DNS-based predictive control of turbulence: an optimal benchmark for feedback algorithms

Direct numerical simulations (DNS) and optimal control theory are used in a predictive control setting to determine controls that effectively reduce the turbulent kinetic energy and drag of a turbulent flow in a plane channel at Re τ = 100 and Re τ = 180. Wall transpiration (unsteady blowing/suction) with zero net mass flux is used as the control. The algorithm used for the control optimization is based solely on the control objective and the nonlinear partial differential equation governing the flow, with no ad hoc assumptions other than the finite prediction horizon, T , over which the control is optimized. Flow relaminarization, accompanied by a drag reduction of over 50%, is obtained in some of the control cases with the predictive control approach in direct numerical simulations of subcritical turbulent channel flows. Such performance far exceeds what has been obtained to date in similar flows (using this type of actuation) via adaptive strategies such as neural networks, intuition-based strategies such as opposition control, and the so-called ‘suboptimal’ strategies, which involve optimizations over a vanishingly small prediction horizon T + → 0. To achieve flow relaminarization in the predictive control approach, it is shown that it is necessary to optimize the controls over a sufficiently long prediction horizon T + gsim 25. Implications of this result are discussed. The predictive control algorithm requires full flow field information and is computationally expensive, involving iterative direct numerical simulations. It is, therefore, impossible to implement this algorithm directly in a practical setting. However, these calculations allow us to quantify the best possible system performance given a certain class of flow actuation and to qualify how optimized controls correlate with the near-wall coherent structures believed to dominate the process of turbulence production in wall-bounded flows. Further, various approaches have been proposed to distil practical feedback schemes from the predictive control approach without the suboptimal approximation, which is shown in the present work to restrict severely the effectiveness of the resulting control algorithm. The present work thus represents a further step towards the determination of optimally effective yet implementable control strategies for the mitigation or enhancement of the consequential effects of turbulence.