Invariant solutions of the Navier–Stokes equations, such as fixed points, traveling waves, and periodic orbits, provide a dynamical systems framework for turbulence. Although dynamically unstable, these solutions organize chaotic dynamics and capture coherent structures observed in experiments and simulations. Over the past three decades, substantial progress has been made in computing invariant solutions, initially focusing on isolated fixed points and traveling waves and, more recently, shifting toward libraries of periodic orbits that offer a more complete description of the dynamics. Computing these solutions is challenging because of their instability, the high dimensionality arising from spatial discretization, and the increasing complexity in more turbulent regimes. This review provides an overview of the numerical methods developed to address these challenges. We discuss matrix-free Newton–Krylov approaches, more recent variational methods designed to improve robustness in the computation of unstable periodic orbits, and emerging strategies that combine equation-based algorithms with data-driven techniques to generate improved initial guesses. Together, these developments provide increasingly powerful tools for exploring the dynamical structure of turbulence.
Ocean circulation is maintained by mechanical and thermodynamical fluxes across its boundaries, primarily imposed by winds and buoyancy forcing at large scales, yet its kinetic energy (KE) is distributed across a vast range of scales and depths. This review synthesizes current understanding of these multiscale pathways from an ocean physics perspective, emphasizing recent progress enabled by scale-aware diagnostics. We summarize global KE spectra, highlighting prominent mesoscale and gyrescale peaks, and a pronounced spectral gap separating them, with implications for parameterizations. Framed using a spatially local, scale-dependent KE budget, we discuss wind work; potential energy (PE) generation and baroclinic release; and KE transfer across gyrescale, mesoscale, and submesoscale ranges. We also examine the unresolved problem of mesoscale KE sinks: A significant upscale mesoscale cascade must ultimately be arrested by processes such as wind damping, bottom drag, loss of balance, and/or submesoscale pathways. Emerging research directions include balanced–unbalanced flow decomposition, conversion and cascade of PE, and coupled ocean–atmosphere feedbacks across scales relevant to climate variability.
Active turbulence grids have reshaped laboratory turbulence research by shifting experiments from passive generation toward controllable forcing. Originally introduced to extend the Reynolds number range of grid-generated turbulence, active grids enable systematic control of turbulence intensity, integral scales, and shear within a single setup. This capability expands the accessible parameter space and reframes the questions that can be addressed experimentally. In canonical flows, active grids permit high–Reynolds number investigations of turbulence scalings and controlled studies of turbulence interacting with walls and free shear flows. In multiphase and Lagrangian systems, they provide well-characterized conditions to probe clustering, settling, and interfacial transfer at previously unattainable Reynolds numbers. In engineering contexts, they enable reproducible coherent inflow structures, e.g., gusts. Together, these advances have expanded the experimental parameter space and changed how universality, forcing, and flow development are investigated in turbulent flows.
Cryogenic carbon capture (CCC) by CO 2 desublimation offers high-purity separation without chemical solvents, but its industrial deployment requires accurate computational models encompassing a wide range of coupled multiphysics phenomena. This article reviews the state of the art in computational modeling of multiphase flows in CCC, covering four complementary frameworks: 1D macroscopic models, Eulerian–Lagrangian and Eulerian–Eulerian computational fluid dynamics approaches, pore-scale lattice Boltzmann simulations, and molecular dynamics studies of nucleation and crystal growth. Together, these span roughly nine orders of magnitude in spatial scale. The physical foundations of CO 2 desublimation—phase equilibria, nucleation kinetics, heat and mass transfer, and frost morphology—are discussed alongside a critical assessment of each framework. Key open challenges include bridging the molecular-to-device-scale modeling gap, addressing cryogenic thermophysical data deficiencies, resolving turbulence–phase-change coupling, and developing reduced-order models for real-time process control.
