In

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 G ...

* Oseen flow
* Poiseuille's law
* Pressure head
* Relativistic Euler equations
* Stokes stream function
*

Encyclopedia: Fluid dynamics

National Committee for Fluid Mechanics Films (NCFMF)

containing films on several subjects in fluid dynamics (in

Gallery of fluid motion

"a visual record of the aesthetic and science of contemporary fluid mechanics," from the

List of Fluid Dynamics books

{{Authority control Piping Aerodynamics Continuum mechanics

physics
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which relat ...

and engineering
Engineering is the use of scientific principles to design and build machines, structures, and other items, including bridges, tunnels, roads, vehicles, and buildings. The discipline of engineering encompasses a broad range of more speciali ...

, fluid dynamics is a subdiscipline of fluid mechanics
Fluid mechanics is the branch of physics concerned with the mechanics of fluids (liquids, gases, and plasmas) and the forces on them.
It has applications in a wide range of disciplines, including mechanical, aerospace, civil, chemical and bio ...

that describes the flow of 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 f ...

s—liquid
A liquid is a nearly incompressible fluid that conforms to the shape of its container but retains a (nearly) constant volume independent of pressure. As such, it is one of the four fundamental states of matter (the others being solid, gas, a ...

s and gas
Gas is one of the four fundamental states of matter (the others being solid, liquid, and plasma).
A pure gas may be made up of individual atoms (e.g. a noble gas like neon), elemental molecules made from one type of atom (e.g. oxygen), or ...

es. It has several subdisciplines, including ''aerodynamics
Aerodynamics, from grc, ἀήρ ''aero'' (air) + grc, δυναμική (dynamics), is the study of the motion of air, particularly when affected by a solid object, such as an airplane wing. It involves topics covered in the field of fluid dyn ...

'' (the study of air and other gases in motion) and hydrodynamics (the study of liquids in motion). Fluid dynamics has a wide range of applications, including calculating force
In physics, a force is an influence that can change the motion of an object. A force can cause an object with mass to change its velocity (e.g. moving from a state of rest), i.e., to accelerate. Force can also be described intuitively as ...

s and moments on aircraft
An aircraft is a vehicle that is able to fly by gaining support from the air. It counters the force of gravity by using either static lift or by using the dynamic lift of an airfoil, or in a few cases the downward thrust from jet engines. ...

, determining the mass flow rate
In physics and engineering, mass flow rate is the mass of a substance which passes per unit of time. Its unit is kilogram per second in SI units, and slug per second or pound per second in US customary units. The common symbol is \dot ('' ...

of petroleum
Petroleum, also known as crude oil, or simply oil, is a naturally occurring yellowish-black liquid mixture of mainly hydrocarbons, and is found in geological formations. The name ''petroleum'' covers both naturally occurring unprocessed crude ...

through pipelines, predicting weather patterns, understanding nebula
A nebula ('cloud' or 'fog' in Latin; pl. nebulae, nebulæ or nebulas) is a distinct luminescent part of interstellar medium, which can consist of ionized, neutral or molecular hydrogen and also cosmic dust. Nebulae are often star-forming regi ...

e in interstellar space
Outer space, commonly shortened to space, is the expanse that exists beyond Earth and its atmosphere and between celestial bodies. Outer space is not completely empty—it is a near-perfect vacuum containing a low density of particles, predo ...

and modelling fission weapon detonation.
Fluid dynamics offers a systematic structure—which underlies these practical disciplines
Applied science is the use of the scientific method and knowledge obtained via conclusions from the method to attain practical goals. It includes a broad range of disciplines such as engineering and medicine. Applied science is often contrasted ...

—that embraces empirical and semi-empirical laws derived from flow measurement
Flow measurement is the quantification of bulk fluid movement. Flow can be measured in a variety of ways. The common types of flowmeters with industrial applications are listed below:
* a) Obstruction type (differential pressure or variable area) ...

and used to solve practical problems. The solution to a fluid dynamics problem typically involves the calculation of various properties of the fluid, such as 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 ...

, pressure
Pressure (symbol: ''p'' or ''P'') is the force applied perpendicular to the surface of an object per unit area over which that force is distributed. Gauge pressure (also spelled ''gage'' pressure)The preferred spelling varies by country a ...

, density
Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematically ...

, and temperature, as functions of space and time.
Before the twentieth century, ''hydrodynamics'' was synonymous with fluid dynamics. This is still reflected in names of some fluid dynamics topics, like magnetohydrodynamics
Magnetohydrodynamics (MHD; also called magneto-fluid dynamics or hydromagnetics) is the study of the magnetic properties and behaviour of electrically conducting fluids. Examples of such magnetofluids include plasmas, liquid metals, ...

and hydrodynamic stability
In fluid dynamics, hydrodynamic stability is the field which analyses the stability and the onset of instability of fluid flows. The study of hydrodynamic stability aims to find out if a given flow is stable or unstable, and if so, how these ins ...

, both of which can also be applied to gases.
Equations

The foundational axioms of fluid dynamics are theconservation law
In physics, a conservation law states that a particular measurable property of an isolated physical system does not change as the system evolves over time. Exact conservation laws include conservation of energy, conservation of linear momentum, c ...

s, specifically, conservation of mass
In physics and chemistry, the law of conservation of mass or principle of mass conservation states that for any system closed to all transfers of matter and energy, the mass of the system must remain constant over time, as the system's mass can ...

, conservation of linear momentum, and conservation of energy
In physics and chemistry, the law of conservation of energy states that the total energy of an isolated system remains constant; it is said to be ''conserved'' over time. This law, first proposed and tested by Émilie du Châtelet, means tha ...

(also known as the First Law of Thermodynamics
The first law of thermodynamics is a formulation of the law of conservation of energy, adapted for thermodynamic processes. It distinguishes in principle two forms of energy transfer, heat and thermodynamic work for a system of a constant amo ...

). These are based on classical mechanics
Classical mechanics is a physical theory describing the motion of macroscopic objects, from projectiles to parts of machinery, and astronomical objects, such as spacecraft, planets, stars, and galaxies. For objects governed by class ...

and are modified in quantum mechanics
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, qua ...

and general relativity
General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...

