TheInfoList

In
physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. "Physical scie ...

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 Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in o ...
that describes the flow of
fluid In physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular s ...
s—
liquid A liquid is a nearly incompressible In fluid mechanics Fluid mechanics is the branch of physics concerned with the mechanics Mechanics (Ancient Greek, Greek: ) is the area of physics concerned with the motions of physical objects, ...

s and
gas Gas is one of the four fundamental states of matter In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space ...

es. It has several subdisciplines, including ''
aerodynamics Aerodynamics, from Greek#REDIRECT Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population is appr ...
'' (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 (physics), motion of an Physical object, object. A force can cause an object with mass to change its velocity (e.g. moving from a Newton's first law, state of rest), i.e., to acce ...

s and moments on
aircraft An aircraft is a vehicle that is able to flight, fly by gaining support from the Atmosphere of Earth, air. It counters the force of gravity by using either Buoyancy, static lift or by using the Lift (force), dynamic lift of an airfoil, or in ...

, 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 of measurement, unit is kilogram per second in SI units, and Slug (unit), slug per second or pound (mass), pound per second in US custo ...
of
petroleum Petroleum, also known as crude oil and oil, is a naturally occurring, yellowish-black liquid A liquid is a nearly incompressible In fluid mechanics or more generally continuum mechanics, incompressible flow (isochoric process, isoc ...

through pipelines, predicting weather patterns, understanding
nebula A nebula ( for 'cloud' or 'fog'; pl. nebulae, nebulæ or nebulas) is a distinct body of s (which can consist of , , , ; possibly as ). Originally, the term was used to describe any diffused , including beyond the . The , for instance, was once ...

e in
interstellar space Outer space, commonly shortened to space, is the expanse that exists beyond Earth and Earth atmosphere, its atmosphere and between astronomical object, celestial bodies. Outer space is not completely empty—it is a hard vacuum containing a ...
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 The scientific method is an Empirical evidence, empirical method of acquiring knowledge that has characterized the development of science since at least the 17th century. It involves caref ...
—that embraces empirical and semi-empirical laws derived from
flow measurement Flow measurement is the quantification of bulk fluid In physics, a fluid is a substance that continually Deformation (mechanics), deforms (flows) under an applied shear stress, or external force. Fluids are a Phase (matter), phase of matter a ...
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 velocityIn continuum mechanics the flow velocity in 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 aerodyna ...
,
pressure Pressure (symbol: ''p'' or ''P'') is the force In physics, a force is an influence that can change the motion (physics), motion of an Physical object, object. A force can cause an object with mass to change its velocity (e.g. moving fr ...

,
density The density (more precisely, the volumetric mass density; also known as specific mass), of a substance is its per unit . The symbol most often used for density is ''ρ'' (the lower case Greek letter ), although the Latin letter ''D'' can also ...

, and
temperature Temperature ( ) is a physical quantity that expresses hot and cold. It is the manifestation of thermal energy Thermal radiation in visible light can be seen on this hot metalwork. Thermal energy refers to several distinct physical concept ...

, 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 magneto-fluid dynamics or hydro­magnetics) is the study of the magnetic properties and behaviour of electrically conducting Electrical resistivity (also called specific electrical resistance or volume res ...
and
hydrodynamic stability In 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 i ...
, both of which can also be applied to gases.

