基于Navier-Stokes方程残差的隐式大涡模拟有限元模型

对应于湍流的大尺度与小尺度流场信息, 本文在有限元的框架下, 假设Navier-Stokes方程的解的形函数由大尺度和不可解尺度形函数叠加组成, 引入对应的权函数, 将Navier-Stokes方程的有限元变分形式分解为大尺度和不可解尺度系统. 根据不可解尺度系统, 构建基于Navier-Stokes大尺度方程残差的不可解尺度模型, 将其代入Navier-Stokes方程的大尺度系统, 进而数值求解大尺度系统得到Navier-Stokes方程的大尺度解. 该方法无需像传统的大涡模拟方法那样对方程的解进行过滤, 通过对形函数进行尺度分解实现解的尺度分解. 本文使用该方法的自编程序代码开展了槽道湍流的数值模拟. 通过与有限差分大涡模拟、DNS计算结果的比较, 发现在使用较少网格情况下该方法预测的平均流向速度在近壁区与DNS数据吻合, 在黏性外层略偏高; 该方法对雷诺应力预测偏低导致从流向向垂向方向上湍动能输运略偏低. 流向速度等值面图显示该方法有效捕捉到了大尺度旋涡结构; 同时在近壁区可以观察到明显的低速条带结构. 

A RESIDUAL-BASED UNRESOLVED-SCALE FINITE ELEMENT MODELLING FOR IMPLICT LARGE EDDY SIMULATION

In consistence with large and small scales in turbulent flows, shape function space can be divided into resolved and unresolved scale spaces in a frame of finite element method. Introducing the same decomposition of the weighting function space, the variational formulations of Navier-Stokes equations can be divided into two systems of equations: resolved- and unresolved-scale equations. Generally, only the resolved-scale equation is computed, and the unresolved scales are modeled. Based on the unresolved-scale equations, an approximate residual-based unresolved-scale modeling is proposed in the present study. The large-scale equations are then computed by substituting the unresolved-scale modeling. The method is called residual-based large eddy simulation, in which unlike in the classical LES a filtering for Navier-Stokes equations is needed, multiscale decomposition is instead used. Numerical simulations of a turbulent channel flow are implemented with in-house codes of the residual-based large eddy simulation. The results show that, with a low number of elements, the mean streamwise velocity obtained using the present method is in agreement with the DNS data in the inner layer, and it is slightly overpredicted in the outer layer. Underprediction of the Reynolds stress by the present method causes a reduction of turbulence intensity transportation from the streamwise direction to the normal direction. Isosurfaces of the streamwise velocity reveals its capability of capturing the large-eddy structures. Meanwhile, low-speed streaks can be clearly observed in the sublayer near the wall.