arXiv subjects

Tim Colonius

Publications and source records attributed to Tim Colonius.

At least 19 recordsLinked to original sources

arXiv2026

Band-Ensemble Spectral Proper Orthogonal Decomposition with Frequency Attribution

This study presents band-ensemble Spectral Proper Orthogonal Decomposition (bSPOD). The approach is inspired by frequency smoothing, a method used to reduce estimator variance in power spectral density estimates, and is here extended to SPOD. The algorithm estimates SPOD modes from consecutive Fourier coefficients obtained from a single Fourier transform of the full time record and thus avoids time segmentation. In this study, bSPOD is applied to artificial test data and to a PIV data set of a broadband-tonal cavity flow. Compared to the more commonly used Welch-based SPOD formulation, bSPOD reduces spectral leakage, permits increased frequency resolution, and retains frequency information of tonal components at comparable computational cost. These features enable reduced estimator variance while maintaining low bias for tonal components, making bSPOD particularly effective for broadband-tonal flows.

physics.flu-dyn↗

arXiv2025

Resolvent4py: a parallel Python package for analysis, model reduction and control of large-scale linear systems

In this paper, we present resolvent4py, a parallel Python package for the analysis, model reduction and control of large-scale linear systems with millions or billions of degrees of freedom. This package provides the user with a friendly Python-like experience (akin to that of well-established libraries such as numpy and scipy), while enabling MPI-based parallelism through mpi4py, petsc4py and slepc4py. In turn, this allows for the development of streamlined and efficient Python code that can be used to solve several problems in fluid mechanics, solid mechanics, graph theory, molecular dynamics and several other fields.

physics.comp-ph↗

arXiv2025

Greedy recursion parameter selection for one-way spatial integration of hyperbolic equations

Solutions to hyperbolic systems comprise waves propagating at finite speeds. When wave propagation is predominantly unidirectional, one-way wave equations can be used to evolve only the right-going solution by removing support for left-going waves. The One-Way Navier-Stokes (OWNS) approach, which was originally developed for systems of first-order hyperbolic equations, constructs one-way approximations to the linearized Navier-Stokes equations using a recursive filter to remove left-going waves. The computational cost scales with the number of recursion parameters, which must be carefully chosen to ensure accuracy and stability of the resulting one-way equation. Previous work has chosen parameters based on heuristic estimates of key eigenvalues, which requires trial-and-error tuning while also yielding slow error convergence. We propose a greedy algorithm for automatic parameter selection, which we show yields faster convergence and a net decrease in computational cost for linear and nonlinear disturbance evolution in boundary-layer flows. We review the OWNS projection (OWNS-P) and recursive (OWNS-R) methods, comparing their convergence properties, and show through our numerical analysis and experiments that OWNS-P yields superior convergence and stability properties. Although we demonstrate the method for Navier-Stokes equations, we perform our analyses on systems of linear first-order hyperbolic equations and emphasize that the greedy algorithm is applicable to such systems.

math.NA↗

arXiv2025

Normality-based analysis of multiscale velocity gradients and energy transfer in direct and large-eddy simulations of isotropic turbulence

Symmetry-based analyses of multiscale velocity gradients highlight that strain self-amplification (SS) and vortex stretching (VS) drive forward energy transfer in turbulent flows. By contrast, a strain-vorticity covariance mechanism produces backscatter that contributes to the bottleneck effect in the subinertial range of the energy cascade. We extend these analyses by using a normality-based decomposition of filtered velocity gradients in forced isotropic turbulence to distinguish contributions from normal straining, pure shearing and rigid rotation at a given scale. Our analysis of direct numerical simulation (DNS) data illuminates the importance of shear layers in the inertial range and (especially) the subinertial range of the cascade. Shear layers contribute significantly to SS and VS and play a dominant role in the backscatter mechanism responsible for the bottleneck effect. Our concurrent analysis of large-eddy simulation (LES) data characterizes how different closure models affect the flow structure and energy transfer throughout the resolved scales. We thoroughly demonstrate that the multiscale flow features produced by a mixed model closely resemble those in a filtered DNS, whereas the features produced by an eddy viscosity model resemble those in an unfiltered DNS at a lower Reynolds number. This analysis helps explain how small-scale shear layers, whose imprint is mitigated upon filtering, amplify the artificial bottleneck effect produced by the eddy viscosity model in the inertial range of the cascade. Altogether, the present results provide a refined interpretation of the flow structures and mechanisms underlying the energy cascade and insight for designing and evaluating LES closure models.

