Nithin Thesis on variable fluid boundary

7/28/2019 Nithin Thesis on variable fluid boundary

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Unified Formulation for Compressible and

Incompressible Unsteady Flow Using The Finite

Element Method

THESIS

Submitted in partial fulfillment of the requirements

of

BITS C422T Thesis

By

Nithin S Poduval

Department of Mechanical Engineering

BITS,PILANI-K.K. BIRLA GOA CAMPUS

Zuarinagar-403 726,

Goa, India

December , 2012


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CERTIFICATE

This is to certify that the Thesis entitled Unified Formulation for Compressible and

Incompressible Unsteady Flow Using the Finite Element Method is submitted by

Nithin S Poduval ID No. 2009A4TS217G in partial fulfillments of the requirements of

BITS C421T Thesis and embodies the work done by him under my supervision.

Signature of the Guide

Place:

Date:


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ACKNOWLEDGEMENT

This dissertation would not be possible without the constant support, guidanceand motivation from my thesis supervisor, Dr. Shibu Clement. I am indebted to him for

his concern and dedication shown in my career throughout the three years I have known

him.It has been a privilege to be working under him. I also thank him for taking the

efforts to provide me an opportunity to work at the Fluid Mechanics and Thermal

Department, VSSC.

I wish to thank Dr. Jayachandran, Group Director, APRG,VSSC, ISRO for

providing the guiding light throughout the duration of the work. He constantly supplied

with me valuable inputs and guidance whenever required. I also thank him for playing his

part in providing me the opportunity to work at the FMTD computational lab.

I express my gratitude to Dr. P. Veeraraghavan, Director, VSSC for accepting my

humble request, thus offering me an opportunity to work at this prestigious organization.

I wish to show my appreciation towards Mr. M. Ajith , Engineer, FMTD for being

my mentor throughout the duration of the thesis. Mr. Ajith is responsible for helping me

acquire the knowledge required to work on this thesis. He has also tirelessly cleared the

many doubts that I raised while I was at the lab.

I also thank all the other members of the FMTD, who have helped me in one way

or the other during my stay.

Lastly, I am thankful to my college for providing a facility for students to do

research for six months in their final year of engineering.


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THESIS ABSTRACT

This thesis work proposes a unified formulation to handle both incompressible andcompressible flows in two dimensions. A new method, the Eulerian velocity correction

method also known as the fractional step method is used to develop the incompressible ,

compressibe and then the unified formulation. This method is seen to have certain

advantages over other methods.The Galerkin weighted method of weighted residuals is

used to discretize the domain.The codes are validated using the test cases-flow over two-

dimensional backward facing step, liquid flow over NACA-0015 hydrofoil and flow over

a circular cylinder.Lastly the code is applied to a practical problem-the orifice meter used

to measure flow rate. An improvement in the design is suggested.


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COVER PAGE 1

CERTIFICATE 2

ABSTRACT 3

ACKNOWLEDGEMENTS 4

TABLE OF CONTENTS

Chapter 1

INTRODUCTION

1.1 Flow Physics 71.2 Overview 8

1.3 Solution Approaches 81.3.1 Artificial Compressibility Method 91.3.2. Pressure Projection method 10

Chapter 2

FINITE ELEMENT DISCRETIZATION 12

2.1 Finite element method 122.2 Taylor - Galerkin Algorithm 14

2.3 Numerical modeling 152.3.1 Convection diffusion equation 15

2.3.1.1.Fluid Energy Equation 15

2.3.1.2. Navier-Stokes equation 152.3.2. Solution procedure for a general

transport equation17

2.3.3 Time Marching Method 212.3.4. Stability criterion 21

Chapter 3

INCOMPRESSIBLE FLOW SOLVER 233.1 Governing Equations 233.2 Velocity correction method 23

Chapter 4

VALIDATION OF THE INCOMPRESSIBLE FLOW SOLVER

4.1. Flow over a backward facing step 26


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Chapter 5

COMPRESSIBLE FLOW SOLVER

32

5.1. Governing Equations 32

5.2. Methodology 33

Chapter 6

UNIFIED FORMULATION

6.1. Methodology 37

Chapter 7

VALIDATION OF UNIFIED FORMULATION42

7.1. Flow over NACA-0015 Hydrofoil 40 427.2. Flow over a Circular Cylinder 467.3. Flow over a Backward facing step 53

Chapter 8

ORIFICE METER 55

Chapter 9

CONCLUSIONS 58NOMENCLATURE 59

APPENDICES

Appendix A: Conjugate Gradient Method 60Appendix B: Evaluation of Matrices 61Appendix C:The two-dimensional heat conduction equation

and its numerical modeling

65

References 76


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1.INTRODUCTION

1.1. Flow Physics

Even though nearly all fluids are compressible in an absolute sense, incompressible flow

approximation can be made when the flow speed is insignificant everywhere in the flow

field compared to the speed of sound of the medium. Following this definition of

incompressibility, a large number of fluid dynamics problems can be classified as

incompressible and, in most cases, viscous. To name a few types of incompressible

flows, there are problems related to low speed aerodynamics, hydrodynamics such as the

flow around submerged vehicles, flow through pumps, mixing of the flow in chemical

reactors, coolant flow in nuclear reactors, and blood flow in the human body. When the

flow is assumed to be incompressible, mathematically the flow field becomes elliptic,

which introduces major challenges in computations.

Additional difficulties arise when the flow is viscous. Most notable, complications

come from predicting flow physics involving turbulence and transition. In flow problems

containing incompressible flow regions, physics involving multi-phase, multi-material,

non-Newtonian and stress-supporting media can add complexities to the incompressible

flow computation in a broad sense. Another challenge may come from resolving multi-

scale dynamics such as those encountered in biomedical applications.

There are several approximations in flow analysis. At a formulation level, the

incompressible Navier Stokes equations care the most commonly accepted governing

equations. The incompressibility assumption in the governing equations is an

approximation for the medium. Other than a small number of laminar flow problems,

most problems of fundamental and engineering interest involves transition and

turbulence. This introduces the question of how to approximate flow physics to resolve

the flow features involved at a reasonable level of accuracy. In most cases computational

approach offers a viable option for flow analysis. The procedure involving computational

modeling, numerical boundary conditions, and algorithms adds another source of

approximation requiring assessment of computational accuracy. Therefore, it is important


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to define the relative contributions from mathematical formulations, physical modeling,

and computational procedures to the accuracy of the analysis results.

1.2. Overview

The Navier-Stokes equations are generally accepted as the equations governing the flow

of Newtonian fluid in a continuum regime. Mathematically, the compressible flow

equations become singular at the limit where the speed of sound of the medium becomes

infinity or the flow speed becomes insignificant relative to the speed of sound. This

singular nature of the governing equations poses the primary difficulty in solving the

incompressible Navier-Stokes equations. Physically the challenge is maintaining

incompressibility during iterative processes for obtaining steady-state solutions or at each

time level for computing time-accurate solutions. Several methods have been developed

in the past where the main differences among various approaches come from the way in

which the incompressibility condition is satisfied computationally.

A traditional approach is to start the computational process directly from an

incompressible Navier-Stokes formulation. The primary concern is then how to satisfy

the continuity equation. One can use the primitive variable, namely, pressure and

velocities or derived quantities like stream function-vorticity and vorticity-velocity. For

general three-dimensional problems, primitive variable formulation poses the least

complications in imposing physical boundary conditions. The primitive variable

formulation has certain advantages over Vorticity - Stream function formulation [Ref.

