In fluid dynamics In physics, fluid dynamics is a sub-discipline of fluid mechanics that deals with fluid flow—the natural science of fluids in motion. It has several subdisciplines itself, including aerodynamics (the study of gases in motion) and hydrodynamics (the study of liquids in motion). Fluid dynamics has a wide range of applications, including, the drag coefficient (commonly denoted as Cd, Cx or Cw) is a dimensionless quantity In dimensional analysis, a dimensionless quantity is a quantity without any physical units and thus a pure number. Such a number is typically defined as a product or ratio of quantities which do have units, in such a way that all the units cancel out which is used to quantify the drag In fluid dynamics, drag refers to forces that oppose the relative motion of an object through a fluid (a liquid or gas). Drag forces act in a direction opposite to the oncoming flow velocity. Unlike other resistive forces such as dry friction, which is nearly independent of velocity, drag forces depend on velocity 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 practical formula used to calculate the force of drag experienced by an object due to movement through a fully-enclosing fluid. The equation is attributed to Lord Rayleigh, who originally used L2 in place of A . The force on a moving object due to a fluid is:, where a lower drag coefficient indicates the object will have less aerodynamic Aerodynamics is a branch of dynamics concerned with studying the motion of air, particularly when it interacts with a moving object. Aerodynamics is a subfield of fluid dynamics and gas dynamics, with much theory shared between them. Aerodynamics is often used synonymously with gas dynamics, with the difference being that gas dynamics applies to or hydrodynamic drag. The drag coefficient is always associated with a particular surface area.[1]

The drag coefficient of any object comprises the effects of the two basic contributors to fluid dynamic In physics, fluid dynamics is a sub-discipline of fluid mechanics that deals with fluid flow—the natural science of fluids in motion. It has several subdisciplines itself, including aerodynamics (the study of gases in motion) and hydrodynamics (the study of liquids in motion). Fluid dynamics has a wide range of applications, including drag: skin friction and form drag Parasitic drag is drag caused by moving a solid object through a fluid medium -- or in the case of aerodynamics, a gaseous medium. Parasitic drag is made up of many components, the most prominent being form drag. Skin friction and interference drag are also major components of parasitic drag. The drag coefficient of a lifting airfoil An airfoil or aerofoil (in British English) is the shape of a wing or blade (of a propeller, rotor or turbine) or sail as seen in cross-section or hydrofoil A hydrofoil is a boat with wing-like foils mounted on struts below the hull. As the craft increases its speed the hydrofoils develop enough lift for the boat to become foilborne - i.e. to raise the hull up and out of the water. This results in a great reduction in drag and a corresponding increase in speed also includes the effects of lift induced drag In aerodynamics, lift-induced drag, induced drag, vortex drag, or sometimes drag due to lift, is a drag force that occurs whenever a moving object redirects the airflow coming at it. This drag force occurs in airplanes due to wings or a lifting body redirecting air to cause lift and also in cars with airfoil wings that redirect air to cause a.[2][3] The drag coefficient of a complete structure such as an aircraft also includes the effects of interference drag Parasitic drag is drag caused by moving a solid object through a fluid medium -- or in the case of aerodynamics, a gaseous medium. Parasitic drag is made up of many components, the most prominent being form drag. Skin friction and interference drag are also major components of parasitic drag.[4][5]

Contents

Definition

The drag coefficient Cd is defined as:

where

Fd is the drag force In fluid dynamics, drag refers to forces that oppose the relative motion of an object through a fluid (a liquid or gas). Drag forces act in a direction opposite to the oncoming flow velocity. Unlike other resistive forces such as dry friction, which is nearly independent of velocity, drag forces depend on velocity, which is by definition the force component in the direction of the flow velocity,[6]
ρ is the mass density The density of a material is defined as its mass per unit volume. The symbol of density is ρ of the fluid, [7]
v is the speed Speed is a scalar quantity with dimensions length/time; the equivalent vector quantity to speed is velocity. Speed is measured in the same physical units of measurement as velocity, but does not contain the element of direction that velocity has. Speed is thus the magnitude component of velocity of the object relative to the fluid, and
A is the reference area Area is a quantity expressing the two-dimensional size of a defined part of a surface, typically a region bounded by a closed curve. The term surface area refers to the total area of the exposed surface of a 3-dimensional solid, such as the sum of the areas of the exposed sides of a polyhedron. Area is an important invariant in the differential.

