The Cauchy-Lagrange method for 3D-axisymmetric wall-bounded and potentially singular incompressible Euler flows

Abstract

We treat the incompressible, and axisymmetric Euler equations for a three-dimensional cylindrical domain with boundaries. The equations are solved by the novel Cauchy-Lagrange algorithm (CLA), which uses the time-analyticity of the Lagrangian trajectories of an incompressible Euler flow and computes the time-Taylor coefficients of the Lagrangian map via recursion relations. This semi-Lagrangian algorithm uses a pseudo-spectral type approach in space by approximating the flow fields by Chebyshev-Fourier polynomials. New methods are presented to solve the resulting Poisson problems directly for their second-order space derivatives. The flow fields, known on the Lagrangian trajectories after one time-step, are interpolated back onto the Eulerian grid to start a new recursion cycle. The time-step is only limited by the radius of convergence and, thus, independent of any Courant-Friedrichs-Lewy (CFL) condition. This allows to advance the flow with larger time-steps independently of the mesh. Stationary and swirl-free flows are used to thoroughly test our implementation for the given geometry. In this work, our ultimate goal is to apply the CLA to a flow, which might develop finite time singularities, resulting in a loss of smoothness. This demands either 3D, or 2D with swirl, and may well require the presence of solid boundaries, as indicated in recent numerical work (Luo and Hou (2014) [45]).