Drag Analysis Of A Supermarine Spitfire Mk V At Cruise Conditions
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Introduction to Flight Aircraft Drag Project β 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___
DRAG ANALYSIS OF A SUPERMARINE SPITFIRE MK V AT CRUISE CONDITIONS
A Project Presented by Nicholas Conde
Introduction to Flight || Nicholas Conde
April 2016
_2___
Table of Contents Introduction ..................................................................................................................... 7 1.1
Project Scope ..................................................................................................... 7
1.2
Project Importance ............................................................................................. 7
1.3
Plane Background ............................................................................................. 7
Flight Data ....................................................................................................................... 8 2.1
Standard Day ..................................................................................................... 8
2.2
Vehicle Dimensions ........................................................................................... 8
2.3
Wing Dimensions ............................................................................................. 10
2.4
Fuselage, Vertical Fin, and Horizontal Stabilizer Dimensions ........................... 12
Calculations................................................................................................................... 13 3.1
Parasite Drag ................................................................................................... 13
3.1.1
Wing, Aerodynamic 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 Drag ............................................................................................. 17
3.4
Compressibility Drag ........................................................................................ 17
3.5
Total Drag ........................................................................................................ 18
Discussion ..................................................................................................................... 19 4.1
Results ............................................................................................................. 19
4.2
Reasonability ................................................................................................... 19
4.3
Conclusion ....................................................................................................... 19
APPENDIX .................................................................................................................... 20 References .................................................................................................................... 21
Introduction to Flight || Nicholas Conde
April 2016
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Table of Figures Figure 1 - Supermarine Spitfire Mk V Scale Drawing ...................................................... 9 Figure 2 - Supermarine Spitfire Mk V Wing ................................................................... 10
Introduction to Flight || Nicholas Conde
April 2016
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Table of Tables Table 1 - Standard 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 ............................................... 20
Introduction to Flight || Nicholas Conde
April 2016
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Table of Equations Equation 1 - Mach Number Formula ............................................................................... 8 Equation 2 - Taper Ratio Calculation............................................................................. 11 Equation 3 - Mean Aerodynamic Chord Calculation ...................................................... 11 Equation 4 - Wetted Surface Area Calculation .............................................................. 11 Equation 5 - Parasitic Drag Equation ............................................................................ 13 Equation 6 - Reynolds Number Calculation................................................................... 13 Equation 7 - Wing Reynolds Number Calculation ......................................................... 14 Equation 8 - Wing Coefficient of Parasitic Drag ............................................................ 14 Equation 9 - Fuselage Reynolds Number...................................................................... 14 Equation 10 - Fuselage Coefficient of Parasitic Drag .................................................... 15 Equation 11 - Induced Drag Coefficient Equation ......................................................... 15 Equation 12 - Coefficient of Lift Equation ...................................................................... 15 Equation 13 - Calculation of Variable "q" ....................................................................... 16 Equation 14 - Calculation of Coefficient of Lift ............................................................... 16 Equation 15 - Efficiency Factor Interpolation ................................................................. 16 Equation 16 - Calculation of Induced Drag Coefficient .................................................. 16 Equation 17 - Calculation of Interference Drag ............................................................. 17 Equation 18 - Compressibility Drag Relationship .......................................................... 17 Equation 19 - Total Drag Coefficient ............................................................................. 18 Equation 20 - Lift to Drag Ratio ..................................................................................... 18 Equation 21 - Drag Calculation ..................................................................................... 18
Introduction to Flight || Nicholas Conde
April 2016
_6___
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 weaponry. With the most accurate data military pilots could feel confident in their vehicle and its capabilities, a lesson that carries through to the modern day.
