Drag Analysis of a Supermarine Spitfire Mk v at Cruise Conditions

Introduction to Flight

Aircraft Drag Project – April 2016

2016

Drag Analysis of a Supermarine Spitfire Mk V at Cruise Conditions Nicholas Conde [email protected] U66182304

Introduction to Flight || Nicholas Conde

April 2016 _ _1 1 ___

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DRAG ANALYSIS ANA LYSIS OF A SUPERMARINE SPITFIRE SPITFIRE MK V AT A T CRUISE CONDITIONS CONDITIONS

A Project Presented Presented by Nicholas Conde









Table Ta ble of Contents Cont ents Introduction Introduction ................................................. ........................................................................... .................................................... .......................................... ................ 7 1.1

Project Scope ......................................... ................................................................... ................................................... .................................. ......... 7

1.2

Project Importance Importance .................................................. ............................................................................ ........................................... ................. 7

1.3

Plane Background Background .................................................. ............................................................................ ........................................... ................. 7

Flight Data ....................................................... ................................................................................. ................................................... ...................................... ............. 8 2.1

Standard Day.................................................. ........................................................................... ................................................... .......................... 8

2.2

Vehicle Dimensions Dimensions ................................................. .......................................................................... .......................................... ................. 8

2.3

Wing Dimensions................................................ ......................................................................... ............................................. .................... 10

2.4

Fuselage, Vertical Fin, Fin, and Horizontal Horizontal Stabilizer Stabilizer Dimensions Dimensions........................... ........................... 12

Calculations...................................................................... ............................................................................................... ............................................. .................... 13 3.1

Parasite Drag.................................................. ........................................................................... ................................................. ........................ 13

3.1.1

Wing, Aerodynamic Aerodynamic Calculation Calculation ................................................. ................................................................ ............... 13

3.1.2

Fuselage, Blunt Body Calculation .................................................. ............................................................. ........... 14

3.1.3

Total Parasitic Drag................................................... ............................................................................. ................................ ...... 15

3.2

Induced Drag .................................................. ........................................................................... ................................................. ........................ 15

3.3

Interference Interference Drag ................................................ ......................................................................... ............................................. .................... 17

3.4

Compressibility Compressibility Drag................................................ ......................................................................... ........................................ ............... 17

3.5

Total Drag .................................................. ............................................................................ .................................................... ............................ .. 18

Discussion Discussion ................................................... ............................................................................. ................................................... ........................................ ............... 19 4.1

Results................................................... ............................................................................. .................................................... ................................ ...... 19

4.2

Reasonability Reasonability .................................................. ........................................................................... ................................................. ........................ 19

4.3

Conclusion Conclusion ................................................. ........................................................................... .................................................... ............................ .. 19

APPENDIX .................................................. ............................................................................ ................................................... ........................................ ............... 20 References References .................................................. ............................................................................ ................................................... ........................................ ............... 21









Table of Figures Figure 1 - Supermarine Spitfire Mk V Scale Drawing ............................ ..................................................... ........................... 9 Figure 2 - Supermarin S upermarine e Spitfire Mk V Wing W ing ............................................... ................................................................... .................... 10









Table Table of Tables Table 1 - Standard S tandard Day Values for Spitfire Mk V at Cruise Conditions............................ ............................ 8 Table 2 - Wing Dimensions ............................. ...................................................... ................................................... ..................................... ........... 10 Table 3 - Fuselage Dimensions................................................. .......................................................................... .................................... ........... 12 Table 4 - Horizontal Stabilizer Dimensions................................................ .................................................................... .................... 12 Table 5 - Vertical Fin Dimensions ................................................. ........................................................................... ................................ ...... 12 Table 6 - Calculated Parasitic Drag for All Components Components ............................................... ............................................... 20









Table Table of Equations Equations Equation 1 - Mach Number Formula ............................................. ....................................................................... .................................. ........ 8 Equation 2 - Taper Ratio Calculation............................................. Calculation....................................................................... ................................ ...... 11 Equation 3 - Mean Aerodynamic Chord Calculation Calculation............................................... ...................................................... ....... 11 Equation 4 - Wetted W etted Surface Area Calculation Calculation .................................. ........................................................... ............................ ... 11 Equation 5 - Parasitic Drag Equation ............................................ ...................................................................... ................................ ...... 13 Equation 6 - Reynolds Number Calculation Calculation............................................... ................................................................... .................... 13 Equation 7 - Wing Reynolds Number Calculation ......................................................... ......................................................... 14 Equation 8 - Wing Coefficient Coefficient of Parasitic Drag ......................................................... ............................................................ ... 14 Equation 9 - Fuselage Fuselage Reynolds Number.................................................. ...................................................................... .................... 14 Equation 10 - Fuselage Fuselage Coefficient Coefficient of Parasitic Drag ................................................. .................................................... ... 15 Equation 11 - Induced Drag Coefficient Equation ......................................................... ......................................................... 15 15 Equation 12 - Coefficient Coefficient of Lift Equation .............................................. ...................................................................... ........................ 15 Equation 13 - Calculation Calculation of Variable "q" ............................................... ....................................................................... ........................ 16 16 Equation 14 - Calculation Calculation of Coefficient of Lift................................................ ............................................................... ............... 16 Equation 15 - Efficiency Factor Interpolation Interpolation .................................................. ................................................................. ............... 16 Equation 16 - Calculation Calculation of Induced Drag Coefficient ................................................ .................................................. .. 16 Equation 17 - Calculation Calculation of Interference Interference Drag .................................................. ............................................................. ........... 17 17 Equation 18 - Compressibility Compressibility Drag Relationship .................................. .......................................................... ........................ 17 Equation 19 - Total Drag Coefficient ..................................................... ............................................................................. ........................ 18 Equation 20 - Lift to Drag Ratio ............................................. ...................................................................... ........................................ ............... 18 18 Equation 21 - Drag Calculation Calculation ................................................. .......................................................................... .................................... ........... 18











