View source for Computational fluid dynamics

{{short description|Analysis and solving of problems that involve fluid flows}}
{{more footnotes needed|date=September 2014}}

{{Computational physics}}

'''Computational fluid dynamics''' ('''CFD''') is a branch of [[fluid mechanics]] that uses [[numerical analysis]] and [[data structure]]s to analyze and solve problems that involve [[fluid dynamics|fluid flows]].  Computers are used to perform the calculations required to simulate the free-stream flow of the fluid, and the interaction of the fluid ([[liquid]]s and [[gas]]es) with surfaces defined by [[Boundary value problem#Boundary value conditions|boundary conditions]]. With high-speed [[supercomputer]]s, better solutions can be achieved, and are often required to solve the largest and most complex problems. Ongoing research yields software that improves the accuracy and speed of complex simulation scenarios such as [[transonic]] or [[turbulence|turbulent]] flows. Initial validation of such software is typically performed using experimental apparatus such as [[wind tunnel]]s.  In addition, previously performed [[Closed-form solution|analytical]] or [[Empirical research|empirical]] analysis of a particular problem can be used for comparison.  A final validation is often performed using full-scale testing, such as [[flight test]]s.

CFD is applied to a range of research and engineering problems in multiple fields of study and industries, including [[aerodynamics]] and aerospace analysis, [[hypersonics]], [[Numerical weather prediction|weather simulation]], natural science and [[environmental engineering]], industrial system design and analysis, [[biological engineering]], fluid flows and [[heat transfer]], [[engine]] and [[combustion]] analysis, and [[visual effects]] for film and games.

==Background and history==
[[File:CFD Shuttle.jpg|thumb|right|A computer simulation of high velocity air flow around the [[Space Shuttle]] during re-entry]]
[[File:X-43A (Hyper - X) Mach 7 computational fluid dynamic (CFD).jpg|thumb|right|A simulation of the [[Hyper-X]] scramjet vehicle in operation at [[Mach number|Mach]]-7]]

The fundamental basis of almost all CFD problems is the [[Navier–Stokes equations]], which define a number of single-phase (gas or liquid, but not both) fluid flows.  These equations can be simplified by removing terms describing [[viscous]] actions to yield the [[Euler equations (fluid dynamics)|Euler equations]].  Further simplification, by removing terms describing [[vorticity]] yields the [[full potential equation]]s. Finally, for small [[Perturbation theory|perturbation]]s in subsonic and [[supersonic]] flows (not [[transonic]] or [[hypersonic]]) these equations can be [[Linearization|linearized]] to yield the linearized potential equations.

Historically, methods were first developed to solve the linearized potential equations. Two-dimensional (2D) methods, using [[conformal transformation]]s of the flow about a [[Cylinder (geometry)|cylinder]] to the flow about an [[airfoil]] were developed in the 1930s.<ref>{{cite book |last1=Milne-Thomson |first1=Louis Melville |title=Theoretical Aerodynamics |date=1973 |publisher=Courier Corporation |isbn=978-0-486-61980-4 }}{{page needed|date=December 2023}}</ref><ref>{{cite journal |last1=McMurtry |first1=Patrick A. |last2=Gansauge |first2=Todd C. |last3=Kerstein |first3=Alan R. |last4=Krueger |first4=Steven K. |title=Linear eddy simulations of mixing in a homogeneous turbulent flow |journal=Physics of Fluids A: Fluid Dynamics |date=April 1993 |volume=5 |issue=4 |pages=1023–1034 |doi=10.1063/1.858667 |bibcode=1993PhFlA...5.1023M }}</ref>

One of the earliest type of calculations resembling modern CFD are those by [[Lewis Fry Richardson]], in the sense that these calculations used finite differences and divided the physical space in cells. Although they failed dramatically, these calculations, together with Richardson's book ''Weather Prediction by Numerical Process'',<ref>{{cite book|title=Weather prediction by numerical process|year=1965|publisher=Dover Publications|author=Richardson, L. F.|author2=Chapman, S.}}</ref> set the basis for modern CFD and numerical meteorology. In fact, early CFD calculations during the 1940s using [[ENIAC]] used methods close to those in Richardson's 1922 book.<ref>{{cite journal |last1=Hunt |first1=J.C.R. |title=Lewis Fry Richardson and his contributions to mathematics, meteorology, and models of conflict |journal=Annual Review of Fluid Mechanics |date=January 1998 |volume=30 |issue=1 |pages=xiii–xxxvi |doi=10.1146/annurev.fluid.30.1.0 |bibcode=1998AnRFM..30D..13H }}</ref>

The computer power available paced development of [[Three-dimensional space|three-dimensional]] methods.  Probably the first work using computers to model fluid flow, as governed by the Navier–Stokes equations, was performed at [[Los Alamos National Lab]], in the T3 group.<ref name="legacy_T3">{{cite web |title=The Legacy of Group T-3 |url=https://www.lanl.gov/engage/organizations/aldsct/theoretical/fdsm |access-date=March 13, 2013}}</ref><ref name=harlow2004fluid>{{cite journal |last1=Harlow |first1=Francis H. |title=Fluid dynamics in Group T-3 Los Alamos National Laboratory |journal=Journal of Computational Physics |date=April 2004 |volume=195 |issue=2 |pages=414–433 |doi=10.1016/j.jcp.2003.09.031 |bibcode=2004JCoPh.195..414H |url=https://zenodo.org/record/1259097 }}</ref> This group was led by [[Francis H. Harlow]], who is widely considered one of the pioneers of CFD. From 1957 to late 1960s, this group developed a variety of numerical methods to simulate transient two-dimensional fluid flows, such as [[particle-in-cell]] method,<ref>{{cite book |last1=Harlow |first1=Francis Harvey |last2=Evans |first2=Martha |last3=Richtmyer |first3=Robert D. |title=A Machine Calculation Method for Hydrodynamic Problems |series=Lams ;1956 |date=1955 |publisher=Los Alamos Scientific Laboratory of the University of California |oclc=1288309947 |hdl=2027/mdp.39015095283399 }}{{page needed|date=December 2023}}</ref> [[fluid-in-cell]] method,<ref>{{cite journal |last1=Gentry |first1=Richard A |last2=Martin |first2=Robert E |last3=Daly |first3=Bart J |title=An Eulerian differencing method for unsteady compressible flow problems |journal=Journal of Computational Physics |date=August 1966 |volume=1 |issue=1 |pages=87–118 |doi=10.1016/0021-9991(66)90014-3 |bibcode=1966JCoPh...1...87G }}</ref> [[vorticity stream function]] method,<ref name=Fromm1963>{{cite journal |last1=Fromm |first1=Jacob E. |last2=Harlow |first2=Francis H. |title=Numerical Solution of the Problem of Vortex Street Development |journal=The Physics of Fluids |date=July 1963 |volume=6 |issue=7 |pages=975–982 |doi=10.1063/1.1706854 |bibcode=1963PhFl....6..975F }}</ref> and
[[marker-and-cell method]].<ref name=harlow_welch>{{cite journal |last1=Harlow |first1=Francis H. |last2=Welch |first2=J. Eddie |title=Numerical Calculation of Time-Dependent Viscous Incompressible Flow of Fluid with Free Surface |journal=The Physics of Fluids |date=December 1965 |volume=8 |issue=12 |pages=2182–2189 |doi=10.1063/1.1761178 |bibcode=1965PhFl....8.2182H }}</ref> Fromm's vorticity-stream-function method for 2D, transient, [[incompressible flow]] was the first treatment of strongly contorting incompressible flows in the world.

