Aerodynamic 1

PART

I

FUNDAMENTAL PRINCIPLES （基本原理）

In part I, we cover some of the basic principles that apply to aerodynamics in general. These are the pillars on which all of aerodynamics is based.

Chapter 1 LIFT AND DRAG Section 1 Characteristics of Airflow 1. Airflow and relative airflow Airflow means moving stream of air. The motion of air relative to an object is called relative airflow, which is equal to the object movement in speed and opposite in direction. The force between an object and the airflow caused by the relative motion is called aerodynamic force

2. Streamline and streamline pattern Streamline means the well-defined continuous paths of the particles of the airflow in a steady air stream.

reamline pattern is a diagram composed of many streamlin e streamline pattern can be directly observed in the smoke nd tunnel test.

The air particles between the two streamlines seem to flow along the tube, usually the tube composed of streamlines is known as stream tube.

Streamline pattern displays the flowing of the air stream. In the low-speed airflow, the streamline pattern depends only on an object’s shape and its relative position in the airflow.

(1) Effect of the shape of an object on the streamline pattern

The feature is that the stream tube is thicker at the front of an object, thinner at the upper and lower convex parts of the object and an eddy zone is formed behind it.

(2) Effect of the relative position of an object in the airflow on the streamline pattern Wing chord: the joining line from the leading edge to the trailing edge of an aerofoil. The angle between the wing chord and the direction of the relative α airflow (or the aircraft moving direction) is called angle of Through )test it can be seen that the streamline attack,( pattern varies with the relative position of an object with the same shape in the airflow.

3. Principle of airflow continuity When the air passes continuously and steadily through a convergent-divergent pipe, because the air in any part of the pipe can neither break off nor squeeze there, the air mass flowing into section 1 and that flowing out of section 2 at the same unit time is equal, that is to say, the air mass(i.e. the airflow)passing through each section per second is equal. This is known as the principle of the airflow continuity.

Where :

ρ=Air density

V=Speed of air passing through the section A=Sectional area

The above formula shows that the density of the air flowing a low speed remains almost constant and the speed of the airflow is in inverse proportion to the area of the section it ha lown, i.e. the air flows faster at the convergent part than the divergent part .

4. Bernoulli’s theorem Bernoulli’s theorem, defining the relationship between the pressure and the flowing speed of the fluid in motion, is one of the basic theorems for studying the characteristics of the airflow. The change of air pressure with flowing speed can also be explained by the relations of the static pressure, dynamic pressure and pitot pressure.

tatic pressure(p): The static pressure means the pressure tha the air acts on the objects surface. For example, the atmospheric pressure is a manifestation of static pressure.

Dynamic pressure(q): It exists in the flowing air, which can be turned into static pressure and applied on the surface of an object only when the airflow is obstructed and its speed ρ reduced. The magnitude of dynamic pressure is directly proportional to the air tot pressure(p0): it is the sum of static pressure and dynamic density( ) and to the square of pressure in the airflow. airflow velocity(V), and its value is .

In the steady airflow, the sum of static pressure and dynamic pressure of air on every section of the same stream tube maintains constant. This constant value is just the pitot pressure of the air. The above relations can be expressed by the following formula:

That is to say, the air pressure is low at the place where the air flowing speed is fast, and vice versa. This is the content of Bernoulli’s theorem.

Section 2

Lift

Lift is a kind of aerodynamic forces, supporting the aircraft in flight.

1. Generation of lift From the streamline pattern of the air flowing over the wing(Fig.1-7), it can be seen that the air is divided into the upper and lower parts at the leading edge of the wing, and flowing over the upper and lower surfaces of the wing respectively and, joining together again at the trailing edge of the wing and flowing backward.

Because the upper surface of the wing is somewhat convex, so the stream tube becomes thinner, the air flowing speeds up and the pressure drops; the stream tube on the lower surface of the wing becomes thicker, the flowing speed decreases and the pressure increases. A pressure differential perpendicular to the moving direction is therefore produced between the upper and lower wing surfaces and forms the lift. Wing lift is thus the sum total of the pressure differential between all the upper and lower sections of the wing. The intersection point of the lift and wing chord is called ----center of pressure.

