Modeling Transitions in Shear Flows Using Reduced-Order Models on Spectral Submanifolds

  • Kaszás, Bálint (ETH Zürich)
  • Cenedese, Mattia (ETH Zürich)
  • Haller, George (ETH Zürich)

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We derive low-dimensional, data-driven models for transitions among exact coherent states in shear flows. We use the theory of spectral submanifolds (SSMs), which has been shown to be effective for reduced-order modeling of phenomena involving coexisting ECSs. These structures, SSMs, are very low dimensional attracting invariant manifolds that emanate from stationary states and have the potential to connect those states to other coexisting stationary states. Recent advances have enabled the identification of SSMs purely from numerical or experimental data. Here we apply these results to construct SSMs of simple ECSs in plane Couette flow. We show that physically relevant quantities, such as the rate of energy input and dissipation rate, can be used to parametrize the most important SSMs connecting coexisting ECSs. By restricting the dynamics to these SSMs, we obtain accurate reduced-order models that capture all asymptotic states of the full Navier-Stokes equations. A similar approach is expected to apply to pipe flow to possibly capture the geometrical aspects of the subcritical transition to turbulence.