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In
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) a ...
, drag (sometimes called air resistance, a type of
friction Friction is the force resisting the relative motion of solid surfaces, fluid layers, and material elements sliding against each other. There are several types of friction: *Dry friction is a force that opposes the relative lateral motion of ...
, or fluid resistance, another type of friction or fluid friction) is a
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 ...
acting opposite to the relative motion of any object moving with respect to a surrounding fluid. This can exist between two fluid layers (or surfaces) or between a fluid and a
solid Solid is one of the four fundamental states of matter (the others being liquid, gas, and plasma). The molecules in a solid are closely packed together and contain the least amount of kinetic energy. A solid is characterized by structur ...
surface. Unlike other resistive forces, such as dry friction, which are nearly independent of velocity, the drag force depends on velocity. Drag force is proportional to the velocity for low-speed flow and the squared velocity for high speed flow, where the distinction between low and high speed is measured by the
Reynolds number In fluid mechanics, the Reynolds number () is a dimensionless quantity that helps predict fluid flow patterns in different situations by measuring the ratio between inertial and viscous forces. At low Reynolds numbers, flows tend to be dom ...
. Even though the ultimate cause of drag is viscous friction, turbulent drag is independent of
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 int ...
. Drag forces always tend to decrease fluid velocity relative to the solid object in the fluid's path.


Examples

Examples of drag include the component of the net
aerodynamic Aerodynamics, from grc, ἀήρ ''aero'' (air) + grc, δυναμική (dynamics), is the study of the motion of air, particularly when affected by a solid object, such as an airplane wing. It involves topics covered in the field of fluid dyn ...
or
hydrodynamic 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) ...
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 ...
acting opposite to the direction of movement of a solid object such as cars ( automobile drag coefficient), aircraft and boat hulls; or acting in the same geographical direction of motion as the solid, as for sails attached to a down wind sail boat, or in intermediate directions on a sail depending on points of sail. In the case of viscous drag of fluid in a pipe, drag force on the immobile pipe decreases fluid velocity relative to the pipe. In the physics of sports, the drag force is necessary to explain the motion of balls, javelins, arrows and frisbees and the performance of runners and swimmers.


Types

Types of drag are generally divided into the following categories: * form drag or
pressure 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 ...
due to the size and shape of a body *
skin friction drag 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 ...
or viscous drag due to the friction between the fluid and a surface which may be the outside of an object or inside such as the bore of a pipe The effect of streamlining on the relative proportions of skin friction and form drag is shown for two different body sections, an airfoil, which is a streamlined body, and a cylinder, which is a bluff body. Also shown is a flat plate illustrating the effect that orientation has on the relative proportions of skin friction and pressure difference between front and back. A body is known as bluff (or blunt) if the source of drag is dominated by pressure forces and streamlined if the drag is dominated by viscous forces. Road vehicles are bluff bodies. For aircraft, pressure and friction drag are included in the definition of parasitic drag. Parasite drag is often expressed in terms of a hypothetical (in so far as there is no edge spillage drag) "equivalent parasite drag area" which is the area of a flat plate perpendicular to the flow. It is used for comparing the drag of different aircraft. For example, the
Douglas DC-3 The Douglas DC-3 is a propeller-driven airliner manufactured by Douglas Aircraft Company, which had a lasting effect on the airline industry in the 1930s to 1940s and World War II. It was developed as a larger, improved 14-bed sleeper versi ...
has an equivalent parasite area of 23.7 sq ft and the
McDonnell Douglas DC-9 The McDonnell Douglas DC-9 is an American five-abreast single-aisle aircraft designed by the Douglas Aircraft Company. It was initially produced by the developer company as the Douglas DC-9 until August 1967 and then by McDonnell Douglas. Afte ...
, with 30 years of advancement in aircraft design, an area of 20.6 sq ft although it carried five times as many passengers. *
lift-induced drag In aerodynamics, lift-induced drag, induced drag, vortex drag, or sometimes drag due to lift, is an aerodynamic drag force that occurs whenever a moving object redirects the airflow coming at it. This drag force occurs in airplanes due to wings o ...
appears with
wing A wing is a type of fin that produces lift while moving through air or some other fluid. Accordingly, wings have streamlined cross-sections that are subject to aerodynamic forces and act as airfoils. A wing's aerodynamic efficiency is e ...
s or a
lifting body A lifting body is a fixed-wing aircraft or spacecraft configuration in which the body itself produces lift. In contrast to a flying wing, which is a wing with minimal or no conventional fuselage, a lifting body can be thought of as a fuselage ...
in aviation and with semi-planing or
planing hull Planing may refer to: * Planing (boat) or hydroplaning, a method by which a hull skims over the surface of the water * Hydroplaning (tires), a loss of traction caused by a layer of water between the tires and the road surface * Using a plane (t ...
s for
watercraft Any vehicle used in or on water as well as underwater, including boats, ships, hovercraft and submarines, is a watercraft, also known as a water vessel or waterborne vessel. A watercraft usually has a propulsive capability (whether by sai ...
*
wave drag In physics, mathematics, and related fields, a wave is a propagating dynamic disturbance (change from equilibrium) of one or more quantities. Waves can be periodic, in which case those quantities oscillate repeatedly about an equilibrium (re ...
(
aerodynamics Aerodynamics, from grc, ἀήρ ''aero'' (air) + grc, δυναμική (dynamics), is the study of the motion of air, particularly when affected by a solid object, such as an airplane wing. It involves topics covered in the field of fluid dy ...
) is caused by the presence of shockwaves and first appears at subsonic aircraft speeds when local flow velocities become supersonic. The wave drag of the supersonic
Concorde The Aérospatiale/BAC Concorde () is a retired Franco-British supersonic airliner jointly developed and manufactured by Sud Aviation (later Aérospatiale) and the British Aircraft Corporation (BAC). Studies started in 1954, and France an ...
prototype aircraft was reduced at Mach 2 by 1.8% by applying the
area rule The Whitcomb area rule, named after NACA engineer Richard Whitcomb and also called the transonic area rule, is a design procedure used to reduce an aircraft's drag at transonic speeds which occur between about Mach 0.75 and 1.2. For superson ...
which extended the rear fuselage 3.73m on the production aircraft. * wave resistance (ship hydrodynamics) or
wave drag In physics, mathematics, and related fields, a wave is a propagating dynamic disturbance (change from equilibrium) of one or more quantities. Waves can be periodic, in which case those quantities oscillate repeatedly about an equilibrium (re ...
occurs when a solid object is moving along a fluid boundary and making surface waves *boat-tail drag on an aircraft is caused by the angle with which the rear fuselage, or engine nacelle, narrows to the engine exhaust diameter. File:Concorde first visit Heathrow Fitzgerald.jpg, Concorde with 'high' wave drag tail File:Aerospatial Concorde (6018513515).jpg, Concorde with 'low' wave drag tail File:BAe Hawk Mk127 76 Sqn RAAF rear view.jpg, Hawk aircraft showing base area above circular engine exhaust


