Optimal mode decomposition for unsteady flows

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

Journal of Fluid Mechanics, vol. 733, pp. 473-503, October 2013.
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  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.

Turbulent separation upstream of a forward-facing step

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

Journal of Fluid Mechanics, vol. 724, pp. 284-304, June 2013.
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  author = {D. Pearson and P. J. Goulart and B. Ganapathisubramani},
  title = {Turbulent separation upstream of a forward-facing step},
  journal = {Journal of Fluid Mechanics},
  year = {2013},
  volume = {724},
  pages = {284-304},
  url = {http://dx.doi.org/10.1017/jfm.2013.113},
  doi = {10.1017/jfm.2013.113}

The turbulent flow over a forward-facing step is studied using two-dimensional time-resolved particle image velocimetry. The structure and behaviour of the separation region in front of the step is investigated using conditional averages based on the area of reverse flow present. The relation between the position of the upstream separation and the two-dimensional shape of the separation region is presented. It is shown that when of ?closed? form, the separation region can become unstable resulting in the ejection of fluid over the corner of the step. The separation region is shown to grow simultaneously in both the wall-normal and streamwise directions, to a point where the maximum extent of the upstream position of separation is limited by the accompanying transfer of mass over the step corner. The conditional averages are traced backwards in time to identify the average behaviour of the boundary-layer displacement thickness leading up to such events. It is shown that these ejections are preceded by the convection of low-velocity regions from upstream, resulting in a three-dimensional interaction within the separation region. The size of the low-velocity regions, and the time scale at which the separation region fluctuates, is shown to be consistent with the large boundary layer structures observed in the literature. Instances of a highly suppressed separation region are accompanied by a steady increase in velocity in the upstream boundary layer.