physics.flu-dyn↗

arXiv2025

MFC 5.0: An exascale many-physics flow solver

Many problems of interest in engineering, medicine, and the fundamental sciences rely on high-fidelity flow simulation, making performant computational fluid dynamics solvers a mainstay of the open-source software community. Previous work, MFC 3.0, was published, documented, and made open-source by Bryngelson et al. CPC (2021) features numerous physical features, numerical methods, and scalable infrastructure. MFC 5.0 is a significant update to MFC 3.0, featuring a broad set of well-established and novel physical models and numerical methods, as well as the introduction of GPU and APU (or superchip) acceleration. We exhibit state-of-the-art performance and ideal scaling on the first two exascale supercomputers, OLCF's Frontier and LLNL's El Capitan. Combined with MFC's single-accelerator performance, MFC achieves exascale computation in practice and has achieved the largest-to-date public CFD simulation at 200 trillion grid points, earning it a 2025 ACM Gordon Bell Prize finalist. New physical features include the immersed boundary method, $N$-fluid phase change, Euler-Euler and Euler-Lagrange sub-grid bubble models, fluid-structure interaction, hypo- and hyper-elastic materials, chemically reacting flow, two-material surface tension, magnetohydrodynamics (MHD), and more. Numerical techniques now represent the current state-of-the-art, including general relaxation characteristic boundary conditions, WENO variants, Strang splitting for stiff sub-grid flow features, and low Mach number treatments. Weak scaling to tens of thousands of GPUs on OLCF's Summit and Frontier, and LLNL's El Capitan, achieves efficiencies within 5% of ideal to over 90% of their respective system sizes. Strong scaling results for a 16-fold increase in device count show parallel efficiencies exceeding 90% on OLCF Frontier.

physics.flu-dyn↗

arXiv2025

Bispectral decomposition and energy transfer in a turbulent jet

We employ bispectral mode decomposition (BMD) to investigate coherent triadic interactions and nonlinear energy transfer in a subsonic turbulent jet. BMD extracts the flow structures corresponding to the dominant triadic interactions. We find a strong triadic correlation among the Kelvin-Helmholtz wavepacket, its conjugate, and the streaks. The most energetic streaks occur at the azimuthal wavenumber $m=2$, with the dominant contributing azimuthal wavenumber triad being $[m_1,m_2,m_3]=[1,1,2]$. The spectral energy budget reveals that nonlinear triadic interactions represent an energy loss to the streaks. Analysis across a wide range of frequencies and azimuthal wavenumbers identifies the direction of nonlinear energy transfer and the spatial regions where these transfers are most active. The turbulent jet exhibits a forward energy cascade in a global sense, though the direction of energy transfer varies locally. In the shear layer near the nozzle exit, triadic interactions between relatively smaller scales are dominant, leading to an inverse energy cascade. Farther downstream, beyond the end of the potential core, triadic interactions between larger scales dominate, resulting in a forward energy cascade.

physics.flu-dyn↗

arXiv2025

Spectral proper orthogonal decomposition of rapid snapshot pairs sampled at sub-Nyquist intervals

Modal decomposition methods are important for characterizing the low-dimensional dynamics of complex systems, including turbulent flows. Different methods have varying data requirements and produce modes with different properties. Spectral proper orthogonal decomposition (SPOD) produces orthogonal, energy-ranked spatial modes at discrete temporal frequencies for statistically stationary flows. However, SPOD requires long stretches of sequential, uniformly sampled, time-resolved data. These data requirements limit SPOD's use in experimental settings where the maximum capture rate of a camera is often slower than the Nyquist sampling rate required to resolve the highest turbulent frequencies. However, if two PIV systems operate in tandem, pairs of data can be acquired that are arbitrarily close in time. The dynamic mode decomposition (DMD) uses this pairwise data to resolve frequencies up to the Nyquist frequency associated with the small time step within a pair. However, these modes do not form a basis and have no set ranking. The present work attempts to compute SPOD modes from pairwise data with a small time step but with large gaps between pairs. We use DMD on pairwise data to estimate segment-wise, uniformly sampled series that can then be used to estimate the SPOD modes, intending to resolve frequencies between the gap and pair Nyquist limits. The method is tested on numerically obtained data of the linearized complex Ginzburg-Landau equation, as well as a Mach 0.4 isothermal turbulent jet. For the jet, pairwise SPOD can accurately de-alias the SPOD spectrum and estimate mode shapes at frequencies up to St = 1.0, while using over 90% less data.

