有理弱非线性理论：具有连续谱的Taylor-Couette不稳定性
A Rational Weakly Non-Linear Theory: Taylor-Couette Instability With a Continuous Spectrum
摘要：
本文研究了圆形Couette流第一分叉点附近不稳定波的连续谱的非线性演化。扰动由所有可能的轴向波数的傅里叶积分表示，并导出了连续谱的振幅密度函数的积分微分方程。描述经典弱非线性不稳定性理论的单色波和慢变波包演化的方程是特殊的极限情况。积分微分方程的数值积分表明，最终的平衡态取决于初始扰动，如实验所观察到的，它不是唯一的。稳定的超临界泰勒涡的波数的预测范围被发现是窄于从线性理论的中性曲线的跨度。泰勒涡流与波数以外的这个范围被发现是不稳定的，衰减，但激发另一个波内的窄带。这个结果与Eckhaus和Benjamin-Feir边带不稳定性一致。自1965年Coles的重大贡献以来，人们就知道在给定几何形状的Taylor-Couette流动中，在固定雷诺数下存在多个解。值得注意的是，Coles发现的多重解的存在性不同于当前流行的分歧理论。除Taylor-Couette流动外，混合对流也有Coles意义下的多解（Yao & Ghosh Moulic 1993，1994）。我们认为，Coles意义的非唯一性，它补充了分歧理论，是所有流体流动的一般属性。
英文摘要：
Nonlinear evolution of a continuous spectrum of unstable waves near the first bifurcation point in circular Couette flow has been investigated. The disturbance is represented by a Fourier integral over all possible axial wavenumbers, and an integrodifferential equation for the amplitude-density function of a continuous spectrum is derived. The equations describing the evolution of monochromatic waves and slowly-varying wave-packets of classical weakly nonlinear instability theories are shown to be special limiting cases. Numerical integration of the integrodifferential equation shows that the final equilibrium state depends on the initial disturbance, as observed experimentally, and it is not unique. The predicted range of wavenumbers for stable supercritical Taylor vortices is found to be narrower than the span of the neutral curve from linear theory. Taylor-vortex flows with wavenumbers outside this range are found to be unstable and to decay, but to excite another wave inside the narrow band. This result is in agreement with the Eckhaus and Benjamin-Feir sideband instability. The presence of multiple solutions at a fixed Reynolds number for a given geometry in Taylor-Couette flows has been known since Coles monumental contribution in 1965. It is worthwhile to note that the existence of multiple solutions, found by Coles, differs from current popular bifurcation theories. The multiple solutions in Coles sense have also been found for mixed-convection flows (Yao & Ghosh Moulic 1993, 1994) besides the Taylor-Couette flows. We believe that the nonuniqueness of Coles sense, which complements the bifurcation theories, is a generic property for all fluid flows.
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