Drag coefficient

Drag coefficient is a link to this article. The pressure loss coefficient is also called the drag coefficient.

Physical key figure
Surname	Drag coefficient, drag coefficient
Formula symbol	${\displaystyle c_{\mathrm {w} }}$
dimension	dimensionless
definition	${\displaystyle c_{\mathrm {w} }={\frac {F_{\mathrm {w} }}{q\cdot A}}}$
${\displaystyle F_{\mathrm {w} }}$ Resilience ${\displaystyle q}$ Dynamic pressure of the flow ${\displaystyle A}$ Reference area
scope of application	Air resistance of bodies

The drag coefficient , drag coefficient , drag coefficient , front drag or c _w value (according to the usual formula symbol ) is a dimensionless measure ( coefficient ) for the flow resistance of a body around which a fluid flows. ${\displaystyle c_{\mathrm {w} }}$

In colloquial terms, the value is a measure of a body's "wind slippage". With additional knowledge of the speed, frontal area , wing area etc. and density of the fluid (e.g. air ), the force of the flow resistance can be calculated from the flow resistance coefficient. ${\displaystyle c_{\mathrm {w} }}$

definition

The drag coefficient is defined by:

${\displaystyle c_{\mathrm {w} }={\frac {F_{\mathrm {w} }}{q\,A}}={\frac {2F_{\mathrm {w} }}{\rho \,v^{2}A}}}$

Here, the drag force is normalized with the dynamic pressure of the flow and a reference surface ${\displaystyle F_{\mathrm {w} }}$ ${\displaystyle q={\frac {\rho }{2}}v^{2}}$ ${\displaystyle A}$

• the density ${\displaystyle \rho }$
• the speed of the undisturbed flow. ${\displaystyle v}$

The reference area or resistance area depends on the definition: ${\displaystyle A}$

• in vehicles, the drag area is the same as the frontal area
• In aircraft aerodynamics , however, the lift area, i.e. the wing area, is used as a reference.

The formula symbol (with w for resistance) is only common in German-speaking countries; in English the drag coefficient is noted as or . ${\displaystyle c_{\mathrm {w} }}$ ${\displaystyle c_{\mathrm {d} }}$ ${\displaystyle c_{\mathrm {x} }}$

Dependencies

With incompressible flow

Drag coefficient of a sphere as a function of the Reynolds number: c _w = f ( Re ). The characteristic length in this case is the sphere diameter  d ; the reference area  A is a circular area with the diameter  d .

In general, the flow resistance coefficient of incompressible flow depends on the Reynolds number : ${\displaystyle {\mathit {Re}}}$

${\displaystyle c_{\mathrm {w} }=f({\mathit {Re}})}$

With

• ${\displaystyle {\mathit {Re}}={\frac {vL\rho }{\eta }}}$
  • the characteristic length whose square in a fixed ratio to the reference surface is ${\displaystyle L}$ ${\displaystyle L^{2}}$ ${\displaystyle A}$
  • the viscosity (tenacity) of the fluid. ${\displaystyle \eta }$

This statement results from the assumption that the drag force of a body in a certain position depends on the flow velocity, the density, the viscosity and a characteristic length of the body:

${\displaystyle F_{\mathrm {w} }=f(v,\,\rho ,\,\eta ,\,L)}$

By means of a dimensional analysis according to Buckingham's Π theorem, it can be deduced that the two similarity indicators flow resistance coefficient and Reynolds number are sufficient to describe the flow resistance of a certain body. This enables a more straightforward, generally applicable representation of the resistance of a specific body shape.

With compressible flow

c _w as a function of the flow velocity

With compressible flows, i.e. with flows with variable density ( ), the flow resistance coefficient is also dependent on the Mach number (see Fig.): ${\displaystyle \rho \neq \mathrm {konst.} }$

• in the transonic range and in the supersonic range, the drag coefficient changes significantly
• in the vicinity of the speed of sound it increases several times
• at very high Mach numbers it drops to about twice the subsonic c _w value.

Above the critical Mach number , partial flows exceed the speed of sound. The flow resistance increases sharply above the resistance divergence Mach number. The behavior in the supersonic range is determined by the geometry of the body; in the drawing, the green curve represents a streamlined body.

Blunt, angular bodies have a largely constant drag coefficient over a large range of the Reynolds number. This is e.g. B. the case with the air resistance of motor vehicles at the relevant speeds.

The drag coefficient determined for Satellite their life in orbit . At an altitude of over 150 km, the atmosphere is so thin that the flow is no longer approximated as a laminar continuum flow , but as a free molecular flow . In this range, the c _w value is typically between 2 and 4, and a value of 2.2 is often used. The influence of the atmosphere decreases with increasing altitude and is negligible above approx. 1000 km.

detection

The drag coefficient is usually determined in a wind tunnel . The body stands on a plate that is equipped with force sensors . The force in the direction of the flow is measured. From this drag force and the known parameters such as air density and frontal area, the flow resistance coefficient is calculated for a given flow velocity. ${\displaystyle F_{\mathrm {w} }}$

In addition, the resistance can also be determined numerically, depending on the complexity of the model shape and the available computer capacity , by integrating the distribution of the coefficient of friction and pressure over the model surface.

application

The drag force is calculated from the drag coefficient as follows:

${\displaystyle F_{\mathrm {w} }={\frac {\rho \,c_{\mathrm {w} }\,A\,v^{2}}{2}}}$

The flow resistance is therefore proportional in each case

• to the density of the flowing fluid (compare air density )
• to the drag coefficient
• to the reference surface (projected front surface)
• to the square of the flow velocity .