. They are expressed using the Reynolds transport theorem.
In addition to the above, fluids are assumed to obey the continuum assumption. Fluids are composed of molecules that collide with one another and solid objects. However, the continuum assumption assumes that fluids are continuous, rather than discrete. Consequently, it is assumed that properties such as density, pressure, temperature, and flow velocity are well-defined at infinitesimal
In mathematics, an infinitesimal number is a quantity that is closer to zero than any standard real number, but that is not zero. The word ''infinitesimal'' comes from a 17th-century Modern Latin coinage ''infinitesimus'', which originally refer ...

ly small points in space and vary continuously from one point to another. The fact that the fluid is made up of discrete molecules is ignored.
For fluids that are sufficiently dense to be a continuum, do not contain ionized species, and have flow velocities that are small in relation to the speed of light, the momentum equations for Newtonian fluid
A Newtonian fluid is a fluid in which the viscous stresses arising from its flow are at every point linearly correlated to the local strain rate — the rate of change of its deformation over time. Stresses are proportional to the rate of chan ...

s are 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 G ...

—which is a non-linear
In mathematics and science, a nonlinear system is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathematicians, and many othe ...

set of differential equations
In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, an ...

that describes the flow of a fluid whose stress depends linearly on flow velocity gradients and pressure. The unsimplified equations do not have a general closed-form solution, so they are primarily of use in 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 t ...

. The equations can be simplified in several ways, all of which make them easier to solve. Some of the simplifications allow some simple fluid dynamics problems to be solved in closed form.
In addition to the mass, momentum, and energy conservation equations, a thermodynamic
Thermodynamics is a branch of physics that deals with heat, work, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these quantities is governed by the four laws of the ...

equation of state that gives the pressure as a function of other thermodynamic variables is required to completely describe the problem. An example of this would be the perfect gas equation of state:
:$p=\; \backslash frac$
where is pressure
Pressure (symbol: ''p'' or ''P'') is the force applied perpendicular to the surface of an object per unit area over which that force is distributed. Gauge pressure (also spelled ''gage'' pressure)The preferred spelling varies by country a ...

, is density
Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematically ...

, and is the absolute temperature
Thermodynamic temperature is a quantity defined in thermodynamics as distinct from kinetic theory or statistical mechanics.
Historically, thermodynamic temperature was defined by Kelvin in terms of a macroscopic relation between thermodynami ...

, while is the gas constant
The molar gas constant (also known as the gas constant, universal gas constant, or ideal gas constant) is denoted by the symbol or . It is the molar equivalent to the Boltzmann constant, expressed in units of energy per temperature increment per ...

and is molar mass
In chemistry, the molar mass of a chemical compound is defined as the mass of a sample of that compound divided by the amount of substance which is the number of moles in that sample, measured in moles. The molar mass is a bulk, not molecul ...

for a particular gas. A constitutive relation may also be useful.
Conservation laws

Three conservation laws are used to solve fluid dynamics problems, and may be written inintegral
In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along with ...

or differential form. The conservation laws may be applied to a region of the flow called a ''control volume''. A control volume is a discrete volume in space through which fluid is assumed to flow. The integral formulations of the conservation laws are used to describe the change of mass, momentum, or energy within the control volume. Differential formulations of the conservation laws apply Stokes' theorem
Stokes's theorem, also known as the Kelvin–Stokes theorem Nagayoshi Iwahori, et al.:"Bi-Bun-Seki-Bun-Gaku" Sho-Ka-Bou(jp) 1983/12Written in Japanese)Atsuo Fujimoto;"Vector-Kai-Seki Gendai su-gaku rekucha zu. C(1)" :ja:培風館, Bai-Fu-Kan( ...

to yield an expression that may be interpreted as the integral form of the law applied to an infinitesimally small volume (at a point) within the flow.
Classifications

Compressible versus incompressible flow

All fluids arecompressible
In thermodynamics and fluid mechanics, the compressibility (also known as the coefficient of compressibility or, if the temperature is held constant, the isothermal compressibility) is a measure of the instantaneous relative volume change of a f ...

to an extent; that is, changes in pressure or temperature cause changes in density. However, in many situations the changes in pressure and temperature are sufficiently small that the changes in density are negligible. In this case the flow can be modelled as an incompressible flow
In fluid mechanics or more generally continuum mechanics, incompressible flow ( isochoric flow) refers to a flow in which the material density is constant within a fluid parcel—an infinitesimal volume that moves with the flow velocity. An ...

. Otherwise the more general compressible flow
Compressible flow (or gas dynamics) 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 (the ...

equations must be used.
Mathematically, incompressibility is expressed by saying that the density of a fluid parcel In fluid dynamics, within the framework of continuum mechanics, a fluid parcel is a very small amount of fluid, identifiable throughout its dynamic history while moving with the fluid flow. As it moves, the mass of a fluid parcel remains constant, ...

does not change as it moves in the flow field, that is,
: $\backslash frac\; =\; 0\; \backslash ,\; ,$
where is the material derivative
In continuum mechanics, the material derivative describes the time rate of change of some physical quantity (like heat or momentum) of a material element that is subjected to a space-and-time-dependent macroscopic velocity field. The material d ...

, which is the sum of local
Local may refer to:
Geography and transportation
* Local (train), a train serving local traffic demand
* Local, Missouri, a community in the United States
* Local government, a form of public administration, usually the lowest tier of administra ...

and convective derivatives. This additional constraint simplifies the governing equations, especially in the case when the fluid has a uniform density.
For flow of gases, to determine whether to use compressible or incompressible fluid dynamics, the 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 = \fr ...

of the flow is evaluated. As a rough guide, compressible effects can be ignored at Mach numbers below approximately 0.3. For liquids, whether the incompressible assumption is valid depends on the fluid properties (specifically the critical pressure and temperature of the fluid) and the flow conditions (how close to the critical pressure the actual flow pressure becomes). Acoustic problems always require allowing compressibility, since sound waves
In physics, sound is a vibration that propagates as an acoustic wave, through a transmission medium such as a gas, liquid or solid.
In human physiology and psychology, sound is the ''reception'' of such waves and their ''perception'' by th ...

are compression waves involving changes in pressure and density of the medium through which they propagate.
Newtonian versus non-Newtonian fluids

All fluids, exceptsuperfluids
Superfluidity is the characteristic property of a fluid with zero viscosity which therefore flows without any loss of kinetic energy. When stirred, a superfluid forms vortices that continue to rotate indefinitely. Superfluidity occurs in two ...

, are viscous, meaning that they exert some resistance to deformation: neighbouring parcels of fluid moving at different velocities exert viscous forces on each other. The velocity gradient is referred to as a strain rate
In materials science, strain rate is the change in strain ( deformation) of a material with respect to time.
The strain rate at some point within the material measures the rate at which the distances of adjacent parcels of the material change ...

