Drag Analysis of a Supermarine Spitfire Mk V at Cruise ... · There were more Spitfire Mk Vs...
Transcript of Drag Analysis of a Supermarine Spitfire Mk V at Cruise ... · There were more Spitfire Mk Vs...
Introduction to Flight || Nicholas Conde April 2016 _1___
Introduction to Flight
Aircraft Drag Project – April 2016
2016
Drag Analysis of a Supermarine
Spitfire Mk V at Cruise Conditions
Nicholas Conde
U66182304
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DRAG ANALYSIS OF A SUPERMARINE SPITFIRE MK V AT CRUISE CONDITIONS
A Project Presented
by
Nicholas Conde
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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
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Table of Figures
Figure 1 - Supermarine Spitfire Mk V Scale Drawing ...................................................... 9
Figure 2 - Supermarine Spitfire Mk V Wing ................................................................... 10
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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
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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
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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]
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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) 20000
Cruise Velocity (ft/sec) 322
Mach Number 0.3104
Pressure (lb/ft2) 973.27
Density (slugs/ft3) 0.0012673
Operational Weight (lbs) 6650
Empty Weight (lbs) 5050
Kinematic Viscosity (ft2/s) 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:
𝑀 =𝑉
√𝛾 ∗ 𝑅 ∗ 𝑇
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.
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Figure 1 - Supermarine Spitfire Mk V Scale Drawing
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2.3 Wing Dimensions
Figure 2 - Supermarine Spitfire Mk V Wing
Table 2 - Wing Dimensions
Wing Dimensions
Specification NACA 2213
Span (ft) 30.833
Wing Area (ft2) 242
t/c 0.13
Taper Ratio 0.4880
Tip Chord (ft) 3.8940
M.A.C (ft) 6.1708
Root Chord (ft) 7.9790
Thickness (ft) 0.8022
Exposed Area 219.6370
Swetted (ft2) 448.0595
Aspect Ratio 5.61
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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.
𝜎 = 𝐶𝑇/𝐶𝑅
0.448 =3.894
7.979
Equation 2 - Taper Ratio Calculation
𝑀. 𝐴. 𝐶. =2
3 𝐶𝑅(1 + 𝜎 −
𝜎
1 + 𝜎)
6.1708 =2
3∗ 7.979 ∗ (1 + 0.448 −
0.448
1 + 0.448)
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.
𝑆𝑤𝑒𝑡𝑡𝑒𝑑 = 𝑆𝑒𝑥𝑝𝑜𝑠𝑒𝑑 ∗ 1.02 ∗ 2
448.0595 = 219.637 ∗ 1.02 ∗ 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.
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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 15.9
M.A.C 3.081576791
Aspect Ratio 0.782718586
Table 5 - Vertical Fin Dimensions
Vertical Fin
Root Chord (ft) 4.827
Tip Chord (ft) 1.1864
Exposed Area (ft2) 13.001
Wetted Area 26.52204
Taper Ratio 0.245784131
Span 3.3668
t/c 0.13
Sweep Angle 36.9
M.A.C 3.374045383
Aspect Ratio 0.871882335
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Calculations
3.1 Parasite Drag
3.1.1 Wing, Aerodynamic Calculation
The calculation for the Parasitic Drag on the Wings is based on the following
formula from Fundamentals of Flight.
𝐶𝐷𝑃 =𝐾 ∗ 𝐶𝑓𝑖 ∗ 𝑆𝑤𝑒𝑡
𝑆𝑅𝐸𝐹
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:
𝑅𝑒 = 𝑣 ∗𝐿
𝜈
Equation 6 - Reynolds Number Calculation
“v” is the velocity of the aircraft at cruise altitude, L is the M.A.C length for wing
calculations and 𝜈 is the kinematic viscosity as originally determined in the standard day
table. Plugging in the appropriate values yields the following results:
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7574083.815 = 322 (ft
sec) ∗
6.1708 (ft)
0.00026234 (ft2
𝑠 )
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:
0.008465 =1.27 ∗ 0.0036 ∗ 448.0595 𝑓𝑡2
242 𝑓𝑡2
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
sec) ∗
29.917 (ft)
0.00026234 (ft2
𝑠 )
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.
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0.0061149 =1.09 ∗ 0.00275 ∗ 183.092 𝑓𝑡2
89.751 𝑓𝑡2
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:
𝐶𝐼𝐷 =𝐶𝐿
2
𝜋 ∗ 𝐴𝑅 ∗ 𝑒
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:
𝐶𝐿 =𝑤
(12) ∗ 𝜌 ∗ 𝑣2 ∗ 𝑠
Equation 12 - Coefficient of Lift Equation
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“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 (1
2) ∗ 𝜌 ∗ 𝑣2 can be
simplified to the single variable q in terms of calculation.
65.6994 𝑙𝑏/𝑓𝑡2 = (1
2) ∗ 0.0012673 ∗ 322
Equation 13 - Calculation of Variable "q"
0.4183 =6650
65.6994 ∗ 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 = − ((0.25 − 0.318
0.25 − 0.2) ∗ (0.84 − 0.88)) − 0.88
Equation 15 - Efficiency Factor Interpolation
With the above information the coefficient of induced drag can be calculated a
follows:
0.0052847 =0.41832
𝜋 ∗ 5.61 ∗ 0.7856
Equation 16 - Calculation of Induced Drag Coefficient
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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:
𝐶𝐼𝑁𝑇 = 𝐶𝑃𝐷 ∗ 0.1
0.00318 = 0.0318 ∗ 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 Λ = 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:
𝛬𝐶𝐷𝑐
cos3𝛬 =
𝑀0
𝑀𝑐𝑐
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.
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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:
𝐿
𝐷= 𝐶𝐿/𝐶𝐷
10.379 = 0.4183/0.0403
Equation 20 - Lift to Drag Ratio
The total Drag can be calculated as:
𝐷 = 𝐶𝑑 ∗ 𝑞 ∗ 𝑠
640.788 𝑙𝑏𝑠 = 0.040303 ∗ 65.6994𝑙𝑏
𝑓𝑡2∗ 242𝑓𝑡2
Equation 21 - Drag Calculation
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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.
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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
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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