow separation is supposed to take place, and

the discrete vortices creation points which are the points in the flow domain without

the body (including the body wall) where the nascent vortices are positioned in order for their induced velocity field to enable the instantaneous satisfaction of the Kutta condition at the separation points.

Sarpkaya [53] provides a very good discussion about this kind of method. It appears well suited to modeling a flow around a body with sharp edges where the determination of the separation location is not a problem. Additionally, there is a feedback from the vortex street to the nascent vortex, dynamically affecting its strength and initial posi- tion. This is a crucial point, as the initial position of the nascent vortices is of major importance on how the vortex sheet rolls up, as stressed by Gerrard [21]. It should be added that the normal force determination is also very sensitive to the distance from the plate of the nascent vortex.

However, some of the inconsistencies of the scheme are that it does not relate the

time step ∆t to both the distance of the nascent vortex to the body and the rate

of vorticity ∆Γ (the latter, indirectly through the determination of an approximated shear layer velocity). Furthermore, there is no explicit treatment of the breakaway an- gle between the separation streamline and the body surface tangent, as well as for the velocity distribution in the vicinity of the separation point after the introduction of the nascent vortex.

Also, it is hard to establish proof of the convergence for this kind of method for a given set of parameters, and such proof have not been found in the literature cited in the current work. Nevertheless, observe that even for more general vortex methods such as the Spalart code, it is not thought possible to prove convergence for a given set of parameters (see Spalart citation in section 2.2.2) despite the existence of demonstra- tion of the convergence of the method (see Koutsoumakos [13] and section 2.2.2). Note that Koutsoumakos [13] clearly stated that even for undersolved system the method is expected to give good qualitative results. This should be considered regarding the sections related to the numerical parameters (notably section 2.3) and the flow results presentation (chapter 3 and section 5.4).

Still, this model produces satisfactory results for a cylinder [49] and for a flat plate [50], despite overestimating the resulting aerodynamic forces. However, for long-time the simulation tends to become unstable due to the singularity of point vortices. Fur- thermore, the force and moment evaluation are dependent on the velocity close to the wall surface. Blob vortices were thus used both for flow and boundary modeling to

help maintain stability by smoothing the velocity field. Indeed, as pointed out by Sarp- kaya [53], this helps to achieve convergence provided that the blobs sufficiently overlap. However, one must keep in mind that this scheme does not ensure a smooth roll-up, and constitutes a mathematical artifice to smooth the velocity field. As stated earlier, the nonlinear nature of the Navier Stokes equations does not formally permit such su- perposition of vorticity fields.

Another modification of the Sarpkaya scheme is that instead of using a complex poten- tial for modeling the body boundaries, a boundary element method was used in which the strength of blob vortex elements is chosen to satisfy a no-penetration boundary condition. The main drawback of this scheme is that the use of blob vortices allows a net flow leakage into the body, although in practice it is small enough so that it does not exert a great influence on the flow. On the other hand, it has some real advantages compared to the use of the complex potential. First of all, it allows the use of other types of vortices than the point vortex. Another benefit is that it avoids the use of image vortices required by the complex potential, which can introduces some artificial flow phenomena [57] when a near-wall vortex is too close to its own image. Finally, despite being of lesser importance for the thin ellipse case, it is more flexible concerning the geometry of the body.

As a last remark, Sarpkaya in his review [53] noted that the Strouhal number does not seem to be overly sensitive to the position of the nascent vortex when introduced into the flow. This is important for the current study, since the emphasis here is on the frequency characteristics of the fluid/structure system, even if it does not guarantee a proper modeling of all of the dynamic characteristics. Again, Sarpkaya [53] provides a simple example whereby the normal-force coefficient on a steady cylinder shows unex- pected (but reasonably small) oscillations compared to experiment. This is also why is a comparison is made of this model against the well-established model from Spalart, in order to assess the behavior of the Sarpkaya based method. Nevertheless, one must keep in mind that this is another vortex method, and as such will presumably have some of the limitations inherent in all vortex-method predictions of bluff-body flow dynamics.

Note that due to the choices made, the simulation developed specifically for the current project is often referred to as “the Sarpkaya-like simulation” by contrast to the “Spalart simulation”.