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The drag coefficient is given by the following expression

$$C_d = \frac{2F_d}{\rho v^2 A}$$

For a race car wing (which is simply an airplane wing upside down to generate downforce), what should we consider as the area? The cross-sectional area perpendicular to the flow or the wing surface area (parallel to the ground)?

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When $C_d$ is an appropriate parameter for discussion, road vehicle aerodynamics uses the planform area of the wing. That's the area formed by the chord (see below) and total width of the wing.

You'd use the same area in lift calculations for $C_L$

For complicated wings, or whole vehicles, where defining an area makes little sense, the common practice is to list the product $C_D* A$ (often written 'CDA') as a parameter, rather than attempting to decouple them. It works perfectly well in computing lift/drag forces and varies as $L^2$ for scale modeling purposes.

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Most race car wings are a lot more complex than an aircraft wing as they are usually designed to manage airflow over the whole body of the car, particularly the wheels and also have to deal with turbulent flow both from preceding aerodynamic surfaces and cars which they are following as well as controlling vortex generation to control airflow over their own surfaces for more complex functions like separating downstream flows between aerodynamic surfaces and intakes for cooling and engine induction.

For example in an F1 car the complex elements of the front wing are not so much to generate downforce as to control the downstream flow, especially over and around the front wheels.

It is also worth noting that race car wings are primarily tuned by varying angle of attack which this formula does not consider at all.

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The area is named "reference area". As long as it is defined by someone, you are free to choose any arbitrary value for A.

To make comparisons easier among different designs for a certain project, engineers define A and they do trade off studies with that reference. This way they can compare the performance of the wings for a certain car/plane model.

On the other hand it's also possible to Not Fix the ref area value. This would allow benchmarking various designs for their aerodynamic efficiency. (A slightly larger wing could, in itself be more efficient). The problem with this approach is that the optimization can easily break the manufacturing interface rules, size rules, weight constraints, etc.

As a result, if the thing has some physical constraints (like car-wing attachment), it's almost certain that you would define the wing area first, as a number, and not change it throughout the design life for practical purposes.

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