Data With Other Aircrafts

Published Date: 02 Nov 2017

Table of Contents

1. Wing Geoemetry 3

1.1 Initial Design 3

1.2 Defining Wing Specifics 4

2. Aerofoil Selection 6

2.1 Wing Section 6

2.2 Tail Section 7

3. 3-D Lift Curve 8

4. Trailing Edge Flaps 9

4.1 2-D Lift Curve 10

4.2 3-D Lift Curve 11

5. Drag Polar 12

5.1 Wing, Horizontal & Vertical Tail 13

5.2 Fuselage & Nacelles 14

5.3 Undercarriage 14

5.4 Flaps 15

5.5 Results 15

6. Lift & Drag Calculations 17

6.1 Take-off Analysis 17

6.2 Cruise Analysis 18

6.3 Approach Analysis 18

7. References 19

1. WING GEOMETRY

1.1 Initial Design

Before the actual area of the wing that was actually needed in order to produce the amount of lift required, a detailed research was carried out and the values required to determine as such were then assumed with logical justification and also by comparing data with other aircrafts. The values that were needed to be assumed in the start are as follows;

Assumed Values

Value

Units

Cruise Height

10000

m

Cruise Velocity

≈ 0.72

Mach

Oswalds Efficiency (e)

0.85

Aspect Ratio (AR)

9

Taper Ratio (λ)

0.25

Sweep Back Angle (Leading Edge)

30

Degrees

MTOW

392400

N

Zero Lift Drag (Cdo)

0.015

Table 1a – Initial Values

For some of the specifics (mentioned as ‘lecturer’) stated above, the assumptions were derived from graphical representations of the aircraft aerodynamic wing geometry. The graph is shown below.

Figure 1a – Derivation of Initial Values (Courtesy of Raymer, 1989)

1.2 Defining wing specifics

The next stage of the process was to determine the wing loading and the wing reference area. This was done by understanding and using the following wing loading formula.

Where;

q = Dynamic Pressure

W = Maximum Take-off Weight (MTOW)

S = Wing reference area

The calculations were carried out using imperial units. The wing loading came out to be 4553 N/m2. By simple mathematics, the reference wing area was then found to be 86.17 m2. The taper ratio was decided using the following graph;

Figure 1b – Derivation of Taper Ratio (Courtesy of Raymer, 1989)

After calculating the wing area, the wing specifics were then defined using the assumptions shown in Table 1. The formulas that were used were all taken from (Raymer, 1989) and were used to define the wing characteristics. The results are recorded in Table 2.

(Courtesy of Raymer, 1989)

Where;

AR = Aspect Ratio

b = Span

S= Wing area

(Courtesy of Raymer, 1989)

Where;

Croot = Root Chord

λ = Taper Ratio

Sref = Wing area

(Courtesy of Raymer, 1989)

Where;

Ctip = Tip Chord

(Courtesy of Raymer, 1989)

Where;

MAC = Mean Aerodynamic Chord

(Courtesy of Raymer, 1989)

Where;

Ó® = MAC distance from centreline

Wing

Units

Area

86.22

m2

Span

27.84

m

Chord Root

4.95

m

Chord Tip

1.237

m

MAC

4.083

m

MAC From Centreline (Y)

5.5713

m

Table 1b – Calculated Values

2. Aerofoil Selection

2.1 Wing

When all of the required values were found, the aerofoil selection for the wing began based on the following conditions;

The required Cl must fall in the drag bucket of the aerofoil.

Must have at least a thickness of between 12% - 16%.

Must be non- symmetric.

The minimum required Cl was calculated using the standard lift formula under cruise condition. This was done in order to ensure that without the help of any high-lift devices the aircraft should be able to remain at a certain amount of altitude. The required Cl was found to be 0.395, therefore the NACA 2415 was chosen to be the aerofoil of the wing.

2311312.jpg

Figure 2a – Aerofoil Drag Polar (Courtesy of http://www.airfoiltools.com)

Figure 2c – Naca 2415 aerofoil (courtesy of http://www.rcgroups.com)

2.2 Tail

The aerofoil selection for the tail began based on the following conditions;

Had no required Cl

Must have a thickness between 9% - 12%

Must be symmetric

The aerofoil chosen for the tail section was the NACA 0009 which perfectly satisfied the above mentioned conditions.

Figure 2d – Naca 0009 aerofoil (courtesy of http://www.geocities.jp)

3. 3D Lift Curve

The original 2-D lift co-efficient versus angle of attack (Cl vs α) was found and is illustrated below;

2-D graph.jpg

Figure 3a - Cl v/s α of NACA 2415 (Courtesy of http://www.airfoiltools.com)

The graph given above was chosen as a base to calculate the 3-D lift curve slope of the chosen aerofoil. The formula used in the process given below.

