A Temporal Proper Orthogonal Decomposition (TPOD) Method for Closed-Loop Flow Control

Abstract

Motivated by closed-loop flow control applications, a new formulation of the proper orthogonal decomposition (POD) is presented which is capable of characterizing not only the controlled and natural states of a given flow but also the transient behavior between these states. This approach, which is termed temporal POD (TPOD) extracts the optimum frameof-reference and the temporal information regarding the dynamics of the system in the presence of the flow control. The TPOD concept is developed in this paper and demonstrated experimentally in an application involving a circular cylinder in cross flow at Re D = 5,000 with active plasma flow control. The resulting model is shown to properly capture the correct dynamics of the first TPOD mode, including the natural, forced and transient regimes. I. Motivation and ObjectivesONVENTIONAL proper orthogonal decomposition (POD)-based modeling extracts an optimal complete set of dominant (in the energy sense) spatial modes, with a minimum number of modes to represent a particular flow state [1][2][3] . These modes are usually projected onto the Navier-Stokes (N-S) equations to obtain a low-dimensional system of ODE equations to describe the temporal evolution of the modes 4,5 . This approach is shown to give satisfactory results, if the flow does not deviate much from a given state, such as the natural evolution of a shear layer or jet, for instance. But when the flow is actively manipulated using some form of flow control, the set of POD modes extracted from the natural flow state typically does a very poor job describing the forced or controlled flow state. From a topological point of view, this can be explained as shown in Figure 1: for the natural state the dynamical system (the flow) occupies a particular finite region in phase space. POD extracts a set of eigenvectors (modes) and provides a low-dimensional frame of reference to describe the system evolution within this region. When the flow control is activated, the system is forced to travel and to occupy a different finite region in the phase space. So, the set of POD modes derived to describe the system evolution in one region with as few eigenvectors as possible becomes quite non-optimal for another region, and, although the set of POD modes is still complete, it results in dramatic increase in the number of POD modes needed to provide a proper frame of reference to describe the forced system.To address this issue, several modifications of the basic POD approach have been proposed to solve this problem. For periodic flows, the double POD or DPOD approach 6, 7 provides a set of POD modes calculated within each period, and performs a second POD-optimization among POD

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“…This type of flow control was studied in more detail 13 using Particle Image Velocimetry (PIV) and Schlieren imagery. A modification of proper orthogonal decomposition 14 has been developed to model closed-loop flow control. This method attempts to model both the controlled and natural states of the flow, as well as the transition between these states.…”

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“…Therefore, we choose instead to work with a set of modes derived directly from the data itself called POD modes. 6,7,8,9,10,11 POD modes are identical to the Karhunen-Loev modes and modes from Principal Component Analysis (PCA). Essentially, POD modes are ordered according to their modal variance.…”

A closed‐loop nonlinear control and sliding mode estimation strategy for fluid flow regulation

“…It has been shown in recent research that ROM accuracy sufficient for flow control can be achieved using only the first few POD modes. 7,[17][18][19][20][21] For the proof of concept of the sliding mode estimation and flow control method presented here, complete flow field reconstruction is not necessary, and it is assumed that the ROM has sufficient accuracy for the flow control objective.…”