High-accuracy methods for frequency-domain flow acoustics

Abstract

Limited by the current computer memory capacities, it is challenging to solve the convected wave equation in heterogeneous media and at high wave number. This thesis is concerned with the investigation of efficient discretisation methods and solvers for frequency-domain flow acoustics. The application of computational modelling to wave propagation problems is hindered by the dispersion error introduced by the discretisation. Two common strategies to address this issue are to use high-order polynomial shape functions, or to use physics-based methods where the shape functions are local solutions of the problem. Both strategies have been actively developed over the past decades and have demonstrated their benefits compared to conventional finite-element methods, but they have yet to be compared. In this work, a high-order polynomial method and the wave-based discontinuous Galerkin method are compared for two-dimensional Helmholtz problems. A number of different benchmark cases are used to perform a detailed and systematic assessment of the relative merits of these two methods. The results indicate that the differences in performance, accuracy and conditioning are more nuanced than generally assumed. The performance of a method relies heavily on efficient solving procedures for the resulting large, sparse, complex linear systems. An alternative to purely iterative or direct solving procedures is to resort to domain decomposition methods. The Finite Element Tearing and Interconnecting method (FETI-2LM) employs Lagrange multipliers to recover the connections between the non-overlapping sub-domains. An iterative solution procedure is formulated in terms of unknowns defined only on the interfaces between sub-domains. The FETI approaches have been used extensively for Helmholtz problems and their performance is well documented for conventional finite elements. In this work, the FETI-2LM formulation is extended to the linearised potential theory for sound waves propagating in a potential base flow. In each sub-domain, a high-order finite element method is used to solve the governing equations. The proposed approach is validated on a number of two-dimensional test cases. In addition to the dependency on the mesh size, frequency, or number of subdomains, the influence of the interpolation order and Mach number on the scalability of the method is also assessed. The memory requirements for solving a simple three-dimensional problem is also evaluated and compared to that of a direct solver. Finally, the proposed method is applied to the problem of propagation of fan noise from the inlet of an engine, considering a realistic three-dimensional geometry and flow field.

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ORCID for Alan Mcalpine: orcid.org/0000-0003-4189-2167

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