The Effect of cylindrical obstructions on the fluid flow ...€¦ · Rochester Institute of Technology RIT Scholar Works Theses Thesis/Dissertation Collections 8-1-1996 The Effect

Rochester Institute of TechnologyRIT Scholar Works

Theses Thesis/Dissertation Collections

8-1-1996

The Effect of cylindrical obstructions on the fluidflow in narrow rectangular channelsGary Compagna

Follow this and additional works at: http://scholarworks.rit.edu/theses

This Thesis is brought to you for free and open access by the Thesis/Dissertation Collections at RIT Scholar Works. It has been accepted for inclusionin Theses by an authorized administrator of RIT Scholar Works. For more information, please contact [email protected].

Recommended CitationCompagna, Gary, "The Effect of cylindrical obstructions on the fluid flow in narrow rectangular channels" (1996). Thesis. RochesterInstitute of Technology. Accessed from

http://scholarworks.rit.edu?utm_source=scholarworks.rit.edu%2Ftheses%2F5877&utm_medium=PDF&utm_campaign=PDFCoverPages

http://scholarworks.rit.edu/theses?utm_source=scholarworks.rit.edu%2Ftheses%2F5877&utm_medium=PDF&utm_campaign=PDFCoverPages

http://scholarworks.rit.edu/etd_collections?utm_source=scholarworks.rit.edu%2Ftheses%2F5877&utm_medium=PDF&utm_campaign=PDFCoverPages

http://scholarworks.rit.edu/theses?utm_source=scholarworks.rit.edu%2Ftheses%2F5877&utm_medium=PDF&utm_campaign=PDFCoverPages

http://scholarworks.rit.edu/theses/5877?utm_source=scholarworks.rit.edu%2Ftheses%2F5877&utm_medium=PDF&utm_campaign=PDFCoverPages

mailto:[email protected]

The Effect of Cylindrical Obstructions on the Fluid Flow

in Narrow Rectangular Channels

by

Gary L. Compagna

A Thesis submitted in partial fulfillmentof the requirements for the degree of

Master of Science in Mechanical Engineering

Approved By:

Professor S. Kandlikar

Professor A. Nye

Professor A. Ogut

Professor C. Haines

DEPARTMENT OF MECHANICAL ENGINEERING

COLLEGE OF ENGINEERING

ROCHESTER INSTITUTE OF TECHNOLOGY

AUGUST 1996

PERMISSION GRANTED

I, Gary L. Compagna, hereby grant pennission to the Wallace Memorial Library of the

Rochester Institute of Technology to reproduce my thesis entitled "The Effect of Cylindrical

Obstructions on the Fluid Flow in Narrow Rectangular Channels" in whole or in part. Any

reproduction will not be for commercial use or profit.

August 31, 1996

Gary L. Compagna

ACKNOWLEDGMENTS

This work could not have been completed without the assistance and understanding of

several outstanding individuals.

Kiran Kumar and Dr. Sung-Eun Kim from Creare, Inc. provided assistance that included

helping with FLUENT software issues and also with understanding of the fluid mechanics

involved in this problem.

A special thanks goes to Satish Kandlikarwithout whose gentle pushing and understanding

this work never would have been completed.

Table of Contents

Page

LIST OF FIGURES iii

LIST OF TABLES vi

LIST OF SYMBOLS vii

ABSTRACT 1

1. INTRODUCTION 2

1.1 APPLICABILITY OF HELE-SHAW FLOW 5

1.2 APPLICATIONS 9

1.2.1 POLYMERASE CHAIN REACTION DETECTION POUCH 9

1.2.2 OTHER APPLICATIONS 10

1.3 PROBLEM DEFINITION 12

1.4 OBJECTIVES OF THE PRESENTWORK 15

2. LITERATURE REVIEW 16

2.1 BASIC GOVERNING EQUATIONS 26

2.1.1 FLUID FLOW GOVERMNG EQUATIONS 26

2.1.2 POTENTIAL FLOW 28

3. THEORETICAL ANALYSIS 37

3.1 FLUID FLOW IN RECTANGULAR DUCTS 37

3.2 INVISCID IRROTATIONAL FLOW IN TWO DIMENSIONAL FLOW 41

3.2. 1 POTENTIAL FLOW FIELD FOR UNIFORM FLOW OVER A CYLINDER.... 42

3.2.2 POTENTIAL FLOW RESULTS USING METHOD OF IMAGES 44

4. NUMERICAL ANALYSIS METHOD 50

4.1 NUMERICALMODEL DEVELOPMENT 50

4.1.1 THREE DIMENSIONAL FLUID FLOW IN NARROW RECTANGULAR

CHANNEL 51

Page

4.1.2 2-D FLUID FLOWWITH CYLINDRICAL OBSTRUCTION 57

4.2 VERIFICATION OF NUMERICAL ANALYSIS 81

4.2.1 2-D FLOW AROUND CYLINDER COMPARISONWITH FLUENT 81

4.2.2 FLUENT COMPARISONS FOR VISCOUS FLOW OVER A CYLINDER 91

4.2.3 VISCOUS FLOW COMPARISON FOR DIFFERENT GRID

FORMULATIONS 97

4.2.4 3-D VISCOUS FLOW COMPARISON TO FLUENT RESULTS IN

NARROW RECTANGULAR DUCTS 99

5. RESULTS FOR 3-D FLOW 104

5.1 BASELINE 3-D RESULT 104

5.2 FLOW UNIFORMITY AS PROBE HEIGHT CHANGES 120

5.3 SENSITIVITY TO CHANNELWIDTH ON THE FLUID FLOW 128

6. CONCLUSIONS 136

7. RECOMMENDATIONS FOR FUTURE WORK 140

REFERENCES 142

n

LIST OF FIGURES

Figure Page

1-1 TYPICAL CHANNEL GEOMETRY FOR PCR DETECTION CHAMBER 4

1-2 TYPICAL HELE-SHAW GEOMETRY 7

1-3 FLUID FLOW PATTERN FOR HELE-SHAW CELL 8

1-4 PCR DETECTION POUCH 1 1

1-5 PCR DETECTION CHAMBER GAP CROSS SECTION 13

2-1 O-H GRID STRUCTURE, Kim and Choudhury (1990) 17

2-2 EXPERIMENTAL FLOW PATTERN FOR LOW REYNOLDS NUMBER 19

2-3 EXPERIMENTAL FLOW PATTERNS AT Re = 19, 26, and 55 20

2-4 STREAMLINES FOR STEADY FLOW PAST A CIRCULAR CYLINDER FOR

Re = 5 AND 40 , Dennis and Chang (1970) 22

2-5 DIMENSIONLESS PRESSURE COEFFICIENT ON THE CYLINDER

SURFACE, Dennis and Chang (1970) 23

2-6 UNIFORM FLOW AROUND A CYLINDER 31

2-7 DOUBLET REFLECTED USINGMETHOD OF IMAGES 36

3-1 VELOCITY PROFILE ACROSS GAP HEIGHT 38

3-2 FLUID VELOCITY ACROSS GAPWIDTH 40

3-3 STREAMLINE PLOTS USING POTENTIAL FLOW THEORY 43

3-4 PRESSURE DISTRIBUTION USING POTENTIAL FLOW THEORY 45

3-5 STREAMLINE PLOTS USING THE METHOD OF IMAGES 46

3-6 VELOCITY PROFILE ON CYLINDERUSINGMETHOD OF IMAGES 48

3-7 PRESSURE DISTRIBUTION USING THE METHOD OF IMAGES 49

4-1 FLUENT VELOCITY IN NARROW RECTANGULAR DUCT 53

4-2 FLUENT 3-D VELOCITY PROFILE 54

4-3 FLUENT VELOCITY ACROSS CHANNEL GAP HEIGHT 55

4-4 FLUENT VELOCITY ACROSSWIDTH 56

4-5 2-D GEOMETRY FOR FLOW OVER A CYLINDER BETWEEN PARALLEL

PLATES 58

in

Figure Page

4-6 POWER LAW SCHEME USED BY FLUENT 61

4-7 PHYSICAL GRID PATTERN FOR 2-D FLOW AROUND CYLINDER 63

4-8 FLUENT COMPUTATIONAL GRID 64

4-9 CELL NOMENCLATURE EMPLOYED IN HIGHER ORDER

INTERPOLATION SCHEMES 65

4-10 FLUENT POTENTIAL FLOW SOLUTION USING POWER LAW

INTERPOLATION SCHEME 67

4- 1 1 FLUENT POTENTIAL FLOW SOLUTION USING QUICK

INTERPOLATION SCHEME 68

4-12 VELOCITY PLOT FOR POTENTIAL FLOW AROUND CYLINDER 69

4- 1 3 FLUENT STREAMLINES FOR VISCOUS FLOWWITH Recy, = 40 AND "NO

SLIP"

BOUNDARY ON THE CYLINDER 72

4-14 GEOMETRY AND GRID LAYOUT FOR SYMMETRIC FLOW AROUND

CYLINDER 73

4-15 VELOCITY FOR VISCOUS FLOWWITH Recyi = 40 AND FREESTREAM

BOUNDARIES ON THE OUTERWALLS 74

4- 1 6 PRESSURE DISTRIBUTION FOR VISCOUS FLOW AROUND A CYLINDER

WITHRecyi = 40 75

4-17 STREAMLINES FOR FLOWWITH Recyi = 40 76

4-18 GEOMETRY AND GRID LAYOUT FOR FLOW 77

4-19 VELOCITY PROFILE FOR FLOWWITH Recyi = 40 78

4-20 PRESSURE PROFILE FOR FLOWWITH Recyi = 40 79

4-2 1 COMPARISON OF PRESSURE PROFILE WITH 180

AND360

CYLINDER . . 80

4-22 LOCATIONS FOR COMPARISON OF FLUENT TO THEORETICAL

RESULTS 83

4-23 VELOCITY PROFILE COMPARISON WITH METHOD OF IMAGES 87

4-24 PRESSURE COEFFICIENT COMPARISON 89

4-25 FLUENT SOLUTION FOR VISCOUS FLOW OVER A CYLINDER 92

4-26 PRESSURE COEFFICIENT RESULTS COMPARED TO BRAZA et al. (1986) ... 94

IV

Figure Page

4-27 FLUENT STATIC PRESSURE DROP IN RECTANGULAR CHANNEL 103

5-1 FLUENT SYMMETRIC GEOMETRY OUTLINE 105

5-2 3-D GEOMETRY FRONT AND TOP VIEWS 106

5-3 CROSS SECTION OF COMPUTATIONAL GRID FOR 3-D FLOW 107

5-4 FLUID VELOCITY ACROSS WIDTH OF PCR DETECTION CHAMBER 109

5-5 PRESSURE DISTRIBUTION ACROSSWIDTH OF PCR DETECTION

CHAMBER 110

5-6 FLUID VELOCITY CLOSE-UPNEAR DETECTION PROBES 1 1 1

5-7 PRESSURE DISTRIBUTION CLOSE-UP NEAR DETECTION PROBES 1 12

5-8 VELOCITY DISTRIBUTION ACROSS CHAMBER THICKNESS 1 13

5-9 PRESSURE DISTRIBUTION ACROSS CHAMBER THICKNESS 114

5-10 CLOSE-UP OF VELOCITY DISTRIBUTION THROUGH THE THICKNESS 1 15

5- 1 1 CLOSE-UP OF THE PRESSURE DISTRIBUTION THROUGH THE

THICKNESS 116

5-12 PRESSURE COEFFICIENTS ON DETECTION PROBES FOR 3-D FLOW 1 18

5-13 PRESSURE ACROSS THE TOP OF THE PCR DETECTION PROBE 1 19

5- 14 GEOMETRY OUTLINES FOR DIFFERENT DETECTOR PROBE HEIGHTS ..121

5-15 FLUID VELOCITY FOR BASELINE HEIGHT DETECTOR PROBE 122

5-16 FLUID VELOCITY FOR INCREASED DETECTOR PROBE HEIGHT 123

5-17 FLUID VELOCITY FOR FULL CHANNEL HEIGHT DETECTOR PROBE 124

5- 18 PRESSURE COEFFICIENT ON FIRST PROBE FOR DIFFERENT PROBE

HEIGHTS 125

5-19 PRESSURE COEFFICIENT FOR DIFFERENT PROBE HEIGHTS 127

5-20 CONFIGURATIONS FOR VARYING CHANNEL WIDTH 129

5-21 FLUID VELOCITY FORNARROW CHANNELWIDTH 130

5-22 FLUID VELOCITY FORWIDER DETECTION CHAMBER 131

5-23 PRESSURE COEFFICIENT FOR DIFFERENT CHAMBERWIDTHS 133

5-24 RELATIVE STATIC PRESSURE ON PROBE #1 134

5-25 SENSITIVITY OF STATIC PRESSURE DROP TO CHAMBERWIDTH 135

LIST OF TABLES

Table Page

4- 1 STREAM LINE COMPARISON TO THEORY UPSTREAM FROM

CYLINDER 84

4-2 STREAM LINE COMPARISON TO THEORY ABOVE CYLINDER 85

4-3 AXIAL VELOCITY COMPARISON TO THEORY ABOVE CYLINDER 86

4-4 PRESSURE COEFFICIENT COMPARISON ON CYLINDER 90

4-5 COMPARISON FOR SEPARATION ANGLE 95

4-6 COMPARISON OFWAKE BUBBLE LENGTH 96

4-7 COMPARISON OF KEY PRESSURE POINTS FOR DIFFERENT SOLUTION

METHODS 98

4-8 COMPARISONWITH FLUENT ACROSS GAPWIDTH 100

4-9 COMPARISONWITH FLUENT ACROSS GAP THICKNESS 101

5-1 SUMMARY OF PRESSURE COEFFICIENTS AT 6 =0

126

5-2 CASE SUMMARY FOR SENSITIVITY TO CHAMBERWIDTH 128

VI

LIST OF SYMBOLS

SYMBOL UNITS

a cylinder diameter m

b 1/2 channel width m

g Gravitym/sec2

h Elevation m

P PressureN/m2

Pa Pressure on cylinder at radius = aN/m2

P- Pressure upstream at inletN/m2

r Distance in radial direction m

u Velocity in axial direction m/sec

V Velocity in vertical direction m/sec

X Distance in axial direction m

y Distance across chamber m

z Distance across height of chamber m

Ac Cross sectional aream2

Cp Pressure coefficient on cylinder

Dh Hydraulic diameter m

H Height of detection chamber m

L Length ofDuct m

Lc Characteristic length for duct m

P Perimeter m

Pe Peclet number

Re Reynolds number

Recyi Reynolds number based on cylinder diameter

um Mean velocity in duct m/sec

u Free Stream Velocity m/sec

Vr Velocity in radial direction m/sec

V6 Velocity in circumferential direction m/sec

w Width of detection chamber m

Vll

Greek Symbols

SYMBOL

P Density

H Viscosity

e Angle

<D Velocity potential

*F Stream function

A Doublet strength

4> flux of variable

UNITS

kg/m3

kg/(m sec)

radians

m2/sec

m2/sec

m /sec

Vlll

The Effect ofCylindrical Obstructions on the Fluid Flow

in Narrow Rectangular Channels

ABSTRACT

This thesis presents a computational fluid flow analysis in narrow rectangular channels, with

regular spaced cylindrical disks acting as obstructions to fluid flow. The problem geometry

is based on an approximation of the configuration used in a PCR detection chamber.

Polymerase chain reaction (PCR) is a method ofDNA analysis that depends on the flow

uniformity within the detection channel. The nominal channel is a narrow channel with a

maximum gap height of 0.13 mm and a maximum width of 5.0 mm. The fluid flow rate and

channel size result in a Reynolds number less than 5.

The effect of oligonucleotide detection probes within the detection channel and the channel

geometry on the fluid pressure is determined. This is accomplished by approximating the

detection probes as short cylinders and using the CFD code FLUENT to calculate the flow

velocity within an idealized rectangular detection chamber.

The CFD results are compared to theoretical potential flow solutions and other published

numerical results in 2-dimensions. This work extends the 2-D solutions to solve the full

3-dimensional flow field within the detection chamber.

1. INTRODUCTION

Fluid flow in micro sized channels with cylindrical obstructions is a geometry of interest in

many applications. Figure 1-1 shows a typical channel for a polymerase chain reaction

(PCR) detection chamber.

PCR is a method ofDNA analysis that requires fluid flow over detection probes in an

enclosed space. Figure 1-1 has dimensions that are much larger in the width (W) direction

than in the height (H) direction. The small size of the channel results in Reynolds Numbers

that are very low (Re < 5). This low Reynolds's numbermeans that the flow will become

fully developed in a very short length compared to the overall channel length.

The low Reynolds Number and the geometry ofWH lead to this fluid flow being a

variation of two types of common families of flow problems.

Stokes Flow: Applicable with very small Reynolds number.When this occurs the viscous

forces overwhelm the inertia forces. This allows the non-linear terms in the Navier Stokes

equations to be neglected.

Hele-Shaw Flow: Involves slow flow of a fluid between parallel flat plates which are fixed

at a small distance (H) apart.

Hele-Shaw flow is often created in experimental test configurations to help understand the

basic flow phenomena. Stokes flow is often assumed because this allows the full Navier-

Stokes equations to be simplified to the point where they can be solved for specific

configurations and boundary conditions. These flow types are very useful for understanding

some fluid flow configurations. However, the assumptions required limit their usefulness to

specific flow conditions.

Using computational fluid dynamics (CFD) codes to solve these types of problems allows a

person to examine a wide range of geometries, boundary conditions, and fluid types in a time

effective manner without the expense of a good experimental test set-up.

