What is CFD ?
	What are Navier-Stokes Equations?
	What is Computational Fluid Dynamics or CFD Discretization?
	What is Computational Fluid Dynamics or CFD Grid?
	What is a Computational Fluid Dynamics or CFD Solver?

What is CFD?

Computational Fluid Dynamics constitutes a new “third approach” in the philosophical study and development of the whole discipline of fluid dynamics. In the 17th century, the foundations for experimental fluid dynamics were laid. The 18th and 19th centuries saw the gradual development of theoretical fluid dynamics. As a result, throughout most of the 20th century, the study and practice of fluid dynamics (indeed, all of physical science and engineering) involved the use of pure theory on the one hand and pure experiment on the other. If you were learning fluid dynamics as recently as, say, 1960, you would have been operating in the “two-approach world” of theory and experiment. However, the advent of the high speed digital computer combined with the development of accurate numerical algorithms for solving physical problems on these computers has revolutionized the way we study and practice fluid dynamics today. It has introduced a fundamentally important new third approach in fluid dynamics – the approach of Computational Fluid Dynamics.

Computational Fluid Dynamics or simply CFD is concerned with obtaining numerical solutions to the fluid flow problems using computers. The advent of high-speed and large-memory computers has enabled CFD to obtain solutions to many flow problems including those that are compressible or incompressible, laminar or turbulent, chemically reacting or non-reacting.

A variety of reasons can be cited for the increased importance simulation techniques have achieved in the recent years:

	Need to forecast performance
	Cost and/or impossibility of experiments
	The desire for increased insight
	Advances in computer speed and memory (1:10 every 5 years)
	Advances in solution algorithms

To understand CFD lets break down the word CFD:

	Computational - having to do with mathematics, computing
	Fluid Dynamics - the dynamics of things that flow

The dynamics of fluid flow is governed by continuity (conservation of mass), the Navier-Stokes (conservation of momentum), and the energy equations (conservation of energy). These equations form a system of coupled non-linear partial differential equations (PDEs). Because of the non-linear terms the in PDEs, analytical methods can yield very few solutions. In general, closed form of the analytical solutions are possible only if these PDEs can be made linear, either because non-linear terms naturally drop out (e.g., fully developed flow in ducts and flows that are inviscid and irrotational everywhere) or because non-linear terms are small compared to other terms so that they can be neglected (e.g., creeping flows, small amplitude sloshing of liquid etc.). If the non-linearity in the governing PDEs cannot be neglected, which is the situation for most engineering flows, then numerical methods are needed to obtain the solutions.

CFD is an art of replacing the differential equation governing the fluid flow, with the set of algebraic equations (the process is called discretization), which in turn can be solved with the aid of a digital computer to get an approximate solution. The well known discretization methods used in CFD are Finite Difference Method (FDM), Finite Volume Method (FVM), Finite Element Method (FEM), and Boundary Element Method (BEM)

Benefits of CFD

Insight of the design

If you have a device or system design which is difficult to prototype or test through experimentation, CFD analysis enables you to virtually crawl inside your design and see how it performs. There are many phenomena that you can witness through CFD, which wouldn't be visible through any other means. CFD gives you a deeper insight into your designs

Foresight of the design

Because CFD is a tool for predicting what will happen under a given set of circumstances, it can quickly answer many 'what if?' questions. You provide a set of boundary conditions, and the software gives you outcomes. In a short time, you can predict how your design will perform, and test many variations until you arrive at an optimal result. All of this can be done before physical prototyping and testing

Efficiency

The foresight you gain from CFD helps you to design better and faster, save money, meet environmental regulations and ensure industry compliance. CFD analysis leads to shorter design cycles and your products get to market faster. In addition, equipment improvements are built and installed with minimal downtime. CFD is a tool for compressing the design and development cycle allowing for rapid prototyping

Numerical Lab or Virtual Wind Tunnel

CFD results are directly analogous to wind tunnel results obtained in a laboratory – they both represent sets of data for given flow configurations at different Mach numbers, Reynolds numbers, etc. However, unlike a wind tunnel, which is generally a heavy, unwieldy device, a computer program (say in the form of CD) is something you can carry around in your hand. Or a source program in the memory of a given computer can be accessed remotely by people on terminals that can be thousands of miles away from the computer itself. A computer program is, therefore, a readily transportable tool, a “transportable wind tunnel”. Just imagine how many experiments you can do with this wind tunnel and at what cost? A countless number of experiments with negligible cost!!!

