Computational fluid dynamics

2733:(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 Spalart-Allmaras model (Philippe R. 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 Spalart–Allmaras model based DES acts as LES with a wall model, DES based on other models (like two equation models) behave as a hybrid RANS-LES model. Grid generation is more complicated than for a simple RANS or LES case due to the RANS-LES switch. DES is a non-zonal approach and provides a single smooth velocity field across the RANS and the LES regions of the solutions. 2402:. 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 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 time-stepping schemes arise in the scientific world. 505: 2585:. 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 time-varying mean flow because these equations are 'time-averaged'. In fact, statistically unsteady (or non-stationary) 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. 2564: 36: 653:(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. It has been used in the development of many 2331:

Purely mathematically, the test functions are completely arbitrary - they belong to an infinite-dimensional function space. Clearly an infinite-dimensional function space cannot be represented on a discrete spectral element mesh; 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

2544: 104: 2840:), while the incoherent parts of the flow composed homogeneous background noise, which exhibited no organized structures. Goldstein and Vasilyev applied the FDV model to large eddy simulation, but did not assume that the wavelet filter 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. 2716:(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. 816:), single-species (i.e., it consists of one chemical species), non-reacting, and (unless said otherwise) compressible. Thermal radiation is neglected, and body forces due to gravity are considered (unless said otherwise). 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. 3165: 723: 2706: 3048: 3000:, the PDF becomes the filtered PDF. PDF methods can also be used to describe chemical reactions, 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 2518:

turbulence make most modeling approaches prohibitively expensive; the resolution required to resolve all scales involved in turbulence is beyond what is computationally possible. The primary approach in such cases is to create numerical models to approximate unresolved phenomena. This section lists some commonly used computational models for turbulent flows.

493: 1504:(SW): 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. 3184:. A 3D model is reconstructed from this data and the fluid flow can be computed. Blood properties such as density and viscosity, and realistic boundary conditions (e.g. systemic pressure) have to be taken into consideration. Therefore, making it possible to analyze and optimize the flow in the cardiovascular system for different applications. 2737: 3022:(VC) method is an Eulerian technique used in the simulation of turbulent wakes. It uses a solitary-wave 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 2996:, 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 2517:

In computational modeling of turbulent flows, one common objective is to obtain a model that can predict quantities of interest, such as fluid velocity, for use in engineering designs of the system being modeled. For turbulent flows, the range of length scales and complexity of phenomena involved in

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The 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. Although FEM must be

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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

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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 (neglected in the EE). To close the Fanno

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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

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CFD can be seen as a group of computational methodologies (discussed below) 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,

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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.

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has the advantage of asymptotically optimal performance on many problems. Traditional solvers and preconditioners are effective at reducing high-frequency components of the residual, but low-frequency components typically require many iterations to reduce. By operating on multiple scales, multigrid

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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, low-pass filter. Instead, the filtering operation is based on wavelets, and the filter can be adapted as the flow field evolves.

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The lattice Boltzmann method (LBM) with its simplified kinetic picture on a lattice provides a computationally efficient description of hydrodynamics. Unlike the traditional CFD methods, which solve the conservation equations of macroscopic properties (i.e., mass, momentum, and energy) numerically,

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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

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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

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Turbulence models can be classified based on computational expense, which corresponds to the range of scales that are modeled versus 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

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particles, these computational elements being called vortices, vortons, or vortex particles. Vortex methods were developed as a grid-free methodology that would not be limited by the fundamental smoothing effects associated with grid-based methods. To be practical, however, vortex methods require

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The finite difference method (FDM) has historical importance and is simple to program. It is currently only used in few specialized codes, which handle complex geometry with high accuracy and efficiency by using embedded boundaries or overlapping grids (with the solution interpolated across each

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Spectral element method is a finite element type method. It requires the mathematical problem (the partial differential equation) to be cast in a weak formulation. This is typically done by multiplying the differential equation by an arbitrary test function and integrating over the whole domain.

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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

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The Linear eddy model is a technique used to simulate the convective mixing that takes place in turbulent flow. Specifically, it provides a mathematical way to describe the interactions of a scalar variable within the vector flow field. It is primarily used in one-dimensional representations of

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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

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Software based on the vortex method offer a new means for solving tough fluid dynamics problems with minimal user intervention. All that is required is specification of problem geometry and setting of boundary and initial conditions. Among the significant advantages of this modern technology;

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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 k-ε

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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 (i.e., the velocity profile does not change along the streamwise direction). Note that this widely-used

2171: 762:, 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. 3195:

In a more recent trend, simulations are also performed on GPUs. These typically contain slower but more processors. For CFD algorithms that feature good parallelism performance (i.e. good speed-up by adding more cores) this can greatly reduce simulation times. Fluid-implicit particle and

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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 (where density multiplies the gravitational acceleration). Also assume that various fluid properties such as

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turbulent flow, since it can be applied across a wide range of length scales and Reynolds numbers. This model is generally used as a building block for more complicated flow representations, as it provides high resolution predictions that hold across a large range of flow conditions.

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LBM models the fluid consisting of fictive particles, and such particles perform consecutive propagation and collision processes over a discrete lattice mesh. In this method, one works with the discrete in space and time version of the kinetic evolution equation in the Boltzmann

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CFD investigations are used to clarify the characteristics of aortic flow in details that are beyond the capabilities of experimental measurements. To analyze these conditions, CAD models of the human vascular system are extracted employing modern imaging techniques such as

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In the finite volume method, the governing partial differential equations (typically the Navier-Stokes equations, the mass and energy conservation equations, and the turbulence equations) are recast in a conservative form, and then solved over discrete control volumes. This

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High-resolution schemes are used where shocks or discontinuities are present. Capturing sharp changes in the solution requires the use of second or higher-order numerical schemes that do not introduce spurious oscillations. This usually necessitates the application of

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equations: Start with the EE. Assume zero fluid-particle rotation (zero vorticity) and zero flow expansion (zero divergence). The resulting flowfield is entirely determined by the geometrical boundaries. Ideal flows can be useful in modern CFD to initialize

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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

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The stability of the selected discretisation is generally established numerically rather than analytically as with simple linear problems. Special care must also be taken to ensure that the discretisation handles discontinuous solutions gracefully. The

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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

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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.

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for unsteady problems and algebraic equations for steady problems. Implicit or semi-implicit methods are generally used to integrate the ordinary differential equations, producing a system of (usually) nonlinear algebraic equations. Applying a

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small perturbation theory by Ballhaus and associates. It uses a Murman-Cole switch algorithm for modeling the moving shock-waves. Later it was extended to 3-D with use of a rotated difference scheme by AFWAL/Boeing that resulted in LTRAN3.

2614:). 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 1510:

equations (BL): Start with the C-NS (I-NS) for compressible (incompressible) 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

3074:. These methods often involve a tradeoff between maintaining a sharp interface or conserving mass . This is crucial since the evaluation of the density, viscosity and surface tension is based on the values averaged over the interface. 3142:

which exhibit mesh-dependent convergence rates, but recent advances based on block LU factorization combined with multigrid for the resulting definite systems have led to preconditioners that deliver mesh-independent convergence rates.

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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. This decomposition is done using

1743: 823:(CL): 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 1498:, 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. 2689:

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

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occupied by the fluid is divided into discrete cells (the mesh). The mesh may be uniform or non-uniform, structured or unstructured, consisting of a combination of hexahedral, tetrahedral, prismatic, pyramidal or polyhedral

2913: 2064: 421:, 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 5574:

Kaufmann, T.A.S., Graefe, R., Hormes, M., Schmitz-Rode, T. and Steinseiferand, U., "Computational Fluid Dynamics in Biomedical Engineering", Computational Fluid Dynamics: Theory, Analysis and Applications, pp.

754:(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 Program H was developed by Bob Melnik and his group at 714:

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

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Samant, S.; Bussoletti, J.; Johnson, F.; Burkhart, R.; Everson, B.; Melvin, R.; Young, D.; Erickson, L.; Madson, M. (1987). "TRANAIR - A computer code for transonic analyses of arbitrary configurations".

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Boundary conditions are defined. This involves specifying the fluid behaviour and properties at all bounding surfaces of the fluid domain. For transient problems, the initial conditions are also defined.

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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.

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Borland, C.J., "XTRAN3S - Transonic Steady and Unsteady Aerodynamics for Aeroelastic Applications,"AFWAL-TR-85-3214, Air Force Wright Aeronautical Laboratories, Wright-Patterson AFB, OH, January, 1986

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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 (ARC3D,

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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 at

605:, 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 582:, 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 5585:

Lao, Shandong; Holt, Aaron; Vaidhynathan, Deepthi; Sitaraman, Hariswaran; Hrenya, Christine M.; Hauser, Thomas (2021). "Performance comparison of CFD-DEM solver MFiX-Exa, on GPUs and CPUs".

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Youngren, H.; Bouchard, E.; Coopersmith, R.; Miranda, L. (1983). "Comparison of panel method formulations and its influence on the development of QUADPAN, an advanced low-order method".

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Tidd, D.; Strash, D.; Epstein, B.; Luntz, A.; Nachshon, A.; Rubin, T. (1991). "Application of an efficient 3-D multigrid Euler method (MGAERO) to complete aircraft configurations".

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Surana, K.A.; Allu, S.; Tenpas, P.W.; Reddy, J.N. (February 2007). "k-version of finite element method in gas dynamics: higher-order global differentiability numerical solutions".

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convection term and a non-linear and non-local pressure gradient term. These nonlinear equations must be solved numerically with the appropriate boundary and initial conditions.

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carefully formulated to be conservative, it is much more stable than the finite volume approach. However, FEM can require more memory and has slower solution times than the FVM.

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Incompressible Navier-Stokes equations (I-NS): 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

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Weakly compressible Navier-Stokes equations (WC-NS): Start with the C-NS. Assume that density variations depend only on temperature and not on pressure. For example, for an

1329: 2758:(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 1542:

One-dimensional Euler equations or one-dimensional gas-dynamic equations (1D-EE): Start with the EE. Assume that all flow quantities depend only on one spatial dimension.

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Carmichael, R.; Erickson, L. (1981). "PAN AIR - A higher order panel method for predicting subsonic or supersonic linear potential flows about arbitrary configurations".

2638: 2678: 4813: 4736: 4679: 4585: 4553: 4521: 4431: 4193: 1496: 1406: 1381: 2786: 5754: 2041: 2014: 1947: 1471: 1433: 1356: 1006: 986: 621:. Fromm's vorticity-stream-function method for 2D, transient, incompressible flow was the first treatment of strongly contorting incompressible flows in the world. 369: 770:

mesh 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 and

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Jameson, A.; Schmidt, Wolfgang; Turkel, ELI (1981). "Numerical solution of the Euler equations by finite volume methods using Runge Kutta time stepping schemes".

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McMurtry, Patrick A.; Gansauge, Todd C.; Kerstein, Alan R.; Krueger, Steven K. (April 1993). "Linear eddy simulations of mixing in a homogeneous turbulent flow".

