Drag equationFrom Wikipedia, the free encyclopedia

In fluid dynamics, the drag equation is a formula used to calculate the force of drag experienced byan object due to movement through a fully enclosing fluid. The formula is accurate only under certainconditions: the objects must have a blunt form factor and the fluid must have a large enoughReynolds number to produce turbulence behind the object. The equation is

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

velocity,[1]

is the mass density of the fluid, [2]

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.

The equation is attributed to Lord Rayleigh, who originally used L2 in place of A (with L being some

linear dimension).[3]

The reference area A is typically defined as the area of the orthographic projection of the object on aplane perpendicular to the direction of motion. For non-hollow objects with simple shape, such as asphere, this is exactly the same as a cross sectional area. For other objects (for instance, a rollingtube or the body of a cyclist), A may be significantly larger than the area of any cross section alongany plane perpendicular to the direction of motion. Airfoils use the square of the chord length as thereference 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 easycomparison to lift. Airships and bodies of revolution use the volumetric coefficient of drag, in whichthe reference area is the square of the cube root of the airship's volume. Sometimes differentreference areas are given for the same object in which case a drag coefficient corresponding to eachof 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.[4] For smooth bodies, like a circular cylinder, the drag coefficient may vary

significantly until Reynolds numbers up to 107 (ten million).[5]

Contents

1 Discussion

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2 Derivation

3 See also

4 Notes

5 References

Discussion

The equation is based on an idealized situation where all of the fluid impinges on the reference areaand comes to a complete stop, building up stagnation pressure over the whole area. No real objectexactly corresponds to this behavior. CD 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 CD around 1, more or less.

Smoother objects can have much lower values of CD. The equation is precise – it simply provides the

definition of CD (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 strikewith twice the flow velocity, but twice the mass of fluid strikes per second. Therefore the change ofmomentum per second is multiplied by four. Force is equivalent to the change of momentum dividedby time. This is in contrast with solid-on-solid friction, which generally has very little flow velocitydependence.

Derivation

The drag equation may be derived to within a multiplicative constant by the method of dimensionalanalysis. If a moving fluid meets an object, it exerts a force on the object. Suppose that the variablesinvolved – under some conditions – are 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, these five variables can be reduced to twodimensionless parameters:

drag coefficient CD and

Reynolds number Re.

Alternatively, the dimensionless parameters via direct manipulation of the underlying differentialequations.

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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.

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 theReynolds 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 afunction of five variables) a much simpler one: the determination of the drag as a function of only onevariable, the Reynolds number.

The analysis also gives other information for free, so to speak. The analysis shows that, other things

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being equal, the drag force will be proportional to the density of the fluid. This kind of informationoften 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 bodieswith fast-flowing fluids (such as real-size airplanes in wind-tunnels), one may just as well experimenton small models with more viscous and higher flow velocity, because these two systems are similar.

See also

Aerodynamic drag

Angle of attack

Morison equation

Stall (flight)

Terminal velocity

Notes

See lift force and vortex induced vibration for a possible force components transverse to the flow direction.1.

Note that for the Earth's atmosphere, the air density can be found using the barometric formula. Air is

1.293 kg/m3 at 0°C and 1 atmosphere

2.

See Section 7 of Book 2 of Newton's Principia Mathematica; in particular Proposition 37.3.

Drag Force (http://www.ac.wwu.edu/~vawter/PhysicsNet/Topics/Dynamics/Forces/DragForce.html)4.

See Batchelor (1967), p. 341.5.

References

Batchelor, G.K. (1967). An Introduction to Fluid Dynamics. Cambridge University Press.

ISBN 0-521-66396-2.

Huntley, H. E. (1967). Dimensional Analysis. Dover. LOC 67-17978.

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