The direction of lift is perpendicular to that of the aircraft movement. And because the left and right wings are symmetric, the lift lift is is therefore within the aircraft symmetry The wing aircraft equal the divides the aircraft into two of the left and plane in (which normal flight. right symmetric sections)as shown in Fig.1-8. To understand further the magnitude of force borne by the each position of the wing, it is necessary to know the pressure distribution on the wing surfaces. The differences between on each point of the wing section and atmospheric pressure can be measured by way of test, which are drawn out along the vertical

direction of wing section by using the line segments with arrows. All the pressure being lower than the atmospheric is called suction or negative pressure, which is indicated with the direction of arrow facing outward. All the pressure being higher than the atmospheric pressure is called positive pressure, which is indicatedtowith pointing thethe wing direction surface.ofAnd then, link the endarrow of all arrows together by using a smooth curved points line and, a pressure distribution diagram is thus constituted. The point at which the pressure is the lowest(i.e. suction is maximum)on the diagram is called minimum pressure point. At the leading edge the flowing speed is zero, i.e. a point at which the pressure

From the pressure distribution diagram of the wing, it can be seen that the aircraft lift is produced primarily by depending on the effect of suction of the upper wing surface but not mainly depending on the effect of the positive pressure on the lower wing surface.

2. The factors affecting lift (1) Effect of angle of attack on lift

Within the extent of the angle of attack, lift increases with the increase in the angle of attack. Because with the increase of the angle of attack, the bending of the streamlines at the front of the upper wing surface increases, causing the stream tube to contract more, the flowing speed to increase and the pressure to reduce further(the suction becomes greater). At the same time, the stream tube under the wing become thicker, the flowing speed decreases and the pressure increases(Fig.1-11 A.B.). The pressure differential between the upper and lower wing surfaces increases, the lift is therefore increased. When the angle of attack increases to a fixed extent, the lift will increase to the maximum(Fig.1-11 C). If the angle of attack is increased beyond that angle, on the contrary, the lift will decrease(the causes will be analysed in section

(2) Effect of flying speed on lift The higher the flying speed, the greater the lift. Experiment proves that if the speed is doubled, the lift will be quadrupled; and the speed is tripled, the lift will be increased to nine times, i.e. the lift is directly proportional to the square of the flying speed.

(3) Effect of air density on lift Lift increases with the increase in air density. Experiment proves that when the air density is doubled, and so will be the lift, i.e. the lift is directly proportional to the air density.

(4) Effect of the wing area on lift The bigger the wing area is, the greater the area for producing the suction on the upper wing surface and the positive pressure on the lower wing surface will be and so is the sum total of pressure differential between the upper and lower wing surfaces, and the lift is therefore great. The lift is directly proportional to the wing area.

(5) Effect of the shape of airfoil on lift

The lift varies with the shape of airfoil(airfoil in short).

For example, comparing the flat-convex wing with biconvex wing, of which their thickness and length of wing chord are the same, under the condition of the same angle of attack, because the bending of upper surface of the flat-convex wing is great, the stream tube on it is thus rather thin, flowing speed is great and the pressure is comparatively low, while the stream tube on the lower surface is rather thick, so the flowing speed is low and the pressure is comparatively high. Therefore, the lift of the flatconvex wing is greater than that of the biconvex wing.(Fig1-12)

3. The formula and coefficient curve of lift In accordance with the above analysis, the lift (L) ρ ), air α depends on the angle of attack( ), flight speed(V density( ),wing area(S ) and airfoil. The summarization of its changing law gives us the following lift formula:

Where :

is the lift coefficient.

The lift coefficient is determined through test. It represents comprehensively the effect of angle of attack and airfoil on the lift.

Usually , an aircraft does not change its airfoil during flight. At this time, the change of the lift coefficient is determined by the magnitude of angle of attack. The lift coefficients of different angles of attack can be obtained through the wind tunnel test. The lift coefficients of different angles of attack for Type-6 Primary Trainer are listed below: α -1 0 2 6 8.9 10 14 16 18 19 20 o

0

o

0.07 3

o

0.22 2

o

0.52

o

0.73 8

o

0.82 2

o

1.12

o

1.27

o

1.36

o

1.37

o

1.35

According to the lift coefficients and angles of attack given in the above table, a curve of the lift coefficient for the aircraft can be drawn, as shown in Fig.1-13.

