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. The equation is:
where
is the drag force, which is by definition the force component in the direction of the flow velocity, is the mass density of the fluid, is the flow velocity relative to the object, is the reference area, and is the drag coefficient – a dimensionless coefficient related to the object's geometry and taking into account both skin friction and form drag. If the fluid is a liquid, depends on the Reynolds number; if the fluid is a gas, depends on both the Reynolds number and the Mach number.
The equation is attributed to Lord Rayleigh, who originally used L2 in place of A (with L being some linear dimension).
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 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. Aircraft use the wing area (or rotor-blade area) as the reference area, which makes for an easy comparison to lift. 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. Sometimes different reference areas are given for the same object in which case a drag coefficient corresponding to each of these different areas must be given.
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. For smooth bodies, like a cylinder, the drag coefficient may vary significantly until Reynolds numbers up to 107 (ten million).
Discussion
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. No real object exactly corresponds to this behavior. is the ratio of drag for any real object to that of the ideal object. In practice a rough un-streamlined body (a bluff body) will have a around 1, more or less. Smoother objects can have much lower values of . The equation is precise – it simply provides the definition of (drag coefficient), which varies with the Reynolds number and is found by experiment.
Of particular importance is the dependence on flow velocity, meaning that fluid drag increases with the square of flow velocity. When flow velocity is doubled, for example, not only does the fluid strike with twice the flow velocity, but twice the mass of fluid strikes per second. 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.
Relation with dynamic pressure
The drag force can also be specified as
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:
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. Suppose that the fluid is a liquid, and the variables involved – under some conditions – are the: speed u, fluid density ρ, kinematic viscosity ν of the fluid, size of the body, expressed in terms of its wetted area A, and drag force Fd. Using the algorithm of the Buckingham π theorem, these five variables can be reduced to two dimensionless groups: drag coefficient cd and Reynolds number Re.
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.