# Wake

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In fluid dynamics, a wake may either be:

• the region of recirculating flow immediately behind a moving or stationary blunt body, caused by viscosity, which may be accompanied by flow separation and turbulence, or
• the wave pattern on the water surface downstream of an object in a flow, or produced by a moving object (e.g. a ship), caused by density differences of the fluids above and below the free surface and gravity (or surface tension).

## Wake effects caused by viscosity

The wake is the region of disturbed flow (often turbulent) downstream of a solid body moving through a fluid, caused by the flow of the fluid around the body.

## Contents

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.

## Waves by density differences, like a water surface

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 non-dimensional parameter of interest is the Froude number.

Wave cloud pattern in the wake of the Île Amsterdam (lower left, at the "tip" of the triangular formation of clouds) in the southern Indian Ocean
Cloud wakes from the Juan Fernández Islands
Wake patterns in cloud cover over Possession Island, East Island, Ile aux Cochons, Île des Pingouins

### Kelvin wake pattern

Waterfowl 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. [1]

This pattern consists of two wake lines that form the arms of a chevron, V, with the source of the wake at the vertex of the V. For sufficiently slow motion, each wake line is offset from the path of the wake source by around arcsin(1/3) = 19.47° and is made up of feathery wavelets angled at roughly 53° to the path.

The inside of the V (of total opening 39° as indicated above) 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. [2]

However, the pattern changes at high speeds (only), viz., above a hull Froude number of approximately 0.5. Then, as the source's speed increases, the transverse waves diminish and the points of maximum amplitude on the wavelets form a second V within the wake pattern, which grows narrower with the increased speed of the source. [3]

The angles in this pattern are not intrinsic properties of merely water: Any isentropic and incompressible liquid with low viscosity will exhibit the same phenomenon. Furthermore, this phenomenon has nothing to do with turbulence. Everything discussed here is based on the linear theory of an ideal fluid, cf. Airy wave theory.

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,

${\displaystyle \omega ={\sqrt {gk}},}$

where

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. This formula implies that the group velocity of a deep water wave is half of its phase velocity, which, in turn, goes as the square root of the wavelength. Two velocity parameters of importance for the wake pattern are:

v is the relative velocity of the water and the surface object that causes the wake.
c is the phase velocity of a wave, varying with wave frequency.

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 v and dissipate into the surrounding water and are not easily observed. Other waves with phase speeds at or below v, however, are amplified through constructive interference and form visible shock waves, stationary in position w.r.t. the boat.

The angle θ between the phase 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 arcsin(1/3) = 19.47°), placing the two arms of the V in the celebrated Kelvin wake pattern.

A concise geometric construction [4] demonstrates that, strikingly, this group shock angle w.r.t. the path of the boat, 19.47°, for any and all of the aboveθ, is actually independent ofv, c, and g; it merely relies on the fact that the group velocity is half of the phase velocity c. On any planet, slow-swimming objects have "effective Mach number" 3!

Concise argument for the universal opening of 39° of the slow-swimmers' Kelvin wake

For slow swimmers, low Froude number, the Lighthill−Whitham geometric argument that the opening of the Kelvin chevron (wedge, V pattern) is universal goes as follows. Consider a boat moving from right to left with constant speed v, emitting waves of varying wavelength, and thus wavenumber k and phase velocity c(k), of interest when < v for a shock wave (cf., e.g., Sonic boom or Cherenkov radiation). Equivalently, and more intuitively, fix the position of the boat and have the water flow in the opposite direction, like a piling in a river.

Focus first on a given k, emitting (phase) wavefronts whose stationary position w.r.t. the boat assemble to the standard shock wedge tangent to all of them, cf. Fig.12.3.

As indicated above, the openings of these chevrons vary with wavenumber, the angle θ between the phase shock wavefront and the path of the boat (the water) being θ = arcsin(c/v) ≡ π/2 − ψ. Evidently, ψ increases with k. However, these phase chevrons are not visible: it is their corresponding group wave manifestations which are observed.

Consider one of the phase circles of Fig.12.3 for a particular k, corresponding to the time t in the past, Fig.12.2. Its radius is QS, and the phase chevron side is the tangent PS to it. Evidently, PQ= vt and SQ = ct = vt cosψ, as the right angle PSQ places S on the semicircle of diameter PQ.

Since the group velocity is half the phase velocity for any and all k, however, the visible (group) disturbance point corresponding to S will be T, the midpoint of SQ. Similarly, it lies on a semicircle now centered on R, where, manifestly, RQ=PQ/4, an effective group wavefront emitted from R, with radius vt/4 now.

