Back

Free Introductory UAV Guide

Receive your free introductory guide on UAV/Small Aircraft Design.

DOWNLOAD

It's a Drag

Issue 2, 25 July 2022

By: Anthony O. Ives

\[ \]

Drag is a force produced as an object moves through a fluid. The picture below illustrates the four forces acting on an aircraft:

Aircraft Forces

In order to determine the thrust required to power your aircraft you need to determine drag. Drag is calculated using a similar equation to the lift coefficient equation:

\[D=q C_D S\]

Where q is dynamic pressure and S is wing area as explained in Ref [2]. Drag is different from lift as it acts on any surface of the aircraft exposed to airflow whereas lift only acts on one side of the wing surface. Hence S is used a reference area for convenience instead of using the wetted area, A which would be the entire airflow exposed surface of the aircraft. So the ratio of wetted area to reference area, S/A is taken into account in calculating the total aircraft drag coefficient.

Drag coefficient equation can be used to give a summary of the main factors influencing drag as was done Ref [2] for lift. It can also be seen that much of opposite is needed to reduce drag compared to what is needed to increase lift:

  1. Low air density means lower drag hence why airliners cruise at higher altitudes to take advantage this, the reduction in drag is of more benefit as the higher speed in cruise produces enough lift.

  2. A small lift coefficient reduces drag for a number of reasons but it especially reduces lift dependent drag as seen in the drag polar equation discussed later.

  3. Less exposed surface area is better for drag as it reduces it.

CD is drag coefficient which is further defined by following equation which is known as the drag polar equation:

\[C_D=C_{D0} + k C_L^2\]

With CD0 being the drag coefficient produced as result of drag which is independent of lift hence referred to as the zero lift drag coefficient. Wings also produce drag as by product of lift hence this is calculated by the term on the right. k is the drag factor which calculated as below:

\[k=\frac{1}{\pi A_r} \]

Ar is the aspect ratio of the wing which is the wingspan divided by its average chord or width.

The drag of an aircraft in cruise could be estimated by assuming it is about 75% of the maximum thrust of the combined thrust of all engines. CD0 commonly referred to as the zero lift drag coefficient which could be estimated by comparing with information and calculations based on similar aircraft as in the following table:

Property General Aviation Aircraft Light Utilty Aircraft Miltary Combat Aircraft Miltary Transport Aircraft Commercial Airliner WW2 Miltary Combat Aircraft
Velocity, V/ms-1 62.8 58.1 388.9 250 242 94.3
Wing Area, S/m2 16.2 16.58 78.8 300 112.3 31.9
Altitude, h/ft 5000 5000 35000 35000 35000 5000
Density, ρ/kgm-3 1.056 1.056 0.379 0.379 0.379 1.056
Dynamic Pressure, q/Pa 2082 1782 28660 11844 11098 4695
Mass, m;/kg 1111 794 35000 195000 63100 4336
Lift Coefficient CL .0.323 0.264 0.152 0.538 0.497 0.284
Thrust, T/N N/A N/A 176600 470800 168200 N/A
Power, P/W 120000 110000 N/A N/A N/A 820000
Drag, D/N 0.75*T 0.75*T 0.75*T 0.75*T 0.75*T 0.75*T
Drag Coefficient, CD 0.042 0.048 0.059 0.099 0.101 0.044
Wing Aspect Ratio, Ar 7.32 7 2.52 8.5 10.97 7.117
Lift Dependent Drag Factor, k 0.043 0.045 0.126 0.037 0.029 0.045
Zero Lift Drag Coefficient CD0 0.037 0.045 0.056 0.088 0.094 0.04

Thrust can estimated from power by dividing it by the cruise speed therefore, CD can be calculated using an assumed value of CD0 from a similar aircraft type as in the following table:

Symbol Property Example Value Units
π Mathematical constant 3.14159265 None
Ar Wing aspect ratio 6 None
k Drag factor 1/(3.14159265x6)=0.0531 None
CL Lift coefficent 0.2 None
CL2 Lift Coefficient squared 0.2x0.2=0.04 None
CD0 Zero Lift Drag coefficent 0.06 None
CD Drag coefficent 0.06+(0.0531x0.04)=0.062 None
q Dynamic pressure 137.88 Pa
S Wing area 0.736 m2
D Drag 0.062x137.88x0.736=6.292 N
V Aircraft velocity 15 ms-1
η Propulsive efficiency 0.7 None
P Propulsive power required (1/0.7)*6.292*15=134 W