1.3
Plane Background There were more Spitfire Mk Vs produced than any other variant of the
Spitfire, with the plane reaching the peak of its popularity in 1942, following its successful use during WW II during the British counterattack over France. The Mk V was the first variant to be used in great volumes outside of Britain, serving as a pivotal asset in securing Malta and campaigns in North Africa. [1] Introduction to Flight || Nicholas Conde
April 2016
_7___
Flight Data 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/ft2)
20000 322 0.3104 973.27
Density (slugs/ft3) Operational Weight (lbs) Empty Weight (lbs) Kinematic Viscosity (ft2/s)
0.0012673 6650 5050 0.00026234
Temperature (Β°R) Ξ³ for air
447.43 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:
π=
π βπΎ β π β π
Equation 1 - Mach Number Formula
2.2
Vehicle Dimensions Dimensions for the vehicle were gained from WW2 Warbirds, with subsequent
dimensions determined through schematic measurements and scaling. Best estimations are used for calculations where sources could not provide further clarification.
Introduction to Flight || Nicholas Conde
April 2016
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Figure 1 - Supermarine Spitfire Mk V Scale Drawing
Introduction to Flight || Nicholas Conde
April 2016
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2.3
Wing Dimensions
Figure 2 - Supermarine Spitfire Mk V Wing
Table 2 - Wing Dimensions
Wing Dimensions Specification Span (ft) Wing Area (ft2) t/c Taper Ratio Tip Chord (ft) M.A.C (ft) Root Chord (ft) Thickness (ft) Exposed Area Swetted (ft2) Aspect Ratio
Introduction to Flight || Nicholas Conde
NACA 2213 30.833 242 0.13 0.4880 3.8940 6.1708 7.9790 0.8022 219.6370 448.0595 5.61
April 2016
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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. π = πΆπ /πΆπ 0.448 =
3.894 7.979
Equation 2 - Taper Ratio Calculation
2 π πΆπ (1 + π β ) 3 1+π 2 0.448 6.1708 = β 7.979 β (1 + 0.448 β ) 3 1 + 0.448 π. π΄. πΆ. =
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 ft2. The wetted area was then calculated using the equation below. ππ€ππ‘π‘ππ = πππ₯πππ ππ β 1.02 β 2 448.0595 = 219.637 β 1.02 β 2 Equation 4 - Wetted Surface Area Calculation
I determined the airfoil thickness based on the NACA 2213 designation. In terms of the numerical notation the 13 at the end of the 2213 signifies a max 13% thickness in relation to the chord length. Based on the M.A.C length I found that the appropriate thickness for the wing is 0.8022 feet or 9.627 inches.
Introduction to Flight || Nicholas Conde
April 2016
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2.4
Fuselage, Vertical Fin, and Horizontal Stabilizer Dimensions
Table 3 - Fuselage Dimensions
Fuselage Dimensions Length (ft) Diameter (ft) Area Wetted Area Fineness Ratio
29.917 3 89.751 183.09204 9.972333333
Table 4 - Horizontal Stabilizer Dimensions
Horizontal Stabilizers Root Chord (ft) Tip Chord (ft)
4.369 1.1864
Exposed Area (ft2) Wetted Area Taper Ratio Span t/c Sweep Angle M.A.C Aspect Ratio
34.387 70.14948 0.271549554 5.188 0.13 15.9 3.081576791 0.782718586
Table 5 - Vertical Fin Dimensions
Vertical Fin Root Chord (ft) Tip Chord (ft) Exposed Area (ft2)
4.827 1.1864 13.001
Wetted Area Taper Ratio Span t/c Sweep Angle M.A.C Aspect Ratio
26.52204 0.245784131 3.3668 0.13 36.9 3.374045383 0.871882335
Introduction to Flight || Nicholas Conde
April 2016
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Calculations 3.1 3.1.1
Parasite Drag Wing, Aerodynamic Calculation The calculation for the Parasitic Drag on the Wings is based on the following
formula from Fundamentals of Flight.
πΆπ·π =
πΎ β πΆππ β ππ€ππ‘ ππ πΈπΉ
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 can be determined from a calculated Reynolds number. Swet is the calculated wetted surface area and the reference area is the initially provided surface area of the component.