Over the last decade CFD solution techniques have improved both in computational

efficiency and numerical accuracy. Also computing hardware has advanced to the point

where individual engineers and scientist have at their disposable relatively low cost

workstations that are capable of analyzing complex fluid flow problems in reasonable time

periods.

These advances have led to the development of numerous commercially available CFD

software codes. FLUENT from Creare, Inc. is one of these commercial CFD codes. The

software includes both pre- and post-processing capability that allow the flow geometry to

be created and modified as required. This allows a wide range of variables to be modified to

determine the sensitivity of the fluid flow result.

Figure 1-1 TYPICAL CHANNEL GEOMETRY FOR PCRDETECTION

CHAMBER

O.OTt.mm

(o. 003>!>0

jzte.8 tvi^

(01.1 in)

(O.Z ir>)

-lO.lfepiM-

-H S.06mtn

(O.ZtJi*') 0.02.^ mrvi

(0.001 in)

1.1 APPLICABILITY OF HELE-SHAW FLOW

Hele-Shaw (1898) used an experimental set up to determine the streamlines for the flow

around bodies of arbitrary shape. He showed that a three-dimensional viscous flow between

two closely spaced flat plates exhibited two dimensional potential flow patterns.

Figure 1-2, from 'ViscousFlows'

by Ockendon and Ockendon (1995) shows a typical

Hele-Shaw cell. Figure 1-3, from 'VisualizedFlow'

compiled by Nakayama (1988) shows

the flow pattern that results when looking down at the top of the Hele-Shaw cell.

There are two aspects of this study that fall under the general Hele-Shaw flow analogies.

The first aspect of the Hele-Shaw flow is that for the motion of a viscous fluid, between two

fixed parallel plates which are sufficiently close together, Saffman and Taylor (1958), define

the mean velocity in the cell as follows.

b2

(dpu=-T27bi+pg| (L1)

b2

dpv =

-m(L2)

The second part of the Hele-Shaw analogy that is applicable to narrow channel flow is when

one fluid of a different density is accelerated perpendicular to the interface of another fluid.

This is the case when one fluid is at rest in the channel and another fluid is pushed into the

channel to "washout"

the previous fluid. When the accelerating fluid

has the higher density the interface between the two will be stable. When the less dense fluid

is accelerated into the higher density fluid the interface will be unstable, Saffman and Taylor

(1958).

The type of fluid flow that will be studied in this case will be a variation of the Hele-Shaw

flow. For Hele-Shaw flow analysis, any obstruction in the channel is assumed to take up the

complete height. This study will look at cases where the cylindrical disk obstructions are

less than the height of the channel.

The second extension ofHele-Shaw flow that will be investigated is the interaction between

the cylindrical disk and the side wall edges. Pure Hele-Shaw cells assume that the flow

around any obstructions are not effected by side walls.

The above two extensions ofHele-Shaw flow can be thought of as taking the Hele-Shaw

flow which is based on the flow being analyzed in 2-dimensions and extending the analysis

to 3-dimensions.

Figure 1-2 TYPICAL HELE-SHAW GEOMETRY

Hele-Shaw flow

Hele-Shaw flow past an obstacle

Figure 1-3 FLUID FLOW PATTERN FOR HELE-SHAW CELL

Slow flow passes a circular cylinder (cylinder diameter 8 cm).

1.2 APPLICATIONS

1.2.1 POLYMERASE CHAIN REACTION DETECTION POUCH

This study was undertaken as an attempt to understand the performance of PCR detection

pouches. The polymerase chain reaction process (PCR) is a method for amplifying and

capturing specific samples ofDNA. PCR amplification may contain 6 x1011

copies of a

particularDNA target strand. One PCR detectionmethod described by Findlay et al (1993)

combines the amplification and detection process in a single closed vessel. The detection

process relies on the specific hybridization to oligonucleotide probes and enzymatic signal

generation. Figure 1-4 shows a drawing of the PCR'pouch'

used for this process. The PCR

pouch shown is an expandable plastic design. To analyze the flow in the narrow detection

chamber the channel geometry will be approximated as a rigid rectangular channel.

The detection process works after hybridization, when biotinylated PCR products are

captured on the discrete detection spots wherever they encounter probes complimentary to

their sequence. After subsequent treatment with different fluids (streptavidin-horseradish

peroxidase conjugate, wash solution, and dye precursor solution) color will develop on the

detection probes

One of the goals in the design of this PCR pouch was to minimize the amount of fluids

required to carry out the process. The second goal was to be able to determine a positive or

negative test result by visual comparison of the detection probes against a color chart or

instrumentally by reflection densitometry. The combination of the two previously mentioned

goals means that the fluid flow across the detection probes is critical to the performance of

the overall system.

The 'color response of the detection probe is based on diffusion from the fluid to the probe

and the process is sensitive to sample concentration in the fluid and the probe. The process is

rate sensitive which leads to the importance ofwell understood fluid and thermal boundary

conditions.

1.2.2 OTHERAPPLICATIONS

Other types of fluid flow where this analysis for flow in narrow channels with obstructions

could be applicable are as follows.

1. Blood flow with blockage; the flow of blood in the body through narrow arteries or

veins with some type ofblockage would fall into this general type of application. The

geometry of the rectangular channel and cylindrical disk blocking the fluid flow would be

modified for this analysis.

2. Flow of fluid in Ink Jet printer; Inkjet printers require the flow of fluid in narrow

channels. Also the insertion of cleaning fluids or different color inks could fall into

Hele-Shaw flow depending on the geometry of the flow path.

3. Electronic cooling in micro-channels; small electronic systems such as multi-chip

modules often require good thermal control to maintain accurate performance. This can

require fluid flow over small electronic components to remove the power being

dissipated within the component.

10

Figure 1-4 PCR Detection Pouch

.110 DIA.OIO'

7 DOTS

? 0 .050% A B C

4- 0 .030% A

2.575*.OlO

11

!-3 PROBLEM DEFINITION

The PCR detection process is based on getting positive or negative readings on each

detection probe. Positive readings are obtained when the fluid passing over the detection

probe diffuses sufficient amounts of the DNA strands, wash fluid, and dye fluid into the

probe. The diffusion process depends on the sample concentration in the detection probe and

in the fluid passing through the detection chamber. The rate the fluid passes over the

detection probe and the pressure the fluid exerts on the probe are significant factors in

obtaining satisfactory machine performance.

Experimentally determining the flow rate and pressure is not an easy process given the

miniature geometry of the PCR detection chamber. Using CFD tools to analytically

determine the flow field is a lower cost approach in terms of time, people, and money.

Figure 1-5 shows a cross section of the detection chamber inside the PCR instrument. This is

the geometry as the fluid is passing through the detection chamber. The PCR pouch is

nominally flat and expands as fluid passes through the detection chamber.

The analysis of this type of fluid problem requires the definition of several variables. These

include the following.

1. Geometry Configuration

2. Fluid type

3. Fluid properties

4. Boundary conditions

Inlet flow rate or velocity

Outlet conditions

Wall properties

5. Solution method

12

Figure 1-5 PCR DETECTION CHAMBER GAP CROSS SECTION

Fixed @+/-.002"

Detection) Pfc.og,E

FtUVCw FuovO Chanmei

13

The overall geometry size was previously shown in Figure 1-1. This shows the problem to

be a full 3-dimensional problem. This work will approach the problem in 3-D because

diffusion into the detection probes depends on flow on both the top and side walls.

The length of the detection chamber and the number of detection probes will be reduced to

minimize the computational time. The 3-D solution will be reduced to 25.4 mm and the

number of detection probes will be reduced to 2. This reduction in geometry size will greatly

reduce the problem run time and still allow the effect of probe and chamber geometry on the

flow field to be determined.

The type of fluid will be assumed to be water for all cases. The actual PCR process uses

several different fluids but they all are largely water based. Using water also means the fluid

properties are readily available in standard publications.

The inlet boundary conditions for this work will use a nominal inlet velocity of 0.01 m/sec in

most cases. This velocity is based on the amount of fluid in the PCR pouch and experimental

results for the time it takes the fluid to transverse the length of the detection chamber.

The outlet boundary conditions for the actual PCR pouch consist of a large fluid reservoir.

The outlet of the detection chamber will be approximated as an infinite reservoir.

The walls and the detection probes will both have a 'noslip'

boundary condition applied

during the solution of this problem. In actual practice the detection probes are porous cells.

Experimental results have shown this to be a secondary effect on the fluid flow within the

detection chamber.

14

This problem will be solved using the CFD software code FLUENT. The problem will be

solved as a steady state solution. Given the finite amount of fluid in the PCR pouch, this

steady state solution would only be valid for a very short amount of time. However, the

steady state solution will provide excellent insight into the flow velocity and pressure

distribution within the detection chamber.

1.4 OBJECTIVES OF THE PRESENTWORK

The objectives of this work are given below. These objectives are based on the desire to

have uniform flow over the maximum possible surface area of the detection probes in the

PCR process. This is because the chemical reaction between the DNA strands and the

detection probes is a rate dependent diffusion process that depends on the concentration of

species in the fluid and the detection probe.

These objectives are based on first verifying that FLUENT is a proper tool for analyzing this

type of flow problem and then using FLUENT to analyze the full 3-D flow field.

1 . Compare the results from potential flow theory to FLUENT for flow over cylinders.

2. Compare the FLUENT results to other published results for flow over a cylinder.

3. Determine if FLUENT will properly predict the effect ofwalls located in close

proximity to a cylinder. Use the potential flowMethod of Images theory for verifying

the FLUENT results.

4. To show that commercial CFD codes such as FLUENT can be used to analyze the three

dimensional fluid flow in narrow channels approximately 0.04 mm thick.

5. Determine the pressure distribution on the detection probes for a typical geometry.

6. Determine the flow field over the cylindrical disk obstruction as the channel width

changes.

7. Determine the flow field within the detection chamber as the probe height is varied.

15

2. LITERATURE REVIEW

There is no literature available to the author's knowledge on a detailed study for 3-D fluid

flow around cylinders within a rectangular duct. There is however a large body ofwork for

simplified versions of this problem. The case of two dimensional fluid flow around a

cylinder is a very popular test case for verifying different CFD codes.

The American Society ofMechanical Engineers published a compilation (FED-Vol. 160) of

different papers from different CFD vendors with solutions to the 2-D cylinder flow

problem. All of these papers were presented at the Fluids Engineering Conference in 1993.

The paper by Kim and Choudhury (1993) is of particular interest as it employs the same

software, FLUENT, which is used in the current study.

Kim and Choudhury (1993) use a unique grid structure that is a combination ofO-type grids

around the cylinder and hexagon grids far upstream and downstream from the cylinder. This

grid structure is shown in Figure 2-1. This grid structure is effective for getting good results

around the cylinder using the O-grid and also far afield from the cylinder using the hexagon

grid. A key simplifying assumption made in this paper is the assumption that a 'freestream'

boundary condition is used to simulate the walls surrounding the flow. The free stream

boundary condition implies that the stream function is equal to zero and also the vorticity is

equal to zero. Kim and Choudhury accomplish this by setting the outer walls as symmetry

boundaries. The assumption of free stream boundary conditions are designed to have the

effect of removing the wall from the solution. Depending on the numerical approach used,

different authors use different techniques to approach a free stream boundary condition at the

wall.

16

Figure 2-1 O-H Grid Structure used by Kim and Choudhury (1990)

CYCLIC

SYMMETRY

INLETZONE 1

/OUTLET

/SYMMETRY

INLETZONE 1

CYCLIC

WALLZONE 1 (CYLINDER SURFACE)

Computational Grid

SYMMETRY

OUTLET

SYMMETRY

Physical Grid

17

Kim and Choudhury (1993) state that in the vicinity of Recyi = 5, the flow starts to separate to

form a pair of recirculating eddies attached to the body. They also state the eddies formed

behind the cylinder remain stationary until another bifurcation takes place around Recyi = 40.

Beyond Recyi = 40, the flow becomes asymmetric and unsteady, being accompanied by

alternate vortex shedding. Kim and Choudhury (1993) present all of their results based on

Recyi= 60.

For the work presented in this report the Recyiwill be equal to 42 or less. The Recyi

approximately equal to 40 was chosen because it represents the actual flow rate for the PCR

process. Unfortunately this is the Reynolds number that is the dividing point between steady

flow and unsteady flow caused by vortex shedding.

The results described by Kim and Choudhury (1993) are consistent with experimental results

available in literature. Shown in Figure 2-2 are some flow experimental flow results from

Nakayama (1988) for very low Reynolds numbers of less than 2. Notice that there are no

eddies on the back side of the cylinder.

As the Reynolds number is increased the eddies do begin to form. Figure 2-3 shows some

more experimental results for Nakayama (1988) for Reynolds numbers of 16 and 26. Once

the Reynolds number is increased higher the eddies become unstable and begin to shed.

Figure 2-3 also shows another experimental result for Nakayama with a Reynolds number of

55. At this point the eddies are not attached to the back of the cylinder, and the flow is now

unsteady.

18

Figure 2-2 Experimental Flow Pattern for Low Reynolds Number

Flow around a circular cylinder at Re = 0.038 (glycerine, flow velocity 0.15 cm/s,

cylinderdiameter 1.0 cm, tankwidth 40 cm, aluminium powder method).

Flow around a circular cylinder at fie = 1.1 (glycerine-^water solution, flowvelocity

0.20 cm/s, cylinderdiameter 1 .0 cm, aluminium powdermethod).

19

Figure 2-3 Experimental Flow Pattern at Reynolds Numbers of 19, 26, and 55

Flow around a circular cylinder at Re

= 1 9 (water, flow velocity 0.20 cm/s, cylinder

diameter 1.0 cm, aluminium powder

method and electrolytic precipitation

method).

Flow around a circular cylinder at Re Flow around a circular cylinder at Re

= 26 (water, flow velocity 0.25 cm/s, cylinder= 55 (water, flowvelocity 0.55 cm/s, cylinder

diameter 1.0 cm, aluminium powder diameter 1.0 cm, aluminium powder

method). method).

20

The numerical results from this analysis will be compared to the results of Braza et al.

(1986), Dennis and Chang (1970), and Fornberg (1980). All these papers because of

simplifying boundary conditions on the walls base the Reynolds number on the cylinder

diameter rather than the channel geometry.

Dennis and Chang (1970) solved the 2-D flow problem for 5 < Recyi< 100 using a finite

difference solution technique. Dennis and Chang (1970) also apply the free stream boundary

condition at the wall. They do discuss other possible boundary conditions but do not give

any results. Figure 2-4 shows the stream line plots from Dennis and Chang (1970) for Recyi

equal to 5 and 40. Notice the difference in the flow pattern behind the cylinder. The

Recyi = 40 plot clearly shows the circulating eddy that were previously discussed. This eddy

does not occur for the lower Recyi .

The other portion of the Dennis and Chang's (1970) result that is of interest is the results

from the pressure coefficient on the cylinder surface. Dennis and Chang (1970) define a

dimensionless pressure coefficient given in equation 2-1 . Figure 2-5 shows the result of this

pressure coefficient for different Reynolds numbers.

cP(e)=^g^ (2.D

21

Figure 2-4 STREAMLINES FOR STEADY FLOW PAST A CIRCULAR

CYLINDER FOR Re = 5 AND 40 , Dennis and Chang (1970)

c-n.

1-223

Ret^= HO

22

Figure 2-5 DIMENSIONLESS PRESSURE COEFFICIENT ON THE CYLINDER

SURFACE, Dennis and Chang (1970)

20

1-5

10

0-5

00

-0-5

-10

-1-5180'

\̂i%\| .R=100___

^(^

150 120 90

e

60 30

23

Fornberg (1980) analyzed the flow over the cylinder in a similar method to Dennis and

Change (1970). The main difference is Fornberg (1980) places much greater emphasis on the

types ofboundary conditions to apply. He points out the calculations for vorticity around the

cylinder can have an error in excess of 20% when using a free stream boundary condition.

This is true even if the boundary condition is applied far away (23 times the radius) from the

cylinder body. Fornberg considers four different boundary conditions.

1 . Free stream

2. One term of the Oseen approximation

3. Normal derivative of stream function = 0

4. A mixed condition of option 1 and 3

The free stream condition implies that the stream function equals 0 at the wall. This will

neglect the effect of the boundary layer on the wall. Combining this with the gradient being

zero does not fully take into account the wall effect. To get the full wall effect one needs to

make the actual velocity equal to zero and allow the boundary layer at the wall to form.

Fornberg presents results very similar to Dennis and Chang (1970) for stream line and

vorticity. The paper presents results for 2 < Recyi< 300. A different solution method is used

for Reynolds numbers less than 10, but no details are given on this solution method except

that it is based on a fast Poisson solver.

The report ofBraza et al. (1986) compares the numerical results to experimental results from

different authors. The solution method used is similar to FLUENT in that the governing

equations are written in a velocity-pressure formulation and in conservative form, are solved

by a predictor-corrector pressure method, a finite volume second order accurate scheme and

an alternating directionimplicit procedure. Braza et al. (1980) concentrates on Reynolds

number above 100 because they are interested primarily in the vortex shedding around the

cylinder. Some results are presented for lower Reynolds numbers of 20 and 40.

24

Braza (1980) also uses a finite volume technique (the same as FLUENT) instead of a

straight finite difference technique like Dennis and Chang (1970). Braza states that the

governing equations integrated over an elementary control volume enhance the local mass

and momentum conservation near the boundaries better than a simple finite difference

approximation scheme. Braza (1980) also rewrites the governing equations and solves them

in a logarithmic-polar coordinate system. This makes the grid configuration conform closer

to the cylinder geometry.

The results ofBraza et al. (1980) show a greater negative pressure coefficient than the

results ofDennis and Chang (1970). For a Reynolds number of 40, Braza et al. (1980) have a

minimum value of -1.19 where Dennis and Chang (1970) have a minimum value of -0.95.