The drag equation

Drag depends on the properties of the fluid and on the size, shape, and speed of the object. One way to express this is by means of the drag equation: :F_D\, =\, \tfrac12\, \rho\, v^2\, C_D\, A where :''F_D'' is the drag force, :\rho is the
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. Mathematicall ...
of the fluid, :''v'' is the speed of the object relative to the fluid, :''A'' is the
cross sectional area In geometry and science, a cross section is the non-empty intersection of a solid body in three-dimensional space with a plane, or the analog in higher-dimensional spaces. Cutting an object into slices creates many parallel cross-sections. ...
, and :''C_D'' 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 number. The drag coefficient depends on the shape of the object and on the
Reynolds number In fluid mechanics, the Reynolds number () is a dimensionless quantity that helps predict fluid flow patterns in different situations by measuring the ratio between inertial and viscous forces. At low Reynolds numbers, flows tend to be dom ...
:R_e=\frac=\frac, where :''D'' is some characteristic diameter or linear
dimension In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coord ...
. Actually ''D'' it is the equivalent
diameter In geometry, a diameter of a circle is any straight line segment that passes through the center of the circle and whose endpoints lie on the circle. It can also be defined as the longest chord of the circle. Both definitions are also valid f ...
''D_''of the object. For a sphere D_ is the D of the sphere itself. :For a rectangular shape cross-section in the motion direction, D_ = 1.30 \cdot \frac , where a and b are the rectangle edges. : is the kinematic viscosity of the fluid (equal to the dynamic viscosity divided by the density ). At low R_e, C_D is asymptotically proportional to R_e^, which means that the drag is linearly proportional to the speed, i.e. the drag force on a small sphere moving through a viscous fluid is given by the
Stokes Law In 1851, George Gabriel Stokes derived an expression, now known as Stokes' law, for the frictional force – also called drag force – exerted on spherical objects with very small Reynolds numbers in a viscous fluid. Stokes' law is derived by ...
: :F_ = 6 \pi \mu R v At high R_e, C_D is more or less constant and drag will vary as the square of the speed. The graph to the right shows how C_D varies with R_e for the case of a sphere. Since the power needed to overcome the drag force is the product of the force times speed, the power needed to overcome drag will vary as the square of the speed at low Reynolds numbers and as the cube of the speed at high numbers. It can be demonstrated that drag force can be expressed as a function of a dimensionless number, which is dimensionally identical to the
Bejan number There are two different Bejan numbers (Be) used in the scientific domains of thermodynamics and fluid mechanics. Bejan numbers are named after Adrian Bejan. Thermodynamics In the field of thermodynamics the Bejan number is the ratio of heat transfe ...
.Liversage, P., and Trancossi, M. (2018). Analysis of triangular sharkskin profiles according to second law, Modelling, Measurement and Control B. 87(3), 188-196. http://www.iieta.org/sites/default/files/Journals/MMC/MMC_B/87.03_11.pdf Consequently, drag force and drag coefficient can be a function of Bejan number. In fact, from the expression of
drag force In fluid dynamics, drag (sometimes called air resistance, a type of friction, or fluid resistance, another type of friction or fluid friction) is a force acting opposite to the relative motion of any object moving with respect to a surrounding flu ...
it has been obtained: :D = \Delta_p A_w = \frac C_D A_f \frac Re_L^2 and consequently allows expressing the drag
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 ...
C_D as a function of
Bejan number There are two different Bejan numbers (Be) used in the scientific domains of thermodynamics and fluid mechanics. Bejan numbers are named after Adrian Bejan. Thermodynamics In the field of thermodynamics the Bejan number is the ratio of heat transfe ...
and the ratio between wet area A_w and front area A_f: :C_D = 2\frac\frac where Re_Lis the Reynolds number related to fluid path length L.


