Editing Computational fluid dynamics (section)

==Hierarchy of fluid flow equations==
{{See also|Computational Fluid Dynamics for Phase Change Materials}}
CFD can be seen as a group of computational methodologies (discussed below) used to solve equations governing fluid flow.  In the application of CFD, a critical step is to decide which set of physical assumptions and related equations need to be used for the problem at hand.<ref name="ferziger">{{cite book|title=Computational methods for fluid dynamics|year=2002|publisher=Springer-Verlag|author=Ferziger, J. H. and Peric, M.}}</ref> To illustrate this step, the following summarizes the physical assumptions/simplifications taken in equations of a flow that is single-phase (see [[multiphase flow]] and [[two-phase flow]]), single-species (i.e., it consists of one chemical species), non-reacting, and (unless said otherwise) compressible.  Thermal radiation is neglected, and body forces due to gravity are considered (unless said otherwise). In addition, for this type of flow, the next discussion highlights the hierarchy of flow equations solved with CFD.  Note that some of the following equations could be derived in more than one way.

* [[Conservation laws]] (CL):  These are the most fundamental equations considered with CFD in the sense that, for example, all the following equations can be derived from them.  For a single-phase, single-species, compressible flow one considers the [[conservation of mass]], [[conservation of linear momentum]], and [[conservation of energy]].
<!-- 
* Liouville equation (LE):
* Boltzmann transport equation (BTE):
<ref name="vincenti">{{cite book|title=Introduction to physical gas dynamics|year=1965|publisher=Krieger Publishing Company|author=Vincenti, W. G. and Kruger, C. H.}}</ref>
* BGK
* Direct Simulation Monte Carlo (DSMC) Method:
<ref name="bird">{{cite book|title=The DSMC method|year=2013|publisher=G. A. Bird|author=Bird, G. A.}}</ref>
-->
* Continuum conservation laws (CCL):  Start with the CL.  Assume that mass, momentum and energy are ''locally'' conserved:  These quantities are conserved and cannot "teleport" from one place to another but can only move by a continuous flow (see [[continuity equation]]).  Another interpretation is that one starts with the CL and assumes a continuum medium (see [[continuum mechanics]]).  The resulting system of equations is unclosed since to solve it one needs further relationships/equations:  (a) constitutive relationships for the [[viscous stress tensor]]; (b) constitutive relationships for the diffusive [[heat flux]]; (c) an [[equation of state]] (EOS), such as the [[ideal gas]] law; and, (d) a caloric equation of state relating temperature with quantities such as [[enthalpy]] or [[internal energy]].
* Compressible [[Navier-Stokes equations]] (C-NS):  Start with the CCL.  Assume a Newtonian viscous stress tensor (see [[Newtonian fluid]]) and a Fourier heat flux (see [[heat flux]]).<ref name="cns">{{cite web |url=https://www.cfd-online.com/Wiki/Navier-Stokes_equations |title=Navier-Stokes equations |access-date=2020-01-07}}</ref><ref name="panton">{{cite book|title=Incompressible Flow|year=1996|publisher=John Wiley and Sons|author=Panton, R. L.}}</ref> The C-NS need to be augmented with an EOS and a caloric EOS to have a closed system of equations.