The reference area depends on what type of drag coefficient is being measured. For automobiles and many other objects, the reference area is the frontal area of the vehicle (i.e., the cross-sectional area when viewed from ahead). For example, for a sphere A = π r2 (note this is not the surface area = 4 π r2).

For airfoils An airfoil or aerofoil (in British English) is the shape of a wing or blade (of a propeller, rotor or turbine) or sail as seen in cross-section, the reference area is the chord In aeronautics, chord refers to the imaginary straight line joining the trailing edge and the center of curvature of the leading edge of the cross-section of an airfoil. The chord length is the distance between the trailing edge and the point on the leading edge where the chord intersects the leading edge of the airfoil multiplied with the length of span, which can be easily related to wing area. Since this tends to be a rather large area compared to the projected frontal area, the resulting drag coefficients tend to be low: much lower than for a car with the same drag, frontal area and at the same speed.

Airships An airship or dirigible is a lighter-than-air aircraft that can be steered and propelled through the air using rudders and propellers or other thrust. Unlike other aerodynamic aircraft such as fixed-wing aircraft and helicopters, which produce lift by moving a wing, or airfoil, through the air, aerostatic aircraft, such as airships and hot air and some bodies of revolution In mathematics, engineering, and manufacturing, a solid of revolution is a solid figure obtained by rotating a plane curve around some straight line that lies on the same plane use the volumetric drag coefficient, in which the reference area is the square In algebra, the square of a number is that number multiplied by itself. To square a quantity is to multiply it by itself. Its notation is a superscripted "2"; a number x squared is written as x². Thus: of the cube root In mathematics, a cube root of a number, denoted or x1/3, is a number a such that a3 = x. All real numbers have exactly one real cube root and a pair of complex conjugate roots, and all nonzero complex numbers have three distinct complex cube roots. For example, the real cube root of 8 is 2, because 23 = 8. All the cube roots of −27i are of the airship volume. 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

Flow around a plate, showing stagnation. Main article: Drag equation In fluid dynamics, the drag equation is a practical formula used to calculate the force of drag experienced by an object due to movement through a fully-enclosing fluid. The equation is attributed to Lord Rayleigh, who originally used L2 in place of A . The force on a moving object due to a fluid is:

The drag equation

is essentially a statement that the drag In fluid dynamics, drag refers to forces that oppose the relative motion of an object through a fluid (a liquid or gas). Drag forces act in a direction opposite to the oncoming flow velocity. Unlike other resistive forces such as dry friction, which is nearly independent of velocity, drag forces depend on velocity force In physics, a force is a push or pull that can cause an object with mass to change its velocity. Force has both magnitude and direction, making it a vector quantity. Newton's second law states that an object with a constant mass will accelerate in proportion to the net force acting upon and in inverse proportion to its mass. Equivalently, the net on any object is proportional to the density of the fluid, and proportional to the square of the relative speed Speed is a scalar quantity with dimensions length/time; the equivalent vector quantity to speed is velocity. Speed is measured in the same physical units of measurement as velocity, but does not contain the element of direction that velocity has. Speed is thus the magnitude component of velocity between the object and the fluid.