Braza et al. (1980) does give a different definition for the pressure coefficient Cp than

Dennis and Chang (1970). It is believed by this author that Braza's definition is a

typographic mistake because the results presented agree well with other published results.

Using Braza's definition as published would result in significantly different results.

One of the reasons for the different results between Braza et al. and Dennis and Chang is

because they each use slightly different governing equations to define the flow field.

Braza et al. have written the governing Navier-Stokes equations in terms of pressure and

velocity. Dennis and Chang have simplified the governing flow equations and written the

equations in terms of stream function and vorticity.

Extensive use was also made through out this report of the classic books that have been

published by Schlichting (1979) andMilne-Thomson (1960). Specific references to their

work and other published books will be discussed latter in the report.

25

2.1 BASIC GOVERNING EQUATIONS

The following section presents some of the top level governing equations. These equations

can be found in one form or another in standard fluid mechanics text books. The following

discussion will include both real fluids with viscosity, and no slip at the solid surface, along

with ideal flow where the flow is allowed to slip and the viscosity is assumed zero or

neglected. Most of the theory summarized here was contained in books by Schlichting

(1979), Churchill (1988) and Fox andMcDonald (1985).

2.1.1 FLUID FLOW GOVERNING EQUATIONS

Given in Figure 1-1 is the geometry for the PCR detection chamber. The small size of the

chamber and the minimal amounts of fluid mean that the Reynolds number will always be

very low .

Re = p_ULc (2.2)

M-

For the geometry of the PCR detection chamber the characteristic length is the hydraulic

diameter.

Lc=Dh=4Ac (2.3)

P

Most of the published literature concentrates on the flow around the cylinder and simplifying

the wall boundary conditions. For this reason the characteristic length used in the published

data is the cylinder diameter. In this report the Reynolds number based on the cylinder will

be shown with a subscript, Recyi.

26

The Navier Stokes equations and the conservation ofmass (or continuity) equation that

define fluid flow in the detection chamber are as follows

continuity: 3p_ + V(pu) = 0 (2.4)

3t

pDu = - Vp + |iV2u (2.5)

Dt

Where

D d d d d= +u- +v +w (2.6)

Dt dt dx dy dz

V=i+J+k (2.7)dx dy dz

u = i + vj + wk (2.8)

This form of the Navier-Stokes equation assumes incompressible flow and variations in the

fluid viscosity can be neglected. Both of these assumptions are valid for the analysis in the

PCR detection chamber because the flow velocity is very low and the chamber is held at a

constant temperature. In the case of frictionless flow, where the viscosity is low and can be

neglected ([J. = 0), the Navier-Stokes equation can be reduced to Euler's Equation.

Du _

p= pg-V/> (2.9)

27

Even though all real fluids have viscosity, there is a significant amount of published work on

ideal fluid flow. Flow with zero viscosity is defined as inviscid fluid flow. There are no

shear stresses present in inviscid fluid flow.

2.1.2 POTENTIAL FLOW

For classical potential flow theory, the flow must be both inviscid (fi = 0) and irrotational.

The key assumption in this type of flow is that fluid friction near the boundary can be

neglected. In real fluids this is never true, but potential flow can give acceptable

understanding of the flow phenomena provided you do not look too close to the boundary

wall. Potential flow analysis is a very popular technique because there are a large number of

analytical solutions representing different types of fluid flow. Potential flow analysis is

currently being used to help in the design of airplanes, boats, and automobiles.

There is a large body of work that falls under the heading of Potential Flow. This report will

concentrate on only two dimensional potential flow. In two dimensions, with constant

density, the conservation ofmass given in equation 2.4 reduces to the following.

+-0 (2.10,dx dy

The stream function is defined such that it also satisfies the continuity equation.

u = (2.11)3y

v =-

(2.12)

28

The same continuity equation and stream function can be defined in cylindrical coordinates.

This will be very useful for looking at flow around a cylinder.

drV dVConservation of mass: + - = 0 (2. 13)

dr dd

Stream function: V=-^- (2.14)r dO

Ve=-d-l (2.15)dr

To be a potential flow the flow must be both inviscid and irrotational. For irrotational flow it

is possible to define a velocity potential as follows.

V = -VO> (2.16)

The above definition for the velocity potential is not consistent across different fluid text

books. Many sources define V = V<E>. This report will use equation 2.16 because it leads to

the positive direction of flow being in the direction of decreasing potential. In cylindrical

coordinates the velocity potentials are define as follows.

dVr=~

(2.17)dr

*-~i

29

For irrotational flow the fluid elements in the flow field do not undergo any rotation. This

leads to the following equation for an irrotational flow.

^-^ = 0 (2.19)dx dy

Substituting the definition for the stream function (eq. 2. 1 1 and 2.12) into the irrotational

flow equation (2.19), and substituting the velocity potential equation (2.16) into the

continuity equation (2.10) it is possible to obtain two equations that are both forms of

Laplace's equation. Also any function *P or O that satisfies Laplace's equation represents a

possible two-dimensional, incompressible, irrotational flow field.

t +^-T = (2-2)3x2 3y2

<92<D (920

^+^= <221)

Part of the reason potential flow analysis is so often used is that different elementary flow

patterns can be added to one another to create a complex flow pattern. Both <1> (velocity

potential) and *F (stream function) satisfy Laplace's equation for flow that is incompressible

and irrotational. Since Laplace equation is linear and homogeneous partial differential

equation solutions may be added together. Using superposition it is possible to simulate the

flow around the cylinder and the walls for inviscid flow. Superposition can be used because

each potential (O3 = Oi + O2) is a unique solution ofLaplace's equation, V2<P = 0. To create

flow around a cylinder, superposition is used for uniform flow past a doublet. A doublet is a

combination of a source and sink ideal flow. Figure 2-6 shows the flow configuration for a

uniform flow around a cylinder.

30

Figure 2-6 UNIFORM FLOW AROUND A CYLINDER

--

LLoO

r-a

Stream liwe.S

31

Uniform Flow;

velocity potential

stream function

O = -Ux= -U^rcosO (2.22)

(2.23)

Doublet;

velocity potential <D = -

Acos0(2.24)

stream function *F = -

Asin0(2.25)

Cylinder; ^cyl = ûniform + ^doublet (2.26)

= U rcosd-

Acosfl(2.27)

Any closed streamline can be taken as the surface of a solid immersed in the fluid flow. This

means the cylinder wall is represented by the streamline *F = 0. For the inviscid flow around

a circular cylinder with radius = a, and A=Uâ2

,Churchill (1988) gives the following

equations for the potential functions and stream functions in cylindrical and rectangular

coordinates.

0 = -u

' a2^

r +

KrJ

,2 A

cost? = -ux\ 1 +2

,2

x +y )(2.28)

T =-uM

( a2^

r

^r J

(sin =

-uy

,2 >

1 +2 . 2x +y

(2.29)

32

The velocity components in cylindrical coordinates are

1 dy/.

a

ur= -- = U,

r d6'l-^lcosfl (2.30)

V

6

dr

fi 2)V r J

sing (2.31)

The velocity at the surface r = a is then

"e,fl=-2LLsin0 (2-32)

u = 0 (2.33)

The velocity is seen to be zero at the forward (0= 7t) and rear (0 = 0), which are called the

points of stagnation. The pressure distribution is given as

Pa=P+^=-[1-4sin2

d] (2.34)

The previous equations have defined how to predict an ideal fluid flow for uniform flow

around a cylinder. These equations do not take into account any effect walls outside the

cylinder would have on the flow. To create the walls the method of images can be used .

"The method of images was introduced by Kelvin for use in electricity and later used by

Helmholtz and Stokes in fluiddynamics."

Granger (1975)

The method of images defines that a rigid planar boundary can be created from a distribution

of potential flows (sources, sinks, etc.) by reflecting the image of all singularities outside the

boundary.

33

To model uniform flow over a cylinder between two walls requires that the singularity inside

the walls, which will be the same ideal flow doublet that was previously discussed, the

doublet will have to be reflected outside the walls. Note that to fully accomplish this each

reflection will have to be reflected itself, which makes the final solution a series of

reflections. Figure 2.7 shows a representation of the reflected doublet. The cylinder is

defined the same as in the previous section by letting the cylinder radius be the point where

the stream function is equal to zero. Chung (1978) gives the stream function and horizontal

velocity using the method of images as follows.

= /_ ?-hr-

H

lit

sinh'

(id>\ . (2jtysin

-

\H) V H

H

,2i nx) i( xycosh cos

'

H

(2.35)

The velocity is determined by taking the derivative of the stream function.

d*F

Vx =^r- = U.x

dy

1-

sinh'Kb^ (2iiycos

-

cosh'

^KX^

VH;

-cos

^7ty^+ -

2 {Hjsm

'27cy

cosh'

TtX

h".cos

'icy"(2.36)

vy= -

dx 2"

Ih,

sin-

sinh

I H J I H Jcosh2

Ih-cos

2f ny

H

(2.37)

34

A nice aspect of potential flow is that after determining the stream function and then taking

the derivatives to get the velocity, the velocities can be input into Bernoulli's equation to get

the pressure profile.

Zj_ + ghi +PL =^L + gh2 +

P^ (2.38)2 P 2 p

The flows that will be discussed in this report will have negligible change in elevation (h)

and will also have constant density. Also on the cylindrical surface the radial velocity is

zero.

Pcy,=P~+^(ui-Ue2) (2.39)

The above equation can be rearranged to put it in the same form as equation 2. 1 which

defines the non-dimensional pressure coefficient.

(0)=PcyL_^~=1_U|_

35

Figure 2-7 DOUBLET REFLECTED USINGMETHOD OF IMAGES

///'//

^-t/ /

H

///// / / /

36

3. THEORETICAL ANALYSIS

3.1 FLUID FLOW IN RECTANGULAR DUCTS

The solution for the axial velocity (u) for fully developed laminar three dimensional flow in

a rectangular duct is as follows.

u =

16Cia2

n~

:1 -\ s n

2

n=l,3,5

cosh

12a

cosh

n Kb

2a

cos

(nnz^

V 2a j

(3.1)

For the above equation C\ is a function of the pressure drop and the viscosity. Shah and

London (1978) give the equation for the mean velocity (Um) related toQ as follows.

Ci= 3Ur

7t5 UJil..n5 I 2a-1

2a j

(3-2)

The results of the above equation show that flow in height direction (H) direction is very

much like the flow between two parallel plates. Figure 3.1 is a plot of the velocity profile

across the gap height.

37

Figure 3-1 VELOCITY PROFILE ACROSS GAP HEIGHT

n =1,3.. 19 C,:=l z:=0.0 a:=l b:=l

0.5

ys o

-0.5

\

\

\

\

\\

1

J

/

\

-^

^"

0 0.05 0.1 0.15 0.2 0.25 0.3

u.

l

38

Looking at the top of the channel, across the width, the flow maintains a uniform velocity

except near the walls. This result gives an understanding ofwhy the Hele-Shaw cell

previously discussed works very well for simulating ideal flow in two dimensions. Except

for a small boundary layer near the edge, the flow velocity looks very much like ideal flow.

Figure 3-2 shows a plot looking down on the top of the rectangular duct.

Figure 3-2 represents the flow looking at the top of the cell, or the velocity gradient across

the width. The flow looks exactly like inviscid irrotational flow in two dimensions (2-D slug

flow), if the edge effects near the walls are neglected. This means we can treat the flow as

ideal flow and use potential equations to model flow around any objects located in this area.

This type of flow has been analyzed experimentally by the use ofHele-Shaw cells. The flow

field for a Hele-Shaw cell can be shown to satisfy the Laplace equation.

When a cylindrical object is placed in the gap of a Hele-Shaw cell, the equations show that

the mean velocity is the gradient of a potential function. This means the flow field past the

cylindrical object will be related to inviscid or irrotational flow in two dimensions.

39

Figure 3-2 FLUID VELOCITY ACROSS GAPWIDTH

n =1,3.. 199 y: = 0 a

- 0.00254 mb:= 3.8098 10

Um=0.01m/s

0.002

z.

l 0

0.002

______-_._____-__-___-.

0 0.005 0.01 0.015 0.02

u.

40

3.2 INVISCID IRROTATIONAL FLOW IN TWO DIMENSIONAL FLOW

This section will present the solution for the two dimensional flow with a cylindrical

obstruction. The two dimensional flow field will be solved in a couple of different methods.

1. 2-D potential flow without walls

2. 2-D potential flow using the method of images to simulate walls

Method 1, 2-D potential flow without walls will simulate the uniform flow around a cylinder

that was discussed in section 2.1.2. Method 2 will determine the effect of adding the walls.

There will be three aspects of the flow solution that will be important.

1. flow velocity

2. streamline pattern

3. pressure distribution

The flow velocity is important because in the actual PCR process one fluid must push out

the previous fluid. Experimental results have shown the current PCR pouch has some trouble

cleaning out the corner areas of the detector chamber where the flow velocity will be a

minimum.

The streamline pattern will be calculated because this is a visual representation that can be

qualitatively compared to the experimental and numerical results shown in section 2.

FLUENT also can provide numerical results for the stream function at particular points. This

will allow the simplified theoretical solution to be compared to the detail numerical solution.

When trying to visualize the flow field it will be more important to determine lines of

constant stream function (streamlines) than to determine the exact value at any point. Also

recall the value of the stream function along the cylinder wall will be zero.

41

The third aspect of the fluid flow that will be investigated will be the pressure distribution.

This is important because in the PCR process the signal is measured using a color reflection

densitometry. How much color is able to diffuse into the detection probe can be optimized

by maximizing the pressure distribution around the detector probe.

3.2.1 POTENTIAL FLOW FIELD FOR UNIFORM FLOW OVER A CYLINDER

Figure 3-3 shows the plot of the stream line function for a particular geometry. These flow

patterns are based on using equation 2.29.

The interesting thing to note is that Figure 3-3 does not show any of the recirculating eddies

that should occur for a Reynolds Number of 40. This is because this plot is based on the 2-D

potential flow theory which neglects the boundary layer around the body. This means this

theory is reasonable on the forward side of the cylinder, but does not do a good job on the aft

side.

The other aspect to consider in this result is what possible effect the addition of walls would

have on the flow pattern. The baseline geometry for this effort considers the outer wall to be

approximately 1.8 times the detection probe radius away from the origin. The top 3 flows

would be impacted by a wall located in this position. This is part of the reason Fornberg

(1980) defines other boundary conditions besides the free stream condition.

42

Figure 3-3 STREAMLINE PLOTS USING POTENTIAL FLOW THEORY

^

"X

.

X

^H)=4TJ

.

/

^- wall locaii'on

*or fcK

.-

-

-

-~

""

~

*N

__

X x

\\f--zo

=s-

-4 -2 0 2 4 -

r-i

"n (*^l)

UM-

10 mm

.sec

rl.4mn

43

Given in Figure 3-4 is a plot of the pressure distribution using the equation 2-34 which has

been non-dimensionahzed to be in the same form as equation 2. 1 . The results show that the

maximum positive pressure occurs at the two stagnation points of0

and 180. When

comparing the result to Dennis and Chang (1970) given in Figure 2-5, the results are similar

in shape on the front side of the cylinder but the pressure fully recovers on the backside of

the cylinder, which does not happen for Dennis and Chang (1970) because they take into

account the boundary layer separation around the cylinder.

3.2.2 POTENTIAL FLOW RESULTS USINGMETHOD OF IMAGES

In section 3.2.1, potential flow around the cylinder was discussed. It did not include the

effect of the walls around the cylinder. Using the method of images, Section 2. 1.2 discuss

how the method of images can account for the walls by reflecting the flow singularities

outside the wall boundary.

Given in Figure 3-5 is a plot of the streamlines using equation 2.35. Unlike the results in

Figure 3-3, these results do take into account the effect of the wall on the fluid flow. This

result shows how the method of images can be used to predict flow patterns within a

channel.

The other item to notice in Figure 3-5 is that it still does not predict the recirculation zone on

the back end of the cylinder. This is because this result is still based on the potential flow

theory that allows the flow to slip on the cylinderwalls and also assumes the fluid has zero

viscosity.

44

Figure 3-4 PRESSURE DISTRIBUTION USING POTENTIAL FLOW THEORY

1

"\/^

0.2

a

P c

\a. 1.*

\~2.2

~3

20 40 60 80 100 120 140 160 180

eAngle aboutCylinder

45

Figure 3-5 STREAMLINE PLOTS USING THEMETHOD OF IMAGES

wal

_jL

_.

.

_-

~~"~~"

~-

.. qjîs

/-- u)= :o

/i

I

\ l|J=. 3

/

X

set.

wru

ft=.08ynm

46

When compared to the fluid flow over the cylinder without walls, the method of images does

show how the wall would cause the velocity of the fluid over the cylinder to increase. Figure

3-6 shows a velocity plot comparing the two cases. For flow over the cylinder, equation 2.3 1

is used. For the method of images equations 2.36 and 2.37 are used with the circumferential

velocity being defined as follows.

x = rcos(0) (3.3)

y = rsin(e) (3.4)

V9 = -Vx sin(0)+ Vy cos(e) (3.5)

Once the velocities are determined it is possible to find the pressure distribution on the

cylinder. Figure 3-7 shows the results for non-dimensionalized pressure coefficient as

defined in equation 2.40.

