Well-posedness of logarithmic spiral vortex sheets

Abstract

We consider a family of 2D logarithmic spiral vortex sheets which include the celebrated spirals introduced by Prandtl (Vorträge aus dem Gebiete der Hydro-und Aerodynamik, 1922) and by Alexander (Phys. Fluids, 1971). We prove that for each such spiral the normal component of the velocity field to any spiral remains continuous across the spiral. Moreover, we give a complete characterization of such spirals in terms of weak solutions of the 2D incompressible Euler equations. Namely, we show that a spiral gives rise to such 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. Our main tools are new explicit formulas for the velocity field and for the pressure function, as well as a notion of a winding number of a spiral, which gives a robust way of localizing the spirals' arms with respect to a given point in the plane.

Logarithmic spirals in 2d perfect fluids

“…Furthermore, we emphasize that the spirals of Prandtl and Alexander are simply very specific solutions to the ODE system (1.8) that we have obtained in this work, and this general approach provides a framework for studying the asymptotic stability of self-similar singularity formation. To illustrate this, we recover some recent results from [4,5,11] on existence and bifurcation of self-similar logarithmic spiral vortex sheets using our formulation in Section 4.…”

“…The mathematical proof of this was done in Elling-Gnann in the m-fold symmetric case with m ≥ 3 [11], using special cancellation which is directly related with the well-posedness theory of 2D Euler under m-fold symmetry which we shall explain below. Without any symmetry hypothesis, the proof was done very recently by Cieślak-Kokocki-Ożański in [4]. The same authors proved the existence of (a variety of) non-symmetric self similar logarithmic vortex spirals in [5].…”

“…In this section, we show how our formulation of logarithmic vortex sheets corresponds to the traditional one which goes back to the work of Prandtl. We follow the notation of [4,5]: in the case of one branch, they write Taking t = 1 to be the initial time for simplicity, the above ansatz corresponds to taking our h function to be h(t = 1, θ) = 2gδ(θ − θ 0 ).…”

The Influence of Vortex Sheet Geometry on the Kelvin-Helmholtz Instability

“…the classic book [11]. This type of spiral roll-up is expected to be self-similar, and a wide range of numerical and analytical work has sought to better understand self-similar solutions to Euler's equation which resemble vortex sheets [7,14,15,36,37,9,10].…”

“…A major motivation for studying the Birkhoff-Rott equation is that one can ignore the dynamics away from shear interfaces, which are likely themselves unstable, and instead focus on a simpler curve evolution equation. Indeed the Birkhoff-Rott equation is observed to admit, with varying degrees of rigor, self-similar spiral solutions [9,10,14,13,19]. However, the Birkhoff-Rott equation is still rather delicate: a more in depth treatment of its analytical properties will be given in Section 1.1.…”