2574: 2552: 2538: 1623: 1077: 2658: 2314: 2294: 2274: 2254: 2234: 2214: 2194: 1987: 1967: 1814: 1786: 1766: 1285: 1249: 1225: 1155: 1098: 1050: 1026: 4207:

Marshall, David; Ruffin, Stephen (2004). "An Embedded Boundary Cartesian Grid Scheme for Viscous Flows Using a New Viscous Wall Boundary Condition Treatment".

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Rubbert, P.; Saaris, G. (1972). "Review and evaluation of a three-dimensional lifting potential flow computational method for arbitrary configurations".

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Harlow, Francis H.; Welch, J. Eddie (December 1965). "Numerical Calculation of Time-Dependent Viscous Incompressible Flow of Fluid with Free Surface".

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fully-developed assumption can be inadequate in some instances, such as some compressible, microchannel flows, in which case it can be supplanted by a

850:). The resulting system of equations is unclosed since to solve it one needs further relationships/equations: (a) constitutive relationships for the 2166:{\displaystyle {\frac {\partial Q}{\partial t}}+{\frac {\partial F}{\partial x}}+{\frac {\partial G}{\partial y}}+{\frac {\partial H}{\partial z}}=0} 5717: 3266: 2858: 2525:

In addition to the wide range of length and time scales and the associated computational cost, the governing equations of fluid dynamics contain a

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code. Both PROFILE and XFOIL incorporate two-dimensional panel codes, with coupled boundary layer codes for airfoil analysis work. PROFILE uses a

2418: 2577:(RANS) equations are the oldest approach to turbulence modeling. An ensemble version of the governing equations is solved, which introduces new 6003: 3246: 1670:

guarantees the conservation of fluxes through a particular control volume. The finite volume equation yields governing equations in the form,

5998: 5690: 5671: 5242: 5064: 5039: 4893: 4226: 3998: 3356: 3306: 3286: 3216: 5198:

Colucci, P.J.; Jaberi, F.A; Givi, P.; Pope, S.B. (1998). "Filtered density function for large eddy simulation of turbulent reacting flows".

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Gentry, Richard A; Martin, Robert E; Daly, Bart J (August 1966). "An Eulerian differencing method for unsteady compressible flow problems".

524:, which define many single-phase (gas or liquid, but not both) fluid flows. These equations can be simplified by removing terms describing 405:. Computers are used to perform the calculations required to simulate the free-stream flow of the fluid, and the interaction of the fluid ( 5977: 3291: 3276: 3134:

perform poorly or fail entirely, so the problem structure must be used for effective preconditioning. Methods commonly used in CFD are the

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IDDES Simulation of the Karel Motorsports BMW. This is a type of DES simulation completed in OpenFOAM. The plot is coefficient of pressure.

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all of the turbulent scales are not modeled, the computational cost is very low, but the tradeoff comes in the form of decreased accuracy.

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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

57: 3105: 677:. Its sister code, USAERO is an unsteady panel method that has also been used for modeling such things as high speed trains and racing 362: 2446:(in which the motion of N objects is tied to their mutual influences). This breakthrough came in the 1980s with the development of the 2442:

means for rapidly computing velocities from the vortex elements – in other words they require the solution to a particular form of the

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conserved: These quantities are conserved and cannot "teleport" from one place to another but can only move by a continuous flow (see

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In the two-dimensional realm, a number of Panel Codes have been developed for airfoil analysis and design. The codes typically have a

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analysis of a particular problem can be used for comparison. A final validation is often performed using full-scale testing, such as

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Program VSAERO Theory Document: A Computer Program for Calculating Nonlinear Aerodynamic Characteristics of Arbitrary Configurations

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methods. Probably the first work using computers to model fluid flow, as governed by the Navier–Stokes equations, was performed at

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method for inverse airfoil design, while XFOIL has both a conformal transformation and an inverse panel method for airfoil design.

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and compressible Favre-averaged Navier-Stokes equations (C-RANS and C-FANS): Start with the C-NS. Assume that any flow variable

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and Schneider tested the CVS method with two flow configurations and showed that the coherent portion of the flow exhibited the

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Hunt, J.C.R. (January 1998). "Lewis Fry Richardson and his contributions to mathematics, meteorology, and models of conflict".

3301: 3083: 828: 771: 747: 681:. The NASA PMARC code from an early version of VSAERO and a derivative of PMARC, named CMARC, is also commercially available. 225: 5367:

Unverdi, Salih Ozen; Tryggvason, Grétar (May 1992). "A front-tracking method for viscous, incompressible, multi-fluid flows".

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CFD made a major break through in late 70s with the introduction of LTRAN2, a 2-D code to model oscillating airfoils based on

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developed the PROFILE code, partly with NASA funding, which became available in the early 1980s. This was soon followed by

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Historically, methods were first developed to solve the linearized potential equations. Two-dimensional (2D) methods, using

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Melton, John; Berger, Marsha; Aftosmis, Michael; Wong, Michael (1995). "3D applications of a Cartesian grid Euler method".

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Wu, Kui; Truong, Nghia; Yuksel, Cem; Hoetzlein, Rama (May 2018). "Fast Fluid Simulations with Sparse Volumes on the GPU".

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Bailly, C., and Daniel J. (2000). "Numerical solution of acoustic propagation problems using Linearized Euler Equations".

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Pinella, David and Garrison, Peter, "Digital Wind Tunnel CMARC; Three-Dimensional Low-Order Panel Codes," Aerologic, 2009.

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Maskew, Brian (February 1982). "Prediction of Subsonic Aerodynamic Characteristics: A Case for Low-Order Panel Methods".

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the following summarizes the physical assumptions/simplifications taken in equations of a flow that is single-phase (see

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Jameson, Antony (May 1974). "Iterative solution of transonic flows over airfoils and wings, including flows at mach 1".

5467:"A taxonomy and comparison of parallel block multi-level preconditioners for the incompressible Navier–Stokes equations" 2848: 457: 133: 4170:

Karman, l, Jr, Steve (1995). "SPLITFLOW - A 3D unstructured Cartesian/Prismatic grid CFD code for complex geometries".

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Katz, Joseph; Maskew, Brian (April 1988). "Unsteady low-speed aerodynamic model for complete aircraft configurations".

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is very low. As a result, the resulting set of equations is linear, something which simplifies greatly their solution.

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CFD is applied to a wide range of research and engineering problems in many fields of study and industries, including

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Giles, M.; Drela, M.; Thompkins, Jr., W. (1985). "Newton solution of direct and inverse transonic Euler equations".

5850: 5799: 5083:; Schneider, Kai (2001). "Coherent Vortex Simulation (CVS), A Semi-Deterministic Turbulence Model Using Wavelets". 4097:

Raj, Pradeep; Brennan, James E. (1989). "Improvements to an Euler aerodynamic method for transonic flow analysis".

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Fromm, Jacob E.; Harlow, Francis H. (July 1963). "Numerical Solution of the Problem of Vortex Street Development".

3097: 3023: 1583:(CAD). From there, data can be suitably processed (cleaned-up) and the fluid volume (or fluid domain) is extracted. 155: 4242:

Jameson, A.; Baker, T.; Weatherill, N. (1986). "Calculation of Inviscid Transonic Flow over a Complete Aircraft".

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speeds. The first description of a means of using the Full Potential equations was published by Earll Murman and

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Bernoulli equation: Start with the EE. Assume that density variations depend only on pressure variations. See

586:, set the basis for modern CFD and numerical meteorology. In fact, early CFD calculations during the 1940s using 461: 210: 4310:

Drela, M. and Youngren H., "A User's Guide to MISES 2.53", MIT Computational Sciences Laboratory, December 1998.

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flow equation, a model for this friction term is needed. Such a closure involves problem-dependent assumptions.

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reduces all components of the residual by similar factors, leading to a mesh-independent number of iterations.

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Developers turned to Full Potential codes, as panel methods could not calculate the non-linear flow present at

708: 642: 614: 564: 190: 1515: 3757:

Hess, J.; Friedman, D. (1983). "Analysis of complex inlet configurations using a higher-order panel method".

5972: 5931: 5926: 5898: 5870: 5814: 5789: 5332:

Hirt, C.W; Nichols, B.D (January 1981). "Volume of fluid (VOF) method for the dynamics of free boundaries".

3236: 3127: 3030:, where conservation laws are satisfied, so that the essential integral quantities are accurately computed. 3027: 2993: 2481: 2325: 1501: 878: 598: 594: 537: 250: 205: 160: 123: 113: 2683: 2454:(FMM) algorithms. These paved the way to practical computation of the velocities from the vortex elements. 1252:

is lost, and the resulting system of equations requires the closure of various unclosed terms, notably the

5921: 5794: 5641: 5429: 5135: 3063: 3052: 2495: 1443: 1228: 1070: 832: 618: 465: 414: 4624: 3112:, operate by minimizing the residual over successive subspaces generated by the preconditioned operator. 1556:

term to recover the effect of wall heat transfer or the effect of the heat sources (neglected in the EE).

5947: 5842: 5779: 5771: 5763: 5412:

Benzi, Michele; Golub, Gene H.; Liesen, Jörg (May 2005). "Numerical solution of saddle point problems".

4807: 4730: 4673: 4014: 3895: 3873: 3793: 3780: 3241: 3092: 3019: 3013: 2997: 2713: 2700: 2451: 2334: 1847: 1580: 851: 200: 95: 5118:

Goldstein, Daniel; Vasilyev, Oleg (1995). "Stochastic coherent adaptive large eddy simulation method".

2940: 2918: 2809: 429:

flows. Initial validation of such software is typically performed using experimental apparatus such as

2543: 914: 5822: 5685:. Hemisphere Series on Computational Methods in Mechanics and Thermal Science. Taylor & Francis. 5478: 5421: 5376: 5341: 5306: 5267: 5207: 5172: 5127: 5001: 4918: 4702: 4653: 4343: 4027:

Jameson, A.; Caughey, D. (1977). "A finite volume method for transonic potential flow calculations".

3956: 3662: 3627: 3592: 3557: 3491: 3431: 3381: 3221: 3211: 1655: 1649: 1062: 824: 778:

also developed the three-dimensional AIRPLANE code which made use of unstructured tetrahedral grids.

638: 579: 434: 326: 195: 5434: 5140: 4772:"Compressible Fanno flows in micro-channels: An enhanced quasi-2D numerical model for laminar flows" 3071: 889:). The C-NS need to be augmented with an EOS and a caloric EOS to have a closed system of equations. 689: 5827: 3311: 3181: 2592: 2548: 2486:

In the boundary element method, the boundary occupied by the fluid is divided into a surface mesh.

1607: 847: 843: 786: 751: 610: 568: 541: 341: 230: 138: 4693:

Harley, J. C. and Huang, Y. and Bau, H. H. and Zemel, J. N. (1995). "Gas flow in micro-channels".

1819: 1267:

Linearized compressible Euler equations (LEE): Start with the EE. Assume that any flow variable

838:

Continuum conservation laws (CCL): Start with the CL. Assume that mass, momentum and energy are

5623: 5586: 5494: 5447: 5100: 4934: 4718: 4579: 4547: 4515: 4425: 4333: 4187: 3947:

Murman, Earll M.; Cole, Julian D. (January 1971). "Calculation of plane steady transonic flows".