# Equations

The foundational axioms of fluid dynamics are the
conservation law In physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular succ ...
s, specifically,
conservation of mass In and , the law of conservation of mass or principle of mass conservation states that for any to all transfers of and , the of the system must remain constant over time, as the system's mass cannot change, so quantity can neither be added n ...
,
conservation of linear momentum In Newtonian mechanics, linear momentum, translational momentum, or simply momentum ( pl. momenta) is the product of the mass Mass is both a property Property (''latin: Res Privata'') in the Abstract and concrete, abstract is what ...
, and
conservation of energy In physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular s ...
(also known as
First Law of Thermodynamics The first law of thermodynamics is a version of the law of conservation of energy, adapted for thermodynamic processes, distinguishing three kinds of transfer of energy, as heat, as thermodynamic work, and as energy associated with matter tran ...
). These are based on classical mechanics and are modified in
quantum mechanics Quantum mechanics is a fundamental theory A theory is a reason, rational type of abstraction, abstract thinking about a phenomenon, or the results of such thinking. The process of contemplative and rational thinking is often associated with ...
and
general relativity General relativity, also known as the general theory of relativity, is the geometric Geometry (from the grc, γεωμετρία; '' geo-'' "earth", '' -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathema ...
. They are expressed using the
Reynolds transport theoremIn differential calculus In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematica ...
. 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, infinitesimals or infinitesimal numbers are quantities that are closer to zero than any standard real number, but are not zero. They do not exist in the standard real number system, but do exist in many other number systems, such a ...
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 small in relation to the speed of light, the momentum equations for
Newtonian fluid A Newtonian fluid is a fluid In physics, a fluid is a substance that continually Deformation (mechanics), deforms (flows) under an applied shear stress, or external force. Fluids are a Phase (matter), phase of matter and include liquids, Gas, ...
s are the
Navier–Stokes equations In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. ...
—which is a
non-linear In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...
set of
differential equations In mathematics, a differential equation is an equation In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), ...
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 In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
, so they are primarily of use in
computational fluid dynamics#REDIRECT 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 dynamics, fluid flows. Computers are used ...
. 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 Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related ent ...

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 : :$p= \frac$ where is
pressure Pressure (symbol: ''p'' or ''P'') is the force In physics, a force is an influence that can change the motion (physics), motion of an Physical object, object. A force can cause an object with mass to change its velocity (e.g. moving fr ...

, is
density The density (more precisely, the volumetric mass density; also known as specific mass), of a substance is its per unit . The symbol most often used for density is ''ρ'' (the lower case Greek letter ), although the Latin letter ''D'' can also ...

, the
absolute temperature Thermodynamic temperature is a quantity defined in thermodynamics Thermodynamics is a branch of physics that deals with heat, Work (thermodynamics), work, and temperature, and their relation to energy, entropy, and the physical properties of ...
, 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 The Boltzmann constant ( or ) is the proportionality fa ...
and is
molar mass In , the molar mass of a is defined as the of a sample of that compound divided by the in that sample, measured in . The molar mass is a bulk, not molecular, property of a substance. The molar mass is an ''average'' of many instances of the co ...
for a particular gas.

## Conservation laws

Three conservation laws are used to solve fluid dynamics problems, and maybe written in
integral In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...

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' theorem, also known as Kelvin–Stokes theorem Nagayoshi Iwahori, et al.:"Bi-Bun-Seki-Bun-Gaku" :ja:裳華房, Sho-Ka-Bou(jp) 1983/12Written in Japanese)Atsuo Fujimoto;"Vector-Kai-Seki Gendai su-gaku rekucha zu. C(1)" :ja:培風館, Ba ...
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 are
compressible In thermodynamics Thermodynamics is a branch of physics that deals with heat, Work (thermodynamics), work, and temperature, and their relation to energy, radiation, and physical properties of matter. The behavior of these quantities is governed ...
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 Fluid mechanics is the branch of physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the o ...

. Otherwise the more general
compressible flow Compressible flow (or gas dynamics) is the branch of fluid mechanics Fluid mechanics is the branch of physics concerned with the mechanics Mechanics (Ancient Greek, Greek: ) is the area of physics concerned with the motions of physical objec ...
equations must be used. Mathematically, incompressibility is expressed by saying that the density of a
fluid parcel In 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 gas ...
does not change as it moves in the flow field, that is, : $\frac = 0 \, ,$ where is the
material derivativeIn continuum mechanics, the material derivative describes the time rate of change of some physical quantity (like heat or momentum In Newtonian mechanics, linear momentum, translational momentum, or simply momentum (plural, pl. momenta) is t ...
, 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 administrati ...
and
convective derivative 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 In physics, a fluid is a substance that continually Deformation (mechan ...
s. 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 #REDIRECT Mach number#REDIRECT Mach number 300px, An F/A-18 Hornet creating a vapor cone at transonic speed">vapor_cone.html" ;"title="F/A-18 Hornet creating a vapor cone">F/A-18 Hornet creating a vapor cone at transonic speed just before reachi ...
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 Acoustic may refer to: Music Albums * Acoustic (Bayside EP), ''Acoustic'' (Bayside EP) * Acoustic (Britt Nicole EP), ''Acoustic'' (Britt Nicole EP) * Acoustic (Joey Cape and Tony Sly album), ''Acoustic'' (Joey Cape and Tony Sly album), 2004 * Aco ...
problems always require allowing compressibility, since
sound waves In physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , is the natural science that studies matter, its Motion (physics), motion and behavior throug ...
are compression waves involving changes in pressure and density of the medium through which they propagate.

