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Construction of Mass Conservative Streamtube

for 3D CFD Velocity Fields

By

Nawin RAJ

A Thesis Submitted in Partial Fulfillment of the

Requirements for the Degree of

Master of Science in Mathematics.

School of Computing, Information and Mathematical Sciences,

Faculty of Science and Technology,

The University of the South Pacific.

2008

Abstract

Streamtube is used to visualize expansion, contraction and various properties of the

fluid flow. These are useful in fluid mechanics, engineering and geophysics. The mass

conservative streamtube constructed in this thesis only reveals the flow expansion and

contraction rates along streamline. The mass conservative streamtube will be

constructed based on the mass conservative streamline, which has shown to be very

accurate. This thesis also presents interesting examples to show the accuracy of this

streamtube.

Declarations

Statement by Author

I hereby declare that the work contained in this thesis is my very own and where I have

used the thoughts and works of others I have clearly indicated this.

Nawin Raj,

S95000756

Date: 22 / 01 / 09

Statement by Supervisor

I hereby confirm that the work contained in this thesis is the work of

Nawin Raj unless otherwise stated.

Dr. Zhenquan Li,

Associate Professor

Date: 22 / 01 / 09

II

Acknowledgements

I am deeply indebted to my supervisor, Associate Professor Zhanquan Li, for his

guidance and support throughout my masters program. I am also very grateful to him for

providing me with the opportunity to carry out this interesting research, and directing and

encouraging me in this area of flow visualization. Without his professional guidance and

help, this work would not have been possible. Additionally, his insights and knowledge

contributing to the conduct of this study have been invaluable. It has been a privilege

and honor to study under his supervision.

I would also like to thank The University of the South Pacific and especially the School

of Computing, Information and Mathematical Sciences.

Finally, I would like to extend a special thanks to my parents, my wife and son, for their

love and emotional support, as well as understanding and patience.

III

Contribution

A part of the research in this thesis has been published in the International Journal of

Mathematical, Physical and Engineering Sciences, Volume 2, Number 1, pp. 41- 45,

2008.

IV

TABLE OF CONTENTS

ABSTRACT i

DECLARATIONS II

ACKNOWLEDGEMENTS.................................................................................................………………………………………………... III

CONTRIBUTION IV

INTRODUCTION 1

CHAPTER 1 Research Background and Importance 3

1.1 Streaklines 3

1.2 Pathlines 3

1.3 What is a Streamline? 5

1.4 The Importance of Streamtube Construction 8

1.5 Construction of Mass Conservative Streamtube for 3D CFD Velocity Fields………………………..….9

1.6 Background 10

1.7 Research Objectives and Methodology 11

1.8 Implications 14

CHAPTER 2 Fluid Mechanics and Streamtube 15

2.1 Computational Fluid Dynamics (CFD) 15

2.2 History of Computational Fluid Dynamics (CFD) 16

2.3 Kinematics of fluids 17

2.4 Steady and Unsteady Flows 18

2.5 Uniform and Non-Uniform Flows 18

2.6 One, Two and Three Dimensional Flows 19

2.7 What is a Streamtube? 19

CHAPTER3 Mass Conservative Streamline Generation 21

3.1 Streamline and Velocity Field 21

3.2 Mass Conservative Streamline 21

3.3 Mass Conservative Streamline Tracking 25

3.4 The Streamline Tracking Method 26

3.5 Examples 35

CHAPTER 4 Generation of Streamtube 41

4.1 Basis of Construction 41

4.2 Radius of Streamtube 42

4.3 Streamtube Construction 44

4.4 A Diagrammatic Perspective of Streamtube Construction 45

4.5 MATLAB Programming of Streamtube Construction 51

4.6 Examples 54

CHAPTER 5 Further Works and Research 67

CONCLUSION 68

REFERENCE…….......................................................................................................................................70

Introduction

This thesis focuses on the successful construction of streamtube and the accuracy of it.

Efficiency and accuracy of construction helps to study fluid properties a lot better hence

contributing in the vast area of computational fluid dynamics (CFD). The process of

streamtube construction involves a lot of steps and the programming is done with

MATLAB.

Firstly, streamline analysis and its properties helps to understand the basic component

of streamtube. The utilization of CFD methods has made its impact in many areas. The

importance and its background reveal past development of different methods. The

research methodology is divided into four major parts. The calculation of radius and

centre has to be done with formulae manipulation. The basic formulae are derived from

Darmofal and Haimes [3].

Secondly, computational fluid dynamics is an important branch of fluid dynamics. The

history of different CFD methods states the importance of computers and how the

development of computers has made these methods easier and faster. The efficiency of

computers has given these methods another dimension. Streamtube is actually not a

tube but a bundle of streamlines which is clearly defined and illustrated.

Thirdly, the mass conservative streamline generation uses the method in [22]. The

method is very accurate and efficient. It contains numerous steps which lead to the

generation of streamline. The first part is the determination of seed point and then the

generation of streamline. Different dimensional views show how accurate and reliable

1

the method is.

Lastly, the mathematical analysis of the streamtube radius and centre is done to

produce the accurate streamtube. The ordinary differential equation is simplified and the

final version is used in the MATLAB program. Different examples are used to view the

generation of the streamtube. The Constructed streamtube is for analytical velocity fields

only. The reasons why we use analytical velocity fields include the fact that we are able

to check the accuracy using the analytical results obtained from the fields. The

combination of this method with other computational scheme (for example, adaptive

finite method) can be used to draw streamtubes for CFD velocity fields that are

calculated from a mathematical model. The linear mass conservative interpolation of

CFD velocity fields has been used to approximate the analytical velocity fields. This

interpolation method that leads to the generation of mass conservative streamline and

streamtube is found to be very accurate. The figures in the examples show not only the

expansion of the flow at the points but also the process of subdivision of the original

mesh. Hence, this thesis generates an accurate mass conservative streamtube which

can be used to provide very accurate information of fluid flow properties. Further

research and study could also be done to consider other aspects of the study. The

successful method and results have been published in the International Journal of

Mathematical, Physical and Engineering Sciences, Vol. 2, No.1, pp. 41-45, 2008.

CHAPTER 1

Research Background and Importance

The construction of a mass conservative streamtube is associated with many important

elements. Streamlines and its properties help to clearly define the streamtube model.

The visualization of flow fields through mass conservative streamtube is very beneficial

to many fields. Many other methods were used to construct the streamtube in order to

visualize fluid flow. The method used in this thesis is based on generation of accurate

mass conservative streamline and circular cross sections normal to the velocity field.

