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Chapter 3
INTRODUCTION TO CFD
3.1 INTRODUCTION TO COMPUTATIONAL FLUID DYNAMICS
What is Computational Fluid Dynamics?
Computational Fluid Dynamics (CFD) is a computer-based tool for simulating the behavior of
systems involving fluid flow, heat transfer, and other related physical processes. It works by
solving the equations of fluid flow (in a special form) over a region of interest, with specified
(known) conditions on the boundary of that region.
Essentially there are three methods for determine the solution to flow problems viz.
Experimental, Analytical and Numerical.
The Analytical methods aim at getting a closed form solution in the entire domain assuming the
process to follow continuum hypothesis. These are generally restricted to simple geometry,
simple physics and generally linear problems. Once the problem becomes complex, the various
assumptions that are needed to be made to obtain a closed form solution, entails loss of accuracy
of the critical parameter of interest. This leads them to be used as a check on the accuracy of a
numerical procedure but makes them mainly unsuitable for the analysis of real engineering
problems but makes them mainly unsuitable for engineering analysis.
However, they give the direction and general nature of the solution. Hence over the years,
scientists and engineers have resorted to experimental techniques concentrating in the regions of
interest. These experimental techniques have their inherent problems viz. that they are equipment
oriented, and they need large resources of hardware, time and operating costs. Their applications
are also limited due to scaling considerations. Further theses involve certain measurement
difficulties and handling of large quantity of data.
Numerical methods have emerged as a third method and have overcome the restrictions in both
experimental and analytical methods. They involve the discretization of the governing
mathematical equations in a way such that the numerical solutions can be obtained. This
approach forms the core of Computational Fluid Dynamics, commonly known as CFD. The
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popularity of CFD has been possible due to great developments in computing algorithms that
have enabled fast Graphic User Interface that makes the interpretation and Visualization of the
results easier.
CFD methods have their own disadvantage in terms of specifications of proper
boundary conditions, truncation errors, convergence problems, right choice of turbulence models
and parameters, right choice of discretization method etc. This applications of CFD to practice
problems need understanding of basic theory to overcome the above mentioned problems.
3.2 COMPARISON OF APPROACHES
The below table 3.1 shows the advantages & disadvantages between different approaches
Table 3.1 Comparison of Approaches
Approach Advantages Disadvantages
Experimental
1. Capable of being most
Realistic
1.Equipment required
2. Scaling problems3.Tunnel corrections
4.Measurement difficulties
5. Operating costs
Theoretical
1.Clean, general
information, which is
usually in formula form.
1. Restricted to simple geometry
and physics.
2. Restricted to linear
Problems
Computational
1. No restriction to linearity
2. Complicated physics can
be treated
3. Time evolution of flow
can be obtained
1. Truncation errors.
2. Boundary condition
problems
3. Computer costs
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3.3 COMPARISION OF COMPUTATIONAL AND EXPERIMENTAL METHODS
Comparison between computational and experimental methods is shown in Table 3.2
Table 3.2 Comparison between Computational & Experimental Methods
Area Computational methods Experimental methods
Capability
1. Software used for all flowtypes.
2. Turbulence rarely
resolved except through useof simpler models.
3. Enable physical situations
to be modeled whereexperiments would be unsafe.
4. Allows geometry
variationto be achieved quickly.
1. Exact simulation if full-scale situation can be used.
2. Experimental situation
also being a model of desired flow situation.
Accuracy1. Depends on algorithms
used.
2. Depends on mesh density.
1. Should be correct withinthe limits of experimental
errors, if geometry and scale
effects are realistic andequipment is appropriately
designed and calibrated.
Detail
1. All variables are
calculated at every mesh pointor cell.2. Variables can be
integrated to find overall
properties.
1. Easy to find overall
properties such as pressure
drop, forces and moments2. Difficulty andexpensive to instrument so
that anything more than a
crude sample of the data isproduced.
Time
1. Solutions can take longtime to iterate. This depends
on the problems being solved
and the speed of computer
being used.
1. Time needed for setupand calibration. Results are
usually quick to gather once
this is done.
Cost
1. Requires relatively cheaphardware but expensivesoftware.
2. Time and care is needed
to get good results.3. Specialists are required to
achieve good results.
1. Instrumentation isexpensive in many cases.
2. Raw experiment is
cheap to carry out but dataachieved is limited.
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3.4 THE HISTORY OF CFD
Computers have been used to solve fluid flow problems for many years. Numerous programs
have been written to solve either specific problems, or specific classes of problem. From the
mid-1970s the complex mathematics required to generalize the algorithms began to beunderstood, and general-purpose CFD solvers were developed. These began to appear in the
early 1980s and required what were then very powerful computers, as well as an in-depth
knowledge of fluid dynamics, and large amounts of time to set up simulations. Consequently
CFD was a tool used almost exclusively in research.