## Newtonian versus non-Newtonian fluids

All fluids 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 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 with time in the neighborh ...
; it has dimensions .
Isaac Newton Sir Isaac Newton (25 December 1642 – 20 March 1726/27) was an English mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Greek: ) includes the study of such topics a ...

showed that for many familiar fluids such as
water Water (chemical formula H2O) is an Inorganic compound, inorganic, transparent, tasteless, odorless, and Color of water, nearly colorless chemical substance, which is the main constituent of Earth's hydrosphere and the fluids of all known li ...

and
air File:Atmosphere gas proportions.svg, Composition of Earth's atmosphere by volume, excluding water vapor. Lower pie represents trace gases that together compose about 0.043391% of the atmosphere (0.04402961% at April 2019 concentration ). Number ...

, 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 stress tensor, viscous stresses arising from its Fluid dynamics, flow, at every point, are linearly correlated to the local strain rate—the derivative (mathematics), rate of change of its deformat ...
. 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 In physics, a fluid is a substance that continually Deformation (mechanics), deforms (flows) under an applied shear stress, or external force. Fluids are a Phase (matter), phase of matter and include liquids, ...

s have a more complicated, non-linear stress-strain behaviour. The sub-discipline of
rheology Rheology (; from Greek#REDIRECT Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population is approxim ...

describes the stress-strain behaviours of such fluids, which include
emulsion An emulsion is a mixture In chemistry, a mixture is a material made up of two or more different chemical substances which are not chemically combined. A mixture is the physical combination of two or more substances in which the identities are ...

s and
slurries A slurry composed of glass beads in silicone oil flowing down an inclined plane A slurry is a mixture of solids denser than water suspended in liquid, usually water. The most common use of slurry is as a means of transporting solids, the liquid ...
, some
viscoelastic In and , viscoelasticity is the property of that exhibit both and characteristics when undergoing . Viscous materials, like water, resist and linearly with time when a is applied. Elastic materials strain when stretched and immediately ret ...
materials such as
blood Blood is a body fluid Body fluids, bodily fluids, or biofluids are liquid A liquid is a nearly incompressible In fluid mechanics or more generally continuum mechanics, incompressible flow (isochoric process, isochoric flow) refers t ...

and some
polymer A polymer (; Greek ''poly- Poly, from the Greek :wikt:πολύς, πολύς meaning "many" or "much", may refer to: Businesses * China Poly Group Corporation, a Chinese business group, and its subsidiaries: ** Poly Property, a Hong Kong inc ...

s, and ''sticky liquids'' such as
latex Latex is a stable dispersion (emulsion An emulsion is a mixture of two or more liquids that are normally Miscibility, immiscible (unmixable or unblendable) owing to liquid-liquid phase separation. Emulsions are part of a more general class o ...

,
honey Honey is a sweet, viscous food substance made by honey bees Honey is a sweet, viscous food substance made by honey bees and some other Bee, bees. Bees produce honey from the sugary secretions of plants (floral nectar) or from secretion ...

and
lubricants A lubricant is a substance that helps to reduce friction Friction is the force In physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , is th ...
.

