# Aerodynamics for Cars and Trucks

## Basic Approach

The following is an aerodynamics concept as proposed by Manfred Mitschke and Henning Wallentowitz 1.

The resultant aerodynamic force and torque is considered to act on a reference point Oc placed on the road surface at the center of the four wheels.

Fa,x = cxL)*P(vr)

Fa,y = cyL)*P(vr)

Fa,z = czL)*P(vr)

Ma,x = cmxL)*l*P(vr)

Ma,y = cmyL)* l*P(vr)

Ma,z = cmzL)* l*P(vr)

Where ci express the aerodynamic coefficients, τL the relative airflow incidence angle, vr the resulting velocity, P the dynamic pressure and l the wheelbase.

The coordinate system in which the above defined force and torque components are to be applied is located in Oc and fixed to the car body.

## On Velocities

A diagonal airflow results in a flow velocity vr with an incidence angle τL to the vehicle’s longitudinal axis. Both parameters vr and τL can be calculated by vector addition of the road speed vector v (negative car speed vector) and the wind velocity vector vw.

## On P(vr)

The dynamic pressure can be calculated using the air density ρ, the resulting velocity vr, and the following formula.

P(vr) = 0.5 * ρ * vr2

The air density is dependent on the ambient temperature and pressure of the environment, and the ideal gas constant for air and it is often regarded as constant.

## Vertical Force and Pitch Torque

The vertical force Fa,z and the pitch torque Ma,y are replaced by two vertical forces acting at the center points of the suspensions Of and Or at ground level introducing cz,f and cz,r.

Fa,zf = cz,fL)*P(vr)

Fa,zr = cz,rL)*P(vr)

The action forces are applied at reference point Oc placed on the road surface at the center of the four wheels and the center points of the suspensions Of and Or at ground level.

This aerodynamic system is the default setting in the Assembly Wizard for the Car and Truck Library.

The complete set of aerodynamic forces and torques can be applied to the rigid car body as follows:

Fa,x = 0.5*cxL)* ρ * A* vr2

Fa,y = 0.5*cyL)* ρ * A* vr2

Fa,zf = 0.5*cz,fL)* ρ * A* vr2

Fa,zr = 0.5*cz,rL)* ρ * A* vr2

Ma,x = cmxL)* l* ρ * A* vr2

Ma,z = cmzL)* l* ρ * A* vr2

Where A is the frontal area of the car. Vehicle frontal areas are typically in the range of 1.5 m2 < A < 2.5 m2 for passenger cars and 4 m2 < A < 9 m2 for trucks and buses.

## How to Determine ci

As a very rough approximation the ci can be regarded as constants, and to disable aerodynamic totally, all ci can be set to zero.

In general, ci should be defined as functions of τ preferably as a table like the following. The data is currently an arbitrary collection of values found in tables and diagrams of the referenced book 1.
τL =0deg τL =10deg τL =20deg τL =30deg
cx 0.3 0.31 0.32 0.33
cy 0 0.4 0.8 1.2
cz,f 0.1 0.2 0.3 0.4
cz,r 0 0.1 0.2 0.3
cmx 0 0.03 0.06 0.09
cmz 0 0.04 0.08 0.12

One limitation of using this table is that is does not allow situations where the car is subjected to diagonal airflow of over 30 degrees (some of the curves go up to 40 degrees). Such conditions, especially pure sidewind conditions, will be considered for implementation in future releases.

ci coefficients could be derived by a virtual wind tunnel experiment. The values measured in such an experiment would be the constrained forces at Oc, Or and Of as functions or vr and τ.