Recent advances in computing power, together with powerful graphics and interactive 3-D
manipulation of models mean that the process of creating a CFD model and analyzing the results
is much less labour-intensive, reducing the time and therefore the cost. Advanced solvers contain
algorithms, which enable robust solution of the flow field in a reasonable time.
As a result of these factors, Computational Fluid Dynamics is now an established industrial
design tool, helping to reduce design timescales and improve processes throughout the
engineering world. CFD provides a cost-effective and accurate alternative to scale model testing,
with variations on the simulation being performed quickly, offering obvious advantages.
3.5 THE MATHEMATICS OF CFDThe set of equations which describe the processes of momentum, heat and mass transfer are
known as the Navier-Stokes equations. These are partial differential equations which were
derived in the early nineteenth century. They have no known general analytical solution but can
be discretised and solved numerically.
Equations describing other processes, such as combustion, can also be solved in conjunction with
the Navier-Stokes equations. Often, an approximating model is used to derive these additional
equations, turbulence models being a particularly important example.
There are a number of different solution methods which are used in CFD codes. The most
common, and the one on which CFX is based, is known as the finite volume technique. In this
technique, the region of interest is divided into small sub-regions, called control volumes. The
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equations are discretised and solved iteratively for each control volume. As a result, an
approximation of the value of each variable at specific points throughout the domain can be
obtained. In this way, one derives a full picture of the behavior of the flow
3.6 CFD METHODOLOGY
CFD may be used to determine the performance of a component at the design stage or it can be
used to analyse difficulties with an existing component and lead to its improved design. For
example, the pressure drop through a component may be considered excessive.
3.7 THE DESIGN OPTIMIZATION PROBLEM
CFD Solver
Figure 3.1. The design optimization flowchart.
At present, in order to shorten product development time, there is a strong tendency to perform
design using computational fluid dynamics (CFD) tools instead of experiments. CFD is a method
that is becoming more and more popular in the modeling of flow systems in many fields,
including reaction Engineering. The block diagram in Fig no: 3.1 explains the optimization
problemit is recognized that experiments remain essential during the final design stages. CFD
based modeling however has many advantages during preliminary design, because it is less time-
consuming than experiments and because it allows greater flexibility.
Early experience with CFD based modeling has shown that these computational tools should be
used carefully. Any kind of CFD computation requires the specification of inlet and boundary
Response
Parameter
Constraint
Design
Parameter
Constraint
Objective
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conditions. Obviously these conditions determine the flow and temperature field resulting from
the CFD computation. The specification of inlet and boundary conditions requires appropriate
measurements or available data onsite.
3.8 GOVERNING EQUATIONS OF CFD
CFD is playing a strong role as a design tool as well as a research tool. In CFD the physical
aspects of any fluid flow is governed by three principles.
The fundamental equations of fluid mechanics are based on the following
Universal laws of conservation:
1. Conservation of mass
2. Conservation of momentum
3. Conservation of energy
These fundamental physical principles can be expressed in terms of basic mathematical
equations. These equations are generally in integral or partial differential form. These equations
and their derivatives are replaced in CFD by discretised algebraic forms, which are in turn solvedto get flow field values at discrete points in space and/or time. The end product is a collection of
numbers, in contrast to closed-form analytical solution. In CFD approach, the equations that
govern a process of interest are solved numerically. Numerical methods have evolved especially
FDM, FVM algorithms for solving ordinary and partial differential equations.
The equation that results from applying the conservation of mass to a fluid is called the
continuity equation. Conservation of momentum is based on application of Newton's Second
Law to a fluid element, which yields a vector equation, which is also called Navier-Stokes
Equation. The conservation of Energy is based on the application of First Law of
Thermodynamics to a fluid element.
In addition to the equations developed from these universal laws, it is necessary to establish
relationships between fluid properties in order to close the system of equations. An example of
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such a relationship is the equation of state, which relates the thermodynamic variables pressure p,
density , and Temperature T. Historically there have been two different approaches taken to
derive the equations of fluid mechanics viz., phenomenological approach and kinetic theory
approach. In the phenomenological approach certain relationship between stress and rate of
strain, heat flux and temperature gradient are postulated, and the fluid dynamic equations are
then developed from the conservation laws. The required constants of proportionality between
stress and rate of strain and heat flux and Temperature gradient (which are called Transport
Coefficients) must be determined experimentally. In the kinetic theory approach also known as
the mathematical theory of non-uniform gases, the fluid dynamic equations are obtained with the
transport coefficients defined in terms of certain integral relations, which involves dynamics of
colliding particles.