The first paper with three-dimensional model was published by John Hess and [[A.M.O. Smith]] of [[Douglas Aircraft]] in 1967.<ref>{{cite journal |last1=Hess |first1=J.L. |last2=Smith |first2=A.M.O. |title=Calculation of potential flow about arbitrary bodies |journal=Progress in Aerospace Sciences |date=1967 |volume=8 |pages=1–138 |doi=10.1016/0376-0421(67)90003-6 |bibcode = 1967PrAeS...8....1H }}</ref> This method discretized the surface of the geometry with panels, giving rise to this class of programs being called Panel Methods.  Their method itself was simplified, in that it did not include lifting flows and hence was mainly applied to ship hulls and aircraft fuselages.  The first lifting Panel Code (A230) was described in a paper written by Paul Rubbert and Gary Saaris of Boeing Aircraft in 1968.<ref>{{cite book |doi=10.2514/6.1972-188 |chapter=Review and evaluation of a three-dimensional lifting potential flow computational method for arbitrary configurations |title=10th Aerospace Sciences Meeting |date=1972 |last1=Rubbert |first1=P. |last2=Saaris |first2=G. }}</ref>  In time, more advanced three-dimensional Panel Codes were developed at [[Boeing]] (PANAIR, A502),<ref>{{cite book |doi=10.2514/6.1981-1255 |chapter=PAN AIR - A higher order panel method for predicting subsonic or supersonic linear potential flows about arbitrary configurations |title=14th Fluid and Plasma Dynamics Conference |date=1981 |last1=Carmichael |first1=R. |last2=Erickson |first2=L. }}</ref> [[Lockheed Corporation|Lockheed]] (Quadpan),<ref>{{cite book |doi=10.2514/6.1983-1827 |chapter=Comparison of panel method formulations and its influence on the development of QUADPAN, an advanced low-order method |title=Applied Aerodynamics Conference |date=1983 |last1=Youngren |first1=H. |last2=Bouchard |first2=E. |last3=Coopersmith |first3=R. |last4=Miranda |first4=L. }}</ref> [[Douglas Aircraft Company|Douglas]] (HESS),<ref>{{cite book |doi=10.2514/6.1983-1828 |chapter=Analysis of complex inlet configurations using a higher-order panel method |title=Applied Aerodynamics Conference |date=1983 |last1=Hess |first1=J. |last2=Friedman |first2=D. }}</ref> [[McDonnell Aircraft]] (MACAERO),<ref>Bristow, D.R., "[https://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/19800007773.pdf Development of Panel Methods for Subsonic Analysis and Design]," NASA CR-3234, 1980.</ref> [[NASA]] (PMARC)<ref>Ashby, Dale L.; Dudley, Michael R.; Iguchi, Steve K.; Browne, Lindsey and Katz, Joseph, "[https://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/19920023178.pdf Potential Flow Theory and Operation Guide for the Panel Code PMARC]", NASA NASA-TM-102851 1991.</ref> and Analytical Methods (WBAERO,<ref>Woodward, F.A., Dvorak, F.A. and Geller, E.W., "[https://web.archive.org/web/20191210001500/http://www.dtic.mil/dtic/tr/fulltext/u2/782202.pdf A Computer Program for Three-Dimensional Lifting Bodies in Subsonic Inviscid Flow]," USAAMRDL Technical Report, TR 74-18, Ft. Eustis, Virginia, April 1974.</ref> USAERO<ref>{{cite journal |last1=Katz |first1=Joseph |last2=Maskew |first2=Brian |title=Unsteady low-speed aerodynamic model for complete aircraft configurations |journal=Journal of Aircraft |date=April 1988 |volume=25 |issue=4 |pages=302–310 |doi=10.2514/3.45564 }}</ref> and VSAERO<ref>{{cite journal |last1=Maskew |first1=Brian |title=Prediction of Subsonic Aerodynamic Characteristics: A Case for Low-Order Panel Methods |journal=Journal of Aircraft |date=February 1982 |volume=19 |issue=2 |pages=157–163 |doi=10.2514/3.57369 }}</ref><ref>Maskew, Brian, "[https://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/19900004884.pdf Program VSAERO Theory Document: A Computer Program for Calculating Nonlinear Aerodynamic Characteristics of Arbitrary Configurations]", NASA CR-4023, 1987.</ref>).  Some (PANAIR, HESS and MACAERO) were higher order codes, using higher order distributions of surface singularities, while others (Quadpan, PMARC, USAERO and VSAERO) used single singularities on each surface panel.  The advantage of the lower order codes was that they ran much faster on the computers of the time.  Today, VSAERO has grown to be a multi-order code and is the most widely used program of this class. It has been used in the development of a number of [[submarine]]s, surface [[ship]]s, [[automobile]]s, [[helicopter]]s, [[aircraft]], and more recently [[wind turbine]]s.  Its sister code, USAERO is an unsteady panel method that has also been used for modeling such things as high speed trains and racing [[yacht]]s.  The NASA PMARC code from an early version of VSAERO and a derivative of PMARC, named CMARC,<ref>Pinella, David and Garrison, Peter, "Digital Wind Tunnel CMARC; Three-Dimensional Low-Order Panel Codes," Aerologic, 2009.</ref> is also commercially available.

In the two-dimensional realm, a number of Panel Codes have been developed for airfoil analysis and design.  The codes typically have a [[boundary layer]] analysis included, so that viscous effects can be modeled. {{ill|Richard Eppler|de}} developed the PROFILE code, partly with NASA funding, which became available in the early 1980s.<ref>Eppler, R.; Somers, D. M., "[https://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/19800020753.pdf A Computer Program for the Design and Analysis of Low-Speed Airfoils]," NASA TM-80210, 1980.</ref>  This was soon followed by [[Mark Drela]]'s [[XFOIL]] code.<ref>Drela, Mark, "[http://www.web.mit.edu/drela/Public/papers/xfoil_sv.pdf XFOIL: An Analysis and Design System for Low Reynolds Number Airfoils]," in Springer-Verlag Lecture Notes in Engineering, No. 54, 1989.</ref> Both PROFILE and XFOIL incorporate two-dimensional panel codes, with coupled boundary layer codes for airfoil analysis work.  PROFILE uses a [[conformal transformation]] method for inverse airfoil design, while XFOIL has both a conformal transformation and an inverse panel method for airfoil design.

