DESIGN AND CALIBRATION OF SEVEN HOLE PROBES FOR
FLOW MEASUREMENT
by
James Douglas Crawford
A thesis submitted to the Department of Mechanical and Materials Engineering
In conformity with the requirements for
the degree of Master of Applied Science
Queen’s University
Kingston, Ontario, Canada
April 2011
Copyright ©James Douglas Crawford, 2011
Abstract
The calibration and use of seven hole pressure probes for hot flow measurement was studied
extensively, and guidelines were developed for the calibration and use of these probes. The
influence of tip shape, Reynolds number, calibration grid density, and curve fit were studied and
reported. Calibration was done using the well established polynomial curve fit method of
Gallington. An improvement to this method was proposed that improved the uniformity and
magnitude of measurement error.
A hemispherical tip was found to be less sensitive to manufacturing defects, and less sensitive to
changes in tip Reynolds number than a conical tip.
The response of the probes was found to be Reynolds number independent over a tip Reynolds
number of 6000 for the entire calibrated range. For flows with an angle of attack less than
approximately 20°, the response of the probe was found to be independent above Re = 3000.
A minimum calibration grid density of 5° was recommended. Error in the measurement of high
angle flows was found to increase significantly when the calibration grid was sparser than this.
The response of the probe was found to contain features that were not properly represented by
third order polynomial terms, and it was found that it was necessary to include fourth order terms
in the polynomial curve fit.
The uniformity of calibration error was found to improve significantly when the high angle
sectors were calibrated using a small number of additional points from adjacent sectors. The
calibration data sorting algorithm was modified to include a calibration point in a given sector if
that sector’s port read the highest pressure, or if that port read within a specified tolerance
(“overlap pressure”) of the highest pressure. An overlap pressure of 15-20% of the calibration
flow dynamic pressure was found to decrease the maximum calibration errors by 10-15%.
ii
Acknowledgements
I would like to first thank Dr. A.M. Birk of the Department of Mechanical and Materials
Engineering for the opportunity to perform this work, and for his guidance and support.
I would like to thank the technical staff at McLaughlin Hall for machining and
assembling the probes used in this study. The contributions of Dr. Mark Cunningham, Dave
Poirier, and Dr. Qi Chen, who wrote and modified the original codes and programs that formed
the basis of this study are also acknowledged and appreciated.
The support of my friends and family is greatly appreciated. I am greatly indebted to my
parents, Anne and George, and my brother Stephen for supporting my pursuit of higher
education. I would also like to thank my labmates, especially Nathon Begg and Grant Armitage
for their help throughout the duration of this program.
This project has been jointly funded by Queen’s University, the Natural Sciences and
Engineering Council of Canada (NSERC), and W.R.Davis Engineering Ltd as a part of a
Collaborative Research and Development initiative.
iii
Table of Contents
Abstract............................................................................................................................................ii
Acknowledgements.........................................................................................................................iii
Table of Contents............................................................................................................................iv
List of Figures.................................................................................................................................ix
List of Tables .................................................................................................................................xii
Chapter 1 Introduction ..................................................................................................................... 1
1.1 Multi-Hole Pressure Probe Concept ...................................................................................... 1
1.2 Rationale ................................................................................................................................ 2
1.3 Motivation and Prior Work.................................................................................................... 4
1.4 Contribution and Scope of Work ........................................................................................... 4
Chapter 2 Theory and Literature Review......................................................................................... 6
2.1 Definitions and Conventions.................................................................................................. 7
2.2 Governing Parameters............................................................................................................ 9
2.2.1 Low Angle Coefficients ................................................................................................ 10
2.2.2 High Angle Coefficients ............................................................................................... 12
2.3 Inviscid Flow Solutions and Limitations ............................................................................. 15
2.4 Calibration Techniques ........................................................................................................ 16
2.4.1 Multi-Variable Polynomial Curve Fits ......................................................................... 16
2.4.2 Direct Interpolation....................................................................................................... 20
2.4.3 Hybrid Models .............................................................................................................. 21
2.5 Reynolds Number Effects .................................................................................................... 22
2.5.1 Flow over a Backward Facing Step .............................................................................. 24
2.5.2 Crossflow over a Cylinder ............................................................................................ 25
iv
2.5.3 Effects of Reynolds Number on Pitot Tube Performance............................................. 27
2.5.4 Effects of Reynolds Number on Seven Hole Probe Response...................................... 27
2.6 Mach Number Effects .......................................................................................................... 28
2.7 Flow Turbulence Effects...................................................................................................... 29
2.8 Velocity Gradient Effects .................................................................................................... 31
Chapter 3 Apparatus and Instrumentation ..................................................................................... 32
3.1 Probe Design........................................................................................................................ 32
3.2 Experimental Apparatus....................................................................................................... 35
3.2.1 Calibration Wind Tunnel .............................................................................................. 35
3.2.2 Rotary Traverse............................................................................................................. 37
3.3 Data Acquisition .................................................................................................................. 39
Chapter 4 Procedures ..................................................................................................................... 40
4.1 Data Collection .................................................................................................................... 40
4.1.1 Alignment and Connection to DAQ.............................................................................. 40
4.1.2 Calibration Grid Requirements and Generation............................................................ 41
4.1.3 Wind Tunnel Operation and Automated Data Collection Setup................................... 44
4.2 Generation of Calibration Curves ........................................................................................ 44
4.2.1 Calibration Data Sorting ............................................................................................... 44
4.2.1.1 Sorting Criteria....................................................................................................... 46
4.2.1.2 Overlap Pressure and the Extent of Calibration Sector Domains .......................... 47
4.2.1.3 Determination of Reference Flow Conditions ....................................................... 49
4.2.2 Calculation of Calibration Coefficients ........................................................................ 50
4.3 Conversion of Measured Probe Pressures from an Arbitrary Flow to Flow Velocity,
Direction, and Pressure .............................................................................................................. 51
4.4 Calibration Verification ....................................................................................................... 53
v
4.4.1 Verification of Flow Separation.................................................................................... 53
4.5 Calibration Validation.......................................................................................................... 56
Chapter 5 Results and Analysis ..................................................................................................... 58
5.1 Data Verification.................................................................................................................. 59
5.2 Factors Affecting Probe Response....................................................................................... 62
5.2.1 Geometry Effects on Pressure Coefficient Distributions .............................................. 62
5.2.1.1 Tip Separation........................................................................................................ 63
5.2.1.2 Downstream Separation ......................................................................................... 67
5.2.2 Reynolds Number Effects ............................................................................................. 71
5.2.2.1 Reynolds Number Effects on Pressure Coefficient Distribution ........................... 71
5.2.2.2 Reynolds Number Effects on Calibration Accuracy.............................................. 76
5.2.2.3 Reynolds Number Effects in Previous Works ....................................................... 79
5.3 Variables Affecting the Representation of Probe Response Using a Curve Fit................... 79
5.3.1 Calibration Grid Independence ..................................................................................... 80
5.3.2 Overlap Pressure ........................................................................................................... 84
5.3.2.1 Proof of Concept .................................................................................................... 85
5.3.2.2 Overlap Pressure in Dense Calibration Grids ........................................................ 93
5.3.2.3 Overlap Pressure in Alternative Tip Geometries ................................................... 95
5.3.3 Order of Polynomial Curve Fit ..................................................................................... 95
5.3.3.1 Grid Independence ................................................................................................. 96
5.3.3.2 Overlap Pressure .................................................................................................. 100
5.4 Method of Lowest Error..................................................................................................... 101
5.5 Summary and Discussion of Findings ............................................................................... 104
Chapter 6 Error Analysis and Propagation .................................................................................. 108
6.1 Sources of Error During Calibration.................................................................................. 108
vi
6.2 Sources of Error in an Arbitrary Flow ............................................................................... 110
6.2.1 Low Angle Flows........................................................................................................ 112
6.2.2 High Angle Flows ....................................................................................................... 114
6.3 Calculation of Total Error in an Arbitrary Flow ................................................................ 116
6.4 Example of Transducer Uncertainty Plots ......................................................................... 117
Chapter 7 Conclusions ................................................................................................................. 120
7.1 Tip Geometry..................................................................................................................... 120
7.2 Reynolds Number Effects .................................................................................................. 121
7.3 Order of Polynomial Curve Fit .......................................................................................... 122
7.4 Calibration Grid Requirements .......................................................................................... 122
7.5 Overlap Pressure ................................................................................................................ 122
7.6 Quantification of Error in Previous Works ........................................................................ 123
Chapter 8 Recommendations and Limitations ............................................................................. 124
References.................................................................................................................................... 126
Appendix A Experimental Apparatus and Calibration ................................................................ 130
A.1 Pressure Transducers......................................................................................................... 130
A.2 X-Y Traverse Tables......................................................................................................... 132
A.3 Sampling Period Sensitivity.............................................................................................. 134
Appendix B Using Seven Hole Probes in the Gas Turbine Lab .................................................. 136
B.1 Assembly and Manufacture............................................................................................... 137
B.2 Storage and Handling ........................................................................................................ 138
B.3 Calibration......................................................................................................................... 139
B.4 Processing of Flow Data ................................................................................................... 142
Appendix C Shop Drawings ........................................................................................................ 143
Appendix D Measurement and Characterization of Calibration Wind Tunnel Outlet Flow ....... 151
vii
D.1 Swirl Characterization....................................................................................................... 151
D.1.1 Direct Port Pressure Comparison ............................................................................... 153
D.2 Static Pressure Profile Uniformity .................................................................................... 154
D.3 Flow Development............................................................................................................ 154
D.4 Wind Tunnel Drift and Unsteadiness................................................................................ 155
D.5 Turbulence Effects ............................................................................................................ 157
viii
List of Figures
Figure 1-1: Typical Seven Hole Probe Tip ...................................................................................... 2
Figure 2-1: Probe Numbering and Angle Conventions................................................................... 7
Figure 2-2: Flow Angle and Coordinate System Conventions [2]................................................... 8
Figure 2-3: Pressure Coefficients on the Surface of a Circular Cylinder in Crossflow ................ 14
Figure 2-4: Downwind Separation at Moderate Angles of Attack ............................................... 23
Figure 2-5: Downwind Separation at High Angles of Attack....................................................... 23
Figure 2-6: Characteristic Lengths of a 7 Hole Probe and a Backward Facing Step.................... 24
Figure 2-7: Flow Structures Downstream of a Cylinder in Laminar and Turbulent Crossflow ... 26
Figure 3-1: Schematic Layout Drawing Showing the Parts of the Seven Hole Probe.................. 33
Figure 3-2: Schematic Drawing of Probe Tip Designs .................................................................. 34
Figure 3-3: Photograph of Tip Shapes as Built.............................................................................. 34
Figure 3-4: Variable Speed Calibration Wind Tunnel ................................................................... 36
Figure 3-5: Position of Probe and Reference Pitot Tube During Calibration ................................ 37
Figure 3-6: Rotary Traverse.......................................................................................................... 38
Figure 4-1: Sample Calibration Grid ............................................................................................ 43
Figure 4-2: Data Sectoring for a Typical Probe............................................................................ 45
Figure 4-3: Data Extracted for Yaw Meter Performance Evaluation ........................................... 54
Figure 4-4: Response of 7 Hole Probe as a Yaw Meter................................................................ 55
Figure 5-1: RMS Average Error Comparison for Low Angle Sector........................................... 60
Figure 5-2: RMS Average Error Comparison for High Angle Sector ........................................... 60
Figure 5-3: Reynolds Number Effect Comparison for all Sectors................................................ 61
Figure 5-4: Yaw Meter Performance of a Seven Hole Probe with a Chamfered Tip ................... 64
Figure 5-5: Yaw Meter Performance of a Seven Hole Probe with a Hemispherical Tip.............. 64
ix
Figure 5-6: Burr Upstream of Port 2............................................................................................. 66
Figure 5-7: Scratched Tip Surface ................................................................................................ 66
Figure 5-8: Pressure Coefficient Distributions around Probe Tip and a Circular Cylinder........... 68
Figure 5-9: High Angle Flow Requiring Attached flow 90° from the Stagnation Point .............. 69
Figure 5-10: Pressure Coefficient Distributions with Data from Three Ports............................... 70
Figure 5-11: Reynolds Number Effects on Conical Tip Yaw Meter Performance........................ 73
Figure 5-12: Reynolds Number Effects on Conical Tip Pressure Coefficient Distribution at 50°
Cone Angle .................................................................................................................................... 73
Figure 5-13: Reynolds Number Effects on Hemispherical Tip Yaw Meter Performance ............ 74
Figure 5-14: Reynolds Number Effects on Hemispherical Tip Pressure Coefficient Distribution at
50° Cone Angle.............................................................................................................................. 74
Figure 5-15: Average Errors in the High and Low Angle Regions for a Conical Tipped Probe .. 77
Figure 5-16: Average Errors in the High and Low Angle Regions for a Hemispherical Tipped
Probe .............................................................................................................................................. 77
Figure 5-17: Average Errors for Both Tip Shapes........................................................................ 79
Figure 5-18: Effects of Grid Density on Conical Tipped Probe Error........................................... 81
Figure 5-19: Effects of Grid Density on Hemispherical Tipped Probe Error ................................ 81
Figure 5-20: Effect of Overlap on High Angle Probe Error ......................................................... 86
Figure 5-21: Error in Calculated Dynamic Pressure with 0 Overlap Pressure.............................. 89
Figure 5-22: Error in Calculated Dynamic Pressure with 15% Overlap Pressure ........................ 89
Figure 5-23: Effects of Grid Density on Probe Error with a 4th Order Polynomial Fit.................. 96
Figure 5-24: Dynamic Pressure Error Distribution with a 3rd Order Curve Fit from 3° Grid
Spacing, 0 Overlap......................................................................................................................... 99
Figure 5-25: Dynamic Pressure Error Distribution with a 4th Order Curve Fit from 3° Grid
Spacing, 0 Overlap......................................................................................................................... 99
x
Figure 5-26: Error Contours for the Optimum Calibration Case ................................................ 103
Figure 6-1: Velocity and Temperature Contours for Sample Mixing Tube Outlet Traverse...... 117
Figure 6-2: Flow Property Uncertainty Resulting from Transducer Uncertainty ....................... 118
Figure A-1: Typical Transducer Casing Arrangement ................................................................ 130
Figure A-2: Pressure Transducer Calibration Arrangement ........................................................ 131
Figure A-3: XY-Positioning Traverse Rig.................................................................................. 133
Figure A-4: Transient Pressure Response to a 45 Degree Change in Flow Angle ..................... 135
Figure D-1: Wind Tunnel Outlet Secondary Flow Vectors ......................................................... 152
Figure D-2: Unbiased Wind Tunnel Outlet Secondary Flow Vectors ........................................ 152
Figure D-3: Wind Tunnel Outlet Static Pressure Contours ......................................................... 154
Figure D-4: Validation of Flow Development............................................................................ 155
Figure D-5: Variation of Measured Pressure over an 8 Hour Period.......................................... 156
Figure D-6: Suction-type Wind Tunnel for Turbulence Testing ................................................. 157
Figure D-7: Test Chamber Bellmouth Inlet ................................................................................ 158
xi
List of Tables
Table 2-1: Flow Angle Conventions................................................................................................ 8
Table 4-1: Error Code Descriptions .............................................................................................. 52
Table 5-1: Calibration Point Distribution for a Conical Tipped Probe......................................... 83
Table 5-2: Calibration Point Distribution for a Hemispherical Tipped Probe ............................... 83
Table 5-3: Total Number of Points in each High Angle Calibration Sector with Overlap ........... 87
Table 5-4: RMS Average High Angle Errors ............................................................................... 92
Table 5-5: Absolute Maximum High Angle Errors ....................................................................... 92
Table 5-6: Changes in High Angle RMS Average Error with the Application of 15% Overlap... 93
Table 5-7: Changes in High Angle Absolute Maximum Error with the Application of 15%
Overlap........................................................................................................................................... 94
Table 5-8: Effects of Increasing Order of Curve fit on Overall RMS Average Error................... 97
Table 5-9: Effects of Increasing Order of Curve fit on Overall Absolute Maximum Error.......... 97
Table 5-10: Average and Maximum Calibration Errors (All Sectors)........................................ 102
Table 5-11: Effects of Overlap Pressure and Direct Interpolation on Calibration Accuracy...... 106
Table 6-1: Calibration Uncertainty for a Sample Probe............................................................... 110
Table D-1: Effect of Stem Position on Probe Response ............................................................. 153
Table D-2: Dimensions of Turbulence Screens ........................................................................... 159
Table D-3: RMS Average Errors at Different Turbulence Levels .............................................. 160
Table D-4: Absolute Maximum Errors at Different Turbulence Levels...................................... 160
xii
Chapter 1
Introduction
The concept of measuring flow velocity using a tube with a hole facing the oncoming flow was
introduced by Henri Pitot in the 1730’s, when he was tasked with measuring water speeds in the
Seine River in France. Multi-hole pressure probes were an extension of the pitot tube concept –
the known relative position of each pressure port allowed calculation of both a flow direction and
a flow magnitude. Three hole probes are capable of measuring a single flow angle – that is,
measuring a two dimensional flow. Five and seven hole probes are capable of determining two
flow angles – allowing a fully three dimensional velocity field to be measured. The two
additional holes allow seven hole probes to measure higher angles of attack than five hole probes.
The seven hole probe is the subject of the current work. Seven hole probes have been widely
shown to be capable of measuring mean flow angles to within 1° and mean flow velocity to
within 1%. The main challenge surrounding these probes is calibration – the response is very
sensitive to small changes in tip geometry, so care must be taken to ensure that the tips are not
damaged or impacted. The present work develops codes and procedures based on the principles
and governing equations from the literature and makes incremental contributions that improve the
accuracy and uniformity of calibration curves.
1.1 Multi-Hole Pressure Probe Concept
A seven hole pressure probe is constructed by surrounding a central pressure port with 6 equally
spaced ports. The central port is on the axis of the probe. The frustal face from which the six
1
peripheral ports are drilled is typically angled between 25° and 35° to the central port. A typical
probe tip is shown in Figure 1-1.
Figure 1-1: Typical Seven Hole Probe Tip
When a seven hole probe is used to measure an arbitrary flow, the pressures in the seven ports are
measured simultaneously. Using the known relative positions of the seven holes, dimensionless
pressure coefficients that represent the direction of flow can be defined based on the difference in
measured pressure across diametrically opposite holes. These directional coefficients can then be
correlated to exact flow angles, and the direction of flow can be established. The flow magnitude
and pressure can be approximated using the dimensionless coefficients derived from the raw
pressure data. The dimensionless directional coefficients are then correlated to a correction factor
which is applied to the pressure magnitude coefficients, and the exact flow pressure and
magnitude can be calculated.
1.2 Rationale
The Queen’s University Gas Turbine lab (GTL) performs experiments on auxiliary gas turbine
exhaust components using a Hot Gas Wind Tunnel (HGWT) to simulate a gas turbine exhaust.
2
This wind tunnel can produce flows of up to 2.5 kg/s of hot air at temperatures up to 600°C. The
flows in the components that are tested are typically highly three dimensional, and have high
velocity and temperature gradients. There is a desire to take velocity, pressure, and temperature
measurements of these flows – both inside the devices and at the device exit plane.
Local velocity measurement is typically performed in one of three main ways – optical, hot
wire/film, and pitot-tube. Optical techniques include laser Doppler anemometry (LDA) and
particle image velocimetry (PIV). These techniques use lasers to locally illuminate the flow, and
the behavior of seeding particles that pass through the area if interest is recorded using either a
high speed camera or a doppler phase shift. Laser techniques are difficult to implement in the
HGWT, however, as the high primary mass flow rates would require enormous amounts of
seeding materials, and the seeding of secondary ejector flows is difficult. Hot wire techniques
measure the current flow through a heated wire and correlate the heat loss from the wire to flow
velocities over the wire. High temperature hot wire and hot film type probes are available,
however they are quite expensive, and the probe tips themselves can be quite fragile. Given the
scale of devices tested in the HGWT, and the manufacturing tolerances in some of the bent and
welded sheet metal ducts, there is a risk of probes colliding with the walls and edges of these
ducts.
The multi-hole pressure probe is the most attractive option for flow measurement in the HGWT
mainly because of its mechanical durability and relative low cost. Hot wire and laser systems can
cost upwards of $30,000, where a seven hole probe can be manufactured in house for less than
$1000. Seven hole probes can be made from stainless steel, which resists corrosion and
degradation due to heating. They are also capable of providing a full 3D velocity and pressure
3
measurement in a single reading – something that would not be possible with optical or hot wire
techniques. Pressure data would have to be taken in a separate traverse and would be subject to
pitot tube error due to a non-axial angle of attack. The scale of flow that can be measured with a
seven hole probe is, however, limited to scales larger than the probe tip, which is around 3mm.
Conventional seven hole probes are also limited to taking time averaged measurements. These
limitations are acceptable, however, for purposes of the GTL.
1.3 Motivation and Prior Work
Seven hole probes have been used in the GTL to measure both hot and cold flows since 2000.
Cunningham [1] wrote the original FORTRAN codes that were used as the basis of the present
study. The probes were first used in hot flow by Chen [2] and Maqsood [3]. The measured flow
angles and total pressures were quite good in their work, however there was concern that there
may have been some error associated with the calculation of static pressure. One purpose of this
work was therefore to investigate and mitigate the causes of this error.
A significant amount of the code that was used in this work was originally written by Dr. Mark
Cunningham, and later revised by David Poirier and Dr. Qi Chen. Their significant contributions
to this work are acknowledged and appreciated.
1.4 Contribution and Scope of Work
This work presents a method, termed “overlap pressure”, which improves the polynomial curve
fit calibration approach by using certain calibration points in multiple probe sectors. This work
also presents a detailed method of error analysis that considers the effects of flow magnitude and
direction on uncertainty.
4
This work also provides a detailed introduction to the calibration and use of seven hole probes
that is valuable to new graduate students and researchers in the GTL. Software for calibration,
flow measurement, and error analysis has also been developed and validated and is available
internally. A section on low level implementation is included, with detailed instructions for the
use of these software tools.
The scope of this work was to study and quantify the key factors that affect the use of seven hole
probes in hot flow. Reynolds number is an important factor, so the design of the tip was studied
to assess its affect on Reynolds number sensitivity. The key variables affecting calibration
accuracy are also studied, and guidance is given for their selection. This study was limited to
time averaged measurements in subsonic, incompressible flow.
5
Chapter 2
Theory and Literature Review
Multi-hole pressure probes have been extensively studied in literature, and standard methods of
calibration and implementation exist and are well documented. The earliest multi-hole probes
were used in a nulling mode. A pitot-static tube was mounted in a flow, and the pressure on
diametrically opposite static pressure ports was measured individually. The probe was
mechanically rotated until the error signal (the difference between the diametrically opposite
ports) reached 0. The inclination angle of the probe was then the flow direction, and the flow
magnitude was captured by the pitot tube. Such an apparatus was described and patented in 1972
by Maiden et al. [4]
Properly implementing a nulled-probe apparatus required a great deal of equipment and
instrumentation. Pressure transducers were required to read all of the individual port pressures.
Motors were required to rotate the probe, and position sensors needed to measure the angle of
attack. The expense and complexity of implementing a nulling setup created a demand for a
method of using a seven hole probe in a non-nulling mode. The accepted approach that is widely
in use today was proposed by Gallington [5] in 1980. His work introduced the concept of
dimensionless directional pressure coefficients that could be used to represent the angularity of a
flow over a large range of flows. The specifics of his methods are reviewed in detail in section
2.2.
6
2.1 Definitions and Conventions
The hole numbering convention is the first that must be established. Figure 2-1 shows a front
view of a 7 hole probe. The probe stem is shown in the view, and the holes numbered relative to
this orientation. The sign conventions of three angles are also defined in this view – pitch, yaw,
and roll.
Figure 2-1: Probe Numbering and Angle Conventions
The flow angle relative to the probe tip can be defined using two coordinate systems. The pitch
and yaw coordinate system defines two flow angles that are referenced to the probe’s X and Y
axis. The cone and roll coordinate system is a polar coordinate system. The cone angle is the
total angle of attack to the probe axis, and the roll angle is a rotation angle that is referenced to a
fixed probe axis. The definition of these angles, as well as the coordinate system that is assumed
is shown schematically in Figure 2-2 and the definitions of each angle are tabulated in Table 2-1.
7
Angles with a subscript T are tangent angles, which are measured between projections of the
velocity vector.
Figure 2-2: Flow Angle and Coordinate System Conventions [2]
Table 2-1: Flow Angle Conventions
Angle
Term
Regime
Definition
α
Pitch
Low Angle
Angle between w and Y-Z projection of velocity
β
Yaw
Low Angle
Angle between velocity vector and Y-Z projection of velocity
θ
Cone
High Angle
Angle between w and the velocity vector
γ
Roll
High Angle
Angle between –v and X-Y projection of velocity
Table 2-1 introduces the concept of a flow regime. The response equations are different for
different angles of attack. At low angles of attack, the flow remains attached over the entire
8
probe tip. As the angle of attack is increased, the flow on the downwind side of the probe
eventually separates. Pressure data in the separated region is unsteady and is not representative of
the flow that is being measured. For this reason, at high angles of attack, data from ports that are
measuring a separated region of the flow are ignored and the flow was calculated based only on
the ports measuring in attached flow. The Low Angle flow regime is therefore the regime in
which the flow is attached over all seven holes, and the High Angle flow regime is the regime in
which flow is attached over only some of the holes. The pitch and yaw coordinate system was
used in the low angle regime, while the cone and roll coordinate system was used in the high
angle regime. Details of the formulation of the governing equations, as well as an explanation of
the choice of different coordinate systems can be found in section 2.2.
2.2 Governing Parameters
Gallington [5] deduced the concept of directional pressure coefficients in 1980, and his definition
remains the preferred method for reducing and non-dimensionalizing seven hole data. The first
step in processing data using the Gallington method is to determine whether the measured flow
falls into the Low or High Angle regime. This was done by indentifying the port reading the
highest pressure – the port reading the highest pressure was indicative of the general direction of
the flow. If port 1, the centre port read the highest pressure, the flow was deemed to be a Low
Angle flow, and the data from all seven pressure ports was used in the calculation. If a peripheral
port (ports 2-7) read the highest pressure, then the flow was deemed to be a High Angle flow, and
there was a significant probability of flow separation over the ports on the downwind side of the
probe. In a high angle flow, the flow was therefore calculated based only on the port with highest
pressure and the three adjacent ports, where the flow was known to be attached. The equations
shown in sections 2.2.1 and 2.2.2 were taken from Gallington [5].
9
The structure and form of the governing parameters is the same in the high and low angle flow
regimes. In both cases, two directional pressure coefficients are defined – this allows the
direction of the flow to be determined directly through various correlation methods, described in
section 2.4. The flow magnitude and pressure is also determined through the use of
dimensionless coefficients.
The total pressure under any flow condition is equal to the sum of the static pressure and the
dynamic pressure. The highest pressure read by any single port on the probe is the best available
approximation of the flow total pressure. The average of the remainder of the peripheral pressure
ports that read in attached flow is the best available approximation of the flow static pressure.
Taking the difference of these two pressures allows the calculation of an approximate flow
dynamic pressure. The error in these approximations is directly related to the angularity of the
flow. As described by Gerner [6], the dimensionless coefficients that are used to define total and
dynamic pressure essentially become correction factors to the approximations of total and
dynamic pressure that are calculated from the pressure data.
2.2.1 Low Angle Coefficients
In a low angle flow, the highest pressure was read in port 1. P1 was therefore the approximate
flow total pressure. In low angle flow, the flow was assumed to be attached over all of the
peripheral pressure ports, so the approximate static pressure was therefore calculated as shown in
Equation (2-1)
10
P=
1 7
∑ Pi
6 i=2
(2-1)
The directional pressure coefficients for the low angle regime are shown in Equation (2-2). The
pressure differences were normalized by the approximate dynamic pressure of the flow.
Ca =
P3 − P2
P − P7
P − P5
, Cb = 4
, Cc = 6
P1 − P
P1 − P
P1 − P
(2-2)
These coefficients were weighted according to their relative positions on the probe tip and used to
generate a further pair of coefficients that are representative of the pitch and yaw. The
calculation of the pitch and yaw coefficients is shown in Equations (2-3) and (2-4). The relative
weighting of the terms in the pitch and yaw coefficients was based solely on the geometry of the
probe tip.
Cα =
2Ca + Cb + Cc
3
(2-3)
Cb + Cc
3
(2-4)
Cβ =
The formulation of the total and dynamic pressure coefficients are shown in Equations (2-5) and
(2-6), respectively. Pt and Pq represent the flow’s true total and dynamic pressure, respectively.
Ct =
P1 − Pt
P1 − P
(2-5)
Cq =
P1 − P
Pq
(2-6)
11
2.2.2 High Angle Coefficients
In a high angle flow, the highest pressure was read in one of the peripheral ports, and this port
was referred to as port n. Pn was therefore taken to be the approximate flow total pressure. As
mentioned, there was a high probability in high angle flow that the downwind side of the probe
would be measuring in separated flow, so only pressures from hole n, the two immediately
adjacent peripheral ports, and the centre port were considered. The pressure in the two adjacent
peripheral ports was termed P+ and P-. The approximate static pressure of the flow was therefore
calculated using Equation (2-7).
Pn =
P+ + P−
2
(2-7)
Directional coefficients in the high angle regime were defined based on the polar coordinate
system. This was convenient because it allowed only a single pair of coefficients to be defined,
and that pair of coefficients was applicable to all six of the peripheral ports. Formulating yaw and
pitch coefficients for the peripheral holes would have involved using unique scalar weightings for
the pressure difference terms at each port. The cone and roll coefficients are introduced in
Equations (2-8) and (2-9). Again, the terms were normalized by the approximate dynamic
pressure of the flow.
Cθ =
Pn − P1
Pn − Pn
(2-8)
Cγ =
P− − P+
Pn − Pn
(2-9)
12
The total and dynamic pressure coefficients were formulated in a way very similar to those in the
low angle regime. The expressions are shown in Equations (2-10) and (2-11).
Ctn =
Pn − Pt
Pn − Pn
(2-10)
Cqn =
Pn − Pn
Pq
(2-11)
It should be reiterated that the equations of flow in the high angle regime are only valid if four
ports were reading in attached flow. As discussed in section 2.3, the assumption that the two
adjacent peripheral ports are in attached flow was very reasonable, and it was highly unlikely that
this assumption could ever be violated. As discussed by Zilliac [7], however, it was possible that
the flow over port 1 could become separated, leading to the possibility of double-valued
directional pressure coefficients, which would render the measurement invalid. The possibility of
double-valued coefficients is illustrated in Figure 2-3, which shows pressure coefficients on the
surface of a cylinder in crossflow as a function of angle of attack. The pressure distribution
around a cylinder is often approximated using equations of flow around a 2-D circular cylinder
[6], [7]. The data for this plot was reproduced from White [8].
13
1
Inviscid
Laminar
Turbulent
CP
0
-1
-2
-3
0
45
90
135
180
Angle of Attack (°)
Figure 2-3: Pressure Coefficients on the Surface of a Circular Cylinder in Crossflow
This plot shows characteristic pressure coefficients for a cylinder in inviscid, laminar, and
turbulent flow. The flow on the downwind side of the cylinder separates in both laminar and
turbulent flow. What is important about these curves is not the separation point, but rather that in
both regimes the flow recovers some pressure before separating. This pressure recovery was
what led to the possibility of double-valued pressure coefficients. The seven hole probe had a
similar characteristic response to changes in angle of attack as the cylinder shown above, but the
angle of attack was not known. In the above case, if all that was known was that the flow was
turbulent and that the Cp value at a port was -1, it would have been impossible to know if the
flow’s angle of attack was 50° or 110°.
A step was added to the data processing procedure to check for this possibility. The
implementation of this step was discussed in detail in section 4.3.
14
2.3 Inviscid Flow Solutions and Limitations
The distribution of pressure around the probe tip can be calculated analytically using inviscid
flow equations. Huffman [9] used slender body theory to define a set of response equations that
would analytically predict the distribution of pressure around the probe tip at an arbitrary angle of
attack. Given the fixed location of the pressure ports, a set of response equations, different from
those outlined in section 2.2 were proposed. The reasoning behind this alternative approach was
that when the governing equations were based on inviscid theory, the equations were more
physically significant. The curve fitting and interpolation process that relates flow properties to
the directional coefficients of the Gallington method was used simply because it produced an
acceptable result – it was not grounded in an expected physical response. It was acknowledged
by Huffman, however, that the inability of inviscid flow theory to predict flow separation, and the
sensitivity of probe response to manufacturing tolerance meant that calibration was still
necessary.
The work of Huffman was continued by Pisdale [10] and Zilliac [7] who both showed that the
response of probes with a simple geometry could be relatively well modeled using potential flow
theory. Pisdale expanded on the potential flow solution approach for a five hole probe by
generating a set of response equations that could be graphically or numerically interpolated.
Again, the justification for this approach was that it was grounded in a potential flow approach
that was physically significant. The implication was that if a polynomial curve fit or a direct
interpolation scheme with directional pressure coefficients was used, it must be proven that the
physical response of the probe was accurately modeled – not simply that the calibration curve fit
15
the data well over the entire response domain, but that all of the physical trends of the probe
response were accurately captured.
2.4 Calibration Techniques
Ultimately, the goal of calibration was to establish a correlation between the directional pressure
coefficients and the flow angles and total and dynamic pressure coefficients. This was done by
placing the probe in a known, axial flow, and moving the probe to a number of known angles.
The independent variables (directional pressure coefficients) could then be related to the
dependant variables (flow angles and total and dynamic pressure coefficients). The method that
was used to relate the independent and dependent variables was extensively studied in literature.
A description of some of these methods is given in the following sections. The present work used
use the multi-variable polynomial curve fit method, so the mathematics of this approach are
presented extensively in section 2.4.1. The detailed mathematics of the other approaches were
omitted for brevity. It should be noted that all of the methods described herein used the same
governing parameters to define the probe response. The methods described below only dealt with
the way in which the dependant and independent variables were related to each other.
2.4.1 Multi-Variable Polynomial Curve Fits
The concept of a polynomial power series fit was first proposed in literature by Gallington [5].
This approach used a bivariant surface polynomial to relate directional pressure coefficients to the
four desired flow properties. The output of the calibration was a set of coefficients that allowed a
flow property to be determined using simple matrix multiplication. Gerner [6] used a similar
approach, but added an additional degree of freedom by defining a compressibility coefficient
that was also included in the calibration. Gerner’s work therefore used a trivariant surface
16
polynomial to relate the directional and compressibility coefficients to the four desired flow
properties. The present study deals with incompressible flows, so the compressibility coefficient
was omitted.
The main advantage of the polynomial surface method is that the number of calibration
coefficients that are required is relatively small. The seven sectors of the probe are calibrated
independently, and each of the four dependant variables requires their own correlation to the
directional coefficients. A fourth order, bivariant polynomial, with 15 terms, therefore requires a
total of 420 calibration coefficients. A similar bivariant polynomial of third order requires 280
calibration coefficients.
The optimal choice of power series is not something that has been studied in the literature.
Gallington [5] used a fourth order expansion. Sumner [11] used a third order expansion. Gerner
[6] used a third order expansion of his trivariant polynomial, as did Everett [12]. Ultimately the
order of power series must be high enough that physical phenomena occurring within the solution
domain are captured. A power series that is too high order, however, is susceptible to noise in the
calibration data, and may generate curves with unphysical peaks and valleys, especially near the
boundaries of the domain [13].
The number of points that are used to calibrate a sector is a very important factor in determining
the appropriate order of fit. Everett [12] and Gerner [6] showed that for a given grid density, as
maximum cone angle considered during calibration was decreased, the standard error in the
calibration was also reduced. The standard error was measured by feeding the calibration data
back through its own calibration curve – a good measure of the quality of the fit to a given set of
17
data. Reducing the maximum cone angle reduces the number of points in a given sector, however
in Everett the number of degrees of freedom in the curve fit was not reduced – so the observed
reduction of standard error could be considered an expected result. When calculating standard
errors, it is therefore important to consider the ratio of the size of the data set to the number of
degrees of freedom in the curve fit to ensure that reductions in error are not due to the effective
increase in the order of polynomial fit.
The mathematics of the polynomial curve fit method are shown in Equations (2-12) and (2-13)
with fourth order terms included. The extension of the method to higher or lower order
polynomials is intuitive, and would be achieved simply by omitting or adding terms that are of
the same form as those shown below. As discussed in sections 2.2.1 and 2.2.2, the high and low
angle regimes are based on different directional coefficients. The exact formulation of the
polynomials is therefore slightly different for the high and low angle regimes. Equations (2-12)
and (2-13) show the exact expressions that are used to calculate a flow property, X, given the two
angular coefficients and a set of probe-dependant calibration coefficients. X represents each of
the dependent variables – two flow angles, and total and dynamic pressure coefficients. All four
of these properties are calculated in the same way.
 K1X + K 2X Cα + K 3X Cβ + K 4X Cα2 + K 5X Cα Cβ + K 6X Cβ2 


