• • • Computational fluid dynamics ( CFD) is a branch of that uses and to solve and analyze problems that involve. Computers are used to perform the calculations required to simulate the interaction of liquids and gases with surfaces defined.

The minimum total potential energy principle is a fundamental concept used in [physics] and [engineering]. It dictates that (at low temperatures) a structure or body. An Introduction to the Finite Element Method by J.N. Reddy (3rd Edition) (2006) - Ebook download as PDF File (.pdf) or read book online. Finite element methods.

With high-speed, better solutions can be achieved. Ongoing research yields software that improves the accuracy and speed of complex simulation scenarios such as or flows. Initial experimental validation of such software is performed using a with the final validation coming in full-scale testing, e.g.. A simulation of the scramjet vehicle in operation at -7 The fundamental basis of almost all CFD problems is the, which define many single-phase (gas or liquid, but not both) fluid flows.

These equations can be simplified by removing terms describing actions to yield the. Further simplification, by removing terms describing yields the. Finally, for small in subsonic and flows (not or ) these equations can be to yield the linearized potential equations. Historically, methods were first developed to solve the linearized potential equations. Two-dimensional (2D) methods, using of the flow about a to the flow about an were developed in the 1930s. One of the earliest type of calculations resembling modern CFD are those by, in the sense that these calculations used finite differences and divided the physical space in cells.

Although they failed dramatically, these calculations, together with Richardson's book 'Weather prediction by numerical process', set the basis for modern CFD and numerical meteorology. In fact, early CFD calculations during the 1940s using used methods close to those in Richardson's 1922 book. The computer power available paced development of methods.

Probably the first work using computers to model fluid flow, as governed by the Navier-Stokes equations, was performed at, in the T3 group. This group was led by, who is widely considered as one of the pioneers of CFD. From 1957 to late 1960s, this group developed a variety of numerical methods to simulate transient two-dimensional fluid flows, such as method (Harlow, 1957), method (Gentry, Martin and Daly, 1966), method (Jake Fromm, 1963), and (Harlow and Welch, 1965). Fromm's vorticity-stream-function method for 2D, transient, incompressible flow was the first treatment of strongly contorting incompressible flows in the world. The first paper with three-dimensional model was published by John Hess and of in 1967. This method discretized the surface of the geometry with panels, giving rise to this class of programs being called Panel Methods.

Their method itself was simplified, in that it did not include lifting flows and hence was mainly applied to ship hulls and aircraft fuselages. The first lifting Panel Code (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 (PANAIR, A502), (Quadpan), (HESS), (MACAERO), (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, surface,,,, and more recently. Its sister code, USAERO is an unsteady panel method that has also been used for modeling such things as high speed trains and racing.

The NASA PMARC code from an early version of VSAERO and a derivative of PMARC, named CMARC, is also commercially available. In the two-dimensional realm, a number of Panel Codes have been developed for airfoil analysis and design. The codes typically have a analysis included, so that viscous effects can be modeled.

Professor of the developed the code, partly with NASA funding, which became available in the early 1980s. This was soon followed by Professor Mark Drela's code.

Both PROFILE and XFOIL incorporate two-dimensional panel codes, with coupled boundary layer codes for airfoil analysis work. PROFILE uses a method for inverse airfoil design, while XFOIL has both a conformal transformation and an inverse panel method for airfoil design. An intermediate step between Panel Codes and Full Potential codes were codes that used the Transonic Small Disturbance equations.

In particular, the three-dimensional WIBCO code, developed by Charlie Boppe of in the early 1980s has seen heavy use. Developers turned to Full Potential codes, as panel methods could not calculate the non-linear flow present at speeds. The first description of a means of using the Full Potential equations was published by Earll Murman and of Boeing in 1970. Frances Bauer, and of the Courant Institute at (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 as Grumfoil., 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 structured 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 's NASCART-GT). Also developed the three-dimensional AIRPLANE code which made use of unstructured tetrahedral grids. In the two-dimensional realm, Mark Drela and Michael Giles, then graduate students at MIT, developed the ISES Euler program (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 development. Two-dimensional codes, such as NASA Ames' ARC2D code first emerged.

A number of three-dimensional codes were developed (ARC3D,, CFL3D are three successful NASA contributions), leading to numerous commercial packages. Methodology [ ] In all of these approaches the same basic procedure is followed. • During • The and physical bounds of the problem can be defined using (CAD). From there, data can be suitably processed (cleaned-up) and the fluid volume (or fluid domain) is extracted. • The 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 elements. • The physical modeling is defined – for example, the equations of fluid motion + + radiation + species conservation • 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. • The is 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. Discretization methods [ ]. Main article: 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 to ensure that the solution is. [ ] Turbulence models [ ] 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 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. 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 all of the turbulent scales are not modeled, the computational cost is very low, but the tradeoff comes in the form of decreased accuracy.

In addition to the wide range of length and time scales and the associated computational cost, the governing equations of fluid dynamics contain a 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. Reynolds-averaged Navier–Stokes [ ]. Volume rendering of a non-premixed swirl flame as simulated by LES. (LES) is a technique in which the smallest scales of the flow are removed through a filtering operation, and their effect modeled using subgrid scale models. This allows the largest and most important scales of the turbulence to be resolved, while greatly reducing the computational cost incurred by the smallest scales. This method requires greater computational resources than RANS methods, but is far cheaper than DNS.

Detached eddy simulation [ ]. Main article: (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 (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.

Direct numerical simulation [ ]. Main article: The (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. VC is similar to, where conservation laws are satisfied, so that the essential integral quantities are accurately computed. Linear eddy model [ ] 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 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. Two-phase flow [ ].

Simulation of bubble swarm using The modeling of is still under development. Different methods have been proposed, including the, the and. 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.

[ ] Lagrangian multiphase models, which are used for dispersed media, are based on solving the Lagrangian equation of motion for the dispersed phase. [ ] Solution algorithms [ ] Discretization in the space produces a system of 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 or 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 or methods. Krylov methods such as, typically used with, operate by minimizing the residual over successive subspaces generated by the preconditioned operator.

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 reduces all components of the residual by similar factors, leading to a mesh-independent number of iterations. [ ] For indefinite systems, preconditioners such as,, and perform poorly or fail entirely, so the problem structure must be used for effective preconditioning. Methods commonly used in CFD are the and 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. Unsteady aerodynamics [ ] CFD made a major break through in late 70s with the introduction of LTRAN2, a 2-D code to model oscillating airfoils based on 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. Biomedical engineering [ ]. Simulation of blood flow in a human CFD investigations are used to clarify the characteristics of aortic flow in detail that are otherwise invisible to experimental measurements. To analyze these conditions, CAD models of the human vascular system are extracted employing modern imaging techniques. A 3D model is reconstructed from this data and the fluid flow can be computed.

Blood properties like Non-Newtonian behavior 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. See also [ ] • • • • • • • • • • • • • • • • • • • • • • • • References [ ]. Theoretical Aerodynamics. Dover Publications..

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