长方腔自然对流第一次分岔突变现象的数值分析

在对网格密度作用进行详细分析的基础上，采用二阶全展开 Euler-Taylor-Galerkin分裂步有限元方法，对封闭水平矩形腔体内流体 (Pr=0.71)自然对流的第一次分岔过程进行了数值预报. 计算结果表明，第一次分岔相应的流动拓扑及临界Rayleigh ($Ra$)数随矩形腔体长宽比(W/B)取值的不同时会发生较大变化. 在所计算的长宽比取值范围内，封闭矩形腔内，流体自然对流第一次分岔拓扑的变化对应两种大的类型: 在较小的长宽比取值范围内(W/B\le 2.5)，临界Ra数两侧，流动从单一涡核的定常流动突变成为具有不对称结构的定常双涡核运动, 在此范围内临界Ra数的取值随W/B取值的增加而减小；当对应长宽比取值2.6 \le W/B \le 4.0时，第一次分岔拓扑结构的变化呈现出更加复杂的特性，临界Ra数两侧流动从定常双涡核突变为定常非对称的三涡核流动，相应的临界Ra 数也随W/B的增加而减少. 而在区间2.5,2.6]两端，临界Ra数的取值发生一次阶跃式突增，将该长宽比取值的区间定义为长方腔内该流体第一次分岔的突变区间. 

NUMERICAL RESEARCH ON THE SUDDEN CHANGE CHARACTERISTIC OF THE FIRST BIFURCATION FOR NATURAL CONVECTION OF AIR ENCLOSED IN 2D RECTANGULAR CAVITY

Based on grid independence analysis, a second order Euler-Taylor-Galerkin finite element method of fractional steps was used to numerically investigate the first bifurcation of natural convection of air enclosed in a 2D rectangular cavity. The characteristics of the first bifurcation of natural convection in 2D cavities were numerically studied with different height-to-width ratios. The corresponding critical Rayleigh number for each case was estimated using the flow topologies varied with Ra and L/B, and the bisection method. It can be concluded that the first bifurcation depends on the values of Ra and L/B. Flow topologies and the first bifurcation experienced a sudden change as L/B varied between 2.5 (from 1 core to 2 cores) and 2.6 (from 2 cores to 3 cores). For each interval of L/B adjacent to the interval of sudden change, the critiacl Ra decreased with the increase in L/B. Furthermore, there is a step increase for Ra_{Cr} for the sudden change interval. It can then be concluded that natural convection of air enclosed in a rectangular cavity experiences local instability more easily with higher value of L/B. According to the given results, it can also be deduced that the variation of the characteristic of the first bifurcation should be more complex with higher L/B.