An intermediate step between Panel Codes and Full Potential codes were codes that used the Transonic Small Disturbance equations.  In particular, the three-dimensional WIBCO code,<ref>{{cite book |doi=10.2514/6.1977-207 |chapter=Calculation of transonic wing flows by grid embedding |title=15th Aerospace Sciences Meeting |date=1977 |last1=Boppe |first1=C. }}</ref> developed by Charlie Boppe of [[Grumman Aircraft]] in the early 1980s has seen heavy use.
[[File:Starship simul 3.png|thumb|220x220px|A simulation of the [[SpaceX Starship]] during re-entry]]
Developers turned to Full Potential codes, as panel methods could not calculate the non-linear flow present at [[transonic]] speeds.  The first description of a means of using the Full Potential equations was published by Earll Murman and [[Julian Cole]] of Boeing in 1970.<ref name="Murman Cole 1971"/> Frances Bauer, [[Paul Garabedian]] and [[David Korn (computer scientist)|David Korn]] of the Courant Institute at [[New York University]] (NYU) wrote a series of two-dimensional Full Potential airfoil codes that were widely used, the most important being named Program H.<ref>{{cite book |doi=10.1007/978-3-642-80678-0 |title=A Theory of Supercritical Wing Sections, with Computer Programs and Examples |series=Lecture Notes in Economics and Mathematical Systems |date=1972 |volume=66 |isbn=978-3-540-05807-6 }}{{page needed|date=December 2023}}</ref>  A further growth of Program H was developed by Bob Melnik and his group at [[Grumman Aerospace]] as Grumfoil.<ref>Mead, H. R.; Melnik, R. E., "[https://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/19860002735.pdf GRUMFOIL: A computer code for the viscous transonic flow over airfoils]," NASA CR-3806, 1985.</ref> [[Antony Jameson]], originally at Grumman Aircraft and the Courant Institute of NYU, worked with David Caughey to develop the important three-dimensional Full Potential code FLO22<ref>{{cite book |doi=10.2514/6.1977-635 |chapter=A finite volume method for transonic potential flow calculations |title=3rd Computational Fluid Dynamics Conference |date=1977 |last1=Jameson |first1=A. |last2=Caughey |first2=D. }}</ref> in 1975.  A number of Full Potential codes emerged after this, culminating in Boeing's Tranair (A633) code,<ref>{{cite book |doi=10.2514/6.1987-34 |chapter=TRANAIR - A computer code for transonic analyses of arbitrary configurations |title=25th AIAA Aerospace Sciences Meeting |date=1987 |last1=Samant |first1=S. |last2=Bussoletti |first2=J. |last3=Johnson |first3=F. |last4=Burkhart |first4=R. |last5=Everson |first5=B. |last6=Melvin |first6=R. |last7=Young |first7=D. |last8=Erickson |first8=L. |last9=Madson |first9=M. }}</ref> which still sees heavy use.

The next step was the Euler equations, which promised to provide more accurate solutions of transonic flows.  The methodology used by Jameson in his three-dimensional FLO57 code<ref>{{cite book |doi=10.2514/6.1981-1259 |chapter=Numerical solution of the Euler equations by finite volume methods using Runge Kutta time stepping schemes |title=14th Fluid and Plasma Dynamics Conference |date=1981 |last1=Jameson |first1=A. |last2=Schmidt |first2=Wolfgang |last3=Turkel |first3=ELI }}</ref> (1981) was used by others to produce such programs as Lockheed's TEAM program<ref>{{Cite journal | doi=10.2514/3.45717|title = Improvements to an Euler aerodynamic method for transonic flow analysis| journal=Journal of Aircraft| volume=26| pages=13–20|year = 1989|last1 = Raj|first1 = Pradeep| last2=Brennan| first2=James E.}}</ref> and IAI/Analytical Methods' MGAERO program.<ref>{{cite book |doi=10.2514/6.1991-3236 |chapter=Application of an efficient 3-D multigrid Euler method (MGAERO) to complete aircraft configurations |title=9th Applied Aerodynamics Conference |date=1991 |last1=Tidd |first1=D. |last2=Strash |first2=D. |last3=Epstein |first3=B. |last4=Luntz |first4=A. |last5=Nachshon |first5=A. |last6=Rubin |first6=T. }}</ref>  MGAERO is unique in being a structured [[cartesian coordinate system|cartesian]] mesh code, while most other such codes use structured body-fitted grids (with the exception of NASA's highly successful CART3D code,<ref>{{cite book |doi=10.2514/6.1995-853 |chapter=3D applications of a Cartesian grid Euler method |title=33rd Aerospace Sciences Meeting and Exhibit |date=1995 |last1=Melton |first1=John |last2=Berger |first2=Marsha |last3=Aftosmis |first3=Michael |last4=Wong |first4=Michael }}</ref> Lockheed's SPLITFLOW code<ref>{{cite book |doi=10.2514/6.1995-343 |chapter=SPLITFLOW - A 3D unstructured Cartesian/Prismatic grid CFD code for complex geometries |title=33rd Aerospace Sciences Meeting and Exhibit |date=1995 |last1=Karman, l, Jr |first1=Steve }}</ref> and [[Georgia Institute of Technology|Georgia Tech]]'s NASCART-GT).<ref>{{cite book |doi=10.2514/6.2004-581 |chapter=An Embedded Boundary Cartesian Grid Scheme for Viscous Flows Using a New Viscous Wall Boundary Condition Treatment |title=42nd AIAA Aerospace Sciences Meeting and Exhibit |date=2004 |last1=Marshall |first1=David |last2=Ruffin |first2=Stephen |isbn=978-1-62410-078-9 |url=https://digitalcommons.calpoly.edu/aero_fac/85 }}</ref> [[Antony Jameson]] also developed the three-dimensional AIRPLANE code<ref>{{cite book |doi=10.2514/6.1986-103 |chapter=Calculation of Inviscid Transonic Flow over a Complete Aircraft |title=24th Aerospace Sciences Meeting |date=1986 |last1=Jameson |first1=A. |last2=Baker |first2=T. |last3=Weatherill |first3=N. }}</ref> which made use of unstructured tetrahedral grids.

In the two-dimensional realm, Mark Drela and Michael Giles, then graduate students at MIT, developed the ISES Euler program<ref>{{cite book |doi=10.2514/6.1985-1530 |chapter=Newton solution of direct and inverse transonic Euler equations |title=7th Computational Physics Conference |date=1985 |last1=Giles |first1=M. |last2=Drela |first2=M. |last3=Thompkins, Jr. |first3=W. }}</ref> (actually a suite of programs) for airfoil design and analysis.  This code first became available in 1986 and has been further developed to design, analyze and optimize single or multi-element airfoils, as the MSES program.<ref>{{cite book |doi=10.2514/6.1990-1470 |chapter=Newton solution of coupled viscous/Inviscid multielement airfoil flows |title=21st Fluid Dynamics, Plasma Dynamics and Lasers Conference |date=1990 |last1=Drela |first1=Mark }}</ref>  MSES sees wide use throughout the world.  A derivative of MSES, for the design and analysis of airfoils in a cascade, is MISES,<ref>Drela, M. and Youngren H., "A User's Guide to MISES 2.53", MIT Computational Sciences Laboratory, December 1998.</ref> developed by Harold Youngren while he was a graduate student at MIT.