3296: 2852: 2617: 2611: 1654:

The finite volume method (FVM) is a common approach used in CFD codes, as it has an advantage in

1570: 1435:

satisfies "its own" equation, such as the EE. The LEE and its many variations are widely used in

646: 438: 394: 288: 243: 5729:, Open access Lectures and Codes for Numerical PDEs, including a modern view of Compressible CFD 3088: 1290: 2663: 2462:

It is practically grid-free, thus eliminating numerous iterations associated with RANS and LES.

5832: 5713: 5686: 5667: 5465:

Elman, Howard; Howle, V.E.; Shadid, John; Shuttleworth, Robert; Tuminaro, Ray (January 2008).

5238: 5060: 5035: 4972: 4889: 4835: 4398: 4359: 4288:

Drela, Mark (1990). "Newton solution of coupled viscous/Inviscid multielement airfoil flows".

4222: 3994: 3530: 3352: 3001: 2837: 2582: 1383:

is a perturbation or fluctuation from this state. Furthermore, assume that this perturbation

859: 846:). Another interpretation is that one starts with the CL and assumes a continuum medium (see 820: 755: 602: 331: 306: 278: 103: 1613:

Finally a postprocessor is used for the analysis and visualization of the resulting solution.

1103: 5615: 5548: 5521: 5486: 5439: 5392: 5384: 5349: 5314: 5275: 5215: 5180: 5145: 5092: 5009: 4926: 4855: 4793: 4783: 4750: 4710: 4661: 4390: 4379:"Unchannelized granular flows: Effect of initial granular column geometry on fluid dynamics" 4351: 4293: 4270: 4247: 4214: 4175: 4152: 4129: 4106: 4079: 4056: 4032: 3986: 3964: 3926: 3855: 3828: 3762: 3739: 3716: 3693: 3670: 3635: 3600: 3565: 3522: 3499: 3439: 3389: 3271: 3135: 3131: 3115: 3067: 2761: 2526: 2430: 716: 629: 606: 316: 283: 165: 17: 5466: 2468:

Time-series simulations, which are crucial for correct analysis of acoustics, are possible.

2019: 1992: 1925: 1449: 1411: 1334: 991: 964: 789:, CFL3D are three successful NASA contributions), leading to numerous commercial packages. 4445: 3908: 3653:

Hess, J.L.; Smith, A.M.O. (1967). "Calculation of potential flow about arbitrary bodies".

3251: 3139: 3101: 1659: 1528: 1253: 882: 871: 809: 743: 727: 390: 170: 3792:

Ashby, Dale L.; Dudley, Michael R.; Iguchi, Steve K.; Browne, Lindsey and Katz, Joseph, "

1185: 1160: 5482: 5425: 5380: 5345: 5310: 5271: 5211: 5176: 5131: 5005: 4922: 4706: 4657: 4347: 3960: 3666: 3631: 3596: 3561: 3495: 3435: 3385: 1476: 1386: 1361: 4827: 3109: 3059: 2643: 2607: 2603: 2568: 2443: 2299: 2279: 2259: 2239: 2219: 2199: 2179: 1972: 1952: 1799: 1771: 1751: 1667: 1507: 1270: 1260: 1234: 1210: 1140: 1083: 1035: 1011: 866:

law; and, (d) a caloric equation of state relating temperature with quantities such as

813: 775: 759: 685: 481: 402: 398: 321: 311: 255: 215: 2836:

energy spectrum exhibited by the total flow, and corresponded to coherent structures (

5992: 5726: 5451: 5388: 5353: 5279: 5013: 4992:

Launder, B.E.; D.B. Spalding (1974). "The Numerical Computation of Turbulent Flows".

4938: 4722: 3674: 3569: 3457: 2599: 2502: 2465:

All problems are treated identically. No modeling or calibration inputs are required.

1738:{\displaystyle {\frac {\partial }{\partial t}}\iiint Q\,dV+\iint F\,d\mathbf {A} =0,} 1552: 1066: 988:

is a conveniently-defined reference pressure that is always and everywhere constant,

694: 557: 497: 469: 418: 5627: 5498: 5104: 904:(EE): Start with the C-NS. Assume a frictionless flow with no diffusive heat flux. 3526: 1610:

is started and the equations are solved iteratively as a steady-state or transient.

1029: 674: 625: 449: 4883: 4208: 3807:

A Computer Program for Three-Dimensional Lifting Bodies in Subsonic Inviscid Flow

3443: 3196:

lattice-Boltzmann methods are typical examples of codes that scale well on GPUs.

1069:

are always and everywhere constant. The Boussinesq equations are widely used in

5080: 3326: 3164: 3062:

is still under development. Different methods have been proposed, including the

2803: 1594:

The physical modeling is defined – for example, the equations of fluid motion +

1524: 893: 739: 513: 453: 442: 430: 336: 4788: 4771: 2788:. DNS is intractable for flows with complex geometries or flow configurations. 2705: 722: 5490: 5443: 5096: 4714: 4394: 4321: 3990: 3534: 3503: 2908:{\displaystyle f_{V}({\boldsymbol {v}};{\boldsymbol {x}},t)d{\boldsymbol {v}}} 1545: 700: 666: 662: 553: 545: 477: 426: 4402: 4363: 1408:

is very small in comparison with some reference value. Finally, assume that

4378: 3983:

A Theory of Supercritical Wing Sections, with Computer Programs and Examples

3152: 2434: 1058: 908: 886: 881:(C-NS): Start with the CCL. Assume a Newtonian viscous stress tensor (see 863: 855: 735: 654: 624:

The first paper with three-dimensional model was published by John Hess and

578:

One of the earliest type of calculations resembling modern CFD are those by

549: 533: 422: 5552: 3985:. Lecture Notes in Economics and Mathematical Systems. Vol. 66. 1972. 3921:

Boppe, C. (1977). "Calculation of transonic wing flows by grid embedding".

3047: 2471:

The small scale and large scale are accurately simulated at the same time.

5709: 5397: 4798: 4338: 3809:," USAAMRDL Technical Report, TR 74-18, Ft. Eustis, Virginia, April 1974. 1595: 1576: 867: 670: 5732: 5525: 4355: 4297: 4274: 4133: 4083: 3766: 3743: 3720: 492: 5163:

Lundgren, T.S. (1969). "Model equation for nonhomogeneous turbulence".

4251: 4218: 4179: 4156: 4036: 3930: 3697: 2798: 2433:

technique for the simulation of incompressible turbulent flows. In it,

1358:

is the value of the flow variable at some reference or base state, and

572: 525: 509: 5619: 5184: 5149: 4060: 4015:

GRUMFOIL: A computer code for the viscous transonic flow over airfoils

3639: 3604: 5318: 5219: 4930: 3909:

XFOIL: An Analysis and Design System for Low Reynolds Number Airfoils

3393: 1587: 1231:) one obtains the Reynolds-averaged Navier–Stokes equations. And if 634: 473: 406: 5294: 4110: 3896:

A Computer Program for the Design and Analysis of Low-Speed Airfoils

3859: 3832: 3521:. Los Alamos Scientific Laboratory of the University of California. 3517:

Harlow, Francis Harvey; Evans, Martha; Richtmyer, Robert D. (1955).

3479: 2736: 1662:

turbulent flows, and source term dominated flows (like combustion).

5591: 5295:"Linear Eddy Simulations Of Mixing In A Homogeneous Turbulent Flow" 4770:

Cavazzuti, M. and Corticelli, M. A. and Karayiannis, T. G. (2019).

3968: 2640:

is a "Two Equation" model because two transport equations (one for

4665: 3794:

Potential Flow Theory and Operation Guide for the Panel Code PMARC

3317:

Unified methods for computing incompressible and compressible flow

3169: 3163: 3046: 2735: 2709:

Volume rendering of a non-premixed swirl flame as simulated by LES

2704: 2562: 704: 678: 587: 503: 491: 3911:," in Springer-Verlag Lecture Notes in Engineering, No. 54, 1989. 1565:

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

2429:

The vortex method, also Lagrangian Vortex Particle Method, is a

1287:, such as density, velocity and pressure, can be represented as 1100:, such as density, velocity and pressure, can be represented as 658: 650: 5736: 5258:

Pope, S.B. (1985). "PDF methods for turbulent reactive flows".

3177: 1658:

usage and solution speed, especially for large problems, high

410: 29: 4320:

Jop, Pierre; Forterre, Yoël; Pouliquen, Olivier (June 2006).

3781:

Development of Panel Methods for Subsonic Analysis and Design

3480:"Fluid dynamics in Group T-3 Los Alamos National Laboratory" 1989:

is the conservation equation expressed on an element basis,

2855:, are based on tracking the one-point PDF of the velocity, 688:

analysis included, so that viscous effects can be modeled.

496:

A computer simulation of high velocity air flow around the

5664:

Computational Fluid Dynamics: The Basics With Applications

4911:

International Journal for Numerical Methods in Engineering

4885:

Computational Fluid Dynamics: The Basics with Applications

4290:

21st Fluid Dynamics, Plasma Dynamics and Lasers Conference

4856:"Detailed Explanation of the Finite Element Method (FEM)" 27:

Analysis and solving of problems that involve fluid flows

1856:

In this method, a weighted residual equation is formed:

1579:

and physical bounds of the problem can be defined using

532:. Further simplification, by removing terms describing 520:

The fundamental basis of almost all CFD problems is the

4967:

Cottet, Georges-Henri; Koumoutsakos, Petros D. (2000).

4952:

Huebner, K.H.; Thornton, E.A.; and Byron, T.D. (1995).

2915:, which gives the probability of the velocity at point 803:

Computational Fluid Dynamics for Phase Change Materials

590:

used methods close to those in Richardson's 1922 book.

3519:

A Machine Calculation Method for Hydrodynamic Problems

3192:

Traditionally, CFD simulations are performed on CPUs.

2588:

RANS models can be divided into two broad approaches:

5727:

Numerical PDE Techniques for Scientists and Engineers

4994:

Computer Methods in Applied Mechanics and Engineering

2965: 2943: 2921: 2861: 2812: 2764: 2666: 2646: 2620: 2337: 2302: 2282: 2262: 2242: 2222: 2202: 2182: 2067: 2022: 1995: 1975: 1955: 1928: 1865: 1822: 1802: 1774: 1754: 1679: 1479: 1452: 1414: 1389: 1364: 1337: 1293: 1273: 1237: 1213: 1188: 1163: 1143: 1106: 1086: 1038: 1014: 994: 967: 917: 2985:{\displaystyle {\boldsymbol {v}}+d{\boldsymbol {v}}} 1182:

is a perturbation or fluctuation from this average.

