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fluid dynamics In physics and engineering, fluid dynamics is a subdiscipline of fluid mechanics that describes the flow of fluids—liquids and gases. It has several subdisciplines, including '' aerodynamics'' (the study of air and other gases in motion) ...
, the drag equation is a formula used to calculate the force of
drag Drag or The Drag may refer to: Places * Drag, Norway, a village in Tysfjord municipality, Nordland, Norway * ''Drág'', the Hungarian name for Dragu Commune in Sălaj County, Romania * Drag (Austin, Texas), the portion of Guadalupe Street adj ...
experienced by an object due to movement through a fully enclosing
fluid In physics, a fluid is a liquid, gas, or other material that continuously deforms (''flows'') under an applied shear stress, or external force. They have zero shear modulus, or, in simpler terms, are substances which cannot resist any shea ...
. The equation is: F_\, =\, \tfrac12\, \rho\, u^2\, c_\, A where *F_ is the drag
force In physics, a force is an influence that can change the motion of an object. A force can cause an object with mass to change its velocity (e.g. moving from a state of rest), i.e., to accelerate. Force can also be described intuitively as a ...
, which is by definition the force component in the direction of the flow velocity, *\rho is the
mass density Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematically ...
of the fluid, *u is the
flow velocity In continuum mechanics the flow velocity in fluid dynamics, also macroscopic velocity in statistical mechanics, or drift velocity in electromagnetism, is a vector field used to mathematically describe the motion of a continuum. The length of the ...
relative to the object, *A is the reference
area Area is the quantity that expresses the extent of a region on the plane or on a curved surface. The area of a plane region or ''plane area'' refers to the area of a shape or planar lamina, while ''surface area'' refers to the area of an open su ...
, and *c_ is the
drag coefficient In fluid dynamics, the drag coefficient (commonly denoted as: c_\mathrm, c_x or c_) is a dimensionless quantity that is used to quantify the drag or resistance of an object in a fluid environment, such as air or water. It is used in the drag e ...
– a
dimensionless A dimensionless quantity (also known as a bare quantity, pure quantity, or scalar quantity as well as quantity of dimension one) is a quantity to which no physical dimension is assigned, with a corresponding SI unit of measurement of one (or 1) ...
coefficient In mathematics, a coefficient is a multiplicative factor in some term of a polynomial, a series, or an expression; it is usually a number, but may be any expression (including variables such as , and ). When the coefficients are themselves ...
related to the object's geometry and taking into account both
skin friction Skin friction drag is a type of aerodynamic or hydrodynamic drag, which is resistant force exerted on an object moving in a fluid. Skin friction drag is caused by the viscosity of fluids and is developed from laminar drag to turbulent drag as a f ...
and
form drag Parasitic drag, also known as profile drag, is a type of aerodynamic drag that acts on any object when the object is moving through a fluid. Parasitic drag is a combination of form drag and skin friction drag. It affects all objects regardless of ...
. If the fluid is a liquid, c_ depends on the Reynolds number; if the fluid is a gas, c_ depends on both the Reynolds number and the
Mach number Mach number (M or Ma) (; ) is a dimensionless quantity in fluid dynamics representing the ratio of flow velocity past a boundary to the local speed of sound. It is named after the Moravian physicist and philosopher Ernst Mach. : \mathrm = \ ...
. The equation is attributed to
Lord Rayleigh John William Strutt, 3rd Baron Rayleigh, (; 12 November 1842 – 30 June 1919) was an English mathematician and physicist who made extensive contributions to science. He spent all of his academic career at the University of Cambridge. Amo ...
, 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 Orthographic projection (also orthogonal projection and analemma) is a means of representing three-dimensional objects in two dimensions. Orthographic projection is a form of parallel projection in which all the projection lines are orthogonal ...
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 Lift or LIFT may refer to: Physical devices * Elevator, or lift, a device used for raising and lowering people or goods ** Paternoster lift, a type of lift using a continuous chain of cars which do not stop ** Patient lift, or Hoyer lift, mobile ...
.
Airship An airship or dirigible balloon is a type of aerostat or lighter-than-air aircraft that can navigate through the air under its own power. Aerostats gain their lift from a lifting gas that is less dense than the surrounding air. In early ...
s 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).See Batchelor (1967), p. 341.


