Variational framework for flow optimization using seminorm constraints

Area of Science:

• Fluid dynamics
• Computational mathematics
• Nonlinear dynamics

Background:

• Constrained optimization and sensitivity analysis are crucial for understanding complex physical systems governed by partial differential equations (PDEs).
• Seminorm constraints frequently arise in optimization problems where the objective function involves contributions from multiple physical mechanisms.
• Existing frameworks may not adequately address the nuanced interplay of complementary seminorm constraints in evolving flows.

Purpose of the Study:

• To introduce a general variational framework for constrained optimization and sensitivity analysis of flows described by PDEs.
• To specifically address optimization problems involving seminorm constraints arising from multiple physical mechanisms.
• To demonstrate the framework's utility in analyzing the stability of evolving flows with dynamic turbulent viscosity.

Main Methods:

• Development of a general variational framework incorporating constrained optimization and sensitivity analysis for PDE-governed flows.
• Introduction of new parameters to quantify the relative magnitudes of complementary seminorm perturbations.
• Application of the framework to the nonmodal stability analysis of a stochastically forced Burgers equation with evolving turbulent viscosity.

Main Results:

• The framework enables quantitative analysis of the influence of individual seminorm components on the total perturbation norm.
• Optimization is achieved by prescribing new parameters as initial conditions on admissible perturbations.
• The study reveals that evolving turbulent viscosity leads to qualitatively different flow dynamics compared to frozen viscosity, including a new perturbation energy production mechanism.

Conclusions:

• The proposed variational framework provides a robust method for analyzing complex fluid flows with seminorm constraints.
• The framework's ability to handle evolving parameters like turbulent viscosity is essential for accurate stability analysis.
• Understanding the interplay between evolving turbulent viscosity and mean flow is critical for predicting flow behavior and identifying novel energy production mechanisms.