18]. In a cavity or channel flow, vorticity - stream function method requires the

specification of vorticity on all boundaries on the other hand the primitive

variable formulation requires .special treatment for the calculation of the

pressure field. Finite element method in conjunction with the primitive variable

formulation have proved efficient in dealing with cavity flow problems.

1.3. Solution Approaches

Depending on the flow features to qualify, the solution method of choice can vary. For

flows involving thin viscous layers, it is advantageous to have large time steps, possibly


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using an implicit method. For time accurate solutions, the physical time step required to

resolve unsteady motion could be very small, in which case, explicit schemes can be

used effectively.

Grid topology and the "goodness" of grid can affect the solution accuracy in significant

ways-not only from a numerical dissipation point of view but also in designing boundary

conditions. Requirements on grid density and distribution are also realistic factors

affecting the order of differencing schemes. All these factors should be considered in

developing a flow solver and implementation guidelines. Solution procedures discussed

below represent different approaches but are not unique combinations of these methods.

1.3.1 Artificial Compressibility Method

Large advances in the state of the art in CFD have been made in conjunction with the

field of aerodynamics. Therefore, it is of significant interest to able to use some of the

compressible flow algorithms. To do this, the artificial compressibility approach of

Chorin (1967) who first introduced the term, can be used.

In this formulation, the continuity equation is modified by adding a time derivative of the

pressure term, resulting in:

1 0i

i

up

t x

(1.1)

where is an artificial compressibility or pseudo compressibility parameter.

Together with the unsteady momentum equations, Equation (1.1) forms a hyperbolic-

parabolic type of time-dependent system of equations. Thus, implicit schemes developed

for compressible flows can be implemented. Note that 't' no longer represents a true

physical time in this formulation.

Physically this means that waves of finite speed are introduced into the incompressible

flow field as a medium to distribute the pressure. For a truly incompressible flow, the

wave speed is infinite, whereas the speed of propagation of the pseudo waves introduced

by this formulation depends on the magnitude of the artificial compressibility parameter.

In a true incompressible flow, the pressure field is affected instantaneously by a


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disturbance in the flow, but with artificial compressibility, there will be a time lag

between the flow disturbances and its effect on the pressure field. Ideally, the chosen

value of the artificial compressibility can be as high as the particular choice of algorithm

allows so that incompressibility is recovered quickly. This must be done without

lessening the accuracy and the stability property of the numerical method implemented.

On the other hand, if the artificial compressibility is chosen such that theses

waves travel too slowly, then the variation of the pressure field accompanying these

waves is very slow. This will interfere with the proper development of the viscous

effects, such as the boundary layer for the wall-bounded flows. In wall-bounded viscous

flows, the behaviour of the boundary layer is very sensitive to the stream wise pressure

gradient, especially when the boundary layer is separated. If the separation is present, a

pressure wave travelling with finite speed will cause a change in the local pressure

gradient, which will affect the location of the flow separation. This change in separated

flow will feed back to the pressure field, possible preventing convergence to a steady

state. Especially for the internal flow, the viscous effect is important for the entire flow

field, and the interaction between the pseudo pressure-waves and the viscous flow field

becomes very important.

1.3.2. Pressure Projection method

The basic idea of this method is to solve the pressure field such that a divergence-free

velocity field is maintained at every time step. This approach was first started with the

Marker-and-cell(MAC) method, first published by Harlow and Welch from Los Alamos

National Laboratory in 1965. In the procedure, pressure is used as a mapping parameter

to satisfy the continuity equation. The usual computational procedure involves choosing

the pressure field at the current time step such that the continuity equation is satisfied at

the next time step. The time step can be advanced in multiple steps (fractional step

method), which is computationally convenient. However the governing equations are not

coupled as in the artificial compressibility approach.


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In a fractional-step procedure, the time-dependant governing equations can be solved in

several steps, which can be convenient for time-dependent computations of the

incompressible Navier-Stokes equations. In this procedure, the time evolution of the flow

field can be approximated through several steps. Operator splitting can be accomplished

in several ways by treating the momentum equations as a combination of convection,

pressure, and viscous terms. The common application of this method to incompressible

Navier-Stokes equations is done in three steps.

The first step is to solve for an auxiliary velocity field using the momentum

equation, in which the pressure-gradient term can be computed from the pressure in the

previous time step. In the second time step, the pressure is computed, which can map the

auxiliary velocity onto a divergence-free velocity field. Finally the corrected velocity

field is obtained by including the pressure-gradient term.

The major drawback of this method is the large amount of computing time required for

solving the Poisson equation for pressure. When the physical problem requires a very

small time step, the penalty paid for an iterative solution procedure for the pressure may

be tolerable.

In the current thesis work, the fractional step method is used to solve the Navier Stokes

equations. After successful validation of the incompressible flow code, the fractional step

procedure is extended to develop a compressible flow code. Finally a unified formulation

to handle both liquids and gases as media is developed.


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2. FINITE ELEMENT DISCRETIZATION

2.1. Finite element method

An analytical solution or exact solution is a mathematical expression that gives

the value of the desired unknown quantity at any location in the domain. Analytical

solutions are found to be practically impossible for complex problems involving

complicated geometric, material properties, boundary conditions, etc. In such cases, a

viable alternative is to resort the numerical methods, which give an approximate but

acceptable solution with the advent of high speed digital computers, the focus of attention

has largely shifted to numerical methods for approximate solutions of complex problems.

Finite element, Finite volume and Finite difference are the most common among

the numerical schemes used for the solution of heat transfer problems. An integral

formulation is used in the Finite elements and Finite volume methods and this allow the

natural applications of the boundary conditions. Finite element technique has certain

distinct advantages over other method. It can easily handle multi-materials configurations

and unstructured grids .complicated geometries can be easily modeled and interface need

not be considered rigorously in Finite element formulation. Higher order variations

within the element are another noticeable advantage.

The finite elements procedure was first proposed by Zienkiewicz and Cheung for

field problems numerous reports on the application of the Finite element method, to

steady ,unsteady, linear and non linear heat conduction problems involving multi

dimensional geometries and complex boundary conditions are available. Weighted

residual method and Galerkin method are incorporated to solve the complex problems.

Pepper and Henrich (ref [13])describe in detail the application of Finite element methodto one, two and three dimensional heat transfer problems. Finite element methods are

used extensively for thermal and structural analysis and a number of general purpose

packages are available for grid generation and analysis of realistic three dimensional

elements that can be used for represent any structure.


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Lary J. Segerlind(ref.[14]) makes an extensive coverage on the applications of

finite element method for the multidimensional problems. The derivation of elemental

matrices for a range of lower and higher order elements is also described. Huebner and

Thornton gives detailed formulation and solutions for various problems of conduction,

convection and radiation using finite element method.Backett and Chu discusses the

application of finite element method to the solution of heat conduction problem involving

non-linear radiation-boundary conditions. The application of finite element space

discretization of unsteady heat conduction equations results in a system of ordinary

differential equation, which are to be integrated in time to obtain the unsteady

temperature response. Various implicit and explicit schemes have been suggested for this

purpose. A comparison of different time marching schemes for transient heat conduction

is presented by Wood and Lewis.

A good discussion on the stability and oscillation characteristics of finite element

methods are given by Mayer. Purely implicit schemes are unconditionally stable and the

calculated values do not oscillate about the correct values. But the disadvantage is that, a

large system of simultaneous equations is to be solved and which require large computer

memory. It may be also noticed that for non-linear problems, the implicit scheme may

numerically oscillates for larger values of time. Explicit schemes are conditionally stable

and are known to be unstable when the time step exceeds a certain value and they do no

require matrix inversions.

