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{{Short description|Retarding force on a body moving in a fluid}}{{Other uses|Drag (disambiguation){{!}}Drag}}{{use dmy dates|date=May 2023}}In fluid dynamics, drag, sometimes referred to as fluid resistance, is a force acting opposite to the relative motion of any object, moving with respect to a surrounding fluid.WEB, Definition of DRAG,www.merriam-webster.com/dictionary/drag, 2023-05-07, Merriam-Webster, This can exist between two fluid layers, two solid surfaces, or between a fluid and solid surface. Drag forces tend to decrease fluid velocity relative to the solid object in the fluid’s path.Unlike other resistive forces, drag force depends on velocity.French (1970), p. 211, Eq. 7-20WEB, What is Drag?,www.grc.nasa.gov/WWW/k-12/airplane/drag1.html, dead,www.grc.nasa.gov/WWW/K-12/airplane/drag1.html," title="web.archive.org/web/20100524003905www.grc.nasa.gov/WWW/K-12/airplane/drag1.html,">web.archive.org/web/20100524003905www.grc.nasa.gov/WWW/K-12/airplane/drag1.html, 2010-05-24, 2011-10-16, This is because drag force is proportional to the velocity of low-speed flow, and the squared velocity for high-speed flow. This distinction between low and high-speed flow is measured by the Reynolds number.

Examples

Examples of drag include:

• Net aerodynamic or hydrodynamic force: Drag acting opposite to the direction of movement of a solid object such as cars, aircraftWEB, What is Drag?,www.grc.nasa.gov/WWW/k-12/airplane/drag1.html, dead,www.grc.nasa.gov/WWW/K-12/airplane/drag1.html," title="web.archive.org/web/20100524003905www.grc.nasa.gov/WWW/K-12/airplane/drag1.html,">web.archive.org/web/20100524003905www.grc.nasa.gov/WWW/K-12/airplane/drag1.html, 2010-05-24, 2011-10-16, and boat hulls.
• Viscous drag of fluid in a pipe: Drag force on the immobile pipe decreases fluid velocity relative to the pipe.WEB, Calculating Viscous Flow: Velocity Profiles in Rivers and Pipes,galileo.phys.virginia.edu/classes/152.mf1i.spring02/RiverViscosity.pdf, 16 October 2011, WEB, Viscous Drag Forces,www.ce.utexas.edu/prof/kinnas/319LAB/Applets/Viscous/viscous.html, 16 October 2011,

• In the physics of sports, drag force is necessary to explain the motion of balls, javelins, arrows and frisbees and the performance of runners and swimmers.JOURNAL, Hernandez-Gomez, J J, Marquina, V, Gomez, R W, 25 July 2013, On the performance of Usain Bolt in the 100 m sprint,www.researchgate.net/publication/236858493, Eur. J. Phys., 34, 5, 1227–1233, 1305.3947, 2013EJPh...34.1227H, 10.1088/0143-0807/34/5/1227, 118693492, 23 April 2016,

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Types of drag are generally divided into the following categories:

• form drag or pressure drag due to the size and shape of a body
• skin friction drag 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 when the source of drag is dominated by pressure forces, and streamlined if the drag is dominated by viscous forces. For example, road vehicles are bluff bodies.Encyclopedia of Automotive Engineering, David Crolla, Paper “Fundamentals, Basic principles in Road vehicle Aerodynamics and Design”, {{ISBN|978 0 470 97402 5}} For aircraft, pressure and friction drag are included in the definition of parasitic drag. Parasite drag is often expressed in terms of a hypothetical.

Parasitic drag experienced by aircraft

This is the area of a flat plate perpendicular to the flow. It is used when comparing the drag of different aircraft For example, the Douglas DC-3 has an equivalent parasite area of {{cvt|23.7|ft2|order=flip}} and the McDonnell Douglas DC-9, with 30 years of advancement in aircraft design, an area of {{cvt|20.6|ft2|order=flip}} although it carried five times as many passengers.Fundamentals of Flight, Second Edition, Richard S. Shevell,{{ISBN|0 13 339060 8}}, p.185

• lift-induced drag appears with wings or a lifting body in aviation and with semi-planing or planing hulls for watercraft
• wave drag (aerodynamics) 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 prototype aircraft was reduced at Mach 2 by 1.8% by applying the area rule which extended the rear fuselage {{cvt|3.73|m}} on the production aircraft.A Case Study By Aerospatiale And British Aerospace On The Concorde By Jean Rech and Clive S. Leyman, AIAA Professional Study Series, Fig. 3.6
• wave resistance (ship hydrodynamics) or wave drag 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.Design For Air Combat, Ray Whitford,{{ISBN|0 7106 0426 2}}, p.212

File:Concorde first visit Heathrow Fitzgerald.jpg|Concorde with ‘high’ wave drag tailFile:Aerospatial Concorde (6018513515).jpg|Concorde with ‘low’ wave drag tail (N.B. rear fuselage spike)File:BAe Hawk Mk127 76 Sqn RAAF rear view.jpg|Hawk aircraft showing base area above circular engine exhaust

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, such as the wing or propeller 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 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 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), wave drag 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.

