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CHAPTER 4

COMPUTATIONAL FLUID DYNAMICS

In this chapter, the concept of computational fluid dynamics and the

usage of this tool for the present work are explained. The required physics for

solving the impeller by this approach is also described. The features available

in the commercial tool and its usage for solving the impeller are discussed in

detail.

4.1 INTRODUCTION

Computational Fluid Dynamics (CFD) has grown from a

mathematical curiosity to become an essential tool in almost every branch of

fluid dynamics, from aerospace propulsion to weather prediction. CFD is

commonly accepted as referring to the broad topic encompassing the

numerical solution, by computational methods. These governing equations,

which describe fluid flow, are the set of Navier-Stokes equation, continuity

equation and any additional conservation equations, for example, energy or

species concentrations.

Since the advent of the digital computer, CFD, as a developing

science, has received extensive attention throughout the international

community. The attraction of the subject is two fold. Firstly, there is the

desire to be able to model physical fluid phenomena that cannot be easily

simulated or measured with a physical experiment, for example, weather

systems. Secondly, there is desire to be able to investigate physical fluid

38

systems more cost effectively and more rapidly than with experimental

procedures.

Traditional restrictions in flow analysis and design limit the

accuracy in solving and visualization of the fluid-flow problems. This applies

to both single and multi-phase flows, and is particularly true of problems that

are three dimensional in nature and involve turbulence, chemical reactions,

and/or heat and mass transfer. All these can be considered together in the

application of CFD, a powerful technique that can help to overcome many

restrictions inherent in traditional analysis.

CFD is a method for solving complex fluid flow and heat transfer

problems on a computer. CFD allows the study of problems that are too

difficult to solve using classical techniques. The flow path inside the impeller

of the centrifugal pump is intricate and this can be analyzed using CFD tool,

which provides an insight into the complex flow behavior.

4.2 CFD SIMULATIONS

The process of performing CFD simulations is split into three

components:

Setting up the simulation : Pre - processing (interactive)

Solving for the flow field : Solver (non - interactive / batch process)

Solving Pre-processing Post-processing

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4.2.1 Pre-processing

The pre-processor contains all the fluid flow inputs for a flow

problem. It can be seen as a user-friendly interface and a conversion of all the

input into the solver in CFD program. At this stage, quite a lot of activities are

carried out before the problem is being solved. These stages are listed below:

Geometry Definition - The region of interests, that is the

computational domain which has to be defined.

Grid generation- It is the process of dividing the domain into a

number of smaller and non-overlapping sub-domains.

Physical and chemical properties - The flow behavior in terms of

physical and chemical characteristics are to be selected.

Fluid property Definition - The fluid properties like density and

viscosity are to be defined.

Boundary conditions - All the necessary boundary conditions have

to be specified on the cell zones.

The solution of the flow problem such as temperature, velocity,

pressure etc. is defined at the nodes insides each cell. The accuracy of the

CFD solution is governed by the number of cells in the grid and is dependent

on the fineness of the grid.

4.2.2 Solution

In the numerical solution technique, there are three different streams

that form the basis of the solver. There are finite differences, finite element

and finite volume methods. The differences between them are the way in

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which the flow variables are approximated and the discretization processes

are done.

4.2.2.1 Finite Difference Element (FDM)

FDM describes the unknown flow variables of the flow problem by

means of point samples at node points of a grid coordinate. By FDM, the

Taylor’s expansion is usually used to generate finite differences

approximation.

4.2.2.2 Finite Element Method (FEM)

FEM uses the simple piecewise functions valid on elements to

describe the local variations of unknown flow variables. Governing equation

is precisely satisfied by the exact solution of flow variables. In FEM, residuals

are used to measure the errors.

4.2.2.3 Finite Volume Method (FVM)

FVM was originally developed as a special finite difference

formulation. The main computational commercial CFD codes packages using

the FVM approaches involves Phoenics, Fluent, Flow 3D and Star-CD.

Basically, the numerical algorithm in these CFD commercial packages

involves the formal integration of the governing equation over all the finite

control volume, the discretization process involves the substitution of a

variety of FDM types to approximate the integration equation of the flow

problem, and the solution is obtained by iterative method. Discretization in

the solver involves the approaches to solve the numerical integration of the

flow problem. Usually, two different approaches are made, one at a time.