The standard Galerkin procedures were found to be oscillating for Pe higher

than 2 and most of the practical problems are in this unstable region. Upwind schemes

have been proposed as a remedy and its extension to the multidimensional problems

using finite element method are not straight forward. In 1984 Donea proposed a Taylor

Galerkin algorithm for connective and transport problems. In 1992 Comini et al

published the Taylor-Galerkin algorithm for convection type problems.Biji Philip

Mathew solved 3-D convection-diffusion equation explicitly by lumping the capacitance

matrix using Taylor-Galerkin algorithm and proved the applicability of Taylor-Galerkin

algorithm for practical 2-D and 3-D problems.


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2.2. Taylor - Galerkin Algorithm

Convection-type problems, frequently encountered in engineering practice,

include the transport of energy and of chemical species. The finite element method,

based on Galerkin formulations, has become a well established procedure for the

solution of these problems, where both advection and diffusion mechanisms must

be accounted for in the mathematical model. On the other hand, it is a well - known

fact that the use of sample.Bubnov - Galerkin schemes for advection-dominated

problems lead to spurious oscillations for Peclet number 'Pe' greater than 2.

The use of "unwinding" or Petrov-Galerkin types of discretization as a treatment

for these node-to-node oscillations. However, the Petrov- Galerkin methods have

been derived for the steady-state form of the transport equation and their extension

to transient solutions is not always possible.

Recently, alternative discretization procedures for the transient convection

equation based on the Taylor-Galerkin method have gained wide acceptance. This

methodology originally developed by Lax and Wendroff [1960] (ref[19]) was

introduced by Donea and coworkers in 1984. In 1992 Comini and coworkers

published Taylor - Galerkin algorithms for Nonlinear Advection - Diffusion problems. In

the frame work of Taylor - Galerkin methods, the solution is achieved by first

approximating the time derivative by means of a forward, time Taylor series expansion,

and by discretizing afterwards the space derivative by means of a Galerkin procedure.

In this context the partial time discretization precedes the space discretization and

allows the direct evaluation of the time derivatives from the original transport

equation. This preliminary approximation naturally provides an unwinding term that is

dependent on the time increment used, while the standard Galerkin process is applied

only afterwards to the time-discretized equation to complete the finite element

formulation.


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2.3. Numerical modeling

2.3.1 Convection diffusion equation

The energy equation for a moving fluid contains the

combined effect of heat transport due to the fluid motion and heat transfer due to

conduction (diffusion). The governing equation known as the convection-diffusion

equation in extensive research work, this peculiar nomenclature is not only because of its

importance in applications but also due to the presence of the convective transport term

which can introduce numerical oscillations both for steady and transient state solutions.

Such numerical oscillations are not generally observed in the solution of diffusion

equation.

2.3.1.1.Fluid Energy Equation

The fluid energy equation for heat transfer in a two dimensional incompressible

flow in is given by

This governing equation for heat transfer analysis is non-linear, interlined second

order, elliptical partial differential equations, which cannot be solved by analytical

method. So an approximate numerical technique is essential

2.3.1.2. Navier-Stokes equation

For the solution of N-S equations and energy equation, finite element method on

Galerkin formulation has gained wide acceptance, but the application of simple Galerkin

finite element method for the above equations leads to numerical oscillations when the

2 2

2 2...........(2.1)T T T T T u v

t x y x y


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peclet number is more than two. Peclet number is the ratio of the convective heat transfer

to the conduction heat transfer, for the most practical problems the peclet number is order

of magnitude above 2 .Discretization procedure for the transient convective equation

based on Taylor-Galerkin methods do not suffer from these oscillations, and hence

gained wide acceptance. In this investigation the numerical model employs forward time

Taylor series expansion for time discretization and standard Galerkin procedure for space

discretization.

The three dimensional unsteady momentum transport equations in X- direction

X- momentum equation

Y-momentum equation

Two dimensional unsteady energy transport equation [convection -diffusion] is

The continuity equation is

)2.2....(1

2

2

2

2

y

u

x

u

x

p

y

uv

x

uu

t

u

)3.2....(12

2

2

2

y

vx

vyp

yvv

xvu

tv

)5.2....(0

y

v

x

u

2 2

2 2.....(2.4)

T T T T T u v

t x y x y


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The solutions of the above equations can be found out numerically when then the

initial conditions and boundary conditions are specified

2.3.2. Solution procedure for a general transport equation

The general form of the transport equation is,

The time derivative is approximated by means of a forward time Taylor series expansion

as follows,

Substitutingt

and 2

2

t

using equation (2.6)

2 2

2 2.....(2.6)u v S

t x y x y

2

221

2 t

t

ttnn

2

21

2 t

t

tt

nn

)7.2....(2

2

2

2

21

yv

xu

t

t

Syxy

vx

ut

nn


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The last term is solved as follows

Substituting fort

from Eqn 2.6

Substituting Eqn. 2.7.a into 2.6 the following equation results

Spatial discretization

The weighted residual form of eqn 3.5 is

( , )x yW (LHS of eqn 3.11) d =0

Note that the weighting functions W(x,y) are chosen to be functions of x and y only.

Now approximated field using the shape functions

1

( , ) ( , )M

j j

j

x y N x y

2

tu v

x t y t

)7.2....(2

ay

vx

uvyy

vx

uux

t

Syxy

vx

uvyy

vx

uux

t

yv

xu

t

TERM

TERM

TERMTERM

nn

4

2

2

2

2

3

21

1

2


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Where M is the number of nodes in the domain , Nj is the shape function ofjth

node and

j is intensive property at jth

node.

+1 + +

2

+

+

+

22 +22 = 0

For convenience integration is done by term by term.At the end it will be assembled

together.

Term 1

+1 = +1

=

+1 =

+1 = .

Where - mass matrixTerm 2

+ =

+

=

+ =


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Where - advection matrix

Term 3

2

+

+

+

=2

+

+

+

+

=2

+

+

+

2 +

+

+

+

ij iU F

where -


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Term 4

22 +22

= + + = +

+

= + +

=

Thus, the formulation becomes fully in explicit form.

1

[ ] [ ] 0...........(2.8)2

n nn n nt

M A U K ft

2.3.3 Time Marching Method

Explicit and implicit time stepping are commonly used for the resulting system of

ordinary differential equations. Implicit schemes are unconditionally stable but involve

inversion of large banded unsymmetrical matrices. As the number of variables increases,

matrix inversion becomes computationally costlier. The explicit schemes are

conditionally stable but do not involve matrix inversion. Explicit methods are of interest

in modern CFD application mainly for time dependent flows.

2.3.4. Stability criterion

For explicit solution, the value of cannot be any arbitrary value, indeed, it mustbe less than or equal to some maximum value. This max value is usually estimated by.


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< [ , ] Or1

*min ,

| | | |cfl

x yN t

u v

This is the famous courant Fredrich-Lewy (1967) or CFL criterion, which means

that, t must be less than the time required for a wave to propagate between to adjacent

grid points .NCFL is known as the CFL number.


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CHAPTER 3

INCOMPRESSIBLE FLOW SOLVER

3.1.Governing Equations

The governing equations for incompressible flow are

X-momentum equation

2 2

2 2

1 ...........(3.1)

u u u P u uu v

t x y x x y

Y-momentum equation

2 2

2 21 ...........(3.2)v v v P v vu v

t x y y x y

The continuity equation is

0.............(3.3)u v

x y

3.2.Velocity correction method

The solution to the incompressible Navier-stokes equations is difficult due to the absence

of an explicit pressure term in the continuity equation. The 2-D NS equation are as

follows.