At high velocity

As mentioned, the drag equation with a constant drag coefficient gives the force experienced by an object moving through a
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 shear ...
at relatively large velocity (i.e. high
Reynolds number In fluid mechanics, the Reynolds number () is a dimensionless quantity that helps predict fluid flow patterns in different situations by measuring the ratio between inertial and viscous forces. At low Reynolds numbers, flows tend to be dom ...
, Re > ~1000). This is also called ''quadratic drag''. 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. A ...
, who originally used ''L''2 in place of ''A'' (''L'' being some length). :F_D\, =\, \tfrac12\, \rho\, v^2\, C_d\, A, see derivation The reference area ''A'' is often
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 orthogona ...
of the object (frontal area)—on a plane perpendicular to the direction of motion—e.g. for objects with a simple shape, such as a sphere, this is the cross sectional area. Sometimes a body is a composite of different parts, each with a different reference areas, in which case a drag coefficient corresponding to each of those different areas must be determined. In the case of a wing the reference areas are the same and the drag force is in the same ratio to the lift force as the ratio of drag coefficient to lift coefficient. Therefore, the reference for a wing is often the lifting area ("wing area") rather than the frontal area. For an object with a smooth surface, and non-fixed separation points—like a sphere or circular cylinder—the drag coefficient may vary with Reynolds number ''Re'', even up to very high values (''Re'' of the
order Order, ORDER or Orders may refer to: * Categorization, the process in which ideas and objects are recognized, differentiated, and understood * Heterarchy, a system of organization wherein the elements have the potential to be ranked a number of ...
107). Batchelor (1967), p. 341. For an object with well-defined fixed separation points, like a circular disk with its plane normal to the flow direction, the drag coefficient is constant for ''Re'' > 3,500. Further the drag coefficient ''Cd'' is, in general, a function of the orientation of the flow with respect to the object (apart from symmetrical objects like a sphere).


Power

Under the assumption that the fluid is not moving relative to the currently used reference system, the power required to overcome the aerodynamic drag is given by: : P_d = \mathbf_d \cdot \mathbf = \tfrac12 \rho v^3 A C_d Note that the power needed to push an object through a fluid increases as the cube of the velocity. A car cruising on a highway at may require only to overcome aerodynamic drag, but that same car at requires . With a doubling of speed the drag (force) quadruples per the formula. Exerting 4 times the force over a fixed distance produces 4 times as much work. At twice the speed the work (resulting in displacement over a fixed distance) is done twice as fast. Since power is the rate of doing work, 4 times the work done in half the time requires 8 times the power. When the fluid is moving relative to the reference system (e.g. a car driving into headwind) the power required to overcome the aerodynamic drag is given by: : P_d = \mathbf_d \cdot \mathbf = \tfrac12 C_d A \rho (v_w + v_o)^3 Where v_w is the wind speed and v_o is the object speed (both relative to ground).