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 In fluid dynamics, stagnation pressure is the static pressure at a stagnation point in a fluid flow.Clancy, L.J., ''Aerodynamics'', Section 3.5 At a stagnation point the fluid velocity is zero. In an incompressible flow, stagnation pressure is equ ...
over the whole area. No real object exactly corresponds to this behavior. c_ 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_ around 1, more or less. Smoother objects can have much lower values of c_. The equation is precise – it simply provides the definition of c_ (
drag coefficient In fluid dynamics, the drag coefficient (commonly denoted as: c_\mathrm, c_x or c_) is a dimensionless quantity that is used to quantify the drag or resistance of an object in a fluid environment, such as air or water. It is used in the drag e ...
), 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 Mass is an intrinsic property of a body. It was traditionally believed to be related to the quantity of matter in a physical body, until the discovery of the atom and particle physics. It was found that different atoms and different element ...
of fluid strikes per second. Therefore, the change of
momentum In Newtonian mechanics, momentum (more specifically linear momentum or translational momentum) is the product of the mass and velocity of an object. It is a vector quantity, possessing a magnitude and a direction. If is an object's mass ...
per time, i.e. the
force In physics, a force is an influence that can change the motion of an object. A force can cause an object with mass to change its velocity (e.g. moving from a state of rest), i.e., to accelerate. Force can also be described intuitively as a ...
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 F_ \propto P_ A where ''P''D is the pressure exerted by the fluid on area ''A''. Here the pressure ''P''D is referred to as
dynamic pressure In fluid dynamics, dynamic pressure (denoted by or and sometimes called velocity pressure) is the quantity defined by:Clancy, L.J., ''Aerodynamics'', Section 3.5 :q = \frac\rho\, u^2 where (in SI units): * is the dynamic pressure in pascals ...
due to the kinetic energy of the fluid experiencing relative flow velocity ''u''. This is defined in similar form as the kinetic energy equation: P_ = \frac12 \rho u^2


Derivation

The drag equation may be derived to within a multiplicative constant by the method of
dimensional analysis In engineering and science, dimensional analysis is the analysis of the relationships between different physical quantities by identifying their base quantities (such as length, mass, time, and electric current) and units of measure (such as ...
. 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 The viscosity of a fluid is a measure of its resistance to deformation at a given rate. For liquids, it corresponds to the informal concept of "thickness": for example, syrup has a higher viscosity than water. Viscosity quantifies the intern ...
''ν'' of the fluid, * size of the body, expressed in terms of its wetted area ''A'', and * drag force ''F''d. Using the algorithm of the
Buckingham π theorem In engineering, applied mathematics, and physics, the Buckingham theorem is a key theorem in dimensional analysis. It is a formalization of Rayleigh's method of dimensional analysis. Loosely, the theorem states that if there is a physically m ...
, these five variables can be reduced to two dimensionless groups: *
drag coefficient In fluid dynamics, the drag coefficient (commonly denoted as: c_\mathrm, c_x or c_) is a dimensionless quantity that is used to quantify the drag or resistance of an object in a fluid environment, such as air or water. It is used in the drag e ...
cd and * Reynolds number Re. 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_, u, A, \rho, \nu) = 0. 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 the Reynolds number, given by \mathrm = \frac and the drag coefficient, given by c_ = \frac. Thus the function of five variables may be replaced by another function of only two variables: f_b\left(\frac, \frac \right) = 0. 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 ''F''d, it is possible to express it as \begin \frac &= f_c\left(\frac \right) \\ F_ &= \tfrac12 \rho A u^2 f_c(\mathrm) \\ c_ &= f_c(\mathrm) \end 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 a function of five variables) a much simpler one: the determination of the drag as a function of only one variable, the Reynolds number. If the fluid is a gas, certain properties of the gas influence the drag and those properties must also be taken into account. Those properties are conventionally considered to be the absolute temperature of the gas, and the ratio of its specific heats. These two properties determine the speed of sound in the gas at its given temperature. The Buckingham pi theorem then leads to a third dimensionless group, the ratio of the relative velocity to the speed of sound, which is known as the
Mach number Mach number (M or Ma) (; ) is a dimensionless quantity in fluid dynamics representing the ratio of flow velocity past a boundary to the local speed of sound. It is named after the Moravian physicist and philosopher Ernst Mach. : \mathrm = \ ...
. Consequently when a body is moving relative to a gas, the drag coefficient varies with the Mach number and 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.


Experimental methods

To empirically determine the Reynolds number dependence, instead of experimenting on a large body with fast-flowing fluids (such as real-size airplanes in
wind tunnel Wind tunnels are large tubes with air blowing through them which are used to replicate the interaction between air and an object flying through the air or moving along the ground. Researchers use wind tunnels to learn more about how an aircraft ...
s), one may just as well experiment using a small model in a flow of higher velocity because these two systems deliver similitude by having the same Reynolds number. If the same Reynolds number and Mach number cannot be achieved just by using a flow of higher velocity it may be advantageous to use a fluid of greater density or lower viscosity.


See also

* Aerodynamic drag *
Angle of attack In fluid dynamics, angle of attack (AOA, α, or \alpha) is the angle between a reference line on a body (often the chord line of an airfoil) and the vector representing the relative motion between the body and the fluid through which it is ...
* Morison equation *
Stall (flight) In fluid dynamics, a stall is a reduction in the lift coefficient generated by a foil as angle of attack increases.Crane, Dale: ''Dictionary of Aeronautical Terms, third edition'', p. 486. Aviation Supplies & Academics, 1997. This occurs when ...
*
Terminal velocity Terminal velocity is the maximum velocity (speed) attainable by an object as it falls through a fluid ( air is the most common example). It occurs when the sum of the drag force (''Fd'') and the buoyancy is equal to the downward force of gravit ...


References


External links

* * * *{{cite web , first=Tom , last=Benson , url=https://www.grc.nasa.gov/www/k-12/rocket/drageq.html , title=The drag equation , publisher=NASA , location=US , access-date=2022-06-09 Drag (physics) Equations of fluid dynamics Aircraft wing design