INTRODUCCIN A LA INGENIERA Y DISEO

AEROESPACIAL

MATERIAL DIDCTICO

QUE PARA OBTENER EL TITULO DE INGENIERO AEROESPACIAL

DESARROLLA:

CARLOS EDUARDO SNCHEZ RAMREZ

DIRECTOR

M.C. ELOY MRQUEZ RODRGUEZ

Chihuahua, Chih. Mxico

Mayo 2014

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NDICE

CAPITULO

I. INTRODUCCIN.3

II. BASE DE CONOCIMIENTOS APLICADOS PARA EL

DESARROLLO DEL TRABAJO4

A. LA CARRERA DE INGENIERO AEROESPACIAL DE LA

UNIVERSIDAD AUTNOMA DE CHIHUAHUA....4

B. UNA ASIGNATURA INTRODUCTORIA..5

1. Beneficios....6

2. Desafos...7

C. EXPERIENCIA DE LA IMPARTICIN DE LA CLASE8

1. Aspectos Positivos..8

2. Oportunidades de Mejora....9

III. DESARROLLO DEL TRABAJO..10

A. LECTURAS..11

B. TAREAS.171

C. EXMENES..184

D. PROYECTOS DE DISEO...202

IV. CONCLUSIONES Y RECOMENDACIONES...204

A. DE LA PARTE TERICA DE LA CLASE..204

B. DE LA PARTE PRACTICA DE LA CLASE204

V. APNDICES Y/O ANEXOS.205

VI. BIBLIOGRAFA...206

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CAPITULO I

INTRODUCCIN

En este trabajo de material didctico se tiene como objetivo la incorporacin de una nueva

asignatura al plan de estudios de Ingeniera Aeroespacial de la Facultad de Ingeniera de la

Universidad Autnoma de Chihuahua. La asignatura tiene como objetivo que lleve el

nombre de Introduccin a la Ingeniera y Diseo Aeroespacial o una versin ms corta de

este nombre como Introduccin a la Ingeniera Aeroespacial. Este trabajo es

estrictamente una propuesta de que la asignatura sea creada y que su contenido aqu

presentado sea utilizado por algn instructor para su enseanza. Es responsabilidad del

Coordinador Acadmico y de la Academia de Ingeniera Aeroespacial su valoracin y voto

para que la asignatura se incorpore oficialmente al plan de estudios.

En el presente trabajo se definen las razones por las cuales la asignatura ha sido creada y

desarrollada, as como las experiencias que el autor de este trabajo ha tenido al impartir la

asignatura durante los pasados dos semestres correspondientes al periodo escolar 2013-

2014.

Adems el resto del trabajo contiene las diferentes partes en las cuales la asignatura fue

dividida y los contenidos temticos de cada parte. Tambin se incluyen ejemplos de tareas,

exmenes, y proyectos de diseo que alumnos realizaron en el semestre agosto-diciembre

de 2013. Cabe destacar que todos los contenidos temticos tambin han sido realizados en

una versin electrnica con el objetivo de diseminar el conocimiento de una manera ms

rpida entre los alumnos y maestros, y que dichos contenidos sean incorporados a la

Plataforma Virtual Moodle de la Facultad de Ingeniera para su fcil acceso.

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CAPITULO II

BASE DE CONOCIMIENTOS APLICADOS PARA EL

DESARROLLO DEL TRABAJO

A. LA CARRERA DE INGENIERO AEROESPACIAL DE LA UNIVERSIDAD

AUTNOMA DE CHIHUAHUA

El programa de estudios de licenciatura en ingeniera aeroespacial de la Facultad de

Ingeniera de la Universidad Autnoma de Chihuahua (UACH) fue creado en 2007 con el

objetivo de incrementar la oferta de recursos humanos calificados para el sector

aeroespacial que se ha encontrado en crecimiento en el estado de Chihuahua desde hace

varios aos. Como una de las industrias de mayor contenido tecnolgico, la industria

aeroespacial requiere de ingenieros bien preparados en las reas tanto de la aeronutica

como espacial para llevar a cabo sus actividades. Y con el objetivo de acelerar el progreso

de la carrera, la Facultad de Ingeniera formo una alianza con el Departamento de

Ingeniera Mecnica y Aeroespacial de la Universidad Estatal de Nuevo Mxico (NMSU)

en Las Cruces, Nuevo Mxico, Estados Unidos, para tener un programa binacional de doble

titulacin entre ambas universidades. Este programa tendra como objetivo formar

estudiantes en la UACH por un periodo de seis semestres y los ltimos tres semestres serian

terminados en NMSU. Durante los primeros seis semestres los alumnos tomaran

asignaturas de ciencias bsicas, matemticas, ciencias de la ingeniera, ciencias sociales e

ingls. Y durante los ltimos tres semestres, los alumnos tomaran nicamente asignaturas

de ingeniera aeroespacial. Al final de los nueve semestres, ambas universidades validaran

el total de los crditos de ambas instituciones y otorgaran cada una un ttulo de Ingeniero

Aeroespacial.

Sin embargo, a partir de enero de 2012 la Facultad de Ingeniera decidi impartir las

asignaturas correspondientes a los ltimos tres semestres para ofrecer a un mayor nmero

de estudiantes la posibilidad de terminar el programa de Ingeniero Aeroespacial sin la

necesidad de transferirse a la Universidad Estatal de Nuevo Mxico. Estos ltimos tres

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semestres contienen las asignaturas de ingeniera aeroespacial nicamente, similar al plan

binacional. La plantilla de profesores de las asignaturas de los ltimos tres semestres ha

sido conformada por maestros de tiempo completo y por ingenieros graduados del

programa binacional actualmente empleados en la industria aeroespacial de la ciudad de

Chihuahua.

El autor de este material didctico ha sido profesor de un total de cuatro asignaturas

aeroespaciales, incluyendo Aerodinmica I, Ingeniera de Sistemas Aeroespaciales,

Estructuras Aeroespaciales, e Ingls Tcnico. A la asignatura de Ingles Tcnico de sexto

semestre se le fue modificado su contenido temtico para ofrecer temas introductorios a las

dems asignaturas de ingeniera aeroespacial. Adems, dado que la asignatura de Ingles

Tcnico tiene sus contenidos en ingls, los temas introductorios se han ofrecido en ingls,

para reforzar las habilidades tcnicas en este idioma extranjero. Esta modificacin fue

hecha con el objetivo de que tanto los alumnos que a partir de sptimo semestre se vallan a

Nuevo Mxico como los que se quedan a terminar el programa de estudios en Chihuahua

tengan un entendimiento elemental de lo que las dems materias consistirn. Al haber

tomado una materia introductoria, se espera que los alumnos obtengan mejores resultados

de aprendizaje en las materias especializadas.

B. UNA ASIGNATURA INTRODUCTORIA

Desde que el autor de este material didctico tomo la asignatura de Ingles Tcnico se ha ido

pensando y analizando la mejor manera de presentar los temas de ingeniera aeroespacial a

un nivel introductorio para que los alumnos de sexto semestre, que para entonces ya

cuentan con conocimientos de matemticas y ciencias bsicas para ingeniera, puedan

comprender los principios fsicos que gobiernan reas como la aerodinmica y la dinmica

del vuelo. Se han investigado diversas universidades con programas en ingeniera

aeroespacial que imparten esta misma asignatura y bibliografa utilizada en la imparticin

de la misma.

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1. Beneficios

Los beneficios de tomar una materia introductoria a las dems materias del programa de

ingeniera aeroespacial son el conocer los principios bsicos que gobiernan ecuaciones

fundamentales como por ejemplo las leyes de la conservacin de la masa, de la

conservacin del momento lineal, de termodinmica o de la conservacin de la energa para

la derivacin de ecuaciones utilizadas en aerodinmica como por ejemplo la ecuacin de

Bernoulli o las ecuaciones del flujo isentrpico. Dado que las materias de aeroespacial

impartidas en los ltimos semestres tienen un alto contenido tcnico y requieren

conocimientos solidos de fsica y matemticas para su entendimiento, el conocer los

principios bsicos y leyes fundamentales junto con problemas de aplicacin tiene como

premisa apoyar al estudiante a que comprenda ms profundamente los contenidos temticos

una vez que se encuentre tomando la asignatura.