2 2

2 2

1...........(3.4)

u u u P u uu v

t x y x x y

2 2

2 2

1...........(3.5)

v v v P v vu v

t x y y x y

0...........(3.6)u v

x y

In the velocity correction method velocity and pressure at a time tn+1

are obtained in three

steps, steps starting from the values at a time tn

as follow


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1) Obtain an intermediate fictitious velocity field after dropping the pressure gradientterm from the momentum equations.

*

...........(3.6)n

nu uRHSu

t

*

...........(3.7)n

nv vRHSv

t

Where,

RHSu=RHS of x-momentum equation 3.4 excluding pressure term

RHSv=RHS of y-momentum equation 3.5 excluding pressure term

3) The final velocities are obtained from the previous steps. Without dropping the

pressure field, find the velocity field.

1 1

...........(3.8)

n nnu u P

RHSut x

1

1...........(3.9)

n nnv v P

RHSvt y

Subtracting 3.7 from 3.9 and 3.8 from 3.10

We get the final velocity field.

1 * 1..........(3.10)

nu u P

t x

1 *

1............(3.11)

nv v P

t y


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2) Compute the pressureThe pressure Poisson equation is obtained by forcing the continuity requirement on

the final velocity field

Differentiating equation 3.10 with respect to x

Differentiating equation 3.11 with respect to y

And adding we get

1 * 2 2

2 2

n

u v u v t P P

x y x y x y

Continuity equation

* 2 2

2 2...........(3.12)

u v t P P

x y x y

Equation 3.12 is termed the pressure poisson equation. This equation is

solved using the iterative conjugate gradient method with tolerance

10E-20


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CHAPTER 4

VALIDATION OF THE INCOMPRESSIBLE FLOW SOLVER

4.1. Flow over a backward facing step

The study of backward facing step flows constitutes an important branch of fundamental

fluid mechanics. The backward step apparatus consists of a single backward- facing step

mounted in a two-dimensional channel. The geometry is exceedingly simple yet it

provides a large amount of fluid mechanical data. This problem is used to validate the

incompressible flow solver. With water as the media, results provided by the code when

applied to the problem of 2-D backward facing step flow are compared with experimental

data. The length of the recirculation zone is used to make the comparison. The results not

only yield the expected primary recirculation zones but also show an additional region of

separation downstream of the step. For obtaining numerical predictions, the two-

dimensional steady differential equations for conservations of mass ad momentum were

solved. Results are reported and are compared with experiments conducted for those

Reynolds numbers for which the flow maintained its two-dimensionality. Under these

circumstances, good agreement is found between experimental and numerical results.

Figure 4.1 Backward-facing step geometry

The open water-driven flow channel that was used for this study is figure. It incorporates

a two-dimensional step that has provided an expansion ratio of 1:1.19423. The expansion

ratio is defined as H/h=1+S/h. The expansion ratio and the Reynolds number are the two


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parameters that affect the reattachment length. In the current study, the expansion ratio is

kept constant and the reattachment length is plotted for varying Reynolds numbers (10E-

3


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Outlet: Atmospheric pressure of one bar is specified at the outlet. The outlet channel is

also sufficiently long so that the flow redevelops at the exit. In the range of Reynolds

numbers investigated, the flow is affected very little by the length of the outflow channel,

assuming that the length of the flow channel was correctly estimated using before

depending on the outlet channel.

The Reynolds number was calculated using the formula where

ReVD

V is two thirds of the maximum measured inlet velocity, which corresponds in the

laminar case to the average inlet velocity, D is the hydraulic diameter of the inlet(small

channel) and is equivalent to twice its height(2h) and is the kinematic viscosity.

The flow over the backward step is two-dimensional and non-oscillatory in the region,

Re


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In the above figures, the streamlines for the Reyolds numbers 200,400,600 and 800 are

presented on the contour plot. In the first two cases, the flow follows the upper corner

without revealing a flow separation. Furthermore, a corner vortex is observed at the

concave corner behind the step. In the range of very small Reynolds numbers,(Re=10E-3)

the size of this vertical structure is nearly a constant varying between x1/S=0.350 and

x1/S=.365. Under these conditions, the effect of inertia forces can be assumed negligible

to the viscous forces often denoted as molecular transport. Hence the flow resembles a

Stoke flow.

In addition to a primary circulation region, there exists a secondary recirculation region

near the upper wall for Re>400. The adverse pressure gradient due to the sudden

expansion at the step induces this separated flow. In the previous figures, the secondary

recirculation zones were clearly visible for Re=600,800. The size of the secondary

recirculation zones increases with increasing Reynolds number, while at the same time,

the flow structure is moving in a stream wise direction. Far downstream, the flow

recovers to a parabolic profile. However, at high Reynolds numbers(800), the flow

recovery takes more than 20 step heights.

The next figure shows the reattachment length plotted against increasing Reynolds

number. The reattachment length is measured as the distance from the step to the point

where the velocity gradient becomes positive. This value is normalized by the step

height.


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As mentioned above, the size of the first eddy is nearly a constant for all Reynolds

numbers below the Re=1. This is depicted in the above figure showing the length x1 of

the first corner eddy behind the step (recirculation region) normalized by the height S as a

function of RE. However, for Re>1, the corner vortex strongly increases in size. As a

direct consequence, the corner vortex reaches upto the corner of the step at Re=10 and

covers the complete face of the step. Hence, a change in the entire flow structure is

observed and the notation of a corner vortex has been replaced by the notation

recirculation region, which for Re>10, better reflects the flow structure. With increasing

Reynolds number, the size of the recirculation region increases.

0

2

4

6

8

10

12

14

0.001 0.01 0.1 1 10 100 1000

Reynolds numbe r

Reattachmentlength

computed results

G.Biswas

Figure 4.4: Variation of reattachment length vs Reynolds number


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31

Figure 4.5: Variation of Reattachment length vs Reynolds Number

The agreement of the present numerical predictions and the experimental results of G.

Biswas is excellent for Re400, a discrepancy is visible between

computations and experiments which increases with Reynolds number. This is because

beyond, Re-400, the flow cannot be accurately modelled by two-dimensional simulations.

The sidewall influences the structure of the laminar flow beyond the step. The measured

reattachment length for the mid-plane does not match with the predictions of the two-

dimesnional predictions. The discrepancy between experimental result and the numerical

results are a consequence of the three-dimensionality affecting the flow structure at the

mid-plane of the channel.

0

2

4

6

8

10

12

14

16

0 200 400 600 800 1000

Reynolds number

Reattachmentlength

Computed_results

experimental_results


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32

CHAPTER 5

COMPRESSIBLE FLOW SOLVER

5.1. Governing Equations

The governing equations for compressible flow in conservative form are

0u u

t x y

20

xx xy

uu P T uv T

t x y

. (5.1)

20

xy yy

vuv T v P T

t x y

.. (5.2)

Substitute

2xxu

T

x

xy

u vT

y x

2yy

vT

y

Expanding the Above equations, we get

2 2

2 2u u u u v u u u v P u v u vt x y x y x y x x y x


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2 2

2 2

v v v u v v v u v P u v v

t x y x y x y y x y y

The various terms of the above equations are approximated as per the scheme given in

the previous chapter

Here (u) and (v) replace in the general transport equation

5.2. Methodology

2 2

2 2

u u u u v u u u v P u v u v

t x y x y x y x x y x

...........(5.3)

2 2

2 2

v v v u v v v u v P u v v

t x y x y x y y x y y

...........(5.4)

The velocity correction method is used to calculate final values of u and v

............(5.5)u PRHSut x

...........(5.6)v P

RHSvt y

1)Obtain an intermediate fictitious velocity field after dropping the pressure gradient term

from the momentum equations.

