6.2.5 Cut-off circulation

As mentioned in subsection 6.2.3, the edges of expanding vorticity distributions are characterized by exponentially small vorticity. Without special care, this would lead to large amounts of computational vortices of extremely small strengths, severely affecting the computational efficiency. To avoid this, some minimum or ``cut-off" vortex strength is chosen; below this strength no new vortices are created. To be precise, in our computations we do not diffuse the vortices if the absolute value of their circulation falls below a chosen cut-off value $\epsilon_\Gamma$. For the computations of the counter-rotating vortex pair of figures 6.1 through 6.3 we set $\epsilon_\Gamma$ to the machine epsilon. Figure 6.2 shows that with this cut-off, the growth of the number of vortices is roughly linear in time. This indicates that the cut-off works well, since for this flow the true area containing vorticity also grows roughly linear in time. Mesh refinements and comparisons with other data in sections 7.1 and 7.2 indicate that the computational accuracy is not affected by the cut-off.

The use of a cut-off circulation amounts to neglecting the exponentially small vorticity field at the outer edges of the region of vorticity; as a result, the exponentially small rotational velocity field induced by that small vorticity fields is also neglected.

Before this study, it was generally felt that ignoring such exponentially small contributions will not affect the solution. For example, for classical boundary layer problems, the exponentially small velocity above the boundary layer does not have to be computed accurately. For such computations, it suffices to simply set the vorticity zero at some position some distance above the boundary layer or to ensure that the velocity remains finite above the boundary layer through some other means.

However, Van Dommelen & Shen [239] made the surprising discovery that this is not necessarily true for the long time behaviour of unsteady boundary layers. They studied the unsteady boundary layer development near stagnation points at the rear of smooth bodies such as circular cylinders. This problem was earlier investigated by Proudman & Johnson [177]; however, these authors found that they could not obtain a unique solution for the long time problem without some ad hoc assumptions which turned out to be only qualitatively correct. Robins & Howarth [183] took the expansions of Proudman & Johnson to higher order, but could not remove the indeterminacy. P. G. Williams [246] further noted that his numerical results did not seem to agree with the predictions of Robins & Howarth. Van Dommelen & Shen [239] discovered that the reason was that the long time solution is completely determined by growth of the exponentially small velocities above the boundary layer; these velocities were ignored in the asymptotic expansions of Proudman & Johnson and Robins & Howarth. Van Dommelen & Shen [239] noted in their conclusion that this must be a concern in numerical schemes: setting the exponentially small velocities to zero is equivalent to eliminating the very information that determines the solution for later times.

Although subsequent computations for flows such as the impulsively translated circular cylinder ignored the work of Van Dommelen & Shen [239] on the rear stagnation point, (as it did their work on the separation singularity), our computations in section 8.2 do show a significant dependence of the results on the cut-off value $\epsilon_\Gamma$. Furthermore, we will present indications that other authors have in fact experienced computational problems because of values of $\epsilon_\Gamma$ that were too optimistic.

In the next subsection we will discuss how our computation evaluates the vorticity at arbitrary points.