Drag Analysis of a Supermarine. Spitfire Mk V at Cruise Conditions
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1 Introduction to Flight Aircraft Drag Project April Drag Analysis of a Supermarine Spitfire Mk V at Cruise Conditions Nicholas Conde nicholasconde@gmail.com U Introduction to Flight Nicholas Conde April 2016 _1
2 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
3 Table of Contents Introduction Project Scope Project Importance Plane Background... 7 Flight Data Standard Day Vehicle Dimensions Wing Dimensions Fuselage, Vertical Fin, and Horizontal Stabilizer Dimensions Calculations Parasite Drag Wing, Aerodynamic Calculation Fuselage, Blunt Body Calculation Total Parasitic Drag Induced Drag Interference Drag Compressibility Drag Total Drag Discussion Results Reasonability Conclusion APPENDIX References Introduction to Flight Nicholas Conde April 2016 _3
4 Table of Figures Figure 1 - Supermarine Spitfire Mk V Scale Drawing... 9 Figure 2 - Supermarine Spitfire Mk V Wing Introduction to Flight Nicholas Conde April 2016 _4
5 Table of Tables Table 1 - Standard Day Values for Spitfire Mk V at Cruise Conditions... 8 Table 2 - Wing Dimensions Table 3 - Fuselage Dimensions Table 4 - Horizontal Stabilizer Dimensions Table 5 - Vertical Fin Dimensions Table 6 - Calculated Parasitic Drag for All Components Introduction to Flight Nicholas Conde April 2016 _5
6 Table of Equations Equation 1 - Mach Number Formula... 8 Equation 2 - Taper Ratio Calculation Equation 3 - Mean Aerodynamic Chord Calculation Equation 4 - Wetted Surface Area Calculation Equation 5 - Parasitic Drag Equation Equation 6 - Reynolds Number Calculation Equation 7 - Wing Reynolds Number Calculation Equation 8 - Wing Coefficient of Parasitic Drag Equation 9 - Fuselage Reynolds Number Equation 10 - Fuselage Coefficient of Parasitic Drag Equation 11 - Induced Drag Coefficient Equation Equation 12 - Coefficient of Lift Equation Equation 13 - Calculation of Variable "q" Equation 14 - Calculation of Coefficient of Lift Equation 15 - Efficiency Factor Interpolation Equation 16 - Calculation of Induced Drag Coefficient Equation 17 - Calculation of Interference Drag Equation 18 - Compressibility Drag Relationship Equation 19 - Total Drag Coefficient Equation 20 - Lift to Drag Ratio Equation 21 - Drag Calculation Introduction to Flight Nicholas Conde April 2016 _6
7 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
8 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) 322 Mach Number Pressure (lb/ft 2 ) Density (slugs/ft 3 ) Operational Weight (lbs) 6650 Empty Weight (lbs) 5050 Kinematic Viscosity (ft 2 /s) Temperature ( R) 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: V M = 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. Introduction to Flight Nicholas Conde April 2016 _8
9 Figure 1 - Supermarine Spitfire Mk V Scale Drawing Introduction to Flight Nicholas Conde April 2016 _9
10 2.3 Wing Dimensions Figure 2 - Supermarine Spitfire Mk V Wing Table 2 - Wing Dimensions Wing Dimensions Specification NACA 2213 Span (ft) Wing Area (ft 2 ) 242 t/c 0.13 Taper Ratio Tip Chord (ft) M.A.C (ft) Root Chord (ft) Thickness (ft) Exposed Area S wetted (ft 2 ) Aspect Ratio 5.61 Introduction to Flight Nicholas Conde April 2016 _10
11 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 = C T /C R = Equation 2 - Taper Ratio Calculation = 2 3 M. A. C. = 2 3 C R(1 + σ σ 1 + σ ) ( ) 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 ft 2. The wetted area was then calculated using the equation below. S wetted = S exposed = 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 feet or inches. Introduction to Flight Nicholas Conde April 2016 _11