47

Figure 3-6 VELOCITY PROFILE ON CYLINDER USINGMETHOD OF

IMAGES

1 ULo

o

0

\\ //

//5

\\\\\\

//

////

10

\ /^

\ \ ,

/Method f XMACES

/ /

'

13 \ \y y /

\

/ /

ER/

/

-20 \

\

\ /

/

/

-25\ y

180 150 120 90

eAngle aboutCylinder

60 30

48

Figure 3-7 PRESSURE DISTRIBUTION USING THEMETHOD OF IMAGES

Pressure 0

-2

NX //\\ //

\ \ / /\ v / /

\ /

\ /

\ \ /PoTtMTlAL FuoW/ / /

\ / A.D*/siD Lvi^lKiOFR,- --

\ V/ /

/ /\ \ / /\ \ / /

\ 7

\ /

\ y

X /

/METHOD 6F

\ / ZMAG-ES /

180 150 120 90

e

60 30

49

4. NUMERICAL ANALYSISMETHOD

FLUENT solves the governing equations of fluid dynamics using a finite volume

formulation. FLUENT is capable of using structured or unstructured grid formulation. For

the purpose of this analysis a structured grid formulation of hexagons will be used to build a

three dimensional grid. Given below is a short summary of the FLUENT solution technique

from Freitas (1995).

Within FLUENT three different spatial discretization schemes are used; Power Law, second

order upwind, and QUICK, which is a bounded third-order accurate method. Pressure and

velocity coupling is achieved by the SIMPLEC (or SIMPLE) algorithm resulting in a set of

algebraic equations which are solved using a line-by-line tridiagonal matrix algorithm,

accelerated by an additive-correction type ofmultigrid method and block correction.

All FLUENT analysis for this work was performed on a DEC ALPHA/3000 workstation.

FLUENT version 4.3 1 was used for all analyses.

4.1 NUMERICALMODEL DEVELOPMENT

The FLUENT CFD software will be used to model the 3-D flow of the fluid over the PCR

detection probes within the narrow rectangular channel. Before determining these results it is

necessary to determine the suitability of using FLUENT for solving this problem. There are

two main issues that need to be addressed. First, can FLUENT accurately predict the flow

field in a very narrow three dimensional rectangular duct that is less than 0. 1 mm thick?

CFD solutions are often susceptible to the grid configuration that is used to solve the

problem.

50

The second issue is to determine if FLUENT can accurately predict the flow over a very

small cylinder with a radius of 1.4 mm? CFD codes have shown the ability to determine the

flow patterns with abrupt changes in geometry. Flow over cylinders and backward facing

steps are often used in the verification of CFD software. However, these geometries and

boundary conditions are often idealized to ensure they will agree with theoretical solutions.

4.1.1 THREE DIMENSIONAL FLUID FLOW IN NARROW RECTANGULAR

CHANNEL

For analyzing the 3-D flow in a narrow rectangular duct the same geometry as shown in

Figure 1-1 will be used, except the cylindrical detection probes will be removed. Given

below are the boundary conditions and physical properties that will be used for this analysis.

Inlet Velocity 10 mm/sec

Fluid Density 994kg/m3

Fluid Kinematic Viscosity6.654xl0"4

kg/m/sec

These values with the narrow duct geometry result in a very small Reynolds number.

pU,Le=(994g)(0.01^)(l.5014Xl0-4m)

ix6.650X10"

msec

$e=yîc=\ nUl **\ ^ = 2.25 (4.1)

^ y-f/-..< z-^-4 kg v '

4*

A 4(0.0006in2)0.00591in = 1.5014x10 m

P 0.406in

Ac = (0.2in)(0.003in) =0.0006in2

P = 2*

(0.2in+ 0.003in) = 0.406in

51

The low Reynolds number means the flow will become fully developed almost instantly. For

this reason all flow comparisons will be done assuming fully developed flow.

Figure 4-1 shows a filled contour plot through a particular slice of the detection channel.

The fluid velocity has a center section with a constant velocity. This result agrees with the

flow field that would be expected from a Hele-Shaw type flow.

To make the visualization of the flow fields easier, the thickness of the detection channel has

been magnified in Figures 4-1 through 4-3.

Figure 4-2 shows a three dimensional velocity profile through a particular slice of the

detection chamber. There are two distinct parts of this 3-D flow field. Figure 4-3 shows the

flow velocity across the detection chamber height. This result looks identical to the velocity

that would occur for flow between parallel plates. This result is expected from the theoretical

plot that was shown in Figure 3-1.

Figure 4-4 shows the other aspect of the 3-D velocity profile. This is the view looking across

the width of the detection chamber. This flow shows the uniform velocity in the center that

is a feature ofHele-Shaw flow. This type of result is expected from the theoretical plot

shown in Figure 3-2. The results from this analysis will be compared to theoretical solutions

later in section 4.

The grid configuration used for this analysis consisted of 500 cells in the chamber length

direction, 7 cells in the chamber width direction, and 13 cells in the chamber height

direction. The 13 cells in the chamber height direction was chosen based on looking forward

to the eventual 3-D solution with the cylindrical detection probes in the rectangular channel.

It was desired to have sufficient cell spacing to be able to determine the pressure and

velocity profiles around the detection probe and in the area above the detection probe.

The 500 cells was chosen because it is the maximum limit available in the version of

FLUENT used for this analysis. The 7 cells in the chamber width direction were selected

based on keeping the total number of cells below 50,000 to minimize computational time.

52

Figure 4-1 Fluent Velocity in Narrow Rectangular Duct

cNCNCNCNCNCNCNCNCNCNcnmcnmmcnmcnmcnen en m en en en en m

o o o o o

a b a u u a3\ * Ji * 3N *

Tf ^r en en cn cs

ooooooooooooooooooooo

Tj-ON^f^t"*^t-^-ir'iUr-|ir'ivCvCvCr~rîôoococONOs

h im h o\ o! 06 06 S f *o vi vi m tn N N h r-i

o

On

39 +

w wr> oON O

ii .

^H

^1"

^

^-3

-3

egE fc

uH^

XH

ei-H

0m 0O +O pqO 0

0^ 0

H 0

U 11

D

Q

<a)

I c

-1

1 (N

O

az

w0

<ON

U 0'""'

w JII

p< w X

Q > |<r> D J

5<

S3

Figure 4-2

rNiNMNNNNrNi(N|(N|m

WWWr- cn f eN

o oen en en en en en en en en en en en en en en ^f

[jJwWWWWWWWWWWWWWWWWWWWWWWWW

* Tt en en eN eN

CN CN m

<o

^-re, CN| N

^ vO O nC

i i On On OC'OC r^ r^ nD

HHI

On On

in

no in

oc *C in intJ-

en, eN eN

m rt

in in

en en fN eN

in

x

+

w

m

Si g,-H Tf ^

*-"S

c/3^ ^

oo

^ 8

U

DQ

<-J

DO

Pi

o

d

n

ci i

S

ci i

s pq.52<

II

X

sm H J

5H

.SP

"S3

aCS

O

"ae

uw

u

"a3

s

S

s>

ONro

O

ON GT-< Tt

><

^c c

^-5

-3

^ E *

CO

o

soo

+-^

H CN

<-> 3? II

Q ffi

^ & 5< o

'c/3 CNen O

<s ON

az<HUw

9^

CO P hJ

o

o

13>

3

CN CN

W Wm r~ en

in rf rf en en

îîr-îrvirNirNrNCNCNenmenenenenenenenenenenenenenenenen

00

SSwwwwwwwwwwwwwwwwwwwwwwwww^t^brtrS^S^^ro^vOT+rNONr^^rfNCtîneNQocinen^

ON

CN

^t no cn r-

CN CN~

ON NO

oc *

CN

NO

On r~ Tt CNr^ en On

o in cn

m nC cn oc

oo in en

en on in

Woo

nC

ONONON0C00i>r^NONONOinin-^--^-enenencN

X

A

ss

C/3

OUu

u

c

3

4)

I

enCN<NCNCNCNCNCNCNCNCNrnenenen

^N ^^ <^ ^^ ^^ ^^ ^^ ^^ ^> *_- >* ^-^ C2 C^J

wwwwwwwwwwwwwwwwwwwwwwwwwwww

en e<~; en en en

oen en en en en en

o o a s+

On on-3-

oc en oc

en en cn cn

oc en ^C m Tf cn ! i On

r- cn r~; cn O;i ! On On

oc r-~ inTj-

en-

_ vC vC

oo oo r~ t~-

nC

oc

nC

nC tm

vOvnm-^-TtenmcNcN in

W W>*

ON -o O

ON ^j G^ ^ HH

G0)

GCD

G1 !T^

C/3

o

^39 wZ Q

z<X

D

az

<H

* a

9>

co D

<

i i

U

o

oo

J

oON

II

X

s

X

JN

S^

4.1.2 2-D FLUID FLOWWITH CYLINDRICAL OBSTRUCTION

Adding the cylindrical disk obstructions to the rectangular channel is necessary to simulate

the PCR detection probes. The first step in this process was to perform this task in two

dimensions. Choosing two dimensions instead of three was selected because using two

dimensions will allow the resulting flow pattern to be compared to the potential flow

solutions that were discussed in section 3.2. There are also several publications cited in the

literature review previously given in section 2 that can be compared to the FLUENT results

for viscous fluids.

Given in Figure 4-5 is the geometry that will be analyzed. The radius of the cylinder was

chosen to match the typical maximum value for the PCR detection probes (05.08 mm). This

2-D geometry was selected because it would be representative of looking at the'top'

of the

PCR detection channel. The 2-D plane selected would also represent the plane in Hele-Shaw

flow the produces an ideal flow pattern.

57

Figure 4-5 2-D GEOMETRY FOR FLOW OVER A CYLINDER BETWEEN

PARALLEL PLATES

02.1 .r^n

c=C>

INLET

IVUvn

58

4.1.2.1 FLUENT DISCRETIZATION SCHEME

In attempting to obtain the FLUENT fluid flow solution, defining the type of discretization

scheme to be used during the solution process became important for flow over a cylinder.

FLUENT reports and stores most variables such as velocity, pressure, etc. at the geometric

center of the control volumes. During the solution process the values of the variables are

required at the control volume boundaries. The values at the control volume faces are

determined via the interpolation schemes mentioned in section 4., Power Law, second order

upwind, or QUICK (quadratic upwind interpolation).

The power-law interpolation is the default method used by FLUENT because it provides

reasonable accuracy with efficient computational speed. The power law interpolation scheme

which is based on a local one-dimensional solution, works well when the flow field is

locally one dimensional. The other two methods work better when you need to resolve

gradients in a flow that is directed at an angle to the computational grid layout. The QUICK

scheme involves a quadratic interpolation in which the value at the control volume face is

based on the adjacent values, and on an additional neighbor node upstream. This quadratic

interpolation provides higher numerical accuracy when the flow is turning and is not aligned

with the computational grid. This minimizes the smearing of gradients due to interpolation

error or "numericaldiffusion"

(FLUENT Users Guide, 1995).

The power law method works by determining whether the change in a variable is determined

by convection or diffusion. The one dimensional equation describing the flux of the variable

(]) is given as:

a(pu*)__a_ra*(Ai^

ax "ax ax{ ]

59

The integration of the above equation yields the following solution according to the Fluent

Users guide, 1995.

<|>(x)-(|>L e^'-l

?L-4>o ^pe,-l

The Peclet number is defined as follows.

nconvection pUL

Pe= = (4.4)diffusion r

Figure 4.6 shows the basis of how the power law scheme works. For large values ofPe,

which are flows dominated by convection, the value of at x = L/2, is approximately equal

to the upstream value. When the Pe = 0, no flow or pure diffusion, cj) may be interpolated by

a linear average between the values at x=0 and x=L. When the Pe number has an

intermediate value, the interpolated value for (j) at x= L/2 is determined by applying the

"powerlaw"

equivalent of equation 4.2.

60

Figure 4-6 Power Law Scheme used by FLUENT

0L

0

00

Variation of a Variable <f> Between x = 0 and x L

61

The reason for going into the details of the FLUENT solution method at this time is the type

of flow around the cylinder requires the higher order solutions, such as QUICK. The

previous solutions shown for the 3-D flow in a rectangular channel will achieve satisfactory

results using the power law solution method. This is not true for the flow around the

cylinder. The reason for this is that the nominal flow down the channel will not be aligned

with computational grid configuration. This can be better understood by comparing the

physical grid configuration to the computational grid configuration.

Given in Figure 4-7 is the grid pattern that was produced for the FLUENT preprocessor,

GEOMESH. Notice that there are no cells within the cylinder area. This area is represented

as a 'deadzone'

in FLUENT. For future analyses the cylinder could be modeled as a porous

flow area or a solid wall. This figure does not represent the computational grid layout.

Shown in Figure 4-8 is the computational grid layout. The (.) symbols represent"live"

cells

for fluid flow. In this figure it is easy to picture how the flow around the back side of the

wall 2 will not be aligned with the primary flow pattern down the channel.

The higher order interpolation scheme QUICK helps to improve the accuracy of the solution

when the flow is not aligned with the computational grid. This scheme computes the face

value of the unknown <|> based on the values stored at the two adjacent cell centers and on a

third cell at an additional upstream point. Figure 4.9 is a simple picture to help understand

this approach. The face value can be written in terms of the neighboring values as:

4>r = eAXr AX,

AXC + AXD AXC + AXD+0-9)

AXu +2AXC

AXu +AXC<l>c +

AXr

AXu +Axc<l>u (4.5)

62

Figure 4-7 PHYSICAL GRID PATTERN FOR 2-D FLOW AROUND

CYLINDER

^

2

///////// .

/ ; ; / / / / / r~T/ //////// /^//////////J

/ / ///// / / / //////// / / / / /

^S/j////// x / / / /Cyv //*-//// / / / ,

s? / / /rT / / / / / / /? ; ; ; s / / / / / / / // / //// / / / / / / ,

<^<SQJ//////// /

/ / /S7si y.ys/y //// / / / / / / / /^y. Â'-V > t / / / / / / / / / ,

. /- j</\?/ ;// j / s ////// /

<yyy/////// / / /////xxy//////////// / / /X?C ( ( ( ( ( ( ( { / t f f / /^\\\\\\\\\\\ \ (^\\\V\\\\\ \ \ \ \ \ \ \\\\\>\\\l\ \ \ \ \ \ \\\\\\\\\ ^ ^ \ \ \ \ \

\\\\\v\\ \ \ \ \ \ \ \

\\\\\\\\ \ \ \ \ \ \\\\\\\\\\\ \ \ \ \

\\\\\\\\\\ \ \ \ \\ \ \ \v_L

_/

I i 1 Jr~t til i i i i i1 1 1 1 i I i 1 I 1 i i j i////////// / / / //////////// / / /////////// / / / / /

/////////// / / / /// / / > / / / / / / / / / /

//// / ////ft/*/ t

//////////// / / /

//////// ////////y\\KK\\\\\\\\\ \ \ \

y\?\\\WW\ \\ \ \ \ \ \ \/\AA\\\ v \\\\\\\\\V

Note: Grid shown is zoomed in around cylinder, entire grid not shown

63

Figure 4-8 FLUENT COMPUTATIONAL GRID

CELL TYPES:

J 1= 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 67 69 71 73 75 77 79

65 - 65

64 64

63 63

62 62

61 61

60 60

59 U2U2U2U2U2U2U2U2U2U2U2U2U2U2U2U2U2U2U2U2U2U2U2U2IiI2U2U21iI2U2 59

58 W2DDDDDDDDDDDDDDDDDDDDDDDDDDDU2 SB

57 UI2DDDDDDDDDEDDDDDDDDDDDDDDDDDU2 57

56 U2DDDDDDDDDDDDDDDDDDDDDDDDDD DU2 56

55 U2DDDDDDDDDDDDDDDDDDDDDDDDDD 0U2 . . . . 55


53 U2DDDDDDDDDDDDDDDDDDDDDDDDDDDU2 53


51 W2DDDDDDDDDDDDDDDDDDDDDDDDDDDU2 51


49 U2BDDDDDDDDDDDDDDDDDDDDDDDDDDU2 49

48 U2DDDDDDDDDDDDDDDDDDDDDDLDDDDU2.... .... 48

47 U2DDDEDDDDDDDDDDDDDDDDDDDDDDDU2 47



44 U2BDDDDDDDDDDDDBDDDDD0DDDDDDEU2 44




40 U2DD0D0DDDDDDD0DDDDDDDDDDDDDDlil2 40

39 U2DDD0DDDDDDDDDDDDDDDDDDDDDDDU2 3938 U2DDDDDDDDDDDDDDDDDDDDDDDDDDDU2 3637 U2DDDDDDDDDDDDDDDDDDDDDDDDDDDU2 37

36 U2DDDDDDDDDDDDDDDDDDDDDDDDDDDU2 3635 U2DDDDDDDDDDDDDD0DDDDDDDDDDDDU2 35


33 U2DDDDDDDDDDDDDDDDDDDDDDDDDD DU2 3332 U2DDDDDDDDDDDDDDDDDDDDDDDDDD DU2 3231 U2U2U2U2U2U2U2U2U2U2U2U2U2U2U2U2U2U2U2U2U2U2U2U2U2U2U2U2U2 3130 3029 29

28 28

27 27

26 26

25 25J 1= 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 67 69 71 73 75 77 79

Note: Only showing portion of grid around cylinder

64

Figure 4-9 CELL NOMENCLATURE EMPLOYED IN HIGHER ORDER

INTERPOLATION SCHEMES

Fluent Users Guide, 1995

u

<l>u

Ax, ,

C

<l>c

AXr

<t>r

D

AXn

Central, Downwind, and UpwindCell Nomenclature

Employed in the Higher Order InterpolationSchemes

Jth

Grid

Line

NODE

X

Cell Storage

Location

J-1*

Grid

Line

l.-lth ,th

Grid Grid

Line Line

65

4.1.2.2 FLUENT Results for Potential Flow Over a Cylinder

Figure 4-10 shows the FLUENT solution for the stream function using the power law

scheme under potential flow assumptions. To obtain potential flow it is necessary to make

some simplifications to the CFD analysis. To be a potential flow the fluid must be both

inviscid (viscosity = 0) and irrotational. Also to satisfy potential flow theory the fluid must

be allowed to'slip'

at the wall and cylinder boundaries.