; it has dimensions . Isaac Newton
Sir Isaac Newton (25 December 1642 – 20 March 1726/27) was an English mathematician, physicist, astronomer, alchemist, theologian, and author (described in his time as a " natural philosopher"), widely recognised as one of the ...

showed that for many familiar fluids such as water and air
The atmosphere of Earth is the layer of gases, known collectively as air, retained by Earth's gravity that surrounds the planet and forms its planetary atmosphere. The atmosphere of Earth protects life on Earth by creating pressure allowing ...

, the stress due to these viscous forces is linearly related to the strain rate. Such fluids are called Newtonian fluids
A Newtonian fluid is a fluid in which the viscous stresses arising from its flow are at every point linearly correlated to the local strain rate — the rate of change of its deformation over time. Stresses are proportional to the rate of chan ...

. The coefficient of proportionality is called the fluid's viscosity; for Newtonian fluids, it is a fluid property that is independent of the strain rate.
Non-Newtonian fluid
A non-Newtonian fluid is a fluid that does not follow Newton's law of viscosity, i.e., constant viscosity independent of stress. In non-Newtonian fluids, viscosity can change when under force to either more liquid or more solid. Ketchup, for ex ...

s have a more complicated, non-linear stress-strain behaviour. The sub-discipline of rheology
Rheology (; ) is the study of the flow of matter, primarily in a fluid (liquid or gas) state, but also as "soft solids" or solids under conditions in which they respond with plastic flow rather than deforming elastically in response to an appl ...

describes the stress-strain behaviours of such fluids, which include emulsion
An emulsion is a mixture of two or more liquids that are normally immiscible (unmixable or unblendable) owing to liquid-liquid phase separation. Emulsions are part of a more general class of two-phase systems of matter called colloids. Alth ...

s and slurries
A slurry is a mixture of denser solids suspended in liquid, usually water. The most common use of slurry is as a means of transporting solids or separating minerals, the liquid being a carrier that is pumped on a device such as a centrifugal pu ...

, some viscoelastic
In materials science and continuum mechanics, viscoelasticity is the property of materials that exhibit both viscous and elastic characteristics when undergoing deformation. Viscous materials, like water, resist shear flow and strain linear ...

materials such as blood and some polymer
A polymer (; Greek '' poly-'', "many" + '' -mer'', "part")
is a substance or material consisting of very large molecules called macromolecules, composed of many repeating subunits. Due to their broad spectrum of properties, both synthetic ...

s, and ''sticky liquids'' such as latex
Latex is an emulsion (stable dispersion) of polymer microparticles in water. Latexes are found in nature, but synthetic latexes are common as well.
In nature, latex is found as a milky fluid found in 10% of all flowering plants (angios ...

, honey
Honey is a sweet and viscous substance made by several bees, the best-known of which are honey bees. Honey is made and stored to nourish bee colonies. Bees produce honey by gathering and then refining the sugary secretions of plants (primaril ...

and lubricants
A lubricant (sometimes shortened to lube) is a substance that helps to reduce friction between surfaces in mutual contact, which ultimately reduces the heat generated when the surfaces move. It may also have the function of transmitting forces, t ...

.
Inviscid versus viscous versus Stokes flow

The dynamic of fluid parcels is described with the help ofNewton's second law
Newton's laws of motion are three basic laws of classical mechanics that describe the relationship between the motion of an object and the forces acting on it. These laws can be paraphrased as follows:
# A body remains at rest, or in motio ...

. An accelerating parcel of fluid is subject to inertial effects.
The Reynolds number
In fluid mechanics, the Reynolds number () is a dimensionless quantity that helps predict fluid flow patterns in different situations by measuring the ratio between inertial and viscous forces. At low Reynolds numbers, flows tend to be d ...

is a dimensionless quantity
A dimensionless quantity (also known as a bare quantity, pure quantity, or scalar quantity as well as quantity of dimension one) is a quantity to which no physical dimension is assigned, with a corresponding SI unit of measurement of one (or ...

which characterises the magnitude of inertial effects compared to the magnitude of viscous effects. A low Reynolds number () indicates that viscous forces are very strong compared to inertial forces. In such cases, inertial forces are sometimes neglected; this flow regime is called Stokes or creeping flow.
In contrast, high Reynolds numbers () indicate that the inertial effects have more effect on the velocity field than the viscous (friction) effects. In high Reynolds number flows, the flow is often modeled as an inviscid flow
In fluid dynamics, inviscid flow is the flow of an inviscid (zero- viscosity) fluid, also known as a superfluid. The Reynolds number of inviscid flow approaches infinity as the viscosity approaches zero. When viscous forces are neglected, suc ...

, an approximation in which viscosity is completely neglected. Eliminating viscosity allows 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 G ...

to be simplified into the Euler equations. The integration of the Euler equations along a streamline in an inviscid flow yields Bernoulli's equation. When, in addition to being inviscid, the flow is irrotational
In vector calculus, a conservative vector field is a vector field that is the gradient of some function. A conservative vector field has the property that its line integral is path independent; the choice of any path between two points does not ...

everywhere, Bernoulli's equation can completely describe the flow everywhere. Such flows are called potential flow
In fluid dynamics, potential flow (or ideal flow) describes the velocity field as the gradient of a scalar function: the velocity potential. As a result, a potential flow is characterized by an irrotational velocity field, which is a valid a ...

s, because the velocity field may be expressed as the gradient
In vector calculus, the gradient of a scalar-valued differentiable function of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p is the "direction and rate of fastest increase". If the grad ...

of a potential energy expression.
This idea can work fairly well when the Reynolds number is high. However, problems such as those involving solid boundaries may require that the viscosity be included. Viscosity cannot be neglected near solid boundaries because the no-slip condition generates a thin region of large strain rate, the boundary layer
In physics and fluid mechanics, a boundary layer is the thin layer of fluid in the immediate vicinity of a bounding surface formed by the fluid flowing along the surface. The fluid's interaction with the wall induces a no-slip boundary cond ...

, in which 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 int ...

effects dominate and which thus generates vorticity
In continuum mechanics, vorticity is a pseudovector field that describes the local spinning motion of a continuum near some point (the tendency of something to rotate), as would be seen by an observer located at that point and traveling along ...

. Therefore, to calculate net forces on bodies (such as wings), viscous flow equations must be used: inviscid flow theory fails to predict drag forces, a limitation known as the d'Alembert's paradox.
A commonly used model, especially in 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 t ...

