Zhang Yong-Tao, Shi Jing, Shu Chi-Wang, Zhou Ye
Department of Mathematics, University of California, Irvine, California 92697, USA.
Phys Rev E Stat Nonlin Soft Matter Phys. 2003 Oct;68(4 Pt 2):046709. doi: 10.1103/PhysRevE.68.046709. Epub 2003 Oct 23.
A quantitative study is carried out in this paper to investigate the size of numerical viscosities and the resolution power of high-order weighted essentially nonoscillatory (WENO) schemes for solving one- and two-dimensional Navier-Stokes equations for compressible gas dynamics with high Reynolds numbers. A one-dimensional shock tube problem, a one-dimensional example with parameters motivated by supernova and laser experiments, and a two-dimensional Rayleigh-Taylor instability problem are used as numerical test problems. For the two-dimensional Rayleigh-Taylor instability problem, or similar problems with small-scale structures, the details of the small structures are determined by the physical viscosity (therefore, the Reynolds number) in the Navier-Stokes equations. Thus, to obtain faithful resolution to these small-scale structures, the numerical viscosity inherent in the scheme must be small enough so that the physical viscosity dominates. A careful mesh refinement study is performed to capture the threshold mesh for full resolution, for specific Reynolds numbers, when WENO schemes of different orders of accuracy are used. It is demonstrated that high-order WENO schemes are more CPU time efficient to reach the same resolution, both for the one-dimensional and two-dimensional test problems.
本文进行了一项定量研究，以研究数值粘性的大小以及高阶加权基本无振荡（WENO）格式求解高雷诺数可压缩气体动力学一维和二维纳维 - 斯托克斯方程的分辨率。一维激波管问题、一个由超新星和激光实验参数驱动的一维示例以及二维瑞利 - 泰勒不稳定性问题被用作数值测试问题。对于二维瑞利 - 泰勒不稳定性问题或具有小尺度结构的类似问题，小尺度结构的细节由纳维 - 斯托克斯方程中的物理粘性（因此由雷诺数）决定。因此，为了获得对这些小尺度结构的忠实分辨率，格式中固有的数值粘性必须足够小，以便物理粘性起主导作用。针对不同精度阶数的WENO格式，进行了仔细的网格细化研究，以捕获特定雷诺数下实现全分辨率的阈值网格。结果表明，对于一维和二维测试问题，高阶WENO格式在达到相同分辨率时具有更高的CPU时间效率。