The Navier–Stokes equations were the ultimate target of development.  Two-dimensional codes, such as NASA Ames' ARC2D code first emerged.  A number of three-dimensional codes were developed (ARC3D, [[Overflow (software)|OVERFLOW]], CFL3D are three successful NASA contributions), leading to numerous commercial packages.

Recently CFD methods have gained traction for modeling the flow behavior of granular materials within various chemical processes in engineering. This approach has emerged as a cost-effective alternative, offering a nuanced understanding of complex flow phenomena while minimizing expenses associated with traditional experimental methods.<ref>{{Cite journal |last1=Jop |first1=Pierre |last2=Forterre |first2=Yoël |last3=Pouliquen |first3=Olivier |date=June 2006 |title=A constitutive law for dense granular flows |url=https://www.nature.com/articles/nature04801 |journal=Nature |language=en |volume=441 |issue=7094 |pages=727–730 |doi=10.1038/nature04801 |pmid=16760972 |arxiv=cond-mat/0612110 |bibcode=2006Natur.441..727J |issn=1476-4687}}</ref><ref>{{Cite journal |last1=Biroun |first1=Mehdi H. |last2=Mazzei |first2=Luca |date=June 2024 |title=Unchannelized granular flows: Effect of initial granular column geometry on fluid dynamics |url=https://doi.org/10.1016/j.ces.2024.119997 |journal=Chemical Engineering Science |volume=292 |article-number=119997 |doi=10.1016/j.ces.2024.119997 |issn=0009-2509|url-access=subscription }}</ref>

==Hierarchy of fluid flow equations==
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 |archive-date=2020-01-12 |archive-url=https://web.archive.org/web/20200112214652/https://home.cscamm.umd.edu/centpack/examples/euler1d.htm |url-status=dead }}</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).

==Methodology==
In all of these approaches the same basic procedure is followed.
* During [[preprocessor (CAE)|preprocessing]]
** The [[geometry]] and physical bounds of the problem can be defined using [[Computer-aided design|computer aided design]] (CAD). From there, data can be suitably processed (cleaned-up) and the fluid volume (or fluid domain) is extracted.
** The [[volume]] occupied by the fluid is divided into discrete cells (the mesh). The mesh may be uniform or non-uniform, structured or unstructured, consisting of a combination of hexahedral, tetrahedral, prismatic, pyramidal or polyhedral elements.
** The physical modeling is defined – for example, the equations of fluid motion + [[enthalpy]] + radiation + species conservation
** Boundary conditions are defined. This involves specifying the fluid behaviour and properties at all bounding surfaces of the fluid domain. For transient problems, the initial conditions are also defined.
* The [[computer simulation|simulation]] is started and the equations are solved iteratively as a steady-state or transient.
* Finally a postprocessor is used for the analysis and visualization of the resulting solution.

===Discretization methods===
{{further|Discretization of Navier–Stokes equations}}

The stability of the selected discretisation is generally established numerically rather than analytically as with simple linear problems.  Special care must also be taken to ensure that the discretisation handles discontinuous solutions gracefully.  The [[Euler equations (fluid dynamics)|Euler equations]] and [[Navier–Stokes equations]] both admit shocks and contact surfaces.

Some of the discretization methods being used are:

==== Finite volume method ====

{{Main|Finite volume method}}

The finite volume method (FVM) is a common approach used in CFD codes, as it has an advantage in [[Random-access memory|memory]] usage and solution speed, especially for large problems, high [[Reynolds number]] turbulent flows, and source term dominated flows (like combustion).<ref>{{cite book|last=Patankar|first=Suhas V.|author-link=Suhas Patankar|title=Numerical Heat Transfer and Fluid FLow|year=1980|publisher=Hemisphere Publishing Corporation
|isbn=978-0891165224}}</ref>

In the finite volume method, the governing partial differential equations (typically the Navier-Stokes equations, the mass and energy conservation equations, and the turbulence equations) are recast in a conservative form, and then solved over discrete control volumes. This [[discretization]] guarantees the conservation of fluxes through a particular control volume. The finite volume equation yields governing equations in the form,
:<math>\frac{\partial}{\partial t}\iiint Q\, dV + \iint F\, d\mathbf{A} = 0,</math>
where <math>Q</math> is the vector of conserved variables, <math>F</math> is the vector of fluxes (see [[Euler equations (fluid dynamics)|Euler equations]] or [[Navier–Stokes equations]]), <math>V</math> is the volume of the control volume element, and <math>\mathbf{A}</math> is the surface area of the control volume element.

==== Finite element method ====

{{Main|Finite element method}}

The finite element method (FEM) is used in structural analysis of solids, but is also applicable to fluids.  However, the FEM formulation requires special care to ensure a conservative solution. The FEM formulation has been adapted for use with fluid dynamics governing equations.<ref>{{Cite web |title=Detailed Explanation of the Finite Element Method (FEM) |url=https://www.comsol.com/multiphysics/finite-element-method |access-date=2022-07-15 |website=www.comsol.com}}</ref><ref name=":0">{{Cite book |last=Anderson |first=John David |url=https://books.google.com/books?id=phG_QgAACAAJ |title=Computational Fluid Dynamics: The Basics with Applications |date=1995 |publisher=McGraw-Hill |isbn=978-0-07-113210-7 |language=en}}</ref> Although FEM must be carefully formulated to be conservative, it is much more stable than the finite volume approach.<ref>{{cite journal| title=k-version of finite element method in gas dynamics: higher-order global differentiability numerical solutions| last1=Surana| first1=K.A.| last2=Allu| first2=S.| last3=Tenpas| first3=P.W.| last4=Reddy| first4=J.N.| journal=International Journal for Numerical Methods in Engineering| volume=69| issue=6| pages=1109–1157|date=February 2007| doi=10.1002/nme.1801|bibcode = 2007IJNME..69.1109S | s2cid=122551159}}</ref> FEM also provides more accurate solutions for smooth problems comparing to FVM.<ref>{{cite journal |last1=Surana |first1=KS |last2=Allu |first2=S |last3=Tenpas |first3=PW |last4=Reddy |first4=JN |title=k-version of finite element method in gas dynamics: higher-order global differentiability numerical solutions |journal=International Journal for Numerical Methods in Engineering |volume=69 |issue=6 |pages=1109–1157 |year=2007 |publisher=Wiley Online Library|doi=10.1002/nme.1801 |bibcode=2007IJNME..69.1109S }}</ref> Another advantage of FEM is that it can handle complex geometries and boundary conditions. However, FEM can require more memory and has slower solution times than the FVM.<ref>{{cite journal |last1=Surana |first1=KS |last2=Allu |first2=S |last3=Tenpas |first3=PW |last4=Reddy |first4=JN |title=k-version of finite element method in gas dynamics: higher-order global differentiability numerical solutions |journal=International Journal for Numerical Methods in Engineering |volume=69 |issue=6 |pages=1109–1157 |year=2007 |publisher=Wiley Online Library|doi=10.1002/nme.1801 |bibcode=2007IJNME..69.1109S }}</ref>

In this method, a weighted residual equation is formed:

:<math>R_i = \iiint W_i Q \, dV^e</math>

where <math>R_i</math> is the equation residual at an element vertex <math>i</math>, <math>Q</math> is the conservation equation expressed on an element basis, <math>W_i</math> is the weight factor, and <math>V^{e}</math> is the volume of the element.

