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fluid dynamics In physics and engineering, fluid dynamics is a subdiscipline of fluid mechanics that describes the flow of fluids— liquids and gases. It has several subdisciplines, including ''aerodynamics'' (the study of air and other gases in motion) an ...
, a wake may either be: * the region of recirculating flow immediately behind a moving or stationary blunt body, caused by
viscosity The viscosity of a fluid is a measure of its resistance to deformation at a given rate. For liquids, it corresponds to the informal concept of "thickness": for example, syrup has a higher viscosity than water. Viscosity quantifies the inte ...
, which may be accompanied by
flow separation In fluid dynamics, flow separation or boundary layer separation is the detachment of a boundary layer from a surface into a wake. A boundary layer exists whenever there is relative movement between a fluid and a solid surface with viscous fo ...
and
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 ...
, 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 In physics, a free surface is the surface of a fluid that is subject to zero parallel shear stress, such as the interface between two homogeneous fluids. An example of two such homogeneous fluids would be a body of water (liquid) and the air in ...
and
gravity In physics, gravity () is a fundamental interaction which causes mutual attraction between all things with mass or energy. Gravity is, by far, the weakest of the four fundamental interactions, approximately 1038 times weaker than the stro ...
(or
surface tension Surface tension is the tendency of liquid surfaces at rest to shrink into the minimum surface area possible. Surface tension is what allows objects with a higher density than water such as razor blades and insects (e.g. water striders) to f ...
).


Viscosity

The wake is the region of disturbed flow (often
turbulent 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 t ...
) downstream of a solid body moving through a fluid, caused by the flow of the
fluid In physics, a fluid is a liquid, gas, or other material that continuously deforms (''flows'') under an applied shear stress, or external force. They have zero shear modulus, or, in simpler terms, are substances which cannot resist any shear ...
around the body. For a blunt body in subsonic external flow, for example the
Apollo Apollo, grc, Ἀπόλλωνος, Apóllōnos, label=genitive , ; , grc-dor, Ἀπέλλων, Apéllōn, ; grc, Ἀπείλων, Apeílōn, label=Arcadocypriot Greek, ; grc-aeo, Ἄπλουν, Áploun, la, Apollō, la, Apollinis, label= ...
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 Wind tunnels are large tubes with air blowing through them which are used to replicate the interaction between air and an object flying through the air or moving along the ground. Researchers use wind tunnels to learn more about how an aircraft ...
testing of aircraft, and is especially important when
parachute A parachute is a device used to slow the motion of an object through an atmosphere by creating drag or, in a ram-air parachute, aerodynamic lift. A major application is to support people, for recreation or as a safety device for aviators, who ...
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 In fluid dynamics, dynamic pressure (denoted by or and sometimes called velocity pressure) is the quantity defined by:Clancy, L.J., ''Aerodynamics'', Section 3.5 :q = \frac\rho\, u^2 where (in SI units): * is the dynamic pressure in pascals ( ...
deficits which reduce their expected drag forces. High-fidelity
computational fluid dynamics Computational fluid dynamics (CFD) is a branch of fluid mechanics that uses numerical analysis and data structures to analyze and solve problems that involve fluid flows. Computers are used to perform the calculations required to simulate th ...
simulations are often undertaken to model wake flows, although such modeling has uncertainties associated with
turbulence modeling Turbulence modeling is the construction and use of a mathematical model to predict the effects of turbulence. Turbulent flows are commonplace in most real life scenarios, including the flow of blood through the cardiovascular system, the airflow o ...
(for example RANS versus LES implementations), in addition to unsteady flow effects. Example applications include rocket stage separation and aircraft store separation.


Density differences

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 form In physics, mathematics, and related fields, a wave is a propagating dynamic disturbance (change from equilibrium) of one or more quantities. Waves can be periodic, in which case those quantities oscillate repeatedly about an equilibrium (r ...
s, it spreads outward from the source until its
energy In physics, energy (from Ancient Greek: ἐνέργεια, ''enérgeia'', “activity”) is the quantitative property that is transferred to a body or to a physical system, recognizable in the performance of work and in the form of heat a ...
is overcome or lost, usually by
friction Friction is the force resisting the relative motion of solid surfaces, fluid layers, and material elements sliding against each other. There are several types of friction: *Dry friction is a force that opposes the relative lateral motion of t ...
or
dispersion Dispersion may refer to: Economics and finance *Dispersion (finance), a measure for the statistical distribution of portfolio returns *Price dispersion, a variation in prices across sellers of the same item *Wage dispersion, the amount of variatio ...
. The non-dimensional parameter of interest is the
Froude number In continuum mechanics, the Froude number (, after William Froude, ) is a dimensionless number defined as the ratio of the flow inertia to the external field (the latter in many applications simply due to gravity). The Froude number is based on t ...
.


