Physics for Aero dynamicsAERODYNAMICS

Dynamic pressure is calculated as half the density multiplied by the velocity squared.. If the density is constant, the dynamic pressure increases sixteen times if the velocity increases

1 Physics for Aerodynamics

The laws of physics that affect the aircraft in flight and on the ground are

de-scribed using the international SI system.The SI system is based on the metric

system and must be used by law throughout the world

You need to use conversion tables for the English or American systems You

can find conversion tables in the appendix of most technical documentation

The laws of physics are described by fundamental units and basic

quanti-ties.The fundamental units can not be defined in other quantiquanti-ties.The basic

quantities are defined in fundamental units

Speed, for example, is a basic quantity It is defined by the fundamental units

distance and time

Speed, denoted by V is distance, denoted by m over time, denoted by s.

There are seven fundamental units in physics mass, length, time,

tempera-ture, current, mol number and the intensity of light.

The fundamental units used in aerodynamics are mass, length, time and

tem-perature.

1.1 Fundamental units

1.1.1 Mass

The unit of measurement for mass is kilograms, denoted by kg The mass of

one kilogram is defined by a piece of platinum alloy at the office of weights and

measurements in Paris

The mass of one kilogram is also the volume of one liter of pure water at a

temperature of four degrees Celsius

Mass is not the same as weight The astronauts flying around in their space

labs have no weight but their bodies have a mass

1.1.2 Length

The unit of measurement for length is meters, denoted by m.

The meter was established as a standard unit of length by a commission set up

by the French government in 1790

A meter is more precisely defined as a certain number of wavelengths of a

par-ticular colour of light

1.1.3 Time

The unit of measurement for time is seconds, denoted by s Originally this was

based on the length of a day However not all days are exactly the same tion so the second is now defined as the time it takes for a certain number ofenergy changes to occur in the caesium atom

dura-1.1.4 Temperature

The unit of measurement for temperature is kelvin, denoted by K Zero kelvin is

called absolute zero because it is the lowest temperature possible

The kelvin scale starts at zero and only has positive numbers

One kelvin is the same size as one degree Celsius

1.2 Speed and acceleration

1.2.1 Speed and velocity

Speed is the distance that a moving object covers in a unit of time For

exam-ple, we can say that an aircraft has a speed of 500 kilometers per hour

Speed is denoted by V.

Velocity is the distance that a moving object covers in a given direction in a unit

of time We can say that an aircraft has a velocity of 500 kilometers per hour

northward

Velocity is also denoted by V.

1.2.2 Acceleration

Acceleration is the change in velocity divided by the time during which the

change takes place

You can see that the velocity changes from 100 m/s to 150 m/s during this ten

second period

In this example the acceleration is 50 m/s per ten seconds This is equal to five

meters per second per one second which is 5 m/s2 Acceleration is measured

in meters per square second ( m/s2 ).

Acceleration is denoted by a.

A special form of acceleration is acceleration due to gravity An object, such as

this ball, which falls freely under the force of gravity has uniform acceleration if

there is no air resistance

Acceleration which is due to gravity is denoted by g.

The value of this acceleration varies across the earth’s surface but on average

it is nine point eight meters per square second For ease of calculation ten

me-ters per square second is often used

1.3 Force and weight

We begin our look at force with an experiment You can see that our friend isstanding on a weighing scale in an elevator and observing his weight ( Fig be-low, left )

There is no change in weight if a body stays at rest or if it moves with uniformvelocity

But what happens to the weight if the elevator accelerates as it moves upward?

As the elevator accelerates there is an additional force which increases theweight

Force is measured in Newtons The term deca Newton is used in all technicalmanuals for force and for weight

Weight is one kind of force It is mass multiplied by the acceleration due togravity You know that gravity is the attraction exerted on any material towardsthe center of the earth

Weight is also measured in Newtons ( Fig below, right )

Work is denoted by joule and is measured in Newton meters.

You can see that the object with a force of six hundred Newton is moved a

dis-tance of thirty meters

The work is six hundred Newton multiplied by thirty meters which is eighteen

thousand Newton meters

Power is work over time or more specifically force multiplied by distance over

time

Power is measured in Watts which is Newton meters per second

You probably know the term horse power When steam engines were first used

their power was compared to the power of horses because they were used for

work which was previously done by horses Now the international SI system

uses watts and kilowatts instead of horsepower

You can see that the object with a force of 600 N is moved a distance of 30 m

in 10 seconds

The power is six hundred Newton multiplied by thirty meters divided by ten

se-conds which is 1800 watts or 1.8 kilowats

Pressure is the force acting on a unit of area.