3-d gradient.jpg (Courtesy of Raymer, 1989)

Where B is a function of mach number. The value of the tangent of the half-chord sweep angle is 0.44 and k had a value of 0.912. The value of the 3-D lift curve slope came out to be around 0.077/degree.

The graph shown below is the 2-D and the 3-D illustration of the Naca 2415 aerofoil.

Figure 3b 2-D & 3-D Wing Lift Curve

4. Trailing Edge Flaps

Triple-slotted fowler flaps were chosen as trailing edge flaps. The ratio of the wing area in the flow path of the flaps to the reference area was at 0.6. For the design the flap to chord ratio was 0.25. According to (Raymer, 1989) the typical trailing edge flap deflection angle is between 15-25 degrees at take-off and around 40-50 degrees during landing. For our design I had chosen the values as 25 and 40 during take-off and landing respectively.

To deduce the performance graphs of our trailing edge flaps the same methodology was adopted as the one used for the Naca 2415 aerofoil above. The steps are illustrated as follows.

2-D lift curve with a specific deflection angle

Plotting the acquired curve.

Conversion of the 2-D graph to 3-D.

4.1. 2-D lift curve.

The formulas for this process are as follows;

(Courtesy of Corke, 2006)

Where;

dα/dδf = change in zero lift angle with respect to deployed angle

δf = Deployed angle

The deployed angle was chosen as 25 and 40 for each respective calculation. The change in zero lift angle with respect to the deployed angle was around 0.5. The value for the change in zero lift angle came out to be -12.5 and -18 for the respective deployment of 25 and 40.

(Courtesy of Corke, 2006)

This formula was used to find the change for the 2-D maximum lift co-efficient from the actual aerofoil lift behaviour curve. As can be seen from the equation there is only a single requirement of a constant value which is 1.4. The change in Clmax for 25 and 40 degrees angle were 3 and 2.66 respectively.

The graphs that were obtained from all the calculations is given below along with the 2-D curve of the aerofoil

Figure 4a – 2-D Wing lift curve with flaps

4.2. 3-D trailing edge flaps.

The formulas used for this process are illustrated as follows.

(Courtesy of Corke, 2006)

The first two notations are the same as discussed before. The third variable is the ratio of the wing area in the flow path of the flap to the reference wing area which was 0.6 and the cosine of the deflection angle of the flaps. This was done for both the deflection angle of 25 and 40 degrees. The resulting maximum lift coefficient was found to be 1.99 and 2.34 for deflection of 25 and 40 degrees respectively. The zero lift angle and stall angle remains similar to the ones found in the 2-D graphs. The following graphs were obtained.

Figure 4b – 3-D wing lift curve

5. Drag Polar

(courtesy of Jason Knight, 2011)

Where;

CD = Drag co-efficient

CDo = Zero lift drag co-efficient

CDi = lift-induced drag

The above stated equation is known as the drag polar equation. This equation enables us to find the drag co-efficient for the whole aircraft during various flight phases mainly to speak of take-off, cruise and approach. The lift induced drag can be calculated by KCL2, where K is a constant function of the wing aspect ratio and oswalds efficiency. The formula can then be written as;

(courtesy of Jason Knight, 2011)

The zero lift drag co-efficient was calculated for each aircraft component. The formulas used to do so are given below;

5.1. Wing, horizontal & vertical tail;

(courtesy of Jason Knight, 2011)

Where;

K = form factor (dependent on quarter-chord angle)

Cf = Friction drag coefficient

Swetted = 2 x planform area (+ ~ 2% for aerofoil curvature)

Sref = Actual area

The friction drag coefficient is a function dependent of the Reynolds number and the form factor is a constant derived from a graph as a correction factor for the pressure drag.

(courtesy of Jason Knight, 2011)

(courtesy of Jason Knight, 2011)

Figure 5a - Form Factor

5.2. Fuselage& Nacelles;

(courtesy of Jason Knight, 2011)

The friction drag coefficient is the same as used in the previous equation for the wing. The form factor K also is the same but this time is dependent on the length and the diameter of the concerned object. The graph for the form factor is given below;

Figure 5b – Form factor

5.3. Undercarriage

(Courtesy of Corke, 2006)

Where;

Slg = frontal area of the wheel, and

S = wing reference area.