X = + K 7X Cα3 + K 8X Cα2Cβ + K 9X Cα Cβ2 + K10X Cβ3 + K11X Cα4 


X 3
X
2 2
X
3
X
4
+ K12 Cα Cβ + K13 Cα Cβ + K14 Cα Cβ + K15 Cβ

(2-12)
 K1X,n + K 2X,nCθn + K 3X,nCγn + K 4X,nCθ2n + K 5X,nCθnCγn + K 6X,nCγ2n 


X = + K 7X,nCθ3n + K 8X,nCθ2nCγn + K 9X,nCθnCγ2n + K10X ,nCγ3n + K11X,nCθ4n 


X
3
X
2
2
X
3
X
4
+ K12,nCθnCγn + K13,nCθnCγn + K14,nCθnCγn + K15,nCγn

(2-13)
18
These complete expansions can also be expressed in matrix form. Only the matrix form of the
low angle expression is shown, for brevity. Equation (2-14) shows that once the calibration
coefficients are known, any number of points (m) can be converted to flow properties quickly and
efficiently using simple matrix multiplication.
 X1  
X  
 2  
 X3  = 
 M  
  

 X m  
1
Cα 1
Cβ1
Cα21
L
1
Cα 2
Cβ 2
Cα22
L
2
1
Cα 3
Cβ 3
Cα 3
L
M
M
M
M
L
Cβ m
2
L
1
Cα m
Cα m
Cβ41   K1 

