Circulation
The Circulation C about a closed curve in a flow field is defined as the negative of the line integral of velocity around that closed curve. It is denoted by the symbol, Γ (Uppercase Gamma). This definition makes it a Scalar Quantity.
It is a kinematic property depending on the velocity of the field and the choice of the curve C. The word 'Circulation' generally means movement in a circle or circuit but in aerodynamics circulation has a very precise technical meaning as per the equation above. It does not necessarily means that fluid elements are moving around in circles in this fluid field rather when circulation exists in a flow it simply means that the line integral taken above is finite. For example, if the airfoil is generating lift, the circulation taken around a closed curve enclosing the airfoil will be finite, although the fluid elements are by no means executing circles around the airfoil.
Vorticity
Vorticity is mathematically defined as the curl of velocity field and hence is a measure of local rotation of the fluid elements in a given flow field. Since curl gives you a vector, vorticity is a vector quantity. It is denoted by ζ (lowercase, Zeta).
Relation Between Circulation and Vorticity
Using Stoke's theorem, the line integral of the velocity field along the closed path, can be expressed as a surface integral of the curl of the velocity field normal to an arbitrary area bounded by the path. But, as already defined, that curl operation is called vorticity. Hence, circulation can be referred to as flux of vorticity. Conversely, it can also be said that that vorticity at a point is essentially circulation per unit area.
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