Development of a

Directional Flow Probe

for the Hypersonic Regime

by

William Thomas O'Gorman

A thesis subrnitted in conforrnity with the requirements for the degree of Master of Applied Science Graduate Department of Aerospace Studies

University of Toronto

O Copyright by William Thomas O'Gorman, 1997

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Flow Probe for the Hypersonic Regime

William Thomas OYGorman Master of Applied Science, 1997

Aerospace Science and Engineering, University of Toronto

Abstract

The hypersonic gun tunnel facilities at the University of Toronto Institute of

Aerospace Studies were used to develop a probe capable of determining the velocity at a

point within hypersonic flows. Investigations into the flow characteristics of the suggested

probe demonstrated that the desired flow structure was established but that some

unknown fiow characteristics around the probe head may be present. A numerical solver

was developed which uses the pressures measured by the probe to calculate the incident

angles of attack and the Mach nurnber of the flow. The nature of the system of equations

makes the solution particularly sensitive to small perturbations in the relationship between

the measured pressures. It was possible to successfully obtain pressure measurernents that

were consistent with theoretical expectations. Recommendations for further development

and refinement of the probe are included.

I would Iike to thank the following people for their contributions to this research

projec t :

- Dr. Philip Sullivan for his guidance and assistance as my supervisor.

- Doug Challenger for his patience in teaching me the operation procedures

for the wind tunnel.

- Andy Logan for his expert machining talents in constructing the models.

- the Staff of UTIAS; particularly Nora Burnett, Gai1 Holiwell, Peter Miras,

Tony Roberts, Ida Abert, and Clara Chang whose generosity allowed me to avoid

many obstacles.

- the other students that helped keep me sane during lunch breaks.

Your assistance was instrumental.

Thank y k i n d l y O

Table of Contents

. . Abstract ........................................................................................................ il

... Acknowledgments ........................................................................................ 111

Table of Contents ......................................................................................... iv . . List of Figures ............................................................................................. vu

List of Tables ............................................................................................... ix

Nomenclature ................................................................................................ x

1 . 0 Introduction ............................................................................................. 1

1.1 Overview ............................................................................................. 1

1.2 Literature Review ................................................................................ 4

...................................................................................... 1.2.1 Duct Flow 4

..................................................... 1.2.2 In-stream Measurement Probes -7 . .

2.0 Experimental Facility ........................................................................... 1 0

3 . 0 Prelirninary Probe Development ..................................................... 1 3

3.1 Sumrnary ............................................................................................ 13

3 -2 Theory .............................................................................................. -14

.............................................................................................. 3.3 ResuIts -18

4.0 Feasibility ............................................................................................. -20

................................................................ 4.1 Angled Probe Investigation 20

4.1.1 Surnmary ..................................................................................... 20

4 . 1.2 Theory ......................................................................................... 21

4.1.3 Results ........................................................................................ -23

7 . A I IUUU r ~ l u y ILIVGZ~LII;ULIUII ................................................................... L3

4.2.1 S u m a r y ..................................................................................... 23

........................................................................................ 4.2.2 Results -24

5 . 0 Numerical Investigation ........................................................................ -27

........................................................................................... 5.1 Summary -27

.............................................................................................. 5.2 Theory -29

5.3 Two Dimensional Solver ................................................................... -34

5.4 Three Dimensional Solver .................................................................. 40

............................................................................................... 6.0 Prototype 43

............................................................................................ 6.1 Surnmary 43

..................................................................................... 6.2 Investigation -43

........................................................................................ 6.3 Calibration -45

6.3.1 Variable a .................................................................................. -45

6.3.2 Variable P ................................................................................... -48

................................................................................. 6.3.3 Verification -51

.............................................................................. 6.3.4 Interpretation -52

............................................................................... 6.4 Shock Generator -54

...................................................................................... 6.5 Tube Length -56

.......................................................................................... 7.0 Conclusions -59

8.0 Recomrnendations ................................................................................ -60

........................................................................................... 9.0 References -61

. ............... ..a...r... 7.. .==....... ...- ............ - -1.1 - ......-......................*.. a A"

Appendix B : Apparatus for Preliminary Probe Development Tests .......... - 1 14

Appendix C: Derivation of System of Equations ....................................... 117

Appendix D: Sample MathCad@ Document ............................................ 125

Appendix E: 2D Probe Code Listing ........................................................ 130

Appendix F: Flowchart of 3D Solver .................................................. 139

Appendix G: 3D Probe Code Listing ........................................................ 144

Appendix H: Prototype Drawings ............................................................. 157

List of Figures

Figure 1:

Figure 2:

Figure 3:

Figure 4:

Figure 5:

Figure 6:

Figure 7:

......... Flow Phenomena Discovered by Davis and Gessner (1989) 63

.............................................................. Transverse Flow Probes 64

.................................................. UTIAS Hypersonic Gun Tunnel -65

Standard Pressure History ............................................................ 66

.................................................... Standard Schlieren Photograph 67

.................................................................. Wedge Mount Design 69

................................................................. Sketch of Probe Types 70

Figure 8: Schlieren Photograph of Preliminary

............................... Probe Development Apparatus 71

.............................................. Figure 9: Notation for Normal Shock-Waves 72

Figure 10:

Figure 1 1 :

Figure 12:

Figure 13:

Figure 14:

Figure 15:

Figure 16:

Figure 17:

Figure 18:

.......................................... Notation for Oblique Shock-Waves -73

...................... Schlieren Photograph of Angled Probe Apparatus 75

........................................................................ Angle Relations -76

............................................................. Angled Probe Pressures 78

.................................................................. Prototype Apparatus 79

................................ Schlieren Photograph of Prototype ($ < 0°) 81

Schlieren Photograph of Prototype ($ > 0°) ............................... 83

.............................................................. Suggested Probe Arrays 84

............................................................ Probe Array Conventions 85

.................................................................. Figure 19: Vector Components -86

Figure 20: Probe Tip Relationships .............................................................. 87

vii

Figure 21: El Solution Space for 2D Probe ................................................. 88

................................................. Figure 22: E2 Solution Space for 2D Probe 89

Figure 23: Ep Solution Space for 2D Probe ................................................. 90

Figure 24: EE Solution Space for 2D Probe ................................................. 92

............................................................... Figure 25: Plot of EE Valley Floor 93

................................................. Figure 26: ET Solution Space for 3D Probe 94

.................... Figure 27: Cross-sections of ET Solution Space at Constant-a 95

................... Figure 28: Cross-sections of ET Solution Space for Constant-p 96

......................................................... Figure 29: 3D Probe Pressure History 97

............................................... Figure 30: Schlieren Photo of 3D Probe Tip 98

.............................. Figure 3 1: 3D Probe Pressure History with Oscillations 99

............................................. Figure 32: 3D Probe Pressures for Varied a 100

..................... Figure 33: Comparison of 3D Probe Pressures for Varied oc 101

............................ Figure 34: 3D Probe Pressure Corrections for Varied OC 102

............................................. Figure 35: 3D Probe Pressures for Varied P 103

...................... Figure 36: Cornparison of 3D Probe Pressures for Varied B 104

............................ Figure 37: 3D Probe Pressure Corrections for Varied P 105

........................................................ Figure 38: Mach Number Sensitivity 106

............... Figure 39: Schlieren Photograph of Shock Generator Apparatus 107

................................................ Figure 40: Tube Length Pressure Changes 108

............................................................ Figure 41: New Probe Geometries 109

viii

List of Tables

Table 1 : Sample Calculations .................................................................... 16

Table 2: Comparison of Pressures ............................................................ 18

................................. Table 3: Probe Pressure Cornparisons for Variable a 46

Table 4: Calibration Constants for Variable a ............................................. 48

Table 5: Program Results for Variable a ..................................................... 48

Table 6: Probe Pressure Comparisons for Variable P ................................... 50

Table 7: Calibration Constants for Variable P .............................................. 51

...................................................... Table 8: Prograrn Results for Variable P 51

............................ Table 9: Probe Pressure Comparisons for Quadrant Tests 52

..................................... Table 10: Calibration Constants for Quadrant Tests 52

Table 1 1 : Program Results for Quadrant Test ............................................. 52

................................................. Table 12: Pressure Sensitivity Comparison 54

Table 13: Probe Pressure Cornparisons for Shock Generator Tests ............. 55

Table 14: Calibration Constants for Shock Generator Tests ....................... -56

............................... Table 15: Program Results for Shock Generator Tests -56

................................................ Table 16: Results from Tube Length Tests 5 8

Nomenclature

Rotation Matrix from Reference i to Reference 1 Probe Error Term Pressure Error Term Each Error Term Total Error Term Free Stream Mach Number Mach Number Vector Cornponent of Mi in x-direction Component of Mi in y-direction Component of Mi in z-direction Effective Mach Number Static Pressure Measured Barre1 Pressure Measured Static Pressure Reservoir Total Pressure Calculated Total Pressure in front of Pitot Tube Calculated Total Pressure in front of Angled Probes Measured Pitot Pressure Measured Angled Probe Pressure Total Length of Tubing from Probe Tips to Transducer

Vertical Angle of Attack (in yz-plane)

Horizontal Angle of Attûck (in xy-plane)

Ratio of Specific Heats

Wedge Deflection Angle for Angled Probe Heads

Effective Wedge Deflection Angle of the Flow

Wedge Deflection Angle of Shock Generator

Rotation Angle between the Reference Frame i and 1

Shock Wave Angle

Effective Angle of Incidence upon Probe Head

Irzdex iderzt~fyirzg the reference frame which the variable is beirzg expressed in.

X

1 .O Introduction

1.7 Overview

There has been little research conducted pertaining hypersonic flows within ducts.

Research efforts on the flow structure within ducts has concentrated on the supersonic

regime with Mach numbers of less than four. Such duct structures are a characteristic

feature of hypersonic wind tunnels and hypersonic engines. An understanding of the flow

structure in ducts at hypersonic velocities is critical to future developments of air-

breathing engines in the aerospace industry. This requirement is most evident in the

development of inlet systems for the supersonic-combustion-ram-jet (SCRAMJET)

engines which will power the new generation of Earth-to-Orbit (EO) or Single-Stage-to-

Orbit (SSO) aircraft. Only from a thorough understanding of the hypersonic flow

structures within these inlet ducts can these engines be successfully constructed for these

trans-atmospheric vehicles. One of the greatest difficulties in developing such

technoIogies arises from the discrepancy bet ween measurement requirements and available

measurement techniques. This discrepancy is further compounded by the separate streams

of advancements being developed for use in the continuous flow wind tunnels and in

impulse flow facilities. The work described here outlines the development of a flow

velocity probe which could be used in either type of tunnel to investigate the interna1 flow

structures of hypersonic ducts.

Toronto Institute of Aerospace Studies (UTIAS) since the late 1980s using a gun tunnel.

Recently, Challenger (1995) established a map of the wall static pressure contours within a

square duct at flows of Mach 7.2 and 8.3. The work also included the use of Schlieren

photography to reconstruct the shock-wave structure. It was found that the wall static

pressure contours and shock geometries are not sufficient to completely determine the

flow interactions present within the square duct. Challenger stated that it was necessary

to take measurements within the flow itself.

One of the dificulties faced in taking such measurements in these high speed fiows

is the lack of suitable equipment. Pitot probes wilI take accurate measurements as long as

the shock-wave remains detached. This allows them to be used over a broad range Mach

numbers and angles of incidence to the free-stream flow. However, pitot probes can only

be used to determine the Mach number when the upstream flow remains isentropic. The

combined measurement of a pitot and a static pressure at a point can be used to infer the

local Mach number. TerziogIu (1994) focused on the development of static pressure

probes that could be used successfully in the UTIAS Hypersonic Gun Tunnel.

Unfortunately, such probes are extremely sensitive to their correct alignment with the flow

vector. This requires that the experimenter be knowledgeable concerning the existing flow

structure in order to make use of this technique. Davis and Gessner (1989) developed the

transverse flow probe shown in Figure 2 to determine the velocity vector in an unknown

flow field. By balancing the pressures across the two angled probe heads, the probe can

be aligned with the direction of the impinging tlow. The design of the probe restricts the

r- --- -u iiiir-iv----.O +.au u ~ . n n y ~ a & u . & L w1 . w I V Y I L J 1 1 1 V l l l J V 1 1 U & J l U l I b . L lllil I l I b L I I U U 1 3 W L l l y

applicable to continuous flow tunnels though since the probe must be rotated in order to

balance the pressure measurements. From the list of probe types that were developed, it

becomes obvious that there are four quantities which must be known in order to fully

describe the flow at a point. Two angles measured in orthogonal planes will define the

direction of the flow vector. Then either a measured pitot and static pressure, or a known

Mach number and pressure will define the magnitude of the vector. Considering the

difficulty in obtaining static measurements, the preferred combination is a measured Mach

number and pitot pressure. The development of an additional type of probe that could

accurately infer these four quantities from pressure measurements in an impulse tunnel is

required in order to reconstruct a flow vector map for the square duct.

This thesis has utilized the hypersonic gun tunnel facilities at UTIAS to investigate

a three dimensional flow probe capable of determining velocity at a point within the flow.

A numerical solver uses the pressures measured by a pitot probe and the developed probe

to calculate the incident angles of attack and the Mach number of the flow. Such a device

will prove instrumental in determining the interna1 flow structures within complex

geornetries.

7.2 Llterature Review

1.2.1 Duct Flow

A significant number of studies have been conducted on the influence of the free-

stream Mach number (Mo), duct size, duct Iength, and boundary layer action on duct

flows having Mo of less than four. Research began in 1950 with the work of Lustwerk

who used Schlieren photography to study the influence of boundary Iayer thickness at the

duct inlet on flow structure at Mo = 2.05. He concluded that there was a progression of

three shock wave structures that evolved. As the boundary layer thickness was increased

the interna1 shock structure shifted from normal shock-waves, to lambda shock-waves,

and finally to oblique shock waves. This work was followed by that of McLafferty in

1955, who explored shock stabilization for application in turbojet inlets. The study was

made over a range of Mo from 1.76 to 2.51. Richmond and Goldstein (1966) then

rneasured the streamwise wall static pressure and the total pressure along the centerline in

a trapezoidal duct at Mo = 1.5. They used the pressure profiles to infer a friction factor as

well as a heat transfer coeffkient for the turbulent channel flow. In 1972, Billig

investigated supersonic combustion in a square duct at Mo = 3.0. Results showed that the

shock train induced by the combustion process affected the upstreain flow. It was

possible for the shock train to unstart the duct, causing the formation of a standing normal

shock-wave positioned at the entrance of the inlet. This work indicated the requirement

to prevent the induced shock train interference from propagating upstream.