Notice that the flow at the back of the cylinder in Figure 4-10 shows some recirculation

zones. This is an erroneous result that is caused by numerical inaccuracies in FLUENT. The

recirculation eddies should only occur in real flow with a viscous boundary layer as

demonstrated in the experimental figures given in section 2. These eddies should not occur

for potential flow as was shown in Figure 3.3. Since the physical diffusion is zero (viscosity

equal to zero), this error in the flow streamlines is created by the numerical diffusion from

the calculation using the power law scheme.

Figure 4-11 shows the FLUENT solution using the QUICK discretization scheme. The input

flow velocity, fluid density, grid configuration are identical to values using to create Figure

4-10. In this case though there are no recirculation eddies. The flow looks very similar to the

potential flow theoretical solution given in Figure 3-3. Based on this result all future

analytical cases will use the QUICK solution scheme. Figure 4-12 shows the velocity plot

for this case which has Recyi equal to 40.

66

ooooooooooooocoooooooooooooo

WrjjLL5LiJtjgrjjLiLijrjgrjjrjJMrjjoaoohôviNtcitNir-

iOO\ooc^vo,^trocs-iOO\oor>Noir)'^-co(NO ONr-~ no

oc r- no m ^ . .

NC O ND NO NO NO ^0ûNîninioioîriNt\t'Nt\t\t\t\t\tCn) CN CN <N CN cn CN CN CN CN CN CN CN CN CN CN CN fsj pg (-vj ^j (-sj jsj f<j ^ (sq ^j (_sj

2o

z

LL.

1<

>-

t/1

in m

o

w wrH O

CN CN

00

X

A

o

a

~*

d c-"

<D O

^ 5

W]

m m in m m m, m in in in m in in in in

o

3Nr- ô

CN CN

-d1

7o

0

U-

I<

(-

C/l

ininminminininininmininin

ooco(-^yi-r;i-^^^^-^^''-['^'^7'î'^r^'Vii.iiiojriiiijagjuuiafflwwwwuwauupjww

'T 5 S S S -v ^-. ,r IA r<": M -N T lfl*INl^C!lCN\nTtn-,SSP3NtmNOO\oo\oinnNH~

vOvCNONONONOininininininin

oc r~ in Tt

* Tf TT TT Tj-Tt^trncncnmmcncn

CN CN CN CN CN CN CN CN CN CN CN CN CN CN CN CN CN CN CN CN CN CN CN CN CN CN CN CN

DO

H

Doo

o

-i

<

Pzw

op.

w

uoo

z

o

H

<

o

PC

w

z

ul-H

Da

az

00

D

X

A

18

Nr^NNNCNNnnggNgNCNNgtNNgggm

Ôr-H

^ Tf HH

O, P,2

OO^ ^

. fi m m ci m

o

SHSKSKnSF-ooOnO^ cn cn rfinNcr~oc'<*'*Tj-inmmvCvOvCNCc-~r-

hC^NtrotN-b^XXr-Oiri^mtN!--

CN CN CN CN CN CN CN CN'

^ ^ V) If) Tl lO

on o cn en <* m

on on oc rS nC in Tt

VO NO t~-

I> 00 NO ON

en cn on no

nC

nC

61

Cn

w

Q

OOW

i e

o

<

w

QDH

o o< El]

s

X

H

ZwH

OP*

4.1.2.3 FLUENT Results for 2-D Viscous Flow Over a Cylinder

Figure 4-11 previously showed the FLUENT streamline results for the case with zero

viscosity and free stream boundary conditions at the wall and cylinder. This section will

show the results after some of these simplifying assumptions are removed.

To correlate the FLUENT results from this study to other published results some changes

need to be made to match the boundary conditions. The first change will be to lengthen the

area behind the cylinder. This is allows the flow around the cylinder to be minimally

impacted by the downstream boundary conditions. The second change is to move the walls

further out than previous analyses. Previous results for the ideal flow have shown that the

wall position will impact the flow around the cylinder. The walls have been moved out a

factor of 10 *cylinder radius for this analysis. The freestream boundary conditions (wall

velocity equal to inlet velocity) will be applied for this case.

One of the first issues that needs to be addressed is whether this flow can be analyzed as a

symmetric flow. The reason for this issue is that at Recyibetween 30 and 40 is when a vortex

sheet behind the cylinder begins to appear, (Bhatia, Rahman, and Agarwal, 1993). If a vortex

sheet does begin to form than the flow essentially becomes non-symmetric. Most authors

Braza et al. (1986), Dennis and Chang (1970), Fornberg (1980) reviewed for this report

analyze their problems with symmetric geometry.

Figure 4-13 show the results once viscosity (ofwater) is added to the FLUENT physical

fluid constants for a symmetric boundary conditions. Figure 4-14 show the outline of the

geometry and a close-up of the grid layout around the cylinder. This case is similar the cases

run by Dennis and Chang (1970) and Fornberg (1980). The results show a wake bubble

formation behind the cylinder.

70

Symmetric boundary conditions make it difficult to tell whether steady state eddies tied to

the cylinder are formed, or if an unsteady vortex sheet that separates from the cylinder is

formed. The symmetric boundary condition by it's nature will make it harder for the vortex

to form because it will minimize any flow gradients at the symmetric boundary. Figure 4-15

shows the velocity profiles for this symmetric flow condition. Figure 4-16 gives the pressure

distribution.

Figure 4-17 shows the streamline plot with a non-symmetric vortex result. The same inlet

conditions and fluid properties are used for this analysis as for the previous symmetric case.

The results clearly show that a vortex sheet is formed behind the cylinder. Figure 4-18 show

the outline of the geometry and a close-up of the grid layout around the cylinder.

Figure 4-19, the velocity profile, and Figure 4-20 the pressure profile also show that the flow

is slightly non-symmetric.

Figure 4-21 shows a plot of the pressure coefficient profile on the cylinder for the two cases.

The results show a higher pressure distribution from the180

symmetric cylinder than the

360

cylinder. The360

cylinder results are closer to previously published results. The next

section of this report will compare these results to previously published results.

71

Figure 4-13 FLUENT STREAMLINES FOR VISCOUS FLOWWITH Recyi = 40

AND "NOSLIP"

BOUNDARY ON THE CYLINDER

72

Figure 4-14 GEOMETRY AND GRID LAYOUT FOR SYMMETRIC FLOW

AROUND CYLINDER

kFLUENTGrid File /* CONFIGURATION =

Grid (171X58)

: checkcase */ Jul 28 1996

Fluent 4.31

Fluent Inc.

73

X

31 < Ti

O)OS<K

HJCD

P

50

O ^

Ip ft)

R O

o

o

o ^

o

3Ui

t/> r

o

3

H

o

3 33 o

CD^

3

HI -P^

oU>

nO

ON

LHUlOiO-J^lOOOOvO*

O Oi o00 O ON L/l

WCdWWWWWWtdWCdCdWtiltdMCdMBtilfd000K> ts3 rO

OOOOOOOOOOOOOOOOO

NC

0 0 1 j ho 'ro Lo LP OJ

4^ <-) t/i 0 './1 0 Ui c > fj\ NC

m m m m m m m ra m m1

+ + + + + + + + +0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0

"7H

CnJ CM

o o

+ +

n en

ON IT)

iri m

CNCNCNCnICNCNCsICNCNCN

OOOOOQ1-

+ + + + + +

u ta BJ u u w3n o <n on nC cn

O nC <Nr- c~, On

u->

C N 3\

O nC

CNCNCNCNCNCNCNCNCNCNCNCNCNCN

OOOOOOOOOOOOOOOOOO

+

. wen r~

^~. < ^ t CD CD O *> ' -

w'

<w < > ^ - ' j > - < - i ^ ^ j > j t_;

î^r^Ts--tcoccsir^csiNCONC-NiNCON(riNCO<r)NCOrn\g> oc <n r~ in

m \t

<ri cn cn cn cn

nC on cn nC on en no

oc cn r~ r-n i/~, o-fl-

in on : cn cn <ri en Tt tt

O <n no

on cn c~-

in wi

oCN

nC

nC

nC

H S h^ 3 d

COMh Mh

X

A

o

IIJ7

O

7S

\\\

V

<_>

On

d d53 a

& 3 d

eghC ^

OO

a><_

CN CN

5 ^r

NNNtNNNtNNNlNNNcNINNtNtNlNCNINtNJCNNr^cNNtMlNI*__ v_* ^_J ^J i-_. - ..

^ *. .-V /". **! 1"^ *N /"*n i-"""*\ f^"N /"""( l-*! i*"^ f^""l i"-"*! J*-***f-

*l r-"^ c""\f-

*"\ ^"^S i-"*,,-*

*nOOOOOOOOOOOOO O O O o o_________- O O O O CD O

( ( |

rjjrjjrjjrjjrjjrjQrjjuarjqrjjULLlM

ddddl OOOOOOOOOOOnOnOnOnOnOnOnOnOnoC^.^.^.^.^j-TtTtNtNt't^tNt * ^ rf ^t ^ "* cn en en en en en cn cn cn en

X

A

7.

Figure 4-18 GEOMETRY AND GRID LAYOUT FOR FLOW

AROUND360

CYLINDER

77

+ + + + + +

w W tu U-l tu _J

in in o in

rt cn e<~. CN CN

o o o oCN cn CN CN

O O O O O

OmiOcnmmcncncncncncNCNCNCNCNCNCNCNcN' cn~ ~

ON O

W^''^-:WS:

Onm

<->

^^]-

hh

^ t3 d

^ J5 ^3

fc fc

ON

z

oN

z

oI (

H

<

DUH*

c z

X

.4

O

d d

p, J2 3

00

CN CN

+ +

w wnC cn

oc *

CNCNCNCNCNCNCNCNCNCNCNCN

^C$?? + + +? + + + + +

[flBJMHBWBJBBWUWW

cn^cNOnit, cNocm, cNocm, OSif)H*NnO\inOVONM

in in mi _ "^f

CNCNCNCNCNCNCNCNCNCN

_- , _ _oooooooooo

+ + + + + + + + + + + + + + +

UJUJPJPJWWWMWWWWtLlWWcNCNNCONCNNCONcnNOOcnNCOcnr^

xtNiritfi*c^-|oC\tocmh

_h o cn CN

+

pqnC

cn cn cn cnr^ cn oc in on h - cn cn cn cn ? t \t in in

zoN

z

o

H

<

Pu

<_

x

7T

Figure 4-21 COMPARISON OF PRESSURE PROFILEWITH180

AND360

CYLINDER

180.00

PRESSURE COEFFICIENT ON CYLINDER - FROM FLUENT

Boundary Layer on Cylinder, Free Stream on OuterWalls

150.00 120.00 90.00

ANGLE (cleg)

60.00 30.00 0.00

80

4.2 VERIFICATION OF NUMERICAL ANALYSIS

The FLUENT results will be compared for both 2-D and 3-D flow regimes. The 2-D flow

will be compared for both potential flow theoretical results and previously published results

for 2-D viscous flow around a cylinder. The 3-D flow results will be compared to viscous

flow in a narrow rectangular channel.

4.2.1 2-D FLOW AROUND CYLINDER COMPARISONWITH FLUENT

The next step was to compare the 2-D fluid flow solution from FLUENT with analytical

solutions. Unlike solutions for flow within a rectangular duct, there is no all encompassing

solution for flow around the cylinder. The potential flow solution is only applicable for flow

away from the cylinder, i.e. outside the boundary layer. Possible solutions that are applicable

near the body cannot be made to merge smoothly with flow far removed.

To gain confidence in the flow solution it was decided to use the potential flow solution and

compare the results to the FLUENT CFD solution.

81

4.2.1.1 COMPARISON FOR STREAM FUNCTION

The first location selected for comparison was a point far upstream from the cylinder. Shown

in Figure 4-22 is the location for points along the channel for comparison oftheoretical and

FLUENT results. Previous results showed that the method of images will be more accurate

for determining flow around a cylinder between parallel plates. For this reason equation 2.35

which was based on the method of images was used for the theoretical baseline. The results

for FLUENT are based on the parameters given in Figure 4-5.

One thing to notice is that the Reynolds number for this flow can not be strictly defined at

this point. This is because the viscosity for the ideal flow is zero, which causes the Reynolds

number, Re = ,to go towards infinity. The inlet velocity (10 mm/sec) is the same as

V-

was used for the Recyi equal to 40 in viscous flow analyses.

82

Figure 4-22 LOCATIONS FOR COMPARISON OF FLUENT TO THEORETICAL

RESULTS

TL-Zb r-ss

83

The results in Table 4-3 show that the potential flow theory using the method of images and

FLUENT are in good agreement with the difference being less than 5.5% at the selected

points.

Table 4-1 STREAM LINE COMPARISON TO THEORY UPSTREAM FROM

CYLINDER

POTENTIAL FLOW USING METHOD OF IMAGES

1 = 20 X-LOCATION Y-LOCATION FLUENT THEORY DIFFERENCE

J= (mm) (mm) (mA2/sec) (mA2/sec) %

80 3.91 4.631 4.57E-05 4.63E-05 -1.2%

60 4.442 3.511 3.45E-05 3.50E-05 -1.4%

40 4.515 2.241 2.18E-05 2.25E-05 -3.2%

20 4.423 1.064 1.01E-05 1.07E-05 -5.4%

The second comparison that was done for the stream function calculation was at a point

approximately aligned with the centerof the cylinder (x= 0.4 inches). The result for this

comparison are given in Table 4-4. The results show a good comparison with the theoretical

values.

84

Table 4-2 STREAM LINE COMPARISON TO THEORY ABOVE CYLINDER

1=55 X-LOCATION Y-LOCATION FLUENT THEORY DIFFERENCE

J= (mm) (mm) (mm/sec) (mA2/sec) %

88 10.133 5.06 5.00E-05 5.04E-05 -0.8%

85 10.136 4.942 4.76E-05 4.81E-05 -1.0%

80 10.14 4.745 4.36E-05 4.42E-05 -1.4%

75 10.145 4.548 3.95E-05 4.02E-05 -1.7%

70 10.15 4.35 3.52E-05 3.60E-05 -2.2%

65 10.155 4.153 3.06E-05 3.16E-05 -3.3%

60 10.16 3.956 2.55E-05 2.67E-05 -4.6%

POTENTIAL FLOW THEORY USING METHOD OF IMAGES

85

4.2.1.2 COMPARISON FOR FLUID VELOCITY

The velocity profile was compared at a location above the cylinder. Equation 2.36 was used

to compare the axial velocity from the theoretical solution using the method of images.

Similar to the stream function comparison discussed in the previous section, the velocity

comparison also showed good agreement between the potential flow theoretical solution

compared to the FLUENT numerical results. The results of this velocity comparison are

given in Table 4-5. Figure 4-19 also shows the velocity profile plot for the two different

cases.

Table 4-3 AXIAL VELOCITY COMPARISON TO THEORY ABOVE

CYLINDER

1=55 X-LOCATION Y-LOCATION FLUENT THEORY DIFFERENCE

J= (mm) (mm) (mm/sec) (mA2/sec) %

88 10.133 5.06 20.071 19.566 2.6%

85 10.136 4.942 20.133 19.635 2.5%

80 10.140 4.745 20.508 19.988 2.6%

75 10.145 4.548 21.249 20.681 2.7%

70 10.150 4.350 22.457 21.815 2.9%

65 10.155 4.153 24.508 23.554 4.1%

60 10.160 3.956 26.931 26.22 2.7%

86

Figure 4-23 Velocity Profile ComparisonWith Method of Images Theory

Axial Velocity above Cylinder

4.94

4.74

c

H 4.54w

oQ.

>

4.34

4.14

3.94

f tI

1

1 11 T

\ \

\ \

\ *

\ \

\ \

1 t

\ \

\ \

\ \

L

\

-

\

\

\ \

\ \

\ \

\ X

V \

\ s

s

s

. s

\ \

\ \

s

\s. \

\ s

\ s

^_ 1

FLUENT

- - THEORETICAL

19.000 21.000 23.000 25.000 27.000

Ux Axial Velocity

87

4.2.1.3 COMPARISON FOR PRESSURE ON CYLINDER

The last comparison that will be used to verify the FLUENT numerical solution will be the

pressure distribution around the cylinder. The results from potential flow theory were

previously shown in Figure 3-7.

Figure 4-24 shows a plot the theoretical and FLUENT results. Table 4-4 give a numerical

summary. The results show good agreement between theoretical predictions and FLUENT.

To this point the flow comparisons around the cylinder have been based on potential flow.

To gain additional confidence in the FLUENT solution, the results with a viscous fluid will

be compared to the previous published results from several authors in the next section.

