In the context of aerodynamics, what is a polar?

Many aerodynamics discussions (including several on this site) talk about "polars" without ever explaining what they actually are.

As with any technical term common to a field, once the community has become accustomed to using it, it's easy to forget that not everyone knows what it means and that a person who never learned the term is unable to follow the discussion. This question hopes to prevent that.

In aerodynamics, what is a polar?

• Does this answer your question? Why are the power and drag curves called polars? That's a duplicate, since reading it will also tell you what a "polar" is Jan 22, 2022 at 1:45
• @DJClayworth - If you already know what a polar is, that question will certainly confirm it, but that's not the point of this question, which is intended to be found by someone who doesn't know and is searching for 'what is a polar' Jan 22, 2022 at 1:59
• IMHO, questions like this are fine and there is no need to close them as duplicate. However, this would be a great example for a self-answer, since you already know the answer. Jan 22, 2022 at 9:21
• Could even be made a community wiki answer if you're not looking for the rep... Feb 15, 2022 at 17:17

A polar is a variable that is tabularized or calculated with respect to an angular value. Examples in aerodynamics include coefficients of lift or drag with respect to angle of attack. The data may be presented in tabular format, on polar axes or on Cartesian axes.

As sophit pointed out, and as is explained in this question Why are the power and drag curves called polars? The name "polar" has been historically applied to the very useful plot which compares the lift and drag information to one another, bypassing the angle of attack information. The etymology of the naming has an obvious explanation, which may be an interesting research project that could be linked by the interested reader to the linked question above.

• The original diagrams of Otto Lilienthal did not bypass angle of attack information but added the angles of attack to points along the curve. When used to reading those diagrams, one can extract more information from them than from a Cartesian plot. Aug 13, 2022 at 19:14

Whenever a body moves in a fluid, its interaction with the fluid generates a force on the body. This force is decomposed in a component perpendicular to the direction of movement plus a component parallel to it. The first component is called lift and the second one is called drag. The lift can be positive or negative while the drag is always positive in the sense that it always act against the movement of the body i.e. it always tends to slow down the body. Both these components depends (among other things) on the relative orientation $$\alpha$$ of the body with the flow. $$\alpha$$ is called angle of attack. If the body has as shape (like an airfoil or even an A380) such that it generates 1)lift in a controlled way and 2)drag in an amount as small as possible, lift and drag have a quite typical trend versus $$\alpha$$:

Lift and drag of NACA 0012 airfoil, from NACA-TR-460. Source: https://ntrs.nasa.gov/citations/19930091108

In this picture I highlighted the drag in red and the lift in blue. As said, drag is always positive and lift can be both negative and positive. The lift has a typical linear trend with $$\alpha$$ while drag has a typical u shape. An important thing to observe here is that for each $$\alpha$$ there is only one value of lift and only one value of drag. That implies that we can get rid of $$\alpha$$ and plot lift versus drag: this plot is exactly the polar (highlighted in yellow in the following picture):

Polar of NACA 0012, from Improving Airfoil Drag Prediction by Ramanujam, Özdemir and Hoeijmakers. Source: https://www.researchgate.net/publication/304104830_Improving_Airfoil_Drag_Prediction/download

The answer to this question: What are these points in this drag curve ("Lilienthal'sches Polardiagramm")? gives a nice explanation of the meaning of some points laying on the polar, highlighting the usefulness of the polar diagrams.