Davis and Gessner ( 1989) were the first to initiate significant studies pertaining to

flow probes in the supersonic regime. Preceding studies had relied on the use of flow

visualisation techniques and static pressure taps. Using in-flow measurernents, Davis and

Gessner studied the corner flow within square ducts at Mo = 2.5. They determined that

the fiow was highly three dimensional. The nature of the flow phenornena they discovered

are depicted in Figure 1. They measured significant transverse flow velocities and found a

number of circulation cells in the plane perpendicular to the duct axis. They also

discovered that both the velocity and the pressure profiles within this plane of the duct are

reflections about lines of symmetry. The duct can be divided into octants which have

identical properties.

Not until 1990 was work on square ducts conducted at free stream Mach numbers

exceeding Mo = 4.0. The first project was conducted at the NASA/Langley Research

Center's Combustion Heated Scramjet Facility on the Generic High Speed Engine

(GHSE) model. The study was designed to produce a database on rectangular combustor

and inlet isolator geometries. Bement (1990) found that the distance between shocks

decreased dong the length of the structure. He also observed that the overall length of

the shock system increased with increasing Mach number and increasing boundary layer

thickness. He concluded that the influence of the isolator duct prevented the combustion

induced shock train from interfering with the inlet fiow, causing an increase in the

combustor heat release.

i - - v - u r w i i v . v u n . v A AI SV VY6UlI y U W l l O l l l L l 6 b L L b I 1 L b 3 U I C i I 111 L l l b 177u3. dI3IidII dliu

Deschambault (1991) investigated the hypersonic flow structure within a circular pipe at

Mach 8.3. The study used wall static pressure taps, as well as in-stream static and pitot

pressure measurements. Preliminary results suggested primarily oblique shock formation

with the presence of normal shocks near the centerline. The report indicated difficulties in

acquiring in-stream measurements.

Chen expanded on this investigation in 1992. Schlieren photographs taken

through the walls of a glass tube proved inconclusive due to the curved glass surface.

Investigation of the shock structure by progressively cutting sections from the rear end of

the duct were successful in demonstrating that the flow formed oblique shock-waves near

the outer edge of the duct and then changed from lambda to normal shock-waves as the

flow approached the centerline. Chen found that in-stream measurements were the most

informative but that current probes experienced flo w instability problems. It was

suggested that the pressure instabilities were caused by a complex shock-boundary layer

interaction generated by the probes themselves.

In 1995, Challenger studied the waIl static pressure distribution in a square duct at

Mo = 7.2 and Mo = 8.3. Pressure profiles were obtained dong cross-sections in both the

streamwise and transverse directions. The study also included the use of Schlieren

photogsaphy to capture the shock-wave structure. The oblique shock-waves which

formed at the inlet intersected dong the centerline of the duct and were reflected from the

duct walls. The results from the two different Mach number tests showed that the flows

maintained these simiIar morphologies. Challenger concluded thnt there are distinct

the pressure profiles and positions of the shock-waves shifted aft. as the Mach number

increased. Challenger's work forms the basis for the continuing research in this field at

UTIAS.

1.2.2 In-stream Measurenient Probes

A significant number of papers have been published on the subject of subsonic flow

probes. Several met hods of constructing such probes have been suggested. The typical

geometry is based upon a spherical head with five pressure taps. The development and

caIibration of five-holed probes is a subject on which A. L. Treaster has published many

papers. Treaster and Yocum (1971), and Treaster and Houtz (1986) are two such

examples. There are some designs which do not make use of a head and simply arrange

the pressure tubes in an array. The major differences in design theory lie in the placement

and number of orifices used to measure the incident pressures. A significant amount of

work has been done by Sitaram (1985) in simplifying the use of four-holed probes.

Research emphasis has now shifted to the miniaturization of the probe arrays and to the

creation of econornical designs. Efforts by Feng (1992) to produce such mini-probes have

proven successful. The smaller frontal section reduces flow disturbances. The probes are

also simpler to calibrate and cheaper to manufacture. Unfortunately, these studies have al1

been conducted at free-stream velocities of the order of 65 ms-'. This corresponds to a

subsonic flow of approximately MO = 0.2.

been undertaken. A paper by Davis ( 1985) details the construction of probes that were

used to measure the transverse flow components of a flow structure within a square duct.

Figure 2 shows the double tube construction of the probes which Davis and Gessner

(1989) used to map the flow field of ü square duct at Mo = 2.5. It is important to note

that al1 of the measurements made by this probe type were made using a nul1 mode inside

of a continuous flow wind tunnel. The probe is used in nul1 mode by rotating the probe

about the axis of interest until the pressure measured by both tubes is equal. Due to the

very short run time of impulse facilities, rotating the probe body until the pressures acting

on the orifices are balanced is not possible. Literature has not been found detailing how to

use these probes in the direct measurement mode which would be required by facilities

such as the hypersonic gun tunnel at UTIAS

In 1994, Terzioglu studied various static pressure probe heads and probe mount

designs at Mo = 7.2 and Mo = 8.3. She focused on the interaction of the probe mounts

with the flow and their consequent effect on the measurement process. She found that

cylindrical mounts generated a detached shock-wave which interacted with the boundary

layer. The disturbances generated from this interaction were capable of affecting the

upstream flow by propagating through the subsonic boundary layer. These disturbances

interfered with the static probe measurements. A mount design which maintained an

attached shock-wave was required. As a result, she suggested that al1 probes be mounted

on a wedge base with an included angle of 35' or less. Terzioglu also determined that the

conventional cone-cylinder probe head was best suited for static pressure measurements.

- - -

alignment caused by the blast during the tunnel initialization procedure. Her conclusions

stated that these measures would minimize interference with the duct flow, and prevent

flow instabilities around the probes.

2.0 Experimental Facility

The experiments were conducted using the UTIAS Hypersonic Gun Tunnel. This

apparatus operates as an impulse wind tunnel. Hypersonic flight conditions are simulated

in the facility by firing bursts of high speed air over a stationary model. A schematic of the

tunnel assembiy is included as Figure 3.

The high pressure air in the driver (1) propels an aluminum piston dong the barre1

(4) when the diaphragms (3) are ruptured. A mass of extra-dry air, which is compressed

by the piston, acts as the reservoir from which the air is to expand into the test section.

This extra-dry air pressurizes the barrel to either 200 kPa, 400 kPa, or 600 kPa. Extra-dry

air is used to avoid condensation during the expansion process. The resulting ice crystals

would scour the model and instruments causing permanent damage. The air is

compressed from its initia1 pressure to a reservoir pressure of approximately 25 MPa.

When the first shock wave strikes the throat of the nozzle (6) , an isolator plug is ejected

through the test section (7) into the dump tank (9). The loss of the plug allows the extra-

dry air to expand isentropically through the nozzle and to accelerate to the desired Mach

number. The final free-stream Mach number of either 7.2 or 8.3 is controlled by changing

the throat section. A piezoelectric transducer measures the barrel pressure at the throüt of

the nozzle. The open-jet test section (7) itself is evacuated preceding the test in order to

increase the available test times to ü duration of 50 ms. The model can also be

instrumented with transducers to measure such properties as the pressures or strains

throughout the run.

conditioners. Each signal received from these conditioners is passed through a 20kHz

low-pass tilter to attenuate any noise created in the expansion process. The filtered signal

is then delivered to transient data recorders which are capable of sampling at 1 MHz for a

duration of 50 ms. A cornputer then downloads the information stored in the data

recorders.

Every measured pressure value is normalized with respect to the barrel pressure.

Normalized pressure readings are calculated by dividing the measured transducer pressure

by the barrel pressure and averaging these values over the steady state region. When

making this calculation it is important to realize that there is a delay between the barrel

pressure history and the transducer pressure histories. This corresponds to the amount of

time required for a pressure wave to propagate the length of the nozzle into the test

section. The delay is a function of the ratio between the pressures used to charge the

barrel and the driver. It varies over a range of 1.2 to 1.4 ms. A standard pressure trace

can be seen in Figure 4 which denotes these regions.

The Mo of the UTIAS Hypersonic Gun tunnel varies. It is dependent upon the

throat section which is being used and the pressures used to initialize the apparatus. All of

the tests within this document were conducted using the Mach 8 throat section. The

driver and barrel sections were pressurized to 20.5 + 0.2 MPa and 400 k 1 kPa

respectively. The test section was evacuated to 40 f 5 Pa. A report by Gordon (1991),

specifies that these initial conditions will produce a flow field that has local variations in

MO ranging from Mo = 8.287 to Mo = 8.373. The arithmetic mean of the variations was

There is also a Schlieren Photograph apparatus available during the experiments. A

timer can be set to trigger the Schlieren camera at the desired time during the run. It is

normally set to fire 19 ms after the experiment begins. This usually corresponds to the

beginning of the quasi-steady-state pressure plateau of the flow. Photographs which are

taken record the density gradients present in the test section. Shock-waves, expansion

waves, and boundary layers are the key characteristics which can be detected. Figure 5

shows such details present during one of the experiments.

The greatest advantage of such facilities is the high degrees of reliability and of

repeatability which have been observed. Deviations in pressure measurements between

independent runs with identical set-ups are typically only 4%. When the normalised

measurements are considered this deviation is reduced to 1.4%.

3.0 Preliminary Probe Development

3.1 Summary

The final design of a wedge base suggested by Terzioglu (1994) is shown in Figure

6 . This design accommodates the old probe types used in the UTIAS facility.

Measurements were taken using the static pressure probes suggested by Terzioglu (1994)

to ascertain whether any interference effects were present when using this new base.

Measurements were also taken with a standard pitot probe to verify the probe readings.

An angled probe similar to those used by Gessner ( 1987) in his transverse flow device was

then investigated to verify agreement wit h theoretical calculations. The various probes are

illustrated in Figure 7. Photographs of al1 of the apparatuses which have been used are

shown in Appendix A. The mechanical apparatus which was used for these tests can be

seen in Appendix B and a Schlieren photograph is shown in Figure 8.

Results from the investigation were encouraging. Measurements showed that

instabilities had been removed and that measured values were consistent with theoretical

calculations. It was possible to take measurements very close to the boundary layes

without establishing a complex shock-wave interference pattern.

3.2 Theory

The theoretical calculations for the probes have al1 been made using the basic

shock-wave relationships. A development of these equations can be found in the report

published by the Ames Research Staff (1953). The nomenclature and its application to

these tests is specified in Figures 9 and 10. The theoretical calculations must be applied in

two steps to determine the pressures measured by the probe transducers. The first step is

to calculate the changes across the first shock-wave, that which is generated by the base

plate. The second is to calculate the changes across the shock-wave generated by the

probe heads t hemselves. The normal shock-wave relationships apply to the pitot probe

while the oblique shock-wave relationships apply to both the angled probe and the base

plate.

The purpose of the base plate is to generate a variable strength shock-wave. As

the flow encounters the base plate it will be diverted through a set deflection angle (6,)

and remain essentially parallel to the surface of the plate. Boundary Iayer and separation

effects can alter the deflection angle of the flow in certain cases. The process of changing

the flow direction produces a shock-wave which varies in strength as a function of 6,.

Since 6, can be adjusted to üny desired angle for each individual test, the local Mach

number behind the first shock-wave (Ml) can be controlled. The MI is contingent on the

Mo and 6, that have been used for the current experiment.

becomes possible to solve either of the following equations for the shock-wave angle ( O ) ,

cot 6' = tan 8, (Y +W,Z 2 ( ~ , ~ sin2 0, - 1) - 11

Now, in the context of the theoretical calculations, it is important to redise that the

barrel pressure history represents the reservoir pressure history. It is assumed that the

expansion process through the nozzle is isentropic. Hence, the averaged value of the

barrel pressure (PRarrel) which is recorded over the steady-state portion of the experirnent is

the reservoir total pressure (P,o) avaihble. Knowing this, it is then possible to find M l , the

total pressure (P,I) and the static pressure (PI) behind the initial shock wave from,

2 y ~ i sin' 8, - ( y - 1) 2

(Y + 1) ( Y - I ) M ; + ~

The conditions to which each of the probes will be subjected can be calculated

from these equations. A sample of tests is shown in Table 1. The only differences arise

from slight deviations in P,o from one run to the next. Runs 1 , 2, and 3 show the

calculated values for tests of the pitot, static, and angled probes respectively.

Run Pm (psi) 8 p (deg) 0 (rads) M i . Pn(psi) Pi(psi) 1 3927.448 O 0.1 2034 8.32 3927.448 0.31 1 2 391 0.641 O 0.1 2034 8.32 391 0.641 0.309 3 3898.464 O O. 1 2034 8.32 3898.464 0.308

Table 1: Sample Calculations

Al1 of the probes which are placed in the wedge mount are held parallel to the base

plate. Since the flow remains approximately parallel with the surface of the plate, the

angle of incidence of the flow impinging on the probe heads remains fixed at OO. The

shock-waves generated by the probes are dependent upon the probe type. The pitot probe

has a detached bow shock similar to that shown in Figure 9, while the angled probe has an

attached oblique shock-wave similar to that shown in Figure 10. This means that the

forward face of the pitot and angled probes are always at 90° and 35" to the flow

respect ively.

The second calculation involves stepping across the probe shock-wave using either

the normal or oblique shock-wave relations as appropriate for the pitot probe or angled

probe respectively. From such cülculations the theoreticül static pressure measured by

each of the probe types can be found. For the theoretical stagnation pressure measured by

relationships is,

The static pressure probes are designed to measure the local free-stream static

pressure (PSI). Terzioglu (1994) explains the function and design of the probes that were

constructed for use in the UTIAS Hypersonic Gun Tunnel. A good design will attain a

ratio of measured pressure to local free-stream static pressure (PSIPI") in the range of 0.9

to 1 . 1 . Ideally, this ratio would be equal to one so that PSI = PI.

