Computational fluid dynamics

Computational fluid dynamics is a branch of fluid mechanics that uses numerical analysis and data structures to analyze and solve problems that involve flows. Computers are used to perform the calculations required to simulate the free-stream flow of the fluid, and the interaction of the fluid with surfaces defined by boundary conditions. With high-speed supercomputers, 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 turbulent flows. Initial validation of such software is typically performed using experimental apparatus such as wind tunnels. In addition, previously performed analytical or empirical analysis of a particular problem can be used for comparison. A final validation is often performed using full-scale testing, such as flight tests.
CFD is applied to a range of research and engineering problems in multiple fields of study and industries, including aerodynamics and aerospace analysis, hypersonics, 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.

Evolution and Development

The fundamental basis of almost all CFD problems is the Navier–Stokes equations, which define a number of single-phase fluid flows. 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 these equations can be linearized to yield the linearized potential equations.
Historically, methods were first developed to solve the linearized potential equations. Two-dimensional methods, using conformal transformations of the flow about a cylinder to the flow about an airfoil were developed in the 1930s.
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, 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.
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. 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, fluid-in-cell method, vorticity stream function method, and
marker-and-cell method. 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. 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 was described in a paper written by Paul Rubbert and Gary Saaris of Boeing Aircraft in 1968. In time, more advanced three-dimensional Panel Codes were developed at Boeing, Lockheed, Douglas, McDonnell Aircraft, NASA and Analytical Methods. Some were higher order codes, using higher order distributions of surface singularities, while others 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 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, 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. developed the PROFILE code, partly with NASA funding, which became available in the early 1980s. This was soon followed by Mark Drela's XFOIL code. 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, 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. Frances Bauer, Paul Garabedian and David Korn of the Courant Institute at New York University wrote a series of two-dimensional Full Potential airfoil codes that were widely used, the most important being named Program H. A further growth of Program H was developed by Bob Melnik and his group at Grumman Aerospace as Grumfoil. 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 in 1975. A number of Full Potential codes emerged after this, culminating in Boeing's Tranair code, 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 was used by others to produce such programs as Lockheed's TEAM program and IAI/Analytical Methods' MGAERO program. MGAERO is unique in being a structured cartesian mesh code, while most other such codes use structured body-fitted grids. Antony Jameson also developed the three-dimensional AIRPLANE code 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 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. MSES sees wide use throughout the world. A derivative of MSES, for the design and analysis of airfoils in a cascade, is MISES, 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, 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.

Hierarchy of Flow Equations and Physical Assumptions

CFD can be seen as a group of computational methodologies 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. To illustrate this step, the following summarizes the physical assumptions/simplifications taken in equations of a flow that is single-phase, single-species, non-reacting, and compressible. Thermal radiation is neglected, and body forces due to gravity are considered. 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 : 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.
Continuum conservation laws : 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. Another interpretation is that one starts with the CL and assumes a continuum medium. The resulting system of equations is unclosed since to solve it one needs further relationships/equations: constitutive relationships for the viscous stress tensor; constitutive relationships for the diffusive heat flux; an equation of state, such as the ideal gas law; and, a caloric equation of state relating temperature with quantities such as enthalpy or internal energy.
Compressible Navier-Stokes equations : Start with the CCL. Assume a Newtonian viscous stress tensor and a Fourier heat flux. 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 : Start with the C-NS. Assume that density is always and everywhere constant. Another way to obtain the I-NS is to assume that the Mach number is very small and that temperature differences in the fluid are very small as well. 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.
Compressible Euler equations : Start with the C-NS. Assume a frictionless flow with no diffusive heat flux.
Weakly compressible Navier-Stokes equations : Start with the C-NS. Assume that density variations depend only on temperature and not on pressure. For example, for an ideal gas, use, where is a conveniently defined reference pressure that is always and everywhere constant, is density, is the specific gas constant, and 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. 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 : Start with the C-NS. Assume that any flow variable, such as density, velocity and pressure, can be represented as, where is the ensemble-average of any flow variable, and is a perturbation or fluctuation from this average. is not necessarily small. If is a classic ensemble-average one obtains the Reynolds-averaged Navier–Stokes equations. And if is a density-weighted ensemble-average one obtains the Favre-averaged Navier-Stokes equations. 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 and zero flow expansion. The resulting flowfield is entirely determined by the geometrical boundaries. Ideal flows can be useful in modern CFD to initialize simulations.
Linearized compressible Euler equations : Start with the EE. Assume that any flow variable, such as density, velocity and pressure, can be represented as, where is the value of the flow variable at some reference or base state, and is a perturbation or fluctuation from this state. Furthermore, assume that this perturbation is very small in comparison with some reference value. Finally, assume that 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 and, and assume that the Mach number at the reference or base state is very small. 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 : 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 : Start with the C-NS for compressible 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.
Bernoulli equation: Start with the EE. Assume that density variations depend only on pressure variations. See Bernoulli's Principle.
Steady Bernoulli equation: Start with the Bernoulli Equation and assume a steady flow. Or start with the EE and assume that the flow is steady and integrate the resulting equation along a streamline.
Stokes Flow or creeping flow equations: Start with the C-NS or I-NS. Neglect the inertia of the flow. 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. 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.
One-dimensional Euler equations or one-dimensional gas-dynamic equations : Start with the EE. Assume that all flow quantities depend only on one spatial dimension.
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. To close the Fanno flow equation, a model for this friction term is needed. Such a closure involves problem-dependent assumptions.
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.