Control-Oriented Model Reduction for Minimizing Transient Energy Growth in Shear Flows

Abstract

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...

Feedback Control for Transition Suppression in Direct Numerical Simulations of Channel Flow

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Reducing transient energy growth in a channel flow using static output feedback control

“…In this work, we overcome the issue with computational complexity by taking advantage of control-oriented reduced-order models (ROM), similar to those described in [15]. Once we have computed the stabilizing SOF controller gain F 0 using the ROM, this can be used to initialize the Anderson-Moore algorithm (see Algorithm 1) for yielding an SOF-LQR control solution.…”

Modal Analysis of Fluid Flows: Applications and Outlook

“…Further, modal analysis techniques can be tailored to and leveraged for feedback flow control synthesis. Within the context of channel flow control, numerous model reduction strategies have been developed around modal decomposition techniques, e.g., global mode truncation [81] and input-output modeling [82][83][84]. Recent efforts have demonstrated that a model reduction approach needs to be selected and tailored carefully with respect to the control objective [83,84].…”

“…Within the context of channel flow control, numerous model reduction strategies have been developed around modal decomposition techniques, e.g., global mode truncation [81] and input-output modeling [82][83][84]. Recent efforts have demonstrated that a model reduction approach needs to be selected and tailored carefully with respect to the control objective [83,84]. For example, within the context of transient energy growth reduction, the performance of feedback controllers designed on reduced-order models (ROMs) can be quite sensitive to the parameters used in generating the underlying ROMs [83,84].…”

“…Recent efforts have demonstrated that a model reduction approach needs to be selected and tailored carefully with respect to the control objective [83,84]. For example, within the context of transient energy growth reduction, the performance of feedback controllers designed on reduced-order models (ROMs) can be quite sensitive to the parameters used in generating the underlying ROMs [83,84].…”