Singular value decomposition analysis of Kelvin-Helmholtz instability in two-phase flow: Temporal mode dynamics and coherent structures

We investigate the spatio-temporal organization of a two-phase Kelvin–Helmholtz (K–H) instability using singular value decomposition (SVD) of the solid-phase vertical velocity field \(w_s\). The analysis quantifies modal energy and reveals that the dominant coherent dynamics are governed by a principal rotating pair. To mitigate coarse snapshot sampling, we introduce an extrema-based harmonic fitting that preserves amplitude and phase while smoothing jitter, enabling clear phase portraits and robust interpretation of modal interactions. We focus on temporal modes \(a_1\)–\(a_6\), report their periods, amplitudes, and phase shifts, and relate these to the emergence and saturation of K–H billows. The results support a low-dimensional, reduced-order description for two-phase shear flows.