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I ALJ.J

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This is to certify that the

dissertation entitled

A NEW APPROACH TO MEASURING INSTANEOUS
FLOW RATES IN UNSTEADY DUCT FLOWS

presented by

Mahmood Ahmad Akhtar Rahi

has been accepted towards fulﬁllment
of the requirements for

Ph.D. degree in MECHANICAL

 

 

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6/01 c:/ClRC/DaIeDue.p65-p.15

A NEW APPROACH TO MEASURING INSTANTANEOUS
FLOW RATES IN UNSTEADY DUCT FLOWS

By

Mahmood Ahmad Akhtar Rabi

A DISSERTATION

Submitted to
Michigan State University
in partial fulﬁllment of the requirements
for the degree of

DOCTOR OF PHILOSPHY
Department of Mechanical Engineering

2002

ABSTRACT

A New Approach to Measuring Instantaneous Flow Rates in Unsteady Duct Flows

By

Mahmood Ahmad Akhtar Rahi

In transient duct flows, there are presently few reliable techniques for measuring
instantaneous flow rates accurately. Although transient ﬂows have been studied
previously the understanding gained has not led to improved unsteady ﬂow metering
devices. The main outcome has been to link steady and unsteady ﬂows through semi-
empirical relations. One possible path to devising an accurate measurement technique is
through relationships from the solved, unsteady ﬂuid mechanics equations; however this
has not been undertaken. In this thesis, solutions to unsteady, fully developed Navier-
Stokes equations and Reynolds averaged momentum equations are found and tested in an
experimental test facility that emulates a prototype instantaneous unsteady ﬂow metering

device.

In the case of laminar ﬂow metering, unsteady, fully developed Navier Stokes
equations are solved to express the ﬂow rate as a functional of pressure drop. The
solution consists of a quasi-steady term which is supplemented by an unsteady correction
term. This unsteady correction term is a convolution integral of the product of momentary
pressure drop and a weighting function. This unsteady correction term is an analytically
exact description of the ﬂow history and is necessary in unsteady duct ﬂows. In order to

ascertain the feasibility of unsteady ﬂow metering, an experimental test facility was

designed and fabricated, with miniature pressure sensors embedded in a long transparent
acrylic pipe to measure the instantaneous pressure drop. The momentary flow rate was
then inferred from the pressure-drop functional and compared with the instantaneous
ﬂow rate as inferred from the cumulative collected ﬂow. In test experiments, the

principle appears to work perfectly for duct ﬂows with arbitrary unsteadiness histories.

A similar approach was adopted for unsteady turbulent ﬂow metering. In this
approach Reynolds averaged momentum equations were solved with a modiﬁed eddy
viscosity model incorporating a time delayed or history response to the turbulence. This
model represents the averaged eddy viscosity across the duct and was tested against the
experimental data for changes in the Reynolds shear stress during imposed transients.
The solution of the Reynolds averaged momentum equation is similar to the laminar ﬂow
result of flow rate as a functional of pressure drop and consists of a quasi steady term
augmented by an unsteady correction term incorporating an eddy viscosity. Using the
same experimental facility, test experiments revealed that this approximate approach
seems to work reasonably accurate in prediction of instantaneous flow rates in unsteady

turbulent duct ﬂows.

This thesis is dedicated to my parents

Bashir Ahmad Akhtar (late)
&
Salima Akhtar

For their love, kindness and prayers for my success

ACKNOWLEDGMENT

I would like to pay special thanks to a few important people:

First, I would like to extend my sincere gratitude to Dr. Giles Brereton, my Major
Professor, for being my adviser, and for his support and patient guidance throughout my
thesis work and studies.

I would also like to extend my sincere thanks to member of my committee Dr.
Harold Schock, Dr. Tom Shih, Dr. Ahmed Naguib and Dr. Milan Miklavcic, for being on
my committee, their helpful comments and thought provoking questions. I would also
like to thank Dr. Kyle Judd for his valuable comments and, Tom Stueken for his help in
fabrication of tallest structure in the lab for this research, and for his cheerful company. I
would also like to thank my fellows at MSUERL, for their day to day guidance and their
wonderful company; Mark Novak, Edward Timm, Anthony Christie, Boon Keat Chui,
Yuan Shen, Andrew Sasak and Andrew Fedewa.

Finally, my wife Rashda, for her patience and support throughout my Masters and
Ph. D. Program, and my mother without whose prayers and encouragements this day

would not have been possible.

TABLE OF CONTENTS

LIST OF TABLES ............................................................................................................. ix
LIST OF FIGURES ............................................................................................................. x
NOMENCLATURE ......................................................................................................... xiv
Chapter I
Introduction ......................................................................................................................... 1
1.1 Statement of the Problem .................................................................................... 1
1.2 Review of Laminar Flow Studies ........................................................................ 3
1.2.1 Unsteady Laminar F low-Theoretical Background ...................................... 3
1.2.2 Development of Transient Response Theory and Experimental
Veriﬁcations ................................................................................................................ 5
1.2.3 Experimental Studies on Flow Rate, Friction Factor and Velocity Proﬁle
Measurements under Transient Flow Conditions ..................................................... 10
1.3 Review of Turbulent Unsteady Flow Studies ................................................... 13
1.3.1 Periodic Turbulent Unsteady Flows .......................................................... 13
1.3.2 Non Periodic Turbulent Unsteady Flows .................................................. 17
1.4 Unsteady Transitional and Other Related Flows ............................................... 18
1.5 Motivation and Objectives of the Present Study ............................................... 22
Chapter 2
Theoretical Background For Laminar Newtonian Flow ................................................... 24
2.1 Laminar Flow Theory ........................................................................................ 24
2.1.1 Navier-Stokes Equations for Fully Developed Flow, Simplifying
Assumptions and Boundary Conditions. ................................................................... 24
2.1.2 Solution Procedure by Laplace Transformation ........................................ 26
2.1.3 Pressure Gradient to Flow Rate Results. ................................................... 34
2.1.4 Flow Rate to Pressure Gradient Results .................................................... 35
2.1.5 Extension of Fully Developed Pipe Flow Relationships to Other
Geometries ................................................................................................................ 40
Chapter 3
Experimental Apparatus .................................................................................................... 41
3.1 Flow Circuit Design .......................................................................................... 41
3.2 Pressure Sensor ................................................................................................. 43
3.3 Load Cells ......................................................................................................... 45
3.4 Viscometer ........................................................................................................ 46
3.5 Data Acquisition and Control Card ................................................................... 47
3.5.1 Data Acquisition and Control .................................................................... 47
3.6 Calibration of Sensors ....................................................................................... 49
3.6.1 Pressure Sensor Laminar Flow Calibration ............................................... 49
3.6.2 Pressure Sensor Turbulent Flow Calibration ............................................ 51
3.6.3 Low Reynolds Number Turbulent Friction Factor Relationship .............. 52
3.6.4 Load Cell Calibration ................................................................................ 54

vi

3.7 Critical Experiment Requirements .................................................................... 55

3.7.1 High Signal to Noise Ratio ....................................................................... 55
3.7.2 Working ﬂuids ........................................................................................... 56
3.8 Experimental Plan ............................................................................................. 58
3.8.1 Laminar Flow Experiments ....................................................................... 59
Chapter 4
Laminar Flow Experimental Results and Discussion ....................................................... 61
4.1 Laminar Flow .................................................................................................... 61
4.2 Experimental Results ......................................................................................... 62
4.2.1 Steady Flow Results for Fluid 1 ................................................................ 62
4.2.2 Accelerating Flow Results for Fluid 1 ...................................................... 63
4.2.3 Decelerating Flow Results for Fluid 1 ...................................................... 65
4.2.4 Oscillating F low Results for Fluid 1 ......................................................... 67
4.2.5 Accelerating Flow Results for Fluid 2 ...................................................... 70
4.2.6 Decelerating Flow Results for Fluid 2 ...................................................... 72
4.2.7 Oscillating Flow Results for Fluid 2 ......................................................... 73
4.2.8 Accelerating and Decelerating Flow Results for Fluid 3 .......................... 74
4.3 Feasibility of Unsteady Pressure Measurement for Flow Metering .................. 76
Chapter 5
Turbulent Flow Modeling ................................................................................................. 77
5.1 Background ....................................................................................................... 77
5.2 Objective ........................................................................................................... 77
5.3 Reynolds Averaged Turbulent Momentum Equation ....................................... 78
5.4 Turbulent Shear Stress Model ........................................................................... 79
5.5 Calibration of the Model Parameters a, B and a ............................................... 81
5.5.1 Calibration of a .......................................................................................... 81
5.5.2 Calibration of B .......................................................................................... 82
5.5.3 Calibration of OL ......................................................................................... 83
5.6 Laplace Transformation and Solutions ............................................................. 85
Chapter 6
Turbulent Flow Experimental Results and Discussion ..................................................... 92
6.1 Turbulent Flow Experiments ............................................................................. 92
6.2 Model Calibration ............................................................................................. 93
6.3 Experimental Results ......................................................................................... 95
6.3.1 Steady Flow Results .................................................................................. 96
6.3.2 Accelerating Flow Experiments ................................................................ 97
6.3.3 Decelerating Flow Experiments .............................................................. 100
Chapter 7
Conclusions and Recommendations ................................................................................ 105

vii

Appendix A

Experimental Apparatus Conﬁgurations ......................................................................... 109
Appendix B
Pressure Sensors .............................................................................................................. 11 l
3.1 Strain Gauge Pressure Sensors ........................................................................ 111
B2 Piezoresistive Integrated Semiconductor ........................................................ 112
B3 Pressure Sensor Parameters ............................................................................. 114
8.3.1 Thermal Zero Shiﬁ (TZS) ....................................................................... 115
B.3.2 Thermal sensitivity Shift (TSS) ............................................................... 116
3.3.3 Zero Offset .............................................................................................. 116
Appendix C
Load Cells ....................................................................................................................... 117
CI Operating Principle and Design ...................................................................... 117
C2 Speciﬁcations of Load Cells ........................................................................... 120
C3 Frequency Response of the Load Cells ........................................................... 121
C.3.1 Impulse Response of the Load Cell ......................................................... 121
C.3.2 Ramp Response of Load Cells ................................................................ 125
Appendix D
Sensor Thermal Sensitivity ............................................................................................. 127
Veriﬁcation of Sensor Temperature Driﬁ due to Heat Loss from Convection Heat
Transfer. ...................................................................................................................... 127
Appendix E
Uncertainty Analysis ....................................................................................................... 130
El Uncertainty Analysis ....................................................................................... 130
E2 Bias Errors ....................................................................................................... 130
E3 Precision Errors ............................................................................................... 131
E4 Propagation of Uncertainty ............................................................................. 131
ES Uncertainty Analysis of Experimental Results ............................................... 132
E51 Uncertainty in Density Measurement ...................................................... 132
E.5.2 Uncertainty in Mean Velocity Measurement .......................................... 133
E.5.3 Uncertainty in Reynolds Number ............................................................ 133
E.5.4 Uncertainty in Pressure Drop and Inferred Accumulated Mass
Measurements for Laminar Flow Experiments ....................................................... 134
E.5.3 Uncertainty in Pressure Drop and Inferred Accumulated Mass
Measurements for Turbulent Flow Experiments ..................................................... 135
List of References ............................................................................................................ 136

viii

LIST OF TABLES

Table 4.1: Properties of ethylene glycol and water solution ............................................. 61
Table B. 1: Different parameters of sensors ..................................................................... 115
Table C. 1: Speciﬁcations of load cells ............................................................................ 121
Table D. 1: Properties of the ﬂuids and sensors used in equation (D.3) .......................... 129
Table E. l: Uncertainty in density, volume and mass ...................................................... 132
Table E.2: Uncertainty in velocity, ﬂow rate and diameter measurements ................... 133
Table E.3: Uncertainty in the Reynolds number and kinematic viscosity ...................... 134

Table E.4: Uncertainty in inferred accumulated mass, pressure drop, and pipe length

measurement in laminar flow experiments ..................................................................... 134

Table E.5: Uncertainty in inferred accumulated mass, pressure drop and a measurement

in turbulent ﬂow experiments ......................................................................................... 135

ix

LIST OF FIGURES

Figure 2.1: Laminar velocity proﬁle in pipe ﬂow ............................................................. 25

Figure 2.2: Weighting function plotted against dimensionless time; p, =3.34E-3 N-s/mz;

y, = 25213-3 N-s/mz; ,u3 = 3.09133 N-s/mz; r = w/R2 ................................................... 36
Figure 3.1: Experimental apparatus .................................................................................. 43
Figure 3.2: Sensor mounted in a pipe wall ........................................................................ 44
Figure 3.3: Cannon-Fenske viscometer for liquids ........................................................... 46
Figure 3.4: Programmable Gain Instrument Ampliﬁer (PGIA) ........................................ 47
Figure 3.5: Pulse train and pulse duty cycles .................................................................... 48

Figure 3.6: Pressure sensor calibration in the laminar ﬂow regime with a 35/65% ethylene
glycol and water mixture in a 9.5mm diameter pipe ........................................................ 50

Figure 3.7: Pressure sensor turbulent calibration curve with 35/65 ethylene glycol and

water solution, in a 9.5mm diameter pipe ......................................................................... 51
Figure 3.8: Friction factor comparison .............................................................................. 53
Figure 3.9: Load cell calibration curve ............................................................................. 54
Figure 3.10: External ampliﬁer to increase signal to noise ratio ..................................... 55
Figure 3.11: Effect of different viscosity ﬂuids on pressure sensor output ...................... 57
Figure 4.1: Steady laminar ﬂow at Reynolds number 1060 .............................................. 63

Figure 4.2: Laminar accelerating ﬂow; Reynolds number ranging from 578 at (=0 secs

to 1321at t: 12 secs with a maximum unsteadiness factor of 1.77 ................................. 64

Figure 4.3: Accelerating ﬂow: Reynolds number increases from 740 at t = 12 to 1517 at
t = 25 sec; with a maximum of unsteadiness factor of 6.85 .............................................. 65
Figure 4.4: Decelerating ﬂow; Reynolds number varies from 1405 at t=0 to 1055
t = 10 sec ; with a maximum of unsteadiness factor of —2.07 .......................................... 66
Figure 4.5: Decelerating ﬂow; Reynolds number decreases from 1311 at t: 13 to 818 at
t = 25 sec; maximum unsteadiness factor 37.8 ................................................................. 67
Figure 4.6: Initially steady, fast decelerating and slow accelerating laminar ﬂow;
Reynolds number 550 to 1133; maximum unsteadiness factor —58 to 40 ........................ 68
Figure 4.7: Initially steady then accelerated, decelerated rapidly above its mean value;
Reynolds number 1307 at t=0 to 123 at t=25 ; unsteadiness factors range from 10.35
to — 25.7 ............................................................................................................................. 69
Figure 4.8: Initially steady, accelerated and decelerated ﬂow; Reynolds number varied
between 1309 to 30; unsteadiness factor varied between 86 and —88 ............................... 70
Figure 4.9: Initially steady then accelerated ﬂow, Reynolds number changed from 1430 at
t = 2 to 191 l at I: 7; maximum unsteadiness factor 2.07 ................................................ 71
Figure 4.10: Initially steady ﬂow then suddenly decelerated ﬂow, Reynolds number
reduced from 1900 at t = 0 to 500 t: 5, maximum unsteadiness factor —25 ................... 72
Figure 4.11: Decelerated and accelerated randomly (ﬂuid-2); Reynolds number varied
from 2050 to 521, unsteadiness factor range 15.22 to —28. 1 8 .......................................... 73
Figure 4.12: Initially steady then suddenly accelerated ﬂow (ﬂuid 3); Reynolds number
increased from 620 at I: 1 to 1440 at t = 4; maximum unsteadiness factor 4.56 ............ 74
Figure 4.13: Initially steady followed by deceleration ﬂow (ﬂuid 3); Reynolds number

reduced from 1520 at t: l to 340 at I: 4; minimum unsteadiness factor —3.90. ............ 75

xi

Figure 5.1: Expected forms of weighting functions in turbulent ﬂow .............................. 80
Figure 5.2: Comparison of development of Reynolds shear stress for a linear ramp
acceleration transient for 5 seconds from Reynolds number of 7000 to 42500 in a 50.4
mm diameter pipe .............................................................................................................. 84
Figure 6.1: Steady ﬂow at a Reynolds number of 5250 .................................................... 96

Figure 6.2: Initially steady ﬂow that is then gradually accelerated, Reynolds number

increased from 4176 to 5128, maximum unsteadiness factor 0.25, Re, (0) = 293 ........... 98

Figure 6.3:- Initially steady ﬂow that is moderately accelerated; Reynolds number

increased from 3800 to 4700, maximum unsteadiness factor 0.53, Re, (0) = 160. .......... 99

Figure 6.4:- Initially steady then suddenly accelerated ﬂow, with Reynolds number

increased from 3750 to 5200, maximum unsteadiness factor 0.90, Re, (0) = 192 ......... 100

Figure 6.5:- Initially steady ﬂow which is then slowly decelerated, with the Reynolds
number reduced from 5255 to 3900, maximum unsteadiness factor 0.25, Re, (0) = 324 .....
......................................................................................................................................... 101
Figure 6.6:- Initially steady ﬂow which is then moderately decelerated, with its Reynolds
number reduced from 5196 to 3900, maximum unsteadiness factor 0.509, Re, (0) = 314...
......................................................................................................................................... 102
Figure 6.7:- Initially steady ﬂow which is then rapidly decelerated, with its Reynolds

number reduced from 5210 to 3500, maximum unsteadiness factor 0.72, Re, (0) = 318 .....

. ........................................................................................................................................ 103
Figure B. 1: Typical layout of the pressure sensor ........................................................... l l 1
Figure B.2: Four element bridge conﬁguration ............................................................... 112

xii

Figure B.3:
Figure B.4:
Figure C. 1:
Figure C.2:
Figure C.3:
Figure C.4:
Figure C.5:

Figure D.l:

Diaphragm of a piezo-resistive pressure sensor ........................................... 1 l3
Deﬂection of diaphragm under pressure ...................................................... l 14
Load cell elastic element with strain gages .................................................. 115
Gage positions in the Wheatstone bridge ..................................................... 120
Impulse response of the load cell ................................................................. 123
Amplitude response of the load cell ............................................................. 124
Ramp response of load cell .......................................................................... 125
Thermal boundary layer and related geometry ............................................ 127

xiii

English Symbols
Name

NOMENCLATURE

Meaning

local duct cross-sectional area
Constant

Diameter of pipe

Sensor diameter

Excitation voltage

Output voltage

Input voltage

Modulus of elasticity (equation (C. 1))

pressure gradient deﬁned as F (t) = lie

p 3x
Acceleration due to gravity
Weighting function deﬁned in equation 2.22
Heaviside step function
Water column height

Bessel ﬁinction of ﬁrst kind order zero
Bessel function of ﬁrst kind of order 1

ratio of Bessel functions, J0/ J1

xiv

 

(plow

‘t

2:

Bulk modulus of ﬂuid

Weighting Function deﬁned in equation 2.19

Thermal conductivity
Length of pipe

Mass ﬂow rate

Initial pressure
Pressure

Transient component of pressure

Axial load

Heat generated deﬁned as Q = (E 2 / lR)xt

Heat loss to the substrate

distance from pipe axis

Radius of the pipe

Sensor impedance

Frequency domain variable
Gauge factor

Time

Time averaged Reynolds Number

Reynolds number based on diameter

Friction velocity

Friction velocity
Transient component of the velocity

Initial velocity

XV

N/m2

W/mzK
rn

Kg/s
Pa

Pa

Pa

N

Joules

Joules
m
m

Ohms

UCL

Vx

Vr

W0)

I
UV

Greek S zmbols
a

Mean velocity

Center line mean velocity
Velocity

Velocity in axial direction

Velocity in radial direction
Velocity in circumferential direction

Weighting function

Reynolds shear stress

Dimensionless frequency as a = R(a)/v)'/2
Dimensionless time parameter 1' = w/R2
Shear Stress at the wall

Shear Stress at the wall

Density

Dynamic viscosity

Kinematic Viscosity

Poisson’s ratio

Instantaneous friction factor
Quasi-Steady friction factor

Quasi-steady friction factor

Oscillating frequency

xvi

N/m?’
N/m2
kg/m3
N-s/m2

,
m‘/s

Rad/s

[I
Acronyms
FFT

LDV
DAC

STC
PGIA
TZS

FSO

TSS

OS

8* = 8 + A8 Time dependent eddy viscosity
Quasi-steady eddy viscosity
Strain

Axial strain
Transverse strain

Oscillatory component of eddy viscosity

Dirac delta function

Inverse Laplace transform

Fast Fourier Transform

Laser Doppler Velocimeter

Data Acquisition and Control

System Timing Controller

Programmable Gain Instrument Ampliﬁer
Thermal Zero Shiﬁ

Full Scale Output

Thermal Sensitivity Shift

Oscillatory components

Time averaged component

Laplace transformed variable

xvii

Chapter 1 Introduction

1.1 Statement of the Problem

In transient duct ﬂows, there are presently no reliable techniques to measure
instantaneous ﬂow rates accurately. The quasi-steady approach is generally used in most
of the unsteady ﬂow measurements, provided that frequencies of pulsation or transients
are small. The common practice has been to relate steady and unsteady ﬂows using semi-
empirical relations. The need for instantaneous ﬂow rate measurement arises for the

following reasons:

i). Unsteady ﬂow with spatial pressure gradients ﬂuctuating around non-zero values
occur frequently in nature and engineering. Similarly transient ﬂows are important and

found in many situations. Accurate measuring techniques are needed for them.

ii). While transient ﬂows have been studied previously, the understanding gained has
not led to improved ﬂow metering devices and devices which use steady ﬂow
approximations are inaccurate. The recent series of laboratory tests carried out by Arasi
[1] indicates that oriﬁce or venturi meters give seriously inaccurate results for pulsating

ﬂows.

iii). As there is no generality in the time-dependence of unsteady ﬂow, it seems that a
transient ﬂow metering solution must be based on time-dependent solutions to the

unsteady ﬂuid mechanics equations, though this has not yet been done.

iv). There are quite a number of theoretical and experimental accounts of steady ﬂow

through ducts and pipes. However, unsteady ﬂow analysis is relatively limited. While

there is a well-developed theory concerning the energy loss in steady pipe ﬂow, very

limited knowledge exists about the nature of energy dissipation in unsteady ﬂow systems.

v). With the ever-increasing development of microelectronics, real time data is easier
to obtain and process. In order to improve the real time measurements of unsteady ﬂow
rates, there is a great demand for improving and elucidating ﬂow metering systems.
There is also a demand for a new real time ﬂow sensor with improved performance, in

terms of high accuracy and fast response under time varying conditions.

vi). In bio—ﬂuid engineering, many ﬂows are pulsating. Many engineering ﬂows are
also unsteady, predominantly in dynamic equipment such as reciprocating engines, turbo-
machinery, and control valves; these kind of equipment create pulsations with
frequencies that are a function of the rate of revolution and its harmonics. Similarly, in
the intake and exhaust manifold of an internal combustion engine, we encounter
accelerating, decelerating and reversing ﬂows. The ﬂuid ﬂow in hydraulic and pneumatic
lines and control systems often pulsate. Unsteady ﬂows are also encountered in
aerodynamics, wind engineering and liquid propellant rocket system. However, there are
no means available to predict the momentary ﬂow rates through these unsteady ﬂow

devices accurately.

Several methods for estimating the unsteady laminar ﬂow rates have been
proposed that uses ﬂow parameters that are functions of the instantaneous mean velocity
and acceleration. In sinusoidally varying ﬂows, the velocity amplitudes, the velocity

phase shiﬁs, and unsteady ﬂow parameters are often expressed or modeled as functions
of the dimensionless parameter a = R(a)/v)'/2 . When ﬂow varies arbitrarily with time,

the instantaneous values of ﬂow parameters can now be expressed as functions of the

instantaneous velocity, pressure gradient or shear stress and the weighted past changes of

these quantities, in which the weighting function depends on the dimensionless time
parameter r = Vt/R2 . In this chapter, a review of the related work in transient or unsteady

ﬂows, both in laminar and turbulent regimes is undertaken. The motivation for the

present study will be discussed in the end of this chapter.

