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### I-SteadyLevel Flight Performance

### II-Steady Climb Performance

Simple Mission Profile for an Airplane

1 Switch on + Worming + Taxi

(Cruising flight)

4

3

Descent

Altitude

Climb

Landing

Takeoff

5

6

1

2

Simple mission profile

Prof. Galal Bahgat Salem Aerospace Dept. Cairo University

Equations of Motions

Static Performance

(Zero acceleration

Dynamic Performance

(Finite acceleration)

Thrust required

Thrust available

Maximum velocity

Takeoff

Power required

Power available

Landing

Maximumvelocity

Rate of climb

Gliding flight

Prof. Galal Bahgat Salem Aerospace Dept. Cairo University

Maximum altitude

Service ceiling

Absolute ceiling

Range and endurance

Road map for Chapter 6

Prof. Galal Bahgat Salem Aerospace Dept. Cairo University

Study the airplane performance requires the derivation of the airplane equations of motion

- As we know the airplane is a rigid body has six degrees of freedom
- But in case of airplane performance we are deal with the calculation of velocities ( e.g.Vmax,Vmin..etc),distances (e.g. range, takeoff distance, landing distance, ceilings …etc), times (e.g. endurance, time to climb,…etc), angles (e.g.climb angle…etc)

Prof. Galal Bahgat Salem Aerospace Dept. Cairo University

So, the rotation of the airplane about its axes during flight in case of performance study is not necessary.

- Therefore, we can assume that the airplane is a point mass concentrated at its c.g.
- Also, the derivation of the airplane’s equations of motion requires the knowledge of the forces acting on the airplane
- The forces acting on an airplane are:

Prof. Galal Bahgat Salem Aerospace Dept. Cairo University

aerodynamic force R

- 1- Lift force L
- 2- Drag force D
- 3- Thrust force T Propulsive force
- 4- Weight W Gravity force
- Thrust T and weight W will be given
- But what about L and D?
- We are in the position that we can’t calculate L and D with our limited knowledge of the airplane aerodynamics

Prof. Galal Bahgat Salem Aerospace Dept. Cairo University

So, the relation between L and D will be given in the form of the so called drag polar

- But before write down the equation of the airplane drag polar it is necessary to know the airplane drag types

Prof. Galal Bahgat Salem Aerospace Dept. Cairo University

■ Drag Types [ Kinds of Drag ]

Skin Friction Drag

Pressure Drag

Form Drag (Drag Due to Flow separation)

Induced Drag

Wave Drag

Note : Profile Drag = Skin Friction Drag + Form Drag

Prof. Galal Bahgat Salem Aerospace Dept. Cairo University

►Skin friction drag

This is the drag due to shear stress at the surface.

►Pressure drag

This is the drag that is generated by the resolved components of the forces due to pressure acting normal to the surface at all points and consists of [ form drag + induced drag + wave drag ].

►Form drag

This can be defined as the difference between profile drag and the skin-friction drag or the drag due to flow separation.

Prof. Galal Bahgat Salem Aerospace Dept. Cairo University

►Profile Drag

● Profile drag is the sum of skin-friction and form drags.

● It is called profile drag because both skin-friction and

form drag [ or drag due to flow separation ] are

ramifications of the shape and size of the body, the

“profile” of the body.

● It is the total drag on an aerodynamic shape due to

viscous effects

Prof. Galal Bahgat Salem Aerospace Dept. Cairo University

►Induced drag ( or vortex drag )

This is the drag generated due to the wing tip vortices , depends on lift, does not depend on viscous effects , and can be estimated by assuming inviscid flow.

Finite wing flow tendencies

Prof. Galal Bahgat Salem Aerospace Dept. Cairo University

Formation of wing tip vortices

Prof. Galal Bahgat Salem Aerospace Dept. Cairo University

Prof. Galal Bahgat Salem Aerospace Dept. Cairo University

The origin of induced drag

Prof. Galal Bahgat Salem Aerospace Dept. Cairo University

►Wave Drag

This is the drag associated with the formation of shock waves in high-speed flight .

