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<p role="presentation">{{Continuum mechanics|fluid}}</p
Fluid dynamics offers a systematic structure—which underlies these [[
Before the twentieth century, ''hydrodynamics'' was synonymous with fluid dynamics. This is still reflected in names of some fluid dynamics topics, like [[
==Equations of fluid dynamics==
The foundational axioms of fluid dynamics are the [[Conservation law (physics)|conservation laws]], specifically, [[
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 considers fluids as continuous, rather than discrete. Consequently, properties such as density, pressure, temperature, and flow velocity are assumed well-defined at [[Infinitesimal|infinitesimally]] small points, and are assumed to 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|Newtonian fluids]] are the [[
In addition to the mass, momentum, and energy conservation equations, a [[Thermodynamics|thermodynamical]] equation of state giving the pressure as a function of other thermodynamic variables for the fluid is required to completely specify the problem. An example of this would be the [[Ideal gas law|perfect gas equation of state]]:
<dd><math>p= \frac{\rho R_u T}{M}</math></dd>
where ''p'' is [[ ===Conservation laws===
Three conservation laws are used to solve fluid dynamics problems, and may be written in [[Integral]] or [[Differential (infinitesimal)|differential]] form. Mathematical formulations of these conservation laws may be interpreted by considering the concept of a ''control volume''. A control volume is a specified volume in space through which air can flow in and out. Integral formulations of the conservation laws consider the change in mass, momentum, or energy within the control volume. Differential formulations of the conservation laws apply [[Stokes'_theorem]] to yield an expression which may be interpreted as the integral form of the law applied to an infinitesimal volume at a point within the flow.
*[[Continuity equation#Fluid dynamics|Mass continuity]] (conservation of mass): The rate of change of fluid mass inside a control volume must be equal to the net rate of fluid flow into the volume. Physically, this statement requires that mass is neither created nor destroyed in the control volume,<ref>Anderson, J.D., ''Fundamentals of Aerodynamics'', 4th Ed., McGraw–Hill, 2007.</ref> and can be translated into the integral form of the continuity equation:
:<dd><math>{\partial \over \partial t} \iiint_V \rho \, dV = - \, {} </math> {{oiint|preintegral=|intsubscpt=<math>{\scriptstyle S}</math>|integrand=<math>{}\,\rho\mathbf{u}\cdot d\mathbf{S}</math>}}</dd>
:Above, <math>\rho</math> is the fluid density, '''u''' is the [[flow velocity]] vector, and ''t'' is time. The left-hand side of the above expression contains a triple integral over the control volume, whereas the right-hand side contains a surface integral over the surface of the control volume. The differential form of the continuity equation is, by the [[Divergence_theorem]]:
::<math>\ {\partial \rho \over \partial t} + \nabla \cdot (\rho \mathbf{u}) = 0 </math>
*[[Momentum|Conservation of momentum]]: This equation applies [[Newton's_second_law_of_motion]] to the control volume, requiring that any change in momentum of the air within a control volume be due to the net flow of air into the volume and the action of external forces on the air within the volume. In the integral formulation of this equation, [[Body force|body forces]] here are represented by ''f''<sub>body</sub>, the body force per unit mass. [[Surface force|Surface forces]], such as viscous forces, are represented by '''<math>\mathbf{F}_\text{surf}</math>''', the net force due to [[Stress (mechanics)|stresses]] on the control volume surface.
:<dd><math> \frac{\partial}{\partial t} \iiint_{\scriptstyle V} \rho\mathbf{u} \, dV = -\, {} </math> {{oiint|intsubscpt=<math>_{\scriptstyle S}</math>|integrand|preintegral=}} <math> (\rho\mathbf{u}\cdot d\mathbf{S}) \mathbf{u} -{}</math> {{oiint|intsubscpt=<math>{\scriptstyle S}</math>|integrand=<math> {}\, p \, d\mathbf{S}</math>}} <math>\displaystyle{}+ \iiint_{\scriptstyle V} \rho \mathbf{f}_\text{body} \, dV + \mathbf{F}_\text{surf}</math></dd>
:The differential form of the momentum conservation equation is as follows. Here, both surface and body forces are accounted for in one total force, ''F''. For example, ''F'' may be expanded into an expression for the frictional and gravitational forces acting on an internal flow.