Ability to Simulate Real Conditions

Many flow and heat transfer processes can not be (easily) tested. Imagine a hypersonic vehicle entering into earth's atmosphere with Mach 20. Creating such a high speed flow in a wind tunnel is very difficult or rather impossible. CFD provides the ability to theoretically simulate any physical condition

Ability to Simulate Ideal Conditions

Can we study effect of viscosity on flow behavior? For example, what are the differences between laminar and turbulent flow over an airfoil for Re = 100,000? If this experiment is done in a wind tunnel, the flow will be viscous always. But CFD allows a great control on physical process, and provides the ability to isolate specific phenomena. For the above case, it’s just a matter of making one run of CFD with turbulence model switched off (laminar flow) and one run of CFD with turbulence model switched on.

There are numerous advantages of such kind and that’s why CFD tool is now-a-days playing major role in the design process of real life engineering applications.

The CFD Process

There are essentially three stages to every CFD simulation process: preprocessing, solving and postprocessing

Preprocessing

This is the first step in building and analyzing a flow model. It includes building the model within a computer-aided design (CAD) package, creating and applying a suitable computational mesh, and entering the flow boundary conditions and fluid materials properties.

There are a large number of commercial CAD packages for creating complex 3D geometries. CATIA, SolidWorks, IDEAS, ProE are few to name. The important and time consuming step in a CFD process is creating good quality grid around or inside the CAD model. A large number of commercial structured and unstructured grid generation softwares are available. GAMBIT, Tgrid, ICEM-CFD, IGG, AUTOGRID, HEXPRESS, GridZ are a few to mention. Each software has its own positive and negative points. Most of the grid generation softwares have geometrical capability which will allow to create moderately complex geometry or to repair the imported geometry.

Solution

The CFD solver does the flow calculations and produces the results by solving the discritized form of governing equations. This stage needs a clear understanding of the flow physics involved in the problem. There is a broad range of physical models present in many commercial softwares like FLUENT, CFX, FINE, FlowZ, etc. All these models have been validated against industrial scale applications, so you can accurately simulate real-world conditions, including:

• Multiphase flows

• Reacting flows

• Rotating equipment

• Moving and deforming objects

• Turbulence

• Radiation

• Acoustics

• Dynamic meshing

As a solution to all the governing equations, solution step will calculate the flow parameters like velocity, pressure, density, temperature etc. at each grid point.

Postprocessing

The enormous amount of data generated by CFD solver can not be analyzed by just looking at the numerical values. The final step in CFD analysis involves the organization and interpretation of the predicted flow data and the production of CFD images and animations. Different post-processing tools like color plots, contour plots and vector plots are used to go into the problem. The interpretation of these results plays an important role in determining the performance of any device being studied.

See the presentation for “Fundamentals of CFD and its industrial applications”

What are Navier-Stokes Equations?

One of the most important ways in which a fluid differs from a rigid solid body is that its motion is almost always associated with a drastic deformation of its own shape and size and under some conditions, also its overall volume. The rigid body dynamics taught at the high school level is insufficient to describe the motion of fluid. Nevertheless, the foundation that one can turn to is the Newton’s Second Law. When the force balance is applied on fluid elements using different approaches, one arrives at the equation that mathematicians in Western Europe namely, Claude-Louis Marie Henri Navier and later George Gabriel Stokes (though there were more than just these two working on the equations) derived in the 19th century that describes the motion of fluid under the action of any applied external field (pressure, body forces, temperature, etc.). Since a fluid can deform in all the four coordinates of the time-space continuum, the force balance when mathematically formulated would show-up in a partial differential equation (PDE) unlike the ordinary differential equation (ODE) obtained for rigid body dynamics due to the F=ma formulation. It may be noted that ODE refers to differential equations with differentiation with respect to only one variable (e.g., t) unlike a PDE where the differentiation of function (s) is with respect to more than one variable (e.g., x, y, z, t).