*

...........(5.7)

n

nu u RHSut

*

............(5.8)n

nv vRHSv

t

Where,


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35

1 *2 2

2 2

n

u v u v P P t

x y x y x y

Substitute from Continuity equation,

* 2 2

2 2...........(5.13)

u v P P t

t x y x y

Assuming that density varies only with pressure,

Using the universal gas equation, =

2

1P P

t P t c t

This equation is solved using the iterative conjugate gradient method

with tolerance 10E-20

*2 2

2 2

1...........(5.14)

2

P u u P Pt

c t x y x y



10E-20

Where

2c RT

To calculate density, assuming adiabatic conditions,

2 2

02

u vT T

cp

2 2

0 2

( ) ( )

2

u vT T

cp


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P

RT

Substituting,

2 22

0 2

( ) ( )

( )2

u v

T T RT cpP

The above quadratic equation is solved to find out temperature values

P

RT

Finally the velocity u and v are found out.

uu

vv

The algorithm for the above procedure is given below

while(time


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37

CHAPTER 6

UNIFIED FORMULATION

A unified procedure to handle both incompressible and compressible fluids is proposed.The flow is assumed to be isothermal at low Mach numbers. Then, the equations of state

are used to calculate density directly from Pressure. Different correlations will be used

for compressible and incompressible fluids to calculate density. The user only needs to

specify whether the fluid is compressible or incompressible through the input files. The

advantage of such a formulation is that both incompressible flows can also be simulated

by making this modification to the compressible flow formulation. The initial values of

velocities, density and pressure are calculated from the input Mach number and adiabatic

relations.

6.1. Methodology

2 2

2 2

u u u u v u u u v P u v u v

t x y x y x y x x y x

.................(6.1)

2 2

2 2

v v v u v v v u v P u v vt x y x y x y y x y y

.................(6.2)

The modified version of the earlier described velocity correction method is used to

calculate final values of u and v

11

.............(6.3)

nn nu u P

RHSu

t x

11

..............(6.4)

nn nv v P

RHSvt y

1)Obtain an intermediate fictitious velocity field after reducing the pressure gradient

term to the previous time step level in the momentum equations.


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38

*

.............(6.5)

nnu u P

RHSut x

*

..............(6.6)

nnv v P

RHSv

t y

Where,

RHSu=RHS of x-momentum equation 6.1 with pressure term in nth

time level

RHSv=RHS of y-momentum equation 6.2 pressure term in nth

time level

3) The final velocities are obtained from the previous steps. Without dropping the

pressure field, find the velocity field.

11

...........(6.8)

nn nv v P

RHSvt y

Subtracting 6.5 from 6.7 and 6..6 from 6.8

We get the final velocity field.

1 * 1( )

............(6.9)n n n

u u P P

t x

1 * 1( )............(6.10)

n n nv v P P

t y

Where1n n

P P P

11

...........(6.7)

nn nu u PRHSu

t x


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39

2)Compute the pressure

The pressure Poisson equation is obtained by forcing the continuity requirement on

the final velocity field

Differentiating equation 6.9 with respect to x

Differentiating equation 6.10 with respect to y

And adding we get

1 * 2 2

2 2

n

u v u v P P t

x y x y x y

Substitute from Continuity equation,* 2 2

2 2...........(6.11)

u v P P t

t x y x y

1n nP P P Assuming that density varies only with pressure,

Using the universal gas equation, =

2

1P P

t P t c t

This equation is solved using the iterative conjugate gradient method

with tolerance 10E-20

* 2 2

2 2

1...........(6.12)

2

P u u P Pt

c t x y x y

* 2 2

2 2

1...........(6.13)

2

P u u P Pt

c t x y x y


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40



10E-20

Where

for air,

2 Pc

for water,

2 7.0 *( 3.0 8)P Ec

To calculate density,

Using Stiffened Equation of State

1.0

infinf

...........(6.14)nP

P

Eqn (6.13) is an empirical correlation that gives the relationship between density and

pressure for any substance by substituting the corresponding values for and n.

For water, = 3.0E8 and n=7

1.0

7.0

inf

inf

3.0 83.0 8

P EP E

For air, =0 and n=


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1

inf

atm

p

P

Finally the velocity u and v are found out.

u= (u)/()v = (v)/()

This formulation solves the compressible Navier Stokes equations for any liquid or gas,

whether the fluid is compressible or incompressible. The next chapter discusses various

validations for the unified formulation.


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CHAPTER 7

VALIDATION OF UNIFIED FORMULATION

7.1. Flow over NACA-0015 Hydrofoil

The NACA-0015 hydrofoil geometry was used to validate the unified formulation for

solving both compressible and incompressible flow. Liquid water was used as a fluid.

At the Mach number, considered in the test case, flow is incompressible. That is, the

density changes with time can be considered a negligible.

The results obtained by the code are compared with experimental results. The pressure

coefficient (Cp) is plotted against the displacement vector non-dimensionalised by the

chord length of the aerofoil.

Computational Domain

Figure 7.1:Computational domain

A mesh of triangular elements of size 0.1 was used.

Nodes 32142 Elements 62178 Chord length 130mm


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Boundary Conditions

The pressure inlet was given as P=59,000Pa. Velocity at inlet is3.11m/s and temperature,

T=298 K. These conditions correspond to an inlet Mach number of 0.0021.

The entire aerofoil is considered as a wall and no slip condition is specified. Further the

top and bottom surfaces are also walls. The boundary conditions are similar to that

present while testing flow over an aerofoil in a wind tunnel.

Inlet : u=uinf, v=0

Walls: u=0, v=0Outlet: P=Patm.

Results

Velocity plot

The contours of the x-directional velocity(u) is shown in figure. Streamlines are plotted

across the contour plot, to indicate the flow direction. From the figure, it is observed that

there is no separation involved and the streamlines smoothly follows the contours. There

is a stagnation point on the nose of the aerofoil, where the u-velocity hits a minimum.


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Figure 7.2:u-velocity contour map

Pressure Plot

Figure 7.3:Pressure contour map


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The Pressure field is displayed in the above contour. It is observed that the region where

the velocity field intensity was at a minimum is the region where the pressure field

intensity is a maximum. This is the stagnation point, at the nose of the aerofoil.

Pressure Coefficient

The pressure coefficient, Cp is defined by

Cp = P-Pinf/(0.5v2)

Pinf is the inlet pressure, 59000Pa.

The pressure coefficient is calculated by the code and these values are compared with the

experimental values, Rapposelli et al,(2003)

The plot has been inverted to depict the upper surface on top. The pressure coefficient is

higher at the lower surface, due to the pressure being higher at the lower surface. This is

what creates lift in an aerofoil shaped wing.

-2

-1.5

-1

-0.5

0

0.5

1

1.5

-0.2 0 0.2 0.4 0.6 0.8 1 1.2

x/c

PressureCoefficient

Computed

resultsExperimental

results

Figure 7.4:Cp vs x


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46

The experimental values give a match with the computed values, validating the code. An

oscillation is observed on the top vertex of the plot. This corresponds to the location of

the stagnation point.

7.2. Flow over a Circular Cylinder

Flow over a circular cylinder has been of interest for many decades, as it offers a

full range of phenomena from the laminar, periodic shedding of vortices, transition to

turbulent, and fully turbulent flow regimes-making it a challenging problem for

computational simulation. Computational studies of flow over a circular cylinder began

as early as the 1930s (e.g., Thom, 1933), and this continues to be a popular subject.

