Dimensional Analysis of Drag

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Engine power is required to move a car through the air.  Lowering the drag force requires lowering the projected area and the drag coefficient.  A significant amount of testing and data analysis is required to reduce the drag on a Formula 1 race car.  Wind tunnels with moving belts to simulate the vehicle traveling over the road are often used to accurately simulate the airflow around, over, and under the car.

By Mark Fosh from Watford, UK (BMW Formula 1 race car) [CC-BY-2.0], via Wikimedia Commons

The drag force equation is one of the most commonly used equations in fluid dynamics. One may ask, “Where did this equation come from, how was it derived?” This page shows that the drag force equation can be obtained from dimensional analysis using the Buckingham P-theorem.

The primary variables affecting the drag force are the size of the object, the relative velocity, and the density and viscosity of the fluid.  The size of the object will be based on the largest projected cross sectional area in the direction of the flow and a characteristic length.  Thus there are six variables (n=6).

 

F      drag force,

V      relative fluid velocity,

A      projected area,

D      characteristic length

m      fluid viscosity, and

r      fluid density.

 

There are three fundamental dimensions (k=3) needed to define these six variables,

 

M     mass,

L      distance, and

t       time.

 

According to the Buckingham P-theorem there are three non-dimensional P-groups.

 

 Choose the density, velocity, and projected area as the base variables.  

 

First P-Group

The first P-Group will be determined using the cross sectional area


Second P-Group

The second P-Group will be determined using the viscosity


 

Third P-Group

The third P-Group will be determined using the viscosity


The relationship between the P-Groups can be written as


The second -Group is the reciprocal of the Reynolds Number.  The Reynolds Number will be used instead of its reciprocal since both are non-dimensional and the functional relationship (reciprocal or non-reciprocal) will be determined by experimental data.  Traditionally, a characteristic distance, D, other than the square root of the area is used to compute the Reynolds number.

The first -Group is defined as the drag coefficient, CD, and the relationship is written as


The functional relationship between the Reynolds number and drag coefficient is determined from experimental data.

Learning Objectives

Learn where the drag force equation comes from.

Learn why the drag coefficient is a function of the Reynolds number.