The Reynolds number for the wings can be calculated using the equation below:
π π = π£ β
πΏ π
Equation 6 - Reynolds Number Calculation
βvβ is the velocity of the aircraft at cruise altitude, L is the M.A.C length for wing calculations and π is the kinematic viscosity as originally determined in the standard day table. Plugging in the appropriate values yields the following results:
Introduction to Flight || Nicholas Conde
April 2016
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7574083.815 = 322 (
ft )β sec
6.1708 (ft)
ft 2 0.00026234 ( π )
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: 1.27 β 0.0036 β 448.0595 ππ‘ 2 0.008465 = 242 ππ‘ 2 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,
where the length is the length of the fuselage:
36720568.73 = 322 (
ft )β sec
29.917 (ft)
ft 2 0.00026234 ( π )
Equation 9 - Fuselage Reynolds Number
Based on the above Reynolds Numbers a skin friction coefficient of 0.00275 is determined. Referring to Figure 11.4 in Fundamentals of Flight a body form factor K is determined as 1.09 based on a fineness ratio of 9.9723. Though this number may be slightly skewed due to the assumption of the chart that the plane is flying at M = 0.5.
Introduction to Flight || Nicholas Conde
April 2016
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1.09 β 0.00275 β 183.092 ππ‘ 2 0.0061149 = 89.751 ππ‘ 2 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
3.2
Induced Drag Now that I have calculated the total Parasitic Drag I am able to determine the
Induced Drag on the Aircraft. The Induced Drag Formula is taken from Fundamentals of Flight and as follows:
πΆπΌπ· =
πΆπΏ2 π β π΄π β π
Equation 11 - Induced Drag Coefficient Equation
In the above equation CL is the Coefficient of Lift, AR is the aspect ratio, and e is the Airplane Efficiency Factor. The coefficient of lift can be calculated using the following equation:
πΆπΏ =
π€ 1 (2) β π β π£ 2 β π
Equation 12 - Coefficient of Lift Equation
Introduction to Flight || Nicholas Conde
April 2016
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βwβ is the operational weight of the aircraft, rho is the density of the air, v is the 1
cruise velocity, and s is the area of the wings. The variables (2) β π β π£ 2 can be simplified to the single variable q in terms of calculation. 1 65.6994 ππ/ππ‘ 2 = ( ) β 0.0012673 β 322 2 Equation 13 - Calculation of Variable "q"
0.4183 =
6650 65.6994 β 242
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 Cdp 0.2 and Cdp 0.25 for the same aspect ratio.
0.7856 = β ((
0.25 β 0.318 ) β (0.84 β 0.88)) β 0.88 0.25 β 0.2
Equation 15 - Efficiency Factor Interpolation
With the above information the coefficient of induced drag can be calculated a follows: 0.41832 0.0052847 = π β 5.61 β 0.7856 Equation 16 - Calculation of Induced Drag Coefficient
Introduction to Flight || Nicholas Conde
April 2016
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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: πΆπΌππ = πΆππ· β 0.1 0.00318 = 0.0318 β 0.1 Equation 17 - Calculation of Interference Drag
3.4
Compressibility Drag The final step in the drag analysis is to find the drag due to compressibility
effects. There are several intermediate steps, the first of which is finding the crest critical Mach number without sweep. This can be determined by using the thickness ratio and the coefficient of lift. Referring to Figure 12.7 in Fundamentals of Flight it can be seen that for the above information, Mcc Ξ = 0 = 0.69. Given the 0 sweep on the wings this is the unmodified value that will be used for continuing calculations. Given the following equation it is clear that next we need the relationship between the free-stream Mach number of crest critical Mach number: π¬πΆπ·π π0 = 3 cos π¬ πππ Equation 18 - Compressibility Drag Relationship
The ratio can be calculated as 0.3104/0.69 which is equal to 0.44986. When this number is found on the chart in Figure 12.13 of Fundamentals of Flight it is found that the value for compressibility drag is found along the asymptote, making the coefficient near negligible. Due to the low Mach number that the Spitfire Mk V cruises at, compressibility effects will be considered trivial.
Introduction to Flight || Nicholas Conde
April 2016
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3.5
Total Drag All of the calculated coefficients of drag can be calculated in order to 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: πΏ = πΆπΏ /πΆπ· π· 10.379 = 0.4183/0.0403 Equation 20 - Lift to Drag Ratio
The total Drag can be calculated as: π· = πΆπ β π β π 640.788 πππ = 0.040303 β 65.6994
ππ β 242ππ‘ 2 ππ‘ 2
Equation 21 - Drag Calculation
Introduction to Flight || Nicholas Conde
April 2016
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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 ratio of 10.379.