A viscous flow is one where transport phenomenon of friction, thermal conduction and/ or mass
diffusion is included. These transport phenomena are dissipative. So they always increase the
entropy of the flow. For this type of viscous flow modeling the Navier-stokes equations are
applied. If these phenomena are neglected, the flow is called inviscid flow and for this Euler
equations are applied.
These are mathematical statements of three fundamental physical principles upon which fluid
dynamics is based shown as flow chart in fig 3.2.
Fundamental physical
Mass is conserved
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Figure 3.2 Block diagram of physical and mathematical basis
3.9 CONTINUITY EQUATION
The basic continuity equation of fluid flow is as follows:
Newtons second
Energy is
Models
of flow
Fixed finite
control
Moving
finite
Fixed
infinitesimall
y small
Moving
infinitesimally
Governing
equations of
fluid flowContinuit
y
equationMoment
um
equation
Energyequation
Forms of these equations
particularly suited forCFD
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Net flow out of control volume = time rate of decrease of mass inside control volume
The continuity equation in partial differential equation form is given by,
/t +. (V)=0 .3.1
= Fluid density
/t = the rate of increase of density in the control volume.
. (V)=the rate of mass flux passing out of control volume.
The first term in this equation represents the rate of increase of density in the control volume and
the second term represents the rate of mass flux passing out of the control surface, which
surrounds the control volume. This equation is based on Eulerian approach. In this approach, afixed control volume is defined and the changes in the fluid are recorded as the fluid passes
through the control volume. In the alternative Lagrangian approach, an observer moving with the
fluid element records the changes in the properties of the fluid element. Eulerian approach is
more commonly used in fluid mechanics. For a Cartesian coordinate system, where u, v, w
represent the x, y, z components of the velocity vector, the continuity equation becomes
( ) ( ) ( ) 0=
+
+
+
w
zv
yu
xt
.3.2
A flow in which the density of fluid assumed to remain constant is called Incompressible flow.
For Incompressible flow, =Constant.
The continuity equation reduces to
0=
+
+
z
w
y
v
x
u.3.3
3. 10 MOMENTUM EQUATION
Newton's Second Law applied to a fluid passing through an infinitesimal, fixed control volume
yields the following momentum equation:
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.3.4
Where,
/t() represents rate of increase of momentum per unit volume.
V represents the rate of momentum lost by convection through
the control volume surface.
f represents the body force per unit volume.
.ij represents the surface force per unit volume and
ij stress tensor.
While solving the equation, the fluid is considered as Newtonian fluid, i.e., stress is directly
proportional to the rate of strain. If the flow is considered as incompressible and the coefficient
of viscosity is assumed constant the equation becomes,
VpfDt
DV 2+=
.3.5
This equation is good approximation for incompressible flow of a gas.
3.11 ENERGY EQUATION
( ) ijfVVvt
.. +=+
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The first law of Thermodynamics applied to a fluid passing through an infinitesimal fixed control
volume yields the energy equation i.e. increase in energy in the system is equal to the heat added
to the system plus the work done on the system.
= +
)..(... VVfqt
QVE
t
Eijt
t
++== .3.6
Where,
Represents the rate of increase of Et in the control volume
Represents the rate of the total energy lost by convection (per unit volume)
through the control surface
Represents the rate of heat produced by external agencies
Represents the rate of heat lost by conduction per unit volume through the control
surface.
Represents the work done on the control volume by the body forces
Represents the work done on the control volume by the surface forces.
In terms of enthalpy, the final form of Energy equation is
++= q
t
Q
Dt
DP
Dt
Dh. .3.7
Where, is known as dissipation function and represents the rate at which mechanical energy is
expended in the process of deformation of the fluid due to viscosity.
Rate of change of energy
inside the fluid elementNet flux of heat in to
element
Rate of work done on
element due to body
and surface forces
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3.12THEORY
Solutions in CFD are obtained by numerically solving a number of balances over a large number
of control volumes or elements. The numerical solution is obtained by supplying boundary
conditions to the model boundaries and iteration of an initially guessed solution.
The balances, dealing with fluid flow, are based on the Navier Stokes Equations for conservation
of mass (continuity) and momentum. These equations are modified per case to solve a specific
problem.
The control volumes (or) elements, the mesh are designed to fill a large scale geometry,
described in a CAD file. The density of these elements in the overall geometry is determined by
the user and affects the final solution. Too coarse a mesh will result in an over simplified flowprofile, possibly obscuring essential flow characteristics. Too fine meshes will unnecessarily
increasing iteration time.
