Data-driven time-dependent bases for turbulent airfoil wake-extreme vortex gust interactions

TL;DR

This work addresses the challenge of characterizing transient, three-dimensional gust wake interactions around a airfoil by developing a data-driven time-varying modal framework. It decomposes the flow into time-dependent in-plane modes and spanwise modes to form a low-rank representation of the wake field, with a rolling-window approach to compute derivatives and evolve the bases without storing the full history. The method yields closed-form evolution equations for the bases and tracks modal energies through the singular values, linking them to lift dynamics under extreme gusts at Re = 5000. Results show that before gust impingement the first mode dominates, after impingement the second mode becomes energetic, and stronger or larger gusts reduce coherence by exciting multiscale content, providing an interpretable, time-resolved picture of wake reorganization and recovery.

Abstract

We analyze interactions between turbulent airfoil wake and an extremely strong gust using a data-driven framework with time-dependent bases. The current approach represents each snapshot with time-varying bases consisting of two-dimensional in-plane modes and one-dimensional spanwise modes, together with a reduced covariance matrix. We derive closed-form evolution equations for these time-varying components and advance them over time, requiring only a small rolling window and avoiding full-history storage. Applied to extreme vortex gust-airfoil interaction at Re=5000, we examine how in-plane modes and their associated energy level evolve across gust conditions of varying intensity and size. Before impingement, the first in-plane mode dominates; after impingement, the second mode gains energy_amplified by stronger/larger gusts. A larger leading-mode energy gap implies coherent structure and faster recovery; a smaller gap with slower decay indicates richer multiscale activity and delayed re-stabilization. These trends follow the transient lift dynamics as well, with higher amplitude and more oscillations indicated by a rise in the leading singular values. This work provides an interpretable, time-varying data-driven modal analysis of extreme gust encounter.

Figures (6)

• Figure 1: Time-dependent bases workflow for the vortex–gust interaction: $(a)$ snapshot field, $(b)$ matricization and initialization via singular value decomposition, $(c)$ time-varying in-plane modes, $(d)$ spanwise modes at $t=1.5$, and $(e)$ singular values.
• Figure 2: Gust–airfoil interaction for $(G,D)=(2,0.5)$. Reference spanwise vorticity field $\omega_z$, time-dependent bases reconstruction (rank $r=37$), rank-2 reconstruction, and in-plane modes are showed, respectively.
• Figure 3: Gust–airfoil interaction for $(G,D)=(-2,0.5)$. Reference spanwise vorticity field $\omega_z$, time-dependent bases reconstruction (rank $r=37$), rank-2 reconstruction, and in-plane modes are showed, respectively.
• Figure 4: Gust--airfoil interaction for $(G,D)=(2,1.5)$. Reference spanwise vorticity field $\omega_z$, time-dependent reconstruction (rank $r=34$), rank-2 reconstruction, and in-plane modes are showed, respectively.
• Figure 5: Gust--airfoil interaction for $(G,D)=(4,0.5)$. Reference spanwise vorticity field $\omega_z$, time-dependent bases reconstruction (rank $r=37$), rank-2 reconstruction, and in-plane modes are showed, respectively.