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A CALIBRATION PROCEDURE FOR
THE MEASUREMENT'OF TRANSVERSE VORTICITY
USING HOTLWIRE ANEMOMETRY

By

Casey Lynn Klewicki

A THESIS

Submitted to
Michigan State University
in partial fulfillment of the requirements

for the degree of

MASTER OF SCIENCE

Department of Mechanical Engineering

1983

ABSTRACT

A CALIBRATION PROCEDURE FOR
THE MEASUREMENT OF TRANSVERSE VORTICITY
USING HOT-WIRE ANEMOMETRY

By

Casey Lynn Klewicki

A technique to obtain an irregular time series of the transverse
vorticity from the four measured hot-wire voltages of a vorticity
probe is presented. The complete technique involves several calcula-
tion schemes. including those for the calibration and the flow field
data. The schemes are presented in detail and demonstration data.

from a large plane shear layer. are used to demonstrate them.

ACKNOWLEDGMENTS

I would like to express my appreciation to my parents for their
continual support and encouragement throughout my entire college

CBIOCI’.

I would like to recognize the support of Lockheed of Georgia Co..
grant monitor: Dr. A. S. W. Thomas and the support of NASA Lang-
ley Research Center. grant monitor: Dr. J. Yu. I am pleased to
acknowledge the substantial contribution of Mr. Peter J. Disimile to
the clarification and organization of flow charts for the vorticity
algorithms during their initial deve10pment stages. I would like to
express my gratitude and appreciation to my thesis advisor. Dr. John

F. Foss, for his guidance throughout my graduate pragram.

TABLE OF CONTENTS

LIST OF TABLES ................................................
LIST OF FIGURES ...............................................
LIST OF SYMBOLS ...............................................
CHAPTER

1 INTRODUCTION ........................................

1.0 Historical Review of Techniques to Measure
vorticity 0.000............OOOOOOOOOCOOOCOO...O.

2 VORTICIIY WASUREWO....0.0.0.0.........OOOOOOOOOOO
2.0 IntIOdnctionO00.0.0.0.........OOOOOOOOOOO0......

2.1 A Qualitative description of the
Micro-Circulation Domain ........................

2.2 no vorticity PrObo O.......OOOOOOOOOOOOOOOOOO...

2.3- Role of the Xrarray in the Vorticity
calcnlntion .-......‘................C.........O.

2.4 no vorticity C‘lcnlation ......OOOOOOOOOOOOOOOOO
2.4.1 Definition of the Micro-Circulation

Domain 0.0......0.........CIOOOOOOOOOOOOOOO

2.4.2 Micro-Domain Average Values ...............
3 SUPPORTING SCHEMES FOR THE VORTICITY CALCULATION . . ...
3.0 Intruduction ......OOOOOOOIOOOOO...0.0.0.0.......

3 .1 De temining Q and Y O O C C O C O O O O O O O O O O O O O O O O O O O O O O 0
3.1.1 Speed-Wire/Angle-Wire
TeChnique: Iterative O . O O O O I C C O O O O C O O O O O O O C

3.2 Three-Dimensional Effects .......................
3.2.1 Effect of the Transverse Velocity

Component on the Q and 7 ..................

3.2.2 Detection of the Transverse Velocity ......

3.2.3 The w' Correction .........................

iii

vi

ix

PAGE

10

10

10
13

17
17
17
19
21
21

22
24

4 CALIBRATION OF THE VORTICITY PROBE AND
PROCESSING AI-IGORImMS O.................ODOOOOOOO....O 27

4.0 Introduction .. ....... C .. .. I C. . ... . . C O. .. ... O . ... 27
4.1 The Data Acquisition Facility ................... 28
4 .2 no calibration Da ta . . . . . . . . . . O . O . . O . C . . . . . . . . I . 29

4.3 A measure of the 'Effective' Angle
Of the Slant Wires 00.0.0.........OOOOOOOOOOOOOOO 30

4.4 Construction of Calibration Functions ........... 31
4.4.1 The 'Smoothed' Calibration Data Base ...... 32
4.4.2 A Computationally Efficient Form for

the Velocity magnitude Evaluation ......... 33
-4.4.3 The Angle-Wire Response Functions ......... 34

4.5 A 'Data-Day' Correction Scheme .................. 45

4.6 Calibration Curves From Experimental

calibration Data 0 O O O O C O O C O O I O O O O O I O O O O O O O O O O O O O O 37
5 Demonstration Data 0 O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O 39
5 .0 IntIOdu-ction O O C O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O 39

5.1 Experimental Results .0.........OOOOOOOOOOOOOOOOO 39

6 SUMMARY 43
APPENDIXA COSLAW 90
APPENDIXB COORDINATE TRANSFORMATION................. 98
APPENDIX c RESPONSE FUNCTION (DEFFICIENTS 101

RHERWCES O.........OOOOCOOOOO......O'.......O......O......O.... 106

iv

LIST OF TABLES

TABLE PAGE

A.1 ACCURACY OF COSLAW (slant wire 1)

a. [(Qc-Qcos)/Qc] ‘ 100. ......OOOOOOOOOOOOOOOOO. 96
b. Yc-Ycos 0.0.0..........OOOOOOOOOOOOOO00.0.00... 96

A.2 ACCURACY OF COSLAW (slant wire 2)

a. [(Qc—Qcos)/Qc] . 100. ......COOOOOOCOOOOOOOOOO 97

b. Yc—Ycos O.....IOC‘OOOI00............COOOOOOOOOO 97

Figure

1.1
2.1a
2.2a
2.3
2.4

2.5

3.1

3.2

0)
0
L0

3.4

4.1

4.2a

LIST OF FIGURES

Page

Organization of the Technique

O......OOOOOOOO‘OOOOOOOO. 46

The Subject Flow Field ' 47

The micro-circulation domain

......OOOOOOCOOOOOOOO0.... 48

The vorticity probe [dimensions in mm] ,,,.,,,,,,,,,,,, 49
A typical hot-wire probe [dimensions in mm] ,,,,,,,,,,, 50

Nomenclature for the cumulative averaging scheme ...... 51

Circulation loop about micro-domain

......OOOOOOOOOO... 52

Pertinent coordinate systems

......OOOOOOOOOOOOOO...... 53

Angle range of validity for cosine and extended

cosine laws

....IO...00...............OOOOOOOO...0.0... 54

Notes:

(i) wire 2 calibration data were used for this

illustration A

(ii) range of validity of cosine law (I):7$I12°'

(iii) estimate for the range of validity of the
extended cosine law (II): -24°Sy$p;
obtained from processed calibration data of
Foss[9]: see Figure 3 of that reference.

Schematic of typical velocity vector/x-array
orientation .............CC............................ 55

Schematic of the Qx—y iteration scheme ................ 56
Frequency Distribution of Qx-Qp ...................... 57
The Calibration Grid .................................. 58
The Calibration Facility; Vorticity Probe/Reference

Probe orientation ..................................... 59

The Vorticity Probe Support Fixture ................... 60
The Vorticity Probe Traverse .......................... 61

vi

4.3a

4.4a

4.5
4.6
4.7
4.8

4.9a

4.10a

4.11a

4.12a

5.1a

5.2a

£5.5a

£5.6a

AQL.1

A.2

Response function for wire 1 = 'angle-wire'-(7$O°) ....
ReSponse function for wire 1 (7200) ...................

Response function for wire 2 (7$0°) ...................
Response function for wire 2 = 'angle-wire' (120°).....

Angular response function for wire 1 (750°) ...........
Angular response function for wire 2 (720°) ...........
Speed function for wire 1 (720°) ......................
Speed function for wire 2 (750°) ......................

Response function for wire 3 (750°) ...................
Response function for wire 3 (120°) ...................

Speed function for “1:03 (7500) ......C...............
Speed function for wire 3 (720°) ......................

Response function for wire 4 (750°)....................
Response function for wire 4 (720°)....................

Speed function for wire 4 (150°) ......................
Speed function for wire 4 (720°) ......................

The Free Shear Layer Flow Facility ....................
Detail View of Test Section.with the Initial and Shear
layer profiles shown schematically .....................
Note: * measurement location

Streamwise and Transverse Velocity Components ..........
Notes: Measurement point: x=1m,y=9.9cm
Streamwise and Transverse Velocity Components...........
Note: Values Corrected for w1 influence

Transverse Vorticity Time Series (see figure 5.2a notes)

. Transverse Vorticity Time Series (see figure 5.2a notes)

Note: Values Corrected for w3 influence

Strain Rate Time Series(see figure 5.2a notes) .........
Strain Rate Time Series (see figure 5.2a notes) ........
Note: Values Corrected for w3 Influence

Velocity gradient (see figure 5.2a notes)...............
Velocity gradient (see figure 5.2a notes)...............

Velocity Gradient (see figure 5.2a notes)...............
Velocity Gradient (see figure 5.2a notes),..............
Note: Values Corrected for w3 Influence

Difference between True Angle and Angle Calculated
“Sing the COSLAW for Wirel 0.00..........C.......OD....

