Drag equation
Formula calculating drag force on an object in a fluid.
In fluid dynamics, the drag equation calculates the force an object feels as it moves through a surrounding fluid. The formula is Fd = ½ ρ u² cd A, where Fd is the drag force (the component of force aligned with the flow’s direction), ρ is the fluid’s mass density, u is the object’s speed relative to the fluid, A is the reference area, and cd is the drag coefficient—a dimensionless number tied to the object’s shape. Lord Rayleigh first proposed the equation, originally using L² (where L is a linear dimension) instead of A.
The reference area A is usually the orthographic projection of the object onto a plane perpendicular to its motion. For simple, non-hollow shapes like a sphere, this equals the maximum cross-sectional area. For other objects—such as a rolling tube or a cyclist’s body—A can be much larger than any cross-section in that plane. Airfoils use the square of the chord length as the reference area (often 1, since chords are typically defined with a length of 1). Aircraft use the wing or rotor-blade area for easy comparison with lift. Airships and bodies of revolution use the volumetric drag coefficient, where the reference area is the square of the cube root of the volume. If different reference areas are used for the same object, a corresponding drag coefficient must be given for each.
The drag coefficient cd works together with the chosen reference area and accounts for both skin friction and form drag. In a liquid, cd depends on the Reynolds number; in a gas, it depends on both the Reynolds number and the Mach number. For sharp-cornered bluff bodies—like square cylinders or flat plates held perpendicular to the flow—the equation holds with a constant cd when the Reynolds number exceeds 1000. For smooth bodies, such as a cylinder, cd can vary significantly until Reynolds numbers reach ten million.
The equation is easiest to grasp in an idealized scenario: all fluid hits the reference area and stops completely, building stagnation pressure over the whole area. No real object behaves exactly this way. The drag coefficient cd is the ratio of a real object’s drag to that of this ideal object. In practice, a rough, unstreamlined (bluff) body has a cd around 1, give or take. Smoother objects can have much lower values. The equation itself is precise—it simply defines cd, which varies with Reynolds number and is determined experimentally.
- Equation
- F_d = 1/2 ρ u^2 c_d A
- Attribution
- Lord Rayleigh
- Original variable
- L^2 in place of A (with L being some linear dimension)
- Drag coefficient range for bluff bodies
- around 1, more or less
- Reynolds number threshold for constant c
- greater than 1000
- Reynolds number threshold for significan
- up to 10^7 (ten million)
Lore & Background
The drag equation is attributed to Lord Rayleigh, who originally used L^2 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, A may be significantly larger than the area of any cross section. Airfoils use the square of the chord length as the reference area; aircraft use the wing area; airships and bodies of revolution use the volumetric coefficient of drag, where the reference area is the square of the cube root of the airship's volume.
The drag coefficient c_d is defined in combination with the choice of reference area and captures both skin friction and form drag. If the fluid is a liquid, c_d depends on the Reynolds number; if the fluid is a gas, c_d depends on both the Reynolds number and the Mach number. For sharp-cornered bluff bodies, like square cylinders and plates held transverse to the flow direction, the 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 ten million.
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. c_d 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 c_d around 1, more or less. Smoother objects can have much lower values of c_d. The equation is precise – it simply provides the definition of c_d, which varies with the Reynolds number and is found by experiment.
Reader's Guide
The drag equation is significant because it provides a precise definition of the drag coefficient, which must be determined experimentally and varies with Reynolds number (and Mach number for gases). Its legacy lies in its u^2 dependence on flow velocity, meaning that fluid drag increases with the square of flow velocity. When flow velocity is doubled, not only does the fluid strike with twice the flow velocity, but twice the mass of fluid strikes per second, so the change of momentum per time (the force) is multiplied by four. This contrasts with solid-on-solid dynamic friction, which generally has very little velocity dependence. The equation also relates drag force to dynamic pressure, defined as 1/2 ρ u^2, similar to the kinetic energy equation. A derivation using Bernoulli's equation is noted, though it includes an additional term for height difference (ρ g Δh) that is usually ignored because it is tiny relative to the other part. The equation is applicable to various objects, with different reference areas defined for airfoils, aircraft, airships, and other bodies, requiring corresponding drag coefficients for each area.
Did You Know?
- For sharp-cornered bluff bodies, the drag coefficient is constant when the Reynolds number is greater than 1000.
- When flow velocity is doubled, the drag force is multiplied by four due to the u^2 dependence.
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