::<math>\ {D \mathbf{u} \over D t} = \mathbf{F} - {\nabla p \over \rho} </math>
<dd>In aerodynamics, air is assumed to be a [[Newtonian_fluid]], which posits a linear relationship between the shear stress (due to internal friction forces) and the rate of strain of the fluid. The equation above is a vector equation: in a three-dimensional flow, it can be expressed as three scalar equations. The conservation of momentum equations for the compressible, viscous flow case are called the Navier–Stokes equations.{{citation needed|date=May 2014}}</dd>
*[[Conservation of energy]]: Although [[Energy]] can be converted from one form to another, the total [[Energy]] in a given closed system remains constant.
::<math>\ \rho {Dh \over Dt} = {D p \over D t} + \nabla \cdot \left( k \nabla T\right) + \Phi </math>
:Above, ''h'' is [[enthalpy]], ''k'' is the [[Thermal_conductivity]] of the fluid, ''T'' is temperature, and <math>\Phi</math> is the viscous dissipation function. The viscous dissipation function governs the rate at which mechanical energy of the flow is converted to heat. The [[Second_law_of_thermodynamics]] requires that the dissipation term is always positive: viscosity cannot create energy within the control volume.<ref>White, F.M., ''Viscous Fluid Flow'', McGraw–Hill, 1974.</ref> The expression on the left side is a [[Material_derivative]].
===Compressible vs incompressible flow===
All fluids are [[Compressibility|compressible]] to some 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]]. Otherwise the more general [[compressible flow]] equations must be used.
Mathematically, incompressibility is expressed by saying that the density ρ of a [[Fluid_parcel]] does not change as it moves in the flow field, i.e.,
:<math>\frac{\mathrm{D} \rho}{\mathrm{D}t} = 0 \, ,</math>
where ''D''/''Dt'' is the [[substantial derivative]], which is the sum of [[Time derivative|local]] and [[Convective derivative|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]] 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). [[Acoustics|Acoustic]] problems always require allowing compressibility, since [[
===Inviscid vs Newtonian and non-Newtonian fluids===
[[Tập_tin:Potential_flow_around_a_wing.gif|nhỏ|Potential flow around a wing]]All fluids are [[Viscosity|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 (materials science)|strain rate]]; it has dimensions <math>T^{-1}</math>. [[Isaac Newton]] showed that for many familiar fluids such as [[water]] and [[Earth's atmosphere|air]], the [[Stress (physics)|stress]] due to these viscous forces is linearly related to the strain rate. Such fluids are called [[Newtonian_fluids]]. The coefficient of proportionality is called the fluid's [[viscosity]]; for Newtonian fluids, it is a fluid property independent of the strain rate.
[[Non-Newtonian fluid|Non-Newtonian fluids]] have a more complicated, non-linear stress-strain behaviour. The sub-discipline of [[rheology]] studies the stress-strain behaviours of these fluids, which include [[Emulsion|emulsions]] and [[slurries]], some [[Viscoelasticity|viscoelastic]] materials such as [[Blood]] and some [[Polymer|polymers]], and ''sticky liquids'' such as [[latex]], [[Honey]] and [[Lubricants]].{{citation needed|date=June 2015}}
The dynamic of fluid parcels is described with the help of [[Newton's second law]]. An accelerating parcel of fluid is subject to inertial effects.
The [[
On the contrary, high Reynolds numbers (''Re''>>1) 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]], an approximation in which [[viscosity]] is completely neglected. The [[Navier–Stokes equations|Navier-Stokes equations]] then simplify into the [[Euler equations (fluid dynamics)|Euler equations]]. Integrating these along a streamline in an inviscid flow yields [[Bernoulli's equation]]. When in addition to being inviscid, the flow is everywhere [[Lamellar field|irrotational]], Bernoulli's equation can be used throughout the flow field. Such flows are called [[Potential flow|potential flows]], because the velocity field may be expressed as the [[gradient]] of a potential.
