Drag Coefficient
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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) an ...
, the drag coefficient (commonly denoted as: c_\mathrm, c_x or c_) is a
dimensionless quantity 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) ...
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 equation In fluid dynamics, the drag equation is a formula used to calculate the force of drag experienced by an object due to movement through a fully enclosing fluid. The equation is: F_\, =\, \tfrac12\, \rho\, u^2\, c_\, A where *F_ is the drag fo ...
in which a lower drag coefficient indicates the object will have less
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) and ...
drag. The drag coefficient is always associated with a particular surface area. The drag coefficient of any object comprises the effects of the two basic contributors to
fluid dynamic 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) an ...
drag:
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 ...
. The drag coefficient of a lifting
airfoil An airfoil (American English) or aerofoil (British English) is the cross-sectional shape of an object whose motion through a gas is capable of generating significant lift, such as a wing, a sail, or the blades of propeller, rotor, or turbine. ...
or
hydrofoil A hydrofoil is a lifting surface, or foil, that operates in water. They are similar in appearance and purpose to aerofoils used by aeroplanes. Boats that use hydrofoil technology are also simply termed hydrofoils. As a hydrofoil craft gains sp ...
also includes the effects of
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 or ...
. The drag coefficient of a complete structure such as an aircraft also includes the effects of interference drag.


Definition

The drag coefficient c_\mathrm d is defined as c_\mathrm d = \dfrac where: * F_\mathrm d is the drag force, which is by definition the force component in the direction of 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 f ...
; * \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 speed of the object relative to the fluid; * 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 A shape or figure is a graphics, graphical representation of an obje ...
The reference area depends on what type of drag coefficient is being measured. For automobiles and many other objects, the reference area is the projected frontal area of the vehicle. This may not necessarily be the cross-sectional area of the vehicle, depending on where the cross-section is taken. For example, for a sphere A = \pi r^2 (note this is not the surface area = 4 \pi r^2). For
airfoil An airfoil (American English) or aerofoil (British English) is the cross-sectional shape of an object whose motion through a gas is capable of generating significant lift, such as a wing, a sail, or the blades of propeller, rotor, or turbine. ...
s, the reference area is the nominal wing area. Since this tends to be large compared to the frontal area, the resulting drag coefficients tend to be low, much lower than for a car with the same drag, frontal area, and speed.
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 some bodies of revolution use the volumetric drag coefficient, in which the reference area is the
square In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90-degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-length adj ...
of the
cube root In mathematics, a cube root of a number is a number such that . All nonzero real numbers, have exactly one real cube root and a pair of complex conjugate cube roots, and all nonzero complex numbers have three distinct complex cube roots. Fo ...
of the airship volume (volume to the two-thirds power). Submerged streamlined bodies use the wetted surface area. Two objects having the same reference area moving at the same speed through a fluid will experience a drag force proportional to their respective drag coefficients. Coefficients for unstreamlined objects can be 1 or more, for streamlined objects much less.