In this section, a circular cylinder is first presented followed by vortex shedding cases

with higher Reynolds numbers up to 1000. This represents a simple case for external

flows, for the purpose of testing the previously described unified pressure projection

formulation without any transition and turbulence modeling.

For external flows in which the computational domain extends a large distance from the

body, the pressure waves originating from the body surface propagate into the far field.

As the Reynolds number increases above 40, a non-symmetric wake develops and

periodic vortex shedding sets in.

The Reynolds number range of periodic vortex shedding is divided into two distinct sub

ranges. At Re=40 to 50, called the stable range, regular vortex streets are formed and noturbulent motion is developed. The range re=150 to 300 is a transition range to a regime

called the irregular range, in which turbulent velocity fluctuations accompany the

periodic formation of vortices. The turbulence is initiated by laminar-turbulence

transition in the free layers which spring from the separation points on the cylinder. This

transition first occurs in the range Re=150 to 300.From spectrum and statistical


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measurements, it is known that in the stable range, the vortices decay by viscous

diffusion. In the irregular range the diffusion is turbulent and the wake becomes fully

turbulent in 40 to 50 diameters downstream.

The shedding frequency is usually expressed in terms of the dimensionless Strouhal

number, S = n1d/U where n1 is the shedding frequency from one side of the cylinder, d is

the circular diameter, and U is the free stream velocity. The Strouhal number S may

depend on Reynolds number, geometry, free-stream turbulence level, cylinder roughness,

and so forth. The principal geometrical parameter is the cylinder shape (for other than the

circular cylinders, d is an appropriate dimension). Usually the geometrical parameters are

fixed, and then S is presented as a function of Reynolds number Re. Instead of Strouhal

number, it is sometimes convenient to use the dimensionless parameter F=n1d2/v where

v is the kinematic viscosity.

Within the lower end of the shedding range, there are three characteristic Reynolds

number ranges, within the lower end of the shedding range. These will be called as

follows

Stable range 40


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48

Computational Domain

.

Figure 7.5:Mesh and computational domain

Nodes 6972 Elements 13598 Cylinder dia 40mm

Boundary conditions

The left vertical edge of the rectangular domain was considered as the inlet. At the inlet

various uniform velocity profiles were described. The vertical velocity (v) in the y-

direction was always kept zero.

The outlet boundary condition was prescribed at the right vertical edge of the domain.

The pressure here was kept constant at atmospheric pressure (1bar).

The edge of the circular cylinder and the top and bottom sides of the domain were kept as

walls. Throughout the circular edge of the cylinder, no slip condition was prescribed.


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However, at the top and bottom walls, a uniform free stream velocity was enforced in the

x-direction. This is to prevent the influence of the boundary layer from affecting the flow

conditions at the cylinder walls. The y-directional velocity was kept zero at all the wall.

Inlet: u=uinf, v=0

Walls (circular cylinder): u=0, v=0

Walls (domain boundaries): u=uinf, v=0

Outlet: P=Patm.

Results

Figure shows the separation length of separation non-dimensionalized by the cylinder

diameter, D for the Reynolds number, Re=40.


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Figure 7.6: Length of vortex vs time

The shedding frequency was calculated using the formula (n1d/u), where n1

is the frequency of vortex shedding. In experimental setups , the fluctuating

velocity or pressure signals were captured using measurement apparatus The

output would be amplified and then fed into an oscilloscope. Typically a

sinusoidal wave is observed for each eddy frequency.

In the present numerical study, pressure variation with time was monitored

at a point behind the cylinder. These values were plotted to get a sinusoidal

wave. The frequency response was obtained after taking Fast Fourier

Ls


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transform of the pressure-time plot. Using the dominant frequency, the

Strouhal number has been calculated for each Reynolds number.

Figure 7.7: Strouhal Number vs Reynolds number


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Re=216

Re=400

Re=1000


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The above figures are contour plots of the vorticity function across the

domain. It can be seen that the vortex shedding is minimum in the first plot,

Re=100. As the Reynolds number increases, vortex shedding increases and

becomes irregular.

7.3. Flow over a Backward facing step

Using the same geometry used in the previous chapter, the unified

formulation code was run for with air as fluid.

The flow phenomenon observed for air as fluid are somewhat different

observed for water.Length of the primary separated-flow region is predicted

to increase nonlinearly with Reynolds number up to Re 420. The ensuing

decrease with Reynolds number is caused by the additional region of

separated flow that occurs on the wall opposite to the step and a t Reynolds

numbers larger than Re > 420. The longitudinal dimensions of this

additional region of separated flow are predicted to increase with Reynolds

number up to Re 980,above which two more regions of recirculating flow

are predicted. Both are located on the channel wall with the step; one inside

the primary region of separated flow and the other downstream of its

reattachment line.

The two-dimensional flow predictions using the code also yielded multiple

regions of separated flow according to experimental observations. In order to

compute the flow inside these regions accurately, the grid distribution would

require further refinements, i.e. more grid points would be needed inside the

computational domain, especially in the region where the additional


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separated-flow regions occur. Converged and grid-independent solutions

could be obtained that are in good agreement with experimental findings up

to a Reynolds number Re 400. For Reynolds numbers in excess of this value

the numerical results also show multiple regions of separated (recirculating)

flow at the wall opposite to the two-dimensional step and on the channel side

where the step is located. Unfortunately,with the occurrence of more than

one separated flow region, the flow in the experiments becomes three-

dimensional in the region downstream of the step, and this prevents direct

comparison between the experimental and theoretical results.

The following are the numerical results compared along with experimental

findings.

Figure 7.8:Reattachment length vs Reynolds number


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CHAPTER 8

ORIFICE METER

An orifice meter is a device used for measuring the volumetric flow rate. It uses the same

principle as a Venturi nozzle, namely Bernoulli's principle which states that there is arelationship between the pressure of the fluid and the velocity of the fluid. When thevelocity increases, the pressure decreases and vice versa.

An orifice plate is a thin plate with a hole in the middle. It is usually placed in a pipe inwhich fluid flows. When the fluid reaches the orifice plate, the fluid is forced to converge

to go through the small hole; the point of maximum convergence actually occurs shortly

downstream of the physical orifice, at the so-called vena contracta point. As it does so,

the velocity and the pressure changes. Beyond the vena contracta, the fluid expands andthe velocity and pressure change once again. By measuring the difference in fluid

pressure between the normal pipe section and at the vena contracta, the volumetric and

mass flow rates can be obtained from Bernoulli's equation.

The code developed was applied to the problem of flow through an orificemeter. To measure the flow rate using an orifice meter, the pressure needs to

be measured between two points upstream and downstream of the plate.

However, it is observed that due to vortex formation immediately after theplate, the pressure cannot be tapped until the flow attains steady state.

An alternative design proposes to use multiple holes in the plate. It isproposed that this arrangement would reduce the vortex formation and hence

enable pressure measurement closer to the plate. This would lead to


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Figure 8.2:U-velocity contour map for triple hole slit

Figure shows the multiple vortices that develop immediately after each hole. Comparingthe above two figures it is seen that the vortex size is considerably smaller in the triple

hole plate case.


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Chapter 8

Conclusions

In this work, a unified formulation to solve incompressible and compressible

flow regardless of the fluid has been suggested. This formulation based on

the fractional step algorithm is a new approach. Many validation test cases

have been done in this thesis work. The results of the test cases suggest that

the unified formulation with the fractional step algorithm is Mach number-

independent regardless of the medium. More comprehensive validation is

necessary to make a definitive conclusion of this approach. However, in

general, a unified approach can be very useful for many engineering

applications where flow speed is in a wide range or where both

incompressible and compressible flow coexist, such as in a multi-material or

multi-phase flow.

