In physics, circulation is the line integral of a vector field around a closed curve. In
fluid dynamics
In physics and engineering, fluid dynamics is a subdiscipline of fluid mechanics that describes the flow of fluids— liquids and gases. It has several subdisciplines, including ''aerodynamics'' (the study of air and other gases in motion) a ...
, the field is the fluid
velocity field. In
electrodynamics
In physics, electromagnetism is an interaction that occurs between particles with electric charge. It is the second-strongest of the four fundamental interactions, after the strong force, and it is the dominant force in the interactions o ...
, it can be the electric or the magnetic field.
Circulation was first used independently by
Frederick Lanchester
Frederick William Lanchester LLD, Hon FRAeS, FRS (23 October 1868 – 8 March 1946), was an English polymath and engineer who made important contributions to automotive engineering and to aerodynamics, and co-invented the topic of operations ...
,
Martin Kutta and
Nikolay Zhukovsky. It is usually denoted Γ (
Greek
Greek may refer to:
Greece
Anything of, from, or related to Greece, a country in Southern Europe:
*Greeks, an ethnic group.
*Greek language, a branch of the Indo-European language family.
**Proto-Greek language, the assumed last common ancestor ...
uppercase
Letter case is the distinction between the letters that are in larger uppercase or capitals (or more formally ''majuscule'') and smaller lowercase (or more formally ''minuscule'') in the written representation of certain languages. The writing ...
gamma
Gamma (uppercase , lowercase ; ''gámma'') is the third letter of the Greek alphabet. In the system of Greek numerals it has a value of 3. In Ancient Greek, the letter gamma represented a voiced velar stop . In Modern Greek, this letter r ...
).
Definition and properties
If V is a vector field and dl is a vector representing the
differential length of a small element of a defined curve, the contribution of that differential length to circulation is dΓ:
:
.
Here, ''θ'' is the angle between the vectors V and dl.
The circulation Γ of a vector field V around a
closed curve
In mathematics, a curve (also called a curved line in older texts) is an object similar to a line, but that does not have to be straight.
Intuitively, a curve may be thought of as the trace left by a moving point. This is the definition that ...
''C'' is the
line integral
In mathematics, a line integral is an integral where the function to be integrated is evaluated along a curve. The terms ''path integral'', ''curve integral'', and ''curvilinear integral'' are also used; '' contour integral'' is used as well, ...
:
:
.
In a
conservative vector field
In vector calculus, a conservative vector field is a vector field that is the gradient of some function. A conservative vector field has the property that its line integral is path independent; the choice of any path between two points does not ...
this integral evaluates to zero for every closed curve. That means that a line integral between any two points in the field is independent of the path taken. It also implies that the vector field can be expressed as the
gradient
In vector calculus, the gradient of a scalar-valued differentiable function of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p is the "direction and rate of fastest increase". If the gr ...
of a scalar function, which is called a
potential
Potential generally refers to a currently unrealized ability. The term is used in a wide variety of fields, from physics to the social sciences to indicate things that are in a state where they are able to change in ways ranging from the simple r ...
.
Relation to vorticity and curl
Circulation can be related to
curl
cURL (pronounced like "curl", UK: , US: ) is a computer software project providing a library (libcurl) and command-line tool (curl) for transferring data using various network protocols. The name stands for "Client URL".
History
cURL was ...
of a vector field V and, more specifically, to
vorticity
In continuum mechanics, vorticity is a pseudovector field that describes the local spinning motion of a continuum near some point (the tendency of something to rotate), as would be seen by an observer located at that point and traveling along wi ...
if the field is a fluid velocity field,
:
.
By
Stokes' theorem
Stokes's theorem, also known as the Kelvin–Stokes theorem Nagayoshi Iwahori, et al.:"Bi-Bun-Seki-Bun-Gaku" Sho-Ka-Bou(jp) 1983/12Written in Japanese)Atsuo Fujimoto;"Vector-Kai-Seki Gendai su-gaku rekucha zu. C(1)" :ja:培風館, Bai-Fu-Kan( ...
