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High order numerical methods for inviscid and viscous flows on unstructured grids.

机译：非结构网格上无粘性和粘性流的高阶数值方法。

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摘要

The spectral volume (SV) method is a newly developed high-order, conservative, and efficient finite volume method for hyperbolic conservation laws on unstructured grids. It has been successfully demonstrated for scalar conservation laws and multi-dimensional Euler equations. In this study, the SV method is compared with another high-order method for hyperbolic conservation laws capable of handling unstructured grids named the discontinuous Galerkin (DG) method. Their overall performance in terms of the efficiency, accuracy and memory requirement is evaluated using the scalar conservation laws and the two-dimensional Euler equations. To measure their accuracy, problems with analytical solutions are used. Both methods are also used to solve problems with strong discontinuities to test their ability in discontinuity capturing. Both the DG and SV methods are capable of achieving the formal order of accuracy while the DG method has a lower error magnitude and takes more memory. They are also similar in efficiency. The SV method appears to have a higher resolution for discontinuities because the data limiting can be done at the sub-element level.; The SV method is also successfully extend to the Navier-Stokes equations. First, the SV method is extended to and tested for the diffusion equation. In this study, three different formulations named Naive SV, Local SV and Penalty SV for the diffusion equation are presented. The Naive SV formulation yields an inconsistent and unstable scheme, while the other two formulations are consistent, convergent and stable. A Fourier type analysis is performed for all the formulations, and the analysis agrees well with the numerical results. Second, the Local SV method is chosen to be extended to solve the Navier-Stokes equations since it gives the optimum accuracy in solving the diffusion equation. The formulation of the Local SV method for the two-dimensional compressible Navier-Stokes equations is described. Accuracy studies are performed on the scalar convection-diffusion and the Navier-Stokes equations using problems with analytical solutions. It is shown that the designed order of accuracy is achieved for 1st, 2nd and 3rd order reconstructions. The solver is then used to solve other viscous laminar flow problems to demonstrate its capability.
机译：频谱体积（SV）方法是一种针对非结构网格上的双曲守恒律的最新开发的高阶，保守且有效的有限体积方法。它已成功地证明了标量守恒定律和多维Euler方程。在这项研究中，将SV方法与另一种能够处理非结构化网格的双曲守恒定律的高阶方法称为不连续Galerkin（DG）方法。使用标量守恒定律和二维Euler方程评估了它们在效率，准确性和存储要求方面的总体性能。为了测量其准确性，使用了分析解决方案中的问题。两种方法也都用于解决具有强不连续性的问题，以测试其在不连续性捕获中的能力。 DG和SV方法都能够达到形式上的精确度，而DG方法具有较低的误差幅度并占用更多内存。它们的效率也相似。 SV方法对于不连续性似乎具有更高的分辨率，因为可以在子元素级别完成数据限制。 SV方法也成功地扩展到Navier-Stokes方程。首先，将SV方法扩展到扩散方程并对其进行测试。在这项研究中，为扩散方程提出了三种不同的公式，分别是Naive SV，Local SV和Penalty SV。朴素的SV公式产生不一致和不稳定的方案，而其他两个公式是一致的，收敛的和稳定的。对所有配方进行傅立叶类型分析，该分析与数值结果非常吻合。其次，选择局部SV方法进行扩展以求解Navier-Stokes方程，因为它在求解扩散方程时提供了最佳精度。描述了用于二维可压缩Navier-Stokes方程的Local SV方法的公式。使用解析解的问题，对标量对流扩散和Navier-Stokes方程进行了精度研究。结果表明，针对一阶，二阶和三阶重构，可以达到设计的精度等级。然后将求解器用于解决其他粘性层流问题，以证明其功能。

著录项

  • 作者

    Sun, Yuzhi.;

  • 作者单位

    Michigan State University.;

  • 授予单位 Michigan State University.;
  • 学科 Engineering Mechanical.
  • 学位 Ph.D.
  • 年度 2004
  • 页码 123 p.
  • 总页数 123
  • 原文格式 PDF
  • 正文语种 eng
  • 中图分类 机械、仪表工业;
  • 关键词

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