Title:

Lowdimensional characterization and control of nonlinear wake flows

Many wake flows exhibit selfexcited flow oscillations which are sustained by the flow itself and are not caused by amplification of external noise. The archetypal example of a selfexcited wake flow is the low Reynolds number flow past a circular cylinder. This flow exhibits selfsustained periodic vortex shedding above a critical Reynolds number. A linear stability analysis of wake flows of this kind shows the presence of a significant region of local absolute instability which admits a temporally growing global mode of oscillation. In general, wake flows may possess multiple global modes, the most unstable of which is the observed oscillation of the wake. Active, closed loop control of such wake flows is of interest within the present study. In single sensor control schemes, flow oscillations may be suppressed at the sensor location but are in general exacerbated elsewhere by the destabilization of further global modes. For complete suppression of the flow oscillations resulting from global flow instability, all of the possible global modes must be attenuated. In general, complete suppression of all possible global modes requires the use of multiple sensors within the control scheme. As the response of the flow to external control forcing is nonlinear, then the most efficient control strategy is also nonlinear. The present work describes a general control strategy for nonlinear selfexcited wakes. Representation of the selfexcited flow field by a finite set of characteristic features, which correspond to the large scale wake components, allows for the efficient design of a closedloop control algorithm. Experimentally, wake flows are seen to be dominated by a finite number of large scale spatial structures and lowdimensional mathematical models of such flows are often adequate. Characterization of the large scale spatial structures of a wake flow can be performed with proper orthogonal decomposition, which selects an orthogonal set of spatial modes that are maximized in terms of retained energy. The low energy modes are neglected and the resulting finite orthogonal basis is used as a finite, lowdimensional representation of the wake flow field. A finite representation of the flow field, afforded by the modes, circumvents the need for a complex control algorithm involving a large number of spatially distributed flow field measurements. An appropriate control strategy is then to provide an external control input to the wake such that the future state of the wake corresponds to a desired set of mode amplitudes. Empirical prediction of the response of the wake to external control is furnished by a nonlinear neural network. Empirical modelling of the wake response avoids the need for explicit representation of the controlwake interaction. Additionally, the neural network structure of the controlwake interaction model allows for the design of a robust nonlinear control algorithm. Furthermore, rearrangement of the mode extraction process into a neural network format provides continuity within the modelling and control scheme. Successful control of a simplified wake flow, which models some of the stability features and spatial complexity of a cylinder wake flow, illustrates the utility of the control scheme.