Prof. Galal Bahgat Salem Aerospace Dept. Cairo University

■ Total DragofAirplane

● An airplane is composed of many components and each will contribute to the total drag of its own.

● Possible airplane components drag include :

1. Drag of wing, wing flaps = Dw

2. Drag of fuselage = Df

3. Drag of tail surfaces = Dt

4. Drag of nacelles = Dn

5. Drag of engines = De

6. Drag of landing gear = Dlg

7. Drag of wing fuel tanks and external stores = Dwt

8. Drag of miscellaneous parts = Dms

Prof. Galal Bahgat Salem Aerospace Dept. Cairo University

● Total dragof an airplane is not simply the sum of the drag of the components.

● This is because when the components are combined into a complete airplane, one component can affect the flow field, and hence, the drag of another.

● these effects are called interference effects, and the change in the sum of the component drags is called interference drag.

● Thus,

(Drag)1+2 = (Drag)1 + (Drag)2 + (Drag)interference

Prof. Galal Bahgat Salem Aerospace Dept. Cairo University

■ Buid-up Technique of Airplae Drag D

● Using the build-up technique, the airplane total drag D is expressed as:

D = Dw + Df + Dt + Dn +De + Dlg + Dwt + Dms + Dinterference

► Interference Drag

● An additional pressure drag caused by the mutual interaction of the flow fields around each component of the airplane.

● Interference drag can be minimized by proper fairing and filleting which induces smooth mixing of air past the components.

● The Figure shows an airplane with large degree of wing filleting.

Prof. Galal Bahgat Salem Aerospace Dept. Cairo University

Prof. Galal Bahgat Salem Aerospace Dept. Cairo University

● Notheoretical method can predict interference drag, thus, it is obtained from wind-tunnel or flight-test measurements.

● For rough drag calculations a figure of 5% to 10% can be attributed to interference drag on a total drag, i.e,

Dinterference ≈ [ 5% – 10% ] of components total drag

■ The Airplane Drag Polar

● For every airplane, there is a relation between CD and CL that can be expressed as an equation or plotted on a graph.

● The equation and the graph are called the drag polar.

Prof. Galal Bahgat Salem Aerospace Dept. Cairo University

For the complete airplane, the drag coefficient is written as

CD = CDo + K CL2

This equation is the drag polar for an airplane.

Where: CDo drag coefficient at zero lift ( or

parasite drag coefficient )

K CL2 = drag coefficient due to lift ( or

induced drag coefficient CDi )

K = 1/π e AR

Prof. Galal Bahgat Salem Aerospace Dept. Cairo University

e Oswald efficiency factor = 0.75 – 0.9

(sometimes known as the airplane efficiency factor)

AR wing aspect ratio = b2/S ,

b wing span and S wing planform area

Schematic of the drag polar

Prof. Galal Bahgat Salem Aerospace Dept. Cairo University

Prof. Galal Bahgat Salem Aerospace Dept. Cairo University

Apply Newton’s 2nd low of motion:

In the direction of the flight path

Perpendicular to the flight path

Prof. Galal Bahgat Salem Aerospace Dept. Cairo University

Prof. Galal Bahgat Salem Aerospace Dept. Cairo University

Un-accelerated (steady) Level Flight Performance (Cruising Flight)

Prof. Galal Bahgat Salem Aerospace Dept. Cairo University

Thrust Required for Level Un-accelerated Flight

(Drag)