* Incompressible Navier-Stokes equations (I-NS):  Start with the C-NS.  Assume that density is always and everywhere constant.<ref name="landau">{{cite book|title=Fluid Mechanics|year=2007|publisher=Elsevier|author=Landau, L. D. and Lifshitz, E. M.}}</ref> Another way to obtain the I-NS is to assume that the [[Mach number]] is very small<ref name="landau"/><ref name="panton"/> and that temperature differences in the fluid are very small as well.<ref name="panton"/>  As a result, the mass-conservation and momentum-conservation equations are decoupled from the energy-conservation equation, so one only needs to solve for the first two equations.<ref name="panton"/>
* Compressible [[Euler equations (fluid dynamics)|Euler equations]] (EE):  Start with the C-NS.  Assume a frictionless flow with no diffusive heat flux.<ref name="fox">{{cite book|title=Introduction to Fluid Mechanics|year=1992|publisher=John Wiley and Sons|author=Fox, R. W. and McDonald, A. T.}}</ref> 
<!-- * Compressible Burnett Equations <ref name="karniadakis">{{cite book|title=Microflows and nanoflows:  fundamentals and simulation|year=2000|publisher=Springer|author=Karniadakis, G. and Beskok, A. and Aluru, N.}}</ref>: -->
* Weakly compressible Navier-Stokes equations (WC-NS):  Start with the C-NS.  Assume that density variations depend only on temperature and not on pressure.<ref name="poinsot">{{cite book|title=Theoretical and numerical combustion|year=2005|publisher=RT Edwards|author=Poinsot, T. and Veynante, D.}}</ref> For example, for an [[ideal gas]], use <math> \rho = p_0 / (R T) </math>, where <math> p_0 </math> is a conveniently-defined reference pressure that is always and everywhere constant, <math> \rho </math> is density, <math> R </math> is the specific [[gas constant]], and <math> T </math> is temperature.  As a result, the WC-NS do not capture acoustic waves.  It is also common in the WC-NS to neglect the pressure-work and viscous-heating terms in the energy-conservation equation.  The WC-NS are also called the C-NS with the low-Mach-number approximation.
* Boussinesq equations:  Start with the C-NS.  Assume that density variations are always and everywhere negligible except in the gravity term of the momentum-conservation equation (where density multiplies the gravitational acceleration).<ref name="kundu">{{cite book|title=Fluid Mechanics|year=1990|publisher=Academic Press|author=Kundu, P.}}</ref> Also assume that various fluid properties such as [[viscosity]], [[thermal conductivity]], and [[heat capacity]] are always and everywhere constant.  The Boussinesq equations are widely used in [[microscale meteorology]].
* Compressible [[Reynolds-averaged Navier–Stokes equations]] and compressible Favre-averaged Navier-Stokes equations (C-RANS and C-FANS):  Start with the C-NS.  Assume that any flow variable <math> f </math>, such as density, velocity and pressure, can be represented as <math> f = F + f'' </math>, where <math> F </math> is the ensemble-average<ref name="panton"/> of any flow variable, and <math> f'' </math> is a perturbation or fluctuation from this average.<ref name="panton"/><ref name="ras">{{cite web |url=https://www.cfd-online.com/Wiki/Favre_averaged_Navier-Stokes_equations |title=Favre averaged Navier-Stokes equations |access-date=2020-01-07}}</ref> <math> f'' </math> is not necessarily small.  If <math> F </math> is a classic ensemble-average (see [[Reynolds decomposition]]) one obtains the Reynolds-averaged Navier–Stokes equations.  And if <math> F </math> is a density-weighted ensemble-average one obtains the Favre-averaged Navier-Stokes equations.<ref name="ras"/>  As a result, and depending on the Reynolds number, the range of scales of motion is greatly reduced, something which leads to much faster solutions in comparison to solving the C-NS.  However, information is lost, and the resulting system of equations requires the closure of various unclosed terms, notably the [[Reynolds stress]].
* Ideal flow or [[potential flow]] equations:  Start with the EE.  Assume zero fluid-particle rotation (zero vorticity) and zero flow expansion (zero divergence).<ref name="panton"/>  The resulting flowfield is entirely determined by the geometrical boundaries.<ref name="panton"/>  Ideal flows can be useful in modern CFD to initialize simulations.
* Linearized compressible Euler equations (LEE):<ref name="bailly">{{cite journal|last=Bailly, C., and Daniel J.|title=Numerical solution of acoustic propagation problems using Linearized Euler Equations|journal=AIAA Journal|year=2000|volume=38|issue=1|pages=22–29|doi=10.2514/2.949|bibcode=2000AIAAJ..38...22B}}</ref>  Start with the EE.  Assume that any flow variable <math> f </math>, such as density, velocity and pressure, can be represented as <math> f = f_0 + f' </math>, where <math> f_0 </math> is the value of the flow variable at some reference or base state, and <math> f' </math> is a perturbation or fluctuation from this state.  Furthermore, assume that this perturbation <math> f' </math> is very small in comparison with some reference value.  Finally, assume that <math> f_0 </math> satisfies "its own" equation, such as the EE. The LEE and its multiple variations are widely used in [[computational aeroacoustics]].