Assuming that zero lift drag is produced only by skin friction in boundary layers on the aircraft surface, CD0 can also be calculated using the graph below using Reynolds number:

CD vs Re Graph

\[Re=\frac{V c}{\nu}\]

ν is the kinematic viscosity which is the dynamic viscosity divided by the air density. Dynamic viscosity, μ is 0.000017 Pas. c is a characteristic dimension. An aircraft speed, V of 15ms-1 is assumed.

The equations given on the graph could also be used directly to determine drag coefficient from Reynolds number. The graph is for airflow on one side of a flat surface or plate, however the CD0 can be very similar for curved surfaces if they are reasonably parallel with the airflow direction. The boundary layer is the name given to a small layer of air flow close to the surface where the majority of friction forces occurs. There is two types of boundary layers laminar and turbulent. As can be seen, drag coefficients for both laminar and turbulent boundary layers can be calculated from the graph based on Reynolds number. Generally speaking for small aircraft you would expect the boundary layers to be laminar. A later article will discuss boundary layers in more detail.

In order to calculate the aircraft CD0, it is necessary to calculate CD0 for each individual part of the aircraft as in the following table:

Property Wing Fuselage Tail Fin Total Aircraft
Characteristic Dimension, c Wing Chord, c=0.35m Fuselage Length, l=1.6m Tailplane Chord, cT=0.136m Tailfin Chord, cF=0.109m Use Wing Chord as Reference Length
Reynolds No, Re 3.088x105 14.12x105 1.2x105 0.962x105 Use Wing Reynolds No as Reference
Area Ratio, A/S 2 1.168 0.299 0.160 1 (As Wing Area is used as Reference Area)
Drag Coefficent, CD0 0.0018 0.00084 0.0029 0.0032 0.00596

After you have determined CD0 for each part of the aircraft you use the following equation to work out the overall CD0:

\[C_{D0(Total)}=C_{D0(Wing)} \frac{A_{Wing}}{S} \]

\[+ C_{D0(Fuselage)} \frac{A_{Fuselage}}{S} \]

\[+ C_{D0(Tail)} \frac{A_{Tail}}{S}\]

\[+ C_{D0(Fin)} \frac{A_{Fin}}{S}\]

CD0(Total)=(0.0018x2)

+(0.00084x1.168)

+(0.0029x0.299)

+( 0.0032x0.160)

Therefore: CD0(Total)=0.00596

The calculated drag coefficient is a lot lower than the assumed value based on a similar aircraft this is possibly due to the calculated value not taking account of separated boundary layer flow which would significantly increase drag. The assumed value is also based on a larger scale aircraft and a 75% thrust assumption so there may an overestimation for it as well, so the actual drag coefficient could be some where in between. Ref [1], [3] and [4] give more detail in estimating drag coefficient. A later article hopes get into more detail in explaining the different types of drag and boundary layers, but you should now have some understanding of the drag coefficient equation and how you can use to estimate thrust.

Please leave a comment on my facebook page or via email and let me know if you understand how to use the drag coefficient equations to estimate thrust.

References:

[1] Fundamentals of Aerodynamics, John D. Anderson Jr., 3rd Edition, 2001, McGraw Hill

[2] http://www.eiteog.com/No1EiteogBlogLiftCL.html

[3] Aircraft Performance & Design, John D. Anderson Jr., 1999, McGraw Hill

[4] Civil Jet Aircraft Design, Lloyd R. Jenkinson, Paul Simpkin, Darren Rhodes, 1999, Butterworth Heinemann

Powered by MathJax

email icon Facebook

Disclaimer: Eiteog makes every effort to provide information which is as accurate as possible. Eiteog will not be responsible for any liability, loss or risk incurred as a result of the use and application of information on its website or in its products. None of the information on Eiteog's website or in its products supersedes any information contained in documents or procedures issued by relevant aviation authorities, manufacturers, flight schools or the operators of aircraft, UAVs.

For any inquires contact: [email protected]

copyright © Eiteog 2022