4.2
Reasonability I believe the ultimately my results are reasonable and fall well within the realm of
expectation. The L/D ratio is the most telling result of the above calculations, giving a value of 10.379. Given the age and size of the aircraft I believe that this result is reasonable and can be compared to the Cessna 172, developed in 1955 and of a similar size, with an L/D ratio of 10.9 [5].
4.3
Conclusion Though the final result falls within the realm of reasonability it is not without error.
Due to limitations of measuring drawings and estimating values from graphs the results are subject to variability. The results herein should be considered a reasonable approximation but for a more exact value a more detailed and inclusive analysis must be performed. If this project were to be reassessed I would seek more readily to take actual measurements of an aircraft or find more complete manuals listing specifications and performances. I would also like to perform a deeper investigation into interference drag due to the effects of rivets and other relatively large protuberances on a smaller aircraft.
Introduction to Flight || Nicholas Conde
April 2016
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APPENDIX Table 6 - Calculated Parasitic Drag for All Components
Component
Length
Wing
6.1708
Reynold Number 7574083
Fuselage Horizontal Stab.
29.917
Vertical Fin
Sweep
K
Cfi
Swet
Sref
0
1.27
0.0036
448.0595
242.0000
0.008464991
36720568
N/A
1.09
0.0027
183.0920
89.751
0.0061149
3.0816
3782372
15.9
1.25
0.0028
70.14948
34.387
0.00714
3.374045
4141353
36.9
1.24
0.004
26.52204
13.001
0.0101184
Total
Introduction to Flight || Nicholas Conde
Cdp
0.031838291
April 2016
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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 http://m-selig.ae.illinois.edu/ads/aircraft.html
[5]
Cessna Skyhawk II Performance Assessment. (n.d.). Retrieved April 20, 2016, from http://temporal.com.au/c172.pdf
Introduction to Flight || Nicholas Conde
April 2016
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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___
DRAG ANALYSIS OF A SUPERMARINE SPITFIRE MK V AT CRUISE CONDITIONS
A Project Presented by Nicholas Conde
Introduction to Flight || Nicholas Conde
April 2016
_2___
Table of Contents Introduction ..................................................................................................................... 7 1.1
Project Scope ..................................................................................................... 7
1.2
Project Importance ............................................................................................. 7
1.3
Plane Background ............................................................................................. 7
Flight Data ....................................................................................................................... 8 2.1
Standard Day ..................................................................................................... 8
2.2
Vehicle Dimensions ........................................................................................... 8
2.3
Wing Dimensions ............................................................................................. 10
2.4
Fuselage, Vertical Fin, and Horizontal Stabilizer Dimensions ........................... 12
Calculations................................................................................................................... 13 3.1
Parasite Drag ................................................................................................... 13
3.1.1
Wing, Aerodynamic 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 Drag ............................................................................................. 17
3.4
Compressibility Drag ........................................................................................ 17
3.5
Total Drag ........................................................................................................ 18
Discussion ..................................................................................................................... 19 4.1
Results ............................................................................................................. 19
4.2
Reasonability ................................................................................................... 19
4.3
Conclusion ....................................................................................................... 19
APPENDIX .................................................................................................................... 20 References .................................................................................................................... 21
Introduction to Flight || Nicholas Conde
April 2016
_3___
Table of Figures Figure 1 - Supermarine Spitfire Mk V Scale Drawing ...................................................... 9 Figure 2 - Supermarine Spitfire Mk V Wing ................................................................... 10
Introduction to Flight || Nicholas Conde
April 2016
