# How do we derive Lift Coefficient of an airplane's wing?

In many references I read that lift coefficient ($$C_l$$) is a product of wind tunnel test. But many times I read the formula to calculate the coefficient of lift. From the lift equation, $$L=\frac{1}{2} \rho v^2 S C_l$$ $$C_l$$ is rearranged to derived the $$C_l$$ equation as below: $$C_l=\frac{2 L}{\rho v^2 S}$$

In my opinion, even it correct mathematically, but is not correct in the engineering process as it is not a product of a mathematic. The Coefficient of lift ($$C_l$$) is required to calculate the lift that will be produced by an airplane's wing with the given speed and the wings' area.

Am I correct with my opinion? Or, is any formula to calculate the coefficient of lift so the lift force can be calculated without any wind tunnel test?

The lift coefficient accounts for complex aerodynamic effects that are impossible to calculate by hand in any practical way. I'm not sure what you mean by "In my opinion, even it correct mathematically, but is not correct in the engineering as the it is not product of mathematic." My best guess is that you're pointing out that even though you can rearrange the lift equation to calculate $$C_L$$, it's not derived from first principles, which is correct. It's a convenient way to represent more complex interactions that are difficult to calculate.

Essentially, we know lift varies linearly with dynamic pressure ($$\frac{1}{2} \rho V^2$$) and it varies linearly with surface area. So the lift equation may be written, in its most reduced form, $$L = C_L q S.$$ Notice the similarity to the drag equation: $$D = C_D q S$$ And the pitching moment equation: $$M = C_M q S c$$

Each of these quantities varies linearly with dynamic pressure and surface area, so it's useful to express them as products of these quantities. Changing either of these variables will probably affect the aerodynamic coefficient as well, but for modeling purposes that don't require extreme precision, in most cases, the aerodynamic coefficient won't change nearly as much as dynamic pressure can.

As a result, knowing $$C_L$$ and $$C_D$$ for a variety of flight conditions, you can model your body of interest with fairly good accuracy without having to go through the complexity of computational fluid dynamics, which is the numerical way to predict aerodynamic forces.

To answer your question about a formula to calculate the coefficient of lift, there are methods that will allow you to get a rough estimate (e.g. lifting line theory), but to get results comparable to wind tunnel testing, you have to use computational fluid dynamics, which will require a computer to achieve meaningful results.

• Apologize for the absence of some words made the passage not easily understandable. – AirCraft Lover Dec 22 '18 at 14:23
• My point is whether is correct to write coefficient of lift ($C_l$) as $$C_l=\frac{2 L}{\rho v^2 S}$$ as the $C_l$ is the thing we have to find so we can calculate the lift ($L$). That $C_l$ is derived from tunnel test or from computer modeling (as you mentioned). But for sure, not from calculation which is involve $L$ that we don't know. That will be like the question, which one is first, chicken or egg. – AirCraft Lover Dec 22 '18 at 14:26
• @AirCraftLover sorry it took me a while to get back to this. You can certainly write the equation that way to solve for $C_L$ if you're taking measurements of the lift (i.e. you know $L$, probably in a wind tunnel) but if both $C_L$ and $L$ are unknown, you can't solve for both in a single equation. In that case, you would likely have to use CFD. – zaen Dec 29 '18 at 3:33
• And should be the $C_L$ is firstly derived before completing the aircraft design, right? That actually my concern as many explaining the mathematical formula which it is seem correct mathematically but wrong in the actual implementation as we have to derive at first hand the $C_L$ before continue to rest. – AirCraft Lover Dec 29 '18 at 3:40
• @AirCraftLover It depends on what you're trying to accomplish. If you have an idea for an aircraft and nothing more, then no, you can't use that formula to find $C_L$. You can get a rough estimate by taking the section lift coefficient $C_l$ of the airfoil and reducing it slightly to account for lift loss from wingtip vortices, but that doesn't take into account contributions or losses from the fuselage, horizontal tail, etc. Before completing an aircraft design, you'd want to have built multiple models and tested them in a wind tunnel, where you could find $C_L$. – zaen Dec 29 '18 at 3:48