Probability of Transition to Turbulence in a Reduced Stochastic Model of Pipe Flow

TL;DR

This work analyzes turbulence initiation in a reduced stochastic model of pipe/plane-Couette flow by estimating lower bounds on metastable transitions from laminar to turbulent states under multiplicative Gaussian noise. It couples a linearized cable-operator framework with Itô white noise and Stratonovich perturbations (white and red in time), using a martingale observable and a Cole-Hopf (KPZ-type) transform to derive probabilistic bounds on turbulence onset. Key contributions include explicit hitting-probability bounds for the Itô case, and analogous Stratonovich bounds that accommodate memory effects via red noise and Ornstein-Uhlenbeck driving, with extensions to spatial heterogeneity. The results advance analytic SPDE techniques for noise-driven transition phenomena and provide tools for predicting rare turbulence events in applied fluid dynamics.

Abstract

We study the phenomenon of turbulence initiation in pipe flow under different noise structures by estimating the probability of initiating metastable transitions. We establish lower bounds on turbulence transition probabilities using linearized models with multiplicative noise near the laminar state. First, we consider the case of stochastic perturbations by Itô white noise; then, through the Stratonovich interpretation, we extend the analysis to noise types such as white and red noise in time. Our findings demonstrate the viability of detecting the onset of turbulence as rare events under diverse noise assumptions. The results also contribute to applied SPDE theory and offer valuable methodologies for understanding turbulence across application areas.

Figures (2)

• Figure 1: $(a)$ and $(b)$ show trajectories of $q$, solution of \ref{['eq:syst_q']}, indicating the onset of turbulence under Itô noise, $0.5=\sigma_{\text{I}}>\sigma_{\text{S}}=\sigma_{\text{R}}=0$; whereas in $(c)$ and $(d)$ it is associated with Stratonovich noise, $0.5=\sigma_{\text{S}}>\sigma_{\text{R}}=\sigma_{\text{R}}=0$. The interpretation of turbulence initiation is associated with an $L^p([0,L])$-norm described below. We consider $\left\{e_i\right\}_{i\in\mathbb{N}}$ the normalized eigenfunctions of the Laplace operator on $[0,L]$ under Neumann boundary conditions. We assume the noise perturbation on the solution along $101$ modes, $b_i=e_i$ for $i\in\{0,\dots,100\}$, with intensity $\zeta_i=\text{exp}\left(-(i-1)^2\right)$. The initial solution is set at $q_0\equiv 0.5$. The Reynolds parameter is $r=\frac{1}{15}$, which implies that $q_-=0.75$ and $q_+=1.25$. We set $L=T=10$ and space and time step as $0.1$ and $0.01$, respectively. In $(a)$ and $(c)$ we observe the rise of $\lvert\lvert q \rvert\rvert_1$ to the value $q_+ L$, while in $(b)$ and $(d)$ we capture the rise of $\lvert\lvert q \rvert\rvert_\infty$ to the value $q_+$. These rare events are computed via the TAMS algorithm, for which we run $50$ simulations each and use the respective norm as a score function. The simulations are achieved through the discretized mild solution formula da2014stochastic. As described in Appendix \ref{['app:A']}, the systems differ in view of the Itô-Stratonovich correction term, which implies an additional heterogeneous positive drift term in the case of Stratonovich white noise.
• Figure 2: $(a)$ and $(b)$ show trajectories of variable $q$, solution of \ref{['eq:syst_q']}, indicating the turbulence onset event under additive red noise (see Appendix \ref{['app:A']}), $1.5=\sigma_{\text{R}}>\sigma_{\text{I}}=\sigma_{\text{S}}=0$ and $F=\operatorname{Id}$; conversely, in $(c)$, $(d)$, $(e)$ and $(f)$ we consider Stratonovich red noise (see Appendix \ref{['app:A']}), $0.5=\sigma_{\text{R}}>\sigma_{\text{I}}=\sigma_{\text{S}}=0$ and $F= \partial_t$. Similarly to Figure \ref{['fig:Fig1']}, the rare events are computed via the TAMS algorithm, for which we obtain $50$ simulations each and use the respective norm as a score function. The size and the discretization of the space-time grid, the operator $Q$, the Reynolds parameter, and the initial condition are chosen as in Figure \ref{['fig:Fig1']}. We set the perturbation intensity of $\xi$, solution of \ref{['eq:syst_z']}, as $\sigma_{\xi}=0.1$ and its dissipation value is indicated under each subfigure. In $(a)$, $(c)$ and $(e)$ we display the rise of $\lvert\lvert q \rvert\rvert_1$ to the value $q_+ L$, whereas in $(b)$, $(d)$ and $(f)$ we capture the rise of $\lvert\lvert q \rvert\rvert_\infty$ to the value $q_+$. The simulations are obtained through the discretized mild solution formula da2014stochastic. The parameter $\kappa$ is associated solely with the dissipation of $\xi$ in the case of additive red noise. This is in contrast with the case $F= \partial_t$, where it also indicates the intensity of a nonlinear perturbation term in \ref{['eq:syst_q']}. In $(c)$ and $(d)$, the solution resembles the case of Stratonovich white noise, which corresponds to $\kappa=0$; whereas in $(e)$ and $(f)$, $\kappa$ assumes a higher value and the solution tends to depart from the turbulent state in a short time scale.

Theorems & Definitions (24)

• Lemma 2.1
• proof
• Lemma 3.1
• proof
• Lemma 3.2
• proof
• Corollary 3.3
• proof
• Lemma 4.1
• proof