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 which [[viscosity]] effects dominate and which thus generates [[vorticity]]. Therefore, to calculate net forces on bodies (such as wings), viscous flow equations must be used: inviscid flow theory fails to predict [[Drag (physics)|drag forces]], a limitation known as the [[
===Steady vs unsteady flow===<!-- [[Steady flow]] redirects here -->
[[Tập_tin:HD-Rayleigh-Taylor.gif|nhỏ|320x320px|Hydrodynamics simulation of the [[Rayleigh–Taylor_instability]] <ref>Shengtai Li, Hui Li "Parallel AMR Code for Compressible MHD or HD Equations" (Los Alamos National Laboratory) [http://math.lanl.gov/Research/Highlights/amrmhd.shtml]</ref>]]When all the time derivatives of a flow field vanish, the flow is considered '''steady flow'''. Steady-state flow refers to the condition where the fluid properties at a point in the system do not change over time. Otherwise, flow is called unsteady (also called transient<ref>[http://www.cfd-online.com/Forums/main/118306-transient-state-unsteady-state.html Transient state or unsteady state?]</ref>). Whether a particular flow is steady or unsteady, can depend on the chosen frame of reference. For instance, laminar flow over a [[sphere]] 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.
[[Turbulence|Turbulent]] flows are unsteady by definition. A turbulent flow can, however, be [[Stationary process|statistically stationary]]. According to Pope:<ref>See Pope (2000), page 75.</ref>{{quote|
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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 vs turbulent flow===
[[Turbulence]] is flow characterized by recirculation, [[Eddy (fluid dynamics)|eddies]], and apparent [[Random|randomness]]. Flow in which turbulence is not exhibited is called [[Laminar flow|laminar]]. It should be noted, however, that 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]] component and a perturbation component.
It is believed that turbulent flows can be described well through the use of the [[Navier–Stokes_equations]]. [[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.<ref>See, for example, Schlatter et al, Phys. Fluids 21, 051702 (2009); {{doi|10.1063/1.3139294}}</ref>
Most flows of interest have Reynolds numbers much too high for DNS to be a viable option,<ref>See Pope (2000), page 344.</ref> given the state of computational power for the next few decades. Any flight vehicle large enough to carry a human (L > 3 m), moving faster than 72 km/h (20 m/s) is well beyond the limit of DNS simulation (Re = 4 million). Transport aircraft wings (such as on an [[Airbus A300]] or [[Boeing 747]]) have Reynolds numbers of 40 million (based on the wing chord). Solving these real-life flow problems requires turbulence models for the foreseeable future. [[Reynolds-averaged Navier–Stokes equations]] (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.
===Subsonic vs transonic, supersonic and hypersonic flows===
While many terrestrial flows (e.g. flow of water through a pipe) occur at low mach numbers, many flows of practical interest (e.g. in aerodynamics) occur at high fractions of the Mach Number M=1 or in excess of it (supersonic flows). New phenomena occur at these Mach number regimes (e.g. shock waves for supersonic flow, transonic instability in a regime of flows with M nearly equal to 1, non-equilibrium chemical behaviour due to ionization in hypersonic flows) and it is necessary to treat each of these flow regimes separately.
===Magnetohydrodynamics===
{{main|Magnetohydrodynamics}}
[[Magnetohydrodynamics]] is the multi-disciplinary 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'
===Other approximations===
There are a large number of other possible approximations to fluid dynamic problems. Some of the more commonly used are listed below.
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==Terminology in fluid dynamics==
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]] 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.
In ''Aerodynamics'', L.J. Clancy writes:<ref>Clancy, L.J. ''Aerodynamics'', page 21</ref> ''To distinguish it from the total and dynamic pressures, the actual pressure of the fluid, which is associated not with its motion but with its state, is often referred to as the static pressure, but where the term pressure alone is used it refers to this static pressure.''
A point in a fluid flow where the flow has come to rest (i.e. 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 [[
===Terminology in compressible fluid dynamics===
In a compressible fluid, such as air, the temperature and density are essential when determining the state of the fluid. In addition to the concept of total pressure (also known as [[stagnation pressure]]), the concepts of total (or stagnation) temperature and total (or stagnation) density are also essential in any study of compressible fluid flows. To avoid potential ambiguity when referring to temperature and density, many authors use the terms static temperature and static density. Static temperature is identical to temperature; and static density is identical to density; and both can be identified for every point in a fluid flow field.
The temperature and density at a [[stagnation point]] are called stagnation temperature and stagnation density.