, is to use two flow models: the Euler equations away from the body, and boundary layer
In physics and fluid mechanics, a boundary layer is the thin layer of fluid in the immediate vicinity of a bounding surface formed by the fluid flowing along the surface. The fluid's interaction with the wall induces a no-slip boundary cond ...

equations in a region close to the body. The two solutions can then be matched with each other, using the method of matched asymptotic expansions.
Steady versus unsteady flow

A flow that is not a function of time is called steady flow. Steady-state flow refers to the condition where the fluid properties at a point in the system do not change over time. Time dependent flow is known as unsteady (also called transient). Whether a particular flow is steady or unsteady, can depend on the chosen frame of reference. For instance, laminar flow over asphere
A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is the ...

is steady in the frame of reference that is stationary with respect to the sphere. In a frame of reference that is stationary with respect to a background flow, the flow is unsteady.
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 ...

flows are unsteady by definition. A turbulent flow can, however, be statistically stationary. The random velocity field is statistically stationary if all statistics are invariant under a shift in time. This roughly means that all statistical properties are constant in time. Often, the mean field is the object of interest, and this is constant too in a statistically stationary flow.
Steady flows are often more tractable than otherwise similar unsteady flows. The governing equations of a steady problem have one dimension fewer (time) than the governing equations of the same problem without taking advantage of the steadiness of the flow field.
Laminar versus turbulent flow

Turbulence is flow characterized by recirculation,eddies
In fluid dynamics, an eddy is the swirling of a fluid and the reverse current created when the fluid is in a turbulent flow regime. The moving fluid creates a space devoid of downstream-flowing fluid on the downstream side of the object. Fluid b ...

, and apparent random
In common usage, randomness is the apparent or actual lack of pattern or predictability in events. A random sequence of events, symbols or steps often has no order and does not follow an intelligible pattern or combination. Individual ra ...

ness. Flow in which turbulence is not exhibited is called laminar
Laminar means "flat". Laminar may refer to:
Terms in science and engineering:
*Laminar electronics or organic electronics, a branch of material sciences dealing with electrically conductive polymers and small molecules
* Laminar armour or "banded ...

. The presence of eddies or recirculation alone does not necessarily indicate turbulent flow—these phenomena may be present in laminar flow as well. Mathematically, turbulent flow is often represented via a Reynolds decomposition, in which the flow is broken down into the sum of an average
In ordinary language, an average is a single number taken as representative of a list of numbers, usually the sum of the numbers divided by how many numbers are in the list (the arithmetic mean). For example, the average of the numbers 2, 3, 4, 7, ...

component and a perturbation component.
It is believed that turbulent flows can be described well through the use of 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 G ...

. Direct numerical simulation (DNS), based on the Navier–Stokes equations, makes it possible to simulate turbulent flows at moderate Reynolds numbers. Restrictions depend on the power of the computer used and the efficiency of the solution algorithm. The results of DNS have been found to agree well with experimental data for some flows.
Most flows of interest have Reynolds numbers much too high for DNS to be a viable option, given the state of computational power for the next few decades. Any flight vehicle large enough to carry a human ( > 3 m), moving faster than is well beyond the limit of DNS simulation ( = 4 million). Transport aircraft wings (such as on an Airbus A300
The Airbus A300 is a wide-body airliner developed and manufactured by Airbus.
In September 1967, aircraft manufacturers in the United Kingdom, France, and West Germany signed a memorandum of understanding to develop a large airliner.
West ...

or Boeing 747
The Boeing 747 is a large, long-range wide-body airliner designed and manufactured by Boeing Commercial Airplanes in the United States between 1968 and 2022.
After introducing the 707 in October 1958, Pan Am wanted a jet times its size, t ...

) have Reynolds numbers of 40 million (based on the wing chord dimension). Solving these real-life flow problems requires turbulence models for the foreseeable future. Reynolds-averaged Navier–Stokes equations
The Reynolds-averaged Navier–Stokes equations (RANS equations) are time-averaged
equations of motion for fluid flow. The idea behind the equations is Reynolds decomposition, whereby an instantaneous quantity is decomposed into its time-average ...

(RANS) combined with turbulence modelling provides a model of the effects of the turbulent flow. Such a modelling mainly provides the additional momentum transfer by the Reynolds stresses, although the turbulence also enhances the heat
In thermodynamics, heat is defined as the form of energy crossing the boundary of a thermodynamic system by virtue of a temperature difference across the boundary. A thermodynamic system does not ''contain'' heat. Nevertheless, the term is ...

and mass transfer
Mass transfer is the net movement of mass from one location (usually meaning stream, phase, fraction or component) to another. Mass transfer occurs in many processes, such as absorption, evaporation, drying, precipitation, membrane filtratio ...

. Another promising methodology is large eddy simulation (LES), especially in the guise of detached eddy simulation (DES)—which is a combination of RANS turbulence modelling and large eddy simulation.
Other approximations

There are a large number of other possible approximations to fluid dynamic problems. Some of the more commonly used are listed below. * The '' Boussinesq approximation'' neglects variations in density except to calculatebuoyancy
Buoyancy (), or upthrust, is an upward force exerted by a fluid that opposes the weight of a partially or fully immersed object. In a column of fluid, pressure increases with depth as a result of the weight of the overlying fluid. Thus the pr ...

forces. It is often used in free convection
Convection is single or multiphase fluid flow that occurs spontaneously due to the combined effects of material property heterogeneity and body forces on a fluid, most commonly density and gravity (see buoyancy). When the cause of the convec ...

problems where density changes are small.
* '' Lubrication theory'' and '' Hele–Shaw flow'' exploits the large aspect ratio of the domain to show that certain terms in the equations are small and so can be neglected.
* '' Slender-body theory'' is a methodology used in Stokes flow problems to estimate the force on, or flow field around, a long slender object in a viscous fluid.
* The '' shallow-water equations'' can be used to describe a layer of relatively inviscid fluid with a 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 ...

, in which surface gradients
In vector calculus, the gradient of a scalar-valued differentiable function of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p is the "direction and rate of fastest increase". If the grad ...

are small.
* ''Darcy's law
Darcy's law is an equation that describes the flow of a fluid through a porous medium. The law was formulated by Henry Darcy based on results of experiments on the flow of water through beds of sand, forming the basis of hydrogeology, a branch of ...

