# How do I choose Cl during cruise flight with a given MTOM and variable S?

Currently I'm challenging myself to build an airplane from scratch. I have a background in aeronautical engineering but I have no experience in actually designing an aircraft from the ground up. (FYI: I want to build an unswept flying wing - yes I know, I can already hear some of you asking why but please just accept the fact that it's the most fun for me).

However, I am struggling to find a starting point. I figured it would be a good idea to start with the facts I have. Although not much, I know that:

1. The MTOM is going to be: $$m_{MTOM} \leq 250 \ \text{g}$$ (for regulation reasons),
2. The cruise speed will (very roughly) be: $$v_{cruise} \approx [40 \dots 50] \ \frac{\text{km}}{\text{h}} = [11 \dots 14] \ \frac{\text{m}}{\text{s}}$$ (I'm going to propel it with a either a T4x2x3 or T4x2.5 propeller powered by a 2S Li-Ion battery and a 1404 3800 KV BLDC motor),
3. The plane should be as efficient as possible with a feasible but scientific design process.

To get started on designing a suitable wing I thought it would make sense to select a fitting airfoil first. I read a lot about flying wings and what stuck with me is that for an unswept flying wing I need an airfoil with reflex. Three candidates I came across are:

Now I ask myself: how do I select at what $$c_l$$ I fly during cruise? What is the best planform for the resulting surface area $$S$$ to lift my plane? I.e. how do I choose $$AR$$ , $$\lambda$$ , etc.? What angle of incidents would make the most sense? Is this even the right starting point?

• You mean something like a Northrop? Commented Apr 14 at 20:08
• I plan on designing an unswept flying wing. More like the Backstrom EPB-1. At RC scale it would probably come close to the Nano Goblin. Commented Apr 14 at 20:33

I'd start calculating the lift curve created by the wing. It can be simply estimated from the 2D curve (the one of the airfoil) using for example the subsonic equation that you find in this answer. Use the correct Reynolds number when you choose the airfoil plots.

What is the best planform for the resulting surface area S to lift my plane? I.e. how do I choose AR, λ, etc.?

You can play with the parameters in that equation to see how the lift changes accordingly. Anyway the biggest limitation of wing geometry is going to arise from the trim analysis (see later).

how do I select at what $$C_l$$ I fly during cruise?

The target lift is the one where your plane is going to fly the most. This really depends on how you are going to fly it. Once you have defined the lift $$L$$, you can get the relevant coefficient simply using its definition: $$C_L=\frac{L}{½\rho V² S}$$. Obviously you have to check that your wing can deliver enough lift to takeoff, land and manoeuvre as well.

Is this even the right starting point?

Afterwards I'd perform a longitudinal trim and stability analysis: this answer should contain all you need for that; just set the contribution from the tailplane and the fuselage to zero (since you don't have them).

• From the trim analysis you'll get that changing the pitching moment of the wing is the only way to trim your airplane: its value and its variation with the deflection of the control surfaces can be estimated with the methods contained in the USAF STABILITY AND CONTROL DATCOM (chapter 4.1.4 and 6.1.5 respectively);
• From the longitudinal stability analysis you'll simply get that the CG must be in front of the quarter chord point.

As said, you can use the results from this last step together with the results from the lift's calculation to revise/design the planform of the wing.

Chapter 4.1.5 of the DATCOM can be used to estimate the wing drag and therefore the needed power. I'd use a rear-mounted propeller since this location stabilises the aircraft in pitch and yaw.

Finally you can do a lateral trim and stability analysis to design the vertical tailplane.

• Thank you very much for your answer! I understand that you want me to basically play with the subsonic equations in the second figure in the answer you quoted. Is capital lambda the same as the lower case one (taper angle)? What are good ways to calculate a factor for Cl for take off and landing once I settled on a cruise Cl? Do you know a good resource to learn about lateral trim and stability analysis? I would also appreciate a source for why a rear prop stabilizes in pitch and yaw. I'm new to this page so please point out if this is too much for a comment, etc. Commented Apr 17 at 18:57
• I'd suggest some standard "general airplane design" textbook, like the one by Raymer or Roskam or Torenbeek. Note that those are all textbooks about full-scale airplanes: all methods/equations based on statistics are therefore strictly valid only for that class of airplanes (MTOW or CG estimation for example). Anyway what is based on physics/aerodynamics is still valid also for model airplanes (lift or drag or stability estimation for example). The DATCOM contains too methods to estimate the wing $C_L$ for straight subsonic wings, you can use it instead of the equation in the other answer. Commented Apr 18 at 6:56
• Thanks! I will look into it. Commented Apr 18 at 8:43
• Sorry, the capital Lambda is actually the aspect ratio. I did not convert the equations fully to that strange US habit of using different symbols when adopting equations. Commented May 19 at 9:01

You state a plank wing but then go on to ask about planform and airfoil options, a plank is a flat rectangle.

So in general: Start with a reasonable average Cl for the class of airfoil you are using to get an approximate required wing area for basic structural design. If you use a pulling propeller the air flow will add airflow over the wing slightly increasing lift.

For subsonic aerodynamics an elliptical planform(more precisely elliptical lift distribution) is considered the most efficient, with the highest aspect ratio practical for the structural limits of materials, airfoil thickness ratio(bending and torsion strengths), construction methods, and mass distribution(eg Tanks/electric batteries inside wings vs fuselage). This minimizes induced drag. However, this is difficult to stabilize in a flying wing form factor, swept wings/delta are preferred for flying wings because the pitch control is easier (effectively blending a wing and tail together).

Use the Cl that occurs at the angle of attack with the lift-to-drag ratio for your complete wing (The 2 dimensional airfoil optimum is minimum parasite drag, whole wing is induced drag plus parasite). Then scale the whole wing slightly up or down from the original area approximation to get the required total lift at the desired cruise speed. The best L/D can be obtained from a drag polar if you have accurate simulation or wind tunnel data that can be adjusted for reynolds number.

Note this method does not necessarily account for other characteristics that may be desired such as takeoff and landing performance(including crosswind), maneuvering and gust safety factors(eg stall warning and spin recovery), oscillations, taxi-area wingspan limits, etc.

• I appreciate your answer, thanks! I thought a "plank" referred to any unswept wing planform. I guess it also rules out taper. I'm going to rephrase the question. I also thought of an elliptical planform or any planform with an elliptical circulation distribution. What do you think are reasons that the nano goblin doesn't have an elliptical planform besides ease of manufacturing? Use the Cl that occurs at the angle of attack with the lift-to-drag ratio for your complete wing. I would appreciate if you could elaborate this point since I don't fully understand it. Commented Apr 17 at 19:10

You might find Lee Nicolai's white paper "Estimating R/C Model Aerodynamics and Performance" a useful guide and reference. Look under "series resources/design event resources" at saeaerodesign.com

• Thanks for the resource! I’m currently reading “Schwanzlose Flugzeuge: Ihre Auslegung und ihre Eigenschaften”. It’s obviously in german but I can recommend it to anyone stumbling over this post. I’m learning a lot. Hope to some day be finally able to answer this question. Commented May 19 at 19:00