1.2 Review of Laminar Flow Studies

Many investigators have studied pulsatile and oscillatory ﬂows theoretically, which
are two representative types of unsteady ﬂows. Analytical and numerical solutions of
continuity, momentum, and energy equations can be found in the laminar regime but are
limited as a result of mathematical complexities involved. Analytical solutions are
particularly difﬁcult to obtain because of the non-linearity of the governing equations when
solving in more than one dimension. However many researchers have attempted numerical

and approximate solutions of these equations when restricted to Newtonian ﬂuids.

1.2.1 Unsteady Laminar Flow-Theoretical Background

Early studies of the Navier-Stokes equations in parallel duct ﬂows focused on those
with speciﬁc time dependent pressure gradients. Sexl [2] developed an analytical solution
to the momentum equation for a Newtonian ﬂuid ﬂow in a horizontal circular pipe
subjected to a pressure gradient of the form dp/dx = Ce’”. He found an exact solution of
the Navier-Stokes equations, which resulted in an expression for the velocity distribution
throughout the cross section. Shortly thereafter, Szymanski [3] in 1932, using a similar
technique, developed an analytical solution to the momentum equation for a ﬂow subjected

to a step change in pressure gradient such that dp/dx = 0 for t S 0 and dp/dx = C fort 2 0 ,

where C is a constant. The time dependence of the axial velocity in both of the solutions

was assumed to be of the formu = F (r) em" , where F (r) was expressed in terms of Bessel

functions. For the non-periodic case, he gave another exact solution to Navier-Stokes

equation with the pressure gradient varying as a Heaveside unit step function.

Uchida [4] developed solutions which were variations on Sexl’s oscillatory pressure
gradient ﬂow, by superimposing the oscillations on a constant pressure gradient with
dp/dx=0 for ISO and dp/dx=Ce"" fort 20. He presented an exact solution of
pulsating laminar ﬂow superposed on the steady ﬂow in a circular pipe. His calculations
showed that in rapid pulsations, ﬂuid velocity at the centerline had a phase lag of 90°
behind the pressure gradients and its amplitude diminished with increasing frequency. In

addition his calculations showed that the phase of the sectional mean velocity is delayed

behind that of the pressure gradient while that of shearing stress is delayed less.

Womersley [5] considered the oscillatory ﬂow of a Newtonian ﬂuid studied by Sexl
and Szymanski from a biological perspective by using both rigid and elastic pipe walls.
Womersley also analyzed the phase relationship between the pressure gradient and
resulting velocity ﬁeld. Womersley showed that, in oscillatory ﬂow, the interaction
between viscous and inertial effects altered the velocity proﬁle so that it no longer
resembled the parabolic shape of a steady ﬂow. He noticed that the resistance and inertance

to ﬂow could be expressed as a function of a single non—dimensional parameter called the

Womersley parameter, a = R (co/v)”2 , where R is the span of the tube, (1) is the oscillation

frequency and v is kinematic viscosity, though the function would clearly depend on the

kind of unsteadiness encountered.

In summary, analytical solution for velocity proﬁles in some unsteady ﬂows with
particular time dependences have long been known. However, these studies have not led to
ways in which the instantaneous ﬂow rate in arbitrarily unsteady ﬂows might be expressed
as a simple function of pressure gradient, as might be useful for design of unsteady ﬂow

meters.

1.2.2 Development of Transient Response Theory and Experimental Veriﬁcations

The early investigations on unsteady ﬂuid ﬂow in straight pipes were based on
linearized two—dimensional Navier-Stokes equations. Their main purpose was either to
analyze the surging phenomena, “water hammer”, or to evaluate the energy loss due to
ﬂuid friction. Hence, most of the attention was given to the transient response of the ﬂuid
lines with frequency dependent friction. Therefore, in the 1950’s most research was
focused on the frequency response characteristics of the ﬂuid lines. It was later reported by
Leonhard [6] that it is theoretically possible to convert ﬁequency response results to

transient response results.

Based on constant viscosity Navier-Stokes equations, Brown [7] derived the

operator forms of the basic transmission parameter, called the “propagation operator” and
characteristic impedance. The propagation operator was expressed as F(s) = IJY (s)Z (s) ,

where l was the distance separating the two stations and Y(s) and 2(5) were the shunt
admittance and series impedance of the ﬂuid transmission line. Given the end conditions,
this propagation operator and the characteristic impedance could be easily applied to
determine either frequency or transient response. With the help of these operators, Brown
was able to describe the effect of frequency dependent viscosity on the propagation of

acoustic waves in ﬂuid transmission lines and provided an analysis for impulse and step

transient decay in lines. However, he neglected the variations of the pressure gradient and

velocity along the length of the line.

Brown and Nelson [8] presented further solutions of step responses to small signal
disturbances for rigid cylindrical lines containing Newtonian, incompressible liquid in
laminar ﬂow. They concluded that the pressure, ﬂow step inputs, and ﬂow outputs for
semi-inﬁnite lines containing constant viscosity ﬂuid can, with the principle of
superposition, be used to estimate the responses of a network of lines for any transient

input.

D’Souza and Oldenburger [9], using the basic Navier-Stokes equations, derived a
transfer matrix relating the dynamic pressure and cross-sectional average velocity for small
diameter hydraulic lines. They then compared the theoretical results with experimental data
obtained from a frequency response run on a half-inch inner diameter test tube. It was then
proposed that the dynamic response of small diameter lines could be predicted correctly by
including the effect of viscosity and compressibility while deriving the transfer function of
the pipe with sinusoidal input. These transfer functions related the pressure and cross-

sectional average velocity variables at two cross sections of the line.

Holmboe and Rouleau [10], performed two experiments to display the effect of
viscous shear on transients in liquid lines. These experiments included a propagation of a
short, rectangular pulse of pressure in a cylindrical tube with initially undisturbed liquid
and a pressure surge resulting from a step change of ﬂow in a pipe. The experimental data
conﬁrmed Brown’s [7] constant viscosity transmission parameter theoretical results. They
also noted that in the pulse propagation and water hammer experiments, the one-

dimensional analysis, utilizing a frequency dependent shear term, is valid and can

accurately predict the distortion and decay of transients in a constant diameter line ﬁlled

with viscous ﬂuid.

The analytical expressions of Brown, D’Souza and Oldenburger were very
complex. In these expressions frequency dependent effects were deduced from the linear
theory of transfer operators for pressure and ﬂow at two cross sections of the pipe. These
transfer operators required all input functions to be expressed as a summation of step
functions, and in addition all reﬂections had to be taken into account individually. This
became somewhat tedious for complicated systems consisting of series, branches or
parallel pipes. In the method of characteristics, all reﬂections could be considered
automatically and complicated systems could be handled without signiﬁcantly increasing

the difﬁculty.

Zielke [l 1], using the method of characteristics, included the effect of the time
varying velocity proﬁle in unsteady laminar ﬂow, deduced from a solution to the one
dimensional Navier-Stokes equation. He solved the problem of transient variation of the
ﬂow exactly relating the wall shear stress to the instantaneous mean velocity and to the
weighted past velocity changes. It constituted a friction law for fully developed pipe ﬂow
undergoing arbitrary unsteadiness from an initially steady state. Therefore exact
calculations of laminar wall friction in transient fully developed pipe ﬂows could be made,
provided that the history of the bulk ﬂow acceleration at all earlier times in the transients

were known. This solution took the form:

 

01‘

8U 3U

ro(r>=ro(o)+ [chew-3(0)] WW 02>

where W(t) is the weighting function, 2'0 is the wall shear stress, U the bulk ﬂow velocity,
[7' an inverse Laplace transform operator, R the pipe radius, and J, the ratio of Bessel

functions, J0 to Jl .

The practical signiﬁcance of Zeilke’s result was that momentary pressure gradients
depend on both local ﬂow conditions and all previous ones, in ﬂows of arbitrary
unsteadiness. Comparison of the theoretical results with experimental data of Holmboe and
Rouleau showed good agreement and an accurate prediction of the distortion effects. His

solution was expressed in terms of the dimensionless time 1:, given by

r=%t (1.3)

Zielke used two distinct series for approximating the inverse Laplace transform in equation

(1.1) or weighting function W(t) , one for t greater than ~ 0.02 and second for values of 1:

smaller than ~ 0.02. This method was also considered expensive in terms of computer
time, since it required excessive computer storage and computation time, and was
considered to be impractical for analysis of large ﬂuid systems.

Trikha [12] developed a method for simulating frequency dependent friction based
on Zeilke’s result, which was claimed to make the inclusion of the frequency dependent
friction in the method of characteristics almost as convenient and practical as the inclusion

of steady state friction. He also proposed an approximate weighting ﬁmction to replace the

inverse Laplace transform in Zielke’s expression, which offered the advantage of not
requiring the storage of mean velocity at all earlier times. Tsang et al. [13] used the FFT
method to predict the dynamic behavior of air transmission lines subjected to an impulse,
step or arbitrary excitation. The calculation procedure was considered relatively simple,

while the numerical results showed good agreement with experimental measurements.

Lisheng and Wylie [14] also proposed a numerical method to compute the
transients in complex systems with various frequency dependent factors. They
demonstrated the application of numerical schemes to solve problems of water hammer and
transients in pipes with frequency dependent ﬁ’iction. Comparisons of computed results
with the standard method of characteristics and with physical experiments of Holmboe and
Rouleau [8] showed good agreement. Both Trikha, and Lisheng and Wylie’s methods are
discretized implementations of Zielke’s result, in which the ‘memory’ of earlier transients

is truncated at a convenient point.

The major disadvantage of Zielke’s [11] solution is that it cannot be implemented
easily for large ﬂuid systems because of large memory requirements. In contrast Trikha’s
weighting function was an approximate expression, constructed by adding three
exponential terms and did not match well with the exact weighting function of Zielke for 1
less than 0.00005. Brereton [15], using Zielke’s technique, solved the reciprocal problem of
deducing the momentary ﬂow rate from wall friction or pressure gradient. The potential
advantage of his solution was its applicability for devising sensors for accurate
measurement of unsteady ﬂow. He also introduced the approximation to Zielke’s weighting
function to reduce the storage requirements, by retaining only the most recent ﬂow rate

history, and thus improving its practicality for predicting the transient behavior of large

ﬂuid systems. Using the numerical inverse Laplace transform techniques, his expressions
provided a means of expressing momentary wall friction in terms of bulk-ﬂow acceleration
history, like Zielke’s expressions and reciprocal expressions for ﬂow rates and cumulative
through ﬂow as functions of recent wall friction and pressure gradient histories. His
solution took the form:

I 2
, R 18p 13p 1' lap , ,
U d =-— —— — )-——0 K d 1.4
I0 (t) 1 8V ()(paxt +1:2(:J‘[paJc(t—t )pax( )] (t) t( )

where K (t) is a different weighting ﬁmction to W(t) in equation (1.2). These expressions

consisted of two terms: ﬁrst term describing the steady ﬂow behavior for viscous,
Newtonian ﬂuid in pipes; and a second term incorporating the unsteady ﬂow corrections

based on the pressure gradient or wall friction history.

1.2.3 Experimental Studies on Flow Rate, Friction Factor and Velocity Profile

Measurements under Transient Flow Conditions

Kataoka, Kawabata and Miki [16] studied the transient start up response of pipe
ﬂow to step inputs of constant ﬂow rate, using an electrochemical technique to measure the
velocity proﬁle with velocity electrodes. They noticed that the velocity proﬁle development
for the step input had a different trend from that of steady ﬂow development at different
positions in the entrance region of the pipe. The velocity proﬁles showed minirna at the
pipe centerline axis and a maximum in the intermediate region between the axis and the

wall as a result of non-uniformity of the acceleration in the central core.

Muto and Nakane [17] derived an expression for the velocity distribution in a
laminar pulsating ﬂow through a rigid circular tube. Transitions of velocity distribution

with respect to time were then measured by a ﬂow visualization technique using aluminum

10

powder suspended in the ﬂuid and photography of the path lines. The experimental data
were in good agreement with their theoretical results. They noticed that the velocity
distribution varied gradually from an initial rectangular like shape to a parabolic one at
steady state. In pulsating ﬂows, they observed that the velocity proﬁle developed in the
positive ﬂow direction at all times provided the ratio of the amplitude of the oscillating
component of the ﬂow to the steady component was near unity. If this ratio was increased

to 2, then the velocity proﬁles during transition developed in the negative direction as well.

Kurokawa and Morikawa [18] studied accelerated and decelerated ﬂows in a
circular pipe in both laminar and turbulent regimes. They noticed different patterns in the
formation of sectional velocity proﬁles and in transition to turbulence. For relatively large
accelerations, the ﬂow was characterized by the velocity proﬁle of a wide potential core
and a narrow boundary layer. In contrast, when deceleration was imposed, the variation of
the velocity proﬁle was comparatively small compared to the quasi-steady flow and this
variation became larger with increased deceleration. They also noted that the friction
coefﬁcient of an accelerated ﬂow was larger than that of the quasi-steady ﬂow, in the

laminar region, and that this trend was opposite in the turbulent regime.

Ohmi and Iguchi [19] investigated the relationship between frictional losses and

dimensionless sinusoidal frequency in terms of the instantaneous friction

factor A, (t) = 8?“. /pl72m . They characterized the ﬂow in three distinct ﬂow pattern regions

based on the dimensionless frequency parameter. These parameters were deﬁned as

/(|Apm.n /’) ¢. =(4 /D)/(IAp.....

and a”, = [(Apm /l) — A r . Here 1'“, is wall shear stress, D is pipe diameter, Ap is the

W.“.\‘ .n

 

 

 

 

 

¢I.n = ponwlumosm Tmusun /l) ’ am : [(Apmm /l) — lumxum ’

ll

pressure drop, wis the angular frequency, and n denotes quantities associated with the nth
harmonic in the ﬁnite Fourier expansion, while as denotes an oscillating component. These
regions were the quasi-steady, intermediate and inertia-dominant regions based on the

relative value of the non-dimensional frequency. They concluded that the instantaneous

ﬁ'iction factor it, (t) in oscillatory ﬂow was almost equal to the quasi—steady ﬁ‘iction factor

xi, in the quasi-steady region. In the intermediate region and inertia-dominant regions

xi (t) was always larger than A, in the acceleration zone of the oscillation cycle and it was

ll

larger than A, in the ﬁrst part of the deceleration zone of the cycle. However this trend was

reversed in the rest of the deceleration zone. Hershey and Song [20] found that above a
certain limiting frequency of oscillation, friction factors were greater than their values for
steady ﬂow. Consequently, energy losses would be greater than would be expected for a

quasi-steady model.

Yokota, Kim, and Nakano [21] proposed a new approach for estimating unsteady
ﬂow rates through pipe lines and other components in real time. They demonstrated that
unsteady ﬂow rate could be tenuously estimated by using pipeline dynamics containing
hydraulic oil and measured pressures at two distant locations along the pipeline. They
measured pressure and ﬂow rates at two distinct locations under the unsteady laminar oil
ﬂow condition, and results were then compared with the cylindrical choke type
instantaneous ﬂow meter. The results were in good agreement with each other. In this
approach, an empirical weighting function used for convolution was estimated
experimentally, by using the Inverse Fast Fourier Transform technique, because,

presumably they were unable to solve the Navier-Stokes equations.

12

1.3 Review of Turbulent Unsteady Flow Studies

The unsteady duct ﬂows are mostly classiﬁed in two main categories: periodic
oscillatory ﬂows and non periodic transient ﬂows. The pulsating periodic ﬂows have
captured much of the attention of researchers because of their practical importance and the
experimental ease with which such ﬂows can be generated. Generally, the investigations on
turbulent unsteady ﬂow have been focused on determining the effect of transients and

pulsations on the velocity proﬁles, skin friction and turbulent intensity.

1.3.1 Periodic Turbulent Unsteady Flows

Relatively little attention has been paid to improving the prediction of skin friction
in transient turbulent ﬂows. In many ﬂows transient effects are either neglected or allowed
for only approximately when representing skin friction. This is understandable because of
the resulting simplicity, but it is nevertheless misleading in some cases. Some experiments
and theoretical results on frictional losses in incompressible pulsating turbulent pipe ﬂows

are reported below.

Schultz-Grunow [26] generated pulsating ﬂow in a 11.5 feet long, 1.95 inch
diameter smooth pipe. The instantaneous velocity was varied between 3.8 feet per second
and 7.5 feet per second for a time period of oscillation of 2.5 seconds, and from 0.16 feet
per second to 7.1 feet per second for 30 seconds. He noticed a strong similarity between
velocity proﬁles of decelerated ﬂow in a uniform pipe and steady ﬂow in a divergent pipe.
A similar resemblance was seen for accelerated ﬂow in a straight pipe and steady ﬂow in a
convergent pipe. He concluded that the time averaged values of ﬂow resistance were close
to the steady state values. It should be noted that in transient ﬂows the instantaneous value

is of great importance.

l3

Daily, Hanky, Olive and Jordaun [27] carried out experiments in a 8.5 feet long,
and one-inch diameter pipe. They established a steady state ﬂow and then subjected it to a
slowly changing acceleration and then a deceleration. In the case of deceleration a rapid
velocity change occurred in the start and became more continuous after some time. The
range of acceleration and deceleration was between 7 and 80 ft/secz. They concluded that
during acceleration, friction is slightly higher than the equivalent steady state value, and
during deceleration friction is appreciably less than the equivalent steady state value. They
also concluded that the initial state from which the ﬂow is initiated affects the subsequent

ﬂow history.

Kinnse [28] carried out measurements of velocities and pressure gradients in a
pulsating turbulent pipe ﬂow by means of directionally sensitive LDV (Laser Doppler
Velocimeter) and inductive pressure transducers. His results suggest that the temporal
average over one period of the measured pressure gradient was always larger than the
spatial pressure gradient of a quasi-steady turbulent ﬂow of the same averaged Reynolds
number. Similarly, results for the Darcy-Weisbach friction factor were between 5 and 10 %
larger than in the corresponding steady ﬂows. He also concluded that pulsating turbulent
ﬂow can not be correctly predicted by existing turbulence models based on the similarity of

the turbulence structure of a stationary boundary layer ﬂow on a ﬂat plate.

Baired, Round and Cardenas [29] measured the frictional pressure drop over one
pulsation cycle in a sinusoidally oscillating pipe ﬂow. They showed that the experimental

values of the pressure gradient term Ap / L agreed well with the values calculated from the

integrated form of the one—dimensional momentum equation of the form:

14

2
ia_p+ﬂl.+ﬂ__=0

1.5
pax at 2D ( )

Here 21, was calculated using the Karman-Nikuradse equation with coefﬁcients adjusted to

ﬁt the data asl/ J1— =1.746ln(Rem Jﬁt— )-9.195. They concluded that the quasi-steady

state model was valid within their experimental range. The limiting value of dimensionless

frequency in this case was a, 2 0.025 Rem where Re, is a time averaged Reynolds number.

Ohmi and Iguchi [30] estimated ﬁiction factors in a pulsating pipe ﬂow from ﬂow

pattern diagrams and compared them with experimental data. They described the

instantaneous friction factor A, (t) =8T'w/pl-J-2m and the time averaged ﬁiction factor

xi,‘m=(8/p(7mm3T) gaﬁmdt as a function of aI/Rei/4 wherea=R((o/v)l/2. The

(0’

instantaneous friction factor 2,,(1) and the quasi-steady ﬁiction factor ,1, were almost

3/4

(0’

equal in the quasi-steady region denoted by (r/Reif4 S 0.145. With an increase in a/ Re
1, (t)became smaller than ’1, in the ﬁrst part of the acceleration zone and larger in the rest
of the acceleration zone. On the other hand, A, (I) became larger than A, in the ﬁrst part
and 2., (t)became smaller than ’1, in the rest of the deceleration zone. In addition, the time

averaged friction factor xi was greater than xi, and it increased with a/ReZ“ . It was

also concluded that pulsatile turbulent ﬂow structure in a pipe differs widely with changing
characteristic ﬂow parameters, i.e. time averaged Reynolds number, amplitude of

oscillation and the dimensionless frequency.

15

Ohmi, Kyomen, and Usui [31] later calculated the velocity distribution in a pulsatile
incompressible turbulent pipe ﬂow by using a time—dependent eddy viscosity 8' to model

the Reynolds stresses. They noticed that 8" varied with friction velocity u‘when

referenced to the instantaneous axial velocity proﬁle in the pipe. They modeled time-

dependent eddy viscosity as e" = e + A8 , where 8 is the quasi-steady component and A8
the oscillatory component. The quantity A8 was considered to arise from the ﬁictional
velocity change corresponding to the instantaneous velocity distribution. They concluded
that there was a good agreement between numerical and experimental results for the time

dependent eddy viscosity in the low and medium dimensionless frequency range.

Kita, Adachi and Hirose [32] also proposed a time-dependent eddy viscosity model

to analyze oscillating turbulent ﬂows in straight pipes. The variation of 8 was modeled on
the basis of similarity with values of e/u'R in steady ﬂow. The cross section of the pipe

was divided into ﬁve layers. The predictions by this model were then compared with
experiments. The comparison between calculation and measurements over a wide range of

Reynolds number and frequency showed good agreement overall.

Brown, Margolis and Shah [33] applied the E-distribution models of
Taylor&Prandtl and Von Karman by dividing the pipe cross-section into three and four
regions. They studied the response of pulsatile turbulent ﬂows to small amplitude
oscillations at acoustic frequencies between 50 Hz and 3000 Hz. At these high frequencies,
the observed attenuation agreed with calculations using a time—independent viscosity
proﬁle. Based on their results, it was concluded that the ﬂow pattern and turbulence did not
have time to adjust to rapid ﬂuctuations in velocity, and at these high frequencies, the

behavior cannot be considered quasi-steady. Based on these ﬁndings they classiﬁed the

16

pulsatile turbulent ﬂow into three kinds with respect to the dimensionless frequency of

oscillation: low, intermediate and high frequency regions with the limit between low and

0.84
(a ’

intermediate frequency regions at a, = 0.014Re and between intermediate and high

frequency regions at a, = 0.00025 Rel," .

1.3.2 Non Periodic Turbulent Unsteady Flows

A number of researchers have studied unsteady turbulent ﬂows which are non-
periodic, with particular emphasis on measurements of wall shear stress. Measurements of
the unsteady wall shear stress are usually based on an indirect method. Shuy [22] canied
out measurements of unsteady wall shear stress in a smooth pipe in accelerating and
decelerating turbulent ﬂows, using two different approaches simultaneously. He measured
wall shear stress directly using a shear tube, and deduced it indirectly from the measured
pressure gradient which is only accurate in quasi-steady ﬂows. He concluded that in slowly
varying ﬂows, both direct and indirect measurements were generally close to their quasi-
steady values. In the case of rapidly accelerating ﬂows, the wall shear deviated increasingly
from quasi-steady values as the rate of acceleration was increased and it was consistently
lower than the equivalent quasi-steady value. During rapidly decelerating ﬂows, unsteady
wall shear stress was consistently higher than the quasi steady values computed from the

Karman- Nikuradse equation for steady smooth pipe ﬂow.

Lefebvre and White [23] conducted accelerating ﬂow experiments in an unsteady
ﬂow loop facility using LDV measurements with ﬂush mounted pressure sensors and hot
ﬁlm wall shear stress sensors. During the experiments, the starting Reynolds number was
about 50,000 at a mean velocity of l m/sec. They noticed that during acceleration wall

shear stress dropped by about 40% at the beginning of the acceleration but then returned

17

almost to its quasi-steady value. The turbulence intensity also dropped about 50 % and then
returned to its quasi-steady value. They also noticed that the shape of the turbulence and
wall shear stress time history curves were strikingly similar. This phenomenon was

attributed to the moderate stabilizing effect of the acceleration on the entire ﬂow ﬁeld.