From the curve of lift coefficient, we can see that: (1) Zero-lift angle of attack: The angle of attack with zero lift is called zero-lift angle of attack, which is equivalent to the angle of attack corresponded by the intersection point between the curve and axis of abscissa in the (2) Critical angle of attack: graph. The angle of attack with maximum lift coefficient is called critical angle of attack, i.e. the angle of attack corresponded by the highest point of the curve in the graph.(19o for Type-6 Primary Trainer).

(3) The changing law of the lift coefficient with angle of attack: The lift coefficient increases with the increase of angle of attack starting from zero-lift angle of attack, and reaches its maximum value at the critical angle of attack, and changes to rapid decrease and the curve bends downward with the increase of angle of attack exceeding the critical angle of attack.

Section 3

Drag

Drag is an aerodynamic force too, which resists the aircraft from moving forward and its direction is opposite to that of the aircraft motion.

1. Generation of drag There are mainly three kinds of drags produced by the aircraft during flight. (1) Friction drag Viscosity is the tendency of air. During flight, the air adhering to the aircraft surfaces over which it is flowing will rub against the aircraft surfaces, resulting in a drag which is called friction drag.

(2) Pressure drag The air must meet obstruction at the leading edge of the wing of an aircraft during flight, its speed is reduced and pressure increased; and at the trailing edge, there is a generation of eddy zone where the air pressure is reduced, therefore a pressure differential is created between the leading and trailing edges, forming a drag which is called pressure drag(Fig.1-14). By analogy, the pressure drag can also be produced on the fuselage and tail, etc.

(2) Lift dependent drag The lift dependent drag is caused by lift, or in another word, it is “induced” by the generation of lift on the wing, which is therefore called lift dependent drag. When the wing is producing lift, the pressure on the lower surface is higher than on the upper surface, so the air caused to spill around the wing tips from the lower surface to the upper surface, the air on the wing tip is thus made to twist, as a result, the tip eddy is formed(Fig.1-15).

The tip eddy makes the air flowing over the wing produce a downward speed which is called downwash speed. The resultant speed of the downwash speed and head-on airflow speed inclines downward. This airflow with the speed direction inclined downward is called downwash flow. Affected by the downwash speed, the head-on airflow will incline downward and the lift produced will slant rearwards accordingly, as L’ shown in Fig.1-16. With regard to the head-on airflow(moving direction), the parallel component force of L’ will also play a role of drag(Dinduced in the figure) with exception of its vertical component L which acts as lift, and this drag is called lift dependent drag.

2. The factors affecting the drag and drag formula Because both lift and drag are the aerodynamic forces, the factors affecting the drag are basically the same as those affecting the lift, that is to say, the drag is directly proportional to the square of the speed, air density and wing area; it is also related to the angle of attack, wing shape, etc. The drag can be expressed by the following formula:

Where:

is the drag coefficient.

3. The drag coefficient and its curve The meaning of the drag coefficient is basically the same as that of the lift coefficient. It expresses comprehensively the effect of factors of the angle of attack and aircraft shape on the drag. To an aircraft, usually its configuration is constant, so the drag coefficient depends on the magnitude of the angle of attack.

The changing law of the drag coefficient with the angle of attack can be expressed by the curve of the drag coefficient as shown in Fig.1-17. From the curve of the drag coefficient, we can see that the same amount of increase in the angle of attack, the drag coefficient increases less in the range of low angle of attack, but in the range of high angle of attack, the drag coefficient increases more. because both After exceeding the and critical angle of attack, the drag lift dependent drag coefficient will would increase much more and, this is because pressure drag also the eddy zone expands rapidly and the pressure drag increase; increases with the increase of angle of attack.

Section 4

Variation of Aerodynamic Force of the Aircraft After Exceeding the Critical Angle of Attack

1. Boundary layer When the air is flowing over the wing, there will be an air flowing layer with the gradual reduction of airflow speed because of the viscous friction between the air and object surface. This flowing layer is known as the boundary layer.