Velocity of a falling object

The
velocity Velocity is the directional speed of an object in motion as an indication of its rate of change in position as observed from a particular frame of reference and as measured by a particular standard of time (e.g. northbound). Velocity i ...
as a function of time for an object falling through a non-dense medium, and released at zero relative-velocity ''v'' = 0 at time ''t'' = 0, is roughly given by a function involving a
hyperbolic tangent In mathematics, hyperbolic functions are analogues of the ordinary trigonometric functions, but defined using the hyperbola rather than the circle. Just as the points form a circle with a unit radius, the points form the right half of the ...
(tanh): : v(t) = \sqrt \tanh \left(t \sqrt \right). \, The hyperbolic tangent has a
limit Limit or Limits may refer to: Arts and media * ''Limit'' (manga), a manga by Keiko Suenobu * ''Limit'' (film), a South Korean film * Limit (music), a way to characterize harmony * "Limit" (song), a 2016 single by Luna Sea * "Limits", a 2019 ...
value of one, for large time ''t''. In other words, velocity
asymptotically In analytic geometry, an asymptote () of a curve is a line such that the distance between the curve and the line approaches zero as one or both of the ''x'' or ''y'' coordinates tends to infinity. In projective geometry and related contexts, ...
approaches a maximum value called the terminal velocity ''vt'': :v_ = \sqrt. \, For an object falling and released at relative-velocity ''v'' = vi at time ''t'' = 0, with vi < vt, is also defined in terms of the hyperbolic tangent function: :v(t) = v_t \tanh \left( t \frac + \operatorname\left( \frac \right) \right). \, For vi > vt, the velocity function is defined in terms of the hyperbolic cotangent function: :v(t) = v_t \coth \left( t \frac + \coth^\left( \frac \right) \right). \, The hyperbolic cotangent has also a
limit Limit or Limits may refer to: Arts and media * ''Limit'' (manga), a manga by Keiko Suenobu * ''Limit'' (film), a South Korean film * Limit (music), a way to characterize harmony * "Limit" (song), a 2016 single by Luna Sea * "Limits", a 2019 ...
value of one, for large time ''t''. Velocity
asymptotically In analytic geometry, an asymptote () of a curve is a line such that the distance between the curve and the line approaches zero as one or both of the ''x'' or ''y'' coordinates tends to infinity. In projective geometry and related contexts, ...
tends to the terminal velocity ''vt'', strictly from above ''vt''. For vi = vt, the velocity is constant: :v(t) = v_t. \, Actually, these functions are defined by the solution of the following
differential equation In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, ...
: :g - \frac v^2 = \frac. \, Or, more generically (where ''F''(''v'') are the forces acting on the object beyond drag): :\frac\sum F(v) - \frac v^2 = \frac. \, For a potato-shaped object of average diameter ''d'' and of density ''ρobj'', terminal velocity is about :v_ = \sqrt. \, For objects of water-like density (raindrops, hail, live objects—mammals, birds, insects, etc.) falling in air near Earth's surface at sea level, the terminal velocity is roughly equal to :v_ = 90 \sqrt, \, with ''d'' in metre and ''vt'' in m/s. For example, for a human body ( \mathbf d ≈0.6 m) \mathbf v_t ≈70 m/s, for a small animal like a cat ( \mathbf d ≈0.2 m) \mathbf v_t ≈40 m/s, for a small bird ( \mathbf d ≈0.05 m) \mathbf v_t ≈20 m/s, for an insect ( \mathbf d ≈0.01 m) \mathbf v_t ≈9 m/s, and so on. Terminal velocity for very small objects (pollen, etc.) at low Reynolds numbers is determined by Stokes law. Terminal velocity is higher for larger creatures, and thus potentially more deadly. A creature such as a mouse falling at its terminal velocity is much more likely to survive impact with the ground than a human falling at its terminal velocity. A small animal such as a
cricket Cricket is a bat-and-ball game played between two teams of eleven players on a field at the centre of which is a pitch with a wicket at each end, each comprising two bails balanced on three stumps. The batting side scores runs by st ...
impacting at its terminal velocity will probably be unharmed. This, combined with the relative ratio of limb cross-sectional area vs. body mass (commonly referred to as the square–cube law), explains why very small animals can fall from a large height and not be harmed.


Very low Reynolds numbers: Stokes' drag

The equation for viscous resistance or linear drag is appropriate for objects or particles moving through a fluid at relatively slow speeds where there is no turbulence (i.e. low
Reynolds number In fluid mechanics, the Reynolds number () is a dimensionless quantity that helps predict fluid flow patterns in different situations by measuring the ratio between inertial and viscous forces. At low Reynolds numbers, flows tend to be dom ...
, R_e < 1). Note that purely laminar flow only exists up to Re = 0.1 under this definition. In this case, the force of drag is approximately proportional to velocity. The equation for viscous resistance is: :\mathbf_d = - b \mathbf \, where: :\mathbf b is a constant that depends on both the material properties of the object and fluid, as well as the geometry of the object; and : \mathbf is the velocity of the object. When an object falls from rest, its velocity will be :v(t) = \frac\left(1-e^\right) where: : \rho is the density of the object, : \rho_0 is density of the fluid, : V is the volume of the object, : g is the acceleration due to gravity (i.e., 9.8 m/s^2), and : m is mass of the object. The velocity asymptotically approaches the terminal velocity \mathbf v_t = \frac. For a given \mathbf b , denser objects fall more quickly. For the special case of small spherical objects moving slowly through a
viscous 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 in ...
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 shear ...
(and thus at small Reynolds number), George Gabriel Stokes derived an expression for the drag constant: :b = 6 \pi \eta r\, where: :\mathbf r is the Stokes radius of the particle, and \mathbf \eta is the fluid viscosity. The resulting expression for the drag is known as Stokes' drag: :\mathbf_d = -6 \pi \eta r\, \mathbf. For example, consider a small sphere with radius \mathbf r = 0.5 micrometre (diameter = 1.0 µm) moving through water at a velocity \mathbf v of 10 µm/s. Using 10−3 Pa·s as the
dynamic 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 int ...
of water in SI units, we find a drag force of 0.09 pN. This is about the drag force that a bacterium experiences as it swims through water. The drag coefficient of a sphere can be determined for the general case of a laminar flow with Reynolds numbers less than 12 \cdot 10^5 using the following formula: C_D = \frac +\frac+0.4 ~\text~~~~~Re<2\cdot 10^5 For Reynolds numbers less than 1, Stokes' law applies and the drag coefficient approaches \frac!