Jackson and He [24] conducted ramp type transient turbulent ﬂow experiments in a
pipe using water as the working ﬂuid. They noticed that after the imposition of accelerating
transients, turbulent shear stresses remain completely unchanged during the early period of
the transients. They then slowly started to relax towards their quasi-steady ﬂow values.
They also observed that the response of the turbulence to imposed excursions of ﬂow rate
was characterized by three distinct time delays: a delay in response of turbulence
production, a delay in turbulence energy redistribution among its three components and a

delay associated with the propagation/diffusion of turbulence radially. They suggested a
time scale of v/ U ,2 for the delay in turbulence production with its value depending on the

initial ﬂow conditions and independent of the nature of the transients. They concluded that
as a result of these delays turbulence intensity is attenuated in accelerating ﬂows and is
increased in decelerating ﬂows. The same trend was earlier observed by Gerrard [25] in
which his ﬁndings revealed that the velocity proﬁles in pulsating turbulent ﬂow change
because of the changes in the turbulent structure of the ﬂow i.e. acceleration reducing the

turbulent intensity and deceleration increasing it.

1.4 Unsteady Transitional and Other Related Flows

There are a number of other studies that are signiﬁcant to understanding unsteady
ﬂow that have some important general conclusions, discussed here. It is well known that a

strong convective acceleration due to a highly favorable free stream pressure gradient has a

18

stabilizing effect on the steady boundary layer ﬂow. In laminar ﬂow, acceleration is known
to delay the transition to turbulence. In turbulent ﬂow, acceleration is believed to thicken
the viscous sublayer, damp the inner layer turbulence, and, if sufficiently strong, induce a
reversion toward laminar ﬂow conditions. Research has been carried out in these areas, and

is described below.

Gilbrech and Combs [34] canied out studies of unsteady pipe ﬂow, in which they
deﬁned a Reynolds number for pulsating ﬂow, relating it to the time average of the cross-
sectional mean velocity. The critical value of the Reynolds number was interpreted as the
value at which the growth rate of turbulent puffs was zero. They suggested that the critical
Reynolds number in unsteady ﬂow is a function of the ratio of the amplitude of the periodic

components of velocity and the time average velocity, and also a function of the
dimensionless frequencya = R(a)/v)'/2. They concluded that the critical value of the

Reynolds number in pulsating ﬂow was greater than that in steady ﬂow and that it
increased with increasing amplitude of pulsation, until it reaches a maximum and decreases

with increasing values of dimensionless frequency.

Lefebvere and White [35] carried out experiments to study the transition to
turbulence in pipe ﬂows started from rest with a linear increase in mean velocity. With the
increase of acceleration rate, the transition Reynolds number Rep increased monotonically
from 2x10S to 5x105. They ﬁrrther noticed that the transition time is nearly independent of
pipe diameter. They also concluded that pipe start up ﬂow has an almost unique transition
time, which is independent of spatial position for a uniform acceleration. However their
results were not valid for non-constant accelerations. Leutheusser and Lan [36] carried out

experiments on the start up of ﬂow in a pipe. They showed that the transition Reynolds

19

number was delayed. Katako, Kawabata and Miki [37] concluded from their start up ﬂow
experiments that transition occurred only after the acceleration had greatly decreased. They
also noticed that the time required for transition from laminar to turbulent ﬂow reduced

with increasing in Reynolds number.

Van der Sande [38] carried out experiments by opening a valve to generate a ﬂow
with slowly decreasing acceleration in a 5 cm diameter water pipe. He noticed that
transition to turbulence occurred at a critical Reynolds number of Rep =58000, far larger

than the value of 2000 normally associated with steady transition.

Moss [39] carried out experiments in accelerating pipe ﬂow at Reynolds number
Rep S 20000. He did not give any correlation for transition Reynolds number but observed
two different modes of transition: 1) a turbulent “slug” arising naturally in the boundary
layer; and 2) a turbulent “puff’ propagating downstream from the pipe inlet. Rarnaprian
and Tu [40] canied out LDV measurements in pulsatile oil ﬂow in a smooth circular pipe
at a mean Reynolds number of about 2100. The ﬂow was subjected to a nominally
sinusoidal ﬂow modulation at frequencies varying from 0.05 to 1.75 Hz. They noticed that
ﬂow oscillations increased the critical Reynolds number of a puff type transitional pipe
ﬂow. They also noticed that under certain conditions, the transitional pipe ﬂow might be

laminarized by periodic pulsation.

Gerrard [25] performed experimental work based on cine photography in pulsatile
ﬂow at Re,a = 3700. He observed that ﬂow is inhibited during the acceleration part of the
pulsation cycle and enhanced during the deceleration part. Hinu, Sawamoto and Takasu
[41] also conducted experiments on transition to turbulence, in a purely oscillatory pipe

ﬂow. The Reynolds number was varied from 105 to 5830. They concluded that at a

20

Reynolds number of 5830 the flow was only partially turbulent. In that partially turbulent
ﬂow, they observed that turbulence is generated suddenly in the decelerating phase when
the proﬁle of the velocity distribution changes drastically. In the acceleration phase, the

ﬂow recovers to a laminar state.

Some more recent information on turbulent ﬂows has come from direct numerical
simulations of the Navier-Stokes equations for ﬂows in simple geometries. Statistics from a
computation of steady turbulent channel ﬂow have been presented by Kim, Moin and
Moser [48]. Although this thesis is focused on the unsteady duct ﬂows, one steady channel
ﬂow numerical study is worth mentioning as it deals with non local momentum transfer.
Sandam [49] proposed a predictive model for velocity proﬁles in a turbulent channel ﬂow.

a(u'v')

3y

 

In this particular case, individual terms for model ejection, transferring low

momentum ﬂuid away from the wall and, sweep, carrying high momentum ﬂuid towards
the wall, with a third term modeling the ﬂuid behavior in the outer region. This model
represents transfer of momentum as a non local process depending on the conditions near
the wall. This model is claimed to show good agreement with experimental data for skin

friction, center line velocity and shape factor.

From the above review, it is concluded that the problem of predicting unsteady
wall shear or friction factors in fully developed turbulent ﬂow is still unsolved, but of
considerable importance. The above literature review indicates that, even in turbulent
ﬂows, wall shear stress or skin ﬁiction is dependent upon the history of the ﬂow.
Therefore, it may be useful to try to extend the approach proposed by Zielke and Brereton

for laminar ﬂow to turbulent ﬂow. However turbulent weighting functions for the velocity

21

or pressure gradient changes would have to be developed in place of laminar ones. Due to
the strong radial diffusion effects of eddies, the time history of the velocity and pressure
gradient changes are much shorter for turbulent ﬂow than for laminar ﬂow. This might
make the proposed approaches of Zielke and Brereton even more potentially useﬁrl, since it

could reduce the computation time further relative to laminar ﬂow calculations.

1.5 Motivation and Objectives of the Present Study

Today’s engineering analyses of transient ﬂows are based on quasi-steady
approximations in which ﬂow characteristics are assumed to be similar to those of steady
pipe ﬂow at the same instantaneous Reynolds number. Existing studies have shown that
such an assumption is invalid particularly during decelerations or when a pulsation
frequency is high. These studies do not give any viable approach for measuring the
instantaneous unsteady ﬂow rates using the ﬂuid dynamics equations. Hence, one must use
truly unsteady approaches for analyzing, measuring and inferring results for unsteady ﬂow

problems.

An exact theoretical solution (Brereton [15]) exists for the instantaneous ﬂow rate
as a ﬁmction of pressure drop history, in laminar ﬂows, but it has not been tested
practically. Hence it is important to check it in controlled unsteady laminar ﬂow
experiments. This theoretical solution might potentially be extended to prediction of
turbulent ﬂows if it could be modiﬁed/tuned against turbulent unsteady-ﬂow data.
However, while there have been numerous unsteady-ﬂow studies of transient turbulent
ﬂows, hardly any include ﬂow rate vs pressure drop data. Therefore it is important to carry
out controlled unsteady turbulent ﬂow experiments to create benchmark data against which

turbulent variations in the laminar ﬂow-rate-to—pressure drop result can be explored.

22

With these motivations an experimental study was carried out to explore the
applicability and limitations of these ﬂow-rate-to-pressure drop expressions in various
incompressible Newtonian ﬂuids for laminar ﬂows. A subsequent series of unsteady
turbulent-ﬂow experiments was then carried out to create target data, against which

unsteady turbulent models for transient duct ﬂows were then compared.

23

Chapter 2 Theoretical Background For Laminar Newtonian Flow

2.1 Laminar Flow Theory

Laminar ﬂow theory is concerned with solution of the continuity and linear
momentum equations together with a ﬂuid stress to rate of strain model such as the
Newtonian “Law of Viscosity”. While many steady ﬂow results relating ﬂow rate to
pressure gradient are well known, few unsteady ﬂow results have been found. For several
speciﬁc cases such as pressure varying sinusoidally, as a Heaviside unit step function or
the Dirac-delta function, the velocity distribution has been calculated and relevant
literature is cited in detail in Chapter 1. In this chapter, an exact solution is derived for the
general case of pressure gradient varying arbitrarily with time. The same approach can be
used to infer the instantaneous shear stress from instantaneous ﬂow rate changes. These
solutions relate the mean velocity to the instantaneous pressure gradient and past pressure
gradient changes. This approach yields a practical approach for measuring unsteady ﬂow
rates instantaneously.

2.1.1 N avier-Stokes Equations for Fully Developed Pipe Flow, Simplifying

Assumptions and Boundary Conditions.

It is convenient to use cylindrical coordinates whose x-axis is identiﬁed with the
centerline of the pipe as shown in the Figure 2.1. Let r be the coordinate in the radial

direction and let t denote time. The following simplifying assumptions are made:
i) The pipe contains an incompressible ﬂuid and is rigid.

ii) The temperature variations are small enough that the ﬂuid viscosity may be

considered constant.

24

x=0

x=L

 

 

 

 

 

\\\\\\\\\\\\\\ ‘R%\\§\\\\x\\ \\\. . 3‘? j~\~

Laminar Boundary
Layer

 

l¢ . 7 Entrance Region
Z/Z/ ///////// ’Z Z///”/Z/”///”{f/{{” ZZZ ’V/ZZZZ/Zn // Z

k ’1 \\
l _ .) \\\

)- \\\. ,

Potential Core \K‘NKMM

F~-",l ' ___’__,,..---';:""'"‘

A ,, f//
i‘ " ,///Ii /
- a ~ A

\ {\\\\\\\\\ \\\\\ \\\\\\

M
TI

'///////// ////, ////, //////
///// ////

.77///

 

 

\\\\\\\\\\\\\\\\\‘\‘ \\\\\\\\
Fully Developed
Flow

Figure 2.1: Laminar velocity proﬁle in pipe ﬂow

The incompressible, constant viscosity equation of motion in the x direction can be

written as:

 

 

 

 

 

 

 

 

 

[av,+v8i vii), v 8v,)__ 3p+ + 13(r av __,2)+_l_a V,+dzv, (21)
p 3: 'ar raa *ax ax pg ” Iar 8r r2362 81:2 '
and the equation of motion in the r-direction as
(an, .339. Maia.” 3v.)
p at ' 3r r 819 r : 8):
(2.2)
—_QE_+ +p— l 1(r av)- Vr LL__2_§KQ a_Z__vx
8r pg, rar or r2 r2 892 r2 30 3x
Similarly, the continuity equation in cylindrical coordinates is written as
130v ) 1309) a(v)
_____+_ + x :0 2.3
r 3r r 36 3x ( )

The following further simplifying assumptions are made:

i)

The velocity and the change of all dependent variables in the 0 (circumferential)

direction are negligible due to rotational symmetry i.e. v9 =0.

25

ii) The ﬂow is laminar. This limits the applicability of the solution to Reynolds

number of about 2100 or less.

iii) The ﬂow is fully developed i.e.8(v, )/8x = 0, so pressure is only a function of x
and t.

iv) v,= 0 at r = R so from continuity 8v, /8r = 0, therefore v,= 0 everywhere.

v) Equation (2.2) then reduces to the hydrostatic pressure equation

0=-3p/3r+pg,

Therefore, for unsteady laminar, axisymmetric ﬂows of Newtonian ﬂuids that are
fully developed in space, the continuity equation and the constant property Navier-Stokes

equation for streamwise linear momentum may then be written as

an

—=0 2.4

8x ( )
Bu 10p Va Bu
—=——— —— — 2.5
at pax+rar[rar) ( )

where v, is replaced by u, with the companion boundary conditions that u = 0 at r = R
and u is ﬁnite at or symmetric about r = 0. The most common use of these equations is
for ﬂows of Newtonians ﬂuids of constant density and viscosity, or at least r- invariant
viscosity. They are equally applicable to variable-density incompressible ﬂows in which

ﬂuid particles maintain their density along their path so thatDp/Dt=0. The same

approach can be extended to the ducts of different geometries.

2.1.2 Solution Procedure by Laplace Transformation

In transient ﬂows it is convenient to decompose the velocity and pressure ﬁelds into

the sum of their values at some initial time t = 0 and subsequent transient changes, i.e.

26

u(x,t)=u(x,0) + u'(x,t) (2.6)

p(x,t)= p(x,0) + p'(x,t) (2.7)

The linearity of the equation (2.5) then ensures that the initial ﬂow ﬁeld (uo, p0) and the
transient component of change in the ﬂow from its initial state are described by equations

of the same form:

 

Buo 1 ap, V a[ duo)
—=——— —— — 2.8
at pax+rarrar ( )
and
21:__l_8_p_+_v__a_(rau) (2.9)
at p 8x rar or

with the same boundary conditions applied to u0 and u'as to u.

Since, in fully developed ﬂows, ap/ax is only a function oft, it is convenient to re-

express equation (2.5) as

dzu lau lau 18p 1
_ _____=——=—F 2.10
dr2+rdr Vat vpax v m ( )

The linearity and constant coefficients of equation (2.10) permit Laplace transformation,

which yields the transformed ordinary differential equation,

dzﬁ 1dr? 3 ..+u0_

1 .
27 rd. v V 3”“) ‘2'“)

and equivalent equations for the initial and transient counterparts. In this equation,ri is the

one sided Laplace transform of u, F the transform of F, and u0 (r) is the velocity ﬁeld at

27

t: 0. The pressure gradient term (l/p)6p/ax = F (t)is assumed continuous, or at least

piecewise continuous, while s is a complex variable, the real part of which is positive. In

the general theory of Laplace transforms, F (t)can be at most of exponential order as

t—>oo, in which case there is range of values of the real part of s over which
I: F (t) e“"’dt is convergent. However since the behavior of the original function as

s —-> co and s —-> 0 is related to the character of the original ﬁmction as t —> 0 and t —> oo ,

respectively, and we anticipate solution to the physical problems to exist for 0 S t < oo ,
we expect that I: F (t) e“”dt is convergent for all possible values of the real part of s.
The transformed boundary conditions are r? = 17' = 0 at r = R and both 1? and 12' are

also ﬁnite at or symmetric about r = 0. For the case of the transient Laplace transformed

velocity ﬁeld 12'(r,s), for which u3(r)=0as a consequence of equation (2.6), the

ordinary differential equation (2.1 l), and its boundary conditions may be solved to give

a’(r,s)=-1: J°(i“(S/V)r)—1 (2.12)
S J0(i (s/V)R)

 

where J0 denotes the Bessel function of the ﬁrst kind, of order zero. Once the Laplace

transformed velocity has been found, expressions for changes in ﬂow rate, changes in
pressure gradients, and relationships between them may be found algebraically in the
Laplace transformed domain, then inverse transformed back to the physical domain.
Hence we can derive expressions for transient ﬂow rates in pipes in terms of pressure

gradients.

28

Changes in the mean velocity, and hence ﬂow rate, in the Laplace transform

domain may be found by integrating the transform of the momentary velocity field over

the cross-sectional area of the duct. Using 0(9) to denote the transform of the transient

part of the mean velocity U '( t) , we deﬁne U'(s) as

. 2 R
U'(s) =-R—2 I0 rii'dr (2.13)

Substitution of ri' from equation (2.12) then leads to

-, F’ 2
U =— —1 (2.14)
(S) s J.(i (s/V)R)

 

with J, (z) = 2210(z)/J|(z).

Unsteady pressure gradient to ﬂow rate relationships may be formed using the
expressions for changes in the ﬂow rate, and pressure gradient, subject to a constraint on
the existence of functions as Laplace transforms, and the corresponding existence of their
inverse transforms. The constraint is the necessary property of Laplace transforms that
the transformed functions F (s)must vanish when 3 tends through real values to +00
(Watson [45]).

The transformed pressure gradient term F ' from (2.14) can be expressed in terms

A

of U'as

1 3,3 ., _
337(8) F (S)‘ 2 _, (2.15)

 

 

 

 

29

or equation (2.14) can be rewritten as

 

ms): 3 F'(s) (2.16)

 

 

 

so, using C' ( )to denote inverse Laplace transformation,

( 2 \

. —1
.7. (2 (3:01?) .. F’(t) (2.17)

 

 

 

 

With the symbol * used to denote the convolution operation, the mean velocity is related to
pressure gradient in convolution integral form as
I
U’(t)=k(t)*F’(t)= J k(v)F’(t-v)dv (2.18)
0

with

k(t)=[' l 2 —1 (2.19)

S ,,7,(i (s/V)R)

 

Thus the transient spatial mean velocity or ﬂow rate can be expressed in terms of the time
history of the transient pressure gradient, weighted by a convolution function that
incorporates the effect of the Navier-Stokes equation and its boundary conditions. An
equivalent reciprocal relationship for the transient pressure gradient as a convolution
integral of the transient average velocity equation (2.15) can not be found because the

response of the transient pressure gradient to unsteadiness always leads that of the velocity

30

ﬁeld. Therefore in this unsteady wall bounded ﬂow, one can represent mean ﬂow as a
functional of the pressure gradient, but not vice versa.

Although the transient pressure gradient cannot be expressed as a ﬁrnctional of
transient ﬂow rate history, it is interesting to consider how a realizable relationship for
transient pressure gradient might be formed. The function given in equation (2.15)

S/((2/j' (,- MR )_ 1)) has no inverse Laplace transform, though it would if it were

divided by s or some higher power of s. If equation (2.15) is rewritten as

 

 

 

 

 

 

 

 

 

/ \
11p _ n _ 1 ~
pax(s)—F(S)— 2 -1 SU (S) (2.20)
J,(i (s/V)R) /
then
f \
. l
E 2
—l
[~7I(i (S/V)R) )
existsand
. ~, _dU' ,
e [sU(s)]— dt +u,5(z),

where 6 ( )is the Dirac delta function, since the Laplace transform of the time derivative

of the average velocity is .C [a U /at] = sU ’(s ) — U 0 . From equation (2.20) the transient

pressure gradient is therefore related to the transient spatially averaged velocity (for which

U5 = 0 in the earlier notation) as

31

 

 

F’(t)=h(t)*dU’(t)= Juh(v)dU,(t—v)dv (2.21)

 

 

 

 

dt 0 dr
with
r \
h (t)= c“ 21 1 (2.22)
K ‘7' om]? ) — J

In this way, the transient pressure gradient is expressed as a functional of the mean ﬂow
acceleration dU/dt .

The extension of these results to form relationships involving time-integrated

quantities is also straightforward. For example, the cumulative volume ﬂow through a duct

during a transient is I; U '(t) dt , multiplied by the duct area. From the Laplace transform

 

 

theory,
, , 1 , I], s
£(IOU (v))=—£{U = ( )
s s
so dividing (2.14) by s,
U’ "
_(_S) = f.- 2 —1 (2.23)

or, after inverse Laplace transformation,
1 t
J‘U'(v)dv = K(t)*F'(t) = I K(v)F'(t—v)dv

0 0

with

32

i2 2 -1 (2.24)

3 .7, (i (s/V)R)

 

K(t)=['

The results developed in the preceding paragraphs can be presented more
completely by reintroducing absolute velocities and pressure gradients in favor of their
transient components. From equation(2.12), the mean velocity to pressure gradient results

for pipe ﬂow can be derived and expressed as

 

 

2 I
—l d 2.25
J] (I‘IS/VR) ] t ( )

OI’

 

t 2 2
U =U0 —— — ————0 t —

 

2 I
-l d 2.26
1.7] (i‘is/VR) ] t ( )

We can re express equation (2.26) as

 

’ e ’ , ’12.._12 1_L 2 _ 1
loUl’l‘”‘i.U(°"”+i.lpax(’ ’1 ,,,<o>)c .2 art-W) 1]“

 

 

(2.27)
or in terms of pressure gradient to ﬂow rate relationship
( )
1 8p 1 3p {(BU , 8U ] 1 l ,
—— =—— 0 — t- —— 0 1C- a’t 2.28
p&0)p&(HL)m(t)aﬂ) 2 4 < >
KJ,(i (s/V)R)

 

 

33

2.1.3 Pressure Gradient to Flow Rate Results.

The pressure gradient to ﬂow rate relationships can be related more readily to well

known steady ﬂow results if expressed as the sum of a steady state and a transient

contribution. The long time or quasi-steady and transient components can be separated by

splitting the inverse Laplace transform as C'[ ]=.C'[ ],_...+(E'[ ]—E'[ L“).

Noting that, as t —) oo

 

 

 

 

 

 

and
r \
h(t)=[' 1:” +8
—1
Kjl(i (S/V)R) )

 

 

When ﬂow is steady prior to a transient, these results simplify to

’dU

J07”—

V

i‘ip _
R2

_8vU(t) +
p 8x

R2

 

(t)—

(t—t’)h(t’)dt’

34

(2.29)

(2.31)

(2.32)

These results illustrate how the unsteady pressure gradient to ﬂow rate expressions departs
from the quasi-steady Hagen-Poiseuille relationship —(l/p)8p/6x = 8VU / R 2 for pipe ﬂow.
By expanding the argument of the inverse Laplace transform for large 5, it can be shown
that the functions tend to minus inﬁnity as t"/2 as the time origin approaches zero.

2.1.4 Flow Rate to Pressure Gradient Results

The relationship reciprocal to the previous equations, i.e. expressing ﬂow rates in
terms of pressure gradient history, can now be devised. The relationships thus found would

be the unsteady counterparts to the Hagen—Poiseuille one. Noting that

E'i 2 —1=R2

s2 J. (I. (SMR) "E17 (2.33)

 

as t —) oo , equation (2.26) can be rewritten as

u(.)=u(0) R2 [193(r)-ia—”(0)-t192—”(0)]

 

"Ev- p ax p ax p axat
7 1 32 32 (2.34)
R‘- 1 p I 1 p I I
+— —— t—t ——-— O K t dt
V Io[paxat( ) paxat( )] ( )
where
v 2 l
K(t)=[' 2 —1 +—
st 171(1 (s/V)R) 8 (2.35)
For the case of steady ﬂow prior to transient, this expression simpliﬁes to
R2 8p R2 tazp , , ,
U I =—-—— t +—- — t—t Kt dt 2.36
() 8pV Bx() v I03x8t( ) ( ) ( )

35

These results express the departure of the unsteady mean velocity from its steady
state value of U = —(R2/8pv)Bp/8x for pipe ﬂow, providing a technique to deduce the
momentary ﬂow rate from the pressure gradient history. They are entirely consistent with

equation (2.25), but can be deduced more conveniently from equation (2.26). The inverse

Laplace transform that determines the weighting function K is shown as a function of

dimensionless time Vt/R2 in Figure 2.1.