Thrust required TR for a given airplane to fly at V∞ is given as : TR = D

Prof. Galal Bahgat Salem Aerospace Dept. Cairo University

● TR as a function of V∞ can be obtained by tow methods

or approaches graphical/analytical

■Graphical Approach

Prof. Galal Bahgat Salem Aerospace Dept. Cairo University

2 - For the chosen V∞ calculate CL

L = W = ½ρ∞V2∞S CL

CL = 2W/ ρ∞V2∞S

3- Calculate CD from the drag polar

CD = CDo + K CL2

4- Calculate drag, hence TR, from

TR = D = ½ρ∞V2∞S CD

5- Repeat for different values of V∞

Prof. Galal Bahgat Salem Aerospace Dept. Cairo University

Prof. Galal Bahgat Salem Aerospace Dept. Cairo University

Prof. Galal Bahgat Salem Aerospace Dept. Cairo University

- It is required to obtain an equation for TR as a function of V∞
- TR = D

Required equation

Prof. Galal Bahgat Salem Aerospace Dept. Cairo University

Note that TR is minimum at the point of intersection of the parasite drag Do and induced drag Di

- Thus Do = Di at [TR]min
- or CDo = CDi
- = KCL2
- Then [CL](TR)min = √CDo/K
- And [CDo](TR)min = 2CDo

Prof. Galal Bahgat Salem Aerospace Dept. Cairo University

Finally, (L/D)max = (CL/CD)max

- = √CDo/K /2CDo
- (CL/CD)max = 1/√4KCDo
- Also,[V∞](TR)min =[V∞](CL/CD)maxisobtained from: W = L
- = ½ρ∞[V]2(TR)minS [CL](TR)min
- Thus:
- [V](TR)min= {2(W/S)(√K/CDo)/ρ∞}½

Prof. Galal Bahgat Salem Aerospace Dept. Cairo University

L/D as function of angle of attack α

L/D as function of velocity V∞

Prof. Galal Bahgat Salem Aerospace Dept. Cairo University

Flight Velocity for a Given TR

- TR = D
- In terms of q∞ = ½ρ∞V2∞we obtain
- Multiplying by q∞and rearranging, we have
- This is quadratic equation in q∞

Prof. Galal Bahgat Salem Aerospace Dept. Cairo University

- By replacing q∞ = ½ρ∞V2∞we get

Prof. Galal Bahgat Salem Aerospace Dept. Cairo University

- Where (TR/W) is the thrust-to-weight-ratio
- (W/S) is the wing loading
- The final expression for velocity is
- This equation has two roots as shown in figure corresponding to point 1 an 2

Prof. Galal Bahgat Salem Aerospace Dept. Cairo University

●When the discriminant equals zero ,then only

one solution for V∞ is obtained

●This corresponds to point 3 in the figure,

namely at (TR)min

Prof. Galal Bahgat Salem Aerospace Dept. Cairo University

- Then the velocity V3 =V(TR)min is
- Substituting for (TR/W)min = √4CDoK we have

Prof. Galal Bahgat Salem Aerospace Dept. Cairo University

- We know that
- (TR/W)min = √4CDoK
- This means that (TR)min is independent of altitude as show in Figure
- (TR)min occurs at higher V∞

V∞1

V∞2

Prof. Galal Bahgat Salem Aerospace Dept. Cairo University

Prof. Galal Bahgat Salem Aerospace Dept. Cairo University

Prof. Galal Bahgat Salem Aerospace Dept. Cairo University

Thrust Available TA and Maximum Velocity Vmax

Prof. Galal Bahgat Salem Aerospace Dept. Cairo University

For turbojet at subsonic speeds, (V∞)max can be obtained from:

- Just substitute (TA)max TR

Prof. Galal Bahgat Salem Aerospace Dept. Cairo University

Prof. Galal Bahgat Salem Aerospace Dept. Cairo University

Prof. Galal Bahgat Salem Aerospace Dept. Cairo University

Prof. Galal Bahgat Salem Aerospace Dept. Cairo University

Prof. Galal Bahgat Salem Aerospace Dept. Cairo University

Power Available PA and Maximum Velocity Vmax

- The high speed

intersection

between PR and

(PA)max gives

Vmax

- Vmax decreases

with altitude

Prof. Galal Bahgat Salem Aerospace Dept. Cairo University

Minimum Velocity: Stall Velocity

- Airplane minimum velocity Vmin is usually dictated by its stall velocity
- Stall velocity Vstall is the velocity corresponds to the maximum lift coefficient (CL)maxof the airplane
- Hence, Vmin = Vstall
- But, L = W = ½ρ∞V2∞S CL