Any simulation stems from the solution of some governing principles of a phenomenon. These governing principles have to be mathematically formulated so as to be interpreted and solved using computational techniques. The N-S equation is in this way the mathematical formulation of the governing principle of momentum conservation.

The Navier-Stokes equation (or the N-S equation) is a vector equation and is also called the momentum equation. It is supplemented by the mass conservation equation, also called continuity equation and the energy equation. In the CFD community, usually the term “Navier-Stokes equations” is used to refer to all of these conservation principles.

These equations can be obtained in the reduced form by applying specific conditions like, incompressible flow, Inviscid flow, Steady flow, etc. A unique solution for the variables u, v, w, T, p, etc. to these equations requires boundary conditions and initial conditions (only for unsteady flows). The solution would be able to describe the value of these unknown variables at each and every point in the domain where the equations are being solved.

Till date no solution technique has been able to obtain a complete solution of the N-S equation. Several mathematical techniques like discretization schemes have approached to obtain a “solvable” form of the N-S equations. CFD solves these “solvable” N-S equations for performing flow analysis

Existence and Uniqueness

The existence and uniqueness of classical solutions of the 3-D Navier-Stokes equations is still an open mathematical problem and is one of the Clay Institute's Millenium Problems. In 2-D, existence and uniqueness of regular solutions for all time have been shown by Jean Leray in 1933. He also gave the theory for the existence of weak solutions in the 3-D case while uniqueness is still an open question.

However, recently, Prof. Penny Smith submitted a paper, Immortal Smooth Solution of the Three Space Dimensional Navier-Stokes System, which may provide a proof of the existence and uniqueness. (It has a serious flaw, so the author withdrew the paper).
 

What is Computational Fluid Dynamics or CFD Discretization?

The word discrete is opposed to the concept of continuity. While at the Quantum level, things are explained with concepts like quantas, discrete packets, etc., in classical physics, continuity holds in describing mathematical formulations and also their solutions. Discretization concerns the process of transferring continuous models and equations into discrete counterparts. This process is usually carried out as a first step toward making them suitable for numerical evaluation and implementation on digital computers.

Discretization can be performed on several continuous entities like, images, equations, geometries, features, etc. In the context of CFD, discretization is performed to solve the Navier-Stokes equations whose direct solution would otherwise be extremely difficult and in most cases impossible. The solution process may involve the usage of digital computers.

Discretization in any CAE analysis involves breaking-down of the given geometry where the computation has to be performed. The result is a discrete distribution of smaller elements which are themselves distinct computational domains where any governing equations (N-S, vibration, species-conservation, etc.) can now be solved in an approximate form. The approximate formulation that is derived from the continuous equation (often PDEs) is dependent on the Discretization scheme/technique which is chosen often based on the nature of the equations to be solved.

Some of the Discretization techniques that are commonly used in the CAE industry are Finite Difference Method (FDM), Finite Volume Method (FVM) and Finite Element Method (FEA).

FDM is the oldest of the three and has been used extensively for solving ODEs. FDM was also employed in initial CFD codes but lost its importance as it was unable to capture sharp gradients in the flow domain (PDEs being of the hyperbolic nature!).

FVM is the commonly used discretization technique in most of the commercial CFD codes. Most of the CFD packages nowadays like ANSYS FLUENT, ANSYS CFX, Star-CD, etc., are based on the Finite Volume formulation for discretizing the governing set of N-S equations and solving them. It is conservative in nature and is stable for a wide range of flow problems.

FEM is the commonly used discretization scheme in almost all of the Structural solvers studying vibration, deformation, and thermo-stress analysis. Some CFD codes also employ the Finite Element code but are not very popular for specific fluid flow analysis especially due to their inability to accurately capture turbulent flows. Generally, the multi-physics software packages are based on FEM.
 

What is a Computational Fluid Dynamics or CFD Grid?

A discretization scheme involves the breaking-down of the geometry into smaller parts (elements, volumes, points, etc.). The resulting discretized geometry which is now a group of several smaller computational regions is termed as a grid. The approximate formulations of the governing equations obtained through the different discretization schemes like FDM, FVM, FEM, etc., are solved on the discretized geometry or grid.

