Fluid dynamics 11034 225964294 2008-07-16T06:11:36Z Khunglongcon 6358314 {{Continuum mechanics}} [[Image:Teardrop shape.svg|thumb|Typical aerodynamic teardrop shape, showing the pressure distribution as the thickness of the black line and showing the velocity in the [[boundary layer]] as the violet triangles. The green [[vortex generator]]s prompt the transition to [[turbulent flow]] and prevent back-flow also called [[flow separation]] from the high pressure region in the back. The surface in front is as smooth as possible or even employ [[Dermal denticle|shark like skin]], as any turbulence here will reduce the energy of the airflow. The [[Kammback]] also prevents back flow from the high pressure region in the back across the [[Spoiler (aeronautics)|spoiler]]s to the convergent part. Putting stuff inside out results in [[Piping|tube]]s, they also face the problem of flow separation in their divergent parts, so called [[diffuser (automotive)|diffuser]]s. Cutting the shape into halfs results in an [[airfoil|aerofoil]] with the low pressure region on top leading to [[lift (force)]].]] '''Fluid dynamics''' is the sub-discipline of [[fluid mechanics]] dealing with '''fluid flow''': [[fluid]]s ([[liquid]]s and [[gas]]es) in motion. It has several subdisciplines itself, including [[aerodynamics]] (the study of gases in motion) and '''hydrodynamics''' (the study of liquids in motion). Fluid dynamics has a wide range of applications, including calculating [[force]]s and [[moment (physics)|moment]]s on [[aircraft]], determining the [[mass flow rate]] of [[petroleum]] through pipelines, predicting [[weather]] patterns, understanding [[nebula]]e in [[interstellar]] space and reportedly modeling fission weapon detonation. Some of its principles are even used in [[traffic engineering (transportation)|traffic engineering]], where traffic is treated as a continuous fluid. Fluid dynamics offers a systematic structure that underlies these practical disciplines and that embraces empirical and semi-empirical laws, derived from [[flow measurement]], used to solve practical problems. The solution of a fluid dynamics problem typically involves calculation of various properties of the fluid, such as [[velocity]], [[pressure]], [[density]], and [[temperature]], as functions of space and time. ==Equations of fluid dynamics== The foundational axioms of fluid dynamics are the [[conservation law]]s, specifically, [[conservation of mass]], [[conservation of momentum|conservation of linear momentum]] (also known as [[Newton's laws of motion|Newton's Second Law of Motion]]), and [[conservation of energy]] (also known as [[First Law of Thermodynamics]]). These are based on [[classical mechanics]] and are modified in [[quantum mechanics]] and [[general relativity]]. They are expressed using the [[Reynolds transport theorem|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 considers fluids to be continuous, rather than discrete. Consequently, properties such as density, pressure, temperature, and velocity are taken to be well-defined at 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 which are sufficiently dense to be a continuum, do not contain ionized species, and have velocities small in relation to the speed of light, the momentum equations for [[Newtonian fluid]]s are the [[Navier-Stokes equations]], which is a [[non-linear]] set of [[differential equations]] that describes the flow of a fluid whose stress depends linearly on velocity gradients and pressure. The unsimplified equations do not have a general [[Solution in closed form|closed-form solution]], so they are only of use in [[Computational Fluid Dynamics]] or when they can be simplified. The equations can be simplified in a number of ways, all of which make them easier to solve. Some of them allow appropriate fluid dynamics problems to be solved in closed form. 