V∞ = (2W/ ρ∞ S CL )½

- Substitute for CL (CL)max

Prof. Galal Bahgat Salem Aerospace Dept. Cairo University

Vmin= Vstall = [2W/ ρ∞ S (CL)max ]½

CL –α curve for an airplane

Prof. Galal Bahgat Salem Aerospace Dept. Cairo University

Prof. Galal Bahgat Salem Aerospace Dept. Cairo University

- Assumptions:

1- dV∞/dt = 0

2- Climb along straight line, V2∞/ r = 0

Prof. Galal Bahgat Salem Aerospace Dept. Cairo University

The equations of motion in this case become:

- T cos ε– D – W sin ϴ = 0
- L + T sin ε– W cos ϴ = 0
- Assuming , ε = 0
- Then, T – D – W sin ϴ = 0
- L– W cos ϴ = 0

Prof. Galal Bahgat Salem Aerospace Dept. Cairo University

Prof. Galal Bahgat Salem Aerospace Dept. Cairo University

Analytical Solution for (R/C)max

- R/C = V∞ sin ϴ
- = (2W/ ρ∞ S CL )½ [ T/W- D/L]
- = (2W/ ρ∞ S CL )½ [T/W-CD/CL]
- = (2W/ ρ∞ S CL )½ [T/W-CDo +KCL2/CL]
- =(2W/ ρ∞ S )½ [CL-½(T/W)-(CDo+KC2L)/CL3/2]
- For turbojet T = const
- For (R/C)max d(R/C)/dCL =0
- So, we get

Prof. Galal Bahgat Salem Aerospace Dept. Cairo University

CL(R/C)max = [ -(T/W) + √ (T/W)2 + 12 K CD0 ] / 2K

- So, we get:
- And, V(R/C)max=[2W/ ρ∞ S CL(R/C)max ]½
- (CD) (R/C)max=CDo +K C2L(R/C)max
- (Sin ϴ) (R/C)max = T/W- (CD/CL) (R/C)max
- (R/C)max = V(R/C)max (sin ϴ) (R/C)max

Prof. Galal Bahgat Salem Aerospace Dept. Cairo University

- For propeller aircraft (R/C)max occurs at
- (PR)min

Propeller aircraft

Prof. Galal Bahgat Salem Aerospace Dept. Cairo University

Analytical Solution for (R/C)max

- V(R/C)max= V(CL3/2/CD)max
- (CD) (R/C)max=CDo +K C2L(R/C)max
- = CDo +K [√3CDo/K ]2 = 4CDo
- (Sin ϴ) (R/C)max = T/W- (CD/CL) (R/C)max
- (R/C)max = V(R/C)max (sin ϴ) (R/C)max

Prof. Galal Bahgat Salem Aerospace Dept. Cairo University

- Assumptions
- 1- Steady gliding
- 2- Along straight line

Prof. Galal Bahgat Salem Aerospace Dept. Cairo University

If PR˃ PA the airplane will descend

In the ultimate situation when T = 0, the

airplane will be in gliding

Prof. Galal Bahgat Salem Aerospace Dept. Cairo University

- For an airplane at a given altitude h, the max. horizontal distance covered over the ground is denoted max. range R

Prof. Galal Bahgat Salem Aerospace Dept. Cairo University

Prof. Galal Bahgat Salem Aerospace Dept. Cairo University

Prof. Galal Bahgat Salem Aerospace Dept. Cairo University

- tmin =
- Assuming linear variation of (R/C)maxwith altitude h, then
- (R/C)max = a + b h
- a = (R/C)max at h = 0
- =1/b[ln(a+bh2)-lna]

max

h

b =slope

0

(R/C)max

Prof. Galal Bahgat Salem Aerospace Dept. Cairo University

Prof. Galal Bahgat Salem Aerospace Dept. Cairo University

Prof. Galal Bahgat Salem Aerospace Dept. Cairo University

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