Optimal mode decomposition for unsteady flows

Abstract

A new method, herein referred to as Optimal Mode Decomposition (OMD), of finding a linear model to describe the evolution of a fluid flow is presented. The method estimates the linear dynamics of a high-dimensional system which is first projected onto a subspace of a user-defined fixed rank. An iterative procedure is used to find the optimal combination of linear model and subspace that minimises the system residual error. The OMD method is shown to be a generalisation of Dynamic Mode Decomposition (DMD), in which the subspace is not optimised but rather fixed to be the Proper Orthogonal Decomposition (POD) modes. Furthermore, OMD is shown to provide an approximation to the Koopman modes and eigenvalues of the underlying system. A comparison between OMD and DMD is made using both a synthetic waveform and an experimental data set. The OMD technique is shown to have lower residual errors than DMD and is shown on a synthetic waveform to provide more accurate estimates of the system eigenvalues. This new method can be used with experimental and numerical data to calculate the "optimal" low-order model with a user-defined rank that best captures the system dynamics of unsteady and turbulent flows.

Interscale energy transfer in the merger of wakes of a multiscale array of rectangular cylinders

“…The methodology is based on the OMD decomposition technique (see Wynn et al [46]) and on phase averaging. Let us briefly review the results of its application to our experimental data.…”

“…The employed triple decomposition method (see Baj et al [1] relies heavily on Optimal Mode Decomposition (OMD, see Wynn et al [46] for an extensive description), which is a snapshot based technique for establishing an optimal linear approximation of the system's dynamics. The underlying assumption is that two consecutive snapshots in a time-resolved sequence, q h and q h+1 (the instantaneous PIV measurements), are approximately linked via linear equation (B.1), where D is a time invariant matrix governing the system's evolution (m is the snapshot size).…”

“…Here both M and L are treated as independent optimization variables. The exact algorithm of solving this problem is described in Wynn et al [46]. …”

“…The adopted triple decomposition method utilizes Optimal Mode Decomposition (OMD, see Wynn et al [46]) to evaluate velocity modes Φ OM D (x) that can be linked to particular coherent fluctuations. These are used to span a basis onto which the total velocity fluctuations are projected.…”

Near field development of artificially generated high Reynolds number turbulent boundary layers

“…This method has been successfully applied to flows in which, small, less energetic flow structures' modes are successfully identified in a multi-scale flow (Baj et al, 2015). A brief summary of the method follows; for further explanations and a deeper mathematical analysis the reader is referred to Wynn et al (2013).…”