亚网格尺度稳定化有限元求解不可压黏性流动

摘要

Started with the concept and fundamental of Subgrid Scale stabilized （SGS） method, an improved SSS finite element formulation is proposed to obtain time independent solutions by solving the unsteady incompressible Navier-Stokes equations. Based on some rational approximation and simplification, a new stabilization parameter which has relation with the selective time step size is deduced. As a result, the numerical instability resulting from high Reynolds and/or small time step size is eliminated. Combined with standard k-ε turbulence model and wall function method to estimate turbulent viscosity, the improved SGS method is applied to solve incompressible turbulent flows. The finite element implementation of wall function along with solution of turbulence transport equations is discussed in detail and the positivity preserving limiter for turbulence variables is introduced. A segregated algorithm for time stepping to solve incompressible turbulent flows is developed. The numerical examples of laminar and turbulent flows over a backward facing step are presented in order to show the possibility, stability and high accuracy of the proposed approach for simulating steady incompressible viscous flows.%从亚网格尺度稳定化方法的基本原理出发，提出了适合时间推进求解非定常Navier—Stokes方程获得定常解的SGS稳定化方法．基于一定程度的近似和简化，获得了与时间步长相关的稳定化参数，从而排除了传统SGS稳定化方法在求解高Re数、小时间步长问题时所引发的数值不稳定性．把SGS稳定化方法应用于求解不可压湍流，结合标准k-ε湍流模型和壁面函数法估计湍流黏性系数，详细讨论了壁面函数法的实施、湍流输运方程的求解和保证湍流变量非负性的限制策略，发展了时间推进求解不可压湍流的分离式算法．二维外掠后台阶层流和湍流计算结果表明，该方法求解不可压黏性流动是可行的，并且具有稳定性好、计算精度高的特点．