'' is used for flow in porous media, and works with variables averaged over several pore-widths.
* In rotating systems, the '' quasi-geostrophic equations'' assume an almost perfect balance between pressure gradient
In atmospheric science, the pressure gradient (typically of air but more generally of any fluid) is a physical quantity that describes in which direction and at what rate the pressure increases the most rapidly around a particular location. The ...

s and the Coriolis force
In physics, the Coriolis force is an inertial or fictitious force that acts on objects in motion within a frame of reference that rotates with respect to an inertial frame. In a reference frame with clockwise rotation, the force acts to the ...

. It is useful in the study of atmospheric dynamics.
Multidisciplinary types

Flows according to Mach regimes

While many flows (such as flow of water through a pipe) occur at lowMach 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 = \fr ...

s (subsonic
Subsonic may refer to:
Motion through a medium
* Any speed lower than the speed of sound within a sound-propagating medium
* Subsonic aircraft, a flying machine that flies at air speeds lower than the speed of sound
* Subsonic ammunition, a type ...

flows), many flows of practical interest in aerodynamics or in turbomachines occur at high fractions of ( transonic flows) or in excess of it (supersonic
Supersonic speed is the speed of an object that exceeds the speed of sound ( Mach 1). For objects traveling in dry air of a temperature of 20 °C (68 °F) at sea level, this speed is approximately . Speeds greater than five times ...

or even hypersonic flows). New phenomena occur at these regimes such as instabilities in transonic flow, shock waves for supersonic flow, or non-equilibrium chemical behaviour due to ionization in hypersonic flows. In practice, each of those flow regimes is treated separately.
Reactive versus non-reactive flows

Reactive flows are flows that are chemically reactive, which finds its applications in many areas, includingcombustion
Combustion, or burning, is a high-temperature exothermic redox chemical reaction between a fuel (the reductant) and an oxidant, usually atmospheric oxygen, that produces oxidized, often gaseous products, in a mixture termed as smoke. Combustio ...

(IC engine
An internal combustion engine (ICE or IC engine) is a heat engine in which the combustion of a fuel occurs with an oxidizer (usually air) in a combustion chamber that is an integral part of the working fluid flow circuit. In an internal combus ...

), propulsion
Propulsion is the generation of force by any combination of pushing or pulling to modify the translational motion of an object, which is typically a rigid body (or an articulated rigid body) but may also concern a fluid. The term is derived fro ...

devices (rockets
A rocket (from it, rocchetto, , bobbin/spool) is a vehicle that uses jet propulsion to accelerate without using the surrounding air. A rocket engine produces thrust by reaction to exhaust expelled at high speed. Rocket engines work entirel ...

, jet engines
A jet engine is a type of reaction engine discharging a fast-moving jet of heated gas (usually air) that generates thrust by jet propulsion. While this broad definition can include rocket, water jet, and hybrid propulsion, the term typic ...

, and so on), detonations
Detonation () is a type of combustion involving a supersonic exothermic front accelerating through a medium that eventually drives a shock front propagating directly in front of it. Detonations propagate supersonically through shock waves with ...

, fire and safety hazards, and astrophysics. In addition to conservation of mass, momentum and energy, conservation of individual species (for example, mass fraction of methane
Methane ( , ) is a chemical compound with the chemical formula (one carbon atom bonded to four hydrogen atoms). It is a group-14 hydride, the simplest alkane, and the main constituent of natural gas. The relative abundance of methane on E ...

in methane combustion) need to be derived, where the production/depletion rate of any species are obtained by simultaneously solving the equations of chemical kinetics
Chemical kinetics, also known as reaction kinetics, is the branch of physical chemistry that is concerned with understanding the rates of chemical reactions. It is to be contrasted with chemical thermodynamics, which deals with the direction in ...

.
Magnetohydrodynamics

Magnetohydrodynamics
Magnetohydrodynamics (MHD; also called magneto-fluid dynamics or hydromagnetics) is the study of the magnetic properties and behaviour of electrically conducting fluids. Examples of such magnetofluids include plasmas, liquid metals, ...

is the multidisciplinary study of the flow of electrically conducting fluids in electromagnetic
In physics, electromagnetism is an interaction that occurs between particles with electric charge. It is the second-strongest of the four fundamental interactions, after the strong force, and it is the dominant force in the interactions ...

fields. Examples of such fluids include plasmas, liquid metals, and salt water
Saline water (more commonly known as salt water) is water that contains a high concentration of dissolved salts (mainly sodium chloride). On the United States Geological Survey (USGS) salinity scale, saline water is saltier than brackish water, ...

. The fluid flow equations are solved simultaneously with Maxwell's equations
Maxwell's equations, or Maxwell–Heaviside equations, are a set of coupled partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, and electric circuits.
...

of electromagnetism.
Relativistic fluid dynamics

Relativistic fluid dynamics studies the macroscopic and microscopic fluid motion at large velocities comparable to the velocity of light. This branch of fluid dynamics accounts for the relativistic effects both from thespecial theory of relativity
In physics, the special theory of relativity, or special relativity for short, is a scientific theory regarding the relationship between space and time. In Albert Einstein's original treatment, the theory is based on two postulates:
# The law ...

and the general theory of relativity
General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physic ...

. The governing equations are derived in Riemannian geometry
Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, smooth manifolds with a ''Riemannian metric'', i.e. with an inner product on the tangent space at each point that varies smoothly from point to point ...

for Minkowski spacetime
In mathematical physics, Minkowski space (or Minkowski spacetime) () is a combination of Three-dimensional space, three-dimensional Euclidean space and time into a four-dimensional manifold where the spacetime interval between any two Event (rel ...

.
Fluctuating hydrodynamics

This branch of fluid dynamics augments the standard hydrodynamic equations with stochastic fluxes that model thermal fluctuations. As formulated byLandau
Landau ( pfl, Landach), officially Landau in der Pfalz, is an autonomous (''kreisfrei'') town surrounded by the Südliche Weinstraße ("Southern Wine Route") district of southern Rhineland-Palatinate, Germany. It is a university town (since 1990 ...

and Lifshitz,
a white noise
In signal processing, white noise is a random signal having equal intensity at different frequencies, giving it a constant power spectral density. The term is used, with this or similar meanings, in many scientific and technical discipline ...

contribution obtained from the fluctuation-dissipation theorem of statistical mechanics
In physics, statistical mechanics is a mathematical framework that applies statistical methods and probability theory to large assemblies of microscopic entities. It does not assume or postulate any natural laws, but explains the macroscopic b ...

is added to the viscous stress tensor and heat flux
Heat flux or thermal flux, sometimes also referred to as ''heat flux density'', heat-flow density or ''heat flow rate intensity'' is a flow of energy per unit area per unit time. In SI its units are watts per square metre (W/m2). It has both a ...

