笛卡尔坐标系下不可压N-S方程在方形通道中的奇点及其影响区域分析

Analysis of Singular Points and Their Influence Regions of Navier-stokes Equation Describing Incompressible Flow in Square Duct under Cartesian Coordinates

在笛卡尔坐标系下,用来描述方形通道中不可压缩流动的N-S方程存在奇点.以方形通道中充分发展层流为研究对象,采用数值方法对动量方程的守恒性和奇点的影响区域进行了分析.数值结果表明:假定沿主流方向的压力梯度为常数时,求解简化的N-S方程而获得的主流方向的速度分布在横截面上的4个角点的附近区域内不满足动量方程;而采用连续性方程构造的压力修正方程的三维数值方法可以获得主流方向的压力梯度恒为常数,但速度分布在横截面上的4个角点的附近区域内也不满足动量方程.得出奇点的影响区域主要集中在角点附近区域,在该区域的壁面上其影响较为显著.为了获得物理意义上的解,在数值计算中需要对奇点作一些特殊处理.

Under Cartesian coordinates, the Navier-Stokes equation describing incompressible flow in a square duct exists singular points. Considering the fully developed laminar flow in the square duct, the momentum equation conservation and the influence regions of the singular points have been analyzed numerically in this paper. The numerical results reveal that when pressure gradient in mean flow direction is assumed constant, the velocity distribution obtained by the simplified Navier-Stokes equation can＇t satisfy the momentum equation at the four corners of the square duct and in their some certain neighboring regions on cross section. On the other hand, by using the three-dimensional numerical method of pressure-correction equation derived from the continuity equation, the constant pressure gradient in main flow direction can be obtained,but velocity distribution also can＇t satisfy the momentum equation at the four corners of the square duct and in their some certain neighboring regions on cross section. Furthermore, the influence regions of the singular points are mainly in the certain neighboring regions of the four corners,and their influence is fairly remarkable on the wall surface in these regions. Thus, to obtain the reasonable solution in physics, it is absolutely necessary to adopt special treatment on these singular points in the computational process.