 Computational fluid dynamics

Computational physics Numerical analysis · Simulation
Data analysis · VisualizationFluid dynamicsParticleScientistsComputational fluid dynamics, usually abbreviated as CFD, is a branch of fluid mechanics that uses numerical methods 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 highspeed 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 validation of such software is performed using a wind tunnel with the final validation coming in flight tests.
Contents
Background and history
The fundamental basis of almost all CFD problems are the Navier–Stokes equations, which define any singlephase fluid flow. These equations can be simplified by removing terms describing viscosity to yield the Euler equations. Further simplification, by removing terms describing vorticity yields the full potential equations. Finally, these equations can be linearized to yield the linearized potential equations.
Historically, methods were first developed to solve the Linearized Potential equations. Twodimensional methods, using conformal transformations of the flow about a cylinder to the flow about an airfoil were developed in the 1930s.^{[1]} The computer power available paced development of threedimensional methods. The first paper on a practical threedimensional method to solve the linearized potential equations was published by John Hess and A.M.O. Smith of Douglas Aircraft in 1967.^{[2]} 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. In time, more advanced threedimensional Panel Codes were developed at Boeing (PANAIR, A502), Lockheed (Quadpan), Douglas (HESS), McDonnell Aircraft (MACAERO), NASA (PMARC) and Analytical Methods (WBAERO, USAERO and VSAERO). 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 multiorder 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, is also commercially available.
In the twodimensional 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 PROFIL code, partly with NASA funding, which became available in the early 1980s. This was soon followed by MIT Professor Mark Drela's XFOIL code. Both PROFIL and XFOIL incorporate twodimensional panel codes, with coupled boundary layer codes for airfoil analysis work. PROFIL uses a conformal transformation method for inverse airfoil design, while XFOIL has both a conformal transformation and an inverse panel method for airfoil design. Both codes are used.
An intermediate step between Panel Codes and Full Potential codes were codes that used the Transonic Small Disturbance equations. In particular, the threedimensional 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 nonlinear 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 (NYU) wrote a series of twodimensional 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 threedimensional Full Potential code FLO22 in 1975. Many Full Potential codes emerged after this, culminating in Boeing's Tranair (A633) 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 threedimensional FLO57 code (1981) 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 bodyfitted grids (with the exception of NASA's highly successful CART3D code, Lockheed's SPLITFLOW code and Georgia Tech's NASCARTGT).^{[3]} Antony Jameson also developed the threedimensional AIRPLANE code (1985) which made use of unstructured tetrahedral grids.
In the twodimensional realm, Mark Drela and Michael Giles, then graduate students at MIT, developed the ISES Euler program (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 multielement 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 "Guppy" Youngren while he was a graduate student at MIT.
The Navier–Stokes equations were the ultimate target of developers. Twodimensional codes, such as NASA Ames' ARC2D code first emerged. A number of threedimensional codes were developed (ARC3D, OVERFLOW, CFL3D are three successful NASA contributions), leading to numerous commercial packages.
Methodology
In all of these approaches the same basic procedure is followed.
 During preprocessing
 The geometry (physical bounds) of the problem is defined.
 The volume occupied by the fluid is divided into discrete cells (the mesh). The mesh may be uniform or non uniform.
 The physical modeling is defined – for example, the equations of motions + enthalpy + radiation + species conservation
 Boundary conditions are defined. This involves specifying the fluid behaviour and properties at the boundaries of the problem. For transient problems, the initial conditions are also defined.
 The simulation is started and the equations are solved iteratively as a steadystate or transient.
 Finally a postprocessor is used for the analysis and visualization of the resulting solution.
Discretization methods
The stability of the chosen discretization is generally established numerically rather than analytically as with simple linear problems. Special care must also be taken to ensure that the discretization handles discontinuous solutions gracefully. The Euler equations and Navier–Stokes equations both admit shocks, and contact surfaces.
Some of the discretization methods being used are:
Finite volume method
Main article: Finite volume methodThe finite volume method (FVM) is a common approach used in CFD codes.^{[citation needed]} The governing equations are solved over discrete control volumes. Finite volume methods recast the governing partial differential equations (typically the NavierStokes equations) in a conservative form, and then discretize the new equation. This guarantees the conservation of fluxes through a particular control volume. The finite volume equation yields governing equations in the form,
where Q is the vector of conserved variables, F is the vector of fluxes (see Euler equations or Navier–Stokes equations), V is the volume of the control volume element, and is the surface area of the control volume element.