1.1 Streaklines

Streaklines are the locus of points of all fluid particles that have passed continuously

through a particular spatial point in the past [9]. Experiments with releasing dye into the

fluid in a time period at a fixed point and then at a later time finding out the path of the

dye shows the streaklines. Another analogy could be of the lights on all cars that pass

through the same check point.

1.2 Pathlines

Pathlines are the trajectory that a fluid particle would make as it moves around with the

flow [9]. The path followed by the fluorescent drop using a long-exposure photograph

can be traced out which is similar to a pathline [15]. It is possible for pathlines to cross

as we imagine car lights to cross as cars change lanes.

Brown - pathline in the shown direction

Green - pathline in the shown direction

Blue - pathline in the shown direction

Red - pathline in the shown direction

Grey - streakline in a particular direction

Black - streakline in a different direction

Fig.1 Streaklines and Pathlines

Fig. 1 shows particles on the same streakline (dashed line) can take different paths (red,

green, blue, brown solid lines) to reach there. The streaklines (dashed lines) originates

from the same point at different time and the pathlines for four different particles on the

streaklines [9].

1.3 What is a Streamline?

A streamline is a line that is tangential to the velocity fields. The paths of mass-less

particles with a steady flow are known as streamlines. Streamlines are a family of curves

that are instantaneously tangent to the velocity vector of the flow and this means if a

point is picked then at that point the flow moves in a certain direction [9]. A small

distance of movement in this direction and finding out where the flow now points would

draw out the streamline.

The construction of this streamline is a very important task because it helps to visualize

the characteristics of flow fields. By definition, streamlines defined at a single instant in a

flow do not intersect. This is because a fluid particle cannot have two different velocities

at the same point [9]. Since the velocity at any point in the flow has a single value (the

flow cannot go in more than one direction at the same time), streamlines cannot cross,

except at points where the velocity magnitude is zero, such as at a stagnation point [15].

In a steady flow, the streamline has a fixed direction at every point and is therefore fixed

in space. A particle always moves tangentially to the streamline and hence in steady

flow the path of a particle is a streamline [2].

Streamlines are frame-dependent. The streamlines observed on one inertial reference

frame are different from those observed in another inertial reference frame, for instance,

the streamlines in the air around an aircraft wing are defined differently for the

passengers in the aircraft than for an observer on the ground [15]. In this example, the

observer on the ground will observe unsteady flow, and the observers in the aircraft will

observe steady flow, with constant streamlines.

Figure 2 shows a streamline.

Y V

dx

dy

dz

u

w. . • • •

X

Fig.2 Streamline and velocity field, V.

A streamline can be defined in mathematics as a graph of the following equation

du dv dwu v w

where u, vand ware velocity components in x, yand z direction respectively.

Streamlines have the following important properties

• The tangent line to a streamline at a point is in the direction of the fluid velocity at

that point.

• The density of streamline in the velocity of a point is proportional to the

magnitude of the velocity at that point

• There is no flow across or no flow normal to it, the streamlines cannot intercept

except at a point of zero velocity.

Streamline simulation had a rapid development and has influenced many fields because

of its effectiveness. Streamline based flow simulation is now accepted as an effective

and complementary technology to more traditional flow modeling approaches such as

finite differences. Marco [13] stresses his point on the effectiveness of the simulation in

solving large, geologically complex and heterogeneous systems, where fluid flow is

dictated by well positions and rates, rock properties, fluid mobility, and gravity. These

types of problems were difficult to solve using the more traditional approach. Streamline

simulation as pertaining to modeling subsurface fluid flow and transport have in the

literature since Muskat and Wyckoff’s 1934 paper and have received repeated attention

since then [13]. For convection driven process, streamline simulation has a tremendous

advantage over conventional finite difference [8]. The advantage of streamline is

associated with its computational speed and efficiency which addresses a lot of

engineering problems.

Knowledge of the streamlines is very useful in fluid dynamics. Bernoulli's principle,

which describes the relationship between pressure and velocity in an inviscid fluid, is

derived for locations along a streamline [9]. The curvature is related to the pressure

gradient acting perpendicular to the streamline and the radius of curvature of the

streamline is in the direction of decreasing radial pressure [9].

1.4 The Importance of Streamtube Construction

The utilization of the CFD methods has greatly helped many fields such as

aerodynamics, industrial processes and geophysics. The study helps optimize chemical

processes, make important changes in weather devices and reduce time frame for

actual constructions of aircrafts.

Gupta [16] states that the turbojet engine develops a thrust by the reaction of hot gases

exhausted at large velocities and it breathes air at the front end through the use of a

compressor driven by a turbine. This information of air pressure and its flow is critical in

the operation of this engine. The information about air pressure and its flow can be

obtained by the study of a streamtube which can show flow velocities at different

regions.

In [1], propellers and windmills are devices that use airfoil sections to change the fluid

pressure in order to produce a force, and the change in pressure occurs because the

fluid momentums change. Streamtube also provides very useful information on the

momentum of particles. The change in pressure at different locations can easily be

analysed using the streamtube.

Baker [11] claims that streamtube modeling had also been very useful in the geological

area. Streamtube technology was originally developed in the 1960s. Two dimensional

streamtube models were initially available for homogeneous permeability regular flow

patterns, such as a five-spot pattern [11]. Streamtube models were later generated for

irregular well positions and heterogeneous reservoirs. Gravity effects and therefore the

8

vertical sweep efficiency were not always accounted for. To account for vertical sweep

efficiency, Chevron developed two-step hybrid streamtube models in which vertical

cross-sections were first simulated and then combined with real streamtube models.

The rotor blade design [17] uses the streamtube model in the industrial process where

the distortion of stream surface through the blading has been observed and a

quantitative analysis of its effects on the blade loading condition is carried out. The

streamlines close to the blades have been modeled fitting to outward and inward conical

streamtubes on the blade suction and pressure sides [17].

A quantitative study of Hundhausen's descriptive model in [19] is based on streamtubes.

Interplanetary magnetic fields of one polarity emanating from north polar cap and fields

of opposite polarity from south polar cap are separated by a neutral sheet at low latitude.

The coronal plasma is accelerated into a high-speed solar wind in streamtubes

emanating from the border of the polar cap which is at low speed [19].

1.5 Construction of Mass Conservative Streamtube for 3D CFD

Velocity Fields.

Streamtube is the surface formed by many streamlines passing through a closed curve.

The construction of streamtube needs efficient and accurate consideration of

streamlines. Streamtube is a fundamental technique to visualize steady flow fields and

the one constructed reveals the flow expansion and contraction rates along the

streamline. Pathlines, streamlines and streaklines are other field lines that show different

properties of flows [9].