In future, the formulation can be extended to simulate even multi-phase,

flow mixing and cavitation.


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59

NOMENCLATURE

Density ,Kg/m3

Cp Specific heat , J/KgK

K Conductivity, W/mK

T Temperature, K

t Time, sec

Thermal Diffusivity,m2/s

Ni, Nj Shape functions

h Heat transfer coefficient, W/m2K

Q Heat generated per unit volume, J/m3

t Time step

L Characteristic length of the element, m

T Ambient temperature, K


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APPENDIX A

Conjugate Gradient method

In Implicit formulation, equation 3.9will takes the form AX=B, Which is a system of

linear equations. The discrete system of equations from the finite element method must

be solved by a numerical solver. The Conjugate Gradient method is chosen as a method

for solving systems of linear equations. The method requires a symmetric positive

definite (SPD) system of equations that is a system of the form:

Algorithm for CG method is given below.

AX=B

x0=Initial guess vector.

Theoretically the Conjugate Gradient method will converge and reach the exact

solution in at most N steps for N degrees of freedom. Each step k projects the exact

solution into the k-dimensional solution space spanned by the A conjugate basis vectors.

In practice, each step will not be solved exactly due to round-off error in the


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computations. The method will however in general converge to an acceptable error-

tolerance in far less than N iterations. This rapid convergence is one of the greatest

strengths of the method.

Appendix B

EVALUATION OF MATRICES

1. Mass matrix, M

The mass matrix is defined as

M =V

NiNjdA

=

1

2 1 2 3

3

L

L L L L dA

L

=

21 1 2 1 3

2 1 2 2 2 3

3 1 3 2 3 3

L L L L L

L L L L L L dA

L L L L L L

=

2!0!0! 1!0!0! 1!0!0!

(2 2 0 0)! (2 2 0 0)! (2 2 0 0)!

1!0!0! 2!0!0! 1!0!0!2

(2 2 0 0)! (2 2 0 0)! (2 2 0 0)!

1!0!0! 1!0!0! 2!0!0!

(2 2 0 0)! (2 2 0 0)! (2 2 0 0)!

A

=

2 1 1

1 2 112

1 1 2

A

Lumping the mass matrix:


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LM =

4 0 0

0 4 012

0 0 4

A

=

1 0 0

0 1 03

0 0 1

A

2. Advection Matrix, A

A =V

j jNi u v dA

x y

A =

1 2 3

1 1 1

1 2 32 2 2

1 2 33 3 3

A

N N NN N Nx x x

N N Nu N N N

x x x

N N NN N N

x x x

1 2 3

1 1 1

1 2 3

2 2 2

1 2 3

3 3 3

A

N N NN N N

y y y

N N N

v N N N y y y

N N NN N N

y y y

=

1 2 3

1 2 33

1 2 3

b b bA

u b b b

b b b

1 2 3

1 2 33

1 2 3

c c cA

v c c c

c c c

3. Upwind matrix, U

The upwind matrix is obtained as:

U = 2 2i j i j i j i j

A

N N N N N N N Nu uv vu v dA

x x x y y x y y


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63

=

1 1 1 2 1 3

2 1 2 2 2 32

3 1 3 2 3 3

A

N N N N N N

x x x x x x

N N N N N Nu dA

x x x x x x

N N N N N N

x x x x x x

+

1 1 1 2 1 3

2 1 2 2 2 3

3 1 3 2 3 3

A

N N N N N N

x y x y x y

N N N N N Nuv dA

x y x y x y

N N N N N N

x y x y x y

+

1 1 1 2 1 3

2 1 2 2 2 32

3 1 3 2 3 3

A

N N N N N Ny y y y y y

N N N N N Nv dA

y y y y y y

N N N N N N

y y y y y y

= 2 21 1 1 2 1 3 1 1 1 2 1 3 1 1 1 2 1 3

2 1 2 2 2 3 2 1 2 2 2 3 2 1 2 2 2 3

3 1 3 2 3 3 3 1 2 2 2 3 3 1 3 2 3 3

b b b b b b b c b c b c c c c c c c

A u b b b b b b uv b c b c b c v c c c c c c

b b b b b b b c b c b c c c c c c c

4. Stiffness Matrix

The stiffness matrix is given as :

K =i j i j

A

N N N NdA

x x x y

=

1 1 1 2 1 3

2 1 2 2 2 3

3 1 3 2 3 3

A

N N N N N N

x x x x x x

N N N N N NdA

x x x x x x

N N N N N N

x x x x x x


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64

1 1 1 2 1 3

2 1 2 2 2 3

3 1 3 2 3 3

A

N N N N N N

y y y y y y

N N N N N NdA

y y y y y y

N N N N N Ny y y y y y

=

1 1 1 2 1 3 1 1 1 2 1 3

2 2 2 2 3 2 1 2 2 2 3

3 1 3 2 3 3 3 1 3 2 3 3

1

b b b b b b c c c c c c




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APPENDIX C

THE TWO-DIMENSIONAL HEAT CODUCTION EQUATION

AND ITS NUMERICAL MODELLING

In this chapter the governing equations for unsteady two dimensional heat

conduction and its boundary conditions are explained first .The finite element

formulation and solution technique is explained next.

C.1. The governing equations and boundary conditions.

The two dimensional unsteady heat conduction governing equation is given by

p

TTKyKx

yT xC Q

x x y

Valid over surface S bounded by a line L with

-T T T

Kx nx Ky ny Kz nz

x y z

=q

Over a portion Lb of the boundary line L, a heat flux q is applied. Over the rest of the

boundary line La,T=Ts

Figure C.1:Two-dimensional surface


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66

Over the portion La of the boundary line where T is specified, the gradient terms

and q are zero. The quantities nx, ny are the direction cosines of the unit vectors normal

to the respective surfaces. Applying Galerkin weak formulation of the method of

weighted residuals to the governing equation the above equation becomes

S S

T Nj T Ni T Ni Cp dxdy Kx Ky dxdy

t x x y y

0S L

T TNiQdxdy Ni Kx nx Ky dL

x y

This equation can be further modified to the generalized Galerkin statement

j ji i

p i j x y j

S S

N NN NC N N dxdy K K dxdy T

x x y y

Q 0i i

S L

N dxdy N qds

Figure C.2:Triangular Element

with Boundary Condition

enforced on 1 side

In the matrix form as

CT + KT+ F = 0

Where C is the capacitance matrix, K is the conductivity matrix and F is the load vector

.


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67

Where C= 1

2 1 2 3

3

p

N

C N N N N dxdy

N

j ji i

x y

S

N NN NK K K dxdy

x x y y

i i

S L

F N Qdxdy N qdL

In this general equation any type of 2-D element can be chosen. We have used

three noded linear triangular elements throughout the work.

The elemental matrices for a typical triangular element can be written as follows.

The shape functions for the linear triangle is given as

N= [N1N2 N3]

The capacitance matrix for constant Cp is given as

C pC NiNjdxdy 1

2 1 2 3

3

N

Cp N N N N dxdy

N

which can be simplified to

C=

2 1 1

1 2 112

1 1 2

pA C

The conductivity matrix is given as

K=j ji i

x y

N NN NK K

x x y y

dxdy


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=area*

1 1 1 2 1 3

2 1 2 2 2 3

3 1 3 3 32

x

b b b b b b

K b b b b b b

b b b b b b

+area*

1 1 1 2 1 3

2 1 2 2 2 3

3 1 3 2 3 3

y

c c c c c c

K c c c c c c

c c c c c c

Where

2 312

y yb

A

3 212

x xc

A

1 2 2 1 3 1 1 3 2 3 3 22 ( ) ( ) ( )A x y x y x y x y x y x y

The load vector is evaluated as

F=FQ+ Fq

FQ=S

NiQdxdy = 113

QA

( )ql

F Niq dl = 112

ql

=

2 1 0

1 2 06

0 0 0

hl

[T-T]

Where l is the length of the edge where the boundary condition is applied.

