Numerical simulations of optimal control problems for the Reynolds averaged Navier–Stokes system closed with a two-equation turbulence model

• Navier–Stokes and k−ωk−ω turbulence optimal distributed control is studied.
• The objective functional includes velocity and turbulence kinetic energy.
• The optimality system together with dual adjoint boundary conditions is derived.
• The turbulence kinetic energy value can be controlled in any point of the domain.
• Numerical results in two- and three-dimensional spaces are reported.

Optimal control theory in fluid dynamics has become popular in the last several years because of its use in the design of new engineering devices and optimization of existing ones. In recent years the optimal control of turbulent flows has gained attention but turbulence modeling and its control is still an open problem. In this work we study a distributed optimal control problem for a flow modeled by the Reynolds averaged Navier–Stokes system closed by a two-equation k−ω turbulence model which takes into account the limits imposed on turbulent viscosity. The complete adjoint system is derived with a Lagrangian multiplier approach allowing to write a cost functional directly based on the turbulence kinetic energy. We assess a cost functional composed of two terms, the first for a velocity matching profile problem and the second for turbulence enhancement or reduction. The optimality system is solved numerically with a finite element code parallelized with standard message passing interface libraries. Numerical results in two and three-dimensional spaces are reported to show the validity of this numerical approach.