Institut du Non-Linéaire de Nice, Université de Nice Sophia Antipolis, UMR CNRS 7010, 1361 route des Lucioles, Valbonne 06560, France.
Phys Rev E. 2018 Feb;97(2-1):023109. doi: 10.1103/PhysRevE.97.023109.
This paper presents a numerical and theoretical study of multistability in two stochastic models of transitional wall flows. An algorithm dedicated to the computation of rare events is adapted on these two stochastic models. The main focus is placed on a stochastic partial differential equation model proposed by Barkley. Three types of events are computed in a systematic and reproducible manner: (i) the collapse of isolated puffs and domains initially containing their steady turbulent fraction; (ii) the puff splitting; (iii) the build-up of turbulence from the laminar base flow under a noise perturbation of vanishing variance. For build-up events, an extreme realization of the vanishing variance noise pushes the state from the laminar base flow to the most probable germ of turbulence which in turn develops into a full blown puff. For collapse events, the Reynolds number and length ranges of the two regimes of collapse of laminar-turbulent pipes, independent collapse or global collapse of puffs, is determined. The mean first passage time before each event is then systematically computed as a function of the Reynolds number r and pipe length L in the laminar-turbulent coexistence range of Reynolds number. In the case of isolated puffs, the faster-than-linear growth with Reynolds number of the logarithm of mean first passage time T before collapse is separated in two. One finds that ln(T)=A_{p}r-B_{p}, with A_{p} and B_{p} positive. Moreover, A_{p} and B_{p} are affine in the spatial integral of turbulence intensity of the puff, with the same slope. In the case of pipes initially containing the steady turbulent fraction, the length L and Reynolds number r dependence of the mean first passage time T before collapse is also separated. The author finds that T≍exp[L(Ar-B)] with A and B positive. The length and Reynolds number dependence of T are then discussed in view of the large deviations theoretical approaches of the study of mean first passage times and multistability, where ln(T) in the limit of small variance noise is studied. Two points of view, local noise of small variance and large length, can be used to discuss the exponential dependence in L of T. In particular, it is shown how a T≍exp[L(A^{'}R-B^{'})] can be derived in a conceptual two degrees of freedom model of a transitional wall flow proposed by Dauchot and Manneville. This is done by identifying a quasipotential in low variance noise, large length limit. This pinpoints the physical effects controlling collapse and build-up trajectories and corresponding passage times with an emphasis on the saddle points between laminar and turbulent states. This analytical analysis also shows that these effects lead to the asymmetric probability density function of kinetic energy of turbulence.
本文对两种过渡壁流的随机模型中的多稳定性进行了数值和理论研究。专门设计了一种用于计算稀有事件的算法来应用于这两种随机模型。主要关注点是 Barkley 提出的随机偏微分方程模型。系统地和可重复地计算了三种类型的事件：（i）最初包含其稳定湍流分数的孤立脉冲和域的崩溃；（ii）脉冲分裂；（iii）在消失方差的噪声扰动下，从层流基流中建立湍流。对于建立事件，消失方差噪声的极端实现将状态从层流基流推到最可能的湍流芽，而后者又发展成完全膨胀的脉冲。对于崩溃事件，确定了层流-湍流管的两种崩溃模式（独立崩溃或脉冲整体崩溃）的崩溃范围的雷诺数和长度范围，以及孤立脉冲崩溃之前每个事件的平均首次通过时间作为雷诺数 r 和层流-湍流共存范围内管长 L 的函数。在孤立脉冲的情况下，崩溃前平均首次通过时间 T 的对数与雷诺数的快速线性增长被分为两部分。人们发现 ln(T)=A_p r-B_p，其中 A_p 和 B_p 为正。此外，A_p 和 B_p 在脉冲的湍流强度的空间积分上是仿射的，斜率相同。对于最初包含稳定湍流分数的管道，崩溃前平均首次通过时间 T 的长度 L 和雷诺数 r 依赖性也被分离。作者发现 T≍exp[L(Ar-B)]，其中 A 和 B 为正。然后根据研究平均首次通过时间和多稳定性的大偏差理论方法讨论 T 的长度和雷诺数依赖性，其中研究了小方差噪声极限下的 ln(T)。可以使用两种观点，即小方差的局部噪声和大长度，来讨论 T 中 L 的指数依赖性。特别是，通过在 Dauchot 和 Manneville 提出的过渡壁流的概念性两自由度模型中识别低方差噪声、大长度极限下的准势能，导出了 T≍exp[L(A^{'}R-B^{'})]。这指出了控制崩溃和建立轨迹以及相应通过时间的物理效应，重点是层流和湍流状态之间的鞍点。这种分析还表明，这些效应导致了湍流动能的非对称概率密度函数。
