A wake is the region of recirculating flow immediately behind a moving or stationary solid body, caused by the flow of surrounding fluid around the body.
In fluid dynamics, a wake is the region of disturbed flow (usually turbulent) downstream of a solid body moving through a fluid, caused by the flow of the fluid around the body. In incompressible fluids (liquids) such as water, a bow wake is created when a watercraft moves through the medium; as the medium cannot be compressed, it must be displaced instead, resulting in a wave. As with all wave forms, it spreads outward from the source until its energy is overcome or lost, usually by friction or dispersion.
The formation of these waves in liquids is analogous to the generation of shockwaves in compressible flow, such as those generated by rockets and aircraft traveling supersonic through air (see also Lighthill equation). The non-dimensional parameter of interest is the Froude number.
For a blunt body in subsonic external flow, for example the Apollo or Orion capsules during descent and landing, the wake is massively separated and behind the body is a reverse flow region where the flow is moving toward the body. This phenomenon is often observed in wind tunnel testing of aircraft, and is especially important when parachute systems are involved, because unless the parachute lines extend the canopy beyond the reverse flow region, the chute can fail to inflate and thus collapse. Parachutes deployed into wakes suffer dynamic pressure deficits which reduce their expected drag forces. High-fidelity computational fluid dynamics simulations are often undertaken to model wake flows, although such modeling has uncertainties associated with turbulence modeling (for example RANS versus LES implementations), in addition to unsteady flow effects. Example applications include rocket stage separation and aircraft store separation.
Wake patterns in water
Waterfowls and boats moving across the surface of water produce a wake pattern, first explained mathematically by Lord Kelvin and known today as the Kelvin wake pattern. This pattern consists of two wake lines that form the arms of a V, with the source of the wake at the point. Each wake line is offset from the path of the wake source by around 19.47° and is made up of feathery wavelets angled at roughly 53° to the path. The inside of the V is filled with transverse curved waves, each of which is an arc of a circle centered at a point lying on the path at a distance twice that of the arc to the wake source. This pattern is independent of the speed and size of the wake source over a significant range of values.
The wave crest wedge is described by the following parametric equations
The angles in this pattern are not intrinsic properties of water; Any isentropic and incompressible liquid with low viscosity will exhibit the same phenomenon. This phenomenon has nothing to do with turbulence. Everything discussed here is based on the linear theory of an ideal fluid.
Parts of the pattern may be obscured by the effects of propeller wash, and tail eddies behind the boat's stern, and by the boat being a large object and not a point source. The water need not be stationary, but may be moving as in a large river, and the important consideration then is the velocity of the water relative to a boat or other object causing a wake.
This pattern follows from the dispersion relation of deep water waves, which is often written as,
- g = the strength of the gravity field
- ω is the angular frequency in radians per second
- k = angular wavenumber in radians per metre
- "deep" means that the depth is greater than half of the wavelength.
- (Here, a radian is 1/(2π) of a wave.)
This formula has two implications:
- The speed of the wave varies as the square root of the wavelength.
- The group velocity of a deep water wave is half of its phase velocity.
- v is the relative velocity of the water and the surface object that causes the wake.
- c is the phase velocity of a wave: it may be the same for all frequencies (e.g. with electromagnetic waves, and sound waves in air), or it may vary with wave frequency (e.g. with waves on water).
As the surface object moves, it continuously generates small disturbances which are the sum of sinusoidal waves with a wide spectrum of wavelengths. Those waves with the longest wavelengths have phase speeds above and dissipate into the surrounding water and are not easily observed. Other waves with phase speeds at or below v are amplified through constructive interference and form visible shock waves.
Where the phase velocity is the same for all frequencies, ω = ck and c is the same for all wavelengths and the group velocity is the same.
The angle θ between the shock wave front and the path of the object is θ = arcsin(c/v). If c/v > 1 or < -1, no later waves can catch up with earlier waves and no shockwave forms.
In deep water, shock waves form even from slow-moving sources, because waves with short enough wavelengths move slower. These shock waves are at sharper angles than one would naively expect, because it is group velocity that dictates the area of constructive interference and, in deep water, the group velocity is half of the phase velocity.
All shock waves that each by itself would have had an angle between 33° and 72°, are compressed into a narrow band of wake with angles between 15° and 19°, with the strongest constructive interference at the outer edge (angle 19°), causing the two arms of the V in the Kelvin wake pattern, because the group velocity is half of the phase velocity. The wavefronts of the wavelets in the wake are at 53°, which is roughly the average of 33° and 72°.
The wave components with would-be shock wave angles between 73° and 90° dominate the interior of the V. They end up half-way between the point of generation and the current location of the wake source. This explains the curvature of the arcs.
Those very short waves with would-be shock wave angles below 33° lack a mechanism to reinforce their amplitudes through constructive interference and are usually seen as small ripples on top of the interior transverse waves.
Kelvin Wake integral
In a polar coordinate system where a ship is at rest, and is the direction of its velocity,the wave height is describe roughly by the following integral:
is the squaredFroude number
is gravity constantis the ship's length.
The integrand oscillates violently, hence this integral was treated by Lord Kelvin with Stationary phase approximation.
Wake from a small motorboat with an outboard motor.
The above describes an ideal wake, where the body's means of propulsion has no other effect on the water. In practice the wave pattern between the V-shaped wavefronts is usually mixed with the effects of propeller backwash and eddying behind the boat's (usually square-ended) stern.
"No wake zones" may prohibit wakes in marinas, near moorings and within some distance of shore in order to facilitate recreation by other boats and reduce the damage wakes cause. Powered narrowboats on British canals are not permitted to create a breaking wash (a wake large enough to create a breaking wave) along the banks, as this erodes them. This rule normally restricts these vessels to 4 statute miles per hour or less.
Wakes are occasionally used recreationally. Swimmers, people riding personal watercraft, and aquatic mammals such as dolphins can ride the leading edge of a wake. In the sport of wakeboarding the wake is used as a jump. The wake is also used to propel a surfer in the sport of wakesurfing. In the sport of water polo, the ball carrier can swim while advancing the ball, propelled ahead with the wake created by alternating armstrokes in crawl stroke, a technique known as dribbling.
Wake of a boat in the Hawaiian Islands
- James LightHill, Waves in Fluid, p274,Cambridge University Press, 1979
- Frank Oliver et al, NIST Handbook of Mathematical Functions, p791,Cambridge Univerity Press, 2010
- BoatWakes.org, Table of distances
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