ii

Cataloging-in-Publication Data

Documentation and Information Division Rolim, Tiago Cavalcanti Experimental Analysis of a Hypersonic Waverider / Tiago Cavalcanti Rolim São José dos Campos, 2009. 120f. Thesis of master in science, Aeronautics and Mechanics Engineering, Aerodynamics, Aeronautics Institute of Technology, 2009. Advisor: Ph.D. Paulo Afonso De Oliveira Soviero, Ph.D. Marco Antônio Sala Minucci.

1. Waveriders. 2. Aerospace vehicles. 3. Hypersonic Aerodynamics. I. General Command for Aerospace Technology. Aeronautics Institute of Technology. Aeronautical Engineering Department. II.Experimental Analysis of a Hypersonic Waverider.

BIBLIOGRAPHIC REFERENCE ROLIM, Tiago Cavalcanti. Experimental Analysis of a Hypersonic Waverider. 2009. 120f. Thesis of master in sciences in Aerodynamics – Aeronautics Institute of Technology, São José dos Campos.

CESSION OF RIGHTS AUTHOR NAME: Tiago Cavalcanti Rolim PUBLICATION TITLE : Experimental Analysis of a Hypersonic Waverider PUBLICATION KIND/YEAR: Master’s Thesis / 2009 It is granted to Aeronautics Institute of Technology permission to reproduce copies of this thesis to only loan or sell copies for academic and scientific purposes. The author reserves other publication rights and no part of this thesis can be reproduced without his authorization.

___________________________

Tiago Cavalcanti Rolim IEAv, Rod. Tamoios Km 5,5. Putim CEP 12228-001 São José dos Campos, SP.

iii

Experimental Analysis of a Hypersonic Waverider

Tiago Cavalcanti Rolim, 1° Ten. Eng.

Thesis Committee Composition:

Prof. Dr. Pedro Teixeira Lacava Chairperson - ITA Prof. Dr. Paulo Afonso de Oliveira Soviero Advisor – ITA Cel. Eng. Dr. Marco Antonio Sala Minucci Co-Advisor – IEAv Dr.ª Valéria Serrano Faillace Oliveira Leite ITA Dr. Demétrio Bastos Netto VSE

ITA

iv

Dedico este trabalho ao meu avô Olívio Barbosa Cavalcanti.

v

Acknowledgements

Thanks God for everything. Special thanks to my parents, Lucia Luzia Cavalcanti Rolim and

Jurandir Bezerra Rolim, and to my whole family. Also, I’d like to express my gratitude to my

fiancée Vanessa for an unstinting support, everyday I think to myself how lucky I am! Thanks

to my advisor, Cel. Marco Antônio whose contribution to this work was of paramount

importance. Dr. Paulo Toro’s strong support was constant throughout the time I dedicated for

this program. Needless to say, Cel. Marco Antônio and Dr. Paulo Toro own a large part of the

authorship of this work. I’m grateful to my advisor on ITA, Prof. Soviero, his frank

dedication in teaching corroborated whatsoever for the conclusion of the task described on the

upcoming pages. Thanks for Francisco Gomes for sharing his valued experience on the T3

tunnel operation, in fact, he was present in almost every test. The staff of the Laboratory

Henry T. Nagamatsu must be mentioned too: José Adeilton, David Romanelli, Thiago

Cordeiro and Felipe Jean, fellows that constantly helped me during the tests. Eng. José

Brosler also gave me very wise technical tips, so I am in debt with him. To Dr.ª Valéria Leite,

Eng. Davi Neves and their staff from the IEAv’s Support Division. Special thanks to Marcelo

who made a tremendous work in the model drawings. Thanks to the OXTIG and MIUS

industries, for the great care they took on the construction of the investigated model and its

parts. I’m grateful too with Dr. Angelo Passaro, 1S Onofre de Lima, and Roberto Yuji, from

the Applied Physics Division, for their support concerning the computer codes and for gently

sharing their laboratory when I was performing computer simulations of the flow over the

investigated model.

vi

Abstract This work presents the results of shock tunnel tests of a Mach 10 waverider with sharp

leading edges. The waverider surface was generated from a conical flowfield with the volume

and the viscous lift-to-drag ratio as optimization parameters. Compression and Expansion

ramps were added to the pure waverider surface in order to simulate the flow over a scramjet

engine. The compression ramp was designed so as to provide the ideal conditions for the

supersonic combustion of the Hydrogen while the expansion section was derived from an

ideal minimum length supersonic nozzle. The experimental data included Schlieren

photographs of the flow and the pressure distribution over the compression surface. These

data were compared with the inviscid theory. During these investigations, the IEAv’s T3

shock tunnel was used to simulate the hypersonic flow. The stagnation conditions as well as

the free stream properties were estimated using numerical codes. The tunnel operated at Mach

number ranges of 8.9 to 10, Reynolds number from 2.25 x 106 to 8.76 x 106 (m-1) and

Knudsen number from 0.06 to 0.19. From the Schlieren photographs it was noted that the

inlet flowfield behaves according to the predictions of the hypersonic viscous interaction

models. Also, the pressure variation along the compression surface centerline was obtained

using piezoelectric pressure sensors. The resulted profile presented the general trend of the

flow described by these models.

vii

Contents

1 Introduction ........................................................................................................................8 2 The T3 Hypersonic Shock Tunnel....................................................................................14

2.1 The Driver section ....................................................................................................15 2.2 The Double-Diaphragm Section (DDS) ...................................................................16 2.3 Driven section...........................................................................................................19 2.4 Convergent-divergent section and Dump Tank........................................................21 2.5 General caveats for T3 operation..............................................................................23

3 Shock Tunnel Flow Modeling ..........................................................................................24 3.1 Ideal shock formation theory....................................................................................24 3.2 Incident shock, reflected mode and equilibrium interface modes of operation........28 3.3 Nozzle flow ..............................................................................................................32 3.4 Pitot pressure-probe..................................................................................................33 3.5 General evaluation of the numerical code ................................................................35

4 Model Design and Construction.......................................................................................39 4.1 Hypersonic Small Disturbance Theory ....................................................................40 4.2 Trade-off analysis .....................................................................................................46 4.3 Surface modeling......................................................................................................52 4.4 Scramjet Integration .................................................................................................53 4.5 Dimensioning ...........................................................................................................70

5 Instrumentation.................................................................................................................75 5.1 Pressure transducers .................................................................................................75 5.2 Yokogawa DL750 ....................................................................................................78 5.3 Schlieren Apparatus..................................................................................................79

6 Results and Discussion .....................................................................................................82 6.1 Shock Tunnel Conditions .........................................................................................82 6.2 Air Flow Investigation..............................................................................................88

7 Conclusions ......................................................................................................................97 8 References ......................................................................................................................100 Appendix A. Shock Tunnel Flow Simulation Codes .......................................................103 Appendix B. Surface Construction Computer Codes ......................................................118

8

1 Introduction

The interest in hypersonic vehicles lies on the promise of an air breathing reusable vehicle

able to deploy a payload into earth orbit. Until now, research has shown that a waverider

vehicle has superior aerodynamic performance compared with other hypersonic aerodynamic

concepts [1]. However, a waverider vehicle features a complex and integrated technology

which was a serious obstacle until recent years. At the present time, researches on high

temperature materials, on supersonic combustion, on computational analysis, and on ground

testing technologies have led to the practical design of such vehicles.

Hypersonic waveriders are being considered for high altitude low flow-density

applications such as accelerators and aerogravity-assisted maneuvering vehicles [2]. They are

also being considered for high-speed long-range cruise vehicles since their high lift-to-drag

ratio becomes important in achieving global range.

Furthermore, with horizontal takeoff and landing capability they could reduce the turn

around time of the current space missions. Using air breathing propulsion, they avoid the need

to carry the oxidizer, which results in weight saving, reduced complexity, and smaller ground

support [3]. The table 1.1 gives some typical values for the weight fraction in present

missions. Regarding these values, one can see that more than a half of a rocket vehicle weight

is due to the oxidizer. Also, comparing a rocket system with an aircraft capable to achieve

earth orbit with the same weight, the aircraft would be able to carry 11% more in payload

weight.

Takeoff Weight Fraction Aircraft Rocket

Payload 15% 4% Empty 55% 7% Fuel 30% 24% Oxygen 0% 65%

Table 1.1. Typical takeoff weight fraction breakdowns of current aircraft and multi-stage rocket transportation systems [4].

9

The concept of a waverider vehicle was introduced by Terence Nonweiler, in 1951, as a

delta shaped vehicle for reentry. While studying the flow over the vehicle’s surface,

Nonweiler realized that the high pressure zone generated by a shock wave could be used to

produce lift. Later, in 1962, Nonweiler [5] described a classic waverider with an anhedral, see

Fig. 1.1. This vehicle was known by caret shaped waverider due to its resemblance to the

typographic symbol ‘^’.

Figure 1.1. Caret shaped waverider.

Nonweiler designed his vehicle based on the flow over a plan top delta shape wedge.

Although this simple configuration produces lift, the delta shape leads to an undesirable cross

flow in the lower surface toward the wing tips which decreases the pressure in that region,

causing lift loss. To circumvent this problem, Nonweiler suggested a shape that featured an

anhedral angle (negative dihedral angle), such as the described in Fig. 1.1.

The sharp leading edge keeps the formed shock wave attached isolating the high pressure

zone from the upper surface. Otherwise, with a blunt leading edge, the flow spillage would

provoke pressure loss and drag increase.

In 1980, Rasmussen [6] presented a new type of aerodynamic surface using the shock

wave to generate lift. It was derived from a supersonic conical flow. The Rasmussen’s surface

obtained a superior overall performance to the classical Nonweiler’s waverider. Since then,

various families of cone derived waveriders as well as their hybrid variations like cone-wedge

and multiple cone derived waveriders have been studied. The objectives of these studies are

10

mainly to increase the low lift-to-drag ratio in viscous flows and to improve the package

capability of such vehicles [7, 8].

For the sake of clarity, the waverider is a supersonic or hypersonic vehicle which uses the

shock wave formed in the fore-body so as to generate lift. In general, the shape of the vehicle

maximizes one or several design parameters, such as the lift coefficient, the lift-to-drag ratio,

the volumetric coefficient and so on. In the case of a waverider integrated with a scramjet

system, the lower surface in the vehicle forebody must provide the scramjet’s inlet with air in

the desirable conditions to maintain a steady supersonic combustion.

Usually, the waverider shape is reversely generated, i.e., a known flow over a base body

is used to derive the surface. In this work, the hypersonic small perturbations theory was

applied in a Mach 10 flow over a conical body, and this will be thoroughly described in

chapter IV.

The only air breathing engine cycle capable to efficiently provide the thrust required for a

hypersonic vehicle is the scramjet engine. As a matter of fact, at hypersonic speeds, a typical

value for the specific impulse of a H2-O2 rocket engine is about 400 s while for a H2 fueled

scramjet is between 2000 s and 3000 s.

Roughly, the scramjet engine, see Fig.1.2, consists of a supersonic inlet that decelerates

the flow, the kinetic energy of the high-speed airflow is converted in pressure – this is referred

as the pressure recovery. In the combustion chamber the flow is heated up and the nozzle

accelerates the combustion products and generates thrust.

Figure 1.2. Schematic diagram of a 2-dimensional scramjet [4].

11

The integration with the propulsive system requires the lower surface to compress the

suctioned air in order to achieve adequate pressure and temperature for combustion and to

maximize the pressure recovery. Furthermore, it’s of paramount importance for the propulsive

efficiency to avoid three-dimensional flow in the inlet because this would lead to the

existence of cross flow and boundary layer increase. Also, the inlet must be large enough so

as to provide the necessary thrust even in high altitudes. In sum, since the flow field in that

region affects the aircraft overall performance, the scramjet inlet geometry is decisive for the

engine performance.

Finally, the nozzle, or expansion ramp, must accelerate the hot air from the combustion

chamber towards the atmosphere. The nozzle design must consider several difficult issues like

the composition of the gaseous products formed in the combustor, which involves the

calculation of the concentration of hundreds chemical species. Moreover, differently from a

typical engine nozzle, a scramjet’s nozzle produces positive net lift, a factor that the practical

design must account to.

The present investigation has the purpose of providing experimental data for the

hypersonic flow over a waverider vehicle’s compression surface and the scramjet combustor

inlet at high Mach numbers and high total enthalpies. The main goals of the research were: i)

To design and build of a waverider model with instrumentation; ii) To Measure surface static

pressures and Pitot pressure for Mach number 10, with low and high reservoir enthalpies; iii)

To take Schlieren photographs to support data analysis; iv) To compare the experimental

results with numerical simulation.

The problem of simulating the hypersonic flight is associated with the high stagnation

enthalpies. The only way to achieve this is using pulse facilities, or shock tunnels, which

unfortunately have a reduced test time compared with subsonic or supersonic wind tunnels,

just a few milliseconds. The tests were performed utilizing the T3 shock tunnel located in the

12

Prof. Henry T. Nagamatsu Laboratory of Aerothermodynamics and Hypersonics of the

Institute for Advanced Studies. Although a multitude of test conditions were investigated, it is

impossible to simulate the whole range of the real flight conditions with ground facilities.

There are several reasons for that. For one, it would require several months, considering that

the time between two tests is still long, about 8 man-hours of work. Another reason is that it is

hard to match the actual flight Reynolds number.

However, the data acquired in ground tests can provide crucial information for further

designs. For instance, the pressure data can be used for validation of numeric predictions.

Moreover, with the Schlieren photographs we can measure the shock wave angle and observe

its interactions with the vehicle’s boundaries.

Figure 1.3. The T3 shock tunnel.

Several works concerning waveriders have been presented. The 1st International

Hypersonic Waverider Symposium [9] gathered very important studies about related subjects

like optimization and design, applications, and experimental investigations. The work of

Rasmussen [6] for example, was the basis for the construction of this work waverider model.

Also important contributions to the area have been published by Maryland’s numerical

group, such as the work of O’neil and Lewis [10]. In that work they study the integration of a

scramjet engine with a waverider vehicle. Another important publication was given by

Anderson and Lewis [11], which presents an overview of the current technologies for a

13

waverider design. The same group has purposed various methods of construction of the

waverider surface, and computer codes that account to viscous effects, turbulence and real gas

effects. Numerical studies in leading-edge shape were performed too [12].

An extremely important experimental work was performed by Cockrell and Huebner [13].

In that investigation, two conical-flow-derived waveriders for a design Mach number of 4.0

that integrate vehicle components, including canopies, engine components, and control

surfaces, were constructed. Experimental data and computational fluid dynamics solutions

were obtained over a Mach number range of 1.60 to 4.63. Each one of models was 26.6 inch

long. Also, the design of the model permitted the testing of a straight-wing and of a cranked-

wing pure waverider configuration.

Another experimental investigation was performed by Gillum and Lewis [14]. In their

work, they used a conical-flow-derived waverider model for Mach 14 with blunt edges.

General performance parameters of the vehicle were analyzed and compared with theoretical

predictions. Summarizing, they concluded that the flow spillage does not cause important

losses to the aerodynamic performance of the vehicle. On the other hand, the blunt edges

increase the total drag and, thus, reduce the vehicle’s performance. A very important

conclusion in that work is that aerodynamic coefficient data were insensitive to changes in

Mach number and Reynolds number.

The upcoming pages give a thorough description about the investigation. The next chapter

provides detailed information about the hypersonic shock tunnel operation. Following, the

shock tunnel flow modeling will be presented in Chapter III. The waverider surface

construction will be discussed in Chapter IV. Next, the instrumentation and experimental

apparatus will be referenced in Chapter V. Chapter VI will be dedicated for results and

discussions. Finally, conclusions and recommendations will be presented in Chapter VII.

14

2 The T3 Hypersonic Shock Tunnel

The large stagnation enthalpy and pressure required for simulation of hypersonic flow

make the shock tunnel facilities the only applicable tools for this purpose. The shock tunnel

consists of a high pressure section (driver section) separated by means of a diaphragm section

from a low pressure section (driven section). See Figure 2.1. When the diaphragm bursts a

shock wave propagates toward the driven, while an expansion wave propagates into the

driver. The shock wave interacts with the cold air in the driven section, increasing pressure

and temperature. Once the shock wave reaches the nozzle diaphragm, it’s reflected and

interacts with the contact surface – the interface between the gases. After some interactions

and reflections, the resulting high temperature and high pressure are used as stagnation

conditions for the supersonic nozzle located in the end of the driven section.

Figure 2.1. Schematic view of a shock tunnel.

The diaphragm rupture is controlled by a Double Diaphragm Section (DDS), using an

electromechanical valve. After the diaphragm rupture, the whole process, including the test,

lasts for just a few milliseconds. For that matter, rapid dynamic response instrumentation is

needed for data acquisition.

Some important geometrical data of the T3 shock tunnel are presented in table 2.1. This

shock tunnel is located in the Laboratory Prof. Henry. T. Nagamatsu of the Institute for

Advanced Studies (IEAv). That hypersonic facility comprises of two other shock tunnels T2

and T1, 11.50 m and 7.80 m long, respectively. Although both can provide a Mach 10

15

Length (mm)

Internal Diameter

(mm) Driver 4080.0 190.5 Driven 10500.0 127.0 Nozzle 1058.3 610.0 Throat 80.0 23.5

Test section 1800.0 1260.0 Table 2.1. Geometrical features of the T3 shock tunnel.

hypersonic flow, they have small test sections and short test times, and cannot reproduce the

required stagnation conditions for the present tests. The following sections are intended to

better clarify the T3 shock tunnel functioning and operation, which were crucial for the

present investigation.

2.1 The Driver section

The driver section is a 4080.0 mm long pressure chamber with 190.5 mm internal

diameter and 69.85 mm thickness, Fig. 2.2. The driver is loaded with the desired gas for the

run at ambient temperature. The gas used for a test is pumped through connections located in

the end of this section. For the present work, air and Helium gas were used in that section.

The driver has a maximum operation pressure at 5,000 psi ( 345 atm). However, during the

tests the maximum pressure used was 3000 psi. When Helium was used, the gas, which is

provided by commercial tanks, was initially directly leaked into the driver. The used

commercial tanks capacity was usually 7.8 m3 (STP) of Helium at 2700 psi. For safety, these

tanks were stored into cages as shown in Figure 2.3. When the pressure in the tanks equals

the driver pressure the flow stops and the compressor type Aerotecnica Coltri model

MCH36/ET was used to pump the remaining gas from the tanks. The compressor working

pressure is 4700 psi and has 600 l/min charge rate. Since the compressor inlet pressure is

relatively low, near by 1.0 atm, a pressure regulator was used between the tanks and the

compressor inlet. If air is to be used in the driver, the compressor pumps air directly from the

ambient. This is possible because the compressor has an internal system for the

dehumidification and filtration of the suctioned air.

16

Figure 2.2. Photograph of the driver section.

The driver is separated from the driven by a Double-Diaphragm Section (DDS), which has

a working operation pressure at approximately half of the driver-to-driven differential

pressure. The rupture of the diaphragm is caused by exhausting the gas of the DDS section.

This gas does not necessarily is the same of the driver gas or of the driven gas. In fact, Argon

was applied in all runs.

The pressures are monitored by the T3’s control panel, see Fig. 2.3. The control panel

consists of four Heise gauges, specifically, a 0-10000 psi gauge, a 0-5000 psi gauge, a 0-5000

psi gauge and a 0-2000mmHg gauge, monitoring the driver, the DDS, the piston (which has

not been installed yet) and the driven, respectively.

