Soft Matter and Complex Systems Seminar
sala 1.40, ul. Pasteura 5
2024-11-22 (09:30) 
Stanisław Gepner (Warsaw University of Technology)
Is the Laminar-Turbulent Edge Crowded? Exploring Multiple Local Attractors in the Edge of Square-Duct Flow
In this work, we present the first streamwise-localized invariant solution for turbulent square duct flow in the moderate Reynolds number range. Through heuristic analysis, we demonstrate that during specific periods within the turbulent time evolution, the flow state approaches the identified localized solution. This finding indicates that the localized solution is embedded within the turbulent attractor, making it the first localized solution identified for square duct flow and a the potential building block of turbulence in this configuration.
We obtain this solution through a bisection process applied within the symmetric subspace of the full state space, which enables the tracking of edge state solutions. Edge states are characterized by a single unstable direction, or a co-dimension one stable manifold, within the symmetric subspace. In the context of the full state space, these solutions are embedded within the turbulent attractor. As relative attractors on the edge of the laminar and turbulent basins, edge states play a significant role in governing the laminar-turbulent transition process. This characteristic makes them particularly interesting for turbulence control applications. In addition to the bisection method, we use Newton-Krylov GMRES-based iterations to converge to invariant solutions. To analyze stability, we apply an Arnoldi-based eigenvalue solver, and an arc-length continuation to track bifurcations. Stability analysis reveals that both branches of our localized solution are unstable in at least one direction. This instability suggests the presence of additional structures that may connect to the branches of the identified solution, indicating that the edge subspace (a co-dimension one subspace of the full space) contains multiple local attractors. Each of these local edge states would have stable manifolds that locally separate initial conditions, leading either toward the laminar attractor, a transient non-laminar excursion or, if it exists, a turbulent attractor. In our ongoing work, we identify and analyze a series of solutions on the edge. We study the positions and potential connections between the lower and upper branches of the identified solutions. By disturbing either the lower or upper branch in the unstable direction, we observe that the system tends either to laminarize smoothly or to experience a transient turbulent excursion. This behavior confirms that both solution branches reside on the edge and that the bifurcation responsible for their creation also lies on the edge. Additionally, we identify a potential heteroclinic connection between these states, which further enriches our understanding of the dynamics governing laminar-turbulent transition in square duct flow.
We obtain this solution through a bisection process applied within the symmetric subspace of the full state space, which enables the tracking of edge state solutions. Edge states are characterized by a single unstable direction, or a co-dimension one stable manifold, within the symmetric subspace. In the context of the full state space, these solutions are embedded within the turbulent attractor. As relative attractors on the edge of the laminar and turbulent basins, edge states play a significant role in governing the laminar-turbulent transition process. This characteristic makes them particularly interesting for turbulence control applications. In addition to the bisection method, we use Newton-Krylov GMRES-based iterations to converge to invariant solutions. To analyze stability, we apply an Arnoldi-based eigenvalue solver, and an arc-length continuation to track bifurcations. Stability analysis reveals that both branches of our localized solution are unstable in at least one direction. This instability suggests the presence of additional structures that may connect to the branches of the identified solution, indicating that the edge subspace (a co-dimension one subspace of the full space) contains multiple local attractors. Each of these local edge states would have stable manifolds that locally separate initial conditions, leading either toward the laminar attractor, a transient non-laminar excursion or, if it exists, a turbulent attractor. In our ongoing work, we identify and analyze a series of solutions on the edge. We study the positions and potential connections between the lower and upper branches of the identified solutions. By disturbing either the lower or upper branch in the unstable direction, we observe that the system tends either to laminarize smoothly or to experience a transient turbulent excursion. This behavior confirms that both solution branches reside on the edge and that the bifurcation responsible for their creation also lies on the edge. Additionally, we identify a potential heteroclinic connection between these states, which further enriches our understanding of the dynamics governing laminar-turbulent transition in square duct flow.