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4.2.2.4 Implicit Versus Explicit Methods

Numerical solution schemes are often referred to as being explicit or

implicit. When a direct computation of the dependent variables can be made

in terms of known quantities, the computation is said to be explicit. When the

dependent variables are defined by coupled sets of equations, and either a

matrix or iterative technique is needed to obtain the solution, the numerical

method is said to be implicit.

In computational fluid dynamics, the governing equations are

nonlinear, and the number of unknown variables is typically very large. Under

these conditions implicitly formulated equations are almost always solved

using iterative techniques.

Iterations are used to advance a solution through a sequence of steps

from a starting state to a final, converged state. This is true whether the

solution sought is either one step in a transient problem or a final steady-state

result. In either case, the iteration steps resemble a time-like process. Of

course, the iteration steps usually do not correspond to a realistic time-

dependent behavior. In fact, it is this aspect of an implicit method that makes

it attractive for steady-state computations, because the number of iterations

required for a solution is often much smaller than the number of time steps

needed for an accurate transient that asymptotically approaches steady

conditions.

On the other hand, it is also this "distorted transient" feature that

leads to the question, "What are the consequences of using an implicit versus

an explicit solution method for a time-dependent problem?" The answer to

this question has two parts. The first part has to do with numerical stability

and the second part with numerical accuracy.

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4.2.2.5 Stability

A numerical solution method is said to be stable if it does not magnify

the errors that appear in the course of numerical solution process. For

temporal problems, stability guarantees that the method produces a bounded

solution whenever the solution of the exact equation is bounded. For iterative

methods, a stable method is one that does not diverge. Stability can be

difficult to investigate, especially when boundary conditions and non-

linearities are present. For this reason, it is common to investigate the stability

of a method for linear problems with constant coefficients without boundary

conditions. Experience shows that the results obtained in this way can often

be applied to more complex problems but there are notable exceptions. The

most widely used approach to studying stability of numerical schemes is the

von Neumann's method. However, when solving complicated, non-linear and

coupled equations with complicated boundary conditions, there are few

stability results so we may have to rely on experience and intuition. Many

solution schemes require that the time step be smaller than a certain limit or

that under-relaxation be used.

4.2.2.6 Accuracy

Numerical solutions of fluid flow and heat transfer problems are only

approximate solutions. In addition to the errors that might be introduced in the

course of the development of the solution algorithm, in programming or

setting up the boundary conditions, numerical solutions always include three

kinds of systematic errors:

Modeling errors, which are defined as the difference between the

actual flow and the exact solution of the mathematical model;

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Discretization errors, defined as the difference between the exact

solution of the conservation equations and the exact solution of the algebraic

system of equations obtained by discretizing these equations.

Iteration errors, defined as the difference between the iterative and

exact solutions of the algebraic equations systems. Iteration errors are often

called convergence errors However, the term convergence is used not only in

conjunction with error reduction in iterative solution methods, but is also

(quite appropriately) often associated with the convergence of numerical

solutions towards a grid-independent solution, in which case it is closely

linked to discretization error.

It is important to be aware of the existence of these errors, and even

more to try to distinguish one from another. Various errors may cancel each

other, so that sometimes a solution obtained on a coarse grid may agree better

with the experiment than a solution on a finer grid - which, by definition,

should be more accurate.

Modeling errors depend on the assumptions made in deriving the

transport equations for the variables. They may be considered negligible when

laminar flows are investigated, since the Navier-Stokes equations represent a

sufficiently accurate model of the flow. However, for turbulent flows, two-

phase flows, combustion etc., the modeling errors may be very large - the

exact solution of the model equations may be qualitatively wrong. Modeling

errors are also introduced by simplifying the geometry of the solution domain,

by simplifying boundary conditions etc. These errors are not known a priori;

they can only be evaluated by comparing solutions in which the discretization

and convergence errors are negligible with accurate experimental data or with

data obtained by more accurate models (e.g. data from direct simulation of

turbulence, etc.). It is essential to control and estimate the convergence and

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discretization errors before the models of physical phenomena (like

turbulence models) can be judged.

As mentioned above that discretization approximations introduce

errors which decrease as the grid is refined, and that the order of the

approximation is a measure of accuracy. However, on a given grid, methods

of the same order may produce solution errors which differ by as much as an

order of magnitude. This is because the order only tells us the rate at which

the error decreases as the mesh spacing is reduced - it gives no information

about the error on a single grid.

Errors due to iterative solution and round-off are easier to control. The

ultimate goal is to obtain desired accuracy with least effort, or the maximum

accuracy with the available resources.