, the
flux
Flux describes any effect that appears to pass or travel (whether it actually moves or not) through a surface or substance. Flux is a concept in applied mathematics and vector calculus which has many applications to physics. For transport ...
of curl or vorticity vectors through a surface ''S'' is equal to the circulation around its perimeter,
:
Here, the closed integration path ''∂S'' is the
boundary or perimeter of an open surface ''S'', whose infinitesimal element
normal dS=ndS is oriented according to the
right-hand rule
In mathematics and physics, the right-hand rule is a common mnemonic for understanding orientation of axes in three-dimensional space. It is also a convenient method for quickly finding the direction of a cross-product of 2 vectors.
Most of ...
. Thus curl and vorticity are the circulation per unit area, taken around a local infinitesimal loop.
In
potential flow
In fluid dynamics, potential flow (or ideal flow) describes the velocity field as the gradient of a scalar function: the velocity potential. As a result, a potential flow is characterized by an irrotational velocity field, which is a valid app ...
of a fluid with a region of
vorticity
In continuum mechanics, vorticity is a pseudovector field that describes the local spinning motion of a continuum near some point (the tendency of something to rotate), as would be seen by an observer located at that point and traveling along wi ...
, all closed curves that enclose the vorticity have the same value for circulation.
[Anderson, John D. (1984), ''Fundamentals of Aerodynamics'', section 3.16. McGraw-Hill. ]
Uses
Kutta–Joukowski theorem in fluid dynamics
In fluid dynamics, the
lift
Lift or LIFT may refer to:
Physical devices
* Elevator, or lift, a device used for raising and lowering people or goods
** Paternoster lift, a type of lift using a continuous chain of cars which do not stop
** Patient lift, or Hoyer lift, mobil ...
per unit span (L') acting on a body in a two-dimensional flow field is directly proportional to the circulation, i.e. it can be expressed as the product of the circulation Γ about the body, the fluid density ''ρ'', and the speed of the body relative to the free-stream V:
:
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
The Magnus effect is an observable phenomenon commonly associated with a spinning object moving through a fluid. The path of the spinning object is deflected in a manner not present when the object is not spinning. The deflection can be expl ...
where the circulation is induced mechanically. In airfoil action, the magnitude of the circulation is determined by the
Kutta condition.
The circulation on every closed curve around the airfoil has the same value, and is related to the lift generated by each unit length of span. Provided the closed curve encloses the airfoil, the choice of curve is arbitrary.
Circulation is often used in
computational fluid dynamics
Computational fluid dynamics (CFD) is a branch of fluid mechanics that uses numerical analysis and data structures to analyze and solve problems that involve fluid flows. Computers are used to perform the calculations required to simulate ...
as an intermediate variable to calculate forces on an
airfoil
An airfoil (American English) or aerofoil (British English) is the cross-sectional shape of an object whose motion through a gas is capable of generating significant lift, such as a wing, a sail, or the blades of propeller, rotor, or turbin ...
or other body.
Fundamental equations of electromagnetism
In electrodynamics, the
Maxwell-Faraday law of induction can be stated in two equivalent forms:
that the curl of the electric field is equal to the negative rate of change of the magnetic field,
:
or that the circulation of the electric field around a loop is equal to the negative rate of change of the magnetic field flux through any surface spanned by the loop, by Stokes' theorem
:
.
:
Circulation of a
static magnetic field is, by
Ampère's law, proportional to the total current enclosed by the loop
:
.
For systems with electric fields that change over time, the law must be modified to include a term known as Maxwell's correction.
See also
*
Maxwell's equations
Maxwell's equations, or Maxwell–Heaviside equations, are a set of coupled partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, and electric circuits ...
*
Biot–Savart law in aerodynamics
*
Kelvin's circulation theorem
References
{{reflist
Fluid dynamics
Physical quantities
Electromagnetism