The angled probe follows theory identical to that used for the base plate since they

both initiate a wedge flow structure. In the case of the angled probe, the front of the

probe has been machined so that &, = 35". The simpler of the angle relations, Equation 2,

can be used:

cot aA = tan02 (Y + 1) Ml2 2 ( ~ , ~ sin2 8, - 1) - 11

The applicable equation from oblique shock-wave relations for the theoretical

static pressure measured by the angled probe transducer ( P w ~ ) is,

The results of these three calculations are included with the experimental resuits in

Table 2.

3.3 Results

The calculated theoretical values for the probe pressures are shown in Table 2

dong with the experimental values that were obtained. It can be seen that there is good

agreement between the results and the theoretical calculations.

Theoretical Measured Measurement Pressure (psi) Pressure (psi) Difference Pitot Probe 27.91 8 27.1 52 2.7% Static Probe 0.309 0.304 1.6% Angled Probe 12.958 10.603 18.2%

Table 2: Cornparison of Pressures

theoretical values. Measurements made using the Schlieren photographs indicate t hat the

growth of the boundary layer along the plate, seen as the white band in Figure I l , can

deflect the flow by up to 1 O. The resulting decrease in P,I caused by the altered values of

liP and 81 is of the order of 10%. This boundary tayer interference could account for the

differences obtained in the pitot and static probe measurements. Unfortunately, the

difference in the angIed probe measurement is much greater. An unknown mechanism is

present which is causing this interference. The probe heads were machined very precisely

to a cut angle of 35" $ 0.2" and carefully deburred to eliminate errors caused by

manufacture. The interference could be attributed to viscous effects present inside the

small diameter hypodermic tubing or complex flow geometries around the probe head.

4.0 Feasibility

An angled probe model and a probe prototype were developed to investigate the

feasibility of constructing a new measurement technology that would be able to discern the

transverse flow cornponents and reconstruct the flow velocity vector. Results suggested

that such a probe could be successfully developed.

4.1 Angled Probe Investigation

4.1.1 Summary

A feasibility study was conducted to determine whether the original design

developed by Davis (1985) could be modified for use in the gun tunnel. A model was

constructed as shown in Figure 1 1 to test the pressure measured by a single probe head

under minor variations in angle of attack (a). The decision to use a 35" cut for the face

angle (aA) was suggested by Chue (1975) for sensitivity considerations. This was

considered acceptable for this case since it is less than the detachment angle associated

with a Mn = 8.32 flow. The tubes used for constructing the probes had an outer diameter

of 0.042 inches. The theory used in the development of the following calculations is based

on the assumption that the shock-wave wilI remain attached. Since the attachment angles

for hypersonic flows (Mo- 6) include any wedge angles of less than 42", a = 35" will

increases then the allowable a will increase as well. In order for the probes to be practical

they had to be able to detect deviations in the flow direction of a magnitude of at least

0.5".

4.1 .2 Theory

The theory which applies to the current apparatus is sirnilar to that developed in

the previous section for the angled probe with the base plate. The only factor which must

be accounted for in the present apparatus which was not previously discussed is the ability

to change the orientation of the probe with respect to the flow. In the previous section,

the probe cylinder and the incident flow vector were always parallel. The current

apparatus alIows the probe to be rotated in the vertical plane, changing the incident angle

of the incorning flow. The angled probe head will still initiate a wedge flow structure,

however, the deflection angle of the flow is no longer constant.

By noting the angle of the incoming flow, the effect of the probe rotation can be

accounted for. The a for the flow impinging on the head of the probe is simply the

negative of the apparatus set angle (Q). This relationship is shown in Figure 12. It is a

simple process to position the apparatus at a desired @ using an inclinometer. From this

information, the a can be readily obtained.

cause the effective deflection angle (6,) to be different for every case. The S, which the

flow must turn through will be a function of the machined 6A and of the chosen a. The

relationship is,

Frorn this point on, the equations that were previously developed to calculate the

theoretical pressure apply as long as 6, is used. The resulting equations are,

It was possible to conduct successful tests to determine the usefulness of the new

probe design. Figure 13 is a pIot of the measured pressures and the theoretical pressures

relative to the offset q. Initial results are encouraging since there is a clearly discernible

variation in PA, over small variations in a. The pressure transducers are fulfy capable of

resolving these differences. It was also found that the probe acted well in terms of

measuring a wedge face pressure. The slopes of the measured and theoretical lines in

Figure 13 are parallel within the bounds of error. The measured pressures are lower than

the calculated values by an average of 16% with a standard deviation of 3%. Notice that

this represents a pressure loss within the probe system similar to those that were

previously detected in Section 3.3. Since the slopes are parallel the loss constitutes a fixed

offset of. PA, pTI-' = 0.000469. The plot shows that the measured and theoretical values

are in good agreement over a large range of a.

4.2 Probe A rray ln vestigation

4.2.1 Summary

Results from the angled probe investigation were promising. They suggested that

this type of probe head could be used accurately within the gun tunnel. However, it was

not certain whether using multiple probes to form an array would be feasible. The greatest

function as a blunt body and produce a detached shock-wave. Should the flow detach

itself in this manner it would eliminate the desired wedge flow and would cause the

deveIoped theory to be inapplicable.

In order to investigate such a possibility, a large scale mode1 of the intended 3D

probe was buiIt and tested in the gun tunnel. This prototype is constructed from three

angIed probe cylinders aligned back to back with their tips touching in the center. A large

scale model was built to allow observation of the shock-wave structure. This model,

shown in Figure 14, was swept through several angles of attack and photographed using

the Schlieren apparatus. There were no pressure transducers mounted on the model

during these tests.

4.2.2 Results

Schlieren photographs of the tests where the probe apparatus had a nose down

attitude (+ < O") showed excellent agreement with theoretical expectations. When the

angle is chosen such that a = 5", then 6, = 30". The corresponding 0 for an Mo = 8.32 is

39". Measurements taken from the Schlieren photograph in Figure 15 show an initial

shock angle of 38 + 0.5". Experiments showed that the shock has rernained attached to

the head of the probe array and that shock angle measurements agree with wedge flow

theory.

displayed some unexpected characteristics. Inspection of Figure 16, suggests that the top

probe cyIinder seems to have swallowed the shock-wave and started. Though the rear of

the cylinders was blocked to simulate the presence of a transducer, the flow appears to

have behavcd as a duct flow. The shock wave is clearly being shed from the trailing edge

of the orifice. Such behaviour does not establish an attached shock and is not in

agreement with wedge flow theory.

The tendency of the large cylinders to exhibit duct flow characteristics during the

tests can be attributed to compressibility effects within the air mass. The large diameter

and extended Iength of the individual tubes used to construct the prototype created a

significant volume within each cyIinder. It is possible that this volume was large enough

to allow the air to enter the tube initially and establish a duct flow. The flow would then

strike the plug at the afi end of the cylinder. As the air compressed, the back pressure

would eventually cause this flow regime to terminate. This back pressure would be

suficient to force the duct to unstart. At this time the expected wedge flow could be

reestablished. Since there were no pressure transducers mounted on the cylinders during

these tests it was not possible to examine the pressure histories for these tell-tale

characteristics.

These results identified the need for a prototype to be constructed. The new

mode1 would ineasure the pressures in each of the probe cylinders and be capable of

sotating the probe head with respect to the tunnel free-strearn flow. This pressure

information could be used to determine the flow characteristics of the probes with

=-- ------- --. -- -..- ---.a--*-- -."..wu""- y.- . * " u " n . II" I.v.v., U l l l b U L 1 3 V I

deterrnining the flow vector from the measured pressures was still required to justify the

construction of the prototype. The next phase was to develop a numerical solver that

could return the flow vector from the pressure information.

5.0 Numerical Investigation

5.1 Summary

The possible probe arrays under consideration are shown in Figure 17. The efforts

of the numerical investigation focus on developing a means to interpret the pressure

readings that would be obtained from such probe arrays during a test in the UTIAS

Hypersonic Gun Tunnel.

Note how the probe arrays are constructed frorn angled probes that have been

placed back-to-back so that their apexes touch in the center. In this orientation each of

the wedge faces are aligned so that their normals are rotated by a known angle. The 2D

probe has two angled probes rotated 180" apart while the 3D probe has three angled

probes rotated at 120° apart. Each angIed probe in the array can be instrumented to

measure the static pressure in the cylinder using a piezoelectric transducer. From this

information it would be desirable to determine the velocity of the impinging fiow. This

would involve stepping backwards through the oblique shock relationships to solve for the

incident Mach number and angles of attack which satisfied the system of equations.

Two key assurnptions were made in deriving the systems of equations to be solved.

The first assumption was that the flow remained attached to the tip of the probe array and

formed oblique shock waves. The second assumption was that the flow could be broken

down into two or tnree distinct wedge-oblique shock-wave interactions. Thus, each

Mach number and effective deflection angle.

Results from the experiments seemed to confirm these assumptions. The Schlieren

photographs of the angled probe and the large three probe array indicated that the flow

remained attached to the probe heads. Experimental measurements showed close

agreement between the measured angle probe pressures and theoretical vaIues caIculated

from oblique shock theory. These results confirmed the basis for the development of the

computer code.

The computer program was developed in two stages using Turbo Pascal. The first

stage was the development of a code to solve the two dimensional case where the probe

array had two angled probes and measured the flow angle constrained in one plane. This

probe array was similar to the original probes used by Davis and Gessner (1989). Such a

case would require measurements in the horizontal and vertical planes to reconstitute the

actual flow vector. Though impractical for experimentation due to the extra

measurements required, it allowed for the code to be developed and verified for a simpler

case. The second stage was an adaptation of the code to a three dimensional probe array

able to measure the impingement angles of the flow vector relative to the probe head.

5.2 Theory

The theory that v is used to establish the system of equations to be solved by the

programs is outlined in this section. The equations are derived from the basic oblique

shock relationships. They are an application of the theories already discussed previously.

For purposes of illustration, the equations are developed assuming the Mo and angles of

attack in the vertical (a) and horizontal (P) planes of the incoming flow were known.

Stepping through the equations in this manner yields the static wedge pressures measured

by each transducer. The program itself would be given the final wedge pressure values as

a user input and would solve for Mo, a, and p. In other words, the program would

progress through the system of equations in the reverse order that they are presented here.

Appendix C offers an alternate derivation.

The first step involved in calculating the wedge pressures is to establish the total

pressure (Pte) in the vicinity of the probe array. This is accomplished through the use of a

pitot probe. The pitot probe measures the total pressure (Ppl) behind the detached shock

formed when a flow of magnitude Mo encounters the blunt face. From normal shock-wave

relationships shown in Section 3.2 recall that:

down into its component vectors in the reference frame of the top probe. Figure 18

defines the conventions that are used. Figure 18(a) shows the numbering sequence for the

angle probes. Al1 of the calculations are made with reference to the top probe which is

identified as probe 1. The normal to the wedge surface of probe 1 is aligned in the vertical

plane as shown in Figure 18(b). The z-axis is perpendicular to the longitudinal axis of the

probe while the y-axis is parailel to its longitudinal axis. The zy-plane is defined such that

the normal to the wedge surface lies within it. A positive a is measured from the y-axis

counter-clockwise to the z-axis. A positive P is measured from the x-axis counter-

clockwise to the y-axis. The angles a = 0" and P = O" both lie dong the y-axis. The flow

vector is decomposed using the simple geometric relationships shown in Figure 19. Note

that the two dimensional case would have only an a value and that P = 0" always. This

occurs since the probe array must be aligned in the axis of interest. The general equation

for the top probe is:

The MI vector can then be transformed into vectors which are expressed in the

frame of reference of the other angled probes using rotation matrices. Note that there are

three separate reference frames, one attached to each probe. The axes for each reference

frame are defined by the geometry of the probe head as described above. Therefore, the

Epsilon (E~) is defined as the angle in the xz-plane between the normal of the probe to

which the vector is being iransformed and the normal of the top probe, as shown in Figure

17. The rotation matrix then becomes:

where i is the probe number.

The flow vector expressed in the new frame of reference then becomes:

For the two dimensional case there is only one rotation to consider, the rotation

from the top probe ( 1 ) to the bottom probe (2). In this case the z-component of the

normals are pointing in opposite directions. The flow vector expressed in terms of the top

probe must be rotated by 180" to be expressed in terms of the second probe. This means

that the indice i, would represent 2 and that E:! = 180'. For the three dimensional case

there are two transformations that are required. The first transformation from probe 1 to

the bottom right probe (2) requires that i represent 2 and that E:! adopt the value of 120".

Similarly, the rotation from the probe 1 reference frame to that of the bottom left probe

(3) would indicate that i = 3 and E~ = - 120'. Note that the designation of Ieft and right are

determined facing the probe array from the front as shown in Figure 1 8(a).

streamline incident on the wedge tip and the amount it is defiected by the wedge face, the

above vectors must be interpreted appropriately before the shock relations can be used

correctly. Figure 20 shows the angles and vectors which will be discussed. The current

set of equations being developed are similar to those developed for wedge flows in

Section 4.1.2. In al1 cases, the x-component of the flow vector is parallel to the resulting

shock-wave. As such, it has no contribution to the formation of the shock-wave and its

ensuing strength. In order to use the wedge flow equations, an effective Mach number

(M,,) and an effective angle of incidence (vei) must be caIculated from the Myi and Mzi

components of each vector using the Pythagorean Theorem. This results in,

!Pei = arctan - [ M: ) Finally, since the angled probes have al1 been machined to a known probe angle

(&), the effective deflection angle (&i) is found from the addition of vci and &. This

means that:

The basic relationship between the Sei and Bi can then be solved for the applicable

values of Oi. Recall that:

It then becomes a simple process to apply the oblique shock-wave pressure

relationship to calculate the static pressures at the wedge surfaces (Pwi). As in Equation

10, the relevant formula is:

A sample MathCada sheet is included for inspection as Appendix D which steps

through the above calculations in the same order which they were developed here. The

MathCad@ document was provided with a known Mo, a, and P, and then given a possible

Pte. The pressure values found from these sample calculations were used later to validate

the computer program.

As noted before, the computer code would be given the values of Pwi, measured

during an experiment, dong with a Ppl, measured previously. The program must then find

Mo, a, and p, which will simultaneously satisfy the resulting system of equations. When

the program guesses a solution it will calculate a different value for the total pressure

which exists in front of each angled probe and the pitot probe. The program finds the

correct solution by attempting to match t hese calculated total pressure values.