Otro de los beneficios apreciables en la adicin de la nueva asignatura es que el alumno

conozca de cerca los temas y tipos de problemas que contiene la ingeniera aeroespacial en

general, y as tenga una mejor perspectiva respecto a continuar con el programa de estudios

o renunciar a l. Esto es importante ya que durante los primeros seis semestres no se

imparten temas directamente asociados a la ingeniera aeroespacial, y por lo tanto, el

alumno llega a conocer de qu se trata la carrera hasta que llega al sptimo semestre.

Desafortunadamente, esto podra provocar algn tipo de desnimo por parte del alumno con

su posterior renuncia a la continuacin del plan de estudios. As, al tomar esta nueva

asignatura durante el sexto semestre, o antes de preferencia, el alumno tendr una toma de

decisiones mejor fundamentada.

Esta asignatura fue tomada como ejemplo a partir del programa de estudios de nivel

licenciatura en Ingeniera Aeroespacial del Instituto Tecnolgico de Georgia (GaTech) y de

la Universidad de Maryland, ambos de Estados Unidos, ya que estas dos universidades se

encuentran dentro de los mejores programas de ingeniera aeroespacial de Estados Unidos

de acuerdo al ranking US News Best Graduate Schools. El Instituto Tecnolgico de Georgia

(GaTech) se encuentra en el lugar # 5 y la Universidad de Maryland en el lugar # 12.

Ambas universidades imparten la materia. La asignatura en el Instituto Tecnolgico de

Georgia es llamada AE 1350: Introduction to AE y en la Universidad de Maryland la

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asignatura es llamada ENAE 283 Introduction to Aerospace Systems. En ambas materias el

contenido temtico es el mismo y la misma bibliografa es utilizada por los instructores.

Cabe resaltar que por ejemplo la asignatura de la Universidad de Maryland tiene como

objetivos de la asignatura ENAE 283 Introduction to Aerospace Systems lo siguiente:

Course Objectives/Student Learning Outcomes:

1. Know the basic principles on which the development of aerodynamics and other

principal subdisciplines of aerospace engineering are based.

2. Use and apply principles from mathematics, physics, and computational

methods to solve beginning level problems in aerodynamics, vehicle

performance, vehicle stability and control, 2-body orbit theory, and propulsion

systems.

Por lo tanto, se puede observar que los objetivos del curso en la Universidad de Maryland

tienen como fin apoyar al estudiante en las asignaturas posteriores que requerirn un mayor

rigor tcnico. Estos mismos objetivos se pretenden alcanzar con la adicin de esta nueva

asignatura. Tambin, ya que en estas dos importantes universidades a nivel internacional se

imparte esta materia, se toma como buena prctica y benchmarking el tomar como

referencia los temas y bibliografa utilizados por dichas instituciones. As, la adicin de esta

nueva asignatura ser clave en el desarrollo acadmico del estudiante y fortalecer el

programa de estudios.

2. Desafos

Uno de los desafos ms grandes en la adicin de una nueva asignatura es el

reconocimiento del personal acadmico y directivo. Para esto es de suma importancia que

tanto los maestros revisores de este material didctico como los dems pertenecientes a la

Academia de Ingeniera Aeroespacial conozcan los contenidos de la asignatura para su

discusin, adicin de contenidos, o en su defecto, la modificacin de los mismos.

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Otro desafo a mediano y largo plazo es el impacto que tendr esta asignatura en el

aprendizaje de los estudiantes mientras y despus de haber cursado las asignaturas de

especialidad. Es importante que se hagan entrevistas a los estudiantes para recabar

retroalimentacin de la relevancia que esta materia les propino al tomar las dems

asignaturas. Todo esto con el fin de enriquecer el contenido de la asignatura y darle forma

de acuerdo a las necesidades tcnicas de los estudiantes y de las materias de especialidad.

C. EXPERIENCIA DE LA IMPARTICIN DE LA CLASE

Esta nueva asignatura ha sido impartida durante los ltimos dos semestres correspondientes

al ciclo escolar 2013-2014 y como prueba piloto hay algunos aspectos positivos a resaltar.

De igual manera se han identificado oportunidades de mejora que se pueden analizar e

implementar conforme la asignatura es impartida. Es importante sealar que de la

experiencia obtenida en la imparticin de esta asignatura se han obtenido estas

observaciones.

1. Aspectos Positivos

Uno de los aspectos que resaltan es la opinin general de los estudiantes del beneficio de

haber cursado una materia introductoria antes de las dems de especialidad. Su opinin

general es que hubiera sido mejor haber tomado algn tipo de materia introductoria antes

del sexto semestre ya que an no haban estudiado contenidos de ndole aeroespacial, y as

ir conociendo su carrera desde semestres iniciales.

Un aspecto positivo apreciado por el autor de este material es que la asignatura contiene

temas de los cursos ms importantes de la carrera de ingeniera aeroespacial como

aerodinmica terica y aplicada, rendimiento de vuelo, estabilidad, propulsin, astronutica

y diseo de aeronaves. Tal amplitud de contenidos temticos si bien no se estudian a

profundidad, cubre lo esencial de cada materia. Se han impartido un total de nueve

unidades ms una nueva correspondiente a los materiales aeroespaciales. Cada unidad

cubre aspectos histricos del tema, teora bsica, y problemas de aplicacin. Durante la

tercera evaluacin, se encarga un proyecto de diseo de aeronaves con la intencin de que

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los alumnos apliquen los conceptos y las formulas estudiadas durante el semestre al diseo

de un avin de uso comercial o militar.

2. Oportunidades de Mejora

Un rea de oportunidad pedaggica importante radica en la inclusin de herramientas de

Tecnologas de la Informacin en la asignatura y de prcticas reales. Actualmente se utiliza

la programacin en MATLAB para la resolucin de algunos problemas, y el modelado en

NX para la realizacin de un proyecto de diseo de aeronaves. Sin embargo, el autor de este

material ha identificado el uso de software especializado en simulacin de fluidos como

una opcin de comprobacin de resultados a distintos problemas de aerodinmica, ya que

diferentes softwares comerciales de fluidos son utilizados en la industria aeroespacial hoy

en da. Tambin, se ha identificado la necesidad de llevar a cabo prcticas en tnel de

viento real con modelos diseados por los mismos estudiantes para obtener coeficientes

aerodinmicos y comparar sus resultados tericos con los obtenidos en la prctica. Se

espera que la inclusin de estos dos mtodos pedaggicos (el uso del software y las

prcticas con tnel), refuerce los contenidos tericos impartidos en clase. Para este fin, se

espera la construccin y puesta en operaciones del nuevo edificio de ingeniera

aeroespacial, prximamente en construccin.

Otra rea de oportunidad dentro del uso de tecnologas de la informacin es la inclusin del

material didctico a la plataforma virtual Moodle de la Facultad de Ingeniera. En este

trabajo de adicin de material didctico se han preparado para cada unidad lecturas en

formato PDF, presentaciones en Power Point, problemas de tarea y sus soluciones para que

el maestro haga uso de ellas. Adicionalmente se han incluido exmenes con sus respuestas

y ejemplos de proyectos de diseo de aeronaves que los alumnos han hecho. Todo este

material puede ser incluido en la plataforma virtual para que el maestro tenga ms control

del mismo y los alumnos accedan de una forma ms cmoda a los materiales en caso de no

poder asistir a la clase.

A manera de sugerencia, futuros maestros de esta clase podran en la plataforma virtual

crear pequeos exmenes de cada unidad para repasar los temas cubiertos y utilizar las

calificaciones como parte de sus calificaciones finales o parciales.

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CAPITULO III

DESARROLLO DEL TRABAJO

El material didctico desarrollado est dividido en nueve unidades, cuyos detalles sern

mostrados a continuacin. Todas las unidades han sido impartidas en ingls. Las nueve

unidades impartidas son:

1. Historia de la Aeronutica e Ideas Fundamentales

2. La Atmosfera Estndar

3. Introduccin a la Aerodinmica

4. Alas y Perfiles

5. Introduccin a la Dinmica de Vuelo: Rendimiento de Vuelo

6. Introduccin a la Dinmica de Vuelo: Estabilidad y Control

7. Introduccin a la Astronutica

8. Introduccin a la Propulsin

9. Introduccin a los Materiales Aeroespaciales

10. Introduccin al Diseo de Aeronaves

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A. LECTURAS

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1. Historia de la Aeronutica e Ideas Fundamentales

1. The First Aeronautical Engineers

It is Kill Devil Hills, 4 miles south of Kitty Hawk, North Carolina, about 10:35 AM on

Thursday, December 17, 1903. Orville and Wilbur Wright are ready to make history. Near

the end of the starting rail, the machine lifts into the air. It is the most historic moment in

aviation history.