After boundary conditions are set on the large scale geometry the CFD code will iterate the entire
mesh using the balances and the boundary conditions to find a converging numerical solution for
the specific case.
3.12.1 FLUID FLOW FUNDAMENTALS
The Physical aspects of any fluid flow are governed by three fundamental principles: Mass is
conserved; momentum and Energy is conserved. These fundamental principles can be expressed
in terms of mathematical equations, which in their most general form are usually non-linear
partial differential equations. Computational Fluid Dynamics (CFD) is the science of
determining a numerical solution to the governing equations of fluid flow whilst advancing the
solution through space or time to obtain a numerical description of the complete flow field of
interest.
The governing equations for Newtonian fluid dynamics, the unsteady Navier-Stokes equations,
have been known for over a century. However, the analytical investigation of reduced forms of
these equations is still an active area of research as is the problem of turbulent closure for the
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Reynolds averaged form of the equations. For non-Newtonian fluid dynamics, chemically
reacting flows and multiphase flows theoretical developments are at a less advanced stage.
Experimental fluid dynamics has played an important role in validating and delineating the limits
of the various approximations to the governing equations. The wind tunnel, for example, as a
piece of experimental equipment, provides an effective means of simulating real flows.
Traditionally this has provided a cost effective alternative to full scale measurement. However,
in the design of equipment that depends critically on the flow behavior, for example the
aerodynamic design of an aircraft, full scale measurement as part of the design process is
economically impractical. This situation has led to an increasing interest in the development of a
numerical wind tunnel.
3.13 THE STRATEGY OF CFD
The strategy of CFD is to replace the continuous domain with a discrete domain using a grid. In
the continuous domain, each flow variable is defined at every point in the domain. In the discrete
domain, each flow variable is defined only at grid points. So in the discrete domain, the variable
would be defined only at N grid points.
Continuous Domain Discrete Domain
0 x 1 x = x1, x2Xn
x = 0 x = 1 x1 xi xN
Grid point
Coupled PDEs + boundary Coupled algebraic equations in
Conditions in continuous variables. Discrete variables
In a CFD solution, one would directly solve for the relevant flow variables only at grid points.
The values at other locations are determined by interpolating the values at the grid points. In the
governing equations define the variables in the discrete form. The discrete system is a large set
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of coupling algebraic equations in the discrete variables. Setting up the discrete system and
solving it involves a very large number of repetitive calculations. This idea can be applied to any
general problem.
3.14 TURBULENT FLOW
Turbulent fluid motion is an irregular condition of flow in which the various quantities
show a random variation with time and space coordinates so statistically distinct average values
can be discerned.
The differences between laminar and turbulence flow: higher values of friction drag and
pressure drop are associated with turbulent flow. The diffusion rate of a scalar quantity is usually
greater in a turbulent flow rather than laminar flow, and turbulent flows are usually noiser. A
turbulent boundary layer can normally negotiate a more extensive region of unfavourable
pressure gradient prior to separation than can a boundary layer. The unsteady Navier-Stokes
equations are generally considered to govern turbulent flows in the continuum regime.
3.15 MODELLING TURBULENT FLOWS
The method to solve turbulent flows by direct numerical simulation (DNS) requires that all
relevant length scales be resolved from the smallest eddies to scales on the order of the physical
domain of the problem domain. The computation needs to be 3-D even if the time-mean aspects
of the flow are 2-D, and the time steps must be small enough that the small-scale motion can be
resolved in a time accurate manner even if the flow is steady in a time-mean sense.
Another approach is large-eddy simulation (LES), in which large-scale structure of the turbulent
flow is computed directly and only the effects of smallest and more nearly isotropic eddies are
modeled. The grid models required for LES is an order of magnitude less than DNS and forpractical engineering problems, even LES is beyond present day computing power. The
computational effort required for LES is less than DNS. Now these are replaced by approximated
modeling methods used as the primary design procedure for engineering applications.
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The main thrust of present day CFD for turbulent flow is through the Time/mass (Favre)
averaged Navier-Stokes equations In computational fluid mechanics and heat transfer in
turbulent flow is through the time averaged Navier-stokes equations. These equations are also
referred as Reynolds-averaged Navier-Stokes equations (RANS). The Reynolds equations are
derived by decomposing the dependent variables in the conservation equations into time-mean
and fluctuating components and then time averaging the entire equation. Two types of averaging
is presently used, the classical Reynolds averaging and the mass-weighted averaging suggested
by Favre. Time averaging the equations of motion gives rise to new terms, which can be
interpreted as "apparent" stress gradients and heat flux quantities associated with the turbulent
motion. These new quantities must be related to the mean flow variables through turbulence
models. This process introduces further assumptions and approximations. Thus this method on
the turbulent flow problem through solving the Reynolds equations of motion does not follow
entirely from first principles, since additional assumptions must be made to close the system of
equations. For flows in which density fluctuations can be neglected, the two formulations
become identical.