88

Figure 4-24 PRESSURE COEFFICIENT COMPARISON

BETWEEN FLUENT AND POTENTIAL FLOW

USING THEMETHOD OF IMAGES

PRESSURE COEFFICIENT ON CYLINDER - FROM FLUENT

"Xy^

\

/FUAEMT

.MSN*0"//

/ OP M

i

fXr*<*S/jL

180 150 120 90

ANGLE (deg)

60 30

-3

PC_

o Ok_

0) V?n ,1

-4

-5

-6

89

Table 4-4 PRESSURE COEFFICIENT COMPARISON ON CYLINDER

MEDIAN 9.54%

(m) (m) FLUENT AVG. 20.70%

I J X-POSIT Y-POSIT theta-global pressure press coeff M of I theory abs % diff

41 45 8.76E-03 2.54E-03 180 5.21E-02 1.05 1 5.0%41 46 8.77E-03 2.62E-03 176.84 5.13E-02 1.03 0.981 5.0%41 47 8.77E-03 2.69E-03 173.67 4.88E-02 0.98 0.925 5.9%41 48 8.78E-03 2.77E-03 170.51 4.46E-02 0.9 0.833 8.0%41 49 8.80E-03 2.84E-03 167.35 3.87E-02 0.78 0.705 10.6%41 50 8.81E-03 2.91E-03 164.18 3.11E-02 0.63 0.541 16.5%41 51 8.84E-03 2.99E-03 161.02 2.19E-02 0.44 0.345 27.5%41 52 8.86E-03 3.06E-03 157.86 1.12E-02 0.22 0.118 86.4%41 53 8.89E-03 3.13E-03 154.7 -1.08E-03 -0.02 -0.139 85.6%41 54 8.93E-03 3.19E-03 151.54 -1.48E-02 -0.3 -0.422 28.9%41 55 8.96E-03 3.26E-03 148.39 -2.97E-02 -0.6 -0.729 17.7%41 56 9.00E-03 3.32E-03 145.23 -4.60E-02 -0.93 -1.058 12.1%41 57 9.05E-03 3.38E-03 142.08 -6.31E-02 -1.27 -1.405 9.6%41 58 9.10E-03 3.44E-03 138.93 -8.35E-02 -1.68 -1.766 4.9%41 59 9.20E-03 3.55E-03 132.63 -1.21E-01 -2.43 -2.518 3.5%42 59 9.26E-03 3.60E-03 129.49 -1.39E-01 -2.79 -2.899 3.8%43 59 9.32E-03 3.65E-03 126.34 -1.59E-01 -3.2 -3.281 2.5%44 59 9.38E-03 3.69E-03 123.21 -1.80E-01 -3.62 -3.654 0.9%45 59 9.44E-03 3.77E-03 119.31 -2.02E-01 -4.06 -4.105 1.1%46 59 9.51E-03 3.81E-03 116.31 -2.23E-01 -4.49 -4.434 1.3%47 59 9.57E-03 3.84E-03 113.31 -2.44E-01 -4.9 -4.743 3.3%48 59 9.64E-03 3.86E-03 110.3 -2.63E-01 -5.3 -5.03 5.4%49 59 9.71E-03 3.86E-03 107.56 -2.81E-01 -5.65 -5.265 7.3%50 59 9.79E-03 3.89E-03 104.44 -2.95E-01 -5.94 -5.5 8.0%51 59 9.86E-03 3.90E-03 101.32 -3.07E-01 -6.18 -5.695 8.5%52 59 9.93E-03 3.92E-03 98.21 -3.16E-01 -6.37 -5.847 8.9%53 59 1.00E-02 3.93E-03 95.1 -3.24E-01 -6.52 -5.952 9.5%54 59 1.01E-02 3.93E-03 92.01 -3.29E-01 -6.63 -6.08 9.0%55 59 1.02E-02 3.94E-03 88.89 -3.32E-01 -6.68 -6.015 11.1%56 59 1.02E-02 3.93E-03 85.78 -3.31E-01 -6.66 -5.973 11.5%57 59 1.03E-02 3.93E-03 82.69 -3.25E-01 -6.53 -5.882 11.0%58 59 1.04E-02 3.92E-03 79.59 -3.16E-01 -6.35 -5.744 10.6%59 59 1.05E-02 3.90E-03 76.51 -3.05E-01 -6.13 -5.564 10.2%60 59 1.05E-02 3.89E-03 73.44 -2.92E-01 -5.87 -5.344 9.8%61 59 1.06E-02 3.86E-03 70.33 -2.76E-01 -5.56 -5.086 9.3%62 59 1.07E-02 3.84E-03 67.25 -2.60E-01 -5.23 -4.799 9.0%63 59 1.07E-02 3.81E-03 64.19 -2.42E-01 -4.86 -4.487 8.3%64 59 1.08E-02 3.77E-03 61.1 -2.21E-01 -4.45 -4.151 7.2%65 59 1.09E-02 3.74E-03 58.02 -1.99E-01 -4.01 -3.799 5.6%66 59 1.09E-02 3.70E-03 54.97 -1.77E-01 -3.55 -3.438 3.3%

67 59 1.10E-02 3.65E-03 51.88 -1.54E-01 -3.1 -3.066 1.1%68 59 1.11E-02 3.60E-03 48.81 -1.33E-01 -2.67 -2.693 0.9%

69 59 1.11E-02 3.55E-03 45.76 -1.11E-01 -2.23 -2.323 4.0%

69 58 1.12E-02 3.44E-03 39.63 -7.30E-02 -1.47 -1.599 8.1%69 57 1.13E-02 3.38E-03 36.57 -5.22E-02 -1.05 -1.254 16.3%

69 56 1.13E-02 3.32E-03 33.52 -3.51E-02 -0.71 -0.926 23.3%69 55 1.14E-02 3.26E-03 30.46 -2.01E-02 -0.4 -0.615 35.0%69 54 1.14E-02 3.19E-03 27.42 -7.18E-03 -0.14 -0.326 57.1%69 53 1.14E-02 3.13E-03 24.37 3.42E-03 0.07 -0.06 216.7%69 52 1.15E-02 3.06E-03 21.32 1.14E-02 0.23 0.18 27.8%69 51 1.15E-02 2.99E-03 18.27 1.69E-02 0.34 0.392 13.3%69 50 1.15E-02 2.91E-03 15.22 2.01E-02 0.41 0.575 28.7%

69 49 1.15E-02 2.84E-03 12.17 2.18E-02 0.44 0.726 39.4%

69 48 1.15E-02 2.77E-03 9.13 2.25E-02 0.45 0.845 46.7%69 47 1.15E-02 2.69E-03 6.09 2.28E-02 0.46 0.931 50.6%69 46 1.16E-02 2.62E-03 3.04 2.28E-02 0.46 0.983 53.2%69 45 1.16E-02 2.54E-03 0 2.29E-02 0.46 1 54.0%

90

4.2.2 FLUENT COMPARISONS FOR VISCOUS FLOW OVER A CYLINDER

The comparisons so far have shown good agreement with the FLUENT results. These

comparisons were limited to potential flow. It is also desirable to get comparisons with real

viscous fluids to gain confidence in the final sensitivity studies that will be performed on the

PCR geometry. The results will be compared to previously published viscous flow results in

several key areas.

1 . pressure coefficient distribution around the cylinder

2. location of flow separation on cylinder

3. length ofwake bubble behind cylinder

4.2.2.1 PRESSURE ON CYLINDER FOR Recy, = 40 WITH VISCOUS FLOW

As previously shown in section 4. 1.2. 1 the viscous flow over a cylinder should be modeled

as a360

cylinder to accurately determine if a non-symmetric vortex sheet will form behind

the cylinder.

Figure 4-25 shows the results form Dennis and Chang (1970) compared to the current results

for Recyi = 40. The results show the same basic shape. The results do show a greater negative

pressure than shown by Dennis and Chang (1970). This is consistent with the results ofKim

and Choudhury (1993) who also showed a higher pressure coefficient for FLUENT results

than was expected from previously authors.

91

Figure 4-25 FLUENT SOLUTION FORVISCOUS FLOW OVER A CYLINDER

-0-5

Pressure coefficient on the cylinder surface.

_enm_ and Cha^ (cl7o')

92

Figure 4-26 shows the FLUENT results compared to the published results ofBraza et al.

(1986). The FLUENT results compare much closer to this work than the results previously

shown in Figure 4-25 ofDennis and Chang (1970). It should be noted thatBraza'

s result for

the pressure coefficient are compared to experimental results for the pressure coefficient.

The results shown in Figure 4-26 have been adjusted to account for the upstream inlet static

pressure. Equation 2.1 does depend on the upstream and free stream pressure. Published

results are very unclear on how this factor is accommodated in their results.

93

Figure 4-26 PRESSURE COEFFICIENT RESULTS COMPARED TO BRAZA et al.

(1986)

94

4.2.2.2 COMPARISON FOR FLOW SEPARATIONANGLE

The results for the flow separation angle are in good agreement with previously published

results. Dennis and Chang give the separation angle as 53.8. Taking the results from the

graph, Fornberg (1980) would have the result at 57. The one problem that FLUENT does

have is whenever you go around the"corner"

of the computational grid, there is a problem

accurately predicting the flow in this area. This is why the QUICK scheme was used in all

solution routines to help minimize this effect. Previously shown in Figure 4-13, FLUENT

predicts a52

separation angle.

Table 4-5 COMPARISON FOR SEPARATION ANGLE

CASE SEPARATION ANGLE

FLUENT; VISCOUS FLOW SOLUTION 52

DENNIS AND CHANG 53.8

FORNBERG 57

95

4.2.2.3 COMPARISON FORWAKE BUBBLE LENGTH

Previously shown in Figure 4-17 was the streamlines for the flow over a cylinder. Given in

Table 4-7 is the comparison of this result to other published results that were given by

Fornberg (1980). The results show that FLUENT tends to underpredict the length of the

wake bubble. The current FLUENT results give a range of values because with the separated

flow it is difficult to determine the length of the wake bubble at any point in time.

Table 4-6 COMPARISON OFWAKE BUBBLE LENGTH

CASE LENGTH OFWAKE BUBBLE

DENNIS AND CHANG 5.69

TAKAMI & KELLER (1969) 5.65

NIEUWSTADT & KELLER (1973) 5.357

TA (1975) 5.27

FORNBERG (1980) 5.48

PRESENTWORK 5.25 to 5.83

Lengths in above table in terms of radius; i.e. 1.3 = 1.3 *Rcyi

96

4.2.3 VISCOUS FLOW COMPARISON FOR DIFFERENT GRID

FORMULATIONS

With any CFD analysis it is important to know how the results are effected by the grid

geometry. As was discussed in previous sections there is always a trade offbetween number

of grid points and computing capability. The results shown in section 4. 1 .2.3 presented the

results for a360

grid configuration and a180

grid configurations. This section will

compare the results from those two models and also see what happens when the number of

grids is reduced. The smaller the number of grids in the two dimensional solution, the

greater the number of grids can be in the third dimension once we go to solving the full 3-D

flow solution.

1. Detail grid (133 x 109)

2. Simplified grid (65 x 50)

3. Symmetric grid (172x59)

The results shown in Table 4-7 also show how the pressure coefficient at key points

compared for different level of model detail. The results show consistent results in most

cases. The symmetric grid formulation does predict a greater negative pressure on the

backside of the cylinder. The comparison of the stagnation pressure on the front of the

cylinder does have the upstream inlet static pressure taken into effect.

The comparison for the back side stagnation pressure and the minimum stagnation point

agree reasonably well with the results ofBraza et al. (1986). The FLUENT results do not

agree as well with the other published values. This results are satisfactory for the type of

sensitivity studies that will be performed on the PCR detection chamber.

97

Table 4-7 COMPARISON OF KEY PRESSURE POINTS FOR DIFFERENT

SOLUTION METHODS

CASE FRONT

STAGNATION

MINIMUM

PRESS

REAR

STAGNATION

FLUENT; DETAIL GRID (CHECKCASE) 1.153 -1.21 -0.704

FLUENT; SIMPLIFIED GRID (CHECKCASE3D2D) 1.114 -1.24 -0.711

FLUENT; SYMMETRIC GRID (CHECKCASE2) 1.20 -1.42 -1.039

BRAZA etal. (1986) 1.0 -1.19 -0.7

DENNIS & CHANG (1970) 1.144 -0.95 -0.509

FORNBERG (1980) 1.14 -0.92 -0.46

98

4.2.4 3-D VISCOUS FLOW COMPARISON TO FLUENT RESULTS IN NARROW

RECTANGULAR DUCTS

Given in Table 4-8 and 4-9 is a comparison ofFLUENT to the theoretical solution for the

3-D flow in a rectangular duct. The theoretical results are from equations 3.1 and 3.2. The

3-D FLUENT results were previously discussed in section 4.1.1.

The results "across the gapwidth"

refer to looking down on the'top'

of the rectangular

channel. The results "across the gapthickness"

refers to the narrow channel height.

This analysis was performed with the dimensions given in Figure 1-1. The case was done

with an inlet velocity of 0.01 m/sec and a Reynolds number of 2.25 for the channel. The low

Reynolds number means that the flow is fully developed almost instantly upon entering the

channel.

99

Table 4-8 COMPARISONWITH FLUENT ACROSS GAPWIDTH

node center

Z position CLOSED-FORM FLUENT DIFFERENCE

K node (mm) (mm/s) (mm/s) %

1 0 0.000 0.000 0.0%

2 0.508 15.13 14.928 -1 .3%

3 1.524 15.14 14.940 -1 .3%

4 2.540 15.14 14.939 -1 .3%

5 3.556 15.14 14.939 -1 .3%

6 4.572 15.13 14.928 -1 .3%

7 5.080 0.000 0.000 0.0%

100

Table 4-9 COMPARISONWITH FLUENT ACROSS GAP THICKNESS

K=4

node center

Y position CLOSED-FORM FLUENT DIFFERENCE

Y node (m xlO*^) (mm/s) (mm/s) %

1 0 0.000 0.000 0.0

2 3.4634 2.627 2.695 2.6

3 10.309 7.13 7.105 -0.4

4 17.317 10.634 10.534 -0.9

5 24.2442 13.136 12.982 -1.2

6 31.1711 14.638 14.450 -1.3

7 38.0980 15.139 14.939 -1.3

8 45.0249 14.638 14.450 -1.3

9 51.9518 13.136 12.981 -1.2

10 58.8787 10.634 10.533 -0.9

11 65.8056 7.13 7.105 -0.4

12 72.7325 2.626 2.695 2.6

13 76.1960 0.000 0.000 0.0

101

The other comparison that needs to be done for the 3-D channel is to determine the pressure

drop down the length of the chamber. For this comparison the 3-D channel will be

approximated as flow between parallel plates. This is a reasonable assumption because for

this geometry the width to thickness ratio for the chamber is 66.7.

The pressure drop for flow between parallel plates is given by Nunn (1989) as

dP=j2uU

dx '_f= 687Pa (4.6)

where

U^ =0.0149m/sec

H=6.544X10^Jl

H = 76196xlO"5m

Shown in Figure 4-27 is the static pressure drop through the detection chamber. The

FLUENT pressure drop is slightly less than predicted by the above equation.

%Difference=FLUENT -Theory^

op _rBPa-7PaV

=

Theory ) { 687Pa J

102

r- (N <N CN CN CN CN CN CN CN CN CN CN CN CN CN CN CN CN CN CN CN CN CN CN CN CN

O O o o O O O o o o O o O O o O O O O o O o o o O o o O O O O

+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + +

W -J UQ PQ w a PQ PQ PQ pq m PQ PQ PQ pq pq m PQ PQ PQ pq PQ PQ PQ PQ PQ PQ PQ PQ PQCN Tt c~ ON <N N

1-

vO oo ^H en m 00 o CN -tf r~ ON ^r v_ 00 o en >n t> o CN *tf *o 'ON

CN CN <- MD on -;en in r-; o CN "- MD ON * i en in l> o CN ^ ON *\ cn in 00 O CN * Ô

t> CN1

vC oc < i <-<

illo

^CN CN CN CN cn cn en ci en <*

1

"* in >n in in"? ^d vC _i

103

OS o O

ON \] G^

^- h-H

^G C

&< J5 5eg^ fe

c--

(N

o

Q

SB +

CD r~-

Pt 0.

e II

s ' fI

oS-H

Q

in

31/3

(N

s

c/5 r~~J

ii

o Xc3

Ph 00 J

jr

5. RESULTS FOR 3-D FLOW

There are several key features of the PCR detection chamber that should be analyzed with a

three dimensional flow analysis.

1 . Determine pressure and velocity profile around the PCR detection probe.

2. Determine the sensitivity to changes in channel width on pressure and flow velocity.

3. Determine the sensitivity to the detection probe height on pressure and flow velocity.

The 3-D flow analysis will be conducted with the following baseline parameters:

Detection chamber geometry See Figure 1-1

reduce length to 1.0 in.

reduce number of detection probes to 2

Inlet Velocity 10 mm/sec

Fluid Density 994kg/m3

Fluid Kinematic Viscosity6.654xl0"4

kg/m/sec

5.1 BASELINE 3-D RESULT

The baseline 3-D case will use a 102 x 15 x 30 symmetric grid in the length, thickness, and

width directions respectively. There are 40 grids located on any one slice of the180

cylinder. Shown in Figure 5-1 is the isometric outline of the flow geometry. Note to make

the gap thickness visible, the height dimension (y) has been magnified by a factor of 20. This

is for visual purposes only, the problem solved used the actual channel thickness of 0.076

mm. This magnification will be true for all future 3-D plots shown in this report.

Figure 5-2 shows the side and top view of the detection chamber geometry. Note to

minimize the number of grids required only 2 detection probes are modeled. Figure 5-3

shows a cross sectional partial view of the computational grid.

104

Vi

*

<_

c/o

c.