Finite element method
Main article: Finite element methodThe 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.^{[citation needed]} Although FEM must be carefully formulated to be conservative, it is much more stable than the finite volume approach^{[4]} However, FEM can require more memory than FVM.^{[5]}
In this method, a weighted residual equation is formed:
where R_{i} is the equation residual at an element vertex i, Q is the conservation equation expressed on an element basis, W_{i} is the weight factor, and V^{e} is the volume of the element.
Finite difference method
Main article: Finite difference methodThe finite difference method (FDM) has historical importance^{[citation needed]} and is simple to program. It is currently only used in few specialized codes.^{[citation needed]} Modern finite difference codes make use of an embedded boundary for handling complex geometries, making these codes highly efficient and accurate.^{[citation needed]} Other ways to handle geometries include use of overlapping grids, where the solution is interpolated across each grid.
where Q is the vector of conserved variables, and F, G, and H are the fluxes in the x, y, and z directions respectively.
Spectral element method
Main article: Spectral element methodSpectral element method is a finite element type method. It requires the mathematical problem (the partial differential equation) to be casted 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 infinitely dimensional function space. Clearly an infinitely dimensional function space cannot be represented on a discrete spectral element mesh. And 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 v(x,y) = ax + by + cxy + d. 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 a 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 timestepping schemes arrise in the scientific world. You can refer to the CCFD website to see movies of incompressible flows in channels simulated with a spectral element solver or to the Numerical Mechanics (see bottom of the page) website to see a movie of the liddriven cavity flow obtained with a compeletely novel unconditionally stable timestepping scheme combined with a spectral element solver.
Boundary element method
Main article: Boundary element methodIn the boundary element method, the boundary occupied by the fluid is divided into a surface mesh.
Highresolution discretization schemes
Main article: Highresolution schemeHighresolution schemes are used where shocks or discontinuities are present. Capturing sharp changes in the solution requires the use of second or higherorder 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]}
Turbulence models
In studying turbulent flows, the objective is to obtain a theory or a model that can yield quantities of interest, such as velocities. For turbulent flow, the range of length scales and complexity of phenomena make most approaches impossible. The primary approach in this case is to create numerical models to calculate the properties of interest. A selection of some commonlyused computational models for turbulent flows are presented in this section.
The chief difficulty in modeling turbulent flows comes from the wide range of length and time scales associated with turbulent flow. As a result, turbulence models can be classified based on the range of these length and time scales that are modeled and the range of length and time scales that are 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 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 convection term and a nonlinear and nonlocal pressure gradient term. These nonlinear equations must be solved numerically with the appropriate boundary and initial conditions.
Main article: Reynoldsaveraged Navier–Stokes equationsReynoldsaveraged NavierStokes (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 timevarying mean flow because these equations are 'timeaveraged'. In fact, statistically unsteady (or nonstationary) 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:
 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 kε (Launder and Spalding),^{[6]} Mixing Length Model (Prandtl),^{[7]} and Zero Equation Model (Cebeci and Smith).^{[7]} 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 is a "Two Equation" model because two transport equations (one for k and one for ) 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]}
Large eddy simulation
Main article: Large eddy simulationLarge 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 article: Detached eddy simulationDetached eddy simulations (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 SpalartAllmaras model (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 SpalartAllmaras model based DES acts as LES with a wall model, DES based on other models (like two equation models) behave as a hybrid RANSLES model. Grid generation is more complicated than for a simple RANS or LES case due to the RANSLES switch. DES is a nonzonal approach and provides a single smooth velocity field across the RANS and the LES regions of the solutions.
Direct numerical simulation
Main article: Direct numerical simulationDirect 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 Re^{3}.^{[8]} 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.^{[9]} 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, lowpass filter. Instead, the filtering operation is based on wavelets, and the filter can be adapted as the flow field evolves. Farge and Schneider tested the CVS method with two flow configurations and showed that the coherent portion of the flow exhibited the energy spectrum exhibited by the total flow, and corresponded to coherent structures (vortex tubes), while the incoherent parts of the flow composed homogeneous background noise, which exhibited no organized structures. Goldstein and Oleg^{[10]} applied the CVS model to large eddy simulation, but did not assume that the wavelet filter completely 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 Lundgren,^{[11]} are based on tracking the onepoint PDF of the velocity, , which gives the probability of the velocity at point being between and . 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.^{[12]} PDF methods can also be used to describe chemical reactions,^{[13]}^{[14]} 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 subfiler particle evolution.