Above is a summary of the moment in which the Wright brothers accomplished what many

had failed before: to fly a heavier-than-air machine. It was the first genuine powered flight

of this kind. After this moment, the world of aviation took a whole new direction, since

many scientific and technical aspects of aviation were applied and controlled. However,

contrary to the common belief, the Wright brothers did not truly invent the airplane; rather,

they represent the milestone of a century of prior aeronautical research and development.

1.1 Very Early Developments

The desire to fly has always been an objective for man since history has a record. We can

witness the early Greek myth of Daedalus and his son Icarus. Imprisoned in the island of

Crete in the Mediterranean Sea, Daedalus is said to have made wings fastened with wax.

Using these wings they both flew and escaped from prison. However, Icarus, against his

fathers warnings, flew to close to the sun; the wax melted and Icarus fell to his death in the

sea.

There were also many ancient and medieval people who tried to fly by attaching wings into

their own arms, jumping from towers or roofs and flapping their arm-wings without success

and sometimes with fatal consequences.

The idea of flying took a slightly different path when people started to build wings that

flapped up and down by various mechanical mechanisms, powered by some type of human

arm, leg, or body movement. These machines are called ornithopters. Among these people

is Leonardo da Vinci, who designed many ornithopters and wrote about 500 sketches that

dealt with flight. However, human-powered flight by flapping wings was always doomed to

failure, and Da Vinci did not make important contributions to the technical advancement of

flight.

Human efforts to fly literally got off the ground on November 21, 1783, when a balloon

designed and built by the Montgolfiers in France, carrying Pilatre de Rozier and the

Marquis dArlandes, ascended into the air and drifted 5 miles across Paris. The balloon was

inflated by hot air from an open fire burning in a large wicker basket underneath. This

flight, which became the first one with human passengers rose into the air and lasted for 25

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minutes. However, balloons made no real technical contributions to human heavier-than-air

flight. They were the only means of human flight for almost 100 years.

1.2 Cayley The True Inventor of the Airplane

The modern airplane has its origin in a design created by George Cayley in 1799. This

design included a fixed wing for generating lift, another separate mechanism for propulsion

(paddles), and a combined horizontal and vertical tail for stability. Cayley engraved his

design in a silver disc, and on the other side he drew a diagram of the lift and drag forces on

an inclined plane (the wing). On the past, people had been thinking that mechanical flight

had to do with flapping the wings of ornithopters, in which the flapping motion would

provide both lift and propulsion. However, Cayley is responsible for breaking with this line

of thought; he separated the concepts of lift from propulsion and started a new era of

aeronautical development that culminated with the Wright brothers success in 1903.

George Cayley is considered the father of modern aviation and the first true aeronautical

engineer.

After experimenting with model helicopters, Cayley engraved his revolutionary concept of

the fixed-wing concept. This was followed by an intense 10-year period of aerodynamic

investigation and development. He built a whirling arm apparatus for testing airfoils; this

was simply a lifting surface (airfoil) mounted on the end of a long rod, which was rotated at

some speed to generate a flow of air over the airfoil. In Cayleys time, the whirling arm was

an important development, which allowed the measurement of aerodynamic forces and the

center of pressure of a lifting surface. However, these measurements were not very

accurate, because after a number of revolutions of the arm, the surrounding air would begin

to rotate with the device. In 1804, Cayley designed, built, and flew a small model glider.

This model glider represents the first modern-configuration airplane of history, with a fixed

wing, and a horizontal and vertical tail that could be adjusted.

Cayleys first outpouring of aeronautical results was documented in his triple paper of

1809-1810 called On Aerial Navigation, which was published in the November 1809,

February 1810, and March 1810 issues of Nicholsons Journal of Natural Philosophy. This

document is one of the most important aeronautical works in history. Cayley documented

many aspects of aerodynamics in his triple paper. It was the first published document on

theoretical and applied aerodynamics in history. In it, Cayley elaborates on his principle of

separation of lift and propulsion and his use of a fixed-wing to generate lift. He states that

the basic aspect of a flying machine is to make a surface support a given weight by the

application of power to the resistance of air He notes that a surface inclined at some

angle to the direction of motion will generate lift and that a cambered surface will do this

more efficiently than a flat surface. He also states that lift is generated by a region of low

pressure on the upper surface of the wing. His triple paper also discussed flight control and

the role of horizontal and vertical tail planes in airplane stability. In 1849, he built and

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tested a full-sized airplane called the boy carrier, which was a human-carrying glider that

lifted several meters off the ground when gliding down a hill. In 1852, in the Mechanics

Magazine, Cayley published the design of a large human-carrying glider which incorporated:

1. A main wing at an angle of incidence for lift, with a dihedral for lateral stability.

2. An adjustable cruciform tail for longitudinal and directional stability.

3. A pilot-operated elevator and rudder.

4. A fuselage in the form of a car, with a pilots seat and a three-wheel undercarriage.

5. A tabular beam and box beam construction.

These combined features were not seen until the Wright brothers designs in the 20th

century. George Cayley died in 1857. During his almost 84 years of life, he laid the basis

for all practical aviation. Unfortunately, the name of George Cayley retreated to the

background after his death. Many subsequent inventors did not make the effort to examine

the literature before forging ahead on their own ideas (This is a problem for engineers

today).

The French aviation historian wrote: The aeroplane is a British invention: it was

conceivedby George Cayleythe greatest genus of aviationhe realized that the

problem of aviation had to be divided between theoretical researchand practical

tests

1.3 Lilienthal The Glider

In 1891 was the year in which a human literally jumped into the air and flew with wings in

any type of controlled fashion. This person was Otto Lilienthal. He designed and flew the

first successful controlled gliders in history. Being a mechanical engineer, Lilienthal went

on to work on designing machinery in his own factory. From early childhood he was

interested in flight and performed some youthful experiments on ornithopters of his own

design. Toward the late 1880s he became interested in fixed-wing gliders.

In 1889, Lilienthal published the book Der Vogelflug als Grundlage der Fliegekunst

(Bird Flight as the Basis of Aviation). This is a classic in aeronautical engineering because

he studied the structure and types of birds wings and he applied the resulting aerodynamic

information to the design of mechanical flight. In 1889 Lilienthal came to the philosophical

conclusion that to learn practical aerodynamics, he had to get up in the air and experience it

himself. He wrote One can get a proper insight into the practice of flying only by actual

flying experiments

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Lilienthal used cambered airfoil shapes on the wing and incorporated vertical and

horizontal tail planes in the back for stability. Lilienthal made over 2500 successful glider

flights. The aerodynamic data he obtained were published in papers circulated around the

world. Such widespread dissemination of his results inspired other pioneers in aviation,

including the Wright brothers.

Lilienthal died after a temporary gust of wind brought Lilienthal monoplane glider to a

standstill; he stalled and crashed to the ground.

1.4 The Wright Brothers Inventors of the First Practical Airplane

The Wright brothers drew on an existing heritage that is part of every aerospace engineer

today. The Wright brothers had mechanical talents and set up a shop in which they started

fixing bicycles, and then designing and constructing their own.

Wilbur and Orville had been following Lilienthals progress intently since Lilienthals

gliders were shown in flight by photographs distributed around the world. His progress and

articles quickly made Wilbur Wright to be interested in human flight. Like several flight

thinkers before him, Wilbur approached mechanical flight by the study of bird flight. He

concluded that birds regain their lateral balance when partly overturned by a gust of

wind, by a torsion of the tips of the wings. With this, it emerged one of the most

important developments in aviation history: the use of wind twist to control airplanes in

lateral (rolling) motion. Ailerons are used on modern airplanes for this purpose. This lateral

motion was called wing warping.

Wilbur wrote to the Smithsonian Institution in 1899 to request books and materials in

aeronautics since he wanted to test his concept of wing warping. He received a vast set of

materials written by earlier pioneers of aviation. Both Wilbur and Orville digested all the

aeronautical literature, which led to the design of a biplane kite. This machine was designed

to test the concept of wing warping, which was accomplished by means of four controlling

strings from the ground. The concept worked!

The Wright brothers thought that by flying they could actually get a better feel of the air

and what needed to be done in order to design their airplanes more efficiently, with the goal

of building a heavier-than-air machine that could actually fly and be controlled. They found

an ideal spot in Kitty Hawk, North Carolina, where there were strong and constant winds.

They designed and built two gliders which were used to test the wing warping concept.