4.2.2.7 Conservation

Since the equations to be solved are conservation laws, the numerical

scheme should also - on both a local and a global basis - respect these laws.

This means that, at steady state and in the absence of sources, the amount of a

conserved quantity leaving a closed volume is equal to the amount entering

that volume. If the strong conservation form of equations and a finite volume

method are used, this is guaranteed for each individual control volume and for

the solution domain as a whole. Other discretization methods can be made

conservative if care is taken in the choice of approximations. The treatment of

sources or sink terms should be consistent so that the total source or sink in

the domain is equal to the net flux of the conserved quantity through the

boundaries.

This is an important property of the solution method, since it

imposes a constraint on the solution error. If the conservation of mass,

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momentum and energy are insured, the error can only improperly distribute

these quantities over the solution domain. Non-conservative schemes can

produce artificial sources and sinks, changing the balance both locally and

globally. However, non-conservative schemes can be consistent and stable

and therefore lead to correct solutions in the limit of very fine grids. The

errors due to non-conservation are in most cases appreciable only on

relatively coarse grids.

The problem is that it is difficult to know on which grid are these

errors small enough. Conservative schemes are preferred in solution

approach.

4.2.2.8 Boundedness

Numerical solutions should lie within proper bounds. Physically non-

negative quantities (like density, kinetic energy of turbulence) must always be

positive; other quantities, such as concentration, must lie between 0% and

100%. In the absence of sources, some equations (e.g. the heat equation for

the temperature when no heat sources are present) require that the minimum

and maximum values of the variable be found on the boundaries of the

domain.

These conditions should be inherited by the numerical

approximation. Boundedness is difficult to guarantee. Some first order

schemes guarantee this property. All higher-order schemes can produce

unbounded solutions; fortunately, this usually happens only on grids that are

too coarse, so a solution with undershoots and overshoots is usually an

indication that the errors in the solution are large and the grid needs some

refinement (at least locally). The problem is that schemes prone to producing

unbounded solutions may have stability and convergence problems.

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4.2.2.9 Realizability

Models of phenomena which are too complex to treat directly (for

example, turbulence, combustion, or multiphase flow) should be designed to

guarantee physically realistic solutions. This is not a numerical issue per se

but models that are not realizable may result in unphysical solutions or cause

numerical methods to diverge.

4.2.2.10 Consistency

The discretization should become exact as the grid spacing tends to zero. The

difference between the discretized equation and the exact one is called the

truncation error. It is usually estimated by replacing all the nodal values in the

discrete approximation by a Taylor series expansion about a single point.

As a result one recovers the original differential equation plus a

remainder, which represents the truncation error. For a method to be

consistent, the truncation error must become zero when the mesh spacing At

t 0 and/or x 0. Truncation error is usually proportional to a power of

the grid spacing xi and/or the time step t. If the most important term is

proportional to (x)n or (t) n . It is called as nth-order approximation; n > 0

is required for consistency. Ideally, all terms should be discretized with

approximations of the same order of accuracy; however, some terms (e.g.

convective terms in high Reynolds number flows or diffusive terms in low

Reynolds number flows) may be dominant in a particular flow and it may be

reasonable to treat them with more accuracy than the others.

Some discretization methods lead to truncation errors which are

functions of the ratio of xi to t or vice versa. In such a case the consistency

requirement is only conditionally fulfilled: xi and t must be reduced in a

way that allows the appropriate ratio to go to zero. Even if the approximations

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are consistent, it does not necessarily mean that the solution of the discretized

equation system will become the exact solution of the differential equation in

the limit of small step size. For this to happen, the solution method has to be

stable.

4.2.2.11 Convergence

When running a well-posed CFD simulation, there are many factors

that govern the decision to declare the solution done. Residual reports, force

monitors, and overall balances are just a few of the factors that can be used to

make this judgment call. More important than the measuring devices is what

is to be achieved by running the simulation in the first place. For example,

deep convergence is required if it is an aerodynamic problem and want to get

accurate predictions of lift and drag coefficients. Rough convergence is

acceptable for simple flow problems.