It is denoted by Pascal ( Pa ) and measured in Newtons per square meter

( N/m 2 ).

Static pressure acts equally in all directions It is denoted by a small ’p’ and

measured in Newtons per square meter ( N/m 2 ).

Static pressure is calculated as height multiplied by density multiplied by

grav-ity P stat. = h x H x g.

1.5.2 Dynamic pressure

Dynamic pressure acts only in the direction of the flow

It is denoted by a small ’q’ and sometimes called q pressure and, like static

pressure, measured in Newtons per square meter ( N/m 2 ).

Dynamic pressure is calculated as half the density multiplied by the velocity

squared q = ½ x H x v 2

The static pressure for aircraft technical systems is denoted by ’bar’ and

mea-sured in decaNewtons per square centimeter ( daN/cm 2 ).

One bar is equal to one hundred thousand PASCAL

The STATIC PRESSURE for technical systems e g for

AIR-CRAFT HYDRAULIC SYSTEMS is denotet by

” bar ” and has the unit

daN

1 bar = 100 000 Pa

Sound waves are the same as pressure waves.

The speed of sound is the speed of the small pressure waves which occur

when you ring the bell

The speed of sound is denoted by ’a’.

In the formula for the speed of sound, the number twenty is an approximation

of the total of all the relevant constant values and ’T’ for temperature

repre-sents the only variable value

Note that the temperature must be expressed in Kelvin!

Now you know that the speed of sound depends on the temperature For ample if the temperature on a Summer day is 15E C, which is 288 K, then wecalculate the speed of sound to be 339.4 m/s

ex-If the temperature decreases in Winter to - 50E C, which is 223 K, then thespeed of sound is 298.6 m/s

The speed of sound is less at high altitudes because the temperature is lower

Now let’s see what happens if the source of the sound moves, for example if

we have an aircraft flying First we see an aircraft flying at a speed which is

below the speed of sound

You can see that the pressure wave moves ahead of the aircraft and also

be-hind it

Next we see an aircraft flying at the same speed as the speed of sound

The pressure wave cannot escape at the front of the aircraft and we get a big

pressure wave forming This pressure wave is known as a shock wave.

Finally we see an aircraft flying at a speed which is above the speed of sound

In this case the pressure waves increase behind the aircraft and shock waves

form outside the periphery of the pressure waves

Now you know that different aircraft speeds affect the sound waves

Below the speed of sound At the speed of sound Above the speed of sound

The pilot must know the relationship between the speed of the aircraft and the

speed of sound

On most aircraft the pilot must make sure that the speed of the aircraft is less

than the speed of sound

Now let’s see what happens when an aircraft flies at a constant speed but in

different temperatures In this example the aircraft is flying at a low altitude with

a speed of 300 m/s

You can see that the aircraft speed is below the speed of sound at this altitude

We assume the speed of sound is 330 m/s

Now the same aircraft is flying at an altitude of 10 km The aircraft continues to

fly with a speed of 300 m/s

At this higher altitude the temperature is lower and the speed of sound creases to 300 m/s

de-Now the aircraft is flying at the speed of sound and you can see that shockwaves are produced

A special indication known as the Mach number, ’M’ is used to keep the pilot

informed of the relationship between the speed of the aircraft and the speed ofsound

The Mach number is the speed of the aircraft divided by the speed of sound

In our example the aircraft flying at an altitude of 10 km has a Mach number of

one ( M = 1 ) A Mach number of one indicates that the aircraft is flying at the

speed of sound

300 M S

These graphics illustrate the three sound regions which are defined by the

Mach numbers In the subsonic region all speeds around the aircraft are below

the speed of sound This is the region up to the critical Mach number

In the transonic region some speeds around the aircraft are below the speed of

sound and some are higher than the speed of sound This is the region

be-tween the critical Mach number and 1.3 Mach

Finally we have the supersonic region Here all speeds around the aircraft are

higher than the speed of sound This is the region at Mach numbers higher

than 1.3 Mach

That’s all we have to say about the speed of sound in this segment You will

see more on this subject in the chapter for high speed flight

To understand aerodynamics we need to know something about the

atmo-sphere where flying happens

The atmosphere is the whole mass of air extending upwards from the surface

of the earth

Air is a mixture of several gases Pure, dry air has approximately 78% nitrogen,

21% oxygen and one percent other gases such as argon and carbon dioxide

For practical purposes it is sufficient to say that air is a mixture of four fifths