Cβ42   K 2 
 
Cβ43   K 3 

M  M 
 
Cβ4m   K15 
(2-14)
This matrix can be further simplified in its expression. The independent variable array is a
function only of the angular pressure coefficients, and hence can be calculated directly from
probe data. The dependent variable vector is known during calibration, but unknown when the
probe is used to measure an arbitrary flow. Similarly, the calibration vector is unknown during
calibration, but must be known when measuring an arbitrary flow. The matrix is expressed in a
simplified form in Equation (2-15).
{X } = [TM ]{K }
(2-15)
From this expression, it is clear that relatively simple matrix algebra can be used to calculate the
calibration vector, K, given that the flow properties are known during calibration. Conversely, it
is clear that the only data required to calculate the flow properties in an arbitrary flow is the
calibration vector. This leads to the main advantage of the polynomial surface method – once the
expressions are formulated, the implementation time and computational expense to calibrate
probes and solve arbitrary flows is quite low.
19
The polynomial surface method has been shown [6], [11] to be capable of measuring flow angles
to within 1° and flow pressures to within 2%. The accuracy of the calibration is of course
dependent on the density of the calibration grid. Accuracy is also somewhat dependant on cone
angle – at high angles of attack, errors tend to be higher. That said, the approach is simple to
implement and has been shown to be capable of producing accurate measurements of flow, which
is why it was selected for the present study.
2.4.2 Direct Interpolation
The direct interpolation method was first proposed by Zilliac [7], and has been shown to improve
the accuracy of flow property calculation, especially at high angles of attack [11],[7], compared
to the polynomial curve method. The increased error in the polynomial curve method was
explained physically in two ways. At high angles of attack, small changes or errors in the
directional coefficient caused large changes in the calculated flow angle. Secondly, the noise in a
polynomial curve is highest near its extents – exacerbating the problem.
There were two main drawbacks to the direct interpolation method. The computational expense
was higher, because the response of the probe could not be represented by a single expression.
The amount of storage required for the calibration data was also much higher, as the complete
calibration data set must be stored.
The actual interpolation procedure was complicated by the non-uniform grid. The spacing of the
pitch and yaw or cone and roll coefficients was non-uniform, meaning that defining the nearest
neighbours could be somewhat complicated. One common solution was to adopt the Akima [14]
20
interpolation scheme, which is capable of interpolating non-uniform grids of multiple
independent variables. This scheme fits a local polynomial to at least five points in each
direction, and uses geometric conditions to ensure local continuity of the function and its
derivatives.
Sumner [11] performed a direct comparison of the direct interpolation method and the polynomial
surface method. The two calibration methods were applied to the same data set, and then a
different, larger data set was processed using each method. In the low angle regime, the
difference in standard error was shown to be negligible. In the high angle regime, there was an
improvement in the on the order of 0.5° in the error in flow angle, but this improvement was only
seen when the calibration grids were quite coarse. Similar improvements were shown by Silva
[15], who compared the polynomial curve method with a simple linear interpolation. This
suggests that the interpolation method (Akima vs linear) may not be responsible for the improved
response – the improvement may simply be due to the nature of the direct interpolation technique.
These results were compared with the present work and discussed in greater detail in section 5.5.
2.4.3 Hybrid Models
A number of hybrid and alternative models have been proposed over the years in an attempt to
reduce the intrinsic error associated with curve fitting. Wenger [16] proposed a combination of
the global polynomial curve fit approach with a local direct interpolation of an error table that
was also output from the calibration. The rationale behind this approach was that the high order
global curve fit would damp out any unsteadiness or noise in the high order derivatives, while the
low order, interpolated error term would allow for local variation of the low order derivatives.
The results were shown to be quite good, reducing interpolation error to approximately 1 order of
21
magnitude below other sources of experimental error. The downside of this approach, however,
was that the accuracy and precision of the calibrator setup become critical. If the transducers
used for calibration were not more accurate than the transducers used for data collection, then the
effect of noise in the calibration data could be significant.
A neural network approach was proposed by Rediniotis [17]. The neural network approach
created a library of nodes at which calibration data (inputs and outputs) are stored. A number of
layers were then created, with each node using a weighting factor on adjacent nodes to determine
its influence. A number of optimization cycles were completed, where the network calculated
expected values and compared with the known, measured values, improving its weighting factors
each time through until errors were minimized.
The issue with this approach was that the architecture of the network – that is, the number and
arrangement of nodes, as well as the definition of the layers, had a significant effect on the
accuracy of the result. Further, the network architecture was highly user defined – meaning that
the user was required to work through a significant number of combinations and network designs
before the optimal design was reached. The advantage of the approach, however, was that
additional calibration data could be quickly and easily added to the network.
2.5 Reynolds Number Effects
As discussed, when the flow attacks the probe at a high angle, the flow separates from the
downwind side of the probe tip. Flow separation is typically highly dependant on Reynolds
number, so understanding the mechanisms of separation and finding representative problems for
22
comparison purposes is important. Figure 2-4 and Figure 2-5 show diagrams of the two main
types of flow separation that were expected.
Figure 2-4: Downwind Separation at Moderate Angles of Attack
Figure 2-5: Downwind Separation at High Angles of Attack
The separation shown in Figure 2-4 is similar to the separation seen downstream of a backward
facing step, which is a problem that has been studied extensively in literature. Similarly, the
separation shown in Figure 2-5 is similar to the separation seen downstream of a cylinder in
crossflow, another well-studied problem. The Reynolds number dependence of separation in
these classic flows was therefore likely to provide some insight into the Reynolds number
dependence of the probes in the present study.
23
2.5.1 Flow over a Backward Facing Step
Separation over a backward facing step is classically studied in a two dimensional test section,
and the flow is classically studied as an internal flow. The problem can still be considered
analogous to separation over the probe tip, however, as long as the step is relatively small – on
the same order of magnitude as the height of the incoming channel. The definition of the length
scale of the Reynolds number is also important. The Reynolds number of the flow over a 7 hole
probe is typically reported in terms of the probe tip diameter [11]. In order to make reasonable
comparisons with data from a backward facing step, the upstream height of the channel was
chosen as the characteristic length. This is shown schematically in Figure 2-6.
Figure 2-6: Characteristic Lengths of a 7 Hole Probe and a Backward Facing Step
Armaly [18] experimentally studied separation downstream of a backward facing step in a
channel with an expansion ratio of 1.94. Data was collected using LDA, and it was found that
there were significant changes in the downstream reattachment length at two critical Reynolds
numbers. Below a Reynolds number of 1200, flow was laminar, and the reattachment length
varied linearly with Reynolds number. Above a Reynolds number of 6600, when the flow was
characterized as fully turbulent, the reattachment length was constant. In the transition region,
24
where Reynolds numbers were between 1200 and 6600, there was a non-linear variation of
approximately 50% in the reattachment length. These results suggest that the 7 hole probe
response could be quite sensitive to Reynolds number in this range of Reynolds numbers. The
choice of transition criteria from the low angle to high angle regime will be critical to mitigating
this potential source of error, as this phenomenon occurs at angles of attack that are very close to
this expected transition.
2.5.2 Crossflow over a Cylinder
When the flow attacks the probe at a very high angle of attack, the flow will separate on the
downwind side of the probe very much in the same way as a cylinder in crossflow. The transition
from laminar to turbulent flow around a cylinder is characterized by a sudden change in the
location of separation. Laminar flows are characterized by a separation that occurs approximately
82° from the stagnation point, while turbulent flows are characterized by a separation that occurs
approximately 110° from the stagnation point. These modes are shown schematically in Figure
2-7.
25
Figure 2-7: Flow Structures Downstream of a Cylinder in Laminar and Turbulent
Crossflow
It has been experimentally shown by Cantwell [19] for a smooth cylinder that this transition to
turbulent flow was expected around a Reynolds number of 2x105, based on the cylinder diameter.
It has also been shown, however, that the transition to turbulence is triggered by surface
roughness or dimples on the surface. This is important to the present work because the pressure
ports themselves act as vortex generators, and trigger an earlier transition to turbulent flow than
would a smooth surface. The transition Reynolds number for a rough surface has been shown to
be as low as 5x104 [20]. It is difficult to characterize a characteristic roughness height of the
pressure ports, so it is possible that Reynolds number effects could affect probe response
anywhere between these two Reynolds numbers in the present work.
26
2.5.3 Effects of Reynolds Number on Pitot Tube Performance
It has been shown in the literature that the pressure measured in the stagnation port of small pitot
tubes can be sensitive to Reynolds number. In low speed flows, the deceleration of the flow may
not be isentropic, and the resulting measured pressure may therefore not truly represent the
stagnation pressure of the flow. Chue [21] related this loss to Reynolds number, and found that it
only had an effect on probe response below Reynolds numbers of about 1000, with the error
diminishing with increasing Reynolds number.. This result was confirmed by Leland [22], who
found that pitot-tube calibrations were Reynolds number independent over 1x105. The Reynolds
numbers considered in the present work were on the order of 103, so the measurements were not
corrected for viscous losses due to Reynolds number.
2.5.4 Effects of Reynolds Number on Seven Hole Probe Response
The net effect of the Reynolds number dependent phenomenon described in the previous sections
could ultimately only be determined through an experimental investigation of seven hole probe
Reynolds number sensitivity. Wenger [16] found Reynolds number independence above 5000.
Wenger observed that an error of 1% in axial velocity and a small offset in flow angle was
introduced when a calibration taken at Re = 3.9x103 was applied to flow data taken at Re =
2.5x103. These observed errors are within the range of experimental error. Sumner [11]
performed a similar study, where a probe was calibrated at Re = 6.5x103. This calibration was
then applied to data from a number of flows ranging from Re = 1x103 to Re = 6.5x103. Error was
found to increase with decreasing Reynolds number. The maximum increase in observed error in
calculated flow angle was 1°, and the maximum increase in the observed error in velocity
magnitude was 4% of the range of Reynolds numbers tested.
27
The literature concludes that Reynolds number effects on seven hole probes can be significant.
The range of Reynolds numbers for which there is dependence is consistent with those of flow
over a backward facing step and crossflow over a cylinder, which suggests there may be some
insight gained through comparison with these flows.
2.6 Mach Number Effects
The Mach number has long been understood and acknowledged as having a significant effect on
the performance of a seven hole probe. Gallington’s [5] original work was deliberately
formulated in a way that would allow an extension to compressible flow. This extension was
proposed by Gerner [6] through the introduction of a compressibility coefficient. This coefficient
is defined for the low angle regime in Equation (2-16) and for the high angle regime in Equation
(2-17). Physically, these terms represent the ratio of approximate dynamic pressure to
approximate total pressure.
CM =
P1 − P
P1
(2-16)
CM , n =
Pn − Pn
Pn
(2-17)
This compressibility coefficient was defined in such a way that it was essentially independent of
flow direction, so its response isolated compressibility effects. In terms of data processing, the
compressibility coefficient simply became a third independent variable, along with the two flow
direction coefficients. The polynomial curve fit then became a function of three variables, which
for a fourth order curve, would result in 35 degrees of freedom instead of 15. The order of fit was
therefore typically reduced to third order [6], which includes only 20 terms. Using a higher order
28
curve was possible, but would require a significantly larger number of calibration points to ensure
that standard errors in the curve fit were reasonable.
The main challenge of adding a third independent variable was to maintain a reasonable size of
calibration grid without losing resolution in any of the three variables. Gerner adopted the
method of Latin Squares to select calibration points, and found that this was an economical way
of selecting grid points in three variables.
The flows studied in the HGWT were all subsonic, with maximum Mach numbers on the order of
0.3, the limit of incompressible flow. For this reason, a compressibility coefficient was not
incorporated into the present work. The influence of Mach number is acknowledged in the
general case, but was ignored for subsonic, incompressible flow.
2.7 Flow Turbulence Effects
Turbulence has long been known to affect the accuracy of pitot tube pressure measurements.
Following Bernoulli’s law, an increase or decrease of the same magnitude in flow velocity will
result in a different magnitude of increase or decrease in pressure, because pressure is
proportional to the square of velocity. Consequently, error can be introduced through the timeaveraging of pitot data in a highly turbulent flow. Further to this, changes in the flow angle due
to local large scale turbulence will result in non-linear probe response, which again, will bias the
time-averaged probe response.
The effect of turbulence on pitot tube response was studied extensively by Becker [23], and it was
found that the pitot tube should be selected to match the flow that was being measured. Four
29
conditions for “ideal” probe response were given. Turbulence must be large scale, with a mean
scale roughly 5 times the probe diameter. The Reynolds number must be large, the Mach number
small, and velocity gradients should be low. It was also noted that many of the detrimental
effects of turbulence could be mitigated through the installation of small flow obstructions inside
the tube itself – effectively damping any oscillations within the pressure lines, without affecting
the mean pressure observed in the stagnant air inside the probe tip. Christiansen [24] performed a
similar study, however this study included yaw meter (3 hole) probes. The results were similar in
terms of pitot tubes. Yaw meters were found to be sensitive to turbulence intensity; however this
sensitivity was less than that of a simple pitot tube’s, especially in terms of the sensitivity to eddy
size – the averaging of pressure over multiple ports suppressed the effect of small scale
turbulence more than a pitot tube.
Developments in pressure transducer technology have led to the development of pressure probes
with transducers that are embedded in the probe tip. This eliminated the need for long pressure
lines, and significantly improved frequency response. It has been shown with total pressure
probes [25], [26] that a frequency response in the kHz scale was possible using these embedded
sensors. At this level of frequency resolution, viscous damping and resonance within the air
cavity in front of the pressure transducer became significant. Care must be taken to design the
probe in such a way that the resonant frequencies of the chamber are above the data acquisition
frequency. The sensor also cannot be installed too close to the tip, or the assumption that the flow
stagnates within the chamber fails.
The potential of such high frequency probes for the measurement of transient flows, even those
on a turbulent length and time scale has been shown. The main limiting factor presently is the
30
pressure transducers – both in terms of the frequency response in micro-scale probes, and in terms
of temperature and environmental limits. This is an area of study however, and transducers
capable of producing a frequency response of up to 20 kHz at temperatures up to 600°C are
currently commercially available and have been successfully implemented in 7 hole probes [27].
Similar high-frequency response transducers have also been implemented in 3 hole probes [28]
for measuring unsteady flows in turbomachines.
2.8 Velocity Gradient Effects
The governing equations for seven hole data reduction were predicated on the assumption that the
flow over the probe tip is uniform. Boundary and shear layer flows have significant gradients,
and thus data collected at points in these regions will not properly represent the flow at that point.
The effect of velocity and pressure gradient on three hole probe response was studied by Sevilla
[29], who used a non dimensional measure of a velocity gradient that was based on a streamline
projection to create a correction factor for velocity measurements. The reduction in error was
found to be on the order of 3° in flow angle in highly shearing flows. The implementation of this
correction was difficult with a seven hole probe because the streamline must be defined in two
dimensions, and the resulting correction factor became a function of two flow angles.
31
Chapter 3
Apparatus and Instrumentation
The study of seven hole probes requires several pieces of equipment, as well as a computerized
Data Acquisition System (DAQ). The following sections describe the detailed design of the
probes themselves, the construction and calibration of the experimental apparatus that was used,
and the setup and verification of the DAQ system.
3.1 Probe Design
The probes considered in the present study were seven hole probes with a tip diameter of
3.87mm. Each individual pressure port had a diameter of 0.406 mm. Stainless steel tubes were
silver soldered to each port, and the tubes were contained in a 6.35 mm diameter tube. This tube
formed the structure of the probe neck. It was bent in an L-shape, with the tip about 100mm from
the bend. This tube was then inserted into a ½” diameter stem that allowed the probe to be
mounted in a rotary or X-Y traverse. A schematic of a typical seven hole probe that was used in
the present work is shown in Figure 3-1.
32
Figure 3-1: Schematic Layout Drawing Showing the Parts of the Seven Hole Probe
The probes used in this study were made in house in the McLaughlin Hall machine shop. The tip
was fabricated by drilling seven holes in a piece of stainless steel rod. Two different tip shapes
were tested – a straight 30° chamfer, and a hemisphere. The straight chamfered tip was turned in
a lathe, while the hemispherical tip was formed by hand filing and sanding a conical tipped probe
by hand. The probe was hand filed because a number of conical tipped probes were available in
33
the lab, and modifying an existing probe was preferred over building a new probe. The tip
shapes are shown schematically in Figure 3-2. The tip shapes are shown as constructed in Figure
3-3. Detailed shop drawings and specifications can be found in Appendix C.
Figure 3-2: Schematic Drawing of Probe Tip Designs
Figure 3-3: Photograph of Tip Shapes as Built
34
3.2 Experimental Apparatus
The two main pieces of equipment that were used in the present study were a wind tunnel and a
rotary traverse. The wind tunnel was used to generate the calibration flow, and the rotary traverse
was used to position the probe tip at known angles to this flow. An X-Y traversing table was also
used briefly. Its setup and calibration is discussed in Appendix A.2.
3.2.1 Calibration Wind Tunnel
The calibration wind tunnel was designed so that the Reynolds and Mach numbers of the
calibration flows would match those that are studied in the HGWT. Flows in the HGWT have
velocities on the order of 50 - 200 m/s at temperatures up to 550°C. The Reynolds number based
on tip diameter was therefore between about 3 x 103 and 8 x 103, and the maximum Mach number
was around 0.35. Achieving tip Reynolds numbers in this range with a room temperature flow
required velocities on the order of 30 m/s. Compressibility effects were ignored.
A Reliance Electric variable speed motor controller was used to drive a 5hp motor, turning a 16
inch centrifugal blower. The blower was connected to a 5” diameter pipe with a length of 13’, for
a length ratio of approximately 30D. This setup was capable of providing velocities up to 60 m/s,
for a maximum tip Reynolds number of 1.1 x 104. Swirl was suppressed by installing an 8” long
section of flow straightener with 1/4” vanes in the middle of the pipe. The wind tunnel is shown
in Figure 3-4.
35
Figure 3-4: Variable Speed Calibration Wind Tunnel
The wind tunnel was instrumented with a 1/8” diameter pitot static tube that was used to record
reference flow conditions during calibration. This pitot tube was mounted opposite the seven
hole probe during calibration, as shown in Figure 3-5.
36
Figure 3-5: Position of Probe and Reference Pitot Tube During Calibration
The fact that reference flow measurements were taken a distance away from the location of the
probe, as well as the fact that the wind tunnel did not have a settling chamber meant that there
was a need to characterize the outlet flow. The requirements, verification, and characterization of
the outlet flow is discussed in Appendix D.
3.2.2 Rotary Traverse
A rotary traverse that was capable of positioning the probe at known flow angles was constructed
for this work. Two stepper motors with a resolution of 1.8° per step were used to rotate the
probe. A gear reduction system was used to increase the torque of the pitch rotation, and the yaw
rotation was connected directly to the stepper motor. The angular resolution of the calibrator was
37
therefore 1.8° per step in pitch angle and 0.32° per step in yaw angle. The assembled rotary
traverse is pictured in Figure 3-6.
Figure 3-6: Rotary Traverse
The probe was installed in the traverse as shown in the figure. The traverse was adjusted so that
the probe tip was located at the intersection of the yaw and pitch axes of rotation. This was done
so that the probe tip would not translate as it was rotated. This ensured that the probe position
relative to the wind tunnel was constant, and any non-uniformity in the exit velocity profile was
mitigated. It should be noted that since the probe was held horizontally, the pitch and yaw angles
were reversed. Changing the yaw angle of the traverse changed the pitch angle seen by the probe,
38
and visa-versa. The convention adopted in this work will be to refer to pitch and yaw in terms of
the angles seen by the probe.
The traverse was able to position the probe at flow angles up to 60° in all directions. At angles
higher than this, the frame of the traverse interfered with the wind tunnel. The calibrator did not
cause any blockage of the outlet in any position.
3.3 Data Acquisition
Dell Optiplex GX620 computers with Pentium 4 processors, running at 3.2 GHz with 3 GB of
RAM were used for data acquisition. LabView was used for data parsing and motor control.
Instrumentation was connected to the PC using Data Translation Inc DT3003-PGL DAQ boards
and DT730-T terminal blocks. These DAQ cards were capable of accepting 32 differential
analog inputs, each with 12 bit resolution. Unused channels were grounded when they were not
in use to reduce noise and cross-talk.
Additional details about other instrumentation, including the selection and calibration of pressure
transducers, can be found in Appendix A. This appendix also details the tests that were
performed to establish sampling periods for time-averaging. Tests showed that a 1.5 second
settling time was sufficient to reposition the probe and damp pressure fluctuations that occur as a
result of the motion, and that a 1 second data acquisition period at 900 Hz was sufficient to
acquire representative time-averaged pressure data.
39
Chapter 4
Procedures
This section outlines the methods and procedures that were followed for each step of the data
collection and analysis. This includes experimental procedures, computational sequences, and
methods of error checking and verification. A discussion of sources and estimates of
experimental error is also included. The discussion in this section is of methods used in
collecting and analyzing data for the present work. A more detailed description and low-level
discussion of the use of the codes and equipment developed in this work is included in Appendix
B. This appendix is intended as a primer for new students and researchers on the specifics of the
implementation of seven hole probes in the Gas Turbine Lab.
4.1 Data Collection
The collection of experimental data required several steps. The following sections detail the
procedures that were followed when collecting raw calibration data.
4.1.1 Alignment and Connection to DAQ
The seven hole probe was installed in the calibrator as shown previously in Figure 3-6. Several
adjustment features of the rotary traverse allowed the location of the probe tip to be adjusted so
that it was at the intersection of the pitch and yaw axis of rotation – and so that the probe tip
would not translate as it was rotated. Error in the probe tip location was estimated as +/- 1mm in
the X and Y directions. The effect of this error was ignored, however, as its effect would be to
40
move the probe to a slightly different position on the outlet. The velocity gradient at the probe
location was small enough that the associated error was neglected.
Once the probe was installed in the traverse, the traverse was aligned with the wind tunnel. The
traverse was positioned so that the probe tip was within 3mm of the exit plane. The axis of the
wind tunnel had been leveled during construction, so the traverse was also leveled using a level
and small wedges. This brought the pitch axis of the probe parallel to the outlet plane. A flat
plate was then placed against the outlet plane of the wind tunnel, and the traverse was rotated
until the yaw axis was parallel with the outlet plane. Once the traverse was aligned, it was
secured with concrete blocks. The error in this alignment procedure was estimated to be +/- 0.5°
in both directions.
With the traverse aligned, the probe was connected to the DAQ system. Silicon tubing was used
to connect the stainless pressure tubes on the probe side to the pressure transducers. The tubing
was connected to the probe, and compressed air was then used to blow any condensation out of
the lines. An eighth pressure line was connected to the static pressure ports on the pitot-static
tube so that reference static pressure data could be collected. Once the probe and the static tube
were connected to the pressure transducers, the pressure transducers were read, and the reported
pressures were taken to be zero offsets for the transducers. All subsequent measurements were
corrected by that amount.
4.1.2 Calibration Grid Requirements and Generation
The LABView program that controlled the position of the traverse was programmed to accept a
sequentially ordered list of pairs of pitch and yaw angles that would define the calibration grid. A
41
FORTRAN code was developed to generate these grids. The calibration grids were uniform in
cone and roll. The grids were a function of two parameters – grid spacing and maximum cone
angle. Grids were generated by moving through cone angles from 0° to the defined maximum,
stepping in the specified increment. At each cone angle, the roll angle was varied from 0° to
360°, again stepping in the specified increment. Each pair of cone and roll angles were converted
to a pitch and yaw angle. Resolution error due to the traverse’s limited resolution was mitigated
by converting the desired pitch and yaw angles to a number of steps, rounding that number of
steps to the nearest whole number, and finally outputting the resulting pitch and yaw angle, which
was necessarily an exact multiple of the traverse’s smallest step size.
Once the list of points to be measured had been generated, the list was sorted to minimize the
number of movements performed by the traverse. The incremental nature of the movement of the
calibrator meant that at low flow angles, a number of points were duplicated. Duplicate points
occurred when a desired incremental change in roll angle translated to less than a full step change
in pitch and yaw. These duplicate points were removed from the calibration grid during the
sorting process.
In light of the removal of a number of points at low angles of attack, the grid resolution was
increased at low cone angles. For cone angles up to 15°, the spacing was halved in cone angle,
and doubled in roll angle. This altered grid density increased the number of unique points that the
traverse was capable of measuring and compensated for the duplicate points that were omitted.
The final step in generating a calibration grid was to add a number of points where the traverse
would return to 0, 0. These reference measurements were needed for two reasons. They were
42
used as a reference for position during the calibration procedure to visually confirm that no steps
had been skipped, and that the traverse was returning to exactly 0, 0 each time. They were also
used to establish the reference total pressure of the flow. With the probe at 0, 0, the probe was
aligned with the flow, and the pressure measured by port 1 was the flow total pressure. Repeating
this measurement multiple times during the course of calibration ensured that the flow was
steady, and also mitigated random transducer error associated with the reference measurement.
The result of this process was a grid that was uniform in cone and roll. A sample grid, with a
maximum cone angle of 55° and a grid spacing of 5° is shown in Figure 4-1.
40
Pitch (°)
20
0
-20
-40
-40
-20
0
Yaw (°)
Figure 4-1: Sample Calibration Grid
43
20
40
4.1.3 Wind Tunnel Operation and Automated Data Collection Setup
The wind tunnel was run for at least 10 minutes before data was collected. This ensured that
transient startup effects, including the mild aerodynamic heating of internal components, did not
affect the flow. The traverse was initialized by moving it a single step in each direction. This
ensured the polarity of the rotor was set correctly for the first calibration step. With the wind
tunnel warmed up and the traverse initialized, the automated data collection process was started.
The calibration grid was read into memory, and the traverse was moved to the first calibration
point. The probe was held in position for 1.5 seconds to allow the damping of transient pressure
effects as well as mechanical vibration caused by the motion of the traverse. The 7 pressure ports
and reference static pressure measurement were then measured simultaneously at 900 Hz for a
period of 1 second. The pitch and yaw angles, and the 8 pressures were then written to a text file.
The traverse then moved to the next calibration point, and the process was repeated.
4.2 Generation of Calibration Curves
Two FORTRAN codes were used to generate the calibration curves. The first code was used to
sort the calibration data, and output a text file for each sector containing the calibration data for
all of the points that fell into that sector. The second code used the data in those text files to
generate a unique set of polynomial coefficients for each sector. The method and approach used
in these codes is described in the following sections.
4.2.1 Calibration Data Sorting
The raw calibration data was read line by line, and a set of sorting criteria was applied to
determine the sectors that the calibration point would fall into. In general, a point would be
44
included in a given sector if that pressure port read the highest pressure. The caveat was that a set
of checks must be performed on the data to ensure that in low angle flow, the flow was attached
over all of the peripheral ports, and in high angle flow, that the flow was attached over the centre
port. The specifics of the implementation of the sorting criteria are discussed in section 4.2.1.1.
The sectoring of data is best represented by the port with the highest pressure plotted against yaw
and pitch angles. This is shown in Figure 4-2 for one of the tested probes. The noise along the
sector boundaries was a result of pressure transducer error – as the pressures measured in adjacent
ports became equal, transducer uncertainty or slight flow unsteadiness could cause a point to
appear to fall in either sector. This noise shows that there was a need to ensure that the
calibration for each sector continued to be valid slightly beyond its expected extents. The method
that was used to extend the extent of each calibration sector’s validity is discussed in section
4.2.1.2.
40
20
3
4
Pitch (°)
5
1
0
-20
6
7
2
-40
-40
-20
0
20
Yaw (°)
Figure 4-2: Data Sectoring for a Typical Probe
45
40
60
4.2.1.1 Sorting Criteria
The first step in sorting the calibration data was to identify the port reading the maximum
pressure. This identified the sector into which the data point should be added. Before the point
could be added, however, an additional check was performed to ensure that the data point did not
violate the assumptions made in the definition of the flow coefficients for that sector.
The low angle flow equations, which were applied to the data in sector 1, were predicated on the
assumption that all seven of the holes on the tip of the probe lay in attached flow. A point could
therefore only be added to sector 1 if it could be shown that the six peripheral ports all lay in
attached flow. An analysis of probe data (see section 5.2.1), and previous work [7] has shown
that it was acceptable to assume that if port 1 was reading the highest pressure, flow over the
peripheral ports would remain reliably attached.
The high angle flow equations, which were applied to the data in sectors 2-7, were predicated on
the assumption that at least 4 ports lay in attached flow. The three peripheral ports that were used
in the calculation would always be in attached flow, because the spacing of the peripheral ports
was such that the ports adjacent to the port of maximum pressure could never be on the
downwind side of the probe. There was a possibility of separation over the centre port, however,
so a test was performed to confirm that the flow over port 1 was not separated. The test that was
implemented was the test described by Zilliac [7]. The pressure at port 1 was compared with the
measured pressure in the separated flow on the downstream side of the tip. If the pressure at port
1 was less than the pressure in the separated flow, port 1 was considered to be measuring
separated flow, and the point was not included in the calibration. No points were rejected in the
46
present work as a result of this check – the present study was limited to a 60° cone angle, which
was well below the seven hole probe’s working limit of 80°.
4.2.1.2 Overlap Pressure and the Extent of Calibration Sector Domains
There are two main reasons that it was desirable to ensure that each sector’s calibration was valid
slightly beyond its expected necessary extents. The first was that when an arbitrary flow was
studied, if the flow angle was such that the point would be very close to a sector boundary,
transducer error may have caused the point to fall into either sector. The second was that if only
calibration points inside a sector were considered, then it was possible that there would be some
parts of the sector’s edge that did not have a near-boundary calibration point. Extrapolated
polynomials tend to infinity outside of their fitted domain, and their derivatives are unbounded, so
high slopes and sudden changes are possible. Significant error could have been introduced in the
regions that were beyond the last calibration point in a sector, but still within the applicable extent
of the sector, in this extrapolated polynomial region. The inclusion of a few points from adjacent
