Keulegan–Carpenter Number

Garbis Keulegan and Lloyd Carpenter, working at the US National Bureau of Standards, published in 1958 the experiments that gave this number its name: a cylinder held in oscillating water, with the forces on it measured through the cycle. What they established is that the character of the force depends on one ratio — how far the water travels in half a cycle, compared with how wide the cylinder is.

Write it as $\mathrm{KC} = vT/D$ and it looks like a velocity times a time over a length. Read it physically and it is a distance over a length: for a sinusoidal orbital motion of semi-amplitude $a$, the velocity amplitude is $v = 2\pi a/T$, so $\mathrm{KC} = 2\pi a / D$. The water sweeps back and forth through a path of $2\pi a$ each cycle, and KC counts how many diameters that is.

That count decides which term of Morison's equation carries the load. Morison, O'Brien, Johnson and Schaaf proposed in 1950 that the force on a slender member in waves is the sum of a drag term, proportional to $u|u|$, and an inertia term, proportional to the acceleration $\dot u$. The two terms are ninety degrees out of phase — drag peaks under the crest where the velocity is greatest, inertia peaks at the zero-crossing where the acceleration is greatest — so the total force is not simply the sum of two maxima, and which one dominates changes the timing of the peak load as well as its size.

• KC below about 2 — inertia dominated. The water barely moves past the member before reversing. No wake forms. The force is essentially the pressure gradient of the undisturbed wave acting on the displaced volume, plus an added-mass term. Large monopiles and gravity structures live here.
• KC above about 20 — drag dominated. The water sweeps many diameters past each half-cycle, a full wake develops, and the member behaves much as it would in a steady current. Bracing, risers and conductors live here.
• KC between roughly 4 and 25 — the difficult middle. Vortices shed on one half-cycle are still in the neighbourhood when the flow reverses and sweeps them back over the member. This produces transverse lift forces at odds with the wave direction, and drag coefficients that swing with KC rather than sitting still.

That middle band is why the drag and inertia coefficients in Morison's equation are functions of KC, not constants, and why they are still the subject of experimental work seventy years on. The coefficients also depend on the Reynolds number and on the roughness — a member with a year of marine growth on it is a different member — and offshore design codes tabulate $C_d$ and $C_m$ against all three.

Three cautions on using the number honestly. First, the orbital velocity decays with depth, so KC at the waterline and KC at the mudline on the same pile are different numbers; compute it where you are applying the coefficients. Second, a KC computed from a significant wave height characterizes the sea state, not the design wave — the biggest waves in a record are steeper and faster, and give higher KC. Third, and most important, Morison's equation itself has a validity limit that KC says nothing about. When the member diameter exceeds roughly a fifth of the wavelength, it is large enough to scatter the wave it stands in; the incident wave field is no longer undisturbed, Morison's premise fails, and the problem belongs to diffraction theory — MacCamy and Fuchs' 1954 solution, or a panel code. A very large monopile in short waves can be both low-KC and outside Morison's range at once, and the low KC will not warn you.

The choice hiding in the group is the diameter, which sounds unambiguous until you meet a member that is not circular, or one with anodes and J-tubes bolted to it, or a lattice whose members shelter one another. Say which diameter you used, as you would say which length you used for a Reynolds number.