Introduction 1.1

Project Scope This project is focused on a complete drag analysis of the Supermarine Spitfire

Mk V at cruise conditions. To begin this assessment I shall first include airplane schematics and dimensions for the Spitfire Mk V in order to establish scale and important variables. I shall further provide any other constants and variables necessary from the flight data in order to calculate the drag on the plane. In terms of the drag analysis I shall provide drag calculations for; parasite drag, induced drag, interference drag, and compressibility drag. With these subsets of drag I will be able to provide the total drag on the plane. This project will be concluded with a discussion of my work. I will assess the reasonability of the data calculated, and discuss any problems that I encountered in my calculations. I shall end with a comparison of my calculated values to a similar airplane.

1.2

Project Importance Drag analysis is an important aspect of overall analysis of a plane. When total

drag is determined then one effectively knows the minimum amount of thrust necessary to move the plane. With the given thrust information engineers can make determinations for engines, or even redesigns of the vehicle in order to reduce drag. For a military designed vehicle like the Spitfire it would be important to know all forces acting on the vehicle, and what changes may occur in the thrust profile with the addition of mounted









Flight Da Data ta 2.1

Standard Day To begin a proper assessment of the Spitfire Mk V I had to first determine the

appropriate values for cruise altitude and speed [2]. With the cruise altitude I was then able to determine the standard day conditions from Fundamentals of Flight .

Table 1 - Standard Day Values for Spitfire Mk V at Cruise Conditions

Standard Day Values Cruise Altitude (ft) Cruise Velocity (ft/sec) Mach Number Pressure (lb/ft 2)

20000 322 0.3104 973.27

Density (slugs/ft 3) Operational Weight (lbs) Empty Weight (lbs) Kinematic Viscosity (ft 2/s)

0.0012673 6650 5050 0.00026234

Temperature ( °R)

447.43

γ for air

1.4

All values in Table 1 were drawn from reference material [2] [3] except for the Mach number, which was calculated using the following formula:

















2.3

Wing Dimensions

Figure 2 - Supermarine Spitfire Mk V Wing

Table 2 - Wing Dimensions Wing Dimensions

Specification

NACA 2213









The Spitfire line, including the Mk V use the NACA 2213 airfoil [4]. I found reference to several dimensions for the Mk V wings including the; span, wing area, and aspect ratio [2]. Using Figure 2 I was able to determine the root chord and tip chord, and subsequently the taper ratio and Mean Aerodynamic Chord (M.A.C) using the wingspan as a scale of reference.

 = / 0.448 = 7.3.987994

Equation 2 - Taper Ratio Calculation

+  ...= 23 1+ 1 +  − 1 + 0.+40.48448 6.1708 = 23 ∗ 7.979 ∗ 1+ 0.448 − 1 +0. Equation 3 - Mean Aerodynamic Chord Calculation

The exposed area was determined by taking the total wing area and subtracting the section that would include fuselage leading to an exposed area of 219.637 ft 2. The wetted area was then calculated using the equation below.

 =  ∗ 1.02 ∗ 2 448.0595 = 219.19.637 ∗ 1.02 ∗ 2











2.4

Fuselage, Vertical Fin, and Horizontal Stabilizer Dimensions

Table 3 - Fuselage Dimensions Fuselage Dimensions

Length (ft)

29.917

Diameter (ft)

3

Area

89.751

Wetted Area

183.09204

Fineness Ratio

9.972333333

Table 4 - Horizontal Stabilizer Dimensions Horizontal Stabilizers

Root Chord (ft)

4.369

Tip Chord (ft)

1.1864

Exposed Area (ft2)

34.387

Wetted Area

70.14948

Taper Ratio

0.271549554

Span

5.188

t/c

0.13

Sweep Angle









Calculations 3.1 3.1.1

Parasite Drag Wing, Aerodyn Aerodynamic amic Calculatio Calculation n The calculation for the Parasitic Drag on the Wings is based on the following

formula from Fundamentals of Flight.