Difference between True Angle and Angle Calculated
“Sing the COSLAW for Wirez 0............O.............O
vii

62
63

64
65

66
67
68

69

70
71

72
73

74
75

76
77

78

79

80

81

82
83

84
85

86
87

88
89

94

95

C.2
C.3

C.4

Variation of ABn Coefficients with y for Wire 1 ,,,,,,,, 102
Variation of ABn Coefficients with 7 for Wire 2 ........ 103
Variation of ABn Coefficients with y for Wire 3 ,,,,,,,, 104

Variation of ABn Coefficients with y for Wire 4 ,,.,.... 105

viii

LIST OF SYMBOLS

Collis and Williams parameter

modified Collis and Williams parameter

Collis and Williams parameter

Collis and Williams parameter

anemometer output voltage

voltage from designated angle-wire (slant wire 1 or 2)
voltage from designated speed-wire (slant wire 1 or 2)
Grashoff number

velocity

velocity determined from output voltage of parallel array
velocity determined from output voltage of x-array
defined by eq. 3.9

Reynolds number

local micro-circulation domain coordinates

free stream velocity

velocity components relative to laboratory coordinates
velocity components relative to micro-domain coordinates
laboratory coordinates

angle between pitch angle 7 and laboratory x-coordinate
angle of hot wire; defined in Figure 2.2a

incremental time between data samples: l/sample rate

incremental convection length

ix

A distance between straight wires of the parallel array (mm)

A7° convergence check for 0,7 iteration scheme

7 pitch angle of velocity vector relative to probe axis
A data-day correction factor

Q angular velocity

1 psuedo-time variable as defined by eq. 2.9

9 angle between probe axis and laboratory x-axis

”z transverse vorticity

“x streamwise vorticity

subscripts

a identified with angle-wire

c identified with (master) calibration of the vorticity probe
d. identified with data-day

1 counter for t

j counter for t (data sampled at tj)

I’ identified with parallel array

5 identified with speed-wire
x identified with x-array

CHAPTER 1

INTRODUCTION

The concept of vorticity is fundamental to the understanding of
turbulent flows. Fluctuating, three-dimensional. components of vorti-
city are a necessary condition for a flow to be labeled turbulent;
hence. vorticity (09) is a primitive variable of turbulent flows.
Corrsin and Kistler[l] considered this in their study of the boundary
region between a turbulent and non-turbulent flow. A signal from a
pyramidal configuration of four hot-wires, responding primarily to the
streamwise component of vorticity, was used to detect the passage of
the turbulent - non-turbulent boundary. More recently Kibens.
Kovasznay and Oswald [2] have developed an anolog circuit for the
detection in real time of the boundary. Several input signals to the
detector were considered. The signal chosen was that used by Corrsin
and Kistler (a signal pr0portional to m!) since it offered the most
contrast between the turbulent and non-turbulent regions. Hardin [3]
has used the concept of vorticity to model and predict the far field
noise associated with turbulent jets. Willmarth and Bogar [4]. in
their investigation of the near wall region of a turbulent boundary
.1ayer, used the concept of pressure gradients at the wall as a source

C>f vorticity. They attempted to measure the streamwise component of

vOrticity in order to gain understanding of the turbulent structures

1

and mechanisms that result in an increase or decrease of drag.
Several other studies. Brown and Roshko[5]. Blackwelder and
Eckelmann[6]. Eckelmann. Nychas. Brodkey. and Wallace[9]. Signor and
Falco[7]. Falco and Lovett[8]. etc have also used vorticity in
attempts to understand phenomena associated with turbulent flows.
i.e.. coherent structures (typical eddies). bursting phenomenbn. mix-
ing. etc. Thus an accurate measurement of the instantaneous vorticity

would be most useful in experimental fluid mechanics.
1.0 Historical Review of Techniques to Measure Vorticity

The measurement of vorticity is much more difficult than measur-
ing velocity. Each component of vorticity involves spatial gradients
in two different directions of two different velocity components.
Hence. the measuring instruments (hot-wire anemometry. hot films. LUV)
tend to be complex. both geometrically and electrically. (Typically if
hot-wires are used, more than one probe is involved. which suggests
multichannels and simultaneous measurements. Even so. the final meas-
urement is usually one or two out of ~the three components of

vorticity.

Kovasznay [10]. 32 years ago, develOped a streamwise vorticity
meter. (Since then. it has been. commonly referred to as the
Ebvasznay-type probe). It consists of four hot-wires mounted on four
Iprongs. which form a Wheatstone Bridge when operated as a constant
<:urrent anemoneter. The output voltage across Opposite prongs is pro-

Portional t0 the streamwise vorticity. (ex. Some of the earliest users

were Uberoi and CorrsinIll] for studies of the propagation of a turbu-
lence into non-turbulent regions. At that time the probe was assumed
to be insensitive to cross-stream velocities. Since that time Kastri-
nakis, Eckelmann and WillmarthllZ] have investigated the influence the
effect of these transverse velocities on the 9:. measured by the
Xovasznay probe. The authors concluded that instantaneous measure-
ments of vorticity in flows of high turbulence levels is impossible
since the influence of the transverse velocity fluctuations (u'. v')
cannot be corrected for; simultaneous knowledge of both u' and v' is
unavailable. Also the effect of these components of velocity can not
be ignored since they may be of the same order of magnitude as the
vorticity signal being observed. Tb allow for the measurement of all
three velocity components and their influence on ”x' thoslavcgvic and
Wallace[13] constructed a probe with the same configuration as the
Kovasznay probe but supported each wire by a separate pair of prongs
(a total of 8). and electrically Operated each wire independently. In
effect the configuration is 2 x-arrays in perpendicular planes which
are parallel to the flow. It was concluded that the instantaneous mi
measurment was badly in error whether transverse velocity components
were accounted for or not. since the effects of the velocity gradients
introduced large errors in their measurement. The errors could be
reduced by decreasing the spacing between wires. but thermal cross

talk then becomes a problem.

In an investigation of the vortex structures associated with the
‘bursting phenomenon. Blackwelder and Echelmannl6] developed a techni-

tlue to obtain a measure two vorticity components in the near wall

region. TWO hot film sensors in a v-configuration were flush mounted
to the wall with a hot-wire located directly above. the signals were
considered to be proportional to ”x and ”z-

The importance of making direct measurements of vorticity is
attested to by the novel concepts that have been developed for this
purpose. Two of these are noted in the following. Frish and Webb[14]
developed an optical method to directly measure vortictiy in fluid
flows. Spherical particles imbedded with crystal mirrors were
suspended in a liquid. The vorticity was obtained directly by measur-
ing the time required for laser reflections from the mirrors to rotate
through small angles. 0=.5w. The technique is limited by sensitivity
of the Optics and electronics to noise and the technique_ can only
determine the vorticity of one sign. Lang and DimotakisllS] have used
a laser dOppler velocimeter technique to measure the circumferential
velocity components at the 4 vertices of a small diamond shaped
region. These velocity components can be related to the curl of V.
i.e. through the use of Stokes theorem. Smoothing and interpolation
procedures are required since the probability of obtaining four simul-
taneously sampled velocities at the four vertices. is relatively

small.

Foss. in a series of publications[l6.l7.l8] presents the develop-
:ment of a technique to obtain a measure of the transverse component of
‘Vorticity using a 4 wire array. The present manuscript reports impor—
tant revisions and improvements of the technique and calculation

Eschemes. The technique lends itself to flows in which large pitch

angles are encountered; free shear flows. outer regions of boundary
layers and wakes. Specifically. this writing presents the theory and
computational schemes involved in obtaining a time series of the
transverse vorticity. strain rate. and velocity components for a small
sample domain in a flow field. The influence of the transverse velo-
city. w. on the measurements is analytically described and a technique
to correct for its existance is presented and it is applied to a body
of data. Experimental data. obtained in a free shear flow. have been
used to demonstrate the complete technique. An organizational flow
chart of the complete technique used to obtain the values for the time
series of vorticity. strain rate and velocity components from the vor-
ticity probe response voltages. E1,Ez,E,,E4, is shown in Figure 1.1.
Each rectangular box represents a supporting calculation scheme used
in determining the time series values. Note that these calculation
schemes utilize various functions which are defined from a complete

calibration data set.

CHAPTER 2

VORTICITY MEASUREMENT

2.0 Introduction

A regular time series of voltages from an array of four hot wires
is used to obtain an irregular time history of vorticity. The compu-
tational scheme used to determine the time history of vorticity has
the same basic structure as the scheme described in Foss [21] but with
important additions and revisions. .The complete method is described

herein.

2.1 A Qualitative Description of the Micro-Circulation Domain

The quantities derived from the four wire probe are to be spa-
tially averaged values over a small domain (i Inn). The domain will
be referred to as the 'micro-circulation' domain. These spatially
averaged values approach 'point measures' in the flow field for situa-
tions where the scale of the micro-circulation.domain is sufficiently
smaller than the scales of the motion being studied. An example of
‘SEch a flow field is shown in Figure 2.1a; the measurements. taken in
tihe intermittent region of the free shear layer with a hot-wire probe.

approach point measures since the length scale of the probe is much

Stasiller than that of the energetic turbulent motion. For the smaller

6

7

scales. the scheme acts as a low pass spatial filter; ie: the smallest
scale that can be measured is limited by the length scale of the

micro-circulation domain.