2. Phenomenon of airflow separation

The air flows from over the minimum pressure point on the upper wing surface to the wing trailing speed goes down progressively, the travelling pressure edge, the airflow will become greater and greater accordingly and so does the pressure in the boundary layer. Hence, under the action of reverse pressure differential, the flowing of air in the boundary layer will be retarded so the speed is caused to reduce gradually, and even the air in the boundary layer at the rear of the wing might be forced to flow back. As a result, the adverse flowing air and rearward flowing air in the boundary layer will collide themselves, causing the airflow in the boundary layer to separate from the wing surface ,an airflow separation phenomenon is formed and many eddies produced.

At the low angle of attack, the effect of reverse pressure differential in the boundary layer is less, the adverse flow is weak and the separation point is therefore close to the trailing edge; but with the increase in the angle of attack, the reverse pressure differential in the boundary layer will also increase and the adverse flow becomes stronger, the separation point is thus made to move forward and the eddy zone to expand forward; after exceeding the critical angle of attack, the reverse pressure differential will become even greater and the separation point will move forward suddenly and greatly and, the eddy zone will expand forward rapidly as well.

1. Reasons for the reduction of the lift coefficient and sudden increase of the drag coefficient after exceeding the critical angle of attack After exceeding the critical angle of attack, the airflow separation point suddenly moves forward to the place near the wing leading edge and the eddy zone becomes greater. At this moment, the stream tubes at the front of the upper surface become thicker, the flowing speed decreases and the suction force reduces (comparing C with D in Fig. 1-11) and this is of a help to reducing the lift coefficient, but in the eddy zone at the rear of the wing, the suction force increases slightly and it is of a help to increasing the lift coefficient. It is obvious that the effects of the above two factors on the lift coefficient are contradictory to each other.

But after exceeding the critical angle of attack, the suction force at the front of the upper wing surface drops rather more, thus its effect on the lift coefficient is principal, the lift coefficient is therefore made to reduce. After exceeding the critical angle of attack, the pressure is reduced slightly within a certain range of the wing trailing edge; but the flowing speed at the front of the lower wing surface is very low and the pressure is increased because of the increase in the angle of attack. Therefore, the pressure differential between the leading and trailing edges of the wing is increased and the drag coefficient will gain a sudden increase.

Section 5

Lift-drag Ratio

Lift-drag ratio (K) is the ratio between lift and drag at the same angle of attack. The higher the ratio is, the less the drag will be in a condition to obtain the same lift. The formula of the lift-drag ratio is:

From the above formula, we can see that the lift-drag ratio is the ratio between the coefficients of lift and drag at the same angle of attack.

Because the coefficients of lift and drag vary mainly with the angle of attack, so does the lift-drag ratio. The proved lift-drag ratio for Type-6 Primary Trainer at different angles of attack is as follows:

α

-1o 0o 0 0.07 3

2o

6o

0.22 2

0.52

0.03 6

0.07 3

0.04 1

0

2

5.43

8 10 14 16 18 19 20 o o o o o o o .9 0.73 0.82 1.12 1.27 1.36 1.37 1.35 8

2

0.06 4

0.08 6

0.09 7

8.08

8.56

8.51

0.14 4 7.78

0.17 8

0.21 6

0.23 7

0.26 1

7.15

6.30

5.78

5.17

From the above table it can be seen that: When the angle of attack increases progressively from low to high, the increased multiples of the lift coefficient are more than that of the drag coefficient, so the lift-drag ratio increases gradually and to the maximum when the angle of attack increases to 8.9o, and any further increase in the angle of attack would

First flight of Wright brothers Dec. 17, 1903

Wilbur and Orville Wright's Wright Flyer was the first successful airplane. On December 17, 1903, at Kitty Hawk, North Carolina, Orville Wright flew the first heavier-than-air machine in a powered, controlled, and sustained flight. The Flyer, constructed of wood, wire, and muslin, went a distance of 120 feet in 12 seconds. It was a tremendous success, coming from a long series of aeronautics experiments that the Wright Brothers started in 1899 with a kite.