To integrate this equation in time, both implicit and explicit schemes can be used.

C.2. Explicit Time integration

In the current chapter explicit time integration is explained ,which is

the most simplest formulation and also easily understandable. A s a continuation from the

previous chapter the unsteady heat conduction equation can be written in matrix form as

0...........( .1)CT KT F C


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Where ,

C, Capacitance or mass matrix

K, Stiffness or conductance matrix

F, Load matrix

1

0...........( .2)n n

nnT TC K T F C

t

Above form can be written in explicit scheme as,

To avoid inversion of the capacitance matrix lumping of that matrix is employed.

Here all the elements in a row are added together and written in the diagonal as explained

below. The original matrix is called the consistent mass matrix and the resulting diagonal

matrix is called the lumped mass matrix.

[C]=

2 1 1

1 2 112

1 1 2

pA C

=

4 0 0

0 4 012

0 0 4

pA C

=

1 0 0

0 1 03

0 0 1

pA C

Using the lumped matrix equation, (3.2) can be written as

1

...........( .3)

nn n n

C T C T t K T F C

From the equation 3.4 it is clear that all the terms in the RHS are known values.

Explicit schemes are restricted by the stability criterion given by

20.5...........( .4)

tC

L

Where L is the characteristic length of the element. The t for the numerical

integration is derived from the above equation.

C.2 Implicit time integration

C.2.1Pure implicit scheme

The equation 2.1 can be written in implicit formulation as


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1

10...........( .5)

n nnnT T

C K T F C t

This scheme is highly stable and gives non-oscillatory solutions for very large

time step. But it is first order accurate.

C.2.2.Crank-Nicholson Scheme

1 1

10...........( .6)

2

n n n nnT T T T

C K F C t

This scheme is second order accurate, but gives oscillatory solution for large time steps.

C.2.3.Three point scheme.

1 1 1 1

10..........( .7)

2 3

n n n n nnT T T T T

C K F C t

This scheme is useful for non-iterative solutions of non-linear equations. But this

scheme is not self starting, since it requires starting solutions at t = n-1 and t = n.

From the pure implicit scheme equation 3.6 can be written as

1 0.............( .8)n n

C CK T T F C

t t

Knowing the initial condition profile at t=0.0,which will be generally ambient

condition, this system can be solved repeatedly solve for different time steps. Since this is

a symmetric linear system of equations, we have used iterative Conjugate Gradient

solver. The details of this solver are explained in appendix.

For nonlinear equation, matrices C and K depends on temperature(T).In implicit

scheme they have to be evaluated at (n+1)th

level. This requires an iterative solution till

convergence. After each iteration the temperature is evaluated as follows.

(1 ) ............( .9)new new old T T T C


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is called as relaxation factor. When1 is

called over relaxation. Their value varies depending on the non-linearity of the problem.

A C++ code has been written based on the above finite element

formulation (explicit as well as implicit) and the flow chart of the same is shown in fig .

Figure C.3:Flow chart of algorithm

C.3. Semi-infinite media

This is a validation problem for the unsteady two-dimensional heat equation solver.

A semi-infinite solid is an idealized body that has a single plane surface and extends to

infinity in all directions, as shown in Fig. 4.16. This idealized body is used to indicate

that the temperature change in the part of the body in which we are interested (the region

close to the surface) is due to the thermal conditions on a single surface. The earth, for


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example, can be considered to be a semi-infinite medium in determining the variation of

temperature near its surface. Also, a thick wall can be modeled as a semi-infinite medium

if all we are interested in is the variation of temperature in the region near one of the

surfaces, and the other surface is too far to have any impact on the region of interest

during the time of observation.

Consider a semi-infinite solid that is at a uniform temperature Ti. At time t=0 , the

surface of the solid at x=0 is exposed to convection by fluid at a constant temperature ,

with a heat transfer coefficient h . This problem can be formulated as a partial differential

equation, which can be solved analytically for the transient temperature distribution

T(x,t) .

The exact solution of the transient one dimensional heat conduction problem in a semi-

infinite medium that is initially at a uniform temperature of Ti and is suddenly subjected

to convection at time t=0 has been obtained, and is expressed as

Where the quantity erfc() is the complementary error function, defined as

Using the two-dimensional heat equation solver described in this chapter, the problem of

semi-infinite media with convective heat flux and convection boundary conditions on one

surface have been solved. All other surfaces are insulated.The various constants used in the formulation are explained above along with their values

C3.1.Convective boundary Condition

Properties used are

Density, =1000 kg/m3


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Specific heat, Cp=2000 J/kgK

Conductivity=20 W/ m2K

Boundary conditi ons

At X=0.0mm

h=1500 W/m2K

T=2200 K

All other faces are insulated.

I nitial condition

T(x,y,0) =T0= 300.0 K

Figure C.3.Temperature contour inside the rod after 1 sec


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Figure C.4. Graph showing solver result with the analytical solution for convective boundary condition.

From the graph it is observed that the code results match with the analytical results. Hence the code is

validated for convection boundary condition

C3.2.Heat flux boundary condition

Properties used are

Density, =1000 kg/m3

Specific heat, Cp=2000 J/kgK

Conductivity=20 W/ m2K

Boundary conditi ons

At X=0.0

q=10000 W/m2

All other faces are insulated

I nitial condition

T(x,y,z,0) =T0= 300.0 K

Temperature vs X

0

100

200

300

400

500

600

700

800

900

0 20 40 60

X(mm)

T(K)

analyticalresults(1sec)

computedresults(1sec)

computedresults(5sec)

analyticalresults(5sec)


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Figure C.5. Graph showing comparison of solver result with analytical solution for heat flux boundary

condition

299.5

300

300.5

301

301.5

302

302.5

303

303.5

304

304.5

0 10 20 30 40 50 60x

Temperature(K)

Computedresults(1sec)

Analyticalresults(1sec)

Computedresults(5sec)

Analyticalresults(5sec)


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REFERENCES

[1] Darell W pepper, Juan C Henrich; Finite element methodHemishere publishing

company[2] Frank P Incropera , David P. Dewitt 'Fundamentals of heat and mass transfer',

WILEY INDIA Edition.

[3] Blazek, Computational Fluid Dynamics

[4]Dochan Cetin, Kwak&Kiris, Computation of Viscous Incompressible Flows, Springer

publications

[5] B.F. Armaly, F.C. Durst, J.C. Pereira and B. Schonung, Experimental and theoretical

investigation of backward facings step flow

[6]G. Biswas, M. Breur, F. Durst, Backward facing step flows for various expansion

ratios at low and moderate Reynolds numbers.

[7]AIAA-85-1689, On the Accuracy of the Pseudo compressibility method in Solving

incompressible Navier-Stokes Equations, S.E. Rogers, D.Kwak and U. Kaul

[8]An upwind differencing scheme for the time-accurate incompressible Navier-Stokes

Equations, Stuart E. Rogers, DochanKwak

[9] Three Dimensional instability in flow over a backard facing step, Dwight Barkley, M.

Gabriela, M . Gomes and Ronald D. Henderson

[10] John D. Anderson, Computational Fluid Dynamics, the Basic applications

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