Aerodynamics

In
aerodynamics Aerodynamics, from grc, ἀήρ ''aero'' (air) + grc, δυναμική (dynamics), is the study of the motion of air, particularly when affected by a solid object, such as an airplane wing. It involves topics covered in the field of fluid dy ...
, aerodynamic drag is the fluid drag force that acts on any moving solid body in the direction of the fluid
freestream The freestream is the air far upstream of an aerodynamic Aerodynamics, from grc, ἀήρ ''aero'' (air) + grc, δυναμική (dynamics), is the study of the motion of air, particularly when affected by a solid object, such as an airplane ...
flow. From the body's perspective (near-field approach), the drag results from forces due to pressure distributions over the body surface, symbolized D_, and forces due to skin friction, which is a result of viscosity, denoted D_. Alternatively, calculated from the flowfield perspective (far-field approach), the drag force results from three natural phenomena:
shock wave In physics, a shock wave (also spelled shockwave), or shock, is a type of propagating disturbance that moves faster than the local speed of sound in the medium. Like an ordinary wave, a shock wave carries energy and can propagate through a me ...
s, vortex sheet, and
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 int ...
.


Overview

The
pressure Pressure (symbol: ''p'' or ''P'') is the force applied perpendicular to the surface of an object per unit area over which that force is distributed. Gauge pressure (also spelled ''gage'' pressure)The preferred spelling varies by country a ...
distribution acting on a body's surface exerts normal forces on the body. Those forces can be summed and the component of that force that acts downstream represents the drag force, D_, due to pressure distribution acting on the body. The nature of these normal forces combines shock wave effects, vortex system generation effects, and wake viscous mechanisms. The
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 int ...
of the fluid has a major effect on drag. In the absence of viscosity, the pressure forces acting to retard the vehicle are canceled by a pressure force further aft that acts to push the vehicle forward; this is called pressure recovery and the result is that the drag is zero. That is to say, the work the body does on the airflow, is reversible and is recovered as there are no frictional effects to convert the flow energy into heat. Pressure recovery acts even in the case of viscous flow. Viscosity, however results in pressure drag and it is the dominant component of drag in the case of vehicles with regions of separated flow, in which the pressure recovery is fairly ineffective. The friction drag force, which is a tangential force on the aircraft surface, depends substantially on
boundary layer In physics and fluid mechanics, a boundary layer is the thin layer of fluid in the immediate vicinity of a bounding surface formed by the fluid flowing along the surface. The fluid's interaction with the wall induces a no-slip boundary cond ...
configuration and viscosity. The net friction drag, D_f, is calculated as the downstream projection of the viscous forces evaluated over the body's surface. The sum of friction drag and pressure (form) drag is called viscous drag. This drag component is due to viscosity. In a thermodynamic perspective, viscous effects represent irreversible phenomena and, therefore, they create entropy. The calculated viscous drag D_v use entropy changes to accurately predict the drag force. When the airplane produces lift, another drag component results.
Induced drag In aerodynamics, lift-induced drag, induced drag, vortex drag, or sometimes drag due to lift, is an aerodynamic drag force that occurs whenever a moving object redirects the airflow coming at it. This drag force occurs in airplanes due to wings ...
, symbolized D_i, is due to a modification of the pressure distribution due to the trailing vortex system that accompanies the lift production. An alternative perspective on lift and drag is gained from considering the change of momentum of the airflow. The wing intercepts the airflow and forces the flow to move downward. This results in an equal and opposite force acting upward on the wing which is the lift force. The change of momentum of the airflow downward results in a reduction of the rearward momentum of the flow which is the result of a force acting forward on the airflow and applied by the wing to the air flow; an equal but opposite force acts on the wing rearward which is the induced drag. Another drag component, namely
wave drag In physics, mathematics, and related fields, a wave is a propagating dynamic disturbance (change from equilibrium) of one or more quantities. Waves can be periodic, in which case those quantities oscillate repeatedly about an equilibrium (re ...
, D_w, results from shock waves in transonic and supersonic flight speeds. The shock waves induce changes in the boundary layer and pressure distribution over the body surface. In summary, there are three ways of categorising drag. # Pressure drag and friction drag # Profile drag and induced drag # Vortex drag, wave drag and wake drag


