Optimal Control of the Wilcox turbulence model with lifting functions for flow injection and boundary control

This paper deals with boundary optimal control problems for the Navier-Stokes equations and Wilcox turbulence model. In this paper we study adjoint optimal control problems for Navier-Stokes equations to improve the advantages of using simulations where turbulence models play a significant role in designing engineering devices. We assess first distributed optimal control problems with the purpose to control the fluid behavior by injecting a flow on the boundary solid region to obtain a desired control over the fluid velocity and the kinetic turbulence energy in specific parts of the domain. Then, with the same purpose, we use lifting functions and boundary control. For this reason we reformulate the boundary optimal control problem into a distributed problem through a lifting function approach. The stronger regularity requirements which are imposed by standard boundary control approaches can then be avoided. The state, adjoint and control equations are derived and the optimality system solved for some simple cases with a finite element. Furthermore, we propose numerical strategies that allow to solve the coupled optimality system in a robust way for a large number of unknowns. The approach presented in this work is general and can be used to assess different objectives and types of control.