Background

The drag equation :F_ = \tfrac12 \rho u^2 c_ A is essentially a statement that 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 p ...
on any object is proportional to the density of the fluid and proportional to the square of the relative flow speed between the object and the fluid. The factor of 1/2 comes from the
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 ( ...
of the fluid, which is equal to the kinetic energy density. The value of c_\mathrm d is not a constant but varies as a function of flow speed, flow direction, object position, object size, fluid density and fluid
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 inte ...
. Speed,
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 inter ...
and a characteristic
length scale In physics, length scale is a particular length or distance determined with the precision of at most a few orders of magnitude. The concept of length scale is particularly important because physical phenomena of different length scales cannot af ...
of the object are incorporated into a dimensionless quantity called 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 domi ...
\scriptstyle Re. \scriptstyle C_\mathrm d is thus a function of \scriptstyle Re. In a compressible flow, the speed of sound is relevant, and c_\mathrm d is also a function of
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 = \frac ...
\mathrm. For certain body shapes, the drag coefficient c_\mathrm d only depends on the Reynolds number \mathrm, Mach number \mathrm and the direction of the flow. For low Mach number \mathrm, the drag coefficient is independent of Mach number. Also, the variation with Reynolds number \mathrm within a practical range of interest is usually small, while for cars at highway speed and aircraft at cruising speed, the incoming flow direction is also more-or-less the same. Therefore, the drag coefficient c_\mathrm d can often be treated as a constant. For a streamlined body to achieve a low drag coefficient, the
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 condi ...
around the body must remain attached to the surface of the body for as long as possible, causing the wake to be narrow. A high ''form drag'' results in a broad wake. The boundary layer will transition from laminar to turbulent if Reynolds number of the flow around the body is sufficiently great. Larger velocities, larger objects, and lower
viscosities The viscosity of a fluid is a measure of its drag (physics), 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 quant ...
contribute to larger Reynolds numbers.Clancy, L. J.: ''Aerodynamics''. Section 4.17 For other objects, such as small particles, one can no longer consider that the drag coefficient c_\mathrm d is constant, but certainly is a function of Reynolds number. At a low Reynolds number, the flow around the object does not transition to turbulent but remains laminar, even up to the point at which it separates from the surface of the object. At very low Reynolds numbers, without flow separation, the drag force F_\mathrm d is proportional to \scriptstyle v instead of v^2; for a sphere this is known as Stokes' law. The Reynolds number will be low for small objects, low velocities, and high viscosity fluids. A c_\mathrm d equal to 1 would be obtained in a case where all of the fluid approaching the object is brought to rest, 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 front surface. The top figure shows a flat plate with the fluid coming from the right and stopping at the plate. The graph to the left of it shows equal pressure across the surface. In a real flat plate, the fluid must turn around the sides, and full stagnation pressure is found only at the center, dropping off toward the edges as in the lower figure and graph. Only considering the front side, the c_\mathrm d of a real flat plate would be less than 1; except that there will be suction on the backside: a negative pressure (relative to ambient). The overall c_\mathrm d of a real square flat plate perpendicular to the flow is often given as 1.17. Flow patterns and therefore \scriptstyle C_\mathrm d for some shapes can change with the Reynolds number and the roughness of the surfaces.


Drag coefficient examples


General

In general, c_\mathrm d is not an absolute constant for a given body shape. It varies with the speed of airflow (or more generally with
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 domi ...
\mathrm). A smooth sphere, for example, has a c_\mathrm d that varies from high values for laminar flow to 0.47 for
turbulent flow In fluid dynamics, turbulence or turbulent flow is fluid motion characterized by chaotic changes in pressure and flow velocity. It is in contrast to a laminar flow, which occurs when a fluid flows in parallel layers, with no disruption between ...
. Although the drag coefficient decreases with increasing \mathrm, the drag force increases.


Aircraft

As noted above, aircraft use their wing area as the reference area when computing c_\mathrm d, while automobiles (and many other objects) use projected frontal area; thus, coefficients are not directly comparable between these classes of vehicles. In the aerospace industry, the drag coefficient is sometimes expressed in drag counts where 1
drag count A drag count is a dimensionless unit used by aerospace engineers. 1 drag count is equal to a C_d of 0.0001. Definition A drag count \Delta C_\mathrm d\, is defined as: :\Delta C_\mathrm d = 10^ \dfrac\, , where: :F_\mathrm d\, is the drag ( ...
= 0.0001 of a c_\mathrm d.Basha, W. A. and Ghaly, W. S., "Drag Prediction in Transitional Flow over Airfoils," Journal of Aircraft, Vol. 44, 2007, p. 824–32.


