Vortex core size

The radius of a vortex core (σ) is set to satisfy a number of conditions, as the blob vortices are used not only as a flow model but also to solve the no-penetration boundary condition. Indeed, all papers related to the convergence of the vortex method such as Hald [22], Beale and Madja [5], Leonard [33], and Chorin [10] show that as the number of vortices increases, the vortices must become closely spaced but also overlap more and more for the method to converge. In practice this means that as the number of vortices per time step increase, the average distance between them (δ0), as well their core size,

should tend to zero, but the core radius σ not as fast as δ0 does.

One must also consider the residual mass flux across the wall which should also be as small as possible. Spalart [57] discovered that in the case of an inviscid boundary, the core radius should be about 0.153 times the average distance between the control points for the algebraic core. In practice, this value can rarely be exactly respected because the spacing of the points is generally uneven.

Furthermore, σ should also be large enough compared to δ0 for the core to overlap

along the boundary, so that the velocity field is as smooth as possible.

Within these limits, the choice of the core size is arbitrary. It was decided, in the case where all the blob vortices had the same core radius, to find a compromise be- tween the boundary leakage and the smoothness of the evolution of the position of the nascent vortices, which is related to the velocity field near the ellipse edges and to the resolution of the Kutta condition as explained in section 2.3.4. It was found that a value of 0.2075 times the average distance between vortices δ0 gave good results. One could use a time-varying core size, which has many advantages including simu- lating the viscous diffusion; an example is given by Rossi [48]. However, as discussed by Spalart [57] and Leonard [33], the convergence is affected and there is of course the extra computational cost. Furthermore, the effect of viscosity was not intended to be simulated here. Therefore, a constant radius core was used. However, with such a small

core radius σ, smoothness can only be ensured on and near the boundary and where

the nascent vortices are. Thus in most of the wake, the vortices tend to act like point vortices, despite the nonsingular core. Nevertheless, this method has shown better sta- bility than the implementation of the Sarpkaya method [50] (use of complex potential flow solution) with the ellipse.

Trials were made using different core radius for the boundary vortices and for the vortices in the wake. The core size defined above was used for the boundary vortices, and a core size of orderU∞.∆t (where ∆t is the timestep value) for the flow vortices in order to have a sufficient number of superposed vortices. This was compared with the uniform core radius flow where the core radius is fixed at 0.2075δ0everywhere, which can be justified in that in the boundary conditions, the term from the boundary elements is

dissociated from the wake vortices term; moreover as the boundary vortices stay still, there is presumably not too much influence on the flow dynamics especially as a blob vortex velocity field is nonsingular. The boundary vortices are simply a nonsingular flow model of the wall, keeping this in mind one can dissociates the flow and boundary vortices core size while still satisfying equation (2.9).

Comparison between the uniform and non-uniform core radius vortices with the freestream flow at 45 degrees of attack, revealed that the uniform vortex core radius produced bet- ter results in terms of smoothness of the solution, in the roundness of the physical vortices and in the symmetry of the vorticity shed (i.e. at both edges of the ellipse the mean vorticity shed must be the same; see Fage and Johanssen (1927) [16]). Ap- parently, this is because in the non-uniform case, as for the vortex panel method, the nascent vortices come too close to the edge, but it is also related to the free vortices core size. Indeed, the radius in the case of the flow vortices was about 100 times larger than the boundary vortices, or abouta/2, thus creating a non-negligible vorticity inside the ellipse which degrades the resolution of the boundary condition.

In conclusion, it was decided to keep a uniform radius throughout the flow in order to maintain normal convergence. This means however that with a high number of boundary vortices – as for the current simulation case, where 1200 vortices are used– the core radius is much smaller than would be required for the flow vortices. In other words, the algebraic vortex velocity field is similar here to that of discrete vortices, even though using the nonsingular core function seems to provide extra stability in the flow.

In the Spalart code, the vortex core radius is uniform everywhere because the boundary vortices are shed into the flow. He also relates the core radiusσ to the average distance between vortices δ0 using a factor of 0.25, i.e. σ =δ0/4.