In fluid dynamics

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 and hydrodynamics...

, disk loading or disc loading is the average pressure

Pressure

Pressure is the force per unit area applied in a direction perpendicular to the surface of an object. Gauge pressure is the pressure relative to the local atmospheric or ambient pressure.- Definition :...

 change across an actuator disk, such as an airscrew. Airscrews with a relatively low disk loading are typically called rotors, including helicopter

Helicopter

A helicopter is a type of rotorcraft in which lift and thrust are supplied by one or more engine-driven rotors. This allows the helicopter to take off and land vertically, to hover, and to fly forwards, backwards, and laterally...

 main rotor

Helicopter rotor

A helicopter main rotor or rotor system is a type of fan that is used to generate both the aerodynamic lift force that supports the weight of the helicopter, and thrust which counteracts aerodynamic drag in forward flight...

s and tail rotor

Tail rotor

The tail rotor, or anti-torque rotor, is a smaller rotor mounted so that it rotates vertically or near-vertically at the end of the tail of a traditional single-rotor helicopter. The tail rotor's position and distance from the center of gravity allow it to develop thrust in the same direction as...

s; propeller

Propeller (aircraft)

Aircraft propellers or airscrews convert rotary motion from piston engines or turboprops to provide propulsive force. They may be fixed or variable pitch. Early aircraft propellers were carved by hand from solid or laminated wood with later propellers being constructed from metal...

s typically have a higher disk loading.
The V-22 Osprey

V-22 Osprey

The Bell Boeing V-22 Osprey is an American multi-mission, military, tiltrotor aircraft with both a vertical takeoff and landing , and short takeoff and landing capability...

 tiltrotor

Tiltrotor

A tiltrotor is an aircraft which uses a pair or more of powered rotors mounted on rotating shafts or nacelles at the end of a fixed wing for lift and propulsion, and combines the vertical lift capability of a helicopter with the speed and range of a conventional fixed-wing aircraft...

 aircraft has a high disk loading relative to a helicopter in the hover mode, but a relatively low disk loading in fixed-wing

Fixed-wing aircraft

A fixed-wing aircraft is an aircraft capable of flight using wings that generate lift due to the vehicle's forward airspeed. Fixed-wing aircraft are distinct from rotary-wing aircraft in which wings rotate about a fixed mast and ornithopters in which lift is generated by flapping wings.A powered...

 mode compared to a turboprop

Turboprop

A turboprop engine is a type of turbine engine which drives an aircraft propeller using a reduction gear.The gas turbine is designed specifically for this application, with almost all of its output being used to drive the propeller...

 aircraft.

Rotors

Disc loading of a hovering helicopter
is the ratio of its weight to the
total main rotor disc area. It is determined by dividing
the total helicopter weight by the rotor disc area,
which is the area swept by the blades of a rotor. Disc
area can be found by using the span of one rotor blade
as the radius of a circle and then determining the area
the blades encompass during a complete rotation. As
the helicopter is maneuvered, disc loading changes.
The higher the loading, the more power needed to
maintain rotor speed.

Increasing the weight of a helicopter increases disk loading. For a given weight, a helicopter with shorter rotors will have higher disk loading, and will require more engine power to hover. A low disk loading improves autorotation

Autorotation

In aviation, autorotation refers to processes in both fixed-wing and rotary-wing aircraft. The term means significantly different things in each context....

 performance in rotorcraft

Rotorcraft

A rotorcraft or rotary wing aircraft is a heavier-than-air flying machine that uses lift generated by wings, called rotor blades, that revolve around a mast. Several rotor blades mounted to a single mast are referred to as a rotor. The International Civil Aviation Organization defines a rotorcraft...

. Typically, an autogyro

Autogyro

An autogyro , also known as gyroplane, gyrocopter, or rotaplane, is a type of rotorcraft which uses an unpowered rotor in autorotation to develop lift, and an engine-powered propeller, similar to that of a fixed-wing aircraft, to provide thrust...