Figure 2.3. On left, dry air, Argon and Helium gas commercial tanks inside the safety cage.

On right, the T3 control panel.

2.2 The Double-Diaphragm Section (DDS)

The DDS, Fig. 2.4, is responsible for controlling the test starting. In that section, argon is

pumped directly from commercial tanks into the DDS. Attention is required because the DDS

17

Figure 2.4. The Double-Diaphragm Section of the T3 shock tunnel.

pressure must be kept as half as the driver-to-driven differential pressure, during the driver

pressurization. This procedure is very important because it avoids an undesirable early rupture

of the diaphragms, caused by non-uniform increments in pressure. The reason for choosing

Argon as the DDS working fluid is that it is necessary a high molecular-weight gas in order to

avoid diffusion of gases through the contact surface. Furthermore, other gases can lead to

unpredictable chemical reactions that could interfere in the flow uniformity and in the

stagnation conditions.

The DDS features two 289.5 mm diameter stainless steel diaphragm, which must be

properly mounted in that section so as to, after the rupture, the resulting petals cover the

electrical valves connections protecting them from the overpressure.

For the rupture happen, a cross-cut must be machined in the DDS diaphragms, see Fig.

2.5. The rupture pressure is determined mainly by three factors: the depth of the cut; the

geometry of the tool used for the cut; and, the diaphragm material. The diaphragm rupture is

caused by a crack which has its origin at the leading edge of the cut, in the central region of

the diaphragm. So, beyond the influence of the cut depth, the sharpness of the cut also

contributes to the severity of the cracking. Indeed, a sharp cut increases the stress

concentration on the vicinities of cut leading edge which lowers the rupture pressure. On the

18

Figure 2.5. DDS diaphragms in detailed view.

other hand, a blunt cut could raise the rupture pressure until levels that are not useful for a

test. For these reasons, the geometry and sharpness of the tool are very important for the

rupture. Moreover, since the tool wear augments the edge radius of the diaphragm cuts, the

number of the machined diaphragms by the same tool is limited. For that matter, each

diaphragm machined must be numbered. A further recommendation is that the diaphragms

must be tightly fastened in the mill’s bed while machining so as to avoid any interference

caused by previous deformations on the cut depth homogeneity. The milling machine adopted

was type Deckel FP4CC with Computer Numeric Control (CNC).

On the course of the present investigation, several stainless steel diaphragms were tested,

i.e., with different thickness and cut depth. For these tests, only the DDS section was

pressurized; air was pumped into it until the burst occurs. The results are presented on Fig.

2.6. To protect the model into the test section from diaphragm petals, the nozzle was blocked

using a circular steel plate. Also, low pressure in driven section ascertained that no high

pressures would be achieved by the reflected shock wave. The ideal rupture pressure in a

19

Figure 2.6. Stainless steel diaphragm tests results.

diaphragm test was found to be between 2000 to 2500 psi (absolute), so that the diaphragm

could properly open during an actual test with a 3000 psi driver pressure. Also, some tests

were performed with the requirement that the diaphragms would not burst after a 30 min

pause in pressurization, standing at 2800 psi. For further compressor restart until 3000 psi is

reached. This sort of test is particularly important to assure a successful test even when some

technical problem requires a delay of the test. The diaphragms that met these requirements are

specified in table 2.2.

Material Cut depth (mm) Thickness (mm) Maximum Pressure (psi) Stainless Steel (AISI-304L)

0.6 2.5 2167 (average)

Table 2.2. Characteristics of the ideal diaphragms for a typical 3000 psi driver pressure test. Machined with a 90° angle tool.

2.3 Driven section

The driven section consists of a 10500 mm long tube with 127 mm internal diameter. See

Fig. 2.7. The driven tube is divided in four sections. The section close to the nozzle is thicker

than the others, for the pressures after several shock wave reflections could be high. That

section also accommodates three pressure transducers, type Kistler, which shall be described

20

Figure 2.7. The driven tube.

on Chapter V. Two of them are used to measure the shock wave velocity, and they are 40 cm

apart. That is, the time between the shock wave arrivals on each station is measured. The third

pressure transducer is used to measure the reservoir pressure, immediately before the throat.

The electrical signal of this transducer is also used to trigger the data acquisition by the

oscilloscope, and to trigger the camera and the light source for the Schlieren apparatus.

In fact, the driven can be loaded with any gas, depending on what atmospheric

composition is intended to be simulated in the test chamber. As a matter of fact,

extraterrestrial atmospheres could be reproduced in order to simulate entry conditions on

space missions. Moreover, for gas dynamic laser studies, the driven is loaded with a particular

combination of gases that under non-equilibrium conditions produces laser phenomena in the

test section.

At the end of the driven tube and before the throat, there is a 219 mm diameter Aluminum

diaphragm with thickness of 4 mm. This third diaphragm isolates the nozzle and test chamber

from the driven because the initial test section pressure is several orders below the ambient

pressure. As required, this diaphragm shall open with the reflected pressure and shall not

release petals or any debris toward the test section. These diaphragms also have a cross-cut

with 1.5 mm depth on the surface that faces the convergent-divergent section.

21

2.4 Convergent-divergent section and Dump Tank

The nozzle-throat system used in this investigation was set to provide a free stream Mach

number 10. That section consisted of a 23.5 mm diameter throat followed by a 1060 mm long

610 mm diameter nozzle composed by four conical sections. See Figure 2.8.

The test section is a 1800 mm long and 1260 mm diameter section with four 749.3 mm

diameter flanges. The flanges allowed the installation of two diametrically opposed quartz

windows of 12 inches diameter for flow observation. The model was placed inside this

section, at the end of the nozzle. The model and its support apparatus was mounted in a 2750

mm long hollow sting, which consisted of a 230 mm diameter stainless steel tube connected

with a screwed rod so that the whole structure can move longitudinally. For that reason, the

model’s distance from the nozzle exit could be changed. One end of the sting is located

outside the dump tank, where a nylon feed-through with 28 microdot electronic connectors is

installed.

The dump tank consists of a 4000 mm high, 2000 mm diameter cylinder. It was designed

to hold, after a test, the driver and driven gases under pressures below 2 bar. The tank has

three operational valves, one for the vacuum pump, other one provides pressure relief after a

test, and one is used for the vacuum gauge. There’s a fourth valve, a safety valve for

overpressures in the tank, located at the top of its structure.

22

Figure 2.8. On the left the dump tank, on the top right the throat section, on the bottom right

the sting’s moving apparatus.

To pump down the dump tank before a test, it was used a BOC Edwards Two Stage

Rotary Vane Vacuum Pump, type E2M80, driven by a Brook Crompton electric motor, type

W-DA90LK-D. Although these devices are capable to achieve even lower pressures, on a

usual test, the dump tank pressure was 0.21 mbar, and the pump took about 40 min to reach

this level. It’s obvious the existence of leaks in the dump tank. Most of them are believed to

be located in the quartz window assembly. That is because during the tests for the present

investigation, previously to the assembly of these windows, on some tests made with a plate

glass window, a typical vacuum pressure was as low as 0.009 mbar. However, despite the

leaks, 0.21 mbar pressure still guarantees an over-expanded Mach 10 nozzle. Thus, the

characteristic lines do not impinge on the model.

The dump tank pressure was measured using a BOC Edwards vacuum gauge, type APG-

L-NW16 Active Pirani Vacuum Gauge, with an Active Gauge Controller. For safety reasons,

the valve for the vacuum gauge was closed before pressurizing the driver, protecting the

gauge from high pressures that could follow the diaphragm opening.

23

2.5 General caveats for T3 operation

Some important technical considerations must be previously regarded when setting up the

tunnel for a test. These recommendations were based on several issues that were faced by the

course of the present investigation.

Firstly, it was noted that due to a large mobility of the driven structure, the nozzle

diaphragm must be changed before the DDS diaphragms. Otherwise, it is very hard to close

the driven diaphragm section because the end of the driven tube needs to be realigned with the

dump tank. The difficult lies on the fact that the three arms which connect the driven and the

dump tank must be regulated in order to perform such realignment.

Secondly, during certain test conditions – driver section pressure 3000 psi and driven

section at 2 bar – the stainless steel diaphragms lose some petals, caused by the reflected

shockwave. Thus, it was necessary to open the driver vessel to check out these lost petals.

Thirdly, the existence of leaks in the driven section was noted too. For that reason, when

driven pressure over 1 bar is required, it is necessary to constantly verify the pressure

indication on the correspondent gage, in spite of the risks of exposing the gages to high

pressures that could follow a non-intended diaphragm opening.

Finally, proper use and maintenance of the tunnel o-rings guarantee against leaks. Every

time, before a test, all o-rings must be inspected in order to find any failure that would cause a

leak. Moreover, the use of vacuum greases combined with a careful positioning of the o-rings

inside their channels are recommended for seemly sealing.

24

3 Shock Tunnel Flow Modeling

The lack of a measurement apparatus for the stagnation and static temperatures for the

hypersonic flow does an accurate modeling of the shock tube flow mandatory. Thus, the flow

conditions in the test section were calculated.

3.1 Ideal shock formation theory

a) Incident shock wave formation in a constant bore tube

Shock tunnel flow study is an application of unsteady, one-dimensional flow. Some

information could be found in any good gas dynamics reference such as the work of Emanuel

[15]. Initially, two quiescent gases are separated by a diaphragm as shown in Figure 3.1.

According to the usual nomenclature, the driver is denominated region 4 and the driven the

region 1. The gases have different values for temperature T, pressure p, and specific energy e.

Figure 3.1. Schematic of the shock tube before diaphragm rupture.

After diaphragm burst, the shock wave moves toward region 1, at a constant speed uw. The

disturbed gas is indicated by the region 2. An expansion fan propagates into region 4, the

rarefied driver gas forms the region 3. The contact surface separates the two gases initially

separated by the diaphragm and it acts as piston pushing the driven gas and pulling the driver

gas. Ideally, the contact surface can be seen as a discontinuity in the thermodynamic

quantities. The Figure 3.2 provides detailed view of the shock tube operation.

25

Figure 3.2. x-t diagram that shows the main flow features in an shock tube.

The expansion fan can be modeled as an inviscid isentropic unsteady one-dimensional

flow. It can be shown that the conservation equations which describe the flow allow the use of

the method of characteristics. In this particular case, it implies that lines exist in the x-t space

along which the solution is a constant. Without further details, the term ua+

−12

γ, is invariant

in the regions 4 and 3. Where a is local speed of sound, γ is heat capacities ratio, and u is the

local flow speed. Accordingly,

34

3

4

4

12

12 uaa

+−

=− γγ

(3.1)

From the isentropic relations, one can find that: the pressure ratio between the regions 4

and 1:

12

3

4

3

4 4

4

−

⎟⎟⎠

⎞⎜⎜⎝

⎛=

γγ

aa

pp

(3.2)

However, it could be written as:

1

3

3

4

1

4

pp

pp

pp

= (3.3)

Although viscous effects and turbulence contributes to accelerate the contact surface and

to mix the gases from the regions 2 and 3, in the ideal case, we can assume the dynamic

equilibrium and uniformity of the contact surface. Therefore, we can state that p3 = p2 and u2

= u3, so using eq. 3.3:

26

1

2

3

4

1

4

pp

pp

pp

= (3.4)

The shock wave moving toward the driven modifies the region 1, unperturbed gas,

properties; the new properties of the perturbed gas, the region 2, could be found by applying

the thermodynamic relations through a shock wave for a perfect gas, in particular, we can

write:

1)1(2

1

12

1

1

2

+−−

=γ

γγ wMpp . (3.5)

Where, 1/ auM ww = .

Furthermore, another important shock wave mathematical relation stands that the induced

flow speed in region 2 could be evaluated according to:

w

w

MM

au1

12 2

11

2−

+=

γ . (3.6)

Finally, using equations 3.1 through 3.6, we can find the so called ideal shock tube

equation:

12

2

4

1

1

41

12

1

1

4

4

4

1111

11

)1(2

−

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

−+−

−+−−

=

γγ

γγγ

γγ

w

w

w

MM

aa

Mpp . (3.7)

The shock tube equation relates the incident shock wave Mach number with the

driver-to-driven pressure ratio 14 pp . It’s worth noting the dependence of wM on the speed

of sound and on the pressure of the driver gas. Indeed, this relation clarifies why low density

gases are preferred as driver gases. Similarly, shock tunnels with a detonation driver could be

used to heat up the gas and elevate the local speed of sound in the region 4. Finally, the higher

the pressure ratios the higher incident shock wave Mach number.

27

b) Area change at the diaphragm section

The shock tube equation previously stated relates to constant diameter shock tubes. For

the T3 shock tunnel, however, which has a monotonic area contraction in the DDS region,

that equation could no longer be applied.

Alpher and White [16] describe theoretical and experimental studies of the effects on

shock tube flows of a monotonic convergence at the diaphragm section. They assume ideal

gas behavior. Their results have shown concordance over a wide range of Mach numbers,

even at high Mach numbers where the shock formation processes are complex.

The model for the flow is shown in Figure 3.3. The diaphragm rupture is followed by an

unsteady isentropic expansion similar to the constant bore case from the state 4 to stated 3a.

Then, a steady nozzle flow is formed in the convergent region 3b’, the flow in the minimum

area AA 13b = is sonic. Finally, the flow in 3b is subjected to a further unsteady expansion to

state 3.

Figure 3.3. Schematic view of the flow in a non-uniform diaphragm section.

Without further details, we can write:

1)1(2

211

1211

1

12

1

)1/(22/)1(4

4

12

11

444

44

+−−

⎥⎦

⎤⎢⎣

⎡ +−+

−=−−

−−

γγγγ

γ

γγγγ w

w

w Mg

aa

MM

gpp (3.8)

This relation can be obtained with process quite similar to that used in the previous item.

The equivalence factor ‘g’ is given by the relation:

)1/(22/1

4

234

1

4

34

44

1)1(2

1)1(2

−−

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛+−+

⎟⎟⎠

⎞⎜⎜⎝

⎛+−+

=

γγ

γγ

γγ aa MM

g (3.9)

28

The Mach number in the region 3a can be found from the nozzle area-Mach relation,

given that bM 3 is sonic.

( )121

4

234

31

4 4

4

1)1(21 −

+

⎟⎟⎠

⎞⎜⎜⎝

⎛+−+

=γ

γ

γγ a

a

MMA

A (3.10)

These three equations, 3.8 through 3.10, could be solved by iteration. Given the area ratio

A4/A1 , the Mach number in 3a is obtained by eq. 3.10. Since we have M3a, the equivalence

factor is directly given by equation 3.9. Then, given the pressure ratio p4/p1, one can get the

incident Mach number - Mw - using eq. 3.8.

3.2 Incident shock, reflected mode and equilibrium interface modes of operation

The incident shock wave, which moves toward the driven section, modifies the quiescent

gas from the state 1 to the state 2. After the shock wave, there’s an increase in both pressure

and temperature of the air. The elevated temperature after the shock means that the perfect gas

mathematical assumption is no longer valid. Therefore, real gas effects must be accounted for

estimation of the state 2 conditions. At high temperatures, the physical and chemical

properties change due to the excitation of molecular vibrational modes, dissociation, and

ionization. Also, several new chemical products are formed in high temperatures such as the

NO and the NO+. The Figure 3.4 depicts the ranges of dissociation and ionization of the air at

a pressure of 1 atm. In calculations, the work of Srinivasan et al [18] provides an excellent

support for the interpolation of equilibrium air properties up to 25,000 K.

29

Figure 3.4. Behavior of the air in various temperature ranges [17].

Yet, besides the real gas behavior, the conservation laws stand that through the incident

shock wave, as shown by Fig. 3.5:

( )( )

( )⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

−++=++

−+=+

−=

22

22

2

22

2

1

11

2222

211

2

uupe

upe

uupup

uuu

ww

ww

ww

ρρ

ρρ

ρρ

(3.11)

Along with the real gas interpolation ),( 2222 ρepp = , this system of equations can be

solved by iteration with the initial values of u2, ρ2, e2 and p2 as the same as those calculated by

the shock relations for a perfect gas.

Figure 3.5. Incident shock wave nomenclature.

The incident shock propagates until the end of the driven section; it reflects and changes

in direction, toward the driver, modifying the state 2. This corroborates to a further increase in

30

pressure and temperature from state 2 to state 2’. In the reflected mode of operation, the

conditions 2’ constitute the stagnation conditions.

Figure 3.6. Schematic view of the reflected shock wave.

Once again, the flow equations through the shock wave are valid and become as follows:

( )( )

( )⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

++=+

++

+=++

=+

22

2

'2

'2'2

22

2

22

2'2'2

2222

'222

rr

rr

rr

upe

uupe

upuup

uuu

ρρ

ρρ

ρρ

(3.12)

Again, this system can be solved by using the real gas curve fits and the solutions for the

ideal case can be used as initial values for ur, ρ2’, e2’ and p2’

Finally, high enthalpy stagnation conditions were only possible in the Equilibrium

Interface mode of operation. In this case, the reservoir conditions are formed after few

reflections of the shock wave in the end wall and in the moving contact surface. After these

interactions, only Mach waves are produced and no changes in the properties are observed.

Fig. 3.8 depicts the Equilibrium Interface mode formation from a single shock-interface

interaction.

31

Figure 3.7. Scheme for shock-interface interaction.

The shock reflected in the interface subjects the state 2’ to the state 2’’, with higher

pressure and temperature. Similarly, the transmitted shock wave moves through the region 3,

constituted from the expanded initial driver gas. The region behind the transmitted shock

wave is named 3’.

Now, the conservation equations for the flow through the transmitted shock wave and

through the reflected shock can be summarized as below:

( ) ( )( ) ( )

( ) ( )⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

+++=

+++

++=++

+=+

22

2'3

'3

'3'3

23

3

33

2'3'3'3

2333

'3'333

uupeuupe

uupuup

uuuu

rtrt

rtrt

rtrt

ρρ

ρρ

ρρ

(3.13)

And,

( )( )

( )⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

−++=++

−+=+

−=

22

21'2

''2

''2''2

2

'2

'2'2

2''2''2''2

2'2'2

''2''2'2

uupeupe

uupup

uuu

rrr

rrrr

rrrr

ρρ

ρρ

ρρ

(3.14)

Actually, there are three possibilities for the one-dimensional head-on collision between

the shock wave and the contact surface. One is a reflected shock wave, as noted above; the

contact surface is then called “hard”. On the other hand, if the incident shock wave is

transmitted through the contact surface, a Mach wave or an expansion wave can be reflected

back, the contact surface is denominated “soft”. Whether situation is achieved depends on the

32

nature of the contact surface, which is partly due to the properties of the driver gas. Using a

gas with a higher acoustic speed - relative to the initial speed of sound of the driven gas -

would lead to a “soft” interface, while a gas with a lower speed of sound would produce a

“hard” contact surface, thus propitiating equilibrium interface conditions. Besides the nature

of the driver gas, one must bear in mind that at high incident Mach numbers the temperatures

of the regions between the contact surface and the end of the tube, states 2 and 2’, are several

times higher than the temperatures of the regions immediately before the contact surface,

states 3 and 3’. Therefore the use of helium gas can also cause the equilibrium interface mode

of operation, suitable for the high enthalpies required for the present investigation.

