Drag (physics)

An object moving through a gas or liquid experiences a force in direction opposite to its motion. Terminal velocity is achieved when the drag force is equal in magnitude but opposite in direction to the force propelling the object. Shown is a sphere in Stokes flow, at very low Reynolds number. F is force, d is drag, and g is gravity. Small arrows show direction of movement of fluid relative to sphere. Large arrows show direction and magnitude of equal and opposite forces on the sphere, which has stopped accelerating and is moving at terminal velocity.

In fluid dynamics, drag (sometimes called fluid resistance) is the force that resists the movement of a solid object through a fluid (a liquid or gas). The most familiar form of drag is made up of friction forces, which act parallel to the object's surface, plus pressure forces, which act in a direction perpendicular to the object's surface. For a solid object moving through a fluid, the drag is the component of the net aerodynamic or hydrodynamic force acting in the direction of the movement. The component perpendicular to this direction is considered lift. Therefore drag acts to oppose the motion of the object, and in a powered vehicle it is overcome by thrust.

In astrodynamics, depending on the situation, atmospheric drag can be regarded as inefficiency requiring expense of additional energy during launch of the space object or as a bonus simplifying return from orbit.

Types of drag are generally divided into the following categories:

parasitic drag, consisting of
- form drag,
- skin friction,
- interference drag,
lift-induced drag, and
wave drag (aerodynamics) or wave resistance (ship hydrodynamics).

The phrase parasitic drag is mainly used in aerodynamics, since for lifting wings drag is in general small compared to lift. For flow around bluff bodies drag is in general dominating, and then form drag, skin friction and interference drag are not coined as being parasitic drag. Further, lift-induced drag is only relevant when wings or a lifting body are present, and is therefore usually discussed either in the aviation perspective of drag, or in the design of either semi-planing or planing hulls. Wave drag occurs when a solid object is moving through a fluid at or near the speed of sound in that fluid — or in case there is a freely-moving fluid surface with surface waves radiating from the object, e.g. from a ship.

For high velocities — or more precisely, at high Reynolds numbers — the overall drag of an object is characterized by a dimensionless number called the drag coefficient, and is calculated using the drag equation. Assuming a more-or-less constant drag coefficient, drag will vary as the square of velocity. Thus, the resultant power needed to overcome this drag will vary as the cube of velocity. The standard equation for drag is one half the coefficient of drag multiplied by the fluid mass density, the cross sectional area of the specified item, and the square of the velocity.

Wind resistance is a layman's term used to describe drag. Its use is often vague, and is usually used in a relative sense (e.g., a badminton shuttlecock has more wind resistance than a squash ball).

1 Drag at high velocity
- 1.1 Power
- 1.2 Velocity of falling object
2 Very low Reynolds numbers — Stokes' drag
3 Drag in aerodynamics
4 See also
5 References
- 5.1 Inline
- 5.2 General
6 External links

Drag at high velocity

Main article: Drag equation

The drag equation calculates the force experienced by an object moving through a fluid at relatively large velocity (i.e. high Reynolds number, R_e > ~1000), also called quadratic drag. The equation is attributed to Lord Rayleigh, who originally used L² in place of A (L being some length). The force on a moving object due to a fluid is:

$\mathbf{F}_d= -{1 \over 2} \rho v^2 A C_d \mathbf{\hat v}$ ^{see derivation}

where

$\mathbf{F}_d$ is the force of drag,

$\mathbf{} \rho$ is the density of the fluid (Note that for the Earth's atmosphere, the density can be found using the barometric formula. It is 1.293 kg/m³ at 0 °C and 1 atmosphere.),

$\mathbf{} v$ is the speed of the object relative to the fluid,

$\mathbf{} A$ is the reference area,

$\mathbf{} C_d$ is the drag coefficient (a dimensionless parameter, e.g. 0.25 to 0.45 for a car), and

$\mathbf{\hat v}$ is the unit vector indicating the direction of the velocity (the negative sign indicating the drag is opposite to that of velocity).

The reference area A is often defined as the area of the orthographic projection of the object — 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 different reference areas are given for the same object in which case a drag coefficient corresponding to each of these different areas must be given.

In case of a wing, comparison of the drag to the lift force is easiest when the reference areas are the same, since then the ratio of drag to lift force is just the ratio of drag to lift coefficient.^[1] Therefore, the reference for a wing often is the planform (or wing) area rather than the frontal area.^[2]

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 R_e, even up to very high values (R_e of the order 10⁷). ^[3] ^[4] 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 R_e > 3,500.^[4] Further the drag coefficient C_d is, in general, a function of the orientation of the flow with respect to the object (apart from symmetrical objects like a sphere).

Power

The power required to overcome the aerodynamic drag is given by:

$P_d = \mathbf{F}_d \cdot \mathbf{v} = {1 \over 2} \rho v^3 A C_d$

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 50 mph (80 km/h) may require only 10 horsepower (7.5 kW) to overcome air drag, but that same car at 100 mph (160 km/h) requires 80 hp (60 kW). With a doubling of speed the drag (force) quadruples per the formula. Exerting four times the force over a fixed distance produces four 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, four times the work done in half the time requires eight times the power.

It should be emphasized here that the drag equation is an approximation, and does not necessarily give a close approximation in every instance. Thus one should be careful when making assumptions using these equations.