 

 

 

 

 

 

 

 

0015 I I I I l I I I I l I I I I I I I I I I I I I I I I I I I :
0.14 .3
1
013 0.12 ﬁiqﬁsﬁvmq... ,........,_: _:
0.11 -: :
0.12 017' "I “Z
1 ...;. a
0.11 a 003 L \‘x -_ '1
0 1 a Aomf- :1 -: E
. i 0.06:- ‘ ‘ - '—
0 09 0.05:L \\‘ ,“3 j E
- \ ‘ x .1 -
' '. ’1' .1 d
. ." \ \ ___ '1 ‘1
E 0.08 002- 'l , 3 \ \Q ‘2‘, “3 .3 ?
M 0.01%- \‘:"-\_‘ -i j
0'07 0" r 1 r 1 l 1 1 1 _ j ‘ I .2:
1 1 2 3 4 5 11 3
0.06 ““1"" 1
:1
0.05 .3
0.04 ‘2
0.03 .5
0.02 _:
0.01 _:
o l l l l l I L l J I l l l I | l I J l l I l l L :

0 0.25 0.5 0.75 1 1.25 1.5

VIIR

Figure 2.2: Weighting function plotted against dimensionless time; p, = 3.34E-3 N-sec/mz;
p, = 2.52E-3 N-sec/mz; p3 = 3.09E-3 N-sec/mz; 2' = Vt/R2

36

The cumulative through ﬂow may be found by manipulating equation (2.27) in the

same way to yield

J‘IUU) dt’= JHU(O) dt’-R—2 [Tl-QUQ-iip-(Oﬂ dt’

02 1 0 (2.37)
R lap , 18p , ,
— —-—- — ———0 K d
+, (1,3,0 .) M1 )] (or

or for an initially steady ﬂow,

1 2 t 2 l
, R 13p , , R 18p , 13p , ,
U d =-— —— d — —— — -——0 K d 2.38
[0 (1)1 8, 0p,,111r+,]0[p,,1~) Mn] 1:): < >

This relation also takes the form of a quasi-steady term supplemented by a term describing
departure from quasi-steady behavior on account of unsteady effects, with weighting
function K the same function given earlier in (2.35). The cumulative through ﬂow in ducts
in which pressure gradient became more adverse over time will exceed that anticipated
using quasi-steady estimates.

In equation (2.38), the quasi-steady term, the ﬁrst term on the right hand side, is
supplemented by an additional integral, which describes the departure from quasi-steady

behavior on account of the unsteady effects. We can re-express equation (2.38), by

replacing -Bp/8x by Ap/L, as

t 2 I 2 l
I mad/=5— iAp-(t’)dt'-—R— lip—ryii‘im) K(t’)dt’(2.39)
0 8V Op L V 0 p L p L
,4 BI 32

In general, there are three different kinds of ﬂow according to the relative magnitudes of

terms B1 and B2 in (2.39):

37

BI=B2: This case corresponds to steady ﬂow. In this case the pressure gradient remains
unchanged during all times resulting in term B. canceling term B2. Hence there is no

unsteady correction in the accumulated mass. Equation 2.39 would then yield the same

results as the Hagen-Poseuille equation U = —(R2/8pV)Bp/ax.

B1>B2: This case corresponds to accelerating ﬂow since the instantaneous
pressure gradient given by B; would be greater than the pressure gradient at the start of
the transient, B2. If used without an unsteady term, the quasi-steady term A, would
overestimate the pressure gradient. This overestimation by term A is therefore
compensated for in unsteady ﬂow by a negative correction term (B1+ Bz).

B1<B2: This case corresponds to decelerating ﬂow since the transient pressure
gradient would be smaller than that at its start. The quasi-steady term would therefore,
underestimate the accumulated mass, which would then be compensated for by a positive

unsteady correction term (Bl+ B2).

The departure from the quasi-steady term at any instant can be estimated by comparing

the following two terms(Brereton [15])

ldu

a“);

V

1

Re

 

(t) 2

 

(2.40)

 

 

This expression requires that the time scale for changes in bulk velocity i.e. u/(du/dt) is

shorter than the viscous time scale R2 / V . This “unsteadiness factor” therefore deﬁnes the
relative importance of the unsteady effects to the quasi-steady ones. Unsteady effects can

be expected to be important when

l du R
—— — l 2.41
d (’v) > ( )

ut(

38

It can be inferred from (2.41) that for a given diameter of the pipe and ﬂuid properties,
higher acceleration would give a higher unsteadiness factor. From (2.41), it can also be
seen that the ﬂuid properties play an important role in determining the size of the

unsteadiness factor. For highly viscous ﬂuids, which would reduce the viscous time scale

R2 / V , much faster transients are required to achieve appreciable unsteadiness factors.

The viscous time scale RZ/V also plays an important role in the weighting
function K (t) used in (2.39). For ﬂuids of different viscosities, the weighting function is

effectively rescaled. These weighting functions, shown in Figure 2.2, decay
monotonically to zero as exp (-constant t) at long times, as a consequence of the inverse
Laplace transforrn’s approximation as a single exponentially decaying term for small 5,
which can be shown from Cauchy’s residue theorem. The weighting ﬁrnction always
leads to positive weighting to prior values of bulk ﬂow acceleration. It is clear that the
pressure gradient generally exceeds its momentary quasi-steady value during and after
the recent bulk ﬂow acceleration. Conversely, the pressure gradient is reduced below its
quasi-steady value during and after the recent bulk ﬂow deceleration.

Zielke [11], using a similar approach calculated the wall shear stress based on the
instantaneous velocity and change in velocity history. In assessing the utility of the
mathematical expression (1.1), he found good agreement with the experimental results of
Holmboe and Rouleau [10]. In a similar manner, the utility of the expressions for unsteady
ﬂow rates as a functional of wall-shear stress or pressure gradient derived here can be
assessed by building a simple pressure gradient meter to measure the momentary mass ﬂow
rate in forced unsteady pipe ﬂows undergoing controlled transients. It will be described in

the following chapter. Since the convolution integral between pressure gradient history and

39

weighting ﬁrnctions, can, in principle, accurately predict the instantaneous ﬂow rate in the
laminar ﬂow regime, this suggests that a laminar unsteady ﬂow-metering device can be

implemented in real world problems.

2.1.5 Extension of Fully Developed Pipe Flow Relationships to Other Geometries
Pressure pulses traveling through a tapered pipe are subject to distortions, which
depend upon the geometry of the pipe and viscous shear forces. The treatment ducts of
arbitrary cross-section is possible if the taper is not so great that it effects the velocity
distribution. Thus the transient analyses of the previous sections can also be applied to
tapered ducts when their motion can be approximated by linear forms of the Navier-Stokes
equations. The one-dimensional form of the linear momentum equation (2.5) then becomes

(Streeter and Wylie [46]),

au au 18p 1 a ,au 18(8u)
— —=—-— —— —— —— — 2.42
at +“Bx p0x+pr" ar(ﬂr 5%)}.pr ﬂax ( )

The parameter n is set to 0 or 1 to signify a Cartesian (channel ﬂow with walls at r = iR)
or axisymmetric (pipe ﬂow) coordinate system. The companion continuity equation can be

written as

a a _
E(pA)+-é;(pUA)—0 (2.43)

where A(x) is the local duct cross-sectional area and U the mean velocity. Using the
simplifying assumptions of section 2.1.1, relations between pressure gradients, friction, and
ﬂow rate in one-dimensional liquid transients in tapered pipes can also be described locally
by the equations (2.36) and (2.3 8), accompanied by the continuity equation (2.43), relating

density and mean velocity changes to area changes.

40

Chapter 3 Experimental Apparatus

3.1 Flow Circuit Design

In order to test the applicability of expression (2.38) derived in Chapter 2, a
simple pressure gradient meter was build to measure unsteady ﬂow of liquid as a function
of pressure gradient and pressure gradient history. An experiment was therefore designed
in which a transient ﬂow could be generated and measured. This facility consisted of 2.5
meters long and 9.5 millimeters diameter pipe fed by a header tank, an upstream unsteady
ﬂow-generating valve, a load cell to measure the true accumulated liquid mass and a
pressure sensor to measure the instantaneous pressure gradient along the pipe as shown in
Figure 3.1. The momentary pressure gradients measured by pressure sensor could then be
utilized to deduce the momentary accumulated mass of the liquid from equation (2.38),
which could then be compared with load cell measurements of the time dependent mass
of the liquid collected. During the design process of the ﬂow circuit design, the following

aspects were considered:

i) The ability to vary ﬂow rates without causing excessive vibrations to the
apparatus.
ii) The ability to use the same apparatus for both laminar and turbulent ﬂow regimes

without any major changes.
iii) A sufficient length of the pipe to ensure fully developed ﬂow.

iv) The pipe material and sensor should be unaffected by the choice of the liquid.

41

v) The pressure sensor should be sensitive enough to detect small pressure changes
with high signal to noise ratio and small enough to be ﬂush mounted with pipe walls so
as not to interfere with the main ﬂow.

vi) Data acquisition should incorporate multiple channel sampling at reasonably fast
rates and with sufﬁcient gain to enable the adequate resolution of sensor outputs.

vii) The load cell should be highly sensitive with minimum signal drift to ensure

repeatability and accurate indication of the momentary mass of the liquid collected.

After careful consideration of the above points an experimental facility was
designed and fabricated. Initially different experimental conﬁgurations were used in
order to achieve optimal results; these conﬁgurations are listed in Appendix A. The ﬁnal
design of the experiment is shown in Figure 3.1. As shown in Figure 3.1, a header tank
was installed on a vertical tower, which could be hoisted or lowered over a range of 3.3
meters elevations. Unsteadiness in the ﬂow was generated by manually opening and
closing a valve controlling ﬂow from the tank to the pipe test section that housed the

pressure SCHSOI‘.

The experiments were conducted using a computer with a dedicated Data
Acquisition and Control (DAC) card. This computer was used for energizing the pressure
sensor, timing signal generation and sampling through the DAC card, and post processing
of the acquired data. The pressure sensor was energized using an analog output from the
DAC card. The data were acquired simultaneously from the load cell and pressure sensor.
The output from pressure sensor was ﬁrst ampliﬁed using an external high gain ampliﬁer
and then passed on to the DAC card for acquisition and streaming of binary data to the

hard drive for later use. Similarly the voltage output from the load cell was ampliﬁed and

42

smoothed in a strain gauge signal conditioner before being fed to the DAC card. The
software package LabVIEW was used for the sensor control, data acquisition and data

post processing.

 

 

 

 

 

 

 

 

 

 

   

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

%
Header
Tank
Upstream Unsteady Pr .
Flow Generating Valve essure Sensor Collecting
/ l
.. l
i 14‘ m... i j“:
H: -3 meters Pressure Sensor
Output Ampliﬁer rim \
£41311
,1 .1...
. Load Cell
LabVIEW s13" “3.3-*5...”
Data Acquisition
and Control D
B Connector Box
D 68 Pin
C D
[ PC1-6023E Data Acquisition
and Control card

 

 

 

 

Figure 3.1: Experimental apparatus

3.2 Pressure Sensor

A pressure sensor was conﬁgured to provide a differential voltage output to the
DAC card and was installed 165 diameters downstream of the unsteady ﬂow-generating
valve, at which point the ﬂow should be fully developed. The pressure drop was

measured along the pipe between the sensor location and the open end of the pipe, where

43

the pressure was considered to be atmospheric pressure. The pressure sensor used in these
experiments was acquired from Entran Devices Inc. and is classiﬁed as an EPE “Low
Pressure” sensor model EPE-541. This sensor was 2.36 mm in diameter and was
designed for use in air, with a typical gauge pressure voltage output in milli-Volts. The
choice of a small sensor allowed it to be almost perfectly ﬂush-mounted in a narrow pipe,
and allowed good spatial resolution in pressure measurements. The pressure sensor was
coated with Paralyne-C to enable its use in water and other ﬂuids. Due to the
piezoresistive nature of this sensor, it was susceptible to zero offset, thermal zero shift
and thermal sensitivity shift effects. A detailed description of the working principle of the
pressure sensor, its parameters, and an explanation of the aforementioned effects are
given in Appendix B. Figure 3.2 illustrate the ﬂush mounting of the sensor in the pipe

wall.

    

0 " l‘ i " Pressure sensor
.36rnm

Figure 3.2: Sensor mounted in a pipe wall
In order to avoid thermal zero shift and zero offset effects, a custom data
acquisition procedure was designed. To avoid zero offset, the sensor was energized for at
least 30 minutes prior to data acquisition, thus giving it sufﬁcient warm up time. The

sensor was also kept in contact with the working ﬂuid during warm up.

44

The highly miniaturized strain sensitive element of the EPE sensor dissipates heat
when energized and the rate at which this heat is transferred is a function of the thermal
conductivity of the surrounding media. The zero offset of the sensor can be related to this
energy dissipation. Therefore reducing the heat generation would reduce the zero thermal

drift of the sensor. The heat generated by the sensor is given by the relation

Q=(E2/IR)><t , where Q is the heat generated in Joules, E is the excitation voltage in

volts provided by the DAC card, I is time in seconds and IR is sensor impedance in
Ohms. Reduction of the excitation voltage by 50% for an input impedance of 1000 Ohms
would reduce the heat generation by 75%, i.e. if the excitation voltage is reduced from 10
volts to 5 volts, it would reduce heat generation from 0.1 Joules to 0.025 Joules in 1
second. In order to reduce heat generation further, the sensor was excited for only 8% of
the duty cycle using the DAC card, further reducing the heat generation by 92%. This
reduction in excitation voltage resulted in a lower voltage output signal from the sensor,

which was then ampliﬁed externally by using the dedicated sensor output ampliﬁer.

3.3 Load Cells

Two types of load cells were used to measure cumulative liquid mass during the
experiments: one for the small masses collected in laminar ﬂow experiments (Interface
Inc. Model SMT1-2.2) and the other for the larger masses in turbulent ﬂow experiments
(Futek model L2357). Each load cell was installed between two support plates thus
enabling the liquid collecting container to be placed on top as shown in Figure 3.1. The
load-measurement errors, which include loading out of plane, lateral loads, and moment
loads were avoided by ensuring the alignment of top and bottom plates with each other

and with the primary axis of the load cells. Care was also taken to apply the load only

45

along the primary axis of the load cells. The technical data and working principle of these

load cells is given at Appendix C.

3.4 Viscometer

Different combinations of ethylene glycol and water solutions were used as the

working ﬂuids. Cannon-Fenske viscometers suspended in a Temp-Trol viscosity bath was

 

 

 

 

Figure 3.3: Cannon-Fenske viscometer for liquids

used to measure the viscosities of these different solutions. A range of 0.3 to 20000
centistokes could be covered with these types of viscometers. The particular viscometer
used for these experiment was viscometer number 100-771A, with a calibration constant
of 0.01489 centistokes/second, which required a charge of approximately 6 milliliters
during the tests. After ﬁlling the charge in the viscometer, the viscometer was placed in a
holder, and inserted in the constant temperature viscosity bath. Approximately ten
minutes were allowed for temperature equilibrium to be reached between the viscosity

bath and the test sample prior to making any viscosity measurements. Then the time was

46

recorded for the liquid meniscus to pass between the prescribed marks at A and B as
shown in the Figure 3.3. The density of the ﬂuid was measured using a graduated

cylinder and a Sentra electronic weight scale.

3.5 Data Acquisition and Control Card

The data acquisition and control card used was a PCI-6023E, with 16 channels of
analog input and eight lines of digital 1/0. This card uses a DAQ-STC System Timing
Controller (STC) for time related functions. The DAQ-STC enables buffered pulse
generation, equivalent time sampling, and seamless changing of the sampling rate. Due to
the use of a Real-Time System Integration (RTSI) bus, several measurement functions
could be synchronized to a common trigger or timing event. The DAC card was
conﬁgured for input of differential voltages with the Programmable Gain Instrumentation

Ampliﬁer (PGIA) as shown in Figure 3.4.

 

Vim

 

Vin- Measured

A - Voltage
6V... =[V...+ - V1... ]*Gain

  

 

 

 

 

 

Figure 3.4: Programmable Gain Instrument Ampliﬁer (PGIA)

3.5.1 Data Acquisition and Control

The data was acquired from two analog input channels at 10 Hz after the
acquisition was triggered by one of the two counters. Acquisition timing was controlled
by a hardware clock for fast and accurate sampling. Data was stored in an intermediate

memory buffer after it was acquired from the analog input channels. The data from the

47

memory buffer was then retrieved while the acquisition was in progress, allowing the

data to be processed and displayed as it was being acquired.

This data was stored in the form of a two-dimensional array; the ﬁrst dimension
being the scan number, the second being the channel. The buffer size was kept at 4000
samples. During the acquisition, data was saved to a binary ﬁle to minimize the storage
requirement. This acquisition could be stopped either manually at any time, or after a

predetermined number of scans.

For the purpose of timing sensor excitation voltages and data sampling, pulses
were generated using dual synchronized continuous pulse trains as shown in Figure 3.5.
The frequency was set as number of scans required per second. Duty cycle control and
frequency control were used to generate the continuous pulse train from counter “0”

‘61,,

output pin and a second pulse train of the same frequency from a second counter

Phase I J Phase 2 I Phase I l Phase 2 l Phase 1

 

 

 

 

Pulse Train
Counter Starts Phase 2
1 Phase 1
Duty Cycle = 0.08
Counter Starts
Phase 1

 

Duty Cycle = 0.05

 

 

 

Figure 3.5: Pulse train and pulse duty cycles

48

output pin, on the data acquisition and control card. The second pulse train had a shorter
duty cycle to ensure the excitation of the pressure sensor prior to the triggering of data
acquisition. The pulse train was stopped at the same time as the acquisition. In order to
avoid any thermal zero shift of the pressure sensor, the sensor was excited for only 8% of
the duty cycle, and the data acquisition was carried out for 5% of the remaining duty

cycle as shown in Figure 3.5.

3.6 Calibration of Sensors

The pressure sensor requirements for the experimental apparatus were that it
should be small compared to the radius of the pipe, with high frequency response. The
smallest suitable commercially available sensor was therefore procured. The limitations
with this sensor were that it was factory calibrated for use in air and could not be used
without recalibration in these experiments with liquids. In addition due to the miniature
sensor design, there is no extra structural support around the sensor. Hence the sensor
was highly sensitive to mounting stress. In order to overcome these limitations, the sensor
was coated with Paralyne-C for use with liquids and isolated from the mounting stresses
during installation with the use of low durometer RTV type materials. This necessitated
its calibration in place with the working ﬂuid to compensate for any effect of mounting

stress on its output.

3.6.1 Pressure Sensor Laminar Flow Calibration

The pressure sensor in the laminar ﬂow regime was calibrated using the following
exact solution to the Navier-Stokes equations for steady, laminar, fully developed ﬂow of

Newtonian ﬂuids:

49

4
12:”pr
8pL

 

(3.1)

With the maximum header tank height set to ensure laminar ﬂow (Reynolds number <
2000), the header tank was raised at equal height increments and mass of liquid
accumulated in the collecting container was recorded at each height for a ﬁxed time

interval. This procedure gave values of AP = P, —Pa,m for a range of different ﬂow rates

with the corresponding pressure sensor output recorded in milli-Volts. Here P, and P are

ulm

 

400 I l I I I I I I I I I I I I l I I I I I I—I I

350
300
250
200

150

Inferred Pressure drop (Pa)

100

01
O

oIIII‘IIIT'IITIIIIII'IIII'IIIIIIIII'IIII
IllIllllllllllllLllLLlJlllLIlllllllllLL

 

 

J l I l I I 1 l l l l l J l I

10 1 5 20
Sensor Output (mV)

.—
1—
p
L.
-
)—
u.
_-
p—

O

 

0'.
N
01

Figure 3.6: Pressure sensor calibration in the laminar ﬂow regime with a 35/65% ethylene

glycol and water mixture in a 9.5mm diameter pipe

50

the pressures at the sensor location and the open end of the pipe. The pressure drop values
were then plotted against the values of the pressure sensor output in milli-Volts as shown in

Figure 3.6, thus calibrating the pressure sensor in laminar ﬂow regime.

3.6.2 Pressure Sensor Turbulent Flow Calibration
In order to conduct the turbulent ﬂow experiments, the pressure sensor was

recalibrated for use in the turbulent ﬂow regime. In contrast to the laminar ﬂow, no such

 

ﬁfﬁTT'IIIIrTIIITTTIIIII

2600 o
2400 d?
2200 o
2000 O

1800

1600

1400

1200 0

1000

800

600

400

200

0 1 l I I l l I I I J l l 1 I J I I I I l I I L 7

0 25 50 75 1 00
Sensor Output (mV)

Measured Pressure Drop (Pa)

Figure 3.7: Pressure sensor turbulent calibration curve with 35/65 ethylene glycol

and water solution, in a 9.5mm diameter pipe

51

exact relationship between pressure drop and ﬂow rate is known which could be used for
the turbulent ﬂow calibration of this sensor. The pressure sensor calibration was therefore
carried out by comparing the sensor output with pressure drop measured by manometer
over a range of steady ﬂow conditions. The pressure taps used to connect the pipe with the
manometer could not be located exactly at the location of the pressure sensor due to the
small diameter of the pipe and ﬁ'agile nature of the sensor. In order to overcome this
shortcoming, an identical test section was made to measure the pressure drop data from
manometer and this data was then compared with the corresponding sensor output under
the identical ﬂow conditions. This gave a calibration for the pressure sensor in the range of
pressures encountered in turbulent ﬂow in Pa/m V. The resulting turbulent ﬂow pressure

sensor calibration curve is shown in Figure 3.7.

3.6.3 Low Reynolds Number Turbulent Friction Factor Relationship

During the calibration of the pressure sensor in the turbulent ﬂow regime, data
obtained from the manometer was also used to determine the friction factor in the low
Reynolds number range. This served two purposes: i) it provided a comparison with the
existing data on friction factors since few empirical relationships exists in the literature for
the turbulent friction factor in the Reynolds number range of 2,300 to 10,000; ii) if
consistent with other ﬁiction factor relationships in fully developed ﬂows, it could indicate
this one has a standard fully developed ﬂow too. Using the relationship for friction factor

and pressure drop

£=A[£)[E’_] (32,
pg D 2g

52

 

 

 

 

l' I I I I j I ﬂ I I I I I Tq
0.04 L- .1.
I 0 3
)- A o —
0.035 T A o O '1,
1— A 0% —1
l: A A MOOQ :
0.03 _- A 2m. 1
F A:
§ 0 025 - -
8 Z I
1.1.. - _
.§ 0.02 :- -:
o _ _
't: _ -
u. _ _
0.015 :- -_
0.01 5- -§
0 005 E- j
' - A Measured d
C O Blasius :

L- I I L l I I I I l I l I I

5000 1 0000 1 5000
Reynolds Number

Figure 3.8: Friction factor comparison

and substituting AP = pgh and U = m/pA , in equation (3.2) and rearranging, we ﬁnd

_ 2ghd

1" 2
LU

 

(3.3)

where 2. is the friction factor, h is the change in the water column height and In is
calculated from the load cell data. Different values of the friction factor for different ﬂow
rates were then plotted against the corresponding Reynolds number, which could be ﬁtted

to the empirical relationship:

2. = 0.0649—6x104’Re (3.4)

53

This friction factor relationship was then compared with the Blasius ﬁiction factor i.e.

2 = 0.3164(Re)4125 (White [45]). The resulting comparison is shown in Figure 3.8.

3.6.4 Load Cell Calibration

Load cell calibration was carried out by placing standard masses ranging from 5
grams to one kilogram on the load cell platform and plotting the load cell output against
mass, as shown in Figure 3.9. Under the present setup, the SMTl-2.2 and Futek load cells

were calibrated to give 3.2198 mV and 1.6399 mV output for one-gram change in the mass.

 

2 000 m I l I I I r I I I I T I T r

_l

llIIIIIIIIIIIIIIIIIJIIIIIIIIIILIIIIIIIIJ

1750

1 500

1250

1 000

Mass(gms)

750

500

250

III'IIII'ﬂﬁITTII'IIIIIIIIIITFTI'TIII

 

 

 

0825
,
k.
r-
)—
-
f.

1 000 2000 3000
Load cell output (mV)

Figure 3.9: Load cell calibration curve

54

Since the DAC card had a resolution of 2.5 mV with no gain, a mass change of 1.0 and

1.5 grams could easily be detected using this setup.

