ABSTRACT
Point vortex is a 2D flow generated by a singular vorticity distribution which is concentrated at a single point, such that the velocity circulation along a contour embracing this point has a finite value Γ. We have already met this solution in questions 6.2.7 and 6.2.8. In question 6.2.8 we obtained the point vortex solution as a limiting case of a Rankine vortex with vorticity Ω(x) which is constant inside a circle of radius a, Ω(x) = κ = const, and zero outside. Then, the limit a → 0 was taken while keeping the circulation constant, Γ = pia2κ = const. This means that the vorticity value inside the circle tends to infinity, κ ∼ 1/a2. In the limit, such a vorticity field will be represented by a Dirac delta function,
Ω(x) = Γ δ(x). (7.1)
This vortex generates the following velocity field,
u = − Γ 2pi
y
x2 + y2 , v =
Γ
2pi
x
x2 + y2 ; (7.2)
see problem 6.2.7. Indeed, for this flow the circulation is zero for all contours which do not encircle the vortex, and equal to Γ for all contours encircling the vortex (please show that!). In question 6.2.7 you were also asked to find the stream function for the point vortex flow.
