Drag equation - Citizendia

The drag equation is a practical formula used to calculate the force of drag experienced by an object due to a fluid that it is moving through. In Fluid dynamics, drag (sometimes called fluid resistance) is the force that resists the movement of a Solid object through a Fluid (a FLUID ( F ast L ight '''U'''ser '''I'''nterface D esigner is a graphical editor that is used to produce FLTK Source code The equation is attributed to Lord Rayleigh, who originally used L² in place of A (L being some linear dimension). John William Strutt 3rd Baron Rayleigh OM (12 November 1842 &ndash 30 June 1919 was an English Physicist who with William Ramsay, discovered The force on a moving object due to a fluid is:

$F_d = {1 \over 2} \rho v^2 C_d A$

where

F_d is the force of drag, which is by definition the force component in the direction of the flow velocity,^[1]

ρ is the mass density of the fluid, ^[2]

v is the velocity of the object relative to the fluid,

A is the reference area, and

C_d is the drag coefficient (a dimensionless constant, e. In Physics, a force is whatever can cause an object with Mass to Accelerate. Mass is a fundamental concept in Physics, roughly corresponding to the Intuitive idea of how much Matter there is in an object The density of a material is defined as its Mass per unit Volume: \rho = \frac{m}{V} Different materials usually have different In Physics, velocity is defined as the rate of change of Position. Area is a Quantity expressing the two- Dimensional size of a defined part of a Surface, typically a region bounded by a closed Curve. The drag coefficient ( Cd, Cx or Cw) is a Dimensionless quantity that describes how streamlined an In Dimensional analysis, a dimensionless quantity (or more precisely a quantity with the dimensions of 1) is a Quantity without any Physical units g. 0. 25 to 0. 45 for a car).

The reference area A is the area of the projection of the object on a plane perpendicular to the direction of motion (ie cross-sectional area). In Geometry, a cross section is the intersection of a body in 2-dimensional space with a line or of a body in 3-dimensional space with a plane etc 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. The reference for a wing would be the plane area rather than the frontal area.

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. In Fluid mechanics and Heat transfer, the Reynolds number \mathrm{Re} is a Dimensionless number that gives a measure of the Ratio ^[3] For smooth bodies, like a circular cylinder, the drag coefficient may vary significantly until Reynolds numbers up to 10⁷ (ten million). ^[4]

1 Discussion
2 Derivation
3 References
4 Notes
5 See also

Discussion

The equation is based on an 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. Pressure (symbol 'p' is the force per unit Area applied to an object in a direction perpendicular to the surface 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 unstreamlined 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. The drag coefficient ( Cd, Cx or Cw) is a Dimensionless quantity that describes how streamlined an In Fluid mechanics and Heat transfer, the Reynolds number \mathrm{Re} is a Dimensionless number that gives a measure of the Ratio

Of particular importance is the v² dependence on velocity, meaning that fluid drag increases with the square of velocity. When velocity is doubled, for example, not only does the fluid strike with twice the velocity, but twice the mass of fluid strikes per second. Mass is a fundamental concept in Physics, roughly corresponding to the Intuitive idea of how much Matter there is in an object Therefore the change of momentum per second is multiplied by four. In Classical mechanics, momentum ( pl momenta SI unit kg · m/s, or equivalently N · s) is the product Force is equivalent to the change of momentum divided by time. In Physics, a force is whatever can cause an object with Mass to Accelerate. This is in contrast with solid-on-solid friction, which generally has very little velocity dependence. Friction is the Force resisting the relative motion of two Surfaces in contact or a surface in contact with a fluid (e

Derivation

The drag equation may be derived to within a multiplicative constant by the method of dimensional analysis. Dimensional analysis is a conceptual tool often applied in Physics, Chemistry, Engineering, Mathematics and Statistics to understand If a moving fluid meets an object, it exerts a force on the object, according to a complicated (and not completely understood) law. We might suppose that the variables involved under some conditions to be the speed, density and viscosity of the fluid, the size of the body (expressed in terms of its frontal area $A$ ), and the drag force. Using the algorithm of the Buckingham π theorem, one can reduce these five variables to two dimensionless parameters: the drag coefficient and the Reynolds number. The Buckingham π theorem is a key Theorem in Dimensional analysis. The drag coefficient ( Cd, Cx or Cw) is a Dimensionless quantity that describes how streamlined an In Fluid mechanics and Heat transfer, the Reynolds number \mathrm{Re} is a Dimensionless number that gives a measure of the Ratio

Alternatively, one can derive the dimensionless parameters via direct manipulation of the underlying differential equations.

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

$f(F,u,A,\rho,\nu)=0. \!$

This rather odd form of expression is used because it does not assume a one-to-one relationship. Here, $f$ is some (as-yet-unknown) function that takes five arguments. We note that the right-hand side is zero in any system of units; so it should be possible to express the relationship described by $f$ in terms of only dimensionless groups.

There are many ways of combining the five arguments of $f$ to form dimensionless groups, but the Buckingham π theorem states that there will be two such groups. The Buckingham π theorem is a key Theorem in Dimensional analysis. The most appropriate are the Reynolds number, given by

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

and the drag coefficient, given by

$C_D=\frac{F}{\rho Au^2}.$

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

$f\left(\frac{F}{\rho Au^2},\frac{u\sqrt{A}}{\nu}\right)=0.$

where $f$ 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 $F$ , it is possible to express it as

$\frac{F}{\rho Au^2}=f\left(\frac{u\sqrt{A}}{\nu}\right)$

$F=\rho Au^2f(Re). \!$

Thus the force is simply $ρ A u 2$ times some (as-yet-unknown) function of the Reynolds number—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. We know 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 slow-flowing, more viscous fluids, because these two systems are similar.

References

Batchelor, G.K. (1967). George Keith Batchelor ( March 8 1920 - March 30 2000) was an Australian Applied mathematician and Fluid dynamicist An Introduction to Fluid Dynamics. Cambridge University Press. ISBN 0521663962.
Huntley, H. E. (1967). Dimensional Analysis. Dover. LOC 67-17978.

Notes

^ See lift force and vortex induced vibration for a possible force components transverse to the flow direction. In the context of a Fluid flow relative to a body the lift force is the component of the Aerodynamic force that is Perpendicular to the flow Vortex-induced vibrations (VIV are motions induced on bodies facing an external flow by periodical irregularities on this flow
^ Note that for the Earth's atmosphere, the air density can be found using the barometric formula. Temperature and layers The temperature of the Earth's atmosphere varies with altitude the mathematical relationship between temperature and altitude varies among five The barometric formula, sometimes called the exponential atmosphere or Isothermal Atmosphere, is a Formula used Air is 1. 293 kg/m³ at 0°C and 1 atmosphere
^ Drag Force
^ See Batchelor (1967), p. The Standard atmosphere is an international reference pressure defined as 101325 Pa and formerly used as unit of Pressure (symbol atm 341.

Contents

Discussion

Derivation

References

Notes

See also