.
Terminology

The concept of pressure is central to the study of both fluid statics and fluid dynamics. A pressure can be identified for every point in a body of fluid, regardless of whether the fluid is in motion or not. Pressure can be measured using an aneroid, Bourdon tube, mercury column, or various other methods. Some of the terminology that is necessary in the study of fluid dynamics is not found in other similar areas of study. In particular, some of the terminology used in fluid dynamics is not used influid statics
Fluid statics or hydrostatics is the branch of fluid mechanics that studies the condition of the equilibrium of a floating body and submerged body "fluids at hydrostatic equilibrium and the pressure in a fluid, or exerted by a fluid, on an im ...

.
Terminology in incompressible fluid dynamics

The concepts of total pressure anddynamic 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 ( ...

arise from Bernoulli's equation and are significant in the study of all fluid flows. (These two pressures are not pressures in the usual sense—they cannot be measured using an aneroid, Bourdon tube or mercury column.) To avoid potential ambiguity when referring to pressure in fluid dynamics, many authors use the term static pressure
In fluid mechanics the term static pressure has several uses:
* In the design and operation of aircraft, ''static pressure'' is the air pressure in the aircraft's static pressure system.
* In fluid dynamics, many authors use the term ''static pr ...

to distinguish it from total pressure and dynamic pressure. Static pressure
In fluid mechanics the term static pressure has several uses:
* In the design and operation of aircraft, ''static pressure'' is the air pressure in the aircraft's static pressure system.
* In fluid dynamics, many authors use the term ''static pr ...

is identical to pressure and can be identified for every point in a fluid flow field.
A point in a fluid flow where the flow has come to rest (that is to say, speed is equal to zero adjacent to some solid body immersed in the fluid flow) is of special significance. It is of such importance that it is given a special name—a stagnation point. The static pressure at the stagnation point is of special significance and is given its own name—stagnation pressure
In fluid dynamics, stagnation pressure is the static pressure at a stagnation point in a fluid flow.Clancy, L.J., ''Aerodynamics'', Section 3.5 At a stagnation point the fluid velocity is zero. In an incompressible flow, stagnation pressure is eq ...

. In incompressible flows, the stagnation pressure at a stagnation point is equal to the total pressure throughout the flow field.
Terminology in compressible fluid dynamics

In a compressible fluid, it is convenient to define the total conditions (also called stagnation conditions) for all thermodynamic state properties (such as total temperature, total enthalpy, total speed of sound). These total flow conditions are a function of the fluid velocity and have different values in frames of reference with different motion. To avoid potential ambiguity when referring to the properties of the fluid associated with the state of the fluid rather than its motion, the prefix "static" is commonly used (such as static temperature and static enthalpy). Where there is no prefix, the fluid property is the static condition (so "density" and "static density" mean the same thing). The static conditions are independent of the frame of reference. Because the total flow conditions are defined byisentropic
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 ...

ally bringing the fluid to rest, there is no need to distinguish between total entropy and static entropy as they are always equal by definition. As such, entropy is most commonly referred to as simply "entropy".
About

Fields of study

*Acoustic theory Acoustic theory is a scientific field that relates to the description of sound waves. It derives from fluid dynamics. See acoustics for the engineering approach.
For sound waves of any magnitude of a disturbance in velocity, pressure, and density w ...

*Aerodynamics
Aerodynamics, from grc, ἀήρ ''aero'' (air) + grc, δυναμική (dynamics), is the study of the motion of air, particularly when affected by a solid object, such as an airplane wing. It involves topics covered in the field of fluid dyn ...

*Aeroelasticity
Aeroelasticity is the branch of physics and engineering studying the interactions between the inertial, elastic, and aerodynamic forces occurring while an elastic body is exposed to a fluid flow. The study of aeroelasticity may be broadly classi ...

*Aeronautics
Aeronautics is the science or art involved with the study, design, and manufacturing of air flight–capable machines, and the techniques of operating aircraft and rockets within the atmosphere. The British Royal Aeronautical Society ident ...

*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 t ...

*Flow measurement
Flow measurement is the quantification of bulk fluid movement. Flow can be measured in a variety of ways. The common types of flowmeters with industrial applications are listed below:
* a) Obstruction type (differential pressure or variable area) ...

*Geophysical fluid dynamics
Geophysical fluid dynamics, in its broadest meaning, refers to the fluid dynamics of naturally occurring flows, such as lava flows, oceans, and planetary atmospheres, on Earth and other planets.
Two physical features that are common to many of th ...

*Hemodynamics
Hemodynamics or haemodynamics are the dynamics of blood flow. The circulatory system is controlled by homeostatic mechanisms of autoregulation, just as hydraulic circuits are controlled by control systems. The hemodynamic response continuousl ...

*Hydraulics
Hydraulics (from Greek: Υδραυλική) is a technology and applied science using engineering, chemistry, and other sciences involving the mechanical properties and use of liquids. At a very basic level, hydraulics is the liquid count ...

*Hydrology
Hydrology () is the scientific study of the movement, distribution, and management of water on Earth and other planets, including the water cycle, water resources, and environmental watershed sustainability. A practitioner of hydrology is cal ...

*Hydrostatics
Fluid statics or hydrostatics is the branch of fluid mechanics that studies the condition of the equilibrium of a floating body and submerged body "fluids at hydrostatic equilibrium and the pressure in a fluid, or exerted by a fluid, on an imm ...

*Electrohydrodynamics
Electrohydrodynamics (EHD), also known as electro-fluid-dynamics (EFD) or electrokinetics, is the study of the dynamics of electrically charged fluids. It is the study of the motions of ionized particles or molecules and their interactions with ...

*Magnetohydrodynamics
Magnetohydrodynamics (MHD; also called magneto-fluid dynamics or hydromagnetics) is the study of the magnetic properties and behaviour of electrically conducting fluids. Examples of such magnetofluids include plasmas, liquid metals, ...

* Quantum hydrodynamics
Mathematical equations and concepts

* Airy wave theory * Benjamin–Bona–Mahony equation * Boussinesq approximation (water waves) * Different types of boundary conditions in fluid dynamics * Elementary flow *Helmholtz's theorems
In fluid mechanics, Helmholtz's theorems, named after Hermann von Helmholtz, describe the three-dimensional motion of fluid in the vicinity of vortex lines. These theorems apply to inviscid flows and flows where the influence of viscous forces are ...

