On deriving nonreflecting boundary conditions in generalized curvilinear coordinates

TL;DR

This paper extends the derivation of nonreflecting boundary conditions from Cartesian to generalized curvilinear coordinates using Fourier analysis of the linearized Euler equations, providing a theoretical framework without application testing.

Contribution

It generalizes the derivation of nonreflecting boundary conditions to three-dimensional curvilinear coordinates based on prior Cartesian analysis.

Findings

Derived boundary conditions for curvilinear coordinates

Performed Fourier analysis to determine eigenvalues and eigenvectors

Identified open questions on well-posedness

Abstract

In this work, nonreflecting boundary conditions in generalized three-dimensional curvilinear coordinates are derived, relying on the original analysis that was done in Cartesian two-dimensional coordinates by Giles (AIAA Journal, 28.12, 2050-2058, 1990). A thorough Fourier analysis of the linearized Euler equation is performed to determine the eigenvalues and the eigenvectors that are then used to derive the appropriate inflow and outflow boundary conditions. The analysis lacks rigorous proof of the well-posedness in the general case, which is open to investigation (a weak assumption is introduced here to complete the boundary conditions). The boundary conditions derived here are not tested on specific applications.