In fluid dynamics, the drag equation is a practical formula used to calculate the force of drag experienced

by an object due to movement through a fully-enclosing fluid. The equation is attributed to Lord Rayleigh,

who originally used L2 in place of A (with L being some linear dimension). The force on a moving object due

to a fluid is:

where

FD is the force of drag, which is by definition the force component in the direction of the flow

velocity,[1]

ρ is the mass density of the fluid, [2]

u is the velocity of the object relative to the fluid,

A is the reference area, and

CD is the drag coefficient — a dimensionless constant, e.g. 0.25 to 0.45 for a car.The reference area A is typically defined as the area of the orthographic projection of the object on a plane perpendicular to the direction of motion. For non-hollow objects with simple shape, such as a sphere, this is exactly the same as a cross sectional area. For other objects (for instance, a rolling tube or the body of a cyclist), A may be significantly larger than the area of any cross section along any plane perpendicular to the direction of motion.

The drag equation may be derived to within a multiplicative constant by the method of dimensional

analysis. If a moving fluid meets an object, it exerts a force on the object, according to a complicated (and

not completely understood) law. We might suppose that the variables involved under some conditions to be

the:

speed u,

fluid density ρ,

viscosity ν of the fluid,

size of the body, expressed in terms of its frontal area A, and

drag force FD.

The Vaschy-Buckingham π theorem is a key theorem in dimensional analysis. The theorem loosely states that if we have a physically meaningful equation involving a certain number, n, of physical variables, and these variables are expressible in terms of k  independent fundamental physical quantities, then the original expression is equivalent to an equation involving a set of p = n − k  dimensionless parameters constructed from the original variables: it is a scheme for nondimensionalization. This provides a method for computing sets of dimensionless parameters from the given variables, even if the form of the equation is still unknown. However, the choice of dimensionless parameters is not unique: Vaschy-Buckingham's theorem only provides a way of generating sets of dimensionless parameters, and will not choose the most 'physically meaningful'.

Derivation

The drag equation may be derived to within a multiplicative constant by the method of dimensional

analysis. If a moving fluid meets an object, it exerts a force on the object, according to a complicated (and

not completely understood) law. We might suppose that the variables involved under some conditions to be

the:

speed u,

fluid density ρ,

viscosity ν of the fluid,

size of the body, expressed in terms of its frontal area A, and

drag force FD.

Using the algorithm of the Buckingham π theorem, one can reduce these five variables to two dimensionless

parameters:

drag coefficient  CD and

Reynolds number  Re.

Alternatively, one can derive the dimensionless parameters via direct manipulation of the underlying

differential equations.

That this is so becomes obvious when the drag force FD is expressed as part of a function of the other

variables in the problem:

This rather odd form of expression is used because it does not assume a one-to-one relationship.

Here, fa is some (as-yet-unknown) function that takes five arguments. We note that the right-hand side

is zero in any system of units; so it should be possible to express the relationship described by fa in

terms of only dimensionless groups.

There are many ways of combining the five arguments of fa to form dimensionless groups, but

the Buckingham π theorem states that there will be two such groups. The most appropriate are the

Reynolds number, given by

and the drag coefficient, given by

Thus the function of five variables may be replaced by another function of only two

variables:

where fb is some function of two arguments. The original law is then reduced to a

law involving only these two numbers.

Because the only unknown in the above equation is the drag force FD, it is possible

to express it as

or

     and with     

Thus the force is simply ½ ρ A u2 times some (as-yet-unknown)

function fc of the Reynolds number Re — a considerably simpler system

than the original five-argument function given above.

Dimensional analysis thus makes a very complex problem (trying to

determine the behavior of a function of five variables) a much simpler

one: the determination of the drag as a function of only one variable, the

Reynolds number.

The analysis also gives other information for free, so to speak. We know

that, other things being equal, the drag force will be proportional to the

density of the fluid. This kind of information often proves to be extremely

valuable, especially in the early stages of a research project.

To empirically determine the Reynolds number dependence, instead of

experimenting on huge bodies with fast-flowing fluids (such as real-size

airplanes in wind-tunnels), one may just as well experiment on small

models with more viscous and higher velocity fluids, because these two

systems are similar.