Cd is not a constant but varies as a function of speed, flow direction, object shape, fluid density and fluid viscosity Viscosity is a measure of the resistance of a fluid which is being deformed by either shear stress or extensional stress. In everyday terms , viscosity is "thickness". Thus, water is "thin", having a lower viscosity, while honey is "thick" having a higher viscosity. Viscosity describes a fluid's internal resistance to. Speed, kinematic viscosity Viscosity is a measure of the resistance of a fluid which is being deformed by either shear stress or extensional stress. In everyday terms , viscosity is "thickness". Thus, water is "thin", having a lower viscosity, while honey is "thick" having a higher viscosity. Viscosity describes a fluid's internal resistance to and a characteristic length scale In physics, length scale is a particular length or distance determined with the precision of one order of magnitude. The concept of length scale is particularly important because physical phenomena of different length scales cannot affect each other and are said to decouple. The decoupling of different length scales makes it possible to have a of the object are incorporated into a dimensionless quantity called the Reynolds number In fluid mechanics and heat transfer, the Reynolds number Re is a dimensionless number that gives a measure of the ratio of inertial forces to viscous forces (μ / L) and, consequently, it quantifies the relative importance of these two types of forces for given flow conditions or Re. Cd is thus a function of Re. In compressible flow, the speed of sound is relevant and Cd is also a function of Mach number Mach number (generally pronounced /ˈmɑːk/, sometimes /ˈmɑːx/ or /ˈmæk/) is the speed of an object moving through air, or any fluid substance, divided by the speed of sound as it is in that substance. It is commonly used to represent an object's (such as an aircraft or missile) speed, when it is travelling at (or at multiples of) the speed Ma.

For a certain body shape the drag coefficient Cd only depends on the Reynolds number Re, Mach number Ma and the direction of the flow. For low Mach number Ma, as usual for automobiles and sports planes, the drag coefficient is independent of Mach number. Also the variation with Reynolds number Re 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 as well more-or-less the same. So the drag coefficient Cd can often be treated as a constant. [8]

For a streamlined body to achieve a low drag coefficient the boundary layer In physics and fluid mechanics, a boundary layer is that layer of fluid in the immediate vicinity of a bounding surface. In the Earth's atmosphere, the planetary boundary layer is the air layer near the ground affected by diurnal heat, moisture or momentum transfer to or from the surface. On an aircraft wing the boundary layer is the part of the around the body must remain attached to the surface of the body for as long as possible, causing the wake In fluid dynamics, a wake is the region of disturbed flow downstream of a solid body moving through a fluid, caused by the flow of the fluid around the body. In incompressible fluids (liquids) such as water, a bow wake is created when a watercraft moves through the medium; as the medium cannot be compressed, it must be displaced instead, resulting to be narrow. A broad wake results in high form drag. The boundary layer will remain attached longer if it is turbulent than if it is laminar Laminar flow, sometimes known as streamline flow, occurs when a fluid flows in parallel layers, with no disruption between the layers. In fluid dynamics, laminar flow is a flow regime characterized by high momentum diffusion, low momentum convection, pressure and velocity independent from time. It is the opposite of turbulent flow. In. The boundary layer will transition from laminar to turbulent providing the Reynolds number In fluid mechanics and heat transfer, the Reynolds number Re is a dimensionless number that gives a measure of the ratio of inertial forces to viscous forces (μ / L) and, consequently, it quantifies the relative importance of these two types of forces for given flow conditions of the flow around the body is high enough. Larger velocities, larger objects, and lower viscosities Viscosity is a measure of the resistance of a fluid which is being deformed by either shear stress or extensional stress. In everyday terms , viscosity is "thickness". Thus, water is "thin", having a lower viscosity, while honey is "thick" having a higher viscosity. Viscosity describes a fluid's internal resistance to contribute to larger Reynolds numbers.[9]

For other objects, such as small particles, one can no longer consider that the drag coefficient Cd is constant, but certainly is a function of Reynolds number. [10][11] [12] 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 Fd is proportional to v instead of v2; for a sphere this is known as Stokes law. Reynolds number will be low for small objects, low velocities, and high viscosity fluids.[9]

A Cd 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. At a stagnation point the fluid velocity is zero and all kinetic energy has been converted into pressure energy. Stagnation pressure is equal to the sum of the free-stream dynamic pressure and free-stream static pressure. Stagnation pressure is 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 size, the Cd of a real flat plate would be less than 1; except that there will be suction on the back side: a negative pressure (relative to ambient). The overall Cd of a real square flat plate perpendicular to the flow is often given as 1.17. Flow patterns and therefore Cd for some shapes can change with the Reynolds number and the roughness of the surfaces.