## Inviscid versus viscous versus Stokes flow

The dynamic of fluid parcels is described with the help of
Newton's second law In classical mechanics, Newton's laws of motion are three laws that describe the relationship between the motion Image:Leaving Yongsan Station.jpg, 300px, Motion involves a change in position In physics, motion is the phenomenon in which a ...
. An accelerating parcel of fluid is subject to inertial effects. The
Reynolds number The Reynolds number () helps predict flow patterns in different fluid flow situations. At low Reynolds numbers, flows tend to be dominated by laminar (sheet-like) flow, while at high Reynolds numbers flows tend to be turbulent In fluid dynam ...
is a
dimensionless quantity In dimensional analysis In engineering and science, dimensional analysis is the analysis of the relationships between different physical quantities by identifying their base quantity, base quantities (such as length, mass, time, and electric cur ...
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 Inviscid flow is the flow of an inviscid fluid, in which the viscosity The viscosity of a is a measure of its to deformation at a given rate. For liquids, it corresponds to the informal concept of "thickness": for example, has a higher vis ...
, an approximation in which viscosity is completely neglected. Eliminating viscosity allows the
Navier–Stokes equations In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. ...
to be simplified into the Euler equations. The integration of the Euler equations along a streamline in an inviscid flow yields
Bernoulli's equation Video of a venturi meter used in a lab experiment In 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, inc ...
. 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 (mathematics), function. Conservative vector fields have the property that the line integral is path independent; the choice of any path between ...
everywhere, Bernoulli's equation can completely describe the flow everywhere. Such flows are called
potential flow around a NACA 0012 airfoil at 11° angle of attack, with upper and lower streamtubes identified. In fluid dynamics, potential flow describes the velocity field as the gradient of a scalar function: the velocity potential. As a result, a potential ...

s, because the velocity field may be expressed as the
gradient In vector calculus Vector calculus, or vector analysis, is concerned with differentiation Differentiation may refer to: Business * Differentiation (economics), the process of making a product different from other similar products * Prod ...

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 conditionIn 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 ...
generates a thin region of large strain rate, the
boundary layer In physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular su ...
, in which
viscosity The viscosity of a fluid In physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, ...

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 with ...

. 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 In 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 ...
. A commonly used model, especially in
computational fluid dynamics#REDIRECT 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 dynamics, fluid flows. Computers are used ...
, is to use two flow models: the Euler equations away from the body, and
boundary layer In physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular su ...
equations in a region close to the body. The two solutions can then be matched with each other, using the
method of matched asymptotic expansionsIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
.

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 a
sphere A sphere (from Greek#REDIRECT Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population is appr ...

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 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 ...

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 Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grassl ...
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 (water), 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 o ...
, and apparent
random In common parlance, randomness is the apparent or actual lack of pattern A pattern is a regularity in the world, in human-made design, or in abstract ideas. As such, the elements of a pattern repeat in a predictable manner. A geometric p ...

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 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 ...
, in which the flow is broken down into the sum of an
average In colloquial language, an average is a single number taken as representative of a non-empty list of numbers. Different concepts of average are used in different contexts. Often "average" refers to the arithmetic mean, the sum of the numbers divide ...

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 Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. ...
.
Direct numerical simulation A direct numerical simulation (DNS)Here the origin of the term ''direct numerical simulation'' (see e.g. p. 385 in ) owes to the fact that, at that time, there were considered to be just two principal ways of getting ''theoretical'' results re ...
(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 A wide-body aircraft, also known as a twin-aisle aircraft, is an with a wide enough to accommodate two passenger s with seven or more seats abreast. The typical diameter is . In the typical wide- ...
or
Boeing 747 The Boeing 747 is a large, long–range wide-body airliner A wide-body aircraft, also known as a twin-aisle aircraft, is an with a wide enough to accommodate two passenger s with seven or more seats abreast. The typical diameter is . ...

) 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 (or 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-averag ...
(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 transfer, heat and mass transfer. 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 (buoyancy), Boussinesq approximation'' neglects variations in density except to calculate buoyancy forces. It is often used in free convection 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 which surface slope, gradients are small. * ''Darcy's law'' is used for flow in porous medium, porous media, and works with variables averaged over several pore-widths. * In rotating systems, the ''quasi-geostrophic equations'' assume an almost Balanced flow#Geostrophic flow, perfect balance between pressure gradients and the Coriolis force. 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 low
Mach number #REDIRECT Mach number#REDIRECT Mach number 300px, An F/A-18 Hornet creating a vapor cone at transonic speed">vapor_cone.html" ;"title="F/A-18 Hornet creating a vapor cone">F/A-18 Hornet creating a vapor cone at transonic speed just before reachi ...
s (Speed of sound, subsonic flows), many flows of practical interest in aerodynamics or in Turbomachinery, turbomachines occur at high fractions of (Transonic, transonic flows) or in excess of it (Supersonic speed, supersonic or even Hypersonic speed, 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, including combustion (Internal Combustion Engine, IC engine), propulsion devices (rockets, jet engines, and so on), detonations, 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 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.