The required drive power is even proportional to the third power of the speed:

${\displaystyle {\begin{aligned}P&={\vec {F}}\cdot {\vec {v}}\\&={\frac {\rho \,c_{\mathrm {w} }\,A\,v^{2}}{2}}\cdot v\\&={\frac {\rho \,c_{\mathrm {w} }\,A\,v^{3}}{2}}\end{aligned}}}$

Therefore, in addition to the drag coefficient (i.e. body shape) and the frontal area, the choice of speed has a particular effect on fuel consumption in motor vehicles .

The air resistance is decisive for the deviation of the actual ballistic curve from the idealized trajectory parabola .

Examples

c _w values ​​of typical body shapes

value	shape
2.3	Half-pipe long, concave side
2.0	long rectangular plate
1.33	Hemispherical shell, concave side, parachute
1.2	Half-tube long, convex side
1.2	long cylinder, wire (Re <1.9 · 10 5 )
1.11	round disc, square plate
0.78	Man, standing
0.6	Paraglider (reference area flow cross-sectional area!)
0.53 ... 0.69	Bicycle (mountain bike, stretched / upright)
0.45	Sphere (Re <1.7 · 10 5 )
0.4	Bicycle (racing bike)
0.35	long cylinder, wire (Re> 6.7 · 10 5 )
0.34	Hemispherical shell, convex side
0.09 ... 0.18	Sphere (Re> 4.1 · 10 5 )
0.08	Aircraft (reference surface wing)
0.04	Streamlined body "teardrop"
0.03	penguin
0.02	optimized spindle shape

${\displaystyle \mathrm {Re} }$refers to the Reynolds number

Drag coefficients of motor vehicles

Published c _w values ​​must be scrutinized extremely critically, as they were and are still often determined today on small models in disregard of the model principles, for example by the German Aviation Research Institute with c _w = 0.244 for the Tatra 87 , which was much later than Original was measured with c _w = 0.36.

The c _w value quantifies the aerodynamic quality of a body. By multiplying it by the reference area (usually the frontal area for vehicles), you get the drag area of a vehicle, sometimes simply referred to as 'air resistance': ${\displaystyle A}$ ${\displaystyle f_{\mathrm {w} }}$

${\displaystyle f_{\mathrm {w} }=c_{\mathrm {w} }A}$.

The power requirement, which determines the fuel consumption of a motor vehicle at high driving speeds, is proportional to the resistance area. The front face is rarely specified by manufacturers.

A comprehensive collection of automotive c _w values for which there is evidence, was on the side "and motorcycle portal Wikipedia Auto / drag coefficient" outsourced.

Remarks

1. ↑ Compressible fluids such as air can also be considered incompressible if the density in the flow field is largely constant. This is generally the case up to a Mach number of 0.3.

literature

• Sighard F. Hoerner: Fluid-Dynamic Drag . Self-published, 1965.
• Horst Stöcker (Ed.): Pocket book of physics . 4th edition. German, Frankfurt am Main 2000, ISBN 3-8171-1628-4 .
• Hans-Hermann Braess, Ulrich Seiffert: Vieweg manual automotive technology . 2nd Edition. Vieweg, Braunschweig 2001, ISBN 3-528-13114-4 .
• Karl-Heinz Dietsche, Thomas Jäger, Robert Bosch GmbH: Automotive pocket book . 25th edition. Vieweg, Wiesbaden 2003, ISBN 3-528-23876-3 .
• Wolfgang Demtröder: Mechanics and Warmth . 4th edition. Springer, Berlin 2005, ISBN 3-540-26034-X ( Experimental Physics , Volume 1).
• Wolf-Heinrich Hucho: Aerodynamics of the automobile . Ed .: Thomas Schütz. 6th edition. Springer Vieweg, Wiesbaden 2013, ISBN 978-3-8348-2316-8 , introduction (over 1000, books.google.de ).

Web links

• Special exhibition in Hamburg: "100 Years Against the Wind". In: Auto, Motor und Sport , December 17, 2008.
• Fuel-saving models from the wind tunnel . auto-motor-und-sport.de, October 18, 2011 (overview article with developer interview); accessed May 30, 2018.
• In the secret chamber of the Luftikusse . FAZ / FAS, September 8, 2014; accessed August 14, 2017.

Individual evidence

1. ↑ Ludwig Prandtl: Results of the aerodynamic research institute in Göttingen, part 1. Universitätsverlag Göttingen 2009 (first published in 1921) ISBN 978-3-941875-35-7 limited preview in the Google book search
2. ↑ Wolfgang Heinrich Hucho: Aerodynamics of the automobile. Springer, Berlin 1999, ISBN 3-540-62160-1 , pp. 111-113.
3. ↑ Jürgen Zierep: Similarity laws and model rules of fluid mechanics. Karlsruhe 1991, ISBN 3-7650-2041-9 .
4. ↑ Drag divergence Mach number on en.wikipedia.org.
5. ↑ case with air resistance , dieter-heidorn.de , material at the rate on Hansa-Kolleg, available May 30, 2018.
6. ↑ ^{a b} http://www.ltam.lu/physique/projekte/reichling/zammlung.pdf ( Memento from October 6, 2014 in the Internet Archive )
7. ^ Wolf-Heinrich Hucho: Aerodynamics of the automobile . Ed .: Thomas Schütz. 6th edition. Springer Vieweg , Wiesbaden 2013, ISBN 978-3-8348-2316-8 , introduction, p. 11-53 ( books.google.de ).
8. ↑ autobild.de: The tops and flops in the wind tunnel