History

The idea that a moving body passing through air or another fluid encounters resistance had been known since the time of
Aristotle Aristotle (; grc-gre, Ἀριστοτέλης ''Aristotélēs'', ; 384–322 BC) was a Greek philosopher and polymath during the Classical period in Ancient Greece. Taught by Plato, he was the founder of the Peripatetic school of ...
. According to
Mervyn O'Gorman Mervyn Joseph Pius O'Gorman (19 December 1871 – 16 March 1958) was a British electrical and aircraft engineer. After working as an electrical engineer, he was appointed Superintendent of what became the Royal Aircraft Factory at Farnborough ...
, this was named "drag" by Archibald Reith Low.https://archive.org/details/Flight_International_Magazine_1913-02-01-pdf/page/n19/mode/2up Flight, 1913, p. 126 Louis Charles Breguet's paper of 1922 began efforts to reduce drag by streamlining. Breguet went on to put his ideas into practice by designing several record-breaking aircraft in the 1920s and 1930s. Ludwig Prandtl's boundary layer theory in the 1920s provided the impetus to minimise skin friction. A further major call for streamlining was made by Sir Melvill Jones who provided the theoretical concepts to demonstrate emphatically the importance of streamlining in
aircraft An aircraft is a vehicle that is able to flight, fly by gaining support from the Atmosphere of Earth, air. It counters the force of gravity by using either Buoyancy, static lift or by using the Lift (force), dynamic lift of an airfoil, or in ...
design. In 1929 his paper ‘The Streamline Airplane’ presented to the
Royal Aeronautical Society The Royal Aeronautical Society, also known as the RAeS, is a British multi-disciplinary professional institution dedicated to the global aerospace community. Founded in 1866, it is the oldest aeronautical society in the world. Members, Fellows, ...
was seminal. He proposed an ideal aircraft that would have minimal drag which led to the concepts of a 'clean' monoplane and retractable undercarriage. The aspect of Jones's paper that most shocked the designers of the time was his plot of the horse power required versus velocity, for an actual and an ideal plane. By looking at a data point for a given aircraft and extrapolating it horizontally to the ideal curve, the velocity gain for the same power can be seen. When Jones finished his presentation, a member of the audience described the results as being of the same level of importance as the
Carnot cycle A Carnot cycle is an ideal thermodynamic cycle proposed by French physicist Sadi Carnot in 1824 and expanded upon by others in the 1830s and 1840s. By Carnot's theorem, it provides an upper limit on the efficiency of any classical thermodyna ...
in thermodynamics.


Lift-induced drag and parasitic drag


Lift-induced drag

Lift-induced drag (also called induced drag) is drag which occurs as the result of the creation of lift on a three-dimensional
lifting body A lifting body is a fixed-wing aircraft or spacecraft configuration in which the body itself produces lift. In contrast to a flying wing, which is a wing with minimal or no conventional fuselage, a lifting body can be thought of as a fuselage ...
, such as the
wing A wing is a type of fin that produces lift while moving through air or some other fluid. Accordingly, wings have streamlined cross-sections that are subject to aerodynamic forces and act as airfoils. A wing's aerodynamic efficiency is e ...
or fuselage of an airplane. Induced drag consists primarily of two components: drag due to the creation of trailing vortices (vortex drag); and the presence of additional viscous drag (lift-induced viscous drag) that is not present when lift is zero. The trailing vortices in the flow-field, present in the wake of a lifting body, derive from the turbulent mixing of air from above and below the body which flows in slightly different directions as a consequence of creation of lift. With other parameters remaining the same, as the lift generated by a body increases, so does the lift-induced drag. This means that as the wing's
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 m ...
increases (up to a maximum called the stalling angle), the lift coefficient also increases, and so too does the lift-induced drag. At the onset of stall, lift is abruptly decreased, as is lift-induced drag, but viscous pressure drag, a component of parasite drag, increases due to the formation of turbulent unattached flow in the wake behind the body.