However, they started to doubt the science behind the literature they received from the

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Smithsonian, and decided to embark in their own aeronautical research program. They built

a wind tunnel and tested over 200 different airfoil shapes. They designed a force balance to

measure accurately lift and drag. This period of aeronautical research and development led

the Wrights to design their glider No. 3, which was flown in 1902. It was so successful that

Orville wrote that our tables of air pressure which we made in our wind tunnel would

enable us to calculate in advance the performance of a machine. The glider No. 3 was

designed with a vertical rudder behind the wings. This rudder was movable, and when

connected to move in unison with the wing warping, it enabled the glider to make a

smooth, banked turn. This combined use of rudder with wing warping (ailerons) was

another major contribution of the Wright brothers to flight control and aeronautics. Using

this glider the Wright brothers made over 1000 perfect flights and became highly skilled

pilots. Powered flight was the next step, and they were very close to achieve it. However,

they faced the problem of propulsion. There was no commercial engine available for this

purpose, so they designed their own engine and propeller.

With all the major obstacles behind them, they designed and built their Wright Flyer 1

during the summer of 1903. It was similar to the glider No. 3, but included a double rudder

behind the wings and a double elevator in front of the wings. And there was the gasoline-

fueled engine, driving two pusher propellers by means of bicycle-type chains.

The Wrights transported the Wright Flier 1 to Kill Devil Hills, North Carolina. On

December 17, 1903, Orville Wright was ready at the controls. A camera was adjusted to

take a picture of the machine as it reached the end of the rail. The engine was put on full

throttle, the holding rope was released, and the machine began to move. The rest is history.

On this date, the world of successful aeronautical engineering was born.

By their persistent efforts, their detailed research, and their superb engineering, the Wrights

had made the worlds first successful heavier-than-air flight, satisfying all the necessary

criteria laid down by aviation historians.

17

2. Fundamental Physical Quantities of a Flowing Gas

The flow of air over the surface of an airplane is the basic source of the lifting force that

allows a heavier-than-air machine to fly. The shape of an airplane is designed to encourage

the airflow over the surface to produce a lifting force in the most efficient manner possible.

The science that deals with the flow of air is aerodynamics. Aerodynamics is applied in

aircraft design, the design of rocket and jet engines, propellers, vehicles entering planetary

atmospheres from space, wind tunnels, and rocket and projectile configurations. Four

fundamental quantities in aerodynamics are pressure, density, temperature, and velocity.

2.1 Pressure

When you are inside a car in motion, if you take out your hand you can feel the air that

strikes your palm. What is happening is that air molecules are transferring some of their

momentum to the surface of your hand.

Pressure is the normal force per unit area exerted on a surface due to the time rate of

change of momentum of the gas molecules impacting on that surface.

Pressure is defined at a point in the gas or a point on a surface and can vary from one point

to another. Let

dA = an incremental area around B

dF = force on one side of dA due to pressure

The pressure P is the limiting form of the force per unit area where the area of interest has

shrunk to zero around the point B.

18

2.2 Density

The density of a substance (including a gas) is the mass per unit volume.

Density () is a point property and can be defined as follows. Let

dv = an elemental volume around B

dm = the mass of gas inside dv

2.3 Temperature

Temperature is a measure of the average kinetic energy of the particles in the gas. If KE is

the mean molecular kinetic energy, then temperature if given by KE = 3/2 kT, where k is

the Boltzmann constant.

We can visualize a high temperature gas as one in which the particles are moving randomly

at high speeds, and a low-temperature gas as one in which the motion of its particles is

relatively low.

2.4 Flow Velocity and Streamlines

Velocity is a vector quantity and has direction and speed. For a flowing gas we can note

that each region of the gas does not necessarily have the same velocity. Therefore, velocity,

like pressure, density, and temperature, is a point property.

We can imagine an infinitesimally small particle of flow and tracing its path as it moves

with time. This traced path is called s streamline of the flow.

2.5 The Source of All Aerodynamic Forces

The previously defined aerodynamic flow quantities will help to define a flow field.

Knowledge of P, , T, and V at each point of a flow fully defines the flow field.

19

How can these four quantities that define a flow field can help in the design of a new

airplane or the shape of a rocket engine? This is defined as follows.

The most practical consequence of the flow of air on an object such as an airplane is that it

exerts an aerodynamic force composed of two sources:

1. Pressure distribution on the surface

2. Shear stress (friction) on the surface

The force exerted by pressure on the surface acts normal to the surface, and the force

exerted by the shear stress acts tangentially to the surface and is due to the frictional effect

of the flow rubbing against the surface as it moves around the body.

A primary function of aerodynamics is to predict and measure the aerodynamic forces on a

body, which includes prediction and measurement of P and w (w means wing).

2.6 Equation of State for a Perfect Gas

In the regular flight of subsonic and supersonic airplanes the air in the atmosphere behaves

very much like a perfect gas. Looking closely at the molecular level, a gas is a collection

of particles in random motion, where each particle is separated a long distance away from

its neighboring particles. Each molecule has an intermolecular force field, which comes

from the complex interactions of the electromagnetic properties of the electrons and

nucleus. The intermolecular force field of a particle extends a long distance and

changes from a strong repulsive force at close range to a weak attractive force at long

range. If the molecules are close (high densities), their motion can be greatly affected by

the intermolecular force field. If they are separated a long distance, the neighboring

particles only feel the tail of the weak attractive force. Therefore,

A perfect gas is one in which intermolecular forces are negligible.

The particles of air in a room are separated an average of 10 molecular diameters from any

other. The same applies to the air around ordinary subsonic and supersonic vehicles.

The equation of state is P = *R*T

Where R is the specific gas constant and its value varies from one type of gas to another.

For normal air it is R = 287 J/kg*K

To measure the deviation of an actual gas in nature from perfect gas behavior the modified

Berthelot equation of state is used:

20

Where a and b are constants of the gas. Therefore, the deviation from a perfect gas behavior

becomes smaller when pressure decreases and temperature increases. If pressure is high, the

intermolecular forces become important and the gas behaves less like a perfect gas.

However, if the temperature increases, the molecules move faster and their distance from

each other is larger, which make the gas behave more like a perfect gas. Furthermore, if air

is heated to above 2500K, oxygen begins to dissociate into oxygen atoms; if it is heated to

above 4000K, nitrogen begins to dissociate and air becomes a chemically reacting gas,

where its chemical composition becomes a function of pressure and temperature. In such a

case, the specific gas constant R becomes a variable, R = R(P, T). The equation of state is

still valid, but it is no longer a constant. This behavior occurs in very high speed flight such

as atmospheric reentry.

2.7 Units

For units it will be used both the SI system and the English system. However, the majority

of the problems in homework and exams will be in SI units.

Unit SI System English System

P

N/m2

kg/m3

lb/ft2

slugs/ft3

1 atm = 2116 lb/ft

2

T

K

R

R (for air)

287 J/kg*K

1716 ft*lb/slug*R

2.8 Specific Volume

Specific volume is the inverse of density. It is volume per unit mass. By definition,

v = 1/ From the equation of state,

P = *R*T = (1/v)*R*T

Units for specific volume are m3/kg and ft

3/slug.

21

2. La Atmosfera Estndar

1. Altitude

Aerospace vehicles can be divided into two basic categories: atmospheric vehicles such as

airplanes and helicopters, and space vehicles such as satellites and deep space probes,

which operate outside the atmosphere. During the design and performance of any aerospace

vehicle, the properties of the atmosphere must be taken into account.

The atmosphere is always in a state of flux. Pressure and temperature depend on altitude,

location on the globe, time of day, season, etc. It is impractical to take all of these variables

into account when designing an aerospace system. Therefore, a standard atmosphere with

average values can be used for validation of flight tests, wind-tunnel results, and general

airplane design and performance analysis.

In this section, six different altitudes will be defined: absolute, geometric, geopotential,

pressure, temperature, and density.

The altitude above the sea is the geometric altitude (hG). The altitude from the center of the

earth to a distance above sea level is called absolute altitude (ha).

If r is the radius of the earth, then ha = r + hG. Absolute altitude is important for space flight

since gravity varies with altitude. From Newtons law of gravitation, we can calculate the

local value of gravity as a function of altitude,

(

) (

)

2. The Hydrostatic Equation

The hydrostatic equation is the basic of a model to obtain values of pressure, density, and

temperature as functions of altitude. The hydrostatic equation can be thought of as a force

balance on an element of fluid at rest.

22

As it can be seen in the diagram above, the cube represents a particle of air suspended in

the atmosphere.

