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...

, the momentum theory or disk actuator theory is a theory describing 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, such as a propeller

Propeller

A propeller is a type of fan that transmits power by converting rotational motion into thrust. A pressure difference is produced between the forward and rear surfaces of the airfoil-shaped blade, and a fluid is accelerated behind the blade. Propeller dynamics can be modeled by both Bernoulli's...

 or 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...

, 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 rotor is modeled as an infinitely thin disc, inducing a constant velocity along the axis of rotation. The basic state of a helicopter is hover

Hover

Hover may refer to:*Hovering , the process by which an object is suspended by a physical force against gravity, in a stable position without solid physical contactIn transport* Hover , nearly stationary flight in a helicopter...

ing. This disc creates a flow around the rotor. Under certain mathematical premises of the fluid, there can be extracted a mathematical connection between power, radius of the rotor, torque

Torque

Torque, moment or moment of force , is the tendency of a force to rotate an object about an axis, fulcrum, or pivot. Just as a force is a push or a pull, a torque can be thought of as a twist....

 and induced velocity. 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:...

 is not included.

The power required to produce a given thrust is:



Where:

• T is the thrust
  Thrust
  Thrust is a reaction force described quantitatively by Newton's second and third laws. When a system expels or accelerates mass in one direction the accelerated mass will cause a force of equal magnitude but opposite direction on that system....
  
  
• is 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...
  
   (or other medium)
• A is the area of the rotor disc

A device which converts the translational energy of the fluid into rotational energy of the axis or vice versa is called a Rankine disk actuator. The real life implementations of such devices are e.g. marine and aviation propeller

Propeller

A propeller is a type of fan that transmits power by converting rotational motion into thrust. A pressure difference is produced between the forward and rear surfaces of the airfoil-shaped blade, and a fluid is accelerated behind the blade. Propeller dynamics can be modeled by both Bernoulli's...

s, windmill

Windmill

A windmill is a machine which converts the energy of wind into rotational energy by means of vanes called sails or blades. Originally windmills were developed for milling grain for food production. In the course of history the windmill was adapted to many other industrial uses. An important...

s, helicopter 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, centrifugal pump

Centrifugal pump

A centrifugal pump is a rotodynamic pump that uses a rotating impeller to create flow by the addition of energy to a fluid. Centrifugal pumps are commonly used to move liquids through piping...

s, wind turbine

Wind turbine

A wind turbine is a device that converts kinetic energy from the wind into mechanical energy. If the mechanical energy is used to produce electricity, the device may be called a wind generator or wind charger. If the mechanical energy is used to drive machinery, such as for grinding grain or...

s, turbocharger

Turbocharger

A turbocharger, or turbo , from the Greek "τύρβη" is a centrifugal compressor powered by a turbine that is driven by an engine's exhaust gases. Its benefit lies with the compressor increasing the mass of air entering the engine , thereby resulting in greater performance...

s and chemical agitator

Agitator (device)

An agitator is a device or mechanism to put something into motion by shaking or stirring.-Manual agitator:* Manual dishwashers* A rock can be a device used to agitate dirt and other solids from fabric in washing* A stirring rod...

s.

See also

• Blade element theory
  Blade element theory
  Blade element theory is a mathematical process originally designed by William Froude , David W. Taylor and Stefan Drzewiecki to determine the behavior of propellers. It involves breaking a blade down into several small parts then determining the forces on each of these small blade elements...
  
  
• Circulation Theory
• Disk loading
  Disk loading
  In fluid dynamics, disk loading or disc loading is the average pressure change across an actuator disk, such as an airscrew. Airscrews with a relatively low disk loading are typically called rotors, including helicopter main rotors and tail rotors; propellers typically have a higher disk...
  
  
• Strip theory for fixed wing aerodynamics
• Conway non linear momentum theory
• Kutta-Joukowski Circulation