In order to simplify the task of developing the cornputer code, the two dimensional

case was considered first. It was unknown whether the complicated system of equations

presented by the oblique shock-wave relations could be solved in the reverse manner to

their standard implementation. These equations are usually applied in the direction of the

flow. The simplifications involved in the two dimensional case would reduce the number

of hurdles to be overcome.

The code was developed considering the case of a two probe array aligned within

the vertical axis in the gun tunnel. In this orientation the probe will measure a variation in

a only. Hence, the value of P will remain zero. Since the tunnel uses dry air as the free

stream fluid the value of y = ' / S . Using the theory derived in the preceding section we can

show that,

O

- M,cosa

- M o sin a

Y=, = arctan - [:;,;] a,, = 35" + Ye,

Resulting in the following cülculations for the total pressure,

The PPI, PW !, and PW2 are the measured pressure values from the pitot probe and

the angled probes respectively. The Pl,, Pt,, and Pt? are the calculated values for the total

pressure in front of the pitot probe and angIed probes respectively.

Equations 3 1 and 32 constitute the system of equations to be solved. A brief

inspection suggests that there are a greater number of unknowns than equations. Many of

the unknowns in the final three equations are not independent variables though. The only

truly independent variables are Mo and a. The remaining variables are al1 intermediary

variables which can be calculated from these two. The correct solution is attained when

al1 three equations for calculating Pt0 agree. This means that the values of Pl,, Pt[ and PL:!

will be equal when the correct Mo and a have been isolated.

The resulting system of equations was rather cornplicated. The presence of the

trigonometric functions and the squared terms introduced the problems of periodicity and

multiplicity. Further compounding the difficulty of solving this set of equations was the

consideration that experimental errors might cause the measurements to suffer minor

routine would best accomplish the task of ascertaining a solution.

In order for the minimization scherne to work correctly, a relevant cost function

had to be defined. The first step involved writing a computer code that would calculate

the total pressures ahead of the probes over a known range of Mo and a given

predetermined inputs Ppi, Pwi, and Pw2. Several cost functions could then be defined

using combinations of the differences in the calculated total pressures P,,, PtI and Pt*.

These differences represent an error in the selection of the velocity vector and will

approach zero when the correct vector has been isolated. A proper understanding of the

behaviour of these error terms within the solution space would allow for the creation of an

efficient solver. An intelligent selection of such an error term would then provide a

solution space that was distinct and easily navigable by the minimization routine.

Three error terms were defined. They were labeled the probe error (Ei), the

pressure error (Ep), and the each error (EE). They were defined as follows:

- - - - - , - - - - - - - - 1 - - - - - r-'--'. - "'-a """. " -W." r-.'Y L..V U...YIV.."V " V L V T V U I l L I l k

total pressure calculated from the pitot probe and an angled probe. The Er term is the

sum of these pressure errors, a cumulative total of the Ei errors for the probe. The EE

term is a comparison between each probe. It equals the difference between the calculated

total pressure in front of each of the angled probes.

The error terms for the two-dimensional case are thus,

Figures 21 through 24 show the results of these errors plotted as a function of Mo

and a. Analysis of these error parameters indicated that there was one term in particular

which was most useful. It had a characteristic shape to its solution space which was

particularly amenable to the minimisation routine. A plot of the solution space generated

by the EE term forms a dual valley which is not discernible from Figure 24. The surface

forms a distinct valley dong the a-axis whose base lies dong the correct cx line. The

valley floor itself has a minimum dong the M-axis which corresponds to the correct MO.

A plot of the valley floor is shown in Figure 25. Note that there are local and global

the pitot probe measurement. Though the EE term gave the best shape for the solution

space and had a distinct minimum, the dificult task of discerning the difference between a

local and a global minimum had to be overcome.

In order to effectively implement the code, the programmer must be aware of the

oscillations upon the floor of the valley dong the M-axis. Care must be taken to ensure

that the value returned corresponds to the global minimum and not to a local minimum.

Note that there is a distinct globaI minimum and several other oscillations dong the vülley

floor. This is particularly evident in the low Mach number regime and can easily trap a

minimisation routine. The first solver which was developed used a gradient analysis

approach for fînding the minimum, subjecting it to this flaw. Due to the steep slope of the

error surface in the Mo direction, any gradient approach will quickly be led towards low

Mach numbers. With further investigation, it was discovered that, when the search grid

was refined finely enough, the error term could actually be driven to zero through the

proper selection of a. There was no unique Mo solution dong that line. The answer was

provided by defining a new error term which combined the characteristics of Ep and EE.

This term, labeled the total error (ET), is shown below,

This information provided the basis for the development of the solver. The solver

is given the pressures measured by a pitot probe and the dual üngled probe array. The

axis using the EE as a cost function. This error term was selected since it has a lower

computational cost. The solver then searches the vaIley floor, dong the M-axis, for the

global minimum using the ET as a cost function. A listing of the code can be found in

Appendix E. An actual ideal value of zero error is not achievable due to the sensitivity of

the exponential and trigonometric terms.

The solver works in essentially the same fashion in which Davis and Gessner

(1989) operated their probes. They rotated the probes physically in the continuous flow

tunnel until a nul1 state was achieved. The program rotates the a until the total pressures

calculated for each probe match. It then finds the Mo with an associated error closest to

zero by minimising the EE term. The final version of the solver returns the correct Mo + 0.005, and the correct a + 0.2". These results were quite satisfactory and proved that a

code could successfully be implemented.

5.4 Three Dimensional Solver

Use of the two dimensional probes suggested above would require that two

measurements be made at the same location in space in order to determine the actual flow

vector. One measurement in each of the x-y and x-z planes would have to be made to be

able to reconstruct the complete vector from the components returned by the program. A

third measurement would still be made with a pitot probe to gather information about the

. " L U 1 y.uLILILa1u UL LllUL F"llC- llla IUYUIIUO JUVWIUI 1 UllO UIIU U W I L I U I I U 3 V U l y UbbUI U L b

placement of the probes. In an effort to reduce the number of runs that are required and

simplify the experimentaI process, a three dimensional probe was investigated. Using the

same principles that were utilized in developing the 2D case, a code was adapted for the

3D measurement process.

Following the sarne manner through which the 2D pressure equations were

developed, the resulting system of equations for the 3D case becomes:

The same error terms that were used in the 2D solver were adopted for the 3D

solver. However, the behaviour of EE changed significantly. A plot of the error surface

for any value of Mo now forms a cone as shown in Figure 26. The apex of the cone lies on

the correct set of angles cx and B. The apex of the cone will drift up and down in an

oscillatory fashion for different values of Mo. The use of the ET term once again provided

for a global minimum located at the correct combination of a, P, and Mo.

The three dimensional solver works similarly to its predecessor. The program

rotates the incident angles until the total pressures calculated for each probe are matched

these plots reveals that the constant-p lines always have a minimum at the correct a, while

the constant-a lines are skewed. Due to this shape of the error space it was best for the

program to isolate the correct a, then proceed to isolate the correct P. It then proceeds to

find the Mo by finding the global minimum of the E~surface. A flow chart of the program

is incIuded in Appendix F and a listing of the code can be found in Appendix G. The final

version of the solver returns the correct Mo f 0.004, and the correct a and P k 0.05".

These results were extremely satisfactory and justified the design of a working prototype

of the three dimensional probe array.

6.0 Prototype

6.1 Summary

The last phase in the development process was the construction and testing of a

working prototype of the 3D probe array. This prototype had to be smaller than the

previous test model since interference is a key issue when measuring interna1 flows. It had

to be instrumented so as to measure the pressures in each of the three angled probe

cylinders. A means of varying the angles of attack of the free Stream flow with respect to

the probe head was required. It was also desirable to have a means of changing the length

of the probe array which extended from the wedge support base in order to accommodate

for interference from the mount. The design should allow for the overall length between

the tip of the probe array and the face of the pressure transducers to be changed in order

to determine the optimal response characteristics. The final design of the apparatus which

was developed to suit these requirements is shown in Appendix H.

6.2 ln ves tiga tion

The first tests using the new model were conducted to confirm the flow

characteristics of the probe array. The model was set at different angles of attack to

obtain the transducer pressure histories and the Schlieren photographs of the probe array

through 3 1.

In Figure 29 the anticipated shape of the pressure curve, as hypothesized in

Section 4.2.2, is evident. As the test begins, there is a distinct spike in the pressure history

which corresponds to the cylinder having started. The flow is entering the cylinder

directiy in the same fashion as a duct flow, and cornpressing itseIf against the rear of the

cylinder where the transducer is mounted. The back pressure then reaches a critical value

of approximately 130% of the quasi-steady-state value, forcing a change in the flow

regime. At this point the wedge flow structure is established and the pressure stabilises to

a quasi-steady-state plateau.

Inspection of the Schlieren photographs taken during these runs show that the flow

has already reestablished the desired wedge flow by the time the film has been exposed.

Figure 30 clearly shows how the shock-waves generated by the tip of the probe array

remain attached to the apex of each of the cylinders. A distinct shock angle is clearly

visible, demonstrating that the desired wedge flow characteristics have been achieved.

It should be noted that some pressure histories, such as in Figure 3 1, displayed

some oscilIatory characteristics. The initial spike, which was attributed to the cylinder

having started, repeats itself. The frequency of these spikes was found to be

approximately 400 Hz. One possibility for this range of frequencies corresponds to a

mechanical resonance. The pressure spikes could be caused by a mechanical oscillation of

unsupported length of the array from the initial 14.0 mm to 9.0 mm.

6.3 Calibra tion

The next step involved calibrating the experimental measurements using the

cornputer code. Through a comparison of the experimental values, the theoretical vaIues

returned frorn the program, and the known angles of attack set during each test run, it is

possible to set up a calibration system for the probe array. This calibration process

consists of two main sets of experiments. In the first set of experiments the probe was

rotated through the vertical plane at discrete values of a while maintaining = O". In the

second set, the array was rotated in the vertical plane at discrete values of P while

maintaining a = 0'. Wence, only one of the angles of attack was changed while both the

Mach number and the remaining angle of attack remained constant. In this way, the

cornparisons could be made with only one of the independent variables changing.

6.3.1 Variable a

Figure 32 shows a plot of the theoretical values that were calculated and a plot of

the pressures measured during the tests. Both sets of pressure curves display the same

behaviour. For both the theoretical case and the measured values, the lower left and right

forms a Iine with an opposite slope. The measured pressures are al1 lower than the

predicted theoretical values. There is a distinct point of intersection. Al1 of the probes

measured an identical pressure at the a = P = 0" point.

A comparison of the theoretical and measured pressures for each specific probe is

shown in Figure 33 and Table 3. It is clear that the measured values and the theoretical

predictions show similar trends. The lower right probe demonstrated the best results,

having the theoretical and rneasured pressure curves parallel. The measured values are

lower than the theoretical values by a constant offset. This shows that the expected

pressure. loss noted in Section 4.1.3 is independent of a. The curves for the lower left

probe and the top probe shared similar trends between measured and theoretical values.

However, both sets of curves displayed dissimilar dopes. The offset is not constant as the

angle a is changed.

1$:&@@g)BI Measured 1 Theoretical 1 Measured 1 Theoretical 1 Measured 1 Theoretical 1 9.0 16.199 NIA 9.904 10.505 8.726 10.600 5.0 13.984 16.620 10.282 1 1.584 9.902 11.584 1 .O 1 1.886 13.549 11.175 12.569 1 1.794 12.569 0.0 1 1.332 13.01 8 11.102 13.01 8 1 1.753 13.01 8 -2 .O 10.579 11.678 1 1 -927 13.652 12.184 13.652 -4.0 9.650 10.451 12.198 14.41 6 12.269 14.41 6 -6.0 9.082 9.264 13.31 2 15.236 13.207 15.236

Table 3: Probe Pressure Cornparisons for Variable cr

some mechanical difficulties that had to be addressed. The epoxies used to bind the three

angled probes together had cracked. Though capable of maintaining the probes in the

correct axial position, the epoxies could no longer hold the probes in the correct facing.

During the experiments the probe heads had shifted orientations. The lower right probe

maintained the correct facing while both of the other probes spun by approximately + IO0.

This means that the E values used to calculate the theoretical pressures were incorrect.

Accounting for the probe rotation in the experiments, a new theoretical prediction was

produced using the altered E values. The results are shown as the green curves in Figure

34. Al1 three curves are now parallel, maintaining a constant offset independent of a.

The calibration constant that is required to correct for the pressure losses in the

measurement system will be the offset between the measured and theoretical curves for

each probe. The calibration constant is calculated as an arithmetic average of the

difference between these two curves. It was expected that each probe would have the

same calibration constant. The calculated constants and their standard deviation are

shown in Table 4. The calibration constants shown are similar, but Vary by up to 16%.

There may be disturbances generated by the geometrical arrangement of the array which

cause slight variations in the pressure losses detected by each probe. It is also possible

that flow spilling off the side of one probe head could interact with the other probe heads.

These effects would alter the calibration constants calculated for each probe head.

Probe Constant Deviation

TOP 1.345 0.848 Left 1.583 0.537

Right 1.606 0.479

Table 4: Calibration Constants for Variable a

Since each experiment was conducted at a known state, the agreement of the

program with the known setting could be verified. Entering the measured values into the

program and applying the calibration constants Iisted above, the results listed in Table 5

were obtained. It can be seen that the solutions for ûi and P agree to within 0.8".

However, the Mo found by the program disagree with the known value by as much as Mo

= 4.

Table 5: Program Results for Variable a

6.3.2 Variable B

1 llC 1-CSUILS OUlillIlCU 1IVlI1 LILC tCSLS VilIyIIlg C> WGlG S l l l l l ld l LU LIIUSt: VULiLlIlCU III

the previous tests at varying a. Figure 35 shows the plot of the theoretical and measured

pressures. Once again, the similarity in trends is apparent and the measured pressures are

lower than the predicted ones. There is a large difference at the intersection point of the

curves though. The measured values for the top probe line are much higher than

expected. As a result, the measured curves do not intersect at a distinct point. The two

lower probes intersect close to the axis. The intersection point should be distinct and lie

on the p = O0 axis. This corresponds to the cc = = O* setting where al1 three probes

should measure equivalent pressures.

