相邻双侧边盖驱动方腔流动的三维线性整体稳定性

Linear Three Dimensional Stability of Two-Sided Non-Facing Lid Driven Cavity Flows

文章考察了相邻双侧边盖驱动方腔流动(即上壁面向右运动和左侧壁面向下运动)的三维线性整体稳定性.首先,采用Taylor-Hood有限元方法并经由Newton迭代过程计算得到双侧边盖驱动方腔流动的二维稳态基本流.其次, Taylor-Hood有限元在Chebyshev Gauss配置点上进行离散,同时Gauss配置点也可以用于线性稳定性方程的高阶有限差分格式离散.然后,离散得到的矩阵形式的广义特征值问题可以结合shift-and-invert算法采用隐式重启Arnoldi方法计算.最后,通过对线性稳定性方程特征值的计算,发现了一个最不稳定的驻定模态和两对对称行波模态.最不稳定的三维驻定模态的临界Reynolds数为Re_c=261.5,远远小于二维不稳定的临界Reynolds数Re_c^(2d)=1 061.7.通过画出这3类三维不稳定模态的流向扰动速度和扰动涡量的空间等值面图像,可以发现不稳定扰动位于稳态基本流的两个主涡区域,因此可以认为主涡区域是三维扰动失稳的主要能量来源地.

The two-sided non-facing lid driven cavity flow for which the upper wall is moved to the right and the left wall to the bottom with equal speeds is investigated numerically for its linear three dimensional stability. The two-dimensional basic steady-state is first obtained by the Taylor-Hood finite element method through Newton iteration process. The triangulation of finite element mesh is based on a transformed Chebyshev Gauss-Lobatto collocation nodes, which is also used for the spatial diseretization of the linear stability equations with a high-order finite-difference scheme. The resulting generalized eigenvalue problem in a matrix form is then solved by the implicitly restarled Arnoldi method with the shift-and-invert algorithm. Through the eigenvalue computation of linear stability equations, the most unstable stationalT mode for long wave instability and two pairs of symmetrical travelling modes for short wave instability are found. The critical Reynolds number of the most unstable stationalT mode occurs at Reynolds number Ree = 261.5 which is far smaller than that of the lwo-dimensional insta- bilily Ree2d = 1 061.7. Through the plotting of spatial iso-surface of streamwise disturbance velocity and streamwise disturbance vorfieily for all three types of eigenmodes, it is also found that the unstable disturbances are located within the region of the two large primary vortices of the basic steady-state, which are considered as the disturbance energy source for the onset of instability.