Bajaj Kapil M S, Ahlers Guenter, Pesch Werner
Department of Physics and Quantum Institute, University of California, Santa Barbara, California 93106, USA.
Phys Rev E Stat Nonlin Soft Matter Phys. 2002 May;65(5 Pt 2):056309. doi: 10.1103/PhysRevE.65.056309. Epub 2002 May 16.
We present experimental and theoretical results near the onset of the Rayleigh-Bénard convection with rotation about a vertical axis in a fluid with a Prandtl number sigma close to 0.18. In the experiment we used a H2-Xe gas mixture with a separation ratio Psi=0.22 and a Lewis number L=1.22 at various pressures and dimensionless rotation rates Omega up to 400. On the basis of a standard weakly nonlinear stability analysis, we found a supercritical, stationary bifurcation for Omega < than approximately 13, which became subcritical over the range 13 < than approximately Omega < or approximately 160. For Omega > than approximately 160 a supercritical Hopf bifurcation precedes the stationary instability of the uniform state. Following the unstable straight-roll fixed point in the subcritical regime by Galerkin methods we determined the location of the saddlenode and the stability of the nonlinear two-dimensional straight-roll state. The rolls were found to be unstable to three-dimensional Küppers-Lortz perturbations for 3.8 < than approximately Omega < than approximately 160. Theoretical results for a pure fluid with the same sigma were qualitatively similar. Measurements using shadowgraph flow visualization yielded a bifurcation line and an Omega range of subcriticality, which agreed with the stability analysis. In the subcritical range the experiment revealed a discontinuity of the pattern amplitude at onset, but was unable to find any hysteresis. Patterns at onset fluctuated irregularly between the ground state and the finite-amplitude state. In this parameter range the convection pattern further above onset was chaotically time dependent. Investigation of the Hopf bifurcation line was difficult because of a wall mode that, for large Omega, preceded the bulk instability. For Omega approximately equal to 400, patterns were found in the sample interior only when the expected Hopf bifurcation was exceeded by about 10%. This is consistent with the convective nature of the bifurcation. However, the observed structure, although time periodic, was spatially disordered and had a frequency that was considerably larger than the expected Hopf frequency. In a separate sample cell with a radial ramp in the spacing no structure was observed at all in the cell interior until the expected stationary instability was reached.
我们给出了在普朗特数σ接近0.18的流体中，围绕垂直轴旋转的瑞利-贝纳德对流起始附近的实验和理论结果。在实验中，我们使用了分离比Ψ = 0.22且刘易斯数L = 1.22的H₂-Xe气体混合物，在不同压力和高达400的无量纲旋转速率Ω下进行实验。基于标准的弱非线性稳定性分析，我们发现对于Ω小于约13时存在超临界的静态分岔，在13小于约Ω小于或约160的范围内变为亚临界。对于Ω大于约160，超临界霍普夫分岔先于均匀态的静态不稳定性出现。通过伽辽金方法追踪亚临界区域中不稳定的直滚不动点，我们确定了鞍结的位置以及非线性二维直滚态的稳定性。发现对于3.8小于约Ω小于约160，这些滚动对三维库珀斯-洛茨扰动是不稳定的。具有相同σ的纯流体的理论结果在定性上相似。使用阴影图流动可视化的测量得到了一条分岔线和亚临界的Ω范围，这与稳定性分析一致。在亚临界范围内，实验揭示了起始时图案振幅的不连续性，但未发现任何滞后现象。起始时的图案在基态和有限振幅态之间不规则地波动。在此参数范围内，起始之上的对流图案在时间上是混沌依赖的。由于壁模式的存在，对霍普夫分岔线的研究很困难，对于大的Ω，壁模式先于整体不稳定性出现。对于Ω约等于400，仅当预期的霍普夫分岔超过约10%时，才在样品内部发现图案。这与分岔的对流性质一致。然而，观察到的结构虽然在时间上是周期性的，但在空间上是无序的，并且其频率远大于预期的霍普夫频率。在一个单独的样品池中，间距有径向斜坡，直到达到预期的静态不稳定性之前，在池内部根本没有观察到任何结构。