Vortex method
The vortex method is a gridfree technique for the simulation of turbulent flows. It uses vortices as the computational elements, mimicking the physical structures in turbulence. Vortex methods were developed as a gridfree methodology that would not be limited by the fundamental smoothing effects associated with gridbased 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 Nbody problem (in which the motion of N objects is tied to their mutual influences). A breakthrough came in the late 1980s with the development of the fast multipole method (FMM), an algorithm by V. Rokhlin (Yale) and L. Greengard (Courant Institute). This breakthrough paved the way to practical computation of the velocities from the vortex elements and is the basis of successful algorithms. They are especially wellsuited to simulating filamentary motion, such as wisps of smoke, in realtime simulations such as video games, because of the fine detail achieved using minimal computation.^{[15]}
Software based on the vortex method offer a new means for solving tough fluid dynamics problems with minimal user intervention.^{[citation needed]} 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 gridfree, thus eliminating numerous iterations associated with RANS and LES.
 All problems are treated identically. No modeling or calibration inputs are required.
 Timeseries simulations, which are crucial for correct analysis of acoustics, are possible.
 The small scale and large scale are accurately simulated at the same time.
Vorticity confinement method
Main article: Vorticity confinementThe vorticity confinement (VC) method is an Eulerian technique used in the simulation of turbulent wakes. It uses a solitarywave 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.
Twophase flow
The modeling of twophase flow is still under development. Different methods have been proposed^{[by whom?]}. The Volume of fluid method has received a lot of attention lately,^{[by whom?]} for problems that do not have dispersed particles,^{[citation needed]} but the Level set method and front tracking are also valuable approaches^{[according to whom?]}. Most of these methods are either good in maintaining a sharp interface or at conserving mass^{[according to whom?]}. This is crucial since the evaluation of the density, viscosity and surface tension is based on the values averaged over the interface.^{[citation needed]} Lagrangian multiphase models, which are used for dispersed media, are based on solving the Lagrangian equation of motion for the dispersed phase.^{[citation needed]}
Solution algorithms
Discretization in space produces a system of ordinary differential equations for unsteady problems and algebraic equations for steady problems. Implicit or semiimplicit methods are generally used to integrate the ordinary differential equations, producing a system of (usually) nonlinear algebraic equations. Applying a Newton or 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 overrelaxation or Krylov subspace methods. Krylov methods such as GMRES, typically used with preconditioning, operate by minimizing the residual over successive subspaces generated by the preconditioned operator.
Multigrid has the advantage of asymptotically optimal performance on many problems. Traditional^{[according to whom?]} solvers and preconditioners are effective at reducing highfrequency components of the residual, but lowfrequency components typically require many iterations to reduce. By operating on multiple scales, multigrid reduces all components of the residual by similar factors, leading to a meshindependent number of iterations.^{[citation needed]}
For indefinite systems, preconditioners such as incomplete LU factorization, additive Schwarz, and multigrid perform poorly or fail entirely, so the problem structure must be used for effective preconditioning.^{[16]} Methods commonly used in CFD are the SIMPLE and Uzawa algorithms which exhibit meshdependent convergence rates, but recent advances based on block LU factorization combined with multigrid for the resulting definite systems have led to preconditioners that deliver meshindependent convergence rates.^{[17]}
See also
 Blade element theory
 Computational magnetohydrodynamics
 Finite element analysis
 Fluid simulation
 Immersed boundary method
 KIVA (Software)
 Lattice Boltzmann methods
 List of finite element software packages
 Multidisciplinary design optimization
 Turbulence modeling
 Visualization
 Wind tunnel
 Meshfree methods
 Smoothedparticle hydrodynamics
 Moving particle semiimplicit method
Notes
 ^ MilneThomson, L.M. (1973). Theoretical Aerodynamics. Dover Publications. ISBN 048661980X.
 ^ Hess, J.L.; A.M.O. Smith (1967). "Calculation of Potential Flow About Arbitrary Bodies". Progress in Aeronautics Sciences 8: 1–138. Bibcode 1967PrAeS...8....1H. doi:10.1016/03760421(67)900036. http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6V3V4811PYT33&_user=10&_coverDate=12%2F31%2F1967&_alid=1339703692&_rdoc=1&_fmt=high&_orig=search&_cdi=5740&_docanchor=&view=c&_ct=176&_acct=C000050221&_version=1&_urlVersion=0&_userid=10&md5=2939d59a5c94a28fcc51ab6d6cec0305.
 ^ "NASCART". Daniel Guggenheim School of Aerospace Engineering. http://www.ae.gatech.edu/people/sruffin/nascart/. Retrieved 20070728.
 ^ Surana, K.A.; Allu, S.; Tenpas, P.W.; Reddy, J.N. (February 2007). "kversion of finite element method in gas dynamics: higherorder global differentiability numerical solutions". International Journal for Numerical Methods in Engineering 69 (6): 1109–1157. doi:10.1002/nme.1801.