nitrogen and one fifth oxygen

The atmosphere has many layers

The troposphere is the lowest of these layers In the troposphere we have

clouds and rain and many different weather conditions

There are no rain clouds in the stratosphere and the temperature does not

change as the altitude increases

The tropopause is the name given to the boundary between the troposphere

and the stratosphere The tropopause has different heights around the earth It

is approximately eight kilometers over the north and south poles and sixteen

kilometers over the equator

21% Oxygen

78% Nitrogen

TROPOSPHERE

You know from watching the weather forecast that temperature, pressure and

density vary quite a lot in the troposphere

These variations must be reduced to a standard so that we have a basis for

comparing aircraft performance in different parts of the world and under varying

atmospheric conditions

In order to have a reference for all aerodynamic computations, the International

Civil Aviation Organisation ( ICAO ) has agreed upon a standard atmosphere

called ISA ( ICAO standard atmosphere) The pressure, temperature and

den-sity in the standard atmosphere serve as a reference only When all

aerody-namic computations are related to this standard, a meaningful comparison of

flight test data between aircraft can be made

Now let’s take a look at the temperature, pressure and density of the ISA at

sea level and at high altitudes

You can see the standard sea level values for temperature, density and

pres-sure Note that the standard altitude for the tropopause is eleven kilometers

Under standard conditions temperature decreases with altitude at a rate of

6,5E C per 1000m, or 2E C ( 3.5E F ) per 1000 foot

This gives a standard temperature of -56,5E C at the tropopause

There is no change in temperature in the stratosphere

The density and pressure decrease gradually with altitude

The graph shows the basic tendencies for temperature, pressure and density

You can find more precise information in the standard atmosphere tables which

you can usually find in the appendix of technical documentations

These are the ISA conditions for sea level:

Temperature T : 288 K = 15E C

Pressure P : 1013,25 hPa

In the subsonic region the speed is so slow that a flying body does not

com-press the air We say that the air is incomcom-pressible in the subsonic region

2.1 Continuity equation

Now let’s have a closer look at the behaviour of the air streamlines

You can see that the streamlines are parallel to each other if there is no

distur-bance

The airflow between the streamlines is similar to the flow in a closed tube You

will see later that we use the term stream tube

Here you see the flow pattern in a tube with different diameters

You can see that as the diameter gets smaller the streamlines move closer to

each other

At the lower picture we isolate the stream tube and identify two cross sections,

A1and A2 Assume that the area of the cross section at point A1is twenty

square centimeters and the velocity of the airflow at this point is 10 m/s

The area of the cross section at point A2is five square centimeters and thevelocity of the airflow at this point is 40 m/s

The continuity equation states that the velocity of the airflow is inversely proportional to the area of the cross section of the tube as long as den- sity remains constant !

For example if the area of the cross section is halved then the velocity of theairflow is doubled or if the area is four times smaller then the velocity is fourtimes greater

We use the term defuser outlet when the diameter increases and the velocitydecreases and the term jet outlet when the diameter decreases and the veloc-ity increases

In this segment we look at another important equation used in aerodynamics,

Bernoulli’s equation Here we will see, how speed effects pressure

We will describe this equation using a tube with a valve

You can see that the valve is closed and that the tube is filled with fluid on the

left side of the valve

Valve closed

The fluid inside the tube has a static pressure The static pressure is

repre-sented by the arrows in the tube and by a line on the graph at the bottom of the

picture

This static pressure acts in all directions

The total pressure is represented by the circle in the tube and by another line

on the graph at the bottom of the picture

You can see on the graph that the total pressure is equal to the static pressure

when the valve is closed

At the next steps, the valve will be opened slightly

Valve half open

When the valve is moved to the half open position the fluid begins to flow

You can see that the static pressure decreases and a new pressure, the

dy-namic pressure, is introduced Remember that the dydy-namic pressure only acts

in the direction of the flow

The dynamic pressure is represented by the horizontal arrows in the tube and a

line on the graph The graph shows the amount of static pressure, dynamic

pressure and total pressure in the half open position

Valve full open

Finally the valve is moved to the fully open position

Did you notice that the total pressure remained constant in all valve positions?

The static pressure decreased every time the valve was opened more and the

dynamic pressure increased as the valve opened

What you have seen is the physical law known as Bernoulli’s principle

The Bernoulli equation states that total pressure is always the sum of static pressure and dynamic pressure or in short hand notation: P tot equals p plus q !

The total pressure remains constant.