On the bottom face the pressure P is felt and exerts a force of P*1*1 = P.

The top face is slightly higher, and it will have a pressure P+dP, since pressure varies with

altitude. This pressure will exert a force of P+dP *1*1 = P+dP on the fluid element.

The volume of the fluid particle would be dhG*1*1 = dhG.

If gravity is g, then the weight of the element is gdhG.

To keep the element in equilibrium, all forces should be zero,

P = (P+dP) + gdhG

Therefore,

This is the hydrostatic equation, and it applies to any fluid of density . This equation states

that for any infinitesimal change in altitude, it corresponds an infinitesimal change in

pressure. To simplify the integration of pressure, the assumption that gravity is constant and

has sea level value can be made to obtain the variation of pressure with altitude. Therefore,

3. Relation Between Geopotential and Geometric Altitudes

We need to relate the pressure with geometric altitude. Dividing dp = -g0dh by

dP = -gdhG, we obtain,

1 = (g0/g)*(dh/dhG) or dh = (g/g0)*dhG Therefore,

( ) . Integrating we have,

( )

( )

(

)

(

)

( ( )

)

(

)

23

where h is the geopotential altitude and the geometric altitude. For low altitudes, there is

little difference between h and . Only at altitudes above 65 km the difference is above 1

percent.

4. Definition of the Standard Atmosphere

We can now obtain values of pressure, temperature, and density as functions of altitude for

the standard atmosphere. The figure below shows the results of temperature as a function of

altitude. The vertical parts of the diagram are called isothermal and the inclined parts are

called gradient regions. To obtain formulas for pressure and density as functions of altitude

we can follow the hydrostatic equation,

If we divide by the equation of state,

For the isothermal part shown below, we can obtain the pressure at any value of altitude by

integrating the above equation,

( )

Taking the e function on both sides,

(

)( )

From the equation of state,

Therefore,

(

)( )

24

These two equations give the variation of pressure and density as functions of geopotential

altitude for isothermal layers of the standard atmosphere.

Now we consider the gradient layers as shown below. The temperature variation is linear

and is given by,

Where a is the lapse rate for the gradient layers and it is a specified constant for each layer.

If we substitute a into

Integrating from to h, it yields,

Therefore,

(

)

25

From the equation of state,

The equation for the variation of pressure obtained above becomes

(

)

(

) ( )

Or,

(

) [( ) ]

The variation of temperature is linear with altitude and is given by,

( )

This equation gives temperature as a function of altitude for the gradient regions and can be

used along with the above squared values to obtain pressure and density. At sea level, the

values of temperature, pressure, and density are given by,

26

These are the base values for the first gradient region until T = 216.66 at h = 11 km. Then,

starting at h = 11 km, the values of pressure and density can be calculated using the

isothermal formulas computed earlier until h = 25 km, and so on. With these calculations

and following the figure of the isothermal and gradient regions of altitude vs temperature,

the values of the standard atmosphere can be tabulated.

5. Pressure, Temperature, and Density Altitudes

With the values of the standard atmosphere we can now define three new altitudes:

pressure, temperature, and density altitudes. The pressure altitude is the altitude that

corresponds to any given value of pressure of the air. For example, imagine you are in an

airplane flying at a given altitude. Then you measure the outside pressure and you find that

it is 6.16x104 N/m

2. From Appendix A, you find that the standard altitude that corresponds

to a pressure of 6.16x104 N/m

2 is 4 km. Therefore, you are flying at a pressure altitude of 4

km. Then if you measure the outside air temperature and find that it is 265.4 K, then in

Appendix A you can find that the value of altitude that corresponds to 265.4 K is 3.5 km.

The density altitude can be calculated in a similar way.

27

3. Introduccin a la Aerodinmica

Aerodynamics has many applications in aeronautics and space technology. Among the most

important are:

- Determination of lift and drag on an airplane, missile, etc.

- Determination of the flow velocity and pressure at the nozzle exit of a propulsion

system.

1. The Continuity Equation (COM) Conservation of Mass

Mass can be neither created nor destroyed

- The mass that flows through the cross section at point 1 must be the same as the

mass that flows through the cross section at point 2.

- After an infinitesimal time dt the fluid elements move a distance V1dt, and have

swept out a volume A1V1dt.

- From the density equation, m = V --- dm = 1*(A1V1dt)

- The mass flow rate through area A is the mass crossing A per unit time.

- Therefore,

- Since mass is conserved,

-

28

2. Incompressible and Compressible Flow

All matter is compressible, but to various degrees. For solids and liquids, compressibility

plays a minor role, since their molecular arrangements permit a low variation of their

volume as a force is applied to them. Therefore, their densities are relatively constant. On

the other hand, for gases, compressibility plays a major role, since their volume can be

decreased considerably by the application of a force. For the study of gases in

aerodynamics, compressible flows are very important in high-speed subsonic aircraft, all

supersonic aircraft, and rocket nozzles. Therefore,

Compressible flows are those in which the density of the fluid elements can change from

point to point. For incompressible flows, we can assume density remains constant.

3. The Momentum Equation (COLM) Conservation of Linear Momentum

The continuity equation does not explain the effect of pressure in air molecules. Differences

in pressure from one point to another create forces that move the fluid elements from one

point to another. We start by Newtons second law,

Force = mass x acceleration

F = ma

29

Consider an infinitesimally small particle of fluid P above along a streamline with velocity

V along the x-axis. The force on this element is a combination of:

- Pressure acting in a normal direction on all faces.

- Frictional shear acting tangentially on all faces.

- Gravity acting on the mass of the element.

For this example, we can ignore the frictional forces and gravity, and assume the only force

acting on the element is pressure. Therefore,

- The pressure on the left face is P

- The area of the left face is dydz

- The force on the left face is P(dydz)

- The change in pressure per unit length is dp/dx

- Therefore, if we move away from the left face by a distance dx along the x-axis, the

change in pressure is (dp/dx)dx.

- Consequently, the pressure on the right face is [P + (dp/dx)dx]*(dp/dx)

- Therefore, the force F in the x-direction is,

(

) or

( )

- Moreover, the mass of the fluid element is,

m = *(dx*dy*dz)

- And the acceleration is a = dV/dt. Since V = dx/dt, therefore,

30

- Combining the above three equations into Newtons second law, we can write,

( ) ( )

Eulers Equation

Eulers equation relates the change of momentum to the force. It relates a small change in

pressure for every small change in velocity. Thats why is called the momentum equation,

or conservation of linear momentum. This equation applies for inviscid flow (frictionless

and neglecting gravity), and steady (invariant with respect to time).

To obtain the value of pressure in Eulers equation, we should integrate between points 1

and 2 along a streamline. For compressible flow, density is a variable. For incompressible

flow, density is a constant.

For the case of incompressible flow,

Integrating, we obtain,

( ) (

)

Bernoullis Equation

31

There should be noted that:

- The equation above holds for inviscid, incompressible flow, not compressible flow.

- It holds for different points along a streamline. If all the streamlines have the same

density and velocity, then Bernoullis equation can be used anywhere in the flow.

- It is Newtons second law applied to fluid dynamics.

There should be noted that the equation of state relates P,T, and to each other at the same

point, and the continuity and Bernoullis equations relate and V and P and V at one point

in the flow to the same quantities an another point in the flow.

4. Review of Thermodynamics

In high speed flows there is an important relation between compressibility and big energy

changes. To study compressible flows, we should first review some fundamentals of

thermodynamics. The pillar of thermodynamics is the first law, and it is an observation of

natural phenomena. Consider a fixed mass of gas (a unit of mass) contained within a

flexible boundary as shown in the figure below. The mass of gas is called the system, and

everything outside the boundary is the surroundings. These gas molecules move with

random motion, and their summed energy is called the internal energy e of the system. The

only ways the internal energy of the system can be increased or decreased is:

- Heat added or taken away from the system.

- Work done on or by the system.

Therefore, it can be stated that,

The equation above is the first law of thermodynamics: the change in internal energy is

equal to the sum of the heat added to and work done on the system.

Lets now obtain a useful formula for so we can use it for aerodynamics calculations.

Consider a small area dA in the system shown below. Let a pressure P be applied on this

area and being pushed a distance s. By definition, work done on this system is,

( )( )

32

Therefore, the total sum of the work done on the whole system is,

( )

If P is constant so there is thermodynamic equilibrium, then,

Since work done on the system (the gas) decreases volume because it is being pushed in,

then the volume is,

Therefore,

If substituted into , then

Alternative form of the first law

Now, well define enthalpy to be,

Differentiating,

Therefore, ( )

The way by which changes of the thermodynamic variables take place is called a process.

