chapter 3, cfd theory

7/28/2019 Chapter 3, CFD Theory

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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.

chapter 3, cfd theory

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