3.7 Critical Experiment Requirements

There were two major requirements which inﬂuenced the experimental procedure
and had a substantial effect on the results. These requirements were to achieve high
pressure-sensor signal to noise ratios for low-pressure signals and that the pressure drops

to be sufﬁciently high to allow accurate measurements, even at low ﬂow rates.

3.7.1 High Signal to Noise Ratio

The low level signal from the pressure sensor was ampliﬁed using an external
high performance ampliﬁer with an adjustable gain of 1000 as shown in Figure 3.10. This
ampliﬁer had a variable gain of 10 to 1000 and the ability to adjust zero offset using a
null control. The use of this ampliﬁer allowed data acquisition at relatively low ﬂow rates
with high signal to noise ratio. The level of the input signal to the data acquisition and

control card was restricted to below i 50 mV so that a gain of 100 could also be applied

 

 

N0“? Instrumentation ampliﬁer
/g / //
\ :\\\‘\\

/>> / ./

+ / /' K
K I my : > ADC
‘ Leadwire E

, A
L°w level 1 W

signal from
pressure sensor External ampliﬁer DAQ board

L2-5 mV)

 

 

 

 

 

 

 

 

 

 

 

Figure 3.10: External ampliﬁer to increase signal to noise ratio
to the incoming signal for better resolution. This not only gave the ability to detect weak

signals but also increased the signal to noise ratio.

55

3.7.2 Working fluids

Different concentrations of ethylene glycol and water solutions were used as the
working ﬂuid. With these solutions, changes in ﬂuid properties could be made easily by
changing the relative concentration of ethylene glycol and water. In addition, a higher
output signal in mV was possible on account of the higher pressure drops caused by greater
viscosities of these ﬂuids. A simple analysis was therefore carried out to estimate the
dependence of pressure drop on the changes in the viscosity of the ﬂuid. From the

deﬁnition of the Reynolds number

 

Re = M (3.5)
p
and from the laminar mass ﬂow rate to pressure drop relationship
4
m = pUer2 = ER—pap (3.6)
8,11L
(7 can be eliminated to yield
4L
= Re 2 3.7
Ap pk, x1 ( )

From (3.7), it is clear that for the ﬁxed geometry and Reynolds number, pressure drop has a

quadratic dependence on the viscosity of the ﬂuid.

Initially, distilled water was used as a working ﬂuid since it was readily available
with non-corrosive properties. However, extensive drift in the pressure sensor signal was
observed during the calibration of the pressure sensor. This was initially attributed to the

higher thermal conductivity of the water and corresponding heat loss to the working ﬂuid

56

 

   

  
   
   

 

 

 

2250 _- I I I I l I I I I l I I I I l I I
: Water I
2000 P Q
E. 15% th I .5
1750 : 35% M?“ my” 3
t _
g 1500 5- -j
s 2 3
21250 _— 1
(I) P _
E ; j
C 1000 r 1
E. - _
a) : -
Of. 750 _- _
500 1' 25% Ethylene glycol '3
: 75% water j
250 E- -:
O : l l l 1 l l L 1 l l 1 1 I l J L l l 1 l l l L :

o 10 20 30 40

Pressure Sensor output (mV)

Figure 3.11: Effect of different viscosity ﬂuids on pressure sensor output
through convective heat transfer from the sensor. Analyses were carried out to estimate
the heat loss due to the temperature gradient between the sensor surface and water, which
was found to be negligible as shown in Appendix D. Typical pressure sensor responses
with water, and two different concentrations of ethylene glycol and water solution

( #:2351343 N-sec/mz; p=2.5213-3 N-sec/mz) are shown in Figure 3.11. It can be

inferred that the voltage output of this sensor is linearly proportional to pressure for ﬂuids

of higher viscosity ,u 22.52E-3 N-sec/mz, but not with ﬂuids of lower viscosity ,1! =

57

2.35E-3 N-sec/m2 in the same Reynolds number range. The reason for this difﬁculty in
sensing pressure in low-viscosity ﬂuids was not fully determined. Therefore all
subsequent unsteady ﬂow experiments were carried out using ethylene glycol-water

solution with viscosities greater than 2.35E-3 N-sec/mz.

3.8 Experimental Plan

In all experimental studies of transient ﬂows, steady ﬂow calibrations were
checked ﬁrst. Once both the accumulated mass indicated by the load cell and deduced
from equation (2.38) were conﬁrmed to be in agreement in steady laminar ﬂow, further
experiments were conducted in transient ﬂow conditions. From time to time these sensors
drifted and behaved erratically and the following observations were made with regard to

their performance.

i) The sensor calibration should be checked in every experiment and should ﬁrst
satisfy the steady ﬂow comparison of the inferred and measured cumulative mass data,
since only then would transient ﬂow experiments yield trustworthy results. Questionable
calibration curves caused by driﬁ in the sensor indicate that the experiment should not be

continued since the sensor’s performance would be untrustworthy.

ii) Only pressure signals within the calibration range of the pressure sensor can be
considered trustworthy. Therefore transient ﬂow experiments must be selected carefully
to fall within the calibration range.

iii) The initial value of the steady pressure gradient 8P(0)/ax is an important
parameter in the overall comparison of the measured and inferred accumulated mass data.
It should be evaluated carefully as it plays an important role in the unsteady ﬂow-to-

pressure gradient relation, as can be observed from the expression

58

t , R2 (lap , , R2 1 lap , 13p , ,
U d =-— ——- d —— —— -— —-—— 0 K d
Jo (1), 8V Iopaxm H v Jill/08x“ t) p8x( )] (I),

iv) The steady ﬂow check for calibration may be repeated more frequently to

minimize the effects of pressure sensor signal driﬂ.

3.8.1 Laminar Flow Experiments
After analyzing the behavior of the pressure sensor under steady ﬂow conditions,

the following set of experiments was carried out in the laminar ﬂow regime:

i). Initially steady, accelerating ﬂows, Reynolds number range 620 to 1440.

ii). Initially steady, decelerating ﬂow, Reynolds number range 1520 to 340.

iii). Initially steady, fast accelerating (ARe/At > 70/sec ), slowly decelerating ﬂows

(A Re/At < 70/ sec ), Reynolds number range 550 to l 140.

iv). Initially steady, fast decelerating (ARe/At > 70/sec ), slowly accelerating ﬂows

(A Re/ At < 70/ sec ), Reynolds number range 1300 to 120.

v). Initially steady, slowly accelerating (ARe/At < 70/sec ), fast decelerating ﬂows

(A Re/ At > 70/ sec ), Reynolds number range 1950 to 520.

vi). Initially steady, slowly decelerating (ARe/ At < 70/sec ), fast accelerating ﬂows

(A Re/ At > 70/sec ), Reynolds number range 500 to 1800.

A similar series of experiments were carried out in the turbulent ﬂow regime with
Reynolds number range of 3800 to 5500. In all the cases, the cumulative mass of liquid

was measured together with the pressure drop along the pipe, so that theoretical

59

expressions relating unsteady ﬂow rate to pressure drop could be evaluated. The results

of these experiments and evaluations are given in Chapters 4 and Chapter 6.

60

 

Chapter 4 Laminar Flow Experimental Results and Discussion

4.1 Laminar Flow

A series of experiments was carried out in the laminar ﬂow regime for both steady
and unsteady ﬂows through pipes. In steady ﬂow experiments, data from a pressure
sensor and a load cell were taken without making any changes to the ﬂow. These steady
ﬂow experiments were then followed by unsteady ﬂow experiments under varying ﬂow
conditions, where unsteadiness was created in the ﬂow by manually operating an
upstream valve. The working ﬂuids were different concentrations of aqueous ethylene
glycol solutions. In total, three solutions were used for laminar ﬂow experiments to
demonstrate the applicability of expression (2.38) to different incompressible ﬂuids. The

properties of these ethylene glycol solutions are tabulated below:

 

 

 

 

 

Properties Solution 1 Solution 2 Solution 3
Density ( kg / m3) 1060 1030 1049
Kinematic viscosity (mz/s) 3.1592 E-6 2.447 E-6 2.95 E-6
Viscous timescale (Rz/V) s 7-18 9-269 7.689

 

 

 

 

Table 4.1: Properties of ethylene glycol and water solutions

The plots shown in the following experimental results section are of the
accumulated mass against time. The accumulated masses are those measured by the load

cell, as well as those inferred from pressure drop measurements using (2.39). The inferred

61

 

masses are also decomposed into quasi-steady part and an unsteady correction to
demonstrate the difference between various components of accumulated masses. Every
fourth data point has been plotted for clarity. The quasi-steady accumulated mass data
indicate the mass accumulated based only on the momentary pressure gradient along the
pipe length, and is given by the term ‘A’ in equation (2.39). The unsteady ﬂow correction
is calculated from the convolution integral term based on the pressure gradient history
and weighting function for that particular ﬂuid (term B in equation (2.39)). The inferred
or completely unsteady accumulated mass refers to the combined values of term A and B
in equation (2.39) and is the sum of the quasi-steady accumulated mass and the unsteady

correction.

4.2 Experimental Results

In these sections, selective results from experiments are presented which include
steady, accelerating, decelerating, and accelerating-decelerating ﬂow experiments. In all
of these experiments, the ﬂow was initially kept steady prior to inducing any transients.
In the case of ﬂuid 1, results from each type of ﬂow are presented whereas, for ﬂuid 2
and ﬂuid 3, only a few representative experimental results are presented for the purposes
of completeness and brevity.

4.2.1 Steady Flow Results for Fluid 1

Figure 4.1 shows the comparison of quasi-steady, unsteady correction, inferred or
completely unsteady, and true cumulative mass in a steady ﬂow experiment. In this
experiment, the Bp/Bx term remains unchanged. Therefore the cumulative mass for

quasi-steady, reference and inferred terms should all indicate the same value if the load

cell and pressure transducer calibration are correct and experimental noise is minimum.

62

 

Moreover the unsteady correction to accumulated mass should show no unsteady

correction. The Reynolds number is based on the time average Reynolds number given

ofOto

 

by DU /V. In Figure 4.1, the time range of 0 to 8 seconds corresponds to
“o

665. From Figure 4.1, it is clear that these requirements are all well met.

 

 

 

 

0.257. . Q" la'si-Ste'ady V . . . - . . V . I . v.41
° Unsteady correction . ‘ ,
° Inferred . l
’ X True I ‘
0.2:- ‘ .
I
I
A l ‘ ,
g 0.15l- xI .
3 z
'5' . I
3 01" x8 -l
'g .’ I
E ' I
3 i '
I
0.057 ‘ ..
j I
I
‘ 4
0oooooooooooooooooooeooo.
.o e i
.050 2 4 6 8
Tlmo(Soc)

Figure 4.1: Steady laminar ﬂow at Reynolds number 1060

4.2.2 Accelerating Flow Results for Fluid 1

Figures 4.2 and 4.3 show comparisons of quasi-steady, unsteady correction,
inferred unsteady and true cumulative mass in an accelerating ﬂow experiment. For the

experiment plotted in Figure 4.2, the ﬂow was steady for 1.5 seconds and then was

63

 

gradually accelerated. During the acceleration phase the quasi-steady value over-
estimates the accumulated mass, whereas the unsteady correction counteracts this
overestimation of the cumulative mass resulting in 3 overall good agreement between the

true and inferred cumulative masses.

In Figure 4.3, the same trend is evident during a more rapidly accelerating ﬂow,

as evidenced by the greater value of the maximum unsteadiness parameter:

 

 

 

 

l du R2
=————t—=6.85.
7 u(t)dt()v
0 35 L ‘ Quasi-Steady ‘ ‘-I
- _ o Unsteadycorrection ‘ 2‘
I o Inferred .‘gﬁ ‘
0.3} x True . , .:
I ‘ '
> A I .
§ 0.25} .‘.' .1
. . I .
v ’ ‘ i ‘
g 0.2" “88 .
a . 3 .
I
g 015.. “I ..
:l I
E , ‘II
. ‘I
g 0.1- ‘I .
. ..
: AI
, .a .
0.05- a” -
. * .
,.l" .ooooOO°°°°°°°°°°°°..°3
Droooo°°° .
41050 A A A A 2A3 A A T A 1A0 A
T|mo(Soc)

Figure 4.2: Laminar accelerating ﬂow; Reynolds number ranging from 578 at t: 0 secs

to 1321at t: 12 secs with a maximum unsteadiness factor of 1.77

64

 

 

 

 

 

0 5 i ‘ Quasi-Steady
' I ° Unsteady correction .
’ A
Q45 :_ ° Inferred .‘ .1
; X True ‘ I. ‘
, .
b I ‘
0.4 ,- “ I 41
b ‘ ‘ J
: ‘ I. 1
A 0.35 " ‘ .- #
Q I ‘ I 1
V : A. I i
3 0.3 _- ‘ llII .
s s
'u 0.25 ,- 4”. -
g E ‘lAl' :
3 0.2 ' AA. .-
E i .‘a' :
I ‘... 1
: .0 .
ll ‘
0.1 " "".I' 1
’ I
n .‘
. '.
005 ." l'.-.-‘ 090001
. 00
' 3".“- oooooooo°°°°°°°°°° i
0 A i!........o«..oooo...ooo.......ooo .1
r 3
-0.05 - _ A , _ - ‘ . . _ ‘ ‘ ‘ ‘ _ . . _ ‘ ‘ A A
0 5 10 15 20 25
Tlme(Sec)

Figure 4.3: Accelerating ﬂow: Reynolds number increases from 740 at t = 12 to 1517 at

t = 25 see; with a maximum of unsteadiness factor of 6.85
In case of the gradual acceleration, the unsteadiness factor is 1.77 (Fig 4.2), compared to
6.85 for fast acceleration (Fig 4.3). In these accelerating ﬂows, the unsteady inferred ﬂow

equation (2.39) matches the experimental data almost perfectly.

4.2.3 Decelerating Flow Results for Fluid 1
Figure 4.4 and 4.5 show comparisons of quasi-steady, unsteady correction,

unsteady and true cumulative mass in decelerating ﬂow experiments. In Figure 4.4 the

65

 

 

 

 

. " ' ' ' ' 5 ' ' ' I A ' ' A A A A ' ' ' ' II .
0.25 _ Quasr-Steady . ' 5 .
. o Unsteady correction . ‘ e .
o Inferred I ' . ‘
. X True I ‘ . .
0.2 - ‘ . . .
. 8 ‘ .
8 8 ‘ ‘ A
a ’ g A ‘ l
5 0.15 - x g . -
a - , .
Q X
E x 9
'6 I
g 01- I -
3 I
5 x
8
0.05 - ‘ d
I .
‘ .
O o o o o o o o o o o o o 4
o 0 ° ° ° ° 0 0 o o o o
-0.05“*“L‘““““““‘*4‘*A
0 2 4 6 8 10

Tlme (See)

Figure 4.4: Decelerating ﬂow; Reynolds number varies from 1405 at t = 0 to 1055 t = 10

sec ; with a maximum of unsteadiness factor of —2.07
ﬂow was initially kept steady for 2 seconds, and then decelerated gradually. During the
deceleration phase, the quasi-steady term underestimates the accumulated mass compared
to the true cumulative mass. This underestimation is corrected by the unsteady correction
term, which results in accurate agreement between inferred and true cumulative mass
values. In ﬁgure 4.5 the deceleration phase is more adverse as is evident from the relative
values of the unsteadiness factor i.e. -2.07 to -37.8. In these ﬂows, the unsteady inferred-

ﬂow equation (2.39) also appears to be matched by experimental data very closely.

66

 

 

 

 

o 5 : . Quasi-Steady ogoggggggggggggggﬂ
. ' - O X "
: Unsteady correction 3293):}...“nunnnn;
, ‘ ‘
0 45 :- ° Infen'ed 8&2! ‘ _l
i X True Rig
o 4 - ,R -I
. g .
: ,1 1
e . a .
v : ’2
3 0.3 _- g -
's' i 2’
.s ‘ 1’
E : 1 .
D x’ 4
g 0.15 _- 2 1
. *2 .
: , .
0.1 ,- a '.
. R .
: *3 :
0.05 " ' '1
. ' q
:I .
0900006000000000 -
0 ooooooooooooo.....°°.°°°.° j
°°°°°°°°oooo A
-0.05 . . - . - . - - - - . . . . - . . . . - - .°°.°°?°-‘l
0 5 10 15 20 25
Time (Soc)

Figure 4.5: Decelerating ﬂow; Reynolds number decreases from 1311 at t: 13 to

818 at t: 25 sec; maximum unsteadiness factor 37.8
4.2.4 Oscillating Flow Results for Fluid 1
After completion of experiments in steady, accelerating and decelerating ﬂow
cases, further experiments were conducted in combined accelerating and decelerating
ﬂows. Figure 4.6 indicates the results from one such experiment in which ﬂow was

initially kept steady for one second, and then was rapidly decelerated and accelerated.

67

 

 

 

 

*""-""'.""-'"'i“'*ﬁ'rr'i"""" J
035L ‘ Qmﬂswmy g!
‘ . o Unsteadycorrection § .
I o Inferred ‘§§ 3
> 4
0.3- X True 1,. 4
. 8‘ .
. 8.
P !§‘ 1
A 0.25- ,5 1
e . 2 1
v l I .
3 L a" .1
~ 0.2? 8“ 4
1: ’ 5 1
8 ‘ ' ‘
_g 0.15- l '1
3 ’ 8 4
s > 22 -
g 01" 3|”! J:
".A
I
C .' I
0.05f g 'j
* .x 1
IL 1
0 °°oo°°°°°oo oooooooooooooo9000.09.09000':
O
-0.05 “““““““““““““““““““

o 2 4 6 8 1o 12 14 AAA16
The (See)

Figure 4.6: Initially steady, fast decelerating and slow accelerating laminar ﬂow;

Reynolds number 550 to 1133; maximum unsteadiness factor —58 to 40
This rapid deceleration and slow acceleration was achieved by partially opening
and closing of the unsteady ﬂew-generating valve. It is evident from the ﬁgure that the
quasi-steady term underestimates accumulated mass during deceleration phases and
overestimates it during acceleration phases. As the accelerations and decelerations are not
of the same levels, a trend towards an overall deceleration dominated ﬂow is observed.
The levels of accelerations and decelerations were estimated as maximum unsteadiness

factors of 40 and -58 respectively. Figure 4.7 also shows the comparison of accelerated

68

 

and decelerated cumulative mass for two complete cycle of oscillation. In this case, the
same trend is observed i.e. quasi-steady overestimation of the accumulated mass during
acceleration phase and underestimation during deceleration phase. In contrast to the
above two cases, Figure 4.8, shows the comparison of quasi-steady, unsteady and true
accumulated mass for accelerating and decelerating ﬂow below the mean value of ﬂow.

In this case, the acceleration and decelerations were more severe as can be inferred by the

 

 

 

 

I Y ' v v v ' I v ' - ' ‘ ' v v I v v v ‘ ' '._1
0 5 i. - Quasr-Steady . 32‘
: o Unsteady correction . g ;
. 5 R .
0.45 ’_ ° Inferred “’ 4
: " True “22 3
C sogx I
r ‘0
A 0.35 L ““AAA“§§§xx d‘
a? i R IIIIII" ‘
‘5 o 3 L 4
a .

: ' 1
s 1 r
'u 0.25 " “I. 1
g I “II I
E 0.2 .' “ a 1
I ‘A .' I
. A”, .
. . .
L 8 ‘
0.1 , ‘39 1
: -§ :
0.05 .- *’Q,R 909990000099900900 0.9000:
’ ll"' oo°°°°° °°°°ooo¢oo°°°°° I
O Ugoooooootw , .1
-0.05 - - . - - - - - 1 - . - . - - . - - . - Ar . 4

0 5 10 15 20

Tlmo(Sec)

Figure 4.7: Initially steady then accelerated, decelerated rapidly above its mean value;
Reynolds number 1307 at t = O to 123 at t = 25 ; unsteadiness factors range from 10.35
to — 25.7

69

|'..l

 

 

 

 

O.5h""'FF.F"'TIF'F"IT"""-w
: - Quas1-Steady I"
i o Unstead correction '31
045: Y ,,.
, Inferred 22 ‘e «
i 2
0.4 r True 85553388883833!“ .3
: a I. 4
’ §‘AL..A.“‘.““‘A 4
0.35 ,- 1}! ~
g 0.3 ’- 2' .
8 I 2‘ .
's' 0.25 f- 2‘2 4
3 ‘ 0,2 *
I O 2 - ox I
E » ox -
: 0°! :
g 0.15 ,- .gx" 1
: 2;: :
0.1 .b ‘ll'gg “ 1
I §§§“““‘ 2
0.05;- ,u‘ -j
> I r
o P'
’90099, .oooooooooooo¢.° '
°°°°¢ooooo°°°°°.° o°°°°oo 090099 I
’9090 000 .
-o 05 °°°
0 5 10 15 20 25

Tlme (See)

Figure 4.8: Initially steady, accelerated and decelerated ﬂow; Reynolds number varied

between 1309 to 30; unsteadiness factor varied between 86 and —88

unsteadiness factors of 86 and —88 for the acceleration and deceleration phases. Again,
the agreement between the theoretical result of (2.39) and the experimental data are very
good.

4.2.5 Accelerating Flow Results for Fluid 2

Figure 4.9 shows the comparison of quasi-steady, unsteady correction, inferred
and true cumulative mass values in an accelerating ﬂow experiment for a ﬂuid of 23%
lower viscosity than ﬂuid-1. The flow is initially steady for 6 seconds and then it is

accelerated. The same trend as fluid 1 is observed in this case i.e. quasi-steady

7O

overestimation of the accumulated mass with the unsteady correction term compensating
for this overestimation. There is a small discrepancy between the true and inferred
accumulated mass, which is present from the start and continues to grow with time. This
discrepancy is partially attributed to the less viscous ﬂuid causing a smaller change in the
sensor output, and possibly affecting the thermal zero shiﬁ of the sensor during the
experiment. In this case the ﬂow was accelerated from a Reynolds number of 1430 to

1911 with a maximum unsteadiness factor of 2.07.

 

 

 

 

025
r A QuaSl'Steady ‘,
. ° Unsteady correction ‘ . ‘
’ ° Inferred ‘x’é‘
‘
0.2'. x True A x0 '
. x0 ‘
Ago
‘ .55 .
3 0.15- x:° .1
V p ‘0 «
Xo ‘
3 ‘0
Q ‘30
S ’ x§° .
3 0.1. h .
to
.Q ’ lo <
a ’ to
lo
a , 85°
3 005' is! -
b ‘ ‘
b “ ‘
‘ 09° ‘
O ooooooooooooooooooooooooooooo .
l
.0005 I A A A J ‘ - - . _ _ ‘ . A

0 2 4 6
Time (See)

Figure 4.9: Initially steady then accelerated flow, Reynolds number changed from 1430 at

t = 2 to 191 1 at I: 7; maximum unsteadiness factor 2.07

71

 

 

 

 

 

. - Quasi-Steady .
o Unsteady correction 2 2 '.
, o Inferred 2 R
R
0.1 ' X True 2 a R .-
. O ‘
o o g x ‘ A
A O O X x A ‘
g 0 : X x ‘ A ‘
a ‘ g ‘ ‘ ‘
'3' 0.05 - . 8 . - .
3 * a ‘ . . ‘
é a
I
3 . '
0 ‘8 o 0 0 0 O -
0
0
o o o o o o 0 ° ° ° ° ° ° ° ° ° 0‘
-0.05 - -
o 1 2 3 A 4 5

Figure 4.10: Initially steady ﬂow then suddenly decelerated ﬂow, Reynolds number

reduced from 1900 at t: 0 to 500 t = 5, maximum unsteadiness factor —25
4.2.6 Decelerating Flow Results for Fluid 2

Figure 4.10 above shows the comparison of quasi-steady, unsteady correction,
inferred and true cumulative mass during a decelerating ﬂow experiment. Flow is initially
steady for the ﬁrst second and then rapidly decelerated from a Reynolds number of 1900
to 500. As shown, even in the case of such an extreme transient, the unsteadiness
correction term was able to compensate and predict the correct accumulated mass. The
discrepancy between the inferred and true masses is comparable to the accelerating-ﬂow

results observed with this lower-viscosity ﬂuid.