sectors helped to improve the response of the surface in these regions of high slope. This was
especially important when coarse calibration grids were used - both because the resolution near
sector boundaries was poor, and because coarser grids have less points, and were therefore more
susceptible to over and undershoot for a given order of polynomial fit.
Including a specific number of additional (overlap) points along a sector boundary was not
straightforward because sectors were not necessarily bounded by lines of constant cone or roll
angle, and even if they were, the specific values of these boundary angles would not be known
until after calibration. The concept of overlap pressure was therefore introduced as an
approximation that would allow a variable number of additional points to be included. There was
47
no analytical relationship that would allow a specific number of additional points to be expressed
as an overlap pressure, rather, the relationship would be empirically determined.
Overlap pressure was defined as a tolerance that was applied when determining the port of
maximum pressure, and thus the sector(s) in which to include a calibration point. A calibration
point was included in a sector if that port was reading either the maximum pressure, or within
overlap of maximum pressure. This meant that points near sector boundaries could be included in
multiple sectors. The relationship between overlap pressure and the actual number of additional
points that are included was unknown, but could be determined during data sorting. The
inclusion of a specific number of points was then possible through iteration. The final criteria that
were used to determine if a point was included in a given sector are shown below in pseudo-code.
Low Angle Sector (Sector 1):
[
IF P1 + POverlap ≥ MAX ( P1− 7 )
]
High Angle Sectors (Sectors 2-7):
[
IF Pn + POverlap ≥ MAX ( P1−7 )
]
AND
IF [P1 ≥ MIN ( P1−7 )]
These criteria were of course applied with the caveat that only points which passed the separation
criteria for that sector could be added to that sector’s calibration. It was expected that increasing
overlap pressure would improve the calibration only up until a certain point – increasing the
48
overlap beyond this ideal value would only begin to consider points that did not meet the
separation criteria.
4.2.1.3 Determination of Reference Flow Conditions
The calibration process correlated the response of the probe to the actual flow conditions, so it
was necessary to measure the freestream flow conditions in the wind tunnel. The flow reference
static pressure was measured with a pitot-static tube, and this value was recorded at each
calibration point. As discussed in section 4.1.2, the probe was returned to (0, 0) periodically
during calibration, and the pressure in port 1 at this point was taken to be the flow total pressure.
The data sorting procedure included identifying (0, 0) points in the calibration, and taking and
averaging the pressure at port 1 over all the times that the traverse moved to (0, 0). This resulted
in an average flow total pressure over the duration of the calibration process. These reference
flow conditions were then written to a file and stored for use in the generation of the calibration
curves.
The use of static pressure as a reference measurement was a change from the procedures followed
by previous students in the Gas Turbine Lab. Two methods had been used previously, and both
methods were found to introduce significant error. The first, the method of Chen [2] and
Masqood [3] was to use a pitot tube traverse of the wind tunnel outlet to collect average static and
dynamic pressure. This introduced error by averaging the flow dynamic pressure, which varied
significantly with position in the outlet plane, over a range of positions that may not have felt the
same flow conditions as the point at which the probe tip was held. The error introduced was on
the order of 5% Q. The second was to mount a fixed pitot tube and measure the flow dynamic
pressure as a reference condition. The total pressure was then taken as the pressure in the centre
49
port when the probe was at (0,0), as described above. This introduced two types of error – firstly,
the direct measurement of dynamic pressure, which was highly spatially varying, introduced error
in the estimate of dynamic pressure. Secondly, when static pressure, which was approximately 0,
was calculated from the difference in total and dynamic pressure, a significant error in static
pressure was introduced. Total and dynamic pressures were approximately equal in this case, and
the combined transducer error in those two measurements led to a significant percentage error in
the calculation of static pressure. When the probe was then used in an arbitrary flow with a
different (non-atmospheric) static pressure, this percentage error in static pressure could have led
to significant absolute error.
4.2.2 Calculation of Calibration Coefficients
Once the data was sorted, and files containing the calibration data had been generated, the
generation of the calibration curves was relatively straightforward. The reference flow total and
static pressure were read into FORTRAN. Raw pressure data for all of the points in each sector
was then read into the program. This data was substituted into the equations of section 2.2, which
calculated all of the terms in the transfer matrix and the dependant variable vector of Equation
(2-15). A multiple linear regression was then performed, which generated the calibration vector
for each of the four flow descriptors (two flow angles, and total and dynamic pressure). This
process was repeated for each of the seven sectors.
It was found during initial testing that the use of single precision floating point numbers during
the regression process led to some error in the resulting curve fit. For this reason, double
precision floating point numbers were used for all floating point numbers. Calibration
coefficients were output and stored with 8 significant digits.
50
4.3 Conversion of Measured Probe Pressures from an Arbitrary Flow to
Flow Velocity, Direction, and Pressure
The first step in converting arbitrary flow data was to read in calibration coefficients for the probe
that was used. Once calibration coefficients were stored, the raw pressure data was processed
line by line. A single data point contained 10 pieces of information – two for the X-Y position of
the point, seven port pressures, and a temperature measurement. From this information, the flow
pressure, direction, density, and velocity was determined.
The sector that would be used to convert the data point was selected using the same criteria that
were applied during the calibration process – the port reading the maximum pressure was
identified, and that sector’s calibration was used to compute the flow properties. A check was
also performed on data points falling into the high angle sectors to ensure that the flow over the
probe tip was not separated – if the separation test was failed, the data point was ignored and an
error code was returned. As during calibration, this test was to compare the pressure at port 1
with the pressure in the separated flow – if port 1 was found to read a lower pressure than the
lowest of the peripheral port pressures, the separation criteria were failed and an error code was
returned.
As during calibration, the directional pressure coefficients were calculated, and all of the terms in
the transfer matrix in Equation (2-15) were computed. The calibration vector was known, so the
dependant variables were calculated directly. This process was repeated for each of the four
dependant variables.
51
Once the flow total and static pressure were known, the measured temperature and the ideal gas
law were used to calculate air density. Compressibility effects on density were ignored. Density
and the flow dynamic pressure were then used to calculate the flow velocity, using the definition
of dynamic pressure. This again assumed incompressible flow. Once the velocity magnitude was
known, geometric relations were used to calculate the vector components of velocity from the
overall magnitude and the flow angles.
Once the flow parameters were calculated and the validity of the raw data was confirmed, the
data was written to an output file. The X-Y coordinate of the data point, the velocity vector, total
and static pressure, density, and temperature were written. There were several error codes,
however, that were output in place of calculated values if an error occurred. The velocity vector
components, total, and static pressures were replaced with the error code if an error occurred.
The error codes are shown in Table 4-1.
Table 4-1: Error Code Descriptions
Error Code
Description
-1
Static pressure higher than total pressure
-2
Cone angle exceeds maximum calibrated cone angle
-3
Negative axial velocity – recirculating flow
-4
Tip separation criteria failed
52
4.4 Calibration Verification
Calibration verification was achieved by first generating calibration coefficients for a set of data,
and then processing that same raw data using those coefficients. The calculated flow pressures
and angles were then compared to the known flow pressures and angles. Errors were reported in
three ways. The root mean square (RMS) average error was computed for each flow parameter
individually as shown in Equation (4-1). The maximum error for each parameter was also found.
Finally, the error was presented graphically in the form of a contour plot, with the X and Y axes
representing pitch and yaw angles. Displaying and analyzing error as a function of flow angle
was particularly instructive, as it allowed for the evaluation of the calibration response at the
boundaries of each sector domain, and aided in choosing optimum overlap pressures.
δ=
1 n
(xi,calculated − xknown )2
∑
n i =1
(4-1)
The verification step, though similar to the validation step, was important because it isolated the
effects of the curve fit. The dataset that was used to generate the curves was the same as the
dataset that was processed using those curves, so there were no sources of error other than error in
the curve fitting. The verification step was particularly important for verifying that a sufficient
number of degrees of freedom had been included.
4.4.1 Verification of Flow Separation
An additional check was performed to verify that the predicate flow assumptions regarding flow
separation had not been violated. The peripheral ports were nominally spaced 60° apart. The
calibration data was processed to extract points along the 3 diameters of constant roll on which
the pressure ports were located. The pressures from the three ports (two peripheral, and the
53
centre port) were converted to pressure coefficients, and output as a function of the angle of
attack of the flow on the probe tip. This essentially evaluated the performance of the seven hole
probe as a three hole probe (yaw meter) for each of its three pairs of holes. A cutaway schematic
of a seven hole probe, showing how data was extracted is shown in Figure 4-3. Holes a and b are
the diametrically opposite holes that are located on the line of constant roll along which data was
extracted.
Figure 4-3: Data Extracted for Yaw Meter Performance Evaluation
54
This data was evaluated graphically to ensure that fundamental assumptions about tip separation
had not been violated. An example of this plot for one of the tested probes is shown in Figure
4-4. Only data for ports 2 and 3 (Roll Angle = 0°/180°) is shown for clarity.
1
Cp = P-PStat / 1/2ρU2
Port 3
Port 2
Port 1
0.5
0
-0.5
-45
-30
-15
0
15
30
45
Angle of Attack (°)
Figure 4-4: Response of 7 Hole Probe as a Yaw Meter
The useful extent of each of the calibration sectors is represented as the domain over which that
port reads the highest pressure. Sector 1, for example, was applicable for angles of attack
between approximately -30° and 20°.
This plot was also used to identify the angle after which the double valued pressure coefficients
discussed in section 2.2 were possible. The pressure data from port 2, for example, showed a
response around an angle of attack of 47° that was characteristic of the onset of flow separation –
pressure was recovered slightly, but was leveling off to a sub-atmospheric stagnation pressure. It
55
was observed graphically that using pressure data from port 2 beyond an angle of attack of
approximately 30° could lead to double-valued coefficients. Port 3 showed a similar characteristic
shape, though the separation had not occurred within the range of angles of attack shown.
Assuming the profile is similar to that of port 2, however, suggests that the limit for port 3 was
around -35°.
Identifying both of the aforementioned types of critical points was the important result of this test.
Following the limit of the sector 1 calibration extent downward, for example, showed that the
flow at the downwind peripheral port was still attached and was not close to reaching the doublevalued region. It was also seen from this plot that the flow over port 1 in the high angle sectors
was not separated and was still moving through the linear decay in pressure with angle of attack
that is characteristic of external flows over bluff bodies.
4.5 Calibration Validation
Validation of the probe calibration was done using the same code that was used for verification,
because the process of error calculation and presentation was the same. Calibration validation
involved creating a new data set, under the same flow conditions, and then using the original set
of calibration coefficients to process that data. This gave a measure of the net effects of all of the
errors associated with the calibration error, including positional uncertainty in the traverse,
pressure transducer error, wind tunnel instability, and of course curve fit error. Again, as during
verification, error was analyzed in terms of the average and maximum magnitudes of error, as
well as the angular distribution.
56
An additional validation step was taken to give a measure of Reynolds number effects. The
rotary traverse was used to collect additional data with the blower speed changed to give different
Reynolds numbers. The effect of Reynolds number on calibration accuracy was then determined
using the same techniques previously described.
57
Chapter 5
Results and Analysis
Several sets of data were collected using different probe tip shapes, different calibration grid
densities, and different calibration flow velocities. These datasets were combined and parsed in
several ways, depending on the type of analysis that was performed. The findings of these
analyses are described in the following sections.
The parametric studies described in this chapter were structured in such a way that each studied
factor could be optimized or analyzed independently. The most important factors were then
identified, and those variables were studied in combination. The factors studied in this work were
broken into two broad categories – parameters affecting the physical response of a probe to
different flows, and parameters affecting the accuracy of curve fitting. The shape of the tip and
Reynolds number of the flow were the parameters affecting the physical response of the probe.
The spacing of the calibration grid, amount of overlap pressure used in data sorting, and order of
polynomial curve fit were parameters that affected the accuracy of curve fitting – changing these
parameters affected the way that the calculated curve fit represented the probe’s physical
response. These two types of parameters were studied virtually independently of each other –
changing the shape of the tip, for example, did not introduce a step physical change in the
response of the probe that required additional degrees of freedom to properly represent.
In terms of measuring the accuracy of a seven hole probe, there were two main goals. The first
was of course, to increase the overall average accuracy of the probe response. The second was to
58
reduce the variation in error with varying angles of attack and flow velocity. Even if the average
error if the probe was low, having large errors at certain angles of attack or flow velocities would
have made measurements of arbitrary flows unreliable. For this reason, error was measured and
presented in three main ways. Average errors were used to represent the overall accuracy of the
probe. Absolute maximum errors were used to show the highest possible error. Contour plots of
error as a function of pitch and yaw angle were used to represent distributions of error. An ideal
calibration had low overall error, a small absolute maximum error, and a uniform distribution of
error. The contour plots were especially important because they captured edge effects –
polynomial extrapolation at sector boundaries could have led to tremendous localized error.
Eliminating these errors was critical, especially for measurement of swirling or shear layer flows.
5.1 Data Verification
The data collected in the present work was verified through a comparison with results in the
literature. Sumner [11] studied the effect of calibration grid density study and of the Reynolds
number of the calibration flow, using third order polynomial curve fits and conical probe tips.
These studies were repeated in the present work and compared with the results of Sumner to give
a degree of confidence in the quality of the data collected. The main difference between the
present work and the work of Sumner was that the present study only considered flow angles up
to 55°, while Sumner calibrated probes to a 72° angle of attack.
The data from the present work was processed in the same way as in Sumner, and the results are
presented below. Figure 5-1 compares the RMS average errors for the low angle sector, while
Figure 5-2 compares the RMS average errors for the high angle sectors.
59
4
4
2
2
0
2
4
6
8
10
12
14
Pressure Error (%)
Angle Error (°)
Yaw Angle
Pitch Angle
PTotal
PDynamic
Sumner
Present Study
0
Grid Spacing (°)
Figure 5-1: RMS Average Error Comparison for Low Angle Sector
Yaw Angle
Pitch Angle
PTotal
PDynamic
Sumner
Present Study
10
8
8
6
6
4
4
2
2
0
2
4
6
8
10
12
14
Grid Spacing (°)
Figure 5-2: RMS Average Error Comparison for High Angle Sector
60
0
Pressure Error (%)
Angle Error (°)
10
The average errors in the low angle sector were very similar to Sumner, both in terms of
magnitude and trend. The average errors in the high angle sectors were also very similar to
Sumner, when the calibration grids were relatively dense. The smaller range of angle of attack
that was used in the present work reduced the total number of calibration points, especially in
sparse grids, in each of the high angle sectors, and thus adversely affected the calibrations in the
present work in a way that would not have been observed by Sumner.
Sumner also studied the effects of Reynolds number on calibration accuracy. The results
presented in his work were limited, however, to the effect of Reynolds number on the overall
error (all sectors) of the calculation of pitch angle and on dynamic pressure. The results of his
work are compared to data from the present work in Figure 5-3.
Pitch
PDynamic
Sumner
Present Work
3
12
2
10
8
1.5
6
1
4
0.5
0
2
0
2000
4000
6000
Reynolds Number
Figure 5-3: Reynolds Number Effect Comparison for all Sectors
61
0
8000
Pressure Error (%)
Angle Error (°)
2.5
14
The trends in the data were very similar. Sumner declared overall Reynolds number
independence at Re = 5000, which was consistent with the results of the present work for the
parameters shown.
These comparisons demonstrated that the data collected for the present study was reasonable, and
gave a degree of confidence that the codes were working as desired. These comparisons also
represent the limit of what quantitative results were presented in the literature. Distributions of
error as an angle of attack, for example, or analysis of pressure coefficient distributions, were not
typically presented. In those papers that did address these topics, the analysis was typically
limited to qualitative observation.
5.2 Factors Affecting Probe Response
This section presents results that deal with the physical response of the probe to changes in
velocity or angle of attack. The ideas in this section are independent of the interpolation scheme
– even if a direct interpolation or neural network calibration scheme were used instead of the
polynomial curve fit, the results presented herein would be instructive.
5.2.1 Geometry Effects on Pressure Coefficient Distributions
The first analysis that was performed was an analysis of the raw pressure data. Simply plotting
raw pressure data was a good first step in ensuring that the dataset is reliable. Qualitative
comparisons were also made between the response of the probe and the characteristic response of
simple shapes, such as circular cylinders. This was important because it isolated the response of
the probe from the response of the calibration scheme – the dynamics of the flow around the
probe tip were investigated directly in this section.
62
5.2.1.1 Tip Separation
The cases of flow over a cylinder and flow over a backward facing step were discussed in section
2.5. The similarities and differences between these cases and the measured flows over the probe
tip were plotted and discussed here. Figure 5-4 shows the yaw-meter performance curves
described in section 4.4.1 for a probe with a chamfered tip. Figure 5-5 shows the same curves for
a probe with a hemispherical tip.
63
0.5
Cp = P-PStat / 1/2ρU
2
1
CP3
CP2
CP1, 2_3
CP5
CP6
CP1, 5_6
CP7
CP4
CP1, 7_4
0
-0.5
-1
-45
-30
-15
0
15
30
45
Angle of Attack (°)
Figure 5-4: Yaw Meter Performance of a Seven Hole Probe with a Chamfered Tip
1
Cp = P-PStat / 1/2ρU
2
0.5
CP3
CP2
CP1, 2_3
CP5
CP6
CP1, 5_6
CP7
CP4
CP1, 7_4
0
-0.5
-1
-45
-30
-15
0
15
30
45
Angle of Attack (°)
Figure 5-5: Yaw Meter Performance of a Seven Hole Probe with a Hemispherical Tip
64
It should be noted that these figures show the centre port pressure plotted as a function of three
different angles of attack. This is because the centre port was not located directly in the centre of
the probe – manufacturing tolerances mean that it may have been slightly offset from the probe
axis. The centre port pressure coefficient therefore varied as a function of the reference roll
angle, so it was presented separately for each of the three reference angles.
Using the analysis techniques described in section 4.4.1, it was observed that none of the
governing equation assumptions had been violated. The peripheral ports were reading in reliably
attached flow through the entire low angle domain. The centre port was also reading in reliably
attached flow through the high angle sectors for the angles of attack tested. No comment could
be made about probe response at angles of attack higher than the limit of these calibration grids,
which was 55°.
There are three curves on Figure 5-4 (chamfered tip) that show a non-linear drop in measured
pressure with increasing angle of attack. The response of the pressure at ports 2, 4, and 7 all
showed separation significantly before the other ports, or step deviations from the linear change
in pressure. The fact that these deviations were not observed in the hemispherical tip’s response
suggests that the cause was probably related to burrs or other physical manufacturing
imperfections. An inspection of the conical probe tip yielded two main defects. There was a
small dimple on the top lip of the probe directly upstream of ports 2 and 7, shown in Figure 5-6.
This dimple was thought to be the cause of the unexpected response of those ports. There was
also a scratch along the side of the probe, next to port 4. The scratch is shown in Figure 5-7, and
was thought to be the cause of the uncharacteristic response of port 4.
65
Burr
Port 2
Port 7
Figure 5-6: Burr Upstream of Port 2
Scratch
Port 4
Figure 5-7: Scratched Tip Surface
66
The response of the hemispherical tip was significantly less symmetrical than the response of the
conical tip. While the characteristic shape of each of the three response curves was very similar
for all three pairs of holes, the angle of attack at which the pressure coefficients peak was quite
different. This was a result of asymmetry in the making of the hemispherical tip. As discussed in
section 3.1, the hemispherical tip was made by hand with a file and sandpaper. The resulting
asymmetry in the tip was clear in these plots. Furthermore, the polishing and deburring that was
the natural result of this modification was also evident – none of the response curves showed the
previously mentioned step changes in the conical-tipped probe’s response.
5.2.1.2 Downstream Separation
It was known that at high angles of attack, the flow would separate on the downwind side of the
probe tip. The number of holes lying in reliably attached flow was determined by plotting
pressure as a function of the angle around the tip surface. The calibration data was filtered and
points of a given cone angle were identified and extracted. The roll angle of the probe was
known, so the position of each of the six peripheral ports relative to the stagnation point was
known. Pressure coefficients were then extracted as a function of the tangential position of the
port.
There was some variation in the pressure coefficient measured at a given tangential position by
each hole because of the tolerances on the location of the ports and the different defects up or
downstream of the ports. Figure 5-8 shows pressure distributions measured by a single hole for
several cases. The distributions are shown over a half cylinder for both the conical and
67
hemispherical tips, and for two different cone angles. The data is compared with data from White
[8] for pressure coefficients around a circular cylinder in different regimes of crossflow.
Inviscid
Laminar
Turbulent
Conical Tip, 32°
Conical Tip, 52°
Round Tip, 32°
Round Tip, 52°
1
Cp = P-PStat / 1/2ρU2
0
-1
-2
-3
0
30
60
90
120
150
180
θ (°)
Figure 5-8: Pressure Coefficient Distributions around Probe Tip and a Circular Cylinder
The pressure coefficient distribution around the probe tip is similar in form to that of the
distribution around a cylinder, though separation occurred further around the probe tip than the
cylinder. This delay in separation occured because the flow around the probe tip was not two
dimensional – the addition of high pressure air flowing over the top of the probe tip aided in
maintaining attached flow as pressure was recovered on the downstream side of the tip.
It was also evident that the distribution of pressure was affected mainly by the tip geometry. At a
52° cone angle the hemispherical tip did not show signs of flow separation, where the flow over
the conical tip separated at about 160°. The cone angle of the flow only affected the magnitude
of the pressure change on the downwind side of the probe. The separation point on the conical tip
68
did not change with the flow angle of attack, and the distribution of pressures around the round
tip inflected at the same point, about 110°, for both angles of attack.
In terms of the calibration, this plot was used to verify that in high angle flows, the two peripheral
ports that neighbour the port reading the highest pressure were measuring in reliably attached
flow. Figure 5-9 shows a flow that impinges on the probe tip at a roll angle that is directly
between two of the peripheral ports. Calculating the flow properties for this roll angle required
pressure data from a port that was nominally 90° from the stagnation point. This was the highest
angular position around the probe tip from which pressure data could be required. Figure 5-8
showed that both probe geometries had attached flow at 90°around the cylinder, for both of the
angles of attack that were shown. This confirmed that the high angle equations would be valid
for both probe geometries.
Figure 5-9: High Angle Flow Requiring Attached flow 90° from the Stagnation Point
Figure 5-8 only showed pressure distributions measured by a single port, for clarity. Figure 5-10
shows the same data, but displays the distributions measured by three of the ports for each probe.
69
This plot shows the effects of asymmetries in the probe tip, as well as the effect of imperfections
and sharp edges.
1
Cp = P-PStat / 1/2ρU2
Conical Tip, 52°
Round Tip, 52°
0
-1
-2
0
30
60
90
120
150
180
θ (°)
Figure 5-10: Pressure Coefficient Distributions with Data from Three Ports
The asymmetry in the hemispherical probe was evident – the differences in the slope of the three
curves showed that the flow was accelerating around the tip differently, depending on the
reference roll angle. None of the ports showed any flow separation, however, so the
characteristic response was the same for all of the ports – the differences in slope were simply
accounted for during calibration. The conical tipped probe had a more symmetric response over
the range of angles for which flow was attached. It showed different performance with respect to
flow separation at each port, however, suggesting that the conical tip was more sensitive to
defects that will trigger separation. These types of step changes can be more difficult to deal with
during calibration, however these pressures measured in the separated regions were not actually
be used in the calculation of any flows. Defects in the conical tip would only affect probe
70
performance in the high angle sectors if they were large enough to trigger separation at or before
90°around the probe tip.
5.2.2 Reynolds Number Effects
Section 2.5 discussed several papers that have studied the effects of Reynolds number on seven
hole probe calibrations, and section 5.1 presented some of these results. These authors have all
followed almost the exact same methodology – calibration data was collected at two or more
Reynolds numbers, calibration curves were calculated based on both sets of data, and then one or
more calibrations were applied to the other sets of data, and errors were calculated. Errors were
typically found to be on the order of 1-5% in pressure and 1°-3° in flow angles. While these
types of results were helpful, they did not paint a complete picture. Typically only average errors
were reported, and there was no discussion of the physical phenomena that were leading to the
dependence. A better understanding of how the flow Reynolds number affected the response of
the probe, especially at different flow angles, would improve the quality of experimental error
analysis.
5.2.2.1 Reynolds Number Effects on Pressure Coefficient Distribution
The pressure coefficient distribution over a body subjected to an external flow was a measure of
local flow acceleration. The local gauge pressure was normalized by the flow dynamic pressure,
so in the absence of viscous effects, a pressure coefficient distribution was only a function of
geometry. Real flows, such as those over a seven hole probe tip, were viscous. The pressure
coefficient distribution over the tip would therefore be a function of both the tip geometry and of
the viscous forces in the flow - hence the flow Reynolds number. When the flow over the probe
became independent of Reynolds number, pressure coefficient curves collapsed on a single line.
71
The dependence of low angle flows on Reynolds number was shown by plotting the yaw-meter
response of the probe to flows of different velocities. The dependence of high angle flows on
Reynolds number was shown by plotting the distribution of pressure coefficients around the
circumference of the probe tip.
Figure 5-11 and Figure 5-13 show the yaw meter response of the conical and hemispherical
tipped probes, respectively, with the response plotted for several Reynolds numbers. Curves from
peripheral ports 2 and 3 are shown – the response of the other pairs of ports is similar. Figure
5-12 and Figure 5-14 show pressure coefficient distributions around the probe tip for the conical
and hemispherical tipped probes. The effects of Reynolds number appear on these plots as the
difference between the distributions for flows at different Reynolds numbers. The difference in
response of the two tip shapes can also be seen by comparing the same plots for the two tip
shapes.
72
1
Cp = P-PStat / 1/2ρU
2
0.5
0
-0.5
Re = 7300
Re = 6000
Re = 5300
Re = 4500
Re = 3800
Re = 3100
Re = 2100
-1
-1.5
-40
-20
0
20
40
Angle of Attack
Figure 5-11: Reynolds Number Effects on Conical Tip Yaw Meter Performance
1
Re = 7300
Re = 6000
Re = 5300
Re = 4500
Re = 3800
Re = 3100
Re = 2100
Cp = P-PStat / 1/2ρU
2
0.5
0
-0.5
-1
-1.5
0
30
60
90
120
150
180
θ (°)
Figure 5-12: Reynolds Number Effects on Conical Tip Pressure Coefficient Distribution at
50° Cone Angle
73
1
Cp = P-PStat / 1/2ρU
2
0.5
0
-0.5
Re = 7300
Re = 6000
Re = 5300
Re = 4500
Re = 3800
Re = 3100
Re = 2100
-1
-1.5
-45
-30
-15
0
15
30
45
Angle of Attack (°)
Figure 5-13: Reynolds Number Effects on Hemispherical Tip Yaw Meter Performance
1
Re = 7300
Re = 6000
Re = 5300
Re = 4500
Re = 3800
Re = 3100
Re = 2100
Cp = P-PStat / 1/2ρU
2
0.5
0
-0.5
-1
-1.5
0
30
60
90
120
150
180
θ (°)
Figure 5-14: Reynolds Number Effects on Hemispherical Tip Pressure Coefficient
Distribution at 50° Cone Angle
74
The yaw meter response plots show that the main Reynolds number effect was to increase the
measured pressure at a port when the port was aligned with the flow – effectively acting as a pitot
tube. The magnitude of this effect was on the order of 2%. This was consistent with the
published experimental data for pitot-tube Reynolds number dependence discussed in section
2.5.3. The curves began to collapse on each other as the yaw angle was varied, up until the point
where the flow began to separate over the downstream peripheral port. The separation, and
therefore the response in this region, was very Reynolds number dependant, although the effects
of separation occurred beyond the limits of where pressure data from that port would be used –
meaning that this Reynolds number effect was of little consequence to probe performance.
In terms of each individual port, the Reynolds number sensitivity was as high as 2% over the
range of flow angles tested. Directional coefficients were calculated based on the difference in
pressure coefficients, however, and there was not a single angle of attack where the response of
all three holes was Reynolds number independent. When the error in the difference was
considered, errors ranged from 2-3%. This gave an indication of the expected error associated
with Reynolds number effects in the low angle flow regime.
The pressure coefficient distribution plots showed that the hemispherical tip was significantly less
Reynolds number dependant than the conical tip. The pressure on the downstream side of the
conical tip showed significant dependence on Reynolds number, while the distribution curves for
the hemispherical tip virtually collapsed to a single line. The consequence of this was somewhat
mitigated by the fact that the affected ports were mostly on the downstream side of the probe in a
high angle flow, so the pressure data from these ports would not be used - as discussed in section
5.2.1.2, only pressure data from a maximum of 90° to the stagnation point could be used in the
75
flow calculation. The conical tip response showed a dependency of about 15% associated with
the pressure at the 90° port, meaning that high angle flows measured with the conical tipped
probe were likely to show a Reynolds number dependence on the order of 15%. The effect on the
hemispherical tip was much smaller – the observed dependence was approximately 5%.
5.2.2.2 Reynolds Number Effects on Calibration Accuracy
While observing the effects of Reynolds number on pressure coefficient gave good insight into
the reasons and the characteristics of the probe’s response to changing Reynolds numbers, it
could not provide a single, quantifiable measure of the associated error. The effect was
quantified by calibrating a probe at Re = 6000, and then applying this calibration to data collected
at five lower Reynolds numbers, and one higher Reynolds number. 200 data points were
collected at each Reynolds number at cone angles up to 55°. A third order polynomial expansion
was used for calibration. No overlap pressure was used during calibration. The calibrations that
were used were verified to be grid independent in section 5.3.1.
The results are presented in two ways. Average errors in the high and low angle regions are
plotted for each of the tip types in Figure 5-15 and Figure 5-16. This shows the relative
sensitivities of the high and low angle regions for each probe.
76
Yaw
Pitch
PTotal
PDynamic
PStatic
Low Angle
High Angle
Angle Error (°)
2.5
15
2
10
1.5
1
5
Pressure Error (%)
3
0.5
0
3000
4000
5000
6000
7000
0
Reynolds Number
Figure 5-15: Average Errors in the High and Low Angle Regions for a Conical Tipped
Probe
Yaw
Pitch
PTotal
PDynamic
PStatic
Low Angle
High Angle
Angle Error (°)
2.5
15
2
10
1.5
1
5
Pressure Error (%)
3
0.5
0
3000
4000
5000
6000
7000
0
Reynolds Number
Figure 5-16: Average Errors in the High and Low Angle Regions for a Hemispherical
Tipped Probe
77
The results showed that the low angle sector was much less sensitive to Reynolds number than
the high angle sectors. Both types of tips were quite insensitive to Reynolds number above 3000
in the low angle sector, while they were only insensitive in the high angle sectors over 6000. This
was consistent with the findings of Sumner [11], who concluded that the response was Reynolds
number insensitive over 5500. It was a useful extension, however, to understand that low angle
sector was still useful at relatively low Reynolds numbers, especially when collecting data from
the HGWT, where many of the flows tested were approximately axial.