5940: 5912: 5889: 5841: 5813: 5770: 3942: 3940: 1838:is the surface area of the control volume element. 1640:Some of the discretization methods being used are: 854:; (b) constitutive relationships for the diffusive 2984: 2951: 2929: 2907: 2851:(PDF) methods for turbulence, first introduced by 2828: 2780: 2672: 2652: 2632: 2394: 2308: 2288: 2268: 2248: 2228: 2208: 2188: 2165: 2035: 2008: 1981: 1961: 1941: 1911: 1830: 1808: 1780: 1760: 1737: 1490: 1465: 1446:: Start with the LEE. Neglect all gradients of 1427: 1400: 1375: 1350: 1323: 1279: 1243: 1219: 1199: 1174: 1157:is the ensemble-average of any flow variable, and 1149: 1129: 1092: 1044: 1020: 1000: 980: 953: 593:The computer power available paced development of 5666:. Science/Engineering/Math. McGraw-Hill Science. 5235:Computational models for turbulent reacting flows 3082:Discretization in the space produces a system of 1816:is the volume of the control volume element, and 4210:42nd AIAA Aerospace Sciences Meeting and Exhibit 3805:Woodward, F.A., Dvorak, F.A. and Geller, E.W., " 3122:For indefinite systems, preconditioners such as 5541:Communications on Pure and Applied Mathematics 1949:is the equation residual at an element vertex 5748: 5512:Adamson, M.R. (January 2006). "Biographies". 4322:"A constitutive law for dense granular flows" 560:to yield the linearized potential equations. 363: 8: 4812:: CS1 maint: multiple names: authors list ( 4735:: CS1 maint: multiple names: authors list ( 4678:: CS1 maint: multiple names: authors list ( 4584:: CS1 maint: multiple names: authors list ( 4552:: CS1 maint: multiple names: authors list ( 4520:: CS1 maint: multiple names: authors list ( 4430:: CS1 maint: multiple names: authors list ( 4377:Biroun, Mehdi H.; Mazzei, Luca (June 2024). 4192:: CS1 maint: multiple names: authors list ( 4172:33rd Aerospace Sciences Meeting and Exhibit 4149:33rd Aerospace Sciences Meeting and Exhibit 4029:3rd Computational Fluid Dynamics Conference 1912:{\displaystyle R_{i}=\iiint W_{i}Q\,dV^{e}} 401:to analyze and solve problems that involve 5755: 5741: 5733: 4565: 4563: 4506:Landau, L. D. and Lifshitz, E. M. (2007). 3207:Application of CFD in thermal power plants 2196:is the vector of conserved variables, and 370: 356: 102: 91: 5590: 5433: 5396: 5260:Progress in Energy and Combustion Science 5139: 5025: 5023: 4797: 4787: 4501: 4499: 4497: 4495: 4479: 4477: 4475: 4473: 4471: 4469: 4467: 4465: 4463: 4461: 4337: 4076:14th Fluid and Plasma Dynamics Conference 3713:14th Fluid and Plasma Dynamics Conference 2977: 2966: 2964: 2944: 2942: 2922: 2920: 2900: 2883: 2875: 2866: 2860: 2816: 2811: 2772: 2763: 2665: 2645: 2619: 2567:A simulation of aerodynamic package of a 2539:Reynolds-averaged Navier–Stokes equations 2336: 2301: 2281: 2261: 2241: 2221: 2201: 2181: 2137: 2114: 2091: 2068: 2066: 2027: 2021: 2000: 1994: 1974: 1954: 1933: 1927: 1903: 1895: 1886: 1870: 1864: 1823: 1821: 1801: 1773: 1753: 1721: 1717: 1701: 1680: 1678: 1624:Discretization of Navier–Stokes equations 1478: 1457: 1451: 1419: 1413: 1388: 1363: 1342: 1336: 1304: 1292: 1272: 1236: 1212: 1187: 1162: 1142: 1105: 1085: 1078:Reynolds-averaged Navier–Stokes equations 1037: 1013: 993: 972: 966: 934: 928: 916: 601:, in the T3 group. This group was led by 464:, industrial system design and analysis, 80:Learn how and when to remove this message 5718:Indian Institute of Technology Kharagpur 4971:. Cambridge, UK: Cambridge Univ. Press. 4776:Thermal Science and Engineering Progress 4625:"Favre averaged Navier-Stokes equations" 4601: 4599: 4597: 4595: 4418:Computational methods for fluid dynamics 3267:List of finite element software packages 2542: 1637:both admit shocks and contact surfaces. 721: 43:This article includes a list of general 5642:"Intersect 360 HPC application Support" 5514:IEEE Annals of the History of Computing 4954:The Finite Element Method for Engineers 4538:Fox, R. W. and McDonald, A. T. (1992). 3409:Weather prediction by numerical process 3407:Richardson, L. F.; Chapman, S. (1965). 3339: 2978: 2967: 2945: 2923: 2901: 2884: 2876: 2610:), and Zero Equation Model (Cebeci and 793:with traditional experimental methods. 766:MGAERO is unique in being a structured 719:in the early 1980s has seen heavy use. 584:Weather Prediction by Numerical Process 94: 5683:Numerical Heat Transfer and Fluid Flow 4832:Numerical Heat Transfer and Fluid FLow 4805: 4728: 4671: 4619: 4617: 4577: 4545: 4533: 4531: 4513: 4423: 4416:Ferziger, J. H. and Peric, M. (2002). 4185: 3347:Milne-Thomson, Louis Melville (1973). 3247:Finite volume method for unsteady flow 2490:High-resolution discretization schemes 1768:is the vector of conserved variables, 4956:(Third ed.). Wiley Interscience. 4877: 4875: 4834:. Hemisphere Publishing Corporation. 4570:Poinsot, T. and Veynante, D. (2005). 3307:Stochastic Eulerian Lagrangian method 3287:Multidisciplinary design optimization 3217:Boundary conditions in fluid dynamics 2992:. This approach is analogous to the 433:. In addition, previously performed 7: 4572:Theoretical and numerical combustion 4267:7th Computational Physics Conference 4053:25th AIAA Aerospace Sciences Meeting 3292:Numerical methods in fluid mechanics 3277:Moving particle semi-implicit method 3168:Simulation of blood flow in a human 5034:(3 ed.). DCW Industries, Inc. 4969:Vortex Methods: Theory and Practice 4126:9th Applied Aerodynamics Conference 3374:Physics of Fluids A: Fluid Dynamics 1227:is a classic ensemble-average (see 3232:Computational magnetohydrodynamics 3004:for subfilter particle evolution. 2395:{\displaystyle v(x,y)=ax+by+cxy+d} 2148: 2140: 2125: 2117: 2102: 2094: 2079: 2071: 1686: 1682: 1598:+ radiation + species conservation 742:of Boeing in 1970. Frances Bauer, 49:it lacks sufficient corresponding 25: 4751:"One-dimensional Euler equations" 3478:Harlow, Francis H. (April 2004). 3282:Multi-particle collision dynamics 3051:Simulation of bubble horde using 2952:{\displaystyle {\boldsymbol {v}}} 2930:{\displaystyle {\boldsymbol {x}}} 2829:{\displaystyle -{\frac {40}{39}}} 797:Hierarchy of fluid flow equations 512:scramjet vehicle in operation at 5471:Journal of Computational Physics 5369:Journal of Computational Physics 5334:Journal of Computational Physics 3550:Journal of Computational Physics 3484:Journal of Computational Physics 3424:Annual Review of Fluid Mechanics 3104:methods. Krylov methods such as 1824: 1722: 954:{\displaystyle \rho =p_{0}/(RT)} 34: 5085:Flow, Turbulence and Combustion 4540:Introduction to Fluid Mechanics 4244:24th Aerospace Sciences Meeting 3923:15th Aerospace Sciences Meeting 3759:Applied Aerodynamics Conference 3736:Applied Aerodynamics Conference 3690:10th Aerospace Sciences Meeting 3302:Smoothed-particle hydrodynamics 3084:ordinary differential equations 2575:Reynolds-averaged Navier–Stokes 2533:Reynolds-averaged Navier–Stokes 2505:to ensure that the solution is 885:) and a Fourier heat flux (see 829:conservation of linear momentum 226:Smoothed particle hydrodynamics 5237:. Cambridge University Press. 5059:. Cambridge University Press. 3655:Progress in Aerospace Sciences 2894: 2872: 2353: 2341: 2043:is the volume of the element. 1207:is not necessarily small. If 948: 939: 1: 6004:Computational fields of study 4882:Anderson, John David (1995). 4013:Mead, H. R.; Melnik, R. E., " 2547:External aerodynamics of the 1788:is the vector of fluxes (see 575:were developed in the 1930s. 221:Dissipative particle dynamics 5999:Computational fluid dynamics 5710:Computational Fluid Dynamics 5389:10.1016/0021-9991(92)90307-K 5354:10.1016/0021-9991(81)90145-5 5280:10.1016/0360-1285(85)90002-4 5014:10.1016/0045-7825(74)90029-2 4383:Chemical Engineering Science 3894:Eppler, R.; Somers, D. M., " 3796:", NASA NASA-TM-102851 1991. 3675:10.1016/0376-0421(67)90003-6 3570:10.1016/0021-9991(66)90014-3 3444:10.1146/annurev.fluid.30.1.0 3008:Vorticity confinement method 2849:Probability density function 1831:{\displaystyle \mathbf {A} } 750:of the Courant Institute at 383:Computational fluid dynamics 18:Computational Fluid Dynamics 5293:Krueger, Steven K. (1993). 