72

4.2.7 Oscillating Flow Results for Fluid 2

Figure 4.11 shows the comparison of quasi-steady, unsteady correction, inferred
and true cumulative mass in a decelerating/accelerating ﬂow experiment. As expected,
we see quasi-steady underestimation of the accumulated mass during deceleration phase
and overestimation during acceleration phase of the ﬂow, with the unsteadiness
correction compensating for this overestimation and underestimation effects. In this case

ﬂow could be varied between Reynolds numbers of 1950 to 521 with maximum unsteadiness

 

VVVVVVVVVVVVVTivvvvvvvvvvvrvwfvvwvv-v

 

 

 

. 3
’ ‘ Quasr-Steady oatg‘
0-2 _' o Unsteady correction at R g x ‘ -
° Inferred tic A
X True 2:0 . .
b *0 ‘A ‘
0.15- Io ‘
. X0 4
are
3 3°
.. i 5
r gggﬂg“
3 0.1? x3 .
s I so ‘
3 r . ‘ A
*0
a * 2° ‘ :
3 b so A.‘
E 0.05? 5 -
3 t a .
t .5 .
“8 .
09090 o°°°°° ‘
0u°°°°° o 0 ° 0‘
0 0 o ‘
O o o
0 O o J
. O 0 O 4
-o.05- °° ° 4
0 1 2 3 4 5 6 7
Tlmo($oc)

Figure 4.11: Decelerated and accelerated randomly (ﬂuid-2); Reynolds number varied
from 2050 to 521, unsteadiness factor range 15.22 to ——28.18

73

factors of 15.22 and -—28.18. The large excursions in this case are due to the fact that the

ﬂuid 2 is less viscous than the other two ﬂuids and can be made to undergo more extreme

transients in simple valve opening experiments.

4.2.8 Accelerating and Decelerating Flow Results for Fluid 3

A ﬁnal series of acceleration and deceleration experiments was carried out with a third

ﬂuid with a viscosity intermediate to that of ﬂuid 1 and 2. The acceleration ﬂow experiment

0.11

0.1
0.09
0.08
0.07

0.05

Accumulated Mass (Kg)

0.02 ’

0.01

-o.o1 I

0.06 i

0.04 '
0.03 '

 

 

X 0 0 D

Quasi-Steady ‘

Unsteady correction .
Inferred .
True .

- o
-:ooooooooo°°°

v

 

 

0

Time (See)

Figure 4.12: Initially steady then suddenly accelerated ﬂow (ﬂuid 3); Reynolds number

increased from 620 at I: l to 1440 at t = 4; maximum unsteadiness factor 4.56

74

is shown in Figure 4.12 and illustrates reasonable agreement between inferred and true
accumulated mass, as was found in ﬂuids 1 and 2. The reason for larger discrepancies
between the two results is not known though it is possibly attributable to sensor drift or
imperfect calibration in the higher part of the pressure range. Figure 4.13 below shows
the results of the corresponding decelerating ﬂow experiment. As we might expect, the
effectiveness of the unsteady correction in inferring the accumulated mass is similar to

that observed in previous two cases of decelerating ﬂow for ﬂuid 1 and ﬂuid 2.

 

f 1 r Y 7 v 1 v r

 

 

 

 

. Quasi-Steady 21's
0 Unsteadycorrection .1!" ...*
I
o Inferred ,3 ﬂ.“
. * ‘.
X True 2 .
0.1” gl'A‘. .
‘
A
. 8!. 4
g3
3 9'
v t 'I
3 . 2' .
z -'
I- ' u
1, 0.05 .-
3 . a
s .-
E l'
: I
3 .-
I
I
O-ooooooooooooo....°°°°° ..
009000..
F 909.96.
0.0.4}
l
-0.05““““““““*“‘
0 1 2 3 4

Time (Sec)

Figure 4.13: Initially steady followed by deceleration ﬂow (ﬂuid 3); Reynolds number

reduced from 1520 at I = l to 340 at t = 4; minimum unsteadiness factor —3.90.

75

4.3 Feasibility of Unsteady Pressure Measurement for Flow Metering

The results presented in the previous section (4.2) validate the unsteady laminar
ﬂow metering theory presented earlier within experimental uncertainty. However exact
matches between true accumulated mass and inferred accumulated mass can only be
achieved under certain restrictive conditions as explained in section 3.8. These
restrictions are partly attributed to the pressure sensor behavior. In terms of the
uncertainty in the experimental data, the main sources of uncertainty are the systematic
and random errors in the pressure sensor data as tabulated in Appendix E. The systematic
and random uncertainties in the inferred accumulated mass are found to be 2.7% and
2.0% respectively with an overall uncertainty of about 3.4% for the steady ﬂow results.
This level is comparable to discrepancies between data and theory in the ﬁgures of this
chapter. It is thought that these uncertainties could be reduced by custom design of the

pressure sensor for unsteady ﬂow metering purposes.

As mentioned in section 3.7.2, this pressure sensor responds differently to the
different pressure changes experienced in different ﬂuids, with a more linear response
observed for the higher viscosity ﬂuids. It would seem that low viscosity ﬂuids, like pure
water, would require extremely detailed, non linear calibrations over small intervals of
the pressure range over which they would be used, for this kind of pressure sensor to be
reliable. On the other hand, higher-viscosity ﬂuids used over small pressure ranges might

allow linear calibrations of this kind of pressure sensor.

76

Chapter 5 Turbulent Flow Modeling

5.1 Background
Modeling of turbulent ﬂows poses a closure problem because of the lack of any
reliable constitutive model for terms in momentum equations known as turbulent shear

stresses. There have been many studies of single point closures such as k — 8 , k — a) , and
ufu; models. These single point models can be complicated and hard to calibrate, and

have rarely been constructed for use with unsteady problems. In unsteady ﬂow problems
these single point closure models do not account for the dynamics of the Reynolds
stresses. In general, closures as complicated as systems of integro-differential equations
have been proposed for accurate modeling of unsteady ﬂows, though very few have ever

been implemented or tested.

5.2 Objective

The main objective of this part of the study is to develop a predictive model for

m(t)as a function of dp/dx(t) for turbulent unsteady incompressible duct ﬂows. These

relationships are the counterparts to the exact laminar ﬂow relations developed in Chapter
2. It is intended to form such relationships by seeking solutions of the Reynolds averaged
equations, in forms such that the property of linearity enables the Laplace transform to be
formed resulting in a convolution form of turbulent ﬂow model, similar to the exact
laminar ﬂow relations derived earlier. As a simple ﬁrst approach, a linear eddy viscosity

model with a memory effect is proposed to close the unsteady momentum equation.

Solutions of the resulting equations relating m(t) to 9%(2-can then be formed and used
x

77

to explore the extent to which this kind of modeling can be applied to prediction of

unsteady ﬂows.

5.3 Reynolds Averaged Turbulent Momentum Equation
The Reynolds averaged turbulent momentum equation for fully-developed

spatially averaged pipe ﬂow in cylindrical coordinates can be written as

 

at pax rar

_ r_ ___

a; 13;? va 3518,,
+ [ 3r) rar(ruv) (5.1)

Equation (5.1) is similar to its counterpart in laminar ﬂow equation (2.5), with an

additional terrnlai(m—’Q7), where Tv'is deﬁned as the apparent turbulent shear stress. In
r r

unsteady ﬂows, the Reynolds averaging is carried out over multiple repetitions of the

same event, at the same phase. This turbulent shear stress is often negatively correlated

with @— in shear ﬂows. The problem of modeling it accurately in terms of 17 and other
r

variables is called the closure problem. This turbulent stress can, in general, be predicted
only with a detailed knowledge of the turbulent structure and is related to the local ﬂow

conditions such as velocity, geometry, surface roughness and upstream history.

For the purpose of this thesis, turbulent shear stress needs to be modeled to
predict both steady and unsteady ﬂow through ducts. We denote u0 , p0 , u'vg as the ﬂuid
quantities representing mean velocity, pressure and turbulent shear stress at some initial
time. If we substitute these initial values into equation (5.1) and subtract it from the
general, time-dependent form of equation (5.1), we can write the transient part of the

equation (5.1) as

78

a(:7-r,)_ 1863—720) v 307-17,) 1a .7 _,_,
at ___}; 8x +7[r 3r -:57(r(uv—uv0)) (52)

Setting i7 = t7 —z'io , b = I)— 50 and 175' =W—7t70 as the transient components of u, p

and u'v’, we have

aa rap va air 13
—— + — 3
rr

r Br —-—(ru v) (5.3)
which is the streamwise momentum equation for the transient part of the fully

developed pipe ﬂow.

5.4 Turbulent Shear Stress Model
The modeling approach we attempt is based on a simple algebraic model of eddy
viscosity accounting for the ﬂow transients through a memory effect. Turbulent shear

stress is usually modeled in terms of eddy viscosity as

because 47127 is generally correlated with Eli/Br in experiments on steady ﬂow.

To incorporate the effect of memory in a 447 to aﬁ/Br relation, we model 4477

2—
u
or

Brat

 

as W (t) convolved with

3217
Brat

 

4:77: W(t)* (5.5)

Setting W(t)=v, ,v, =V, (y); recovers the relation: —;717=V, aa/ar in steady ﬂow, and

setting W(t) to some function of time allows an effect of memory to be introduced in a

79

simple way. Since we expect the effect of changes in the shear, BIT/Br , to lead to a

delayed change in the turbulence as observed by Jackson and He [24], we might expect

W(t) to be a function sketched like

 

 

 

 

 

 

 

2 yY T ‘ T‘ V T V VI I YT V I Y YTTTY T Tﬁ 771777'7 rT 2 1 V YT I U I V I l I I I 1 I Y I' Y Y I Y Y I Y] 77 Y TTTj’YT
: i 1
175 _- 1 1.75 _- J
: 1 I j
P 4 v- 4
15 L- 1 1.5 L- -4
125- -: 125 "- -
g 1 E. : a ‘ _ _.
3 : , 3
075} 5 07s :- -
0.5 :- J 05 :- -
P 1 'h
.4
.1
0.25 - 1 o 25 - -
. 1 .
o A l l A l l I A 1 l A 1+LJ A A 1 1 1) LL L l l 1 L J l A A 1 14 o '- A
n 5 1o 15 20 25 30 35 n 5 10 15 20 25 30 35
Tlm(Sec) rams»)

or
Figure 5.1: Expected forms of weighting functions in turbulent ﬂow

It is convenient to model the adjustment of W(t) to its long time value, or

W(t)—W(oo) as scaling on e'”, where T is a turbulent timescale like ior -R—. The
ur “(I

long time value of W(t) aﬁer effects of a transient have died out should be the steady state
value of V, consistent with the equilibrium basis for eddy viscosity theories. Therefore
W(t) is modeled as the sum of an immediate or quasi-steady part and a delayed response,

as W (t) = a[H (t)+ae‘ﬂ’ ]. In this case, the transient part of the turbulent shear stress is

modeled as

~

azu

Brat

 

4’17 = [H (t)+ae‘l"]a* (5.6)

80

where a has the units of eddy viscosity, and H is the Heaviside step function, and

~ denotes the transient component. Equation (5.6) can be expressed as the

changes in —u v from some initial value as

—W(t)-(—W(0)) -_- aH(t)[a:£t) - 378:") ).....—w (3:5) - 3:53)]

 

 

(5.7)

The ﬁrst part of the RHS of the equation (5.7) represents an instantaneous or
quasi steady response of the turbulence, whereas the second part represents a time
delayed or history response to the turbulence. The relative initial magnitude of the overall
turbulence response depends on the values of the parameter a and a, with parameter

,6 controlling the rate at which transients in shear stress reach a steady state value. For
any initial time! = 0, the value of awould signify the quasi-steady change in the —W
for any change in Eli/Br. Depending on the values of a and ,6 , there can be an

immediate or delayed response of turbulent shear stress which would only approach to its
steady state value at long times. The contribution of a is to determine the immediate

offset in the initial steady turbulent shear value, whereas ,6 determines how fast this
offset relaxes to its steady state value. The higher the value of ,B, the faster it would

approach the steady state value.

5.5 Calibration of the Model Parameters a, ,B and a

5.5.1 Calibration of a
The parameter a has the units of kinematic viscosity and, because, for simplicity,

it is set to a single value for all r-values, it represents the average value of the apparent

81

eddy viscosity across the duct. Its value is chosen such that it provides a match with
steady ﬂow rate to pressure drop data in ducts. In equation (5.7) a can be expressed in

terms of Laplace-transformed variables as

a = V' , (5.8)

C' [It/(S)]

 

l-)°°

where W( (s): £{W W(=t)} £(H (t)+ae‘ﬂ’}=l/s+(a/s+,6). For ﬂows at different
initial Reynolds numbers, it is expected that a might change, just as v, does in

measurements made in steady ﬂows. v, (apparent eddy viscosity) is then modeled as
i = xRe, (5.9)
v

where K‘ is a calibration constant dependent upon the Reynolds number. Arpaci and
Larsen [42] have found a similar expression for turbulent ﬂow through tubes at Reynolds

number of as high as 50,000, as v,/V=Re,/15 , where Rer =u,R/V.Here u, is the

friction velocity of the ﬂow and is deﬁned as Jr”. / p and R is the radius of the pipe. It

follows that setting It to 1/15 times the molecular kinematic viscosity allows a to be

Re

I

E'[::-+%+,B].

calibrated as a =

5.5.2 Calibration of ,6
The parameter ,6 is a time constant which causes the delay as the instantaneous

shear stress relaxes to its steady state shear stress value. This is chosen to scale on the

characteristic velocity and characteristic length scale of the ﬂow, i.e. uT/R or

82

u0 (t)/R,based on dimensional reasoning. Its value, as a multiple of ur/R or u(t)/R,

must be found empirically from transient ﬂow experiments.

5.5.3 Calibration of a

The parameter a determines the magnitude of the unsteadiness response of the

. Bu .
turbulent shear stress to the changes 1n a—, as a part of the memory effect. Prror to
r

assessing model performance against experimental data, it is not clear if a may be
calibrated as a constant along with the value of 0 depending on the initial ﬂow conditions

or if it must be represented as a measure of turbulence response to imposed transients

time scale such asa = (u,/(dUCL /dt))(u,/R). This calibration must also be carried out

empirically, using data from unsteady ﬂow experiments.

In order to check the reasonableness of the proposed shear stress model in
equation (5.7), the model was compared with the ramp up transient experimental data of
Jackson and He [24]. The parameters of equation (5.7) were calculated from the
experimental data as a =4.5 a was taken as -8, since this value seems to give a
comparable delay in the response to shear transients of u'v’as can be seen from Figure

5.1. This value was used subsequently to test the model against the experimental data of
this study. Parameterﬂ =%was calculated from the initial ﬂow conditions of Jackson

and He [24]. Two data series were selected from their experimental data to indicate the
variation of Reynolds shear stresses at y = 1.9mm (y+ = 17.1) and y =12mm ()2+ = 120)
in a 50.4 mm diameter pipe during a ramp up transient in ﬂow rate. These data series for

measured values of —u'v'(t)—(—u'v’(0)) are plotted in Figure 5.1 for the ﬁrst 5 seconds

83

of this constant acceleration experiment together with the model prediction for the span-

averaged value of —u'v'(t) (5.7). The shear stress model of (5.7) shows a reasonable

agreement with the experimental data insofar as it represents an average eddy viscosity
behavior across the pipe quite well. It is evident from the Figure 5.1 that ‘average’ u'v'
predicted by the model shows the same order of magnitude and history as that of the

experimental data of Jackson and He [24] for a ramp up acceleration.

 

   
   

 

 

 

0.“)16 LIJuI‘TI I ly-I121o I r l I I I I l I T T I 1 I I I;
l- ' "I" A 0 _
E MW y-12.0mn__v.__ :
0m14 h Jill-”random with!“ + _:
E mold-sandy y-1.9m e E
.. WWW _
QW‘IZ '_- .3
0.001 :— .3
NA '- ‘ :1
8 E .1
ﬁ 0.0008 _- _
E - I
T; : :
’: 0.m r -‘
L- —:
0.0004 [- 0:
0.0002 5- q
E 3
O —
‘O.W2 I l l l J L I l l I L l l I I I J l l I l l l I l d

0 1 2 3 4 5

t(sec)

Figure 5.2: Comparison of development of Reynolds shear stress for a linear ramp
acceleration transient for 5 seconds from Reynolds number of 7000 to 42500 in a 50.4

mm diameter pipe.

84

5.6 Laplace Transformation and Solutions

The Laplace transform technique elegantly transforms the PDE in equation (5.3)
to a linear ODE, for which a solution relating m(t) to Bp/ax can be found in
convolution integral form, as was done in the unsteady Iarninar ﬂow case. Laplace

transforming equation (5.3) with .C [817/ at] = 31') —- u0 and rearranging,

 

2 .. A A 3A .
dr r dr r 3r p ﬁx
The model proposed in equation (5.6) can be rewritten as
’7"! a~ at
—uv (t)=W(t)* ff): ) (5.11)

where 7(r,t)=aaﬁ/Br. In the Laplace transformed domain, equation (5.11) can be

rewritten as
—17;'=W(s)s](r,s) (5.12)

Here W(s) is a function of s only i.e. turbulence is assumed to be well mixed with the

same average turbulent memory across the entire ﬂow. Similarly

d fr, d A A d“ 1
El} v]=E[W(s)sf(s)]=sW(s)f—(d:—L)- (5-13)
Thuswehave
—t7;'=SW(s)[ad—ﬁ) (5.14)
dr

85

and

”d (17;) A all:
T=SW(S)(G;{—r-2-] (5.15)

Substituting W (s) , equation (5.14) and equation (5.15) into equation (5.10), we have

 

deii :9: l s+ﬂ+as aﬂ
s+ﬂ dr

dr2 r dr r

 

 

 

 

(5.16)
s+ﬂ+as dzu . 1 313
+ —— a— —su=——
s+,6 dr2 pax
Substituting 1:" =(1/p)(af2/ax) in equation (5.16) and rearranging
d—“2’+lﬂ— S a: F (5.17)
dr rdr v 1+(s+ﬂ+as)a v l+(s+,6+as)a
‘ V(S+ﬁ) V(S+ﬁ)

The ordinary differential equation (5.17) is then solved with the same boundary

conditions as for the laminar case and yields the following results:

JO 4“,:—
Xv

.-

 

 

 

 

 

 

ﬁ(r,s)=f- : 1 -l (5.18)
S s
.1, -iR —
_ XVJ -
where
X=1+3[1+ as J (5.19)
V s+ﬂ

Changes in the mean velocity, and hence ﬂow rate, in the Laplace transform

domain may be found by integrating the transform of the momentary velocity ﬁeld over

86

the cross sectional area of the pipe. Using equation (2.13) and substituting (5.18) into

(2. 13) and simplifying,

 

 

 

 

 

U(s)=F—

 

 

 

 

 

 

 

 

. . \I[1+3[”s:sﬂ)}. .

 

(5.20)

 

where II is a modiﬁed Bessel function of ﬁrst kind of order one and J0 is the Bessel
function of ﬁrst kind of order zero. Multiplying and dividing by s to allow inverse

Laplace transformation, equation (5.20) becomes

. . l

21. R [
“Hg—[1+ as Dv
_ v s+ﬂ . _1 (5.21)

1 s H. s
R¢[1+§[1+Sisﬂ))vj RJ[I+§[1+S‘SB)}‘ .

Here 313' = GPA/8t + 132,6 (t). Taking the inverse Laplace transform of the equation (5.21)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

87

U(t)=%—f*£'

 

 

21I

 

 

 

 

 

 

R

_ (that

 

 

_,R l

 

S

 

 

 

II

a
1+—
v

[1+

as

s+ﬂ

 

 

l}.

 

 

(5.22)

with the symbol * denoting the convolution operation. We can rewrite this equation as

3F
5*710)

U(t)

where H(t)= E' [7765)] and ﬁ(s)is given by

21l

 

 

 

 

 

 

 

1'1

 

S

 

S

 

a
1+—
v

 

(1+
S

 

as

+ﬂ

 

) 5"”1’11

as

s+ﬂ

 

 

 

ll- .

(5.23)

(5.24)

The mean velocity to pressure gradient results can be derived from equation (5.22) as

U(t)=U(0)+J

0

t 1 32p
La“

1 82 p
p axat

—t’)———(O)]E' [52(3)] dt’

Cumulative through-ﬂow may be found by integrating equation (5.25) as

r I ( 18p ’ lap
Ut dt= U 0 dt+ —— t—t __—
l, 0 l, 0 1w max

88

(5.25)

(0)]2' [771 (3)]dt' (5.26)

The ﬂow rate to pressure gradient relationship can be expressed as the sum of a steady

state and a transient contribution by rewriting

C'[]=E'I],_...+(f'[]-C'l1......)

Then equation (5.26) can be expressed as

0 0 pax

"LN/19%? t'-—) iip(o)](c'[H(s)j—c'[ﬁ(s)]w)dt'
(5.27)

JuU(t)dt=J-IU(O)dI+J‘ [£30— z'—) ia—Pm) E'[7\i(s)]l_mdt'

 

Using the residue theorem, at s = 0 or at I: oo , we have

E'[72(s)] =—- R2 (5.28)

H... 8(a+v)

 

Now for steady ﬂow,

Jul/(0) day-[139539) £'[7A1(s )],_...= 8p (aw) I 8” (0)21: (5.29)

0 0

 

For initially steady ﬂow, equation (5.27) becomes

i030” T—pwa? zap—Lt)”

+J: [pg—:(t— t-') 13:10))(c'[ﬁ(s)]—c'[ﬁ(s)]w)dr

(5.30)

where E'[’H (s )]— C'[7’:{(s)]

is a weighting function and [' [7?(s)] is given by

l—)00

89

 

 

 

 

 

 

 

 

 

 

 

 

     

 

 

 

 

 

Zesjr 1
m+2s+R2_m+4s+2Rza a _ (its+ s?”
.7? _R, 2:» 2v 2v2 (swﬁ) (Suﬁ ’
[ (S)] 8(a+V) 23’7’
+
m"2s"R2_m"‘s‘2Rzas a _ as‘ s’
2v 2v2 (s‘+,6) (s'+ﬂ)2 ’
(5.31)d
with
j=222[— [ﬂ—szﬁ. (1+3 (HOOD)
“Ital—+1) R2 11“” (1+a)]]-4nfﬁ—W(1+3]
V v
:= CH” f[1+3(l+a)]]]
‘ 2R v v
—\/['6R+n R2 ”(1+ (1+a))]—417f—’6£(1+-a-]
V V
r )%
m+= 1
1+£[1+ fly ]
K V s +ﬂ)
and

90

l
MEI—[1+ _aS ]
K V s +ﬂ )

The behavior of the weighting function E' [ﬁ(s)]—C' [712(5)] is controlled

l—no

 

 

 

 

by the three parameters: a, ,6 and a. The parameter a represents the overall eddy
viscosity and governs the initial magnitude of the weighting function. As explained in
5.5.1, it should scale in proportion to Reynolds number. The effect of the magnitude of

,6 is not very signiﬁcant during the initial stage of the weighting function because of its

(243’ dependence. The initial response remains the same for short times and then it starts to

relax to zero at faster rates with increasing ,6. In comparison to the effect of parameter [3 ,

the effect of a is more pronounced. It is more significant during the initial phase of the

weighting function, for ﬁxed values of a and ﬂ In this phase, the weighting function

decays faster for lower values of a ,and decays slower for higher values of a. In essence,

for higher Reynolds numbers and values of [3, the lower values of a will cause transient

turbulent shear stress to reach their quasi-steady turbulent shear stress values faster.