At the rear of the 1903 Wright Flyer one finds a pair of pusher propellers. The propellers are long, thin, twisted pieces of wood which are spun at high speed. Control of roll: WING WARP

Overview of Wright Brothers Discoveries

Aerodynamic heating of the reentry vehicle ICBMs reentry the atmosphere at the speeds of from 6 to 6.7km/s. The aerodynamic heating of the reentry vehicles becomes severe, the cover of the war head will be heated up to 10,000K. Blunt reentry body design can minimize the aerodynamic heating problem.

1.2 Aerodynamics:Classification and Practical Objectives ( 空气动力学：分类和应用目标） Distinction of solids, liquids, and gases Practical applications in engineering

Solids, liquids, and gases in a container

The solid object will not change: its shape and boundaries will remind the same.

The liquid will change its shape to conform to that of the container and will take on the same boundaries as the container up to the maximum depth of the liquid.

The gas will completely fill the container, taking on the same boundaries as the container.

Solid and “fluid”(a liquid or a gas) under a tangential force == deformation 固体和 流体在 受到 剪应力 时， 各自形 状所 发生的 变 化方式 截然不 同。 Under a force applied tangentially to the surface of a solid body, the solid body will undergo a finite deformation, and the tangential force per unit area— the shear stress—will usually be proportional to the amount of deformation. If the case happens for a fluid, then, the fluid will experience a continuously increasing deformation and the shear stress will usually be proportional to the rate of the deformation.

Solid : fluid:

τ ∝θ τ ∝ θ

τ: θ: θ :

Shear stress 剪应力 Deformation 变形 Rate of deformation 变

Mechanics distinction of solids, liquids, and gases

Distinction of solids, liquids, and gases respects to the intermolecular forces Fluid dynamics is subdivided into three areas: Hydrodynamics --- flow of liquids Gas dynamics --- flow of gases Aerodynamics --- flow of air

Practical objectives of Aerodynamics 1. The prediction of forces and moments on and heat transfer to, bodies moving through a fluid. 2. Determination of flows moving internally through ducts 3. External aerodynamics 4. Internal aerodynamics

1.3 Road Map of this chapter What’s the usage of the road map  At the beginning of each chapter,

road map give you the sense for you get to know where you are, where you are going, and how can you get there  Show the interrelationship of the materials in the chapter  At the end of the chapter, after you look back over the road map, you

1.4 Some fundamental Aerodynamic Variables 1. Aerodynamic variables are something like technical vocabulary for the physical science and engineering understanding 2. First introduced aerodynamic variables: pressure,density,temperature, and flow velocity

The velocity description of a fluid is quite different to that of a solid body.

Velocity of a flowing gas at any fixed point B in space is the velocity of a small fluid element as it sweeps through B.

1.5 Aerodynamic forces and moments Aerodynamic forces and moments on a moving body are due to only two basic sources: 1. Pressure distribution over the body surface 2. Shear stress distribution over the body surface Both pressure and shear stress have dimensions of force per unit area.

pressure acts normal to the body surface. shear stress acts tangential to the surface.

The net effect of the pressure and shear stress distribution results in a aerodynamic force R and moment M on the body.

 The resultant force R can be split into components

L = lift : component of R perpendicularV∞to D = drag : components of R parallel V to ∞ (wind system) N = normal force : component of R perpendicular to c A = axial force : components of R parallel to c (body system)

After the pressure and shear stress distributions being defined, and the geometry shape of the body being known, the resultant aerodynamic force can be obtained by the integration of the pressure and shear stress distributions along the surface of the body.

From Eqs. (1.7),(1.8) and (1.11), we can see clearly, that the sources of the aerodynamic lift, drag, and moments on a body are the pressure and shear stress distribution integrated over the body. The basic task of theoretical aerodynamics is to calculate p(s) and τ(s) for a given body shape and freestream conditions, and then obtain the aerodynamic forces and moments with the use of Eqs. (1.7),(1.8) and (1.11)

 Dimensionless aerodynamic force and moment coefficients are even more important than the aerodynamic forces and moments. Definition ofρ ∞ and V∞ density and velocity in the freestream, which is far ahead of the body. Definition of dynamic pressure

1 2 q∞ = ρ ∞V∞ 2 The dynamic pressure has the unit of pressure

Definition of dimensionless force and moment coefficients Lift coefficient:

L CL = q∞ S

D Drag coefficient: C D = q∞ S N Normal force coefficient: C N = q∞ S

N Axial force coefficient: C N = q∞ S Moment coefficient:

S : reference area l : reference length

M CM = q∞ Sl

Definition ofS andl may be different for different shapes of the body being concerned.

