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Concept
Circulation (physics)
Summary
In physics, circulation is the line integral of a vector field around a closed curve. In fluid dynamics, the field is the fluid velocity field. In electrodynamics, it can be the electric or the magnetic field.
Circulation was first used independently by Frederick Lanchester, Martin Kutta and Nikolay Zhukovsky. It is usually denoted Γ (Greek uppercase gamma).
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 C is the line integral:
In a conservative vector field 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 of a scalar function, which is called a potential.
Circulation can be related to curl of a vector field V and, more specifically, to vorticity if the field is a fluid velocity field,
By Stokes' theorem, the flux 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. Thus curl and vorticity are the circulation per unit area, taken around a local infinitesimal loop.
In potential flow of a fluid with a region of vorticity, all closed curves that enclose the vorticity have the same value for circulation.
Kutta–Joukowski theorem
In fluid dynamics, the lift 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 , and the speed of the body relative to the free-stream :
This is known as the Kutta–Joukowski theorem.
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Related courses (1)
ME-344: Incompressible fluid mechanics
Basic lecture in incompressible fluid mechanics
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Flow velocity
In continuum mechanics the flow velocity in fluid dynamics, also macroscopic velocity in statistical mechanics, or drift velocity in electromagnetism, is a vector field used to mathematically describe the motion of a continuum. The length of the flow velocity vector is the flow speed and is a scalar. It is also called velocity field; when evaluated along a line, it is called a velocity profile (as in, e.g., law of the wall).
Helmholtz's theorems
In fluid mechanics, Helmholtz's theorems, named after Hermann von Helmholtz, describe the three-dimensional motion of fluid in the vicinity of vortex lines. These theorems apply to inviscid flows and flows where the influence of viscous forces are small and can be ignored. Helmholtz's three theorems are as follows: Helmholtz's first theorem The strength of a vortex line is constant along its length. Helmholtz's second theorem A vortex line cannot end in a fluid; it must extend to the boundaries of the fluid or form a closed path.
Stokes' theorem
Stokes' theorem, also known as the Kelvin–Stokes theorem after Lord Kelvin and George Stokes, the fundamental theorem for curls or simply the curl theorem, is a theorem in vector calculus on . Given a vector field, the theorem relates the integral of the curl of the vector field over some surface, to the line integral of the vector field around the boundary of the surface. The classical theorem of Stokes can be stated in one sentence: The line integral of a vector field over a loop is equal to the flux of its curl through the enclosed surface.