Dynamic mode decomposition analysis of rotating detonation waves
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ORIGINAL ARTICLE
Dynamic mode decomposition analysis of rotating detonation waves M. D. Bohon1
· A. Orchini1
· R. Bluemner1
· C. O. Paschereit1
· E. J. Gutmark2
Received: 9 December 2019 / Revised: 23 September 2020 / Accepted: 21 October 2020 © The Author(s) 2020
Abstract A rotating detonation combustor (RDC) is a novel approach to achieving pressure gain combustion. Due to the steady propagation of the detonation wave around the perimeter of the annular combustion chamber, the RDC dynamic behavior is well suited to analysis with reduced-order techniques. For flow fields with such coherent aspects, the dynamic mode decomposition (DMD) has been shown to capture well the dominant oscillatory features corresponding to stable limit-cycle or quasi-periodic behavior within its dynamic modes. Details regarding the application of the technique to RDC—such as the number of frames, the effect of subtracting the temporal mean from the processed dataset, the resulting dynamic mode shapes, and the reconstruction of the dynamics from a reduced set of dynamic modes—are analyzed and interpreted in this study. The DMD analysis is applied to two commonly observed operating conditions of rotating detonation combustion, viz., (1) a single spinning wave with weak counter-rotating waves and (2) a clapping operating mode with two counter-propagating waves at equal speed and strength. We show that care must be taken when applying DMD to RDC datasets due to the presence of standing waves (expressed as either counter-propagating azimuthal waves or longitudinal pulsations). Without accounting for these effects, the reduced-order reconstruction fails using the standard DMD approach. However, successful application of the DMD allows for the reconstruction and separation of specific wave modes, from which models of the stabilization and propagation of the primary and counter-rotating waves can be derived. Keywords Dynamic mode decomposition · Rotating detonation · Pressure gain combustion · Reduced-order dynamics
List of symbols A Linear mapping between frames b Initial condition coefficients c Linear mapping coefficients D Eigenvalue matrix of A and S i Imaginary unit r Residual vector S Companion-type mapping matrix Reconstructed snapshot matrix V1N Reduced snapshot matrix V1N v Vectorized image v˜ Reconstruction of vectorized image x Eigenvectors of S Communicated by F. Lu.
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M. D. Bohon [email protected]
1
Technische Universität Berlin, Müller-Breslau-Str. 8, Berlin, Germany
2
Department of Aerospace Engineering, University of Cincinnati, Cincinnati, OH 45220, USA
y z ε ϕ σ
Eigenvectors of A Mode shape of eigenvector pair Residual of reconstructed snapshots Angle of discrete mapping eigenvalues Growth rate of eigenvalue
Subscripts i Index of image j Index of eigenvalue
Image acquisition Data acquisition rate, frames per second fs IM Snapshot image m Number of rows in image n Number of columns in image N Number of images pxm,n Pixel at index (m, n) Exposure time (µs) te
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1 Introduction Rotating
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