3.3 Nozzle flow

Since the reservoir conditions can be found from the relations and arguments stated above,

we can also estimate the free stream conditions downstream the nozzle. This estimation is

very important since only the impact pressure - the pressure from the Pitot- is measured

during each test.

As mentioned before, the nozzle is placed at the end of the driven section and it is

responsible for the expansion of the reservoir gases to the test section. For the sake of

simplicity, the expansion phenomena can be assumed as an isentropic, one-dimensional, non-

viscous flow, and at thermodynamic equilibrium. However, the high reservoir temperatures

demand that no perfect gas assumption is valid for the air.

For the sake of completeness, when the nozzle flow is fed by a gas at a high temperature,

the air properties throughout the flow are dominated by the properties of its chemical

composition. A multitude of chemical reactions dictate the concentration of an equally high

quantity of chemical compounds. If the relation times of these reactions - the time to reach the

equilibrium state- are of the order of the test time, then the phenomenon is referred as non-

equilibrium flow. On the other hand, for relaxation times much larger than the test time, the

33

flow is said to be “frozen”. It’s quite obvious that the situation when the test time is much

larger than the relaxation times leads to the equilibrium flow.

Although the enthalpies required for this investigation are very high, Nagamatsu and

Sheer [19, 20] noted on experimental investigations performed in a Mach 10 conical nozzle

that for stagnation pressures of 500 psi and temperatures up to 4,500K the flow in the nozzle

can be assumed as in equilibrium. As long as the reservoir temperature for the present

investigation doesn’t exceed that value, and the stagnation pressures are far greater than 500

psi the hypersonic flow can be considered to be in equilibrium.

The process to find the free stream solution consists in firstly encounter the conditions in

the throat, which is sonic, then these conditions are used to calculate the free stream

properties.

Without further physical or mathematical details, in the convergent section of the nozzle

the conservation relations reduce to:

.entropy specific theis s , *ss

2*

***

0

2

0

00

⎪⎭

⎪⎬

⎫

=

++=+upep

e ρρ (3.15)

Where the subscript 0 indicates the stagnation conditions and the superscript * relates to

the throat conditions.

Also, the assumption of a sonic throat leads to ** au = . Once again, the interpolations for

equilibrium air were used.

For the isentropic supersonic flow expansion from the throat to the nozzle exit we have:

⎪⎪⎭

⎪⎪⎬

⎫

=

++=++

=

∞

∞

∞

∞∞

∞∞∞

ss

upeupe

AuAu

*22

****

***22

ρρ

ρρ

(3.16)

3.4 Pitot pressure-probe

34

The impact pressure, acquired from a Pitot probe, was measured during the tests of the

present investigation. This value along with the stagnation pressure, measured at the end of

the driven section, permit a more accurate prediction of the real flow conditions in the test

section. That is because they can be used as correctors of the numerical code in order to

compensate the viscous and non-equilibrium effects in the nozzle region.

Regarding Fig. 3.8, one must consider that the flow along a nozzle centered streamline,

between F1 – a point inside the non-perturbed flow - and F2 - immediately after the bow

shock wave formed in front of the Pitot probe -, can be modeled as a supersonic flow across a

normal shock wave. The numerical scheme then consists of the mass, momentum and energy

conservation through a normal shock wave:

⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

++=++

+=+

=

22

2

22

21

1

11

2222

2111

222111

22F

F

FF

F

F

FF

FFFFFF

FFFFFF

upeupe

upup

AuAu

ρρ

ρρ

ρρ

(3.17)

For the subsonic flow, F2 to P – the assumed stagnation point - , the process can be

considered as an isentropic compression. Thus, the flow equations stand that the total energy

and entropy must be conserved along the streamline:

⎪⎪⎭

⎪⎪⎬

⎫

=

++=++

=

PF

P

P

PP

F

F

FF

PPPFFF

ss

upeupe

AuAu

2

222

2

22

222

22 ρρ

ρρ

(3.18)

Figure 3.8. An illustration of the bow shock formed on a Pitot probe. The flow properties are

calculated on centered streamline [21].

35

3.5 General evaluation of the numerical code

From the above considerations, numerical evaluation of the stagnation conditions and free

stream conditions were made. The computer code was implemented using the MatLab@

platform. The choice of this software was mainly based on its simple user interface and the

existence of various useful built-in subroutines.

As stated before, various factors lead to the failure of the ideal shock tunnel theory in

giving a good estimative for the values of the incident shock wave Mach number. To

circumvent this problem, in order to increase the accuracy of the numerical code, the actual

shock wave speed, measured by the two pressure transducers located at the driven section, can

be used as entry on the code. In fact, this approach was firstly used by Minucci [21] for shock

tunnel simulations. The computer program made by Minucci is largely used during the

present investigations.

The implemented code presented in Minucci’s work also simulates the flow in a shock

tube with real gas corrections. The real gas effects for the air were accounted for through the

use of tabulated air properties assuming thermodynamic equilibrium. The code is divided in

three main programs, hstr, equint and freestream. The first one provides the stagnation

conditions in the reflected mode of operation. The inputs are the shock wave speed, measured

by the time interval of excitation of two pressure transducers with a known separation; the

initial pressure and temperature of driven section; plus, the cross section driven and throat

areas. The second one, equint, calculates the reservoir conditions in the equilibrium interface

mode. For that code, the calculated reflected pressure and temperature are required together

with the measured equilibrium interface pressure. It considers that the few interactions which

occur between the reflected shock wave and the contact surface can be modeled as isentropic

compressions. Finally, the computer code freestream entries are the equilibrium interface

pressure and calculated stagnation temperature along with the impact pressure (from the Pitot

36

probe). Also, it requires the nominal Mach number for the nozzle. The freestream code

calculates the free stream conditions in the test section.

To evaluate the precision of the numerical code, the output was compared with Minucci’s

codes under several different inputs. The inputs were extracted from the five first tests

performed during the present investigation. In all of them it was used Helium as the driver

gas.

Firstly, the reflected mode conditions were calculated using both codes. The table 3.1

shows the obtained results. As one can see, a very good agreement is achieved for the

conditions in region 2, after the incident shock wave, as well as for the conditions after the

reflected shock wave, region 2’. This result indicates the reliability of the implemented code

t3_sing for the reservoir conditions in the reflected mode. For further detailed information,

typical inputs for t3_sing code and the codes proposed in this work are shown in Appendix A.

INPUT OUTPUT Run # p4(psi) p1(atm) Mw T1=T4(°C) p2(psi) T2(K) p2’(psi) T2’(K) Source

287.6 1163.6 1783.5 2061.7 t3_sing1 2700 1.00 4.08 22.0 285.6 1147.8 1757.6 2052.1 hstr 398.4 910.7 2205.5 1573.5 t3_sing2 2700 2.00 3.41 23.5 397.5 899.3 2179.2 1552.3 hstr 374.8 876.8 2031.4 1505.7 t3_sing3 2300 2.00 3.31 24.0 373.9 866.1 2005.1 1483.2 hstr 521.7 1079.5 3147.0 1906.4 t3_sing4 2700 2.00 3.89 20.0 519.6 1064.2 3109.9 1890.4 hstr 533.3 910.8 2970.3 1577.7 t3_sing5 2700 2.63 3.44 20.0 530.7 899.9 2926.3 1556.9 hstr

Table 3.1 Validation of Numerical Simulations for the reflected mode conditions.

Secondly, in order to evaluate the accuracy of the equilibrium interface temperature, the

reflected pressure and temperature, together with the equilibrium interface pressure p2’’, both

calculated in the t3_sing code, were used as entries in the equint code. The results are shown

in table 3.2. It can be noted that the results present a very high level of accuracy with the

Minucci’s code.

37

INPUT for equint OUTPUT Run # p2’(psi) T2’(K) p2’’(psi) T2’’(K) Source 2230.2 t3_sing 1 1783.5 2061.7 2569.5 2231.1 equint 1588.2 t3_sing 2 2205.5 1573.5 2295.6 1588.5 equint 1509.9 t3_sing 3 2031.4 1505.7 2055.3 1509.9 equint 1936.2 t3_sing 4 3147.0 1906.4 3374.9 1936.9 equint 1579.9 t3_sing 5 2970.3 1577.7 2986.6 1579.8 equint

Table 3.2. Reservoir temperature comparison for the equilibrium interface mode.

Finally, to better assess the reliability of the numerical code concerning the free stream

conditions, the reservoir conditions for the equilibrium interface mode (p2’’ and T2’’ and the

Pitot probe pressure, outputs from the t3_sing, were utilized as entries for the freestream

code. The results are summarized in table 3.3. Again, an indisputable accuracy for t3_sing

results can be noted for the Mach number, temperature and pressure of the free stream flow.

INPUT for freestream OUTPUT Run # T2’’(K) p2’’(psi) Pitot(psi) M∞ T∞ (K) p∞ (psi) Source 9.77 128.1 0.0518 t3_sing 1 2230.2 2569.5 6.6 9.51 134.5 0.0557 freestream9.92 84.5 0.0446 t3_sing 2 1588.2 2295.6 5.7 9.93 84.7 0.0444 freestream9.94 79.5 0.0397 t3_sing 3 1509.9 2055.3 5.1 9.96 79.4 0.0394 freestream9.81 108.0 0.0677 t3_sing 4 1936.2 3374.9 8.5 9.72 109.8 0.0688 freestream9.89 84.6 0.0587 t3_sing 5 1579.9 2986.6 7.5 9.91 84.4 0.0586 freestream

Table 3.3. Validation of the Implemented Code for the free stream properties. The reservoir conditions, with subscript 2’’, were entries for the freestream code, as well as the impact

pressure( Pitot).

It should be pointed out that the importance of the t3_sing code lies on its ability to

provide estimations for the free stream flow conditions even if some input parameters from

the experimental apparatus are absent. This is because the code can calculate the incident

shock wave Mach number, the reservoir pressure for the equilibrium interface mode as well as

38

the impact pressure, from the Pitot probe. This fact was of paramount importance for the

present investigations since it didn’t permit that a technical failure occurred on whatever T3

tunnel pressure transducer to compromise the whole experimental data.

39

4 Model Design and Construction

For the present investigation, the waverider model was built according to the Rasmussen

method [6]. In that work, the hypersonic small disturbance theory is applied to analyze

waveriders derived from axisymmetric flows past circular cones. The waverider “raw” shape

does not constitute the entire vehicle setup but it represents only one part. That is because

there are three parts in which an aerospace air breathing vehicle can be divided: the forebody,

the scramjet combustor and the free-expansion region. The waverider configuration can be

used as the forebody, for it generates both lift and it yields compressed air for the scramjet

engine.

For the choice of the waverider geometry, anyway it was generated, there are several

parameters that could be used so as to evaluate the best configuration. In general, the design

intends to maximize the lift-to-drag ratio, to minimize drag and to maximize the payload

volume. From the drag analysis, it was found that at hypersonic flows the viscous drag is

large enough to influence the shape design too. Therefore, it is required the wet-surface area

to be minimized.

Using the Rasmussen method, the design was based on a known flow field over a conical

body. The Fig. 4.1 shows the coordinates and angles for the applied method. Given a conical

body, or base body, and a parabolic upper surface trailing edge, the supersonic flow

streamlines are traced back until they reach the shock wave formed by the base body. The

generated curve is the vehicle’s leading edge. The localization of the leading edge then

permits the flow stream lines to be traced downward defining the lower surface trailing edge.

Finally, the lower and upper trailing edges along with the leading edge are used to calculate

the entire lower surface; the upper surface is derived from the free stream flow.

40

Figure 4.1. Coordinates and angles for flow past a cone [6].

The most appropriate coordinate system for the conical flow is the spherical coordinate

system r, θ, φ. Where θ is measured from the z axis, φ is the azimuthal angle measured from

the x axis and r is the radial distance from the origin. The basic cone semi-angle is denoted

by δ, the shock wave angle by β, and the ratio between them is σ=β/δ.

4.1 Hypersonic Small Disturbance Theory

The high Mach number inviscid flow over a slender body permits the flow equations to be

simplified according to the hypersonic small disturbance theory. For that reason, it is

convenient to define the hypersonic similarity parameter as Kδ = M∞δ. Accordingly, the ratio

of the shock wave angle to the cone semi-angle is given by the relation:

2/1

2

12

1⎥⎦

⎤⎢⎣

⎡+

+==

δ

γδβσ

K (4. 1)

Once the shock angle is obtained, the entire waverider configuration can be derived from

the stream lines of the conical flow. Figure 4.2 shows a waverider derived from the

41

Figure. 4.2. Construction of a general cone-derived waverider [6].

axisymmetric flow over a conical body as well as detailed explanation of the parameters

involved in its design.

The free stream surface and the compression surface are given by the following

approximated expressions, for small values of θ, β and δ:

βφθ ).(. srr = , free stream surface. (4.2)

and

2/12222 )).(().( δβφδθ −=− srr , compression surface. (4. 3)

Where rs(φ) curve defines the leading edge of the vehicle, when θ =β.

From the coordinate system analysis, the radial distance projected in the base plane is

given by Rb = r sin θb. For small angles, Rb≈ r θb and z = r cos θ ≈ r. In the base plane, z = l.

Thus, one can define the non-dimensional distance of a point in the free stream trailing edge

projected in the base plane, R∞b = θ∞ (φ)/δ.

:

42

Figure 4.3. Trailing edge

Furthermore, through equation 4.3 evaluated in the base plane, the leading edge curve can

be obtained by the expression:

)()( φσ

φ bs Rlr ∞= (4. 4)

The free stream trailing edge curve in the base plane is an arbitrary function, represented

by R∞b(φ) in cylindrical coordinate system, or by X(Y), in Cartesian coordinate system,.

However, it must satisfy several conditions. Firstly, when φ = φl, it must intercept the shock

wave formed, thus R∞b(φl) = σ. Secondly, R∞b(φ) must be single-valued in 0<φ<σ. Thirdly,

it’s almost mandatory the upper surface to be smooth on the symmetry plane(φ=0 or Y=0), so

that dR∞b(φ)/dφ=0 at φ=0 or dX/dY=0 at Y=0. Finally, it is obvious that the functions

describing the upper surface are symmetrical in respect to the x axis.

From the above qualifications, the upper surface could be represented by polynomials of

the form:

...)( 420 +++= BYAYRYX (4. 5)

Nevertheless, we considered the simplest case of eq. 4.5, the quadratic form:

20)( AYRYX += (4. 6)

Where Ro is the distance of the free stream surface to the base body axis of symmetry. See

Figure 4.3.

Now, applying the condition that this relation is also valid in the shock location (Xs,Ys):

43

20

20 /)( ssss YRXAAYRX −=⇒+= (4. 7)

Regarding the cylindrical coordinate system adopted, one must remember that:

φcosbRX ∞= (4. 8)

and

φsinbRY ∞= (4. 9)

Now, algebraic manipulation of Eq. 4.6, along with the shock location constraint, and the

relations between X,Y and R∞b,φ, permit the free stream trailing edge to be represented as:

φφφφ

20

2

0

sin4coscos

2)(

AR

RR b

−+=∞ (4. 10)

Also, when σ≤≤ 02/ RX s the curve is single-valued.

For the waverider shape analysis, it’s useful to calculate some features of the generated

shape. The waverider length, lw, the plan form area, Sp, and the volume, V, can be evaluated

by means of:

⎟⎠⎞

⎜⎝⎛ −=

σ01

Rl

lw (4. 11)

φφσ

δσφ

dR

lS l

bp ∫ ⎥⎦

⎤⎢⎣

⎡−= ∞

02

2

2 cos1 (4. 12)

φσσ

δφ

dRR

lV

lbb∫ ⎥

⎦

⎤⎢⎣

⎡+−= ∞∞

03

3

2

22

3 32

31 (4. 13)

Moreover, the expressions for the lift, the pressure drag, and the momentum were

obtained by using the integral forms of the conservation equations with hypersonic small-

disturbance approximations. For the lift, p

L SqLC

∞= , by definition. Thus,

φφσσ

σδ φ

dR

SlC

lcb

pL cos1

1 02

322

∫ ⎥⎦⎤

⎢⎣⎡ −

−= (4. 14)

44

For the drag calculation, defined asp

D SqDC

∞= , it is composed by the drag due to the

shock wave plus the friction drag, so it can be written as:

fp DDD += (4. 15)

P

wfpD

p

fpD S

SCC

SqDD

C +=+

=∞

(4. 16)

Where,

φσσσ

σδ φ

dRR

SlC

lcbcb

ppD ∫

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛−−

−=

02

2

2

2

2

242

ln11

(4. 17)

And, the total wet area is given by the sum of the free stream surface wetted area and the

compression surface wetted area:

wcww SSS += ∞ (4. 18)

The friction drag is dependent on the local Reynolds number. The average friction drag,

however, can be expressed in terms of the free stream Reynolds number based on the

waverider length ∞

∞∞=μ

ρ wl

lVw

Re , given by the expression:

( ) 2/10

Re664.0

wl

af

FFC = (4. 19)

Where,

φφσ

δ φ

dd

dRR

Rl

lSlF b

bbw

wo

l2/12

2

0

2/12/12

14⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛+⎥⎦

⎤⎢⎣⎡ −⎟

⎠⎞

⎜⎝⎛= ∞

∞∞

∞∫ , (4. 20)

and

w

wcwa S

SKSF

2/12 )1( δγ++= ∞ . (4. 21)

The wet areas are calculated as following:

45

φφσ

δφ

dIR

lS cb

wc

l

)(20

22 ∫ ∞= , (4. 22)

φφσ

δφ

dd

dRR

RlS b

bb

w

l2/12

2

0

2 12⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛+⎥⎦

⎤⎢⎣⎡ −= ∞

∞∞

∞ ∫ (4. 23)

With,

3

2/1

14

2222

)(

1)1()()(

udu

uK

ud

dRR

uKIc

a

b

b∫⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛−+

=

∞

∞ φσ

φ , (4. 24)

and

σφ /)(bRa ∞= (4. 25)

Given that

22 )1(1)( uuK −+= σ . (4. 26)

For the aerodynamic momentum calculation, the following relations were used:

bmsmm CCC ,, += (4. 27)

Where,

φφσ

σδφ

dR

lSlC

lb

wpsm ∫ ⎥

⎦

⎤⎢⎣

⎡−−= ∞

03

333

3

, cos134 (4. 28)

[ ]∫ −−−−

= φφσσσ

σδ dRRlS

lC cbcbwp

bm cos)(313

4 332

233

, (4. 29)

The moment is calculated about the y-axis from the vertex of the basic cone.

The pitching moment can be represented in terms of the center-of-pressure. By definition,

its location, Zcp, satisfies the relation:

wlL

mcp C

C- Z = (4. 30)

46

4.2 Trade-off analysis

Observing the previous section, the parameters involved in the generation of the

waverider shape are the free stream the Mach number M∞, the base body semi-angle δ, the

anhedral angle φl and R0. The parametric studies helped the on the design of the best

waverider configuration regarding both the aerodynamic CL/CD and the volumetric V2/3/Sp

efficiencies. Also, these studies are comprehensive for generic waverider configurations.

The basic geometry adopted for these studies features M=10, φl=30°, δ=5.5°, and

Ro/Xs=3/4, considering Relw=107. To perform these calculations, a simple Excel® file,

dados.xls, was used.

To better assess the influence of the base cone semi-angle δ, various geometries were

analyzed varying this parameter. The results for the lift-to-drag ratio and volume

efficiency are shown in Figures 4.4 and 4.5, respectively. Regarding those figures, one can

Figure 4.4. Lift-to-drag ratio for various base-cone semi-angles, δ.

