Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 1305.7259

Maximum Palinstrophy Growth in 2D Incompressible Flows

Diego Ayala·Bartosz Protas
View original source ↗Read source document ↗Export metadata ↓

Abstract

In this study we investigate vortex structures which lead to the maximum possible growth of palinstrophy in two-dimensional incompressible flows on a periodic domain. The issue of palinstrophy growth is related to a broader research program focusing on extreme amplification of vorticity-related quantities which may signal singularity formation in different flow models. Such extreme vortex flows are found systematically via numerical solution of suitable variational optimization problems. We identify several families of maximizing solutions parameterized by their palinstrophy, palinstrophy and energy and palinstrophy and enstrophy. Evidence is shown that some of these families saturate estimates for the instantaneous rate of growth of palinstrophy obtained using rigorous methods of mathematical analysis, thereby demonstrating that this analysis is in fact sharp. In the limit of small palinstrophies the optimal vortex structures are found analytically, whereas for large palinstrophies they exhibit a self-similar multipolar structure. It is also shown that the time evolution obtained using the instantaneously optimal states with fixed energy and palinstrophy as the initial data saturates the upper bound for the maximum growth of palinstrophy in finite time. Possible implications of this finding for the questions concerning extreme behavior of flows are discussed.

Explore related subjects

physics.flu-dyncs.NAmath.NA

Keep this discovery

BibTeXRIS

Diego Ayala, Bartosz Protas. 2013-05-30. Maximum Palinstrophy Growth in 2D Incompressible Flows. https://doi.org/10.1017/jfm.2013.685

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Correlative effects of induced magnetic field-buoyancy on reactive solute dispersion dynamics in couple-stress fluids

We investigate the dispersion of a reactive solute in a couple-stress fluid flowing between two parallel plates under the combined effects of pressure-driven flow, buoyancy, and an induced magnetic field. The model incorporates first-order heterogeneous reactions at both channel walls alongside a bulk reaction. Using Mei's multiscale homogenization technique accurate to third order, we develop a higher-order asymptotic formulation to determine the effective longitudinal dispersion coefficient and concentration field. Analytical predictions are complemented by Brownian dynamics simulations and finite-difference solutions, while the Aris method of moments quantifies transient mean displacement, spatial variance, and effective dispersivity. The hydrodynamic analysis reveals a singular branch in the velocity solution when the Hartmann number equals half the couple-stress parameter and identifies a characteristic quarter-power scaling between the Hartmann number and couple-stress parameter, separating couple-stress- and magnetically dominated regimes. The model recovers classical Taylor-dispersion behavior in the non-reactive Newtonian limit and agrees well with experimental measurements. Couple-stress rheology and magnetic damping suppress shear-induced dispersion, whereas buoyancy enhances dispersion through additional transverse velocity gradients. A distinct saturation regime of the dispersion coefficient emerges with an increasing couple-stress parameter, while unequal wall absorption induces persistent transverse asymmetry, and stronger absorption enhances solute removal near the source. Numerical and stochastic results validate the analytical framework while resolving higher-order concentration structures and particle-scale wall adsorption.

physics.flu-dyn↗
arXiv2026

DiffSWE2d: a differentiable Shallow Water Equations solver for end-to-end flood and tsunami modelling

Solving inverse and optimisation problems with traditional shallow water equations (SWE) solvers can be computationally expensive, particularly when gradients with respect to model inputs or parameters must be estimated through repeated forward simulations. In this paper, we introduce DiffSWE2d, an open-source differentiable shallow water equations solver for end-to-end flood and tsunami modelling implemented in PyTorch. By leveraging automatic differentiation, DiffSWE2d represents the time-marching physics as a differentiable computational graph, enabling gradients to be propagated directly through the numerical solver. We validate the solver against two established benchmark cases and demonstrate its application to tsunami waveform inversion, showing its ability to infer model inputs through gradient-based optimisation. DiffSWE2d provides a flexible framework for integrating physics-based hydrodynamic modelling with modern optimisation and machine learning methods. The source code and reproducible examples are publicly available at: https://github.com/ZhonghouXu/DiffSWE2d

physics.flu-dyn↗
arXiv2026

Low inertia limit of elasto-inertial turbulence

Pipe and channel flows of viscoelastic fluids display chaotic dynamics at unusually low speeds, a phenomenon referred to as elasto-inertial turbulence, EIT. First reported in experiments a century ago, recent theoretical studies and model computations predict a variety of scenarios for the phenomenon's origin, ranging from hoop stress modes to center modes and to Tollmien-Schlichting waves. Lacking experimental confirmation, the relevant scenario in actual flows of polymer solutions remains unknown. We here determine the transition threshold of EIT in pipe experiments, covering three decades in elasticity number. Across this entire parameter range, the transition features center mode structures at onset. Eventually the instability diverges at a lower inertia (upper elasticity) limit, which is a robust signature of this center mode scenario. Finally, we report the first experimental observation of a traveling wave in viscoelastic pipe flow, and the sequences of localized structures found, are in excellent agreement with a center mode traveling wave, the "arrowhead" solution, discovered in model simulations.

physics.flu-dyn↗