==== Finite difference method ====

{{Main|Finite difference method}}

The finite difference method (FDM) has historical importance<ref name=":0" /> and is simple to program.  It is currently only used in few specialized codes, which handle complex geometry with high accuracy and efficiency by using embedded boundaries or overlapping grids (with the solution interpolated across each grid).{{Citation needed|date=November 2010}} 
:<math> 
\frac{\partial Q}{\partial t}+
\frac{\partial F}{\partial x}+
\frac{\partial G}{\partial y}+
\frac{\partial H}{\partial z}=0
</math>
where <math>Q</math> is the vector of conserved variables, and <math>F</math>, <math>G</math>, and <math>H</math> are the fluxes in the <math>x</math>, <math>y</math>, and <math>z</math> directions respectively.

==== Spectral element method ====

{{Main|Spectral element method}}

Spectral element method is a finite element type method. It requires the mathematical problem (the [[partial differential equation]]) to be cast in a weak formulation. This is typically done by multiplying the differential equation by an arbitrary test function and integrating over the whole domain. Purely mathematically, the test functions are completely arbitrary - they belong to an infinite-dimensional function space. Clearly an infinite-dimensional function space cannot be represented on a discrete spectral element mesh; this is where the spectral element discretization begins. The most crucial thing is the choice of interpolating and testing functions. In a standard, low order FEM in 2D, for quadrilateral elements the most typical choice is the bilinear test or interpolating function of the form <math>v(x,y) = ax+by+cxy+d</math>. In a spectral element method however, the interpolating and test functions are chosen to be polynomials of a very high order (typically e.g. of the 10th order in CFD applications). This guarantees the rapid convergence of the method. Furthermore, very efficient integration procedures must be used, since the number of integrations to be performed in numerical codes is big. Thus, high order Gauss integration quadratures are employed, since they achieve the highest accuracy with the smallest number of computations to be carried out.
At the time there are some academic CFD codes based on the spectral element method and some more are currently under development, since the new time-stepping schemes arise in the scientific world.

==== Lattice Boltzmann method ====

{{Main|Lattice Boltzmann methods}}

The lattice Boltzmann method (LBM) with its simplified kinetic picture on a lattice provides a computationally efficient description of hydrodynamics.
Unlike the traditional CFD methods, which solve the conservation equations of macroscopic properties (i.e., mass, momentum, and energy) numerically, LBM models the fluid consisting of fictive particles, and such particles perform consecutive propagation and collision processes over a discrete lattice mesh. In this method, one works with the discrete in space and time version of the kinetic evolution equation in the Boltzmann [[Bhatnagar–Gross–Krook operator|Bhatnagar-Gross-Krook (BGK)]] form.

==== Vortex method ====

The vortex method, also Lagrangian Vortex Particle Method, is a [[Meshfree methods|meshfree]] technique for the simulation of incompressible turbulent flows. In it, [[vorticity]] is discretized onto [[Lagrangian and Eulerian specification of the flow field|Lagrangian]] particles, these computational elements being called vortices, vortons, or vortex particles.<ref>{{cite book |last1=Cottet |first1=Georges-Henri |last2=Koumoutsakos |first2=Petros D. |date=2000 |title=Vortex Methods: Theory and Practice |location=Cambridge, UK |publisher=Cambridge Univ. Press |isbn=0-521-62186-0
}}</ref> Vortex methods were developed as a grid-free methodology that would not be limited by the fundamental smoothing effects associated with grid-based methods. To be practical, however, vortex methods require means for rapidly computing velocities from the vortex elements – in other words they require the solution to a particular form of the [[N-body problem]] (in which the motion of N objects is tied to their mutual influences). This breakthrough came in the 1980s with the development of the [[Barnes–Hut simulation|Barnes-Hut]] and [[fast multipole method]] (FMM) algorithms. These paved the way to practical computation of the velocities from the vortex elements.

Software based on the vortex method offer a new means for solving tough fluid dynamics problems with minimal user intervention.{{Citation needed|date=November 2010}}  All that is required is specification of problem geometry and setting of boundary and initial conditions. Among the significant advantages of this modern technology;
* It is practically grid-free, thus eliminating numerous iterations associated with RANS and LES.
* All problems are treated identically. No modeling or calibration inputs are required.
* Time-series simulations, which are crucial for correct analysis of acoustics, are possible.
* The small scale and large scale are accurately simulated at the same time.

==== Boundary element method ====

{{Main|Boundary element method}}

In the boundary element method, the boundary occupied by the fluid is divided into a surface mesh.

==== High-resolution discretization schemes ====

{{Main|High-resolution scheme}}

High-resolution schemes are used where shocks or discontinuities are present. Capturing sharp changes in the solution requires the use of second or higher-order numerical schemes that do not introduce spurious oscillations. This usually necessitates the application of [[flux limiters]] to ensure that the solution is [[total variation diminishing]].{{Citation needed|date=November 2010}}

===Turbulence models===

In computational modeling of turbulent flows, one common objective is to obtain a model that can predict quantities of interest, such as fluid velocity, for use in engineering designs of the system being modeled.  For turbulent flows, the range of length scales and complexity of phenomena involved in turbulence make most modeling approaches prohibitively expensive; the resolution required to resolve all scales involved in turbulence is beyond what is computationally possible.  The primary approach in such cases is to create numerical models to approximate unresolved phenomena.  This section lists some commonly used computational models for turbulent flows.

Turbulence models can be classified based on computational expense, which corresponds to the range of scales that are modeled versus resolved (the more turbulent scales that are resolved, the finer the resolution of the simulation, and therefore the higher the computational cost). If a majority or all of the turbulent scales are not modeled, the computational cost is very low, but the tradeoff comes in the form of decreased accuracy.

In addition to the wide range of length and time scales and the associated computational cost, the governing equations of fluid dynamics contain a [[Nonlinear system|non-linear]] convection term and a non-linear and non-local pressure gradient term.  These nonlinear equations must be solved numerically with the appropriate boundary and initial conditions.

==== Reynolds-averaged Navier–Stokes ====
{{main|Reynolds-averaged Navier–Stokes equations}}
[[File:DrivAer SST-URANS-DDES Comparison.png|thumb|258x258px|External aerodynamics of the [https://www.mw.tum.de/en/aer/research-groups/automotive/drivaer/ DrivAer] model, computed using [[Reynolds-averaged Navier–Stokes equations|URANS]] (top) and [[Detached eddy simulation|DDES]] (bottom)]]

[[File:Verus Engineering Porsche 987.2 Ventus 2 Package.png|thumb|A simulation of aerodynamic package of a [[Porsche Cayman|Porsche Cayman (987.2)]]]]

[[Reynolds-averaged Navier–Stokes equations|Reynolds-averaged Navier–Stokes]] (RANS) equations are the oldest approach to turbulence modeling. An ensemble version of the governing equations is solved, which introduces new ''apparent stresses'' known as [[Reynolds stresses]]. This adds a second-order tensor of unknowns for which various models can provide different levels of closure. It is a common misconception that the RANS equations do not apply to flows with a time-varying mean flow because these equations are 'time-averaged'. In fact, statistically unsteady (or non-stationary) flows can equally be treated. This is sometimes referred to as URANS. There is nothing inherent in Reynolds averaging to preclude this, but the turbulence models used to close the equations are valid only as long as the time over which these changes in the mean occur is large compared to the time scales of the turbulent motion containing most of the energy.

