Coefficients in Aerodynamic Engineering

If you read anything online about aerodynamics, you will come across something called a drag coefficient. Often, articles on blogs, magazine websites, or Youtube videos will explain this drag coefficient as a measure for comparing the aerodynamic efficiency of two cars or trucks. But what is a “drag coefficient” exactly?
 
Coefficients in Engineering
 
To answer that, we need to step back and look at the concept of coefficients more generally.
 
Coefficients are nondimensional numbers—that is, they have no units and don’t represent a measure of something physical like speed or force. For example, engineers working with compressible gases use several coefficients to determine properties of these gases, including reduced pressure (p_R), reduced temperature (T_R), and reduced specific volume (v’_R), where

Coefficients of performance are used to calculate the efficiency of refrigeration or heat pump cycles, or the thermal efficiency of power cycles (like the homework problem above):

In aerodynamics, one common coefficient is Reynolds number, which is the ratio of inertial force to viscous force in a flow:

Another useful aeronautical coefficient is the coefficient of lift-induced drag:

Something you may have noticed about all these coefficients is: they are all ratios (of pressure, temperature, specific volume, energy, force, and area, respectively, for the examples above). The reason for this is simple. If you compare two measures with the same units in a ratio (fraction), the units cancel and you will be left with a nondimensional number—a coefficient. Coefficients are useful for summarizing a comparison of one parameter with another of the same units in one number.
 
Aerodynamic Coefficients
 
Coefficients in aerodynamics work exactly the same way; they are ratios. Aerodynamic drag force can be described as the product of pressure and area:

Pressure here is symbolized by q rather than p because it is the dynamic pressure of the airflow (dynamic pressure is given by static pressure subtracted from total pressure, and it is derived from the kinetic energy of the air in motion), and A here is something we call “drag area.” You can think of the drag force measured on a car in a wind tunnel as being represented by the dynamic pressure of the air acting as a pressure difference between two sides of an imaginary plate of area A, which produces the same force acting on the plate as acts on the car.
 
So how do we get a drag coefficient out of that? We use a ratio of the drag area to some reference area (S) that is based on the actual dimensions of the car; in car aerodynamics, this is typically the projected frontal area, or the area covered by the car’s shadow if a uniform light source directly behind it shines onto a wall in front of it. Note that projected frontal area is simply the maximum cross section area and has nothing to do with the shape of the front surfaces of a car—something even experienced engineers get wrong.

"Frontal area" does not refer to the shape or surface area of the bumper, hood, and windshield. The new Prius has ~10% higher aerodynamic drag than the previous generation, based on Toyota's published dimensions and drag coefficients.

Our reference area doesn’t have to be frontal area, either; in aeronautics, reference area is usually wing plan area (as in the lift-induced drag equation above; S in the denominator is wing plan area, and the coefficient is proportional to the ratio of plan area and span, b, squared). But conventionally, frontal area is used to calculate a drag coefficient for an automobile:

Drag coefficient is properly equated to a ratio of forces (total drag divided by [dynamic pressure times reference area]; this is what a proper dimensional analysis will produce), but it simplifies to a ratio of areas. You can also think of the drag coefficient as a percentage; that is, how big a car's drag area is as a percentage of its reference area. A car like my Prius (C_D = 0.25) has a drag area 25% as large as its reference area; a Rivian R1T (C_D = 0.31) has a drag area 31% as large as its reference area; a Formula SAE car (C_D = as high as 1.30) can have a drag area 130% as large as its reference area.

To get an accurate frontal area for your car, you’ll need a better picture than this. Take a photo from as far away as possible to minimize distortion.

Thus, the smaller C_D is, the smaller the drag area is compared to the actual dimensions of the car, and the smaller the aerodynamic force is referenced to the dimensions of the car. This last part is important, because the actual drag force acting on a car is not described by its drag coefficient alone but by its drag area. And, following from that, drag coefficient by itself is insufficient for comparing the aerodynamic drag of two cars unless their reference areas are roughly the same size. As Goro Tamai puts it in The Leading Edge, “The conclusion is, to compare the aerodynamic drag of vehicle bodies, we must compare the drag areas—C_DA’s” (15; emphasis original).
 
Drag Comparisons
 
For example, say we have a car of C_D = 0.30. Is that good? Who knows; we need more information. If this is a small car with S = 2.10 m² (Mitsubishi Mirage-sized), say, then the drag force acting on it at, for example, 100 kph and STP will be:

But if that car is a big SUV with S = 3.38 m² (Chevrolet Suburban-sized), its drag force at the same speed will be:

…or 64% greater despite having the same drag coefficient.
 
Where drag coefficients are useful is in comparing vehicles of similar size/class e.g. full-size trucks or mid-size SUVs, etc. where vehicles in the same size class tend to have similar dimensions. Where they are not useful is in comparing vehicles of differing size because this leads to incorrect conclusions. And the size difference doesn’t need to be extreme to throw things off. For example, one might conclude based on drag coefficients that the current Toyota Prius (C_D = 0.27) is “more aerodynamic” than the GT86 (C_D = 0.28). In reality the GT86 has lower aerodynamic drag due to its smaller size and thus smaller reference area used to calculate its drag coefficient, which we infer from the relationship between drag area and reference area will make C_D appear larger for the same drag force.

Modifications like this large air dam that reduce drag force but increase S mean the percent change in drag coefficient does not scale with percent change in drag area. In this case, since S has increased, C_D will decrease more than if S remained constant for the same reduction in drag.

Another example: in 2014, Car & Driver did a comparison test of 5 low-drag cars in the A2 wind tunnel.* The frontal area of each car was used to calculate its drag coefficient based on drag force measured at 70 mph. The Tesla Model S had the lowest drag coefficient, at C_D = 0.24, while the Nissan Leaf had the highest, C_D = 0.32. Using these numbers, the Leaf should have 33% higher drag than the Model S, right? Wrong! Car & Driver published measured drag force as well, and the Leaf (F_D = 97 lbf) developed only 26% higher drag force than the Model S (F_D = 77 lbf). The two winners of the test, the Model S and Prius, had virtually the same drag force (77 lbf and 78 lbf, respectively) but different drag coefficients (0.24 and 0.26) because of their differing reference areas (25.2 ft² and 23.9 ft²).

(*This test wasn’t without its problems: the tunnel used has no moving ground plane and is on the small side, giving a high blockage ratio for full-size cars tested in it. But it is useful here in comparing the drag force vs. coefficient of several cars measured in the same tunnel and same conditions; just take all these numbers with a grain of salt if you use them to compare to other tests).

This tailgate spoiler, unlike the air dam above, does not change the reference area of the truck since it sits entirely within its projected frontal area already—so here, percent change in drag coefficient will equal percent change in drag area (+8% in this configuration).

So, be careful when you draw conclusions about aerodynamic drag—and especially when comparing drag between two cars—based solely on drag coefficients. This can lead to erroneous conclusions. Unfortunately, manufacturers don’t usually publish drag areas and instead trumpet drag coefficients; these may even be used in marketing to influence people’s beliefs about the drag of a certain model compared with its competitors. But drag coefficients alone don’t tell the whole story. Always look for more information so you can compare drag areas if possible.