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Nonmodal amplitude equations.

Yves-Marie Ducimetière1, François Gallaire2

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This study introduces a new analytical method to derive weakly nonlinear amplitude equations for fluid flows, simplifying analysis of nonmodal responses to perturbations. The method efficiently predicts flow behavior near the linear regime but may oversimplify dynamics at higher excitation amplitudes.

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Area of Science:

  • Fluid Dynamics
  • Nonlinear Systems
  • Non-normal Operators

Background:

  • Fluid flows with strongly non-normal linearized Navier-Stokes operators exhibit complex responses.
  • Traditional modal reduction techniques are insufficient for describing these "nonmodal" responses.
  • Existing methods for analyzing these flows are computationally intensive and complex.

Purpose of the Study:

  • To develop a simplified analytical method for deriving weakly nonlinear amplitude equations.
  • To accurately describe nonmodal responses of fluid flows to various perturbations.
  • To reduce the computational cost associated with analyzing nonlinear fluid dynamics.

Main Methods:

  • Proposed a general method to analytically derive weakly nonlinear amplitude equations for nonmodal responses.
  • Focused on reducing the system to a low-dimensional representation based on singular modes.
  • Applied the method to harmonic forcing, stochastic forcing, and initial perturbations in parallel base flows.

Main Results:

  • Successfully derived three distinct amplitude equations for different perturbation types.
  • The derived equations accurately predict weakly nonlinear modifications of flow gains at low excitation amplitudes.
  • The method offers a significant reduction in numerical cost compared to fully nonlinear techniques.

Conclusions:

  • The proposed analytical approach provides a computationally efficient way to study weakly nonlinear fluid flow dynamics.
  • The method excels at capturing leading-order nonmodal responses but may not fully describe complex phenomena like subcritical transitions.
  • This work offers a valuable tool for analyzing fluid flows governed by non-normal operators, bridging linear and nonlinear regimes.