### Table of Contents

- Definition
- Formula
- Drag coefficient
- Frontal area
- Aerodynamic drag force
- Aerodynamic drag power
- Example
- Calculator
- References

### Definition

Any body which moves through a fluid (gas or liquid) is opposed by a resistance force called **aerodynamic drag force**. The same principle applies for road vehicles, when moving, due to the interaction with the surrounding air, a resistive force develops which tries to stop the vehicle from moving. This force, called aerodynamic drag force, increases in magnitude with the speed of the vehicle and depends on the shape and size of the car body.

Aerodynamic drag is an important aspect of the vehicle design because it has direct impact on the vehicle’s energy consumption and performance (especially at high speed).

### Formula

When the vehicle is moving through the air, it creates a dynamic pressure in front of the vehicle, which further develops into a drag force. The **aerodynamic drag force** acting on a vehicle while moving is approximated by the formula [2]:

_{ad}= 0.5 · C

_{d}· A · ρ

_{air}· (v – v

_{wind})

^{2}

where:

F_{ad} [N] – aerodynamic drag force

C_{d} [-] – aerodynamic drag coefficient

A [m^{2}] – maximum cross-section area of the vehicle

ρ_{air} [kg/m^{3}] – air density (equal to 1.202 kg/ m^{3} for dry air at 20 °C and 101.325 kPa)

v [m/s] – vehicle speed

v_{wind} [m/s] – wind speed

If we assume that there is no wind, equation (1) becomes:

_{ad}= 0.5 · C

_{d}· A · ρ

_{air}· v

^{2}

From equation (2), we can see that the aerodynamic drag force increases with the square of speed, therefore it becomes critically important at higher vehicle speeds.

### Drag coefficient

The **aerodynamic drag coefficient** has a major influence on the aerodynamic drag force. In general, the drag coefficient can be defined as the aerodynamic quality of the shape of a body in a flow. Depending on the shape of the body, the drag coefficient can vary a lot.

The lower the drag coefficient, the lower the aerodynamic drag force of the vehicle, the higher the energy efficiency.

The optimal aerodynamic shape, which has the lower drag coefficient is the **shape of a water drop** (Cd = 0.04). Due to the air friction, the water drop is forced to take the shape which has lowest resistance in contact with the air, hence it has the lowest drag coefficient.

The drag coefficient depends on the body type of the vehicle and on the parts of the vehicle mounted on the exterior surfaces, like: roof rack, mud flaps, rear spoiler, side mirrors, radio antenna and windshield wipers. These parts disrupt the streamline shape of the vehicle and increase its drag coefficient.

From [4] we can list the drag coefficient for some common vehicles:

Vehicle | Model Year | C_{d} [-] |

Ford Escape Hybrid | 2005 | 0.400 |

Toyota Camry | 1992 | 0.330 |

Smart Roadster Coupé | 2003 | 0.380 |

Toyota Prius | 2014 | 0.260 |

Chevrolet Volt | 2014 | 0.281 |

### Frontal area

The aerodynamic drag force is also influenced by the maximum frontal area of the vehicle A [m^{2}]. The frontal area of the vehicle is often given by the manufacturer or it can be approximated using a reference grid as in the picture below.

Vehicle | Model Year | A [m^{2}] |

Ford Escape Hybrid | 2005 | 1.080 |

Toyota Camry | 1992 | 0.703 |

Smart Roadster Coupé | 2003 | 0.596 |

Toyota Prius | 2014 | 0.576 |

Chevrolet Volt | 2014 | 0.622 |

### Aerodynamic drag force

Using equation (2) and data from the tables above, we can plot the aerodynamic drag force for vehicle speeds between 0 and 250 kph.

The aero drag force increases with the square of the vehicle speed. For this reason, especially at high vehicle speed (> 100 kph) aerodynamics has a critical importance in terms of vehicle performance and energy efficiency.

If we compare the aerodynamic drag force with the rolling resistance force, we can see that up to 100 kph they have similar value. At higher vehicle speeds the aerodynamic losses are far greater and they take a lot of traction force from the powertrain.

### Aerodynamic drag power

A better way of understanding the magnitude of the aerodynamic drag is by looking at the power consumption for aerodynamic losses. The **aerodynamic drag power** P_{ad} [W] is calculated by multiplying the aerodynamic drag force F_{ad} [N] with the vehicle speed v [m/s]:

_{ad}= F

_{ad}· v

Using equation (3), we can plot the aerodynamic drag power function of vehicle speed for example above:

As expected, at high vehicle speed, the power losses due to air drag is significant. In many cases, the maximum vehicle speed is limited by the aerodynamic drag, since it consumes most of the wheel power and there is no power reserve for acceleration.

### Example

Calculate the aerodynamic drag force and power for a vehicle with the drag coefficient of 0.4 and frontal area of 1.08 m^{2}, which is travelling at a speed of 100 kph.

**Step 1**. Convert vehicle speed from kph to m/s.

**Step 2**. Calculate the aerodynamic drag force using equation (2).

_{ad}= 0.5 · 0.4 · 1.08 · 1.202 · 27.78

^{2}= 200 N

**Step 3**. Calculate the aerodynamic drag power using equation (3).

_{ad}= 200 · 27.78 = 5556 W = 5.556 kW

### Calculator

C_{d} [-] = | A [m^{2}] = | ρ [kg/m^{3}] = | V [kph] = |

F_{ad} [N] = | P_{ad} [kW] = |

### References

[1] Henning Wallentowitz, Longitudinal Dynamics of Vehicles – Lecture, IKA RWTH, Aachen, 2004.

[2] Lars Eriksson, Lars Nielsen, Modeling and Control of Engines and Drivelines, Wiley, 2014.

[3] Automotive Handbook, 9th Edition, Bosch, 2014.

[4] https://en.wikipedia.org/wiki/Automobile_drag_coefficient

[5] https://www.motortrend.com/how-to/hdrp-0609-aero-tricks-tips

## Matt

Something is way off, likely the frontal areas

otherwise those cars could do well above 240 km/h as the drag at 240 km/h is way less than their Power outputs even accounting for drivetrain losses, and the other main loss is rolling resistance which is nowhere near the loss that aero drag is

Think about it.

## mike

a dimensional analysis of the Drag equation (2) does bit yield Force (i.e., pounds or kilograms)? what am I missing?

## Vilivili

This is cool stuff, thanks. But looks like Your reference 4, the frontal area, is actually the drag area. So no need to multiply by the drag co-efficient for the second time. To correct Your results those should be divided by the drag co-efficient. Eg. In Your example case the aerodynamic drag force @100km/h would be 200 N / 0,4 = 500 N