 (or gyroplane) has a lower rotor disc loading than a helicopter, which provides a slower rate of descent in autorotation.

Propellers

In reciprocating and propeller engines, disk loading can be defined as the ratio between propeller-induced velocity and freestream velocity. Lower disk loading will increase efficiency, so it is generally desirable to have larger propellers from an efficiency standpoint. Maximum efficiency is reduced as disk loading is increased due to the rotating slipstream; using contra-rotating propellers

Contra-rotating propellers

Aircraft equipped with contra-rotating propellers, also referred to as coaxial contra-rotating propellers, apply the maximum power of usually a single piston or turboprop engine to drive two propellers in contra-rotation...

 can alleviate this problem allowing high maximum efficiency even at relatively high disc loadings.

The Airbus A400M

Airbus A400M

The Airbus A400M, also known as the Atlas, is a multi-national four-engine turboprop military transport aircraft. It was designed by Airbus Military as a tactical airlifter with strategic capabilities. The aircraft's maiden flight, originally planned for 2008, took place on 11 December 2009 in...

 fixed-wing aircraft will have a very high disk loading on its propellers, and this had to be factored into the design.

Theory

The momentum theory or disk actuator theory describes a mathematical model

Mathematical model

A mathematical model is a description of a system using mathematical concepts and language. The process of developing a mathematical model is termed mathematical modeling. Mathematical models are used not only in the natural sciences and engineering disciplines A mathematical model is a...

 of an ideal actuator disk, developed by W.J.M. Rankine

William John Macquorn Rankine

William John Macquorn Rankine was a Scottish civil engineer, physicist and mathematician. He was a founding contributor, with Rudolf Clausius and William Thomson , to the science of thermodynamics....

 (1865), Alfred George Greenhill

Alfred George Greenhill

Sir George Greenhill, F.R.S. , was a British mathematician.George Greenhill was educated at Christ's Hospital School and from there he went up to St John's College, Cambridge in 1866. In 1876, Greenhill was appointed professor of mathematics at the Royal Military Academy at Woolwich, London, UK...

 (1888) and R.E. Froude (1889). The helicopter

Helicopter

A helicopter is a type of rotorcraft in which lift and thrust are supplied by one or more engine-driven rotors. This allows the helicopter to take off and land vertically, to hover, and to fly forwards, backwards, and laterally...

 rotor

Helicopter rotor

A helicopter main rotor or rotor system is a type of fan that is used to generate both the aerodynamic lift force that supports the weight of the helicopter, and thrust which counteracts aerodynamic drag in forward flight...

 is modeled as an infinitely thin disc with an infinite number of blades that induce a constant pressure jump over the disk area and along the axis of rotation. For a helicopter that is hovering, the aerodynamic force is vertical and exactly balances the helicopter weight, with no lateral force.

The upward action on the helicopter results in a downward reaction on the air flowing through the rotor. The downward reaction produces a downward velocity on the air, increasing its kinetic energy

Kinetic energy

The kinetic energy of an object is the energy which it possesses due to its motion.It is defined as the work needed to accelerate a body of a given mass from rest to its stated velocity. Having gained this energy during its acceleration, the body maintains this kinetic energy unless its speed changes...

. This energy transfer from the rotor to the air is the induced power loss of the rotary wing, which is analogous to the lift-induced drag

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...

 of a fixed-wing aircraft.

Conservation of linear momentum relates the induced velocity downstream in the far wake field to the rotor thrust per unit of mass flow

Mass flow

Mass flow, also known as mass transfer and bulk flow, is the movement of material matter. In physics, mass flow occurs in open systems and is often measured as occurring when moving across a certain boundary characterized by its cross-sectional area and a flow rate. In engineering and biology it...