Drag coefficient Cd examples

General

In general, Cd is not an absolute constant for a given body shape. It varies with the speed of airflow (or more generally with Reynolds number). A smooth sphere, for example, has a Cd that varies from about 0.47 for laminar (slow) flow to 0.1 for turbulent (faster) flow.

Shapes [1]
Cd Item
0.9 a typical bicycle A bicycle, bike, or cycle is a pedal-driven, human-powered vehicle with two wheels attached to a frame, one behind the other. A person who rides a bicycle is called a cyclist or a bicyclist plus cyclist
0.4 rough sphere (Re In fluid mechanics and heat transfer, the Reynolds number Re is a dimensionless number that gives a measure of the ratio of inertial forces to viscous forces (μ / L) and, consequently, it quantifies the relative importance of these two types of forces for given flow conditions = 106)
0.1 smooth sphere (Re = 106)
0.001 laminar flat plate parallel to the flow (Re = 106)
0.005 turbulent flat plate parallel to the flow (Re = 106)
0.26 modern car (e.g. Toyota Prius[13])
0.295 bullet (not ogive, at subsonic velocity)
1.0–1.3 man (upright position)
1.28 flat plate perpendicular to flow
1.0–1.1 skier Snow skiing is a group of sports using skis as primary equipment. Skis are used in conjunction with boots that connect to the ski with use of a binding. Skiing can be grouped into two general categories. Nordic skiing is the oldest and includes sport that evolved from skiing as done in Scandinavia. Nordic style bindings attach at the toes of the
1.0–1.3 wires and cables
1.1-1.3 ski jumper[13]
1.3–1.5 Empire State Building The Empire State Building is a 102-story Art Deco skyscraper in New York City at the intersection of Fifth Avenue and West 34th Street. Its name is derived from the nickname for the state of New York. It stood as the world's tallest building for more than forty years, from its completion in 1931 until construction of the World Trade Center's North
1.8–2.0 Eiffel Tower The Eiffel Tower is an iron tower built on the Champ de Mars beside the Seine River in Paris. The tower has become a global icon of France and is one of the most recognizable structures in the world
2.1 a smooth brick
>2.5 spacecraft in LEO A low Earth orbit is generally defined as an orbit within the locus extending from the Earth’s surface up to an altitude of 2,000 km. Given the rapid orbital decay of objects below approximately 200 km, the commonly accepted definition for LEO is between 160 - 2,000 km (100 - 1,240 miles) above the Earth's surface

Aircraft

As noted above, aircraft use wing area as the reference area when computing Cd, while automobiles (and many other objects) use frontal cross sectional area; thus, coefficients are not directly comparable between these classes of vehicles.

