Editing Fluid dynamics (section)

== Classifications ==

===Compressible versus incompressible flow===
All fluids are [[compressibility|compressible]] 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]]. Otherwise the more general [[compressible flow]] equations must be used.

Mathematically, incompressibility is expressed by saying that the density {{mvar|ρ}} of a [[fluid parcel]] does not change as it moves in the flow field, that is,
<math display="block">\frac{\mathrm{D} \rho}{\mathrm{D}t} = 0 \, ,</math>
where {{math|{{sfrac|D|D''t''}}}} is the [[material derivative]], which is the sum of [[time derivative|local]] and [[convective derivative]]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]] 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 [[sound waves]] are compression waves involving changes in pressure and density of the medium through which they propagate.

===Newtonian versus non-Newtonian fluids===
[[File:Flow around a wing.gif|thumb|Flow around an [[airfoil]]]]

All fluids, except [[superfluids]], 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 (materials science)|strain rate]]; it has dimensions {{math|''T''{{isup|−1}}}}. [[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 that is independent of the strain rate.

[[Non-Newtonian fluid]]s have a more complicated, non-linear stress-strain behaviour. The sub-discipline of [[rheology]] describes the stress-strain behaviours of such fluids, which include [[emulsion]]s and [[slurries]], some [[viscoelasticity|viscoelastic]] materials such as [[blood]] and some [[polymer]]s, and ''sticky liquids'' such as [[latex]], [[honey]] and [[lubricants]].<ref>{{cite journal |last1=Wilson | first1=DI |title=What is Rheology? |journal=Eye |date=February 2018 |volume=32 |issue=2 |pages=179–183 |doi=10.1038/eye.2017.267 |pmid= 29271417 |pmc=5811736}}</ref>

===Inviscid versus viscous versus Stokes flow===

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 [[Reynolds number]] is a [[dimensionless quantity]] which characterises the magnitude of inertial effects compared to the magnitude of viscous effects. A low Reynolds number ({{math|''Re'' ≪ 1}}) 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 flow|Stokes or creeping flow]].

In contrast, high Reynolds numbers ({{math|''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. Eliminating viscosity allows the [[Navier–Stokes equations]] to be simplified into the [[Euler equations (fluid dynamics)|Euler equations]]. The integration of the Euler equations along a streamline in an inviscid flow yields [[Bernoulli's equation]]. When, in addition to being inviscid, the flow is [[Lamellar field|irrotational]] everywhere, Bernoulli's equation can completely describe the flow everywhere. Such flows are called [[potential flow]]s, because the velocity field may be expressed as the [[gradient]] of a potential energy expression.

This idea can work fairly well when the Reynolds number is high. However, problems such as those involving solid boundaries may require that the viscosity be included. Viscosity cannot be neglected near solid boundaries because the [[no-slip condition]] generates a thin region of large strain rate, the [[boundary layer]], in 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 [[d'Alembert's paradox]].

A commonly used<ref>{{Cite journal|last=Platzer|first=B.|date=2006-12-01|title=Book Review: Cebeci, T. and Cousteix, J., Modeling and Computation of Boundary-Layer Flows|url=http://dx.doi.org/10.1002/zamm.200690053|journal=ZAMM|volume=86|issue=12|pages=981–982|doi=10.1002/zamm.200690053|bibcode=2006ZaMM...86..981P |issn=0044-2267}}</ref> model, especially in [[computational fluid dynamics]], is to use two flow models: the Euler equations away from the body, and [[boundary layer]] equations in a region close to the body. The two solutions can then be matched with each other, using the [[method of matched asymptotic expansions]].

==={{Anchor|Steady vs unsteady flow}} Steady versus unsteady flow===<!-- [[Steady flow]] redirects here -->

[[File: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] {{Webarchive|url=https://web.archive.org/web/20160303182548/http://math.lanl.gov/Research/Highlights/amrmhd.shtml|date=2016-03-03}}</ref> ]]
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<ref>{{Cite web|url=https://www.cfd-online.com/Forums/main/118306-transient-state-unsteady-state.html|title=Transient state or unsteady state? -- CFD Online Discussion Forums|website=www.cfd-online.com}}</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]]. The random velocity field {{math|''U''(''x'', ''t'')}} is statistically stationary if all statistics are invariant under a shift in time.<ref name=pope >{{cite book|last=Pope|first=Stephen B.|title=Turbulent Flows|publisher=Cambridge University Press|year=2000|isbn=0-521-59886-9}}</ref>{{rp| 75}} This roughly means that all statistical properties are constant in time. Often, the mean [[Field (physics)|field]] is the object of interest, and this is constant too in a statistically stationary flow.

Steady flows are often more tractable than otherwise similar unsteady flows. The governing equations of a steady problem have one dimension fewer (time) than the governing equations of the same problem without taking advantage of the steadiness of the flow field.

===Laminar versus turbulent flow===
[[File:Laminar-turbulent transition.jpg|thumb|The transition from laminar to turbulent flow]]
Turbulence is flow characterized by recirculation, [[Eddy (fluid dynamics)|eddies]], and apparent [[random]]ness. Flow in which turbulence is not exhibited is called [[laminar flow|laminar]]. 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 name=pope/>{{rp|344}} given the state of computational power for the next few decades. Any flight vehicle large enough to carry a human ({{mvar|L}} > 3&nbsp;m), moving faster than {{cvt|20|m/s|km/h mph}} is well beyond the limit of DNS simulation ({{mvar|Re}} = 4&nbsp;million). Transport aircraft wings (such as on an [[Airbus A300]] or [[Boeing 747]]) 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]] (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 form of [[detached eddy simulation]] (DES) — a combination of LES and RANS turbulence modelling.

===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 gradient]]s and the [[Coriolis force]]. It is useful in the study of [[atmospheric dynamics]].