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]]: :<math>p= \frac{\rho R_u T}{M}</math> where <math>p</math> is [[pressure]], <math>\rho</math> is [[density]], <math>R_u</math> is the [[gas constant]], <math>M</math> is the [[molecular mass]] and <math>T</math> is [[temperature]]. ===Compressible vs incompressible flow=== All fluids are [[compressibility|compressible]] to some extent, that is changes in pressure or temperature will result in 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 modeled as an [[incompressible flow]]. Otherwise the more general [[compressible flow]] equations must be used. Mathematically, incompressibility is expressed by saying that the density <math>\rho</math> 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 <math>\mathrm{D}/\mathrm{D}t</math> is the [[substantial derivative]], which is the sum of 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 to be 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 [[sound waves]] are compression waves involving changes in pressure and density of the medium through which they propagate. ===Viscous vs inviscid flow=== [[viscosity|Viscous]] problems are those in which fluid friction has significant effects on the fluid motion. The [[Reynolds number]] can be used to evaluate whether viscous or inviscid equations are appropriate to the problem. [[Stokes flow]] is flow at very low Reynolds numbers, such that inertial forces can be neglected compared to viscous forces. On the contrary, high Reynolds numbers indicate that the inertial forces are more significant than the viscous (friction) forces. Therefore, we may assume the flow to be an [[inviscid flow]], an approximation in which we neglect [[viscosity]] at all, compared to inertial terms. This idea can work fairly well when the Reynolds number is high. However, certain problems such as those involving solid boundaries, may require that the viscosity be included. Viscosity often cannot be neglected near solid boundaries because the [[no-slip condition]] can generate a thin region of large strain rate (known as [[Boundary layer]]) which enhances the effect of even a small amount of [[viscosity]], and thus generating [[vorticity]]. Therefore, to calculate net forces on bodies (such as wings) we should use viscous flow equations. As illustrated by [[d'Alembert's paradox]], a body in an inviscid fluid will experience no drag force. The standard equations of inviscid flow are the [[Euler equations]]. Another often used model, especially in computational fluid dynamics, is to use the [[Euler equations]] away from the body and the [[boundary layer]] equations, which incorporates viscosity, in a region close to the body. The [[Euler equations]] can be integrated along a streamline to get [[Bernoulli's equation]]. When the flow is everywhere [[Lamellar field|irrotational]] and inviscid, Bernoulli's equation can be used throughout the flow field. Such flows are called [[Potential flow|potential flows]]. ===Steady vs unsteady flow===<!-- [[Steady flow]] redirects here --> [[Image:HD-Rayleigh-Taylor.gif|thumb|320px|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 to be a '''steady flow'''. Otherwise, it is called unsteady. 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 than the governing equations of the same problem without taking advantage of the steadiness of the flow field. Although strictly unsteady flows, time-periodic problems can often be solved by the same techniques as steady flows. For this reason, they can be considered to be somewhere between steady and unsteady. <!-- {{disputed-section|Steady vs unsteady flow}}. In the absence of turbulence, fluid flow can be described by streamlines. A streamline is a curve in the flowing fluid along which the fluid's velocity is constant in space. If, in addition, the particles making up the fluid flow along the streamlines, the flow is called steady. (See Landau & Lifshitz "Fluid Mechanics.") Steady flow is applicable to a large class of problems, such as lift and drag on a wing or flow through a pipe. Both the Navier-Stokes equations and the Euler equations become simpler when the steady flow approximation is valid. Problems where the flow is steady and for which all the streamlines have the same velocity over a cross-section of the flow, [[potential flow]], are the most easily tractable. The flow properties mentioned above, potential, steady, and unsteady, are not coordinate dependent and so cannot be transformed away by global coordinate transformations. For example Poisieulle flow, the flow of a viscous fluid in a tube, is not potential because the streamlines have different velocities and at the tube wall the velocity actually vanishes. It may be assumed steady in some cases but there is no coordinate system in which it is potential flow. Similarly, the flow in the vicinity of a surface is not steady because molecules in the flow that encounter the surface itself, i.e. in the boundary layer, are slowed by collisions with the surface and with molecules