In constructing a value for the transverse vorticity. m two

2'
spatial velocity gradients are required; wz = avlax - Bu/ay. A value
for the cross-stream spatial gradient au/ay may be obtained by using
measurements from two hot wire probes that are parallel to the z-axis
and separated by a distance Ay. The measurement of av/ax. by using
two probes that are displaced in the streamwise direction. would be
disallowed due to probe interference effects. Hence a streamwise
length prOportional to (velocity) x (time) is used instead. Foss [16]
used [(1/u)a/at]=>a/ax; however this formulation does not account for
the contribution of val llay to the BI llat value. A.more accurate
description would utilize the total velocity component in the x-y'
plane as the convection or translation velocity: (1/Q)8[ ]/at=-8[
llas. An incremental streamwise length: 83. may be defined in this

manner and an apprOpriately defined sum of such lengths may be used to

create a micro-domain over which the vorticity (mz) is defined as:

(NZ) AA = [AA wz dA ; where AA = AsAn (eq. 2.1)

A schematic representation of the micro-domain is shown in Figure
2.1b. Note that the width of the domain: An. depends upon the spa-
tial orientation of the parallel array with respect to an average of
the streamwise directions for the time segment used to define the

xmicro-domain. The time segment is chosen such that the micro-domain

8
is nominally square: AsgAn. As a result of these procedures. the
reqularly sampled voltages: { E1(tj)....E4(tj)], are converted to an
irregular time series of spatially averaged values over the
micro-domain: <u(ti)).(v(ti)).(m(ti)). and (sxy(ri)>. The irregular
time series: Ti, reflects the variable speed of translation as well

as the variable dimensions of the micro-domain.
2.2 The Vorticity Probe

The vorticity probe consists Of four hot wires or two arrays: an
x-array and a parallel array. A schematic representation of the probe
is presented in Figures 2.2a and 2.2b. The slant wires of the x-array
are. nominally at an angle of 450 with respect to the probe axis. The
distance between them is of the order 1mm. The parallel array is
located below the xrarray and consists of‘2 straight wires which are
parallel to the z-axis. They are separated by a distance Of nominally
1mm. The placement of the parallel array is such that the wires are
directly below the active region of the x-array wires: hence. the
measurements from the parallel and xrarrays are at the same streamwise
location. The distance between centers of the 2 arrays is approxima-

tlely 3.8 mm.

The fundamental. and the most limiting. assumption in the use of
the four wire array to define ml is that:

37/62 = 0 for each t- value

J

9
The (62) separation between the two arrays requires that this assump-
tion be, made; its validity is dependent upon the instantaneous
character of the velocity field. One motivation for the present
algorithm. and that of Foss [17] is that the error in: 37/8220, is
presumed to be much less than the error in the alternative assumption:

3(3V/31)/az=0. a la Foss [18].

Each of the 4 wires of the vorticity probe is fabricated using
the same technique. A representative wire is shown in Figure 2.2.b
The wire is 5 pm tungsten which is copper plated on the ends. The
cOpper plating enables the wire to be soft-soldered to the ends of the
jeweler's broaches (prongs) and it aerodynamically isolates the active
portion of the wire from the prongs. The total wire length is nomi-
nally 3mm with an active portion of 1mm. ie 1/d 2 200. The minimum
spatial resolution that can be expected is greater than or equal to
1mm. No wire length corrections. (6.8.. Wyngaardll9l). were utilized
in the present algorithms. According to Collis and Williams[20] and
others. the effects of bouyancy forces may be neglected if Gr < Re’.
For the condition: Grlke’. the velocity components adjacent to the
wire induced by the buoyancy forces become comparable in magnitude
with the flow velocity being measured. The relation Gr<Re’ therefore
defines the lowest speed at which the hot wire is capable of measuring
‘without ambiguity. For the flow conditions and hot-wires used in the
gpresent study the effect of buoyancy forces was insignificant. That
:is the Grashoff number was of the order 3.5E-06 (using an overheat
Iratio of 1.7) and the Reynolds number based on the 5pm diameter was of

‘tlle order .59 for Doo=2m/s. The free convection effects for these

10

conditions would become significant at a flow speed of .OSm/s or less.

2.3 Role of the x-array in the Vorticity Calculation

The primary role of the x-array is to provide a measure of the
pitch angle 7 at the location of the vorticity probe. This pitch
angle information is used at the parallel array location in defining
the velocity components and the orientation of the micro-circulation
domain with respect to the probe axis. It is assumed that the spatial

variations of the pitch angle are sufficiently small that:

ll

“1.352%“ 7(x.y + A/2.z)

ll

y(x.y - A/2.z)

A secondary role of the x-array is to provide the requisite informa-
tion to allow a correction for the influence of the transverse
velocity. Namely. if the velocity magnitude determined from the
x-array exceeds the velocity magnitude defined by the parallel array.
the difference may be attributed to the existance of a (w’) influence
on the hot wire voltages of the x-array. A technique to correct for

this w’ influence is presented in a later section.

2 .4 The Vorticity Calculation

2 ~4 .1 Definiton of the Micro-Circulation Domain

The vorticity calculation scheme yields a time series of spatial-

13’ averaged quantities; the Velocity components (<u>.<v)) and their

11
derivatives from which the transverse vorticity. and the strain rate
are found. The spatial averaging is performed in a region referred to
as the micro-circulation domain. This domain has been defined quali-
tively in section 2.1. The present section provides an operational
definition of this domain and specifies the procedures which are used

to compute the locally averaged quantities.

A key element in the computational procedure is the use of cumu-
lative averaged quantities for the identification of the length (As)
and the width (An) of the micrOfcirculation domain. The cumulative
averaging procedure can be defined using the terms introduced in Fig-

ure 2.3 and the following Operational steps.

For a given time step (tj). the previous and present quantities
are used to define an incremental value: e.g.. the convection speed

for the increment: tj-£9tj is:

(2

convection

(tj) = o.s[[<a,+q.)/21(tj) + [(Q,+Q.)/2](tj_1)] (eq. 2.2)

Trhe locally averaged quantities at time (ti) are defined by the incre-
mental values existing between ri_1 and 1:1. For purposes of

iJllustration. consider a time counter: k. such that k=0 at ti-1° The
«cranvection speed: Qc(tk). and the cumulative average of y:

<7<tk)> = (y. + 71 + y, + ... + 1k) / (k+1). (eq. 2.3)

8are used to define an incremental length in the streamwise direction:

12
58(tk) = Qc(tk) COS(Y(tk) ‘ (7(tk)))(tk”tk_1) (eq. 2.4)

where 7(tk) is the incremental value for tk-1-9 tk'

The correSponding value of the transverse dimension is:

An(tk) = A cos(1(tk) - (7(tk)>). (eq. 2.5)

Note that the An(tk) values will form a convergent series for the

expected. i.e.. smoothly varying. 7(tk) values.

The nominally square micro-domain is established by comparing. at
each time step value. An with the cumulative sum of the 8s values.

Namely, for

k
As(tk) = 2k'=1 85(tk'). (eq. 2.6)

the computational scheme allow k to increase until the cumulative

length first exceeds the current value of the width:

As(tk) l An(tk) (eq. 2.7)

13hr prOper number of time steps (N) is defined as that value which

Causes As to most nearly equal An. Specifically. N=k if

[As(tk) - An(tk)] ( [An(tk_1) - As(tk_1)]. (eq. 2.8)

Similarly. N=k-l if the inequality is reversed. These operations

13

define the number of time steps to proceed from ri_1 to Ti; viz.

"i = 11-. + Nfit- (eq. 2.9)

J

and the quantities at these two limits will be identified as. for

example. nn(ri) or “n(Ti-1)' It is pertinent to note that the nature

of the variable in question defines whether it represents an instan-
taneous (e.g..un(ri)). an incremental (e.g..83(tk)). or a cumulative

(6-8-:<7(Ti)>) value. The procedures to evaluate the spatial average

quantities given the correct N value are presented in the following.

2.4.2 Micro-Domain Average Values

The spatial average value for the transverse vorticity over the

micro-circulation domain [AnAs](ti), can be expressed as:

-1
(wz)(ti) = (AsAn) IAmzdnds. (eq. 2.10)

7Ihe area integral in eq. 2.10 is transformed into a contour integral
.around the circumference of the paralleIOgram AsAn by applying Stokes

lrheorem:

_ -1 +An/z
(mz) = (AnAs) [J:-An/z[un(s+A8/2) - un(s-As/2))cos<y)]dn/cos<y)

- js+Asl2

S_As/2[us(n+An/2) - us(n-An/2)]ds] (eq. 2.11)

I?igure 2.4 shows a representation of the defined circulation 100p

ilround the circumference of the micro domain. The first integral in

14
equation 2.11 may be interpreted as (Bun/as) and the second integral
BS (ans/an). Hence equation 2.11 may be rewritten in terms of these

spatial velocity gradients;

(NZ) = (Bun/as) - (Bus/an) (eq. 2.12)

The strain rate. “xy' may be readily evaluated using these spatial

velocity gradients: viz..

(81y) = (Bun/as) + (ans/an) (eq. 2013)

The evaluation 0f (Bun/as) is estimated using the difference between

the cross-stream velocity “n calculated at s and s+As and then divid-

ing by Asi, ie.;

(Bun/690:1) = [un(ri_1) " un(ti)] I AS(‘Ci) (eq. 2.14)
where

nn(ri) = Qp(ri)sin(7(ri)-<y(ti>) (eq. 2.15)

un(ti-1) = Qp(ti_1)sin(7(Ti_1)-<7(ti>) . (eq.2.16)

Qp = (Q3 + Q‘)/2. (eq. 2.17)

15
The factor sin(7(ti)-<y(ri)>) accounts for the difference between the
direction of the instantaneous streamwise velocity [Qp(tj)] and the

mean flow direction: (7061)). for theaveraging time 1:1.1 to ‘Ci of

the current micro-circulation domain.