1.6 Background

The accuracy of construction is extremely important for reliable analysis of fluid flows.

Different methods are used for the constructions that are described. Generally, the

methods used are based on either sweeping a stream polygon or using the radius of the

flow to construct the streamtube.

Different methods are:

In [18], Schroeder describes a technique called stream polygon. The generation of

streamtube is done by sweeping the stream polygon along a streamline. The radius of

the streamtube varies with the velocity magnitude, such that the mass flow is constant

along the streamline.

In [10], Ma and Smith describe the visualization of both flow convection and diffusion,

statistical dispersion of the fluid elements about a streamline is computed by rising

added scalar information about the mean square root value of the vector field and its

Lagrangian tune scale. The result defines the radius of the cross flow section and also

forms a tube like surface (Streamtube).

In [3], Darmofal and Haimes created a streamtube by connecting the circular cross flow

sections along a streamline. The radius of a cross flow section is determined by the local

cross flow expansion rate [14]. The streamtube constructed in this manner reveals the

flow expansion rate along the streamline.

10

The streamtube construction involved in this research is using the Darmofal and Haimes

method. It is computationally more feasible. In [15], test results show saving in both

computational cost and memory requirements. Explicit solutions of ordinary differential

equations that govern the streamtube radius are derived to speed up the construction.

The streamlines used in this research is generated by the adaptive streamline tracking

method in [22].

1.7 Research Objectives and Methodology

Streamtube is very useful because it helps to gain information about expansion,

contraction and various properties of the fluid flow. An accurate and computational

feasible method is much needed in practice. The current most accurate mass

conservative streamline tracking method and Darmofal and Haimes method are

combined to construct the mass conservative streamtube. The following objectives of

this research are achieved:

A. A new streamtube construction method is introduced. From the analysis, it would

be more accurate than the existing methods.

B. Some applications are conducted to check expansion and contraction of the

flows.

The procedures involved in construction of streamline and then streamtube involves a

lot of steps. The construction process involves:

1. The generation of mass conservative streamlines. Li [22] considers the

11

multiplication of a scalar function and a linearly interpolated vector field over a

tetrahedron. It satisfies the law of mass conservation and gives the expressions of the

scalar function for all possible Jacobeans which the coefficient matrices in the linear

vector field may have. A hexahedron of hexahedral meshes is subdivided into smaller

hexahedra when there are points in the hexahedron at which the scalar function

approaches infinity and then seeks more data of the CFD velocity fields at the vertices of

the smaller hexahedra if the values of the CFD velocity fields are unknown. The

subdivision can be an infinity process. A threshold number is introduced to measure how

many times the hexahedra will be subdivided in the initial mesh. The accuracy of

computation depends on the initial mesh and the threshold number. The accuracy of the

constructed streamlines increases with the increase in the value of threshold number.

Exact tangent curves for linear vector fields are used to draw streamline segments in

tetrahedral that are obtained from subdividing hexahedra. It has been shown that the

method generates much more accurate streamlines than the existing methods.

2. The construction of streamtube by Darmofal and Haimes method using the

streamlines generated. In [3], the circular cross flow sections along the streamlines are

connected. The centres of streamtube constructed are the points of a streamline

following the method in [3].

The tube radius is calculated using the following equation:

r 1 —= - vT. u

r t 2 T

12

u is the local cross flow divergence.

u.* - *'VT. u .u .

uwhere oe, is the change of velocity magnitude along the streamline.

This ordinary differential equation will be solved analytically to determine the radius of

streamtube.

3. The calculation is done at each step along the streamlines and the circular cross-

sections are connected.

4. Analytical velocity fields will be used to check the accuracy of the method

introduced in this research.

13

1.8 Implications

The results of this research will provide a more accurate streamtube construction

method than the existing ones, as the method of mass conservative streamline

generation method is the latest and most accurate. There are wide applications in

practice of streamtube as mentioned before. This initiative can be of a lot of help as the

process is faster and accurate. The improved accurate model of mass conservative

streamtube will give more accurate analysis of fluids. It will greatly benefit the field of

flow visualization.

Summary

The background information reveals how important the analysis of fluid flow is and the

scope of historical research in this area. The method used in this thesis is based on the

accurate generation of mass conservative streamline and connection of circular cross

sections to construct the streamtube. The efficiency and accuracy of the construction is

very significant for accurate analysis of fluid visualization. The next chapter emphasizes

the aspects and development of computational fluid dynamics.

14

CHAPTER 2

Fluid Mechanics and Streamtube

Computational fluid dynamics helps to study fluid flow. The development of computer

technology has greatly influenced the analysis of fluid flow. Most methods were based

on the discretisation of surface of geometry which led to the use of computer programs.

The conditions of steady flow and unsteady flow are different. Similarly, the

characteristics of uniform flow and non uniform flow are also different. Streamtube is a

bundle of streamlines and can be used to analyze the flow properties in it.

2.1 Computational Fluid Dynamics (CFD)

CFD is a computational technology that helps to study the dynamics of things that flow.

Using CFD, a computational model can be built that represents a system or device that

is under study, then the fluid flow physics and chemistry is applied to this virtual

prototype, and the software will output a prediction of the fluid dynamics and related

physical phenomena [7]. Hence, CFD is a sophisticated computationally-based design

and analysis technique and it gives the power to simulate flows of gases and liquids,

heat and mass transfer, moving bodies, multiphase physics, chemical reaction, fluid

structure interaction and acoustics through computer modeling [7].

Flow visualization has become an important measure to explore the properties of flows

due to the development of computer capacity and software. In the last decade, the need

for visualization has grown exponentially. The development of more and more powerful

15

supercomputers have enabled researchers in disciplines such as computational fluid

dynamics to perform increasingly complex three dimensional simulations using fine grids

[6]. The basic feature of CFD remains with how the fluid is treated and it uses computer

models to predict the properties of fluids. In this thesis, the construction of mass

conservative streamtube will make flow visualization more accurate and efficient.

2.2 History of Computational Fluid Dynamics (CFD)

The basis of CFD problems are the Navier- Stokes equations. This equation defines any

single-phase fluid flow [7]. Further simplification of the equation yields the linearised

potential equations. Different two dimensional methods were developed to solve these

equations. The development of computer led to three dimensional methods. The first

paper on a practical three dimensional method to solve the linearised potential

equations was published by John Hess and A. M. Smith of Douglas Aircraft in 1966.

This method discretised the surface of the geometry with panels and this has led to the

use of programs called the panel method [7]. Later, more advanced three dimensional

methods were developed.