. Conservation of energy

Conservation of energy

The nineteenth century law of conservation of energy is a law of physics. It states that the total amount of energy in an isolated system remains constant over time. The total energy is said to be conserved over time...

 considers these parameters as well as the induced velocity at the rotor disk. Conservation of mass

Conservation of mass

The law of conservation of mass, also known as the principle of mass/matter conservation, states that the mass of an isolated system will remain constant over time...

 relates the mass flow to the induced velocity. The momentum theory applied to a helicopter gives the relationship between induced power loss and rotor thrust, which can be used to analyze the performance of the aircraft. Viscosity

Viscosity

Viscosity is a measure of the resistance of a fluid which is being deformed by either shear or tensile stress. In everyday terms , viscosity is "thickness" or "internal friction". Thus, water is "thin", having a lower viscosity, while honey is "thick", having a higher viscosity...

 and compressibility of the air, friction

Friction

Friction is the force resisting the relative motion of solid surfaces, fluid layers, and/or material elements sliding against each other. There are several types of friction:...

al losses, and rotation of the slipstream in the wake are not considered.

Momentum theory

For an actuator disk of area , with uniform induced velocity at the rotor disk, and with as the density of air

Density of air

The density of air, ρ , is the mass per unit volume of Earth's atmosphere, and is a useful value in aeronautics and other sciences. Air density decreases with increasing altitude, as does air pressure. It also changes with variances in temperature or humidity...

, the mass flow rate

Mass flow rate

Mass flow rate is the mass of substance which passes through a given surface per unit time. Its unit is mass divided by time, so kilogram per second in SI units, and slug per second or pound per second in US customary units...

  through the disk area is:


By conservation of mass, the mass flow rate is constant across the slipstream

Slipstream

A slipstream is a region behind a moving object in which a wake of fluid is moving at velocities comparable to the moving object . The term slipstream also applies to the similar region adjacent to an object with a fluid moving around it...

 both upstream and downstream of the disk (regardless of velocity). Since the flow far upstream of a helicopter in a level hover is at rest, the starting velocity, momentum, and energy are zero. If the homogeneous slipstream

Slipstream

A slipstream is a region behind a moving object in which a wake of fluid is moving at velocities comparable to the moving object . The term slipstream also applies to the similar region adjacent to an object with a fluid moving around it...

 far downstream of the disk has velocity , by conservation of momentum the total thrust developed over the disk is equal to the rate of change of momentum, which assuming zero starting velocity is:


By conservation of energy, the work done by the rotor must equal the energy change in the slipstream:


Substituting for and eliminating terms, we get:


So the velocity of the wake far downstream is twice the velocity at the disk, which is the same result for an elliptically loaded fixed wing predicted by lifting-line theory

Lifting-line theory

Lifting-line theory or Lanchester-Prandtl wing theory was published by Ludwig Prandtl in 1918–1919 after working with Albert Betz and Max Munk on the problem of a useful mathematical tool for examining lift from "real world" wings....

.

Bernoulli's principle

To compute the disk loading using Bernoulli's principle

Bernoulli's principle

In fluid dynamics, Bernoulli's principle states that for an inviscid flow, an increase in the speed of the fluid occurs simultaneously with a decrease in pressure or a decrease in the fluid's potential energy...

, we assume the pressure in the slipstream far downstream is equal to the starting pressure , which is equal to the atmospheric pressure

Atmospheric pressure

Atmospheric pressure is the force per unit area exerted into a surface by the weight of air above that surface in the atmosphere of Earth . In most circumstances atmospheric pressure is closely approximated by the hydrostatic pressure caused by the weight of air above the measurement point...

. From the starting point to the disk we have:


Between the disk and the distant wake, we have:


Combining equations, the disk loading is:


The total pressure in the distant wake is:


So the pressure change across the disk is equal to the disk loading. Above the disk the pressure change is:


Below the disk, the pressure change is:


The pressure along the slipstream is always falling downstream, except for the positive pressure jump across the disk.

Power required

From the momentum theory, thrust is:

The induced velocity is:


Where is the disk loading as before, and the power required in hover (in the ideal case) is:


Therefore the induced velocity can be expressed as:


So, the induced velocity is inversely proportional to the power loading .

Examples