The relative sensitivity of the two tip shapes is compared in Figure 5-17. The average error over
the entire calibration range is plotted against Reynolds number. The results show that the
hemispherical tip was slightly less Reynolds number dependant than the conical tip. The
calculation of dynamic pressure showed the most error, and the most sensitivity to Reynolds
number. Static pressure was derived from a calculated total and dynamic pressure, so it also
showed significant error.
78
Yaw
Pitch
PTotal
PDynamic
PStatic
Conical
Hemispherical
Angle Error (°)
8
15
10
6
4
5
Pressure Error (%)
10
2
0
3000
4000
5000
6000
7000
0
Reynolds Number
Figure 5-17: Average Errors for Both Tip Shapes
5.2.2.3 Reynolds Number Effects in Previous Works
Chen [2] and Maqsood [3] reported that they calibrated their probes at a Reynolds number of
2x105. Their hot flow measurements were taken at Reynolds numbers between 2000 and 8000,
which led to an uncertainty on the order of 5-10% for the conical tipped probes used in their
work.
5.3 Variables Affecting the Representation of Probe Response Using a
Curve Fit
The results described in this section deal only with how well the fitted curve represents the
physical response of the probe to changes in velocity and angle of attack, and are virtually
independent of the actual physical response. Changes to the physical response will only affect the
79
accuracy of the curve fitting scheme if there are significant changes to the slopes of the
directional pressure coefficients with respect to angle of attack and flow velocity. Provided that
the response follows a relatively consistent characteristic shape, the effect of the interpolation and
curve fitting scheme can be studied virtually independently of the probe response parameters.
This section demonstrates this independence and then carries out a parametric study of the
variables affecting the curve fit.
5.3.1 Calibration Grid Independence
The effect of grid density was studied by collecting calibration data for the conical tipped and the
hemispherical tipped probe, using different angular spacing of the calibration points. Calibration
grids were generated using the methodology described in section 4.1.2. Nominal angular
spacings of 3°, 5°, 6.5°, 8°, and 10° were tested for cone angles up to 55°.
The data from all five of these grids was aggregated to form a dataset of 3822 points that could be
used to validate the calibrations. The data from each different grid density was used to generate
calibration curves. Each of these calibrations was then used to convert the entire dataset, and
standard errors were calculated. Errors are reported for the entire dataset, and then separately for
the high and low angle regimes. A third-order polynomial curve was fit to the data. An overlap
pressure of 0 was used for each of the curve fits. Figure 5-18 shows the effect of the calibration
grid on solution accuracy for a conical tipped probe, while Figure 5-19 shows the same curves for
the hemispherical tipped probe.
80
Yaw Angle
Pitch Angle
PTotal
PDynamic
PStatic
Overall
Low Angle
High Angle
Angle Error (°)
8
10
8
6
6
4
4
2
2
0
2
4
6
8
10
Pressure Error (%)
10
0
Grid Spacing (°)
Figure 5-18: Effects of Grid Density on Conical Tipped Probe Error
Yaw Angle
Pitch Angle
PTotal
PDynamic
PStatic
Overall
Low Angle
High Angle
Angle Error (°)
8
10
8
6
6
4
4
2
2
0
2
4
6
8
10
Pressure Error (%)
10
0
Grid Spacing (°)
Figure 5-19: Effects of Grid Density on Hemispherical Tipped Probe Error
81
Considering first only the overall grid sensitivity, the plots show that the conical tipped probe
calibration could be considered grid independent at 6.5°spacing, while the hemispherical tipped
probe was independent at a 5° spacing. The hemispherical tipped probe’s need for additional
calibration points was attributed to the asymmetry in the probe tip – because the tip was handfiled, the response was less predictable, and more calibration points were required.
The calibration of the low angle sector was quite insensitive to grid density – there was very little
increase in error with increasing grid density. The overall error was primarily driven by the high
angle error, which was consistent with the findings of Sumner [11]. The magnitude of the
observed errors was also consistent with his findings for grid densities up to 6.5°.
Above 6.5° grid spacings there was a very significant increase in high angle error, higher than
was observed in the literature. This was a result of the relatively small range of cone angle that
was used in the present work. Seven hole probes were typically calibrated to at least a 70° cone
angle. The probes in the present study were calibrated to a maximum cone angle of 55°, which
was the mechanical limit of the rotary traverse. This reduced the number of calibration points
that fall into the high angle calibration sectors. When the grid was sparse, the number of degrees
of freedom in the polynomial curve fit became significant compared to the number of calibration
points. The number of calibration points in each sector is tabulated against grid density for each
tip shape in Table 5-1 and Table 5-2.
82
Table 5-1: Calibration Point Distribution for a Conical Tipped Probe
Sector
3°
5°
6.5°
8°
10°
1
883
371
215
149
98
2
170
61
37
21
17
3
249
92
50
35
20
4
193
70
40
25
17
5
206
78
42
27
17
6
160
60
33
21
17
7
194
71
39
24
20
Total
2055
803
456
302
206
Approximate Duration
2:45
1:15
0:45
0:30
0:20
Table 5-2: Calibration Point Distribution for a Hemispherical Tipped Probe
Sector
3°
5°
6.5°
8°
10°
1
698
296
181
120
79
2
288
111
61
40
29
3
192
70
34
25
18
4
190
70
40
27
17
5
209
80
44
29
17
6
262
98
52
33
24
7
216
78
44
28
22
Total
2055
803
456
302
206
Approximate Duration
2:45
1:15
0:45
0:30
0:20
83
A third order polynomial fit was used in this study, which meant there were 10 degrees of
freedom in each sector. The massive errors observed at high angles were a result of fitting 10
degrees of freedom to as few as 17 points – the calibration became very sensitive to error in the
raw data at small numbers of points. The points were also unlikely to be near the sector
boundaries, so unphysical edge effects were expected to influence near-boundary regions. At
6.5° spacing, the conical tipped probe had between 30 and 50 points in each of the high angle
sectors, while the hemispherical tipped probe had between 35 and 60. The hemispherical tipped
probe showed significantly less error in the high angle sectors, and this difference was attributed
to the larger number of calibration points.
The approximate time to complete each traverse was also tabulated, in hours. There was a
significant reduction in the time to calibrate when a 5° grid is chosen over a 3° grid. The time
savings associated with moving to a 6.5° grid were much less significant. The 5° grid was
therefore recommended, as the results showed that the resulting calibration was grid independent,
and the time to complete data collection was reasonable.
5.3.2 Overlap Pressure
The concept of overlap pressure was introduced in section 4.2.1.2 as a method for improving the
near-boundary response of the calibration curves, especially with sparse grids. When there was
not a calibration point near a sector boundary, a significant portion of the sector could have been
calibrated with an extrapolated curve, rather than with an interpolated curve. Extrapolated
polynomial curves tend to infinity, and their derivatives in the extrapolated regions are
uncontrolled – meaning that significant, sudden, unphysical changes in response are possible.
The concept of overlap pressure was therefore introduced as a controlled method of including a
84
small number of points from an adjacent calibration region, so that the response would continue
to be interpolated, rather than extrapolated, through the entire calibration domain.
The results presented in this section are based on third order polynomial fits, which have 10
degrees of freedom. The tip Reynolds number is 6000, and the flow total pressure is
approximately 335 Pa in all cases.
5.3.2.1 Proof of Concept
The viability of using overlap pressure as a method of improving calibration accuracy was first
investigated with the 8° grid for the conical tipped probe. This calibration spacing was shown in
section 5.3.1 to have a significant increase in error compared to the 6.5° grid, especially in the
high angle sectors. Overlap pressure was presented as a percentage of the flow dynamic pressure.
Based on the pressure coefficient distributions in Figure 5-4 and Figure 5-8, it was estimated that
an overlap pressure of up to approximately 25% would include adjacent calibration points that
would satisfy the separation criteria. Errors were calculated and compared for overlap pressures
of 0, 3%, 7.5%, 15%, 22.5%, and 30% of the known flow dynamic pressure.
The results showed that no amount of overlap improved the low angle calibration. The low angle
region contained enough points, and sufficiently few degrees of freedom that there was no
improvement in the response of the low angle sector through the use of additional calibration
points. For this reason, only the effects of overlap pressure on high angle probe response were
shown.
85
The effect of overlap pressure is shown in terms of changes to the average high angle RMS error
and to the maximum error in Figure 5-20. Table 5-3 shows how changes in the overlap pressure
affected the number of data points used in the calibration of each sector. The data that is
presented is from the conical tip, 8° grid density case.
Angle Error (°)
10
25
8
20
6
15
4
10
2
5
0
0
10
20
30
Overlap Pressure
Figure 5-20: Effect of Overlap on High Angle Probe Error
86
0
40
Pressure Error (%)
Yaw Angle
Pitch Angle
PTotal
PDynamic
PStatic
RMS Average
Absolute Maximum
Table 5-3: Total Number of Points in each High Angle Calibration Sector with Overlap
Overlap
0.0%
3.0%
7.5%
15.0%
22.5%
30.0%
Sector 2
21
22
22
30
34
38
Sector 3
35
36
37
43
50
58
Sector 4
25
28
31
35
39
46
Sector 5
27
28
31
34
41
50
Sector 6
21
21
22
28
33
38
Sector 7
24
28
29
34
37
41
The results show that the application of a small amount of overlap quickly reduced the average
error in all five parameters. The response of the yaw, pitch, and total pressure calibration showed
dramatic decreases in their maximum error through the inclusion of only two or three additional
calibration points. This was an expected result – the first adjacent points to be included as the
overlap pressure was increased were those closest to the sector boundaries, and the highest errors
were expected at sector boundaries where there was not a calibration point near the boundary.
The RMS error was minimized at a 15% overlap pressure in this case. Increasing the overlap
pressure beyond this forced unphysical trends that increased error by considering data points that
were too far into adjacent sectors.
Dynamic pressure showed the highest error, both in average and in its maximum error, of the four
flow properties that were directly calculated. Static pressure was derived from the difference in
87
total and dynamic pressure, so the error in static pressure was of course higher. The effect of
overlap pressure on the spatial distribution of error was therefore seen most easily on a contour
plot of the error in dynamic pressure. Figure 5-21 shows percentage error in dynamic pressure as
a function of pitch and yaw. Figure 5-22 shows the same plot of error, but with an overlap
pressure of 15%. The locus of points that was used to calibrate sector 7 in each case is also
shown for reference. The solid black squares represent the points that actually lie inside sector 7,
while the outlined squares represent the points that were included by the overlap process.
88
40
PDynamic Error
20
16
12
8
4
0
-4
-8
-12
-16
-20
Pitch (°)
20
0
-20
-40
-40
-20
0
20
40
Yaw (°)
Figure 5-21: Error in Calculated Dynamic Pressure with 0 Overlap Pressure
40
PDynamic Error
20
16
12
8
4
0
-4
-8
-12
-16
-20
Pitch (°)
20
0
-20
-40
-40
-20
0
20
40
Yaw (°)
Figure 5-22: Error in Calculated Dynamic Pressure with 15% Overlap Pressure
89
These plots clearly showed the effect of overlap pressure. The sharp increases in error around the
sector boundaries were largely eliminated through the inclusion of the overlap points. The effect
on sector 7 was quite dramatic – with no overlap, there was clearly a significant polynomial
extrapolation effect at the low-cone boundary. This error was effectively eliminated by the
inclusion of a line of points from sector 1.
The spatial representation of error in the above figures showed that error was not uniform across
the functional range of the probe. Uniform error was desirable, especially when measuring
swirling flows that impinged on the probe at a variety of angles of attack. These results showed
that overlap pressure was able to achieve this goal as well.
The previous figures showed that the use of additional overlap calibration points improved the
accuracy of the calibration generated by a coarse calibration grid – a grid sparser than the grid
independent calibration. These figures have not, however, compared the resulting calibration
with one generated by a denser grid with no overlap.
90
Table 5-4 and Table 5-5 compare the RMS and absolute maximum errors for the 8° grid with
overlap to the 6.5° grid with no overlap. The 6.5° grid had a total of 456 points, while the 8° grid
had a total of 302 points. In terms of the high angle sectors only, the 6.5° grid with no overlap
used 241 points for calibration. The 8° grid with overlap used 204 calibration points in the high
angle sectors, and some of those points were actually the same data point, but used in more than
one sector.
91
Table 5-4: RMS Average High Angle Errors
6.5°Grid
8° Grid
0 Overlap
15% Overlap
Yaw (°)
0.96
0.92
Pitch (°)
0.73
0.75
PTotal (%)
2.4
2.2
PDynamic (%)
3.9
3.6
PStatic (%)
4.7
4.2
Table 5-5: Absolute Maximum High Angle Errors
6.5°Grid
8° Grid
0 Overlap
15% Overlap
Yaw (°)
6.8
6.1
Pitch (°)
4.3
4.4
PTotal (%)
11
7.6
PDynamic (%)
33
18
PStatic (%)
34
20
The results showed that the RMS average error was virtually the same for the two grids, but that
the coarse grid with overlap had lower maximum error, especially in dynamic pressure. This
suggested that while grid density is important to calibration accuracy, an equivalently accurate,
but more uniform calibration could be obtained by using fewer points in the core of the sector,
and including extra points near or across sector boundaries. The reduced absolute maximum
92
error also confirmed that the main source of non-uniformity in probe error was sector edge effects
caused by polynomial extrapolation.
5.3.2.2 Overlap Pressure in Dense Calibration Grids
The previous section showed that overlap pressure can be used to improve the calibration
generated by a coarse grid. The next step was to determine the effect of overlap pressure on
denser calibration grids. If it could be shown that applying overlap to dense calibration grids did
not negatively affect the resulting calibration, then it could be recommended that overlap always
be used, no matter what the grid spacing.
The effect of 15% overlap is presented in Table 5-6 and Table 5-7. These tables show the
difference in high angle error between a no-overlap calibration and a 15% overlap calibration.
The effects on RMS average and absolute maximum are shown, respectively. A negative value
indicates that the error has been reduced, while a positive value indicates that the value has
increased through the application of overlap. Data was from a conical tipped probe.
Table 5-6: Changes in High Angle RMS Average Error with the Application of 15%
Overlap
3°
5°
6.5°
8°
10°
Yaw (°)
0.04
0.07
0.03
-2.4
-8.7
Pitch (°)
0.08
0.08
0.07
-0.83
-9.2
PTotal (%)
0.01
0.05
0.06
-3.2
-18
PDynamic (%)
0.1
0.13
-0.40
-47
-93
PStatic (%)
0.02
0.11
-0.31
-46
-93
93
Table 5-7: Changes in High Angle Absolute Maximum Error with the Application of 15%
Overlap
3°
5°
6.5°
8°
10°
Yaw (°)
-0.16
0.11
-0.04
-62
-66
Pitch (°)
0.77
1.1
0.79
-30
-64
PTotal (%)
0.48
-0.38
-0.33
-92
-92
PDynamic (%)
-14
-15
-18
-81
-9.7
PStatic (%)
-6.3
-8.8
-13
-80
-11
The results showed that the average error of dense grids was statistically insensitive to the
application of a small amount of overlap pressure. Although some of the errors increased slightly
when overlap was used, the increase in error was not statistically significant. For most properties,
especially dynamic pressure, which showed the most spatial variation of error, the maximum
error was reduced significantly. This was a significant finding, as the maximum errors were
typically on the boundary of the high angle sectors, which coincided approximately with the 25°30°cone angle line, which was right in the middle of the probe’s useful range. Errors at these
angles of attacks may not be apparent in experimental investigations of highly swirling flows, so
minimizing these errors to obtain a uniform spatial distribution of error was desirable.
It was concluded, based on this significant reduction in absolute maximum high angle error, that
the application of 15% overlap in the high angle was always beneficial for the present case.
Without additional data this conclusion could not, however, be extended to calibration flows
having a different total pressure, or to calibrations at a different Reynolds number. Furthermore,
94
simply analyzing the current dataset could not demonstrate that normalizing the overlap pressure
by the flow dynamic pressure was the most appropriate guidance.
5.3.2.3 Overlap Pressure in Alternative Tip Geometries
The data analysis that was performed in the proceeding two sections on the conical tipped probe
was repeated for the hemispherical tipped probe, and the results were very similar. The
application of overlap pressure decreased the absolute maximum error on all grids, had virtually
no effect on the RMS average error of dense grids, and improved the RMS average error of
coarse grids. The numerical results were omitted for brevity as very little additional insight is
gained from their inclusion.
5.3.3 Order of Polynomial Curve Fit
The order of polynomial curve fit affected the accuracy of calibration in a number of ways.
Increasing the order of fit allowed for more variation in the shape of the surface, which was
beneficial if there were characteristics of the response with varying high order derivatives. It
could also introduce unphysical error, however – if the physical response of the probe was
smooth, then introducing additional degrees of freedom increased the noise in the calibration.
The effect of a fourth order polynomial curve fit was tested by repeating the data analysis
procedures followed in the previous sections. A grid independence study was carried out using
no overlap pressure. The effect of overlap pressure was then studied again to see whether the
same improvement could be found by applying overlap to a non-grid independent grid density.
Only the data from the conical tipped probe was presented, as the hemispherical tipped probe
showed an almost identical response.
95
5.3.3.1 Grid Independence
The grid independence study was repeated on the conical tipped probe, and the results are
presented in Figure 5-23 in the same way that they were presented for a 3rd order polynomial fit in
Figure 5-18.
Figure 5-23: Effects of Grid Density on Probe Error with a 4th Order Polynomial Fit
The trends in this figure were virtually the same as those for the 3rd order calibration, suggesting
that the 4th order terms did not improve probe calibration on sparse grids. The difference in
overall error (high and low angle sectors) using third and fourth order fits are shown in Table 5-8
and Table 5-9. The effects on RMS average and absolute maximum are shown, respectively. A
negative value indicates that the error has been reduced, while a positive value indicates that the
value has increased through the increase in the order of curve fit.
96
Table 5-8: Effects of Increasing Order of Curve fit on Overall RMS Average Error
3°
5°
6.5°
8°
10°
Yaw (°)
-0.06
-0.05
-0.02
0.95
0.02
Pitch (°)
-0.05
-0.02
-0.05
0.55
-1.4
PTotal (%)
-0.13
-0.09
0.17
11
-0.83
PDynamic (%)
-0.74
-0.66
-0.42
-17
-31
PStatic (%)
-0.65
-0.53
-0.54
-18
-32
Table 5-9: Effects of Increasing Order of Curve fit on Overall Absolute Maximum Error
3°
5°
6.5°
8°
10°
Yaw (°)
-0.59
-1.7
-0.38
1.8
1.8
Pitch (°)
-0.49
-0.4
-1.4
-13
-18
PTotal (%)
0.80
0.3
3.9
N/A
N/A
PDynamic (%)
-12
0.03
6.3
N/A
N/A
PStatic (%)
-10
-2.3
0.26
N/A
N/A
The results show that there was a reduction in average error and a significant reduction in
absolute maximum error when a higher order polynomial curve was used on a dense calibration
grid. The reduction in maximum error on a dense grid suggested that there may have been
characteristics of the probe response that were fourth order – which in itself justified the inclusion
of fourth order terms. Comparing Table 5-9 with Table 5-7 (pg 94) , which showed changes in
absolute maximum error when overlap was applied to a 3rd order fit, showed that using a 4th order
97
fit decreased the absolute maximum error approximately the same amount as applying 15%
overlap to a 3rd order fit., while also decreasing average error.
The effect of the order of fit on the distribution of error was best shown on a contour plot of error.
Dynamic pressure showed the highest sensitivity to angle of attack, so contours of dynamic
pressure error are shown in Figure 5-24 and Figure 5-25 for 3rd and 4th order fits, respectively.
98
PDynamic Error (%)
6
4
2
0
-2
-4
-6
-8
-10
-12
40
Pitch (°)
20
0
-20
-40
-40
-20
0
20
40
Yaw (°)
Figure 5-24: Dynamic Pressure Error Distribution with a 3rd Order Curve Fit from 3° Grid
Spacing, 0 Overlap
PDynamic Error (%)
6
4
2
0
-2
-4
-6
-8
-10
-12
40
Pitch (°)
20
0
-20
-40
-40
-20
0
20
40
Yaw (°)
Figure 5-25: Dynamic Pressure Error Distribution with a 4th Order Curve Fit from 3° Grid
Spacing, 0 Overlap
99
The contour plots clearly show that increasing the number of degrees of freedom significantly
improved the uniformity of error in the curve fit. The large spikes just beyond the low angle
sector boundary were greatly reduced. The 3rd order curve showed an over-calculation of velocity
in the centre of the high angle sectors and an under-calculation of velocity at some of the high
angle sector boundaries. This was indicative of a higher order physical response that was not
properly captured with a 3rd order curve – especially since these errors were not observed in the
4th order plot.
In terms of declaring grid independence, the fact that the effect of grid density on average error
followed the same trend for the 4th order curve and the 3rd order curve meant that there was no
reason to think that grid density affected a 4th order fit any differently than it did a 3rd order fit.
As in section 5.3.1 then, a minimum of 5°grid spacing was recommended for a 4th order curve fit
as well.
5.3.3.2 Overlap Pressure
The effect of overlap pressure was found to be the same as for the third order fit, but the optimum
overlap pressure was slightly different. The analysis of the effect of overlap pressure for the
fourth order fit was carried out in the same way as the analysis of the third order fit – the 8 degree
grid calibration was repeated using incrementally higher amounts of overlap pressure. As overlap
was increased, the RMS average error decreased to a point, and then began to rise again as points
that were very far outside the sector domain were included. The optimum overlap was found to
be 22.5% for the fourth order fit, which is more than the 15% that was found to be optimal for the
third order fit. This 7.5% increase in overlap caused 4-7 additional points per sector to be
100
included, which was reasonable given that the increased order of curve fit resulted in an
additional 5 degrees of freedom in the fit.
The 22.5% overlap was then applied to all of the calibration grids, and the result was found to be
the same as that found or the third order fit. The average error in denser calibration grids was
statistically insensitive to overlap, but the absolute maximum error was reduced through the
inclusion of overlap points in each case. Again, the inclusion of overlap points did not reduce the
accuracy of the calibration in any case.
The results presented in this section showed that fourth order terms were not negligible. The
spatial distributions of error showed that there were features of the probe response that required
fourth order terms to properly represent. The results also showed, however, that the order of
curve fit was largely independent of other calibration variables – that is, regardless of the order of
curve fit used, a minimum of 5°calibration grid spacing and an overlap pressure on the order of
15-20% was recommended.
5.4 Method of Lowest Error
The results have shown that the best calibration was obtained using a 3° calibration grid density,
15-20% overlap, and a fourth order polynomial curve fit. The errors arising from this optimum
calibration are presented in the following figures. Table 5-10 shows the magnitude of the average
and maximum errors. Figure 5-26 shows error contours for the four directly calculated flow
properties, as well as for static pressure, which was derived from total and dynamic pressure.
101
Table 5-10: Average and Maximum Calibration Errors (All Sectors)
RMS Average
Absolute Maximum
Yaw (°)
0.59
4.1
Pitch (°)
0.48
4.1
PTotal (%)
1.7
10
PDynamic (%)
2.3
12
PStatic (%)
2.7
16
102
Yaw Error (°)
3
2.5
2
1.5
1
0.5
0
-0.5
-1
-1.5
-2
-2.5
-3
Pitch (°)
20
0
20
0
-20
-20
-40
-40
-40
-20
0
20
3
2.5
2
1.5
1
0.5
0
-0.5
-1
-1.5
-2
-2.5
-3
40
Pitch (°)
40
Pitch Error (°)
40
-40
-20
Yaw (°)
0
20
40
Yaw (°)
PTotal Error (%)
8
6
4
2
0
-2
-4
-6
-8
-10
-12
Pitch (°)
20
20
0
0
-20
-20
-40
-40
-40
-20
0
20
8
6
4
2
0
-2
-4
-6
-8
-10
-12
40
Pitch (°)
40
PDynamic Error (%)
40
-40
Yaw (°)
-20
0
Yaw (°)
PStatic Error (%)
8
6
4
2
0
-2
-4
-6
-8
-10
-12
40
Pitch (°)
20
0
-20
-40
-40
-20
0
20
40
Yaw (°)
Figure 5-26: Error Contours for the Optimum Calibration Case
103
20
40
The distributions of error showed that the use of a fourth order curve fit and the application of
high angle overlap gave very good uniformity in error, and the tabulated average and maximum
errors showed that the magnitude of the error was very reasonable. The highest errors were
concentrated in sector 3, and a substantial amount of noise was observed in the error plots in this
region. During data collection, the traverse was moved from low pitch to high pitch, stopping at
each desired pitch angle to traverse all of the corresponding yaw angles. The data in sector 3 was
therefore collected towards the end of the traverse. It is possible that there was some transient
effect that caused additional noise towards the end of data collection. Given that this was the
greatest source of error, and that the contour plots are free of polynomial extrapolation edge
effects, the results showed that the proposed calibration method was reasonable and gave good
accuracy across the entire calibrated range.
5.5 Summary and Discussion of Findings
As discussed in section 5.1, the results of the present study agreed quite well with results
available in the literature. The results that were presented in the following sections have provided
a great deal more data than was available in literature – especially in terms of the level of detail in
the presented data.
In terms of probe geometry effects, the study of Reynolds number influence on seven hole probe
error at different angles of attack provided a level of detail that was not available in the literature.
The closest available results were presented by Lee [30], who found that Reynolds number effects
on five hole probes were more significant at higher angles of attack. The findings of the present
work were consistent with this trend, although the magnitude of the effect was different for the
five and seven hole probes.
104
In terms of curve fitting schemes, the study of the influence of the order of curve fit provided
data and supported conclusions on the required number of degrees of freedom that were not
available in the literature. The concept of overlap pressure was a novel concept, and introduced a
significant reduction in the absolute maximum calibration errors and a corresponding
improvement in uniformity.
Sumner [11] and Silva [15] compared the accuracy of the polynomial curve fit and direct
interpolation data reduction schemes. Sumner used the Zilliac [7] direct interpolation method,
while Silva used a simple linear interpolation. In both cases, the direct interpolation scheme was
found to improve calibration accuracy in high angle sectors, and to have no effect on calibration
accuracy in the low angle sector. In the present work, a similar improvement in high angle
calibration accuracy was obtained through the application of overlap pressure in data sorting.
The findings of these previous works are shown and compared with the improvements seen
through the use of overlap pressure in Table 5-11. For Sumner and Silva, the values shown were
the reported reduction in high angle RMS average error from using the direct interpolation
scheme. For the present work, the values shown are the reduction in high angle RMS average
error from applying overlap pressure. A negative value indicated that the direct interpolation
scheme was actually less accurate than the polynomial curve fit. The observed reduction in error
associated with the use of direct interpolation is similar in magnitude to the reduction in error
associated with the use of overlap pressure.
105
Table 5-11: Effects of Overlap Pressure and Direct Interpolation on Calibration Accuracy
Pitch Angle (°)
Yaw Angle (°)
PTotal (%Q)
PDynamic (%Q)
Sumner
0.3
0.4
-0.3
-0.3
Silva
-0.1
0.7
0.3
1.6
Present Work
0.1
0.1
0.2
0.02
Silva also reported that the direct interpolation scheme reduced the standard deviation, and thus
improved the uniformity of the response error as a func tion of angle of attack. As mentioned, the
application of overlap pressure also improved the uniformity of response error, mainly due to the
elimination of polynomial extrapolation at sector boundaries. No detail is given in any of
Sumner, Silva, or Zilliac of how near-sector boundary points were interpolated – that is, whether
the points nearest the sector boundary were simply extrapolated, or whether points in an adjacent
sector were considered and interpolated.
The present work showed that points near sector boundaries were a significant source of error.
The choice of calibration data near sector boundaries (overlap pressure) was shown to reduce the
overall error of the calibration by the same magnitude as the reduction achieved by Sumner and
Silva by using a direct interpolation scheme. None of Sumner, Silva, or Zilliac gave specific
details of their treatment of points near sector boundaries. Given the lack of detail about this
important aspect of calibration, there may not be evidence to support the conclusion that the
direct interpolation method improves calibration accuracy. If in the direct interpolation schemes,
106
near-boundary points were calculated by interpolating points from adjacent sectors, then the
reduction in error may simply be due to the inadvertent elimination of edge effects.
107
Chapter 6
Error Analysis and Propagation
The error analyses presented in the previous chapter were calculated by taking the difference
between the measured magnitudes and direction of the calibration flow and the calculated
magnitudes and directions using the calibration curve. These errors dealt only with the
calibration side, and did not account for certain types of bias errors. While the effects of random
transducer error and very low frequency (oscillations on the order of minutes and hours) wind
tunnel unsteadiness were accounted for in this calculation, errors in the alignment and positioning
of the probe were ignored, because the probe was not removed and the rotary traverse was not
adjusted between tests. These bias errors must be considered.
Error must also be considered on the experimental data collection side. When data is collected
from an arbitrary flow there is not a reference measurement that allows the error to be directly
calculated, so the error must be propagated through the governing equations from first principles.
The procedure and mathematics of this propagation are presented in this section.
6.1 Sources of Error During Calibration
The four main sources of error on the calibration side were unsteadiness and non-uniformity in
the calibration flow, uncertainty in the angle of the probe in the rotary traverse, random
transducer error, and residual error in curve fitting. As mentioned, three of these sources of error
were accounted for when the quality of the curve fit was calculated, because the effects of those
errors appeared as noise in the calibration data. If the calibration grid and order of curve fit were
108
sufficient to represent all of the phenomenon that are physically present, then the residual in the
curve fit was only a function of noise in the data, which was only a function of measurement and
flow uncertainty.
The nature of the calibration did, however, make it sensitive to errors in the alignment and motion
of the probe during calibration. There was no feedback from the rotary traverse to the DAQ, so it
was simply assumed that when the probe was moved to a desired position that it was actually in
that position. The largest error in a stepper motor, by nature, is cyclical – the error in a complete
revolution is zero, because the position of the shaft at a given step is controlled by the relative
location of each magnet and winding. This means that while some steps may be larger and some
may be smaller, over the entire rotation, the motor will return to exactly where it started. This
assumes, of course, that steps were not skipped. This was confirmed during each calibration by
periodically returning the traverse to (0,0) and visually confirming that it was physically in the
same location.
The uncertainty associated with the exact size of each step was small, especially when compared
to the error associated with the positioning of the tip of the probe in the calibrator. The error in a
single step was therefore ignored, and the only source of bias error that was considered was the
misalignment during installation.
The probe tip was positioned based on setting it a fixed distance in space from surfaces of the
rotary traverse. Based on the uncertainty in the distance measurements, a total bias error of +/0.6° in yaw and +/- 0.15° in pitch was possible. This error would manifest itself as an offset to all
of the flow angles – the probe would be calibrated to return (0°,0°) when the position of the probe
109
was actually (+/- 0.6°, +/- 0.15°). A similar DC offset bias would be applied to flows at all
angles. The total error in the flow angle was therefore the sum of these two components of error
– the worst case scenario being that noise was biasing the flow angle in the same direction as the
alignment offset bias.
Considering this offset error, the error in the calibration that was reported in Table 5-10 of section
5.4 was refined. The refined estimate of error is shown in Table 6-1. These uncertainties
represent the error associated with the curve fit – they represent the average error associated with
the calculated flow properties for a data point where the seven pressures were known exactly.
Table 6-1: Calibration Uncertainty for a Sample Probe
RMS Average
Yaw (°)
1.2
Pitch (°)
0.63
PTotal (%)
1.7
PDynamic (%)
2.3
PStatic (%)
2.7
6.2 Sources of Error in an Arbitrary Flow
The calculation of error for data that is collected in an arbitrary flow is somewhat more involved,
because the actual flow conditions are not known for comparison purposes. Uncertainty must
therefore be computed from first principles. There are three sources of error when collecting data
in an arbitrary flow – probe misalignment and mislocation, transducer uncertainty, and calibration
110
uncertainty. Calibration uncertainty is dependant on the calibration scheme, and its calculation
has been discussed extensively in previous sections.
Probe misalignment and mislocation is highly application dependant, so it is difficult to quantify
in the general case. In the Gas Turbine Lab, probes are typically mounted in an X-Y traverse
table, and the traverse moves the probe across a plane, recording data periodically. Misalignment
of the probe would lead to a uniform offset bias across all measured points, much the same way
that misalignment in the rotary traverse led to a uniform bias during calibration. The degree of
misalignment is dependant on the technique used to align the probe, and cannot be quantified
here, because that technique is left up to the end user. The mislocation of the probe is a function
of the accuracy of the origin of the traverse and of the precision of the traverse’s movements.
The error that is introduced as a result of mislocation is then a function of the velocity and angle
gradients in the flow – in a highly swirling flow, or one with significant shear layers, the partial
derivative of angle or velocity with respect to X-Y position is very high, and small errors in the
probe location will lead to large uncertainties in measured angles and velocities. Again, this error
is highly application dependant and cannot be quantified in the present work.
Pressure transducer uncertainty can, however, be propagated analytically with a constant odd
combination. In the general case, Moffat [31] uses a root-sum-square of the product of the
uncertainty in each constituent term, X, with the partial derivative of the calculated property, R,
with respect to X. The general equation is shown as (6-1) for a governing equation of N variables.
 ∂R
∑  ∂X δX i
i =1
i
N
δR =