CDlg = Constant 0.15

5.4. Flaps.

(Courtesy of Corke, 2006)

Where;

Cf/C = Flap to aerofoil ratio

A & B = Specific aerofoil constants

δf = Flap deployment angle

5.5. Results

These were the formulas used to calculate the zero lift drag coefficients of various components of our aircraft. The values found for these each component. The overall zero lift drag is the sum of all the individual zero lift drags. The overall drag polar quation is then given below;

 

Cdo-takeoff

Cdo-cruise

Cdo-approach

Fuselage

0.002272

0.002263

0.002192

Wing

0.007155

0.007125

0.006873

Horizontal Tail

0.007218

0.007185

0.00692

Vertical Tail

0.007058

0.007027

0.006769

Nacelles

0.005451

0.005426

0.005226

Undercarriage

0.005328

 

0.005328

Flaps

0.00335

 

0.009487

 

 

 

 

Total Cd0

0.037832

0.029026

0.042795

Table 5a – Zero lift drag coefficients

The Induced drag coefficient was a constant value of 0.0416CL2. Therefore the final overall drag polar equation is given below

 

Drag-polar Equation

Take-off

 0.037832 + 0.0416CL2

Cruise

 0.029026 + 0.0416CL2

Approach

 0.042795 + 0.0416CL2

Table 5b – Drag Polar Equation

Using the graphs for the 3-D lift curve in graph , the performance of the aricraft at three main stages were plotted. The graphs of each performance stage is given on the following page.

Figure 5c – Take-off Performance

Figure 5d – Cruise Performance

Figure 5e – Approach Performance

6. Lift & Drag Calculations

As discussed earlier the total drag of an aircraft consists of two major components, the zero lift drag and the lift induced drag. The zero lift drag coefficients (Cdo) that were all stated previously were used to find the zero lift drag of each component at each flight phase. The lift induced drag was calculated using the angle at which the flight phase would take place, then taking the corresponding lift-coefficient value, the drag coefficient value was used to find the induced drag.

The values that were obtained by the adopted method are illustrated below with all the values used in the equation mentioned as well;

6.1. Take-off Analysis

 

Air density (Kg/m3)

Speed (m/s)

Area (m2)

Coefficient

Drag (N)

Fuselage

1.225

60.95

317.719

0.002272

1445.86

Wing

1.225

60.95

86.22

0.007155

1373.28

Horizontal Tail

1.225

60.95

15.5196

0.007218

247.19

Vertical Tail

1.225

60.95

6.6876

0.007058

106.518

Nacelles

1.225

60.95

9.009

0.005451

102.49

Undercarriage

1.225

60.95

18.823

0.005328

115.481

Flaps

1.225

60.95

6.78

0.00335

51.6807

Lift-Induced Drag

 

 

 

 

22257.48

TOTAL

 

 

 

 

25699.98

Table 6a – Take-Off Analysis

6.2. Cruise Analysis

 

Air density (Kg/m3)

Speed (m/s)

Area (m2)

Coefficient

Drag (N)

Fuselage

0.4128

223.15

317.719

0.002272

6530.96

Wing

0.4128

223.15

86.22

0.007155

6203.11

Horizontal Tail

0.4128

223.15

15.5196

0.007218

1116.56

Vertical Tail

0.4128

223.15

6.6876

0.007058

481.14

Nacelles

0.4128

223.15

9.009

0.005451

462.95

Lift-Induced Drag

 

 

 

 

16752.55

TOTAL

 

 

 

 

31547.25

Table 6b – Cruise Analysis

6.3. Approach Analysis

 

Air density (Kg/m3)

Speed (m/s)

Area (m2)

Coefficient

Drag (N)

Fuselage

1.225

79.235

317.719

0.002272

2443.51

Wing

1.225

79.235

86.22

0.007155

2320.85

Horizontal Tail

1.225

79.235

15.5196

0.007218

417.752

Vertical Tail

1.225

79.235

6.6876

0.007058

180.015

Nacelles

1.225

79.235

9.009

0.005451

173.211

Undercarriage

1.225

79.235

18.823

0.005328

195.162

Flaps

1.225

79.235

7.58

0.00949

276.527

Lift-Induced Drag

 

 

 

 

27498.074

TOTAL

 

 

 

 

35302.094

Table 6c – Approach Analysis

A verification process is carried out in order to check whether the lift and drag calculations are in accordance to specified range of sensible values. This is done by deriving the lift/drag ratio of each phase. The value of lift/drag ratio should be between the range of 11-17. The results are illustrated in the following table.

 

L/D

Take-off

15.502

Cruise

15.01

Approach

12.9629

Table 6d – Lift/Drag Ratio