Maximizing Vortex for the Navier--Stokes Flow with a Convective Boundary Condition: A Shape Design Problem

TL;DR

This paper addresses a shape optimization problem for 2D Navier--Stokes flow, aiming to maximize vorticity by shaping an obstacle within a channel, using regularization and volume constraints to ensure well-posedness and practical applicability.

Contribution

It introduces a novel shape derivative formulation for maximizing vorticity in Navier--Stokes flow with artificial boundary conditions, incorporating regularization and volume constraints.

Findings

Gradient descent successfully maximizes vorticity.

Volume preservation methods maintain obstacle size.

Numerical examples demonstrate method effectiveness.

Abstract

In this study, a shape optimization problem for the two-dimensional stationary Navier--Stokes equations with an artificial boundary condition is considered. The fluid is assumed to be flowing through a rectangular channel, and the artificial boundary condition is formulated so as to take into account the possibility of ill-posedness caused by the usual do-nothing boundary condition. The goal of the optimization problem is to maximize the vorticity of the said fluid by determining the shape of an obstacle inside the channel. Meanwhile, the shape variation is limited by a perimeter functional and a volume constraint. The perimeter functional was considered to act as a Tikhonov regularizer and the volume constraint is added to exempt us from topological changes in the domain. The shape derivative of the objective functional was formulated using the rearrangement method, and this derivative was later on used for gradient descent methods. Additionally, an augmented Lagrangian method and a class of solenoidal deformation fields were considered to take into account the goal of volume preservation. Lastly, numerical examples based on the gradient descent and the volume preservation methods are presented.