Drag equation

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 formula is accurate only under certain conditions: the objects must have a blunt form factor and the fluid must have a large enough Reynolds number to produce turbulence behind the object. The equation is

$F_{D}\,=\,{\tfrac 12}\,\rho \,u^{2}\,C_{D}\,A$

$F_{D}$ is the drag force, which is by definition the force component in the direction of the flow velocity,^[1]

$\rho$ is the mass density of the fluid, ^[2]

$u$ is the flow velocity relative to the object,

$A$ is the reference area, and

$C_{D}$ is the drag coefficient – a dimensionless coefficient related to the object's geometry and taking into account both skin friction and form drag, in general $C_{D}$ depends on the Reynolds number.

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 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. 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.^[4] For smooth bodies, like a circular cylinder, the drag coefficient may vary significantly until Reynolds numbers up to 10⁷ (ten million).^[5]

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. 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 (drag coefficient), which varies with the Reynolds number and is found by experiment.

Of particular importance is the $u^{2}$ 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 second is multiplied by four. Force is equivalent to the change of momentum divided by time. This is in contrast with solid-on-solid friction, which generally has very little flow velocity dependence.

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 variables involved – 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 F_D.

Using the algorithm of the Buckingham π theorem, these five variables can be reduced to two dimensionless parameters:

• drag coefficient C_D and
• Reynolds number R_e.

Alternatively, the dimensionless parameters via direct manipulation of the variables.

That this is so becomes apparent when the drag force F_D is expressed as part of a function of the other variables in the problem:

$f_{a}(F_{D},\,u,\,A,\,\rho ,\,\nu )\,=\,0.\,$

This rather odd form of expression is used because it does not assume a one-to-one relationship. Here, f_a 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 f_a in terms of only dimensionless groups.

There are many ways of combining the five arguments of f_a 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

${\mathrm {Re}}\,=\,{\frac {u\,{\sqrt {A}}}{\nu }}$

and the drag coefficient, given by

$C_{D}\,=\,{\frac {F_{D}}{{\frac 12}\,\rho \,A\,u^{2}}}.$

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

$f_{b}\left({\frac {F_{D}}{{\frac 12}\,\rho \,A\,u^{2}}},\,{\frac {u\,{\sqrt {A}}}{\nu }}\right)\,=\,0.$

where f_b 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 F_D, it is possible to express it as

${\frac {F_{D}}{{\frac 12}\,\rho \,A\,u^{2}}}\,=\,f_{c}\left({\frac {u\,{\sqrt {A}}}{\nu }}\right)$

or

$F_{D}\,=\,{\tfrac 12}\,\rho \,A\,u^{2}\,f_{c}(R_{e}),\,$     and with     $C_{D}\,=\,f_{c}(R_{e}).$

Thus the force is simply ½ ρ A u² times some (as-yet-unknown) function f_c of the Reynolds number R_e – 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. 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.

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 flow velocity, because these two systems are similar.

See also

• Aerodynamic drag
• Angle of attack
• Morison equation
• Stall (flight)
• Terminal velocity

Notes

1. See lift force and vortex induced vibration for a possible force components transverse to the flow direction.
2. 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
3. See Section 7 of Book 2 of Newton's Principia Mathematica; in particular Proposition 37.
4. Drag Force
5. See Batchelor (1967), p. 341.

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.