Figure 36 and Table 6 show the measured and theoretical pressures for each probe

separately for cornparison. The lower left probe demonstrated the best results. The

measured pressure is lower than the theoretical pressure by a constant offset. The curves

for the lower right probe do not maintain parallel slopes. The curves for the lower right

probe show the same divergence characteristics seen in the previous tests for the rotüted

probe heads. The curves for the top probe are neariy coincident and diverge for large

angles P.

I.$f('iieg)&( Measured 1 Theoretical ( Measured 1 Theoretical 1 Measured 1 Theoretical 1

Table 6: Probe Pressure Cornparisons for Variable B

The probe array was inspected again after completion of the tests to determine

whether the same orientation problems detected before had recurred. A close scrutiny of

the probe heads showed that an epoxy had once again failed. The lower right probe had

rotated by 35" from the correct orientation. Accounting for this probe rotation produces

the curves found in Figure 37. The top probe itself had received structural damage to the

lip of the opening. Part of the wall was collapsed inward and multiple metal spurs were in

evidence. These flaws may have been responsible for the elevated pressure readings that

were obtained.

The application of the same procedure outlined in Section 6.3.1 to the

experimental data produced the calibration constants shown in Table 7. The values of the

calibration constants for the lower left and right probe are nearly the same and correspond

to the values that were calculated in the previous section. The constant for the top probe

is much smaller since the measured pressures were much higher.

Probe Constant Deviation

TOP 0.831 0.605 Left 2.01 O 0.41 6 Right 2.587 0.382

Table 7: Calibration Constants for Variable P

The measured values were then entered into the program with the corresponding

calibration constants listed above. The results are listed in Table 8. It can be seen that the

solutions for a and B agree to within 0.9". Once again, the Mo found by the program

disagrees with the known value by as much as Mo = 4.

Table 8: Program Results for Variable P 6.3.3 Verification

probe positioned in each of the quadrants. This was done as a final verification of the

calibration constants and validation of the program. The comparison of the pressures are

found in Table 9. The new calculation for the calibration constants are listed in Table 10.

These are similar to those obtained in the variable angle tests. A comparison of the results

of the program solution to the initial settings of the apparatus is seen in Table 1 1 .

Table 9: Probe Pressure Cornparisons for Quadrant Tests

Averaged Standard Probe Constant Deviation

TOP 2.274 1.324 Left 2.372 1.138

Right 2.31 5 1 .O01

Table 10: Calibration Constants for Quadrant Tests

1 Set 1 Calculated 1 Set 1 Calculated 1 Set 1 Calculatedipl

Table 11: Program Results for Quadrant Test

6.3.4 lnterpretation

was troubling. When the program was tested using theoretka1 values, the results that

were returned were correct. Yet, when measured values are input into the program using

the calibration constants, the answers that are returned are unacceptable.

In order to discern the source of this difficulty a sensitivity analysis of the system

of equations was undertaken. The correct initial values for the probe pressures were

entered into the program to test whether it functioned correctly. The theoretical values

used corresponded to an Mo = 8.32 with a = 3.0° and fl = 1.0". Then, each of the

pressures entered in the program were changed by minute amounts independently of the

other pressure values. By changing only one pressure input at a time from the theoretica!

value, the response of the program to these perturbations could be monitored.

Table 12 shows that in response to perturbations in the wedge face pressures, the

program produces little change in the angles which are discerned, while any changes in the

pitot pressure produce drastic changes in the Mo solution. The perturbations of 0.2 psi in

the Pwi values do produce changes in the measured angle. These changes in angle of + 0.2", and of Mo* 0.2 are of the same magnitude as the numericai accuracy of the program.

A deviation of only 0.2 psi in the Ppl value is al1 that is required to cause the solution

returned by the solver to Vary drastically frorn the correct solution. When a11 of the Pwi

values where altered in the same positive or negative direction, similar drastic changes in

the correct solution were observed. The important characteristic is the difference between

the average wedge face pressure and the pitot pressure. Figure 38 shows how minor

changes in this difference produce large changes in the Mo solution. Unfortunately, the

deviation that was obtained for the calibration constants. The current mode1 has an

experimental error and a calibration error that are both up to four times the 0.2 psi

sensitivity limit. As a result, the measured pressures that are input into the computer code

have perturbations greater than the sensitivity limit, causing erroneous Mo solutions.

Probe Pressure (psi) Mo a P + 0.2 8.30 3.20 0.90

1 TOP Theoretical 8.32 2.90 1.20

1 ~ e f t Theoretical 8.32 3.10 1.10

+ 0.2 8.29 2.70 0.80 1 Righf Theoretical 8.31 3.00 1.10

1 Pitot Theoretical 8.33 3.20 1 .O0

Table 12: Pressure Sensitivity Comparison

6.4 Shock Generator

wave produced by a shock generator plate. A Schlieren photograph of the apparatus

during a test is shown in Figure 39. This configuration placed the probe array in a known

flow field that could be altered as desired by changing the angle of deflection of the shock

generator (6,). The resulting flow is parallel to the plate having an Mi which is a function

of 6, as seen in Section 3.3. This allows for the probe to be used in varying Mach

numbers at changing angles of attack.

Table 13 shows the cornparison of the probe pressures frorn the three runs that

were conducted. Table 14 lists the results of the calculation for the calibration constants.

These calibrat ion constants are slightly smaller t han t hose previously determined. The

solutions obtained by entering the pressure measurements into the program with the

calculated calibration constants are shown in Table 15. Results indicate there is agreement

for the values of a and B which are returned by the solver. The difficulty in obtaining the

correct Mois once again observed.

Table 13: Probe Pressure Cornparisons for

Shock Generator Tests

Probe Constant Deviation

TOP 1.343 1.380 Left 1.744 1.41 5

Right 1.754 0.883

Table 14: Calibration Constants for Shock Generator Tests

4 -7 -7.6 0.0 -0.8 7.35 6.48 6 -6 -6.3 0.0 0.2 6.87 4.56 8 -4 -3.8 0.0 0.6 6.39 4.23

Table 15: Program Results for Shock Generator Tests

6.5 Tube Length

As a final consideration in the development process, the length of the tubes from

the tip of the probe array to the transducer face was changed to study the effect of tube

length on the response characteristics of the angled probe. It is important to ascertain the

change in response tinie and accuracy with respect to the length of the tubes in order to be

able to construct an effective final design. It was expected that increasing the length of the

tubes would lead to a decrease in the response time which would eventually lead to a loss

of precision in the instrument.

Mo = 8.32. The length of the plastic tubing that joined the probe cylinders in the array to

the transducer mounts was then varied. The fength of the plastic tubes was varied from

the minimum design length of 55 mm to a maximum length of 350 mm. The probe

cylinders themselves are 75 mm in Iength. The resulting range in the total length of tubing

from probe tip to transducer face (Xt) was 130 mm to 425 mm.

The results from this investigation are shown in Table 16 and Figure 40. The

initial increases in X, of up to 325 mm cause no discernible effect on the pressure histories.

The variations in the measured pressures for these X, values lie within the 1.4% scatter

which is common to the facility. It is clear that any subsequent increase in Xt causes a

marked dehy in the pressure response of the transducers. This increase in the time

constant of the transducers causes the measured Pwi to be higher. The increased response

times cause the pressure histories to be progressively more attenuated, smoothing and

drawing out the peaks in the signal. Note that the standard normalisation interval of 11

ms could be reduced to omit the drawn out initial pressure spike. Adjusting the

normalisation interval in this fashion would bring the measured Pwi into agreement with

the preceding ones. However, the required reduction in the size of the normalisation

interval to 5 ms was considered too iimiting.

Table 16: Results from Tube Length Tests

Since the system of equations that is solved by the program is very sensitive to

small changes in Pwi, slight variations in the measurement due to X, can cause a significant

change in the converged answer. Hence, any deviation in the pressure measurement was

considered unacceptable. The length of the tubes should, therefore, be limited to a

maximum total length of 325 mm from the probe tip to the transducer. However, when

the final mode1 is designed, it may make use of smaller diarneter tubing to reduce the size

of the probe head. Should this be the case, the volume of air between the probe tip and

the transducer face for an equal X, will decrease. This reduction in the affected air mass

will presurnably allow for a greater overall length before degradation of the response time

is experienced. The limit suggested above could possibly be increased without

experiencing any negative effects.

7.0 Conclusions

Investigations into the flow characteristics of the suggested 3D probe array

demonstrated that the desired flow structure was established. Some experiments have

shown that there may be unknown flow characteristics present at the mode1 head due to

the gap between the probes. The current design has dificulties maintaining the correct

orientation when subjected to the harsh environment of the gun tunnel. Even with such

difficulties, it was possible to successfully obtain pressure measurements that were

consistent with theoretical expectations.

The program functions as desired when given theoretical values. However, the

nature of the solving routine makes the system of equations particularly sensitive to small

perturbations in the relationship between the rneasured pressures. The current prototype

is not sensitive enough to suficiently minimise these perturbations, causing the program to

reach erroneous solutions for Mo during operation.

8.0 Recommendations

The greatest difficulties have come from the perturbations in the pressure

measurements. A new design which would minimise these perturbations is required.

There are two possible geometries that could be investigated to achieve this end. The first

is to build the tri-diagonal pyramid head that is shown in Figure 4 1 (a). Such a probe array

essentially places a pyramidal cap on the old probe head. There are three distinct faces,

each of which have a pressure tap. This probe head would eliminate the tip effects caused

by the gap between the apexes of the cylinders present in the previous cylindrical probe

array. The next geornetry to investigate is a conical probe head as displayed in Figure

4 1 (b). This modification would replace the pyramidal cap with a conical one. This

geometry would also eliminate the tip effects caused by the gap. It is not known whether

the conical tip would maintain three independent flow structures, nor whether the flow

would remain distinct enough to have three independent sets of equations. In either case,

the effect of the location and size of the pressure taps must be considered. This is

particularly important in the conical head where the taps will create a missing ellipse on

the curved surface which can cause significant interference patterns.

9.0 References

Arnes Research Staff. "Equations, Tables, and Charts for Compressible Flow". Report 1 135, National Advisory Committee for Aeronautics, 1953.

Bement D.A. et al. "Measured Operating Characteristics of a Rectangular Combustor / Inlet Isolator" AIAA, SAE, ASME 26Ih oint Propulsion Conference, Orlando, July 1990.

Billig, F. S . et al. "Inlet-Combust ion Interface Problems in SCRAMJET Engines" International Airbreathing Propulsion Committee, Marseilles, France, 1 st

International Symposium on Air Breathing Engines, June 1972.

Bryer D.W. and Pankhurst R.C. "Pressure-Probe Methods for Determining Wind Speed and Flow Direction" National Physical Laboratory, Her Majesty's Stationery Office, London, 197 1.

Challenger D. "An Experimental Study of Hypersonic Flow In a Square Duct" MASc. Thesis, University of Toronto Institute for Aerospace Studies, 1995.

Chen L. P. "An Experimental Study of Flow within a Circular Duct" MASc Thesis, University of Toronto Institute for Aerospace Studies, 1992.

Chue S.H. " Pressure Probes for Fluid Measurements " Progressive Aerospace Science, Volume 16, No 2, pp 147-223, 1 975.

Davis D. O. "Experimental and Numerical Investigation of Steady, Supersonic, Turbulent flow Through a Square Duct" MASc. Thesis, University of Washington, 1985.

Davis D. 0. and Gessner F.B. "Further Experirnents on Supersonic Turbulent Flow Development in a Square Duct" AIAA Journal, Vol. 27, No 8, August, 1989.

Feng Lin. "Prelirninary Investigation on the Technique of Mini-Four-Hole Probe" Nanjing Aeronautical Institute. Journal of Propulsion Technology. August 1992.

Gessner F. B. et al. "Experiments on Supersonic Turbulent Flow Development in a Square Duct" AIAA Journal, Vol. 25, May, 1987.

Gordon K. A. "An Investigation of Local Mach Numbers in the UTIAS Hypersonice Gun Tunnel" BASc Thesis, Faculty of Applied Science and Engineering, University of Toronto, 199 1.

Richmond J. K. and Goldstein R. "Fully Developped Turbulent Supersonic Flow in a Rectangular Channel." AIAA Journal No. 8, p. 133 1 - 1336, 1966.

Lustwerk, F. "The Influence of Boundary Layer on the Normal Shock Configuration" MIT, Cambridge, Massachusetts, Meteor Report #966 1, September 1950.

McLafierty, G.H. et al. "Investigation of Turbojet Inlet Design Parameters" United Aircraft Corp., East Harford, Connecticut, Report R-0790- 1 3, December 1955.

Sisiian, J.P, He, Z. and Deschambault R.L. "An Experimental study of Hyperosnic Flow Developmwnt in a Pipe" UTIAS Technical Note #275, August 199 1.

Sitaram N. and Treaster A. L. " A Simplified Method of Using Four-Hole Probes to Measure Three-Dimensional Flow Fields7' Journal of Fluids Engineering, Vol. 107, March 1985.

Terzioglu L. "An Experimental Study of Static Pressure Probe Geometries and Probe Mounts in Hypersonic Flow" MASc. Thesis, University of Toronto Institute for Aerospace S tudies, 1994.

Treaster A. L. and Houtz, H. E. "Fabricating and Calibrating Five-Hole Probes" Fluid Measurements and Instrumentation Forum, American Society of Mechanical Engineers, Atlanta, 1986.

Treaster A. L. and Yocum A. M. ''The Calibration and Application of Five-Hole Probes" ISA Transactions, Vol. 18, No. 3, p 23-24, 197 1.

