Predictive control of spiral vortex breakdown

Abstract

The predictive control of the self-sustained single spiral vortex breakdown mode is addressed in the three-dimensional flow geometry of Ruithet al.(2003) for a constant swirl number$S=1.095$. Based on adjoint optimization algorithms, two different control strategies have been designed. First, a quadratic objective function minimizing the radial velocity intensity, taking advantage of the physical mechanism underpinning spiral vortex breakdown. The second strategy focuses on the hydrodynamic instability properties using as objective function the growth rate of the most unstable global eigenmode. These minimization algorithms seek for an optimal volume force in an axisymmetric domain avoiding therefore expensive three-dimensional computations. In addition to considering eigenvalues around the base flow, we also investigate the stability around the mean flow and we find that it correctly predicts the frequency of the self-sustained single spiral vortex breakdown mode for Reynolds numbers up to$Re=500$. Close to the instability threshold, at a Reynolds value of$Re=180$, all these control strategies successfully quench the spiral vortex breakdown. The related volume force is found identical for the base and mean flow eigenvalue control even if the uncontrolled growth rates differ significantly. The control of the least unstable eigenvalue of the mean flow is not only found optimal at$Re=180$, it also stabilizes the flow at a Reynolds value as large as$Re=300$, which opens promising extensions to industrial applications.