47

Figure 4.5. Volume efficiency (V2/3/Sp) vs δ, for M=9, 10 and 11.

conclude that the maximum CL/CD occurs at 5.2°, with almost no variation near that

maximum. On the other hand, Figure 4.5 shows that V2/3/Sp increases for large values of δ. In

both figures, the curves were calculated for free stream Mach numbers 9,10 and 11. As will be

depicted latter, increasing M reduces the CL/CD while increases the value of V2/3/Sp. These

presented characteristics permit the designer to infer that the best choice for δ would be in the

region between 5° and 6°.

Following, the influence of Ro, referred in Figure 4.3, on the waverider characteristics was

analyzed. Figures 4.6 and 4.7 clearly show that V2/3/Sp and CL/CD behave inversely under Ro

variation. However, the volume efficiency is much more sensible to Ro variation. As a matter

of fact, in the range of study of Ro/Xs, 0.65 to 0.85, V2/3/Sp relative variation was -18.9%.

Meanwhile, in this same range, the CL/CD increased by 6.5%. For these reasons, one can

naïvely figure out that low values of Ro/Xs would represent good configuration choice,

nevertheless, the drag coefficient for such configurations would be large, as shown in Fig. 4.8.

It shows that the drag falls sharply as Ro/Xs is increased. For the range studied, the drag

relative variation was about -12%. Moreover, there is a fourth important waverider

48

Figure 4.6. V2/3/Sp variation with Ro/Xs.

characteristic that should be considered, the maximum-thickness to length ratio, tmax/lw. As

pointed out by Figure 4.9, tmax/lw diminishes with Ro/Xs. It is important to note that low

vehicle thickness could make the design impracticable, without enough space to place the

required flight systems.

Figure 4.7. Lift-to-drag ratio vs Ro/Xs.

49

Figure 4.8. Drag coefficient vs Ro/Xs.

Figure 4.9. Maximum thickness ratio vs Ro/Xs.

Figure 4.10 shows how the lift-to-drag ratio behaves as the anhedral angle increases. It

can be seen that CL/CD is maximum about φl=30°, with other parameters fixed. Also, when

analyzing the effect of the anhedral angle in the waverider performance, it is of paramount

importance to verify its influence on the wetted area, thus, in the friction drag. Figure 4.11 is a

plot of wet-to-plan area ratio (Sw/Sp) as function of φl from where it can be noted that the ratio

reaches a minimum near φl=30°. Although the existence of a minimum near φl=30° in both

curves are related, since the friction drag contributes largely for the total drag, it is

50

Figure 4.10. Lift-to-drag ratio vs the anhedral angle, φl.

important to keep in mind that the calculated friction drag was a simple approximation.

However, regardless the approach used to calculate the friction drag, its behavior with the

anhedral angle could be readily related to the Sw/Sp behavior, presented on Figure 4.11.

Figure 4.11. Sw/Sp vs φl.

Also, according to Figure 4.12, it was found that Mach number increases substantially

denigrate the aerodynamic efficiency of the waverider. That is because the lift decreases while

the drag augments, exacerbating the efficiency loss. The Figure 4.13 shows the drag

51

Figure 4.12. Lift-to-drag ratio dependence on M.

coefficient variation as a function of the free stream Mach number. Besides this drawback,

larger Mach numbers benefit V2/3/Sp, as one can observe in Figure 4.14. For these reasons,

allied to the fact that larger Mach numbers correspond to larger global range, design Mach

number 10 seems to be a good choice for hypersonic cruise vehicles.

Figure 4.13. Drag coefficient (CD) vs Mach number.

52

Figure 4.14. Volume efficiency vs Mach number.

For the sake of completeness, the table 4.1 presents the waverider parameters chosen for

the present investigation, based on the above tradeoff studies. This final configuration was

used for further surface modeling in CAD.

φ Ro/Xs M δ 30° 0.75 10 5.5°

Table 4.1. Waverider parameters correspondent to the best configuration, resulted from the tradeoff analysis.

4.3 Surface modeling

Once the waverider parameters were chosen, the leading edge curve was evaluated using

equation 4.4. With knowledge of this curve, the entire waverider surface could be calculated

by applying equations 4.2 and 4.3. For that matter, a computer code using MATLAB®

platform was implemented. Appendix B provides the reader with more details concerning

these calculations. The code surfaces.m calculates and prints on file the entire point cloud that

represents the lower and the upper surfaces of the waverider. That program only requires an

entry file, entries.m, - also inserted in Appendix B - with the values of the construction

parameters, from the Excel® file dados.xls.

53

Figure 4.15. Generated waverider, for φ=30°, M=10, δ=5.5° and Ro/Xs=0.75.

The waverider surfaces are recorded in the files: waverider_”configuration

number”_fs.dat and waverider_”configuration number”_cs.dat, corresponding to the free

stream surface and to the compression surface, respectively. Both files use the ascii format to

represent the Cartesian coordinates of each point. These files were further exported to a CAD

software in order to reconstruct the waverider shape. The shape used for this investigation is

presented by Figure 4.15.

4.4 Scramjet Integration

As stated before, in Chapter I, the scramjet engine consists of a supersonic inlet that

decelerates the flow, a combustor chamber where the flow is heated up and the nozzle, in

which the heated flow accelerates and generates thrust.

a) Inlet

The integration with the propulsive system requires the lower surface to compress the

suctioned air in order to achieve adequate pressure and temperature for the combustion.

Bearing this fact in mind, one must consider that after the conical shock wave produced by the

54

leading edge and after the compression ramp, the flow must reach the adequate conditions, or

close to them, for auto ignition of the scramjet fuel -Hydrogen- , with pressures between 25-

100kPa and temperatures between 1000-2000K[10], to occur. Therefore, the change of the air

properties with flight altitude must be accounted.

Figure 4.16. Altitude vs pressure, standard atmosphere data.

Figure 4.17. Altitude vs temperature, standard atmosphere data.

The inlet ramp adopted was a constant slope ramp. Also, it was assumed two-dimensional,

calorically perfect and non-viscous flow in that region. Although, these hypotheses do not

correspond to the complete flow modeling, they substantially simplify calculations for the

ramp geometry to be matter of primary investigation. The Figure 4.18 illustrates the flow

55

direction before and after each shock wave interaction. In that figure, μ and μ2 angles

represent the flow direction changes after the primary conical shock wave and after the shock

caused by the compression ramp, respectively, measured from the z axis. The ramp geometry

was omitted for the sake of clarity, even though, it’s reasonable to assume that flow is parallel

to the vehicle’s walls. With these considerations in mind, the ramp inclination measured from

the z axis can be evaluated as (μ+μ2).

Figure 4.18. Inlet shock waves schematic views.

The flow conditions along with the direction angles were obtained from the oblique shock

relations for supersonic flows, which can be found in any good aerodynamics reference. For

the first shock wave interaction, the free stream flow faces an oblique shock wave, generated

by the vehicle leading edge, with an inclination angle β, measured from the z axis. The flow

relations stand that [17]:

56

( )[ ]2/)1(sin

sin2/11sin 221

22122

2 −−++

=γβγ

βγμ

MMM (4. 31)

)1sin(1

21 221

1

2 −+

+= βγ

γ Mpp (4. 32)

( ) ( )( )

1

221

22122

11

2

sin12sin11sin

121

−

⎥⎦

⎤⎢⎣

⎡−+

+⎥⎦

⎤⎢⎣

⎡−

++=

βγβγ

βγ

γM

MMTT (4. 33)

Moreover,

( ) ⎥⎦

⎤⎢⎣

⎡++

−= −

22cos1sin

tan2tan 2

1

2211

βγβ

βμ

MM (4. 34)

The second shock wave interaction occurs in the compression ramp. Now, equations 4.31

to 4.34 could be modified as functions of the new conditions. Thus,

( )[ ]2/)1(sin

sin2/11sin 222

222

222

3 −−++

=γκγ

κγμ

MMM (4. 35)

)1sin(1

21 222

2

3 −+

+= κγ

γ Mpp

(4. 36)

( ) ( )( )

1

222

22222

22

3

sin12sin11sin

121

−

⎥⎦

⎤⎢⎣

⎡−+

+⎥⎦

⎤⎢⎣

⎡−

++=

κγκγ

κγ

γM

MMTT

(4. 37)

And,

( ) ⎥⎦

⎤⎢⎣

⎡++

−= −

22cos1sin

tan2tan 2

2

2221

2 κγκ

κμ

MM (4. 38)

Using these equations, it’s convenient to calculate the ratio between the inlet pressure and

the free stream pressure and as a function of the shock wave angle κ. The results are shown in

Figure 4.19. It’s obvious that increasing the ramp inclination angle strengthens the oblique

shock wave formed in that region. By comparing these values with the required inlet pressure

range one can relate the allowed values of the free stream pressure with the values of κ.

57

Furthermore, the correspondent altitude for these limiting values of free stream pressures

could be obtained according to the International Standard Atmosphere (ISA) data. After these

procedures, one can get the results shown in Figure 4.20. That figure shows for each ramp

angle one can find the correspondent altitude ceiling and floor.

Figure 4.19. Ramp compression ratio as a function of the shock wave angle κ.

Figure 4.20. Relationship between the shock wave angle and the required altitude condition.

According to the International Standard Atmosphere model.

58

Similarly, from equations 4.31 to 4.38, the ratio between the free stream temperature and

the inlet temperature were evaluated for various values of the oblique shock wave angle κ.

The results are shown in Figure 4.21.

Figure 4.21. Inlet temperature for various shock wave angles.

Furthermore, the range of free stream temperatures allowed as function of κ can be seen in

Figure 4.22. In this case, the temperature limits from the atmospheric data - 196.5K and

288.15K - were added to the graph. With this information, it was possible to determine that

κ>0.43 in order to make combustion possible.

Figure 4.22. Temperature range versus shock wave angle.

59

According to the information presented in figures 4.17, 4.20 and 4.22, an oblique shock

wave angle κ=0.45 (25°) was pertinent for the present investigation, which corresponds to a

ramp inclination of 20°. That is due to several reasons. Firstly, one can conclude that the

correspondent altitude limits are 40km and 50km. Secondly, ramps with larger inclination

angles produces higher wave drag along with higher entropy losses, both consequences are

undesirable for a practical design.

b) Supersonic nozzle

The design of an efficient combustion chamber was beyond the purpose of this work.

However, it was possible to estimate the nozzle geometry based on the Method of

Characteristics (MOC). In fact, the obtained curve and the tools used for its calculation could

be used for further optimization of the geometry in future works.

First of all, the exhaust nozzles for hypersonic vehicles are variations of the classical

supersonic nozzles. However, differently from the classical case, which the flow is confined

in the nozzle, in hypersonic vehicles the flow undergoes a free expansion. That is because

other geometries could imply heavier configurations [17].

On the nozzle design, the entry flow Mach number must be perfectly expanded to the free

stream static pressure at exit, where perfectly means in a uniform and parallel way. Allied to

this fact, it was assumed that the exhaust nozzle flow were two-dimensional, isentropic and

calorically perfect. These assumptions permit the use of the Method of Characteristics

(MOC), a very strong tool to deal with hyperbolic partial differential equations, which gives

the exact solutions under these assumptions.

Also, the design was a Minimum Length Nozzle (MLN). For that matter, sharp corners

placed in the nozzle entry produces centered Prandtl-Meyer expansion fans. It is assumed too

that no reflection occurs, each of the produced characteristics is canceled in the opposite wall

60

contour by a weak compression wave. For a deeper understanding, Figure 4.23 depicts the

most important features of the applied method.

Figure 4.23. Main parameters used in the Method of Characteristics.

In Figure 4.23. A, B and C are internal points of a flow field. L- characteristic line

emanates from the point B, while L+ characteristic line starts in C. From the theory, the

quantity )(MK υθ +=− is constant along a L- characteristic line and )(MK υθ −=+ is

constant along a characteristic line L+. Where θ is the flow direction and ν(M) is the Prandtl-

Meyer function. Therefore, since the point A lies on the intersection of these two lines, one

can get:

2,, CB

A

KK +− +=θ (4. 39)

2,, CB

A

KK +− −=ν (4. 40)

The Prandtl-Meyer function is given by [17]:

( ) 1tan111tan

11)( 2121 −−−

+−

−+

= −− MMMγγ

γγν (4. 41)

61

For γ=1.4 and 1 < M < 10, in order to calculate the local Mach number, the inverse

Prandtl-Meyer function can be approximated by the polynomial:

44

33

2210)( ννννν AAAAAM ++++= (4. 42)

With coefficients given by:

A0 1,13855A1 0,00899A2 0,00151A3 -2,89E-05A4 2,10E-07

Also, according to ref. [17], the Mach angle could be evaluated as following:

⎟⎠⎞

⎜⎝⎛= −

M1sin 1μ (4. 43)

Finally, the slope of each characteristic line is given from the relation:

)tan( μθ ±=⎟⎠⎞

⎜⎝⎛

±Ldxdy (4. 44)

Now, in order to allow the readers a complete understanding of the Method of

Characteristics as well as it was applied in the present investigation, one should consider the

two-dimensional nozzle shown in Figure 4.24. The entry Mach number Mc>1 increases to

Me>1 in the exit. Centered expansion fans emanate from the first portions of the nozzle,

points 0 and 1. These expansion fans are discretized in three characteristics lines on each side.

Those characteristic lines which start in point 0 are negative slope lines. On the other hand,

those which emanates from 1 are positive slope lines.

62

Figure 4.24. Schematic view of the Method of Characteristics.

Along the lines that pass through 0, for the ith characteristic line, −Li0 , the characteristic

−Ki0 is constant. Where i=1,2,3. Meanwhile, those characteristic lines that come from 1,

+Lj0 , conserve the quantity +Kj

0 . With j=1,2,3.

So far, it’s important to note that when a flow element originated from the region c passes

through a 0-emanated-characteristic line, the characteristic is increased by −Δ K0 . Likewise,

when a flow element passes through a 1-emanated-characteristic line, the characteristic is

increased by +Δ K1 . Thus,

−−− Δ+= KiKK ci

0,0 (4. 45)

And,

+++ Δ+= KjKK cj

1,1 (4. 46)

But, once the flow in region c is considered uniform and parallel to x axis one can find:

)(, cc MK ν−=+ (4. 47)

Also,

)(, cc MvK =− (4. 48)

Finally, at the end of the expansion fan in region e, the flow is parallel to x axis:

63

)(1230 eMKK ν== −− (4. 49)

Similarly,

)(31 eMK ν−=+ (4. 50)

From the equations above, the increments in characteristic values are given by:

3)()(

0ce MM

Kνν −

=Δ − (4. 51)

With,

−+ Δ−=Δ KK 01 (4. 52)

Also, from the information given by figures 4.23 and 4.24, plus the Equations 4.39 and

4.40, one can conclude by tracking the characteristic line −L30 that for the point 2 the

following relations are valid:

2

10

11

2−+ +

=KK

θ (4. 53)

2

11

10

2+− −

=KK

ν (4. 54)

Similarly, for the point 0, after the flow past through −L30 , the maximum expansion angle

0θ is calculated as means of:

212

0−+ +

=KK cθ (4. 55)

Thus, using relations 4.44, 4.40 and 4.38:

2)()(

0ce MM νν

θ−

= (4. 56)

Furthermore, as the evaluations proceed, all internal point - 3, 4, 6, 7, 8, 10, 11 and 12 -

properties can be evaluated using similar relations. For that matter, Figure 4.25 depicts an

auxiliary scheme from which internal flow parameters could be calculated. In that figure, in

the first row are placed the correspondent characteristic lines emanated from 1, +Lj1 , while in

64

the first column the related characteristic line emanated from 0, −Li0 , can be found. The

elements are the nodes, numbered as shown by Figure 4.24.

128411731062

30

20

10

31

21

11

−

−

−

+++

LLL

LLL

Figure 4.25. Auxiliary scheme for obtaining internal point flow conditions θ and ν.

Thus, regarding figures 4.25 and 4.24, one can easily infer the node placed in row i and

column j has the properties:

2, 01 −+ +

=KK

jiatnodeij

θ (4. 57)

2, 10 +− −

=KK

jiatnodeji

ν (4. 58)

In addition to that, one can obtain the node coordinates for every internal point, since it

was assumed that between connected nodes the characteristics lines are straight lines. For

instance, analyzing the node 2 – see Figure 4.26 -, starting from points 0 and 1, which have

known localizations, its coordinates (x2,y2) are obtained from the following set of equations:

)( 02

1

002 xx

dxdyyy −⎟

⎠⎞

⎜⎝⎛=−

−

(4. 59)

)( 12

1

112 xx

dxdyyy −⎟

⎠⎞

⎜⎝⎛=−

+

(4. 60)

Where ⎟⎠⎞

⎜⎝⎛

dxdy can be found by means of equation 4.44.

65

Figure 4.26. Auxiliary scheme for point 2 coordinates calculations.

For the wall contour calculation, it is assumed too that no reflection occurs in the wall

nodes, as stated before. Hence, the characteristic line connecting a particular wall node to its

closest internal node is a straight line, along which θ is constant. Thus, in the present case, θ4

= θ5, θ8 = θ9 and θ12 = θ13.

Also, as the contour wall must be aligned to the flow in order to avoid compression waves

to be formed, the lines connecting two wall nodes - in this case, 0, 5, 9 and 13 – are

considered as straight lines with average slopes. For instance, one should consider the region

0-5 in the wall, in this region, the slope of the line connecting these nodes is given by:

250

50

θθ +=⎟

⎠⎞

⎜⎝⎛

−walldxdy (4. 61)

Where 0θ is obtained via Equation 4.56.

Following, one should consider the general case when N characteristic lines emanate from

each point, 0 and 1. In this case, similarly to Figure 4.25, Figure 4.27 illustrates how the

calculations of the properties of the internal flow points were made. The elements are the

internal points.

66

)1(33221

362542524314232

0

230

220

210

131

21

11

++++

+++++++++

↓

→

−

−

−

−

++++

NNNNNL

NNNLNNNLNNNL

LLLL

i

j

N

N

L

MMMMMM

L

L

L

L

Figure 4.27. Auxiliary scheme for obtaining internal point flow conditions θ and ν.

Yet, from Figure 4.25, one can conclude that the node placed on the row i and column j is

the node i)(jN1)-(j ++ . So, analyzing the relations found for the simple case which N=3,

previously discussed, we can conclude that:

)()1(, eNN MK ν−=++ , (4. 62)

)()1(, eNN MK ν+=+− (4. 63)

And, from equations 4.57 and 4.58:

210

)()1(+−

++−

+=

KK ji

ijNjθ , (4. 64)

210

)()1(+−

++−

−=

KK ji

ijNjν (4. 65)

Also, equations 4.45 and 4.46 permit one to infer that:

−++− Δ−

= KjiijNj 0)()1( 2

θ (4. 66)

−++− Δ+

+= KjiMv cijNj 0)()1( 2)(ν (4. 67)

Where,

NMM

K ce )()(0

νν −=Δ − (4. 68)

To calculate the internal point coordinates, one could conclude that equations 4.59 and

4.60 lead to:

67

.)( 010

01 xxdxdyyy i

i

i −⎟⎠⎞

⎜⎝⎛=− +

−+ (4. 69)

)( 11)1(1

11)1( xxdxdyyy jNj

j

jNj −⎟⎠⎞

⎜⎝⎛=− ++−

+++− (4. 70)

Also,

( )ijNjijNjijNj

ijNjijNj xxdxdyyy ++−−++−

−−++−++−−++− −⎟

⎠⎞

⎜⎝⎛=− )1(1)1(

1)1()1(1)1( (4. 71)

( )ijNjijNjijNj

ijNjijNj xxdxdyyy ++−−++−

+−++−++−−++− −⎟

⎠⎞

⎜⎝⎛=− )1(1)2(

1)2()1(1)2( (4. 72)

For i,j =1,2,3…N.