* Kirchhoff equations
* Knudsen equation
*Manning equation The Manning formula or Manning's equation is an empirical formula estimating the average velocity of a liquid flowing in a conduit that does not completely enclose the liquid, i.e., open channel flow. However, this equation is also used for calcula ...

* Mild-slope equation
*Morison equation
In fluid dynamics the Morison equation is a semi-empirical equation for the inline force on a body in oscillatory flow. It is sometimes called the MOJS equation after all four authors—Morison, O'Brien, Johnson and Schaaf—of the 1950 paper in ...

*Stream function
The stream function is defined for incompressible ( divergence-free) flows in two dimensions – as well as in three dimensions with axisymmetry. The flow velocity components can be expressed as the derivatives of the scalar stream function. T ...

* Streamlines, streaklines and pathlines
* Torricelli's Law
Types of fluid flow

*Aerodynamic force
In fluid mechanics, an aerodynamic force is a force exerted on a body by the air (or other gas) in which the body is immersed, and is due to the relative motion between the body and the gas.
Force
There are two causes of aerodynamic force:
...

*Convection
Convection is single or multiphase fluid flow that occurs spontaneously due to the combined effects of material property heterogeneity and body forces on a fluid, most commonly density and gravity (see buoyancy). When the cause of the convec ...

*Cavitation
Cavitation is a phenomenon in which the static pressure of a liquid reduces to below the liquid's vapour pressure, leading to the formation of small vapor-filled cavities in the liquid. When subjected to higher pressure, these cavities, cal ...

*Compressible flow
Compressible flow (or gas dynamics) 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 (the ...

* Couette flow
* Effusive limit
* Free molecular flow
*Incompressible flow
In fluid mechanics or more generally continuum mechanics, incompressible flow ( isochoric flow) refers to a flow in which the material density is constant within a fluid parcel—an infinitesimal volume that moves with the flow velocity. An ...

*Inviscid flow
In fluid dynamics, inviscid flow is the flow of an inviscid (zero- viscosity) fluid, also known as a superfluid. The Reynolds number of inviscid flow approaches infinity as the viscosity approaches zero. When viscous forces are neglected, suc ...

* Isothermal flow
* Open channel flow
* Pipe flow
* Pressure-driven flow
* Secondary flow
* Stream thrust averaging
*Superfluidity
Superfluidity is the characteristic property of a fluid with zero viscosity which therefore flows without any loss of kinetic energy. When stirred, a superfluid forms vortices that continue to rotate indefinitely. Superfluidity occurs in two ...

* Transient flow
*Two-phase flow
In fluid mechanics, two-phase flow is a flow of gas and liquid — a particular example of multiphase flow. Two-phase flow can occur in various forms, such as flows transitioning from pure liquid to vapor as a result of external heating, sepa ...

Fluid properties

* List of hydrodynamic instabilities *Newtonian fluid
A Newtonian fluid is a fluid in which the viscous stresses arising from its flow are at every point linearly correlated to the local strain rate — the rate of change of its deformation over time. Stresses are proportional to the rate of chan ...

*Non-Newtonian fluid
A non-Newtonian fluid is a fluid that does not follow Newton's law of viscosity, i.e., constant viscosity independent of stress. In non-Newtonian fluids, viscosity can change when under force to either more liquid or more solid. Ketchup, for ex ...

*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) t ...

* Vapour pressure
Fluid phenomena

* Balanced flow *Boundary layer
In physics and fluid mechanics, a boundary layer is the thin layer of fluid in the immediate vicinity of a bounding surface formed by the fluid flowing along the surface. The fluid's interaction with the wall induces a no-slip boundary cond ...

* Coanda effect
*Convection cell
In the field of fluid dynamics, a convection cell is the phenomenon that occurs when density differences exist within a body of liquid or gas. These density differences result in rising and/or falling currents, which are the key characteristics ...

* Convergence/Bifurcation
* Darwin drift
*Drag (force)
In fluid dynamics, drag (sometimes called air resistance, a type of friction, or fluid resistance, another type of friction or fluid friction) is a force acting opposite to the relative motion of any object moving with respect to a surrounding flu ...

* Droplet vaporization
*Hydrodynamic stability
In fluid dynamics, hydrodynamic stability is the field which analyses the stability and the onset of instability of fluid flows. The study of hydrodynamic stability aims to find out if a given flow is stable or unstable, and if so, how these ins ...

* Kaye effect
*Lift (force)
A fluid flowing around an object exerts a force on it. Lift is the component of this force that is perpendicular to the oncoming flow direction. It contrasts with the drag force, which is the component of the force parallel to the flow dire ...

*Magnus effect
The Magnus effect is an observable phenomenon commonly associated with a spinning object moving through a fluid. The path of the spinning object is deflected in a manner not present when the object is not spinning. The deflection can be ex ...

*Ocean current
An ocean current is a continuous, directed movement of sea water generated by a number of forces acting upon the water, including wind, the Coriolis effect, breaking waves, cabbeling, and temperature and salinity differences. Depth contours ...

*Ocean surface waves
In fluid dynamics, a wind wave, water wave, or wind-generated water wave, is a surface wave that occurs on the free surface of bodies of water as a result from the wind blowing over the water surface. The contact distance in the direction o ...

*Rossby wave
Rossby waves, also known as planetary waves, are a type of inertial wave naturally occurring in rotating fluids. They were first identified by Sweden-born American meteorologist Carl-Gustaf Arvid Rossby. They are observed in the atmospheres and ...

*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 me ...

*Soliton
In mathematics and physics, a soliton or solitary wave is a self-reinforcing wave packet that maintains its shape while it propagates at a constant velocity. Solitons are caused by a cancellation of nonlinear and dispersive effects in the mediu ...

* Stokes drift
* Teapot effect
* Thread breakup
* Turbulent jet breakup
* Upstream contamination
*Venturi effect
The Venturi effect is the reduction in fluid pressure that results when a fluid flows through a constricted section (or choke) of a pipe. The Venturi effect is named after its discoverer, the 18th century Italian physicist, Giovanni Battista ...

*Vortex
In fluid dynamics, a vortex ( : vortices or vortexes) is a region in a fluid in which the flow revolves around an axis line, which may be straight or curved. Vortices form in stirred fluids, and may be observed in smoke rings, whirlpools in ...

