Transition to disordered front propagation

自19世纪末雷诺的开创性实验以来，压力驱动管流向湍流的转变一直是流体力学中最大的悬而未决的问题。尽管柱状管道中的泊松埃流动对于所有雷诺数(惯性力与粘性力的比率)都是线性稳定的，但对于Re&gt；2000，如果扰动超过有限幅度阈值，则可能会以沿管道平流的局部喷雾的形式出现湍流。在过去的二十年里，通过关注管流的非线性动力学，人们在理解管流以及更一般的剪切流中的湍流转变方面取得了重大进展。这个提议背后的中心问题是，剪切流的复杂过渡场景是否会出现在其他流体力学系统中。我们关注的是正则流，Saffman-Taylor指进受限通道-由窄缝分隔的平行板，从而使宽深长宽比非常大-这是锋面传播和图案形成的原型。在恒定体积流量(或压头)下，粘度较高的流体(油)被粘度较低的流体(空气)驱替，产生的模式从单个空气手指的稳定传播到非稳定的锋面传播--通过重复的尖端分裂事件和手指竞争而产生的高度分支的模式。这种转变与剪切流转变有着惊人的相似之处，因为(A)深度平均模型的单传播指解已知在很大的驱动参数范围内是线性稳定的，以及(B)实验发现转变的驱动参数的阈值对系统中的扰动水平非常敏感。在剪切流湍流中，一个关键的理论概念是将局域湍流烟团解释为边缘状态-具有稳定流形的弱不稳定状态，这决定了盆地边界将初始条件从增长到湍流衰减到层流。我们要研究的基本假设是，Saffman-Taylor流的不稳定解是边缘状态，它既是从稳定传播的Saffman-Taylor指向实验观察到的复杂图案转变的基础，也是图案本身的动态。这一假设源于初步的实验观测和对深度平均模型的随时间变化的数值模拟，该模型表明，当从REST施加较大的参数值时，通过对Saffman-Taylor问题的弱不稳定解的瞬时探索，气泡是不稳定的。剪切流转变也表现出可激发的动力学，在阈值以下，由局部扰动激发的湍流喷雾是层流的瞬时漂移，最终在长时间尺度上衰减。在阈值之外，出现了一个湍流固定点，使得局部的湍流斑块得以生长。我们将研究Saffman-Taylor问题中的跃迁背后是否存在可激发的动力学。我们将应用一系列已知幅度的局部或空间分布的地形扰动，以探索界面的动力学响应，并建立作为扰动和驱动参数的函数的转变阈值。最后，剪切流转变的一个尚未得到证实的假设是，湍流可以由不稳定解之间的混沌曲折来表征。萨夫曼-泰勒指法问题表现出一个简单得多的空间结构，部分原因是非线性只在界面条件下发生。因此，我们将尝试刻画无序前锋传播的特征，并为萨夫曼-泰勒相变情景评估上述假设。

The transition to turbulence in pressure-driven pipe flow has remained the greatest unsolved problem in fluid mechanics since Reynolds' pioneering experiments in the late nineteenth century. Although Poiseuille flow in a cylindrical pipe is linearly stable for all values of the Reynolds number - the ratio of inertial to viscous forces - turbulence can appear for Re > 2000 in the form of localised puffs advected down the pipe, if perturbations exceed a finite-amplitude threshold. In the last twenty years, significant progress in the understanding of the transition to turbulence in pipe flow, and more generally shear flows, has been achieved by focusing on the nonlinear dynamics of these flows. The central question underlying this proposal is whether the complex transition scenario uncovered for shear flows may arise in other fluid mechanical systems.We focus on a canonical flow, Saffman-Taylor fingering in a confined channel - parallel plates separated by a narrow gap so that the width to depth aspect ratio is very large - which is an archetype for front propagation and pattern formation. The displacement of a more viscous fluid (oil) by a less viscous fluid (air) under constant volume-flux (or pressure head) yields patterns ranging from to the steady propagation of a single air finger to unsteady front propagation - highly branched patterns which arise through repeated tip-splitting events and finger competition. This transition exhibits striking similarities with shear flow transition in that (a) the single propagating finger solution of a depth-averaged model is known to be linearly stable up to very large values of the driving parameter, and (b) the threshold value of the driving parameter for transition was found experimentally to be very sensitive to the level of perturbations in the system.In shear flow turbulence, a key theoretical concept is the interpretation of localised turbulent puffs as edge states - weakly unstable states with a stable manifold that determines the basin boundary separating initial conditions decaying to laminar flow from those growing to turbulence. The fundamental hypothesis to be investigated in the proposed research is that unstable solutions of the Saffman-Taylor flow are edge states that underlie both the transition from the steadily propagating Saffman-Taylor finger to the experimentally observed complex patterns, and the dynamics of the patterns themselves. This hypothesis stems from preliminary experimental observations and time-dependent numerical simulations of a depth averaged model, which indicates destabilisation of a bubble through the transient exploration of weakly unstable solutions of the Saffman-Taylor problem, when a large value of parameter is applied from rest.The shear flow transition also exhibits excitable dynamics, in that below threshold a turbulent puff excited by a localised perturbation is a transient excursion from laminar flow, which eventually decays on long time scales. Beyond threshold, a turbulent fixed point appears that enables localised patches of turbulence to grow. We will investigate whether excitable dynamics underlie transition in the Saffman-Taylor problem. We will apply a range of localised or spatially-distributed topographical perturbations of known amplitude in order to probe the dynamical response of the interface and establish the transition threshold as a function of perturbation and driving parameter.Finally, a yet unproven hypothesis of shear flow transition is that turbulence can be characterised by a chaotic meandering between unstable solutions. The Saffman-Taylor fingering problem exhibits a much simpler spatial structure partly because nonlinearities only occur within interfacial conditions. Hence, we will attempt to to characterise disordered front propagation and assess the above hypothesis for the Saffman-Taylor transition scenario.