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Table of Tables Table 1 - Standard 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 ............................................... 20
Introduction to Flight || Nicholas Conde
April 2016
_5___
Table of Equations Equation 1 - Mach Number Formula ............................................................................... 8 Equation 2 - Taper Ratio Calculation............................................................................. 11 Equation 3 - Mean Aerodynamic Chord Calculation ...................................................... 11 Equation 4 - Wetted Surface Area Calculation .............................................................. 11 Equation 5 - Parasitic Drag Equation ............................................................................ 13 Equation 6 - Reynolds Number Calculation................................................................... 13 Equation 7 - Wing Reynolds Number Calculation ......................................................... 14 Equation 8 - Wing Coefficient of Parasitic Drag ............................................................ 14 Equation 9 - Fuselage Reynolds Number...................................................................... 14 Equation 10 - Fuselage Coefficient of Parasitic Drag .................................................... 15 Equation 11 - Induced Drag Coefficient Equation ......................................................... 15 Equation 12 - Coefficient of Lift Equation ...................................................................... 15 Equation 13 - Calculation of Variable "q" ....................................................................... 16 Equation 14 - Calculation of Coefficient of Lift ............................................................... 16 Equation 15 - Efficiency Factor Interpolation ................................................................. 16 Equation 16 - Calculation of Induced Drag Coefficient .................................................. 16 Equation 17 - Calculation of Interference Drag ............................................................. 17 Equation 18 - Compressibility Drag Relationship .......................................................... 17 Equation 19 - Total Drag Coefficient ............................................................................. 18 Equation 20 - Lift to Drag Ratio ..................................................................................... 18 Equation 21 - Drag Calculation ..................................................................................... 18
Introduction to Flight || Nicholas Conde
April 2016
_6___
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 weaponry. With the most accurate data military pilots could feel confident in their vehicle and its capabilities, a lesson that carries through to the modern day.
1.3
Plane Background There were more Spitfire Mk Vs produced than any other variant of the
Spitfire, with the plane reaching the peak of its popularity in 1942, following its successful use during WW II during the British counterattack over France. The Mk V was the first variant to be used in great volumes outside of Britain, serving as a pivotal asset in securing Malta and campaigns in North Africa. [1] Introduction to Flight || Nicholas Conde
April 2016
_7___
Flight Data 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/ft2)
20000 322 0.3104 973.27
Density (slugs/ft3) Operational Weight (lbs) Empty Weight (lbs) Kinematic Viscosity (ft2/s)
0.0012673 6650 5050 0.00026234
Temperature (Β°R) Ξ³ for air
447.43 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:
π=
π βπΎ β π β π
Equation 1 - Mach Number Formula
2.2
Vehicle Dimensions Dimensions for the vehicle were gained from WW2 Warbirds, with subsequent
dimensions determined through schematic measurements and scaling. Best estimations are used for calculations where sources could not provide further clarification.
Introduction to Flight || Nicholas Conde
April 2016
_8___
Figure 1 - Supermarine Spitfire Mk V Scale Drawing
Introduction to Flight || Nicholas Conde
April 2016
_9___
2.3
Wing Dimensions
Figure 2 - Supermarine Spitfire Mk V Wing
Table 2 - Wing Dimensions
Wing Dimensions Specification Span (ft) Wing Area (ft2) t/c Taper Ratio Tip Chord (ft) M.A.C (ft) Root Chord (ft) Thickness (ft) Exposed Area Swetted (ft2) Aspect Ratio
Introduction to Flight || Nicholas Conde
NACA 2213 30.833 242 0.13 0.4880 3.8940 6.1708 7.9790 0.8022 219.6370 448.0595 5.61
April 2016
_10___
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. π = πΆπ /πΆπ 0.448 =
3.894 7.979
Equation 2 - Taper Ratio Calculation
2 π πΆπ (1 + π β ) 3 1+π 2 0.448 6.1708 = β 7.979 β (1 + 0.448 β ) 3 1 + 0.448 π. π΄. πΆ. =
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 ft2. The wetted area was then calculated using the equation below. ππ€ππ‘π‘ππ = πππ₯πππ ππ β 1.02 β 2 448.0595 = 219.637 β 1.02 β 2 Equation 4 - Wetted Surface Area Calculation
I determined the airfoil thickness based on the NACA 2213 designation. In terms of the numerical notation the 13 at the end of the 2213 signifies a max 13% thickness in relation to the chord length. Based on the M.A.C length I found that the appropriate thickness for the wing is 0.8022 feet or 9.627 inches.
