Birkhoff-Rott equation

A planar vortex sheet is a curve in a two-dimensional inviscid incompressible flow across which the tangential velocity is discontinuous (cf. also Von Kármán vortex shedding). The vortex sheet is described by its complex position $z ( \Gamma , t ) = x + i y$. For simplicity, assume that the vorticity on the sheet is all positive and that the flow outside the sheet is irrotational. The sheet is parameterized by a real variable $\Gamma$ which represents the circulation, i.e. $\gamma = | \partial z / \partial \Gamma | ^ { - 1 }$ is the vorticity density along the sheet. Vortex sheet evolution is then described by the Birkhoff–Rott equation [a1], [a11]:

$$\begin{equation} \tag{a1} \partial _ { t } \overline{z( \Gamma , t )} = ( 2 \pi i ) ^ { - 1 } \operatorname{PV} \int _ { - \infty } ^ { \infty } \frac { d \Gamma ^ { \prime } } { z ( \Gamma , t ) - z ( \Gamma ^ { \prime } , t ) }. \end{equation}$$

Because of the singularity of the integral at $\Gamma ^ { \prime } = \Gamma$, the integral in (a1) is understood as a Cauchy principal value integral (cf. also Cauchy integral).

Perturbations of a flat sheet of uniform strength grow due to the linear Kelvin–Helmholtz instability and at some time later the sheet begins to roll-up. D. Moore [a8], [a9] showed by asymptotic analysis that a singularity could develop along the sheet at finite time starting from smooth initial data. The singularity found by Moore has the form $z _ { \Gamma } = \mathcal{O} ( \Gamma ^ { - 1 / 2 } )$ in which $z ( \Gamma ) = x + i y$ is the position and $\Gamma$ is the circulation variable. This singularity form was later found to be generic [a16]. Exact singular solutions of the non-linear Birkhoff–Rott equation, corresponding to Moore's singularity, have been constructed in [a3], [a4].

Numerical simulations of the vortex sheet problem [a5], [a7], [a12] have produced singular solutions which are in agreement with Moore's theory. Krasny's method [a5] used a non-linear filter to remove the numerical noise generated by the physical instability, the convergence of which was proved in [a15] for analytic initial data. R. Krasny [a6] also computed roll-up of a sheet, using a desingularized equation, and found that the sheet begins to roll-up immediately after the appearance of the first singularity. A general set of similarity solutions for a rolled-up vortex sheet were constructed numerically in [a10].

Existence results almost up to the singularity time have been proved [a2], [a13], using the abstract Cauchy–Kovalevskaya theorem. The results for existence and for singularity formation use an extension of the Birkhoff–Rott equation (a1) into the complex $\Gamma$-plane for analytic initial data. Since the linearization of (a1) is elliptic in $\Gamma$ and $t$ (cf. also Elliptic partial differential equation), it is hyperbolic in the imaginary $\Gamma$ direction (cf. also Hyperbolic partial differential equation). Singularities in the initial data at complex values of $\Gamma$ travel towards the real axis at a finite speed.

The Birkhoff–Rott equation has been extended to three-dimensional sheets in [a14]. Short-time existence theory for the three-dimensional equations has been established in [a13]. A computational method for the three-dimensional equations was implemented in [a17].

Open questions as of 2000 include the well-posedness for continuation after Moore's singularity and the form of singularities in three dimensions.

References

[a1]	G. Birkhoff, "Helmholtz and Taylor instability" , Proc. Symp. Appl. Math. , XII , Amer. Math. Soc. (1962) pp. 55–76
[a2]	R.E. Caflisch, O.F. Orellana, "Long time existence for a slightly perturbed vortex sheet" Commun. Pure Appl. Math. , 39 (1986) pp. 807–838
[a3]	R.E. Caflisch, O.F. Orellana, "Singularity formulation and ill-posedness for vortex sheets" SIAM J. Math. Anal. , 20 (1989) pp. 293–307
[a4]	J. Duchon, R. Robert, "Global vortex sheet solutions of Euler equations in the plane" J. Diff. Eqs. , 73 (1988) pp. 215–224
[a5]	R. Krasny, "On singularity formation in a vortex sheet and the point vortex approximation" J. Fluid Mech. , 167 (1986) pp. 65–93
[a6]	R. Krasny, "Desingularization of periodic vortex sheet roll-up" J. Comput. Phys. , 65 (1986) pp. 292–313
[a7]	D.I. Meiron, G.R. Baker, S.A. Orszag, "Analytic structure of vortex sheet dynamics, Part 1, Kelvin–Helmholtz instability" J. Fluid Mech. , 114 (1982) pp. 283–298
[a8]	D.W. Moore, "The spontaneous appearance of a singularity in the shape of an evolving vortex sheet" Proc. Royal Soc. London A , 365 (1979) pp. 105–119
[a9]	D.W. Moore, "Numerical and analytical aspects of Helmholtz instability" F.I. Niordson (ed.) N. Olhoff (ed.) , Theoretical and Applied Mechanics (Proc. XVI ICTAM) , North-Holland (1984) pp. 629–633
[a10]	D.I. Pullin, W.R.C. Phillips, "On a generalization of Kaden's problem" J. Fluid Mech. , 104 (1981) pp. 45–53
[a11]	N. Rott, "Diffraction of a weak shock with vortex generation" JFM , 1 (1956) pp. 111
[a12]	M. Shelley, "A study of singularity formation in vortex-sheet motion by a spectrally accurate vortex method" J. Fluid Mech. , 244 (1992) pp. 493–526
[a13]	P. Sulem, C. Sulem, C. Bardos, U. Frisch, "Finite time analyticity for the two and three dimensional Kelvin–Helmoltz instability" Comm. Math. Phys. , 80 (1981) pp. 485–516
[a14]	R.E. Caflisch, X. Li, "Lagrangian theory for 3D vortex sheets with axial or helical symmetry" Transport Th. Statist. Phys. , 21 (1992) pp. 559–578
[a15]	R.E. Caflisch, T.Y. Hou, J. Lowengrub, "Almost optimal convergence of the point vortex method for vortex sheets using numerical filtering" Math. Comput. , 68 (1999) pp. 1465–1496
[a16]	S.J. Cowley, G.R. Baker, S. Tanveer, "On the formation of Moore curvature singularities in vortex sheets" J. Fluid Mech. , 378 (1999) pp. 233–267
[a17]	M. Brady, A. Leonard, D.I. Pullin, "Regularized vortex sheet evolution in three dimensions" J. Comput. Phys. , 146 (1998) pp. 520–45