Keulegan–Carpenter number

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The Keulegan–Carpenter number is important for the computation of the wave forces on oil platforms.

In fluid dynamics, the Keulegan–Carpenter number, also called the period number, is a dimensionless quantity describing the relative importance of the drag forces over inertia for bluff objects in an oscillatory fluid flow. Or similarly, for objects that oscillate in a fluid at rest. For small Keulegan–Carpenter number inertia dominates, while for large numbers the (turbulence) drag forces are important.

The Keulegan–Carpenter number K_C is defined as:^[1]

$K_C = \frac{V\,T}{L},$

where:

V is the amplitude of the flow velocity oscillation (or the amplitude of the object's velocity, in case of an oscillating object),
T is the period of the oscillation, and
L is a characteristic length scale of the object, for instance the diameter for a cylinder under wave loading.

A closely related parameter, also often used for sediment transport under water waves, is the displacement parameter δ:^[1]

$\delta = \frac{A}{L},$

with A the excursion amplitude of fluid particles in oscillatory flow. For sinusoidal motion of the fluid, A is related to V and T as A = VT/(2π), and:

$K_C = 2\pi\, \delta.\,$

The Keulegan–Carpenter number can be directly related to the Navier–Stokes equations, by looking at characteristic scales for the acceleration terms:

temporal acceleration: $\frac{\partial \mathbf{u}}{\partial t} \sim \frac{V}{T},$
convective acceleration: $(\mathbf{u}\cdot\nabla)\mathbf{u} \sim \frac{V^2}{L}.$

Dividing these two acceleration scales gives the Keulegan–Carpenter number.

[edit] References

Keulegan, G. H. & Carpenter, L. H. (1958), “Forces on cylinders and plates in an oscillating fluid”, Journal of Research of the National Bureau of Standards 60 (5): 423–440
Dean, R.G. & Dalrymple, R.A. (1991), Water wave mechanics for engineers and scientists, vol. 2, Advanced Series on Ocean Engineering, World Scientific, Singapore, ISBN 978 981 02 0420 4