Equation (5.30) is similar to equation (2.43) and offers a similar solution to the
unsteady pipe flow problem in the turbulent regime. This equation represents a quasi-
steady term supplemented by an unsteady correction term to account for the ﬂow
memory during transients. As can be inferred from equation (5.30), the quasi—steady term
underestimates the ﬂow rate during deceleration and overestimates it during acceleration,

as was the case for laminar ﬂow.

91

Chapter 6 Turbulent Flow Experimental Results and Discussion

6.1 Turbulent Flow Experiments

The main objective of the turbulent ﬂow experiments was to determine the
applicability of the turbulent shear stress model developed in Chapter 5 in unsteady
turbulent duct ﬂows. In this series of turbulent ﬂow experiments, the methodology used
in laminar ﬂow experiments was followed. However, due to limitations of the
experimental setup for turbulent ﬂows, only steady, accelerating and decelerating
turbulent ﬂows with varying accelerating and decelerating flow rates were studied. In
these experiments two working ﬂuids were used. One ﬂuid had a kinematic viscosity 0 of
1.414><10'6 mz/s and a density p of 1040 kg/m3 and the second had a kinematic viscosity 0

of 1.515X104’ mz/s and a densityp of 1030 kg/m3.

In accelerating ﬂow experiments, the acceleration was varied from 0.04 to 0.26
m/s2 and, in decelerating ﬂow experiments, the deceleration was varied from 0.11 to
0.325 m/sz. All transient turbulent experiments were designed to ensure that Reynolds
numbers remained above 3500, thereby avoiding relaminarization and the transition
region during acceleration and deceleration transients. The departure from the quasi-
steady state was ascertained using the turbulent unsteadiness parameter, a turbulent

counter part to the laminar unsteadiness parameter, deﬁned as

l du R2
mE(t)v(ﬂ+1) (6.1)
V

which is large during very severe transients.
92

6.2 Model Calibration

The parameters a, a and ,6 given in the turbulent shear stress model

2~
,7, _ u . . .
—u V’=[H (t)+ae ﬁ’]a*— must be calibrated from unsteady flow experiments if
r 1

they are to be used to determine the accumulated mass according to the unsteady

turbulence model developed in Chapter 5, which is

 

 

 

 

 

 

0 (5.30)

( lap I lap I A | A I

”Ea—x" t —p3_x(0)] (E ”Ml—E [H(S)]lﬁw)dt

where
" r — _
21l R l S
«Hangs/3D,,

ﬁ(s)=i ' ~ — __1 (5.24)

 

 

 

 

RI 3 .r 3

J0 —1R
Jil+g(l+sisﬂijv _ dilﬁLSiHsisﬂDi/d

Initially a rough range of parameters was explored by keeping one parameter ﬁxed and

 

 

 

 

 

 

b

changing the other two. This gave us an insight into how parameters such as a and ,B
affect the magnitude and decay rate of the weighting function, thereby controlling the
ﬁnal values of the accumulated mass inferred from pressure drop measurements in the
turbulent flow experiments. These parameters should ideally be modeled in terms of the

ﬂow properties. The dimensionless parameter a was initially modeled

93

as(u, / ( dUCL / dt))(ur / R) , as a scaling of the degree of unsteadiness, but this dependence

was not consistent with experimental results. When tested against Jackson and He’s [24]
experimental data, a value of a = 8.0 yielded a comparable delay in the adjustment of the
Reynolds shear stresses after imposing transients in their pipe ﬂows. at was then
tentatively modeled as if a constant for all types of turbulent transient pipe ﬂows,

including the ones measured experimentally in this study.

The reciprocal time scale ,6 was initially scaled on ad / R. The relatively large
values of um, and small pipe radius made changes in this parameter too extreme to match
the data of this study so this was later rescaled on u, / R where u, is the friction velocity at
the start of the transient. It was also noticed that values of ,6 of 30 or above do not affect
the weighting function signiﬁcantly and the decay rate remained almost unchanged at
higher values. The data of this study could be matched more closely by setting [3 equal

to a multiple of u,/ R with u, calculated from initial flow conditions. For the turbulent

ﬂow experiments of this study, ur (0) / R varied from 9 to about 19 sec". The data could

be matched by settingﬂ=1.0%)—, where the 1.0 is the calibration factor,

u, = ,[rw/p ,1, = 0.0395pm”"Isa—"35a''75 and Re, = utd/V.

Parameter a, which has the units of eddy viscosity and represent the average eddy
viscosity across the pipe, was also calibrated against the steady ﬂow. For each transient
ﬂow, the initial Reynolds number during steady ﬂow was used to determine the value of

K'VRe

T

a i.e. a = 1 . for that particular experiment, which is also consistent with the
(I
E' — + — + ﬂ]
s s

94

expression V, / V = KRe, given by (Arpaci and Larsen [42]). It was noticed that calibration
of a during steady ﬂow i.e. 0(0) has a vital effect during the transients and it needs to be
calibrated precisely. Ideally, the value of a(t) should be based on the momentary value of

u,(t)at each time step in a transient ﬂow to model the quasi-steady eddy viscosity

accurately. A more convenient way of including the dependence of eddy viscosity on

u,(t)is to note that u, ~,/3p/ax in steady ﬂows (White [47]), so we can write

v-
d 1 dx
a(t) = 404%] , where dp(t)/dx is the pressure gradient during transients and

dp(0)/dx is the pressure gradient before the start of transients. In this way, we can scale

the eddy viscosity only at the start of the transients and on the measured instantaneous

pressure gradient. In summary, calibration against experimental data led to the following
values for the model coefﬁcients: a = 8.0; ,B =1.0%9-)-,a(0)=0.0145 VRe,(0) which

depends only on initial flow conditions. These values of calibration coefﬁcients are used

in all predictions given in the following sections.

6.3 Experimental Results

In the subsequent sections, selective results from a range of experiments are
presented, with slow, moderate, and severe acceleration and deceleration rates. In all
experiments transients were imposed on an initially steady ﬂow, as was done in the
laminar case, to allow calibration of a(0). The pressure sensors were calibrated prior to
these experiments against a manometer over a range of steady ﬂow rates that bounded the

pressures experienced in the transient flow experiments.

95

6.3.1 Steady Flow Results

Before conducting the accelerating and decelerating ﬂow experiments, the

applicability of equation (5.30) was checked against steady turbulent ﬂow experimental

 

 

 

 

data.
I 2 !a t!
JU(t)dt=—§'—(-R—+—SJ-£é(——)dt’
a V x
° p ° (5.30)
' lap , lap |A IA '
+ —— t—t -—— 0 (C ”H —[ ”H )dt
[1”; > pa; >) [ m] [ (s)],_,,
0.4,-
’ ‘ QuaSI-Steady . .
° Unsteadycorrection I
’ o Inferred l
0.3b " True I n .
’ l
. I i
A h I 4
an » .
if; 02- , ' .
g i I l
'3' r .
5 0.14 _ ' .
2 . . .
r I 4
: - ‘
0": o o o o o o o o o o o o o o o o o o o-
0.10.-..i...-é_...é...,4
Time(Sec)

Figure 6.1: Steady ﬂow at a Reynolds number of 5250
96

In the steady ﬂow experiments, the Bp/Bx term remains unchanged so that
(Bp(t—t')/Bx)-(8p(0)/8x) = 0. Obviously there is no unsteady correction value so the

accumulated mass predicted by the pressure gradient data, and measured by the load cell
data can be compared directly. Figure 6.1 shows the cumulative liquid mass in this
turbulent steady ﬂow case with quasi-steady, inferred and true accumulated masses
yielding exactly the same results. The signiﬁcance of this agreement between the data is

that the eddy viscosity V, and the pressure transducers are correctly calibrated at this

ﬂow rate, and that the unsteady correction algorithm performs correctly. In this ﬁgure,

t

of 0 to 1050.
R/uo

 

the time range of 0 to 4 seconds corresponds to

6.3.2 Accelerating Flow Experiments

The accelerating ﬂow experiments can be characterized by the unsteadiness factor
(6.1) imposed during the transients with varying magnitudes of acceleration ranging from
0.25 to 0.9. In this series of experiments, the flow rate was controlled manually by
opening and closing the valve, as in the laminar ﬂow experiments. The initial Reynolds
number at the start of the transient was chosen to be around 3900 and it was increased to
about 5300 over different time periods. Ideally a much broader range of minimum and
maximum Reynolds numbers would be desired, but these experiments were restricted to
this Reynolds number range because of the limited elevation of the header tank and the

corresponding maximum ﬂow rate through pipe of this length and diameter.

In the slow acceleration experiment shown in Figure 6.2, the ﬂow was initially
kept steady for about one second and then was gradually accelerated. In this ﬁgure, the

cumulative mass is plotted as a function of time, together with the quasi-steady, unsteady
97

0.4

0.35

0.3

0.15

Accumulated Mass (Kg)

0.1

0.05

Figure 6.2: Initially steady ﬂow that is then gradually accelerated, Reynolds number

increased from 4176 to 5128, maximum unsteadiness factor 0.25, Re, (0) = 293

correction, and unsteady inferred cumulative masses determined using the model and
calibration coefﬁcients presented in 6.2. It is evident from this ﬁgure that the quasi—
steady term overestimates the accumulated mass during acceleration and that this
overestimation is corrected almost perfectly by the unsteady correction data, resulting in
a good agreement with the true accumulated liquid mass measured by the load cell data.
A similar trend is found in the moderate acceleration experiment, the results of which are
shown in Figure 6.3. We observe a signiﬁcant deviation between the quasi-steady

accumulated mass and the true accumulated mass. This over-prediction by the quasi- steady

 

0.25

0.2

X00

 

' v v v - v v

v w — v v v v w v v v v w v v v v v f v v

A A A A-A A A A-LAAA‘A A A AAA—LJ_A-AAAA;J A A A -AAA AAA A A

X0

4 m A

A A L . A

 

 

Quasi-Steady
Unsteady correction
Inferred
True
A I
A I
, 3
‘ s
. u
a
8
I
I
o o o o o 0 ° ° ° °
1 15 2 2 5
Time (Sec)

98

o v r v v1 rf‘virv v v v v v v v v v v v v v w v v v v v v
I

 

 

 

 

e Quasi-Steady ‘ 1
0 35 i. Unsteady correction .‘
' ; ° Inferred ‘ l
I x True A 1
0.3 _- ‘ ' 9 1
E . . ;
a 0.257 ‘ , .1
35 : 2 .
m _ A 2
g 0'2 T . 2 1
g : - R :
_ ’ A g .
g 0.157 ‘ I .‘
8 P A 1
< 01 - - 8 .‘
. a 1
_ 2 . .
> . ‘
0.05 :- ‘ o o o o '2
Z ' o 9 ° 1
. 0 ° 0 0 °
0 n: o 0 ° 0 .
-o.05’ ------------------------------------ ‘
0 0.5 1 1.5 2 2.5 3 3.5

Time (See)

Figure 6.3:- Initially steady ﬂow that is moderately accelerated; Reynolds number

increased from 3800 to 4700, maximum unsteadiness factor 0.53, Ref (0) = 160

measurement is compensated for by the unsteady correction term resulting in a good
agreement between the predicted and true accumulated mass data.

Figure 6.4 shows the results of a severe acceleration case, in which a larger spread
in accumulated mass data between quasi—steady, and true accumulated mass data is
observed. This deviation from the quasi-steady value is completely compensated for by
the unsteady correction term, resulting in good agreement between the true and inferred

accumulated mass data.

99

0.35

 

 

 

 

i . Quasi-Steady .
i o Unsteady correction ‘
0,3 .. o Inferred ‘ J
i X True 1
0.25 b A "
A o
50 t A ‘
as o 2 ’ 2 " ‘
a ' T . .
E a
g : A I i
8 1 5
< . 3
p o s
0.1 1' ‘ 8 o o -
I 0 °
C ‘ I ° 1
0.05 ' , a 0 ° .1
’ g o
Q o 0 ‘
o L 2 9 ° J
0 0 5 1 1.5 2 2 5
Time (See)

Figure 6.4:- Initially steady then suddenly accelerated ﬂow, with Reynolds number

increased from 3750 to 5200, maximum unsteadiness factor 0.90, Ref (0) = 192

6.3.3 Decelerating Flow Experiments

The decelerating ﬂow experiments were conducted in a similar manner to the
laminar decelerating ﬂow experiments. The ﬂow was initially kept steady before
decelerating from a higher Reynolds number to lower Reynolds number by partially
closing the valve. In this case again varying magnitude of deceleration rate were imposed
characterized by the unsteadiness factor ranging from 0.25 to 0.72. Figure 6.5 shows

experimental data for a slow deceleration case, in which ﬂow is initially kept steady for 3

100

seconds and then gradually decelerated. During steady ﬂow there is no deviation among
the quasi-steady, inferred, and reference accumulated mass since the unsteady correction
is zero. Aﬁer this phase, the quasi-steady term underestimates the accumulated mass which
is in tum compensated for by the unsteady correction term. The resulting agreement

between the inferred and true accumulated mass is within a few percent and is very good.

 

 

 

 

008? vvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvv
» ‘ Quasr-Steady
0 7’ ° Unsteady correction
. P
: ° Inferred 858‘
I X True “8.‘A‘:
0.6h "“‘ u
C I A l
’ ﬁ‘A .
. '2‘. .
A 005. 2“ 1
O0 ’ gA .
a 04' I. 4
2 . ’ l' .
. ‘ ‘
g I ‘3
'5 0.3- ; .1
E I ,3 ,
3 i ‘ :
002- “ I
i ‘ 1
. I
. ‘5 t
0.1” 3
. ' ,
P.‘ J
l
O ooooooooooooooooo°°°°
0.000909999000000:
o1 --------------------------------------- ‘
0 1 2 3 4 5 6 7 8
Time(Sec)

Figure 6.5:- Initially steady ﬂow which is then slowly decelerated, with the Reynolds

number reduced from 5255 to 3900, maximum unsteadiness factor 0.25, Re, (0) = 324

Figure 6.6 shows experimental data from a moderate deceleration case with initially

steady ﬂow. In this case, a fairly good agreement is observed between the inferred and

101

true accumulated mass with the quasi-steady term underestimating the accumulated mass.
Figure 6.7 shows data from a more severe deceleration case, as can be seen from the
larger departure of quasi-steady accumulated mass data from the true accumulated
mass data. The same trend of underestimation of accumulated mass is seen for the quasi-
steady data, with the unsteady correction yielding a good agreement between the true and

inferred accumulated masses.

 

 

 

 

005 > """"""""""""""""""""""""""""""""""" 4
; ‘ QuaSI-Steady I, 3 51
0,45 _- ° Unsteady correction I I 1
I Inferred . I ‘ A - ‘1
0-4 1' x True I ‘ e ‘ 1
L I ' A I
b ‘ 1
0.35 _' I A d
. . A ‘
A : - ‘
on 0.3 - . -
Eﬁ I '.'
m I l
g 0.25 r I -
, I .
3 i I 1
.g 0.2 "' I 'i
g . I .
o r
o .1 b ' u
< 0 5 i I ,
P 4
0.1 C- . ' -:
i I I
0.05 - i -*
r - 4
1
0 -o o o o o o o o o o o o o o e o o o o o d
i 0 0 0 1
r 0 0 0 0 4
-0.05 - ..................................................... 2-
0 0 5 1 1 5 2 2 5 3 3 5 4 4 5 5 5 5
Time (See)

Figure 6.6:- Initially steady ﬂow which is then moderately decelerated, with its Reynolds

number reduced from 5196 to 3900, maximum unsteadiness factor 0.509, Re, (0) = 314

102

 

0.4 p """"""""""""""""""""""""
» ‘ QuaSI-Steady x;
. . X
0 35 . o Unsteady correction x o c.
' : o Inferred 5 ° ‘
i X T1116 8 g A ‘
0.3 f- 3 . e .
1 " . ‘
. I . I
M . , .
V t g .
VJ . A 4
8 0.2 u- 2 A "
2 » 2 . .
'8 I .
a- » 2
= 0.15 ' g '-
5 1
8 i 8 .
< 0.1 - ' .
i ' I
, I 1
0.05 f I ".
. I .
’I .
0 no 9 o o o o o o o '
o o o o 4
. O 9 0 Q 0 1
t o O o o .4
-0I05 AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA

 

 

 

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5
Time (Sec)

Figure 6.7:- Initially steady flow which is then rapidly decelerated, with its Reynolds

number reduced from 5210 to 3500, maximum unsteadiness factor 0.72, Re, (0) = 318

Based on the excellent agreement between experimental and modeled cumulative
mass in accelerating and decelerating turbulent ﬂows, it appears that the time-dependent
model for the effect of Reynolds shear stress on ﬂow rate shows promise for prediction of
fully developed ﬂows of arbitrary unsteadiness. For a simple linear model of a non linear
effect like turbulence, it is signiﬁcant that it is equally applicable to both accelerating and

decelerating ﬂows, without requiring further modeling or calibration. This ﬁnding is

103

interesting as some decelerating ﬂows have also been observed to be de-stabilizing and

the model can not account for such behavior.

The model requires three calibration constants, one of which is for the average eddy
viscosity of the equivalent steady ﬂow at the same pressure gradient/ﬂow rate, and the
other two are concerned with the unsteady response. The calibration constant a
determines the delay in the development of turbulent shear stresses aﬁer imposing the

transients and ﬂ effectively determines the short time response of the average Reynolds

stresses. According to this calibration, the turbulent ‘memory’ time scale is

~ 1.0 R/u, compared with the memory time scale of laminar ﬂow of ~ 0.7 R2 / V (Brereton

[15]).

104

Chapter 7 Conclusions and Recommendations

In this dissertation, a new approach for estimating unsteady ﬂow rates in real time
is proposed and experimentally investigated. We ﬁrst presented an exact analytical
solution to the unsteady laminar Navier-Stokes equations for incompressible ﬂuids in
ducts. The applicability of these solutions was then assessed in an experimental test
facility mimicking an unsteady ﬂow metering device. Using a time-history turbulence

model, these laminar ﬂow solutions were then extended to turbulent ﬂows in ducts.

In the development of turbulent ﬂow solutions, Reynolds shear stresses were ﬁrst
modeled in terms of eddy viscosity with memory effects. This model was then compared
with the other researchers experimental data for validation. The Reynolds averaged
momentum equation for duct ﬂows was then solved analytically using the inverse
Laplace transform technique. This turbulent ﬂow solution was then calibrated and applied
in the same experimental facility to test its applicability in real ﬂow problems. The results
show that reliable instantaneous unsteady ﬂow metering is possible for laminar ﬂows and
that a simple time-dependent turbulence model appears to extend the technique to

unsteady turbulent ﬂows.

The main ﬁndings of the present study are summarized as follows:
0 Unsteady ﬂow metering in ducts, based on pressure drop measurements and
an exact solution of the laminar, fully developed unsteady Navier Stokes

equations, appears to be practical in pipe ﬂows.

105

0 The metering technique was demonstrated in a range of laminar unsteady pipe
ﬂows, in which it predicted almost perfectly the cumulative mass of liquid in a

range of different transients.

0 The successful implementation of this metering technique requires
design/selection of pressure sensors with high sensitivity, high frequency
response, high signal to noise ratio, and minimal zero drift, that can be ﬂush-
mounted in the pipe wall. There may be better choices than miniaturized piezo-

resistive sensors.

0 A simple turbulence model for unsteady bulk ﬂow as a function of pressure
gradient, which essentially replaces a laminar/molecular viscosity with eddy
viscosity in the ‘ﬂow-memory’ term, appears to provide a comparable way of

predicting unsteady ﬂow rates in turbulent duct ﬂows.

0 A three-coefﬁcient model for unsteady turbulent bulk ﬂow was calibrated and
used to predict cumulative mass as a functional of pressure gradient in a range of
accelerating and decelerating ﬂows, using a single set of coefﬁcients, with good

accuracy.

0 Unsteady turbulent ﬂow rates under— and over-shoot the quasi-steady ﬂow
rates during decelerating and accelerating ﬂow in the same way that laminar ﬂows

do.

0 The success of a simple time history model for the average eddy viscosity in
transient pipe ﬂows suggests that this modeling approach might be useful in other

turbulence modeling explorations.

106

The following recommendations are made for future work:

0 Miniature pressure transducers (with good spatial resolution) that do not drift

over time should be selected for unsteady ﬂows over longer time periods.

0 The applicability of the turbulence model should be tested against other

unsteady ﬂows like pulsatile or oscillatory ﬂow.

0 Similar metering approaches could be developed using m ~ rfunctional

results derived in a similar manner.

107

Appendices

108

Appendix A

Experimental Conﬁgurations

In order to achieve the objectives of this research, a number of variations in the
experimental instrumentations and setup were implemented before reaching the ﬁnal

conﬁguration. These conﬁgurations are listed below in order of implementation.

Conﬁguration 1 Initially an ACL 8112 HG data acquisition card was used for the
purpose of data acquisition from the pressure sensors and Sentra electronic weigh scale.
This card was discarded since it was not straightforward to program for continuous data

acquisition.

Configuration 2 The data acquisition card was replaced with PCI-6023E card from
National Instruments. This card enabled us to acquire buffered, continuous data and could
be fully conﬁgured with the operating software of LabVIEW 4.0. However, a Sentra
electronic weigh scale, which was connected to the serial port through an RS 233 cable,
had a much slower data rate than the pressure drop data from pressure sensors, which was

unsuitable for simultaneous mass and pressure-drop measurement.

Conﬁguration 3 The Sentra electronic weigh scale was replaced with load cell of 10
lbs capacity, which enabled faster acquisition rate. It was noticed that the two pressure
sensors had different thermal zero shift and zero offset characteristics under the same
ﬂow conditions. This resulted in different pressure sensor readings for the same ﬂow

rates.

Conﬁguration 4 The two pressure sensors were replaced with one sensor. The

pressure drop was then recorded with the single pressure sensor upstream with ambient

109

atmospheric pressure at the exit of the ﬂow. The 10 lbs load cell also had a lower
sensitivity. Therefore it was replaced with a load cell with a range of 2 lbs giving higher

sensitivity.

Configuration 5 In conﬁguration 4, the pressure sensor was excited with 10 volts
DC continuously, which resulted in the thermal zero drift in its output. The data
acquisition was then modiﬁed to excite the pressure sensor at 10 volts for a 5% of the
duty cycle, at the end of which data was acquired. The excitation voltage was later on
reduced to 5 volts DC with the output of the sensor ampliﬁed using Entran’s IEM-
15/05/1000R-WW ampliﬁer. In turbulent ﬂow, the same conﬁguration was kept except

for using a load cell with a capacity of 5 lbs to accommodate larger accumulated masses.

110

Appendix B Pressure Sensors

B.1 Strain Gauge Pressure Sensors

Strain gauge pressure sensors originally used a metal diaphragm with strain
gauges bonded to it. A strain gauge measures the strain in a material subjected to applied
stress. Metallic strain gauges depend only on dimensional change in resistance. A stress
applied to the strip causes it to become slightly longer, narrower and thinner, thereby
increasing its resistance.

Semiconductor strain gauges are widely used, both bonded and integrated into
silicon diaphragms, because the response to applied stress is an order of magnitude larger
than for metallic strain gauges. When the crystal lattice structure of silicon is deformed
under applied stress, the resistance changes. This is called a piezoresistive effect. The
most commonly strain gauges used in pressure sensors are: deposited strain gauges,
bonded semiconductor strain gauges and piezoresistive integrated semiconductor strain

gauges. The layout of the Entran EPE low pressure piezoresistive sensor is shown in

  
 

 

 

Figure B. l.
Epoxy Steel Casmg
Layer \. A//Diaphragm and
Protective \ Strain Gauges
Screen ‘ ,.,//

Figure B. 1: Typical layout of the pressure sensor

111

B.2 Piezoresistive Integrated Semiconductor

Integrated circuitry (IC) processing is used to form the piezoresistors on the
surface of a silicon wafer. There are four piezoresistors within the diaphragm area on the
sensor. Two are subjected to tangential stress and two are subjected to radial stress when
the diaphragm is deﬂected. They are connected in a four element bridge conﬁguration as

shown in Figure B2 and provide the following output:

V... /V... = 13% (B. 1)

where: V“. is supply voltage; R is base resistance of the piezoresistor and AR is change

in R with applied pressure and is at least 2.5 % of the full R.