5032:Turbulence Modeling for CFD 3227:Central differencing scheme 3124:incomplete LU factorization 2756:Direct numerical simulation 2751:Direct numerical simulation 2745:Direct numerical simulation 2633:{\displaystyle k-\epsilon } 2507:total variation diminishing 2419:Bhatnagar-Gross-Krook (BGK) 1539:fully-developed assumption. 1437:computational aeroacoustics 413:) with surfaces defined by 6020: 5662:Anderson, John D. (1995). 4789:10.1016/j.tsep.2019.01.003 4695:Journal of Fluid Mechanics 3024:finite difference equation 3011: 2792:Coherent vortex simulation 2748: 2723: 2698: 2536: 2493: 2479: 2409: 2323: 2050: 2016:is the weight factor, and 1845: 1647: 1621: 1324:{\displaystyle f=f_{0}+f'} 800: 156:Morse/Long-range potential 5491:10.1016/j.jcp.2007.09.026 5444:10.1017/S0962492904000212 5030:Wilcox, David C. (2006). 4715:10.1017/S0022112095000358 4446:"Navier-Stokes equations" 4395:10.1016/j.ces.2024.119997 3991:10.1007/978-3-642-80678-0 3504:10.1016/j.jcp.2003.09.031 3458:"The Legacy of Group T-3" 3262:Lattice Boltzmann methods 3098:successive overrelaxation 2731:Detached eddy simulations 2673:{\displaystyle \epsilon } 2412:Lattice Boltzmann methods 2316:directions respectively. 615:vorticity stream function 565:conformal transformations 556:) these equations can be 462:environmental engineering 5681:Patankar, Suhas (1980). 3349:Theoretical Aerodynamics 3322:Visualization (graphics) 3257:Immersed boundary method 2726:Detached eddy simulation 2720:Detached eddy simulation 2606:), Mixing Length Model ( 2406:Lattice Boltzmann method 2053:Finite difference method 2047:Finite difference method 709:conformal transformation 538:full potential equations 452:and aerospace analysis, 5608:Computer Graphics Forum 5097:10.1023/A:1013512726409 3898:," NASA TM-80210, 1980. 3527:2027/mdp.39015095283399 3351:. Courier Corporation. 3237:Discrete element method 3028:shock capturing methods 2994:kinetic theory of gases 2482:Boundary element method 2476:Boundary element method 2326:Spectral element method 2320:Spectral element method 1794:Navier–Stokes equations 1635:Navier–Stokes equations 1502:Shallow water equations 1130:{\displaystyle f=F+f''} 879:Navier-Stokes equations 599:Los Alamos National Lab 522:Navier–Stokes equations 161:Lennard-Jones potential 64:more precise citations. 5978:Transportation science 5553:10.1002/cpa.3160270302 4542:. John Wiley and Sons. 4488:. John Wiley and Sons. 4484:Panton, R. L. (1996). 4017:," NASA CR-3806, 1985. 3876:", NASA CR-4023, 1987. 3783:," NASA CR-3234, 1980. 3172: 3160:Biomedical engineering 3108:, typically used with 3064:Volume of fluid method 3055: 3053:volume of fluid method 2986: 2953: 2931: 2909: 2830: 2782: 2781:{\displaystyle Re^{3}} 2741: 2710: 2674: 2654: 2634: 2571: 2569:Porsche Cayman (987.2) 2560: 2551:model, computed using 2496:High-resolution scheme 2396: 2310: 2290: 2270: 2256:are the fluxes in the 2250: 2230: 2210: 2190: 2167: 2037: 2010: 1983: 1963: 1943: 1913: 1832: 1810: 1782: 1762: 1739: 1618:Discretization methods 1492: 1467: 1444:acoustic wave equation 1429: 1402: 1377: 1352: 1325: 1281: 1245: 1229:Reynolds decomposition 1221: 1201: 1176: 1151: 1131: 1094: 1071:microscale meteorology 1046: 1022: 1002: 982: 955: 833:conservation of energy 731: 619:marker-and-cell method 517: 501: 488:Background and history 466:biological engineering 460:, natural science and 5764:Computational science 3620:The Physics of Fluids 3585:The Physics of Fluids 3411:. Dover Publications. 3242:Finite element method 3167: 3147:Unsteady aerodynamics 3050: 3020:vorticity confinement 3014:Vorticity confinement 2998:large eddy simulation 2987: 2954: 2932: 2910: 2831: 2783: 2739: 2714:Large eddy simulation 2708: 2701:Large eddy simulation 2695:Large eddy simulation 2684:Reynolds stress model 2675: 2655: 2635: 2593:Boussinesq hypothesis 2566: 2546: 2452:fast multipole method 2397: 2311: 2291: 2271: 2251: 2231: 2211: 2191: 2168: 2038: 2036:{\displaystyle V^{e}} 2011: 2009:{\displaystyle W_{i}} 1984: 1964: 1944: 1942:{\displaystyle R_{i}} 1914: 1848:Finite element method 1842:Finite element method 1833: 1811: 1783: 1763: 1740: 1622:Further information: 1581:computer aided design 1516:Bernoulli's Principle 1493: 1468: 1466:{\displaystyle f_{0}} 1430: 1428:{\displaystyle f_{0}} 1403: 1378: 1353: 1351:{\displaystyle f_{0}} 1326: 1282: 1246: 1222: 1202: 1177: 1152: 1132: 1095: 1047: 1023: 1003: 1001:{\displaystyle \rho } 983: 981:{\displaystyle p_{0}} 956: 852:viscous stress tensor 725: 571:to the flow about an 540:. Finally, for small 528:actions to yield the 507: 495: 96:Computational physics 5823:Electronic structure 5233:Fox, Rodney (2003). 3222:Cavitation modelling 3212:Blade element theory 2963: 2941: 2919: 2859: 2810: 2762: 2664: 2644: 2618: 2437:is discretized onto 2335: 2300: 2280: 2260: 2240: 2220: 2200: 2180: 2065: 2020: 1993: 1973: 1953: 1926: 1863: 1820: 1800: 1772: 1752: 1677: 1650:Finite volume method 1644:Finite volume method 1477: 1450: 1412: 1387: 1362: 1335: 1291: 1271: 1235: 1211: 1186: 1161: 1141: 1104: 1084: 1063:thermal conductivity 1036: 1012: 992: 965: 915: 825:conservation of mass 726:A simulation of the 673:, and more recently 580:Lewis Fry Richardson 567:of the flow about a 508:A simulation of the 484:for film and games. 261:Metropolis algorithm 5828:Molecular mechanics 5526:10.1109/MAHC.2006.5 5483:2008JCoPh.227.1790E 5426:2005AcNum..14....1B 5381:1992JCoPh.100...25U 5346:1981JCoPh..39..201H 5311:1993PhFlA...5.1023M 5272:1985PrECS..11..119P 5212:1998PhFl...10..499C 5200:Physics of Fluids A 5177:1969PhFl...12..485L 5165:Physics of Fluids A 5132:2004PhFl...16.2497G 5120:Physics of Fluids A 5055:Pope, S.B. (2000). 5006:1974CMAME...3..269L 4923:2007IJNME..69.1109S 4707:1995JFM...284..257H 4658:2000AIAAJ..38...22B 4486:Incompressible Flow 4356:10.1038/nature04801 4348:2006Natur.441..727J 4298:10.2514/6.1990-1470 4275:10.2514/6.1985-1530 4134:10.2514/6.1991-3236 4099:Journal of Aircraft 4084:10.2514/6.1981-1259 3961:1971AIAAJ...9..114C 3848:Journal of Aircraft 3821:Journal of Aircraft 3767:10.2514/6.1983-1828 3744:10.2514/6.1983-1827 3721:10.2514/6.1981-1255 3667:1967PrAeS...8....1H 3632:1965PhFl....8.2182H 3597:1963PhFl....6..975F 3562:1966JCoPh...1...87G 3496:2004JCoPh.195..414H 3436:1998AnRFM..30D..13H 3386:1993PhFlA...5.1023M 3312:Turbulence modeling 3182:Computed Tomography 3078:Solution algorithms 3026:. VC is similar to 1200:{\displaystyle f''} 1175:{\displaystyle f''} 862:(EOS), such as the 848:continuum mechanics 844:continuity equation 752:New York University 415:boundary conditions 244:Monte Carlo methods 5785:Biological systems 4828:Patankar, Suhas V. 4606:Kundu, P. (1990). 4420:. Springer-Verlag. 4252:10.2514/6.1986-103 4219:10.2514/6.2004-581 4180:10.2514/6.1995-343 4157:10.2514/6.1995-853 4037:10.2514/6.1977-635 3931:10.2514/6.1977-207 3698:10.2514/6.1972-188 3297:Shape optimization 3173: 3056: 2982: 2949: 2927: 2905: 2826: 2778: 2742: 2711: 2670: 2650: 2630: 2572: 2561: 2392: 2306: 2286: 2266: 2246: 2226: 2206: 2186: 2163: 2033: 2006: 1979: 1959: 1939: 1909: 1828: 1806: 1778: 1758: 1735: 1491:{\displaystyle f'} 1488: 1463: 1425: 1401:{\displaystyle f'} 1398: 1376:{\displaystyle f'} 1373: 1348: 1321: 1277: 1241: 1217: 1197: 1172: 1147: 1127: 1090: 1042: 1018: 998: 978: 951: 732: 647:McDonnell Aircraft 518: 502: 468:, fluid flows and 458:weather simulation 417:. With high-speed 395:numerical analysis 289:Molecular dynamics 5986: 5985: 5953:Materials science 5833:Quantum mechanics 5714:Suman Chakraborty 5692:978-0-89116-522-4 5673:978-0-07-001685-9 5620:10.1111/cgf.13350 5299:Physics of Fluids 5244:978-0-521-65049-6 5185:10.1063/1.1692511 5150:10.1063/1.1736671 5066:978-0-521-59886-6 5041:978-1-928729-08-2 4895:978-0-07-113210-7 4610:. Academic Press. 4332:(7094): 727–730. 4228:978-1-62410-078-9 4061:10.2514/6.1987-34 4000:978-3-540-05807-6 3640:10.1063/1.1761178 3626:(12): 2182–2189. 3605:10.1063/1.1706854 3430:(1): xiii–xxxvi. 