 =  ∗  ∗ 

Equation 5 - Parasitic Drag Equation

In the above equation K is the correction factor for pressure drag and increased local velocities, where it can be determined by either referencing Figure 11.3 or 11.4 in Fundamentals of Flight providing either the thickness ratio (t/c) and sweep angle, or

fineness ratio respectively. Cfi references the skin friction coefficient, which for all purposes shall be considered typical transport aircraft roughness, the value of which









6.1708 ft  7574083.815 = 322 secft  ∗ 0.00026234 ft  Equation 7 - Wing Reynolds Number Calculation

Based on the above Reynolds number the K correction factor was determined from Figure 11.3 of Fundamentals of Flight to be 1.27 based on a 0 degree sweep angle of the wings and a thickness ratio of 0.13. The skin friction coefficient was found to be 0.0036 given the Reynolds Number. Using the Parasitic Drag Equation the Wing Parasitic Drag Coefficient can thereby be calculated as:

 1 . 2 7∗ 0. 0 036 ∗ 448. 0 595  0.008465 = 242  Equation 8 - Wing Coefficient of Parasitic Drag

3.1.2

Fuselage, Blunt Body Calculation For the fuselage a new Reynolds number must be calculated using Equation 6,









 1 . 0 9 ∗0. 0 0275 ∗ 183. 0 92  0.0061149 = 89.751  Equation 10 - Fuselage Coefficient of Parasitic Drag

3.1.3

Total Parasitic Drag Refer to Appendix for subsequent calculation of Parasitic Drag for the remaining

surfaces. Below the components of Parasitic Drag Coefficients are summed in order to determine the total Parasitic Drag of the aircraft.

CDP WING =

0.008467

CDP FUSEALGE =

0.0061149

CDP VERTICAL FIN =

0.0101184

CDP HOR. STAB. =

0.00714

CDP =

0.0318383









“w” is the operational weight of the aircraft, rho is the density of the air, v is the

cruise velocity, and s is the area of the wings. The variables simplified to the single variable q in terms of calculation.

 ∗  ∗ 

can be

65.6994 / = (12) ∗0.0012673 ∗ 322 Equation 13 - Calculation of Variable "q"

6650∗ 242 0.4183 = 65. 6994

Equation 14 - Calculation of Coefficient of Lift

Based on the Coefficient of Drag, and the Aspect Ratio the efficiency factor can be determined from Figure 11.8 in Fundamentals of Flight . Interpolation was required in order to determine the appropriate value, based on the efficiency factors determined at











3.3

Interference Drag Interference drag is drag caused by numerous effects including small

protuberances, surface gaps, and base drag. This interference drag can be modeled as a percentage of the parasitic drag coefficient. As an aircraft powered by a reciprocating, piston engine the Spitfire Mk V in the Rolls-Royce Merlin, has an interference drag of around 10% of the parasitic drag, thereby:

 =  ∗ 0.1 0.00318 = 0.0318 ∗0.1









3.5

Total Drag All of the calculate calculated d coefficients coefficients of drag drag can be calculated calculated in order order to determine determine a

total drag coefficient and total drag in pounds. The total drag coefficient can be found in the following equation:

0.040303 = 0.0318383 + 0.0052847 + 0.00318383 Equation 19 - Total Drag Coefficient

The Lift to Drag Ratio can be calculated given the total coefficient of drag:









Discussion 4.1

Results Ultimately it was found that overall the Supermarine Spitfire Mk V has a

Coefficient of Drag of 0.040303. Its largest contributing factor is parasitic drag, of which the largest contributing factor therein is the wing parasitic drag. There is some interference drag due to unaccounted variables and its reciprocating engine. There is induced drag, simplified due to its unswept wing design. At its cruise speed of around 0.31 Mach it was ultimately found that compressibility drag would be negligible. The total drag acting on the aircraft was determined to be 640.788 pounds with a Lift to Drag









APPENDIX Table 6 - Calculated Parasitic Drag for All Components

Component

Length

Reynold Number

Sweep

K

Cfi

Swet

Sref

Cdp

Wing

6.1708

7574083

0

1.27

0.0036

448.0595

242.0000

0.008464991

Fuselage

29.917

36720568

N/A

1.09

0.0027

183.0920

89.751

0.0061149

Horizontal Stab.

3.0816

3782372

15.9

1.25

0.0028

70.14948

34.387

0.00714

Vertical Fin

3.374045

4141353

36.9

1.24

0.004

26.52204

13.001

0.0101184

Total

0.031838291









References [1]

Supermarine Spitfire Mk V. (n.d.). Retrieved April 20, 2016, from http://www.historyofwar.org/articles/weapons_spitfire_mkV.html

[2]

The Supermarine Spitfire. (n.d.). Retrieved April 20, 2016, from http://www.ww2warbirds.net/ww2htmls/supespitfire.html

[3]

Shevell, R. S. (1989). Fundamentals of Flight. Englewood Cliffs, NJ: Prentice Hall.

[4]

The Incomplete Guide to Airfoil Usage. (n.d.). Retrieved April 20, 2016, from

Drag Analysis of a Supermarine Spitfire Mk v at Cruise Conditions

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