Full potential codes were developed as the panel codes could not calculate the non

linear flow present at transonic speeds. Full potential airfoil codes were widely used, the

most important being named Program H. A further growth of program H was developed

by Bob Melnik and his group at Grumman Aerospace as Grumfoil [7].

Euler equations provided more accurate solutions of transonic flows hence taking the

development to another level. The one dimensional numerical method is used to solve

16

problems of transport along lines. The multiphase simulation is solved by either

Lagrangian or Eulerian method. The Lagrangian discretisation has no obvious time step

limitation and the Eulerian calculation along each line are de-coupled, and so each may

be performed optionally [7]. Euler's method uses both explicit and implicit techniques.

Eulerian methods have been successfully applied for multi-component systems because

of the difficulty of formulating an appropriate Lagrangian approach [8].

In the two dimensional realm, Mark Drela and Michael Giles, and later graduate students

at MIT, developed the ISES Euler program (actually a suite of programs) for airfoil

design and analysis. This code first became available in 1986 and has been further

developed to design, analyze and optimize single or multi-element airfoils, as the MSES

program [7]. The development followed with more design and analysis.

2.3 Kinematics of Fluids

The kinematics of fluids deal with space-time relationships for fluids in motion and the

Lagrangian method of describing the fluid motion is concerned with the path tracing of

individual fluid particles (elements) [2]. Different properties such as velocities and

pressures are found with the passage of time. The coordinates of a particle A (x, y, z) at

any time, t, are dependent on its initial coordinates (a, b, c) at the instant t, and can be

written as functions of a, b, c and t [2], that is,

x = Ф1 (a, b, c, t)

y = Ф2 (a, b, c, t)

z = Ф3 (a, b, c, t)

17

2.4 Steady and Unsteady Flows

The flow parameters such as velocity, pressure and density of a fluid flow are

independent of time in a steady flow whereas they depend on time in unsteady flows. At

a point, in reality these parameters are generally time dependent but often remain

constant on average over a time period T [2]. In steady flow (when the velocity vector-

field does not change with time), the streamlines, pathlines, and streaklines coincide and

this is because when a particle on a streamline reaches a point, A0, further on that

streamline the equations governing the flow will send it in a certain direction of vector V,

as the equations that govern the flow remain the same when another particle reaches

A0, it will also go in the direction of vector V [9]. When the flow is not steady, the flow

would have changed where the particle will go in a different direction when the next

particle reaches position A0 [9]. This fact is very important as it is difficult to look at

streamlines in an experiment but if the flow is steady, streaklines can be used to

describe the streamline pattern.

2.5 Uniform and Non-uniform Flows

A flow is uniform if its characteristics at any given instant remain the same at different

points in a flow; otherwise it is termed as non-uniform flow [2]. The flow through a long

pipe at a constant rate is steady uniform flow and at a varying rate is unsteady uniform

flow whereas flow through a diverging pipe at a constant rate is steady non-uniform flow

and at a varying rate is unsteady non-uniform flow [2].

18

2.6 One, Two and Three Dimensional Flows

Flow through a pipe may usually be characteristics of one dimensional. In two

dimensional flows, the velocity vector is a function of two coordinates and time, and the

wide river can be considered as two dimensional. Three dimensional flows are the most

general type of flows in which the velocity vector varies with space and time [2].

2.7 What is a Streamtube?

If we pull out a bundle of streamlines from inside of a general flow for analysis, we refer

this bundle as a streamtube. The following illustration shows a streamtube in a fluid

Fig. 3 Illustration of a streamtube.

Formally, a streamtube is defined as the surface formed by all streamlines passing

through a given closed curve in the flow. A region of connected streamlines is called a

streamtube and because the streamlines are tangent to the flow velocity, fluid that is

inside a streamtube must remain forever within that same streamtube [9]. It is a three

dimensional bundle of streamlines with the connections of circular cross flow sections.

Streamtube is used to visualize expansion, contraction and various properties of the

fluid flow. Streamtube provides very valuable information about the fluid flow, which in

19

other ways may not be possible. These are useful in fluid mechanics, engineering and

geophysics. Streamtube and streamline technology, to a large extent, have been driven

by the realization that heterogeneity controls recovery methods for many fields [11].

Summary

Flow visualization using the computational method has been done to solve many

problems in different fields. All methods used depend on the accuracy of fluid modeling

and computational efficiency of computers. The analysis of fluid flow is done with the

consideration of various conditions. The study of mass conservative streamtube reveals

very valuable information of fluid flow such as expansion and contraction. The next

chapter explains the method of mass conservative streamline generation.

20

CHAPTER 3

Mass Conservative Streamline Generation

There are some mass conservative streamline generation methods using two stream

functions [4, 5]. The papers show that the satisfaction of law of mass conservation is

important in streamline construction. Different mass conservative streamline generation

methods are introduced by Li [20, 23]. These methods verify the mass conservation

using multiplication of an unknown scalar function and the linear interpolation of CFD

velocity fields. The streamline tracking is done with subdivision of a hexahedron of

hexahedral meshes into smaller hexahedra if more data is needed for accurate

streamlines in three-dimension. This subdivision may be done many times in the

tracking process. The number of subdivisions depends on the pre-specified accuracy.

3.1 Streamline and Velocity Field

Streamline is an important tool to analyze and understand velocity fields. It is a

fundamental technique to study flow fields. CFD velocity fields are expressed as

numerical solutions of mathematical models. It is assumed that the CFD velocity field is

an approximation to a continuous mass conservative velocity field in the same domain

[23]. In this thesis, the approximation means linear interpolation l + = B A X V at the four

vertices of a tetrahedron to the continuous velocity field in the tetrahedron. A is a

constant matrix and B is a vertical vector.

3.2 Mass Conservative Streamline

CFD velocity field is an approximation to a continuous mass conservative velocity field in

21

the same domain. Mass conservation is important in order to get accurate streamline

visualization of flow fields. Most construction methods of mass conservative streamline

consider two stream functions f and g for a steady compressible fluid whereby the

momentum is given by pV = V / x V g (p is fluid density). Other methods generate

streamline without considering conditions on the stream functions. The adaptive

streamline tracking methods used in [20, 23] are more accurate than all other methods.

We take adaptive streamline tracking method for three-dimensional CFD velocity fields

as an example in the introduction of mass conservative streamline tracking. Taking f a s

a scalar function, assume that f V l satisfies the law of mass conservation on a

tetrahedron. Different expressions of f are found for different Jacobean forms of matrix

A. This helps to give conditions about when more data of velocity fields are needed in a

hexahedron if we apply the conditions to the tetrahedral that compose the hexahedron.

