Matthew King (University of Warwick): The Hydrodynamic Instability in Quadratic Sheared Flow over Acoustic Linings.
Abstract: Jet engines, with increasing noise restrictions, make use of acoustic linings to reduce sound attenuation. Modelling these reduces the engine to a duct and the lining to an impedance boundary condition.
Making use of the Fourier transformed Euler equations, one can locate ‘wave modes’ that act as poles in contribution to Fourier inversion by residues. These wave modes can be located by numerical methods and in many cases include an unstable mode with a growing contribution. Under a sheared flow a branch cut, known as the critical layer, is also present but is often ignored.
Solving the problem analytically using Frobenius series solutions for a quadratic shear allows us to track the ​poles as we vary the system parameters and find that the unstable mode may become stable. This occurs by the pole moving through the branch cut and onto another Riemann sheet, where numerical methods would no longer be able to locate it. The pole’s contribution when this occurs is absorbed by the critical layer branch cut, suggesting the critical layer branch cut cannot be ignored.
Prof. Euan Spence (University of Bath): The Helmholtz boundary element method does not suffer from the pollution effect.
Abstract: In d dimensions, approximating an arbitrary function oscillating with frequency less than or equal to k requires ~ k^d degrees of freedom. A numerical method for solving the Helmholtz equation (with wavenumber k and in d dimensions) suffers from the pollution effect if, as k increases, the total number of degrees of freedom needed to maintain accuracy grows faster than this natural threshold (i.e., faster than k^d for domain-based formulations, such as finite element methods, and k^{d-1} for boundary-based formulations, such as boundary element methods).
It is well known that the h-version of the finite element method (FEM) suffers from the pollution effect. In contrast, at least empirically, the h-version of the boundary element method (BEM) does not suffer from the pollution effect, but this has not been proved up till now.
In this talk, I will discuss recent results (obtained with Jeffrey Galkowski) showing that the h-BEM does not suffer from the pollution effect in certain common situations.