The symbols in capital letters, such as

C L , C D , C M , C N and C A

represents the force and moment coefficients for a three-dimensional body. The symbols in lowercase letters

cl , cd and cm denote the force and moment coefficients for a two-dimensional body '

'

'

L D M cl = , cd = , cm = 2 q∞ c q∞ c q∞ c '

'

'

S = c(1)

L , D , M are force and moments per unit span

Two additional dimensionless quantities of immediate use are Pressure coefficient

Skin friction coefficient

p − p∞ Cp = q∞

τ cf = q∞

Where p∞ is the free stream pressure

1.6 Center of pressure （压力 中心） The center of the pressure is a point on the body about which the aerodynamic moment contributed by the pressure and shear stress distributions is equal to zero.

' M If LE is defined as the moment generated by ' the distributed loads,Nand is the

component of the resultant force, then the xcpcenter must be located pressure ' edge downstream of the leading M

xcp = −

LE '

N

L' ≈ N ' If the angle of attack is small, ' LE '

M xcp = − L

,thus

It is clear to see that as lift approaches to zero, the center of pressure moves to infinity. So, the center of pressure is not always a convenient concept in aerodynamics. There are other ways to define the force-andmoment system on an airfoil

M

' LE

= −L c 4 + M '

' c4

= − xcp L

'

1.7 Dimensional analysis: The Buchingham PI theorem （量纲分析： PI 定理） ※What physical quantities determine the variation of the aerodynamic forces and moments? On a physical, intuitive basis, we expect R is depend on: 1. Freestream velocity 2. Freestream density 3. Viscosity of the fluid 4. The size of the body 5. The compressibility of the fluid

R = f ( ρ ∞ , V∞ , c, µ ∞ , a∞ )

(1.23)

※ How to find a precise functional relation for the equation above? Execute huge amount of wind tunnel experiment might be one way. Is there any other way can do more effectively? Method of dimensional analysis ※An obvious fact for the dimensional analysis

ψ +η + ζ = φ All the terms in this physical relation must have the same dimensions

※Buckingham PI theorem 1. Let K to be the number of fundamental dimensions required to describe the physical P1 , P2 , , PN variables 2. Let represent N physical variables in the physical relation

f1 ( P1 , P2 ,  PN ) = 0

3. Then the physical relation can be reexpressed as a relation of (N-K) dimensionless products.

f 2 (Π1 , Π 2 , Π N −K ) = 0

4. Every product is a dimensionless product of a set of K physical variables plus one other physical variable.

Π1 = f 3 ( P1 , P2 , PK , PK +1 ) Π 2 = f 4 ( P1 , P2 ,  PK , PK + 2 )  Π N − K = f 5 ( P1 , P2 ,  PK , PN ) 5. P1 , P2 , PK is called repeating variables. These variables should include all the K dimensions used in the problem.

※Aerodynamic force on a given body at a given angle of attack. 1. Eq. (1.23)

R = f ( ρ ∞ , V∞ , c, µ ∞ , a∞ )

(1.23)

can be expressed as

g ( R, ρ ∞ , V∞ , c, µ ∞ , a∞ ) = 0

(1.27)

2. Following Buckingham theorem and our physical intuition, the fundamental dimensions are m,l and t . Hence, K=3

3. The physical variables and their dimensions are −2 −3

[ R ] = mlt , [ ρ ∞ ] = ml , [V∞ ] = lt , −1

[c] = l , [ µ ∞ ] = ml t , [a∞ ] = lt −1 −1

−1

and N=6 4. As explained by Buckingham theorem, Eq.(1.27) can be reexpressed in terms of NΠ K=3 dimensionless products, that is

f 2 (Π 1 , Π 2 , Π 3 ) = 0

(1.28)

5. Now, we choseρ ∞ ,V∞ , c as repeating Π variables, from Eq.(1.26), these products are

Π1 = f 3 ( ρ ∞ , V∞ , c, R ) Π 2 = f 4 ( ρ ∞ , V∞ , c, µ ∞ ) Π 3 = f 5 ( ρ ∞ , V∞ , c, a∞ ) 5. Assume