The drag equation

File:Drag coefficient on a sphere vs. Reynolds number - main trends.svg|upright=1.3|thumb|Drag coefficient Cd for a sphere as a function of Reynolds numberReynolds numberDrag 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, Awhere

• F_D is the drag force,
• rho is the density of the fluid,For Earth’s atmosphere, the air density can be found using the barometric formula. It is 1.293 kg/m3 at 0 °C and 1 atmosphere.
• v is the speed of the object relative to the fluid,
• A is the cross sectional area, and
• C_D is the drag coefficient – a dimensionless number.

The drag coefficient depends on the shape of the object and on the Reynolds numberR_e=frac{vD}{nu} = frac{rho vD}{mu},where

• D is some characteristic diameter or linear dimension. Actually, D is the equivalent diameter D_{e} of the object. For a sphere, D_{e} is the D of the sphere itself.
• For a rectangular shape cross-section in the motion direction, D_{e} = 1.30 cdot frac{(a cdot b)^{0.625}} {(a+b)^{0.25}}, where a and b are the rectangle edges.
• {nu} is the kinematic viscosity of the fluid (equal to the dynamic viscosity {mu} divided by the density {rho} ).

At low R_e, C_D is asymptotically proportional to R_e^{-1}, 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:F_{rm d} = 3 pi mu D vAt high R_e, C_D is more or less constant, but drag will vary as the square of the speed varies. 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.Liversage, P., and Trancossi, M. (2018). “Analysis of triangular sharkskin profiles according to second law”, Modelling, Measurement and Control B. 87(3), 188-196. Consequently, drag force and drag coefficient can be a function of Bejan number. In fact, from the expression of drag force it has been obtained:F_{rm d} = Delta_p A_w = frac{1}{2} C_D A_f frac {nu mu}{l^2}Re_L^2and consequently allows expressing the drag coefficient C_D as a function of Bejan number and the ratio between wet area A_w and front area A_f:C_D = 2frac{A_w}{A_f}frac{Be}{Re_L^2}where Re_Lis the Reynolds number related to fluid path length L.

At high velocity

File:194144main 022 drag.ogv|thumb|320px|Explanation of drag by NASANASAAs mentioned, the drag equation with a constant drag coefficient gives the force moving through fluid a relatively large velocity, i.e. high Reynolds number, Re > ~1000. This is also called quadratic drag. F_D, =, tfrac12, rho, v^2, C_d, A,The derivation of this equation is presented at {{slink|Drag equation#Derivation}}.The reference area A is often the orthographic projection of the object, or the frontal area, on a plane perpendicular to the direction of motion. 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 area (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 as the lift force.Size effects on drag {{Webarchive|url=https://web.archive.org/web/20161109102323www.grc.nasa.gov/WWW/K-12/airplane/sized.html |date=2016-11-09 }}, from NASA Glenn Research Center. Therefore, the reference for a wing is often the lifting area, sometimes referred to as “wing area” rather than the frontal area.Wing geometry definitions {{Webarchive|url=https://web.archive.org/web/20110307125108www.grc.nasa.gov/WWW/k-12/airplane/geom.html |date=2011-03-07 }}, from NASA Glenn Research Center.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, up to extremely high values (Re of the order 107).JOURNAL, Roshko, Anatol, Experiments on the flow past a circular cylinder at very high Reynolds number, Journal of Fluid Mechanics, 10, 3, 1961, 345–356, 10.1017/S0022112061000950, 1961JFM....10..345R, 11816281,authors.library.caltech.edu/10105/1/ROSjfm61.pdf, 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.The 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: The power needed to push an object through a fluid increases as the cube of the velocity increases. For example, a car cruising on a highway at {{convert|50|mph|km/h|abbr=on}} may require only {{convert|10|hp|kW|lk=in}} to overcome aerodynamic drag, but that same car at {{convert|100|mph|km/h|abbr=on}} requires {{convert|80|hp|kW|abbr=on}}.{{citation |url=http://phors.locost7.info/phors06.htm |title=Part 6: Speed and Horsepower |author=Brian Beckman|date=1991 |access-date=18 May 2016 |archive-url=https://web.archive.org/web/20190616063447phors.locost7.info:80/phors06.htm |archive-date=2019-06-16}} With a doubling of speeds, 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, for example, a car driving into headwind, the power required to overcome the aerodynamic drag is given by the following formula: Where v_w is the wind speed and v_o is the object speed (both relative to ground).

