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Bifurcation and stability of forced convection in curved ducts of square cross-section

机译：方形截面弯管中强制对流的分叉与稳定性

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摘要

A numerical study is made on the fully developed bifurcation structure and stability of the forced convection in a curved duct of square cross-section (Dean problem). In addition to the extension of three known solution branches to the high Dean number region, three new asymmetric solution branches are found from three symmetry-breaking bifurcation points on the isolated symmetric branch. The flows on these new branches are either an asymmetric two-cell state or an asymmetric seven-cell structure. The linear stability of multiple solutions are conclusively determined by solving the eigenvalue system for all eigenvalues. Only two-cell flows on the primary symmetric branch and on the part of isolated symmetric branch are linearly stable. The symmetric six-cell flow is also linearly unstable to asymmetric disturbances although it was ascertained to be stable to symmetric disturbances in the literature. The linear stability is observed to change along some solution branch even without passing any bifurcation or limit points. Furthermore, dynamic responses of the multiple solutions to finite random disturbances are also examined by the direct transient computation. It is found that possible physically realizable fully developed flows evolve, as the Dean number increases, from a stable steady two-cell state at lower Dean number to a temporal periodic oscillation state, another stable steady two-cell state, a temporal intermittent oscillation, and a chaotic temporal oscillation. Among them, three temporal oscillation states have not been reported in the literature. A temporal periodic oscillation between symmetric/asymmetric two-cell flows and symmetric/asymmetric four-cell flows are found in the range where there are no stable steady fully developed solutions. The symmetry-breaking point on the primary solution branch is determined to be a sub-critical Hopf point by the transient computation.
机译：对充分发展的分叉结构和方形截面弯曲管道中强制对流的稳定性进行了数值研究（Dean问题）。除了将三个已知解分支扩展到高Dean数区域外，还从隔离的对称分支上的三个对称破坏分叉点中发现了三个新的不对称解分支。这些新分支上的流要么是不对称的两单元状态，要么是不对称的七单元结构。通过求解所有特征值的特征值系统，可以最终确定多个解决方案的线性稳定性。在主要对称分支和孤立的对称分支的一部分上只有两单元流是线性稳定的。尽管在文献中已确定对称六单元流动对对称扰动稳定，但它对非对称扰动也线性不稳定。即使没有通过任何分叉点或极限点，也观察到线性稳定性会沿着某些溶液分支变化。此外，还通过直接瞬态计算检查了多种解决方案对有限随机扰动的动态响应。发现随着Dean数的增加，可能的物理上可实现的充分发展的流量从较低Dean数的稳定稳态两单元状态发展到时间周期性振荡状态，另一稳定稳态两单元状态即时间间歇振荡，和混乱的时间振荡。其中，文献中尚未报道三种时间振荡状态。在没有稳定稳定的充分发展解决方案的范围内，发现对称/非对称两单元流和对称/非对称四单元流之间的时间周期振荡。通过瞬态计算将主解分支上的对称破坏点确定为次临界Hopf点。

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