The second integral represents (ans/3n)(ti) which is analogous to
(Bu/8y) in laboratory coordinates. The strategy to obtain a value for
(ans/GnHti) utilizes a summation of the incremental values of

Bus(tk); name ly. for

Bus(tk) ==0.5[(0u-Q,)cos(7(tk)-<7(tk)>) +

(CL-Q,)cos(7(tk_1)-<7(tk_1)))] (eq. 2.13)
the (ans/6n) value can be written as
N N
(ans/8n(ri)) == 2k=1[8us(tk)][bs(tk)] / [An(ti)§k=188(tk)] (eq. 2.19)

Note that the factor c08(7(tk)-<1(tk))) aligns each incremental con-
vective length: 53(tk)' with the average convection velocity vector
of the fluid element. . In a similar manner. the spatial average for
the two velocity components. <u8(ri)> and (un(-ci)> are also defined

Over the micro-circulation domain: AsAn(1:i). as:

< "1N
‘13(ti)> = (As) 21:1 0.5[Qp(tk)cos(7(tk)-(7(tk)>) +

Qp(tk_1)cos(1(tk-1)-<7(tk_1)>)]Ss(tk) (eq. 2.20)

16

N
(un(ri)> = (As)’1§k_1 0.5[Qp(tk)sin(7(tk)-<y(tk))) +

Qp(tk-1)sin(7(tk_1)-<7(tk_1)))]fis(tk) (eq. 2.21)

Thus equations 2.12-21 are used to Obtain values for the irregular
time series of the Spatially averaged quantities: (us(ti)),<un(ti)),
(ans/6n(ri)).<aun/as(ri)>.(wz(ri)>.<axy(ri)). These quantities are in
terms of the s-n or micro-circulation domain coordinates. A coordi-
nate transformation converts these. quantities to laboratory
coordinates: x-y: the transformation equations are presented in
Appendix B. The vorticity values are independent of this transforma-
tion but the strain rate. velocity components and their derivatives
are not. Figure 2.5 shows the relation between the s-n coordinates.
probe coordinates and the laboratory coordinates. The relative angle
between the x-y and s-n coordinate systems of a given

inicro-circulation domain is given by

whereB is the angle between the vorticity probe axis and the
:xr-laboratory coordinate and (1(ri)> is the angle between the velocity

Vector <Q(ti)) and the vorticity probe axis.

CHAPTER 3

SUPPORTING SCHEMES FOR THE VORTICIT'Y CALCULATION

3.0 Introduction

The complete scheme to determine the time history of the
transverse vorticity involves several intermediate calculations: spec-
ifically. the regular time series: [Qx(tj).y(tj)]. is computed from
the slant wires of the x-array. and a calculation to correct for the
influence of the transverse velocity component on the computedQx and
'7 values may be performed. The present section describes these calcu-
llation procedures and the extensive processing of the calibration data

that is required to support them.
3 .1 Determining Q and 7 From the x-array Voltages

The response of the x-array. [31.E,](tj). is used to construct
tlle regular ‘time series 0f Qx(tj).1(tj). Basically there are three
SChemes that have been evolved from which the Qx and 7 information may
‘16 extracted from the x-array measurements: (1) a
t“"‘r'O-equation/two—-unknown scheme technique (e.g.. Bradshaw [21]). (2) a
talble look-mp (e.g. Willmarth and Bogar [4]) and (3) an iterative
speed wire/angle wire procedure (Foss [18]). The data base for the

iterative scheme is similar to that required for the table look-up

17

18

technique. Smoothing and interpolation operations that are made pos-
sible by the use of analytic relationships. are employed in the
iterative technique; hence. this method gains accuracy. at the expense

of computation time and complexity. with respect to the table look-up

scheme.

The two-equation/two-unknown scheme is characterized by ,the
inethod described by Bradshaw [21]. For E2=A+BQ§ff and Qeff=QCOS(B"7)n

the resulting velocity component equations are:

11/n1 (eq. 3.1)

u 00881 + v sinfl; [(Ei-A1)/Bl

[(Ezz-A,)/B,]1/nz (eq. 3.2)

u c0583 + v sinfi3

where
‘y = TAN-1(v/u) (ea 3-3)

:Thus the regular time series of sampled voltages [E1,E3](tj) from the
x-array are used to obtain a regular time series of u(tj),v(tj),y(tj).
711115 calculation scheme is referred to (herein)as OOSLAW and is des-
caribed more fully in Appendix A. Implicitly. it is assumed that the
<:C>sine relationship is uniformly valid over a range of angles from the
OIfzientation at which the calibration is executed (i.e.; y=0 for the

Pt esent study).

19

A correction to the basic cosine relationship can. extend the
range of validity for a given slant wire; for example. Foss [16] con-
siders the available analytical forms. The pitch angle range of
validity for the cosine and extended cosine laws for a given slant
wire is shown schematically in Figure 3.1. Note that the viable range
for an x-array to deliver accurate Q and 1 values is limited by the
inaccuracy of the analytical form at large l8-7I values. The range of
pitch angles where the analytical form fails to describe E=f(Q.7)
relationship is referred to as the 'outer range'. This inability of a
single analytic form to accurately represent E(Q.7) in the 'outer
range' motivates the development of an alternative calculation strate-
gy. The strategy involves the designation. for a given data pair
[El.E,]. of one slant wire as the angle wire and one slant wire as the
speed wire. An iterative calculation is used to determine Q1 and y
for each sample time t- as noted below.

J

3.1.1 Speed-Wire/Angle-Wire Technique: Iterative

At a given sample time. tj. one of the slant wires will be
oriented at a relatively large value of [8-7] while the other will be
tit a relatively small value of [8-7]. The wire at the small [8-yl
awalue is designated the 'speed wire' since it is predominantly sensi-
txive to the speed Q and minimally sensitive to the pitch angle 7.
I.ikewise. the wire at large [8-7] is designated the 'angle wire' and
iws most sensitive to 7 with a reduced sensitivity to Q. Figure 3.2
depicts wire 1 as the speed wire and wire 2 as the angle wire (i.e.,

7>0°).

20

Given the state shown in Figure 3.2. the pitch angle and flow

speed are determined as follows. From the calibration of the x-array

the functions: fa and £5. are available; specifically,

7 = fa(Ea(7)/Ea(0)3QjJ-) (eq. 3.4)

Qs = fs(Es‘70) (eq. 3.5)

where the convention: (azb) is used to distinguish between an inde-
pendent variable: 'a'. and a parameter: 'b'. Figures 3.3arc show
schematically how these functions are used to obtain the_ 01 and 7
values. Note that wire 1 is the speed wire and wire 2 is the angle

wire for this illustration.

The calculation scheme is initiated with an estimate for the
speed ’Qx' via the GDSLAW technique. Using Qx as a parameter. the
pitch angle 7 is determined from the functional form given in eq.
3.4. for which lQi-ijl is minimized. Then. using 7 as a parameter. a
:mew value for the speed is determined from the functional form given
:in. eq. 3.5. for which [7-16] is minimized. (Note. both ij and 7c
are members of the calibration grid: see Figure 4.1.) For values of 7

’E 13 the speed is adjusted accordingly by (locally) employing the con-

cept of an effective cooling velocity:

Qx(Es;7) = Q8 [cos(B-7c)/cos(B-7)] (eq. 3.6)

21
The value of Qx' found using eq. 3.6. may then) be used as the
parameter to re-evaluate 7. The process iterates to convergence
wherein |7n+1’7n'$A70' A value of A7°=.5o was used for the present
study. but the value may be arbitrarily established. Note that as
A7q90 the Q.7 values converge to the true values. This convergence
was observed numerically and it follows from considering the effect of

a perturbation. from the true values, in the sequence of steps shown

in Figures 3 .3 a-c.

The above iterative scheme accurately determines GI and y for
large [8-7) values. At the smaller ”d” values the accuracy is also
maintained but the convergence time is significantly increased with
respect to using an explicit two-equation/two-unknown scheme. To max-
imize the accuracy and efficiency in determining an and 7 from the
[Eszl data pairs. the calculation algorithm employs both methods.
The initial value of 7 is determined using the (DSLAW; if it exceeds
the 2:120 range. then the cOrresponding Qx value is used as the start-
ing value for the iteration technique. If the initial value of 7 is
within the I7I$12° range. then this initial value is accepted and the
computation continues with the Ox and 7 values provided by the (DSLAW

calculation.

3 -2 Three-Dimensional Effects

3 -2.1 Effect of the Transverse Velocity Component on the Qx and 7

Ev aluation

Neither the COSLAW nor the iterative scheme accounts for the

22

effects of a transverse velocity component (w)on the response of slant

wires. Neglecting this effect may introduce significant errors in the

Q1 and 7 values calculated from the "contaminated" x-array voltages.

These errors have been observed. for example. by Vukoslavcevic and
Wallace [13]. Bruun [23] and Kastrinakis. Eckelmann and Willmarth [12]
in flows where large turbulence intensities are present. In the ana-
lyses of their data. these authors have accounted for the effect of
the transverse velocity by including higher order terms in the model
for the hot wire response. The following section presents a different
technique for determining the magnitude of the transverse velocity (w)

and a method to correct for its influence on the calculated values of

Qx and 7.
3.2.2 Detection of the Transverse Velocity

Consider that a z-component velocity (w) is added to the xry
plane velocity magnitude (0!). The change in the parallel array vol-
tages would be relatively small. since this component is parallel to
these wires: however. the speed-wire and the angle-wire would experi-
ence non-neglible changes in their voltages. The . speed-wire.
angle-wire voltages that would be, created by [01.7] will retain their

designations 83: Es and Ea respectively. The measured voltages. that
:ixxclude flhe effect of w. will be designated as Eslm and Balm respec-

tiively. The effects of w are then.designated as the difference (8E)

”Values as:

_E = E + SE and E

s'm s s alm Ba + 5Ea- (eq. 3.7)

23

It is pertinent to note that a given w value represents a relatively
larger effect on the angle-wire since its additive cooling effect is a
larger fraction that that which is provided by (01.7) acting alone.