A similar approach is also taken with the thermodynamic properties of compressible fluids. Many authors use the terms total (or stagnation) [[
===Fields of study===
{{columns-list|3|
*[[Acoustic theory]]
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*[[Rheology]]
}}
===Mathematical equations and concepts===
{{columns-list|3|
*[[Airy wave theory]]
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*[[Torricelli's Law]]
}}
===Types of fluid flow===
{{columns-list|3|
*[[Aerodynamic force]]
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*[[Two-phase flow]]
}}
===Fluid properties===
{{columns-list|3|
*[[Density]]
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*[[Compressibility]]
}}
===Fluid phenomena===
{{columns-list|3|
*[[Boundary layer]]
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*[[Wave drag]]
}}
===Applications===
{{columns-list|3|
*[[Acoustics]]
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*[[3D computer graphics]]
}}
===Fluid dynamics journals===
{{columns-list|3|
* ''[[Annual Review of Fluid Mechanics]]''
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* ''[[Flow, Turbulence and Combustion]]''
}}
===Miscellaneous===
{{columns-list|3|
*[[List of publications in physics#Fluid dynamics|Important publications in fluid dynamics]]
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* [[Finite volume method for unsteady flow]]
}}
===See also===
{{div col|4}}
* [[Aileron]]
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* [[Wing]]
* [[Wingtip vortices]]
{{div col end}}<h2>References</h2>{{reflist|2}}
==Further reading==
<ul><li>{{cite book|last=Acheson|first=D. J.|title=Elementary Fluid Dynamics|publisher=Clarendon Press|year=1990|isbn=0-19-859679-0}}</li><li>{{cite book|last=Batchelor|first=G. K.|authorlink=George Batchelor|title=An Introduction to Fluid Dynamics|publisher=Cambridge University Press|year=1967|isbn=0-521-66396-2}}</li><li>{{cite book|last=Chanson|first=H.|authorlink=Hubert Chanson|title=Applied Hydrodynamics: An Introduction to Ideal and Real Fluid Flows|publisher=CRC Press, Taylor & Francis Group, Leiden, The Netherlands, 478 pages|year=2009|isbn=978-0-415-49271-3}}</li><li>{{cite book|last=Clancy|first=L. J.|title=Aerodynamics|publisher=Pitman Publishing Limited|location=London|year=1975|isbn=0-273-01120-0}}</li><li>{{cite book|last=Lamb|first=Horace|authorlink=Horace Lamb|title=Hydrodynamics|edition=6th|publisher=Cambridge University Press|year=1994|isbn=0-521-45868-4}} Originally published in 1879, the 6th extended edition appeared first in 1932.</li><li>{{cite book|last1=Landau|first1=L. D.|author1-link=Lev Landau|last2=Lifshitz|first2=E. M.|author2-link=Evgeny Lifshitz|title=Fluid Mechanics|edition=2nd|series=[[Course of Theoretical Physics]]|publisher=Pergamon Press|year=1987|isbn=0-7506-2767-0}}</li><li>{{cite book|last=Milne-Thompson|first=L. M.|title=Theoretical Hydrodynamics|edition=5th|publisher=Macmillan|year=1968}} Originally published in 1938.</li><li>{{cite book|last=Pope|first=Stephen B.|title=Turbulent Flows|publisher=Cambridge University Press|year=2000|isbn=0-521-59886-9}}</li><li>{{cite book|last=Shinbrot|first=M.|title=Lectures on Fluid Mechanics|publisher=Gordon and Breach|year=1973|isbn=0-677-01710-3}}</li><li>{{citation |last1=Nazarenko|first1=Sergey|year=2014|title=Fluid Dynamics via Examples and Solutions|publisher=CRC Press (Taylor & Francis group)|isbn=978-1-43-988882-7}}</li><li href="Scholarpedia">[http://www.scholarpedia.org/article/Encyclopedia:Fluid_dynamics Encyclopedia: Fluid dynamics] [[Scholarpedia]]</li></ul><h2>External links</h2>{{Commons category|Fluid dynamics}}{{Commons category|Fluid mechanics}}
*[http://www.efluids.com/ eFluids], containing several galleries of fluid motion
*[http://web.mit.edu/hml/ncfmf.html National Committee for Fluid Mechanics Films (NCFMF)], containing films on several subjects in fluid dynamics (in [[RealMedia]] format)
*[http://www.salihnet.freeservers.com/engineering/fm/fm_books.html List of Fluid Dynamics books]
{{NonDimFluMech}} {{physics-footer|continuum='''[[Continuum mechanics]]'''}}<p role="presentation"></p>
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