Circulation (fluid dynamics)

Circulation (fluid dynamics)

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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...

, circulation is the line integral
Line integral
In mathematics, a line integral is an integral where the function to be integrated is evaluated along a curve.The function to be integrated may be a scalar field or a vector field...

around a closed curve of the fluid
Fluid
In physics, a fluid is a substance that continually deforms under an applied shear stress. Fluids are a subset of the phases of matter and include liquids, gases, plasmas and, to some extent, plastic solids....

velocity
Velocity
In physics, velocity is speed in a given direction. Speed describes only how fast an object is moving, whereas velocity gives both the speed and direction of the object's motion. To have a constant velocity, an object must have a constant speed and motion in a constant direction. Constant ...

. Circulation is normally denoted .

If is the fluid velocity on a small element of a defined curve, and is a vector representing the differential length of that small element, the contribution of that differential length to circulation is : where is the angle between the vectors and .

The circulation around a closed curve is the line integral:

The dimensions of circulation are length squared, divided by time.

Circulation was first used independently by Frederick Lanchester
Frederick Lanchester
Frederick William Lanchester, Hon FRAeS was an English polymath and engineer who made important contributions to automotive engineering, aerodynamics and co-invented the field of operations research....

, Wilhelm Kutta
Martin Wilhelm Kutta
Martin Wilhelm Kutta was a German mathematician.Kutta was born in Pitschen, Upper Silesia . He attended the University of Breslau from 1885 to 1890, and continued his studies in Munich until 1894, where he became the assistant of Walther Franz Anton von Dyck. From 1898, he spent a year at the...

, and Nikolai Zhukovsky.

Kutta–Joukowski theorem

The lift
Lift (force)
A fluid flowing past the surface of a body exerts a surface force on it. Lift is the component of this force that is perpendicular to the oncoming flow direction. It contrasts with the drag force, which is the component of the surface force parallel to the flow direction...

force acting per unit span on a body in a two-dimensional inviscid flow field can be expressed as the product of the circulation () about the body, the fluid density (), and the speed of the body relative to the free-stream (). Thus,

This is known as the Kutta–Joukowski theorem.

This equation applies around airfoils, where the circulation is generated by airfoil action, and around spinning objects, experiencing the Magnus effect
Magnus effect
The Magnus effect is the phenomenon whereby a spinning object flying in a fluid creates a whirlpool of fluid around itself, and experiences a force perpendicular to the line of motion...

, where the circulation is induced mechanically.

Circulation is often used in computational fluid dynamics
Computational fluid dynamics
Computational fluid dynamics, usually abbreviated as CFD, is a branch of fluid mechanics that uses numerical methods and algorithms to solve and analyze problems that involve fluid flows. Computers are used to perform the calculations required to simulate the interaction of liquids and gases with...

as an intermediate variable to calculate forces on an airfoil
Airfoil
An airfoil or aerofoil is the shape of a wing or blade or sail as seen in cross-section....

or other body. When an airfoil is generating lift the circulation around the airfoil is finite, and is related to the vorticity of the boundary layer
Boundary layer
In physics and fluid mechanics, a boundary layer is that layer of fluid in the immediate vicinity of a bounding surface where effects of viscosity of the fluid are considered in detail. In the Earth's atmosphere, the planetary boundary layer is the air layer near the ground affected by diurnal...

. Outside the boundary layer the vorticity is zero everywhere and therefore the circulation is the same around every circuit, regardless of the length of the circumference of the circuit.

Relation to vorticity

Circulation can be related to vorticity by Stokes' theorem
Stokes' theorem
In differential geometry, Stokes' theorem is a statement about the integration of differential forms on manifolds, which both simplifies and generalizes several theorems from vector calculus. Lord Kelvin first discovered the result and communicated it to George Stokes in July 1850...

:

but only if the integration path is a boundary, not just a closed curve. Thus vorticity is the circulation per unit area, taken around an infinitesimal loop.