Specifically,

Mia/F.a > OBS/Es . (eq. 3.8)

Since the x-array voltages are significantly more responsive to
the presence of a z-component velocity that are the vlotages of the
parallel array. an inequality of the form: QX)QP may signal the pres-
ence of a z-component of velocity. This inequality is not. however.

uniquely related to the presence of a non-zero w value as noted below.

The physical separation (Oz) between the two arrays and the exis-
tance of flhreeedimensional effects in the flow are sufficient to
produce an inequality in the Q1 and GP values. These effects alone
would create a symmetric distribution for the velocity magnitude
difference: OQ=IQi-Qp]. BOwever. the existance of non-zero w values
‘will give a positive bias to this distribution. A first order techni-I
«Ice. to discriminate between. the effect of w and the effect of

aQ/azfi'o. can be established using the measured frequency distribution
‘tlaat approximates the probability density function (p.d.f.) of the
Velocity difference: p(80). A typical frequency distribution is
Presented in Figure 3.4. The area of the frequency distribution that
i-8 associated with OQ<0 represents the dominance of a negative value

<>f acvaz. If it is assumed that wzo, then a symmetrically distributed

24

area for SQ>0 would exist since the average value of aQ/ax is equal to
zero. For computational simplicity. the magnitude of a "cut-off"

value: Scht-off=5pr’ is defined from the negative values of the SQ

distribution. Specifically.

I0 ap(80)du = kcjo ap(OQ)da (eq. 3.9)
-OQx -°
P .
where kc may be selected arbitrarily. A nominal value of 0.8 is used
in order that extreme values of 80 are not inapprOpriately weighted.
Hence. if OQ is positive and greater than Spr' the value of y at the
time of the [E1,Ez] measurement is assumed non-zero. The procedure to
correct the measured voltages. for the influence of this w value. is

described below.
3.2.3 The w3 Correction

The strategy for the correction scheme is that for an "inverse
problem": Given 80>801p. what value of the transverse velocity (w).
is required to cause 80 to be equal to zero. To answer this question.
the effect of w on the x-array response must be investigated. (It is
zissumed that the magnitude of w is identical at both of the slant

Wires.)

The concept of an effective cooling velocity in the x-y plane is
liiavoked in order to account for the effect of w. The effective cool-
illg velocity may be thought of as a velocity that is perpendicular to

'tlle wire and that provides a cooling effect equal to that of the actu-

25

al velocity. The effective velocity may therefore be described as:

.. 3 1 1/2
Qeff'total — [Q eff'x—y + W ] (eq. 3.10)

’where: Qeff'total - cooling effect on the wire including w
Qefflx—y - coolingeffect on the wire from a velocity

in the x-y plane only.

An explicit analytic expression for Qefflx—y is not required; however.
the response from each of the slant wires may be used to determine its

own effective cooling velocity.

(
Qefflspeed-wire = ([((Es+5Es)z-A(B))/B]3/n B) ' [w’])1/’ (eq. 3.11)
and
/ ( ) I
Qefflanglrwire = (I((E:a+513&)’-A(13))IB]3 '1 B - [1:31)1 3 (eq. 3.12)

The above coeffients: A.B n. were determined by calibrating the wire
at the respective B values for each slant wire. If an initial esti-
:mate for w2 is arbitraily selected. then effective velocities for the
speed and angle wires may be computed. NOte that these effective
‘velocities are in the x-y plane: hence. the initial estimates for the
¢:orrected speed and angle wire voltages may be evaluated by using the
Equa ti ons:

(BS)

B’s = A(Bs) + B<ps)[oefflsln (eq. 3.13)

26

n(fi )
E’a = A(Ba) + B(Ba)[Qeffla] a

(eq. 3.14)
The corrected voltages represent the response of the x—array given the
cooling effect of a velocity that is totally in the x-y plane. These
voltages are then used to determine 0x and 7 in the same manner
presented previously. i.e.. using equations 3.4 and 3.5 and the itera-
tive calculation scheme. A complete utilization of the previously

described scheme also accounts for the second order effect of 7 on the

Qp calculation; viz.:
Q1: = °-5[Q=(3337) + (110307)] (eq. 3.15)

The resulting 0x value is then compared with the Qp value determined
from the parallel array response. If the two are in agreement. it is
inferred that the postulated w value is correct; if not. a new w value

is selected and the above computation is repeated until saeo.

The above describes an overview of the strategy for determining
(ix and 7 from the response of an x-array; this strategy assumes that
tlie prOper calibration functions are available. The next section pro-
1vides the detailed considerations for Obtaining the calibration
fhanctions and the associated processing algorithms that are required

iv): the complete 'vorticity calculation'.

CHAPTER 4

CALIBRATION OF THE VORTICITY PROBE AND PROCESSING ALGORITHMS

4.0 Introduction

The objective of the vorticity probe calibration is to Obtain
analytical forms that accurately describe the response of each wire
for a range of pitch angles and flow speeds. Specifically the

required analytical forms are:

1 = fa(Ea(7)/Ea(0) ; ac) ‘ (eq. 4.1)
Qx = fs(Es 3 7c) 7 (eq. 4.2)
Q." = fpfli,’4 3 70) ‘ (eq. 4.3)

Equa tions 4.1 and 4.2 utilize the voltages from the xearray response
and equation 4.3 utilizes the parallel array voltages for the range of
Pitch angles and flow speeds established for the calibration grid.
The calibration grid is represented in Figure 4.1. For [7[.(12° the
COef ficients determined from a calibration at 7=0° are sufficient to
determine (.1x and 7; recall that the OOSLAW was shown to provide accu-
rate results for this range of 7 values. For ]7[)12° a more elaborate
cfilli‘btation is required to obtain the functions given in equations

27

28

4.1—4.3. The following sections present the details of the complete
calibration: i.e. the calibration facility and equipment, the data
aquisition procedure and the processing of the calibration data to

obtain the required analytical functions.

4.1 The Data Acquisition Facility

A.complete set of calibration data is obtained by sampling
[E1,E,,E,,E4] at each of the grid points: [Qc'Ycl’ shown in Figure
4.1. For the present study. 7 flow speeds: =2mls$Qc$§13m/s, and 15
pitch angles, ranging from -42° to 42° in increments of 6°. were used.
The calibration facility is shown in Figures 4.2a-b. The x and paral-
lel arrays of the-vorticity probe are supported by the fixture shown
in figure 4.2b. This fixture is mounted on a vertical stem which is
attached to a (2250m) horizontal arm. The active portions of the
hot-wires are positioned on the axis of rotation of the stem/arm
assembly; i.e. the hot wires spatial location (x,y,z) remains con-
stant as the probe body is rotated through the calibration pitch
angles. The constant temperature hot wire probes are operated with
DISA.55591 anemometers and the data acquisition system is based upon a
TSI 1075K (12-bit AID , 0-5 volts. matched SOkHz low pas filters) and
a Charles River Data LSI-11/23 computer (hereafter referred to as 'the
computer'). The rotation of the probe through the set of calibration
pitch angles is computer controlled. which insures the precision and
repeatability of the orientation for each calibration grid point. The
flow Speed is monitored using an additional straight wire (referred to

‘as the 'reference wire') which is located near the vorticity probe.

29
Prior to calibrating the vorticity probe. the reference wire is cali-
brated using a Validyne DP45-22 pressure transducer and CD12 carrier
demodulator over the same range of speeds as that to be used for the

probe calibration.
4.2 The Calibration Data

The calibration data set is acquired by sampling the 4 wires of
the vorticity probe at each of the 15 angles for the 7 flow speeds.
The reference wire voltage is simultaneously recorded. The final data

set consists of : [E1,E,,E,,E4,Erefl taken at each of the conditions

defined by the grid points in Figure 4.1.