33

Specific heat is heat added per unit change in temperature

1.-Specific heat at constant volume: (

)

2.-Specific heat at constant pressure: (

)

Lets now take a look at each of those two processes:

1. - Consider a constant-volume process in which,

Therefore,

And, is constant for air at normal conditions

If if we integrate

for a perfect gas

2. - Consider a pressure-constant process,

Then, if . If we integrate,

For a perfect gas

Therefore,

Relation of internal energy

and enthalpy to

temperature

34

5. Isentropic Flow

The concept that bridges thermodynamics and compressible aerodynamics is isentropic

flow.

1. Adiabatic process No heat added / removed

2. Reversible process No friction on dissipative forces

3. Isentropic process adiabatic and reversible

Flow of fluid elements through wind-tunnel nozzles and docket engines are isentropic.

For compressible flows, temperature may not be constant. As the flow moves to

lower or higher density regions volume changes, because work is done, and the

internal energy changes, so temperature changes.

For incompressible flows, density is constant, no work done no dT.

For isentropic process,

Likewise,

If we divide,

Or

(

)

For compressible flows,

. For air at normal conditions cv and cp are constants, and

for air.

Therefore,

Integrating,

35

(

)

Since

(

) Isentropic flow

From

(

)

(

)

(

) (

)

(

)

Isentropic flow

And

(

) (

)

Isentropic flow

6. The Energy Equation

Fundamental principle: Energy can neither be created nor destroyed. It can only change in

form 1st law of thermodynamics.

From the 1st law, . Also,

For adiabatic flows,

From Eulers equation,

Therefore, since

Integrating between two points along the streamline, we obtain:

36

( ) (

) (

)

Energy equation for frictionless, adiabatic flow.

If

7. Summary of Equations

1. For steady, incompressible flow of a frictionless fluid.

Continuity (density is constant)

Bernoullis equation

2. For steady isentropic compressible flow.

Continuity

(

) (

)

Isentropic relations

Energy

Equation of state

37

8. The Speed of Sound

Sound waves travel through the air at a definite speed: The speed of sound.

Stagnant gas.

The sound wave is a thin region of disturbance in the air across which the pressure,

temperature, and density change slightly.

The change in pressure activates your eardrum to hear the sound wave.

If you imagine you ride on the wave, the air in front appears to be coming at you with

velocity a. It looks like:

The air behind you (at 2) is going away from you and its properties change slightly:

and the wave velocity also changes da. The velocity behind the wave is

Objective: to obtain an equation for a.

With the continuity equation:

Or

( ) ( )

Blast Source of

sound wave

Ahead of

wave

Behind the wave 1 2

38

The Area of the stream tube through the wave is constant Therefore:

( )( )

Or

The product is very small and can be ignored.

We apply the momentum equation, in Eulers formula/

...

Therefore

The flow through a sound wave is isentropic (no heat addition) and no frictional forces.

Therefore,

(

)

Moreover for isentropic flow,

(

) or

Therefore, the ratio

is constant in an isentropic flow.

And (

)

=

( ) =

Therefore (

)

39

Therefore

but for a perfect gas,

...

The propagation of a sound wave through a gas takes place via molecular collisions. When

something explodes, some of its energy is transferred to the neighboring molecules, thus

increasing their kinetic energy. These molecules then transfer some extra energy to the

other molecules. Therefore, the energy of a sound wave is transmitted through air by

molecules which collide with each other.

Temperature is a measure of the mean kinetic energy; hence of the mean molecular velocity

and temperature is also a measure of velocity.

A mean average velocity can be defined for the whole gas. Therefore, the energy of sound

waves will be transmitted at this velocity.

The Mach number at a point in the gas is the velocity of the speed of sound.

Ratio of velocity to the speed of sound

The regimens in aerodunamics are:

1. the flow is subsonic

2. the flow is sonic

3. the flow is super sonic

4. If M , it is transonic flow

5. If , it is hypersonic flow

The speed of sound in a perfect gas depends only

on the temperature of the gas.

40

9. Low Speed Subsonic Wind Tunnels

Wind tunnels - They are ground-based experimental facilities designed to produce flows of

air, which simulate natural flows. Simulation of flows for airplanes, missiles, and space

vehicles are carried out in wind tunnels.

Airflow with pressure enters the nozzle at a velocity where the area is . The nozzle

converges to a smaller area at the test section. Since the flow is incompressible, the

velocity u the test section is:

After flowing over an aerodynamic model, the air passes through a diverging duck called a

diffuser, where the area increases to and can be found by:

The pressure is related to the velocity through Bernoullis equation

From Bernoullis equation, as v increases p decreases( ).

In many subsonic wind tunnels all or part of the test section is open; therefore, the test

section pressure

In the diffuser, the pressure increases as velocity decreases . If 1 then

and .

41

In real wind tunnels, the flow that passes through the model loses some momentum.

Therefore is slightly less than .

In practical operation the test section velocity is governed by the pressure difference

and the area of ratio .

...

( )

, and, if (

) , then

( ) (

)

Solving for V2 we have,

( )

[ (

) ]

10. Measurement of Airspeed

If the flow is not constant over a given cross-section or if the flow in the middle of the

test section is higher than near the walls, then obtained earlier is only a mean value.

Therefore, we need a point measurement at a given location in the flow. This can be

done by the Pitot-static tube.

Static pressure at a given point is the pressure we would feel if we were moving along

with the flow at that point. They are molecules moving about with random motion and

transferring their momentum to a across surfaces.

Total pressure at a given point in a flow is that pressure that would exist if the flow

were slowed down isentropically to zero velocity. total pressure

Total pressure is a property of the gas flow at a given point. Therefore there are 2

pressures we can consider at a given point in the flow: static and total pressure, where

.

For the case of a gas that is not moving, . Examples are a gas confined in a

cylinder or stagnant air in the room.

Where variable to control in a wind tunnel

42

The Pitot tube measures total pressure. It consists of a tube placed parallel to the flow

and open to the floor at one end (A). The other end of the tube is closed (B). As the

flow gets into the tube, the gas will pile up inside the tube.

After a few moments, there will be no motion inside the tube, and the gas will stagnate.

Therefore, any flow molecule at point (C) will stop at point (A). Since theres no

friction or heat exchanged, the process will be isentropic.

Therefore, a fluid element moving along streamline (C) will be isentropically brought to

rest at point (A). So the pressure at point (A) is the total pressure .

Any point where is called a stagnation point.

A uniform flow with velocity flows parallel to the surface.

The pressure at point A is the static pressure

At point (B), the total pressure is measured.

The differential pressure gauge will measure the difference , which gives a

measure of the flow velocity V1.

However, the way to obtain differs depending on whether the flow is

incompressible, high-speed subsonic, or supersonic.

10.1. Incompressible Flow

Consider the diagram above. At point (A) the pressure is and the velocity . At

point (B) the pressure is and the velocity zero.

Applying Bernoullis equation

Dynamic pressure: its applied from incompressible aerodynamics to

hypersonic.

State

pressure Dynamic

pressure

Total

pressure

43

Therefore Incompressible flows only

Therefore ( )

To obtain the correct value of the engineer must decide what value of to

use. If is the true value of density measured in the air around the airplane, then,

( )

However, the measurement of the atmospheric air density at the airplanes location is

difficult. For practical reasons, the value of used is the sea-level value .

and are different by the factor ( )

At near sea-level, this difference is small.

10.2. Subsonic Compressible Flow

Flight regime of Boeing and Airbus commercial and military aircraft.

Flight velocities greater than 0.3 Mach but less than .

If ,

If we divide

by Cp, then

...

( )

Therefore, the equivalent airspeed is,

44

Therefore

Equation for a perfect gas with constant specific heats.

When the flow in a Pitot tube is brought to zero velocity isentropically, its pressure

and temperature are .

Using the energy equation applied to the incoming flow and the stagnation point at

the Pitot tube,

If we substitute the value of in the equation, we have,

[ ( ) ]

If the value of the speed of sound is given by

Then

then

This equation holds between the freestream and the stagnation point.

If

(

) (

) ( ) then

(

)

(

) ( )

Now, to obtain the measurement of airspeed, we use the pressure ratio, and solve for

the Mach number,

,

[( ) ]

Important for

compressible,

isentropic flow.