RANS models can be divided into two broad approaches:

; [[Turbulence modeling|Boussinesq hypothesis]]: This method involves using an algebraic equation for the Reynolds stresses which include determining the turbulent viscosity, and depending on the level of sophistication of the model, solving transport equations for determining the turbulent kinetic energy and dissipation. Models include k-ε ([[Brian Launder|Launder]] and [[Brian Spalding|Spalding]]),<ref>{{cite journal|last=Launder|first=B.E.|author2=D.B. Spalding|year=1974|title= The Numerical Computation of Turbulent Flows|journal=Computer Methods in Applied Mechanics and Engineering|pages=269–289|doi = 10.1016/0045-7825(74)90029-2|bibcode = 1974CMAME...3..269L|volume=3|issue=2 }}</ref> Mixing Length Model ([[Ludwig Prandtl|Prandtl]]),<ref name=wilcox>{{cite book|last=Wilcox|first=David C.|title=Turbulence Modeling for CFD|year=2006|publisher=DCW Industries, Inc.|isbn=978-1-928729-08-2|edition=3}}</ref> and Zero Equation Model (Cebeci and [[Apollo M. O. Smith|Smith]]).<ref name=wilcox /> The models available in this approach are often referred to by the number of transport equations associated with the method. For example, the Mixing Length model is a "Zero Equation" model because no transport equations are solved; the <math>k-\epsilon</math> is a "Two Equation" model because two transport equations (one for <math>k</math> and one for <math>\epsilon</math>) are solved.
; [[Reynolds stress model]] (RSM): This approach attempts to actually solve transport equations for the Reynolds stresses. This means introduction of several transport equations for all the Reynolds stresses and hence this approach is much more costly in CPU effort.{{Citation needed|date=November 2010}}

====Large eddy simulation====
{{main|Large eddy simulation}}

[[Image:LESPremixedFlame.jpg|thumb|right|250px|Volume rendering of a non-premixed swirl flame as simulated by LES]]
[[Large eddy simulation]] (LES) is a technique in which the smallest scales of the flow are removed through a filtering operation, and their effect modeled using subgrid scale models.  This allows the largest and most important scales of the turbulence to be resolved, while greatly reducing the computational cost incurred by the smallest scales. This method requires greater computational resources than RANS methods, but is far cheaper than DNS.

====Detached eddy simulation====
{{main|Detached eddy simulation}}

[[Detached eddy simulation]]s (DES) is a modification of a RANS model in which the model switches to a subgrid scale formulation in regions fine enough for LES calculations. Regions near solid boundaries and where the turbulent length scale is less than the maximum grid dimension are assigned the RANS mode of solution. As the turbulent length scale exceeds the grid dimension, the regions are solved using the LES mode. Therefore, the grid resolution for DES is not as demanding as pure LES, thereby considerably cutting down the cost of the computation. Though DES was initially formulated for the Spalart-Allmaras model (Philippe R. Spalart et al., 1997), it can be implemented with other RANS models (Strelets, 2001), by appropriately modifying the length scale which is explicitly or implicitly involved in the RANS model. So while Spalart–Allmaras model based DES acts as LES with a wall model, DES based on other models (like two equation models) behave as a hybrid RANS-LES model. Grid generation is more complicated than for a simple RANS or LES case due to the RANS-LES switch. DES is a non-zonal approach and provides a single smooth velocity field across the RANS and the LES regions of the solutions.
[[File:Cp IDDES.gif|thumb|IDDES Simulation of the Karel Motorsports BMW. This is a type of DES simulation completed in OpenFOAM. The plot is coefficient of pressure.]]

====Direct numerical simulation====
{{main|Direct numerical simulation}}

[[Direct numerical simulation]] (DNS) resolves the entire range of turbulent length scales.  This marginalizes the effect of models, but is extremely expensive.  The computational cost is proportional to <math>Re^{3}</math>.<ref name="Pope_2000">{{cite book|title=Turbulent Flows|author=Pope, S.B.|publisher=Cambridge University Press|year=2000|isbn=978-0-521-59886-6}}</ref> DNS is intractable for flows with complex geometries or flow configurations.

====Coherent vortex simulation====

The coherent vortex simulation approach decomposes the turbulent flow field into a coherent part, consisting of organized vortical motion, and the incoherent part, which is the random background flow.<ref name="Farge_2001">{{cite journal|title=Coherent Vortex Simulation (CVS), A Semi-Deterministic Turbulence Model Using Wavelets|last1=Farge | first1= Marie | author1-link = Marie Farge|author2=Schneider, Kai|journal=Flow, Turbulence and Combustion|volume=66|issue=4|pages=393–426|doi=10.1023/A:1013512726409|year=2001|bibcode=2001FTC....66..393F |s2cid=53464243 }}</ref>  This decomposition is done using [[wavelet]] filtering.  The approach has much in common with LES, since it uses decomposition and resolves only the filtered portion, but different in that it does not use a linear, low-pass filter.  Instead, the filtering operation is based on wavelets, and the filter can be adapted as the flow field evolves. [[Marie Farge|Farge]] and Schneider tested the CVS method with two flow configurations and showed that the coherent portion of the flow exhibited the <math>-\frac{40}{39}</math> energy spectrum exhibited by the total flow, and corresponded to coherent structures ([[vortex stretching|vortex tubes]]), while the incoherent parts of the flow composed homogeneous background noise, which exhibited no organized structures.  Goldstein and Vasilyev<ref name="Goldstein_2004">{{cite journal|author1=Goldstein, Daniel|author2=Vasilyev, Oleg|title=Stochastic coherent adaptive large eddy simulation method|journal=Physics of Fluids A|year=1995|volume=24|page=2497|doi=10.1063/1.1736671|bibcode = 2004PhFl...16.2497G|issue=7 |citeseerx=10.1.1.415.6540}}</ref> applied the FDV model to large eddy simulation, but did not assume that the wavelet filter eliminated all coherent motions from the subfilter scales.  By employing both LES and CVS filtering, they showed that the SFS dissipation was dominated by the SFS flow field's coherent portion.

