Input-output inspired method for permissible perturbation amplitude of transitional wall-bounded shear flows

Area of Science:

• Fluid dynamics
• Turbulence theory
• Nonlinear dynamics

Background:

• The exact parameters dictating the transition to turbulence in wall-bounded shear flows are not fully understood.
• Existing theoretical bounds often lack consensus with experimental or simulation data.

Purpose of the Study:

• To develop a method for calculating a provable, Reynolds-number-dependent bound on perturbation amplitudes that maintain laminar flow.
• To offer a computationally efficient alternative to existing nonlinear approaches for analyzing flow stability.

Main Methods:

• Utilized an input-output approach, modeling nonlinear dynamics as static feedback within a Lur'e system.
• Constructed quadratic constraints for the nonlinear term, ensuring energy conservation and bounded input-output energy.
• Reformulated stability analysis and perturbation amplitude calculations as linear matrix inequalities (LMIs).

Main Results:

• The derived analytical bounds are consistent with results from extensive simulations.
• The LMI-based approach significantly reduces computational cost compared to sum-of-squares programming.
• The method provides a provable guarantee for the permissible level of perturbations.

Conclusions:

• The proposed framework offers a computationally efficient and provably rigorous method for analyzing laminar flow stability.
• This approach advances the understanding of turbulence transition by providing reliable bounds on flow perturbations.
• The framework is versatile and applicable to energy method computations and linear stability analysis.