# Conundrum about designing a glider relating to coefficient of lift

I'm designing a high-altitude glider. Just to clarify a few points before my question:

• The glider will weigh close to 200 grams.
• It will be using the A18 airfoil.
• The glider is dropped from 80,000 feet.
• The pitch should not need to change in flight, it should be equal to the glider ratio.
• Because of this, the elevator should not be needed to be used in flight.
• Correct me if I'm wrong on the last two points!

I'm doing some CFD analysis in SimScale to calculate the lift coefficient and the drag coefficient to calculate the glide ratio of my glider. SimScale requires a speed for the glider, which makes a lot of sense. This is all good because the coefficient of drag and lift shouldn't change with different speeds, so I don't need to know the speed beforehand. That being said, I understand that you can commonly find the speed that your glider should be flying at by multiplying the stall speed by 1.3. However, something that does change the coefficient of lift is the AoA, which is determined from Reynolds number which is derived from number of things, including speed. So, to calculate the coefficient of lift I need to know the AoA, but to calculate the AoA I need to know the coefficient of lift to calculate the stall speed.

How do I get around this?

I'm probably missing something, please excuse my ignorance because I'm new here!

• This is all hypothetical, yes? Or is somebody actually taking a 200 gm (7 oz) model airplane up to 80,000' to drop it into imprecisely known winds, almost certainly never to be seen again? While certainly possible, that seems a bit expensive, and to what purpose? So is this intended to actually fly, or just design it & that's that?
– Ralph J
Apr 8 at 3:31
• Highly related: aviation.stackexchange.com/q/104486/7532 and aviation.stackexchange.com/q/104580/7532 It's not clear how this question is materially different from these others.
– Ralph J
Apr 8 at 3:34
• @RalphJ This is certainly happening. No, it isn't very expensive either. Everything is under a thousand dollars. Apr 8 at 16:58
• Before putting something up to 80 000 feet in the hope of flying it all the way down, I would start with removing all the zeros and work your way up from there. Build the thing, get it flying from 8ft. If that goes well, next step is to master 80ft. Then 800ft, 8000ft and finally FL800. Somewhere between 8 and 80ft you'll figure out the trim setting, without CFD, AoA and Reynolds Apr 9 at 20:31
• @DeltaLima That is a good idea, but I would like to figure out everything with math to find the most efficient way possible to do this. Apr 10 at 11:03

To start with, you can ignore changes due to compressibility (Mach number) and viscosity (Reynolds number).

Your aircraft will have a drag polar approximately of the form:

$$C_D=C_{D,0}+K\,{C_L}^2$$

You can do your aerodynamic analysis to determine the $$C_{D,0}$$ and $$K$$ coefficients.

This form of drag polar achieves best L/D when $$C_{D,0}$$ = $$K\,{C_L}^2$$. You should be able to derive this, it is fun. For now, you can just trust me.

From that, you can solve for the lift coefficient at best L/D. This will be:

$${C_L}^*= \sqrt{\frac{C_{D,0}}{K}}$$

This tells you the lift coefficient you must fly at to achieve best L/D. This corresponds to some angle of attack, but I don't care about the numeric value of angle of attack just yet.

In steady flight, the lift coefficient of an airplane is:

$$C_L=\frac{W\,cos(\theta)}{q\,S_{ref}}$$

Your glide slope will be shallow, so we can make the small angle approximation $$cos(\theta)=1$$. If you want, we can revisit this later.

$$C_L=\frac{W}{q\,S_{ref}}$$

Here, $$q$$ is the dynamic pressure $$q=0.5*\rho*V^2$$, $$W$$ is the weight, and $$S_{ref}$$ is the reference area of the wing.

For a given wing loading ($$W/S_{ref}$$), your aircraft will fly at best L/D at a specific value of dynamic pressure $$q$$.

This $$q$$ corresponds to a single value of equivalent airspeed. Or, if you want to work in terms of true airspeed, the TAS will vary with altitude.

I suggest you pick a typical altitude to do you calculations around. Later, you can come back and check how much changing altitude changes things.

(For the record, changing Mach and Reynolds number (which are altitude effects) will change the $$C_{D,0}$$ and $$K$$ coefficients slightly.)

OK, so now we know that we want to match a desired $${C_L}^*$$ with a speed and altitude in the form of dynamic pressure $$q$$ and wing loading $$W/S_{ref}$$.

While we are designing the aircraft, we have control of $$S_{ref}$$ in order to get the desired cruise $$q$$ to match $${C_L}^*$$.

Once the aircraft is designed and built, we can't change wing loading any more (except by adding ballast). So, then we must choose $$q$$ to get the desired $${C_L}^*$$.

Now we'll assume you have all this worked out -- your wing is sized ($$S_{ref}$$) to make your aircraft $$W$$, $$C_{D,0}$$ and $$K$$ reach best $$L/D$$ at the desired dynamic pressure $$q$$.

The next step is to trim the aircraft to actually fly at the desired $$C_L$$. For this you need a horizontal tail / elevator or control of the CG.

