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New Multi-Layer Compact High-Order Finite Difference Methods with Spectral-Like Resolution for Compressible Flow Simulations
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| 자료유형 | 학위논문 |
|---|---|
| 서명/저자사항 | New Multi-Layer Compact High-Order Finite Difference Methods with Spectral-Like Resolution for Compressible Flow Simulations. |
| 개인저자 | Bai, Zeyu. |
| 단체저자명 | University of California, Los Angeles. Mechanical Engineering. |
| 발행사항 | [S.l.]: University of California, Los Angeles., 2019. |
| 발행사항 | Ann Arbor: ProQuest Dissertations & Theses, 2019. |
| 형태사항 | 289 p. |
| 기본자료 저록 | Dissertations Abstracts International 80-12B. Dissertation Abstract International |
| ISBN | 9781392235119 |
| 학위논문주기 | Thesis (Ph.D.)--University of California, Los Angeles, 2019. |
| 일반주기 |
Source: Dissertations Abstracts International, Volume: 80-12, Section: B.
Publisher info.: Dissertation/Thesis. Advisor: Zhong, Xiaolin. |
| 이용제한사항 | This item must not be sold to any third party vendors. |
| 요약 | Numerical simulations of multi-scale flow problems such as hypersonic boundary layer transition, turbulent flows, computational aeroacoustics and other flow problems with complex physics require high-order methods with high spectral resolutions. For instance, the receptivity mechanisms in the hypersonic boundary layer are the resonant interactions between forcing waves and boundary-layer waves, and the complex wave interactions are difficult to be accurately predicted by conventional low-order numerical methods. High-order methods, which are robust and accurate in resolving a wide range of time and length scales, are required. The objective of this dissertation is to develop and analyze new very high-order numerical methods with spectral-like resolution for flow simulations on structured grids, with focus on smooth flow problems involving multiple scales. These numerical methods include: the multi-layer compact (MLC) scheme, the directional multi-layer compact (DMLC) scheme, and the least square multi-layer compact (LSMLC) scheme.In the first place, a new upwind multi-layer compact (MLC) scheme up to seventh order is derived in a finite difference framework. By using the 'multi-layer' idea, which introduces first derivatives into the MLC schemes and approximates the second derivatives, the resolution of the MLC schemes can be significantly improved within a compact grid stencil. The auxiliary equations are introduced, and they are the only nontrivial equations. The original equation requires no approximation which contributes to good computational efficiency. In addition, the upwind MLC schemes are derived on centered stencils with adjustable parameters to control the dissipation. Fourier analysis is performed to show that the new MLC schemes have very small dissipation and dispersion in a wide range of wavenumbers in both one and two-dimensional cases, and the anisotropic error is much smaller than conventional finite difference methods in the two-dimensional case. Comparison with discontinuous-Galerkin methods is performed with Fourier analysis as well. Furthermore, stability analysis with matrix method shows that high-order boundary closure schemes are stable because of compactness of the stencils. The accuracies and rates of convergence of the new schemes are validated by numerical experiments of the linear advection equation, the nonlinear Euler equations, and the Navier-Stokes equations in both one and two-dimensional settings. The numerical results show that good computational efficiency, very high-order accuracies, and high spectral resolutions especially on coarse meshes can be attained with the MLC scheme. On the other hand, even though the MLC scheme is promising in most test cases, it shows weak numerical instabilities for a small range of wavenumbers when it is applied to multi-dimensional flows, which are mainly triggered by the inconsistency between its one and two-dimensional formulations. The instability could lead to divergence in long-time multi-dimensional simulations. Moreover, the cross-derivative approximation in the MLC scheme requires an ad-hoc selection of supporting grid points, and the cross-derivative approximation is relatively inefficient for very high-order cases. To address the remaining challenges of the MLC scheme and achieve better performance for multi-dimensional flow simulations, another two new schemes are developed - the directional multi-layer compact (DMLC) scheme, and the least square multi-layer compact (LSMLC) scheme. In the second place, a new upwind directional multi-layer compact (DMLC) scheme is developed for multi-dimensional simulations. The main idea of the DMLC scheme is to introduce auxiliary equation for cross derivative in multi-dimensional cases. Consequently, the spatial discretization can be fulfilled along each dimension independently. With this directional discretization technique, the one-dimensional formulation of the MLC scheme can be applied to all spatial derivatives in a multi-dimensional governing equation. Therefore, the DMLC scheme overcomes the inconsistency between one and two-dimensional formulations of the MLC scheme, and it also avoids the ad-hoc cross-derivative approximations. Two-dimensional Fourier analysis demonstrates that all modes of the DMLC scheme are stable in the full range of wavenumbers, and it has better spectral resolution and smaller anisotropic error than the MLC scheme. Stability analysis with matrix method indicates that stable boundary closure schemes are much easier to be obtained in the DMLC scheme. Numerical tests in the linear advection equation and the nonlinear Euler equations validate that the DMLC scheme are more accurate and require less CPU time than the MLC scheme on the same mesh. In particular, the long-time simulation results reveal that the DMLC scheme is always stable for both periodic and non-periodic boundary conditions in two-dimensional cases. In the third place, a new upwind least square multi-layer compact (LSMLC) scheme is developed for multi-dimensional simulations. The main idea of the LSMLC scheme is using the weighted least square approximation to redesign the two-dimensional formulation for cross derivatives. It avoids the ad-hoc selection of grid points in the MLC scheme. Meanwhile, the two-dimensional upwind scheme can be derived by introducing upwind correction into the weight function. The upwind factor 棺 can adjust the dissipation and stability of the LSMLC scheme. Lagrange multiplier is used to ensure that the LSMLC scheme satisfies both the consistency constraint at the base point and the one-dimensional constraint from the MLC scheme. |
| 일반주제명 | Fluid mechanics. Computational physics. Applied Mathematics. |
| 언어 | 영어 |
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