12 2.4 Fuselage, Vertical Fin, and Horizontal Stabilizer Dimensions Table 3 - Fuselage Dimensions Fuselage Dimensions Length (ft) Diameter (ft) 3 Area Wetted Area Fineness Ratio Table 4 - Horizontal Stabilizer Dimensions Horizontal Stabilizers Root Chord (ft) Tip Chord (ft) Exposed Area (ft 2 ) Wetted Area Taper Ratio Span t/c 0.13 Sweep Angle 15.9 M.A.C Aspect Ratio Table 5 - Vertical Fin Dimensions Vertical Fin Root Chord (ft) Tip Chord (ft) Exposed Area (ft 2 ) Wetted Area Taper Ratio Span t/c 0.13 Sweep Angle 36.9 M.A.C Aspect Ratio Introduction to Flight Nicholas Conde April 2016 _12
13 Calculations 3.1 Parasite Drag Wing, Aerodynamic Calculation The calculation for the Parasitic Drag on the Wings is based on the following formula from Fundamentals of Flight. C DP = K C fi S wet S REF 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: Introduction to Flight Nicholas Conde April 2016 _13
14 = 322 ( ft sec ) (ft) ( 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 The skin friction coefficient was found to be given the Reynolds Number. Using the Parasitic Drag Equation the Wing Parasitic Drag Coefficient can thereby be calculated as: = ft2 242 ft 2 Equation 8 - Wing Coefficient of Parasitic Drag 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: = 322 ( ft sec ) (ft) ( ft2 s ) Equation 9 - Fuselage Reynolds Number Based on the above Reynolds Numbers a skin friction coefficient of 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 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 _14
15 = ft ft 2 Equation 10 - Fuselage Coefficient of Parasitic Drag 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 = CDP FUSEALGE = CDP VERTICAL FIN = CDP HOR. STAB. = CDP = 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: C ID = C L 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: w C L = ( 1 2 ) ρ v2 s Equation 12 - Coefficient of Lift Equation Introduction to Flight Nicholas Conde April 2016 _15
16 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 lb/ft 2 = ( 1 ) Equation 13 - Calculation of Variable "q" = 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 = (( ) ( )) Equation 15 - Efficiency Factor Interpolation With the above information the coefficient of induced drag can be calculated a follows: = π Equation 16 - Calculation of Induced Drag Coefficient Introduction to Flight Nicholas Conde April 2016 _16
17 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: C INT = C PD = 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 = 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: ΛC Dc cos 3 Λ = M 0 Mcc Equation 18 - Compressibility Drag Relationship The ratio can be calculated as /0.69 which is equal to When this number is found on the chart in Figure 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 _17
18 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: = Equation 19 - Total Drag Coefficient The Lift to Drag Ratio can be calculated given the total coefficient of drag: L D = C L/C D = / Equation 20 - Lift to Drag Ratio The total Drag can be calculated as: D = C d q s lbs = lb ft 2 242ft2 Equation 21 - Drag Calculation Introduction to Flight Nicholas Conde April 2016 _18
19 Discussion 4.1 Results Ultimately it was found that overall the Supermarine Spitfire Mk V has a Coefficient of Drag of 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 pounds with a Lift to Drag ratio of 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 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 _19
20 APPENDIX Table 6 - Calculated Parasitic Drag for All Components Component Length Reynold Number Sweep K Cfi Swet Sref Cdp Wing Fuselage N/A Horizontal Stab Vertical Fin Total Introduction to Flight Nicholas Conde April 2016 _20
21 References [1] Supermarine Spitfire Mk V. (n.d.). Retrieved April 20, 2016, from [2] The Supermarine Spitfire. (n.d.). Retrieved April 20, 2016, from [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 [5] Cessna Skyhawk II Performance Assessment. (n.d.). Retrieved April 20, 2016, from Introduction to Flight Nicholas Conde April 2016 _21
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