Computational fluid dynamics Computational fluid dynamics, usually abbreviated as The fundamental basis of almost all CFD problems are CFD, is a branch of fluid mechanics that uses numerical the Navier–Stokes equations, which define any single- analysis and algorithms to solve and analyze problems that phase (gas or liquid, but not both) fluid flow. These involve fluid flows. Computers are used to perform the equations can be simplified by removing terms de- calculations required to simulate the interaction of liquids scribing viscous actions to yield the Euler equations. and gases with surfaces defined by boundary conditions. Further simplification, by removing terms describing With high-speed supercomputers, better solutions can be vorticity yields the full potential equations. Finally, for achieved. Ongoing research yields software that improves small perturbations in subsonic and supersonic flows (not the accuracy and speed of complex simulation scenarios transonic or hypersonic) these equations can be linearized such as transonic or turbulent flows. Initial experimen- to yield the linearized potential equations. tal validation of such software is performed using a wind Historically, methods were first developed to solve the tunnel with the final validation coming in full-scale test- linearized potential equations. Two-dimensional (2D) ing, e.g. flight tests. methods, using conformal transformations of the flow about a cylinder to the flow about an airfoil were devel- oped in the 1930s.[1] 1 Background and history One of the earliest type of calculations resembling mod- ern 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 dra- matically, these calculations, together with Richardson’s book “Weather prediction by numerical process”,[2] 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.[3] The computer power available paced development of 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.[4][5] This group was led by Francis H. Harlow, who is widely considered as one of the pio- neers of CFD. From 1957 to late 1960s, this group de- A computer simulation of high velocity air flow around the Space Shuttle during re-entry. veloped a variety of numerical methods to simulate tran- sient two-dimensional fluid flows, such as Particle-in-cell method (Harlow, 1957),[6] Fluid-in-cell method (Gen- try, Martin and Daly, 1966),[7] Vorticity stream func- tion method (Jake Fromm, 1963),[8] and Marker-and-cell method (Harlow and Welch, 1965).[9] Fromm’s vorticity- stream-function method for 2D, transient, incompressible flow was the first treatment of strongly contorting incom- pressible 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.[10] 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 A simulation of the Hyper-X scramjet vehicle in operation at flows and hence was mainly applied to ship hulls and Mach−7 1 2 2 METHODOLOGY aircraft fuselages. The first lifting Panel Code (A230) at Grumman Aircraft and the Courant Institute of NYU, was described in a paper written by Paul Rubbert and worked with David Caughey to develop the important Gary Saaris of Boeing Aircraft in 1968.[11] In time, three-dimensional Full Potential code FLO22[28] in 1975. more advanced three-dimensional Panel Codes were Many Full Potential codes emerged after this, culminat- developed at Boeing (PANAIR, A502),[12] Lockheed ing in Boeing’s Tranair (A633) code,[29] which still sees (Quadpan),[13] Douglas (HESS),[14] McDonnell Air- heavy use. craft (MACAERO),[15] NASA (PMARC)[16] and [17] [18] The next step was the Euler equations, which promised to Analytical Methods (WBAERO, USAERO provide more accurate solutions of transonic flows. The and VSAERO[19][20]). Some (PANAIR, HESS and methodology used by Jameson in his three-dimensional MACAERO) were higher order codes, using higher FLO57 code[30] (1981) was used by others to pro- order distributions of surface singularities, while others duce such programs as Lockheed’s TEAM program[31] (Quadpan, PMARC, USAERO and VSAERO) used and IAI/Analytical Methods’ MGAERO program.[32] single singularities on each surface panel. The advantage MGAERO is unique in being a structured cartesian of the lower order codes was that they ran much faster mesh code, while most other such codes use struc- on the computers of the time. Today, VSAERO has tured body-fitted grids (with the exception of NASA’s grown to be a multi-order code and is the most widely highly successful CART3D code,[33] Lockheed’s SPLIT- used program of this class. It has been used in the devel- FLOW code[34] and Georgia Tech's NASCART-GT).