A novel data-driven modal decomposition of fluid flow is proposed comprising key features of POD and DMD. The first mode is the normalized real or imaginary part of the DMD mode which minimizes the time-averaged residual. The N -th mode is defined recursively in an analogous manner based on the residual of an expansion using the first N 1 modes. The resulting recursive DMD (RDMD) modes are orthogonal by construction, retain pure frequency content and aim at low residual. RDMD is applied to transient cylinder wake data and is benchmarked against POD and optimized DMD (Chen et al. 2012) for the same snapshot sequence. Unlike POD modes, RDMD structures are shown to have pure frequency content while retaining a residual of comparable order as POD. In contrast to DMD with exponentially growing or decaying oscillatory amplitudes, RDMD clearly identifies initial, maximum and final fluctuation levels. Intriguingly, RDMD outperforms both POD and DMD in the limit cycle resolution from the same snaphots. Robustness of these observations demonstrated for other parameters of the cylinder wake and for a more complex wake behind three rotating cylinders. RDMD is proposed as an attractive alternative to POD and DMD for empirical Galerkin models, with nonlinear transient dynamics as a niche application.

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In the current study, a generalization of POD Galerkin models is proposed targeting strategies for experimental feedback flow control. For practical reasons, that model should incorporate a range of flow operating conditions with small number of degrees of freedom. Standard POD Galerkin models are challenged by the over-optimization at one operating condition (Deane et al. 1991). Recent successful developments to extend the dynamic range require additional modes. This leads to a control design which is less online-capable and less robust. These side constraints for control-oriented ROMs are taken into account by a 'least-dimensional' Galerkin approximation based on a novel technique for continuous mode interpolation (Morzyński et al. 2006). This interpolation allows to preserve the model dimension of a single state while covering several states by adjusting (interpolated) modes. The resulting 3-dimensional Galerkin model is presented for the transient flow around the circular cylinder and shown to be in good agreement with the corresponding direct numerical simulation. The 3-dimensional model is based on the shift mode and interpolates between the two most energetic POD modes of natural limit cycle and the complex stability eigenmode representing the fixed-point dynamics. The interpolation technique is also used to approximate the POD modes with suitable stability eigenmodes. These efforts shall lead to a priori models from first principles without the need for empirical flow data.

The construction of low‐dimensional models of the flow, containing only reduced number of degrees of freedom, is the essential prerequisite of closed‐loop control of that flow. Presently used models usually base on the Galerkin method, where the flow is approximated by the number of modes and coefficients. The velocities are computed from a system of ordinary differential equations, called Galerkin System, instead of Navier‐Stokes equation. In this paper, reduced order models of the flow around NACA‐0012 airfoil are presented. The chosen mode sets include POD modes from Karhunen‐Loeve decomposition (which require previous direct numerical simulation), as well as different eigenmodes from global stability analysis of the flow.

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