Fluid Flow & Aerodynamics
why things fly, and why the usual explanation is wrong
Series

Keulegan–Carpenter number — the series

3 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. Morison's two terms over one cycle, at KC = 10. The drag term, in phase with the velocity and going as its square; the inertia term, in phase with the acceleration and ninety degrees ahead of it; and their sum, which is what a load cell records. The peak of the total is not the peak of either, and its position in the cycle is the only thing in the record that says how the two are divided.

    Long enough to make a wake

    A Reynolds number cannot ask whether an oscillating flow gets round a body before it turns and comes back, because it has no time in it. The Keulegan–Carpenter number can, and it decides which of Morison's two terms is the force. What it discards is the phase — and a peak force measurement cannot recover it.

    part 1 · regimes
  2. Two forces on one cylinder, a quarter of a cycle apart. The inertia and drag terms of Morison's equation over one wave period, at a Keulegan-Carpenter number of ten. The inertia term follows the acceleration and peaks where the velocity is zero; the drag term follows the velocity and peaks where the acceleration is. They are a quarter of a cycle apart and they are different kinds of quantity.

    Two forces, and only one of them remembers

    Morison's equation adds an inertia term to a drag term and is usually presented as an empirical patch. It is not: the two terms are two different kinds of memory, one a function of the present acceleration and one a function of the wake left by the previous half cycle.

    part 2 · regimes
  3. Below separation Morison's drag coefficient is a Stokes layer's, and it grows without limit as the motion shrinks. Wang's drag coefficient for a circular cylinder in an attached oscillatory flow against the Keulegan–Carpenter number, for β of 10², 10³, 10⁴ and 10⁵, each drawn up to Hall's onset of the Honji instability. β = 10²: 2.77 at KC = 1, onset at KC 1.83; β = 10³: 0.845 at KC = 1, onset at KC 1.03; β = 10⁴: 0.264 at KC = 1, onset at KC 0.578; β = 10⁵: 0.0831 at KC = 1, onset at KC 0.325. At fixed β the coefficient falls as 1/KC.

    Below separation a cylinder's drag is a Stokes layer, and KC stops deciding the force

    Morison's equation splits the force on a cylinder in an oscillating flow into inertia and drag, and the Keulegan–Carpenter number says which dominates — once the flow separates and leaves a wake. Below that, nothing separates, and the drag term describes the thin oscillating boundary layer on the cylinder's surface. Its coefficient then grows without limit as the motion shrinks, as one over the Keulegan–Carpenter number, and that exactly cancels the number's power to decide anything: the drag's share of the force is fixed by the frequency parameter alone, a few per cent at most, until Honji's instability ends the regime.

    part 3 · regimes

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