Vcc = Supply voltage

 

GND

Figure B.2: Four element bridge conﬁguration

The almost linear elasticity of the single crystal silicon is utilized by etching the
back of the wafer to form the diaphragm as shown in Figure 8.3. In the integrated
construction, the high output of the bonded strain gauge is combined with the low

hysteresis of the deposited strain gauge.

Bond Buried
pads piezoresistors

/— w "“—" “\

Etched cavity

      

 

_/

 

Figure B.3: Diaphragm of a piezoresistive pressure sensor

These piezoresistive pressure sensors can be conﬁgured to provide the absolute,
differential or gauge pressure readings, depending on the reference. In Figure B.4, the

diaphragm is shown deﬂecting under applied differential pressures.

(A)

 

 

113

(B)

 

 

 

P2>P1

P2

Figure B.4: Deﬂection of diaphragm under pressure

The circuitry needed for ampliﬁcation, temperature compensation, and calibration
is also included on the same IC. Various pressure ranges are accommodated by varying
the diaphragm thickness and, for very low pressures, by selecting a large diaphragm
diameter. Also, the small size means that it has a wide frequency response and may be
used for dynamic pressure measurements without concerns about errors. Mechanical

vibration and acceleration also have a negligible effect due to its small size.

8.3 Pressure Sensor Parameters

The pressure sensor used was the Entran EPE-N21-541 whose parameters are

listed in Table B.1 below.

 

Pressure Range 2 PSI

 

 

Resonant Frequency 80KHz

 

 

 

114

 

 

 

 

 

 

 

Excitation 10 Volts DC
Thermal Sensitivity Shift i4%/50 °C
Output (nom/ min) 300/ 120
Thermal i3% FSO
Zero Shift/50°C
Zero Offset at 20°C i 10 mV typical
Pressure Reference Vented (gauge/relative)

 

 

 

Table 8.]: Different parameters of sensors

13.3.1 Thermal Zero Shift (TZS)

The change of zero offset as a ﬁmction of temperature is the thermal zero shiﬁ. It
is expressed as either a %F SO for a speciﬁc temperature change such as :tl%FSO/50°C
or in voltage units as i lmV/50°C and is a non-linear function. The speciﬁcations
provided concern thermal equilibrium conditions for slow and stable changes of
temperature within the compensated temperature range of the sensor. Thermal shocks or
temperature gradients around the sensing surface of the sensor may cause zero offset
changes greater than those speciﬁed. Thermal zero shift speciﬁcations are also a function
of the heat sinking of the sensing diaphragm which results from the heat dissipation
characteristics of the working ﬂuid with the time allowed for the sensing element to

warm up from the beginning of the excitation. During the experiments it was routine to

115

permit a 30 minute warm up before taking any experimental data. Powering a sensor with
an excitation other than the one for which it was compensated for may also cause the
thermal zero shift to vary from the stated speciﬁcations.

8.3.2 Thermal Sensitivity Shift (TSS)

The expected change in the sensitivity of the sensor as a function of temperature
is the thermal sensitivity shift. It is usually expressed as a percent bandwidth change in
sensitivity for a speciﬁed change in temperature such as i2%FSO/50°C and is generally
linear with moderate temperature changes. The thermal sensitivity shift speciﬁcation is
intended to apply within the compensated temperature range of the sensor when it has
had adequate time to warm up. As the sensor is calibrated at the speciﬁc temperature of
use, any errors due to thermal sensitivity shiﬁ can be eliminated or minimized by using
sensitivity numbers determined at or near to the temperature of use. In our case the
temperature of the working ﬂuid was nearly same as the calibration temperature.

8.3.3 Zero Offset

The electrical output of the sensor when there is no applied pressure is the zero
offset. For vented side sensors for gauge or relative pressure measurement, the zero offset

is determined when the sensing side is at the same pressure as the vented side.

116

Appendix C Load Cells

C.1 Operating Principle and Design

Load cells are utilized in nearly every electronic weighing system. An entire
system consists of load cells, cables, a junction box, signal conditioners and a data
acquisition system. A load cell is classiﬁed as a force transducer since it converts force or

weight into an electrical signal.

The strain gauge is the heart of a load cell; a device that changes resistance when
a force is applied. The gauges are developed from an ultra-thin heat-treated metallic foil
and are chemically bonded to a thin dielectric layer. “Gauge patches” are then mounted to
the strain element with specially formulated adhesives. The precise positioning of the
gauge, the mounting procedure and the materials used all have a measurable effect on the
overall performance of the load cell. Each gauge patch consists of one or more ﬁne wires
cemented to the surface of the beam, ring or column (the strain element) within the load
cell. As the surface to which the gauge is attached becomes strained, the wires stretch or

compress changing their resistance in proportion to the applied load.

One or more strain gauges are used in a load cell. Multiple strain gauges are
connected to create the four legs of a Wheatstone-bridge conﬁguration. When an input
voltage is applied to the bridge, the output is a voltage proportional to the applied force
on the cell. The output can be ampliﬁed and processed by conventional electrical

instrumentation. A detailed mathematical derivation is given in the following paragraphs.

A simple uni-axial link type load cell with strain gauges is shown in Figure CI.

The load P can be either a tensile load or a compressive load. The four strain gauges are

117

bonded to the link such that two are in the axial direction and two are in the transverse
direction. The four gauges are wired to a Wheatstone bridge with the axial gauges in arms
1 and 3, and the transverse gauges in arms 2 and 4, as shown in Figure C.2.

When the load P is applied to the link, axial and transverse strains 8,, and 8,

develop in the link and are related to the load by expressions

P
8 =— Cl
and
VP
8 =—— C2
. AE ( )

where A is the cross-sectional area of the link, E is the modulus of elasticity of the link
material, and V is the Poisson’s ratio of the link material. A strain gauge exhibits a

resistance change AR/ R that is related to the strain E as

9R3 = 5,8 (C.3)

where 58 is the gauge factor or calibration constant for the gauge. The response of the

gauges to the applied load P is given by using equations (C. 1 ),(C.2) and (C3) as

 

 

S P
ﬂ = AR: = 5880 = s ((3.4)
, R, AE
and
VS P
AR2:AR"=S£= g (C.5)

 

 

 

118

The output voltage E0 from the Wheatstone bridge can be expressed in terms of the load P

by substituting equations (C4) and (C5) into the constant voltage Wheatstone bridge

equation. Assuming that all the strain gauges on the link are identical i.e. Rl = R2 we get

: SgP(1+V)E,
° 2AE

 

OI'

ZAE

 

 

 

 

p = — E = CE
5, (1+ V)E,. ° °
P
Axial gage 3
Transverse {/6 on the back
gage 4 on face
the left face

 

Figure C. 1: Load cell elastic element with strain gages

(C6)

(C7)

Equation (C.7) indicates that the load P is linearly proportional to the output voltage E0 and

the constant of proportionality is C = 2AE/Sg (1+V)E, . The sensitivity of the load cell-

Wheatstone bridge combination is given by S = E, / P = 1/ C .

119

 

 

Y3} $062.6
(.90
E0 0
,9 <2
0‘91» a

 

 

E1

 

 

 

___V—_

Figure C.2: Gage positions in the Wheatstone bridge.
C.2 Specifications of Load Cells

The technical speciﬁcations of Interface model SMT1-2.2 load cell and Futek
model L2357 are given in Table CI. The load cell speciﬁcations do not include the
frequency response of these sensors. A frequency response analysis of each load cell was

deduced and is described in the following paragraphs.

120

 

 

 

 

 

 

 

 

 

 

Specification SM T 1 -2.2 L235 7

Capacity 2.2 Lbs 5 Lbs

Rated Output (R.O.) l mV/V nom 1 mV/V nom

Excitation VDC 10 max 10 max

Non-Linearity i 0.05% of R.O. :t 0.1% of R.O.

Hysteresis i 0.03% of R.O. i 0.1% of R.O.

Zero Balance 3% of R.O. 3% of R.O.

Temp. Shift Zero :1: 0.15/100°F :h 0.004 - 0.01% of R.O./°F

 

 

 

C.3 Frequency Response of the Load Cells

Table C.1: Speciﬁcations of load cells

In order to analyze the quality of measurement under dynamic loading conditions,

the response of the load cells to impulse and ramp inputs was analyzed. The frequency

response for the two load cells was needed in order to evaluate the operating bandwidth

of these sensors, as no data was available from the technical details provided by the

respective manufacturers.

C.3.l Impulse Response of the Load Cell

The most widely useful mathematical model for the study of measurement system

dynamic response is the ordinary linear differential equation with constant coefficients.

The relationship between any particular input q, and output q0 can be written in the form

dnqo "—1 dn-lq0

a"——+a

dt" d!"—1

m m-l
+...+aidqti+a0q0 =bmd—$+bm"ﬂ—i+...+b%+boq, (C8)

611’" dt”"I

121

 

For a second order system, equation(C.8) reduces to

2
a 1611+al -C:1q—t°+a0q0 = boq, (C.9)

Dividing equation (C9) by a0 and simplifying yields

 

2
[57+ 2:0 +1 ]q0 = kq, (C. 10)

where k = b0 /a0 is the static sensitivity, a), = ,lao /a2 is the undamped natural frequency

and g" = al /2,/aoa2 is the damping ratio. For a step input of the size q, , the solution in the
non-dimensional form is

en—{wr

kg“, :1"; _

 

Sin(,/1-{2mz+¢)+1(c.11)

where ¢ = Sin‘l ‘11 — g2 . In equation (Cl 1), the static sensitivity k was evaluated from the

static sensitivity calibration curve. An experiment was carried out to evaluate the remaining
parameters. In the particular experiment, a mass in the midrange of each sensor was
dropped from a height of one inch from the load cell and its response was recorded at a
sampling rate of about 1 KHz. From the known load and the steady output of the load cell,
the parameters on the left-hand side of the equation (C.11) was determined. The damping

ratio was calculated using the relationship

6=1n[ﬂ]=—; (C12)

122

This gave the damping ratio 339' = 0.51. The natural frequency of the load cell was then

calculated using the expression

1,, 2” (c.13)

:amll-g’2

Xl

Load Cell Output (mV)

9.75 10 10.25 10.5
Time (sec)
Period

 

Figure C.3: Impulse response of the load cell.

where I, is the damped period between two successive peaks as shown in ﬁgure C-4.

The amplitude response of the system was determined using the relationship

f:- = —lOlog[(l—(a)/a), )2 )2 +4;2 (co/co, )2]dB (CM)

123

As can be observed from the Figure C4, the amplitude response of the load cell is
nearly ﬂat for the lower frequencies, followed by an increase in the amplitude ratio a), ,
then attenuation. The bandwidth of the load cell was determined to be one radian per

second for 95% accuracy. This value restricted the oscillating experiments that could be

performed to within the limitation of 0. l 6 cycle per second or less.

 

4 TTTIIIII T ITFIIII' I WITHTII' I T111111

[TIP
1

 

IIII

 

o 4_.—~"’/’f\\ ‘

 

a

N
IFTI
1111

 

 

 

 

 

 

 

 

E . l —
<3" I I
if -4:: :
.6 — —J
— a
t :
-8:: :
:, Bﬂﬂdﬂidlh .l L :
_.1 I 111111] 1 1111111] L llllllll LLJIIIF

’10 10'2 10'T 10° 101

who

Figure C.4: Amplitude response of the load cell

124

C.3.2 Ramp Response of Load Cells

The ramp response of the load cell was determined by gradually increasing the
load while recording the load cell output. The purpose of this experiment was to record
the steady state time lag when collecting liquid mass during accelerating and decelerating

ﬂows.

 

QOTITITTIIT'IIITITII

_1
q

80

70

60

50

40

30

Mass Accumulated (gms)

20

10

 

ﬂIIII'IIlIIIIII'IIIIIIIIIITIITTTTIFITIIIIIIII
IlllI'llllllllllLllllllllllilllLllillllLlJlll

 

 

O
_s
N
h

Time(sec)

Figure C.5: Ramp response of load cell

125

The ramp response of the second order system is obtained by applying the initial
conditions q0 = dq0 /dt = 0 at t: 0+ in equation (C.lO). The solution of equation (C.lO)

is

qo _- 2§__q._1.~ Cw",
7-11,,1— a)” 1—2——;J:Z31n(,/1 g wt+¢) (015)

where

tan¢ =£__ H (C.l6)

In the above expression, the steady state error is 2:4,, / a)" , where 4,, is the mass ﬂow rate

during accelerating and decelerating ﬂow. The estimated steady state time delay was found

as 0.1 second. The ramp response curve during accelerating ﬂow is shown in Figure C5.

126

Appendix D Sensor Thermal Sensitivity

Veriﬁcation of Sensor Temperature Drift due to Heat Loss from Convection Heat

Transfer

The thermal boundary layer develops within a laminar or turbulent hydrodynamic
boundary layer over the sensor surface of diameter D, in the streamwise directions. The
thermal boundary layer is produced by a sudden jump in the surface temperature from a
constant value of ambient temperature T e to the higher constant value of T 0 as shown in

the Figure D.l.

Flow direction

 
 
 
 

 

Hydrodynamic
Boundary Layer
02-0 ‘

    
   

Thermal Boundary -
Layer Edge xxx,“ Te X

//7//Z‘/=71‘.77777//7//

 
     

r/////7////////7

 

<—-»- Ds—>
Figure D. 1: Thermal boundary layer and related geometry

Due to high thermal drift of the sensors, it was decided to evaluate the
temperature difference between the sensor surface and the ﬂuid in contact with it. Higher
thermal gradients would cause higher heat losses that might affect pressure sensor data.

The analytical solution proposed by Ramaprian and Menendez [43] was used as a starting

127

point for this analysis. Ramaprian and Menendez gave the following results relating skin

friction to the ﬂush mounted rectangular hot-ﬁlm gauge sensor heat transfer as

 

3 2
ms: .2 5; +4132) —“‘° +,_.,.,_.___.
pp 0 a x L Qw/ 0k A70 at (l n) le

here L is the length of the sensor, 1'“, is the wall shear stress, Q“, is the heat transfer from
wall to the ﬂuid at x = x0 + L and A7}, = I], — 7;. Equation (D.1) can be simpliﬁed using

the assumptions that ﬂow is steady with negligible pressure gradient along the sensor
surface. Replacing the area of the hot ﬁlm WL by the surface area of the pressure sensor,

equation (D.1) becomes

542...] (D2)

 

r, =
R

(1"?)2ﬂ [

achkzlt'rfATo3

where Q,- is the heat loss to the substrate, considered to be much less than the heat

generated from the sensor. Replacing the wall shear stress term by the ﬂow rate using the

relations Q =7tD‘Ap/ 1281/1 and 1'“, = ApD/ 41 and rearranging for A7},

1— 2R3 2 3
AT 3: ( n). —-E (D.3)
0 .. 3
8acppk Itl‘sQ IR

 

Using the values shown in the Table E.l, values of A7}, were calculated for water and

ethylene glycol solutions as 0.86 K and 0.687 K respectively at a Reynolds number of
1900. It was therefore concluded that a temperature difference of less than ldegree
Kelvin was unlikely to have much effect in terms of convection heat transfer that might

cause erratic output of the sensor.

128

 

 

 

 

 

 

 

 

 

 

 

 

Properties Ethylene Glycol Water
Shape factor (a) 0.23 0.23
n for laminar ﬂow 0.29 0.29
pDensity (kg/m3) 1077 998
CI) constant speciﬁc heat 3400 4181
(J/Kg.K)
K Thermal conductivity 0.389 0.602
( W/m.K)
/1 Dynamic viscosity 5.6x10'3 9.68x104
(kg/mSec)
r, Sensor radius (m) 1.18x10'3 1.18x10‘3
E supply voltage (Volts) 5 5
R radius of the tube (m) 4.7625x10'3 4.7625x10‘3
1R sensor resistance (ohms) 1000 1000

 

 

 

Table D. 1: Properties of the ﬂuids and sensors used in equation (D.3)

129

 

E.1

unc
exp

pen

unc:
and

E2

instr

blaSI

Appendix E Uncertainty Analysis

E.1 Uncertainty Analysis

Every measurement includes some level of error, and this error can never be
known exactly. However a probable bound on the error can be estimated, known as its
uncertainty. The purpose of this appendix is to quantify the uncertainty in the
experimental results presented earlier. The uncertainties are stated at a conﬁdence

percentage of 95% or odds of 19:1. The bound typically has the form

X ,. = X, (measured) idX, (n: 1 ), where 6X, is the uncertainty estimated at odds of 11:1

i.e. only one measurement in n will have an error greater than 6X,. When estimating

uncertainty, two types of errors are normally considered: precision error (random error)
and bias error (systematic error).

E.2 Bias Errors

These errors occur the same way each time a measurement is made i.e. if an
instrument reads high by a certain percentage, then the entire set of measurements are
biased by that percentage above the true value. Fixed offset errors also contribute towards
the biased errors. The only direct method for uncovering bias error in a measurement is
by comparison with measurement made using a separate and presumably more accurate
apparatus. Bias uncertainty is also estimated using experience and understanding of
calibrations and manufacturing tolerances. The estimate of the bias uncertainty is

presented with a conﬁdence level of 95%.

130

 

E.3 Precision Errors

Precision errors are also called the random errors. They may be different for each
successive measurement but have an average value of zero. In this case readings of the
same quantity differ slightly, creating a distribution of values surrounding the true value.
In such cases, a statistical analysis can be carried out to estimate the likely size of the
error. The precision error is presumed to behave randomly, with zero mean. The precision
error in repeat-sampled data reveals its own distribution, enabling us to bound its
magnitude using statistical methods. The potential precision error underlying a single
measurement of a random variable can be estimated from the resolution of the
instrument, the precision with which the instrument can repeat the reading and how much
its reading ﬂuctuates during the test.
E.4 Propagation of Uncertainty

If a calculated result is a function of several independent measured variables, then
each measured value has some uncertainty and these uncertainties lead to an overall
uncertainty in calculated results. Uncertainty in results due to uncertainty in independent
variables is called the propagation of uncertainty, expressed mathematically (Beckwith et

al. [44]) as

2 2 2
ay ay 8v
U , = —u + -—u + ....... + ——'-—u E]
y (83‘: I] (3x, 2] [8x n] ( )
In the above expression all the uncertainties have the same odds and are independent of

each other. The uncertainties U, are either bias uncertainties or precision uncertainties.
Both of these uncertainties propagate separately. The overall uncertainty U, is calculated

by combining B, and R as

131

(E.2)

E.5 Uncertainty Analysis of Experimental Results

The uncertainty analysis of the data was carried out based on manufacturer’s data
sheets, calibration results and knowledge of the experimental procedures. In this research,
the objective was to compare the cumulative mass inferred from pressure sensor data with
true accumulated mass indicated by the load cell. The uncertainty of each measurement
was recorded ﬁrst and then the propagation of uncertainty was estimated based on these

individual uncertainty measurements.

E.5.1 Uncertainty in Density Measurement
For the density of the working ﬂuid, the volume of the ﬂuid was measured using a
graduated cylinder and the mass of the ﬂuid contained in the cylinder was measured using

an electronic weigh scale. The uncertainty in density measurements is tabulated below.

 

 

 

 

 

Parameter Systematic Random Overall
Uncertainty (%) Uncertainty (%) Uncertainty (%)
Mass 0.5 0.1 0.51
Volume 0.1 1 1.005
Density 0.5 l 1.12

 

 

 

 

Table E. l: Uncertainty in density, volume and mass

132

 

E.5.2 Uncertainty in Mean Velocity Measurement

The velocity of the ﬂuid was measured using the relationship U = m/pA , where m
was calculated from the load cell data. The cross sectional area of the tube was determined

as A = (n/ 4) d 2 . The calculated uncertainty in velocity measurement is tabulated below.

 

 

 

 

 

Parameter Systematic Random Overall
Uncertainty (%) Uncertainty (%) Uncertainty (%)
Mass ﬂow rate (load 0.06 0.04 0.07
cell)
Pipe diameter 0.25 0 0.25
Velocity 0.1 1 1 .004

 

 

 

 

Table E.2: Uncertainty in velocity, ﬂow rate and diameter measurements

E.5.3 Uncertainty in Reynolds Number

The Reynolds number was an important parameter used to indicate whether ﬂow
was in the laminar or turbulent regime. The Reynolds number was calculated using the
relationship Re: Ud/ V , where the uncertainty in the velocity and pipe diameter has already
been shown in table E.2. The kinematic viscosity was measured using a Cannon-Fenske
viscometer. Although uncertainty measurements in viscosity involve the uncertainty in time
due to errors associated with using a stopwatch, this is ignored since the time involved was
relatively long, of the order of 80 seconds. The calculated uncertainty in the kinematic

viscosity and the Reynolds number is tabulated overleaf.

133

 

 

 

inf

rel

Th

ca‘

1117

UT

LIT

FL?

 

Parameter Systematic Random Overall
Uncertainty (%) Uncertainty (%) Uncertainty (%)

 

Kinematic viscosity 1 0.15 1.011

 

 

Reynolds number 1.3 1.4 1.91

 

 

 

 

Table E.3: Uncertainty in the Reynolds number and kinematic viscosity
E.5.4 Uncertainty in Pressure Drop and Inferred Accumulated Mass
Measurements for Laminar Flow Experiments

Using the exact laminar ﬂow relationshipAp =8Vln'r/71'R4 , the pressure drop

inferred from mass ﬂow rate was used for the calibration of the pressure sensor. In this
relationship, the uncertainty in all the parameters except length has already been discussed.
The overall uncertainty in the pressure sensor calibration involves uncertainty in the
calibration curve and uncertainty in the other factors in the above relationship. These
uncertainties are combined in the overall uncertainty in the pressure drop data. The overall
uncertainty in the inferred accumulated mass is tabulated below along with the

uncertainties in length and pressure drop data.

 

 

 

 

 

Parameter Systematic Random Overall
Uncertainty (%) Uncertainty (%) Uncertainty (%)
Pipe length 0 0.25 0.25
Pressure drop 2.25 2.0 3.01
Inferred 2.7 2.0 3.36
accumulated mass

 

 

 

 

Table E.4: Uncertainty in inferred accumulated mass, pressure drop, and pipe length

measurement in laminar ﬂow experiments

134

 

 

T1

 

U1]
to

UT

1..

7

 

E.5.S Uncertainty in Pressure Drop and Inferred Accumulated Mass

Measurements for Turbulent Flow Experiments
In this case the uncertainty in the pressure drop measurement was calculated ﬁrst.

2
The relationship dp = 2(3)? was used for calibrating the pressure sensor. The

uncertainty in It is associated with the Reynolds number since A = 0.316 Rego'25 was used

to determine)“ The uncertainty in L,d, Re, p and U has been discussed earlier, when the
uncertainty in pressure drop data was evaluated. For accumulated mass in turbulent ﬂow
experiments, the uncertainty in the evaluation of a or V, (0),and the uncertainty
contribution due to R,p and V was also considered. The uncertainties calculated in

pressure drop and inferred accumulated mass in turbulent ﬂow experiments are tabulated

 

 

 

 

 

below.
Parameter Systematic Random Overall
Uncertainty % Uncertainty % Uncertainty %
a l 0.25 1.03
Pressure drop 2.67 2.97 4.0
Inferred
accumulated mass 307 3.15 4.4

 

 

 

 

Table E.5: Uncertainty in inferred accumulated mass, pressure drop and a measurement in

turbulent ﬂow experiments.

 

 

 

[6

[1].

[2]-

[3]-

[4]-

[5].

l6].

[7].

[8].

[9]-

[10].

[11].

[12].

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