physics.flu-dyn↗

arXiv2024

Boundary layer stability analysis using the nonlinear One-Way Navier-Stokes approach

We extend the One-Way Navier Stokes (OWNS) approach to support nonlinear interactions between waves of different frequencies, which will enable nonlinear analysis of instability and transition. In OWNS, the linearized Navier-Stokes equations are parabolized and solved in the frequency domain as a spatial initial-value (marching) problem. OWNS yields a reduced computational cost compared to global solvers, while also conferring numerous advantages over the parabolized stability equations (PSE), despite its higher computational cost relative to PSE, that we seek to extend to nonlinear analysis. We validate the nonlinear OWNS (NOWNS) method by examining nonlinear evolution of two- and three-dimensional disturbances in a low-speed Blasius boundary layer compared to nonlinear PSE (NPSE) and direct numerical simulation (DNS) results from the literature. We demonstrate that NOWNS can be used to simulate flows involve blowing/suction strips, is more robust to numerical noise, and converges for stronger nonlinearities, as compared to NPSE.

physics.flu-dyn↗

arXiv2024

Nonlinear dynamics of vortex pairing in transitional jets

This study investigates the onset of linear instabilities and their later nonlinear interactions in the shear layer of an initially-laminar jet using a combination of stability analysis and data from high-fidelity flow simulations. We provide a complete picture of the vortex-pairing process. Hydrodynamic instabilities initiate the transition to turbulence, causing the shear layer to spread rapidly. In this process, the shear layer rolls up to form vortices, accompanied by the exponential growth of the fundamental frequency. As the fundamental frequency grows, it gains energy from the mean flow. Subsequently, as it saturates and begins to decay, the fundamental vortices start to pair. During this vortex pairing process, the subharmonic vortex acquires energy both linearly from the mean flow and nonlinearly through a reverse cascade from the fundamental. The process concludes when the subharmonic vortex eventually saturates. Similarly, two subharmonic vortices merge to form a second subharmonic vortex. Our results confirm Kelly's (1967) hypothesis of a resonance mechanism between the fundamental and subharmonic, which supplies energy to the subharmonic. In this multi-tonal, convective-dominated flow, we clarify the ambiguity surrounding the fundamental frequency by demonstrating that the spatially most amplified frequency should be considered fundamental, rather than the structure associated with the spectral energy peak. For the initially-laminar jet considered here, the fundamental frequency corresponds to the fourth largest spectral peak, highlighting the important distinction between the energetically and dynamical significance of a tone. Despite its low energy, the fundamental frequency is dynamically dominant as it determines all other spectral peaks and supplies energy to the subharmonics through a reverse energy cascade.

physics.flu-dyn↗

arXiv2024

Velocity gradient partitioning in turbulent flows

The velocity gradient tensor can be decomposed into normal straining, pure shearing and rigid rotation tensors, each with distinct symmetry and normality properties. We partition the strength of turbulent velocity gradients based on the relative contributions of these constituents in several canonical flows. These flows include forced isotropic turbulence, turbulent channels and turbulent boundary layers. For forced isotropic turbulence, the partitioning is in excellent agreement with previous results. For wall-bounded turbulence, the partitioning collapses onto the isotropic partitioning far from the wall, where the mean shearing is relatively weak. By contrast, the near-wall partitioning is dominated by shearing. Between these two regimes, the partitioning collapses well at sufficiently high friction Reynolds numbers and its variations in the buffer layer and the log-law region can be reasonably modelled as a function of the mean shearing strength. Altogether, our results highlight the expressivity and broad applicability of the velocity gradient partitioning as advantages for turbulence modelling.

physics.flu-dyn↗

arXiv2024

Fast and robust method for screened Poisson lattice Green's function using asymptotic expansion and Fast Fourier Transform