Finally, after thorough examination of equations 4.59, 4.60 and 4.61, in the light of figures

24 and 27, one can obtain the wall point coordinates from the set of equations:

( )0202

02 2tan xxyy N

NN −⎟

⎠⎞

⎜⎝⎛ +

=− ++

+

θθ (4. 73)

.1),(2

tan 1)1)(1(1)1(1)1)(1(1)1(

1)1)(1(1)1( ≠−⎟⎟⎠

⎞⎜⎜⎝

⎛ +=− ++−++

++−++++−++ jwithxxyy NjNj

NjNjNjNj

θθ (4. 74)

)(2

tan )1(1)1()1(1)1(

)1(1)1( ++++++

+++ −⎟⎟⎠

⎞⎜⎜⎝

⎛ +=− NjNj

NjNjNjNj xxyy

θθ (4. 75)

For j = 1, 2, 3...N.

Since the flow throughout the nozzle is homoentropic, the pressure distribution can be

evaluated as means of:

( )[ ]( )[ ][ ]

1

2

2

)(2/112/11)( −

⎟⎟⎠

⎞⎜⎜⎝

⎛

−+−+

=γ

γ

γγ

xMM

pxp c

c

(4. 76)

Although the mathematical apparatus discussed above does not consist a complete

modeling of the flow in the nozzle section, the results found on the Method of Characteristics

were enough for a primary investigation of the flow characteristics and properties in the

68

expansion region. For instance, high pressure in the combustion chamber can be important to

the combustor efficiency, since it benefits the reaction time; however, higher values of Pc

results in larger nozzles, see Figure 4.28. On the other hand, as the flow is heated inside the

combustor, the gases temperature increases, decreasing the flow Mach number in that region.

Meanwhile, Figure 4.28 depicts that lower entry Mach numbers generates shorter nozzles

geometries. These two conclusions allow one to infer that for maximum nozzle efficiency –

only -, the flow in the combustor exit must have two features: low pressure and low Mach

number.

Figure 4.28. Nozzle length versus Pressure ratio, for different entry Mach numbers.

For the Hydrogen combustion, at altitudes between 40 and 50 km, one can conclude that

1.001.0 <<c

e

pp

. Although more information about the flow conditions at the scramjet

combustor is necessary, it was assumed that the flow pressure at the nozzle entry could be

averaged, so 05.0=c

e

pp

.

69

Figure 4.29. Resulted nozzle contour for entry Mach number 3 and pressure ratio 0.05.

The Figure 4.29 illustrates the wall contour for the case of entry Mach number of 3 and

05.0=c

e

pp

. With these parameters fixed, one can conclude that Me = 5.23 and L/y1 = 35.15.

The correspondent pressure distribution can be found in Figure 4.30. Regarding these two

figures, one can observe that the divergent region, 0 < x/L < 0.15, is related to an abrupt

pressure decrease. That is because the obtained divergent region is a 15°-slope straight line

segment.

Figure 4.30. Pressure variation along the nozzle length.

70

4.5 Dimensioning

Since the shape of the pure waverider configuration, the inlet ramp angle, and the nozzle

contour were obtained in the previous sections, the conditions for a full integrated waverider

vehicle design were reached. Therefore, a 781.2 mm stainless steel model of a waverider was

machined for investigation of the pressure distribution and for flow visualization. The time for

the complete design of the model was about one year. Then, the machining time took about

six months. The model was machined under contract awarded by the Oxtig company. Because

of the complexity of the pure waverider surface, the model was almost entirely machined

using Computational Numeric Control (CNC). The material, stainless steel, was chosen for its

durability and hardness, as the model was expected to meet very severe conditions during

tests.

The model comprised a pure waverider surface part, with a cavity made in order to

accommodate the engine geometry; the scramjet ramps; and the support adapter. Also, the

model was instrumented with various pressure transducers located in the compression surface.

Thus, it featured several cavities and channels used for properly accommodation of the sensor

cables. During the machining process, extreme care was taken so as to maintain the leading

edge thickness of the order of 0.05 mm.

So as to simulate the propulsion system geometry, the inlet consisted of a 66.8 mm-length

20°-constant-inclination ramp. Two longitudinal fins were placed in the entry to mitigate

three-dimensional effects. The inlet was followed by a 270.1 mm flat region which ends in the

nozzle expansion ramp. Then, adopting the obtained results using the Method of

Characteristics (MOC), described previously, it was noted that for practical model lengths to

be used in the T3 shock tunnel, the MOC nozzle contour would follow a very short combustor

exit, in the order of a few millimeters. For this reason, a 15° constant inclination expansion

ramp was adopted. A conic-surface then was added too, joining the pure waverider surface

71

and the expansion ramp, in order to eliminate vortex formation near the pressure transducers.

The scramjet modulus was fixed to the pure waverider part by means of ten bolts.

All in all, the configuration used for the current investigations, without the support

adapter, can be seen in Figures 4.31 and 4.32.

Figure 4.31. Model’s views.

Figure 4.32. Model plan-form and some dimensions.

72

Furthermore, the diamond-like-adapter was attached to the model by four Allen bolts. It

permitted the model to be fixed in the sting structure, which consisted by a 738 mm long

circular tube; however, another circular nylon tube insulated the adapter from the sting

apparatus. That was necessary because electrical noise or structural vibration could denigrate

the pressure transducer signals. The Figure 4.33 shows the model along with the sting and the

support structure. Also, Figure 4.34 presents actual photographs of the machined model.

Pressure taps were positioned along the center line of the vehicle and in other particular

places. These taps featured the same dimensions as those specified by the PCB®, for types

132A31 and 132A35 pressure transducers. Following the factory recommendations is very

important to the proper functioning of these devices. Otherwise, the existence of pre-stresses

on the pressure-sensitive transducer surface would follow a non-linear electronic response of

such device. Similarly, the adequate housing of the transducers reduces structural vibration.

Chapter V provides the reader with more information about the sensors.

Also, to the model was added a Pitot pressure probe, not inserted in the presented figures.

It consisted of a 485 mm long tube attached to the model’s adapter. In the free extremity was

located a pressure transducer type PCB® 112A22 with a proper housing. The semi-spherical

shape of the probe causes the formation of a bow shock wave ahead, which can be modeled as

a normal shock for estimations of the stagnation and the free stream conditions during each

test. Also, the Pitot data was used to determine the test time.

Special attention was required to best place the model inside the test section. Firstly, it

was precisely aligned with the horizontal plane otherwise some deformation and shadows

would appear in the Schlieren photographs. Secondly, for several runs, the foremost portion

of the leading edge was placed about 10 mm apart from the nozzle exit plane. Thus, the angle

of the formed shock wave could be easily visualized.

73

Figure 4.33. Waverider model and its mounting, detailed views.

74

Figure 4.34. Photographs of the investigated waverider model.

75

5 Instrumentation

As stated before, during the present investigation measurements of the pressure

distribution on the model surface along with Schlieren photographs were made. The provided

data was used to better assess the flow properties on large part of the model surface, such as

the shock wave angle and the inlet compression ratio.

5.1 Pressure transducers

A set of seven pressure transducers types 132A31 and 132A35 were disposed on the

model’s compression surface, they were located as shown by Figure 5.1.

Figure 5.1. Location of pressure sensors.

Given the short duration of the test time, the piezoelectric pressure transducers are the

more appropriate types of pressures gages. That is because they have a very short rise time, in

the order of a few microseconds. In these transducers, preloaded ceramic plates are used as

the sensing elements [22]; they are separated from the environment by a diaphragm. Once

under stress, the ceramic generates charge. However, besides the good electrical insulation of

the ceramic housing, the charge eventually goes to zero, following a RC-circuit charge

76

exponential decay. For those reasons, piezoelectric transducers can only measure dynamic

pressure. The high impedance output is converted by a built-in Integrated Circuit Amplifier

(ICP®) to a low impedance output, usable for reading purposes. This conversion avoids the

use of low-impedance cables and allows the output signal to be transmitted by long cables

without substantial noise interference. Table 5.1 shows detailed information about the main

characteristics of applied sensors.

For good pressure measurements, the pressure sensors must be mounted seemly. Flush

mounting was adopted, in this case, according to the supplier recommendations, the sensors

were mounting as depicted in Fig. 5.2. The internal diameter of the mounting hole was coated

with cyanoacrylate fast-acting glue.

Figure 5.2. Correct installation of pressure sensors [23].

Measurement Range 350 kPa (50psi) Sensitivity 20.3 mV/kPa (140 mV/psi) Maximum Dynamic Pressure 5516 kPa (800 psi) Resolution 0.007 kPa (1mpsi) Rise Time ≤ 3 microseconds Discharge Time Constant ≥45 microseconds Output Impedance ≤100 ohms

Table 5.1. Some performance characteristics of the PCB 132A31 and PCB123A35 pressure sensor [23].

77

The microelectronic was powered by a 24 to 27 VDC, 2 to 20 mA constant-current

supply. The signal conditioner used was type PCB 481A02, which featured 16 channels,

programmable gains of 1, 10 or 100, with BNC input/output. Figure 5.3 shows a typical trace

of a piezoelectric transducer.

Figure 5.3. Piezoelectric sensor typical output.

In addition, the end of the T3 driven section features three pressure transducers. Two of

them are separated by 40 cm and they are used to mete the incident shock wave speed. The

other one measures the reservoir pressure and is used to trigger the oscilloscope and the

Schlieren photographic apparatus. Both of them operate in the charge mode, thus, they are

connected to charge amplifiers. The table 5.2 shows the set up of each amplifier along with

the correspondent pressure sensor.

Sensor Measurement Type Sensitivity

(pC/bar) Serial

Number

Charge Amplifier

type

Scale (mechanical

units/V) 1 Reservoir pressure 7005 -47.0 582222 5008 20 or 0.2* 2 Incident shock wave, first 701A -81.1 629999 5007 5 3 Incident shock wave, second 701A -81.0 598603 5001 5

Table 5.2. Pressure sensors data. *during hammer tests.

Before each test run, the hammer test, a fast and simple method for checking sensitivity of

the triggering sensor and the electrical connections, was applied. When the sensitivity scale of

the charge amplifier is reduced to 0.2 mechanical units/V, the impact of the hammer on the

78

vicinities of the sensor 1 produces a Dirac pulse that can be detected by the sensor. The

electrical impulse is then used for triggering the data acquisition system and the Schlieren

system, with a triggering threshold of 1.0V.

5.2 Yokogawa DL750

The data acquisition system used in this work was an oscilloscope type Yokogawa DL750

[24]. It features 16 channels (8 slots, 2 channels per slot), with several functions for real-time

data manipulation, including filtering, addition, subtraction, multiplication, division, and

others. Also, the equipment can be operated via USB by a computer. The data can be saved in

ASCII, waveform, and jpeg formats, and it can be stored in the 20 GB internal hard disk or in

a USB drive, among other types of storage. The 12-bit module with two input channels can

perform up to 10 million samples per second. Furthermore, the software for data manipulation

can export the recorded to various file formats, including xls. Also, the DL750 has a 10.4"

color LCD monitor. The Figure 5.4 shows the device.

Figure 5.4. Oscilloscope Yokogawa DL750.

As stated before, the oscilloscope recording mode was triggered by the sensor which

measures the reservoir pressure. Although the sampling interval was not the same for all the

test runs, it was noted that 1 microsecond of time resolution was adequate to observe recorded

79

data. With this sampling rate, the correspondent recording time interval was 10 milliseconds

(0.1 Mega Samples per second).

5.3 Schlieren Apparatus

Finally, to better assess the flow characteristics over the model, a complete Schlieren

optical set up was used. The system was previously mounted in the T3 test section for

investigation of the hypersonic flow over blunt bodies [25]. The Figure 5.5 shows the used

system.

Figure 5.5. Schlieren mounting on T3 tunnel.

Figure 5.6. Schematic view of the Schlieren mounting.

80

The theory underlying the Schlieren visualization method can be found in any good

reference about compressible flows. Basically, when the light path crosses a shock wave, the

density variation makes the light to be refracted accordingly, therefore, the shock layer

becomes visible. The light from a flash is collimated and driven through the test section; then

the image is deviated to the camera. See Figure 5.6. Before the light reaches the camera optics

it is filtered by a sharp knife that partly blocks non-disturbed light rays, then the disturbances

are made visible in the captured image.

The camera, type Cordin 550, uses a complex optical system centered on a multi-faceted

mirror that spins at very high speeds [26]. This action distributes the image to individual CCD

channels which record the frames. The camera captures a total of 32 frames in the exposure

time. The standard mirror-drive is a compressed gas driven turbine which allows speeds up to

2 million frames per second. Although the turbine can be driven by air or Helium too,

Nitrogen was used as the driver gas. In order to avoid overheating, the turbine must be turned

on just a few seconds before a test. Also, the camera provides a very high resolution of 1000 x

1000 pixels. Besides, a USB interface allows the full control through a computer along with

instantaneous image transfer for immediate analysis. During the tests, it was found that 50000

frames per second were appropriate to visualize the whole air flow development.

Figure 5.7. The camera Cordin 550 placed just beside the T3 test section.

81

Also, it was verified that the use of quartz windows in the test section was of paramount

importance to better visualize the thin shock wave caused by the Mach 10 flow. The weak

main shock wave formed on the model nose produced low density gradients which reduce the

image contrast. However, due to its higher optical quality, the image’s information losses

using quartz windows were lower than using standard glass windows.

82

6 Results and Discussion

The present chapter will discuss the investigation of the Mach 10 air flow over a

waverider model. This investigation consisted of several shock tunnel tests with stagnation

conditions up to 2150 K and 2946 psi. The pressure distribution over the model compression

surface was measured using piezoelectric sensors located as described by the section 1 of

Chapter V. Also, some Schlieren photographs were taken in order to visualize the main flow

characteristics in the fore body. To perform such tests, the shock tunnel used was the T3, cited

in Chapter II.

6.1 Shock Tunnel Conditions

The stagnation conditions were estimated using the data from the three pressure sensors

located at the end of the driven section, as mentioned in sections 2, in Chapter II, and 1, in

Chapter V, and in Chapter III. Firstly, these sensors permitted direct measuring of the shock

wave speed, which was used as an input to the computer codes t3_sing and hstr - for numeric

validation – which provided the user with the reflected mode conditions. Secondly, using the

same t3_sing code or equint with the reservoir pressure as input data, it was possible to

estimate the stagnation conditions in the equilibrium interface mode.

After the procedure described above the test reservoir conditions during the current

investigation were obtained, the results are shown in Table 6.1. The reader shall note that in

some runs Helium was applied as the driver gas, while in others air was used. As Table 6.1

shows, the former gas led to higher stagnation conditions, correspondent to the equilibrium

interface mode. On the other hand, the reflected mode conditions were possible by using air in

the driver section.

83

Initial conditions Stagnation Conditions

Run # Date

Gas p4 (psi) p1 (atm) T1 (°C) p4/p1

Incident Mach

number P0(psi) T0(K) 1 02/04/08 He 2700 1.00 22.0 183.7 4.08 2176 2150 2 14/04/08 He 2700 2.00 23.5 91.8 3.41 2639 1624 3 17/04/08 He 2300 2.00 24.0 78.2 3.31 2319 1535 4 07/05/08 He 2700 2.00 20.0 91.8 3.89 2639 1820 5 09/05/08 He 2700 2.63 20.0 69.8 3.44 2938 1558 6 27/05/08 Air 3000 2.67 23.0 76.4 2.09 764 795 7 27/06/08 Air 3000 2.67 19.0 76.4 2.03 790 715 8 04/07/08 Air 3000 2.67 20.5 76.4 2.26 725 834 9 08/0708 He 3000 2.00 19.0 102.0 3.83 2900 1840 10 14/07/08 He 3000 2.00 20.0 102.0 3.64 2778 1730 11 25/07/08 He 3000 2.00 23.0 102.0 3.64 2946 1757 12 30/07/08 He 3000 2.00 23.0 102.0 3.49 2850 1702 13 06/08/08 He 3000 2.00 24.0 102.0 3.47 2938 1706

Table 6.1. Initial conditions and reservoir conditions for all tests.

Figure 6.1 shows typical pressure traces from the three sensors placed in the driven

section, for high enthalpy tests. During these tests, the time interval between the signal of the

two sensors assembled to measure the shock wave speed ranged from 285 to 339

microseconds, as the incident Mach number varied from 3.31 to 4.08. In these cases, from

Table 6.1, it can be noted that for each run, the reservoir pressure was near the initial driver

pressure. The maximum stagnation pressure found was 2946 psi, while the maximum

stagnation temperature was 2150 K, related with the fastest incident shock wave.

84

Figure 6.1. Typical traces for the pressure sensors installed in the T3 tunnel, for high enthalpy

tests. Run #12.

For the high enthalpy cases, the incident Mach number as a function of the driver-to-

driven pressure ratio p4/p1 calculated by the ideal shock tunnel theory (see Chapter II) is

compared with the experimental data in Fig. 6.2.

As expected, according to the Fig. 6.2, the theoretical forecasts yield larger values for the

incident Mach number than the actual ones. This is because the applied mathematical model

discards viscous effects, the complex shock formation from compression waves, and non-

ideal opening of the diaphragms. The attenuation of the shock wave, caused by the viscous

effects, strongly contributes to the presented results as the shock wave speed measurement in

the T3 is made at the end of the driven tube. As a matter of fact, the shock strength, which

means how much the pressure increased after the shock, decreases with increasing distances

from the diaphragm region. Thus, the 32 feet long T3 driven exacerbates this effect.

85

Figure 6.2. Incident Mach number as a function of the driver-to-driven pressure ratio, for high

enthalpy tests.

Finally, there’s another technical issue: the measurement of the driven pressure p1. The

valve of the pressure gauge in that region must be closed before high pressure levels in the

driver are reached –about 2000 psi- in order to protect the gauge from elevated pressures

produced by the arrival of an incident shock wave from a non-intentional diaphragm rupture.

Thus, due to leaks in the driven tube, it’s hard to evaluate the precision of the driven pressure.

That could exacerbate the differences between the measurements and calculations of the

incident Mach number as seen in Fig. 6.2.

For the low enthalpy tests - air as driver gas- the typical shock wave time interval was

about 515-575 microseconds, which means a very slow incident shock wave. See Figure 6.3.

This result agrees with the predictions made on section 1 of the Chapter III: higher density

gases produce slower incident shock waves. Thus, in this case, the temperature and pressure

conditions are considerable lower than they were in the Helium driven tests.

86

Figure 6.3. Typical traces for the pressure sensors installed in the T3 tunnel, for low enthalpy

tests. Run #6.

Finally, the Pitot pressure probe, which was exposed to the free stream conditions,

provided the impact pressure data that, used as input for the t3_sing or freestream, could give

the estimated free stream conditions for each test. The complete results, outputs from the

t3_sing, are shown in table 6.2.

