Optimal control of a transitional jet using a continuous adjoint method

The use of active flow control with unsteady means gains increasing interest in engineering designs. The main bottleneck of the methodology is the strong dependence on trial and error to find the right set of control parameters. In this context, adjoint-based control using high-fidelity simulations is a promising method to explore optimal values in large parameter spaces. However, the applicability of the methodology to complex engineering geometries remains extremely limited. In this work, we employ adjoint-based optimal control using unsteady high-fidelity simulations in a generic unstructured grid framework. To this end, an optimal flow control study is conducted in OpenFOAM® using the continuous adjoint method and DNS simulations. To demonstrate the methodology, we study control of an incompressible axisymmetric jet at ReD=2000, with focus on improving its mixing properties. The gradient of the cost functional is calculated with a newly developed unsteady-adjoint solver based on a classical incremental projection scheme. Particular attention is paid into the presentation of mathematical and algorithmic details. Moreover, we address three main issues that remained relatively undiscussed in common practise: the choice of adjoint boundary conditions on computational boundaries, the failure of the adjoint methodology for long optimization horizons in turbulent flows and the treatment of the additional transposed convective term in the adjoint equations. Practical solutions are employed for these issues. Two optimization cases with different initial conditions are designed. To this end, we considered maximization of enstrophy in the near field, for which increments of 10.5% and 5.6% are obtained.