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What is Computational Fluid
Dynamics or 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:
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Need to
forecast performance |
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Cost and/or
impossibility of experiments |
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The desire
for increased insight |
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Advances in
computer speed and memory (1:10 every 5 years) |
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Advances in
solution algorithms |
To
understand CFD lets break down the word CFD:
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Computational - having to do with mathematics, computing |
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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
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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
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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
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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
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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!!!
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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
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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
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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.
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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:
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Multiphase flows |
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Reacting
flows |
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Rotating
equipment |
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Moving
and deforming objects |
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Turbulence |
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Radiation |
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Acoustics |
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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.
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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:
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Different
mechanisms and models (e.g., turbulence models,
moving/deforming mesh) |
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Incorporate
different physics (e.g., phase change, real-gas
formulations) |
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Adjust the
numerical parameters (e.g., face-approximation techniques,
under-relaxation factors) |
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Convergence
criteria (e.g., residuals accepted, lift and drag
coefficients monitors) and |
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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. |
