Editing Computational fluid dynamics (section)

==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 |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 |pages=119997 |doi=10.1016/j.ces.2024.119997 |issn=0009-2509}}</ref>