Velocity of falling object

Main article: Terminal velocity

The velocity as a function of time for an object falling through a non-dense medium is roughly given by a function involving a hyperbolic tangent:

$v(t) = \sqrt{ \frac{2mg}{\rho A C_d} } \tanh \left(t \sqrt{\frac{g \rho C_d A}{2 m}} \right) \,$

In other words, velocity asymptotically approaches a maximum value called the terminal velocity:

$v_{t} = \sqrt{ \frac{2mg}{\rho A C_d} } \,$

For a potato-shaped object of average diameter d and of density ρ_obj terminal velocity is about

$v_{t} = \sqrt{ gd \frac{ \rho_{obj} }{\rho} } \,$

For objects of water-like density (raindrops, hail, live objects — animals, birds, insects, etc.) falling in air near the surface of the Earth at sea level, terminal velocity is roughly equal to

$v_{t} = 90 \sqrt{ d } ,$

For example, for 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, for a fog droplet ( $\mathbf{} d$ ~ 0.0001 m) $\mathbf{} v_t$ ~ 0.9 m/s, for a pollen or bacteria ( $\mathbf{} d$ ~ 0.00001 m) $\mathbf{} v_t$ ~ 0.3 m/s and so on. Actual terminal velocity for very small objects (pollen, etc) is even smaller due to the viscosity of air.

Terminal velocity is higher for larger creatures, and thus 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 impacting at its terminal velocity will probably be unharmed. This explains why small animals can fall from a large height and not be harmed.

Very low Reynolds numbers — Stokes' drag

Main article: Stokes' law

The equation for viscous resistance or linear drag is appropriate for small objects or particles moving through a fluid at relatively slow speeds where there is no turbulence (i.e. low Reynolds number, $R_e < 1$ ).^[5] In this case, the force of drag is approximately proportional to velocity, but opposite in direction. The equation for viscous resistance is:^[6]

$\mathbf{F}_d = - b \mathbf{v} \,$

where:

$\mathbf{} b$ is a constant that depends on the properties of the fluid and the dimensions of the object, and

$\mathbf{v}$ is the velocity of the object.

When an object falls from rest, its velocity will be

$v(t) = \frac{(\rho-\rho_0)Vg}{b}\left(1-e^{-bt/m}\right)$

which asymptotically approaches the terminal velocity $\mathbf{} v_t = \frac{(\rho-\rho_0)Vg}{b}$ . For a given $\mathbf{} b$ , heavier objects fall faster.

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:

$\mathbf{} r$ is the Stokes radius of the particle, and $\mathbf{} \eta$ is the fluid viscosity.

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 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.

Drag in aerodynamics

Parasitic drag

Main article: parasitic drag

Parasitic drag (also called parasite drag) is drag caused by moving a solid object through a fluid. 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.

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 induced drag becomes much less, but parasitic drag increases because the fluid is flowing faster around protruding objects increasing friction or drag. At even higher speeds in the 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 maximise gliding range in the event of an engine failure.

Lift induced drag

Main article: lift-induced drag

In aerodynamics, lift-induced drag, induced drag, vortex drag, or sometimes drag due to lift, is a drag force which occurs whenever a lifting body or a wing of finite span generates lift. With other parameters remaining the same, as the angle of attack increases, induced drag increases.

Wave drag in transonic and supersonic flow

Main article: wave drag

The general form of the high speed equation applies fairly well even at speeds approaching or exceeding the speed of sound, however, the Cd factor varies with speed, in a way dependent on the nature of the object.

In general, above Mach 0.85 the drag coefficient climbs to a value several times higher at Mach 1.0, and then comes down again at higher speeds, tending to a value perhaps 30% higher than that at subsonic speeds. This is due to the creation of shockwaves which generates wave drag.

For aircraft, the minimum practical wavedrag is generated by the Sears-Haack body or variations thereof. Busemann's Biplane is not, in principle, subject to wave drag at all when operated at its design speed, but is incapable of generating lift.

References

Inline

↑ Size effects on drag, from NASA Glenn Research Center.
↑ Wing geometry definitions, from NASA Glenn Research Center.
↑ Roshko, Anatol (1961). "Experiments on the flow past a circular cylinder at very high Reynolds number". Journal of Fluid Mechanics 10 (3): 345–356.
↑ ^4.0 ^4.1 Batchelor (1967), p. 341.
↑ Drag Force
↑ Air friction, from Department of Physics and Astronomy, Georgia State University

General

Serway, Raymond A.; Jewett, John W. (2004). Physics for Scientists and Engineers (6th ed. ed.). Brooks/Cole. ISBN 0-534-40842-7.
Tipler, Paul (2004). Physics for Scientists and Engineers: Mechanics, Oscillations and Waves, Thermodynamics (5th ed. ed.). W. H. Freeman. ISBN 0-7167-0809-4.
Huntley, H. E. (1967). Dimensional Analysis. Dover. LOC 67-17978.
Batchelor, George (2000). An introduction to fluid dynamics. Cambridge Mathematical Library (2nd ed. ed.). Cambridge University Press. MR1744638. ISBN 978-0-521-66396-0.

External links

Educational materials on air resistance
Aerodynamic Drag and its effect on the acceleration and top speed of a vehicle.