Steady nonlinear waves in diverging channel flow
Steady nonlinear waves in diverging channel flow
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DOI：
10.1017/s0022112003007572
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发表时间：
2004-02
影响因子：
3.7
通讯作者：
R. Kerswell;O. Tutty;P. Drazin
中科院分区：
文献类型：
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作者：
R. Kerswell;O. Tutty;P. Drazin
顶点处有流体线源的无限发散通道是有限通道扩展中流动的自然理想状态。在相关几何（Tutty 1996）中获得的数值结果的激励下，我们在这个理论模型中表明，对于某些通道半角$\alpha$和雷诺数$\hbox{\it Re}\,{:=}\,Q/2\nu$ （$Q$是单位长度的体积通量，$\nu$是运动粘度），存在一个稳定的、空间周期性的二维波，它在空间上是稳定的，因此在物理系统中似乎是可以实现的。这个空间波（或极限环）是由亚临界干草叉臂上的异斜分叉产生的，它起源于众所周知的杰弗瑞-哈默尔分叉点$\alpha\,{=}\,\alpha_2(\hbox{\it Re})$。这些波在$5\,{\leq}\,\hbox{\it Re}\,{\leq}\,5000$范围内被发现，值得注意的是，这些波存在于半角$\alpha$，超过了jeffrey - hamel理论已经被证明是哑巴的$\alpha_2$点。然而，在有限$Re$处没有达到$\alpha\,{\rightarrow}\, 0$的极限，因此这些波与平面泊泽维尔流无关。
An infinitely diverging channel with a line source of fluid at its vertex is a natural idealization of flow in a finite channel expansion. Motivated by numerical results obtained in an associated geometry (Tutty 1996), we show in this theoretical model that for certain channel semi-angles $\alpha$ and Reynolds numbers $\hbox{\it Re}\,{:=}\,Q/2\nu$ ($Q$ is the volume flux per unit length and $\nu$ the kinematic viscosity) a steady, spatially periodic, two-dimensional wave exists which appears spatially stable and hence plausibly realizable in the physical system. This spatial wave (or limit cycle) is born out of a heteroclinic bifurcation across the subcritical pitchfork arms which originate out of the well known Jeffery-Hamel bifurcation point at $\alpha\,{=}\,\alpha_2(\hbox{\it Re})$. These waves have been found over the range $5\,{\leq}\,\hbox{\it Re}\,{\leq}\,5000$ and, significantly, exist for semi-angles $\alpha$ beyond the point $\alpha_2$ where Jeffery-Hamel theory has been shown to be mute. However, the limit of $\alpha\,{\rightarrow}\, 0$ at finite $Re$ is not reached and so these waves have no relevance to plane Poiseuille flow.