The mass conservation for incompressible flow is given as

V-(/"V)= d Q i ) | d(/v2) ,V ' xydx dy dz

where V = ()vvv 12 3 T

If V= V l , then the above expression can be written as

22

where — d f + V r — + V VV is the material derivative.dt dt x dx y dy z dz

Different expressions of f are calculated from — = ++(aa a 112 2 33)f, these are

expressed as solutions of — = V l .Since l = +BAXV B , the Jacobeans of matrix A and

expressions of f for any possible case are calculated so that f V l holds the law of

mass conservation. Table 1 show all the cases and each case is programmed to

generate the accurate mass conservative streamline.

23

Table 1: Jacobeans and expressions of f for all possible cases of a non-conservative

3D linear field (where C is a constant).

Case

1

2

3

4

5

Jacobean

(

(

(

\ 0 0

0 r2 0

0 0 r3

0 * rx * r2 ?

' n x oN

-A // 0

0 0 ry

u δ 0

0 a 0

0 0 rV /

\

/

- r3 ^ 0 )

°)

o)

> X 0N

-A // 0

0 0 0v /

V δ o0 r 0

0 0 0 y

Oorl)

f

f/ N2 , S2J-1 !

[ l 1 ' ^+;l2 J Y " ' "2+;l2 J J V ' ' r)

( h \~2( h V1

\( Xb f f xb + f l " 1

| [ > J ' / / 2 + A 2 J ' [ > 2 ' / / 2 + A 2 J J

I 2 r )

24

r0

V

8r

0

08

rj

(r * 0,8 = 0 or l)

r

r0

0

0

0

0 0,-1

(r ^ 0,8 = 0 or l)

8

0

0

0r2

0

o0

0

(Source: From [22])

3.3 Mass Conservative Streamline Tracking

A mass conservative streamline tracking method for two dimensional CFD velocity fields

[20] was done whereby the method used was for quadrilateral mesh only. The method

subdivides a quadrilateral into smaller quadrilaterals if f in Table 1 equals infinity at

some points inside any of two triangles obtained from the subdivision of the

quadrilateral. Next subsection briefly introduces the procedure of extended adaptive

streamline tracking for three-dimensional CFD velocity fields [23].

25

3.4 The Streamline Tracking Method

The seed point is given before the process of streamline tracking. The hexahedron

containing the seed point can be found which can be subdivided into five tetrahedra.

The CFD velocity field is calculated at the vertices of each tetrahedron from the

analytical velocity field. The tetrahedron which contains the seed point is found by the

procedure given in this section. The streamline segment that goes through this point in

the tetrahedron is drawn. When the streamline intersects with the boundary of the

tetrahedron, this point is taken as the next seed point and the streamline is drawn in

other tetrahedron using the same formulae. This process continues until the streamline

intersects with the hexahedron boundary. The subdivision process is done many times

and a pre-specified threshold number T is used to achieve the desired accuracy.

The following steps describe the process of seed point determination [22]:

1) The hexahedron is subdivided into five tetrahedra as shown in Fig. 5.

2) For each te t rahedron, let Y1123 = (y111,,yy)T, Y2 = (y222,,yy)T, Y3123 =()y333,,yyT,

and Y4123=(y444,,yy)T b e t h e f o u r vertices of the tetrahedron and

Y0123 = (y000,,yy)T be the coordinates of the seed point in a Cartesian coordinate

system. All the following four values are calculated:

Y - Y Y - Y Y - Y, _ *Q *2 *0 *3 *0 *4 r _

1 "%7 TKT TKT TKT TKT TKT ' 2Y-Y Y-Y Y-Y*1 *2 *1 *3 *1 X4

Y - '

Y —'* 2

Y —'

Y - 'X2

Y - Y

Y - YX2 X4

26

Y

Y3

-Yx

" Y i

Yo

Y3

-Y 2

"Y 2

Y

Y3

-Y 4

"Y 4b4 =

Y ' _ • "

n V —" Y ' _ • "

Y,-1Y 4 - 1 Y,-1

3) The seed point is in a tetrahedron if all four values bb 123 ,,, bb4 are positive. The

seed point is in a hexahedron if it is in one of its five tetrahedra.

After the seed point is determined, the following algorithm [22] is used for adaptive

streamline tracking:

1. Set i = 0.

2. Find the hexahedron that contains the seed point and divide the hexahedron into

five tetrahedra as shown in Fig. 2. If the CSFD conditions are not satisfied in all

five tetrahedra, draw the streamline segment that goes through the seed point in

the hexahedron; otherwise go to Step 3. Take the intersection of the streamline

segment with the boundary of the hexahedron as seed point or end point and

repeat this step (for a new hexahedron).

3. Subdivide hexahedron ABCDEFGH as shown in Fig. 3 and calculate the values

of the velocity fields at AB , BC, , , , AD, AEBF, , , , CG , DH, EF, FG , GH,

EH and O1, O2, , , O3, O5 , O6, O, and then let i = i + 1 and go to step 4.

4. Take the elements (smaller hexahedra in Fig. 3) in the subdivided hexahedron as

new elements of the mesh by replacing the initial element and go to step 2 if

i < T; Otherwise go to Step 1.

27

The following figures show the tetrahedral subdivision and the tracking process.

•

•*

*

/

/

Fig. 4. A hexahedral element.

Fig 5. Tetrahedral subdivision of a hexahedron.

28

D CD

AD

AE -

-, CG

G

FG

EF

Fig. 6 An example of subdivision of a hexahedron

B

B

D

E

Fig. 7 shows the connecting of points in the tracking process.

29

A two dimension view of streamline generation on a mesh grid

A two dimensional view of a

quadrilateral which will be

subdivided.

The quadrilateral is subdivided

into four smaller quadrilaterals.

The quadrilateral is subdivided

even further into more smaller

quadrilaterals.

30

•

•

The dots show the generated

points that constitute the

streamline.

A three dimensional view of streamline construction

A hexahedron is shown with a

three dimensional view.

y—; ;""x 1 1

1 I

l •

**

*The hexahedral is divided into

four smaller hexahedral.

31

Each hexahedron is divided into five

tetrahedra. Here is one of them

The tetrahedron which contains the seed

point is found.

The next point of streamline is

calculated and is plotted.

32

The process of streamline point

generation continues.

The mass conservative streamline

segment is generated after more

points are plotted.

As the line reaches close to the

boundary of tetrahedron, the next

tetrahedron is found which contains

the new seed point and then the

streamline segment in it can be

generated.