3.16 THE GENERAL DIFFERENTIAL EQUATION
A generalized conservation principle is obeyed by all the independent variables of interest, so the
basic balance or conservation equation is
(Outflow from cell) (inflow into the cell) = (net source within the cell.)
The quantities being balanced are the dependent variables like mass of a phase, mass of a
chemical species, energy, momentum, turbulence quantities, electric charge etc.
The terms appearing in the balance equation are convection, diffusion, time variation and sourceterms.
If the dependent variable is denoted by, the general differential equation or the general
purpose CFD equation is given as
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.3.8
Where,
p, u, v, w, h, k, =Dependent variable, ()
t, x, y, z =independent variable
=Exchange coefficient
=Scalars
S =source terms
= Boundary conditions sources
Div =divergence (V. J)
Grad =gradient (V)
3.17 REYNOLDS AVERAGED NAVIER-STOKES EOUATION
In the conventional averaging procedure, following Reynolds, we define a time averaged
quantity f as
.3.9
We require that t be large compared to the period of the random fluctuations associated with the
turbulence, but small with respect to the time constant for any slow variations in the flow field
associated ordinary unsteady flows.
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In the conventional Reynolds decomposition, the randomly changing flow variables are replaced
by time averages plus fluctuations about the average.
For Cartesian coordinate system, we may write
Fluctuations in other fluid properties such as viscosity, thermal conductivity, and specific
heat are usually small and will be neglected here.
By definition, the average of a fluctuating quantity is zero
.3.10
It should be clear from these definitions that for symbolic flow variable f and g, the following
relations hold:
.3.11
It should also be clear that, where f' = 0, the time average of the product of two fluctuating
quantities is, in general, not equal to zero, i.e., f, f' 0 .In fact, the root mean square of the
velocity fluctuations is known as the turbulence intensity.
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For treatment of compressible flows and mixtures of gases in particular, mass-weighed averaging
is convenient. In this approach we define mass-averaged variables according to f.
.3.12
We note that only the velocity components and thermal variables are mass averaged. Fluid
properties such as density and pressure are treated as before. To substitute into conservation
equations, we define new fluctuating quantities by
.3.13
It is very important to note that the time averages of the doubly primed fluctuating quantities (u,
v, etc.) are not equal to zero, in general, unless p'=0. In fact, it can be shown that,
.3.14
Instead, the time average of the doubly primed fluctuation multiplied by the density is equal to
zero.
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3.18 REYNOLDS FORM OF CONTINUITY EQUATION
Reynolds form of the average equations of continuity for incompressible flow is as follows.
.3.15
For compressible flow the continuity equation becomes
.3.16
Reynolds form of the momentum equation for incompressible flow is
.3.17
For compressible flows the momentum equation becomes
.3.18
If we compare the original N-S equations with dependent variables based on instantaneous
velocities with Reynolds average N-S (RANS) equations with dependent variable based on time
averaged/mass averaged velocities we find an additional term namely,
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These apparent stress gradients due to transport of momentum by turbulent fluctuations are
called Reynolds stresses. Similar correlation functions for turbulent heat flux will correspond to
the averaged energy equation.
These Reynolds stresses and other correlation functions need to be modeled for closure of the
RANS. Modeling these is the subject of turbulence.
3.19 K-EPSILON MODEL
Boussinesq suggested that the apparent turbulent shearing stresses might be related to the rate of
mean strain through an apparent scalar turbulent or "eddy" viscosity. For the general Reynolds
stress tensor the Boussinesq assumption gives
.3.19
Where the turbulent viscosity, k is is the kinetic energy of turbulence given by,
.3.20
By analogy with kinetic theory, by which molecular (laminar) viscosity for gases be evaluated
with reasonable accuracy, we might expect that the turbulent viscosity can be modeled as
.3.21
Where V and l are characteristic velocity and length scale of turbulence respectively. The
problem is to find suitable means of evaluating them.
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Algebraic turbulence models invariably utilize Boussinesq assumption. One of the most
successful of this type of model was suggested by Prandtl and is known as "mixing length
hypothesis".
.3.22
Where a mixing length can be thought of as a transverse distance over which particles maintain
their original momentum, somewhat on the order of a mean free path for the collision or mixing
of globules of fluid. The product can be interpreted as the characteristic
velocity of turbulence, V. In the above equation, u is the component of velocity in the primary
flow direction, and y is the coordinate transverse to the primary flow direction.
