Classical free-streamline flow over a polygonal obstacle

In classical Kirchhoff flow, an ideal incompressible fluid flows past an obstacle and around a motionless wake bounded by free streamlines. Since 1869 it has been known that in principle, the two-dimensional Kirchhoff flow over a polygonal obstacle can be determined by constructing a conformal map onto a polygon in the log-hodograph plane. In practice, however, this idea has rarely been put to use except for very simple obstacles, because the conformal mapping problem has been too difficult. This paper presents a practical method for computing flows over arbitrary polygonal obstacles to high accuracy in a few seconds of computer time. We achieve this high speed and flexibility by working with a modified Schwarz-Christoffel integral that maps onto the flow region directly rather than onto the log-hodograph polygon. This integral and its associated parameter problem are treated numerically by methods developed earlier by Trefethen for standard Schwarz-Christoffel maps

Regularization of point vortices for the Euler equation in dimension two

In this paper, we construct stationary classical solutions of the incompressible Euler equation approximating singular stationary solutions of this equation. This procedure is carried out by constructing solutions to the following elliptic problem [ -\ep^2 \Delta u=(u-q-\frac{\kappa}{2\pi}\ln\frac{1}{\ep})_+^p, \quad & x\in\Omega, u=0, \quad & x\in\partial\Omega, ] where $p>1$, $\Omega\subset\mathbb{R}^2$ is a bounded domain, $q$ is a harmonic function. We showed that if $\Omega$ is simply-connected smooth domain, then for any given non-degenerate critical point of Kirchhoff-Routh function $\mathcal{W}(x_1,...,x_m)$ with the same strength $\kappa>0$, there is a stationary classical solution approximating stationary $m$ points vortex solution of incompressible Euler equations with vorticity $m\kappa$. Existence and asymptotic behavior of single point non-vanishing vortex solutions were studied by D. Smets and J. Van Schaftingen (2010).Comment: 32page

Computable analysis of linear rearrangement optimization

Optimization problems over rearrangement classes arise in various areas such as mathematics, fluid mechanics, biology, and finance. When the generator of the rearrangement class is two-valued, they reduce to shape optimization and free boundary problems which can exhibit intriguing symmetry breaking phenomena. A robust framework is required for computable analysis of these problems. In this paper, as a first step towards such a robust framework, we provide oracle Turing machines that compute the distribution function, decreasing rearrangement, and linear rearrangement optimizers, with respect to functions that are continuous and have no significant flat zones. This assumption on the reference function is necessary, as otherwise, the aforementioned operations may not be computable. We prove that the results can be computed to within any degree of accuracy, conforming to the framework of Type-II Theory of Effectivity

Models for inviscid wakes past a normal plate

Closed and open hollow wakes are considered as analytic models for the twodimensional inviscid steady flow past a plate normal to the stream. It is shown that only open configurations which satisfy the Kutta condition exist. The main argument is based on considering a plate located on the edge of a step with varying height. It is shown that solutions for open wakes exist for backward-, null and forward-facing steps, while closed wakes only exist for backward-facing steps. The occurrence of secondary separation has been modelled by adding a hollow region attached to the downstream corner. Peculiar accuracy issues of the problem are pointed out which may explain other contradictory results from the literature. It is shown how the Kirchhoff wake is a limiting solution for certain values of the governing parameter