Drag Analysis of a Supermarine Spitfire Mk V at Cruise ... · There were more Spitfire Mk Vs produced than any other variant of the Spitfire, with the plane reaching the peak of its

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

[email protected]

U66182304


DRAG ANALYSIS OF A SUPERMARINE SPITFIRE MK V AT CRUISE CONDITIONS

A Project Presented

by

Nicholas Conde


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


Table of Figures

Figure 1 - Supermarine Spitfire Mk V Scale Drawing ...................................................... 9

Figure 2 - Supermarine Spitfire Mk V Wing ................................................................... 10


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


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

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]


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.


Figure 1 - Supermarine Spitfire Mk V Scale Drawing


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


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.


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



Exposed Area (ft2) 34.387


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



Exposed Area (ft2) 13.001


Taper Ratio 0.245784131

Span 3.3668

t/c 0.13

Sweep Angle 36.9

M.A.C 3.374045383

Aspect Ratio 0.871882335


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:


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.


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


“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


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.


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


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.


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

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

Documents

Drag Analysis of a Supermarine Spitfire Mk V at Cruise ... · There were more Spitfire Mk Vs produced than any other variant of the Spitfire, with the plane reaching the peak of its