in the boundary layer. If a problem is incompressible, irrotational, inviscid, and steady, it can be solved using [[Laplace's equation]]. Problems in this class have elegant solutions which are linear combinations of well-studied elementary flows. --> ===Laminar vs turbulent flow=== [[Turbulence]] is flow dominated by recirculation, [[Eddy (fluid dynamics)|eddies]], and apparent [[random]]ness. Flow in which turbulence is not exhibited is called [[laminar flow|laminar]]. It should be noted, however, that the presence of eddies or recirculation does not necessarily indicate turbulent flow--these phenomena may be present in laminar flow as well. Mathematically, turbulent flow is often represented via [[Reynolds decomposition]], in which the flow is broken down into the sum of a steady component and a perturbation component. It is believed that turbulent flows obey the [[Navier-Stokes equations]]. [[Direct numerical simulation|Direct Numerical Simulation]] (DNS), based on the incompressible Navier-Stokes equations, makes it possible to simulate turbulent flows with moderate Reynolds numbers (restrictions depend on the power of computer and efficiency of solution algorithm). The results of DNS agree with the experimental data. Most flows of interest have Reynolds numbers too high for DNS to be a viable option (see: Pope), 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). In order to solve these real life flow problems, turbulence models will be a necessity for the foreseeable future. [[Reynolds-averaged Navier-Stokes equations|Reynolds-Averaged Navier-Stokes equations]] (RANS) combined with [[turbulence modeling]] provides a model of the effects of the turbulent flow, mainly the additional momentum transfer provided by the [[Reynolds stresses]], although the turbulence also enhances the [[heat transfer|heat]] and [[mass transfer]]. [[Large eddy simulation|Large Eddy Simulation]] (LES) also holds promise as a simulation methodology, especially in the guise of [[detached eddy simulation|Detached Eddy Simulation]] (DES), which is a combination of [[turbulence modeling]] and large eddy simulation. ===Newtonian vs non-Newtonian fluids=== Sir [[Isaac Newton]] showed how [[stress (physics)|stress]] and the rate of [[Strain (materials science)|strain]] are very close to linearly related for many familiar fluids, such as [[water]] and [[Earth's atmosphere|air]]. These [[Newtonian fluid]]s are modeled by a coefficient called [[viscosity]], which depends on the specific fluid. However, some of the other materials, such as emulsions and slurries and some visco-elastic materials (eg. [[blood]], some [[polymer]]s), have more complicated ''[[Non-Newtonian fluid|non-Newtonian]]'' stress-strain behaviours. These materials include ''sticky liquids'' such as [[latex]], [[honey]], and lubricants which are studied in the sub-discipline of [[rheology]]. ===Magnetohydrodynamics=== {{main|Magnetohydrodynamics}} [[Magnetohydrodynamics]] is the multi-disciplinary study of the flow of [[electrical conduction|electrically conducting]] fluids in [[electromagnetic]] fields. Examples of such fluids include [[plasma]]s, liquid metals, and [[salt water]]. The fluid flow equations are solved simultaneously with [[Maxwell's equations]] of electromagnetism. ===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]]''' 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. * The '''[[Boussinesq equations (water waves)|Boussinesq equations]]''' are applicable to [[surface waves]] on thicker layers of fluid and with steeper surface [[slope]]s. * '''[[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 '''[[Balanced flow#Geostrophic Flow|quasi-geostrophic approximation]]''' assumes an almost perfect balance between [[pressure gradient]]s and the [[Coriolis force]]. It is useful in the study of [[atmospheric dynamics]]. == 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 (also known as [[stagnation 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 (page 21): "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 [[stagnation point]]. The [[pressure]] at the stagnation point is of special significance and is given its own name - [[stagnation pressure]], which is equal to the total pressure. === 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. Readers might wonder if there are such concepts as dynamic temperature or dynamic density. There aren't. A similar approach is also taken with the