Π1 = ρ V c R d b e ∞ ∞

in dimensional form

[ Π1 ] = (ml

−3 d

−1 b

e

−2

) (lt ) (l ) (mlt )

6. As Π1 is dimensionless, then

for m : for l : for t :

d +1 = 0 − 3d + b + e + 1 = 0 −b−2 = 0

7. The above Equations give d=-1,b=-2,and e=2 , then we have

Π 1 = Rρ V c

−1 − 2 − 2 ∞ ∞

R = 2 2 ρ ∞V∞ c

or

R

R Π1 = = 1 q S 2 ∞ ρ ∞V∞ S 2

where S is defined as reference area Π1 is a force coefficient, definedCas R

8. In the same way, we can obtain the remaining products as follows

ρ ∞V∞ c Π2 = µ∞

Reynolds Number 雷 诺数

V∞ Π3 = a∞

Mach Number 马赫数

9. Inserting all theΠ (1.28) R

products into Eq.

ρ ∞V∞ c V∞ f2 ( , , )=0 2 0.5 ρ ∞V∞ S µ ∞ a∞

or or

f 2 (C R , Re, M ∞ ) = 0 C R = f 6 (Re, M ∞ )

10. Important conclusion: In the general function form, R is expressed with five independent physical variables After our dimensional analysis, R can be expressed with only two independent variables • R can be expressed in terms of a CR dimensionless force coefficientM ∞ • is a function of only Re and M∞ 11. Important applications of Re and . similarity parameters

12. As lift and drag are components of the resultant force, then the lift and drag M∞ coefficients are also functions of only Re and .

C L = f 7 (Re, M ∞ )

C D = f 8 (Re, M ∞ ) Moreover, a relation similar to aerodynamic forces holds for aerodynamic moments, and dimension analysis yields

CM = f 9 (Re, M ∞ )

13. If the angle of attack is allowed to vary, then, the lift, drag and moment coefficients will in general depend αon the value of .

C L = f10 (Re, M ∞ , α ) C D = f11 (Re, M ∞ , α ) CM = f12 (Re, M ∞ , α )

14. Other similarity parameters associated with thermodynamics and heat transfer. Physical variables should be added temperature, specific heat , thermal conductivity, temperature of the body surface Fundamental dimension should be added unit of the temperature(K) Similarity parameters created

c p cv

Tw T∞

Pr = µ ∞ c p k∞

1.8 Flow similarity （流动相似） ※Definition of flow similarity Different flows are dynamically similar if: 3. The streamline patterns are geometrically V V∞ , p p∞ , T T∞ , similar 2. The distributions of etc., throughout the flow field are the same when plotted against common nondimensional coordinates. 3. The force coefficients are the same

※Criteria to ensure flow similarity 2. The bodies and any other solid boundaries are geometrically similar for both flows. 2. The similarity parameters are identical for both flows. 3. Reynolds and Mach number are the most dominant similarity parameters for many aerodynamic problems. ※Examples 1.4 and 1.5

1.9 Fluid Statics ： Buoyancy Force ( 流体静力学：浮力）

Skipped over

1.10 Types of Flow （流动类型 ） 1. The purpose for categorizing different types of flow. 2. The strategy to simplify the flow problems. 3. Itemization and comparison of different types of flow, and brief description of their most important physical phenomena.

1.10.1 Continuum versus free molecule flow 1. 2. 3. 4.

Definition of mean-free path . λ Continuum flow . λ << d Free molecule flow λ ≈ d In most aerodynamic problems, we will always treat the fluid as continuum flow.

1.10.2 Inviscid versus viscous flow 1. The random motion of the molecule will transport their mass, momentum, and energy from one location to another in the fluid. This transport on a molecule scale gives rise to the phenomena of mass diffusion, viscosity, and thermal conduction. All real flows exhibit the effect of these transport phenomena; such flows are call viscous flows. 2. A flow that is assumed free with all these phenomena above is called inviscid flow . 3. Inviscid flow is approached in the limit as the Reynolds number goes to infinity.

friction, thermal conduction, and diffusion is limited in the boundary layer.