33

The above process is continued in

the five tetrahedra which constitute

the hexahedron.

y

After streamline is generated in all

hexahedra in the domain, the

streamline is seen in a three-

dimensional space.

34

3.5 Examples

Figures 8-10 shows the tracked streamline for saddle-spiral flow

as seen in [23].

N

V= (xz—y,yz+x—z2)

-510 -10

Fig. 8 Exact and tracked streamlines for Example 1 and T=3.

35

-10 r

Tracked streamline

Exact streamline

Fig. 9 Projection of exact and tracked streamlines in Fig. 8 on xy-plane.

1

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

Tracked streamlineExact streamline

: /

J/

/

-10 -8 -6 -4 -2 0

y

8 10

Fig. 10 Projection of exact and tracked streamlines in Fig. 8 on yz-plane.

36

Figures 11-15 shows the exact and tracked streamlines for the toroidal flow

y=[cx{z-zo)jrl -csyjr, +<ux/r, -c(r-ro)/r

at different view points. The seed point is (5.9996 6.2867 1.3923), where r x y= + 22.

The red line is the exact and blue line is the tracked streamline. The accuracy of tracked

streamline can be verified from these figures.

10 10

Fig. 11 Exact and tracked streamlines for Example 2 for T=5 with torus.

37

Fig. 12 Exact and tracked streamlines for Example 2 for T=5 without torus.

Effi B

Tracked streamline

Exact streamline

s

-10 -S -6 -4 -2 0 2 4 6

y

10

Fig.13 Projection of exact and tracked streamlines in Fig. 12 on xy-plane.

38

4

3

2

1

N 0

-1

-2

-3

jf

Tracked streamline

-Exact streamline

-10 - 8 - 6 - 4 - 2 0 2 4 6 10

Fig. 14 Projection of exact and tracked streamlines in Fig. 12 on yz-plane.

4

3

2

1

0

1

2

3

1

T \/... . ' = ; : ; • •

Tracked streamline

Exact streamline

• - - 1 .

K , T ,:

ft '

vJ

-10 -8 -G -4 -2 2 4 6 8 10

Fig. 15 Projection of exact and tracked streamlines in Fig. 12 on xz-plane.

39

Summary

This adaptive streamline tracking method is efficient and very accurate. The accuracy of

the method can always be improved with the choice of threshold value, T. This method

satisfies the law of mass conservation. The tracked streamline drawn is exact for linear

vector fields. This mass conservative streamlines will be used to generate the mass

conservative streamtube. The next chapter explains the accurate method of mass

conservative streamtube generation.

40

CHAPTER 4

Generation of Mass Conservative Streamtube

The construction of mass conservative streamtube is done with the connection of

circular cross flow sections along a streamline. The local flow expansion radii are

calculated at each step and the circular cross section is everywhere normal to the

streamline which is the centre. The radii are numerical solutions of a differential

equation. This process is channelled into MATLAB program to visualise the

construction. There are many objects of a main program which carry out specific tasks.

The examples of saddle spiral flow and toroidal flow show the results.

4.1 Basis of Construction

The streamtube construction method in this thesis is based on the adaptive streamline

tracking method in [23] and is for incompressible flows but the method can also be used

to compressible steady flows by replacing the CFD velocity fields with CFD momentum

fields. The method followed is the streamtube construction in [3, 14] where a streamtube

is created by connecting the circular cross flow sections along a streamline. The radius

of a cross flow section is determined by the local cross flow expansion rate. The process

of creating a streamtube is: (1) generating a streamline; (2) connecting the circular cross

flow sections along the streamline.

41

4.2 Radius of Streamtube

The streamtube is generated by taking a mass conservative streamline as the centres

and the local flow expansion radii. This section derives the formulae for the radius of

streamtube. The radius of a streamtube, r, is governed by the following ordinary

differential equation [14]:

\dr 1= —VT u

r dt 2 (4.2.1)

where v r u is the local cross flow divergence and is defined as

V . . u = V . u -du

du *

in which represents the change of the velocity magnitude along the streamline.dξ

For the velocity field given,

u =

u

V

V W J

The magnitude of the velocity field:

w =

= + X + B

42

The change of velocity magnitude along the streamline:

where Txsuysvzsw)

Further calculation for the change of velocity magnitude gives

Dum8u 8u 8u

U -\ VH W \U +

m y dx 8y 8zdv dv dv— u - \ VH w \v8x By dz

8w 8w 8wU H V H W I W

8x 8y 8z

m1 T A—T Aum

U 2122 V ++33 ) ww

Since the streamtube created is based on the mass conservative streamline constructed

in [23], the streamline has been viewed as the lines generated by a piecewise linear

vector field that satisfy the law of mass conservation subject to the preset tolerance, i.e.

43

even though the linear vector field u does not satisfy the law of mass conservation at

some points in the domain, V u is small enough to be considered as zero. Thus

Equation (1) can be written as

du *r dt 2

Take integral on both sides of (4.2.2) with respect to t gives

r = r e' r0^ (4.2.3)

4.3 Streamtube Construction

Theoretically, the streamtube can be created by drawing circles with radius given by

equation 4.2.3 and centre at the points on the streamline constructed by the method in

[23]. A physical construction is introduced in [14] by drawing cylinders with the two ends

of circles at two instants with the radii that are numerical solutions of a differential

equation and centres at the points on streamlines. In this thesis, the circles are drawn on

the planes that are orthogonal to the vector field at the points that are the first point

outside of the hexahedra. These hexahedra may be the original mesh elements or

smaller hexahedra that are obtained by subdividing the larger ones. Each circular

44

section consists of many points generated around the streamline point and the section is

in a plane that is normal to the streamline point. These sections are generated at many

points on the streamline. The blue circular sections in the examples show the continuity

which results in the mass conservative streamtube. The figures in the examples show

not only the expansion of the flow at the points but also the process of subdivision of the

original mesh.

4.4 A Diagrammatic Perspective of Streamtube Construction

A two dimensional view of streamtube construction

1

#

ft

4

•

•

• ] • • • •A

/,*

.*

The dots show the generated

points that constitute the

streamline.

The circles are the generated

circular cross flow section of the

streamtube.

45

A three dimensional view of streamtube construction

Many points around the streamline point are generated of the same radius which is in

the plane that is normal to the velocity field at that point.

More circular cross sections are generated around other streamline points.

46

The tetrahedra share a common surface as streamline points and circular cross sections

are generated.

47

The process is continued to other tetrahedra.

The same process of circular cross flow sections are generated around other streamline

points in other tetrahedra. This diagram shows the subdivided tetrahedra share a

common surface as the streamline point and circular cross flow sections are generated.