The figures 6.4 and 6.5 show typical values for the Pitot pressure probe sensor along with

the reservoir pressure data, for high enthalpy tests and low enthalpy tests, respectively.

Regarding those figures, the reader should note that the time interval between the rising of the

reservoir pressure and the rising of the Pitot sensor was about 1.0 millisecond for both

conditions. This evaluation permitted to set up the optical apparatus so as to the recording

begin just after the flow inside test section achieves steady state. The relative large difference

87

Free Stream Conditions

Run # Date PPitot (psi) p∞ (psi) T∞(K) M∞ Re∞(m-1) Kn∞ 1 2/4/2008 5.7 0.043 121.0 10.0 2.25 x 106 0.19 2 14/4/2008 6.8 0.051 87.9 10.0 4.36 x 106 0.11 3 17/4/2008 5.9 0.045 82.5 10.0 4.20 x 106 0.12 4 7/5/2008 8.0 0.069 109.2 9.4 3.96 x 106 0.11 5 9/5/2008 8.5 0.070 87.9 9.6 5.73 x 106 0.08 6 27/5/2008 3.0 0.025 41.6 9.7 7.03 x 106 0.11 7 27/6/2008 3.2 0.027 38.0 9.6 8.76 x 106 0.09 8 4/7/2008 2.5 0.019 41.3 10.0 5.58 x 106 0.15 9 08/0708 11.2 0.107 122.0 8.9 4.91 x 106 0.08 10 14/7/2008 11.0 0.106 114.3 8.9 5.35 x 106 0.07 11 25/7/2008 11.3 0.107 114.8 9.0 5.41 x 106 0.07 12 30/07/08 12.9 0.129 116.9 8.7 6.17 x 106 0.06 13 6/8/2008 7.6 0.058 92.8 10.0 4.48 x 106 0.11

Table 6.2. Free stream conditions during the tests.

Figure 6.4. Typical pressure sensor data from a high enthalpy test. Run #10.

on the value of the free stream conditions for the same initial conditions are due to the change

in the model’s position inside the test section. During the runs from 1 to 5, and 13, the model

was placed such as the distance from the nose to nozzle-plane was 10 millimeters. On the

88

Figure 6.5. Typical pressure sensor data from a low enthalpy test. Run #8.

other hand, during the runs 6 to 12, the model’s nose was placed inside the nozzle for the

Schlieren photographs of the ramp section to be possible.

6.2 Air Flow Investigation

As shown in table 6.2, the free stream conditions achieved during the present

investigations were quite homogeneous. Thus, for all tests we might expect similar flow

behavior on several aspects. For instance, the low Reynolds numbers indicates that the flow is

laminar over some distance from the leading edge of the model. Furthermore, the low

Knudsen numbers suggest continuum flow conditions to be expected throughout the

investigated model, except inside the slip region. Also, at very high Mach numbers, the strong

viscous interaction phenomenon is likely to affect the downstream conditions.

Although the forebody compression surface presents some longitudinal angle variation,

detailed observation of the model sketches can show that air flow turning is so slight – in fact

the turning angle reaches 3° as a maximum at the inlet - that the flow in this region can be

compared with the flow over a flat plate under the same conditions.

Several works [27,28] have shown the existence of the three different flow field regions

downstream the leading edge of a flat plate. Although these works refer to the flow

investigation over a flat plate, the flow development over the flat portions of the model, where

89

Figure 6.6. Flowfield on flat plate at hypersonic flow. Adapted from Toro [29].

the flow was not perturbed by a corner or a ramp, is expected to be the same. The distinct

flow regions found are sketched in Figure 6.6. Even under continuum flow conditions

(Kn<<1) it was reported by Nagamatsu and Sheer [27] the existence of slip near leading edge

for very large Mach numbers. According to the theory developed by these authors, the length

of the slip region can be assumed as proportional to M∞. Indeed, following the same

procedure as taken by Minucci [21] it was possible to estimate the slip region length as

between 0.03 and 0.1mm. In this region, which can be considered a rarefied flow, the

Boltzmann equations are more suitable. As it can be noted in the Schlieren photograph in

Figure 6.7, the flow pattern around the leading edge no longer presents a bow shock as in the

continuum case, instead, near the leading edge the shock wave starting point was almost

imperceptible, a main characteristic of a slip condition.

Furthermore, the region that follows the slip region presents shock-wave/boundary-layer

interactions, the extent of the influence of this region downstream depends on the size of the

subsonic portion of the boundary layer and on the strength of the shock wave [30]. When the

rate of growth of the boundary layer is large, the boundary layer and the shock wave are

merged within a limited region. In this situation, the outer inviscid flow is strongly affected

by the displacement thickness, which in turn substantially affects the boundary layer. This

mutual process is called strong viscous interaction [17]. As stated before, because of the low

Reynolds number, one can assume that the boundary layer is laminar.

90

Figure 6.7. Mach 10.0 flow past the model leading edge. Reservoir conditions: P0 = 2938 psi

T0 = 1706 K. Run #13.

The similarity parameter that governs laminar viscous interactions is given by:

CM

xRe

3∞=χ (6.1)

Where Rex is based on the free stream properties, calculated at a distance x from the leading

edge. While M∞ is the free stream Mach number, and ee

wwCρμρμ

= , where μ is the dynamic

viscosity ρ is the density and the subscript w relate to the wall and e to the edge of the

boundary layer, respectively. The effects of the hypersonic viscous interactions on the

pressure distribution over a flat plate as function of the parameter χ were presented in several

works [21, 28, 31]. A common result is that the induced pressure change varies linearly with

χ - the linear variation has been also theoretically predicted by Anderson [17] – thus, one can

write:

χFp

p=

Δ

∞

(6.2)

91

∞p is the free stream static pressure, Δp is the static pressure perturbation and F is a function

of the ratio of the specific heats and the wall temperature condition. Another important result

relates the leading-edge shock-wave angle β with the local Reynolds number – thus with the

leading edge distance x - inside the strong interaction region, as below:

2/14/1

Re*

∞⎟⎟⎠

⎞⎜⎜⎝

⎛∝≅ MC

dxd

x

δβ (6.3)

Where δ* is the displacement thickness. For the above equation, the reader must recall that

the shock wave and the boundary layer are merged within the considered region.

Following in the weak interaction region, the displacement effects are small enough that

the inviscid flow does not interact with the boundary layer. Also, small pressure gradients

inside the boundary layer permit the use of the Blausius solution for a viscous flow over a flat

plate, adapted for compressible flows. Thus, the displacement thickness variation with the

local Reynolds number within the weak interaction region is so as [32]:

x

Gdx

dRe86.0*

=δ (6.4)

Where G accounts for the compressibility effects, but also it is a function of M∞, the wall

temperature, the Prandtl number and the boundary-layer-edge temperature.

After careful data analysis of Figure 6.7, the shock wave angle was measured by using a

computational grid on the image, thus its dependence on the viscous interaction parameter

was investigated. The results are shown in Fig. 6.8. It is evident that the curve follows the

same pattern as seen in a strong interaction region. In fact, as previously stated, the small

shock-wave angle and the low Reynolds number in the conditions indicated in that figure

imply a merged shock-wave/boundary-layer. Also, the inviscid value of the shock-wave angle

is indicated in the Fig. 6.7. The measured angles, however, are quite larger than the predicted

92

Figure 6.8. Primary shock wave angle variation with the parameter C/χχ = . Same flow

conditions indicated in Figure 6.7.

by the inviscid theory. This result is reasonable since the displacement thickness δ* modifies

the effective body, the effect is exacerbated by the adverse pressure gradient due to the slight

turning of the flow along the forebody region. Besides, the author believes that the nozzle‘s

non-equilibrium effects certainly impart some deviations from the inviscid value. Another

aspect of the hypersonic flow which was observed concerns the viscous flow over the inlet

ramp. Like the flow over a flat plate, the shock-wave/boundary-layer interaction depends

largely on the length of the subsonic portion of the boundary layer and on the shock wave

strength. The adverse pressure gradient causes the boundary layer to thicken as it approaches

the ramp deflection. If the conditions are such that the boundary-layer separates, as depicted

in Figure 6.9, a series of compression waves is formed in the separation point and they

coalesce into a single curved shock-wave. Downstream that point, the region of separated

flow features unsteady nature and large gradients. In the reattachment point, the inviscid flow

93

Figure 6.9. Main characteristics of a separated flow over a compression ramp. Extracted from

[30].

along the effective ramp encounters the actual ramp, a phenomenon which causes an

incremental compressive turning. The compression waves formed in the reattachment point

coalesce to another shock wave. As noted by Roshko and Thomke [33] that the location of the

separation and reattachment points correspond to local pressure rises. Also, the attachment

point presents a peak in the heating rate, as the boundary layer is the thinnest [34].

Finally, the separation and the recompression shock waves interact downstream. A simple

model of this shock/shock interaction can be found on Fig. 6.10. In this model, the inviscid

flow impinges two successive compression ramps, the first one represents the separated

region and the second one the region of reattached flow. The shock waves intersect at point I,

the flow field downstream I presents a curved shock wave, a free-shear layer, and an

expansion fan that is reflected at the wall. The shock wave is curved due to its interaction with

the reflected expansion fan. Since the stream lines of the flow which go through this shock

wave have different properties from those which faced the two original shock waves, a free-

shear layer is formed, along where the pressure is constant. Figure 6.11 shows a Schlieren

photograph of the flow over the rear region of the compressive ramp. In that run, a plastic

bump was fixed on the upper surface of the model in order to achieve better contrast

94

Figure 6.10. Schematic view of a shock/shock interaction downstream separation point, over a

ramp. Adapted from [30].

Figure 6.11. Mach 8.71 flow past the inlet ramp. Reservoir conditions: P0 = 2850 psi T0 =

1702 K. Run #12. For contrast comparison and trigger timing, a bump was placed on the upper surface.

comparison between the oblique shock wave over the ramp and the bow shock around the

bump. In fact, this mounting was used to check the ability of the Schlieren apparatus to detect

the oblique shock wave formed in the compression ramp. With the fins mounted, the complete

flow development over the inlet ramp could not be observed. However, it can be seen the

coalescent compression waves that seem to intersect downstream the frame region. Their

extensions indicate a large separated region compared with the ramp length. In addition to

that, besides the fact that the inlet shock-wave is initially curved it was possible to infer the

95

Figure 6.12. Pressure distribution along the model center line.

angle of its linear portion as being 27°, only 8% larger than the inviscid value, 25°. Once

again, this author believes that this result is due to viscous effects taking place along the ramp

inlet ramp, as stated previously.

Figure 6.12 shows the static pressure variation with the distance from the nose leading

edge along the centerline of the model during the first five high enthalpy tests. The

correspondent parameter ∞

∞

Re

3M was indicated in that figure. The inviscid solution for

the pressure distribution was also inserted. Regarding that figure it was found that the

pressure decreases from x/Lw = 0.52 to = 0.58, just after the compression ramp. This result is

believed to be consequence of the existence of a non-centered expansion fan formed at the

end of the ramp, caused by a separated region. Thus, where the flow is fully expanded, such

as in station x/Lw = 0.58, the pressure is slightly lower than at the initial part of the expansion

fan, x/Lw = 0.52. On the other hand, from stations x/Lw =0.58 to 0.86, there is a pressure

increase despite the fact that station x/Lw = 0.86 was situated over the expansion

96

Figure 6.13. Pressure variation on station x/Lw =0.58 with the parameter ∞

∞

Re

3M .

ramp. The author believes that the viscous effects cause the pressure increase along the

centerline, so the expansion produces larger pressure values than expected. Indeed, it can be

noted that as the parameter ∞

∞

Re

3M grows, the pressure increase is exacerbated. To better

clarify such influence, one can refer to Fig. 6.13, in which the effect of such parameter was

investigated on station x/Lw = 0.58 static pressure measurements. Flow separation is also

possible to take place on the flat portions between stations x/Lw =0.58 and x/Lw =0.86, where

large local Reynolds number could be achieved. Thus, it is entirely possible that a bubble

shock forms between these stations. Also, from stations x/Lw = 0.86 to 0.93, located over the

expansion ramp, it can be noted a pressure increase. In that region, where there is no adverse

pressure gradient, no separation is expected, however, the author notes that the boundary layer

increase could elevate the local pressure on station x/Lw= 0.93.

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7 Conclusions

An experimental investigation of a waverider configuration was performed in the IEAv T3

shock tunnel. Static pressure measurements along with Schlieren Photographs were made for

further analysis of the hypersonic flow over the waverider model.

Due to the lack of measurement systems on the T3 tunnel to acquire the stagnation

temperature or static temperature inside the test section, the shock tunnel flow had to be

modeled so as to provide the free stream conditions. Thereby a computer code, t3_sing, based

on the MATLAB® platform was implemented. It simulated the flow expansion from the

driver to the driven sections, the shock-wave interactions that occur inside the driven section

as well as the nozzle expansion. In the last two processes the code accounted for real gas

behavior, based on the Srinivasan curve fits for the air properties [18]. The code t3_sing was

validated using the results of the computer codes developed by Minucci [21], presenting

excellent data accuracy. Three Kistler pressure sensors located in the end of the driven section

of the T3 tunnel allowed the direct measurement of the incident shock wave speed and the

reservoir pressure. These measurements were used as correction-entries for the t3_sing code

likewise they are used for the Minucci codes [21]. Further, the outputs from these codes gave

the free stream conditions for each test run.

The pure waverider surface was constructed according to the Hypersonic Small

Disturbances Theory, as developed by Rasmussen [6]. To achieve the final geometry, several

tradeoff analyses were made. Even though the applied theory does not enclose the entire

complexity of the hypersonic flow, it was possible to investigate some general properties of

these aerodynamic surfaces. The resulting surface combined high lift-to-drag ratio and

volume efficiency. Further, a scramjet ramp was integrated to the pure waverider surface. It

was designed so as to provide the ideal pressure and temperature conditions for supersonic

Hydrogen combustion [10]. For the Mach 10 free stream flow condition, a 20° inclination

98

ramp resulted from the calculation of an inviscid hypersonic flow past a ramp, after a primary

oblique shock generated by the waverider leading edge. Also, the work presented all the steps

for the construction of the free-expansion surface, an ideal hypersonic nozzle for a waverider

configuration. However, due to its large length, the real model nozzle was truncated in order

to avoid the appearance of undesirable compression waves or even shock waves over the

expansion surface.

A 781 mm stainless steel model was machined using a Computational Numeric Control

(CNC) based milling machine. The leading edge radius was about 0.05 mm. The model was

composed by two parts: the waverider surface; and, the compression and expansion ramps

module. The modulated assembly of the model permitted the placement of the 07 PCB®

pressure sensors on the compression surface. Hollow tunnel sting was connected to the model

by means of a diamond shaped adapter and it allowed electrical connections between sensors

and the data acquisition system.

The stagnation conditions as well as the free stream properties were estimated using the

numerical codes. The tunnel operated at Mach number ranges of 8.9 to 10, Re = 2.25 x 106 to

8.76 x 106 (m-1) and Kn = 0.06 to 0.19. During the high enthalpy tests, the reservoir conditions

varied up to 2150 K and 2946 psi.

As a result, the Schlieren photographs demonstrated that flowfield complex structure is

similar to the one predicted by the hypersonic viscous interaction model, namely, the

formation of the compression waves generated in separation region. It was also possible to

identify the existence of the expansion fan. Moreover, the shock wave angle variation with the

hypersonic similarity parameter agreed with the theory.

Furthermore, the pressure distribution over the compression surface was measured and the

results confirmed the theoretical predictions concerning the inlet flow structure by viscous

models. The found pressure rise downstream the centerline is believed to have been caused by

99

the boundary layer thickness increase, which resulted in larger pressure values over the

expansion surface than it would be expected in the inviscid theory.

As final recommendations, this author points out that the shock tunnel flow modeling

must be improved based on test data, so as to account for viscous effects, as previously

discussed. This would permit a complete simulation of the flow which would diminish the

strong dependence on sensors data.

Following, the optimization of fully integrated waverider surfaces demands truly complex

mathematical models, and as a consequence, powerful numeric fluid simulations are needed.

These simulations would account for viscous effects, real gas properties of the air, and

chemical reactions inside the scramjet. They shall be subjects of further works.

In addition, the author remarks that a complete understanding of the flow over the

compression surface would be possible increasing the number of the pressure sensors in that

region. However, the model volume frankly restricts this procedure.

The presented study will also be used for the design of a scramjet engine, to be integrated

with an experimental waverider model, named 14-X, which shall be tested under real flight

conditions. The model will be boosted by a modified rocket engine VS-40 until it reaches

Mach 6, when separation takes place. After that, the engine will start and perform some

aerodynamic-parameter-identification manuevers. During the test time, the data acquisition

system will record informations about the pressure distribution, temperature, acceleration, etc.

100

8 References

[1] HAGSETH, P.E.; ISAIAH, M. Current Technologies for Waverider Aircraft. AIAA Paper No. 93-0400, presented at the AIAA 31st Aerospace Sciences Meeting, Reno, Nevada, January 11-14, 1993. [2] RAULT, D.F.G. Aerodynamic Characteristics of a Hypersonic Viscous Optimized waverider at high altitudes. Journal of Spacecraft and Rockets, Vol. 31, No. 5, 1994, pp. 719–727. [3] JAVAID, K.H.; SERGHIDES, V.C. Airframe-Propulsion Integration Methodology for Waverider Derived Hypersonic Cruise Aircraft Design Concepts. Journal of Spacecraft and Rockets, Vol. 42, No. 5, 2005, pp. 663-667. [4] HEISER ,W.H.; PRATT, D.T.; DALEY, D.H.; MEHTA, U.B. Hypersonic Airbreathing Propulsion. AIAA Education Series. Published by AIAA, 594 pages, 1994. [5] NONWEILER, T.R.F. Delta Wings of Shape Amenable to Exact Shock Wave Theory. Journal of the Royal Aeronautical Society, Vol. 67, 1963, p. 39. [6] RASMUSSEN, M.L.; HE, X. Analysis of Cone - Derived Waveriders by Hypersonic Small-Disturbance Theory. Proceedings of the First International Hypersonic Waverider Symposium, College Park, Maryland, USA, October 17-19, 1990. [7] MUNDY, J.A; HASEN, G.A. A Numerical Study of Conically Derived Waveriders. AIAA-1994-765. Aerospace Sciences Meeting and Exhibit, 32nd, Reno, NV, Jan 10-13, 1994. [8] WANG, Y.; ZHANG, D.; DENG, X. Design of Waverider Configuration with High Lift-Drag Ratio .Journal of Aircraft, Vol.44, No.1, 2007, pp. 144-148. [9] UNIVERSITY OF MARYLAND. Proceedings of the First International Hypersonic Waverider Symposium. College Park, Maryland, USA, October 17-19, 1990. [10] O'NEILL, M.K.; LEWIS, M.D. Optimized Scramjet Integration on a Waverider. Journal of Aircraft, Vol. 29, No. 6, 1992, pp. 1114-1121 [11] ANDERSON, J.D.;LEWIS, M.D. Hypersonic Waveriders - Where Do We Stand ? AIAA Paper. 93-0399, Jan. 1993. [12] SANTOS, W.F.; LEWIS, M. Aerothermodynamic Performance Analysis of Hypersonic Flow on Power Law Leading Edges. Journal of Spacecraft and Rockets, Vol. 42, No. 4, pp. 588-597. [13] COCKRELL, C.E.; HUEBNER, L.D. Aerodynamic Characteristics of Two Waverider-Derived Hypersonic Cruise Configurations. NASA TP-3559, 1996. [14] GILLUM, M. J.; LEWIS, M. J. Experimental Results on a Mach 14 Waverider with Blunt Leading Edges. AIAA Journal of Aircraft, Vol.34, No. 3, May-June 1997, pp. 296-303.