Automobile


Blunt and streamlined body flows


Concept

The force between a fluid and a body, when there is relative motion, can only be transmitted by normal pressure and tangential friction stresses. So, for the whole body, the drag part of the force, which is in-line with the approaching fluid motion, is composed of frictional drag (viscous drag) and pressure drag (form drag). The total drag and component drag forces can be related as follows: \begin c_\mathrm d &= \dfrac\\ &= c_\mathrm p + c_\mathrm f \\ &= \underbrace_+ \underbrace_ \end where: *''A'' is the planform area of the body, *''S'' is the wet surface of the body, *c_\mathrm p is 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 and e ...
drag coefficient, *c_\mathrm f is the
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 t ...
drag coefficient, *\hat \mathbf is the unit vector in the direction of the shear stress acting on the body surface d''S'', *\hat \mathbf is the unit vector in the direction perpendicular to the body surface d''S'', pointing from the fluid to the solid, *T_\mathrm w magnitude of the
shear stress Shear stress, often denoted by (Greek: tau), is the component of stress coplanar with a material cross section. It arises from the shear force, the component of force vector parallel to the material cross section. ''Normal stress'', on the ot ...
acting on the body surface d''S'', *p_\mathrm o is the pressure far away from the body (note that this constant does not affect the final result), *p is pressure at surface d''S'', *\hat is the unit vector in direction of free stream flow Therefore, when the drag is dominated by a frictional component, the body is called a streamlined body; whereas in the case of dominant pressure drag, the body is called a blunt or bluff body. Thus, the shape of the body and the angle of attack determine the type of drag. For example, an airfoil is considered as a body with a small angle of attack by the fluid flowing across it. This means that it has attached
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 condi ...
s, which produce much less pressure drag. The wake produced is very small and drag is dominated by the friction component. Therefore, such a body (here an airfoil) is described as streamlined, whereas for bodies with fluid flow at high angles of attack, boundary layer separation takes place. This mainly occurs due to adverse
pressure gradient In atmospheric science, the pressure gradient (typically of air but more generally of any fluid) is a physical quantity that describes in which direction and at what rate the pressure increases the most rapidly around a particular location. The p ...
s at the top and rear parts of an
airfoil An airfoil (American English) or aerofoil (British English) is the cross-sectional shape of an object whose motion through a gas is capable of generating significant lift, such as a wing, a sail, or the blades of propeller, rotor, or turbine. ...
. Due to this, wake formation takes place, which consequently leads to eddy formation and pressure loss due to pressure drag. In such situations, the airfoil is stalled and has higher pressure drag than friction drag. In this case, the body is described as a blunt body. A streamlined body looks like a fish (
Tuna A tuna is a saltwater fish that belongs to the tribe Thunnini, a subgrouping of the Scombridae (mackerel) family. The Thunnini comprise 15 species across five genera, the sizes of which vary greatly, ranging from the bullet tuna (max length: ...
), Oropesa, etc. or an airfoil with small angle of attack, whereas a blunt body looks like a brick, a cylinder or an airfoil with high angle of attack. For a given frontal area and velocity, a streamlined body will have lower resistance than a blunt body. Cylinders and spheres are taken as blunt bodies because the drag is dominated by the pressure component in the wake region at 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 domi ...
. To reduce this drag, either the flow separation could be reduced or the surface area in contact with the fluid could be reduced (to reduce friction drag). This reduction is necessary in devices like cars, bicycle, etc. to avoid vibration and noise production.


Practical example

The
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 ...
design of cars has evolved from the 1920s to the end of the 20th century. This change in design from a blunt body to a more streamlined body reduced the drag coefficient from about 0.95 to 0.30.


See also

*
Automotive aerodynamics Automotive aerodynamics is the study of the aerodynamics of road vehicles. Its main goals are reducing drag and wind noise, minimizing noise emission, and preventing undesired lift forces and other causes of aerodynamic instability at high spee ...
*
Automobile drag coefficient The drag coefficient is a common measure in automotive design as it pertains to aerodynamics. Drag is a force that acts parallel to and in the same direction as the airflow. The drag coefficient of an automobile measures the way the automobile ...
*
Ballistic coefficient In ballistics, the ballistic coefficient (BC, ''C'') of a body is a measure of its ability to overcome air resistance in flight. It is inversely proportional to the negative acceleration: a high number indicates a low negative acceleration—the ...
* Drag crisis *
Zero-lift drag coefficient In aerodynamics, the zero-lift drag coefficient C_ is a dimensionless parameter which relates an aircraft's zero-lift drag force to its size, speed, and flying altitude. Mathematically, zero-lift drag coefficient is defined as C_ = C_D - C_, wher ...


Notes


References

*
L. J. Clancy Laurence Joseph Clancy (15 March 1929 - 16 October 2014) was an Education Officer in aerodynamics at Royal Air Force College Cranwell whose textbook ''Aerodynamics'' became standard. He was born in Egypt to Alfred Joseph Clancy and Agnes Hunter. I ...
(1975): ''Aerodynamics''. Pitman Publishing Limited, London, {{ISBN, 0-273-01120-0 * Abbott, Ira H., and Von Doenhoff, Albert E. (1959): ''Theory of Wing Sections''. Dover Publications Inc., New York, Standard Book Number 486-60586-8 * Hoerner, Dr. Sighard F., Fluid-Dynamic Drag, Hoerner Fluid Dynamics, Bricktown New Jersey, 1965. * Bluff Body
http://user.engineering.uiowa.edu/~me_160/lecture_notes/Bluff%20Body2.pdf
* Drag of Blunt Bodies and Streamlined Bodies: http://www.princeton.edu/~asmits/Bicycle_web/blunt.html * Hucho, W.H., Janssen, L.J., Emmelmann, H.J. 6(1975): ''The optimization of body details-A method for reducing the aerodynamics drag''. SAE 760185. Drag (physics) Aerospace engineering Dimensionless numbers of fluid mechanics