[35] opment of many submarines, surface ships, automobiles, Antony Jameson also developed the three-dimensional helicopters, aircraft, and more recently wind turbines. AIRPLANE code[36] which made use of unstructured Its sister code, USAERO is an unsteady panel method tetrahedral grids. that has also been used for modeling such things as high speed trains and racing yachts. The NASA PMARC In the two-dimensional realm, Mark Drela and Michael code from an early version of VSAERO and a derivative Giles, then graduate students at MIT, developed the ISES of PMARC, named CMARC,[21] is also commercially Euler program[37] (actually a suite of programs) for air- available. foil design and analysis. This code first became available in 1986 and has been further developed to design, ana- In the two-dimensional realm, a number of Panel Codes lyze and optimize single or multi-element airfoils, as the have been developed for airfoil analysis and design. MSES program.[38] MSES sees wide use throughout the The codes typically have a boundary layer analysis in- world. A derivative of MSES, for the design and anal- cluded, so that viscous effects can be modeled. Pro- ysis of airfoils in a cascade, is MISES,[39] developed by fessor Richard Eppler of the University of Stuttgart de- Harold “Guppy” Youngren while he was a graduate stu- veloped the PROFILE code, partly with NASA fund- dent at MIT. ing, which became available in the early 1980s.[22] This was soon followed by MIT Professor Mark Drela’s The Navier–Stokes equations were the ultimate tar- XFOIL code.[23] Both PROFILE and XFOIL incorpo- get of development. Two-dimensional codes, such as rate two-dimensional panel codes, with coupled bound- NASA Ames’ ARC2D code first emerged. A number ary layer codes for airfoil analysis work. PROFILE uses of three-dimensional codes were developed (ARC3D, a conformal transformation method for inverse airfoil de- OVERFLOW, CFL3D are three successful NASA con- sign, while XFOIL has both a conformal transformation tributions), leading to numerous commercial packages. and an inverse panel method for airfoil design. An intermediate step between Panel Codes and Full Potential codes were codes that used the Transonic 2 Methodology Small Disturbance equations. In particular, the three- dimensional WIBCO code,[24] developed by Charlie In all of these approaches the same basic procedure is Boppe of Grumman Aircraft in the early 1980s has seen followed. heavy use. • During preprocessing Developers turned to Full Potential codes, as panel meth- ods could not calculate the non-linear flow present at • The geometry (physical bounds) of the prob- transonic speeds. The first description of a means of us- lem is defined. ing the Full Potential equations was published by Earll • The volume occupied by the fluid is divided Murman and Julian Cole of Boeing in 1970.[25] Frances into discrete cells (the mesh). The mesh may Bauer, Paul Garabedian and David Korn of the Courant be uniform or non-uniform. Institute at New York University (NYU) wrote a se- • The physical modeling is defined – for exam- ries of two-dimensional Full Potential airfoil codes that ple, the equations of motion + enthalpy + ra- were widely used, the most important being named Pro- diation + species conservation [26] gram H. A further growth of Program H was de- • Boundary conditions are defined. This in- veloped by Bob Melnik and his group at Grumman volves specifying the fluid behaviour and prop- [27] Aerospace as Grumfoil. Antony Jameson, originally erties at the boundaries of the problem. For 2.1 Discretization methods 3 transient problems, the initial conditions are a conservative solution. The FEM formulation has been also defined. adapted for use with fluid dynamics governing equations. Although FEM must be carefully formulated to be con- • The simulation is started and the equations are servative, it is much more stable than the finite volume solved iteratively as a steady-state or transient. approach.[41] However, FEM can require more memory [42] • Finally a postprocessor is used for the analysis and and has slower solution times than the FVM. visualization of the resulting solution. In this method, a weighted residual equation is formed: 2.1 Discretization methods ZZZ e Ri = WiQ dV Further information: Discretization of Navier–Stokes equations where Ri is the equation residual at an element vertex i , Q is the conservation equation expressed on an element basis, W is the weight factor, and V e is the volume of The stability of the selected discretisation is generally es- i tablished numerically rather than analytically as with sim- the element. ple linear problems. Special care must also be taken to ensure that the discretisation handles discontinuous solu- 2.1.3 Finite difference method tions gracefully.