有限差分格式的一个通用色散-耗散条件

Dispersion-Dissipation Condition for Finite Difference Schemes

通过分析显式有限差分格式的数值色散和数值耗散,导出一个适于有限差分格式的通用色散-耗散条件.根据群速度和耗散率之间的物理关系,确定了用以抑制数值解中伪高波数波所需要的适度耗散.在以往发展的低耗散加权基本无振荡格式WENO-CU6-M2上的应用表明,该条件可用作优化线性或非线性有限差分格式的色散和耗散的通用指导准则.此外,满足色散-耗散条件的改进WENO-CU6-M2格式还可选作低分辨率数值模拟,以三维Taylor-Green涡向湍流转捩和自相似能量衰减问题展现了它的这种能力.与经典的动态Smagorinsky亚网格尺度模型相比,在Reynolds数Re=400～3000条件下,无黏和黏性Taylor-Green涡的数值模拟结果均得到明显改善.在保持激波捕捉特性同时,与最新的隐式大涡模拟模型的计算效果相当.

A general dispersion-dissipation condition for finite difference schemes is derived by analyzing the numerical dispersion and dissipation of explicit finite difference schemes. The proper dissipation required to damp spurious high wavenumber waves in the solution is detennined from a physically motivated relation between group velocily and dissipation rate. The application to a previously developed low-dissipation weighted essentially non-oscillatory scheme （WENO-CU6-M2） demonstrates that this condition can serve as a general guideline for optimizing the dispersion and dissipation of linear and non-linear finite difference schemes. Moreover, the improved WENO-CU6-M2 scheme which satisfies the dispersion-dissipation condition can be used for under-resolved simulations. This capabilily is demonstrated by considering transition to turbulence and self-similar energT decay of the three-dimensional Taylor-Green vortex. Simulations of the inviscid and the viscous Taylor-Green vortex at Reynolds numbers ranging from Re =400 to Re = 3000 show a significant improvement over the classical dynamic Smagorinsky model and demonstrate competitiveness with state-of-the-art implicit LES models, while preserving shock-capturing properties.