Identifying efficient routes to laminarization: an optimization approach

TL;DR

This work defines and computes the minimal seed for relaminarization—the closest perturbation in the turbulent region that directly yields laminar flow—using a fully nonlinear optimization framework augmented by a multi-step penalty method. By applying the approach to the nine-mode Moehlis–Faisst–Eckhardt model at $Re=345$, the authors identify two distinct minimal seeds: the transition seed, dominated by the $a_3$ streamwise-vortex mode, and the relaminarization seed, which distributes across multiple modes. Both seeds lie near the edge of chaos, but their trajectories are organized by different edge structures, with the laminarizing path converging toward a periodic orbit on the edge and then exiting along its unstable manifold. The relaminarization pathway provides a practical reference trajectory for control, demonstrated by a brief actuation that directs the flow onto the laminarizing route and allows natural cessation of actuation as the flow laminarizes. Overall, the framework clarifies nonlinear mechanisms governing laminarization, enabling interpretable, efficient control strategies and setting the stage for extensions to more complex flow configurations and data-driven low-order models.

Abstract

The nonlinear and chaotic nature of turbulent flows poses a major challenge for designing effective control strategies to maintain or induce low-drag laminar states. Traditional linear methods often fail to capture the complex dynamics governing transitions between laminar and turbulent regimes. In this work, we introduce the concept of the minimal seed for relaminarization-the closest point to a reference state in the turbulent region of the state space that triggers a direct transition to laminar flow without a chaotic transient. We formulate the identification of this optimal perturbation as a fully nonlinear optimization problem and develop a numerical framework based on a multi-step penalty method to compute it. Applying this framework to a nine-mode model of a sinusoidal shear flow, we compute the minimal seeds for both transition to turbulence and relaminarization. While both of these minimal seeds lie infinitesimally close to the laminar-turbulent boundary-the edge of chaos-they are generally unrelated and lie in distant and qualitatively distinct regions of state space, thereby providing different insights into the flow's underlying structure. We find that the optimal perturbation for triggering transition is primarily in the direction of the mode representing streamwise vortices (rolls), whereas the optimal perturbation for relaminarization is distributed across multiple modes without strong contributions in the roll or streak directions. By analyzing trajectories originating from these minimal seeds, we find that both transition and laminarization behavior are controlled by the stable and unstable manifolds of a periodic orbit on the edge of chaos. The laminarizing trajectory obtained from the minimal seed for relaminarization provides an efficient pathway out of turbulence and can inform the design and evaluation of flow control strategies aimed at inducing laminarization.

Figures (13)

• Figure 1: Schematic depiction of the minimal seed for transition, the minimal seed for relaminarization, and the laminar-turbulent basin boundary.
• Figure 2: Schematic figure illustrating the multi-step optimization. For chaotic systems, standard gradient-based optimization methods suffer from the exponential growth of sensitivity backward in time, which leads to prohibitively large gradients. The multi-step penalty approach combats this by introducing breakpoints at which continuity is enforced by additional constraints, thereby reducing the magnitude of the gradients chung_optimization_2022.
• Figure 3: Energetic quantities over time for a typical trajectory of the MFE system. From top to bottom: viscous dissipation rate, $D$; power input rate, $I$; kinetic energy, $E$; disturbance kinetic energy, $E_d$. $D$ and $I$ values are normalized by their laminar value, $D_L$. Chaotic transient is denoted by gray (), and the trajectory on the chaotic attractor is shown in cyan ().
• Figure 4: Input-Dissipation projection for the trajectory of the MFE model showing sustained turbulent behavior shown as a time-series in Fig. \ref{['fig:mfe_energies_ex']}. A chaotic initial transient is denoted by gray (), and the trajectory on the chaotic attractor is shown in cyan (). The initial condition for this trajectory is indicated by the black marker.
• Figure 5: Minimal seed for transition: (a) optimized final distance from the laminar state, $d_L(\tau)$ for varying initial perturbation magnitudes $d_L(0)$. The sharp increase in $d_L(\tau)$ indicates that the basin boundary has been crossed. (b) evaluation of optimized initial conditions for extended trajectories up to $t=800$. Results for the $\tau=200$ case in (a).