(a) Representation of Constant Pressure Contours

(b) Representation of Circulation Cells

Figure 1: Flow Phenomena Discovered by Davis and Gessner(l989)

DIMENSIONS IN M ILLIMETERS

Figure 2: Transverse Flow Probes

I Testing Time

-Shock Waves /'

Legend:

1. Driver 2. Bal1 Valve 3. Breech with Diaphragms

4. Barrel 5. Barrel Lock 6. Nozzle

7. Test Section 8. Diffuser 9. Dump Tank

Figure 3: UTIAS Hypersonic Gun Tunnel

Figure 4: Standard Pressure History

Figure 5: Standard Schlieren Photograph

Figure 6: Wedge Mount Design

(A) Pitot Probe

(B) Static Pressure Probe

(C) Angled Probe

Figure 7: Sketch of Probe Types

Figure 8: Schlieren Photograph of Preliminary Probe Development Apparatus

Shock-Wave f

Figure 9: Notation for Normal Shock-Waves

Shock-Wave

Figure 10: Notation for Oblique Shock-Waves

Figure 11: Schlieren Photograph of Angled Probe Apparatus

Figure 12: Angle Relations

Figure 13: Angled Probe Pressures

Figure 14: Prototype Apparatus

Figure 15: Schlieren Photograph of Prototype (41 < 0°)

Figure 16: Schlieren Photograph of Prototype (4 > 0")

(a) Sketch o f 2D Array

(b) Sketch of 3D Array

Figure 17: Suggested Probe Arrays

(As seen facing the probe arrays fiom the front)

(a) Numbering Conventions

(b) Axis Conventions

Figure 18: Probe Array Conventions

My = - Mm COS^ = - M o cosa

Mx = Mx, s inP

= M , c o s a

c o s p

s in p

Figure 19: Vector Components

Figure 20: Probe Tip Relationships

Figure 21: El Solution Space for 2D Probe

Figure 22: Ez Solution Space for 2D Probe

Figure 23: Ep Solution Space for 2D Probe

Figure 24: EE Solution Space for 2D Probe

Figure 25: Plot of EE Valley Floor

, - _.?.. :: - 7 . Ci.. .C

i - + : . . . * . -. .'. -.

,"-"-t.... ai.. .L-. * ... c:.. ,2:..- . ... ,..W..

Figure 26: ET Solution Space for 3D Probe

Figure 27: Cross-sections of ET Solution Space at Constant-a

Figure 28: Cross-sections of ET Solution Space for Constant+

Figure 29: 3D Probe Pressure History

Figure 30: Schlieren Photo of 3D Probe Tip

Figure 31: 3D Probe Pressure History with Oscillations

1 neoretlcal Pressures

a Top Probe

i Lower Lefl

A Lower Right

Experimental Pressures

Top Probe Lower Left

A Lower Right

Figure 32: 3 D Probe Pressures for Varied a

Lower Left Probe

Lower Rlght Probe

Figure 33: Comparison of 3D Probe Pressures for Varied a

Lower Left Probe

1 . _ _ _ ___. ._ - -- -- . A-.-. - -.A -. .. -- -A-- .- - -- - - . - ... -- - . -- - - -.

Figure 34: 3D Probe Pressure Corrections for Varied a

4 Top Probe

Ii Lower Lefi

A Lower Right

Experimental Pressures

I I 16 O00

I 14 O00 + - - a - - - * 12.000

A i o o o o A

8 O00 1 Top Probe

O Lower Left

A Lower Right

Figure 35: 3D Probe Pressures for Varied f!

Lower Lett Probe

Lower Rlght Probe

Figure 36: Comparison of 3D Probe Pressures for Varied

Lower LeR Probe

Lower RIgM Probe

Figure 37: 3D Probe Pressure Corrections for Varied

O O z P

O O - z

t!sd) 13

Figure 38: Mach Number Sensitivity

Figure 39: Schlieren Photograph of Shock Generator Apparatus

Figure 40: Tube Length Pressure Changes

(a) Trigond Pyrarnid Probe Tip

(b) Conical Probe Tip

Figure 41: New Probe Geometries

Appendix A

Photographs of Apparatuses

{Above) Prototype Open to Show Tube Length

{Below) Shock Generator Apparatus

{Above] Angled Probe Investigation Apparatus

{Below f Probe Array Investigation Apparatus

{Above] Prelirninary Probe Development Apparatus

{Bqlow ) 3D Probe Apparatus

Appendix 6

Apparatus for Preliminary Probe

Development Tests

Appendix C

Derivation of System of Equations

To obtain normal component of Mach number in probe fixed frarne, calculate a unit vectorfm opposite the local velocity vector Vo, and then calculate a unit vector normal to the presumed wedge-shock Tsi.

Example: SingIe wedge with shock nomal in yz-plane

P c' c z, = COS aly + sin m,

2.2 Calculation of id

For probe # 1, by definitionTsl lies in the yz-plane. That is,

? A

I , , = sin B,< + cos B,i,

For other two probes, consider a rotated CO-ordinate system which maintains the same relationship geometricaliy to the probe as x, y, z does toTs2.

l)Rotate/l;, about T, axis to get î,.

2)Rotate Td about 7,. to get îx7+ = ',

?

Zr,I = cos a sin & + cos a cospc + sin a<

=iS, = cos a sinpcos 8, sin E~ + COS^ cosflsin 4 + sin a cos 8, cos gi

? A

zsi = sin 8,c1 + cos qi,,

Here

t' z, = cos 8, sin gi; + sin B,;, + cos cos gi;.

3.0 Systern of Equations

The right hand column of nurnbers listed with a slash represent the number of unknown variables in the system compared to the nurnber of equations whîch have been currently developed. For a system of equations with 3 unknowns and 4 equations the notation would be 3/4.

The variables Iisted following each equation indicate the unknowns which were introduced within that equatîon.

It is assumed that Mo, Pte, a, and p are unknown.

The pressures Ppi, Pwl, Pwz, PW3 are measured by the probes.

3.1 Measure Pitot Pressure

Introduces: Pte, Mo

3.2 Measure Three Tube Pressures

From geometry of array = O". [sin€, = 0, coscl = I l

Introduces: Mn,

Introduces: 8

M , , = ~ , ( c o s a c o s p s i n ~ , + s i n a c o s ~ , ) 614

Introduces: cc, f3

Introduces: Md

Introduces: €17

1 M ,, = M , ( G c o s a s i n p c o s & + c o ç a s i n p s i n 49, - T s i n a c o s $ , 1 Introduces: None

Introduces: Mn3

Introduces: e3

f i 1 M n , = M o ( - cosa a. s i n p c o s 0 , + c o s a c o s p s i n 6, - sin a c o s t

10/10

Introduces: None

Closu re m

Appendix D

Sample MathCada Document

This MathCad@ Document will return the wedge pressure values for the 30 probe given the M 0, Pts and incident angles of attack.. . . .

Known Values:

Enter Mach number test was conducted ai,

Mo =8.3

Enter Barrei sensor measurements,

Pt, 3 897.65

Enter angles of attack in degrees,

a - 5.0 p - - 5 . 0

Calculations:

Normal shock relation.. .

Decomposition into vectors gives:

Rotation matrices are:

Applying the these matices,

Effective Mach number and angle:

Effective deflection angle.. .

6a ; 0.61 O865

61 =6a + y 1 62 =6a+ y2

61 = 0.69846 62 = 0.64271

Primer values to initialise solver.. . 01 ' - 0.7 02 . = 0.7

Solve for 01

Given

Solve for 02 Given

Solve for 83 Given

Oblique relationships.. .

for a measured static pressure on the wedge face:

Appendix E

2D Probe Code Listing

- - - - - - - - - - a - - -

{ Probe Solution ) { 1 { Name : WilliamOIGorman 3 { College Number : 009558515 1 ( Supervisor : Dr. P . A . Sullivan 1 { 1 { . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1

PROGRAM SolProbe (output) ;

CONST

TYPE

VAR

PHigh PLow

Pts

C

Tol

N

NN

MaxNum

DoubleArray = Array [1..NI of Double; SearchArray = Array [ l . . M \ J ] of Double;

Outfile

Line, MachMat

Alpha, Guess , ax, bx, Last, Pwl, Pw2, Ptl, Pt2

Counter , Point

: Text;

: DoubleArray;

: Double;

: Integer;

{Measured values. 1 {Upper Wedge pressure. ) {Lower Wedge pressure. 1

{Pitot probe pressure. )

{Calibration Constant. 1

{Fractional accuraccy. 1

{~inesearch resolution. 1

{Search vector size. 1

{Maximum iterations of M)

{Error matrix. 1 {Mach number matrix. )

{Angle of attack. 1

{Limiting bounds. 1

{Wedge pressure for #. ) {Total pressure for # . 1

(Index marker for array}

{ This module evaluates the coded function at the point AA passed into the} {sub-routine. i.e. Func = f(AA) The formula coded is the oblique shock } {relationship between deflection angle and shock angle written to equal 0. }

FUNCTION FuncAng (SAngle, Delta, SMach :Double):Double;

VAR Temp : Double;

BEGIN

Ternp : = 6*SMach*SMach / ( 5 * ( SMach*SMach*sin(SAngle)*sin(SAngle)-1 ) ) ;

END ;

{- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 1 { This is the secant solver method to find the Theta root of FuncAng. It} {needs the effective deflection angle and Mach number as well as lirniting } {bounds on the angle Theta. It has been adapted fxom the "Numerical Recipes 1 (in Pascal Text". The value of the function is the root. 1

Function RootSec ( Delta, SMach, xl,x2 :Double; var Found : Boolean 1 :Double;

CONST MaxIt = 100; X e r r = 1E-4;

VAR dx, f, fl, swap,xl,rts: Double;

j : integer;

B E G I N Found : = false;

£1 := FuncAng(x1, Delta, SMach); f := FuncAng(x2, Delta, SMach);

If abs(f1) < abs(f) then begin rts:= xl; x1 := x2; swap :=fl; £l:=f; f := swap;

end Else Begin

x1:= x1; rts :=x2;

end ;

While j < MaxIt do begin

dx : = (xl-rts)*f/(f-£1); xl: =rts; fl:=f; rts : =rts+dx; f:=FuncAng(rts, Delta, SMach);

{Maximum iterations. } {Tolerance to satisfy. }

(Function to evaluate. }

{Set bracketing. 1

{Do iterations.

If (abs(dx) < xerr) OR (f = 0.0) then {Acceptable conditions.} Begin

j : = Maxit; Found : = true;

End Else If (abs (dx) > 1E8) then Begin

j : = Maxit End ;

{When system fails to 1 {converge, stop looking}

end;

If Found then

END ;

{- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 1 { ~ h i s procedure will calculate the pressures in front of the oblique 1 {shock waves caused by {and Pressure values.

PROCEDURE Anglepress (

VAR Dell, De12,

Thetal, Theta2,

Intl, Int2, Up, Down

Found

BEGIN up := 1.57; Down := 0.2;

the wedge tube rneasurements. It needs the Mach, kgle} 1

MM, AA : Double ; var Pwl, Pw2, Ptl, Pt2 :Double ) ;

{Effective deflection. )

: Double;

: Boolean;

Del1 := 0.610865 + AA; Del2 := 0.610865 - AA;

Thetal := RootSec (Dell, MM, Down, Up, Found);

IF Found {AND (Thetal > O)} then Begin

Intl := sin(Theta1);

{Shock Angle matrix. }

(Intermediary values. } (Bounds on guess. 1

(Error trap variable. 1

( Limits for the guess 1 {of Theta in solver. }

{Calculate the pressure}

Pt1 := Pwl * exp(-7/2 *ln( 5/ (MM*MM + 5) ) ; Pt1 := Pt1 / ( (7*MM*MM*Intl*Intl - 1) /6);

End ELSE Begin {or set an error flag. 1

Pt1 := l.lE9; End;

Theta2 : = RootSec (De12, MM, Down, Up, Found) ;

IF Found {AND (Theta2 > O) 1 then Begin

Int2 := sin(Theta2) ;

Pt2 := Pw2 * exp(-7/2 *ln( 5/ (MM*MM + 5 ) 1 ) ; Pt2 := Pt2 / ( (7*MM*MM*Int2*Int2 - 1) /6);

End ELSE Begin

Pt2 := 2.239; End;

END ;

{- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 1 { This Function will calculate the Each error profile of the system. The } {ErrE term consisls of the difference between the Pt calculated £rom the 1 {oblique shock relations for each of the Angled probes at the current Mach 1 {number and Alpha. 1

BEGIN

Anglepress ( Mach, Alpha, Pwl, Pw2, Ptl, Pt2 ) ;

ErrE := abs ( Pt1 - Pt2 ) ;

END ;

{ This procedure will calculate the array of points surrounding the } { current guess. The array is spaced at a distance Delta from the guess ) { stored in [Il. The array has two points, forward and backward. 1

Procedure Angles ( var Alphus :DoubleArray; Del ta : Double) ;

BEGIN

Alphus[S] := Alphus[ll + Delta;

Alphus[3] := Alphus[ll - Delta;

{ UP

{ Down

END ;

{--------------------------------------------------------------------------- 1 { This procedure will take the current mach number and find the flow angles) {which satisfy the wedge pressure values. It matches the PT# terms for al1 ) {probes using the ErrE. It must be passed the Mach#, Alpha, and al1 of the 1 {Wedge and Pitot pressures. 3

PROCEDURE AngleFind (var Alpha, ax, bx, Mach, Pwl , Pw2, Pt1 , Pt2 : Double ) ;

CONST MaxIt= 500;

VAR Alphus, Grid : DoubleArray;

Error, Current , Steepest, Delta : Double;

i l Count , Choice : Integer;

{Maximum iterations. 1

{ G r i d of surrounding A.) {Function evaluations. )

{Error tracking. 1 {Current gradient. 1 {Largest gradient. 1 {Change in Angle. 1

{Number of iterations. 1 {Best new point to take)

BEGIN

Count := 0 ; {Initialise variables. ) Error := 1 ; {ErrE( Mach, Alpha, Pwl, Pw2, Ptl, Pt2);) Delta :=0.1;

While ((Exxor*î0000) > Tol) AND (Count <= MaxIt) Do Begin

Alphus [ 1 1 : = Alpha ; {Set location of guess.)

Choice := 1 ; Steepest := 0.0;

{Reset criterion.