====PDF methods====

[[Probability density function]] (PDF) methods for turbulence, first introduced by [[Thomas S. Lundgren|Lundgren]],<ref name="Lundgren_1969">{{cite journal|title=Model equation for nonhomogeneous turbulence|author=Lundgren, T.S.|journal=Physics of Fluids A|volume=12|issue=3|year=1969|pages=485–497|doi=10.1063/1.1692511|bibcode = 1969PhFl...12..485L }}</ref> are based on tracking the one-point PDF of the velocity, <math>f_{V}(\boldsymbol{v};\boldsymbol{x},t) d\boldsymbol{v}</math>, which gives the probability of the velocity at point <math>\boldsymbol{x}</math> being between <math>\boldsymbol{v}</math> and <math>\boldsymbol{v}+d\boldsymbol{v}</math>.  This approach is analogous to the [[kinetic theory of gases]], in which the macroscopic properties of a gas are described by a large number of particles.  PDF methods are unique in that they can be applied in the framework of a number of different turbulence models; the main differences occur in the form of the PDF transport equation.  For example, in the context of [[large eddy simulation]], the PDF becomes the filtered PDF.<ref name="Colucci_1998">{{cite journal|title=Filtered density function for large eddy simulation of turbulent reacting flows|author1=Colucci, P.J.|author2=Jaberi, F.A|author3=Givi, P.|author4=Pope, S.B.|journal=Physics of Fluids A|year=1998|volume=10|issue=2|pages=499–515|doi=10.1063/1.869537|bibcode = 1998PhFl...10..499C }}</ref>  PDF methods can also be used to describe chemical reactions,<ref name="Fox_2003">{{cite book|author=Fox, Rodney|title=Computational models for turbulent reacting flows|year=2003|publisher=Cambridge University Press|isbn=978-0-521-65049-6}}</ref><ref name="Pope_1985">{{cite journal|title=PDF methods for turbulent reactive flows|author=Pope, S.B.|journal=Progress in Energy and Combustion Science|year=1985|volume=11|pages=119–192|bibcode = 1985PECS...11..119P|doi=10.1016/0360-1285(85)90002-4|issue=2 }}</ref> and are particularly useful for simulating chemically reacting flows because the chemical source term is closed and does not require a model.  The PDF is commonly tracked by using Lagrangian particle methods; when combined with large eddy simulation, this leads to a [[Langevin equation]] for subfilter particle evolution.

====Vorticity confinement method====

{{Main|Vorticity confinement}}

The [[vorticity confinement]] (VC) method is an Eulerian technique used in the simulation of turbulent wakes. It uses a solitary-wave like approach to produce a stable solution with no numerical spreading. VC can capture the small-scale features to within as few as 2 grid cells. Within these features, a nonlinear difference equation is solved as opposed to the [[finite difference equation]]. VC is similar to [[shock capturing methods]], where conservation laws are satisfied, so that the essential integral quantities are accurately computed.

====Linear eddy model====

The Linear eddy model is a technique used to simulate the convective mixing that takes place in turbulent flow.<ref>{{cite journal|last=Krueger|first=Steven K.|title=Linear Eddy Simulations Of Mixing In A Homogeneous Turbulent Flow|journal=Physics of Fluids|year=1993|volume=5|issue=4|pages=1023–1034|bibcode = 1993PhFlA...5.1023M |doi = 10.1063/1.858667 |url=https://zenodo.org/record/1232081}}</ref> Specifically, it provides a mathematical way to describe the interactions of a scalar variable within the vector flow field. It is primarily used in one-dimensional representations of turbulent flow, since it can be applied across a wide range of length scales and Reynolds numbers. This model is generally used as a building block for more complicated flow representations, as it provides high resolution predictions that hold across a large range of flow conditions.

===Two-phase flow===

[[File:Bubble-rising.jpg|thumb|right|Simulation of bubble horde using [[volume of fluid method]]]]

The modeling of [[two-phase flow]] is still under development. Different methods have been proposed, including the [[Volume of fluid method]], the [[level-set method]] and [[front tracking]].<ref>{{cite journal |last1=Hirt |first1=C.W |last2=Nichols |first2=B.D |title=Volume of fluid (VOF) method for the dynamics of free boundaries |journal=Journal of Computational Physics |date=January 1981 |volume=39 |issue=1 |pages=201–225 |doi=10.1016/0021-9991(81)90145-5 |bibcode=1981JCoPh..39..201H }}</ref><ref>{{cite journal |last1=Unverdi |first1=Salih Ozen |last2=Tryggvason |first2=Grétar |title=A front-tracking method for viscous, incompressible, multi-fluid flows |journal=Journal of Computational Physics |date=May 1992 |volume=100 |issue=1 |pages=25–37 |doi=10.1016/0021-9991(92)90307-K |bibcode=1992JCoPh.100...25U |hdl=2027.42/30059 |hdl-access=free }}</ref>  These methods often involve a tradeoff between maintaining a sharp interface or conserving mass {{According to whom|date=November 2010}}.  This is crucial since the evaluation of the density, viscosity and surface tension is based on the values averaged over the interface.{{Citation needed|date=November 2010}}

===Solution algorithms===

Discretization in the space produces a system of [[ordinary differential equations]] for unsteady problems and algebraic equations for steady problems. Implicit or semi-implicit methods are generally used to integrate the ordinary differential equations, producing a system of (usually) nonlinear algebraic equations.  Applying a [[Newton's method#Nonlinear systems of equations|Newton]] or [[Fixed point iteration|Picard]] iteration produces a system of linear equations which is nonsymmetric in the presence of advection and indefinite in the presence of incompressibility. Such systems, particularly in 3D, are frequently too large for direct solvers, so iterative methods are used, either stationary methods such as [[Successive over-relaxation|successive overrelaxation]] or [[Krylov subspace]] methods. Krylov methods such as [[Generalized minimal residual method|GMRES]], typically used with [[Preconditioner|preconditioning]], operate by minimizing the residual over successive subspaces generated by the preconditioned operator.

[[Multigrid method|Multigrid]] has the advantage of asymptotically optimal performance on a number of problems. Traditional{{According to whom|date=November 2010}} solvers and preconditioners are effective at reducing high-frequency components of the residual, but low-frequency components typically require a number of iterations to reduce. By operating on multiple scales, multigrid reduces all components of the residual by similar factors, leading to a mesh-independent number of iterations.{{Citation needed|date=November 2010}}

For indefinite systems, preconditioners such as [[incomplete LU factorization]], [[Additive Schwarz method|additive Schwarz]], and [[Multigrid method|multigrid]] perform poorly or fail entirely, so the problem structure must be used for effective preconditioning.<ref>{{cite journal |last1=Benzi |first1=Michele |last2=Golub |first2=Gene H. |last3=Liesen |first3=Jörg |title=Numerical solution of saddle point problems |journal=Acta Numerica |date=May 2005 |volume=14 |pages=1–137 |doi=10.1017/S0962492904000212 |bibcode=2005AcNum..14....1B |s2cid=122717775 |citeseerx=10.1.1.409.4160 }}</ref> Methods commonly used in CFD are the [[SIMPLE algorithm|SIMPLE]] and [[Uzawa iteration|Uzawa algorithms]] which exhibit mesh-dependent convergence rates, but recent advances based on block LU factorization combined with multigrid for the resulting definite systems have led to preconditioners that deliver mesh-independent convergence rates.<ref>{{cite journal |last1=Elman |first1=Howard |last2=Howle |first2=V.E. |last3=Shadid |first3=John |last4=Shuttleworth |first4=Robert |last5=Tuminaro |first5=Ray |title=A taxonomy and comparison of parallel block multi-level preconditioners for the incompressible Navier–Stokes equations |journal=Journal of Computational Physics |date=January 2008 |volume=227 |issue=3 |pages=1790–1808 |doi=10.1016/j.jcp.2007.09.026 |bibcode=2008JCoPh.227.1790E |osti=920807 |s2cid=16365489 |url=https://digital.library.unt.edu/ark:/67531/metadc902332/ }}</ref>