2
(6-1)
111
The response equations are known, so they can be differentiated directly with respect to pressure
to obtain an estimate of uncertainty in the ultimate output with uncertainty in measured pressure.
The response equations are non-linear with respect to the directional pressure coefficients,
however, so the uncertainty at a data point depends on the value of these coefficients. It also
depends on the derivative of the calibration surface with respect to these coefficients, which
means that the uncertainty is not constant at all flow angles – at flow angles near sector
boundaries, where the slope of the calibration curve is higher, the uncertainty will also be higher.
6.2.1 Low Angle Flows
The equations used in the calculation of uncertainty at low angles are presented below. The
equations are processed in the same order as they are during data conversion. Equations (6-2)
through (6-5) through are partial derivatives of the governing equations described in section 2.2.1.
 ∂C 
δC A = ∑  δPi A 
∂Pi 
i =1 
2
 ∂C 
δC B = ∑  δPi B 
∂Pi 
i =1 
2
7
7
 ∂C 
δCC = ∑  δPi C 
∂Pi 
i =1 
7
(6-2)
2
These equations are the basic directional pressure coefficients. The uncertainty in each measured
pressure is just the transducer error. The partial derivatives of these terms depend on the
magnitude of the measured pressures, so the error in these terms is a function of flow velocity and
pressure.
112