The flow speeds indicated by the reference wire are used to cali-i
brate one of the straight wires of the parallel array. The selected
wire is designated as the 'master—wire'. The remaining 3 wires of the
probe are then calibrated using the speed indicated by the master-wire
for the particular 7c value. The selection is based on the best fit
(i.e. ' smallest standard deviation) to the modified Collis and Willi-
ams equation:

n(yc)

E“ = M70) + B(7c)Q (eq. 4.4)

s'm [0mm - a] = [(N—l)‘1§1:=1[Qmeas.-Q]:11]: (eq. 4.5)

where: Qmeas. is determined from the reference wire and Q is

30
evaluated from the measured E and the coefficients of equation 4.4.
This procedure relates 3 of the measured voltages to the 4th’ voltage
from the designated master-wire. Note that the vorticity is deter-
mined as the difference between two quantities which themselves are
differences. the relative (and not the absolute) values of the vol-
tages at a given time are of principal importance. Hence. a
calibration scheme which maximizes the relative accuracy of the wire
responses is both recommended and utilized. At each of the 7c values.
the [E'Q(Emaster)] pairs are fit to the modified Collis and Williams
equation (eq. 4.4). This results in 15 sets of the coefficients

(ABn) for each of the 4 wires.
4.3 A measure of the 'Effective' Angle of the Slant Wires

The angle between the wire and the probe axis. (B). is be deter-
mined operationally for each of the slant wires of the xrarray. That
is. when the probe is pitched at 7c=112° the data provides redundant

P

measures of B1 and B3. The cosine law can be used to obtain a rela-

tion for B as a function of 7:

3%,) = A(0) + h(0)Q§§°f) (eq. 4.6)

where Qeff = Q cos(B-7).

lEquation 4.6 represents a four parameter family of equations; viz.
&%(0). b(0).n(0) and B. The previously described calibration pro-
<3edures provide bestefit values for A(O) and n(0): however. b(0) and B

Suist be evaluated using additional information. Note that the compan-

31
ion value. B(O). may be used with the calibration data from one other

angular position to evaluate b(0) and B; specifically,

n(°)

B(O) = b(0) cos (5) (eq. 4.7)
and. for example.
E’(12°) = MO) + b(0)cosn(0)(B-12°)Qn(0) (eq. 4.8)

The second of these equations. i.e.(4.8). may be subjected to an aver-
aging procedure to evaluate the coefficient:
N n(°)
x = (1/Ns)§s:1[b(0)cos (5-12°)]s
where Ns = number of speeds used for the calibration (eq. 4.9)

Since Ez.A(0).n(O) and Q are known values. D can be solved for

explicitly as that value which satisfies:
K = [B(O)/cosn(°)B] cosn(°)(B-12°) (eq. 4.10)

The values of B found using equation 4.10 are approximately 3° less
than the measured values. vukoslavcevic and Wallace[13] have also
Observed this difference and attribute it to the hydrodynamic upstream

Effect of' the prongs on the flow and nonfuniform prOperties of the

Wires.

‘4.4 Construction.of Calibration Functions

32

4.4.1 The 'Smoothed' Calibration Data Base

The hon-wire voltages for wires 1 and 2 represent surfaces above
the Q.7 plane. In principle. these surfaces monotonically increase in
"height" as 793 and as Q increases. Similarly. the voltages from
wires 3 and 4 represent surfaces that monotonically increase in height
with increasing Q but are. in principle. not dependent upon 7. (The
acceleration around the support prongs does provide a weak dependence
upon 7. as shown comparatively in Figures 4.10arb and 4.12a-b. This

effect is also noted in the section 3.2.1 for the w3 correction

scheme.)

The calibration data are acquired at discrete positions in the
(Q.7) plane: see Figure 4.1. Tb insure the implied differentiability
of the E(Q.7) surface. the discrete data samples are replaced with an
apprepriately selected analytic function. The modified Collis and
Williams [20] relationship (eq. 4.4) is used for this purpose. The
coefficients are selected by minimizing the standard deviation (eq.
4.5)for discrete values of» n:.20..21..22. .70. Recall that a is
derived from the voltage value of the master-wire and its A.B.n coef-

ficients for that 7 value.

The above procedure is uniformly valid for wires 3 or 4 (i.e..
tile non—master—wire for a given 7) and for wires 1 and 2 in their
fipeed-wire range of 7 values. For the angle range. and especially for
llarge B~7 values. the calibration data may be better fit to equation

‘4.4 in a piecewise manner. That is. the coefficients A,B.n may vary

33
for different segments of the velocity range: QminTQmax° This can
result in a non-differentiable condition at the juncture points of the
Ea(Q37c) distributions. waever. this condition does not present any
difficulty since the analytical fit in the angle-wire range need only
provide for interpolation between measured velocity values at a given

7. These considerations are explored more fully below.

4.4.2 A Computationally Efficient Form for the Velocity bhgnitude

Evaluation

A smoothed calibration "data" base is constructed using equation
4.4. for the speed-wire range of wires 1 and 2 and for all 7 values
of the non-master-wire: 3 or 4. Each wires response (E)is analyti-
cally evaluated (using eq. 4.4) at the grid point conditions:
(Qc.7c) of Figure 4.1. The smoothness of these "data" is insured by

the use of this analytical form.

Drubka and Wlezian [22] have shown that a polynomial fit to such
data. can dramatically increase the computational efficiency of evalu-
ating Q from a measured E and with a negligible effect on the accuracy
Of the ErQ relationship. That is. although eq. 4.4 also offers a

Irelation for Q=f(E 7°). the computation.time is significantly greater.

 

"It is pertinent to note that a polynomial fit to the orginal (i.e..
tile non-smoothed) data base led to a condition in which the higher

0 I D1 0 c
<3er1vat1ves: ( n3/MQnI2G for n 3. did not show a montonlc decrease
VVith increasing Q values. In some cases. this led to problems of

(Hanvergence when the iterative method was employed.

34

on the computer than using a polynomial of integer powers. The polyb

nomial form is suggested by the inverse form of eq. 4.4: "for a

function that is defined and continuous on a closed interval. there

exists a polynomial that is close to the given function

as

desired"(Burden [25]). A.4th order Chebychev polynomial was chosen

since it allowed for maximum accuracy at a reduced degree of the

approximating polynomial. Specifically.

5 .
cc...) = 2M .i Bar-1) «eq. 4.12)

where ai are the polynomial coefficients

For each of the slant wires. the approximating polynomial

.required only over the range of 7c for which that wire is designated

the speed-wire:

wire 1 - speed-wire: 7c 2 00

wire 2 — speed-wire: 7c S 00

P41: the straight wires. the polynomial coefficients are determined

3‘7611' 7c. even though their responses are a very weak function of the

Pi tch angle.

4-.4.3 The Angle-Wire Response Functions

is

at

The iterative calculation scheme requires that 7 be determined

from a function of the form:

35

Y = f3(Ea(7)/Ea(0) 3 ij) . (eq. 4°13)

where Ea(7) represents the measured voltage. ij is a reference velo-
city that is "close" to that existing at the instant of the
measurement and Ea(0) is the corresponding voltage at 7=0o for the
angle-wire.‘ The calibration data provide the necessary information to

develop the function ”fa".

The procedure to interpolate: B(stc). between the measured Q
values at a given 7c was described in section 4.4.1: from that discus-

sion it is apparent that the subdivision of the velocity range into a

set of ij values can be accomplished. The ij values are described

as:

len 1 ij S Qmax ; 8ij = (omax-Qmin)l36 (eq. 4.14)

I‘or.convenience. fa is expressed as a function 0f n with a parametric

dependence upon ij as

7 = fe<n ; ij) where n = Ea(7)/Ea(0) (eq. 4.15)
The function: fa is known from the smoothed calibration data. at the
discrete locations described by the 7 values of the calibration pro-

CC 85: 70: viz..

lvcl = 12.18.24.30.36.42 degrees.

36
Cubic splines are used to interpolate between these discrete 7c
values. Figures 4.5 and 4.6 are representative sets of such functions

that were fitted to the smoothed and doubly interpolated data set.
4.5 A 'Data-Day' Correction Scheme

From the time of calibration to the time of a data day run. the
wires response character may be altered due to corrosion. dust. etc.
The calibration coefficients for the Q and 7 functions (eq. 4.12 and
4.13) may then become invalid. If the overall changes are small and
have only the effect of causing a minor drift in each wire's response.
a linear correction may be applied to the data-day voltages. The mag-
nitude of the drift is estimated by assuming that. the data-day
voltages have shifted by SE"

Eda = BC2 + 8E3: (eq 4.16)

where: Ed - data-day voltage and
Ec - voltage measured at the time of the master calibration

time
'Ijais expression may be expressed in terms of the coefficient A as:
Ed2 = EC2 (1 + A) ; where i = eB’IEc’. (eq. 4.17)
15 'minor' drift condition is represented by i=0. For this condition

tile master calibration coefficients are considered to remain valid for

tize corrected data-day voltages:

37
Ec’ = Ed’lu + 1) (eq. 4.18)
The correction factor (1+1) is established by calibrating the vortici-
ty probe at the 70:00 condition prior to a data run. The set of
[Ed'od] pairs are sampled over the same range of flow speeds used for
the master calibration: Qc-minquiqc-max' At each ad the (1+1) factor
is determined for each wire using eq. 4.17:

( )
1 + l. = E2/(Ac(0) + Bc(O)an° o ) (eq. 4.19)

The average correction factor is then found for each wire. If the
((1+).)> factors are sufficiently different than 1. the vorticity probe
must be recalibrated; otherwise the data-day voltages are corrected

according to eq. 4.19 before being used in the vorticity algorithm.
4.6 Calibration Curves From Experimental Calibration Data

Actual experimental data was used to produce the curves shown in
I7igures 4.3-4.8. These curves are of the same form as those shown
Previously in Figures 3.3a-c. The curves that are used in the itera-
tive scheme for determining Q1 and 7 are designated by the angle or
speed-wire condition noted for wires 1 and 2. Lack of this designa-
tion indicates determination of Q.x and 7 by the (DSLAW technique (see
Appendix A). (two-equation/two—unknown). Figures 4.9-4.12 are cali-
bration curves for the straight wires 3 and 4. Note the slight pitch
atlgle dependence for the Q=f(E2) curves as identified in section

4 -4.1: see Figures 4.10a-b and 4.12a-b. This slight 7 dependence is

38

also apparent in Figures 4.9a-b and 4.11a-b for the E2=f(Qn) curves.

CHAPTER 5

DE l-DN STR ATION DATA

5.0 Introduction

A limited body of experimental data has been acquired to demon-
strate the computational procedures described in Chapters 2 and 3.
These experimental results. and the relevant observations that are

inferred from them. are presented in this chapter.