Count := Count + 1;

{Find e r r o r values. 1 G r i d [ l l := E r r E ( Mach, Alphus[l l , Pwl, Pw2, P t l , P t 2 ) ;

For i := 2 t o 3 Do Begin

Gridii.1 := E r r E ( Mach, Alphus [ i l , Pwl, Pw2, P t l , Pt2) ;

Current := Grid[ i l - G r i d [ l ] ; {Take gradient i n f o . 1

I f Current < Steepest then Begin

Steepest : = Current; Choice : = i;

End;

End ;

I f Steepest = 0.0 then Begin

Delta := Delta / 2 ; End Else Begin

Alpha := Alphus[Choicel;

Error : = Grid [Choice 1 ; Delta := Delta * 0.90;

End ;

{Compare t o o thers i n } { t h i s a r ray f o r b e s t . }

{If none found reduce 3 { s t ep s i z e used. 3

{Take bes t gradient as 3 { the new guess. 1

w r i t e ( ' . ' 1 ;

End ;

END ;

{ This procedure w i l l c a lcu la te the pressure i n f ront of the normal } {shock wave caused by the p i t o t tube measurement. I t needs the Mach number ) {and a v a r i a b l e t o r e tu rn as t o t a l pressure. The measured p i t o t pressure i s } {taken a s a Global value named Pts s e t i n the program constants . 1

PROCEDURE ~ i t o t p r e s s ( var MM, PT :Double ) ;

BEGIN PT := Pts* exp(-7/2*ln( 6*MM*MM / ( M M * M M + ~ ) ) ) ;

PT : = PT * exp(-5/2*ln( 6 / (7*MM*MM-1) ) ) ; END :

{- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 3 { This procedure w i l l c a lcu la te the pressure er ror f o r the system. The } { ErrP term c o n ç i s t s of the cununulative t o t a l of d i f ferences between the 3 { P t ca lcu la ted £rom the P i t o t tube and the Angled probe r e l a t i o n s a t the } { current angle of a t t ack and Mach number. The e r r o r term i s returned a s the} { value of the function. 3

Function E r r P ( var Mach, P i t o t , P t l , Pt2 :Double ):Double;

BEGIN

Pi to tPress ( Mach, ~ i t o t ) ; {Find Pitot pressure . 1

END ;

(--------------------------------------------------------------------------- 3 ( This procedure will initialise the search vector and the Mach number } (matrix. The error is the ErrE value along the Alpha line for Mach numbers 3 (from ax -> bx with a number of points equal to N. 3

PROCEDURE Ini tLine ( Alpha, ax, bx, Pwl, Pw2, Pt 1, Pt2 : Double; var MachMat, Line : Doublearray ) ;

VAR i : Integer; {Loop variable. 3

Temp, Pitot : Double;

B E G I N

Pitot := 0 ;

For i : = 1 to N do Begin

MachMat [ i l : = abs (ax-bx) * (i-1) / (N-1) ; {Incremental Mach #. ) MachMat[il := MachMatii] + ax;

{Calculate Error term. } Line[il := ErrE ( MachMat[il, Alpha, Pwl,PwS, Pt1,PtS ) ; Line [ i 1 := Line[i] i ErrP ( MachMat[il, Pitot, Ptl, Pt2);

End ;

END ;

{-------- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 1 { This procedure will find the area of the global minimum as a function} {of Mach number for the predetermined angle of incidence. It returns the 1 (location of the lowest error in the array Line as the integer Point. 1

PROCEDURE LowLine ( var Line :DoubleArray; var Point :Integer ) ;

VAR

i : Integer; {Loop variable. 1

Smalles t : Double; {Minimum value f ound. )

B E G I N

Smallest : = 1E6; {Initialise variable. )

For i := 1 to N do {Compare. 3 Begin

If (Line[i] < Smallest) (AND (Line[i] <z 0.0)) then Begin

Srnallest : = Line [il ; {Assign values if less.} Point : = i;

End ; End ;

END ;

PROCEDURE SetLine ( Point : Integer ; var ax, bx : Double ) ;

VAR Centre, Temp : Double;

BEGIN

Temp : = abs (ax-bx) ;

Centre := ((Point-1)/N) * Temp + ax;

ax := Centre - Temp/4; bx := Centre + Temp/4;

{Current Mach range. 1

{Make MachMat centre to) { v a l u e at minimum ErrE.1

{Reduce range O£ array.}

END ;

{--------------------------------------------------------------------------- 1 { This function will write the values passed to it into a datafile named) {in the Outfile string. It is currently set up to write the data values in } {a structured formnat. It also includes the input pressures. 1

PROCEDURE Output (var Outfile :text; Mach, Alpha, ErrorMach : Double) ;

VAR il j : Integer; {Loop variables.

BEGIN

writeln (Outf ile) ; writeln (Outfile, 'For pressures of: ',Pwl:10:3,' and1,Pw2:10:3); write (Outfile, 'Alpha found to be: ',Alpha:lO:5 ) ; write (Outfile, 'Mach found to be: ',Mach:lO:2 ) ; writeln (Outf ile, ' with ' , ErrorMach) ; writeln (Outf ile) ; writeln (Outf ile) ;

END ;

{++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++) BEGIN

assign (Outfile, 'PrbSol.outl); append (Outf ile) ;

Pwl : = PHigh + C ; Pw2 := Plow + C ;

Guess := 7.50; Alpha := 0.04 ;

{Ready datafile for program.)

{Initial guess. 1

{Get pressure values. 1 ax := ErrE ( Guess, Alpha, Pwl, Pw2, Ptl, Pt2 1 ;

{Initial angle bounds. 3

{Find the angle Alpha. 3

C o u n t e r := 0 ; L a s t := 1;

{ S e t l o o p p a r a m e t e r s . 1

{ I n i t i a l Mach b o u n d s . 1

{ L o c a t e the M a c h n u m b e r . }

While ( L a s t > T o l AND ( C o u n t e r <= MaxNum) AND (ax <> bx) Do B e g i n

C o u n t e r := C o u n t e r + 1;

~ n i t ~ i n e ( ~ l p h a , a x , b x , P w l , P w 2 , P t l , P t 2 , M a c h M a t , L i n e ) ;

L o w L i n e ( L i n e , P o i n t 1 ; {Find greatest m i n i m u m . 1

L a s t : = L i n e [ P o i n t 1 ;

S e t L i n e ( P o i n t , ax, bx ) ;

End;

w r i t e l n ;

{ R e c o r d n e w e r ror v a l u e . )

{ R e a d j u s t f o r r e s o l u t i o n )

{ F i l e t o Save data i n . } O u t p u t ( O u t f i l e , M a c h M a t [ P o i n t l , A l p h a , L i n e [ P o i n t ] ) ; close ( O u t f i l e ) ;

END.

Appendix F

Flowchart of 3D Solver

I ! Begin l

1

i Read lnput ,/ /

from User j'

,/' /

r-- t

Check lnput 1 I

1 -L -. .

I I

1 l

1 Convert input string \ i into a number i i !

Apply Calibration Constants

1 Initialise Parameters

1 Initialise Mach Maû-ix

Determine the Angles of Attack

i; The procedure "AngleFind" is outlined in greater detail

,.. ' in the following pages.

1 Determine the

l Mach Number

Output / Solution /'

procedure MachFind is outlined in greater detail

1.'

__,, , L---- in the following pages.

l ---- - - -- -f_ . - . . - - -

( ' End \

I f - [ Initialise 1

1 Parayeters 1 l

, t--\

/ Establish Search Grid

Find point in Grid with least Error I

Reassign this point as centre of Grid i

1 Store the value of least / i Error I

-- -- - - - . ..t . . . - --

;. ,, [ 1 End While

Initialise Parameters 1

( While ErrT 100 large

w ,'

/ Determine ErrT I

Appendix G

30 Probe Code Listing

1 r r o ~ e so~u~ion 1 1

( Name : William O ' Gorman 1 ( College Number : 009558515 1 ( Supervisor : Dr. P.A. Sullivan 1 E 1 ( . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1

PROGRAM SolProbe (output);

( This program requires the three pressure measurements £rom the 1 2 0 ' probe) (array and the pitot pressure measurement at the same location, to be input ) {by the user. It will then proceed to isolate the incoming flow direction ) {and the Mach number. The flow direction is specified by the angles Alpha ) {and Beta. They measure the angle within the ZY and XY planes respectively. ) (The geometry is set up specifically with the Top probe axis vertical, 1 {aligned with the z-axis. The y-axis lies along the probe axis. 1

CONST Cl = 0 . 8 3 0 9 ; C 2 = 2.0097 ; C 3 = 2 . 5 8 7 0 ;

Tol = 1E-10 ;

TYPE VectorArray = Array [1..3] of Double; DoubleArray = Array [l..N] of Double;

VAR Outf ile : Text;

Test : String;

Pitot, Ptl, Pt2, P t 3 ,

Alpha, Beta,

Pts,

Pwl , Pw2, Pw3,

MLow, MHigh,

PPt , PTOP, PLeft, PRight : Double;

Point : Integer;

{Calibration Constant. 1

(Fractional accuraccy. )

(Linesearch resolution, 1 (determines gridpoints.)

{Array of error terms. 1

{Working rnatrix for M. 1

{Four calculated total 1 {pressure terms. 1

{Angles of attack.)

{Pitot pressure on face)

{Wedge pressures on the) (probe faces. 1

{Boundaries on the Mach) inumber search. 1

{~easured pressures 1 {£rom the experiments. }

{Place holder. 1

{ This procedure will take the string of numbers, including the possible ) {decimal point, passed in list UserInput and return the numerical equivalent) {as a double precision number in Value. The length of the input string and 1 {position of the decimal point are required. The string characters other ) {than the period must be al1 numbers. 1

PROCEDURE ValNumber ( var UserInput :String; LineLn, Place : Integer; var Value : Double) ;

VAR Start, Finish, i, Temp : Integer ;

BEGIN Value := 0 ;

If Place = O then Begin

{Account for no decimal.)

Start := LineLn; Finish := 1;

For i := Start downto Finish Do {Recompose number . 1 Begin

Temp := Ord( UserInput[i] ) - Ord( 'O' ) ; Value := Value + Temp * exp( (Start - i) * ln(10) ) ;

End ;

End Else Begin

Start := Place - 1; Finish := 1;

{For # with decimals. 1

For i := Start downto Finish Do {Recompose large portion) Begin

Temp := Ord( UserInput[i] ) - Ord( 'O' ) ; Value := Value + Temp * exp( (Start - i) * ln(l.0) ) ;

End ;

Start := Place + 1; Finish := LineLn;

For i : = Start to Finish Do { Include decimal portion) Begin

Temp := Ord( ~serInput[il ) - Ord( 'O' ) ; Value := Value + Temp / (10 * exp( - (Start - il * ln(10) ) ) ;

En6 ; End ;

END ;

{ This procedure will check the UserInput passed to it to ensure that 1 {the characters correspond to a valid number. If the string represents a } {valid number then it calls ValNumber to establish a numerical equivalent 1 {which is returned in Value and Done is set to True. If the string is not } {a valid number then the Done flag is set to False. 1

PROCEDURE CheckIn ( var UserInput :String; var Done : Boolean; var Value :Double ) ;

VAR LineLn, i, Decimal, Place : Integer;

Numbers , Ended : Boolean ;

luuilluers :- 1rue; Ended := False; LineLn : = Length (UserInput) ; Decimal : = 0; Place := 0; i := 1;

Repeat {Compare to digit or . 3

If (UserInput[il < '0') OR (UserInputli] >'9') then Begin

Numbers := False;

If UserTnput[i] = ' . ' then Begin

Numbers : = True; Place : = i; Decimal : = Decimal + 1;

End; End;

If i > Lineln then Ended : = True;

Until not Numbers or Ended;

If Decimal > 1 then Numbers : = False; {Ensure no double " . " 1

If Numbers and Ended then {For a number convert it) ValNumber (UserInput, LineLn, Place, Value) ;

Done : = Numbers; END ;

{-------- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 1 { This procedure will get the user to input the measured pressures and ) { Save the entered values in the passed variables. It is set up for the 1 {three probe array with a seperate Pitot measurement. It uses CheckIn to 1 {verify that the user inputs numbers. 3

PROCEDURE UserIn ( var Test :String; var Pts, PTop, PLeft, PRight :Double ) ;

VAR Done : Boolean ;

UserInput :String;

BEGIN wri teln; writeln ('Please type the experiment name:'); readln ( Test 1 ;

writeln; writeln ( 'Please enter only numbers. ' ) ;

Done : = False; Repeat

writeln ('Enter the pressure measured by the PITOT probe....'); readln ( UserInput ) ; CheckIn ( UserInput, Done, Pts ) ;

Until Done;

writeln; writeln ('Enter the pressures measured by the 3-probe array.');

Y V A I G . - L U A = = ,

Repeat writeln; writeln ('The value £rom the TOP probe was . . . . O ) ;

readln ( UserInput ; CheckIn ( UserInput, Done, PTop ) ;

Until Done;

Done := False; Repeat

writeln; writeln ('The value from the BOTTOM LEFT probe was . . . . ' ) ; readln ( UserInput 1 ; CheckIn ( UserInput, Done, PLef t ) ;

Until Done;

Done := False; Repea t

writeln; writeln ('The value £rom the BOTTOM RIGHT probe was . . . . ' ) ; readln ( UserInput ) ; CheckIn ( UserInput, Done, PRight ) ;

Until Done;

writeln; writeln ('Thank You : ) ' ) ; write ('Computing . . . ' ) ;

END ;

PROCEDURE InitMach ( MLow, MHigh var MachMat

VAR i : Integer;

: Double ; :DoubleArray ) ;

{Loop variable.

BEGIN

For i := 1 to N do Begin

MachMat[i] := abs(MLow-MHigh) * (i-l)/(N-1); {Incremental Mach # . MachMat[i] : = MachMatliI t MLow;

End ;

END ;

PROCEDURE CalcVector ( MM, AngleYZ, AngleXY :Double; var VectorA :VectorArray ) ;

VAR Hypotenuse : Double;

B E G I N VectorA[3] : = - MM * sin( AngleYZ ) ; Hypotenuse : = MM * cos ( AngleYZ ) ;

{ z-axis cornponent. 1

VectorAE2I : = - Hypotenuse * cos( AngleXY ) ; { y-axis component. 1

{--------------------------------------------------------------------------- 1 { This procedure will rotate the given vectorA components around the y 1 Caxis through the provided angle (rneasured in degrees) and then will return } {the components in the new reference frame as vectorB. 1

PROCEDURE RotateVector ( Angle :Double; Vec torA :VectorArray;

var VectorB :VectorArray ) ;

VAR RAngle : Double ;

BEGIN RAngle := Angle / 180 * Pi;

{Angle in radiians. 1

{Convert to radians. 1

VectorBIlI := cos( RAngle * VectorAIl] - sin( RAngle ) * VectorA[3];

VectorB[3] := sin( RAngle ) * VectorA[l] + cos ( angle ) * VectorA[31 ; END ;

{--------------------------------------------------------------------------- 3 { This procedure will find the effective angle of attack and Mach 1 {number in the current frarne of reference given the vector components in 1 {that frame. The x-axis component is pârallel to the shock wave and hence 3 {makes no contribution to the oblique shock. 1

PROCEDURE Effective (var EEfMach, Chi :Double; Vec t orA :VectorArray ) ;

VAR RAngle : Double; {Convert to radians. 3

BEGIN Chi := arctan( VectorA[3] / VectorA[2] ) ;

EffMach := Sqrt( VectorA[3l*VectorA[3j + VectorA[S]*VectorA[2] ) ;

{ This module evaluates the coded function at the point AA passed into the} {sub-routine. i.e. Func = £ ( A A ) The formula coded is the oblique shock 3 {relationship between deflection angle and shock angle written to equal O. 1

FUNCTION FuncAng (SAngle, Delta, SMach :Double) :Double;

VAR Temp : Double;

B E G I N

Temp : = 6*SMach*SMach / ( 5 * ( SMach*SMach*sin(SAngle)*sin(SAngle)-1 ) ) ; Temp : = (Ternp -1 ) * sin ( SAngle) / cos (SAngle) ; FuncAng : = cos (Delta) / sin(De1ta) - Temp;

END ;

{ - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - > { This is the secant solver method to find the Theta root of FuncAng. It)

{in Pascal Text". The value of the function is the root.