X)

II

o

H

ID

oII

ou

a

a

H

pd o

>%% -N

<V

ior

5:e_

OH

Q2:^3

H55O

HW

ow

o

Q

I

10,

5f

t/3

_>

co

II

oIt

H

<Pt

o

ou

Figure 5-3 CROSS SECTION OF COMPUTATIONAL GRID FOR 3-D FLOW

CELL TYPES: J = 4

K 1= 26 28 30 32 34 36 38 40 42 44 46 48 50 52 54 56 58 60 62 64 66 68 70 72 74

30 SSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSS 30

29 2U2U2U2_2U2U2lil2lJ2lil2U2U2_2U2U3W3W3lJ3lJ3U3U3U3lil3W3ld3U3U3UI3 ... 29

28 U2U2U2U2U2U2U2U2U2U2U2U2U2U2 U3ld3L)3U3_3W3lJ3U3tiJ3ld31J3U3_3liJ3 ... 28

27 ..... .U2UI2U2W2U2U2lil2U2U2lil2liJ2lil2LI2U2 ..............ld3W3U3U3W3W3W3U3_33W3U3_31J3... 27

26 U2U2U2U2U2U2U2U2U2U2U2U2U2U2 W33L03U3U3W333ld3W33ld3W3U3 ... 26

25 U2UI2U12U2U2UI2IJ2U)2U)2UI2U)2U2U2U2 UI3W3U3U3U3W3W3W3W3W3W3U3U3U3 ... 25

24 UI2UI2UI2U2U2U21J2U2U2U12U2_2U2U2 W3U3U3U3U3UI3U3td3LI3ld3W3U3lJ3W3 ... 24

23 U2U2U2lJ2UI2_2U2U2liJ2l>12U2lJ2W2W2 U3W3U3_3U3U3IJ3U3LI33W3ld3lJ33 ... 23

22 UI2LI2ld2U2_2U2W2LI2ld2lil2U2W2UI2U2 UI3-3U13U3W3W3W3U3W3W3UI3l>J3ld3U3 ... 22

21 U2ld2U2ld2U12U2td2UI2U12W2W2U2W2UI2 U13U3ld3ld3U3U3U3U3_3U3U3U3U3U3 ... 21

20 2U2U2U2UI2U2W2U2_2_2U2W2UI21J2W33U3_3U3U3U3LJ3liJ3W33U3U3IJ3 ... 20

19 U2U2U2U)2U2U2U2_2U2U2U21J2UI2U2 U3U3U3U3U3U3UJ3_3U3y3U3U3lJ3U3 ... 19

18 U2U2U2U2U2U2U2U2U2U2U2U2U2U2 .U3LJ3IJ3_3U3UI3lJ3U3ld3U!3_3_3lJ3IJ3... 18

17 ..... .U2U2_2U2ld2_2W2_2W2U2l>J2UI2U2lJ2 ..U)3W3U3IJ3IJ3U3U3ld3UI33lJ3U3W3U3... 17

16 U2U2U)2U2Ui2U2U)2U)2U2UI2liJ22U2U2 3W3W3W3liJ3W3U3W33W3W3U3UI3lJ3... 16

15 15

14 14

13 13

12 12

11 11

10 10

9

107

9

8 B

1 UI1_1U1U1U1U1U11J11J1U1UI1U1U1U11J1U1U1IJ1U1U1U1_1U1U1U1Ii]1U1UI11J1U1U1IJ11J1U11J11J1U1U1U1U1U11J1_1W1U1U1U]1U1U1U)1U1 1

K 1= 26 28 30 32 34 36 38 40 42 44 46 48 50 52 54 56 58 60 62 64 66 68 70 72 74

Figure 5-4 shows the fluid velocity through a cross section of the detection chamber across

the width. Figure 5-5 shows the static pressure profile through this same cross section.

Figure 5-6 and 5-7 are close-ups of the flow around the detection probes. Figures 5-4

through 5-7 have been"mirrored"

for visual representation. The problem was solved with a

180

symmetric model.

Figure 5-8 shows a velocity profile cross section across the chamber thickness. Note that this

figure has been increased by a factor of 20 in the y-direction to make the results easier to

understand. Figure 5-9 shows the pressure profile across this width cross section. Figure

5-10 and Figure 5-11 show close-ups of the velocity and pressure around the detection

probes.

The results in Figures 5-4, 5-7, 5-9, and 5-11 show that the pressure distribution is

dominated by the channel geometry. This is caused by the very narrow channel thickness.

The static pressure drop of 395.7 Pa is slightly higher than 2 times the straight rectangular

duct pressure drop given in section 4.2.4. The factor of 2 is present because the full 3-D

problem with the detection probes was solved as a180

model so it only contained 1/2 of the

channel width.

The maximum fluid velocity of 0.0196 m/sec is also slightly higher than the straight

rectangular duct flow. The maximum fluid velocity can be seen in Figure 5-4 and Figure 5-6

to be located locally around the cylinder. The velocity in the unobstructed chamber area

agrees with the velocity of 0.0148 m/sec that was predicted in the 3-D rectangular duct flow.

This result agreed very well with theoretical predictions.

108

CN CN CN

o o o

CN CN CN CN

inoccNmONCN^CONcn

CNCNCNCNCNCNCNCNcncncncncncncncncnccicncncncn

OpopppppppppppppppQPQO

PQoen

PQPQPQPQPQPQPQPQPQPQPQUQOQPQPQPQPQPQPQPQPQPQPQPQPQPQPQPQ

en en

r- o rf_, ^- o

moocnoomONTi-ONr- o

^t- r~ ôo

^f On Tf ON in

in oo cn in on

oo

+

PQ PQOn

ONON0cr^r^^cinin'*cncncN vo

ON ^ fl,-H Tt HH

CM ^ C

^ J2 -3

Hh

W

X

-N

85 nH Tf HH

^ J3 -3

fe

'

o o o O(-r'

+ + + +

PQ PQ PQ PQ PQ

vO cn m r~

CN m 00 CN

ON CN cn m

o o o o

+ + + +

PQ PQ PQ PQ

rNirxICNCNCNCNCNCNCNCNCNCNCNCNCNCNCNCNCNCNCNCN

_ o

+ + + +o+o

+ +o o

+ +o+o

+o o

+ +o+o o

+ +

CN>_> m oc

ih oo ; p ;

vcpQ on

PJPQPQPQPQPQPQPQPQPQPQPQPQPQPQPQPQPQPQPQ

OC

in p~ 00 ON

rj- c~~ o cn i> O , . -

cNcnin'Or-'ONO*

en vo

oO cn vo On cn

cn rf in _> oo

CN

O

+

PQ

On

-Q cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn

XH

s

oo

oo

OPt

< Pt

w Su ^J <

y\S

*zo

m cl, W

D H p.

O ^ Q

pc-i oo m

x

-N

CN)

CN CN

oCN CN CN

OCN

oCN.

)

CN

o

PQin

ON

PQoo

OC

PQCN

OC

PQm

PQOn

PQCN

PQ

in

PQON

PQcn

CNCNCNCNCNCNcncncncncncncncncncncncncncn-^t-_?

0000000000000000000009

vOOcnr~0-^-cnoccnoocnoN^fON-^-ON^-ONinoo\Sen en CN

' "r~- ~ -* ' -*

-_.----..-_-

m

zo

uwHW

Q

<

z

H

o

w

>

o

OQ

<

Xu

zo

HW

Q

tf w u

D g ^

2 3 Q

E u c^

X

t N

CNCNCNCNCNCNCNCNCNCNCNCNCNCNCNCNCNCNCNCNCNCNCNCNCNCNCNCNCNCNCnIooooooooooooooooooooooooooooooo+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +PQpQpQpQPQPQPQPQPQPQPQPQPQPQPQPQPQPQPQPQPQPQPQpQPQpQpQpqpJLqrn

Or^cnor^cnor^cnor^cnor^cnot^cnor^cnor--cnor--cnoF^cno00|CNCNcn^învqorôqoooNOO^cNcNcn^îniMD_>r---oooooNO

^Q r-I - t-H h h < H CN CN CN CN CN CN CN CN CN CN CN CN CN CN CN cn

111,

X

-N

^cn 2

as Gr-l Tf

I'

^ 3 -3

& it *

CN CN

"5 O

PQ PQO IT)

in *.

(^CviCNCNCNCNc^cncncncncncn

OOOOOOOOOOOO

pjpQPQPQPQPQPQPQPQPQWW

motnOinOmOOOOO

cncncNCN opinpinp_j _Q ,i ,i i-I i I <-Q

> ' on on oo oo

cncncncncncncncncncncn'^-

o o o o o o

pjpQPQPQPQPQpQUQoooooooo

inoinpinpinpt-~! r~ \_ in in *fr ",

o o o o o o o

PQ PQ PQ PQ PQ PQ PQo o o o o o oin p in o in p o

cn cn cn cn -Q ^Q in

oo

+

PQoo

o

uII

00

00

OO

w

p<

113

00

1

ôo H w

1

in u Q

pj o p<

2 J u

D w Oh

O > Qh1

Pi

cn

i/

v^

o o

+ +

o o

+ +o

+o

+o

+

CNCNCNCNCNCNCNCNCNCNCNCNCNCNCNCNCNCNCNCNCN

WPQPQPQPQPQPQPQPQin<nt-~1_'_>_>C"--'~~'

~

o rr oc

o+

PQ

o

+o

+

PQ PQm

~

o o o o o

+ + + + +o o

+ +o

+o+o+o

+o+o o

+ +o

+o o

+ +

PQpQPQPQPQPQPQPQPQpQpQpQp_|pJ[_!PQPQ

CN

o

+

PQin

Os

Os

(N

C<~)

J

o

G

G 3

^-3

% E ^

CO

o o o o

rxicNr^rsicNCNCNCNcncncncncncncncncncncncncncncncncncn'^-xi_ i_ i-.: >^ , , . , , . , .

- ^ . * r i ' > r 1[

1 r"N c 5 c > ; > f > c * c > r~^ ; 1 f 1C"

"i I l r~> _-

ooooooppo o o o O O O O o o o o <qf

PQ PQo m

in <_

PQo

papjpjpQPQPQPQPQPQPQPQPQPQPQPQPQPQPQPQPQPQPQPQPQPQPQPQPQ

En S tn o in o <n ooooooooooooooooooooocncncNCN^ÔOinoinOiniOinOiriOinoinoinoinooo

OOOOOOOOOOOOOOOOPOOOomoinOinpinpinpinpmoinoinooo

ONONododrî^NCNdinin^TtcncncNcN-Qîn'o

. ..'...

OO

00

/

CNCNCNCNCNCNCNCNCNCNCNCNCNCNCNCNCNCNCNCNCNCNCNCNCNCNCNCNCNCN

o

+o

+o

+o

+PQ PQ PQ PQoo

p- cn

o

o

+

. PQO P~

CN CN

+o+ +

o

+o

+

W UQ PQ PQ PQp~ cn OO

o o

+ +o

+o+o

+o+o+o

+o

+o o

+ +o+o

+o+o

+o

+

pQpQPQpQpQpQPQPQPQpQPQPQPQPQpQpQpQpQpJtqcn o p~ cn

<tm--c-ooo\

p~ cn

oO P~ cn

ih cn cn cnOP-cnOp-cn'pp-cnOTl-TflnVOVOP-OOOOONO

CNCNCNCNCNCNCNCNCNCNCNCNCNCNCN

i I ^t

OO

00

oo

/

o

G

G C

ft 3 3

E ^

in

Ph U cn

Another aspect of the results investigated is the local pressure profile around the detection

probes. There were two basic questions that were addressed.

1. How do the results compare to the 2-D cases?

2. What is the difference between the pressure profile on the first detection probe and the

profile on the second detection probe?

Shown in Figure 5-12 are the pressure profiles around the side walls of the both detection

probes (at J=4). The results look similar in some respects to the 2-D results, but there is also

a significant difference. Similar to the 2-D results there is a pressure drop as the flow goes

around the cylinder. Unlike the 2-D results there is no "pressurerecovery"

on the back side

of the cylinder. This is caused by the static pressure drop from the narrow detection

chamber overwhelming the flow pattern around the cylinder. The results also show that there

is negligible pressure difference along the height of the detection probe.

The other important feature ofFigure 5-12 is that the pressure coefficient drop on the second

detection probe is relatively the same as the first probe, -1505. However, the absolute

pressure coefficient is much lower because of the static pressure drop that occurs in between

the location of the two detection probes. This result would be expected to continue for any

succeeding detection probes that were included in the model.

Figure 5-13 shows the pressure drop across the top of the detection probe. The result shows

the same axial pressure drop that is expected from the basic duct flow. When determining

how the fluids will interact with the detection probes, this pressure across the top and flow

velocity are more important than the pressure on the sides of the cylinder. This is because for

the basic detection probe geometry there is over 20 times the surface area on the top of the

detection probe compared to the side walls.

117

Figure 5-12 PRESSURE COEFFICIENTS ON DETECTION PROBES FOR 3-D

FLOW

PRESSURE COEFFICIENT ON CYLINDERS - FROM FLUENT

3-D BASELINE

2000

6000

180.00 150.00 120.00 90.00

ANGLE (deg)

60.00 30.00 0.00

118

CN CN

O o

+ + +

CNCNCNCNCNCNCNCNCNCNCN

o

+o

+o

+

o

+o o

+ +

CN

o

+

CNCNCNCNCNCNCNCNCNCNCNCNCNCNCNCN

o o

+ +o

+o

+o

+o+o

+o

+o

+o

+o

+o

+o

+o

+

PQlipQPQPQPQPQPQPQPQPQPQQQPQpQPQPQPQPQPQPQPQPQPQPQPQPQPQPQPQpQ

p- cn

o

p~ en

CN cn

p~ cn

Tt mi

O P~ en O P~ cn

kC o p- oc oo ON

5 "> gr^ ^T HH

^G

ft 3 3

egfc fc

X

-N

5.2 FLOW UNIFORMITY AS PROBE HEIGHT CHANGES

One aspect of the PCR detection chamber that was investigated as part of this work was to

determine how the height of the detection probe would effect the flow within the chamber

and the pressure distribution on the probes.

Shown in Figure 5-14 are the outlines for the geometry with different probe heights. Again

these heights have been magnified by 20 for visual purposes. There were three different

cases analyzed.

1. Baseline case; probe height = 0.029 mm

2. Probe 1OH; probe height = 0.050 mm

3. Full channel height; probe height = 0.076 mm

The inlet velocity, fluid viscosity, and fluid density were the same as for the baseline case.

These values were previously given in section 5.0.

Figures 5-15 through 5-17 show the fluid velocity through a channel cross section for the

different detector probe heights. The results show that the maximum velocity will increase

as the probe height is increased.

Figure 5-18 shows the pressure coefficient distribution around the first detection probe for

the three different cases. The results show that as the probe height is increased the change in

pressure coefficient from the front of the detector probe to the rear is increased. This result

occurs because as the probe height is increased the area for fluid flow in the channel is

decreased. This reduction in area causes the fluid velocity to increase which increases the

shear forces on the chamber walls which causes the pressure drop to increase.

120

Figure 5-14 GEOMETRY OUTLINES FOR DIFFERENT DETECTOR PROBE

HEIGHTS

II

121

Osm O

ON fiH Nt H

*^ 3 -3

egE fc

\____/

CN CNo o

PQ PQmi oo

On oo

CNCNCNCNCNCNCNcNcNCNCNCNcNcncncncncncncncncncncncncnrn-rt-

o o o o o o o- - - - - - -000000000000000000000

PQPQPQpQpgpQPQPQPQPQPQPQPQPQPQPQPQPQPQpQpJWM

^.îi;^nnrNj-.-.q^q'i;is;^rtinooNinoifriôQHrtHrtHHHHHHCtiaOdS^lOln^^Mt'iciHrt.d

cn m

00 p~

u,-3

00

nI-H

_=

w

Q X>

HoS-H

Xa,

o a.11 CO

w C.

on

W PQ00 .

J

<CQ

in

a

Pft

o

CO

w O2 j

s

Oi

<D

NO

ON

Xpq

>

6 3

c

cN cn

o

I1/qn

PQp~

cn

es CN

CN CN CN CN CN CN CN

o o o o o o oCNCNCNCNCNCNCNCNcncncncncncncncncncncnTfOOOOOOOOOOOOOOOOOOOO

WWWBBBfflfflWUWWBBWHBnBmm_r_>

r

* /^l _t- >n r*-% .-ns ry^ m I . T\ >^l ^+ ^r~* .Tv i in b ^ . ^T Z-J r_rT1 T- rrSN * . OC 3N00 O

CNCN"_OONOCP~p-~Oin

CN CN CN CN

en in P~ On CN

t cn cn CN OONp-vo^tcn--HON0d\_incnS

on oo od pQ_! \r,i , Tt cn cni i-J oo

Oo

+

PQoo

o

ON ^ OON ^J Gr-( <nT HH

^3 3

^ J2 -3

cÊ fc

NO

oo+

pqooo

d

00O

pq

in > ^.rT, -ti <n

S o II

O >HH ,_L G

Pft H^ hJ

CN CN CN CN

PQ PJON O\C vC

CN CN

PQ PQ" CN

in >,

cNCNcNcNCNCNCNcNCNCNCNcNcNCNCNcncncncncncncncncncn^t-

OOOOOOOOOOOOOOOOOOOOOOOOOO

aBWBUBBWUBBUUBWBBUBWWBBBWWcn^f-mNCp~ocoNO"CNcnin>_p-oc>op--p~p-ooococoNONONp-

rncN OONOOP-;P-;vCin'*cncN-pocONO cNcn^-in^Cp-ON

cn cn cn cn ' > i i-Q on oo oc pQ v_ in <_ cn cn < oc

oo

+

PQoo

o

WM

g; nr-H ^t

2 *

O

G

GCD

s^,3

h3

egPh Ph

OO

oo+

pqooo

d

n

G

(N

O

o

in

^ o

o llpq o

2^ 5 c_

fc D nq

!_ N

JV

Figure 5-18 PRESSURE COEFFICIENT ON FIRST PROBE FOR DIFFERENT

PROBE HEIGHTS

luapjiieoo ajnssajd

COo+LUoooo

cvi

coo+LUOOOm

cvi

coo+LUOOOO

CO

coo+LUOOom

co

CO

o+

LUOOOo

CO

o+

LUOOOin

jj=

=

=

4 ft

oo

o

dCM

oo

d"3-

oo

dCD

oo

d00

oo

do

oo

dCM

oo

d

oo

dco

oo

dco

0)

2,

LUI

(5Z

<

125

Figure 5-19 shows the pressure coefficient for both detector probes for the different probe

heights. The results show that the pressure coefficient on the second detector probe is

increased as the probe height is increased. Also the delta pressure coefficient between the

first and second detector probe also increases as the probe height increases. This result is

summarized below in Table 5-1.

