# Computational fluid dynamics

### Computational fluid dynamics

Computational fluid dynamics, usually abbreviated as CFD, is a branch of fluid mechanics that uses numerical analysis and algorithms to solve and analyze problems that involve fluid flows. Computers are used to perform the calculations required to simulate the interaction of liquids and gases with surfaces defined by boundary conditions. With high-speed supercomputers, better solutions can be achieved. Ongoing research yields software that improves the accuracy and speed of complex simulation scenarios such as transonic or turbulent flows. Initial experimental validation of such software is performed using a wind tunnel with the final validation coming in full-scale testing, e.g. flight tests.

## Contents

• Background and history 1
• Methodology 2
• Discretization methods 2.1
• Finite volume method 2.1.1
• Finite element method 2.1.2
• Finite difference method 2.1.3
• Spectral element method 2.1.4
• Boundary element method 2.1.5
• High-resolution discretization schemes 2.1.6
• Turbulence models 2.2
• Reynolds-averaged Navier–Stokes 2.2.1
• Large eddy simulation 2.2.2
• Detached eddy simulation 2.2.3
• Direct numerical simulation 2.2.4
• Coherent vortex simulation 2.2.5
• PDF methods 2.2.6
• Vortex method 2.2.7
• Vorticity confinement method 2.2.8
• Linear eddy model 2.2.9
• Two-phase flow 2.3
• Solution algorithms 2.4
• References 4
• Notes 5

## Background and history

A computer simulation of high velocity air flow around the Space Shuttle during re-entry.
A simulation of the Hyper-X scramjet vehicle in operation at Mach-7

The fundamental basis of almost all CFD problems are the Navier–Stokes equations, which define any single-phase (gas or liquid, but not both) fluid flow. These equations can be simplified by removing terms describing viscous actions to yield the Euler equations. Further simplification, by removing terms describing vorticity yields the full potential equations. Finally, for small perturbations in subsonic and supersonic flows (not transonic or hypersonic) these equations can be linearized to yield the linearized potential equations.

Historically, methods were first developed to solve the linearized potential equations. Two-dimensional (2D) methods, using conformal transformations of the flow about a cylinder to the flow about an airfoil were developed in the 1930s.[1]

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",[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 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 (Harlow, 1957),[6] Fluid-in-cell method (Gentry, Martin and Daly, 1966),[7] Vorticity stream function 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 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.[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 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.[11] In time, more advanced three-dimensional Panel Codes were developed at Boeing (PANAIR, A502),[12] Lockheed (Quadpan),[13] Douglas (HESS),[14] McDonnell Aircraft (MACAERO),[15] NASA (PMARC)[16] and Analytical Methods (WBAERO,[17] USAERO[18] and VSAERO[19][20]). 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 many submarines, surface ships, automobiles, helicopters, aircraft, and more recently wind turbines. Its sister code, USAERO is an unsteady panel method that has also been used for modeling such things as high speed trains and racing yachts. The NASA PMARC code from an early version of VSAERO and a derivative of PMARC, named CMARC,[21] 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. Professor Richard Eppler of the University of Stuttgart developed the PROFILE code, partly with NASA funding, which became available in the early 1980s.[22] This was soon followed by MIT Professor Mark Drela's XFOIL code.[23] 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,[24] developed by Charlie Boppe of Grumman Aircraft in the early 1980s has seen heavy use.

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.[25] Frances Bauer, Paul Garabedian and 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.[26] A further growth of Program H was developed by Bob Melnik and his group at Grumman Aerospace as Grumfoil.[27] 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[28] in 1975. Many Full Potential codes emerged after this, culminating in Boeing's Tranair (A633) code,[29] 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[30] (1981) was used by others to produce such programs as Lockheed's TEAM program[31] and IAI/Analytical Methods' MGAERO program.[32] MGAERO is unique in being a structured

• CFD Tutorial Many examples and images, with references to robotic fish.
• CFD-Wiki
• biomedical use
• Course: Introduction to CFD – Dmitri Kuzmin (Dortmund University of Technology)
• Course: Numerical PDE Techniques for Scientists and Engineers, Open access Lectures and Codes for Numerical PDEs, including a modern view of Compressible CFD