The reason that solution convergence is an issue with all CFD

software is due to the iterative nature of the solution procedures used. In

particular, iteration is necessary to handle the non-linearity of the equations

that govern fluid flow, heat transfer, and related processes. For any given

conservation equation, an approximate solution is obtained at each iteration

that results in a small imbalance in the conservation statement. During the

course of the iterative solution algorithm, the imbalance in each cell is a

small, non-zero value that, under normal circumstances, decreases as the

solution progresses. This imbalance is called the residual. The total residual

for each variable across the entire solution domain is the sum of the absolute

values of the individual cell residuals. This total residual is often scaled so

that the residuals of different variables can be compared or combined. Scaling

factors are taken from the bulk flow quantities or from the error at the start of

the calculation.

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The convergence criteria are pre-set conditions on the residuals that

indicate that a certain level of convergence has been achieved. For example, a

criterion that the scaled residual for the x-momentum drop to 0.001 indicates

that the overall error in this variable is about three orders of magnitude less

than the bulk x-momentum in the system. Variations on this definition exist

for certain variables or solver techniques, but a common thread for all is that

as the net residual declines, so does the error in the solution.

Unfortunately, the reduction in residuals is not the only indicator of

convergence. A truly converged solution is one that is no longer changing

with successive iteration. If the residuals for all problem variables fall below

the convergence criteria but are still in decline, the solution is still changing,

to a greater or lesser degree. A better indicator occurs when the residuals

flatten in a traditional residual plot (of residual value vs. iteration). This point,

sometimes referred to as convergence at the level of machine accuracy, takes

time to reach, however, and may be beyond the needs. For this reason,

alternative tools such as reports of forces, heat balances, or mass balances can

be used instead.

4.2.3 Post-Processing

The CFD package provides the data visualisation tools to visualise

the results of the flow problem. This includes – vectors plots, domain

geometry and grid display, line and shaded counter plots, particle tracking etc.

Recent facilities are aided with animation for dynamic result display and they

also have data export facilities for further manipulation external to the code.

Determining the convergence, whether the solution is consistent and

stable for all range of flow variables, is important.

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Convergence is a property of a numerical method to produce a

solution that approaches the exact solution by which the grid spacing and

control volume size are reduced to a specific value or to zero value.

Consistency is to produce the system of algebraic equations that can

be equivalent to the original governing equation.

Stability associates with the damping of errors as a numerical

method proceeds. If a technique chosen is not stable, even the round-off error

in the initial data can lead to wild oscillations or divergence.

4.3 BASIC EQUATIONS AND ASSUMPTIONS

In an inertial frame of reference the flow of an isothermal

Newtonian fluid is described by the continuity equation

( ) 0v.t

=ρ∇+∂

ρ∂

(4.1)

and by the Navier-Stokes equation

Fv.3

vvvp

1v.v

t

v 2 +∇∇+∇+∇ρ

−=∇+∂

∂ (4.2)

With ρ the density, v the velocity vector, t the time, p the pressure, v the

kinematic viscosity, and F an external force.

A number of assumptions can be made for the flow in hydraulic

pumps:

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1) Mach-numbers are usually small enough to justify the

assumption of incompressible flow (Ma 2 << 1). The continuity

equation (4.1) then reduces to

0v. =∇ (4.3)

2) This means that viscous forces can be neglected when compared

to inertia forces, except in boundary layers and wakes. With this

assumption, the Navier-Stokes equation (4.2) for the main flow

reduces to

Fp1

-v.vt

v+∇

ρ=∇+

∂

∂ (4.4)

3) The final assumption which can be made in case of hydraulic

pumps is the flow being irrotational,

0v =×∇ (4.5)

4) Vorticity is generated by viscous shear forces or non-

conservative external forces. Thus, in the absence of non-

conservative forces and viscous effects confined to thin (and

attached) boundary layers and wakes, the core of the flow can be

assumed irrotational, provided that the incoming fluid is free of

rotation. A velocity potential φ can now be defined as

φ∇=v (4.6)

which can be substituted into equation (4.3) to obtain the

Laplace equation

02 =φ∇ (4.7)

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Potential is solved from the above equation. The Navier-Stokes

equation in this case with irrotational flow and force of gravity reduces to

unsteady Bernoulli’s equation,

)t(cgZP

v.v2

1

t=+

ρ++

∂

φ∂ (4.8)

Bernoulli’s equation in steady state is,

0gZp

v.v2

1=+

ρ+ (4.9)

Potential theory shows the simplification of the complex flow

physics in to the solvable form analytically that is used for the calculation.

The need of commercial tool is understood from the simplification made in

the equation, as it neglects the viscous effects and related phenomena. By

introducing the viscous effect with the turbulence models improves the

solution accuracy in the commercial solvers.

