Networked-oscillator-based modeling and control of unsteady wake flows

A networked-oscillator-based analysis is performed to examine and control the transfer of kinetic energy for periodic bluff body flows. The dynamics of energy fluctuations in the flow field are described by a set of oscillators defined by conjugate pairs of spatial proper orthogonal decomposition (POD) modes. To extract the network of interactions among oscillators, impulse responses of the oscillators to amplitude and phase perturbations are tracked. Tracking small energy inputs and using linear regression, a networked-oscillator model is constructed that reveals energy exchange among the modes. The model captures the nonlinear interactions among the modal oscillators through a linear approximation. A large collection of system responses is aggregated to capture the general network structure of oscillator interactions. The present networked-oscillator model describes the modal perturbation dynamics more accurately than the empirical Galerkin reduced-order model. The linear network model for nonlinear dynamics is subsequently utilized to design a model-based feedback controller. The controller suppresses the modal amplitudes that result in wake unsteadiness leading to drag reduction. The strength of the proposed approach is demonstrated for a canonical example of two-dimensional unsteady flow over a circular cylinder. The present formulation enables the characterization of modal interactions to control fundamental energy transfers in unsteady bluff body flows.

Figure 1

Modal oscillator model in the complex plane and oscillator network. The circle in the oscillator model describes the limit cycle trajectory of the oscillators ($z_m^b$) and the perturbed temporal coefficients ($z_m$). The nodes of the network correspond to oscillators with directed edges showing interactions. Modal interactions corresponding to edges from oscillator I to II highlighted in red (bottom shaded edge) are further expanded (top right).

Figure 2

(a) Instantaneous and (b) time-averaged (mean) vorticity fields. Proper orthogonal decomposition (POD) applied to the cylinder flow problem results in (c) spatial modes and (d) temporal coefficients of oscillators in the complex plane. The colorbar of the spatial modes indicates the contour level and the colorbar of the temporal coefficients and oscillators indicates the phase of the oscillators varying from $[-\pi ,\pi ]$ over the periodic limit cycles.

Figure 3

Response of the system to amplitude perturbation introduced to oscillator II. (a) Adjacency matrix of the networked-oscillator model and (b) corresponding network of oscillator interactions. The color and thickness of the network edges in the network represent the magnitude of the elements of ${\mathit{A}}_{\mathcal{G}}$. (c) Oscillator dynamics from DNS, the networked-oscillator model, and the POD-Galerkin model (color of temporal dynamics of oscillators indicates phase). The red dot in oscillator II dynamics indicate the initial perturbation.

Figure 4

(a) Modal amplitude and (b) energy tracking for amplitude perturbation introduced to oscillator II. Network model and Galerkin model dynamics are shown by red (darker gray) and green (lighter gray) dashed lines, respectively.

Figure 5

(a) Network degrees for varying fractions of training data. (b) Adjacency matrix of the aggregate network model and (c) the associated model performance. (d) Aggregate network structure of oscillator interactions. The color and thickness of the edges in the network represent the magnitude of the elements of ${\mathit{A}}_{\mathcal{G}}$.

Figure 6

Shift mode temporal coefficient (red) and drag coefficient (blue) variation (indicated by arrows) with application of control. Parabolic inertial manifold shown in gray.

Figure 7

(a) Pole trajectories with application of different control inputs to the aggregate network model for a range of $\sigma$. Stars indicate $\lambda (-{\mathit{L}}_{\mathcal{G}})$. (b) Movement of $\max [Re(\lambda \left)\right]$ for the multiple oscillator input cases. (c) Network structure with control. The oscillators to which control is added are indicated by red (filled) circles. The color and thickness of the network edges in the network represent the magnitude of elements of ($-{\mathit{L}}_{\mathcal{G}}-\mathit{B}\mathit{K}$).

Figure 8

Oscillator dynamics without control and with control implemented in DNS for suppressing perturbations to the limit cycle.

Figure 9

Oscillator dynamics without control and with control implemented in DNS for suppressing modal oscillations.

Figure 10

(a) Shift mode temporal dynamics on the parabolic inertial manifold for a range of $\sigma$ and (b) shift mode temporal dynamics in the time domain. (c) Drag force compared to the baseline.