5.1 Expermental Results

The four-wire array was placed in the intermittent region of a
Jlarge plane Shear layer. The shear layer measurements were taken in
tile test section of the Free Shear Flow Facility that is shown in Fig-
letes 5.1a-b. The probe position: =lm. y=.099m. was selected to
Provide an intermittent condition wherein vortical and non-vortical
leuid would occupy the probe location. The nonrdimensional probe
31<>cation may be described in terms of y1/2 (i.e.. the y value such
‘tllat u/U°=.5) and the apparant origin (x,) of the linearly growing
Sllear layer: 8w=C(x-xo). The vorticity thickness: 5m’ is defined
3.31.

em = 00 / (Bu/6y)max.

39

40
The non-dimensional: “=(Y'Y1/,)/(x-xo). location of the probe was

0.076 and the correSponding value of E/Uo was 0.17.

The data were acquired with an imposed probe angle of: 9=-20°:
this insured that the large angles of the entrainment stream. with
respect to the x-axis. would not exceed the (1)42o pitch angle of the

calibration grid.

The four. hot-wire voltages were simultaneously sampled at a rate
of 15.625 hz and the data acquisition computer (see section 4.1) was
able to store a continuous record of 8.125 samples per wire. The ini-
tial processing made use of the scheme. described in Chapter 3. to
convert these voltages to Q1 and 7 values at each time value. A value
of 0.25 degrees was used for the 7-convergence criterion. A.computa-
tion time of 42 minutes was required on the 11-23 micro-computer(RT=11
operating system). These (Qx'V) values were then combined with the
measured (E,.E4) values as the inputs to the computational procedures
described in Chapter 2. This computation time was 16 minutes. The Q1
and Qp values were then used to correct E1 and E, for the presence of
a transverse velocity: w’. as described in Chapter 3. The corrected
'7 values were then used to recompute the micro-domain quantities. The
resulting time series for the transverse vorticity. the strain rate
sand velocity components are presented in Figures 5.2a-5.6a. The

ilifluence of the transverse velocity. w on each of these quantities
veers also determined. The time series using the corrected values are

Presented in Figures 5.2b-5.6b.

41
In principle. the transition from the vortical to the
non-vortical state can be characterized by the magnitude of the vorti-
city. In practice. this transition is obscured by the difficulty of

providing a measurement of d)

that is sufficiently free from uncer-
tainties. For the present demonstration data. it is encouraging that
the effect of the w1 correction has very little influence on the

inferred location of the transition. It is also encouraging that the

02(1) time series appears to be qualitatively reasonable.

Given the w2(r) signal. it is of interest to note that the vorti-
cal/non-vortical transition is not readily apparent in the
corresponding u and v timeseries record. This observation is compa-
tible with the motivation for the present effort: flhat the direct
ineasurement of the transverse vorticity consitutes a significant

experimental capability for fluid mechanic investigations.

The frequency distribution in Figure 3.4 showed the occurance of
both Qx>Qp and Qx<Qp in the processed data. it is speculated that for
‘3x>Qp’ the difference is in part due to the presence of a transverse
‘velocity (w) and in part due to dIIIBZfO. A correction for the w3
:influence on the x-array voltages was made (kc=o,8 from eq. 3,9) and
'the) resulting time series for the vorticity and velocity components
sire Shown in Figures 5.3b-5.4b. For the present data. the qualitative
<3haracter of the signals remain unchanged by the application of the w3
c<xrrection. Quantitatively. the magnitude of the vorticity values in
tile highly fluctuating regions is increased. by approximately

400(l/sec). The velocity components. for the corresponding times.

42

showed a slight decrease (approximately .6 m/s) in magnitude for the

large fluctuations of (u) and (v). The corresponding time series for

the strain rate and velocity derivatives are presented in Figures 5.4

to 5.6.

CHAPTER 6

SUMMARY

A method to obtain time series for the transverse vorticity. and
the in-plane velocity components from 4 hot-wire voltages has been
presented. The method includes several calculation schemes. The
iterative scheme. which determines the velocity magnitude (Q!) and (7)
from the voltages of the xrarray. is basically the same as FossIlB];
however several significant improvements have been.made in this compu-
tation procedure. Specifically. the present scheme utilizes the
two-equation two-unknown method (COSLAW) for pitch angles that-are
‘within :120 of the probe axis. This modification significantly
decreases the computation time. without decreasing the accuracy. and
it provides a viable first estimate of the flow speed as required for
the iterative procedure (I7))12°). An evaluation of the iterative
Inethod subsequent to the preparation of [18]. revealed a very slow
(convergence at small 7 values; hence. the used of the OOSLAW solves an
linherent problem with the iterative method. The iterative method is
13ound to be relatively efficient at large pitch angle values: when the
(:alibration data are inserted in the (01.7) calculation scheme. Using

cExand 7 values determined from OOSLAW as initial estimates the method

<=c>nverged to the correct values within 3 interations at the large

I?i.tch angles for a convergence criterion of A7=.25°.

43

44

The use of a temporally aligned (s-n). micro-domain in the
present computational scheme is considered to be a significant
improvement compared with the (x-y) orientation of flhe previous
method. Specifically. the basis for the space (Ss)/time (5tk) core
respondence of the convected (no) fluid element is more rational if
the velocity vector in the x—y plane is used for the convective speed.
The variable size and orientation of the micro-domains is compatible

with the reconstruction of the time series (ti).

The locally defined s—n coordinates require that coordinate
transformation techniques be used to evaluate the velocity components
and their spatial derivatives in the laboratory. or (x-y). coordi-
nates. These considerations have been used for the demonstration data
of the present study and the coordinate transformations account for
the pitched probe orientation (9=-20°) as well as the s-n orientation

with respect to the probe axis.

The calibration of the vorticity probe and the subsequent cali-
bration functions. (7=f(n;055). Q=f(E3.7c))' have revealed some
interesting characteristics of the vorticity probe. hhmely at large
Ifi-7' values the aerodynamic influence of the parallel array on the
slant wire adjacent to it. is suggested by the consistent appearance
Of a steepening in the 7=f(n:ij) function for 7)36°. In the present
configuration of the vorticity probe. wire 1 shows this steepening
‘trend. see Figure 4.3a. The same steepening character was found to

texist in two previous calibration curves for different wires but in

‘tlie same position. that is the trend appeared in the x-array wire

45
directly above the parallel array. A study to determine the minimum

distance for which no significant effect is observed would be useful.

The calibration speed functions for the parallel wires showed a
slight dependence for specific ranges of pitch angle. These ranges
corresponded alternately for the large '7' values in which one wire
affected ‘the flow on the other wire. Specifically wire 4 appeared to
show the pitch angle dependence for large positive 7 and negligible-

dependence for 7(0. see Figures 4.12a-b. The opposite trend was

observed for wire 3.

The designation of a 'master wire' for the evaluation of the flow
speeds during calibration was introduced by Foss[l7]. The present
calibration scheme utilizes this concept in defining one of the wires
()f the parallel array as a master wire. for each pitch angle used dur-
ing the vorticity probe calibration. The straight wire chosen. wire 3
or 4. is based on the minimum STD of the data fit to the response

function at a given 7c. This is in constrast with the technique of

[18] which defined on straight wire as the master wire for the entire

c a 1 ibration.

FIGURES

46

I START )

‘1

 

 

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YES

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HOT WIRE VOLTAGES

no _ J

DETERMINE: 0,7 FROM X- ARRAY RESPONSE I

 

 

 

 

 

 

 

 

   

DETERMINEI EFFECT OF w
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YES

 

 

 

 

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DETERMINE: TIME SERIES OF TRANSVERSE VORTICITY,
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Ix

STOP

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APPENDI CES

APPENDIX A

COSLAW

The two-equation two-unknown technique, OOSLAW, uses the voltages
from the x—array to obtain estimates for Qx and 7. These estimates
are either accepted as valid measures of Q1 and 7 (lyli12°) or are

utilized as initial values for the iterative scheme.

The COSLAW technique employs the following form of the Collis and
Williams equation,
n(°)

E“ = A(0) + b(0)Qeff (eq. A.1)
where

Qeff = Qx 00803-7) - (eq. A.1a)
and

b(0)=B(o)/cosn(°)B

Note that B(O) is the coefficient determined for the modified Collis

and Williams form (eq. 4.4) and the calibration data at 7=0°.
9O

91

Equation A.1 may then be written for both slant wires in an expanded

form, as:

l/ (0)
[(Elz-A1(O))/b1] n = Qxcos(B1-7)
= Qx(cosfilcosy + siuplsiny) (eq. A.2)
1/ (°)
[(E,’-A,<0))/b,] ” = Qxcos(B,-7)
= Qx(cosfi,cosy + sinfizsiny) (eq. A.3)

Recognizing that u=Qxcosy and v=stiny equations A.2 and A.3 may be

rewritten as:

C
II

1 u c0381 + v sinB; (eq. A.4)

C
II

2 u cosB2 + v sinB2 (eq. A.5)

where

]l/nk(°)

II

[(Ek’-Ak(o))/bk(o> ; for k=1.2.

Ck

Equations A.4 and A.5 are then solved simultaneously for the velocity

components u and v;

u = A'—1 [ClsinB2 - Czsinflll (eq. A.6)

92

and

v = A-1 [Czcosfi1 - Clcosfizl (eq. A.7)

where

D.
II

cosfilsinfiz - sinBlcosBZ,

from which Qx and y are readily determined as:

1/2
01 = [u2 + V3]

7 = TAN"1 (v/u).