Significantly, the resulting wavefront angle with the boat's path, the angle of the tangent from P to this smaller circle, obviously has a sine of TR/PR=1/3, for any and all k, c, ψ, g, etc.: Strikingly, virtually all parameters of the problem have dropped out, except for the deep-water group-to-phase-velocity relation! Note the (highly notional) effective group disturbance emitter moves slower, at 3v/4.

Thus, summing over all relevant k and ts to flesh out an effective Fig.12.3 shock pattern, the universal Kelvin wake pattern arises: the full visible chevron angle is twice that, 2arcsin(1/3) ≈ 39°.

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.

## Other effects

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.

The Kelvin angle is also derived for the case of deep water in which the fluid is not flowing in different speed or directions as a function of depth ("shear"). In cases where the water (or fluid) has sheer, the results may be more complicated. [5]

## Recreation

"No wake zones" may prohibit wakes in marinas, near moorings and within some distance of shore [6] 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 (3.5 knots or 6.4 km/h) 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.

## Related Research Articles

Diffraction refers to various phenomena that occur when a wave encounters an obstacle or opening. It is defined as the bending of waves around the corners of an obstacle or through an aperture into the region of geometrical shadow of the obstacle/aperture. The diffracting object or aperture effectively becomes a secondary source of the propagating wave. Italian scientist Francesco Maria Grimaldi coined the word diffraction and was the first to record accurate observations of the phenomenon in 1660.

The Doppler effect or Doppler shift is the change in frequency of a wave in relation to an observer who is moving relative to the wave source. It is named after the Austrian physicist Christian Doppler, who described the phenomenon in 1842.

The group velocity of a wave is the velocity with which the overall envelope shape of the wave's amplitudes—known as the modulation or envelope of the wave—propagates through space.

In physics, refraction is the change in direction of a wave passing from one medium to another or from a gradual change in the medium. Refraction of light is the most commonly observed phenomenon, but other waves such as sound waves and water waves also experience refraction. How much a wave is refracted is determined by the change in wave speed and the initial direction of wave propagation relative to the direction of change in speed.

Total internal reflection (TIR) is the optical phenomenon when waves travelling in one medium strike at sufficiently oblique incident angle against the boundary with another medium of lower refractive index, instead of transmitting into the second ("external") medium at a refracted angle, the waves all get reflected back into the first ("internal") medium. An example would be the water surface in a fish tank, which when viewed obliquely from below often reflects the underwater scenery like a mirror with no loss of brightness.

In physics, the wavelength is the spatial period of a periodic wave—the distance over which the wave's shape repeats. It is the distance between consecutive corresponding points of the same phase on the wave, such as two adjacent crests, troughs, or zero crossings, and is a characteristic of both traveling waves and standing waves, as well as other spatial wave patterns. The inverse of the wavelength is called the spatial frequency. Wavelength is commonly designated by the Greek letter lambda (λ). The term wavelength is also sometimes applied to modulated waves, and to the sinusoidal envelopes of modulated waves or waves formed by interference of several sinusoids.

Snell's law is a formula used to describe the relationship between the angles of incidence and refraction, when referring to light or other waves passing through a boundary between two different isotropic media, such as water, glass, or air.

In physics, a shock wave, or shock, is a type of propagating disturbance that moves faster than the local speed of sound in the medium. Like an ordinary wave, a shock wave carries energy and can propagate through a medium but is characterized by an abrupt, nearly discontinuous, change in pressure, temperature, and density of the medium.

Compressible flow is the branch of fluid mechanics that deals with flows having significant changes in fluid density. While all flows are compressible, flows are usually treated as being incompressible when the Mach number is smaller than 0.3. The study of compressible flow is relevant to high-speed aircraft, jet engines, rocket motors, high-speed entry into a planetary atmosphere, gas pipelines, commercial applications such as abrasive blasting, and many other fields.

A hydraulic jump is a phenomenon in the science of hydraulics which is frequently observed in open channel flow such as rivers and spillways. When liquid at high velocity discharges into a zone of lower velocity, a rather abrupt rise occurs in the liquid surface. The rapidly flowing liquid is abruptly slowed and increases in height, converting some of the flow's initial kinetic energy into an increase in potential energy, with some energy irreversibly lost through turbulence to heat. In an open channel flow, this manifests as the fast flow rapidly slowing and piling up on top of itself similar to how a shockwave forms.

In continuum mechanics, the Froude number is a dimensionless number defined as the ratio of the flow inertia to the external field. Named after William Froude (;), the Froude number is based on the speed–length ratio which he defined as:

In fluid dynamics, a Mach wave is a pressure wave traveling with the speed of sound caused by a slight change of pressure added to a compressible flow. These weak waves can combine in supersonic flow to become a shock wave if sufficient Mach waves are present at any location. Such a shock wave is called a Mach stem or Mach front. Thus, it is possible to have shockless compression or expansion in a supersonic flow by having the production of Mach waves sufficiently spaced. A Mach wave is the weak limit of an oblique shock wave where time averages of flow quantities don't change;. If the size of the object moving at the speed of sound is near 0, then this domain of influence of the wave is called Mach cone.