Ptot= p + q = const.

p = pstat; q = ½ H V2

VALVE CLOSED

Now let’s see how pressure is measured You know that the airflow around the

surface of this object has static pressure and dynamic pressure

At the point of stagnation the velocity of the airflow falls to zero and the static

pressure equals the total pressure You know that there is no dynamic pressure

if there is no flow

At the picture below you can see how we measure the static and dynamic

pres-sure when there is a velocity

The actual static pressure is sensed directly at the static port

The static pressure line and the total pressure line are attached to a differential

pressure gauge

The net pressure indicated on the gauge is the dynamic pressure As you know

the dynamic pressure is the total pressure minus the static pressure

The dynamic pressure varies directly with changes in density and with the

square of the change in velocity

If the density is constant, the dynamic pressure increases sixteen times if the

velocity increases four times

The dynamic pressure is the indicated air speed

In this segment we see how lift is produced We begin by looking at a special

design of tube known as a venturi tube

You can see that the inlet and the outlet of the venturi tube are the same size

The velocity of the airflow increases until it reaches the narrowest point in the

tube

You know that as the velocity increases the static pressure decreases and the

dynamic pressure increases

The velocity decreases again after the narrowest point and returns to the inlet

level by the time the airflow reaches the outlet

During this phase the static pressure increases again and the dynamic

If we remove the upper surface we find that the streamlines themselves

pro-vide the upper boundary The next step is to change the lower surface of the venturi tube into a profileand to add some streamlines below it

Now we have a surface with an area of low static pressure above it and area ofunchanged static pressure below it

This difference in static pressure acts on the surface to create the force which

we call lift

2.4 Magnus Effect and Circulation

Here you see the side view of a cylinder in an airstream

The static pressure on the upper surface of the cylinder is the same as the

static pressure on the lower surface

If there is no differential pressure, there is no lift !

Let’s see what happens if we rotate the cylinder

When the cylinder rotates the circulatory flow causes an increase in local

veloc-ity on the upper surface of the cylinder and a decrease in local velocveloc-ity on the

lower surface

This generates lift

This mechanically induced circulation is called the Magnus effect

You can see that the circulatory flow produces what we call an up wash mediately in front of the cylinder and a down wash immediately behind the cyl-inder

im-You can also see that the fore and aft neutral streamlines are lowered

Circulation around a profile

If the cylinder in the flow will be replaced by a profile, we will get the same

ef-fect as for the cylinder with circulation

A velocity difference between the upper and lower profile surface will be

ob-tained and lift will be created

This lift will be normal to the direction of flow, as for the Cylinder

This profile also generates a circulation which produces an up wash and a

down wash

There is no lift without circulation !

3 Profile and wing geometry

In this chapter we look at the geometry of a wing and a profile This is

impor-tant for our understanding of lift and drag

In the first segment we look at profile geometry and in the second segment we

look at wing geometry

3.1 Geometry of a profile

As you can see a profile is a cross section of a wing

It is sometimes called an airfoil

Cord line, Leading edge, Trailing edge

The profile has a leading edge and a trailing edge

The cord line is a straight line connecting the leading edge and the trailingedge

The mean camber line is a line drawn half way between the upper and the

lower surfaces of the profile

The shape of the mean camber line is very important in determining the

aero-dynamic characteristics of a profile

The end points of the mean camber line are the same as the end points of the

The maximum camber and the location of the maximum camber help to define

the shape of the mean camber line

These quantities are expressed as a fraction or a percentage of the basic cord

dimension

A typical low speed profile might have a maximum camber of 5 % located 45 %

aft of the leading edge

For example a typical low speed profile might have a maximum thickness of

18 % located 30 % aft of the leading edge

Thickness

The relative wind is the speed and direction of the air acting on the aircraft

which is passing through it

You can see that the relative wind is opposite in direction to the flight path

In this segment we look at wing geometry The wing area is the plan surface

area of the wings

It includes the area of the fuselage which is between the wings

On this simplified graphic the wing area S, is the wing span b, multiplied by the

cord of the wing c

On this more realistic tapered wing we have different wing cords You can seethat the root cord Cr, is the cord at the wing centerline and the tip cord Ct, is thecord at the wing tip

C

Taper ratio λ

The taper ratio λ ( lambda ), is the ratio of the tip cord to the root cord

l = Ct/Cr

The wing area is the average cord multiplied by the wing span

The average cord C, is the geometric average of all the cords and the wingspan b, is measured from tip to tip

The aspect ratio is the wing span b, divided by the average cord C.

Typical aspects ratios vary from 35 for a high performance sail plane, to 3.5 for

a jet fighter plane

You can see, that the aspect ratio can also be expressed as the wing span

squared divided by the wing area

The sweep angle is the angle between the quarter cord, or the 25 % line and

the pitch axis

Positive sweep = Backwards !

Negative sweep = Forewards !