Aircraft[14]
Cd Aircraft model
0.021 F-4 Phantom II The McDonnell Douglas F-4 Phantom II is a tandem two-seat, twin-engined, all-weather, long-range supersonic jet interceptor fighter/fighter-bomber originally developed for the U.S. Navy by McDonnell Aircraft. Proving highly adaptable, it became a major part of the air wings of the United States Navy, Marine Corps, and Air Force. It was used (subsonic)
0.022 Learjet 24
0.024 Boeing 787 The Boeing 787 Dreamliner is a mid-sized, wide-body, twin-engine jet airliner currently under development by Boeing Commercial Airplanes. It will carry between 210 and 330 passengers depending on variant and seating configuration. Boeing stated that it will be more fuel-efficient than earlier Boeing airliners and will be the first major airliner [15]
0.027 Cessna 172 The Cessna 172 Skyhawk is a four-seat, single-engine, high-wing fixed-wing aircraft. First flown in 1955 and still in production, more Cessna 172s have been built than any other aircraft/182
0.027 Cessna 310 The 310 first flew on January 3, 1953 with deliveries starting in late 1954. The sleek modern lines of the new twin were backed up by innovative features such as engine exhaust thrust augmenter tubes and the storage of all fuel in tip tanks in early models. In 1964, the engine exhaust was changed to flow under the wing instead of the augmenter
0.031 Boeing 747 The Boeing 747 is a widebody commercial airliner, often referred to by the nickname "Jumbo Jet". It is among the world's most recognizable aircraft, and was the first widebody ever produced. Manufactured by Boeing's Commercial Airplane unit in the US, the original version of the 747 was two and a half times the size of the Boeing 707,
0.044 F-4 Phantom II (supersonic)
0.048 F-104 Starfighter The Lockheed F-104 Starfighter was an American single-engined, high-performance, supersonic interceptor aircraft that served with the United States Air Force from 1958 until 1967. One of the Century Series of aircraft, it continued in service with Air National Guard units until it was phased out in 1975. The National Aeronautics and Space
0.095 X-15 The North American X-15 rocket-powered aircraft was part of the X-series of experimental aircraft, initiated with the Bell X-1, that were made for the USAF, NASA, and the USN. The X-15 set speed and altitude records in the early 1960s, reaching the edge of outer space and returning with valuable data used in aircraft and spacecraft design. It (Not confirmed)

Automobile

See also: Automobile drag coefficient

Notes

  1. ^ McCormick, Barnes W. (1979), Aerodynamics, Aeronautics, and Flight Mechanics, p.24, John Wiley & Sons, Inc., New York ISBN 0-471-03032-5
  2. ^ Clancy, L.J., Aerodynamics, Section 5.18
  3. ^ Abbott, Ira H., and Von Doenhoff, Albert E., Theory of Wing Sections, Sections 1.2 and 1.3
  4. ^ NASA’s Modern Drag Equation
  5. ^ Clancy, L.J., Aerodynamics, Section 11.17
  6. ^ See lift force A fluid flowing relative to a body exerts a force on it, called an aerodynamic force if the fluid is air. Lift is the component of this force which is perpendicular to the oncoming flow direction. It contrasts with the drag force, which is the component of the aerodynamic force parallel to the flow direction. An airfoil is a streamlined body that and vortex induced vibration for a possible force components transverse to the flow direction.
  7. ^ Note that for the Earth's atmosphere, the air density can be found using the barometric formula. Air is 1.293 kg/m3 at 0°C and 1 atmosphere
  8. ^ Clancy, L.J., Aerodynamics, Sections 4.15 and 5.4
  9. ^ a b Clancy, L.J., Aerodynamics, Section 4.17
  10. ^ Clift R., Grace J.R., Weber M.E., "Bubbles, drops, and particles", Academic Press NY (1978).
  11. ^ Briens C.L., Powder Technology, 67, 1991, 87-91.
  12. ^ Haider A., Levenspiel O., Powder Technology, 58, 1989, 63-70.
  13. ^ a b http://www.engineeringtoolbox.com/drag-coefficient-d_627.html
  14. ^ http://www.aerospaceweb.org/question/aerodynamics/q0184.shtml
  15. ^ http://www.lissys.demon.co.uk/samp1/index.html

References

  • Clancy, L. J. (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, S. F. (1965), Fluid-Dynamic Drag, Hoerner Fluid Dynamics, Brick Town, N. J. USA

See also

External links

Categories: Aerodynamics | Aerospace engineering | Dimensionless numbers

 

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