There are other models, which use one partial differential equation for the transport of turbulent
kinetic energy (TKE) from which velocity scales are obtained. The length scale is prescribed by
an algebraic formulation.
The most common turbulence model generally used is the two-equation turbulence model or k-
model. There are so many variants of this model. In these models the length scale is also
obtained from solving a partial differential equation.
The most commonly used variable for obtaining the length scale is dissipation rate of turbulent
kinetic energy denoted by E. Generally the turbulent kinetic energy is expressed as turbulent
intensity as defined below.
3.23
K= (Actual K.E in Flow) (mean K.E in Flow)
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3.24, 3.25
The transport PDE used in standard k- model is as follows
.3.26
Thus for any turbulent flow problem, we have to solve in addition to continuity, momentum and
energy equations, two equations for transport of TKE and its dissipation rate.
.3.27
3.20 TURBULENCE MODELING
Special attention needs to be paid to accurate modeling of turbulence. The presence of turbulent
fluctuations, which are functions of time and position, contribute a mean momentum flux or
Reynolds stress for which analytical solutions are nonexistent. These Reynolds stresses govern
the transport of momentum due to turbulence and are described by additional terms in the
Reynolds-averaged Navier-Stokes equations. The purpose of a turbulence model is to provide
numerical values for the Reynolds stresses at each point in the flow. The objective is to represent
the Reynolds stresses as realistically as possible, while maintaining a low level of complexity.
The turbulence model chosen should be best suited to the particular flow problem. A wide range
of models is available, and type of model that is chosen must be done so with care. It is
understood that these models are not used when modeling laminar flows.
The final result of the flow, turbulence, reaction, heat transfer, and multiphase calculations will
be a detailed map of the local liquid velocities, temperatures, chemical reactant concentrations,
reaction rates, and volume fractions of the various phases. These outcomes can be analyzed in
detail using graphical visualization, calculation of overall parameters and integral volume or
surface averages, and comparison with experimental or plant data. This analysis phase is referred
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to as post processing. Because of improvements in computer power and enhanced graphics
software, it is now much easier for CFD analysts to create animations of their data. These often
help in understanding complex flow phenomena that are sometimes difficult to see from static
plots.
3.21 DISCRETIZATION OF GOVERNING EOUATIONS
The above governing partial differential equations are continuous functions of x, y, z. In the
finite difference approach, the continuous problem domain "discretised", so that the dependent
variables are considered to exist only at discrete points. Derivatives are approximated by
differences, resulting in algebraic representation of the PDE. Thus a problem involving
differential calculus has been transformed into algebraic problem.
The nature of the resulting algebraic system depends upon the character of the problem posed the
original PDE. Equilibrium problems usually result in a system of algebraic equations that must
be solved simultaneously throughout the domain in conjunction with specified boundary values.
These are mathematically known as elliptic problems. Marchingproblems result in algebraic
equations that usually solved one at a time. These are known as parabolic or hyperbolic
problems.
Three methods are generally used for discretization,
1. Finite difference method.
2. Finite control volume method.
3. Finite element method.
The discretization (numerical simulation) techniques used in CFD are shown in Fig 3.3
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Figure 3.3. Block Diagram of Numerical Solution Techniques in CFD
Discretization
Finite
differenc
Finite
volume
Finite
element
Basic derivations
of finite
Basic derivations
of finite-volume
Finite-difference
equations:
Types of solutions:
explicit and
Stability
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3.21.1DISCRETIZATION
To solve the non-linear partial differential equations from the previous section, it is necessary to
impose a grid on the flow domain of interest, see Fig. 3.4. Discrete values of fluid velocities,
properties, pressure and temperature, are stored at each grid point (the intersection of two grid
lines). To obtain a matrix of algebraic equations, a control volume is constructed (shaded area in
the figure) whose boundaries (shown by dashed lines) lie midway between grid points P and its
neighbors N, S, E, W. A complex process of formal integration of the differential equations over
the control volume, followed by interpolation schemes to determine flow quantities at the control
volume boundaries (n, s, e, w) in Fig. 3.4, finally yield a set of algebraic equations for each grid
point P:
(AP B) p - AC c = C 3.28
where the subscript c on , A and refers to a summation over neighbor nodes N, S, E and W,
is a general symbol for the quantity being solved for (u, v or t), AP, etc. are the combined
convection-diffusion coefficients (obtained from integration and interpolation), and B and C are,
respectively, the implicit and explicit source terms (and generally represent the force(s) which
drive the flow, e.g. a pressure difference).