## Magnetohydrodynamics

Magnetohydrodynamics is the multidisciplinary study of the flow of electrical conduction, electrically conducting fluids in Electromagnetism, electromagnetic fields. Examples of such fluids include Plasma (physics), plasmas, liquid metals, and Saline water, salt water. The fluid flow equations are solved simultaneously with Maxwell's equations 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 the special theory of relativity and the general theory of relativity. The governing equations are derived in Riemannian geometry for Minkowski spacetime.

# 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 Pressure measurement, 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 in fluid statics.

## Terminology in incompressible fluid dynamics

The concepts of total pressure and dynamic pressure arise from
Bernoulli's equation Video of a venturi meter used in a lab experiment In 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, inc ...
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 to distinguish it from total pressure and dynamic pressure. Static pressure 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 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 by isentropically 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".

## Fields of study

*Acoustic theory *Aerodynamics *Aeroelasticity *Aeronautics *Computational fluid dynamics *Flow measurement *Geophysical fluid dynamics *haemodynamics, Hemodynamics *Hydraulics *Hydrology *Hydrostatics *Electrohydrodynamics *Magnetohydrodynamics *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 *Helmholtz's theorems *Kirchhoff equations *Knudsen equation *Manning equation *Mild-slope equation *Morison equation *
Navier–Stokes equations In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. ...
*Oseen flow *Poiseuille's law *Pressure head *Relativistic Euler equations *Stokes stream function *Stream function *Streamlines, streaklines and pathlines *Torricelli's Law

## Types of fluid flow

*Aerodynamic force *Convection *Cavitation *Compressible flow *Couette flow *Effusive limit *Free molecular flow *Incompressible flow *Inviscid flow *Isothermal flow *Open channel flow *Pipe flow *Secondary flow *Stream thrust averaging *Superfluidity *Transient flow *Two-phase flow

## Fluid properties

*List of hydrodynamic instabilities *
Newtonian fluid A Newtonian fluid is a fluid In physics, a fluid is a substance that continually Deformation (mechanics), deforms (flows) under an applied shear stress, or external force. Fluids are a Phase (matter), phase of matter and include liquids, Gas, ...
*
Non-Newtonian fluid A non-Newtonian fluid is a fluid In physics, a fluid is a substance that continually Deformation (mechanics), deforms (flows) under an applied shear stress, or external force. Fluids are a Phase (matter), phase of matter and include liquids, ...

*Surface tension *Vapour pressure

## Fluid phenomena

*Balanced flow *Boundary layer *Coanda effect *Convection cell *squeeze mapping#Corner flow, Convergence/Bifurcation *Darwin drift *Drag (force) *Droplet vaporization *Hydrodynamic stability *Kaye effect *Lift (force) *Magnus effect *Ocean current *Ocean surface waves *Rossby wave *Shock wave *Soliton *Stokes drift *Fluid thread breakup, Thread breakup *Turbulent jet breakup *Upstream contamination *Venturi effect *Vortex *Water hammer *Wave drag *Wind

## Applications

*Acoustics *Aerodynamics *Cryosphere science *Fluidics *Fluid power *Geodynamics *Hydraulic machinery *Meteorology *Naval architecture *Oceanography *Plasma physics *Pneumatics *3D computer graphics

## Fluid dynamics journals

* ''Annual Review of Fluid Mechanics'' * ''Journal of Fluid Mechanics'' * ''Physics of 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

*List of publications in physics#Fluid dynamics, Important publications in fluid dynamics *Isosurface *Keulegan–Carpenter number *Rotating tank *Sound barrier *Beta plane *Immersed boundary method *Bridge scour *Finite volume method for unsteady flow

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

# References

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

National Committee for Fluid Mechanics Films (NCFMF)
containing films on several subjects in fluid dynamics (in RealMedia format)
List of Fluid Dynamics books
{{Authority control Fluid dynamics, Piping Aerodynamics Continuum mechanics