A study of aliasing error in DGM solutions to turbofan exhaust noise problems

Highly non-uniform flows such as shear layers are an intrinsic feature of turbofan exhaust noise problems. Modeling the sound radiation from turbofan exhausts with the linearized Euler equations raises the issue of accurately representing strongly spatially-varying mean flows numerically, while ensuring that numerical solutions are not polluted by spurious solutions such as aliasing errors. This paper investigates the behavior of aliasing instabilities in time domain solutions obtained by the discontinuous Galerkin method. A model exhaust noise problem is studied to demonstrate the growth of unphysical temporal instabilities. A new fully-discrete dispersion analysis technique is developed that permits non-uniform mean flows. The dispersion analysis is used to study the spectral behavior of aliasing instabilities and the impact of polynomial order on their formation and growth. The results of this study indicate that aliasing errors are largely absolute instabilities which build up in the solution over time and are highly sensitive to the polynomial order

Performance of the DGM for the linearized Euler equations with non-uniform mean-flow

A dispersion analysis of the fully-discrete, nodal discontinuous Galerkin method (DGM) for the solution of the time-domain linearized Euler equations (LEE) is performed. Two dispersion analysis methods are developed, considering both uniform and non-uniform mean-flow effects. Convergence studies are performed for the dispersion, dissipation, and nodal solution errors of the acoustic, entropy, and vorticity modes. The accuracy and stability of the DGM are analyzed in the context of aeroacoustic applications, and guidelines are proposed for the choice of optimal discretizations. Computational costs are estimated for a model problem and related to the choice of the element size, polynomial order, and time step. Results indicate that temporal error can become a dominant source of error for high accuracy requirements and long distance wave propagation. The stability of the scheme is analyzed for a shear layer mean flow profile. Aliasing-type errors are found to contribute to the formation of numerical instabilities which are further strengthened by increases in the polynomial order

Time-domain DGM for turbofan exhaust noise predictions

Over the past fifty years, large strides have been made in reducing aircraft noise pollution. To continue on this trend, and to meet more aggressive noise regulations, higher-fidelity numerical models must be used early in the design process to accurately characterize the noise signature of prospective aircraft and engine designs. Predicting turbofan exhaust noise propagation requires a numerical scheme that can model a highly non-uniform flow-field, has low dispersion and dissipation error, has high computational efficiency, and can handle complex geometries. Time-domain nodal discontinuous Galerkin methods (DGM) have shown success in applications requiring high spatial accuracy, computational efficiency, and complex geometry representation. This thesis further develops the DGM for turbofan exhaust noise applications using a hybrid approach and solving the three-dimensional (3D) linearized Euler equations (LEE) in the time-domain. A parallel, 3D implementation of the above scheme is outlined and the accuracy and efficiency are verified. Challenges in applying the scheme to engineering applications are addressed. The relationship between accuracy and computational cost is investigated using a dispersion analysis of the scheme. Complications involving modeling of the highly non-uniform exhaust flow-field are addressed, including developing a new dispersion analysis of the LEE to study the formation and growth of aliasing-driven instabilities in shear layers, and the impact of the mean flow representation accuracy on the acoustic solution. A mapping procedure used to interface the mean flow solver with the aeroacoustic solver is discussed, and extended to the treatment of mean-flow boundary-layers that are unresolved on the acoustic mesh. Addressing the robustness and efficiency of the method, compact analytical source terms are developed that exploit the numerical flux between elements and band-limited source terms are investigated to reduce the number of computations required for an analysis. The numerical scheme is applied to the problem of scattering of fan tonal noise by noise-reducing chevrons on the bypass duct, considering a realistic geometry and flow-field

Parallel Anisotropic Block-based Adaptive Mesh Refinement Algorithm For Three-dimensional Flows

A three-dimensional, parallel, anisotropic, block-based, adaptive mesh refinement (AMR) algorithm is proposed and described for the solution of fluid flows on body-fitted, multi-block, hexahedral meshes. Refinement and de-refinement in any grid block computational direction, or combination of directions, allows the mesh to rapidly adapt to anisotropic flow features such as shocks, boundary layers, or flame fronts, common to complex flow physics. Anisotropic refinements and an efficient and highly scalable parallel implementation lead to a potential for significant reduction in computational cost as compared to a more typical isotropic approach. Unstructured root-block topology allows for greater flexibility in the treatment of complex geometries. The AMR algorithm is coupled with an upwind finite-volume scheme for the solution of the Euler equations governing inviscid, compressible, gaseous flow. Steady-state and time varying, three-dimensional, flow problems are investigated for various geometries, including the cubed-sphere mesh.MAS

Interface Source Terms for High-Order Aeroacoustics

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