Introduction to Flight || Nicholas Conde
April 2016
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2.4
Fuselage, Vertical Fin, and Horizontal Stabilizer Dimensions
Table 3 - Fuselage Dimensions
Fuselage Dimensions Length (ft) Diameter (ft) Area Wetted Area Fineness Ratio
29.917 3 89.751 183.09204 9.972333333
Table 4 - Horizontal Stabilizer Dimensions
Horizontal Stabilizers Root Chord (ft) Tip Chord (ft)
4.369 1.1864
Exposed Area (ft2) Wetted Area Taper Ratio Span t/c Sweep Angle M.A.C Aspect Ratio
34.387 70.14948 0.271549554 5.188 0.13 15.9 3.081576791 0.782718586
Table 5 - Vertical Fin Dimensions
Vertical Fin Root Chord (ft) Tip Chord (ft) Exposed Area (ft2)
4.827 1.1864 13.001
Wetted Area Taper Ratio Span t/c Sweep Angle M.A.C Aspect Ratio
26.52204 0.245784131 3.3668 0.13 36.9 3.374045383 0.871882335
Introduction to Flight || Nicholas Conde
April 2016
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Calculations 3.1 3.1.1
Parasite Drag Wing, Aerodynamic Calculation The calculation for the Parasitic Drag on the Wings is based on the following
formula from Fundamentals of Flight.
πΆπ·π =
πΎ β πΆππ β ππ€ππ‘ ππ πΈπΉ
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 can be determined from a calculated Reynolds number. Swet is the calculated wetted surface area and the reference area is the initially provided surface area of the component.
The Reynolds number for the wings can be calculated using the equation below:
π π = π£ β
πΏ π
Equation 6 - Reynolds Number Calculation
βvβ is the velocity of the aircraft at cruise altitude, L is the M.A.C length for wing calculations and π is the kinematic viscosity as originally determined in the standard day table. Plugging in the appropriate values yields the following results:
Introduction to Flight || Nicholas Conde
April 2016
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7574083.815 = 322 (
ft )β sec
6.1708 (ft)
ft 2 0.00026234 ( π )
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: 1.27 β 0.0036 β 448.0595 ππ‘ 2 0.008465 = 242 ππ‘ 2 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,
where the length is the length of the fuselage:
36720568.73 = 322 (
ft )β sec
29.917 (ft)
ft 2 0.00026234 ( π )
Equation 9 - Fuselage Reynolds Number
Based on the above Reynolds Numbers a skin friction coefficient of 0.00275 is determined. Referring to Figure 11.4 in Fundamentals of Flight a body form factor K is determined as 1.09 based on a fineness ratio of 9.9723. Though this number may be slightly skewed due to the assumption of the chart that the plane is flying at M = 0.5.
Introduction to Flight || Nicholas Conde
April 2016
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1.09 β 0.00275 β 183.092 ππ‘ 2 0.0061149 = 89.751 ππ‘ 2 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
3.2
Induced Drag Now that I have calculated the total Parasitic Drag I am able to determine the
Induced Drag on the Aircraft. The Induced Drag Formula is taken from Fundamentals of Flight and as follows:
πΆπΌπ· =
πΆπΏ2 π β π΄π β π
Equation 11 - Induced Drag Coefficient Equation
In the above equation CL is the Coefficient of Lift, AR is the aspect ratio, and e is the Airplane Efficiency Factor. The coefficient of lift can be calculated using the following equation:
πΆπΏ =
π€ 1 (2) β π β π£ 2 β π
Equation 12 - Coefficient of Lift Equation
Introduction to Flight || Nicholas Conde
April 2016
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βwβ is the operational weight of the aircraft, rho is the density of the air, v is the 1
cruise velocity, and s is the area of the wings. The variables (2) β π β π£ 2 can be simplified to the single variable q in terms of calculation. 1 65.6994 ππ/ππ‘ 2 = ( ) β 0.0012673 β 322 2 Equation 13 - Calculation of Variable "q"
0.4183 =
6650 65.6994 β 242
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 Cdp 0.2 and Cdp 0.25 for the same aspect ratio.