 ^ Huebner, K. H., Thornton, E. A., and Byron, T. D. (1995). The Finite Element Method for Engineers (Third ed.). Wiley Interscience.
 ^ Launder, B. E.; D. B. Spalding (1974). "The Numerical Computation of Turbulent Flows". Computer Methods in Applied Mechanics and Engineering 3 (2): 269–289. Bibcode 1974CMAME...3..269L. doi:10.1016/00457825(74)900292.
 ^ ^{a} ^{b} Wilcox, David C. (2006). Turbulence Modeling for CFD (3 ed.). DCW Industries, Inc.. ISBN 9781928729082.
 ^ Pope, S. B. (2000). Turbulent Flows. Cambridge University Press. ISBN 9780521598866.
 ^ Farge, Marie; Schneider, Kai (2001). "Coherent Vortex Simulation (CVS), A SemiDeterministic Turbulence Model Using Wavelets". Flow, Turbulence and Combustion 66 (4): 393–426. doi:10.1023/A:1013512726409.
 ^ Goldstein, Daniel; Vasilyev, Oleg (1994). "Stochastic coherent adaptive large eddy simulation method". Physics of Fluids A 24 (7): 2497. Bibcode 2004PhFl...16.2497G. doi:10.1063/1.1736671.
 ^ Lundgren, T.S. (1969). "Model equation for nonhomogeneous turbulence". Physics of Fluids A 12 (3): 485–497. Bibcode 1969PhFl...12..485L. doi:10.1063/1.1692511.
 ^ Colucci, P. J.; Jaberi, F. A; Givi, P.; Pope, S. B. (1998). "Filtered density function for large eddy simulation of turbulent reacting flows". Physics of Fluids A 10 (2): 499–515. Bibcode 1998PhFl...10..499C. doi:10.1063/1.869537.
 ^ Fox, Rodney (2003). Computational models for turbulent reacting flows. Cambridge University Press. ISBN 9780521650496.
 ^ Pope, S. B. (1985). "PDF methods for turbulent reactive flows". Progress in Energy and Combustion Science 11 (2): 119–192. Bibcode 1985PrECS..11..119P. doi:10.1016/03601285(85)900024.
 ^ Gourlay, Michael J. (July 2009). "Fluid Simulation for Video Games". Intel Software Network. http://software.intel.com/enus/articles/fluidsimulationforvideogamespart1/.
 ^ Benzi, Golub, Liesen (2005). "Numerical solution of saddlepoint problems". Acta Numerica 14: 1–137. doi:10.1017/S0962492904000212.
 ^ Elman et al.; Howle, V; Shadid, J; Shuttleworth, R; Tuminaro, R (January 2008). "A taxonomy and comparison of parallel block multilevel preconditioners for the incompressible Navier–Stokes equations". Journal of Computational Physics 227 (3): 1790–1808. doi:10.1016/j.jcp.2007.09.026.
References
 Anderson, John D. (1995). Computational Fluid Dynamics: The Basics With Applications. Science/Engineering/Math. McGrawHill Science. ISBN 0070016852
 Patankar, Suhas (1980). Numerical Heat Transfer and Fluid Flow. Hemisphere Series on Computational Methods in Mechanics and Thermal Science. Taylor & Francis. ISBN 0891165223
 Shah, Tasneem M.; Sadaf Siddiq, Zafar U. Koreshi. "An analysis and comparison of tube natural frequency modes with fluctuating force frequency from the thermal crossflow fluid in 300 MWe PWR". International Journal of Engineering and Technology 9 (9): 201–205. http://docs.google.com/viewer?a=v&q=cache:4gQM2Vj9itIJ:www.ijens.org/1936091%2520IJET.pdf+tasneem+M.+Shah&hl=en&gl=us&pid=bl&srcid=ADGEESglxYkvf5xfu4Nv6UfFxgO2yzh2O6wa7EzNVurfVgtXdi_bkvvSAijrk7DvwDqWJE9KetrzYKGEM5GCvi5wqmbo5nvDaGt4kvmBrXL6in2xo0WRyHVX3o4roKLkcZgTAk&sig=AHIEtbQ4XElNn3nmOsUXoM0JdkNeY9cMxg.
External links
 CFD Tutorial Many examples and images, with references to robotic fish.
 CFDWiki
 Course: Introduction to CFD – Dmitri Kuzmin (Dortmund University of Technology)
 CFD Success Stories Examples of Successful CFD simulation examples
Categories:  During preprocessing
Wikimedia Foundation. 2010.