Velocity of a falling object

(File:Speed vs time for objects with drag.png|upright=2.5|thumb|An object falling through viscous medium accelerates quickly towards its terminal speed, approaching gradually as the speed gets nearer to the terminal speed. Whether the object experiences turbulent or laminar drag changes the characteristic shape of the graph with turbulent flow resulting in a constant acceleration for a larger fraction of its accelerating time.|center)Velocity 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 (tanh): The hyperbolic tangent has a limit value of one, for large time t. In other words, velocity asymptotically approaches a maximum value called the terminal velocity vt:v_ = sqrt{ frac{2mg}{rho A C_d} }. ,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{ g }{ v_t } + operatorname{arctanh}left( frac{ v_i}{ v_t} 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{ g }{ v_t } + coth^{-1}left( frac{ v_i}{ v_t} right) right). ,The hyperbolic cotangent also has a limit value of one, for large time t. Velocity asymptotically tends to the terminal velocity vt, strictly from above vt.For v’i = v’t, the velocity is constant:v(t) = v_t. These functions are defined by the solution of the following differential equation:g - frac{rho A C_d}{2m} v^2 = frac{dv}{dt}. ,Or, more generically (where F(v) are the forces acting on the object beyond drag):frac{1}{m}sum F(v) - frac{rho A C_d}{2m} v^2 = frac{dv}{dt}. ,For a potato-shaped object of average diameter d and of density ρobj, terminal velocity is aboutv_ = sqrt{ gd frac{ rho_{obj} }{rho} }. ,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 with d in metre and vt in m/s. v_ = 90 sqrt{ d }, ,For example, for a human body ( d ≈0.6 m) v_t ≈70 m/s, for a small animal like a cat ( d ≈0.2 m) v_t ≈40 m/s, for a small bird ( d ≈0.05 m) v_t ≈20 m/s, for an insect ( d ≈0.01 m) 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.In short, 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.Haldane, J.B.S., “On Being the Right Size” {{Webarchive|url=https://web.archive.org/web/20110822151104irl.cs.ucla.edu/papers/right-size.html |date=2011-08-22 }}

Low Reynolds numbers: Stokes’ drag

File:Inclinedthrow.gif|thumb|upright=1.3|Trajectories of three objects thrown at the same angle (70°). The black object does not experience any form of drag and moves along a parabola. The blue object experiences Stokes’ drag, and the green object Newton drag.]]The equation for viscous resistance or linear drag is appropriate for objects or particles moving through a fluid at relatively slow speeds (assuming there is no turbulence). 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:Air friction, from Department of Physics and Astronomy, Georgia State Universitymathbf{F}_d = - b mathbf{v} ,where:

• 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{v} is the velocity of the object.

When an object falls from rest, its velocity will bev(t) = frac{(rho-rho_0),V,g}{b}left(1-e^{-b,t/m}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 v_t = frac{(rho-rho_0)Vg}{b}. For a given b , denser objects fall more quickly.For the special case of small spherical objects moving slowly through a viscous fluid (and thus at small Reynolds number), George Gabriel Stokes derived an expression for the drag constant:b = 6 pi eta r,where r is the Stokes radius of the particle, and eta is the fluid viscosity.The resulting expression for the drag is known as Stokes’ drag:BOOK, Butterworth-Heinemann, 9780080928593, Collinson, Chris, Roper, Tom, Particle Mechanics, 1995, 30, mathbf{F}_d = -6 pi eta r, mathbf{v}.For example, consider a small sphere with radius r = 0.5 micrometre (diameter = 1.0 Î¼m) moving through water at a velocity v of 10 Î¼m/s. Using 10−3 Pa·s as the dynamic viscosity 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 2 cdot 10^5 using the following formula:WEB, tec-science, 2020-05-31, Drag coefficient (friction and pressure drag),www.tec-science.com/mechanics/gases-and-liquids/drag-coefficient-friction-and-pressure-drag/, 2020-06-25, tec-science, en-US, C_D = frac{24}{Re} +frac{4}{sqrt{Re}}+0.4 ~text{;}~~~~~Re