0.7856 = β ((
0.25 β 0.318 ) β (0.84 β 0.88)) β 0.88 0.25 β 0.2
Equation 15 - Efficiency Factor Interpolation
With the above information the coefficient of induced drag can be calculated a follows: 0.41832 0.0052847 = π β 5.61 β 0.7856 Equation 16 - Calculation of Induced Drag Coefficient
Introduction to Flight || Nicholas Conde
April 2016
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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: πΆπΌππ = πΆππ· β 0.1 0.00318 = 0.0318 β 0.1 Equation 17 - Calculation of Interference Drag
3.4
Compressibility Drag The final step in the drag analysis is to find the drag due to compressibility
effects. There are several intermediate steps, the first of which is finding the crest critical Mach number without sweep. This can be determined by using the thickness ratio and the coefficient of lift. Referring to Figure 12.7 in Fundamentals of Flight it can be seen that for the above information, Mcc Ξ = 0 = 0.69. Given the 0 sweep on the wings this is the unmodified value that will be used for continuing calculations. Given the following equation it is clear that next we need the relationship between the free-stream Mach number of crest critical Mach number: π¬πΆπ·π π0 = 3 cos π¬ πππ Equation 18 - Compressibility Drag Relationship
The ratio can be calculated as 0.3104/0.69 which is equal to 0.44986. When this number is found on the chart in Figure 12.13 of Fundamentals of Flight it is found that the value for compressibility drag is found along the asymptote, making the coefficient near negligible. Due to the low Mach number that the Spitfire Mk V cruises at, compressibility effects will be considered trivial.
Introduction to Flight || Nicholas Conde
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3.5
Total Drag All of the calculated coefficients of drag can be calculated in order to 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: πΏ = πΆπΏ /πΆπ· π· 10.379 = 0.4183/0.0403 Equation 20 - Lift to Drag Ratio
The total Drag can be calculated as: π· = πΆπ β π β π 640.788 πππ = 0.040303 β 65.6994
ππ β 242ππ‘ 2 ππ‘ 2
Equation 21 - Drag Calculation
Introduction to Flight || Nicholas Conde
April 2016
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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 ratio of 10.379.
4.2
Reasonability I believe the ultimately my results are reasonable and fall well within the realm of
expectation. The L/D ratio is the most telling result of the above calculations, giving a value of 10.379. Given the age and size of the aircraft I believe that this result is reasonable and can be compared to the Cessna 172, developed in 1955 and of a similar size, with an L/D ratio of 10.9 [5].
4.3
Conclusion Though the final result falls within the realm of reasonability it is not without error.
Due to limitations of measuring drawings and estimating values from graphs the results are subject to variability. The results herein should be considered a reasonable approximation but for a more exact value a more detailed and inclusive analysis must be performed. If this project were to be reassessed I would seek more readily to take actual measurements of an aircraft or find more complete manuals listing specifications and performances. I would also like to perform a deeper investigation into interference drag due to the effects of rivets and other relatively large protuberances on a smaller aircraft.
Introduction to Flight || Nicholas Conde
April 2016
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APPENDIX Table 6 - Calculated Parasitic Drag for All Components
Component
Length
Wing
6.1708
Reynold Number 7574083
Fuselage Horizontal Stab.
29.917
Vertical Fin
Sweep
K
Cfi
Swet
Sref
0
1.27
0.0036
448.0595
242.0000
0.008464991
36720568
N/A
1.09
0.0027
183.0920
89.751
0.0061149
3.0816
3782372
15.9
1.25
0.0028
70.14948
34.387
0.00714
3.374045
4141353
36.9
1.24
0.004
26.52204
13.001
0.0101184
Total
Introduction to Flight || Nicholas Conde
Cdp
0.031838291
April 2016
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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 http://m-selig.ae.illinois.edu/ads/aircraft.html
[5]
Cessna Skyhawk II Performance Assessment. (n.d.). Retrieved April 20, 2016, from http://temporal.com.au/c172.pdf
Introduction to Flight || Nicholas Conde
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