5. The inviscid theory can be used to predicts the pressure distribution and lift. However, it cannot predicts total drag. 6. Flows dominated by viscous effects.

Flow around airfoil at high angle of attack

Flow around blunt body

7. No inviscid theory can independently predict the aerodynamics of such flows.

1.10.3 Incompressible versus compressible Flows ρ • A flow in which the density is constant is called incompressible. In contrast, a flow where the density is variable is called compressible. 2. All the flows are compressible, more or less 3. There are a number of aerodynamic problems that can be modeled as being incompressible without any detrimental loss of accuracy.

the Mach number of the flow.

1.10.4 Mach number regimes 1. Local definition

Subsonic if

M <1

Sonic if

M =1

Supersonic if M > 1 M Where is the local Mach number at an arbitrary point in a flow field.

2. Definition for whole flow field

3. Block diagram categorizing the types of aerodynamic flows

1.11 Applied aerodynamics: The aerodynamic coefficients — Their magnitude and variations • Difference between the fundamentals and applications of aerodynamics. 2. Aerodynamic coefficients, such as lift, drag, and moment coefficients, are the primary language of application external aerodynamics. 3. Typical values for the aerodynamic coefficients for some common aerodynamic shapes and it’s variation with Mach number and Reynolds number.

4. Some typical drag coefficients for various aerodynamic configurations in low speed flows.

C D = D q∞ S '

S = d (1)

 Comparison through case a to c : the Reynolds numbers for all these three cases are the same based on d (diameter). the wakes are getting smaller in size from a to c C D also becomes smaller from case a to c

 Comparison between case b and d : the Reynolds number in case10b5 : the Reynolds number in case10d4 : C D is the same for case b to d C D for a circular cylinder is relatively independent of Reynolds number 10 4 between 105 Re= and

 Comparison between case b to e : the Reynolds number in case b10:5 the Reynolds number in case e10:7 C D in case e is 0.6 smaller wake behind the cylinder in case e compared to that in case b . Note: With based on the frontal projected area (S=d(1) per unit span), the value of range from a maximum 2 to numbers as low as 0.12. Magnitude of Reynolds number of a flow around a circular cylinder at standard sea 3 5 d =level, 1m, ρ ∞ where, = 1.23kg / m , µ ∞ = 1.789 ×10 kg / m ⋅ s, V∞ = 45m / s

Then the Reynolds number is :

ρ ∞ V∞ d (1.23)(45)(1) 6 Re = = = 3 . 09 × 10 µ∞ 1.789 ×105 for practical applications in aerodynamics, the values of Re are in millions. Pressure drag and skin friction drag: The total drag exerted on the bodies are combined with pressure drag and skin friction drag. the drag of the vertical flat plate and the circular cylinder is dominated by pressure drag, whereas, in contrast, most of the drag of the streamlined body is due to skin

Drag on a flat plate at zero angle of attack. Here, the drag is completely due to shear stress, there is no pressure force in the drag direction. '

C f = D q∞ S = D' q∞ c(1)

The reference area is the planform area

From the above figure, we can conclude that 1. C D is a strong function of Re 2. The value ofC D depends on whether the flow over the plate surface is laminar or CD turbulent. 3. The magnitudes of range typically from 0.001 to 0.01 over a large range of Re.

Drag coefficient of a complete low-speed aircraft.

Drag coefficient of a complete high-speed aircraft.

Lift coefficient of an airfoil.

Lift coefficient increases linearly with angle α of attack until reaches near 14 degrees. And beyond this angle of attack, lift coefficient decreases precipitously. The ratio of lift to drag is a very important characteristic for flight performance. The L/D ratio for NACA 63-210 α at= 20 is 130. This is much lager than that of a complete aircraft. Application of flap

Application of flap (High-lift device).

L = C L q∞ S In the take-off and landing phases, the flight speed is very slow compared with cruise phases. And, as we know, the lift is proportional to the square of the flight speed. So, with the same shape and angle of attack, the lift at take-off and landing phases will be much smaller than that of the cruise phase. Flaps mounted at the trailing edge of the wing are used to increase the lift or lift coefficient during the take-off and landing of an aircraft.

Moment coefficient.