3358:978-0-486-61980-4 3034:Linear eddy model 3002:Langevin equation 2824: 2653:{\displaystyle k} 2583:Reynolds stresses 2579:apparent stresses 2513:Turbulence models 2309:{\displaystyle z} 2289:{\displaystyle y} 2269:{\displaystyle x} 2249:{\displaystyle H} 2229:{\displaystyle G} 2209:{\displaystyle F} 2189:{\displaystyle Q} 2155: 2132: 2109: 2086: 1982:{\displaystyle Q} 1962:{\displaystyle i} 1809:{\displaystyle V} 1781:{\displaystyle F} 1761:{\displaystyle Q} 1693: 1280:{\displaystyle f} 1244:{\displaystyle F} 1220:{\displaystyle F} 1150:{\displaystyle F} 1093:{\displaystyle f} 1045:{\displaystyle T} 1021:{\displaystyle R} 860:equation of state 821:Conservation laws 756:Grumman Aerospace 603:Francis H. Harlow 595:three-dimensional 389:) is a branch of 380: 379: 231:Turbulence models 211:Lattice Boltzmann 191:Finite difference 90: 89: 82: 16:(Redirected from 6011: 5876:Particle physics 5856:Electromagnetics 5757: 5750: 5743: 5734: 5696: 5677: 5649: 5648: 5646: 5638: 5632: 5631: 5603: 5597: 5596: 5594: 5582: 5576: 5572: 5566: 5563: 5557: 5556: 5536: 5530: 5529: 5509: 5503: 5502: 5477:(3): 1790–1808. 5462: 5456: 5455: 5437: 5409: 5403: 5402: 5400: 5364: 5358: 5357: 5329: 5323: 5322: 5319:10.1063/1.858667 5305:(4): 1023–1034. 5290: 5284: 5283: 5255: 5249: 5248: 5230: 5224: 5223: 5220:10.1063/1.869537 5195: 5189: 5188: 5160: 5154: 5153: 5143: 5115: 5109: 5108: 5077: 5071: 5070: 5052: 5046: 5045: 5027: 5018: 5017: 4989: 4983: 4982: 4964: 4958: 4957: 4949: 4943: 4942: 4931:10.1002/nme.1801 4917:(6): 1109–1157. 4906: 4900: 4899: 4879: 4870: 4869: 4867: 4866: 4852: 4846: 4845: 4824: 4818: 4817: 4811: 4803: 4801: 4791: 4767: 4761: 4760: 4758: 4757: 4747: 4741: 4740: 4734: 4726: 4690: 4684: 4683: 4677: 4669: 4641: 4635: 4634: 4632: 4631: 4621: 4612: 4611: 4603: 4590: 4589: 4583: 4575: 4567: 4558: 4557: 4551: 4543: 4535: 4526: 4525: 4519: 4511: 4503: 4490: 4489: 4481: 4456: 4455: 4453: 4452: 4442: 4436: 4435: 4429: 4421: 4413: 4407: 4406: 4374: 4368: 4367: 4341: 4339:cond-mat/0612110 4317: 4311: 4308: 4302: 4301: 4285: 4279: 4278: 4262: 4256: 4255: 4239: 4233: 4232: 4204: 4198: 4197: 4191: 4183: 4167: 4161: 4160: 4144: 4138: 4137: 4121: 4115: 4114: 4094: 4088: 4087: 4071: 4065: 4064: 4047: 4041: 4040: 4024: 4018: 4011: 4005: 4004: 3979: 3973: 3972: 3944: 3935: 3934: 3918: 3912: 3905: 3899: 3892: 3886: 3883: 3877: 3872:Maskew, Brian, " 3870: 3864: 3863: 3843: 3837: 3836: 3816: 3810: 3803: 3797: 3790: 3784: 3779:Bristow, D.R., " 3777: 3771: 3770: 3754: 3748: 3747: 3731: 3725: 3724: 3708: 3702: 3701: 3685: 3679: 3678: 3650: 3644: 3643: 3615: 3609: 3608: 3580: 3574: 3573: 3545: 3539: 3538: 3514: 3508: 3507: 3475: 3469: 3468: 3466: 3464: 3454: 3448: 3447: 3419: 3413: 3412: 3404: 3398: 3397: 3394:10.1063/1.858667 3380:(4): 1023–1034. 3369: 3363: 3362: 3344: 3272:Meshfree methods 3140:Uzawa algorithms 3128:additive Schwarz 3068:level-set method 3058:The modeling of 2991: 2989: 2988: 2983: 2981: 2970: 2958: 2956: 2955: 2950: 2948: 2936: 2934: 2933: 2928: 2926: 2914: 2912: 2911: 2906: 2904: 2887: 2879: 2871: 2870: 2835: 2833: 2832: 2827: 2825: 2817: 2787: 2785: 2784: 2779: 2777: 2776: 2679: 2677: 2676: 2671: 2659: 2657: 2656: 2651: 2639: 2637: 2636: 2631: 2401: 2399: 2398: 2393: 2315: 2313: 2312: 2307: 2295: 2293: 2292: 2287: 2275: 2273: 2272: 2267: 2255: 2253: 2252: 2247: 2235: 2233: 2232: 2227: 2215: 2213: 2212: 2207: 2195: 2193: 2192: 2187: 2172: 2170: 2169: 2164: 2156: 2154: 2146: 2138: 2133: 2131: 2123: 2115: 2110: 2108: 2100: 2092: 2087: 2085: 2077: 2069: 2042: 2040: 2039: 2034: 2032: 2031: 2015: 2013: 2012: 2007: 2005: 2004: 1988: 1986: 1985: 1980: 1968: 1966: 1965: 1960: 1948: 1946: 1945: 1940: 1938: 1937: 1918: 1916: 1915: 1910: 1908: 1907: 1891: 1890: 1875: 1874: 1837: 1835: 1834: 1829: 1827: 1815: 1813: 1812: 1807: 1787: 1785: 1784: 1779: 1767: 1765: 1764: 1759: 1744: 1742: 1741: 1736: 1725: 1694: 1692: 1681: 1497: 1495: 1494: 1489: 1487: 1472: 1470: 1469: 1464: 1462: 1461: 1434: 1432: 1431: 1426: 1424: 1423: 1407: 1405: 1404: 1399: 1397: 1382: 1380: 1379: 1374: 1372: 1357: 1355: 1354: 1349: 1347: 1346: 1330: 1328: 1327: 1322: 1320: 1309: 1308: 1286: 1284: 1283: 1278: 1250: 1248: 1247: 1242: 1226: 1224: 1223: 1218: 1206: 1204: 1203: 1198: 1196: 1181: 1179: 1178: 1173: 1171: 1156: 1154: 1153: 1148: 1136: 1134: 1133: 1128: 1126: 1099: 1097: 1096: 1091: 1051: 1049: 1048: 1043: 1028:is the specific 1027: 1025: 1024: 1019: 1007: 1005: 1004: 999: 987: 985: 984: 979: 977: 976: 960: 958: 957: 952: 938: 933: 932: 774:'s NASCART-GT). 717:Grumman Aircraft 698: 637:(PANAIR, A502), 630:Douglas Aircraft 607:particle-in-cell 544:in subsonic and 372: 365: 358: 284:Particle-in-cell 206:Boundary element 166:Yukawa potential 129:Particle physics 119:Electromagnetics 106: 92: 85: 78: 74: 71: 65: 60:this article by 51:inline citations 38: 37: 30: 21: 6019: 6018: 6014: 6013: 6012: 6010: 6009: 6008: 5989: 5988: 5987: 5982: 5936: 5908: 5885: 5837: 5809: 5766: 5761: 5703: 5693: 5680: 5674: 5661: 5658: 5653: 5652: 5644: 5640: 5639: 5635: 5605: 5604: 5600: 5584: 5583: 5579: 5573: 5569: 5564: 5560: 5538: 5537: 5533: 5511: 5510: 5506: 5464: 5463: 5459: 5435:10.1.1.409.4160 5411: 5410: 5406: 5366: 5365: 5361: 5331: 5330: 5326: 5292: 5291: 5287: 5257: 5256: 5252: 5245: 5232: 5231: 5227: 5197: 5196: 5192: 5162: 5161: 5157: 5141:10.1.1.415.6540 5117: 5116: 5112: 5079: 5078: 5074: 5067: 5057:Turbulent Flows 5054: 5053: 5049: 5042: 5029: 5028: 5021: 4991: 4990: 4986: 4979: 4966: 4965: 4961: 4951: 4950: 4946: 4908: 4907: 4903: 4896: 4888:. McGraw-Hill. 4881: 4880: 4873: 4864: 4862: 4854: 4853: 4849: 4842: 4826: 4825: 4821: 4804: 4769: 4768: 4764: 4755: 4753: 4749: 4748: 4744: 4727: 4692: 4691: 4687: 4670: 4643: 4642: 4638: 4629: 4627: 4623: 4622: 4615: 4608:Fluid Mechanics 4605: 4604: 4593: 4576: 4569: 4568: 4561: 4544: 4537: 4536: 4529: 4512: 4508:Fluid Mechanics 4505: 4504: 4493: 4483: 4482: 4459: 4450: 4448: 4444: 4443: 4439: 4422: 4415: 4414: 4410: 4376: 4375: 4371: 4319: 4318: 4314: 4309: 4305: 4287: 4286: 4282: 4264: 4263: 4259: 4241: 4240: 4236: 4229: 4206: 4205: 4201: 4184: 4169: 4168: 4164: 4146: 4145: 4141: 4123: 4122: 4118: 4111:10.2514/3.45717 4096: 4095: 4091: 4073: 4072: 4068: 4049: 4048: 4044: 4026: 4025: 4021: 4012: 4008: 4001: 3981: 3980: 3976: 3946: 3945: 3938: 3920: 3919: 3915: 3906: 3902: 3893: 3889: 3884: 3880: 3871: 3867: 3860:10.2514/3.57369 3845: 3844: 3840: 3833:10.2514/3.45564 3818: 3817: 3813: 3804: 3800: 3791: 3787: 3778: 3774: 3756: 3755: 3751: 3733: 3732: 3728: 3710: 3709: 3705: 3687: 3686: 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1631:Euler equations 1626: 1620: 1563: 1529:Reynolds number 1480: 1475: 1474: 1453: 1448: 1447: 1415: 1410: 1409: 1390: 1385: 1384: 1365: 1360: 1359: 1338: 1333: 1332: 1313: 1300: 1289: 1288: 1269: 1268: 1254:Reynolds stress 1233: 1232: 1209: 1208: 1189: 1184: 1183: 1164: 1159: 1158: 1139: 1138: 1119: 1102: 1101: 1082: 1081: 1034: 1033: 1010: 1009: 990: 989: 968: 963: 962: 924: 913: 912: 902:Euler equations 883:Newtonian fluid 872:internal energy 810:multiphase flow 805: 799: 744:Paul Garabedian 730:during re-entry 728:SpaceX Starship 692: 530:Euler equations 500:during re-entry 490: 399:data structures 391:fluid mechanics 376: 347: 346: 302: 294: 293: 274: 266: 265: 246: 236: 235: 186: 176: 175: 171:Morse potential 151: 141: 86: 75: 69: 66: 56:Please help to 55: 39: 35: 28: 23: 22: 15: 12: 11: 5: 6017: 6015: 6007: 6006: 6001: 5991: 5990: 5984: 5983: 5981: 5980: 5975: 5970: 5965: 5960: 5955: 5950: 5944: 5942: 5938: 5937: 5935: 5934: 5929: 5924: 5918: 5916: 5914:Social science 5910: 5909: 5907: 5906: 5901: 5895: 5893: 5887: 5886: 5884: 5883: 5881:Thermodynamics 5878: 5873: 5868: 5863: 5861:Fluid dynamics 5858: 5853: 5847: 5845: 5839: 5838: 5836: 5835: 5830: 5825: 5819: 5817: 5811: 5810: 5808: 5807: 5802: 5797: 5792: 5787: 5782: 5776: 5774: 5768: 5767: 5762: 5760: 5759: 5752: 5745: 5737: 5731: 5730: 5721: 5702: 5701:External links 5699: 5698: 5697: 5691: 5678: 5672: 5657: 5654: 5651: 5650: 5633: 5614:(2): 157–167. 