45

Therefore,

[(

)

]

[(

)

]

If standard-sea-level values for and are used then,

[(

)

( )

]

10.3. Supersonic Flow

For supersonic flows, , the airspeed measurement is different.

A shock wave will form ahead of the Pitot-tube.

Shockwaves are very thin regions of flow ( ) across which very severe

changes in the flow properties take place.

How do the shock waves form?

For flows where , molecules that collide with the probe set up a disturbance

which is communicated to other regions of the flow at the local speed of sound.

True values of static

temperature in the air

surrounding the airplane is

difficult to measure.

Used to obtain true

speed.

As a fluid element flows through a shock

wave:

1. The Mach number decreases

2. The static pressure increases

3. The static temperature increases

4. The flow velocity decreases

5. The total pressure decreases

6. The total temperature To stays the

same for a perfect gas.

46

For flows where , the molecules that collide with the probe cannot move away

and pile up at a finite distance from the probe.

In supersonic flows, air molecules dont have the time to move away and pile up

forming a shockwave.

Shove waves are made visible by means of a Schlieren system.

To calibrate a Mach meter for supersonic flow,

(( )

( )

)

It relates measured behind the shock wave and value of the freestream static

pressure obtained by an orifice somewhere in the surface of the airplane.

11. Supersonic Wind Tunnels and Rocket Engines

The main aerodynamic interest in supersonic flows occurred after World War II with

the advent of jet aircraft and rocket-propelled guides missiles.

The flow through rocket engine nozzles is an example of the application of the

fundamental laws of aerodynamics.

Consider isentropic flow in a stream tube,

From the continuity equation,

Or ( )

Differentiating, we obtain

Rayleigh

pitot tube

formula

47

In the momentum equation, we obtain,

Substituting, we obtain

Since the flow is isentropic

(

)

Therefore,

Rearranging, we obtain,

(

)

Or

( )

Area velocity relation

Associated phenomena

1) When the flow is subsonic( ), the area must converge for the velocity to

increase.

2) If the flow is supersonic, the area must diverge for the velocity to increase.

3) If the flow is sonic ( ), then

Looks like infinity

48

1 2 3

3) The only way for

to be finite is to have

, so

Finite number

If

, the streamtube has a minimum area at .This minimum area is called

the throat.

In order to expand a gas to supersonic speeds starting with a stagnant gas in a reservoir,

a duct of sufficiently converging diverging shape must be used.

For both supersonic wind tunnels nozzles and rocket engine nozzles the flow starts out

with a very low velocity in the reservoir, expands to high subsonic speeds in the

convergent section, reaches at the throat, and then goes supersonic in the

diverging section.

Smooth uniform flow is designed; C-D nozzles are long.

Flow quality is not so important, but the weight of the nozzle is a concern.

The real flow through nozzles is closely approximated by isentropic flow because

almost no heat is added or taken away and its almost frictionless.

Therefore, the static temperature density and pressure( ) can be found if M is

known.

From differential

calculus theory

49

[

( )

]

[

( )

]

( )

[

( )

]

( )

Also the variation of Mach number through the nozzle is a function of the ratio of

the cross-sectional area to the throat area,

(

)

[

(

)]

( )( )

50

4. Alas y Perfiles

1. Airfoil Nomenclature

An airfoil is the cross sectional shape obtained by the intersection of the wing with the

perpendicular plane.

The mean cumber line is the locus of points halfway between the upper and lower

surfaces.

The camber is the maximum distance between the mean camber line and the chord line.

The camber shape controls the lift and moment characteristics of the airfoil.

The angle of attack () is the angle between the relative wind and the chord line.

(R) is the resultant force by pressure and shear stress distributions.

(D) Component of R parallel to the relative wind.

51

Lift is the component of R perpendicular to the relative wind.

Surface pressure and shear stress distributions create a moment M which tends to rotate

the wing.

Will create a moment which will tend to rotate the airfoil.

Lift, drag and moments will change as changes.

Aerodynamic center is the point about which moments dont vary with .

The moment about the aerodynamic center is

{

For subsonic low-speed aircraft, the ac is located at about the quarter-chard point.

2. Lift, Drag, and Moment Coefficients

Coefficients are dimensions less quantities that simplify our aerodynamics calculations.

For an airfoil of given shape the dimension less lift, drag, and moments coefficients are

defined as,

The moment coefficient differs because M has dimensions of a force-length product.

All the physical complexity of the flow field around an aerodynamic body is implicitly

burred in

To obtain lift, drag, and moment forces on a body, we should first obtain the respective

coefficients.

52

3. Airfoil Data

A goal of theoretical aerodynamics is to predict the values of from

basic equations.

The practical aerodynamicist has to rely upon direct experimental measurements of

The NACA measured lift, drag, and moment coefficients of many airfoils in low speed,

subsonic wind tunnels.

The experimental data were obtained for infinite wings.

Appendix D lists these airfoil graphs.

The slope of the linear portion of the lift is the lift slope.

The lift curve for a symmetric airfoil goes through the origin

53

As is increased beyond the maximum value for the lift coefficient drops

rapidly. This angle is the stall angle.

Stall is caused by flow separation on the upper surface of the airfoil.

Example 5.1

A model wing of constant chord length is placed in a low-speed subsonic wind tunnel,

spanning the test section. The wing has a NACA 2412 airfoil and a chord length of 1.3 m.

the flow in the test section is at a velocity of 50 m/s at standard sea-level conditions. If the

wing is at a 4 angle of attack calculate

a)

b) The lift, drag, and moments about the quarter chord, per unit span.

SOLUTION

a) From Appendix D, for a NACA 2412 at a 4 angle of attack

To obtain we must first check the value of the Reynolds number:

( )( )( )

( )( )

For this value of Re and for from Appendix D

54

b) Since the chord is 1.3 , and we want the aerodynamic forces and moments per unit

span (a unit length along the wing, perpendicular to the flow), then ( )

( )

Also

( )( )

From Eq. (5.20)

( )( )

Since 1 N=0.2248 lb, then also

( )( )

( )( )

( )

Note: The radio of lift to drag, which is an important aerodynamic quantity, is

( )( )

Example 5.2

The same wing in the same flow as in example 5.1 is pitched to an angle of attack such that

the lift per unit span is 700 N (157 lb).

a) What is the angle of attack?

b) To what angle of attack must the wing be pitched to obtain zero lift?

SOLUTION

a) From the previous example,

Thus

( )

55

From appendix D for the NACA 2412 the angle of attack corresponding to is

b) Also from Appendix D, for zero lift, that is,

4. Infinite vs Finite Wings

The data obtained in Appendix D were measured in low-speed subsonic wind tunnels

where the model wing spanned the test section.

The flow about this wing is two-dimensional.

All real airplanes wings are finite,

The wing span b is the distance between the 2 wingtips.

Infinite wings

2-D flow

The area of the wing on the

platinum view (top) is S

The flow field about a finite

wing is three-dimensional

For finite wings, use CL, CD,

and CM.

56

5. Pressure Coefficient

The pressure coefficient is defined as

is positive at the leading edge. ( )

As the flow expands around the top surface, decreases rapidly, is negative.

The distribution of over the airfoil surface leads directly to

Considerations of lead directly to the calculation of the effect of on the lift

coefficient.

At M > 0.3, Cp increases dramatically.

An approximation of this relation is

Prandtl-Glauert rule

Cp is constant for M

57

value at point of an airfoil of fixed shape and angle of attack. Good for

Example 5.3

The pressure at a point on the wing of an airplane is the airplane is flying

with a velocity of at conditions associated with a standard altitude of 2000 m.

Calculate the pressure coefficient at this point on the wing.

SOLUTION

For a standard altitude of 2000 m,

Thus

( )( ) From eq. (5.23),

( )

Example 5.4

Consider an airfoil mounted in a low-speed subsonic wind tunnel. The flow velocity in the

test section is 100 ft/s, and the conditions are standard sea level. If the pressure at a point on

the airfoil is 2102 lb/f , what is the pressure coefficient?

SOLUTION

( )( )

Form eq. (5.23),

58

Example 5.5

In the example 5.4, if the flow velocity is increased such that the freestream Mach number

is 0.6, what is the pressure coefficient at the same point on the airfoil?

SOLUTION

First of all, what is the Mach number of the flow in Example 5.4? At standard sea level,

Hence, ( )( )

Thus, in example 5.4, a very low value. Hence, the

flow in Example 5.4 is essentially incompressible, and the pressure coefficient is a low-

speed value, that is, thus, if the flow Mach number is increased to 0.6, from

the Prandtl-Glauert rule, Eq. (5.24),

( )

( )

6. Obtaining Lift Coefficient from Cp

If the distribution of over the top and bottom surfaces of an airfoil is given, can be

calculated easily.

