8.2.8 The drag near the Van Dommelen & Shen time

During the boundary layer separation process described by Van Dommelen & Shen [241], the upper part of the boundary layer is `ejected' away from the wall. It might be thought that this singular process would be reflected in the net forces experienced by the cylinder.

In fact, in fluid dynamics there is a close relationship between boundary layer separation and drag forces. For a nonseparated boundary layer such as exists at, say, $t=1$, boundary layer scalings predict a drag coefficient that vanishes for high Reynolds numbers. Since that is in contradiction with the finite values of the drag observed experimentally on cylinders under steady conditions, it is known as D'Alembert's paradox. For steady flows, the drag is due to boundary layer separation which causes the boundary layer vorticity to move far from the wall, changing the overall flow field and inducing adverse pressure forces.

In the unsteady case the situation is different. While previously it had been believed that drag cannot directly be predicted from the standard first order boundary layer theory, (hence from the Van Dommelen & Shen singularity), Van Dommelen & Shankar [229] showed that this is in fact possible. Ignoring the details of their derivations, their final conclusion was that the leading order drag is not affected by the initial unsteady separation! Although the separation does introduce pressure forces, the adverse and favorable forces cancel, leaving the net force initially unaffected. This prediction seems in good agreement with our results. Our computed drag shows no sign of singular behavior until well after $t=1.50$, when the rolled-up vortex forms. Even then, in spite of the ejection of vorticity, the drag goes down, rather than up.

There is a practical implication of the above conclusion that the Van Dommelen and Shen singularity does not affect the net force initially. It is that the control of unsteady separation in two-dimensional flows based on monitoring the net forces may not be very effective, since the separation has occurred before the forces change.