In fluid dynamics, wind waves are water surface waves that occur on the free surface of bodies of water. They result from the wind blowing over a fluid surface, where the contact distance in the direction of the wind is known as the fetch. Waves in the oceans can travel thousands of miles before reaching land. Wind waves on Earth range in size from small ripples, to waves over 100 ft (30 m) high, being limited by wind speed, duration, fetch, and water depth.

Internal waves are gravity waves that oscillate within a fluid medium, rather than on its surface. To exist, the fluid must be stratified: the density must change with depth/height due to changes, for example, in temperature and/or salinity. If the density changes over a small vertical distance, the waves propagate horizontally like surface waves, but do so at slower speeds as determined by the density difference of the fluid below and above the interface. If the density changes continuously, the waves can propagate vertically as well as horizontally through the fluid.

A bow wave is the wave that forms at the bow of a ship when it moves through the water. As the bow wave spreads out, it defines the outer limits of a ship's wake. A large bow wave slows the ship down, is a risk to smaller boats, and in a harbor can damage shore facilities and moored ships. Therefore, ship hulls are generally designed to produce as small a bow wave as possible.

In fluid dynamics, dispersion of water waves generally refers to frequency dispersion, which means that waves of different wavelengths travel at different phase speeds. Water waves, in this context, are waves propagating on the water surface, with gravity and surface tension as the restoring forces. As a result, water with a free surface is generally considered to be a dispersive medium.

In fluid dynamics, drag is a force acting opposite to the relative motion of any object moving with respect to a surrounding fluid. This can exist between two fluid layers or a fluid and a solid surface. Unlike other resistive forces, such as dry friction, which are nearly independent of velocity, drag force depends on velocity.

A ship must be designed to move efficiently through the water with a minimum of external force. For thousands of years ship designers and builders of sailing vessels used rules of thumb based on the midship-section area to size the sails for a given vessel. The hull form and sail plan for the clipper ships, for example, evolved from experience, not from theory. It was not until the advent of steam power and the construction of large iron ships in the mid-19th century that it became clear to ship owners and builders that a more rigorous approach was needed.

A supersonic expansion fan, technically known as Prandtl–Meyer expansion fan, a two-dimensional simple wave, is a centered expansion process that occurs when a supersonic flow turns around a convex corner. The fan consists of an infinite number of Mach waves, diverging from a sharp corner. When a flow turns around a smooth and circular corner, these waves can be extended backwards to meet at a point.

In fluid dynamics, a Stokes wave is a nonlinear and periodic surface wave on an inviscid fluid layer of constant mean depth. This type of modelling has its origins in the mid 19th century when Sir George Stokes – using a perturbation series approach, now known as the Stokes expansion – obtained approximate solutions for nonlinear wave motion.

## References

1. William Thomson (1887) "On ship waves," Institution of Mechanical Engineers, Proceedings, 38 : 409–34; illustrations, pp. 641–49.
2. The corresponding theory for the transient Kelvin wakes has further been derived for the Marangoni (Shu, Jian-Jun (2004). "Transient Marangoni waves due to impulsive motion of a submerged body". International Applied Mechanics. 40 (6): 709–14. arXiv:. Bibcode:2004IAM....40..709S. doi:10.1023/B:INAM.0000041400.70961.1b. S2CID   30003915.) and free-surface (Shu, Jian-Jun (2006). "Transient free-surface waves due to impulsive motion of a submerged source". Underwater Technology. 26 (4): 133–37. arXiv:. doi:10.3723/175605406782725023. S2CID   118527101.) waves.
3. The "hull Froude number" (Fr) of a ship is Fr = U / gL, where U is the ship's speed, g is the acceleration of gravity at the earth's surface, and L is the length of the ship's hull, a characteristic wavelength. See Marc Rabaud and Frédéric Moisy (2013) "Ship wakes: Kelvin or Mach angle?," Physical Review Letters, 110 (21) : 214503. Available on-line at: University of Paris, Sud; Alexandre Darmon, Michael Benzaquen, and Elie Raphaël (2014) "Kelvin wake pattern at large Froude numbers," Journal of Fluid Mechanics, 738 : R3-1–R3-8. Available on-line at: ESPCI ParisTech
4. G.B. Whitham (1974). Linear and Nonlinear Waves (John Wiley & Sons Inc., 1974) pp. 409–10 Online scan
5. Norwegian University of Science and Technology, "A 127-year-old physics riddle solved", Phys.org, Aug 21, 2019. Retrieved 22 August 2019
6. BoatWakes.org, Table of distances