Function RootSec ( Delta,SMach,xl,x2 :Double; var Found :Boolean ) :Double;

CONST MaxIt = 100; Xerr = 1E-8;

VAR d~,f,fl,swap,xl,rts: Double;

j : integer;

BEGf N Found := false;

£1 := FuncAng(x1, Delta, SMach); f : = FuncAng(x2, Delta, SMach);

If abs ( f 1) < abs ( f ) then begin rts:= xl; x1 := x2; swap :=fl; fl:=f; f := swap;

end Else Begin

xl:= xl; rts :=x2;

end;

While j < MaxIt do begin

dx := (xl-rts)*f/(f-£1); xl: =rts; fl:=f; rts: =rts+dx; f:=FuncAng(rts, Delta, SMach);

{Maximum iterations . } {Tolerance to satisfy. 1

{Function to evaluate. }

{Set bracketing. 1

{Do iterations.

If (abs(dx) < xerr) OR (f = 0.0) then {Acceptable conditions.} Begin

j : = Maxit; Found : = true;

End Else If (abs(dx) > 1E8) then Begin

j := Maxit End ;

{When system fails to } {converge, stop looking}

end;

If Found then Begin

rootsec := rts; End ;

END ;

{- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 1 { This procedure will calculate the pressures in front of the oblique }

~ L ~ L U L I I S LIE GCUGULCLL~U LUCCLI pressure Derore izne snocK wave as r'r. ~t usesj {the subroutine RootSec. 1

PROCEDURE AnglePress ( EffMach, Chi, PW :Double; var PT :Double ) ;

VAR Del, {Effective deflection. 1

Theta, {Shock Angle. 1

Int, Up, Down

Found

BEGIN up := 1.57; Down := 0.2;

: Double;

: Boolean;

{Intermediary value. ) {Bounds on guess. 1

{Error trap variable. 1

( Lirnits for the guess 1 (of Theta in solver. }

Del := 0.610865 + Chi; {Effective deflection. }

Theta := RootSec (Del, EEfMach, Down, Up, Found);

IF Found then Begin

ïnt : = sin (Theta) ;

{Calculate the pressure)

PT := PW * exp(-7/2 *ln( 5/ (EffMach*EffMach + 5) ) ) ; PT := PT / ( (7*EffMach*EffMachXInt*Int - 1) /6);

End Else Begin {or set an error f lag. 3

PT := l.lE9; End ;

END ;

{ This procedure w311 calculate the compared error for the system given) { the current Mach number and angles of incidence. It will return the 1 { corresponding calculated total pressure in front of each wedge shock as } { Pt#. It uses routines CalcVector, RotateVector, Effective, and Anglepress.) { It must be passed the Mach#, Alpha, Beta, Wedge and Pitot pressures. The ) { value of the error is returned as the function. 1

FUNCTION ErrE ( var M, A, B, Pwl , Pw2, Pw3, Ptl, Pt2, Pt3 :Double ) :Double;

VAR Temp, Ef £Mach, Chi

Vectorl , Vector2, Vector3 : Vec torArray ;

BEGIN CalcVector (Ml A, B, Vectorl);

RotateVector ( 120.0, Vectorl, VectorS);

RotateVector (-150.0, Vectorl, Vector3);

Effective ( EffMach, Chi, Vectorl); AnglePress ( EffMach, Chi, Pwl, Pt1 ) ;

{Get reference vector. }

{Find equivalnet in the) {other xeference £rames)

{Determine effectives. 3 {Calculate pressures. 1

Effective ( EffMach, Chi, VectorS); AnglePress ( Ef £Mach, Chi, Pw2, Pt2 ) ;

{Do error calculation. 1

END ;

{- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 1 { This procedure will calculate the array of points surrounding the } { current guess. The array is spaced at a distance Delta £rom the guess } { stored in il]. The array finds eight points at the cardinal directions. ) { The directions are stored clockwise starting with [21 as NE. 1

Procedure Angles ( var Alphus, Betus :DoubleArray; Delta : Double) ;

BEGf N Alphus[2] := Alphus[ll + 0.707106 * Delta; { NE Betus [2] : = Betus[l] + 0.707106 * Delta;

Alphus[3] := Alphus[ll + Delta; Betus[3] : = Betus[ll;

Alphus[4] : = Alphus[ll + 0.707106 * Delta; { SE Betus 141 := Betus[ll - 0.707106 * Delta;

Alphus[5] : = Alphus[ll; Betus [51 := Betus[ll - Delta;

ALphus[6] : = Alphus[ll - 0.707106 * Delta; { SW Betus [61 := Betus[l] - 0.707106 * Delta;

Alphus[7] : = Alphus[l] - Delta; Betus [7] : = Betus [1] ;

Alphus[81 : = Alphus[ll - 0.707106 * Delta; { NW Betus[8] := Betus[ll + 0.707106 * Delta;

Alphuç[91 : = Alphus il] + Delta; Betus[91 : = Betus[l];

END ;

{ This procedure will take the current mach number and find the fLow angles) {which satisfy the wedge pressure values. It matches the PT# terms for al1 ) {probes. It rnust be passed the Mach#, Alpha, Beta, Wedge and Pitot pressures) {and records the Error term. 1

PROCEDURE AngleFind (var Mach, Alpha, Beta, Pwl, Pw2, Pw3, Ptl, Pt2, Pt3 :Double ) ;

CONST MaxIt = 500;

VAR Alphus, Betus , Grld : ~oubleArray;

Error , Current , Steepest , Delta : Double;

{Maximum iterations. 1

{Grid of surrounding A.) {Grid of surrounding B.) {Function evaluations. )

{Error tracking. 1 (Current gradient. 1 {Largest gradient. 1 {Change in Angle. 1

L I I V I G ~ ~ : A r l L e g e r ; iBest new point to take}

BEGIN

Count := O ; {Initialise variables. 1 Error := ErrE( Mach, Alpha, Beta, Pwl, Pw2, Pw3, PtZ, Pt2, Pt3) ; Delta := 0.1;

While ( (Error*10000) > Tol) AND (Count c= MaxIt) Do Begin

Alphus [ 11 : = Alpha ; {Set location of guess.3 Betus[l] := Beta;

Choice := 1 ; Steepest := 0.0;

{Reset criterion.

Count := Count + 1;

Angles ( Alphus, Betus, Delta ) ; {Determine new array. 1

{Find error values. 3 Grid[ll := ErrE( Mach, Alphus[ll, Betus[l], Pwl, Pw2, Pw3, Ptl, Pt2, Pt3);

For i := 2 to 9 Do Begin

Grid[il := ErrE( Mach, Alphus[il, Betus[il, Pwl, Pw2, Pw3, Ptl, Pt2, Pt3);

Current := Gridiil - Grid[ll ; {Take gradient info. 1

If Current c Steepest then Begin

S teepes t : = Current ; Choice : = i;

End ;

End;

If Steepest = 0.0 then Begin

Delta := Delta /2; End Else Begin

Alpha : = Alphus [Choice] ; Beta : = Betus [Choice] ; Error := Grid[Choicel ; Delta := Delta * 0.90;

End ;

{Compare to others in 1 {this array for best. 1

{If none found reduce 1 {step size used. 3

{Take best gradient as 1 {the new guess. 1

End ;

END ;

{---- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 1 { This procedure will calculate the pressure in front of the normal 3 Cshock wave caused by the pitot tube measurement. It needs the Mach number,) {and a variable to return as total pressure. The measured pitot pressure is) (taken as a Global value named Pts. 1

PROCEDURE PitotPress ( var MM, PT :Double ) ;

BEGIN PT := Pts* exp(-7/2*ln( 6*MM*MM / (MM*MM+5)));

{ This procedure will calculate the pressure error for the systern at } { the Mach number given al1 the calculated total pressures. Pressure Error ) { consists of the cummulative total of differences between the Total £rom } { the Pitot and the Angled probes at the current angles of attack and Mach. } ( The error term is returned as the value of the function. 3

Function ErrP ( var Mach, Pitot, Ptl, Pt2, Pt3 :Double ) :Double;

VAR T emp : Double;

BEGIN

PitotPress ( Mach, Pitot ) ; {Find Pitot pressuxe. 1

Temp := abs (Pitot - Ptl) + abs (Pitot - Pt2) ; {Calculate error term. ) ErrP := abs(Pitot - Pt3) + Temp;

END ;

{ This procedure will find the area around of the global minimum as a ) {function of Mach number for the predetermined angles of incidence. It will} {return the location of the lowest error as the integer Point. 1

PROCEDURE LowLine ( var ErrLine, MachMat : DoubleArray; var Point :Integer ) ;

VAR Smallest : Double ; {Minimum value found. 1

i : Integer ; {Loop variable. 1

BEGIN

Smallest : = 1E6; {Initialise variable. 1

For i : = 1 to N do Begin

If ~rr~ine[il < Smallest then Begin

Smallest : = ErrLine [ i 1 ; Point := i;

End ; End;

{Compare. 1

{Assign values if less.)

END ;

PROCEDURE SetMach ( Point : Integer; var MLow, MHigh :Double 1 ;

VAR Temp, Centre : Double;

BEGIN

Centre : = ( (Point-1) /N) * Temp + MLow; {Make MachMat centre to) {value at minimum ErrE.)

MLow := Centre - Temp/5; MHigh := Centre + Temp/5;

{Reduce bounds of array)

END;

{ - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 1 { This procedure will take the determined Angles of Attack and complete a ) {line search over the range of Mach numbers £rom MLow to MHigh to determine ) {a global minimum. The minimum is based on the ErrT function. The ErrLine ) {and MachMat matrices are updated as the search is refined. The position of) {the global minimum is stored in the placeholder Point. 1

PROCEDURE MachFind ( var ErrLine, MachMat :DoubleArray; var Alpha, Be ta, Pwl , Pw2, Pw3,

PtlIPt2,Pt3, MLow,Mhigh :Double; var Point :Integer ) ;

CONST MaxIt = 300; {Maximum iterations. }

VAR

Current : Double; {Current minimum error.)

il Count : Integer ;

BEGIN

Count := 0; Current : = 1;

{Number of iterations. 1

{Initialise variables. 1

While (Current > Tol) AND (Count <= MaxIt) AND (MLow <> migh) Do Begin

Count := Count + 1;

For i := 1 to N Do (Fil1 error matrix. 1 Begin

{Force Pt# recalculated) Errline [il := ErrE ( MachMat [il, Alpha, Beta,

Pwl, Pw2, Pw3, Ptl, Pt2, Pt3);

(Get pressure error. 1 ErrLineLiI := ErrP ( MachMat[il, Pitot, Ptl, Pt2, Pt3 ) ;

End;

LowLine ( ErrLine, MachMat, Point ) ; {Find greatest minimum. )

Current : = ErrLine [Point 1 ; {Record new error value.)

SetMach ( Point, MLow, MHigh ) ; {Readjust for resolution)

InitMach ( MLow, MHigh, MachMat ) ; {Reset array of Mach # S . )

write ( ' . ' ) ;

End ;

END ;

PROCEDURE Output (var Outfile :text; Test :String; Pts, Mach, Alpha, Beta, Error :Double);

VAR Factor :Double;

BEGIN Factor := 180/Pi;

writeln (Outfile, 'In experiment ' , Test 1 ;

writeln (Outfile, 'with a Pitot pressure of: ' , Pts:10:3 ) ;

writeln (Outfile, ' For pressures of: ' , PTop: 10 : 3, and ',PLeft:lO:3, ' and ',PRight:lO:3 1 ;

writeln (Outfile, ' Alpha was found to be: ' , (Alpha*Factor) :10:5, ' and Beta: ' , (Beta*Factor) : 10: 5 ;

writeln (Outfile, ' At a Mach number of: ' , Mach:lO:S ) ;

writeln (Outfile, ' with an error term: ' , Error 1 ;

writeln (Outfile) ; writeln (Outfile) ; writeln (OutfiLe) ;

END ;

{+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++} BEGIN

aççign (Outfile, 'prote-out' ) ; append (Outf ile) ;

{Ready datafile for program.}

If IOResult o O then rewrite (Outfile);

UserIn ( Test, PPt , PTop, PLef t , PRight ) ; { G e t user input. 1

pwl : = PTop + Cl; pw2 := PLeft + C2; ~ w 3 := PRight + C3; Pts := PPt t

MLOW : = 5.0; MHigh := 10.0;

Alpha := 0.0 ; Beta := 0.0 ;

{Apply calibration. 1

{Initial Mach bounds. 1

{Initial guesses.

InitMach ( MLow, MtIigh, MachMat ) ; {Get array of Mach # S . ?

AngleFind ( MachMat[l3], Alpha, Beta, Pwl,PwS,Pw3, PtlJpt2,Pt3 ) ;

MachFind ( ErrLine, MachMat, Alpha, Beta, Pwl, Pw2, Pw3, Ptl, Pt2, Pt3, MLow, MHigh, Point);

{File to Save data in. 1 Output (Outfile, Test, PPt, MachMat [Point],

Alpha, Beta, ErrLine [Point 1 ) ;

close (Outfile) ;

END.

Appendix H

Prototype Drawings

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