48

The circular sections are seen around the streamline.

<&*'

1 : • \..\-£c

This diagram shows the mass conservative streamtube around the mass conservative

streamline.

49

«'»"<"H

X

The streamtube diagram shows a toroidal flow. The circular sections follow the

streamline in an enclosed pattern.

50

X

y

The above diagram show the circular pattern following the streamline in a spiral flow.

The arrangement of circles indicates the importance of their presence in a normal plane

to the velocity field.

4.5 MATLAB Programming of Streamtube Construction

The process of MATLAB programming required the breakdown of method into small

objects. Each of this function is able to communicate with the main program and among

each other to transfer evaluated values.

51

The subdivide function

The larger hexahedron is divided into smaller hexahedron using the subdivide program.

It checks the layers and collects the x, y and z coordinates. Starting from layers, i = 2,

and threshold number, T, the coordinates of hexahedral vertices are used to subdivide

the hexahedra. As a hexahedron is subdivided, the seed point is located and returned.

For each subdivided hexahedron, tetrahedronsub.m and checkintetrahedron.m

programs are used to find the seed point. The interpolation and draw streamline objects

are also communicated to facilitate the subdivision process.

The function of locating seed point in tetrahedra

Four values of b1, b2, b3, b4 are calculated in this function. If all these four values are

positive in a tetrahedron, then the seed point is located in the tetrahedron. These four

values depend on the coordinates of seed point and also the coordinates of the vertices

of the tetrahedra. This function is also used to check when a point on tracked streamline

is out of a tetrahedron.

The interpolation function

The calculation of coefficient matrices A and B in linear interpolation of CFD velocity

field is carried out in this program. The function interpolates P1, P2, P3, P4 which are

the tetrahedral vertices ((P1=x1, y1,z1), (P2=x2, y2,z2), (P3=x3, y3,z3) and (P4=x4, y4,z4)).

52

The velocity fields at P1, P2, P3, P4 make up matrix V (V = V1 - V4, V2 - V4, V3 -V4). This

matrix is multiplied with the inverse of matrix P (P = P1 - P4, P2 - P4, P3 - P4) to obtain

matrix A. Matrix B is calculated by subtracting the product of matrices A and P4(x4,y4,z4)

from V4 (velocity field of P4).

The condition checking function

The tetrahedron is checked for the nine conditions to subdivide using the condition.m

program. The eigenvalues of the coefficient matrix A is used to check the condition.

Each case is thoroughly checked and it confirms tetrahedron subdivision.

The velocity field function

Velocity.m calculates the velocity field at the four vertices of tetrahedron using the

coordinates of a point. P is the coordinate of the given point and V is the velocity field of

point P.

The circular construction function

The drawcylinder.m object constructs the circle which is normal to the velocity field on

the streamline. The formulated streamtube construction formula is processed into this

MATLAB program.

The streamline construction function

The streamline is generated using the drawstreamline.m program. The expressions of

matrix A are checked for each condition. If the eigenvalues are not real, then the

program under 'complex' is run. When the eigenvalues are real and different, then the

53

program under 'three different' is run. There are eight cases corresponding to the eight

cases in Table 1. The coordinates of points on streamline are plotted. The line width

and colour are specified in this program.

The main program

All these objects are associated with a main program. Values from other objects are

collected and sent to facilitate the generation process. The time step, threshold number

and initial radius value are declared at the beginning.

4.6 Examples

The following two examples show the adaptively tracked streamline by the method in

[23] and the streamtube constructed based on the streamlines. Some of the figures are

not of high quality due to the current computing facilities available. The examples shown

reveal important flow properties. The circular sections are normal to the streamline at

different points. One example shown is a saddle-spiral flow and the other one is a

toroidal flow velocity field. The tubes reveal the flow expansion rate along the

streamline.

The tubes produced are very accurate as the streamline shown is very accurate. These

tubes are shown as strings of blue colored rings. The string of red dots forms the

streamline. The meshes generated give the three dimensional views of the examples

shown. The two dimensional views also show the tubes and reveal the behavior of the

tubes.

54

The streamlines are drawn to the stated threshold numbers which are significant to their

accuracy. The initial r0 (radius) is taken as 1 and the time step as 0.01 for the

generations. The results shown from the examples are very efficient in flow visualization

modeling. The analysis of the velocity fields in the examples also reveals flow properties

in different regions.

Example 1

Saddle-spiral flow

V= [xz—y,yz +x,—z2

with seed point (-0.8, 0.8, 1).

The variation of the expansion rate for the saddle-spiral flow in this example is very

small in the time period shown in the figures. The figure below shows circular flow

sections in blue. They are produced at right angles to the streamline point. Numerous

streamline points are generated to construct the continuous curve in red.

The xy plane is the asymptotic plane for the velocity field. The streamline is generated

with very careful consideration so that it does not intersect with the asymptotic plane.

More data on velocity field is needed when the tracked streamline is close to the z-axis.

The subdivision of first hexahedron begins with points (0,0,0.5), (-2,0,0.5), (-2,2,0.5),

(0,2,0.5) (0,0,1), (-2,0,1), (-2,2,1) and (0,2,1). Twenty hexahedral are subdivided in this

55

construction process. The initial radius is taken as 0.1 and the threshold value, T is

taken as 10. The following figure gives a three dimensional view of Saddle-spiral flow.

Fig. 17 Streamtube for Example 1 in 3 Dimension.

56

The following figure gives a two dimensional view on yz-plane. The blue dots are the

circular cross flow sections which collectively form the streamtube. The red continuous

curve is the streamline.

- 0 -8 -6 -4

Fig. 18 Projection of the streamtube in Fig. 17 on yz-plane.

57

The following figure gives a different two dimensional view on xy-plane. The apparent

blue dots arrange together to form the streamtube around the streamline in red.

Figure 1

dit View Insert Tools Desktop Window Help

D

- n x

10 r

8

6

4

2

0

-2

-4

-6

-8

-10

'" r" /

$?

• ' " " • • • •

" • /•-•• -\

: ; \

; • • -

' ' .

: . . . ^ , ,

i ^ ^ •"

* , • _ ,

^ •• ' \

v . • • • ' • .

• • •

' ' •

i

' . • . . , . ™ "

v- ' • • ' ' ••-

^iL " ""i " ' 1

<'

. - • . : • ' • • ' ,

»• • • ' \ L

L.r \

r

' , / y ' +

' / ' _."'f"

•

j

-10 -8 -6 -4 -2 8 10

Fig. 19 Projection of the streamtube in Fig. 17 on xy-plane.

