  We consider a family of 2D logarithmic spiral vortex sheets which include the
celebrated spirals introduced by Prandtl (Vortr\"age aus dem Gebiete der Hydro-
und Aero-dynamik, 1922) and by Alexander (Phys. Fluids, 1971). We prove that
for each such spiral the normal component of the velocity field remains
continuous across the spiral. We give sufficient conditions for spiral vortex
sheets to be weak solutions of the 2D incompressible Euler equations. Namely,
we show that a spiral gives rise to such a solution if and only if two
conditions hold across every spiral: a velocity matching condition and a
pressure matching condition. Furthermore we show that these two conditions are
equivalent to the imaginary part and the real part, respectively, of a single
complex constraint on the coefficients of the spirals. This in particular
provides a rigorous mathematical framework for logarithmic spirals, an issue
that has remained open since their introduction by Prandtl in 1922. Another
consequence of the main result is well-posedness of the symmetric Alexander
spiral with two branches, despite recent evidence for the contrary. Moreover,
our result implies a sharpness result of Delort's theorem on global existence
of solutions to the Euler equations with initial vorticity measure. Our main
tools are new explicit formulas for the velocity field and for the pressure
function, as well as a notion of a \emph{winding number} of a spiral, which not
only gives a robust way of localizing the spirals' arms with respect to a given
point in the plane, but also ensures correct asymptotic behaviour near $0$.