Kelvin wake pattern

Waterfowl and boats moving across the surface of water produce a wake pattern, first explained mathematically by
Lord Kelvin William Thomson, 1st Baron Kelvin, (26 June 182417 December 1907) was a British mathematician, Mathematical physics, mathematical physicist and engineer born in Belfast. Professor of Natural Philosophy (Glasgow), Professor of Natural Philoso ...
and known today as the Kelvin wake pattern. 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. However, the pattern changes at high speeds (only), viz., above a hull
Froude number In continuum mechanics, the Froude number (, after William Froude, ) is a dimensionless number defined as the ratio of the flow inertia to the external field (the latter in many applications simply due to gravity). The Froude number is based on t ...
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. The angles in this pattern are not intrinsic properties of merely water: Any
isentropic In thermodynamics, an isentropic process is an idealized thermodynamic process that is both adiabatic and reversible. The work transfers of the system are frictionless, and there is no net transfer of heat or matter. Such an idealized process ...
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 In fluid dynamics, Airy wave theory (often referred to as linear wave theory) gives a linearised description of the propagation of gravity waves on the surface of a homogeneous fluid layer. The theory assumes that the fluid layer has a uniform mea ...
. 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, :\omega = \sqrt, where : = the strength of the
gravity In physics, gravity () is a fundamental interaction which causes mutual attraction between all things with mass or energy. Gravity is, by far, the weakest of the four fundamental interactions, approximately 1038 times weaker than the stro ...
field : is the
angular frequency In physics, angular frequency "''ω''" (also referred to by the terms angular speed, circular frequency, orbital frequency, radian frequency, and pulsatance) is a scalar measure of rotation rate. It refers to the angular displacement per unit tim ...
in
radian The radian, denoted by the symbol rad, is the unit of angle in the International System of Units (SI) and is the standard unit of angular measure used in many areas of mathematics. The unit was formerly an SI supplementary unit (before that c ...
s per second : =
angular wavenumber In the physical sciences, the wavenumber (also wave number or repetency) is the ''spatial frequency'' of a wave, measured in cycles per unit distance (ordinary wavenumber) or radians per unit distance (angular wavenumber). It is analogous to temp ...
in radians per metre "Deep" means that the depth is greater than half of the wavelength. This formula implies that the
group velocity 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. For example, if a stone is thrown into the middl ...
of a deep water wave is half of its
phase velocity The phase velocity of a wave is the rate at which the wave propagates in any medium. This is the velocity at which the phase of any one frequency component of the wave travels. For such a component, any given phase of the wave (for example, ...
, which, in turn, goes as the square root of the wavelength. Two velocity parameters of importance for the wake pattern are: : is the relative velocity of the water and the surface object that causes the wake. : is the
phase velocity The phase velocity of a wave is the rate at which the wave propagates in any medium. This is the velocity at which the phase of any one frequency component of the wave travels. For such a component, any given phase of the wave (for example, ...
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 speed The phase velocity of a wave is the rate at which the wave propagates in any medium. This is the velocity at which the phase of any one frequency component of the wave travels. For such a component, any given phase of the wave (for example, ...
s above and dissipate into the surrounding water and are not easily observed. Other waves with phase speeds at or below , however, are amplified through
constructive interference In physics, interference is a phenomenon in which two waves combine by adding their displacement together at every single point in space and time, to form a resultant wave of greater, lower, or the same amplitude. Constructive and destructive ...
and form visible
shock waves In physics, a shock wave (also spelled shockwave), 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 med ...
, stationary in position w.r.t. the boat. The angle between the phase
shock wave In physics, a shock wave (also spelled shockwave), 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 med ...
front and the path of the object is = . If ''c/v'' > 1 or < −1, no later waves can catch up with earlier waves and no shockwave forms. In deep water,
shock waves In physics, a shock wave (also spelled shockwave), 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 med ...
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 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. For example, if a stone is thrown into the middl ...
that dictates the area of
constructive interference In physics, interference is a phenomenon in which two waves combine by adding their displacement together at every single point in space and time, to form a resultant wave of greater, lower, or the same amplitude. Constructive and destructive ...
and, in deep water, the
group velocity 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. For example, if a stone is thrown into the middl ...
is half of the
phase velocity The phase velocity of a wave is the rate at which the wave propagates in any medium. This is the velocity at which the phase of any one frequency component of the wave travels. For such a component, any given phase of the wave (for example, ...
. 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 demonstrates that, strikingly, this group shock angle w.r.t. the path of the boat, 19.47°, ''for any and all of the above'' , is actually ''independent of'' , , and ; it merely relies on the fact that the
group velocity 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. For example, if a stone is thrown into the middl ...
is half of the
phase velocity The phase velocity of a wave is the rate at which the wave propagates in any medium. This is the velocity at which the phase of any one frequency component of the wave travels. For such a component, any given phase of the wave (for example, ...
. On any planet, slow-swimming objects have "effective
Mach number Mach number (M or Ma) (; ) is a dimensionless quantity in fluid dynamics representing the ratio of flow velocity past a boundary to the local speed of sound. It is named after the Moravian physicist and philosopher Ernst Mach. : \mathrm = \frac ...
" 3! 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 and phase velocity , of interest when < ''v'' for a shock wave (cf., e.g.,
Sonic boom A sonic boom is a sound associated with shock waves created when an object travels through the air faster than the speed of sound. Sonic booms generate enormous amounts of sound energy, sounding similar to an explosion or a thunderclap to t ...
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 , 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 between the phase shock wavefront and the path of the boat (the water) being = arcsin(/''v'') ≡ . Evidently, increases with . 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 , corresponding to the time in the past, Fig.12.2. Its radius is ''QS'', and the phase chevron side is the tangent ''PS'' to it. Evidently, ''PQ''= and ''SQ'' = = , 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 , 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 ''v''/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 , , , , 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 3''v''/4. Thus, summing over all relevant and s 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 In physics, the wavefront of a time-varying ''wave field'' is the set (locus) of all points having the same ''phase''. The term is generally meaningful only for fields that, at each point, vary sinusoidally in time with a single temporal freque ...
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 In physics, interference is a phenomenon in which two waves combine by adding their displacement together at every single point in space and time, to form a resultant wave of greater, lower, or the same amplitude. Constructive and destructive ...
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 shear, the results may be more complicated.