Since you want to fly at the same $$C_L$$ for the entire flight, you won't need to change this in-flight (ignoring Mach and Reynolds number effects), but you do need to figure out how to get the airplane to trim where you want it.

This is another question, so ask for more if you need it.

The short version is to do a simple linear trim problem to achieve the desired $$C_L$$ at $$C_M=0$$

Edit:

Summary:

1. The drag polar of the aircraft ($$C_{D,0}$$, $$K$$), has a single $${C_L}^*$$ where it achieves best L/D.

2. Start by estimating the drag polar for an assumed wing area / design.

3. Once you know the polar, $${C_L}^*$$ is easy to calculate.

4. The flight condition can be thought of as a combination of Velocity $$V_{TAS}$$ and density $$\rho$$ (two quantities) or as the dynamic pressure $$q$$ (one quantity) or equivalently $$V_{EAS}$$. I will use $$q$$.

5. The $$C_L$$ that you fly at is determined by the weight $$W$$, the wing area $$S_{ref}$$ (combined into the wing loading $$W/S_{ref}$$), and the flight condition $$q$$.

6. As a designer, I can have a target flight condition $$q$$ and I can control the wing loading $$W/S_{ref}$$ to make that flight condition align with $${C_L}^*$$.

7. Once the drag polar and wing loading are determined, I do not have choice in flight condition -- there is a single flight condition where the aircraft will achieve $${C_L}^*$$. We're stuck with what you get there.

8. Even after all this, you must also trim the aircraft such that $$C_M=0$$ at the desired $${C_L}^*$$. Trim can be achieved by deflecting control surfaces, moving the wing, moving the CG, etc.

• Great post, thank you for going in depth. I'm a bit confused though; what does the outcome of all this mean? Apr 10 at 20:56
• "Start by estimating the drag polar for an assumed wing area / design." - How does one do that? (Thanks for the edit, BTW). Apr 11 at 0:09
• You could run your CFD code at three angles of attack and fit a parabola to CD vs. CL and a line to CL vs. Alpha. Everything I said above also applies for a polar of the form $C_D=C_{D,0}+K_1\,C_L+K_2\,{C_L}^2$ -- which is more obviously a parabola and takes three points. While you're at it, fit a line to $C_L$ vs $\alpha$ and $C_M$ vs $C_L$, you'll need those later. Pick moderate $\alpha$, stay in the linear range of lift, avoid stall. There are less computationally expensive ways, but if you're set up with CFD, this way is conceptually simple. Apr 11 at 0:41

However, something that does change the coefficient of lift is the AoA, which is determined from Reynolds number.

The AoA is determined solely by the relative angle between the lifting surface and the airflow, it's a pure geometric matter (picture from Wikipedia):

The Reynolds number has only an impact on the aerodynamic coefficients. For example the following picture shows the drag coefficient for a sphere as a function of the Reynolds number (plot source):

I don't know at which Reynolds numbers your glider is going to fly, but within a more or less ample range of Reynolds numbers the changes in the coefficients can be neglected.

So, to calculate the coefficient of lift I need to know the AoA, but to calculate the AoA I need to know the coefficient of lift to calculate the stall speed.

The process is much easier: according to the flight condition you want to achieve, you determine the needed lift and lift coefficient and the relevant AoA as a consequence.

• "The AoA is determined solely by the relative angle between the lifting surface and the airflow, it's a pure geometric matter" So how do you explain this airfoil plotter, where the AoA with the best Cl/Cd changes with Reynolds number: airfoiltools.com/airfoil/details?airfoil=a18-il? Apr 8 at 17:10
• As said in the answer, the aerodynamic coefficients change with Re and as a consequence their ratio as well, that's why $C_l/C_d$ changes.with Re. Apr 8 at 17:47

I would start with Reynolds number and evaluate your anticipated speed envelope.

The wing you have chosen is good for low Re number aircraft. Why not use the same airfoil for you horizontal stabilator?

Re number = (chord meters × velocity meters/second ÷ kinematic viscosity 1.48 meters$$^2$$/second) × 10$$^5$$

A 200g glider with a chord of 0.14 m at sea level will glide at around 5 meters/second, giving a Re value of around 50000.

Using the True Air Speed Calculator we find TAS at 80,000 feet will be around 35 meters/second, for a Re value of around 350,000.

All other airfoiltools graphs can be discarded.

The increase in coefficient of lift at the higher Re number will result in a slower glide speed. As lift is proportional to V$$^2$$, the change in airspeed will be the square root of the change in coefficient of lift.

When a similar analysis is done for your stabilator airfoil, you may find that trim changes in flight may not be necessary.

Find you gliders Vbg at sea level (or where you live). To be safe, trim it to a slightly higher speed (lower AoA) for your first flight.

Based on your glide ratio and prevailing winds, your glider may easily travel over 100 miles. It might be wise to add some rudder and ailerons to let it spiral down, yet another reason to go with a higher (safer but less efficient) glide speed.