thermodynamic properties of compressible fluids. Many authors use the terms total (or stagnation) [[enthalpy]] and total (or stagnation) [[entropy]]. The terms static enthalpy and static entropy appear to be less common, but where they are used they mean nothing more than enthalpy and entropy respectively, and the prefix 'static' is being used to avoid ambiguity with their 'total' or 'stagnation' counterparts. == References == * Acheson, D.J. (1990) "Elementary Fluid Dynamics" (Clarendon Press). * [[George Batchelor|Batchelor, G.K.]] (1967) "An Introduction to Fluid Dynamics" (Cambridge University Press). * Clancy, L.J. (1975) "Aerodynamics" (Pitman Publishing Limited). * [[Horace Lamb|Lamb, H.]] (1994) "Hydrodynamics" (Cambridge University Press, 6<sup>th</sup> ed.). Originally published in 1879, the 6<sup>th</sup> extended edition appeared first in 1932. * [[Lev Landau|Landau, L.D.]] and [[Evgeny Lifshitz|Lifshitz, E.M.]] (1987) "Fluid Mechanics" (Pergamon Press, 2<sup>nd</sup> ed.). * Milne-Thompson, L.M. (1968) "Theoretical Hydrodynamics" (Macmillan, 5<sup>th</sup> ed.). Originally published in 1938. * Pope, S.B. (2000) "Turbulent Flows" (Cambridge University Press). * Shinbrot, Marvin (1973) "Lectures on Fluid Mechanics" (Gordon and Breach) ==Notes== {{reflist}} ==See also== {{col-begin}} {{col-break}} === Fields of study === *[[Acoustic theory]] *[[Aerodynamics]] *[[Aeroelasticity]] *[[Aeronautics]] *[[Computational fluid dynamics]] *[[Flow measurement]] *[[haemodynamics|Hemodynamics]] *[[Hydraulics]] *[[Hydrology]] *[[Hydrostatics]] *[[Hydrodynamics]] *[[Electrohydrodynamics]] *[[Magnetohydrodynamics]] *[[Rheology]] *[[Quantum hydrodynamics]] === Mathematical equations and concepts=== *[[Bernoulli's equation]] *[[Reynolds transport theorem]] *[[Boussinesq approximation]] *[[Conservation laws]] *[[Euler equations]] *[[Darcy's law]] *[[Dynamic pressure]] *[[Fluid statics]] *[[Helmholtz's theorems]] *[[Kirchhoff equations]] *[[Manning equation]] *[[Navier-Stokes equations]] *[[Pascal's law]] *[[Poiseuille's law]] *[[Pressure]] *[[Static pressure]] *[[Pressure head]] *[[Relativistic Euler equations]] *[[Reynolds decomposition]] *[[Stokes stream function]] *[[Stream function]] *[[Streamlines, streaklines and pathlines]] {{col-break}} === Types of fluid flow === *[[Cavitation]] *[[Compressible flow]] *[[Couette flow]] *[[Free molecular flow]] *[[Incompressible flow]] *[[Inviscid flow]] *[[Laminar flow]] *[[Open channel flow]] *[[Potential flow]] *[[Secondary flow]] *[[Stokes flow]] *[[Superfluidity]] *[[Supersonic]] *[[Transient flow]] *[[Transonic]] *[[Turbulence|Turbulent flow]] *[[Two-phase flow]] === Fluid properties === *[[Density]] *[[Newtonian fluid]] *[[Non-Newtonian fluid]] *[[Surface tension]] *[[Viscosity]] *[[Vapor pressure]] *[[Compressibility]] {{col-break}} ===Fluid phenomena=== *[[Boundary layer]] *[[Coanda effect]] *[[Convection cell]] *[[squeeze mapping#Fluid flow|Convergence/Bifurcation]] *[[Drag (force)]] *[[Lift (force)]] *[[Ocean surface waves]] *[[Rossby wave]] *[[Shock wave]] *[[Soliton]] *[[Stokes drift]] *[[Turbulence]] *[[Venturi effect]] *[[Vortex]] *[[Vorticity]] *[[Wave drag]] ===Applications=== *[[Acoustics]] *[[Aerodynamics]] *[[Fluid power]] *[[Meteorology]] *[[Naval Architecture]] *[[Oceanography]] *[[Plasma physics]] *[[Pneumatics]] *[[Pump]] === 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]] {{col-end}} {{NonDimFluMech}} == External links == *[http://www.chemengg.com/FM/FM_Gen.htm Fluid Mechanics @ Chemical Engineering Information Exchange] * [http://www.tandf.co.uk/journals/titles/03091929.asp Geophysical and Astrophysical Fluid Dynamics] * [http://www.salihnet.freeservers.com/engineering/fm/fm_books.html List of Fluid Dynamics books] {{physics-footer|continuum='''[[Continuum mechanics]]'''}} [[Category:Aerodynamics]] [[Category:Chemical engineering]] [[Category:Continuum mechanics]] [[Category:Fluid dynamics| ]] [[Category:Fluid mechanics| ]] [[Category:Piping]] <!-- interwiki --> [[de:Strömungslehre]] [[et:Hüdrodünaamika]] [[cs:Proudění]] [[eo:Fluidaĵ-Dinamiko]] [[fa:دینامیک سیالات]] [[fr:Dynamique des fluides]] [[he:הידרודינמיקה]] [[ko:유체동역학]] [[id:Dinamika fluida]] [[io:Fluido dinamiko]] [[it:Fluidodinamica]] [[no:Væskedynamikk]] [[nn:Væskedynamikk]] [[om:Fluid dynamics]] [[pl:Dynamika płynów]] [[ru:Гидродинамика]] [[simple:Fluid dynamics]] [[fi:Virtausdynamiikka]] [[tr:Akışkanlar dinamiği]] [[vi:Thủy động lực học]] [[zh:流體動力學]]