N
W
S
E
Pw e
s
n
x y
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Fig 3.4 Control Volume on Grid Point
3.21.2 DISCRETIZATION USING FINITE-VOLUME METHOD
In the finite-volume method, quadrilateral/triangle is commonly referred to as a cell and a grid
point as a node. In 2D, one could also have triangular cells. In 3D, cells are usually hexahedral,
tetrahedral, or prisms. In the finite-volume approach, the integral form of the conservation
equations are applied to the control volume defined by a cell to get the discrete equations for the
cell. For example, the integral form of the continuity equation for steady, incompressible flow is
S V .
n dS = 0 3.29
The integration is over the surface S of the control volume and
n is the outward normal at the
surface. Physically, this equation means that the net volume flow into the control volume is zero.
Consider the rectangular cell shown below in fig 3.5
face 4 (u4,v4)
face 1 face 3
y (u1, v1) (u3, v3)
face 2 (u2,v2)
Y x
X
Fig 3.5 Rectangular Cell
Cell center
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The velocity at face i is taken to beiV
= ui
i + vi
j . Applying the mass conservation equation
(3.29) to the control volume defined by the cell gives
-u1 y - v2 x + u3 y +v4 x = 0 3.30
This is the discrete form of the continuity equation for the cell. It is equivalent to summing up
the net mass flow into the control volume and setting it to zero. So it ensures that the net mass
flow into the cell is zero i.e. that mass is conserved for the cell. Usually the values at the cell
centers are stored. The face values u1, v2, etc. are obtained by suitably interpolating the cell-
center values for adjacent cells.
Similarly, one can obtain discrete equations for the conservation of momentum and energy for
the cell. One can readily extend these ideas to any general cell shape in 2D or 3D and any
conservation equation.
3.21.3 SALIENT FEATURES OF FINITE VOLUME METHOD
1. Integral forms of governing equations are discretised in space.
2. Can be used on arbitrary mesh.
3. Definition of control volume arbitrary.
4. Basic qualities such as mass, momentum etc. are conserved at discrete level.
5. Flexible and fundamentally conservative for complicated geometry.
6. Conservative discretization.
=+
QdSdFUd
tS
. 3.31
- Control volume
S- Surface enveloping
U- Conserved scalar
F- Diffusive and convective flux
Q- Volumetric source of U
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3.22 BENEFITS OF CARRYING OUT CFD ANALYSIS
Low cost: The most important advantage of computational prediction is its low cost. In most
applications, the cost of a computer run is many orders of magnitude lower than the cost of a
corresponding experimentation investigation. This can reduce or even eliminate the need for
expensive or large-scale physical test facilities. This factor assumes increasing importance as the
physical situation to be studied becomes larger and more complicated. Further whereas the prices
of most items are increasing, computing cost is likely to be even lower in the future.
0 Speed: A computational investigation can be performed with remarkable speed. A
designer can study the implication of hundreds of different configurations in less than a day and
choose the optimum design. With the ability to reuse information generated in other stages of the
design, rapid evaluation of design alternatives can be made. On the other hand, a corresponding
experimental investigation would take a long time.
Complete information: A computer solution of problem gives detailed and complete information
.It can provide the values of all relevant variables (such as velocity, pressure, temperature,
concentration, turbulence intensity) throughout the domain of interest. This provides a better
understanding of the flow phenomenon and the product performance because knowledge of such
values is not restricted to those areas that can be instruments during testing. For this reason, even
when an experiment is performed, there is great value in obtaining a companion computer
solution to supplement the experimental information.
Ability to simulate realistic conditions: In a theoretical calculation, realistic conditions can be
easily simulated. There is no need to resort to small scale or cold models. Through a computer
program, there is little difficulty in having very large or very small dimensions, in treating very
low or very high temperature, in handling toxic or flammable substances, or in following very
fast or very slow processes.
Ability to simulate ideal conditions: A prediction method is sometimes used to study a basic
phenomenon, rather than a complex engineering application. In the study of phenomenon, one
wants to focus attention on a few essential parameters and eliminates all irrelevant features. Thus
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many idealizations are desirable for example, two dimensionality, constant density, an adiabatic
surface, or infinite reaction rate. In a computation, such conditions can be easily and exactly
setup, whereas even careful experimental can barely approximate the idealization.
Reduction of failure risks: CFD can also be used to investigate configurations that may be too
large to test or which pose a significant safety risk, including pollutant spread nuclear accident
scenarios. This can often provide confidence in operation, reduce or eliminate the cost of
problem solving during installations, reduce product liability risks.
3.23 APPLICATIONS OF CFD
The major applications of CFD are in the following fields of engineering to simulate
Various parameters.