∂C 
δCα = ∑  δCi α 
∂Ci 
i= A 
C
C

∑  δC
δCβ =
i= A
i

2
∂Cβ 

∂Ci 
(6-3)
2
These equations are the pitch and yaw coefficients, and they are calculated directly from the
directional pressure coefficients. Their magnitude depends only on the uncertainty in the
directional pressure coefficients, and not on their magnitude.

∂X 

δX = ∑  δCi
∂Ci 
i =α 
β
2
(6-4)
This equation is the most labour intensive to derive, because it involves a 15 term polynomial
expression. The calculation of the partial derivatives is straightforward, but time consuming.
The uncertainty in the flow property X is a function of the magnitude of the errors in the pitch and
yaw coefficients, as well as the magnitude of those coefficients. It is also scaled by the
magnitude of each calibration coefficient, which means that the slope of the surface is important.
This equation is solved 4 times – once for each of the two flow angles and the two pressure
coefficients.
2



2
2

∂P  
∂P
δPT = ∑  δPi T  +  δCT T
∂Pi  
∂CT
i =1 
7
2
(6-5)
7
δQ =


∂Q
+  δCq

∂C q
 
∂Q 
∑  δP ∂P 
i =1

i
i




113
Finally, since total and dynamic pressures are calculated from total and dynamic pressure
coefficients that are obtained through calibration, a final derivative step is required to calculate
their uncertainty. These equations depend on the magnitude of the measured pressures, as well as
the value of the pressure coefficient – so the uncertainty of these terms varies with the velocity
and angle of attack of the oncoming flow.
6.2.2 High Angle Flows
The equations used in the calculation of uncertainty in high angle flows are virtually the same as
the low angle flow equations, except that an additional step is needed to convert cone and roll
uncertainties to pitch and yaw uncertainties. Equations (6-6) through (6-9) are the partial
derivatives of the governing equations in section 2.2.2.



2
 ∂C 
δCγ = ∑  δPi γ 
∂Pi 
i =1 
2
 ∂C
δCθ = ∑  δPi θ
∂Pi
i =1 
7
7
(6-6)
These equations are the cone and roll coefficients, which are calculated directly from pressure
data. The uncertainty in these terms depends on the uncertainty in pressure transducer readings,
as well as the magnitudes of the measured pressures.

∂X
δX = ∑  δCi
∂Ci
i =θ 
γ



2
(6-7)
114
This equation is of the exact same form as the low angle flow equation. It is labour intensive but
simple to derive. Uncertainties in roll and cone angles, and total and dynamic pressure
coefficients are returned.
2



2
2

∂P  
∂P
δPT = ∑  δPi T  +  δCT T
∂Pi  
∂CT
i =1 
7
2
(6-8)
7
δQ =


∂Q
+  δCq

∂C q
 
∂Q 
∑  δP ∂P 
i =1

i
i




As with low angle flows, these equations are used to relate uncertainties in total and dynamic
pressure coefficients to uncertainties in calculated total and dynamic pressure. These equations
depend on the measured pressures, so the uncertainty will vary with the velocity and angle of
attack of the oncoming flow.
γ
2
γ
2
 ∂α 
δα = ∑  δi