Table 5-1 SUMMARY OF PRESSURE COEFFICIENTS AT 6 =0

Probe Height Probe #1 Probe #2

CASE (mm) Cp Cp delta Cp

BASELINE 0.029 -3767.0 -5663.6 -1896.6

PROBE 10H 0.050 -4104.4 -6310.1 -2205.7

FULL HEIGHT 0.076 -4226.8 -6547.5 -2320.7

126

Figure 5-19 PRESSURE COEFFICIENT ON DETECTION PROBE #1 AND #2

luajoiiiaoo ajnssajd

</)\-

X

(5

in

x

UJ

CD

OOCQ.

CC

OHOLUI-

UJ

Q

UJ

CCUJLLLU

Q

CC

OLL

t-

ZUJ

o

LLLL

UJ

OO

UJ

CC

(0c/>UJ

CCQ.

5.3 SENSITIVITY TO CHANNEL WIDTH ON THE FLUID FLOW

Given the manufacturing technique of the PCR detection pouch there can be significant

variation in the chamber width. This change in chamber width will also cause the inlet

velocity to vary because the PCR pouch only contains a finite amount of fluid volume.

Given in Figure 5-20 are three different channel width's that will be analyzed to determine

the sensitivity to this parameter. Table 5-2 summarizes the basic parameters.

Table 5-2 CASE SUMMARY FOR SENSITIVITY TO CHAMBERWIDTH

CHANNELWIDTH INLET VELOCITY

CASE (mm) (m/sec)

BASELINE 5.08 0.01

SYM070 3.56 0.0143

SYM2W 10.16 0.005

Figure 5-21 shows the velocity plot for the narrow channel width. Figure 5-22 shows the

velocity profile for the wider detection chamber. The figures shown are the plan view from

the top of the detection chamber. These cases were solved in 3-dimensions but the results are

plotted in 2-D for visual purposes only. The nominal baseline velocity plot was previously

shown in Figure 5-15. The results show that the velocity will decreased as the chamber

width is increased.

128

Figure 5-20 CONFIGURATIONS FOR VARYING CHANNEL WIDTH

129

ON fr> O

ON i G^ ^t ^

^33

^ ^ ^3

fe fe

CNCNCNCNCNCNCNCNCNCNCNCNCNCNCNCNCNCNCNCNCNCNCNcncncncncncncn

oooooooo oooooooo o o o o o o o o o o o

p^"*OP~cno\OCNON

O On oo vC m cn CN

cn cn cn cn cn en

w cn oo ^t p-^fOP~cnONvCcNON inoNcnp^cN^co^ÔNOCpîn'rtcn OONPô^-cncNpin p~-^-op~cno

cn CN CN CN CN Cvi CN CN CN ' ' ---3ioC>OIO Tfr CN r-J O

u

-J

X

II

-N

cncncncnrncncncncncncncncncncncncncncncncncncncncncncn

oopooopoooooooooooooooopppo_ ~ ,

cn vo on cn in oc -^ ^c on cn in oo^pôcNinoc ^p-Ocn^coccncN^-ot rV^ 1/^ r*~ f s 1 I/-, /^l TN ^ *T+ rVl LO r*~i r 1 n^ l/~. f0 IN 1^^ ^t"

< T\ JN rt^ f 1 ^*- I > \

o o o P

cn vo on (N in 6c . _ , .......

; oc in cn o r~ m cn on ^d^3-

"*tj-

tT

o oc in CN On P-;cn cn cn cn cn

rt on vO cn

CN CN i t-i i i| "3; p^

OC in CN

ONrr)

OON ^ G^H Tf

H-1

^33

^ 5 -3

^ Ph Ph

oo+

wooo

d

n

-J

X & ^

u J

(N

in >>

<; >

5-i ro

S o

o >

X

co

^ oo

II

X

Ph J

-N

isy

Figure 5-23 shows the pressure coefficient sensitivity for the first detector probe. This plot

shows a much greater negative pressure coefficient for the wide channel width. The reason

for this is the pressure coefficient is also a function of the inlet velocity (see equation 2.1).

Figure 5-24 shows the relative static pressure drop for detection probe #1. This result shows

that the static pressure drop is much lower for the wide channel than for the narrow channel.

Figure 5-25 shows the static pressure drop between the first and second detection probe for

the narrow and wide channel cases. The result shows that there is amuch greater pressure

drop from probe #1 to probe #2 for the narrow chamber width. This results leads to a very

interesting design trade which should be verified by experiment.

Previous PCR pouch designs have been based on maximizing the pressure on the detection

probe. This often caused the first probe to have significantly different results than the last

probe. The pressure difference between the probes is probably a significant factor in this

result. To get uniform results for each probe the detection chamber width should be made

wider. However, there are some practical drawbacks to this result. If the chamber is made

wider the detection system will need to be made more sensitive because the pressure forces

on each probe will be lower for a wider chamber. To increase the flow rate for a wider

chamber would require more sample fluids which would have an impact on the pouch size

and the cost of each patient test.

132

Figure 5-23 PRESSURE COEFFICIENT FOR DIFFERENT CHAMBERWIDTHS

luajoiuaoo ajnssajd

CO COo o+ +LU LUO OO OO OO O

CM CO

COo+

LUOOOO

COo+LUOOOO

lO

CO

O+LUOOOO

CO

coo+LUOOOO

CCUJ

DO

O

o

>

c/>

UJ

m

oocQ.

UJ *z

Siz "

UJ

o

LLLL

UJ

OO

UJ

OC

D

</></>UJ

oca.

E E

E

CO

CO

o

lO

b

CO

IO

dII CO

x: II

n TJ .c

_c T3

T3fit

Cc 0

C

c

CO

O

_C

CO(-

CO

co O

o c 5

CO CO o

T> (0 1_

CO

co

<0

+ H

oo

oo

dCVJ

oo

d

oo

dCD

o

O m

CO -g

OO

do

oo

dCM

oo

d

oo

dCD

oo

dco

LU_J

(3Z

<

133

Figure 5-24 RELATIVE STATIC PRESSURE ON PROBE #1

RELATIVE STATIC PRESSURE SENSITIVITY TO CHAMBER WIDTH

0.0000E+00i i i l

--Wide Channel Width = 10.16 mm

--

Baseline; Channel Width = 5.08 mm

- Narrow Channel Width = 3.56 mm

i

"

;

-5.0000E+01

-1.0000E+02

-1.5000E+02

-2.0000E+02

-2.5000E+02

Oi-

3(0

CO

0)

-3.0000E+02

180.00 160.00 140.00 120.00 100.00 80.00 60.00 40.00 20.00 0.00

Angle (deg)

134

Figure 5-25 SENSITIVITY OF STATIC PRESSURE DROP TO CHAMBER

WIDTH

PROBE #1 AND PROBE #2 STATIC PRESSURE

50

-100

-150

-200 "toa.**

CO

-2501_

-3

(0

(0

CDw.

-300 a.

350

-400

-450

-500

180.00 160.00 140.00 120.00 100.00 80.00 60.00 40.00 20.00 0.00

Angle (deg)

135

6. CONCLUSIONS

FLUENT has demonstrated the capability to be an excellent CFD analysis tool. The software

was able to handle the narrow detection chamber geometry with the cylindrical flow

obstructions, and give accurate results in a reasonable computational time period. The results

achieved with this analysis have been excellent. All of the major objectives given in section

1.4 have been satisfied.

1 . Compare the results from potential flow theory to FLUENT for flow over cylinders.

FLUENT did a very good job of predicting the flow around the cylinder. The key item

that was learned during this particular analysis was the need to use the QUICK

interpolation scheme instead of the standard power law scheme used by FLUENT. The

author recommends that a person using FLUENT understand the relationship between

the geometry grid and the computational grid. If accurate resultsof the flow pattern are

required around a corner of the computational grid, then the higher order interpolation

schemes should be used.

2. Compare the FLUENT results to other published results for flow over a cylinder.

The results comparing FLUENT to otherpublished results was very good for some

results and marginal for others. This current work compared well with the work ofBraza

et al. (1986). The results in Figure 4-26 show good agreement for the pressure coefficient

around the cylinder. The FLUENT results compared to similar work ofDennis and

Chang (1970) in Figure 4-25 are not verygood.

136

Because the work ofBraza et al. compares well with experimental results, the author has

good confidence in the FLUENT results.

The comparisons to published results for the flow separation angle around the cylinder

given in Table 4-5 are good. The results also compare well for the wake bubble length

given in Table 4-6.

3. Determine if FLUENT will properly predict the effect ofwalls located in close

proximity to a cylinder. Use the potential flowMethod of Images theory for verifying

the FLUENT results.

This result was really outstanding. The potential flow theory using theMethod of Images

did an excellent job of predicting the flow field around the cylinder. The result for the

pressure coefficient given in Figure 4-24 shows excellent agreement except at the

0 =90

and the rear stagnation point.

TheMethod of Images could be used to develop design trends for any future changes.

Once I realized how good this theory is at predicting the flow field, I was surprised that

other authors had not used this to validate their flow solutions. Simulating the walls

outside the cylinder with theMethod of Images certainty more accurately represents flow

fields that exist in actual practice than assuming the free stream boundary conditions

chosen by the authors cited in this report.

137

4. To show that commercial CFD codes such as FLUENT can be used to analyze the three

dimensional fluid flow in narrow channels less than 0.1 mm thick.

FLUENT did a very good job at solving the fluid flow in the narrow chamber. The result

given in section 4.2.4.1 show very small differences between the theoretical solution and

the FLUENT results for both the flow velocities and static pressure drop in the 3-D

rectangular channel.

5. Determine the pressure distribution on the detection probes for a typical 3-D geometry.

The results in section 5 give a very good understanding of the flow field within the PCR

detection chamber. The flow characteristics are dominated by the small size of the

detection chamber. This is clearly shown in Figure 5-5 which gives the pressure drop in

the detection chamber.

The results also show that because of the pressure drop along the channel, it is

unreasonable to expect the first detection probe to have the same interaction with the

sample fluid as the 2nd, 3rd, 4th, etc. Also the variation in pressure drop around the

cylinder means that any positive or negative test can not expect uniform color for the

entire detection probe.

138

6. Determine the flow field over the cylindrical disk obstruction as the channel width

changes.

These results show that as the channel is widened the pressure will become more

uniform around the detection probe. Also the result shown in Figure 5-25 is that the 2nd

detection probe will have a response closer to the first probe because of the lower

pressure drop in the detection chamber. The negative side of getting this more uniform

pressure distribution is that the relative static pressure on the detection probe is reduced.

This was shown in Figure 5-23. This result would lead to the requirement for a more

sensitive detection system.

7. Determine the flow field within the detection chamber as the probe height is varied.

These results showed that the flow velocity will be increase as the probe height is

increased. This result is caused by minimizing the flow area within the chamber.

This results in a higher pressure drop within the detection chamber. The other result that

comes from this is shown in Table 5-1 is that the difference in pressure coefficient from

the front of the probe to the rear is increased. This would result in a less uniform color

response across the detection probe.

139

7. RECOMMENDATIONS FOR FUTURE WORK

There are several aspects of this problem that should be investigated further. Most of these

are based on minimizing the number of simplifying assumptions that were made for this

work.

a. The results of literature review show that at Recyi = 40 the flow field is on the edge of

having vortex shedding. The flow field should be analyzed as a transient solution to

more accurately determine this effect.

b. Solve the 3-D problem as a360

model instead of a 180. This could easily be done

given enough time and computing power. This would also be true for expanding the size

of the problem to include the entire length of the detection chamber. The number of grids

can get very large for 3-D problems. To obtain reasonable run times the 3-D problem

with the cylinder was solved as a symmetric problem. The baseline model had 45,900

cells. Depending on the case being run, and how much time was spent tweaking the

FLUENT solver parameters, it took between 300 and 1500 iterations to converge to a

solution. This usually took from 4 to 8 hours to solve.

c. The effect of detector probe spacing should also be considered. It was not considered in

this report because the current spacing is more a function of the number of tests required,

physical space on the machine, and manufacturing capability for applying the detector

probe. Itwould be desirable to minimize this spacing to minimize the size of the PCR

pouch. However, the results in this work show that the current spacing between the

detection probes restricts the fluid flow on the front side of downstream detection

probes.

140

d. Another aspect of the PCR detection chamber that was not addressed in this report was

the effect of a secondary fluid pushing out the previous fluid. The actual PCR pouch only

contains a finite amount of fluid. Once one fluid was been'pushed'

through the system,

the next fluid will hopefully push out the previous fluid. Experimental results with PCR

pouches has shown there can be problems cleaning out the corners of the detection

chamber in some cases. This analysis should also be performed as a transient analysis.

e. The big step that is beyond the capability of this author is to treat the detection chamber

as a non-rigid geometry. The actual chamber is a plastic material that nominally lies flat

until the fluid causes the chamber to expand. This analysis would probably require a

transient non-linear structural analysis of the pouch deformation that is based on the

pressure created by the fluid flow internal to the pouch.

141

REFERENCES

Barnes, A.F., Hardey, M.J., and Lever, T.J., 1993, August, "Review of the Applications of

ThermalMethods within The Pharmaceutical Industry ", J. ofThermal Analysis

Batchelor, G.K., 1967, Introduction to Fluid Dynamics, Cambridge University Press

Bhatia, K., Rahman, M., and Agarwal, B., 1993, "Analysis ofThree Benchmark Problems

using NISA/3D-FLUID", ASME FED-Vol. 160, pg. 55 - 60

Braza, M., Chassaing, P., and HaMinh, H., 1986, "Numerical study and physical analysis of

the pressure and velocity fields in the near wake of a circular cylinder", J. FluidMech.,

Vol. 165, pp. 79-130

Chirgwin, B.H. and Plumpton C, 1967, Elementary Classical Hydrodynamics, Pergamon

Press

Chung, T.J., 1978, Finite Element Analysis in Fluid Dynamics, McGraw-Hill, Inc.

Churchill, S.W., (1988), Viscous Flows The Practical Use ofTheory,

Butterworh Publishers

Dennis, S.C.R. and Chang, G-Z., 1970, "Numerical solutions for steady flow past a circular

cylinder at Reynolds numbers up to 100", J. Fluid Mech. Vol. 42, part 3, pp. 471-489

FED-Vol. 160, 1993, The CFD Triathlon Three Laminar Flow Simulations by Commercial

CFD Codes, American Society ofMechanical Engineers

Findlay, J.B., et al, 1993, "AutomatedClosed-Vessel System for in Vitro Diagnostics Based

on Polymerase Chain Reaction", Clinical Chemistry, vol. 39, no. 9, pg. 1927-19

FLUENT Users Guide, 1995, Fluent Inc., Version 4.3

Fornberg, B., 1980, "A numerical study of viscous flow past a circular cylinder",

J. FluidMech. Vol. 98, part 4, pp. 819-855

Fox, R.W. andMcDonald, A.T., 1985, Introduction to FluidMechanics,

JohnWiley & Sons

Freitas, C.J., 1995, "Perspective: Selected Benchmarks From Commercial CFD Codes",

Journal of Fluids Engineering, Vol. 1 17, pp. 208 - 218

Granger, R.A., 1975, Incompressible Fluid Dynamics, United States naval Academy

142

Hele-Shaw, H.J.S., 1898,"

On the motion of a viscous fluid between two parallel plates",

Nature, 58, 34-36

Kim, S-E. and Choudhury D., 1993, "A Computational Study ofLaminar Flows: Benchmark

Calculations Using FLUENT", ASME FED-Vol. 160

Lu, PC, 1973, Introduction to theMechanics ofViscous Flows, Hemisphere Publishing

Ma, S.W. and Gerner, F.M., November 1993, "Forced Convection Heat Transfer fromMicro

Structures", Journal ofHeat Transfer

Milne-Thomson, L.M., 1960, Theoretical Hydrodynamics, TheMacmillan Company

Mullis, K.B., 1990, April "The Unusual Origin of the Polymerase Chain Reaction",

Scientific American, pg. 56-65

Mutama, K.R. and Iacovides, H., 1993, November, "Investigation ofDeveloping Flow and

Heat Transfer in a Long Converging Duct", Journal ofHeat Transfer

Nakayama, Y., (1988), Visualized Flow, Pergamon Press

Nunn, R.H., 1989, Intermediate FluidMechanics, Hemisphere Publishing Corporation

Ockendon H. and Ockendon, J.B., 1995, Viscous Flows, Cambridge University Press

Patankar, S.V., 1980, Numerical Heat Transfer and Fluid Flow, Hemisphere Publishing

Company

Patankar and Spalding, 1970, Heat andMass Transfer in Boundary Layers, 2nd ed. Interleaf

Pratap, V.S. and Spalding, D.B., 1976, "Fluid Flow and Heat Transfer in Three Dimensional

Duct Flows", Int. J. Heat Mass Transfer, vol. 19, pi 183

Saffman, P.G. and TaylorG, 1958, January"

The penetration of a fluid into a porous

medium or Hele-Shaw cell containing amore viscous liquid", Jan 1958

Schlichting, H., 1979, Boundary Layer Theory,McGraw-Hill Book Company

143

Documents

The Effect of cylindrical obstructions on the fluid flow ...€¦ · Rochester Institute of Technology RIT Scholar Works Theses Thesis/Dissertation Collections 8-1-1996 The Effect