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 Inertia is the idea that an object will continue its current motion until some force causes its speed or direction to change. The term is properly understood as shorthand for "the principle of inertia" as described by Newton in his first law ...

forces 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 In fluid dynamics, turbulence or turbulent flow is fluid motion characterized by chaotic changes in pressure and flow velocity. It is in contrast to a laminar flow, which occurs when a fluid flows in parallel layers, with no disruption between ...

) drag forces are important. The Keulegan–Carpenter number ''K_C'' is defined as:Dean & Dalrymple (1991), p. 232. :

K_C = \frac,

where: *''V'' is the

amplitude The amplitude of a periodic variable is a measure of its change in a single period (such as time or spatial period). The amplitude of a non-periodic signal is its magnitude compared with a reference value. There are various definitions of am ...

of the

flow velocity In continuum mechanics the flow velocity in fluid dynamics, also macroscopic velocity in statistical mechanics, or drift velocity in electromagnetism, is a vector field used to mathematically describe the motion of a continuum. The length of the f ...

oscillation (or the amplitude of the object's velocity, in case of an oscillating object), *''T'' is the

period Period may refer to: Common uses * Era, a length or span of time * Full stop (or period), a punctuation mark Arts, entertainment, and media * Period (music), a concept in musical composition * Periodic sentence (or rhetorical period), a concept ...

of the oscillation, and *''L'' is a characteristic length scale of the object, for instance the

diameter In geometry, a diameter of a circle is any straight line segment that passes through the center of the circle and whose endpoints lie on the circle. It can also be defined as the longest chord of the circle. Both definitions are also valid fo ...

for a

cylinder A cylinder (from ) has traditionally been a three-dimensional solid, one of the most basic of curvilinear geometric shapes. In elementary geometry, it is considered a prism with a circle as its base. A cylinder may also be defined as an infin ...

under wave loading. The Keulegan–Carpenter number is named after Garbis H. Keulegan (1890–1989) and Lloyd H. Carpenter. A closely related parameter, also often used for sediment transport under water waves, is the displacement parameter ''δ'': :

\delta = \frac,

with ''A'' the excursion amplitude of fluid particles in oscillatory flow and ''L'' a characteristic diameter of the sediment material. 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 In physics, the Navier–Stokes equations ( ) are partial differential equations which describe the motion of viscous fluid substances, named after French engineer and physicist Claude-Louis Navier and Anglo-Irish physicist and mathematician Geo ...

, by looking at characteristic scales for the

acceleration In mechanics, acceleration is the rate of change of the velocity of an object with respect to time. Accelerations are vector quantities (in that they have magnitude and direction). The orientation of an object's acceleration is given by t ...

terms: *convective acceleration:

(\mathbf\cdot\nabla)\mathbf \sim \frac,

*local acceleration:

\frac \sim \frac.

Dividing these two acceleration scales gives the Keulegan–Carpenter number. A somewhat similar parameter is the Strouhal number, in form equal to the

reciprocal Reciprocal may refer to: In mathematics * Multiplicative inverse, in mathematics, the number 1/''x'', which multiplied by ''x'' gives the product 1, also known as a ''reciprocal'' * Reciprocal polynomial, a polynomial obtained from another pol ...

of the Keulegan–Carpenter number. The Strouhal number gives the vortex shedding

frequency Frequency is the number of occurrences of a repeating event per unit of time. It is also occasionally referred to as ''temporal frequency'' for clarity, and is distinct from ''angular frequency''. Frequency is measured in hertz (Hz) which is eq ...

resulting from placing an object in a steady flow, so it describes the flow unsteadiness as a result of an instability of the flow downstream of the object. Conversely, the Keulegan–Carpenter number is related to the oscillation frequency of an unsteady flow into which the object is placed.

Notes

Bibliography

* * {{DEFAULTSORT:Keulegan-Carpenter number Dimensionless numbers of fluid mechanics Fluid dynamics Water waves Marine engineering

See also

Notes

Bibliography