*Water hammer
Hydraulic shock ( colloquial: water hammer; fluid hammer) is a pressure surge or wave caused when a fluid in motion, usually a liquid but sometimes also a gas is forced to stop or change direction suddenly; a momentum change. This phenomenon c ...

*Wave drag
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 (re ...

*Wind
Wind is the natural movement of air or other gases relative to a planet's surface. Winds occur on a range of scales, from thunderstorm flows lasting tens of minutes, to local breezes generated by heating of land surfaces and lasting a few ...

Applications

*Acoustics
Acoustics is a branch of physics that deals with the study of mechanical waves in gases, liquids, and solids including topics such as vibration, sound, ultrasound and infrasound. A scientist who works in the field of acoustics is an acousticia ...

*Aerodynamics
Aerodynamics, from grc, ἀήρ ''aero'' (air) + grc, δυναμική (dynamics), is the study of the motion of air, particularly when affected by a solid object, such as an airplane wing. It involves topics covered in the field of fluid dyn ...

* Cryosphere science
* EFDC Explorer
*Fluidics
Fluidics, or fluidic logic, is the use of a fluid to perform analog or digital operations similar to those performed with electronics.
The physical basis of fluidics is pneumatics and hydraulics, based on the theoretical foundation of fluid dy ...

*Fluid power
Fluid power is the use of fluids under pressure to generate, control, and transmit power. Fluid power is subdivided into hydraulics using a liquid such as mineral oil or water, and pneumatics using a gas such as air or other gases. Compressed-a ...

*Geodynamics
Geodynamics is a subfield of geophysics dealing with dynamics of the Earth. It applies physics, chemistry and mathematics to the understanding of how mantle convection leads to plate tectonics and geologic phenomena such as seafloor spreading, ...

*Hydraulic machinery
Hydraulic machines use liquid fluid power to perform work. Heavy construction vehicles are a common example. In this type of machine, hydraulic fluid is pumped to various hydraulic motors and hydraulic cylinders throughout the machine and ...

*Meteorology
Meteorology is a branch of the atmospheric sciences (which include atmospheric chemistry and physics) with a major focus on weather forecasting. The study of meteorology dates back millennia, though significant progress in meteorology did no ...

*Naval architecture
Naval architecture, or naval engineering, is an engineering discipline incorporating elements of mechanical, electrical, electronic, software and safety engineering as applied to the engineering design process, shipbuilding, maintenance, and o ...

*Oceanography
Oceanography (), also known as oceanology and ocean science, is the scientific study of the oceans. It is an Earth science, which covers a wide range of topics, including ecosystem dynamics; ocean currents, waves, and geophysical fluid dynami ...

*
*Pneumatics
Pneumatics (from Greek ‘wind, breath’) is a branch of engineering that makes use of gas or pressurized air.
Pneumatic systems used in industry are commonly powered by compressed air or compressed inert gases. A centrally located and ...

* 3D computer graphics
Fluid dynamics journals

* ''Annual Review of Fluid Mechanics
''Annual Review of Fluid Mechanics'' is a peer-reviewed scientific journal covering research on fluid mechanics. It is published once a year by Annual Reviews and the editors are Parviz Moin and Howard Stone.
As of 2022, ''Journal Citation Rep ...

''
* ''Journal of Fluid Mechanics
The ''Journal of Fluid Mechanics'' is a peer-reviewed scientific journal in the field of fluid mechanics. It publishes original work on theoretical, computational, and experimental aspects of the subject.
The journal is published by Cambridge Uni ...

''
* '' Physics of Fluids''
* '' Physical Review Fluids''
* '' Experiments in Fluids''
* ''European Journal of Mechanics B: Fluids''
* ''Theoretical and Computational Fluid Dynamics''
* ''Computers and Fluids''
* '' International Journal for Numerical Methods in Fluids''
* '' Flow, Turbulence and Combustion''
Miscellaneous

* Important publications in fluid dynamics *Isosurface
An isosurface is a three-dimensional analog of an isoline. It is a surface that represents points of a constant value (e.g. pressure, temperature, velocity, density) within a volume of space; in other words, it is a level set of a continuous ...

* Keulegan–Carpenter number
* Rotating tank
*Sound barrier
The sound barrier or sonic barrier is the large increase in aerodynamic drag and other undesirable effects experienced by an aircraft or other object when it approaches the speed of sound. When aircraft first approached the speed of sound, ...

*Beta plane
In geophysical fluid dynamics, an approximation whereby the Coriolis parameter, ''f'', is set to vary linearly in space is called a beta plane approximation.
On a rotating sphere such as the Earth, ''f'' varies with the sine of latitude; in the so ...

* Immersed boundary method
*Bridge scour
Bridge scour is the removal of sediment such as sand and gravel from around bridge abutments or piers. Hydrodynamic scour, caused by fast flowing water, can carve out ''scour holes'', compromising the integrity of a structure.
In the United Stat ...

* Finite volume method for unsteady flow
* Flow visualization
Flow visualization or flow visualisation in fluid dynamics is used to make the flow patterns visible, in order to get qualitative or quantitative information on them.
Overview
Flow visualization is the art of making flow patterns visible. ...

See also

* * * * * * * * * * * * * * * * * * * * * * (hydrodynamic) * * * * * * * * * * * * * (aerodynamics) * * * * * * * * *References

Further reading

* * * * * Originally published in 1879, the 6th extended edition appeared first in 1932. * Originally published in 1938. * *Encyclopedia: Fluid dynamics

Scholarpedia
''Scholarpedia'' is an English-language wiki-based online encyclopedia with features commonly associated with open-access online academic journals, which aims to have quality content in science and medicine.
''Scholarpedia'' articles are wri ...

External links

National Committee for Fluid Mechanics Films (NCFMF)

containing films on several subjects in fluid dynamics (in

RealMedia
RealMedia is a proprietary multimedia container format created by RealNetworks with the filename extension . RealMedia is generally used in conjunction with RealVideo and RealAudio, while also being used for streaming content over the Internet ...

format)
Gallery of fluid motion

"a visual record of the aesthetic and science of contemporary fluid mechanics," from the

American Physical Society
The American Physical Society (APS) is a not-for-profit membership organization of professionals in physics and related disciplines, comprising nearly fifty divisions, sections, and other units. Its mission is the advancement and diffusion of k ...

List of Fluid Dynamics books

{{Authority control Piping Aerodynamics Continuum mechanics