In fluid dynamics, the drag equation is a formula used to calculate the force of drag experienced by an object due to movement through a fully enclosing fluid.
where The equation is attributed to Lord Rayleigh, who originally used L2 in place of A (with L being some linear dimension).
For non-hollow objects with simple shape, such as a sphere, this is exactly the same as the maximal 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.
Airfoils use the square of the chord length as the reference area; since airfoil chords are usually defined with a length of 1, the reference area is also 1.
Airships and bodies of revolution use the volumetric coefficient of drag, in which the reference area is the square of the cube root of the airship's volume.
For sharp-cornered bluff bodies, like square cylinders and plates held transverse to the flow direction, this equation is applicable with the drag coefficient as a constant value when the Reynolds number is greater than 1000.
[3] For smooth bodies, like a cylinder, the drag coefficient may vary significantly until Reynolds numbers up to 107 (ten million).
[4] The equation is easier understood for the idealized situation where all of the fluid impinges on the reference area and comes to a complete stop, building up stagnation pressure over the whole area.
(drag coefficient), which varies with the Reynolds number and is found by experiment.
Therefore, the change of momentum per time, i.e. the force experienced, is multiplied by four.
This is in contrast with solid-on-solid dynamic friction, which generally has very little velocity dependence.
where PD is the pressure exerted by the fluid on area A.
Here the pressure PD is referred to as dynamic pressure due to the kinetic energy of the fluid experiencing relative flow velocity u.
This is defined in similar form as the kinetic energy equation:
The drag equation may be derived to within a multiplicative constant by the method of dimensional analysis.
Suppose that the fluid is a liquid, and the variables involved – under some conditions – are the: Using the algorithm of the Buckingham π theorem, these five variables can be reduced to two dimensionless groups: That this is so becomes apparent 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.
Now 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.
Because the only unknown in the above equation is the drag force Fd, it is possible to express it as
Thus the force is simply 1/2 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.
Those properties are conventionally considered to be the absolute temperature of the gas, and the ratio of its specific heats.
These two properties determine the speed of sound in the gas at its given temperature.
The Buckingham pi theorem then leads to a third dimensionless group, the ratio of the relative velocity to the speed of sound, which is known as the Mach number.
The analysis shows 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.
[5] To empirically determine the Reynolds number dependence, instead of experimenting on a large body with fast-flowing fluids (such as real-size airplanes in wind tunnels), one may just as well experiment using a small model in a flow of higher velocity because these two systems deliver similitude by having the same Reynolds number.
If the same Reynolds number and Mach number cannot be achieved just by using a flow of higher velocity it may be advantageous to use a fluid of greater density or lower viscosity.