∂i 
i =θ 
δβ =

∂β 
∑θ  δi ∂i 
(6-9)
i=
These equations relate the uncertainty in cone and roll to the uncertainty in pitch and yaw. The
partial derivatives also depend on the actual cone and roll angle, so the response of these
equations varies with cone and roll.
115
6.3 Calculation of Total Error in an Arbitrary Flow
The calculation of the overall uncertainty in the measurement of an arbitrary can be somewhat
involved, because there are a number of different approaches, depending on the desired accuracy.
As discussed in the previous section, the three main sources of error are probe misalignment in
the traverse, pressure transducer uncertainty, and calibration uncertainty. The overall uncertainty
at a data point would be the sum of all of these components of error.
The highest level of accuracy in uncertainty estimation is obtained using the following procedure:
1. Calculate the uncertainty resulting from transducer at each data point
2. Based on the angle of attack of the flow, interpolate the calibration error plots to calculate
the contribution of calibration uncertainty at that flow angle
3. Estimate the uncertainty in probe alignment based on the process through which it was
aligned
4. Sum the above uncertainties at each measured data point to obtain a distribution of error
across the traverse plane
A lower level of accuracy, but perhaps a more concise estimate of uncertainty can be obtained
more quickly using the following procedure:
1. Average the uncertainties resulting from transducer error at all points
2. Average the calibration uncertainty of the probe across all angles of attack
3. Estimate the uncertainty in probe alignment based on the process through which it was
aligned
4. Sum the above uncertainties to obtain an overall average uncertainty for the entire exit
plane
116
While this second method neglects the fact that uncertainty is not constant with angle of attack
the result still may be a good estimate of uncertainty if there are no significant secondary flows.
The method that is chosen is again application dependant, and depends on both the nature of the
flow and the desired accuracy of the uncertainty estimate.
6.4 Example of Transducer Uncertainty Plots
The form and use of calibration uncertainty plots has been shown throughout Chapter 5, so
examples of those plots are not shown here. This section shows an example of transducer
uncertainty plots for a flow at the exit plane of an ejector system. In this flow, a hot primary jet
passively entrains, and then mixes with, a cool secondary supply of air to reduce the overall
exhaust plume temperature and velocity. The flow is axial. Data is taken from the preliminary
work of Begg [32]. Figure 6-1 shows velocity and temperature contours for this flow. Secondary
flow vectors are not shown because they are not significant.
w, m/s
Temp, °C
140
130
120
110
100
90
80
70
60
50
40
30
20
10
0
360
340
320
300
280
260
240
220
200
180
160
140
120
100
80
Figure 6-1: Velocity and Temperature Contours for Sample Mixing Tube Outlet Traverse
117
In this flow, the spatial variation in uncertainty in flow properties that results from transducer
uncertainty is primarily a function of velocity and temperature gradients – the hot, high velocity
core flow results in very high port pressures, which minimizes the effect of transducer
uncertainty. The error increases in the regions of cooler, slower moving flow that occur away
from the jet core. Figure 6-2 shows the spatial variation of uncertainties for each of the four
calculated flow properties.
Yaw (°)
Pitch (°)
5
4.5
4
3.5
3
2.5
2
1.5
1
0.5
0
5
4.5
4
3.5
3
2.5
2
1.5
1
0.5
0
PTotal (%)
PDynamic (%)
20
18
16
14
12
10
8
6
4
2
0
20
18
16
14
12
10
8
6
4
2
0
Figure 6-2: Flow Property Uncertainty Resulting from Transducer Uncertainty
118
The uncertainty is highest around the perimeter of the mixing tube, where data has been collected
in a boundary layer. The velocity gradient across the tip of the probe is significant in this region,
and causes an apparent flow that is at an extremely high angle. This is an unphysical result,
which is typically ignored during testing.
The majority of the flow, however, is relatively unaffected by transducer error. In this particular
case, because the flow angles are all very similar, a uniform calibration uncertainty could be
applied to this result to give an indication of the total error.
119
Chapter 7
Conclusions
The goal of the present work was to study the calibration and use of seven hole probes to
establish guidelines for their use in a hot flow wind tunnel. To this end, several parameters that
were known to affect probe response were studied and their effects were quantified. The effects
of each parameter are discussed individually in the following sections. The parameters that affect
hot flow measurements differently from cold flow measurements are then discussed again, in the
context of the results of Chen [2] and Maqsood [3], who were the first in the Gas Turbine Lab to
use seven hole probes in hot flow. The findings of the present work are used to explain
anomalies and suspected errors in their results.
7.1 Tip Geometry
The main effect of tip geometry was to change the uniformity of response across the sectors. The
response of the hemispherical tip was much more uniform than the conical tip – that is, the angle
of attack at which flow began to separate, and the manner in which it separated on the downwind
side of the probe was much more consistent across the 6 peripheral holes. The hemispherical tip,
or at least a conical tip with the sharp edges rounded, was therefore recommended, as the range of
angles of attack for which the calibration is valid is more uniform for a rounded tip.
The response of the conical tip was very sensitive to burrs and detents. If a conical tip is used, the
sharp edges should be broken with very fine sandpaper, and the pressure ports should be carefully
deburred and cleaned. A cap is recommended to protect the tip from damage.
120
Changes in the tip geometry did not affect the characteristic shape of the calibration curves to the
point that additional degrees of freedom or different numbers of calibration points were required
to properly capture and model the response. The requirements for a good curve fit were
unaffected by changes to tip geometry.
7.2 Reynolds Number Effects
Reynolds number effects were found to be significant. Both the hemispherical and chamfered
tips showed Reynolds number independence above Reynolds numbers of 6000 for the entire
calibrated range. The low angle sector was much less sensitive, showing Reynolds number
independence above a Reynolds number of 3000 for both tip shapes. The user must be aware of
Reynolds number effects and possibly perform additional calibrations when studying flows at or
below these observed limits.
The error associated with the use of the hemispherical probe beyond its Reynolds number limit
was significantly less than the error associated with using the conical tip beyond its limit,
especially in the calculation of dynamic pressure, which was found to be the most sensitive flow
property to Reynolds number. An additional error of 1.5° in angle measurements and 3% in total
pressure was introduced by using both probe shapes at a Reynolds number of 2000. For the same
Reynolds number however, the conical tip showed a 15% increase in dynamic pressure error,
while the hemispherical tip only showed a 5% increase in error. For this reason, the
hemispherical tip is recommended for measuring flows in this range of Reynolds numbers.
121
7.3 Order of Polynomial Curve Fit
A fourth order bivariant polynomial surface model, with 15 degrees of freedom, was able to
accurately capture the important features of the probe’s response curve. Residual plots showed
no trend when a fourth order fit was applied, indicating that the only source of error was noise in
the calibration data. A third order polynomial fit with 10 terms was observed to leave a residual
with a parabolic shape, indicating that there were physical features of the response that were not
adequately modeled with a third order surface fit. For this reason, the fourth order terms were not
found to be negligible, and a fourth order curve is recommended.
7.4 Calibration Grid Requirements
A calibration grid with points spaced at 5° in cone and roll was found to sufficiently resolve the
response curve across all sectors. With grid spacing greater than 5°, the resolution in the middle
of the sectors was not sufficient to resolve and represent the important features of the probe
response, and the resulting curve fits were therefore not properly modeling the physical response.
7.5 Overlap Pressure
An overlap pressure of between 15 and 20% in the high angle sectors was found to give the most
uniform response, with the lowest absolute maximum errors. The application of overlap pressure
in the high angle sectors was shown to reduce absolute maximum errors by up to 4% of the flow
dynamic pressure. There was virtually no effect on the RMS average error. Overlap pressure
was not shown to improve the accuracy of low-angle sector calibrations.
122
7.6 Quantification of Error in Previous Works
It was found that the uncertainty in static pressure reported in the works of Chen [2]and Maqsood
[3] appears to be understated. Errors of up to 5% in the determination of calibration flow
reference pressures and of 5-10% due to Reynolds number effects were not acknowledged. These
additional uncertainties explain the static pressure readings in previous works.
123
Chapter 8
Recommendations and Limitations
There are a few limitations of the present work, and of seven hole probes that are recognized and
acknowledged in this section. Recommendations for improvements to the calibration process and
equipment are also discussed in this section.
The study of overlap pressure considered only a single speed of calibration flow. The conclusion
that 15-20% overlap pressure is ideal is therefore only demonstrated for a calibration jet flow
issuing to atmosphere with a 330 Pa total pressure. Measured pressures scale with the flow
dynamic pressure, however, so it is reasonable to assume that this recommended value will
remain valid for a wide range of calibration flows. Care must still be taken, however, through
examination of plots of the points included in each calibration region, to ensure that an
unreasonable number of additional points are being included.
The results presented are only applicable to calibrations and testing in subsonic flow. The effects
of compressibility were ignored in the present work, and can be significant at Mach numbers
greater than 0.3. Compressible flows must be tested and modeled with additional coefficients and
additional tip geometry considerations.
The effect of flow turbulence was not considered or quantified in the present work. Significant
turbulence is known to affect the accuracy of pressure measurements, but those effects were not
measured or quantified. The agreement between observed errors and repeatability of calibration
124
with the literature suggests that freestream turbulence did not significantly affect the results. It is
recommended, however, that a settling chamber be added to the calibration wind tunnel for the
purposes of turbulence suppression so that this can be verified.
In terms of seven hole probes, it is important to recognize that the probes themselves are limited
in their capability. The 3.87mm size of the probe means that small scale flow structures on the
order of the probe size cannot be measured accurately. Seven hole probes also cannot be used to
accurately measure boundary layers or significant shear layers. Any gradient large enough that
the change in the flow across the thickness of the probe is significant cannot be accurately
measured, as the seven measured pressures do not represent the flow at a particular point. The
frequency response of the probes in the current work is very low, so transient data is unreliable –
these probes in their current state may only be used for the measurement of time-averaged mean
properties.
125
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[33]
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[36]
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129
Appendix A
Experimental Apparatus and Calibration
A.1 Pressure Transducers
Two types of transducers were used for collecting pressure data – the Omega PX139-001D4V
and the Omega PX143-2.5BD5V. These models are both differential transducers, with +/- 1 psi
and +/- 2.5 psi ranges, respectively. The transducer that was used for each measurement was
chosen based on the magnitude of the pressure that was being measured.
Transducers were arranged in boxes of eight transducers with a 25-pin connector to attach the
unit to the data acquisition system. A typical transducer box is shown in Figure A-1.
Figure A-1: Typical Transducer Casing Arrangement
130
The transducers were calibrated using a water manometer, shown in Figure A-2. One end of the
manometer was left open to atmosphere, while the other was connected to all eight of the
transducers in each box. The tube was pressurized and the pressure and output voltage were
measured and recorded. The transducers were calibrated through their entire rated operating
range with 14 data points, including three readings at zero pressure to confirm repeatability. A
linear curve fit was then applied to the transducer output.
Figure A-2: Pressure Transducer Calibration Arrangement
A linear fit was chosen over a higher order polynomial curve fit for two main reasons. The first is
that the non-linearity of the transducer is rated at 0.75% of full scale, so the error in a linear fit is
well quantified. The second is that applying a higher-order curve fit to a data set that is expected
to be linear results in unnecessary degrees of freedom in the fit, which makes the result highly
susceptible to outlier data points. The higher order curve fits were found to be sensitive to small
errors in manometer reading during calibration. It was also found that the difference between a
131
linear and a cubic fit for the worst transducer was less than the rated hysteresis error of the
transducer – meaning that the choice of curve fit was not significant.
The error in the transducer output was taken to be the manufacturer’s reported value for nonlinearity, which was 0.75% of the full scale output. A hysteresis error was also provided,
however this error was ignored because several thousand data points were averaged each time a
pressure was measured. The number of points that were averaged was studied for independence,
and was increased until the result was independent of the number of points – that is, until the net
effect of random error was negligible. The hysteresis error was therefore systemically
suppressed, and the error in the transducer output is only due to the non-linearity of the
calibration. Error due to atmospheric conditions was controlled by recalibration when there were
significant changes in ambient temperature and humidity.
A.2 X-Y Traverse Tables
An XY traverse table manufactured by Arrick robotics was used to position the 7-hole probe in
the flow. The unit is pictured in Figure A-3.
132
Figure A-3: XY-Positioning Traverse Rig
The X and Y position of the probe was controlled with VEXPA 12V stepper motors with a 1.8°
step size. A pulley reducer was used on the motor controlling Y-position to increase torque. The
motors were controlled with a Salem Controls A200SMC stepper motor controller and a
LabView program.
The traverse was calibrated by moving the motor a set number of steps and measuring the
resulting displacement of the probe. The constants were found to be 0.000254m/step in the X-
133
direction and 0.0000635m/step in the Y-direction. The movements of the motor were found to be
repeatable to within 1mm/1000 steps in both directions.
The repeatability was confirmed by traversing the outlet of a 30 cm x 40 cm duct in a square 1 cm
x 1 cm pattern. This required over 375,000 steps in the Y-direction. When the probe was
returned to 0, it was within 3 mm of its original position. It required over 3000 steps in the Xdirection to complete this traverse, and the probe returned to within 0.5mm of its original
position.
A.3 Sampling Period Sensitivity
The appropriate sampling period was determined by taking transient pressure data at 900 Hz, the
maximum sampling rate of the data acquisition system, and measuring the response of the seven
pressures to a significant change in flow angle. The seven transient pressure profiles were then
plotted and a settling time was determined qualitatively. The settling time is the length of time
that is allowed after the probe moves, but before sampling is started. The sampling period was
then determined by plotting the change in the moving average of the seven pressures. This plot is
shown in Figure A-4.
134
Change in Moving Average Pressure (Pa)
-100
P1
P2
P3
P4
P5
P6
P7
-50
0
50
100
0
1
2
3
4
5
Time (s)
Figure A-4: Transient Pressure Response to a 45 Degree Change in Flow Angle
A settling time of 1.5 seconds was chosen because the plot shows that the most significant peaks
and fluctuations have ended by this time. A sampling period of 1 second was chosen because the
change in moving average has decreased to less than 0.005% per additional sample by this time.
135
Appendix B
Using Seven Hole Probes in the Gas Turbine Lab
This appendix is intended as a primer for new students and researchers in the Gas Turbine Lab.
The intent of this section is to introduce the specifics of the use of the computer codes and probes
that are used in the lab.
There are several sections of the body of this thesis that should be reviewed and fully understood
before taking data or calibrating probes. Sections 2.1 through 2.4.1 of the theory and literature
review chapter introduce all of the governing equations and formulations that are used in the
calibration and processing of seven hole data. Understanding these equations and parameters is
critical to understanding the way that the seven hole probe deals with data, and thus
troubleshooting and interpreting experimental data. Sections 2.6 through 2.8 should also be
reviewed, as they deal with the effects of parameters that were not studied or included in this
work – Mach number, flow turbulence, and velocity gradients.
In terms of using seven hole probes in the HGWT, the most important effect that heating the
probe has on probe response is to drop the Reynolds number significantly. The Reynolds number
of the hot flow will typically be an order of magnitude lower than the cold flow generated by the
same inlet restriction and exhaust device. Knowing the Reynolds number of the flow under test,
and knowing the Reynolds number that was used during calibration are therefore paramount.
Section 5.2.2 of this work should be reviewed for a discussion of how Reynolds number affects
probe response at different angles of attack.
136
It is recommended that all of the aforementioned reading is completed before using seven hole
probes – those readings are applicable to everyone using seven hole probes. The following
subsections describe or reference more specific tasks to do with seven hole probe use and
calibration.
B.1 Assembly and Manufacture
Part drawings and a high level assembly drawing for the probes used in the GTL are provided in
Appendix C. These drawings are nearly complete; they do not, however, describe the process of
installing the pressure tubes and bending the probe neck.
The probe tip must be machined. The machining of the tip is very difficult, because stainless is a
difficult material to cut, and small diameter drill bits are quite delicate. The lead time on probe
tips is therefore quite significant. The only other machining that must be done is to make the
collars, and to ream the probe stem to accept the collars.
Once the probe tip is machined, small diameter stainless tubes are inserted into each of the seven
counterbored holes on the backside of the probe tip. These tubes must be carefully brazed to the
probe tip, using Braze 380 and the appropriate flux. Care must be taken to ensure that the
connection is airtight, but at the same time, that the tubes do not become plugged with solder.
The probe tip is then inserted into the Tip Holder, and the resulting joint is again sealed with
Braze 380. Any step or bulge that results from the soldering should be carefully filed and sanded
away. The Tip Holder is then soldered to the Probe Neck, again filling the resulting shoulder to a
smooth chamfer.
137
A tubing bender is then used to bend the Probe Neck. Care must be taken during this step to
ensure that the pressure tubes inside the probe do not crack. It may be necessary to use a torch to
heat the bend.
Once the neck is bent, the probe can be installed in the stem. The neck of the probe should be
polished at the stem end so that the collars slide relatively smoothly over the neck. The slit collar
is installed first. The bottom set of setscrews is then installed in the probe stem, protruding very
slightly into the inside face. The probe neck is inserted into the probe stem until the bottom collar
contacts the setscrews. Ensure that the slit collar does not enter the probe stem yet. The neck can
then be pushed into the probe stem to the desired depth. When the neck is at the desired depth,
the slit collar is pushed into the stem until it is underneath the first set of setscrews. The first set
of setscrews are then installed and tightened, followed by the bottom set of setscrews. The slit
collar should be captured by the setscrews, and pinned from sliding axially. The second collar is
held in place between the slit collar and the bottom setscrews. It may have up to 1/8” of axial
slop, but this is acceptable.
Once the probe is assembled, the tubes should be labeled with their corresponding hole number.
The holes can be identified by putting the tip in a dish of water and blowing gently through each
tube.
B.2 Storage and Handling
Seven hole probes are very sensitive to damage, and damage to probe tips is often difficult to
diagnose or recognize because it will not cause step changes in response. Rather, a slight offset
138
or a slowly varying error will be introduced, and these errors may not be immediately apparent
from contour plots.
The tip should be protected from marring or chipping when not in use. Small scratches and
abrasions can cause changes in the nature of downstream flow separation, which can cause errors
at high angles of attack. Larger chips and defects can trigger sudden separation over peripheral
holes, which will render the data meaningless. It is recommended that a soft cap, such as a
silicone tube, is placed over the probe tip when not in use.
The probe neck must also be handled carefully, as it can be bent. Bending the probe neck will
cause a bias error in the calculation of the pitch angle. This type of error is difficult to detect,
because the probe neck bend angle is not typically consistent between probes. Seven hole probes
should be handled by gripping the mounting stem, and should not be hung from the probe neck
when stored.
B.3 Calibration
The procedure for mounting a probe in the rotary traverse is described in detail in section 4.1.
Note that there are two sets of arms for the traverse, one is for short probes, and one is for longer
probes. Following the procedures for the mechanical alignment of the probe in the rotary
traverse, and the alignment of the rotary traverse with the wind tunnel is critical, as the accuracy
of the calibration depends on aligning the probe tip with the flow, and on minimizing translation
of the tip during rotation.
139
The generation of a calibration grid is done using two programs. “CalGrid.exe” generates a text
file containing all of the calibration points within the specified limit, and at the specified grid
spacing. The output of this program is “xy.txt”, which must be imported into excel and sorted to
minimize the number of traverse movements. This file should be sorted by pitch angle, then by
yaw angle. Once the sorted list of points is saved, “AddZeros.exe” should be run to add (0,0)
points to the file periodically. As outlined in section 4.2.1.3, these (0,0) points are necessary to
determine the average flow total pressure during the calibration.
Once the probe is aligned and the calibration grid is established, LABView is used to run the
rotary traverse, and to collect pressure data at each calibration point. The LABView programs
will generate text files that contain the pitch and yaw angles of each calibration point, along with
the corresponding seven port pressures, and an eighth reference static pressure. Once data is
collected, a calibration is generated using the following procedure:
1. Run PressureCoefficients.exe. This program generates data files that characterize the
performance of each pair of holes on the probe as a yaw meter. These data files are
formatted for direct import to Tecplot 360, though they can also be loaded into excel for
visualization. Plot the data generated by this program to ensure that burrs or defects in
the tip are not tripping flow separation – see section 5.2.1.
2. Run MakeCorrelateData.exe. This program sorts the raw calibration data into data files
for each individual sector. The sorting and rejection criteria discussed in section 4.2.1 are
implemented here.
3. Run CorrelateM3.exe. This program generates the calibration coefficients for each
sector.
140
4. Run ConverteM3.exe, using the calibration data the input pressure data. This program
calculates flow properties from the raw pressure data.
5. Run AngleConvert.exe, using the results file as the input. This program calculates the
error in the calculated flow properties. The error files that are generated are formatted for
direct import to Tecplot 360.
6. Examine the average and maximum errors in the calibration, and plot the calibration
errors in terms of pitch and yaw angle on a 2-D contour plot. Look for spikes in error
around the sector boundaries. Large increases in error around sector boundaries may
indicate discontinuities in neighbouring calibration surfaces. Adjusting the overlap
pressure may improve calibration accuracy.
7. For improved confidence in calibration accuracy, validate the calibration by collecting
additional pressure data and repeating steps 4 through 6.
MakeCorrelateData.exe is the only program that interfaces with the Labview generated files. If a
different DAQ system or a different Labview interface is used, the output should be formatted as
follows:
•
No header lines
•
10 columns, tab-delimited
•
YawT, Pitch, Pressures 1-7, reference flow static pressure
The formatting of the files that are passed between the executable programs described above is
controlled by those programs and is independent of the data collection interface.
141
B.4 Processing of Flow Data
Seven hole data from measurements of an arbitrary flow can be converted to flow properties
simply by running ConverteM3.exe. This program requires that the files containing the
calibration coefficients for the probe that was used are in the same folder as the raw pressure data.
Programs are also available to generate traverse grids, and to perform useful integrations of the
converted pressure data. “OutletTraverse.exe” generates a set of points that will traverse a
specified size of circular or rectangular outlet at a specified uniform grid interval.
“ExtensiveProperties.exe” will process a results file and compute useful bulk flow properties,
such as total mass flow, and mass averaged velocity and temperature.
142
Appendix C
Shop Drawings
143
144
1
1
2
1
3
1
A
6
1
4
1
Align Holes as Shown
when Soldering
Drawing Notes:
1) Connect all Inserted Parts with Braze 380 and
Appropriate Flux
2) Use 10-24UNC Setscrews to retain collars
3) Perform Bending of Probe Neck after Soldering
of Pressure Tubes
4) 7 x Stainless Steel Tubes are Silver Soldered in
the Back of the Probe Tip and Passed Thru Other
Parts
5) File Brazed Connections Smooth and Clean Away
Residual Flux
DETAIL A
PROBE-002
PROBE-003
PROBE-004
PROBE-005
PROBE-006
3
4
5*
6
PROBE-001
1
2
Document
Number
Item
Number
Setscrews - One Set
contacts the slit collar, the
other is below the collar,
preventing it from sliding
down
Probe Stem
Collar
Slit Collar
Probe Neck
Tip Holder
Probe Tip
REVISION HISTORY
DESCRIPTION
Title
DATE
B
B
1
1
1
1
1
1
Quantity
APPROVED
Queen's University
NAME
DATE
DRAWN
jcrawford 03/09/11
Gas Turbine Lab
CHECKED
TITLE
ENG APPR
Probe Assembly
MGR APPR
SIZE DWG NO
REV
UNLESS OTHERWISE SPECIFIED
A4
PROBE-101
DIMENSIONS ARE IN INCHES
FILE NAME: PROBE-101 - Probe Assembly.dft
ANGLES ±0.5°
0 PL± 0.1 1 PL± 0. 2PL ±0.05 3 PL ±0.005 SCALE:
WEIGHT:
SHEET 1 OF 1
SECTION B-B
Collar
Slit Collar
REV
145
Polish These Faces
60°
TYP
7 x O .02300 THRU ALL
c O 0.03550 ` 0.10
Round These Edges
Drawing Notes:
1) Material: Stainless Steel
2) Break Sharp Edges
3) Hand File Chamfer Edges and Polish Indicated
Surfaces on Lathe with Emery Cloth
.45250
25°
.11000
DATE
APPROVED
Queen's University
A
A
SECTION A-A
10:1
REVISION HISTORY
DESCRIPTION
NAME
DATE
DRAWN
jcrawford 03/09/11
Gas Turbine Lab
CHECKED
TITLE
ENG APPR
Probe Tip
MGR APPR
SIZE DWG NO
REV
UNLESS OTHERWISE SPECIFIED
A4
PROBE-001
1
DIMENSIONS ARE IN INCHES
FILE
NAME:
PROBE-001
Probe
Tip.dft
ANGLES ±0.5°
0 PL± 0.1 1 PL± 0. 2PL ±0.05 3 PL ±0.005 SCALE:
WEIGHT:
SHEET 1 OF 1
.00000
.06400 .07400
.04088
REV
146
Drawing Notes:
1) Material: Stainless Steel
2) Make from COTS 3/16" OD x 1/8" ID Tubing
3) Break Sharp Edges
4) Leave no Burrs or Flashing
O .188
O .125
1.500
DATE
APPROVED
Queen's University
REVISION HISTORY
DESCRIPTION
NAME
DATE
DRAWN
jcrawford 03/09/11
Gas Turbine Lab
CHECKED
TITLE
ENG APPR
Tip Holder
MGR APPR
SIZE DWG NO
REV
UNLESS OTHERWISE SPECIFIED
A4
PROBE-002
1
DIMENSIONS ARE IN INCHES
FILE
NAME:
PROBE-002
Tip
Holder.dft
ANGLES ±0.5°
0 PL± 0.1 1 PL± 0. 2PL ±0.05 3 PL ±0.005 SCALE:
WEIGHT:
SHEET 1 OF 1
REV
147
Drawing Notes:
1) Material: Stainless Steel
2) Make from COTS 1/4" OD x 0.035" Wall Tubing
3) Break Sharp Edges
4) Leave no Burrs or Flashing
5) Bend After Assembly
6) Unbent length 10"
R .875
MIN
8.125
DATE
APPROVED
Queen's University
2.125
REVISION HISTORY
DESCRIPTION
NAME
DATE
DRAWN
jcrawford 03/09/11
Gas Turbine Lab
CHECKED
TITLE
ENG APPR
Probe Neck
MGR APPR
SIZE DWG NO
REV
UNLESS OTHERWISE SPECIFIED
A4
PROBE-003
1
DIMENSIONS ARE IN INCHES
FILE
NAME:
PROBE-003
Probe
Neck.dft
ANGLES ±0.5°
0 PL± 0.1 1 PL± 0. 2PL ±0.05 3 PL ±0.005 SCALE:
WEIGHT:
SHEET 1 OF 1
Sand and Polish at
least 3" Length to
Sliding Fit with
0.250/0.255 ID Collar
O .188
O .250
REV
148
Drawing Notes:
1) Material: Stainless Steel
2) Make from PROBE-005 - Collar - See PROBE-005 for
dimsensions
3) Break Sharp Edges
4) Leave no Burrs or Flashing
5) Slit can be hand-cut
.022
MIN
.500
REVISION HISTORY
DESCRIPTION
DATE
APPROVED
Queen's University
NAME
DATE
DRAWN
jcrawford 03/09/11
Gas Turbine Lab
CHECKED
TITLE
ENG APPR
Slit Collar
MGR APPR
SIZE DWG NO
REV
UNLESS OTHERWISE SPECIFIED
A4
PROBE-004
1
DIMENSIONS ARE IN INCHES
FILE
NAME:
PROBE-004
Slit
Collar.dft
ANGLES ±0.5°
0 PL± 0.1 1 PL± 0. 2PL ±0.05 3 PL ±0.005 SCALE:
WEIGHT:
SHEET 1 OF 1
REV
149
Drawing Notes:
1) Material: Stainless Steel
2) Break Sharp Edges
3) Leave no Burrs or Flashing
O .255
.250
O .375
.370
.500
REVISION HISTORY
DESCRIPTION
DATE
APPROVED
Queen's University
NAME
DATE
DRAWN
jcrawford 03/09/11
Gas Turbine Lab
CHECKED
TITLE
ENG APPR
Collar
MGR APPR
SIZE DWG NO
REV
UNLESS OTHERWISE SPECIFIED
A4
PROBE-005
1
DIMENSIONS ARE IN INCHES
FILE
NAME:
PROBE-005
Collar.dft
ANGLES ±0.5°
0 PL± 0.1 1 PL± 0. 2PL ±0.05 3 PL ±0.005 SCALE:
WEIGHT:
SHEET 1 OF 1
REV
150
SECTION B-B
120°
TYP
O .375
O .500
Drawing Notes:
1) Material: Stainless Steel
2) Make from COTS 1/2" OD x 3/8" ID Tubing
3) Break Sharp Edges
4) Leave no Burrs or Flashing
Ream to 0.375
to at least 2"
Depth after
Tapping Holes
24
B
DATE
.25
#10-24 UNC
THRU WALL
6 PLCS
APPROVED
Queen's University
DETAIL A
REVISION HISTORY
DESCRIPTION
NAME
DATE
DRAWN
jcrawford 03/09/11
Gas Turbine Lab
CHECKED
TITLE
ENG APPR
Probe Stem
MGR APPR
SIZE DWG NO
REV
UNLESS OTHERWISE SPECIFIED
A4
PROBE-006
1
DIMENSIONS ARE IN INCHES
FILE
NAME:
PROBE-006
Probe
Stem.dft
ANGLES ±0.5°
0 PL± 0.1 1 PL± 0. 2PL ±0.05 3 PL ±0.005 SCALE:
WEIGHT:
SHEET 1 OF 1
A
B
REV
Appendix D
Measurement and Characterization of Calibration Wind
Tunnel Outlet Flow
The wind tunnel that was used for the present study was constructed specifically for this project,
so there was a need to characterize the outlet flow – its uniformity, swirl, and steadiness.
Equipment for measuring flow turbulence was n\ot available, so the blower was reconfigured to
generate a jet on the suction side that would have inherently low turbulence, and the results were
compared with the high turbulence calibration flow that was used during data collection.
D.1 Swirl Characterization
The simplest way to characterize flow swirl was to simply traverse the outlet plane and analyze
the results. This was not straightforward, however, as the results of the traverse depended on the
probe calibration. An existing wind tunnel with known axial outlet flow [33] was used to obtain a
calibration for verification purposes. This wind tunnel was much larger, however, providing a
calibration flow with a Reynolds number of 33000. The Reynolds number of the new wind
tunnel’s outlet flow during the traverse was 5500– an order of magnitude lower. As discussed in
section 7.2, however, both of these Reynolds numbers were sufficiently high that both flows were
Reynolds number independent.
The results of the traverse are shown in Figure D-1. The vectors are normalized by the axial
velocity, so the vectors represent a total flow angle.
151
10 % Axial Velocity
Figure D-1: Wind Tunnel Outlet Secondary Flow Vectors
The vector fields were very uniform, and the offset in velocity was the same direction. This
indicated that the error was likely due to misalignment of the probe in the traversing apparatus,
rather than due to a swirling outlet flow. If the outlet flow were swirling, the vector fields would
have been larger on one side than the other, and they are not. The bias error resulting from probe
misalignment was removed from the dataset by subtracting the mean secondary vector from each
point. The resulting vector field, shown in Figure D-2, shows no indication of significant swirl or
asymmetry in the calibration flow.
1 % Axial Velocity
Figure D-2: Unbiased Wind Tunnel Outlet Secondary Flow Vectors
152
D.1.1 Direct Port Pressure Comparison
Another check to verify that the oncoming flow was axial was to directly compare 7-hole
pressure data taken from the same point in the exit plane, but with the stem held in two different
orientations. For a perfectly axial flow, the response of the probe was independent of the stem
orientation – the pressure at each of the seven ports was the same with the stem left of the tip and
the stem above the tip. The measured response of the probe with the stem held in these two
positions was shown in Table D-1. Pressures were normalized by the flow dynamic pressure,
which was 333 Pa.
Table D-1: Effect of Stem Position on Probe Response
Port 1
Port 2
Port 3
Port 4
Port 5
Port 6
Port 7
Stem Left (%)
100%
22%
36%
32%
32%
28%
25%
Stem Above (%)
99%
24%
36%
29%
38%
32%
29%
Difference (%)
1%
-2%
0%
3%
-6%
-2%
-4%
There was very little change in the measured pressures. A change in apparent flow direction
would have manifested itself in these results as a change in the difference between diametrically
opposite pressure port pairs - ports 2 and 3, 6 and 5, and 4 and 7. There was not a significant
trend here – the difference between 6 and 5 and 4 and 7, which were both indicators of yaw angle,
changes in opposite directions – indicating that there is no discernable change in yaw angle.. The
changes are also within transducer error, which is 20 Pa. This further confirms that the flow over
the probe tip can be considered axial for the present study.
153
D.2 Static Pressure Profile Uniformity
The uniformity of the outlet static pressure profiles was investigated using traverse data. The
uniformity of the static pressure profile was important because reference measurements were
taken at a different point on the outlet plane from the probe tip, and thus there was a possibility of
error. Figure D-3 shows static pressure contours taken from the outlet traverse
Ps, Pa
30
20
10
0
-10
-20
-30
Figure D-3: Wind Tunnel Outlet Static Pressure Contours
The contours of static pressure show that the flow static pressure is quite uniform, and thus that
there was very little streamline curvature at the jet exit. A small variation in space was observed,
but this variation was random and thus attributed to noise.
D.3 Flow Development
The flow was driven by a centrifugal blower, which is known to produce a non-uniform outlet
flow. It was therefore necessary to investigate the development of the flow at the pipe exit to
ensure that these asymmetries were not significantly affecting the flow. This was investigated by
comparing velocity data with data from Sharan [34] for pipe flow development after 27
154
diameters. Velocity profiles on horizontal and vertical rakes were compared with Sharan in
Figure D-4. The wind tunnel in the present study had a length of 30 diameters.
Exp - Vertical Rake
Exp - Horizontal Rake
Sharan
U/UMass Average
1
0.8
0.6
0.4
0.2
0
0.2
0.4
0.6
0.8
1
y/R
Figure D-4: Validation of Flow Development
The results show good agreement between the expected and observed velocity profile. There was
a slight asymmetry observed in the vertical rake of the wind tunnel, but the horizontal rake was
quite symmetrical. The flow was developing as expected, and the assumption that the outlet flow
was developing normally was reasonable.
D.4 Wind Tunnel Drift and Unsteadiness
Extremely low frequency flow unsteadiness was quantified using a subset of calibration data.
The traverse returns to a zero pitch, zero yaw position periodically during calibration to take flow
reference measurements. These measurements were extracted from a calibration that was
155
performed over an 8 hour period. The measured pressure at each port is plotted against time in
Figure D-5.
350
300
Measured Pressure (Pa)
P1
P2
P3
P4
P5
P6
P7
PStatic
250
200
150
100
50
0
Figure D-5: Variation of Measured Pressure over an 8 Hour Period
The spike that appears was the result of a significant change in temperature in the lab. A door
was opened and the HGWT was started, which changed the temperature of the room from about
20°C to about 5°C. This temperature change causes a zero-offset error. The transducers were rezeroed at the new ambient temperature, and the tests were continued. The results show no
discernable time-based variation in the observed flow pressure – that is, the unsteadiness in the
flow is random. The maximum large scale unsteadiness of the flow was calculated to be 1.72%
on a 95% confidence interval. It should be noted that this 1.72% includes the effects of random
transducer error - the actual unsteadiness of the flow is lower.
156
D.5 Turbulence Effects
While it was not possible to measure the turbulence in the calibration flow directly, the effect of
turbulence was investigated by using wires and screens to trip additional turbulence in the
calibration flow. A suction-type wind tunnel was constructed for this test, and different screens
were installed over the inlet to introduce various levels of turbulence. The test chamber is shown
in Figure D-6. The inlet with no screens attached is shown in Figure D-7.
Figure D-6: Suction-type Wind Tunnel for Turbulence Testing
157
Figure D-7: Test Chamber Bellmouth Inlet
Turbulence was generated using three different types of mesh. The details of each mesh are
given in Table D-2. The coarse chicken wire mesh was folded three times to increase wire
density.
158
Table D-2: Dimensions of Turbulence Screens
Wire Size (mm)
Gap Dimensions
(mm)
0.7
27
0.8
5.7
2.25
8.5 - 22
Chicken Wire
(Sparse)
Chicken Wire
(Dense)
Expanded Metal
159
Picture
The probe was calibrated using a data set that was collected with no screen on the inlet. An
additional 206 data points that were spaced a uniform 10° in cone and roll were then collected
with no screen, and with each of the screens shown. The RMS average and absolute maximum
errors in each of the data sets was calculated and is presented in Table D-3 and Table D-4. The
same errors were also calculated for a set of data that was collected from the blown wind tunnel,
and these errors are shown for comparison purposes. Pressure errors are normalized by the flow
dynamic pressure.
Table D-3: RMS Average Errors at Different Turbulence Levels
Yaw (°)
1.7
Chicken Wire
(Coarse)
1.6
Pitch (°)
1.9
1.7
1.4
1.3
0.86
PTotal (%)
2.0
2.3
7.6
5.7
2.2
PDynamic (%)
3.9
3.6
9.0
7.4
4.4
PStatic (%)
3.2
3.2
4.4
4.8
3.5
No Screen
Chicken Wire
(Dense)
6.9
Expanded
Metal
8.4
Blown Wind
Tunnel
2.0
Table D-4: Absolute Maximum Errors at Different Turbulence Levels
Yaw (°)
5.4
Chicken Wire
(Coarse)
5.0
Pitch (°)
6.0
5.0
4.3
3.8
2.6
PTotal (%)
12
14
15
13
6.2
PDynamic (%)
14
13
27
24
14
PStatic (%)
13
17
26
26
12
No Screen
Chicken Wire
(Dense)
9.6
Expanded
Metal
12
Blown Wind
Tunnel
4.9
160
The results showed that increasing turbulence by using larger, denser screens, reduces the
accuracy of the calibration. Based on the observed RMS error, the turbulence level in the blown
wind tunnel was approximated to be equal to the turbulence generated by the coarse chicken wire
grid. No measurements were performed on turbulence level, scale, or energy distribution.
Though it was difficult to quantify the turbulence generated by a grid, there were a number of
design guidance points for the implementation of turbulence generating grids available in the
literature. The installation of a screen had two main effects – firstly, to limit the scale of the
largest eddies in the flow to the largest gap diameter, and secondly, to generate eddies on the
scale of the screen wire diameter. The inlet flow was stagnant, so the effect of the gap size was
minimal in this case – there were no large scale eddies in the inlet flow. The most important
feature of the screens was therefore the wire diameter.
In terms of the wire diameter, the transition from laminar to turbulent wakes from the wires
occurred when the Reynolds number based on wire diameter was between 40 and 80 [35],[36].
The Reynolds number was much greater than 80 for all of the screens in the present study. The
turbulence generated by a grid has also been shown to become isotropic at between 20 and 40 gap
lengths downstream of the grid [35],[36]. The coarse chicken wire, which had the largest gaps,
was 23 grid diameters upstream of the calibrator, which means that the turbulence at the probe tip
could be considered virtually isotropic for all test cases. Beyond the point at which the
turbulence became isotropic, the decay has been found to be linear [35], and the exponent of the
standard power law decay rate for axial turbulence can be taken as approximately 1. The
turbulence intensity would then be given by an equation of the form of Equation (D-1) [35].
161
2
 U 
 x '− x '0 

 = C

 u RMS 
 M 
n
(D-1)
The coefficients C and n can then be determined from empirical data, such as that found in Gadel-Hak [37]. The difficulty with this was that the value of the coefficient C was a function of the
geometrical solidity, and of the Reynolds number of the flow through the grid gaps. The
coefficient C was very sensitive to variations in both the solidity and the Reynolds number, and it
was not possible to find empirical data at the solidities and Reynolds numbers of the flows
described in this section.
162