Implications of the selection of a particular modal decomposition technique for the analysis of shallow flows

Abstract

This work deals with the capabilities of two synoptic modal decomposition techniques for the identification of the spatial patterns and temporal dynamics of coherent structures in shallow flows. Using two different experimental datasets it is shown that due to the linear behaviour of large-scale, quasi-two-dimensional flow structures, there are almost no differences in the identification of dominant modes between the results obtained from a traditional proper orthogonal decomposition and the more recently developed dynamic mode decomposition. However, it is also shown that nonlinear dynamics can arise in the transition of these structures to a quasi-two-dimensional behaviour, which can result in the proper orthogonal decomposition identifying structures composed of multi-frequencies, a sign of a convoluted dynamics. Thus dynamic mode decomposition is recommended instead for the analysis of such phenomena. In addition, this paper introduces a simple ranking methodology for the use of the dynamic mode decomposition technique in shallow flows, which is based on the results of the proper orthogonal decomposition.

Numerical Analysis of the Flow around Two Square Cylinders in a Tandem Arrangement with Different Spacing Ratios Based on POD and DMD Methods

“…Therefore, the first few order modes obtained by the POD method reflect the main flow characteristics of the flow field, and usually the dominant character of flow fields can be reconstructed well using these modes. In fact, the POD method is basically consistent with the singular value decomposition (SVD) and principal component analysis (PCA) methods [25]. The SVD formula is shown in Equation (2), where W is the variable matrix of interest, and each column of vectors represents a time sample.…”

“…Suppose W B can be obtained by W A transformation, as shown in Equation ( 4), and the singular value vector of W A is decomposed into W A = USV * . The solution of the conversion matrix F can be further simplified to Equation (5) [25], and then the eigenvalue solution of the matrix F is derived to obtain the eigenvalue µ j and the eigenvector Λ j .…”

“…Through DMD, the flow field can be decomposed based on frequency information, that is, each DMD mode has a single and different frequency value, and these modes are sorted in descending order of pulsating energy at different frequencies [12]. Compared to the POD algorithm, the DMD algorithm considers both the temporal (spectral) and spatial orthogonalities, resulting in the phase/frequency information and the corresponding coherent structures, while POD only decomposes the modes sorted by energy, and these modes may contain multiple frequency components [24,25]. Yu [26] used DMD to decompose the calculated flow field around an NACA0012 airfoil, and analyzed the evolution of the vortex structure of the airfoil in the shear flow.…”

Forecasting short-term dynamics of shallow cumuli using dynamic mode decomposition

“…The singular value decomposition (SVD) X = UΣV * (where * means complex conjugate transpose, U ∈ R N ×N and V ∈ R M ×M are unitary, and Σ ∈ R N ×M , diagonal with nonnegative entries σ k , ordered from largest to smallest magnitude) separates the time dynamics which are captured by the sampled states into a set of spatial modes embodied in the columns of U and temporal modes given by the columns of V. The SVD, when computed on a matrix so constructed, is the POD of X [6].…”

Using a proper orthogonal decomposition to elucidate features in granular flows

“…There have been numerous efforts to successfully implement POD in this area of research and application [5,19,22,24,43,44]. In essence, POD extracts the following: a set of spatial modes ranked by their variance (here we analyze velocity, and variance corresponds to fluctuating kinetic energy); a set of singular values which describe contribution of each spatial mode; and, a set of temporal coefficients which describe the time evolution of spatial modes as show by Higham et al [23]. Periodicity in these spatial modes can be obtained using decomposition techniques such as Fourier transform, wavelet transform or Wigner distribution.…”