Stability of N-front and N-back solutions in the Barkley model

TL;DR

This work addresses the stability of $N$-front and $N$-back traveling waves in the Barkley pipe-flow model at intermediate Reynolds numbers, linking existence results to local linear stability through Sandstede's traveling-wave framework. By performing a slow-fast decomposition, Melnikov analysis, and a detailed verification of hypotheses $H_0$–$H7$, the authors establish the existence and linear stability of $N$-front and $N$-back solutions for each $N>1$ in the regime $r>\tfrac{2}{3}$, with a nondegeneracy window $(\tfrac{2}{3},\beta)$ where $\beta\approx 0.72946$. They show that the front and back heteroclinic loops are twisted (double twist for $r\in(\tfrac{2}{3},\beta)$; single twist for $r>\beta$) and that the Melnikov data are nonzero, leading to a rigorous stability statement via Sandstede's theory. The results provide a rigorous explanation for the variety of puff/slug patterns observed during pipe-flow transition and demonstrate how a reduced Barkley model captures key dynamical mechanisms of turbulence onset. Nonlinear stability remains an open problem due to the advective coupling $-uu_x$ in the centerline equation, suggesting directions for future work.

Abstract

In this paper we establish for an intermediate Reynolds number domain the stability of N-front and N-back solutions for each N > 1 corresponding to traveling waves, in an experimentally validated model for the transition to turbulence in pipe flow proposed in [Barkley et al., Nature 526(7574):550-553, 2015]. We base our work on the existence analysis of a heteroclinic loop between a turbulent and a laminar equilibrium proved by Engel, Kuehn and de Rijk in [Engel, Kuehn, de Rijk, Nonlinearity 35:5903, 2022], as well as some results from this work. The stability proof follows the verification of a set of abstract stability hypotheses stated by Sandstede in [SIAM Journal on Mathematical Analysis 29.1 (1998), pp. 183-207] for traveling waves motivated by the FitzHugh-Nagumo equations. In particular, this completes the first detailed analysis of Engel, Kuehn and de Rijk in [Engel, Kuehn, de Rijk, Nonlinearity 35:5903, 2022] leading to a complete existence and stability statement that nicely fits within the abstract framework of waves generated by twisted heteroclinic loops.

Figures (3)

• Figure 2: Representation of $\widehat{M}$ as a function of the model Reynolds number $r$ for $r\in(\frac{2}{3},\frac{5}{2})$ from EKR2022. Note that the grey curve just indicates that $\widehat{M}$ is given by the blue curve.
• Figure 3: Representation of the singular heteroclinic loop (black) and actual heteroclinic connections (red) between the equilibria $X_1$ and $X_2$. The singular heteroclinic loop is made up of the heteroclinic connections $X_f$ and $X_b$ in the fast subsystem \ref{['fastss']} and orbit segments lying on the manifold $M_0$ in the slow subsystem \ref{['slowsubsys']}.
• Figure 4: Representation of a 4-front wave solution. Note the constant distance for even layers and the decreasing distance for odd layers. The return times with respect to $\Sigma_f$ and $\Sigma_b$ are explicitly given in \ref{['Fallunt']}.

Theorems & Definitions (24)

• Theorem 3.1
• proof
• Corollary 3.2
• proof
• Lemma 3.3
• proof
• Theorem 3.4
• proof
• Theorem 3.5
• proof