Global resolvent analysis of three-dimensional jets using randomized linear algebra and time stepping

Resolvent analysis is a powerful tool for modeling coherent structures in turbulent flows. Resolvent forcing and response modes are defined in terms of the singular value decomposition (SVD) of the resolvent operator, but the computational cost of computing these modes scales poorly with problem size and becomes computationally expensive for large systems, typically limiting resolvent analysis to flows with one or two inhomogeneous directions. Recently, we have developed an improved algorithm (called RSVDt) by combining randomized SVD with an optimized direct and adjoint time-stepping routine, which achieves linear cost scaling with problem size and enables efficient resolvent analysis of three-dimensional flows. Time stepping constitutes the majority of the cost of our algorithm, and we propose a new approach to minimize this cost by eliminating undesired transients that otherwise lengthen the time interval to be computed. We use a series of jets to demonstrate our algorithm. First, we use two jet models to show that RSVDt decreases memory and CPU cost by two and three orders of magnitude, respectively, compared to other state-of-the-art algorithms. Second, we use RSVDt to compute resolvent modes for a previously intractable three-dimensional jet and use this new capability to explore the impact of steady streaks on Kelvin-Helmholtz wavepackets.