We study the lattice Green's function (LGF) of the screened Poisson equation on a two-dimensional rectangular lattice. This LGF arises in numerical analysis, random walks, solid-state physics, and other fields. Its defining characteristic is the screening term, which defines different regimes. When its coefficient is large, we can accurately approximate the LGF with an exponentially converging asymptotic expansion, and its convergence rate monotonically increases with the coefficient of the screening term. To tabulate the LGF when the coefficient is not large, we derive a one-dimensional integral representation of the LGF. We show that the trapezoidal rule can approximate this integral with exponential convergence, and we propose an efficient algorithm for its evaluation via the Fast Fourier Transform. We discuss applications including computing the LGF of the three-dimensional Poisson equation with one periodic direction and the return probability of a two-dimensional random walk with killing.

math.NA↗

arXiv2024

Optimal frequency resolution for spectral proper orthogonal decomposition

We demonstrate that accurate computation of the spectral proper orthogonal decomposition (SPOD) critically depends on the choice of frequency resolution. Using both artificially generated data and large-eddy simulation data of a turbulent subsonic jet, we show that the optimal choice depends on how rapidly the SPOD modes change in space at adjacent frequencies. Previously employed values are found to be too high, resulting in unnecessarily biased results at physically important frequencies. A physics-informed adaptive frequency-resolution SPOD algorithm is developed that provides substantially less biased SPOD modes than the standard constant resolution method.

physics.flu-dyn↗

arXiv2024

An adaptive lattice Green's function method for external flows with two unbounded and one homogeneous directions

We solve the incompressible Navier-Stokes equations using a lattice Green's function (LGF) approach, including immersed boundaries (IB) and adaptive mesh refinement (AMR), for external flows with one homogeneous direction (e.g. infinite cylinders of arbitrary cross-section). We hybridize a Fourier collocation (pseudo-spectral) method for the homogeneous direction with a specially designed, staggered-grid finite-volume scheme on an AMR grid. The Fourier series is also truncated variably according to the refinement level in the other directions. We derive new algorithms to tabulate the LGF of the screened Poisson operator and viscous integrating factor. After adapting other algorithmic details from the fully inhomogeneous case, we validate and demonstrate the new method with transitional and turbulent flows over a circular cylinder at $Re=300$ and $Re=12,000$, respectively.

physics.flu-dyn↗

arXiv2023

The effect of flight on a turbulent jet: coherent structure eduction and resolvent analysis

We study coherent structures in subsonic turbulent jets subject to a flight stream. A thorough characterisation of the effects of a flight stream on the turbulent field was recently performed by Maia et al. (2023) and fluctuation energy attenuations were observed over a broad range of frequencies and azimuthal wavenumbers. The Kelvin-Helmholtz, Orr and lift-up mecahnisms were all shown to be weakened by the flight stream. Here we expand upon that study and model the changes in the dynamics of jets in flight using global resolvent analysis. The resolvent model is found to correctly capture the main effects of the flight stream on the dynamics of coherent structures, which are educed from a large-eddy simulation database using spectral proper orthogonal decomposition (SPOD). Three modifications of note are: the damping of low-frequency streaky/Orr structures that carry most of the fluctuation energy; a degradation of the low-rank behaviour of the jet in frequencies where modal instability mechanisms are dominant; and a rank decrease at very low Strouhal numbers. The latter effect is underpinned by larger gain separations predicted by the resolvent analysis, due to a reduction in the wavelength of associated flow structures. This leads to a clearer relative dominance of streaky structures generated by the lift-up mechanism, despite the fact that the lift-up mechanism has been weakened with respect to the static jet.

physics.flu-dyn↗

arXiv2023

Filtering Dynamical Systems Using Observations of Statistics

We consider the problem of filtering dynamical systems, possibly stochastic, using observations of statistics. Thus, the computational task is to estimate a time-evolving density $\rho(v, t)$ given noisy observations of the true density $\rho^\dagger$; this contrasts with the standard filtering problem based on observations of the state $v$. The task is naturally formulated as an infinite-dimensional filtering problem in the space of densities $\rho$. However, for the purposes of tractability, we seek algorithms in state space; specifically, we introduce a mean-field state-space model, and using interacting particle system approximations to this model, we propose an ensemble method. We refer to the resulting methodology as the ensemble Fokker-Planck filter (EnFPF). Under certain restrictive assumptions, we show that the EnFPF approximates the Kalman-Bucy filter for the Fokker-Planck equation, which is the exact solution to the infinite-dimensional filtering problem. Furthermore, our numerical experiments show that the methodology is useful beyond this restrictive setting. Specifically, the experiments show that the EnFPF is able to correct ensemble statistics, to accelerate convergence to the invariant density for autonomous systems, and to accelerate convergence to time-dependent invariant densities for non-autonomous systems. We discuss possible applications of the EnFPF to climate ensembles and to turbulence modeling.