As shown in the figure, the grid itself has more entities associated with it, namely, node, edge, face and cell. The definitions differ for two dimensional and three dimensional grids. Grids are generated using pre-processing software packages.

Grids can be of different shapes as shown in the figure below:

Grids can also be of different types based on the strategy of creation. They are classified as:

1. Structured – Single block
2. Structured – Multi block
3. Unstructured
 

CFD solutions are best obtained on structured grids than unstructured grids. The structured grids allow the connectivity information for the entire mesh to be accessed by three index variables: i, j and k. Due to this constraint, if structured meshes are created using only a single-block, the meshes may include 180 degree corners (candidate for negative element!). Multi-block structured grids on the other hand can give the accuracy of structured grid while preventing such negative elements.

All triangle or tetrahedral elements are unstructured grids. Such elements should be avoided at least in those regions where a huge gradient of variables (velocity, pressure, temperature, etc.) is expected. At the walls, as much as possible, there should be at least some layers of quadrilateral (2D) or prism/hexahedral (3D).

The solution on unstructured grids is limited to only some kinds of solvers and even if the solver can solve on unstructured grids, the solution is often not as accurate as in the case if a structured grid is used. Another great advantage of having structured grids is the huge reduction in the number of grid elements as opposed to the unstructured grids. Structured grids however, cannot be applied always over very complicated geometries.

The Finite Difference Method solves the partial differential equations at the nodes of the grids. Therefore, FDM requires the grids to be structured (having i, j, k information access).

The Finite Volume Method can solve the partial differential equations on structured as well as unstructured grids since, the variables are solved for at the cell-centers, independent of the number of edges/faces the cell has.

The Finite Element Method can also solve the governing equations on any kind of grids as the formulation is in terms of a weak function definition at the nodes.
 

What is a Computational Fluid Dynamics or CFD Solver?

The CFD process involves discretization of the geometry, solving the discretized governing equations on the discretized geometry and finally viewing the results obtained at the discrete points (cell centers, nodes, etc). The solution at the discrete points for the governing equations is obtained by the solver. Therefore, it is the solver that has the governing equations transformed into the discretized form using one of the discretization techniques (FDM, FVM or FEM) and performing the solution of these equations.

Depending on which discretization technique is employed in the solver, the PDEs stating the governing principles in mathematical form are converted into algebraic equations using the initial and boundary conditions. These algebraic equations are obtained at each of the points (cell-centers, nodes, etc) where the solution for the variables (u, v, w, T, etc) is to be obtained. The final algebraic equation set results in a matrix which has to be solved for obtaining values of the variables at each of those points. Due to the non-linearity of the governing equations which are converted into algebraic equations through some approximation techniques, this procedure of obtaining the algebraic equations and solving the resulting matrix is an iterative procedure. The bound on the number of iterations is enforced through convergence criteria (defined by the user). This whole process involves computing at a large number of points (depending on the size of the grid) for each of the governing equations. The task becomes especially daunting when the individual equations are closely coupled to each other, due to which they cannot be solved separately, but have to be considered together. Such problems on complicated geometries with huge cell count call for high capacity RAM (Random Access Memory) from the computer. Besides, solving these equations iteratively to the required convergence criteria demands high speed RAM. It is a widely accepted fact that the requirements of the simulation industry have been accelerating the growth in the computer hardware industry. It is because of these reasons that nowadays, CFD simulation is possible on Desktop computers. With more physics models being incorporated into CFD, there is still a huge demand for high-speed computational capabilities. The advancement in fast numerical techniques may compensate for the lag that the computer hardware industry has in catering to the CFD requirements.

The Solver offers the user to define:

1. Different mechanisms and models (e.g., turbulence models, moving/deforming mesh)

2. Incorporate different physics (e.g., phase change, real-gas formulations)

3. Adjust the numerical parameters (e.g., face-approximation techniques, under-relaxation factors)

4. Convergence criteria (e.g., residuals accepted, lift and drag coefficients monitors) and

5. Time-step for calculations (not valid for steady-state analysis).

The ability to choose the appropriate models for capturing the actual physics and to fiddle with the numerical parameters is an art that one learns through experience in CFD. Different Solver packages offer customized solutions for specific industrial problems.