The aerodynamic force due pressure is ( ) which is normal.

LE

59

Its component in the lift direction is ( ) Therefore, the contribution to lift of

the pressure is

To obtain the contribution from LE to TE we integrate,

} Lift due to pressure distribution on the upper surface

A similar integral can be obtained on the lower surface. Therefore, the total lift can be

found by,

adding and substating we get,

( ) ( )

( )

Then,

Therefore,

( )

The lift coefficient can be obtained by integrating over the airfoil surface.

On the plots of upper and lower surfaces, the is equal to the area between both

curbes, divided by the chord length C.

This equation is valid only for and small values of

Cp Cp

60

Example 5.6

Consider an airfoil with chord length c and the running distance x measured along the

chord. The leading edge is located at and the trailing edge at The

pressure-coefficient variations over the upper and lower surfaces are given respectively as

(

)

Calculate the lift coefficient.

SOLUTION

From eq. (5.30)

( )

( ) (

)

(

) (

) [ (

)

]

(

) (

)

|

(

)

|

|

(

)

|

|

(

)

|

61

It is interesting to note that, when and are plotted on the same graph versus

( )

(

)

7. Compressibility Correction for Lift Coefficient

The pressure coefficient can be replaced by the compressibility correction,

( )

( )

The subscript denotes low-speed incompressible flow.

Now, since

( )

Therefore,

Example 5.7

Consider a NACA 4412 airfoil at an angle of attack of 4, if the freestream Mach number is

0.7, what is the lift coefficient?

SOLUTION

From Appendix D, for However, the data in Appendix D were obtained

at low speeds, hence the lift coefficient value obtained above, namely 0.83, is really

62

For high Mach number, this must be corrected according to Eq. (5.32):

( )

( )

8. Critical Mach Number and Critical Pressure Coefficient

Consider the flow of air over an airfoil. As the gas expands around the top surface near

the leading edge, the velocity and Mach number increase.

There are regions on the airfoil surface where the local Mach number is greater

than .

The freestream Mach number at which sonic flow is first obtained somewhere on the

airfoil surface is the critical Mach number.

At some freestream Mach number above , the drag will increase.

63

On the thin airfoil:

The flow is more streamlined and slightly perturbed.

The pressure on the top surface decreases a small amount

can be increased to a large subsonic valve before sonic flow is encountered

on the airfoil

On the medium-thickness airfoil:

Flow expansion over leading edge sill be stronger

The specific value of Cp

that corresponds to sonic

flow is the critical pressure

coefficient, Cp,cr

The specific value of Cp

that corresponds to sonic

flow is the critical pressure

coefficient, Cp1cr

64

Velocity will increase on the top surface

Pressure will decrease to lower valves will be more negative

Because flow expansion is stronger, sonic conditions will happen sooner.

The same for the thick airfoil

Therefore, thin airfoils are desirable to expand the , and are used on all high-speed

airplanes.

Derivation of the critical pressure coefficient and Mach number

(

)

From dynamic pressure

( )

( )

If

therefore,

= Equation A.

Now, for isentropic flow,

(

)

( )

And,

(

)

( )

Now, dividing both equation, we get,

(

( )

( )

)

( )

Substituting with equation A, we obtain for

(

)

[(

( )

( )

) ]

65

[(

( )

( )

) ]

Relation of local value of Cp to a local M at any given point, on the surface of the airfoil

If , then , and we get:

[( ( )

)

( )

] ( )

As increases, decreases.

Example 5.8

Given a specific airfoil, how can you estimate its critical Mach number?

SOLUTION

There are several steps to this process, as follows:

a) Obtain a plot of versus M from Eq. (5.37).This is illustrated by curve A in

Figure 5.18 this is a fixed universal curve, which you can keep for all such

problems.

66

b) From measurement or theory for low-speed flow, obtain the minimum pressure

coefficient on the top surface of the airfoil. This is shown as point Bin Figure

5.18.

c) Using Eq. (5.24), plot the variation of this minimum pressure coefficient versus

.This is illustrated by curve C.

d) When curve C intersects curve A, then the minimum pressure coefficient on the top

surface o the airfoil is equal to the critical pressure coefficient, and the

corresponding is the critical Mach number. Hence, point D is the solution for

9. Drag Divergence Mach Number

At low Mach numbers, less than is almost constant and is equal to the low-

speed valve in Appendix D. a

If is increased slightly above a bubble of supersonic flow will occur

surrounding the minimum pressure point b. will still be reasonably low.

If is still increased, a sudden increase in the drag coefficient will take place, and

shock waves appear.

The increase in pressure creates a strong pressure gradient, and the flow separates from

the surface, which increases the drag.

The freestream Mach number at which increases rapidly is the drag-divergence

Mach number.

67

This behavior is called transonic and is characterized when

Supercritical airfoils have been designed to increase the drag-divergence Mach number.

They discourage the formation of shock-waves.

10. Wave Drag at Supersonic Speeds

Shock waves in supersonic flow create wave drag.

Beeper emits a sound wave at speed (a) and moves at Velocity V, where V

68

Now the beeper is moving at supersonic speed V>a. at time t the sound wave will have

moved a distance at.

After the same time t, the beeper will have moved a distance Vt.

If the beeper is constantly emitting sound waves as it moves along, the waves will pile

up inside an envelope from point S tangent to the sound wave circle.

The tangent line from S is called Mach wave.

The pilling up of pressure waves in supersonic flight can create sharply defined waves.

The Mach wave makes an angle with direction of movement of the beeper. This angle

is called Mach angle.

Objects such as needle will create a very weak disturbance in the flow, limited to a

Mach wave.

M

SUPERSONIC

69

If a thicker object like a wedge is moving at supersonic speeds, it will create a strong

disturbance, shock waves. The SW will be inclined at a oblique angle , where > .

Across the SW, pressure increases and creates wave drag.

In order to minimize the strength of the shockwave, all supersonic airfoils are thin, with

relatively sharp leading edges.

Approximation of thin supersonic airfoil by a flat plate

An Expansion Wave (EW) is fan-shaped region through which the pressure decreases.

The air is turned away with the EW and turned back with the SW, which increases

pressure, which creates lift.

70

From this pressure, wave drag is also created.

( )

( )

As increases, both lift and drag increase.

Example 5.9

Consider a thin supersonic airfoil with chord length c=5 ft in Mach 3 freestream at a

standard altitude of 20,000 ft. the airfoil is at an angle of attack of 5.

a) Calculate the lift and wave drag coefficients and the lift and wave drag per unit

span.

b) Compare these results with the same airfoil at the same conditions, exep at Mach 2.

SOLUTION

a) In Eqs. (5.39) and (5.40) the angle of attack must be in radians. Hence

Also

( )

( )

At 20,000 ft, Hence

( )( )

( )

( )( )

Useful for thin airfoils at small to

moderate angles of attack in

supersonic flow

71

( ) ( )( )

( ) ( )( )

b)

( )

( )

Note: at Mach 2, are higher than at Mach 3. This is general result; both

decrease with increasing Mach number, as clearly seen from Eqs. (5.39)

and (5.40) . Does this mean that L and also decrease with increasing Mach number?

Intuitively this does not seem correct. Let us find out.

( )

( )( )

( ) ( )( )

( ) ( )( )

Hence, there is no conflict with our intuition. As the supersonic Mach numbers

increase, L and also increase although the lift and drag coefficients decrease.

11. Finite Wings

72

Higher-pressure air at the bottom leaks toward lower-pressure regions, creating a

circulator motion called a vortex.

Vortices include a small downward component of air velocity called downwash w.

Consequences of this local relative wind:

1. The angle of attack is reduced in comparison to the angle of attack of the wing

referenced to

2. There is an increase in drag, called induced drag.

Interpretations of induced drag

1. The wing tips alter the flow field and create drag.

2. The lift vector is tilted back and creates a drag component.

3. The vortices contain rotational kinetic energy which comes from the aircraft

propulsion system, where extra power has to be added to overcome the extra

increment in drag due to induced drag.

12. Calculation of Induced Drag

73

Geometric angle of attack.- angle between the mean chord of the sing and the direction of

(relative wind).

The local flow is deflected downward by the angle blc of downwash.

Induced angle of attack.- difference between beat flow direction