5598: 5577: 5567: 5558: 5547:(3): 283–309. 5531: 5504: 5457: 5404: 5359: 5340:(1): 201–225. 5324: 5285: 5266:(2): 119–192. 5250: 5243: 5225: 5206:(2): 499–515. 5190: 5171:(3): 485–497. 5155: 5110: 5091:(4): 393–426. 5072: 5065: 5047: 5040: 5019: 5000:(2): 269–289. 4984: 4977: 4959: 4944: 4901: 4894: 4871: 4860:www.comsol.com 4847: 4841:978-0891165224 4840: 4819: 4762: 4742: 4685: 4636: 4613: 4591: 4559: 4527: 4491: 4457: 4437: 4408: 4369: 4312: 4303: 4280: 4257: 4234: 4227: 4199: 4162: 4139: 4116: 4089: 4066: 4042: 4019: 4006: 3999: 3974: 3969:10.2514/3.6131 3955:(1): 114–121. 3936: 3913: 3907:Drela, Mark, " 3900: 3887: 3878: 3865: 3854:(2): 157–163. 3838: 3827:(4): 302–310. 3811: 3798: 3785: 3772: 3749: 3726: 3703: 3680: 3645: 3610: 3591:(7): 975–982. 3575: 3540: 3509: 3490:(2): 414–433. 3470: 3449: 3414: 3399: 3364: 3357: 3338: 3337: 3335: 3332: 3330: 3329: 3324: 3319: 3314: 3309: 3304: 3299: 3294: 3289: 3284: 3279: 3274: 3269: 3264: 3259: 3254: 3249: 3244: 3239: 3234: 3229: 3224: 3219: 3214: 3209: 3203: 3201: 3198: 3189: 3188:CPU versus GPU 3186: 3161: 3158: 3148: 3145: 3079: 3076: 3072:front tracking 3060:two-phase flow 3044: 3043:Two-phase flow 3041: 3035: 3032: 3012:Main article: 3009: 3006: 2980: 2976: 2973: 2969: 2947: 2937:being between 2925: 2903: 2899: 2896: 2893: 2890: 2886: 2882: 2878: 2874: 2869: 2865: 2845: 2842: 2823: 2820: 2815: 2793: 2790: 2775: 2771: 2767: 2749:Main article: 2746: 2743: 2724:Main article: 2721: 2718: 2699:Main article: 2696: 2693: 2692: 2691: 2687: 2681: 2669: 2649: 2629: 2626: 2623: 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1522: 1519: 1512: 1508:Boundary layer 1505: 1499: 1486: 1483: 1460: 1456: 1442:Sound wave or 1440: 1422: 1418: 1396: 1393: 1371: 1368: 1345: 1341: 1319: 1316: 1312: 1307: 1303: 1299: 1296: 1276: 1265: 1261:potential flow 1259:Ideal flow or 1257: 1240: 1216: 1195: 1192: 1170: 1167: 1146: 1125: 1122: 1118: 1115: 1112: 1109: 1089: 1074: 1054: 1053:approximation. 1041: 1017: 997: 975: 971: 950: 947: 944: 941: 937: 931: 927: 923: 920: 905: 898: 890: 875: 836: 814:two-phase flow 798: 795: 776:Antony Jameson 760:Antony Jameson 690:Richard Eppler 686:boundary layer 489: 486: 482:visual effects 480:analysis, and 419:supercomputers 378: 377: 375: 374: 367: 360: 352: 349: 348: 345: 344: 339: 334: 329: 324: 319: 314: 309: 303: 300: 299: 296: 295: 292: 291: 286: 281: 275: 272: 271: 268: 267: 264: 263: 258: 256:Gibbs sampling 253: 247: 242: 241: 238: 237: 234: 233: 228: 223: 218: 216:Riemann solver 213: 208: 203: 201:Finite element 198: 193: 187: 184:Fluid dynamics 182: 181: 178: 177: 174: 173: 168: 163: 158: 152: 149: 148: 145: 144: 143: 142: 136: 134:Thermodynamics 131: 126: 121: 116: 108: 107: 99: 98: 88: 87: 70:September 2014 42: 40: 33: 26: 24: 14: 13: 10: 9: 6: 4: 3: 2: 6016: 6005: 6002: 6000: 5997: 5996: 5994: 5979: 5976: 5974: 5971: 5969: 5966: 5964: 5961: 5959: 5956: 5954: 5951: 5949: 5946: 5945: 5943: 5939: 5933: 5930: 5928: 5925: 5923: 5920: 5919: 5917: 5915: 5911: 5905: 5902: 5900: 5897: 5896: 5894: 5892: 5888: 5882: 5879: 5877: 5874: 5872: 5869: 5867: 5864: 5862: 5859: 5857: 5854: 5852: 5849: 5848: 5846: 5844: 5840: 5834: 5831: 5829: 5826: 5824: 5821: 5820: 5818: 5816: 5812: 5806: 5805:Phylogenetics 5803: 5801: 5798: 5796: 5793: 5791: 5788: 5786: 5783: 5781: 5778: 5777: 5775: 5773: 5769: 5765: 5758: 5753: 5751: 5746: 5744: 5739: 5738: 5735: 5728: 5725: 5722: 5719: 5715: 5711: 5708: 5705: 5704: 5700: 5694: 5688: 5684: 5679: 5675: 5669: 5665: 5660: 5659: 5655: 5643: 5637: 5634: 5629: 5625: 5621: 5617: 5613: 5609: 5602: 5599: 5593: 5588: 5581: 5578: 5571: 5568: 5562: 5559: 5554: 5550: 5546: 5542: 5535: 5532: 5527: 5523: 5520:(1): 99–103. 5519: 5515: 5508: 5505: 5500: 5496: 5492: 5488: 5484: 5480: 5476: 5472: 5468: 5461: 5458: 5453: 5449: 5445: 5441: 5436: 5431: 5427: 5423: 5419: 5415: 5414:Acta Numerica 5408: 5405: 5399: 5398:2027.42/30059 5394: 5390: 5386: 5382: 5378: 5374: 5370: 5363: 5360: 5355: 5351: 5347: 5343: 5339: 5335: 5328: 5325: 5320: 5316: 5312: 5308: 5304: 5300: 5296: 5289: 5286: 5281: 5277: 5273: 5269: 5265: 5261: 5254: 5251: 5246: 5240: 5236: 5229: 5226: 5221: 5217: 5213: 5209: 5205: 5201: 5194: 5191: 5186: 5182: 5178: 5174: 5170: 5166: 5159: 5156: 5151: 5147: 5142: 5137: 5133: 5129: 5125: 5121: 5114: 5111: 5106: 5102: 5098: 5094: 5090: 5086: 5082: 5076: 5073: 5068: 5062: 5058: 5051: 5048: 5043: 5037: 5033: 5026: 5024: 5020: 5015: 5011: 5007: 5003: 4999: 4995: 4988: 4985: 4980: 4978:0-521-62186-0 4974: 4970: 4963: 4960: 4955: 4948: 4945: 4940: 4936: 4932: 4928: 4924: 4920: 4916: 4912: 4905: 4902: 4897: 4891: 4887: 4886: 4878: 4876: 4872: 4861: 4857: 4851: 4848: 4843: 4837: 4833: 4829: 4823: 4820: 4815: 4809: 4800: 4799:11392/2414220 4795: 4790: 4785: 4781: 4777: 4773: 4766: 4763: 4752: 4746: 4743: 4738: 4732: 4724: 4720: 4716: 4712: 4708: 4704: 4700: 4696: 4689: 4686: 4681: 4675: 4667: 4666:10.2514/2.949 4663: 4659: 4655: 4651: 4647: 4640: 4637: 4626: 4620: 4618: 4614: 4609: 4602: 4600: 4598: 4596: 4592: 4587: 4581: 4574:. RT Edwards. 4573: 4566: 4564: 4560: 4555: 4549: 4541: 4534: 4532: 4528: 4523: 4517: 4509: 4502: 4500: 4498: 4496: 4492: 4487: 4480: 4478: 4476: 4474: 4472: 4470: 4468: 4466: 4464: 4462: 4458: 4447: 4441: 4438: 4433: 4427: 4419: 4412: 4409: 4404: 4400: 4396: 4392: 4388: 4384: 4380: 4373: 4370: 4365: 4361: 4357: 4353: 4349: 4345: 4340: 4335: 4331: 4327: 4323: 4316: 4313: 4307: 4304: 4299: 4295: 4291: 4284: 4281: 4276: 4272: 4268: 4261: 4258: 4253: 4249: 4245: 4238: 4235: 4230: 4224: 4220: 4216: 4212: 4211: 4203: 4200: 4195: 4189: 4181: 4177: 4173: 4166: 4163: 4158: 4154: 4150: 4143: 4140: 4135: 4131: 4127: 4120: 4117: 4112: 4108: 4104: 4100: 4093: 4090: 4085: 4081: 4077: 4070: 4067: 4062: 4058: 4054: 4046: 4043: 4038: 4034: 4030: 4023: 4020: 4016: 4010: 4007: 4002: 3996: 3992: 3988: 3984: 3978: 3975: 3970: 3966: 3962: 3958: 3954: 3950: 3943: 3941: 3937: 3932: 3928: 3924: 3917: 3914: 3910: 3904: 3901: 3897: 3891: 3888: 3882: 3879: 3875: 3869: 3866: 3861: 3857: 3853: 3849: 3842: 3839: 3834: 3830: 3826: 3822: 3815: 3812: 3808: 3802: 3799: 3795: 3789: 3786: 3782: 3776: 3773: 3768: 3764: 3760: 3753: 3750: 3745: 3741: 3737: 3730: 3727: 3722: 3718: 3714: 3707: 3704: 3699: 3695: 3691: 3684: 3681: 3676: 3672: 3668: 3664: 3660: 3656: 3649: 3646: 3641: 3637: 3633: 3629: 3625: 3621: 3614: 3611: 3606: 3602: 3598: 3594: 3590: 3586: 3579: 3576: 3571: 3567: 3563: 3559: 3556:(1): 87–118. 3555: 3551: 3544: 3541: 3536: 3532: 3528: 3524: 3520: 3513: 3510: 3505: 3501: 3497: 3493: 3489: 3485: 3481: 3474: 3471: 3459: 3453: 3450: 3445: 3441: 3437: 3433: 3429: 3425: 3418: 3415: 3410: 3403: 3400: 3395: 3391: 3387: 3383: 3379: 3375: 3368: 3365: 3360: 3354: 3350: 3343: 3340: 3333: 3328: 3325: 3323: 3320: 3318: 3315: 3313: 3310: 3308: 3305: 3303: 3300: 3298: 3295: 3293: 3290: 3288: 3285: 3283: 3280: 3278: 3275: 3273: 3270: 3268: 3265: 3263: 3260: 3258: 3255: 3253: 3250: 3248: 3245: 3243: 3240: 3238: 3235: 3233: 3230: 3228: 3225: 3223: 3220: 3218: 3215: 3213: 3210: 3208: 3205: 3204: 3199: 3197: 3193: 3187: 3185: 3183: 3179: 3171: 3166: 3159: 3157: 3154: 3146: 3144: 3141: 3137: 3133: 3129: 3125: 3120: 3117: 3113: 3111: 3107: 3103: 3099: 3094: 3090: 3085: 3077: 3075: 3073: 3069: 3065: 3061: 3054: 3049: 3042: 3040: 3033: 3031: 3029: 3025: 3021: 3015: 3007: 3005: 3003: 2999: 2995: 2974: 2971: 2897: 2891: 2888: 2880: 2867: 2863: 2854: 2850: 2843: 2841: 2839: 2821: 2818: 2813: 2805: 2800: 2791: 2789: 2773: 2769: 2765: 2757: 2752: 2744: 2738: 2734: 2732: 2727: 2719: 2717: 2715: 2707: 2702: 2694: 2688: 2685: 2682: 2680:) are solved. 