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حل مسائل علم الموائع بالكومبيوتر Computational Fluid Dynamic

حل مسائل الموائع بإستخدام برامج الكومبيوتر طبعا بإستخدام

أساسيات الطرق المعروفة في الرياضيات

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Computational fluid dynamics

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Please helpimprove this articleby addingreliable references. Unsourced material may bechallengedandremoved.(June 2008)Computational physicsNumerical analysis·Simulation

Data analysis·Visualization

[show]Fluid dynamicsFinite element·Riemann solver

Smoothed particle hydrodynamics

[show]Monte Carlo methodsIntegration·Gibbs sampling·Metropolis algorithm[show]ParticleN-body·Particle-in-cell

Molecular dynamics

[show]Scientistsvon Neumann·GodunovThis box:view•talk•edit

Computational fluid dynamics (CFD)is one of the branches offluid mechanicsthat usesnumerical methodsandalgorithmsto solve and analyze problems that involve fluid flows. Computers are used to perform the millions of calculations required to simulate the interaction of liquids and gases with surfaces defined by boundary conditions. Even with high-speedsupercomputersonly approximate solutions can be achieved in many cases. Ongoing research, however, may yield software that improves the accuracy and speed of complex simulation scenarios such as transonic orturbulentflows. Initial validation of such software is often performed using awind tunnelwith the final validation coming inflight test.

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- <LI class=toclevel-1>
1 Background and history<LI class=toclevel-1>2 Technicalities<LI class=toclevel-1>3 Methodology

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- <LI class=toclevel-2>
3.1 Discretization methods<LI class=toclevel-2>3.2 Turbulence models

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- <LI class=toclevel-3>
3.2.1 Direct numerical simulation<LI class=toclevel-3>3.2.2 Reynolds-averaged Navier–Stokes<LI class=toclevel-3>3.2.3 Large eddy simulation<LI class=toclevel-3>3.2.4 Detached eddy simulation3.2.5 Vortex method3.3 Two phase flow3.4 Solution algorithms4 See also<LI class=toclevel-1>5 Notes6 External links[edit] Background and history

A computer simulation of high velocity air flow around theSpace Shuttleduring re-entry.

A simulation of theHyper-Xscramjet vehicle in operation atMach-7

The fundamental basis of almost all CFD problems are theNavier-Stokes equations, which define any single-phase fluid flow. These equations can be simplified by removing terms describing viscosity to yield theEuler equations. Further simplification, by removing terms describing vorticity yields thefull potential equations. Finally, these equations can be linearized to yield thelinearized potential equations.

Historically, methods were first developed to solve the Linearized Potential equations. Two-dimensional methods, using conformal transformations of the flow about acylinderto the flow about anairfoilwere developed in the 1930s. The computer power available paced development ofthree-dimensionalmethods. The first paper on a practical three-dimensional method to solve the linearized potential equations was published by John Hess andA.M.O. SmithofDouglas Aircraftin 1966. 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 three-dimensional Panel Codes were developed atBoeing(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 multi-order code and is the most widely used program of this class. This program has been used in the development of manysubmarines, surfaceships,automobiles,helicopters,aircraft, and more recentlywind turbines. Its sister code, USAERO is an unsteady panel method that has also been used for modeling such things as high speed trains and racingyachts. 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, quite a number of Panel Codes have been developed for airfoil analysis and design. These codes typically have aboundary layeranalysis included, so that viscous effects can be modeled. ProfessorRichard Epplerof theUniversity of Stuttgartdeveloped thePROFILcode, partly with NASA funding, which became available in the early 1980s. This was soon followed byMITProfessorMark Drela's Xfoil code. Both PROFIL and Xfoil incorporate two-dimensional panel codes, with coupled boundary layer codes for airfoil analysis work. PROFIL uses aconformal transformationmethod for inverse airfoil design, while Xfoil has both a conformal transformation and an inverse panel method for airfoil design. Both codes are widely used.

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 ofGrumman Aircraftin the early 1980s has seen heavy use.

Developers next turned to Full Potential codes, as panel methods could not calculate the non-linear flow present attransonicspeeds. 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 GarabedianandDavid Kornof the Courant Institute atNew York University(NYU) 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 Progam H was developed by Bob Melnik and his group atGrumman Aerospaceas 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. 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 three-dimensional 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 structuredcartesianmesh code, while most other such codes use structured body-fitted grids (with the exception of NASA's highly successful CART3D code, Lockheed's SPLITFLOW code andGeorgia Tech's NASCART-GT).[1]Antony Jamesonalso developed the three-dimensional AIRPLANE code (1985) 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 (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. 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. Two-dimensional codes, such as NASA Ames' ARC2D code first emerged. A number of three-dimensional codes were developed (OVERFLOW, CFL3D are two successful NASA contributions), leading to numerous commercial packages.

[edit] Technicalities

The most fundamental consideration in CFD is how one treats a continuous fluid in a discretized fashion on a computer. One method is to discretize the spatial domain into small cells to form a volumemeshorgrid, and then apply a suitablealgorithmto solve the equations of motion (Euler equationsfor inviscid, andNavier-Stokes equationsfor viscous flow). In addition, such a mesh can be either irregular (for instance consisting of triangles in 2D, or pyramidal solids in 3D) or regular; the distinguishing characteristic of the former is that each cell must be stored separately in memory. Where shocks or discontinuities are present,high resolution schemessuch asTotal Variation Diminishing(TVD),Flux Corrected Transport(FCT),Essentially NonOscillatory(ENO), orMUSCLschemes are needed to avoid spurious oscillations (Gibbs phenomenon) in the solution.

If one chooses not to proceed with a mesh-based method, a number of alternatives exist, notably :

It is possible to directly solve the

Smoothed particle hydrodynamics(SPH), a Lagrangian method of solving fluid problems,Spectral methods, a technique where the equations are projected onto basis functions like thespherical harmonicsandChebyshev polynomials,Lattice Boltzmann methods(LBM), which simulate an equivalentmesoscopicsystem on a Cartesian grid, instead of solving the macroscopic system (or the real microscopic physics).Navier-Stokes equationsforlaminar flowsand forturbulent flowswhen all of the relevant length scales can be resolved by the grid (aDirect numerical simulation). In general however, the range of length scales appropriate to the problem is larger than even today's massivelyparallel computerscan model. In these cases, turbulent flow simulations require the introduction of a turbulence model.Large eddy simulations(LES) and theReynolds-averaged Navier-Stokes equations(RANS) formulation, with thek-εmodel or the Reynolds stress model, are two techniques for dealing with these scales.

In many instances, other equations are solved simultaneously with theNavier-Stokes equations. These other equations can include those describing speciesconcentration(mass transfer),chemical reactions,heat transfer, etc. More advanced codes allow the simulation of more complex cases involving multi-phase flows (e.g. liquid/gas, solid/gas, liquid/solid),non-Newtonian fluids(such asblood), orchemically reactingflows (such ascombustion).

[edit] Methodology

In all of these approaches the same basic procedure is followed.

- During
preprocessing

- The
geometry(physical bounds) of the problem is defined.- The
volumeoccupied 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
simulationis started and the equations are solved iteratively as a steady-state or transient.- Finally a postprocessor is used for the analysis and visualization of the resulting solution.

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