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A linear non-modal mechanism for transient amplification of perturbation energy is known to trigger sub-critical transition to turbulence in many shear flows. Feedback control strategies for minimizing this transient energy growth can be formulated as convex optimization problems based on linear matrix inequalities. Unfortunately, solving the requisite linear matrix inequality problem can be computationally prohibitive within the context of high-dimensional fluid flows.In this work, we investigate the utility of control-oriented reduced-order models to facilitate the design of feedback flow control strategies that minimize the maximum transient energy growth. An output projection onto proper orthogonal decomposition modes is used to faithfully capture the system energy. Subsequently, a balanced truncation is performed to reduce the state dimension, while preserving the system's input-output properties. The model reduction and control approaches are studied within the context of a linearized channel flow with blowing and suction actuation at the walls. Controller synthesis for this linearized channel flow system becomes tractable through the use of the proposed control-oriented reduced-order models. Further, the resulting controllers are found to reduce the maximum transient energy growth compared with more conventional linear quadratic optimal control strategies. Nomenclature Θ max = maximum transient energy growth Φ r = matrix of r dominant proper orthogonal decomposition modes (α, β) = streamwise and spanwise wave number pair A = linear time invariant state matrix of plant A = reduced-order state matrix after balanced-truncation B = linear time invariant input matrix of plant B = reduced-order input matrix after balanced-truncation * Graduate Student. AIAA Student Member. † Assistant Professor. AIAA Senior Member. arXiv:1907.01664v1 [physics.flu-dyn] 2 Jul 2019 E = energy of the full-order plant E = reduced-order approximation of E K = controller feedback gain Re = Reynolds number T s = matrix of s dominant balanced modes U cl = channel laminar base flow center-line velocity h = channel half-height n = dimension of the full-order plant q u , q l = wall-normal velocities at upper and lower wall respectively r, s = number of proper orthogonal decomposition modes and balanced modes, respectively. u = input vector x = state vector z = proper orthogonal decomposition coefficients I. IntroductionThe transition of flows from a laminar to turbulent regime has been extensively studied, and remains a topic of continuing interest. It is well known that turbulent flows exhibit specific detrimental effects on systems, e.g., an increase in skin friction drag in wall-bounded shear flows [1]. It has been observed that transition to turbulence in many shear flows occurs at a Reynolds number (Re) much below the critical Re predicted by linear (modal) stability analysis of a steady laminar base flow [2,3]. This sub-critical transition is associated with non-modal amplification mechanisms that cause small disturbances to grow befo...

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Feedback flow control is developed to suppress the transient energy growth of flow disturbances in a linearized channel flow. Specifically, we seek a controller that minimizes the maximum transient energy growth, which can be formulated as a linear matrix inequality problem. Solving linear matrix inequality problems can be computationally prohibitive for high-dimensional systems encountered in flow control applications. Thus, we develop reduced-order fluids models using balance truncation and proper orthogonal decomposition techniques. These models are designed to optimally approximate system energy while preserving the input-output dynamics that are essential for controller synthesis. Controllers developed based on these reduced-order models are found to reduce transient energy growth and to outperform linear quadratic controllers in the context of a linearized channel flow.

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