===Unsteady aerodynamics===
CFD made a major break through in late 70s with the introduction of LTRAN2, a 2-D code to model oscillating airfoils based on [[transonic]] small perturbation theory by Ballhaus and associates.<ref>{{cite journal |last1=Adamson |first1=M.R. |title=Biographies |journal=IEEE Annals of the History of Computing |date=January 2006 |volume=28 |issue=1 |pages=99–103 |doi=10.1109/MAHC.2006.5 |bibcode=2006IAHC...28a..99A }}</ref> It uses a Murman-Cole switch algorithm for modeling the moving shock-waves.<ref name="Murman Cole 1971">{{cite journal |last1=Murman |first1=Earll M. |last2=Cole |first2=Julian D. |title=Calculation of plane steady transonic flows |journal=AIAA Journal |date=January 1971 |volume=9 |issue=1 |pages=114–121 |doi=10.2514/3.6131 |bibcode=1971AIAAJ...9..114C }}</ref> Later it was extended to 3-D with use of a rotated difference scheme by AFWAL/Boeing that resulted in LTRAN3.<ref>{{cite journal |last1=Jameson |first1=Antony |title=Iterative solution of transonic flows over airfoils and wings, including flows at mach 1 |journal=Communications on Pure and Applied Mathematics |date=May 1974 |volume=27 |issue=3 |pages=283–309 |doi=10.1002/cpa.3160270302 }}</ref><ref>Borland, C.J., "XTRAN3S - Transonic Steady and Unsteady Aerodynamics for Aeroelastic Applications,"AFWAL-TR-85-3214, Air Force Wright Aeronautical Laboratories, Wright-Patterson AFB, OH, January, 1986</ref>

===Biomedical engineering===

[[File:Vel-Streamline-FC.jpg|thumb|right|Simulation of blood flow in a human [[aorta]]]]
CFD investigations are used to clarify the characteristics of aortic flow in details that are beyond the capabilities of experimental measurements. To analyze these conditions, CAD models of the human vascular system are extracted employing modern imaging techniques such as [[MRI]] or [[Computed Tomography]]. A 3D model is reconstructed from this data and the fluid flow can be computed. Blood properties such as density and viscosity, and realistic boundary conditions (e.g.  systemic pressure) have to be taken into consideration. Therefore, making it possible to analyze and optimize the flow in the cardiovascular system for different applications.<ref>Kaufmann, T.A.S., Graefe, R., Hormes, M., Schmitz-Rode, T. and Steinseiferand, U., "Computational Fluid Dynamics in Biomedical Engineering", Computational Fluid Dynamics: Theory, Analysis and Applications, pp. 109–136</ref>

=== CPU versus GPU ===
Traditionally, CFD simulations are performed on CPUs.<ref>{{cite arXiv |last1=Lao |first1=Shandong |last2=Holt |first2=Aaron |last3=Vaidhynathan |first3=Deepthi |last4=Sitaraman |first4=Hariswaran |last5=Hrenya |first5=Christine M. |last6=Hauser |first6=Thomas |title=Performance comparison of CFD-DEM solver MFiX-Exa, on GPUs and CPUs |date=2021 |class=cs.DC |eprint=2108.08821 }}</ref>

In a more recent trend, simulations are also performed on GPUs. These typically contain slower but more processors. For CFD algorithms that feature good parallelism performance (i.e. good speed-up by adding more cores) this can greatly reduce simulation times. Fluid-implicit particle<ref>{{cite journal |last1=Wu |first1=Kui |last2=Truong |first2=Nghia |last3=Yuksel |first3=Cem |last4=Hoetzlein |first4=Rama |title=Fast Fluid Simulations with Sparse Volumes on the GPU |journal=Computer Graphics Forum |date=May 2018 |volume=37 |issue=2 |pages=157–167 |doi=10.1111/cgf.13350 |s2cid=43945038 }}</ref> and lattice-Boltzmann methods<ref>{{Cite web|url=http://www.nvidia.com/content/intersect-360-HPC-application-support.pdf|title=Intersect 360 HPC application Support}}</ref> are typical examples of codes that scale well on GPUs.

== See also ==
{{Div col|colwidth=25em}}
* [[Application of CFD in thermal power plants]]
* [[Blade element theory]]
* [[Boundary conditions in fluid dynamics]]
* [[Cavitation modelling]]
* [[Central differencing scheme]]
* [[Computational magnetohydrodynamics]]
* [[Discrete element method]]
* [[Fictitious domain method]]
* [[Finite element method]]
* [[Finite volume method for unsteady flow]]
* [[Fluid animation]]
* [[Immersed boundary method]]
* [[Lattice Boltzmann methods]]
* [[List of finite element software packages]]
* [[Meshfree methods]]
* [[Moving particle semi-implicit method]]
* [[Multi-particle collision dynamics]]
* [[Multidisciplinary design optimization]]
* [[Numerical methods in fluid mechanics]]
* [[Shape optimization]]
* [[Smoothed-particle hydrodynamics]]
* [[Stochastic Eulerian Lagrangian method]]
* [[Turbulence modeling]]
* Unified methods for computing incompressible and compressible flow
* [[Visualization (graphics)]]
* [[Wind tunnel]]
{{Div col end}}

==References==
{{reflist}}

==Notes==

*{{Cite book | last = Anderson | first = John D. | title = Computational Fluid Dynamics: The Basics With Applications | publisher = McGraw-Hill Science | series = Science/Engineering/Math | year = 1995 | isbn = 978-0-07-001685-9 }}
*{{Cite book | last = Patankar | first = Suhas | title = Numerical Heat Transfer and Fluid Flow | publisher = Taylor & Francis | series = Hemisphere Series on Computational Methods in Mechanics and Thermal Science | year = 1980 | isbn = 978-0-89116-522-4 }}

==External links==
{{Sister project links| wikt=no | commons=Category:Computational fluid dynamics | b=no | n=no | q=Computational fluid dynamics | s=no | v=no | voy=no | species=no | d=no}}

* ''Course:'' [https://nptel.ac.in/courses/112105045/ Computational Fluid Dynamics] – [[Suman Chakraborty]] ([[Indian Institute of Technology Kharagpur]])
* ''Course:'' [http://www3.nd.edu/~dbalsara/Numerical-PDE-Course/ Numerical PDE Techniques for Scientists and Engineers], Open access Lectures and Codes for Numerical PDEs, including a modern view of Compressible CFD

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