4 Lift and drag

In this Chapter we look in more detail at the factors affecting the lift first the

angle of attack and then the shape of the profile

After that we will have a look at the factors affecting the drag

At the end we will see how lift and drag are represented in the polar diagram

You know that the main function of a profile is to provide lift so that the aircraft

can overcome the force of gravity and rise into the air

You will see that the design of the profile is very important

4.1 Introduction

Here you see the distribution of static pressure on a profile The dark area in

front of the leading edge, is where the static pressure is higher than the

ambi-ent static pressure

This is because the velocity of the air approaching the leading edge, slows to

less than the flight path velocity The static pressure is highest at the point of

stagnation where the air comes to a stop

In the lighter areas above and below the profile, the static pressure is lower

than the ambient static pressure This is because the air speeds up again as it

passes above and below the profile so that the local air velocity is greater than

the flight path velocity

We have maximum air velocity and minimum static pressure at a point near the

maximum thickness of the profile

The air velocity decreases and the static pressure increases after this point

In the dark area at the trailing edge the static pressure is higher than the ent static pressure

ambi-This is caused by low velocity turbulent air in this area

The aerodynamic force is the resultant of all forces on a profile in an airflowacting on the center of pressure

The aerodynamic force has two components lift which is perpendicular to therelative wind and drag which is parallel to the relative wind Here the center ofpressure is identified This is the point on which all pressures and all forces act.This point is located where the cord of a profile intersects with the resultant ofthe aerodynamic forces lift and drag

Aerodynamic Force

The aerodynamic forces of lift and drag depend on the combined effect of

many variables the dynamic pressure the surface area of the profile the

shape of the profile and the angle of attack

Aerodynamic Force

Now we look at how to calculate the lift You might think that this is simple all

we need to know about is the surface and the pressure

However it’s not as easy as you might think In reality a profile has different

pressures because of different angles of attack

First let’s look at the simple calculation of theoretical lift

The theoretical lift is the dynamic pressure multiplied by the surface area You

know from an earlier lesson that the dynamic pressure is half the air density

multiplied by the velocity squared

Theoretical Lift = ½ xH x V2 x A

In this example we assume that the air density is 1,225 kg/m3and the air locity is 28 m/s and the surface area of the profile is 0,05 m2and we get atheoretical lift of 24 N

ve-H = 1,225 kg/m 3

V = 28 m/s

A = 0,05 m 2

Theoretical Lift = ½ x 1,225 x 28 2 x 0,05 = 24 N

You can see that a universal joint provides the bearing for this construction.

There are two scales attached to the support arm a horizontal scale to

mea-sure the drag and a vertical scale to meamea-sure the lift

Now let’s see what happens when we switch on the wind tunnel

You can see that the measured lift is only 8,4 N This is much less than the

theoretical lift of 24 N

The theoretical lift must therefore be adjusted

A coefficient of lift CL, is introduced to the lift equation to account for the

differ-ence between the measured lift and the theoretical lift

The coefficient of lift is the measured lift divided by the theoretical lift In our

example it is 0,34

The lift equation is now the coefficient of lift multiplied by the dynamic pressure

multiplied by the surface area

Coefficient of Lift = Measured Lift Theoretical Lift

Dynamic Pressure q

V

4.1.2 Drag Equation

For the same reasons a coefficient of drag CD, is introduced to the drag

equa-tion to account for the difference between measured drag and theoretical drag.The coefficient of drag is the measured drag divided by the theoretical drag.The drag equation becomes the coefficient of drag multiplied by the dynamicpressure multiplied by the surface area

Dynamic Pressure q

Coefficient of Drag = Theoretical Drag Measured Drag

V

4.2 Factors Affecting Lift

4.2.1 Angle of Attack ( AOA ) =

You know that the coefficient of lift is the ratio of the measured lift to the

theoretical lift

The coefficient of lift is a function of the angle of attack and of the shape of the

profile

We look at the effect of the angle of attack in this segment

In this wind tunnel experiment you will see that each angle of attack produces a

different measured lift and therefore a different coefficient of lift

The vertical scale will show the coefficient of lift as the angle of attack changes

The relationship between the angle of attack and the coefficient of lift will be

plotted on the graph

Now you can see what will hapen, when the angle of attack varies between

−8E to 20E

Remember to observe the coefficient of lift on the scale and the relationship

between the angle of attack and the coefficient of lift on the graph

You can see on the graph that the coefficient of lift increases up to the

maxi-mum coefficient of lift, CL max, and then decreases again

The maximum coefficient of lift corresponds to the maximum angle of attack,

αmax

If the angle of attack increases above=max,the airflow cannot follow the upper

surface of the profile and an airflow separation, known as stall occurs

= = − 8E

= = 0E