Parasitic drag

Parasitic drag, or profile drag, is drag caused by moving a solid object through a fluid. Parasitic drag is made up of multiple components including viscous pressure drag (form drag), and drag due to surface roughness (skin friction drag). Additionally, the presence of multiple bodies in relative proximity may incur so called interference drag, which is sometimes described as a component of parasitic drag. In aviation, induced drag tends to be greater at lower speeds because a high
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 m ...
is required to maintain lift, creating more drag. However, as speed increases the angle of attack can be reduced and the induced drag decreases. Parasitic drag, however, increases because the fluid is flowing more quickly around protruding objects increasing friction or drag. At even higher speeds (
transonic Transonic (or transsonic) flow is air flowing around an object at a speed that generates regions of both subsonic and supersonic airflow around that object. The exact range of speeds depends on the object's critical Mach number, but transoni ...
),
wave drag In physics, mathematics, and related fields, a wave is a propagating dynamic disturbance (change from equilibrium) of one or more quantities. Waves can be periodic, in which case those quantities oscillate repeatedly about an equilibrium (re ...
enters the picture. Each of these forms of drag changes in proportion to the others based on speed. The combined overall drag curve therefore shows a minimum at some airspeed - an aircraft flying at this speed will be at or close to its optimal efficiency. Pilots will use this speed to maximize endurance (minimum fuel consumption), or maximize gliding range in the event of an engine failure.


Power curve in aviation

The interaction of parasitic and induced drag ''vs.'' airspeed can be plotted as a characteristic curve, illustrated here. In aviation, this is often referred to as the ''power curve'', and is important to pilots because it shows that, below a certain airspeed, maintaining airspeed counterintuitively requires ''more'' thrust as speed decreases, rather than less. The consequences of being "behind the curve" in flight are important and are taught as part of pilot training. At the subsonic airspeeds where the "U" shape of this curve is significant, wave drag has not yet become a factor, and so it is not shown in the curve.


Wave drag in transonic and supersonic flow

Wave drag (also called compressibility drag) is drag that is created when a body moves in a compressible fluid and at speeds that are close to the speed of sound in that fluid. In
aerodynamics Aerodynamics, from grc, ἀήρ ''aero'' (air) + grc, δυναμική (dynamics), is the study of the motion of air, particularly when affected by a solid object, such as an airplane wing. It involves topics covered in the field of fluid dy ...
, wave drag consists of multiple components depending on the speed regime of the flight. In transonic flight (Mach numbers greater than about 0.8 and less than about 1.4), wave drag is the result of the formation of shockwaves in the fluid, formed when local areas of supersonic (Mach number greater than 1.0) flow are created. In practice, supersonic flow occurs on bodies traveling well below the speed of sound, as the local speed of air increases as it accelerates over the body to speeds above Mach 1.0. However, full supersonic flow over the vehicle will not develop until well past Mach 1.0. Aircraft flying at transonic speed often incur wave drag through the normal course of operation. In transonic flight, wave drag is commonly referred to as transonic compressibility drag. Transonic compressibility drag increases significantly as the speed of flight increases towards Mach 1.0, dominating other forms of drag at those speeds. In supersonic flight (Mach numbers greater than 1.0), wave drag is the result of shockwaves present in the fluid and attached to the body, typically oblique shockwaves formed at the leading and trailing edges of the body. In highly supersonic flows, or in bodies with turning angles sufficiently large, unattached shockwaves, or bow waves will instead form. Additionally, local areas of transonic flow behind the initial shockwave may occur at lower supersonic speeds, and can lead to the development of additional, smaller shockwaves present on the surfaces of other lifting bodies, similar to those found in transonic flows. In supersonic flow regimes, wave drag is commonly separated into two components, supersonic lift-dependent wave drag and supersonic volume-dependent wave drag. The closed form solution for the minimum wave drag of a body of revolution with a fixed length was found by Sears and Haack, and is known as the Sears-Haack Distribution. Similarly, for a fixed volume, the shape for minimum wave drag is the Von Karman Ogive. The
Busemann biplane The Busemann biplane is a theoretical aircraft configuration invented by Adolf Busemann, which avoids the formation of N-type shock waves and thus does not create a sonic boom or the associated wave drag. However in its original form it does not ...
theoretical concept is not subject to wave drag when operated at its design speed, but is incapable of generating lift in this condition.


d'Alembert's paradox

In 1752 d'Alembert proved that potential flow, the 18th century state-of-the-art
inviscid flow In fluid dynamics, inviscid flow is the flow of an inviscid (zero-viscosity) fluid, also known as a superfluid. The Reynolds number of inviscid flow approaches infinity as the viscosity approaches zero. When viscous forces are neglected, suc ...
theory amenable to mathematical solutions, resulted in the prediction of zero drag. This was in contradiction with experimental evidence, and became known as d'Alembert's paradox. In the 19th century the
Navier–Stokes equations In physics, the Navier–Stokes equations ( ) are partial differential equations which describe the motion of viscous fluid substances, named after French engineer and physicist Claude-Louis Navier and Anglo-Irish physicist and mathematician Geo ...
for the description of
viscous 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 in ...
flow were developed by Saint-Venant, Navier and Stokes. Stokes derived the drag around a sphere at very low
Reynolds number In fluid mechanics, the Reynolds number () is a dimensionless quantity that helps predict fluid flow patterns in different situations by measuring the ratio between inertial and viscous forces. At low Reynolds numbers, flows tend to be dom ...
s, the result of which is called
Stokes' law In 1851, George Gabriel Stokes derived an expression, now known as Stokes' law, for the frictional force – also called drag force – exerted on spherical objects with very small Reynolds numbers in a viscous fluid. Stokes' law is derived by ...
.Batchelor (2000), pp. 337–343. In the limit of high Reynolds numbers, the Navier–Stokes equations approach the inviscid Euler equations, of which the potential-flow solutions considered by d'Alembert are solutions. However, all experiments at high Reynolds numbers showed there is drag. Attempts to construct inviscid steady flow solutions to the Euler equations, other than the potential flow solutions, did not result in realistic results. The notion of
boundary layer In physics and fluid mechanics, a boundary layer is the thin layer of fluid in the immediate vicinity of a bounding surface formed by the fluid flowing along the surface. The fluid's interaction with the wall induces a no-slip boundary cond ...
s—introduced by Prandtl in 1904, founded on both theory and experiments—explained the causes of drag at high Reynolds numbers. The boundary layer is the thin layer of fluid close to the object's boundary, where viscous effects remain important even when the viscosity is very small (or equivalently the Reynolds number is very large).