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[15] EMANUEL, G. Gasdynamics: Theory and Applications. AIAA Education Series, AIAA, 1986. [16] ALPHER, R. A.; WHITE D. R. Flow in Shock Tubes with Area Change at the Diaphragm Section. Journal of Fluid Mechanics, Vol. 3, No. 5, 1958, pp. 457-470. [17] ANDERSON Jr., J. D. Hypersonic and High Temperature Gas Dynamics. McGraw-Hill, New York, 1989, 690 p. [18] SRINIVASAN, S., TANNEHILL, J.C.; WEILMUENSTER K.J.. Simplified Curved Fits for the Thermodynamic Properties of Equilibrium Air. NASA-RP-1181, 1987. [19] NAGAMATSU, H. T.; SHEER, R. E. Hypersonic Nozzle Expansion with Air Atom Recombination. Present. Journal of Aerospace Sciences, vol. 28, 1961, pp. 833-837. [20] NAGAMATSU, H. T.; SHEER, R. E., Jr., “Vibrational Relaxation and Recombination of Nitrogen and Air in Hypersonic Nozzle Flows,” AIAA Journal, Vol. 3, , 1965, p.1386. [21] MINUCCI, M.A.S.. Experimental Investigation of a 2-D Scramjet Inlet at Flow Mach Numbers of 8 to 25 and Stagnation Temperatures of 800K to 4,100K. Doctoral thesis, Ressenlaer Polytechnique Institute, New York, Troy, 1991. [22] PCB® PIEZOTRONICS, INC. General Piezoelectric Theory. Available at http://www.pcb.com/techsupport/tech_pres.php . [23] PCB® PIEZOTRONICS, INC. Model 132A35 ICP® Dynamic Pressure Sensor Installation and Operating Manual. Available at http://www.pcb.com/contentstore/docs/PCB_Corporate/Pressure/products/Manuals/132A35.pdf . [24] YOKOGAWA ELECTRIC CORPORATION. DL-750 Scopecorder User’s Manual. [25] OLIVEIRA, A.C. Investigação Experimental da Adição de Energia por Laser em Escoamento Hipersônico de Baixa Densidade. Doctoral Thesis, Instituto de Pesquisas Espaciais, São José dos Campos, São Paulo, Brasil. [26] CORDIN SCIENTIFIC IMAGING. 500-Series Cameras. Available at: http://www.cordin.com/productsie.html. [27] NAGAMATSU, H. T. ; SHEER, R. E., Jr. Hypersonic Shock Wave-Boundary layer Interaction and Leading Edge Slip. American Rocket Society Journal, Vol. 30, No.5, Jul 1961, pp.454-462. [28] HAYES, W. D; PROBSTEIN, R. F. Hypersonic Flow Theory. Academic Press, New York, 1959. 464 p.

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[29] TORO, P.G.P.; RUSAK, Z.; NAGAMATSU, H.T.; MYRABO, L.N. Hypersonic Flow Over a Flat Plate. AIAA Paper 98-0683, AIAA 36th Aerospace Sciences Meeting and Exhibit, Reno, Nevada, January 12-15, 1998. [30] BERTIN, J.J. Hypersonic Aerothermodynamics. AIAA Educational series, AIAA, NewYork, 1994. 608p. [31] NAGAMATSU, H. T.; LI, T. Y. Hypersonic Viscous Flow Near the Leading Edge of a Flat Plate. Phys. Fluids, Vol. 3, No. 1, Jan. 1960, pp.140-141. [32] ANDERSON, J.D. Fundamentals of Aerodynamics. McGraw-Hill, 2nd Ed., 1991. 892 p. [33] ROSHKO, A.; THOMKE, G. J. Supersonic Turbulent Boundary-Layer Interaction with a Compression Corner at Very High Reynolds Number. Proceedings of the Symposium Viscous Interaction Phenomena in Supersonic and Hypersonic Flows. USAF Aerospace Research Labs., Wright-Patterson AFB, Ohio, University of Dayton Press, May 1969. [34] MARKARIAN, C. F. Heat Transfer in Shock Wave Boundary Layer Interactions. Naval Weapons. Center, China Lake, Calif., NWC TP 4485, November 1968.

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Appendix A. Shock Tunnel Flow Simulation Codes

Main code: t3_sing.m Entries for t3_sing.m: Apender resultados à arquivo existente?(s/n)n Digite o nome do arquivo de saida: teste Gas do Driver? 2-Ar, qq outro valor para He: 4 Temperatura ambiente em celsius: 23 p4 em psi: 3000 p1 em atm: 1 Mediu numero de Mach?(s/n) s numero de Mach medido: 4 Mediu pressão de reservatório?(s/n) s Digite o valor em psi: 2979 Mediu pressão de impacto (Pitot)?(s/n) s Digite o valor em psi: 11 Output files: teste_stag.txt %contains the initial, incident and stagnation conditions teste_freestr.txt%contains the freestream conditions t3_sing.m function t3_sing % simulates the shock tunnel flow global gama1 gama2 R cv cp ad bar patm gama1=1.4;R1=287;gama2=1.667;R2=2078.5; patm=101330; bar = 0.986923267; tic %Áreas A4=190.5^2;%driver A1=127^2;%driven %Para imprimir em arquivo fileformat = input('Apender resultados à arquivo existente?(s/n)','s'); filename = input('Digite o nome do arquivo de saida: ','s'); filename1 = [filename,'_stag','.txt']; filename2 = [filename,'_freestr','.txt']; if fileformat=='s' z='a'; elseif fileformat=='n' z='w'; else '******opção inválida******' return ; end fid=fopen(filename1,z); fie=fopen(filename2,z);

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gasdriver = input('Gas do Driver? 2-Ar, qq outro valor para He: '); if gasdriver==2 gama2=gama1 R2=R1 end tempamb = input('Temperatura ambiente em celsius: ');t4=tempamb+273;t1=t4; a4=(gama2*R2*t4)^0.5; a1=(gama1*R1*t1)^0.5; p4= input('p4 em psi: '); p4=p4/14.7;p4=p4*patm; p1= input('p1 em atm: ');p1=p1*patm; str1=input('Mediu numero de Mach?(s/n) ','s'); if str1=='s' Mw= input('numero de Mach medido: '); elseif str1=='n' Mw=expansao_drr(p4,p1,a4,a1,A4,A1,gama1,gama2); else '******opção inválida******' return ; end str2=input('Mediu pressão de reservatório?(s/n) ','s'); if str2=='s' p0_meas= input('Digite o valor em psi: '); flag_0=1; p0_meas=p0_meas/14.7*patm; elseif str2=='n' p0_meas=0; flag_0=0; else '******opção inválida******' return ; end str3=input('Mediu pressão de impacto (Pitot)?(s/n) ','s'); if str3=='s' p_impact= input('Digite o valor em psi: '); p_impact=p_impact/14.7*patm; flag_1=1; elseif str3=='n' p_impact=0; flag_1=0;

105

else '******opção inválida******' return ; end %1)Onda de choque incidente %Gás ideal-temperatura baixa e1=R1*t1/(gama1-1); rho1=p1/t1/R1; uw=a1*Mw; %initial guess for conditions after shockwave [p2,e2,t2,rho2,uw_menos_u2]=chuteinicial(p1,t1,uw,gama1,R1); X0=[uw_menos_u2,rho2,e2,p2]; X=fminsearch(@NavierStokes,X0,optimset('TolX',1e-12,'TolFun',1e-12),uw,rho1,e1); %-function tgas_bk is the Tannehil’s interpolation M1=uw/a1; u2=uw-X(1); %Laboratory frame rho2=X(2); e2=X(3); clear tgas_bk; [p2,a2,t2,s2]=tgas_bk(e2,rho2); p2atm=p2/patm; MX=X(1)/a2; M2=u2/a2; t2/t1; p2/p1; %unsteady expansion fan t3=t4*(p2/p4)^((gama2-1)/gama2); a3=(gama2*R2*t3)^0.5; p3=p2; u3=u2; rho3=p3/R2/t3; e3=R2*t3/(gama2-1); fprintf(fid,'%3.2f\t%3.2f\t%3.2f\t%3.2f\t%3.2f',p4,p1,p2,t2,M1); %-------------------------------------------------------------------------- %2)Reflected shockwave %ideal gas as initial guess Mrr=2;%inicialização i=-1;%contador de interações com a interface while Mrr>1 i=i+1 ; aux=M2*(gama1+1)/2; Mr=aux/2+(aux^2+4)^0.5/2; ur=Mr*a2; %chute inicial das condições atrás da onda de choque

106

[p2l,e2l,t2l,rho2l,uaux]=chuteinicial(p2,t2,ur,gama1,R1); X0r=[ur,rho2l,e2l,p2l]; X=fminsearch(@NavierStokes_ref,X0r,optimset('TolX',1e-12),u2,rho2,e2); Mr=X(1)/a2; urw=X(1)-u2; %referencial do Laboratório rho2l=X(2); e2l=X(3); clear tgas_bk; [p2l,a2l,t2l,s2l]=tgas_bk(e2l,rho2l); if i==0 % verifica se o modo é refletido (i=0) ou não (i=!0) fprintf(fid,'\t%3.2f\t%3.2f\t%3.2f\t%3.2f\t%3.2f\t%3.2f\t%3.2f',Mr,rho2l,e2l,p2l,a2l,t2l,s2l); if flag_0==1 X0is_stg=[e2l,rho2l]; X=fminsearch(@isent_stag,X0is_stg,optimset('TolX',1e-12,'TolFun',1e-12'),s2l,p0_meas); e2ll=X(1); rho2ll=X(2); clear tgas_bk; [p2ll,a2ll,t2ll,s2ll]=tgas_bk(e2ll,rho2ll); break; end%if flag_0==1 end %if i==0 % 3)Modo de equilíbrio de interface %i-ésima interação, i>0. [urt0,ux0,rho3l0,px0,e3l0,urr0,rho2ll0,e2ll0]=equint_ideal(gama1,gama2,R1,R2,a2l,p2l,t2l,p3,u3,t3,a3);% função equi_ideal calcula o modo de equilíbrio de interface para ar ideal X0int=[urt0,ux0,rho3l0,px0,e3l0,urr0,rho2ll0,e2ll0]; for j=1:5 X=fminsearch(@NavierStokes_equint,X0int,optimset('TolX',1e-12,'TolFun',1e-12'),u3,rho3,e3,rho2l,e2l,p2l,urt0,ux0,rho3l0,px0,e3l0,urr0,rho2ll0,e2ll0); X0int=X; end if X(6)/a2l>1 urt=X(1) ux=X(2) rho3l=X(3) px=X(4) e3l=X(5) urr=X(6) rho2ll= X(7) e2ll=X(8) Mrr=urr/a2l

107

clear tgas_bk; [p2ll,a2ll,t2ll,s2ll]=tgas_bk(e2ll,rho2ll) if abs(ux)<1e-1 break; end else break; end%if X(6)/a2l>1 u2=ux; a2=a2ll; M2=u2/a2; p2=p2ll; t2=t2ll; rho2=rho2ll; e2=e2ll; end %while if i>=1 | flag_0==1 fprintf(fid,'\t%d\t%3.2f\t%3.2f\t%3.2f\t%3.2f\t%3.2f\t%3.2f\n',i,rho2ll,e2ll,p2ll,a2ll,t2ll,s2ll); isentest_new(p2ll,t2ll,e2ll,rho2ll,23.5,610,gama1,R1,p_impact,flag_1,fie); else fprintf(fid,'\t%d\t%3.2f\t%3.2f\t%3.2f\t%3.2f\t%3.2f\t%3.2f\t%3.2f\t%3.2f\n',i,0,0,0,0,0,0,0,0); isentest_new(p2l,t2l,e2l,rho2l,23.5,610,gama1,R1,p_impact,flag_1,fie); end%if fclose(fid); fclose(fie); toc %------------------------------------------------ function erro = NavierStokes(X,ua,rhoa,ea) %equações de continuidade+q. de movimento+ conserv. energia para onda de choque propagada ud=X(1);rhod=X(2);ed=X(3); pd=X(4); clear tgas_bk; [pa,aa,ta,sa]=tgas_bk(ea,rhoa);%-função tgas_bk é a interpolação de Tannehil- %para normalizar(sufixo ad): rhoad=rhoa; uad=ua; ead=ea; pad=pa;

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f1=(rhoa*ua-ud*rhod)/rhoad/uad; f2=(ua^2*rhoa+pa-pd-ud^2*rhod)/(pad+rhoad*uad^2); f3=(ea+pa/rhoa+ua^2/2-ed-pd/rhod-ud^2/2)/(ead+pad/rhoad+uad^2/2); clear tgas_bk; [pdt,ad,td,sd]=tgas_bk(ed,rhod); f4=(pdt-pd)/pad; erro=f1^2+f2^2+f3^2+f4^2; %------------------------------------------------ function erro = NavierStokes_ref(X,u2,rhoa,ea) %equações de continuidade+q. de movimento+ conserv. energia para onda de choque refletida ua=X(1); ud=ua-u2; rhod=X(2);ed=X(3); pd=X(4); clear tgas_bk; [pa,aa,ta,sa]=tgas_bk(ea,rhoa); %para normalizar(sufixo ad): rhoad=rhoa; uad=ua; ead=ea; pad=ea; f1=(rhoa*ua-ud*rhod)/rhoad/uad ; f2=(ua^2*rhoa+pa-pd-ud^2*rhod)/(pad+rhoad*uad^2); f3=(ea+pa/rhoa+ua^2/2-ed-pd/rhod-ud^2/2)/(ead+pad/rhoad+uad^2/2); clear tgas_bk; [pdt,ad,td,sd]=tgas_bk(ed,rhod); f4=(pdt-pd)/pad; erro=f1^2+f2^2+f3^2+f4^2; %------------------------------------------------ function erro = NavierStokes_equint(X,u3,rho3,e3,rho2l,e2l,p2l,urt0,ux0,rho3l0,px0,e3l0,urr0,rho2ll0,e2ll0); %equações de continuidade+q. de movimento+ conserv. energia para interação entre onda de choque e a interface dos gases do driver e do driven urt=X(1); ux=X(2); rho3l=X(3); px=X(4); e3l=X(5); urr=X(6); rho2ll= X(7); e2ll=X(8); %condição na interface p2ll=px;p2ll0=px0; p3l=px;p3l0=px0; u3l=ux;u3l0=ux0; u2ll=ux;u2ll0=ux0; %transmitida na interface

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[p3,a3,t3]= tgas_ideal(e3,rho3); f1=(rho3*(u3+urt)-(u3l+urt)*rho3l)/rho3/(u3+urt0); f2=((u3+urt)^2*rho3+p3-p3l-(u3l+urt)^2*rho3l)/(p3+rho3*(u3+urt0)^2) ; f3=(e3+p3/rho3+(u3+urt)^2/2-e3l-p3l/rho3l-(u3l+urt)^2/2)/(e3+p3/rho3+(u3+urt0)^2/2) ; [p3lt,a3l,t3l]=tgas_ideal(e3l,rho3l); f4=(p3lt-p3l)/p3l0; %refletida na interface f5=(rho2l*urr-(urr-u2ll)*rho2ll)/(rho2l*urr0); f6=(urr^2*rho2l+p2l-p2ll-(urr-u2ll)^2*rho2ll)/(urr0^2*rho2l+p2l); f7=(e2l+p2l/rho2l+urr^2/2-e2ll-p2ll/rho2ll-(urr-u2ll)^2/2)/(e2l+p2l/rho2l+urr0^2/2); clear tgas_bk; [p2llt,a2ll,t2ll,s2ll]=tgas_bk(e2ll,rho2ll); f8=(p2llt-p2ll)/p2ll0; erro=f1^2+f2^2+f3^2+f4^2+f5^2+f6^2+f7^2+f8^2; %------------------------------------------------ function [pc,ec,tc,rhoc,uc]=chuteinicial(p,t,u,g,r) % Calcula propriedades após a onda de choque para o caso ideal a=(g*r*t)^0.5; rho=p/r/t; e=r*t/(g-1); M=u/a; pc=p*(1+2*g/(g+1)*(M^2-1)); rhoc=rho*((g+1)*M^2/(2+(g-1)*M^2)); ec=pc/rhoc/(g-1); tc=(g-1)*ec/r; uc=u*r/rhoc; %------------------------------------------------ function Mw=expansao_drr(p4,p1,a4,a1,A4,A1,gama1,gama2) %Expansão isentrópica que occore no driver, considerando variação de área e gás ideal m3b=1;%m3>=1 m3a0=0.1; m3a=fminsearch(@fm3a,m3a0,optimset('TolX',1e-20,'TolFun',1e-20),m3b,A4/A1,gama2); m3a g=(((2+(gama2-1)*m3a^2)/(2+(gama2-1)*m3b^2))^(0.5)*(2+(gama2-1)*m3b)/(2+(gama2-1)*m3a))^(2*gama2/(gama2-1)) %g=1;%modificado em 03/09/2008 Mw0=2;

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Mw=fminsearch(@fmw,Mw0,optimset('TolX',1e-12,'TolFun',1e-12),p4/p1,g,a4,a1,gama1,gama2); %----------------- function erro=fm3a(m3a,m3b,ra,gama2)%Subrotina da função expansao_drr rax= m3b/m3a*((2+(gama2-1)*m3a^2)/(2+(gama2-1)*m3b^2))^((gama2+1)/2/(gama2-1)); erro=(ra-rax)^2/ra^2; %----------------- function erro=fmw(mw,rp,g,a4,a1,gama1,gama2)%Subrotina da função expansao_dr u2sobrea1=2/(gama1+1)*(mw^2-1)/mw; p2sobrep1=(2*gama1*mw^2-(gama1-1))/(gama1+1); m3=1/(u2sobrea1^(-1)*a4/a1*g^((gama2-1)/2/gama2)-(gama2-1)/2); rpx=1/g*(1+(gama2-1)/2*m3)^(2*gama2/(gama2-1))*p2sobrep1; erro=(rp-rpx)^2/rp^2; %------------------------------------------------ function [p,a,t]= tgas_ideal(e,rho) % Calcula propriedades do gás ideal, gas frio tanto he qto ar global gama2 R2 p=(gama2-1)*rho*e; t=p/rho/R2; a=(gama2*R2*t)^0.5; %------------------------------------------------ function erro=isent_stag(X,scte,p0_meas) e2ll=X(1); rho2ll=X(2); clear tgas_bk; [p2ll,a2ll,t2ll,s2ll]=tgas_bk(e2ll,rho2ll); e1=(p0_meas-p2ll)/p0_meas; e2=(scte-s2ll)/scte; erro=e1^2+e2^2; =================================================================== equint_ideal.m function [urt,ux,rho3l,px,e3l,urr,rho2ll,e2ll]= equint_ideal(gamaa,gamab,Ra,Rb,a2l,p2l,t2l,p3,u3,t3,a3) %Modo de equilíbrio de interface no caso ideal global gama1 gama2 R1 R2 gama1=gamaa;R1=Ra; gama2=gamab;R2=Rb; p2ll0=p2l;