In the case of viscous flow k- model is taken into account.

Belonging to the family of eddy-viscosity models, this is one of the most

prominent turbulence prediction tools implemented in many general purpose

CFD codes. It has proven to be stable and numerically robust having a well

established predictive capability. The k- model introduces two new variables

into the system of conservation equations. The values of k and are directly

calculated from the differential transport equations for the turbulence kinetic

energy (k) and turbulence dissipation rate ().

4.4 COMPUTATIONAL DOMAIN AND BOUNDARY

CONDITIONS

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A typical pump consists of inlet pipe, the impeller, the discharge

region and exit pipe. Different types of conditions apply at its boundaries:

Inlet and outlet surfaces: Possible inlet and outlet boundary

conditions are following;

Inlet Boundary Condition

Normal Speed Inlet:

The magnitude of the inlet velocity is specified and the direction is

taken to be normal to the boundary. The direction constraint requires that the

flow direction is parallel to the boundary surface normal, which is calculated

at each element face on the inlet boundary.

Cartesian Velocity Components:

The boundary velocity components are specified, with a non-zero

resultant into the domain.

UInlet = Ui+Vj+Wk (4.10)

U, V, W are the velocity component in x, y and z directions.

Cylindrical Velocity Components:

In this case the velocity boundary condition is specified in a local

cylindrical coordinate system. Only the axial direction of the local coordinate

system needs to be given and the components of velocity in the r, theta and z

directions are automatically transformed by the Solver into Cartesian velocity

components.

UInlet = Ur, rˆ+ U, ˆ+Uz, zˆ (4.11)

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Ur, U, Uz are the velocity components in r, and z directions.

And the solver will compute the rotation matrix which transforms these

components from the cylindrical components to the Cartesian components

such that the boundary condition is the same as if Cartesian components were

specified.

Total Pressure

The Total Pressure, , is specified at an inlet boundary condition and the

Solver computes the static pressure needed to properly close the boundary

condition. For rotating frames of reference one usually specifies the stationary

frame total pressure instead.

The direction constraint for the Normal To Boundary option is the

same as that for the Normal Speed Inlet option. Alternatively, the direction

vector can be specified explicitly in terms of its three components. In both

cases, the boundary mass flow is an implicit result of the flow simulation.

Mass Flow Rate

The boundary mass flow rate is specified along with a direction

component. If the flow direction is specified as normal to the boundary, a

uniform mass influx is assumed to exist over the entire boundary. Also, if the

flow direction is set using Cartesian or cylindrical components, the

component normal to the boundary condition is ignored and, again, a uniform

mass influx is assumed. The mass influx is calculated using:

ρU = m / s dA (4.12)

where s dA is the integrated boundary surface area at a given

mesh resolution. The area varies with mesh resolution because the

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resolution determines how well resolved the boundary surfaces are. The value

of is held constant over the entire boundary surface.

Outlet Boundary Condition

Static Pressure (Uniform)

Relative Static Pressure is specified over the outlet boundary:

Pstatic, Outlet = Pspecified (4.13)

Normal Speed

The magnitude of the outlet velocity is specified and the direction is

taken to be normal to the boundary at mesh resolution.

Cartesian Velocity Components

The boundary velocity components are specified, with a non-zero

resultant out of the domain.

Uoutlet = Ui+Vj+Wk (4.14)

Average Static Pressure: Overall

The Outlet Relative Static Pressure is constrained such that the average

is the specified value:

P spec = 1/A s PndA (4.15)

where the integral is over the entire outlet boundary surface.

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To enforce this condition, pressure at each boundary integration point

is set as:

Pip = Pspec avg + (Pnode – Pnode avg) (4.16)

So, the integration point pressure in this case is set to the specified

value plus the difference between the local nodal value and the average outlet

boundary pressure. In this way the exit boundary condition pressure profile

can float, but the average value is constrained to the specified value.

Impeller blade surfaces: At the impeller blade surfaces (both high

pressure and suction side) the Neumann boundary condition takes the form

( ) .rn

×Ω=∂

φ∂ n (4.17)

where is the rotational velocity of the impeller and n is the outward unit

normal vector.

At boundaries, which do not move, normal velocity components vanish

0n

=∂

φ∂ (4.18)

The flow region of interest is divided into non-overlapping

sub-domains so as to have greater ease in creating a good mesh for

complex 3D geometries, considerable reduction of computing time and

greater accuracy in results.