See also

*
Added mass In fluid mechanics, added mass or virtual mass is the inertia added to a system because an accelerating or decelerating body must move (or deflect) some volume of surrounding fluid as it moves through it. Added mass is a common issue because the ...
* Aerodynamic force *
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 m ...
* Atmospheric density * Automobile drag coefficient *
Boundary layer In physics and fluid mechanics, a boundary layer is the thin layer of fluid in the immediate vicinity of a bounding surface formed by the fluid flowing along the surface. The fluid's interaction with the wall induces a no-slip boundary cond ...
*
Coandă effect The Coandă effect ( or ) is the tendency of a fluid jet to stay attached to a convex surface. ''Merriam-Webster'' describes it as "the tendency of a jet of fluid emerging from an orifice to follow an adjacent flat or curved surface and to en ...
*
Drag crisis In fluid dynamics, drag crisis (also known as the Eiffel paradox) is a phenomenon in which drag coefficient drops off suddenly as Reynolds number increases. This has been well studied for round bodies like spheres and cylinders. The drag coeffi ...
*
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 ...
* Drag equation * Gravity drag * Keulegan–Carpenter number * Lift (force) * Morison equation * Nose cone design * Parasitic drag * Projectile motion#Trajectory of a projectile with air resistance *
Ram pressure Ram pressure is a pressure exerted on a body moving through a fluid medium, caused by relative bulk motion of the fluid rather than random thermal motion. It causes a drag force to be exerted on the body. Ram pressure is given in tensor form as ...
*
Reynolds number In fluid mechanics, the Reynolds number () is a dimensionless quantity that helps predict fluid flow patterns in different situations by measuring the ratio between inertial and viscous forces. At low Reynolds numbers, flows tend to be dom ...
*
Stall (fluid mechanics) 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 the ...
*
Stokes' law In 1851, George Gabriel Stokes derived an expression, now known as Stokes' law, for the frictional force – also called drag force – exerted on spherical objects with very small Reynolds numbers in a viscous fluid. Stokes' law is derived by ...
* Terminal velocity *
Wave drag In physics, mathematics, and related fields, a wave is a propagating dynamic disturbance (change from equilibrium) of one or more quantities. Waves can be periodic, in which case those quantities oscillate repeatedly about an equilibrium (re ...
*
Windage Windage is a term used in aerodynamics, firearms ballistics, and automobiles. Usage Aerodynamics Windage is a force created on an object by friction when there is relative movement between air and the object. Windage loss is the reduction in ...


References

*'Improved Empirical Model for Base Drag Prediction on Missile Configurations, based on New Wind Tunnel Data', Frank G Moore et al. NASA Langley Center *'Computational Investigation of Base Drag Reduction for a Projectile at Different Flight Regimes', M A Suliman et al. Proceedings of 13th International Conference on Aerospace Sciences & Aviation Technology, ASAT- 13, May 26 – 28, 2009 *'Base Drag and Thick Trailing Edges', Sighard F. Hoerner, Air Materiel Command, in: Journal of the Aeronautical Sciences, Oct 1950, pp 622–628


Bibliography

* * * * * * * L. J. Clancy (1975), ''Aerodynamics'', Pitman Publishing Limited, London. * Anderson, John D. Jr. (2000); ''Introduction to Flight'', Fourth Edition, McGraw Hill Higher Education, Boston, Massachusetts, USA. 8th ed. 2015, .


External links


Educational materials on air resistanceAerodynamic Drag
and its effect on the acceleration and top speed of a vehicle.
Vehicle Aerodynamic Drag calculator
based on drag coefficient, frontal area and speed.
Smithsonian National Air and Space Museum's How Things Fly websiteEffect of dimples on a golf ball and a car
{{Authority control Articles containing video clips Force