CFD has become a powerful influence on the way fluid dynamicists and aero dynamicists.
Aerodynamic design of transportation vehicles likes cars, aircraft, etc.
Fluid flow pattern and conditions in common engineering equipment like, Heat
Exchangers, Stirred reactors, Ducts, Pulverizes, Boilers, Turbo machinery viz. Steam, Gas and
Hydro turbines.
Fluid flow in electric equipment like computers, control panels etc.
Heat transfer equipment including reactions and radiative modeling like burners, NOx
estimation. Cooling of Generators, motors, Transformers etc.
Metropolitan authorities can determine where pollutant-emitting industrial plant may be
safely located, and under what conditions motor vehicle access must be restricted so as to
preserve air quality.
Meteorologists and oceanographers to foretell wind and water currents. Hydrologists and
others concerned with ground water to forecast the effects of changes to ground-surface cover, of
the creation of dams and aqueducts on the quantity and quality of water supplies.
Petroleum engineers to design optimum oil-recovery strategies and the equipment for
putting them into practice.
Automobile and engine applications: To improve performance means environmental
quality, fuel economy of modern trucks and cars. It is study of the external flow over the body of
a vehicle, or the internal flow through the internal combustion engines.
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Industrial manufacturing applications: A mold being filled with liquid modular cast iron.
The liquid flow field is calculated as a function of time. Another example is the manufacture of
ceramic materials.
Civil engineering applications: Problems involving the theology of rivers, lakes etc are
also subject of investigations using CFD. Example is filling of mud from an underwater mud
capture reservoir.
Environmental engineering applications: The discipline of heating, air conditioning and
general air circulation through buildings. Another example is fluid burning in furnaces.
Bio-medical Engineering applications: Used to analyze the blood flow through
grafted blood vessels
3.24 OVERVIEW OF FLUENT
There are many CFD packages in the market now, FLUENT is most widely used and this
package has been used in this project for the simulation.
FLUENT, Inc. is the world's largest computational fluid dynamics (CFD) software provider,
enabling solutions for a broad array of fluid flow and heat transfer phenomenon. It uses the
finite-volume method to solve the governing equations for a fluid. It provides the capability to
use different physical models such as incompressible or compressible, inviscid or viscous,
laminar or turbulent, etc. Geometry and grid generation is done using GAMBIT which is the pre-
processor bundled with FLUENT.
GAMBIT is a software package designed to help analysts and designers build and mesh models
for computational fluid dynamics (CFD) and other scientific applications. GAMBIT receives
user input by means of its graphical user interface (GUI). The GAMBIT GUI makes the basic
steps of building, meshing, and assigning zone types to a model simple and intuitive, yet it is
versatile enough to accommodate a wide range of modeling applications. It also provides tools
for checking the quality of the mesh.
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FLUENT is written in the C language and makes full use of the flexibility and power offered by
the language. Consequently, true dynamics memory allocation, efficient data structures and
flexible solver control are all made possible in addition, FLUENT uses client/server architecture.
FLUENTs CFD solvers provide a wide range of physical models and numerical techniques.
From combustion to plastic extrusion, from supersonic airfoils to fluidized beds, FLUENT
provides the physics and numeric needed to get accurate answers and stable calculations. The
benefits of using FLUENT and CFD are better designs, lower risk and faster time to the market
place for your product or process.
Each simulation using CFD, including FLUENT, consists of five basic, but important steps.
These steps are described below. In each of the following steps, the user has to specify the input
parameters, which control the execution of the code and post processing of the results.
Step 1 Preliminary Inputs
During this step the user allocates memory for the CFD simulation that is going to be
performed. At this point it is also helpful to gather the inputs needed for the rest of the simulation
and prepare them to be entered into the CFD software.
Step 2 Grid Generation:
This step is used to specify the geometry of the system, such as radius of a pipe that is to be
modeled. It is also during this step that the user sets the boundary conditions.
Step 3 Flow Parameters
The fluid characteristics, such as density and viscosity are very important to the CFD simulation.
It is during this step that these two parameters, as well at the mass flow rate, will be set.
Step 4 Solve
This step many times proves to be the easiest for the user. The user simply tells the program how
many calculations to perform and activates the solver.
Step 5 Post Processing
The final step consists of the analysis of the results as well as interpretation. FLUENT provides
output in both visual and numerical form. Both are key in understanding the flow results. In
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every CFD simulation whether simple or complex, these five basic steps are followed. It is of the
utmost importance that care be taken while entering the input in each ofthese steps to ensure
quality results. CFD, if used correctly, is as very useful and powerful tool.