stat.ME↗

arXiv2023

Velocity gradient analysis of a head-on vortex ring collision

We simulate the head-on collision between vortex rings with circulation Reynolds numbers of 4000 using an adaptive, multiresolution solver based on the lattice Green's function. The simulation fidelity is established with integral metrics representing symmetries and discretization errors. Using the velocity gradient tensor and structural features of local streamlines, we characterize the evolution of the flow with a particular focus on its transition and turbulent decay. Transition is excited by the development of the elliptic instability, which grows during the mutual interaction of the rings as they expand radially at the collision plane. The development of antiparallel secondary vortex filaments along the circumference mediates the proliferation of small-scale turbulence. During turbulent decay, the partitioning of the velocity gradients approaches an equilibrium that is dominated by shearing and agrees well with previous results for forced isotropic turbulence. We also introduce new phase spaces for the velocity gradients that reflect the interplay between shearing and rigid rotation and highlight geometric features of local streamlines. In conjunction with our other analyses, these phase spaces suggest that, while the elliptic instability is the predominant mechanism driving the initial transition, its interplay with other mechanisms, e.g. the Crow instability, becomes more important during turbulent decay. Our analysis also suggests that the geometry-based phase space may be promising for identifying the effects of the elliptic instability and other mechanisms using the structure of local streamlines. Moving forward, characterizing the organization of these mechanisms within vortices and universal features of velocity gradients may aid in modelling turbulent flows.

physics.flu-dyn↗

arXiv2023

Method for portable, scalable, and performant GPU-accelerated simulation of multiphase compressible flow

Multiphase compressible flows are often characterized by a broad range of space and time scales. Thus entailing large grids and small time steps, simulations of these flows on CPU-based clusters can thus take several wall-clock days. Offloading the compute kernels to GPUs appears attractive but is memory-bound for standard finite-volume and -difference methods, damping speed-ups. Even when realized, faster GPU-based kernels lead to more intrusive communication and I/O times. We present a portable strategy for GPU acceleration of multiphase compressible flow solvers that addresses these challenges and obtains large speedups at scale. We use OpenACC for portable offloading of all compute kernels while maintaining low-level control when needed. An established Fortran preprocessor and metaprogramming tool, Fypp, enables otherwise hidden compile-time optimizations. This strategy exposes compile-time optimizations and high memory reuse while retaining readable, maintainable, and compact code. Remote direct memory access, realized via CUDA-aware MPI, reduces communication times. We implement this approach in the open-source solver MFC. Metaprogramming-based preprocessing results in an 8-times speedup of the most expensive kernels, 46% of peak FLOPs on NVIDIA GPUs, and high arithmetic intensity (about 10 FLOPs/byte). In representative simulations, a single A100 GPU is 300-times faster than an Intel Xeon CPU core, corresponding to a 9-times speedup for a single A100 compared to the entire CPU die. At the same time, near-ideal (97%) weak scaling is observed for at least 13824 GPUs on Summit. A strong scaling efficiency of 84% is retained for an 8-times increase in GPU count. Collective I/O, implemented via MPI3, helps ensure negligible contribution of data transfers. Large many-GPU simulations of compressible (solid-)liquid-gas flows demonstrate the practical utility of this strategy.

physics.flu-dyn↗

arXiv2023

Spectral proper orthogonal decomposition of harmonically forced turbulent flows

Many turbulent flows exhibit time-periodic statistics. These include turbomachinery flows, flows with external harmonic forcing, and the wakes of bluff bodies. Many existing techniques for identifying turbulent coherent structures, however, assume the statistics are statistically stationary. In this paper, we leverage cyclostationary analysis, an extension of the statistically stationary framework to processes with periodically varying statistics, to generalize the spectral proper orthogonal decomposition (SPOD) to the cyclostationary case. The resulting properties of the cyclostationary SPOD (CS-SPOD for short) are explored, a theoretical connection between CS-SPOD and the harmonic resolvent analysis is provided, simplifications for the low and high forcing frequency limits are discussed, and an efficient algorithm to compute CS-SPOD with SPOD-like cost is presented. We illustrate the utility of CS-SPOD using two example problems: a modified complex linearized Ginzburg-Landau model and a high-Reynolds-number turbulent jet.

physics.flu-dyn↗