Optimal mode decomposition for unsteady flows

A. Wynn, D. Pearson, B. Ganapathisubramani and P. J. Goulart

@article{WPGG:2013,
  author = {A. Wynn and D. Pearson and B. Ganapathisubramani and P. J. Goulart},
  title = {Optimal mode decomposition for unsteady flows},
  journal = {Journal of Fluid Mechanics},
  publisher = {Cambridge Univ Press},
  year = {2013},
  volume = {733},
  pages = {473-503},
  url = {http://dx.doi.org/10.1017/jfm.2013.426},
  doi = {10.1017/jfm.2013.426}
}

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 minimizes the system residual error. The OMD method is shown to be a generalization of dynamic mode decomposition (DMD), in which the subspace is not optimized 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.