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p2ll=fminsearch(@equideal,p2ll0,optimset('TolX',1e-12),a2l,p2l,t2l,p3,u3,a3); %gas drr (refletida pela interface) Mrt=sqrt((gama2+1)*((p2ll/p3)-1)/(2*gama2)+1) ; urt=Mrt*a3-u3; u3l=(urt+u3) /((gama2+1)/2*Mrt^2/(1+(gama2-1)/2*Mrt^2))- urt; [p3l,e3l,t3l,rho3l]=chuteinicial(p3,t3,urt+u3,gama2,R2); px=p3l; ux=u3l; %gas drn (transmitida através da interface) Mrr=sqrt((gama1+1)*((p2ll/p2l)-1)/(2*gama1)+1) ; urr=Mrr*a2l ; u2ll=-urr/((gama1+1)/2*Mrr^2/(1+(gama1-1)/2*Mrr^2))+ urr; [p2ll,e2ll,t2ll,rho2ll]=chuteinicial(p2l,t2l,urr,gama1,R1); %------------------------------------------------ function [pc,ec,tc,rhoc]=chuteinicial(p,t,u,g,r) %chute inicial das condições atrás da onda de choque a=(g*r*t)^0.5; rho=p/r/t; e=r*t/(g-1); M=u/a; pc=p*(1+2*g/(g+1)*(M^2-1)); rhoc=rho*((g+1)*M^2/(2+(g-1)*M^2)); ec=pc/rhoc/(g-1); tc=(g-1)*ec/r; %------------------------------------------------ function erro = equideal(X,a2l,p2l,t2l,p3,u3,a3) %Procura a velocidade induzida pela onda refletida na interface global gama1 gama2 R1 R2 px=X(1); Mrt=sqrt((gama2+1)*((px/p3)-1)/(2*gama2)+1) ; urt=Mrt*a3-u3; u3l=(urt+u3) /((gama2+1)/2*Mrt^2/(1+(gama2-1)/2*Mrt^2))- urt; Mrr=sqrt((gama1+1)*((px/p2l)-1)/(2*gama1)+1) ; urr=Mrr*a2l ; u2ll=-urr/((gama1+1)/2*Mrr^2/(1+(gama1-1)/2*Mrr^2))+ urr; erro=(u3l-u2ll)^2/u3^2; %------------------------------------------------- =================================================================== isentest_new.m function isentest_new(P0,T0,E0,RHO0,dt,dn,gama1,R1,p_impact_meas,flag_1,fie) %nozzle expansion At=dt^2;%garganta An=dn^2;%na seção de teste

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%1-Expansão isentrópica até a garganta clear tgas_bk; [P0,A0,T0,S0]=tgas_bk(E0,RHO0); H0=E0+P0/RHO0; U0=0; [pc,ec,tc,rhoc]=chuteinicial_is(1,P0,T0,RHO0,gama1,R1); X0=[(gama1*R1*tc)^0.5,rhoc,ec,pc]; for i=1:1 X=fminsearch(@NavierStokes_star,X0,optimset('TolX',1e-25,'TolFun',1e-20),U0,RHO0,E0,P0,A0,S0); X0=X; end ustar=X(1); rhostar=X(2); estar=X(3); [pstar,astar,tstar]=tgas(estar,rhostar); clear tgas; star=S0; %2-Expansão isentrópica da garganta até a seção de testes, total expansao if flag_1==0 [pc,ec,tc,rhoc]=chuteinicial_is(10,pstar,tstar,rhostar,gama1,R1);%é 10 para a tubeira presente no T3, rAAstar=(2/(gama+1))^((gama+1)/2/(gama-1))/M*(1+(gama-1)/2*M^2)^((gama+1)/2/(gama-1)); X0=[10*(gama1*R1*tc)^0.5,rhoc,ec,pc]; for i=1:1 X=fminsearch(@NavierStokes_nz,X0,optimset('TolX',1e-25,'TolFun',1e-20),ustar,rhostar,estar,pstar,star,At,An,gama1,R1); X0=X; end uf1=X(1); rhof1=X(2); ef1=X(3); clear tgas_bk; [pf1,af1,tf1,sf1]=tgas_bk(ef1,rhof1); Mf1=uf1/af1; end%if flag_1==1 %ou 3-Expansão isentrópica da garganta até a seção de testes,com pitot dado if flag_1==1 [pcf1,ecf1,tcf1,rhocf1]=chuteinicial_is(5,P0,T0,RHO0,gama1,R1);%é 5(um bom número) ucf1=5*(gama1*R1*tcf1)^0.5; [pcf2,ecf2,tcf2,rhocf2,ucf2]=chuteinicial(pcf1,tcf1,ucf1,gama1,R1);

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[pcp,ecp,tcp,rhocp] =chuteinicial_is2(ucf2/(gama1*R1*tcf2)^0.5,pcf2,tcf2,rhocf2,gama1,R1); X0=[rhocp,ecp]; for i=1:1 X=fminsearch(@NavierStokes_nz2,X0,optimset('TolX',1e-25,'TolFun',1e-20),H0,gama1,R1,p_impact_meas); X0=X; end rhop=X(1); ep=X(2); clear tgas_bk; [pp,ap,tp,sp]=tgas_bk(ep,rhop); Y0=[ucf1,rhocf1,ecf1,ucf2,rhocf2,ecf2]; for i=1:5 Y=fminsearch(@NavierStokes_nz3,Y0,optimset('TolX',1e-25,'TolFun',1e-20),ucf2,rhocf2,ecf2,pcf2,sp,H0,S0,gama1,R1); Y0=Y; end uf1=Y(1); rhof1=Y(2); ef1=Y(3); clear tgas_bk; [pf1,af1,tf1,sf1]=tgas_bk(ef1,rhof1); Mf1=uf1/af1; end%if flag_1==0 p_pitot=pitot(pf1,tf1,rhof1,Mf1,uf1,ef1,H0,gama1,R1); fprintf(fie,'%3.3f\t%3.3f\t%3.3f\t%3.3f\t%3.3f\t%3.3f\t%3.3f\n',pf1,tf1,rhof1,Mf1,uf1,ef1,p_pitot); %------------------------------------------------ function [pc,ec,tc,rhoc]=chuteinicial_is(M,p0,t0,rho0,gama,R) tc=t0/(1+(gama-1)/2*M^2); pc=p0*(tc/t0)^(gama/(gama-1)); rhoc=rho0*(tc/t0)^(1/(gama-1)); ec=pc/rhoc/(gama-1); uc=M*(gama*R*tc)^0.5; %------------------------------------------------ function [p0,e0,t0,rho0]=chuteinicial_is2(M,p,t,rho,gama,R) t0=t*(1+(gama-1)/2*M^2);

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p0=p/(t/t0)^(gama/(gama-1)); rho0=rho/(t/t0)^(1/(gama-1)); e0=p0/rho0/(gama-1); %------------------------------------------------ function erro = NavierStokes_star(X,ua,rhoa,ea,pa,aa,sa) ud=X(1);rhod=X(2);ed=X(3); pd=X(4); clear tgas_bk; [pdt,ad,td,sd]=tgas_bk(ed,rhod); f1=(ea+pa/rhoa-ed-pd/rhod-ud^2/2)/(ea+pa/rhoa+ua^2/2); f2=(pdt-pd)/pa; f3=(sd-sa)/sa; f4=(ud-ad)/aa; erro=f1^2+f2^2+f3^2+f4^2; %------------------------------------------------ function erro = NavierStokes_nz(X,ua,rhoa,ea,pa,sa,At,An,gama,R) ud=X(1);rhod=X(2);ed=X(3); pd=X(4); clear tgas_bk; [pdt,ad,td,sd]=tgas_bk(ed,rhod); f1=(rhoa*ua*At-rhod*ud*An)/ua/rhoa/An ; f2=(ea+pa/rhoa+ua^2/2-ed-pd/rhod-ud^2/2)/(ea+pa/rhoa+ua^2/2); f3=(pdt-pd)/pa; f4=(sd-sa)/sa; erro=f1^2+f2^2+f3^2+f4^2; %------------------------------------------------ function erro = NavierStokes_nz2(X,h0,gama,R,p_impact_meas) rhop=X(1);ep=X(2); clear tgas_bk; [ppt,ap,tp,sp]= tgas_bk(ep,rhop); f1=(ppt-p_impact_meas)/p_impact_meas; f2=(h0-ep-ppt/rhop)/h0; erro=f1^2+f2^2; %------------------------------------------------ function erro = NavierStokes_nz3(X,ucf2,rhocf2,ecf2,pcf2,sp,h0,s0,gama,R) uf1=X(1);rhof1=X(2);ef1=X(3); uf2=X(4);rhof2=X(5);ef2=X(6); clear tgas_bk; [pf1,af1,tf1,sf1]=tgas_bk(ef1,rhof1);

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clear tgas_bk; [pf2,af2,tf2,sf2]=tgas_bk(ef2,rhof2); f1=(rhof1*uf1-rhof2*uf2)/rhocf2/ucf2; f2=(ef1+pf1/rhof1+uf1^2/2-ef2-pf2/rhof2-uf2^2)/h0; f3=(sf2-sp)/sp; f4=(ef1+pf1/rhof1+uf1^2/2-h0)/h0; f5=(sf1-s0)/s0; erro=f1^2+f2^2+f3^2+f4^2+f5^2; %------------------------------------------------ function [pc,ec,tc,rhoc,uc]=chuteinicial(p,t,u,g,r) % Calcula propriedades após a onda de choque para o caso ideal a=(g*r*t)^0.5; rho=p/r/t; e=r*t/(g-1); M=u/a; pc=p*(1+2*g/(g+1)*(M^2-1)); rhoc=rho*((g+1)*M^2/(2+(g-1)*M^2)); ec=pc/rhoc/(g-1); tc=(g-1)*ec/r; uc=u*rho/rhoc;

p_pitot.m function p_pitot=pitot(pinf,tinf,rhoinf,Minf,uinf,einf,h0,gama,R)% calculates the pitot pressure %1)incident shock wave- entries:p1,t1,uw (wave speed) outputs:u2-uw,p2,t2 [p2fi,e2fi,t2fi,rho2fi,u2fi]=chuteinicial(pinf,tinf,uinf,gama,R); X0=[u2fi,rho2fi,e2fi,p2fi]; for i=1:2 X=fminsearch(@NavierStokes,X0,optimset('TolX',1e-25,'TolFun',1e-20),uinf,rhoinf,einf,h0); X0=X; end u2f=X(1); rho2f=X(2); e2f=X(3); clear tgas_bk; [p2f,a2f,t2f,s2f]=tgas_bk(e2f,rho2f); M2f=u2f/a2f;

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[p20i,e20i,t20i,rho20i]=chuteinicial_is2(M2f,p2f,t2f,rho2f,gama,R); Y0=[0,rho20i,e20i,p20i]; for i=1:2 Y=fminsearch(@NavierStokes_compisen,Y0,optimset('TolX',1e-25,'TolFun',1e-20),u2f,rho2f,e2f,p2f,s2f,h0); Y0=Y; end p_pitot=Y(4); %------------------------------------------------ function [pc,ec,tc,rhoc,uc]=chuteinicial(p,t,u,g,r) % Calcula propriedades após a onda de choque para o caso ideal a=(g*r*t)^0.5; rho=p/r/t; e=r*t/(g-1); M=u/a; pc=p*(1+2*g/(g+1)*(M^2-1)); rhoc=rho*((g+1)*M^2/(2+(g-1)*M^2)); ec=pc/rhoc/(g-1); tc=(g-1)*ec/r; uc=u*rho/rhoc; %------------------------------------------------ function erro = NavierStokes(X,ua,rhoa,ea,h0) %equações de continuidade+q. de movimento+ conserv. energia para onda de %choque ud=X(1);rhod=X(2);ed=X(3); pd=X(4); clear tgas_bk; [pa,aa,ta,sa]=tgas_bk(ea,rhoa);%-função tgas_bk é a interpolação de Tannehil- %para normalizar(sufixo ad): rhoad=rhoa; uad=ua; ead=ea; pad=pa; f1=(rhoa*ua-ud*rhod)/rhoad/uad; f2=(ua^2*rhoa+pa-pd-ud^2*rhod)/(pad+rhoad*uad^2); f3=(h0-ed-pd/rhod-ud^2/2)/h0; clear tgas_bk; [pdt,ad,td,sd]=tgas_bk(ed,rhod); f4=(pdt-pd)/pad; erro=f1^2+f2^2+f3^2+f4^2;

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%------------------------------------------------ function erro = NavierStokes_compisen(X,ua,rhoa,ea,pa,sa,h0) ud=X(1);rhod=X(2);ed=X(3); pd=X(4); clear tgas_bk; [pdt,ad,td,sd]=tgas_bk(ed,rhod); f1=ud/ua; f2=(h0-ed-pd/rhod-ud^2/2)/h0; f3=(pdt-pd)/pa; f4=(sd-sa)/sa; erro=f1^2+f2^2+f3^2+f4^2; %------------------------------------------------ function [p0,e0,t0,rho0]=chuteinicial_is2(M,p,t,rho,gama,R) t0=t*(1+(gama-1)/2*M^2); p0=p/(t/t0)^(gama/(gama-1)); rho0=rho/(t/t0)^(1/(gama-1)); e0=p0/rho0/(gama-1);

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Appendix B. Surface Construction Computer Codes

Main code: surfaces.m Entries for surfaces.m: file entries.m =================================================================== entries.m –example- global M delta phil Lw L b beta A0 A2 B0 B2 B4 B6 B8 p pc flag M= 10 ; delta= 5.500000 ; phil= 30 ; Lw= 944.04 ; L= 2729.54 ; b= 200 ; beta= 8.3370000 ; A0= 0.276495 ; A2= 2.061955 ; B0= 1.894898 ; B2= 1.899633 ; B4= 6.087163 ; B6= -23.152877 ; B8= 92.697088 ; thetamin= 0.119784413; thetamax= 0.146394732; flag=1; =================================================================== surfaces.m function surfaces global M delta phil Lw L b beta A0 A2 B0 B2 B4 B6 B8 thetamin thetamax flag run=input('Digite o # da configuracao:'); runs=int2str(run); str1=['C:\Documents and Settings\tiagorolim\Desktop\Tese\modelos\waverider_#' runs '.dat']; str3=['C:\Documents and Settings\tiagorolim\Desktop\Tese\modelos\waverider_#' runs '_fs.dat']; str4=['C:\Documents and Settings\tiagorolim\Desktop\Tese\modelos\waverider_#' runs '_cs.dat']; fic=fopen(str1,'w'); fie=fopen(str3,'w'); fif=fopen(str4,'w'); entries; fprintf(fic,'M=%d;\ndelta=%3.6f;\nphil=%3.6f;\nL=%3.6f;\nLw=%3.6f;\nb=%3.6f;\nbeta=%3.6f;\nA0=%3.6f;\nA2=%3.6f;\nB0=%3.6f;\nB2=%3.6f;\nB4=%3.6f;\nB6=%3.6f;\nB8=%3.6f;',M,delta,phil,L,Lw,b,beta,A0, A2, B0, B2, B4, B6, B8); fclose(fic);

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beta=beta*pi/180; delta=delta*pi/180; %+ freestream for ya=-b/Lw:0.005:b/Lw xa=(A0+ A2*ya^2); za=(xa^2+ya^2)^0.5/tan(beta); for z1=L/Lw:-0.05:za if flag==1 plot3(xa,ya,z1,'+');hold on; end fprintf(fie,'%2.3f\t%2.3f\t%2.3f\n ', xa*Lw,ya*Lw,z1*Lw); end end fclose(fie); phil=phil*pi/180; for phi=-phil:0.05:phil rs=(B0+B2*phi^2+B4*phi^4+B6*phi^6+B8*phi^8)*Lw; for theta=thetamin:0.0005:thetamax r=rs*(beta^2-delta^2)^0.5/(theta^2-delta^2)^0.5; X=r*theta*cos(phi); Y=r*theta*sin(phi); Z=r; %free stream surface as a boundary Yl=Y; Xl=(A0+ A2*(Yl/Lw)^2)*Lw; Zl=(Xl^2+Yl^2)^0.5/tan(beta); if X>Xl if Z>Zl& Z<L if flag==1 plot3(X/Lw,Y/Lw,Z/Lw,'+r');hold on; end%ifflag fprintf(fif,'%3.3f\t%3.3f\t%3.3f\n ', X,Y,Z); end%Z end%x end%fortheta end%forphi fclose(fif);

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FOLHA DE REGISTRO DO DOCUMENTO

1. CLASSIFICAÇÃO/TIPO

DM

2. DATA

07 de maio de 2009

3. REGISTRO N°

CTA/ITA/DM-023/2009

4. N° DE PÁGINAS

119 5. TÍTULO E SUBTÍTULO:

Experimental analyisis of a hypersonic waverider

6. AUTOR(ES):

Tiago Cavalcanti Rolim 7. INSTITUIÇÃO(ÕES)/ÓRGÃO(S) INTERNO(S)/DIVISÃO(ÕES): Instituto Tecnológico de Aeronáutica – ITA 8. PALAVRAS-CHAVE SUGERIDAS PELO AUTOR:

1. Waveriders. 2. Aerospace vehicles. 3. Hypersonic Aerodynamics 9.PALAVRAS-CHAVE RESULTANTES DE INDEXAÇÃO:

Veículos de sustentação por ondas de choque; Veículos aeroespaciais; Configurações aerodinâmicas; Escoamento hipersônico; Dinâmica dos fluidos; Aerodinâmica; Engenharia aeroespacial

10. APRESENTAÇÃO: X Nacional Internacional ITA, São José dos Campos. Curso de Mestrado. Programa de Pós-Graduação em Engenharia Aeronáutica e Mecânica. Área de Aerodinâmica, Propulsão e Energia. Orientador: Prof. Paulo Afonso de Oliveira Soviero; co-orientador: Marco Antonio Sala Minucci – IEAv/CTA. Defesa em 08/04/2009. Publicada em 2009. 11. RESUMO:

This work presents the results of shock tunnel tests of a Mach 10 waverider with sharp leading edges. The waverider surface was generated from a conical flowfield with the volume and the viscous lift-to-drag ratio as optimization parameters. A compression and expansion ramps were added to the pure waverider surface in order to simulate the flow over a scramjet engine. The compression ramp was designed so as to provide the ideal conditions for the supersonic combustion of the Hydrogen while the expansion section was derived from an ideal minimum length supersonic nozzle. The experimental data included Schlieren photographs of the flow and the pressure distribution over the compression surface. These data were compared with the inviscid theory. During these investigations, the IEAv’s T3 shock tunnel was used to simulate the hypersonic flow. The stagnation conditions as well as the free stream properties were estimated using numerical codes. The tunnel operated at Mach number ranges of 8.9 to 10, Reynolds number from 2.25 x 106 to 8.76 x 106 (m-1) and Knudsen number from 0.06 to 0.19. From the Schlieren photographs it was noted that the inlet flowfield behaves according to the predictions of the hypersonic viscous interaction models. Also, the pressure variation along the compression surface centerline was obtained using piezoelectric pressure sensors. The resulted profile presented the general trend of the flow described by these models.

12. GRAU DE SIGILO:

(X ) OSTENSIVO ( ) RESERVADO ( ) CONFIDENCIAL ( ) SECRETO