A variational framework for flow optimization using semi-norm constraints

TL;DR

This paper introduces a variational framework extension using semi-norm constraints to improve flow optimization, especially when traditional norm-based constraints are insufficient, demonstrated on a turbulent flow stability problem.

Contribution

The paper develops a novel variational framework extension with semi-norm constraints for flow optimization, addressing issues of unbounded state components and divergence in existing methods.

Findings

Framework successfully regularizes flow optimization problems.

Application to turbulent flow stability demonstrates effectiveness.

Potential for controlling specific state components in fluid dynamics.

Abstract

When considering a general system of equations describing the space-time evolution (flow) of one or several variables, the problem of the optimization over a finite period of time of a measure of the state variable at the final time is a problem of great interest in many fields. Methods already exist in order to solve this kind of optimization problem, but sometimes fail when the constraint bounding the state vector at the initial time is not a norm, meaning that some part of the state vector remains unbounded and might cause the optimization procedure to diverge. In order to regularize this problem, we propose a general method which extends the existing optimization framework in a self-consistent manner. We first derive this framework extension, and then apply it to a problem of interest. Our demonstration problem considers the transient stability properties of a one-dimensional (in space) averaged turbulent model with a space- and time-dependent model "turbulent viscosity". We believe this work has a lot of potential applications in the fluid dynamics domain for problems in which we want to control the influence of separate components of the state vector in the optimization process.