D'Alembert's Paradox - Inviscid Separated Flow: Kirchhoff and Rayleigh

Inviscid Separated Flow: Kirchhoff and Rayleigh

In the second half of the 19th century, focus shifted again towards using inviscid flow theory for the description of fluid drag—assuming that viscosity becomes less important at high Reynolds numbers. The model proposed by Kirchhoff and Rayleigh was based on the free-streamline theory of Helmholtz and consists of a steady wake behind the body. Assumptions applied to the wake region include: flow velocities equal to the body velocity, and a constant pressure. This wake region is separated from the potential flow outside the body and wake by vortex sheets with discontinuous jumps in the tangential velocity across the interface. In order to have a non-zero drag on the body, the wake region must extend to infinity. This condition is indeed fulfilled for the Kirchhoff flow perpendicular to a plate. The theory correctly states the drag force to be proportional to the square of the velocity. In first instance, the theory could only be applied to flows separating at sharp edges. Later, in 1907, it was extended by Levi-Civita to flows separating from a smooth curved boundary.

It was readily known that such steady flows are not stable, since the vortex sheets develop so-called Kelvin–Helmholtz instabilities. But this steady-flow model was studied further in the hope it still could give a reasonable estimate of drag. Rayleigh asks "... whether the calculations of resistance are materially affected by this circumstance as the pressures experienced must be nearly independent of what happens at some distance in the rear of the obstacle, where the instability would first begin to manifest itself."

However, fundamental objections arose against this approach: Kelvin observed that if a plate is moving with constant velocity through the fluid, the velocity in the wake is equal to that of the plate. The infinite extent of the wake—widening with the distance from the plate, as obtained from the theory—results in an infinite kinetic energy in the wake, which must be rejected on physical grounds. Moreover, the observed pressure differences between front and back of the plate, and resulting drag forces, are much larger than predicted: for a flat plate perpendicular to the flow the predicted drag coefficient is C_D=0.88, while in experiments C_D=2.0 is found. This is mainly due to suction at the wake side of the plate, induced by the unsteady flow in the real wake (as opposed to the theory which assumes a constant flow velocity equal to the plate's velocity).

So, this theory is found to be unsatisfactory as an explanation of drag on a body moving through a fluid. Although it can be applied to so-called cavity flows where, instead of a wake filled with fluid, a vacuum cavity is assumed to exist behind the body.