Deconvolution of reacting-flow dynamics using proper orthogonal and dynamic mode decompositions

Abstract

Analytical and computational studies of reacting flows are extremely challenging due in part to nonlinearities of the underlying system of equations and long-range coupling mediated by heat and pressure fluctuations. However, many dynamical features of the flow can be inferred through low-order models if the flow constituents (e.g., eddies or vortices) and their symmetries, as well as the interactions among constituents, are established. Modal decompositions of high-frequency, high-resolution imaging, such as measurements of species-concentration fields through planar laser-induced florescence and of velocity fields through particle-image velocimetry, are the first step in the process. A methodology is introduced for deducing the flow constituents and their dynamics following modal decomposition. Proper orthogonal (POD) and dynamic mode (DMD) decompositions of two classes of problems are performed and their strengths compared. The first problem involves a cellular state generated in a flat circular flame front through symmetry breaking. The state contains two rings of cells that rotate clockwise at different rates. Both POD and DMD can be used to deconvolve the state into the two rings. In POD the contribution of each mode to the flow is quantified using the energy. Each DMD mode can be associated with an energy as well as a unique complex growth rate. Dynamic modes with the same spatial symmetry but different growth rates are found to be combined into a single POD mode. Thus, a flow can be approximated by a smaller number of POD modes. On the other hand, DMD provides a more detailed resolution of the dynamics. Two classes of reacting flows behind symmetric bluff bodies are also analyzed. In the first, symmetric pairs of vortices are released periodically from the two ends of the bluff body. The second flow contains von Karman vortices also, with a vortex being shed from one end of the bluff body followed by a second shedding from the opposite end. The way in which DMD can be used to deconvolve the second flow into symmetric and von Karman vortices is demonstrated. The analyses performed illustrate two distinct advantages of DMD: (1) Unlike proper orthogonal modes, each dynamic mode is associated with a unique complex growth rate. By comparing DMD spectra from multiple nominally identical experiments, it is possible to identify "reproducible" modes in a flow. We also find that although most high-energy modes are reproducible, some are not common between experimental realizations; in the examples considered, energy fails to differentiate between reproducible and nonreproducible modes. Consequently, it may not be possible to differentiate reproducible and nonreproducible modes in POD. (2) Time-dependent coefficients of dynamic modes are complex. Even in noisy experimental data, the dynamics of the phase of these coefficients (but not their magnitude) are highly regular. The phase represents the angular position of a rotating ring of cells and quantifies the downstream displacement of vortices in reacting flows. Thu...

Dynamic-mode decomposition based analysis of shear coaxial jets with and without transverse acoustic driving

“…The corresponding differentiation in POD based methods relies on irregular features having lower energies than the regular flow facets. The results of the injector flow analysis, outlined in the next section, indicate that this differentiation may not be valid (Roy et al 2015). It should also be noted that in constructing reduced-order models or in developing techniques to control the flow, one must focus on the robust features of the flow.…”

“…Previous DMDs of sprays (Tirunagari et al 2012b) have also not addressed the issue. The analysis reported here is motivated by an approach introduced in Roy et al (2015) and outlined below which differentiates robust and non-robust constituents of a flow using Koopman mode analysis (Rowley et al 2009;Schmid 2010;Mezic 2013). As we observe from figures 15 and 24 below, robust and non-robust constituents of injector flows cannot be differentiated using their contributions to the flow alone.…”

“…One can use this observation to address an important issue: a constituent of the injector flow may not be captured by a single dynamic mode; rather, several such modes may be required to reconstruct a typical constituent (Roy et al 2015). Analysis of simulated and experimental data suggests approaches for identifying multiple modes that may be associated with a single flow constituent.…”

“…Experimental noise and irregularities in the flow cause fluctuations in the coefficients of the dynamic modes. Interestingly, angular changes of the phase-space orbit are highly robust, and the irregularities are restricted to the radial component (Roy et al 2015). The phase of the coefficient a k (t) of a dynamic mode Φ k (x) is defined as The relationship between the phases of two dynamic modes can be illustrated using Lissajous figures, such as figure 9, which relate phases of a pair of modes through their sine functions (Roy et al 2015).…”

“…These eigenvalues can be used to address a critical issue in the analysis of nonlinear flows, namely how to differentiate robust features of the flow from noise and other non-robust characteristics. As we contend (Roy et al 2015), comparison of Koopman spectra from nominally identical experiments can be the basis of this differentiation. The starting point is a search for reproducible Koopman modes, i.e.…”