The COSLAW technique is considered to adequately determine Qx and
7 for values of 7$I12°I. Figures A.1 and A.2 show the difference
between calculated values of 7 (using OOSLAW) and the Ye value of the
calibration data. Note that this difference is both speed and angle
dependent. Tables 1 and 2 present the numerical values of these
differences as well as the percent difference between the calculated
and measured speed values for both slant wires. For 7<I12°I these
differences for both wires are relatively constant and adequately
close to zero at all flow speeds. This is the angle-range for ‘which
the OOSLAW technique is considered valid in determining values for Oz
and 7. It is interesting to note that the COSLAW probides a relative-
1y accurate Qx value for a much larger range of 7 values; the relative

errors in the u and v calculations are self-compensating since

Q=(u2+v2)1/3.

93

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ACCURACY OF COSLAW

96

TABLE 1

8.755

119.238

25.156
7.563
2.378
0.881
0.123
0.282

“0.098

“0.214

“0.234

“0.185

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0.015

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“0.489
“0.183
0.062
0.000
0.226
0.364
0.472
0.547
1.319
0.608

a. [(Qc - Qcos)/Qc] * 100.
\ m/s 12.489 10.405
7
“42.0 “469.914 “494.155 “527.021
“36.0 94.331 105.821
-30.0 18.240 21.143
“24.0 4.533 6.206
“18.0 1.345 1.926
“12.0 0.263 0.483
“6.0 “0.269 0.420
0.0 “0.160 0.048
6.0 “0.064 “0.029
12.0 0.094 “0.289
18.0 “0.220 0.327
24.0 0.025 “0.076
30.0 0.149 0.248
36.0 “0.189 0.079
42.0 “0.164 “0.408
b' 7c 7 7cos
“42.0 “8.580 “9.011
'“36.0 “3.976 “4.443
“30.0 “1.902 “2.177
“24.0 ‘“0.783 “1.026
“18.0 “0.353 “0.439
“12.0 “0.130 “0.133
“6.0 0.042 “0.145
0.0 0.000 0.000
6.0 -0.057 0.046
12.0 “0.188 0.248
18.0 0.055 “0.259
24.0 “0.220 0.146
30.0 “0.498 “0.321
36.0 0.069 “0.072
42.0 0.017 1.713

3.099

7.357

“562.641

132.778

29.918
9.685
3.434
1.136
0.590

“0.074

“0.396

“0.529

“0.414

“0.552

-0.400

“0.214

“1.005

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“3.104
“1.630
“0.822
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0.000
0.192
0.335
0.313
0.561
0.512
70.323
3.227

(slant wire 1)

5.503

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162.127

36.607
12.847
4.271
1.592
0.347
0.098
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-1.184
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-2.129
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“0.097
0.000
0.335
0.627
0.764
1.297
1.944
2.923
4.323

3.779

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200.397

48.257
17.368

6.363

2.169

0.143
-0.342
—0.926
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-1.606
-2.112
-2.136
-2.165

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-5.064
-2.978
-1.531
-0.776
-0.189
0.000
0.349
0.659
1.374
1.464
2.654
73.703
5.629

2.100

“902.556
272.802

73.879
29.623
13.032

5.004

1.448

0.146
“1.250
“1.612
-2.585
-2.577
“2.733
-2.596
“2.464

“16.570
-11.562

“7.690
“4.961
“3.040
“1.500
“0.507
0.000
0.826
1.277
2.436
3.054
4.155
5.365
7.410

97

TABLE 2

ACCURACY OF COSLAW (slant wire 2)

a. [(Qc - Qcos)/Qc] * 100.
\ m/s 12.489 10.405 8.755 7.357 5.503 3.779 2.100
“7’
—42.0 —0.121 —0.216 -0.720 -1.046 —1.272 -1.619 -2.806
-36.0 -O.418 0.021 -0.382 -0.690 -0.862 -1.370 -2.949
-3o.0 “0.182 -0.102 -0.457 —0.534 ,-0.947 —1.541 —2.661
~24.0 —0.445 0.025 -0.768 —o.513 -0.271 -1.011 —2.181
~13.0 0.154 -0.035 —0.774 -0.555 -0.540 -0.907 -1.547
-12.0 0.113 —0.106 —0.529 -0.569 ~0.223 —0.637 ~1.801
—6.0 0.112 0.318 —0.477 -0.239 -0.130 -O.660 —0.895
0.0 0.125 0.064 -0.203 -0.325 0.350 0.028 ~0.030
6.0 0.118 -0.034 “0.282 -0.154 0.371 0.649 1.150
12.0 0.303 -0.073 -0.240 0.060 0.780 1.502 3.016
18.0 0.762 0.929 0.722 1.330 2.720 3.834 6.867
24.0 2.748 2.675 3.101 4.240 6.454 9.706 16.391
30.0 7.950 6.015 9.450 12.230 16.398 22.673 36.206
36.0 25.541 27.844 32.855 38.611 47.778 62.488 90.421
42.0 137.173 148.778 167.202 187.173 223.572 273.954 363.872
b' 70 - 7008
-42-0 -2.189 -2.418 -3.793 —4.775 -8.041 -8.135 -11.185
-36.0 -1.858 -0.164 -0.657 -1.292 -3.764 -4.259 ~7.688
-3o.o —o.661 -0.360 ~0.550 -0.456 -2.616 ~3.126 -4.972
-24.0 -0.852 -0.059 -0.847 -0.286 -0.924 -1.531 -3.065
~18.0 0.033 -0.113 -0.648 -0.263 ~0.998 -1.051 -1.688
~12.0 —o.011 -0.152 -0.291 —0.218 -0.508 —0.591 -1.555
-6.0 —0.010 0.182 -0.197 0.062 -0.342 -o.491 -0.617
0.0 -0.000 -0.000 -0.000 -0.000 , —0.000 -0.000 -0.000
6.0 -0.003 -0.046 -0.037 0.081 0.010 0.293 0.558
12.0 0.067 -o.052 —0.014 0.146 0.162 0.559 1.159
18.0 0.189 0.257 0.276 0.494 0.704 1.136 2.070
24.0 0.590 0.587 0.746 1.032 1.373 2.190 3.741
33.0 1.238 0.941. 1.533 1.999 2.542 3.609 5.815
36.0 2.419 2.647 3.161 3.732 4.522 5.992 8.735
42.0 4.689 5.094 5.753 6.457 7.647 9.437 12.610

APPENDIX.B

COORDINATE TRANSFORMATION

The spatially averaged values. associated with a given
micro-domain, are in terms of the local s-n (micro-domain) coordi-
nates. A coordinate transformation is used to express the values in
terms of the laboratory coordinates; x-y. From Spencer[29]. the
transformation for the velocity components in tensor notation is given

by;

1:5. = ‘5 e51 (eq. 3.1)

where * quantities represent the value in the new corrdinates; z“y.
non-starred values are in terms of the original coordinates;
s-n,

and eki and elk are direction cosines.

By setting i=1 and i=2. the velocity components u and v may be

expressed in terms of u and v as;

S n

t

u = 118 case + un sina (eq. B.2)

98

99

* .
V = “u. 51nd + un cosa. (eq. 3.3)

5
Similarly, the transformation for the spatial derivatives in tensor

notation is given by;

*
Ai,j = eki elj Ak,1 (eq. 8.4)
To express one spatial derivative in the new coordinates. four spatial
derivatives in the s-n system must be converted to x-y corrdinates.
The following sequence presents the transformations to obtain au/ay

and avlax. Specifically, Bulay may be written as;

33;/8x, = ene12 dullax1 + ene12 au,/ax1 +

enen Gui/axz + e31e12 au,/ax, (eq. B.5)

which is determined from setting i=1 and j=2 in eq. 8.4.

au/ay = “cosa sina Bus/as - sinza aun/as +
cos’a ans/an + sina cosa dun/an (eq.-B.6)
au/By = 00820 ans/an “ sinza Bun/as +

sina cosa (Gun/an — ans/as) (eq. B.7)

Similarly by setting i=2 and j=1 in eq. B.4 the equation for avlax is

auz/ax1 = e1,e,,au1/ax1 + enenauz/ax1 +

100

612e218u1/ax2 + 622e213u3/6x, (eq. B.8)

av/ax = “sina cosu Bus/as + cosza dun/as -
sinza Bus/an + cosa sina dun/6n (eq. B.9)
Bv/ax = 00520 Bun/as - sinza Bus/6n +
sina cosa (dun/an - ans/as) (eq. 8.10)
where
a = 9 + 7. (eq. 8.11)

Since the vorticity is coordinate independent (perpendicular to the
x-y plane). the equation for its value should be the same in terms of
s-n and x~y. By the addition of eqs. B.7 and B.10. it can be shown

that the vorticity vectors in the s-n and x-y coordinates. to have the

same form:

8v/ax - au/ay = Bun/as - ans/6n. (eq. B.12)

APPENDIX C

RESPONSE FUNCTION COEFFICIENTS

For each of the four wires of the vorticity probe the response
function. i.e. modified Collis and Williams equation. was determined
at every calibration pitch angle 1c. The set of coefficients, [ABn].

from each of these functions are presented in Figures C.1-C.4 as a

function of 70, for each of the wires.

101

102

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8 4
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886
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1. 1.
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1. + + opxcue QGQH
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...... 8.3

REFERENCES

p...
to
0

REFERENCES

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107

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Bruun, H. H. "Hot-wire Data Corrections in a Low and High
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108

27. Comte-Bellot, G. "Hot-Wire Anemometry." Ann. Rev. Fluid Mech.
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