58

The following figure is another two dimensional view on xz-plane. The uniformity of the

circular cross flow can be very clearly seen in terms of the streamtube radius in blue

around the streamline in red.

File Edit View Insert Tools DeskJtop Window Help

Fig. 20 Projection of the streamtube in Fig. 17 on xz-plane.

59

Example 2

Toroidal flow

vv =

129xz(

velocity

z - l )22

field

5r

2y(z 2x

5r'

2

r )

V xV 2 2x +y

Fig. 21 shows the exact streamline in blue and tracked streamline in red generated by

method in [23]. This figure indicates that the tracked streamline is very accurate which is

shown by the comparison with the exact streamline. When we drew the streamtube for

this example, we were not allowed to draw circles like Example 1 due to the constrains

of the computing facilities available. We drew two dots instead of one circle.

The first hexahedron has points (8,4,2), (6,4,2), (6,6,2), (8,6,2) (8,4,0), (6,4,0), (6,6,0)

and (8,6,0). The subdivision is done till the eight tetrahedron which has the coordinates

as (-2,8,2), (-4,8,2), (-4,10,2), (-2,10,2) (-2,8,0), (-4,8,0), (-4,10,0) and (-1,10,0). Each of

the eight tetrahedral is checked and subdivided to generate the streamline and the

streamtube.

60

Fig. 21 shows the streamtube in three-dimensions or more precisely, cross-section of

the streamtube. The expansion rate for this example varies significantly from Fig. 22,

and the projections of Fig.21 on yz-plane in Fig. 23, on xy-plane in Fig. 24, and on xz-

plane in Fig. 25.

Fig. 21 Projection of the streamtube for Example 2 on xy-plane.

61

The following view shows a three dimensional streamtube appearance in blue. It shows

the arrangement and parallel behavior with respect to the streamline in red.

Fig. 22 Streamtube for Example 2 in 3 Dimension.

62

The following figure gives a two dimensional rope like view of the streamtube in blue.

The red curve is the streamline.

-4-10 -8 -6 -4 -2

Fig. 23 Projection of the streamtube in Fig. 22 on yz-plane.

63

The following two dimensional view appears like an ellipse. The streamtube is in blue

and the streamline is in red.

Edit View Insert Tools DeskJtop Window Help

e i n n•i nIU

8

4

2

>, 0

-2

-4

-6

-8

•in

l J

/ /

•' /

I\ \\ \

\ •

••

•• y

V,/ . / •

-

\

\ l" .

• • 1

'1

-

-1

l ^ . - i

'-

1 •

r , •

""1. ^

•-.

! T •• •'

• h . t

-

/

/ J

, ' • • •

\ • ,

\ • •

, \

/ /

/ /

/

-10 -8 -6 -4 -2 0 8 10

Fig. 24 Projection of the streamtube in Fig. 22 on xy-plane.

64

The following is also a two dimensional rope like view where the streamtube is in blue

and the streamline in red.

N 0

-2

-3

-4

-10 -3 -6 -4 -2 0 2 4 6 3 10x

j i

Fig. 25 Projection of the streamtube in Fig. 22 on xz-plane.

65

Summary

The accurate mass conservative streamtube was successfully generated. The

construction process involved generating a streamline and then connecting the circular

cross flow sections along this streamline. The radius of a cross flow section was

determined by the local cross flow expansion rate whereby the centre was the point on

the streamline. The mathematically acquired information was efficiently programmed

using MATLAB. The diagrams helped to show the different stages of streamtube

generation. The examples of saddle spiral flow and toroidal flow showed the high

accuracy of the result.

66

CHAPTER 5

Further Works and Research

The accuracy of created streamtube is dependent on both the accuracies of the

streamline and the circular cross flow sections. Li [23] introduced an adaptive streamline

tracking method for three-dimensional CFD velocity fields based on the law of mass

conservation. This method suits to the CFD velocity fields whereby more data is

available. The advantages of the method introduced in [23] include that the accuracy of

the tracked streamlines for a given mesh can be controlled by the threshold number T.

The larger the threshold number T, the more accurate the tracked streamlines are, and

the tracking process is stopped when not enough data is provided. The overall accuracy

of tracked streamline depends on the initial mesh and the threshold number T. When

CFD velocity fields are given as numerical solutions of mathematical models, more data

of the velocity fields are possible to calculate.

This thesis has introduced a streamtube construction technique for flow visualization.

The circular cross flow sections along the accurate streamline were connected. An

analysis of this streamtube successfully reveals the local cross flow expansion rate. The

use of MATLAB programming involved many objects which were part of a parent

program. Each object performs a specific task to facilitate the construction process.

Further work in this area could be carried out to discover other factors in a flow

visualisation. Expansion rate could be very well shown using colours on the figures.

Better computer with higher visual memory will reveal figures of even much better

quality.

67

Conclusion

This research has introduced a new idea of streamtube generation. It effectively

combined the streamline generation method [23] with circular cross flow connection

method [3]. Streamlines were accurately generated and satisfied the law of mass

conservation. The overall accuracy of tracked streamline depended on the initial mesh

and the threshold number, T. The formula of streamtube radius was successfully

evaluated to connect the circular cross flow sections. The circles are perpendicular to

the velocity field at the points on streamlines. The points on the mass conservative

streamline forms the centre of the mass conservative streamtube.

Accurate and efficient streamtube was constructed using the accurate streamlines and

exact solution of the radius equation for linear velocity field. The projection of the

streamtube show different views of the generation. These dimensional views help to

show the accuracy. However, powerful and faster computers will improve the picture

generation.

These streamtube gives reliable analysis of fluid flows. The examples shown indicate

the high level of accuracy. The examples of spiral saddle flow and toroidal flow show the

streamtube on the mass conservative streamline. The figures are shown in three

dimension and projection on different planes. An increased accuracy of the model will

help in various fields of study such as aerodynamics, industrial process and geophysics.

It effectively visualizes the steady flow fields and reveals the flow expansion rate.

The streamtubes were effectively generated using MATLAB programming. The process

68

was broken down into small steps to facilitate the generation. It consists of many small

programs and required good visual memory because of high computational demand.

Various parts of streamtube could reveal different properties on the streamtube using

colors which will require more MATLAB functions and can be future research work.

69

Reference

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Wiley & Sons, USA.

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http://www. blackwellpublishing. com/book. asp ?ref=9 780632055142&site=1

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70

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term2004/2003/ICDF/streamlinesimulationA4.doc

[9] King, Osaka, and Gupta, 2004, "Streamlines, Streaklines, and Pathlines", 2007

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