封闭水平矩形腔内流体自然对流第一次逆叉形分岔的数值研究

Numerical studies of the first inverse pitchfork bifurcations for natural convections enclosed in a 2D horizontal rectangular cavity

本文采用二阶全展开ETG(Euler-Taylor-Galerkin)分裂步有限元方法,对长宽比为3.5(L/B=3.5,如图1所示)的封闭矩形腔体内,三种Pr数条件下,定常层流范围内,流体自然对流叉形分岔随Rayleigh数的演化过程进行了数值模拟。研究结果表明,该矩形腔内对应三种Pr数条件下,流体的叉形分岔的演化过程中,在第二次模态Ⅱ型叉形分岔之后,均会出现两个较小尺度涡旋合并,突变为一个较大尺度涡旋的全新叉形分岔模态。即在某临界Ra数两侧,存在定常四涡结构和定常三涡结构两个定常解支,当系统控制参数Ra越过临界值,前者被后者突发性取代,这是完全不同于传统叉形分岔的逆叉形分岔。其数值预报,则采用分半法结合流动拓扑结构及典型截面处速度扩线上鞍点的变化来确定。计算结果表明,在计算的Pr数条件下,随Pr数的增加逆叉形分岔对应临界Ra数的取值也会提高。

A second order Euler-Taylor-Galerkin finite element method of fractional steps was used in the numerical study of the evolution processes of bifurcations for natural convections of water at three different Pr enclosed in a rectangular cavity with aspect ratio L/B=3.5（plotted in Fig.1）.A new phenomenon of vortex merging in laminar flow has been found for all the three Pr.The vortex merging phenomenon discovered in the present paper is a new mode pitchfork bifurcation,the inverse pitchfork bifurcation.Moreover,aided by the variation of flow topologies and velocity profiles of velocity v vs.x at y=0.5 for each cavity,corresponding critical Rayleigh numbers were numerical predicted by using the bisection method.It can be deduced f rom the presented results that the critical Ra increased with the increase in Pr.