The analysis of the impeller flow passage is done using software

packages. The flow passage is modeled, meshed and analyzed in different

software packages. The pressure and velocity distributions in the flow passage

is the desired output with which comparisons and inferences can be made in

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order to achieve the goal of redesigning the passage to increase the efficiency

of the pump.

4.5 USAGE OF COMMERCIAL PACKAGE

4.5.1 Modeling

The impeller has five flow passages and six vanes. The flow in

all the five flow passages are considered to be uniform, and only one

flow

passage is considered for analysis. The flow passage is modeled in

Pro/ENGINEER- 2001 as shown in Figure 4.1. The flow passage is

modeled in mm. The passage is modeled along with a part of the eye

to depict the existing inlet passage. The model has an axial entry and

radial exit with a converging flow path. The vane profile has four radii

of curvature. The model is then converted into STEP file which will

then be exported for meshing and analysis.

(a) Impeller (b) Impeller Fluid Domain

Figure 4.1 Pro E Model

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4.5.2 Discretization

Meshing is dividing the whole model into smaller cells by choosing

an appropriate element size and type. The STEP file is imported into

GAMBIT. The solver is to be selected in GAMBIT to give the appropriate

boundary conditions. The solver selected is Fluent 5/6. The volume is then

meshed using the mesh tool with tetrahedral elements as shown in Figure 4.2.

Figure 4.2 Discretized Fluid Domain

4.5.3 Solution

Fluent 6.1 is a software package in which the analysis of fluid flows,

heat related problems and other turbulent flow problems are analyzed to

obtain pressure, velocity, temperature and other distributions. The solver

selected for the impeller analysis is a segregated type, which provides

flexibility in solution procedure. The implicit cell based gradient with

absolute velocity formulation is followed.

The viscous k- (k-Epsilon) two equation model with standard wall

function which solves the continuity and the momentum equation is chosen

for the purpose of capturing the kinetic energy and the dissipative energy

58

effects encountered in the turbo machine. Here k represents the kinetic energy

and represents the dissipative energy.

The working fluid is chosen from the database and copied. The

material chosen is water with constant default density and constant default

viscosity corresponding to single phase, single species flow. As the pumping

system is operating in atmosphere, the operating pressure is set as 101325 Pa.

The boundary conditions reflect the physics of the prevalent

conditions. The conditions specified play a critical role in determining the

final output.

The boundary conditions are specified as follows.

• Flow - material – water-liquid

• Motion type - moving reference frame with rotational

speed (rad/s)

• Inlet - pressure inlet

• Outlet - velocity inlet (negative value is specified

to signify outward flow)

• Wall - stationary wall

Before solving the problem, the solution parameters have to be set.

First the solver controls are set to the default under relaxation values for

pressure, density, momentum and body forces. First order upwind criteria are

maintained to achieve convergence faster.

The solution has to be initialized to eliminate any garbage values.

The residual monitors are set, and the convergence criteria for continuity,

x-velocity, y-velocity, z-velocity, k and are set at third decimal convergence

of 0.001.

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The grid has to be adapted for attaining faster convergence. Usage

of k- viscous model necessitates two types of adaptation namely boundary

adaptation and Y+/Y

* adaptation. Adaptation increases the number of cells

and adapts the cells adjacent to the walls.

The problem is now ready to be iterated and the number of iterations

is specified and the reporting interval is set as one so as to monitor the trend

every time. The iteration continues till the convergence criteria in all the six

parameters namely continuity, x, y, z velocity, k and are attained. The plot

of the convergence can be viewed simultaneously by activating the display

window.

The output of the solution can be viewed as projected areas, surface

integrals etc. For better understanding, area-weighted average of parameters

like pressure and velocity are taken.

The mass flow rate and volume flow rate can also be obtained. The

whole results can also be viewed graphically in the graphic display window.

The display can be in the mode of contours or vectors. The display will be self

explanatory as different ranges of pressure and velocity will support a

different color.

Mesh adoption is carried out in the fluent solver. It adapts the mesh

according to the flux parameter. Boundary adaptation technique is utilized in

this analysis. Adaptation increases the number of cells and adapts the cells

adjacent to the walls. It is adapted in the solver automatically and the results

are obtained for the adapted condition.

Rotating flows is treated using the moving reference frame capability

in Fluent. The Figure 4.3 depicts the flow in a moving reference frame.

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Figure 4.3 Flow in a moving reference frame

In this work the grid convergence is ensured and the same pattern is applied

for all analysis