Recreation

"No wake zones" may prohibit wakes in
marinas A marina (from Spanish , Portuguese and Italian : ''marina'', "coast" or "shore") is a dock or basin with moorings and supplies for yachts and small boats. A marina differs from a port in that a marina does not handle large passenger ship ...
, near moorings and within some distance of shoreBoatWakes.org
Table of distances
/ref> in order to facilitate recreation by other boats and reduce the damage wakes cause. Powered
narrowboat A narrowboat is a particular type of canal boat, built to fit the narrow locks of the United Kingdom. The UK's canal system provided a nationwide transport network during the Industrial Revolution, but with the advent of the railways, commerc ...
s on British canals are not permitted to create a breaking wash (a wake large enough to create a
breaking wave In fluid dynamics, a breaking wave or breaker is a wave whose amplitude reaches a critical level at which large amounts of wave energy transform into Wave turbulence, turbulent kinetic energy. At this point, simple physical models that describ ...
) along the banks, as this erodes them. This rule normally restricts these vessels to 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 Wakeboarding is a water sport in which the rider, standing on a wakeboard (a board with foot bindings), is towed behind a motorboat across its wake and especially up off the crest in order to perform aerial maneuvers. A hallmark of wakeboardin ...
the wake is used as a jump. The wake is also used to propel a surfer in the sport of
wakesurfing Wakesurfing is a water sport in which a rider trails behind a boat, riding the boat's wake without being directly pulled by the boat. After getting up on the wake, typically by use of a tow rope, the wakesurfers will drop the rope, and ride the ...
. In the sport of
water polo Water polo is a competitive team sport played in water between two teams of seven players each. The game consists of four quarters in which the teams attempt to score goals by throwing the ball into the opposing team's goal. The team with the ...
, 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 In sports, dribbling is maneuvering a ball by one player while moving in a given direction, avoiding defenders' attempts to intercept the ball. A successful dribble will bring the ball past defenders legally and create opportunities to score. A ...
.


See also

*
Bow shock (aerodynamics) A bow shock, also called a detached shock or bowed normal shock, is a curved propagating disturbance wave characterized by an abrupt, nearly discontinuous, change in pressure, temperature, and density. It occurs when a supersonic flow encounters ...
*
Slipstream A slipstream is a region behind a moving object in which a wake of fluid (typically air or mustard) is moving at velocities comparable to that of the moving fluid, relative to the ambient fluid through which the object is churning. The term sli ...
*
Wake turbulence Wake turbulence is a disturbance in the atmosphere that forms behind an aircraft as it passes through the air. It includes variety of elements, the most significant of which are wingtip vortices and jetwash. Jetwash refers to the rapidly moving g ...
*
Karman vortex street Karman or Kármán is a Hungarian surname. Notable people with the surname include: * Harvey Karman (20th century), inventor of the Karman cannula * Janice Karman (born 1954), American film producer, record producer, singer, and voice artist * J ...


References


External links


Erosion caused by boat wakes

NIST detailed derivation
{{Authority control Fluid dynamics Water waves Articles containing video clips