2667: 2647: 2627: 2624: 2621: 2613: 2609: 2605: 2601: 2596: 2594: 2591: 2590: 2589: 2586: 2584: 2580: 2576: 2570: 2565: 2558: 2554: 2550: 2545: 2540: 2532: 2530: 2528: 2523: 2519: 2512: 2510: 2508: 2504: 2503:flux limiters 2497: 2489: 2487: 2483: 2475: 2470: 2467: 2464: 2461: 2460: 2459: 2455: 2453: 2449: 2445: 2440: 2436: 2432: 2425:Vortex method 2424: 2422: 2420: 2413: 2405: 2403: 2389: 2386: 2383: 2380: 2377: 2374: 2371: 2368: 2365: 2362: 2359: 2356: 2350: 2347: 2344: 2338: 2327: 2319: 2317: 2303: 2283: 2263: 2243: 2223: 2203: 2183: 2160: 2157: 2151: 2143: 2134: 2128: 2120: 2111: 2105: 2097: 2088: 2082: 2074: 2061: 2060: 2059: 2054: 2046: 2044: 2028: 2024: 2001: 1997: 1976: 1956: 1934: 1930: 1904: 1900: 1896: 1892: 1887: 1883: 1879: 1876: 1871: 1867: 1859: 1858: 1857: 1854: 1849: 1841: 1839: 1803: 1795: 1791: 1775: 1755: 1732: 1729: 1726: 1718: 1714: 1711: 1708: 1705: 1702: 1698: 1695: 1689: 1673: 1672: 1671: 1669: 1663: 1661: 1657: 1651: 1643: 1641: 1638: 1636: 1632: 1625: 1617: 1612: 1609: 1605: 1600: 1597: 1593: 1589: 1585: 1582: 1578: 1574: 1573: 1572: 1571:preprocessing 1568: 1567: 1566: 1560: 1554: 1553:Rayleigh flow 1551: 1547: 1544: 1541: 1538: 1533: 1530: 1526: 1523: 1520: 1517: 1513: 1509: 1506: 1503: 1500: 1484: 1481: 1458: 1454: 1445: 1441: 1438: 1420: 1416: 1394: 1391: 1369: 1366: 1343: 1339: 1317: 1314: 1310: 1305: 1301: 1297: 1294: 1274: 1266: 1262: 1258: 1255: 1238: 1230: 1214: 1193: 1190: 1168: 1165: 1144: 1123: 1120: 1116: 1113: 1110: 1107: 1087: 1079: 1076:Compressible 1075: 1072: 1068: 1067:heat capacity 1064: 1060: 1055: 1039: 1031: 1015: 995: 973: 969: 945: 942: 935: 929: 925: 921: 918: 910: 906: 903: 900:Compressible 899: 895: 891: 888: 884: 880: 877:Compressible 876: 873: 869: 865: 861: 857: 853: 849: 845: 841: 837: 834: 830: 826: 822: 819: 818: 817: 815: 811: 804: 796: 794: 790: 788: 783: 779: 777: 773: 769: 763: 761: 758:as Grumfoil. 757: 753: 749: 745: 741: 737: 729: 724: 720: 718: 712: 710: 706: 702: 696: 691: 687: 682: 680: 676: 675:wind turbines 672: 668: 664: 660: 656: 652: 648: 644: 640: 636: 631: 627: 622: 620: 616: 612: 611:fluid-in-cell 608: 604: 600: 596: 591: 589: 585: 581: 576: 574: 570: 566: 561: 559: 555: 551: 547: 543: 542:perturbations 539: 535: 531: 527: 523: 515: 511: 506: 499: 498:Space Shuttle 494: 487: 485: 483: 479: 475: 471: 470:heat transfer 467: 463: 459: 455: 451: 446: 444: 440: 436: 432: 428: 424: 420: 416: 412: 408: 404: 400: 396: 392: 388: 384: 373: 368: 366: 361: 359: 354: 353: 351: 350: 343: 340: 338: 335: 333: 330: 328: 325: 323: 320: 318: 315: 313: 310: 308: 305: 304: 298: 297: 290: 287: 285: 282: 280: 277: 276: 270: 269: 262: 259: 257: 254: 252: 249: 248: 245: 240: 239: 232: 229: 227: 224: 222: 219: 217: 214: 212: 209: 207: 204: 202: 199: 197: 196:Finite volume 194: 192: 189: 188: 185: 180: 179: 172: 169: 167: 164: 162: 159: 157: 154: 153: 147: 146: 140: 137: 135: 132: 130: 127: 125: 122: 120: 117: 115: 112: 111: 110: 109: 105: 101: 100: 97: 93: 84: 81: 73: 63: 59: 53: 52: 46: 41: 32: 31: 19: 5860: 5851:Astrophysics 5800:Neuroscience 5723: 5706: 5682: 5663: 5636: 5611: 5607: 5601: 5580: 5570: 5561: 5544: 5540: 5534: 5517: 5513: 5507: 5474: 5470: 5460: 5417: 5413: 5407: 5375:(1): 25–37. 5372: 5368: 5362: 5337: 5333: 5327: 5302: 5298: 5288: 5263: 5259: 5253: 5234: 5228: 5203: 5199: 5193: 5168: 5164: 5158: 5123: 5119: 5113: 5088: 5084: 5081:Farge, Marie 5075: 5056: 5050: 5031: 4997: 4993: 4987: 4968: 4962: 4953: 4947: 4914: 4910: 4904: 4884: 4863:. Retrieved 4859: 4850: 4831: 4822: 4808:cite journal 4779: 4775: 4765: 4754:. Retrieved 4745: 4731:cite journal 4698: 4694: 4688: 4674:cite journal 4652:(1): 22–29. 4649: 4646:AIAA Journal 4645: 4639: 4628:. Retrieved 4607: 4571: 4539: 4507: 4485: 4449:. Retrieved 4440: 4417: 4411: 4386: 4382: 4372: 4329: 4325: 4315: 4306: 4289: 4283: 4266: 4260: 4243: 4237: 4209: 4202: 4171: 4165: 4148: 4142: 4125: 4119: 4102: 4098: 4092: 4075: 4069: 4052: 4045: 4028: 4022: 4009: 3982: 3977: 3952: 3949:AIAA Journal 3948: 3922: 3916: 3903: 3890: 3881: 3868: 3851: 3847: 3841: 3824: 3820: 3814: 3801: 3788: 3775: 3758: 3752: 3735: 3729: 3712: 3706: 3689: 3683: 3658: 3654: 3648: 3623: 3619: 3613: 3588: 3584: 3578: 3553: 3549: 3543: 3518: 3512: 3487: 3483: 3473: 3461:. 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Elsevier. 3327:Wind tunnel 2844:PDF methods 1561:Methodology 1525:Stokes Flow 894:Mach number 740:Julian Cole 693: [ 667:helicopters 663:automobiles 649:(MACAERO), 641:(Quadpan), 548:flows (not 536:yields the 454:hypersonics 403:fluid flows 317:von Neumann 251:Integration 62:introducing 5993:Categories 5963:Statistics 5904:Lexicology 5866:Geophysics 5592:2108.08821 4865:2022-07-15 4756:2020-01-12 4630:2020-01-07 4451:2020-01-07 4389:: 119997. 3535:1288309947 3334:References 2555:(top) and 2527:non-linear 2448:Barnes-Hut 2439:Lagrangian 1608:simulation 1546:Fanno flow 897:equations. 801:See also: 748:David Korn 701:Mark Drela 657:, surface 655:submarines 558:linearized 554:hypersonic 546:supersonic 478:combustion 435:analytical 393:that uses 301:Scientists 150:Potentials 139:Simulation 45:references 5973:Semiotics 5932:Economics 5927:Sociology 5899:Semantics 5871:Mechanics 5815:Chemistry 5790:Cognition 5452:122717775 5430:CiteSeerX 5420:: 1–137. 5136:CiteSeerX 4939:122551159 4782:: 10–26. 4723:122833857 4580:cite book 4548:cite book 4516:cite book 4426:cite book 4403:0009-2509 4364:1476-4687 4188:cite book 4105:: 13–20. 3661:: 1–138. 3463:March 13, 3153:transonic 3132:multigrid 3116:Multigrid 2814:− 2668:ϵ 2628:ϵ 2625:− 2581:known as 2435:vorticity 2149:∂ 2141:∂ 2126:∂ 2118:∂ 2103:∂ 2095:∂ 2080:∂ 2072:∂ 1880:∭ 1712:∬ 1696:∭ 1687:∂ 1683:∂ 1591:elements. 1059:viscosity 996:ρ 919:ρ 909:ideal gas 887:heat flux 864:ideal gas 858:; (c) an 856:heat flux 768:cartesian 736:transonic 550:transonic 534:vorticity 439:empirical 427:turbulent 423:transonic 342:Richtmyer 114:Mechanics 5922:Politics 5795:Genomics 5628:43945038 5499:16365489 5105:53464243 4830:(1980). 3200:See also 2853:Lundgren 2604:Spalding 2559:(bottom) 2431:meshfree 2058:grid). 1596:enthalpy 1577:geometry 1485:′ 1395:′ 1370:′ 1331:, where 1318:′ 1194:″ 1169:″ 1137:, where 1124:″ 961:, where 868:enthalpy 787:OVERFLOW 671:aircraft 645:(HESS), 639:Lockheed 613:method, 609:method, 569:cylinder 322:Galerkin 273:Particle 5948:Finance 5843:Physics 5780:Anatomy 5772:Biology 5724:Course: 5707:Course: 5575:109–136 5479:Bibcode 5422:Bibcode 5377:Bibcode 5342:Bibcode 5307:Bibcode 5268:Bibcode 5208:Bibcode 5173:Bibcode 5128:Bibcode 5002:Bibcode 4919:Bibcode 4703:Bibcode 4654:Bibcode 4344:Bibcode 3957:Bibcode 3663:Bibcode 3628:Bibcode 3593:Bibcode 3558:Bibcode 3492:Bibcode 3432:Bibcode 3382:Bibcode 2799:wavelet 2690:effort. 2608:Prandtl 2600:Launder 2549:DrivAer 1569:During 1537:locally 840:locally 643:Douglas 573:airfoil 526:viscous 510:Hyper-X 407:liquids 307:Godunov 58:improve 5689: 5670: 5626: 5497: 5450: 5432: 5241: 5138: 5103: 5063: 5038: 4975: 4937: 4892: 4838: 4721: 4401: 4362: 4326:Nature 4225: 3997: 3533: 3355: 3136:SIMPLE 3130:, and 3093:Picard 3089:Newton 3066:, the 2421:form. 2296:, and 2236:, and 2176:where 1922:where 1748:where 1656:memory 1588:volume 1065:, and 1032:, and 911:, use 831:, and 679:yachts 635:Boeing 474:engine 332:Wilson 327:Lorenz 279:N-body 47:, but 5941:Other 5656:Notes 5645:(PDF) 5624:S2CID 5587:arXiv 5495:S2CID 5448:S2CID 5101:S2CID 4935:S2CID 4719:S2CID 4334:arXiv 3170:aorta 3106:GMRES 2804:Farge 2686:(RSM) 2612:Smith 2553:URANS 1511:wall. 705:XFOIL 697:] 659:ships 588:ENIAC 411:gases 337:Alder 5687:ISBN 5668:ISBN 5239:ISBN 5061:ISBN 5036:ISBN 4973:ISBN 4890:ISBN 4836:ISBN 4814:link 4737:link 4680:link 4586:link 4554:link 4522:link 4432:link 4399:ISSN 4360:ISSN 4223:ISBN 4194:link 3995:ISBN 3531:OCLC 3465:2013 3353:ISBN 3138:and 3070:and 3018:The 2959:and 2602:and 2557:DDES 2450:and 1633:and 1606:The 1586:The 1575:The 1473:and 812:and 746:and 651:NASA 514:Mach 476:and 409:and 397:and 312:Ulam 5616:doi 5549:doi 5522:doi 5487:doi 5475:227 5440:doi 5393:hdl 5385:doi 5373:100 5350:doi 5315:doi 5276:doi 5216:doi 5181:doi 5146:doi 5093:doi 5010:doi 4927:doi 4794:hdl 4784:doi 4711:doi 4699:284 4662:doi 4391:doi 4387:292 4352:doi 4330:441 4294:doi 4271:doi 4248:doi 4215:doi 4176:doi 4153:doi 4130:doi 4107:doi 4080:doi 4057:doi 4033:doi 3987:doi 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Knowledge (XXG)

Computational fluid dynamics

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