* Sound wave or [[acoustic wave equation]]:   Start with the LEE.  Neglect all gradients of <math> f_0 </math> and <math> f' </math>, and assume that the Mach number at the reference or base state is very small.<ref name="poinsot"/>  The resulting equations for density, momentum and energy can be manipulated into a pressure equation, giving the well-known sound wave equation.
* [[Shallow water equations]] (SW):  Consider a flow near a wall where the wall-parallel length-scale of interest is much larger than the wall-normal length-scale of interest.  Start with the EE.  Assume that density is always and everywhere constant, neglect the velocity component perpendicular to the wall, and consider the velocity parallel to the wall to be spatially-constant.
* [[Boundary layer]] equations (BL):  Start with the C-NS (I-NS) for compressible (incompressible) boundary layers.  Assume that there are thin regions next to walls where spatial gradients perpendicular to the wall are much larger than those parallel to the wall.<ref name="kundu"/>
* Bernoulli equation:  Start with the EE.  Assume that density variations depend only on pressure variations.<ref name="kundu"/>  See [[Bernoulli's Principle]].
* Steady Bernoulli equation:  Start with the Bernoulli Equation and assume a steady flow.<ref name="kundu"/>  Or start with the EE and assume that the flow is steady and integrate the resulting equation along a streamline.<ref name="fox"/><ref name="landau"/>
* [[Stokes Flow]] or creeping flow equations:  Start with the C-NS or I-NS.  Neglect the inertia of the flow.<ref name="panton"/><ref name="landau"/>  Such an assumption can be justified when the [[Reynolds number]] is very low.  As a result, the resulting set of equations is linear, something which simplifies greatly their solution.
* Two-dimensional channel flow equation:  Consider the flow between two infinite parallel plates.  Start with the C-NS.  Assume that the flow is steady, two-dimensional, and fully developed (i.e., the velocity profile does not change along the streamwise direction).<ref name="panton"/>  Note that this widely-used fully-developed assumption can be inadequate in some instances, such as some compressible, microchannel flows, in which case it can be supplanted by a ''locally'' fully-developed assumption.<ref name="harley">{{cite journal|last=Harley, J. C. and Huang, Y. and Bau, H. H. and Zemel, J. N.|title=Gas flow in micro-channels|journal=Journal of Fluid Mechanics|year=1995|volume=284|pages=257–274|doi=10.1017/S0022112095000358|bibcode=1995JFM...284..257H|s2cid=122833857 }}</ref>
* One-dimensional Euler equations or one-dimensional gas-dynamic equations (1D-EE):  Start with the EE.  Assume that all flow quantities depend only on one spatial dimension.<ref name="1d-ee">{{cite web |url=https://home.cscamm.umd.edu/centpack/examples/euler1d.htm |title=One-dimensional Euler equations |access-date=2020-01-12}}</ref>
* [[Fanno flow]] equation:  Consider the flow inside a duct with constant area and adiabatic walls.  Start with the 1D-EE.  Assume a steady flow, no gravity effects, and introduce in the momentum-conservation equation an empirical term to recover the effect of wall friction (neglected in the EE).  To close the Fanno flow equation, a model for this friction term is needed.  Such a closure involves problem-dependent assumptions.<ref name="cavazzuti">{{cite journal|last=Cavazzuti, M. and Corticelli, M. A. and Karayiannis, T. G.|title=Compressible Fanno flows in micro-channels: An enhanced quasi-2D numerical model for laminar flows|journal=Thermal Science and Engineering Progress|year=2019|volume=10|pages=10–26|doi=10.1016/j.tsep.2019.01.003|doi-access=free|bibcode=2019TSEP...10...10C |hdl=11392/2414220|hdl-access=free}}</ref>
* [[Rayleigh flow]] equation.  Consider the flow inside a duct with constant area and either non-adiabatic walls without volumetric heat sources or adiabatic walls with volumetric heat sources.  Start with the 1D-EE. Assume a steady flow, no gravity effects, and introduce in the energy-conservation equation an empirical term to recover the effect of wall heat transfer or the effect of the heat sources (neglected in the EE).