Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 1802.07582

Axisymmetric inertial modes in a spherical shell at low Ekman numbers

Michel Rieutord·Lorenzo Valdettaro
View original source ↗Read source document ↗Export metadata ↓

Abstract

We investigate the asymptotic properties of axisymmetric inertial modes propagating in a spherical shell when viscosity tends to zero. We identify three kinds of eigenmodes whose eigenvalues follow very different laws as the Ekman number $E$ becomes very small. First are modes associated with attractors of characteristics that are made of thin shear layers closely following the periodic orbit traced by the characteristic attractor. Second are modes made of shear layers that connect the critical latitude singularities of the two hemispheres of the inner boundary of the spherical shell. Third are quasi-regular modes associated with the frequency of neutral periodic orbits of characteristics. We thoroughly analyse a subset of attractor modes for which numerical solutions point to an asymptotic law governing the eigenvalues. We show that three length scales proportional to $E^{1/6}$, $E^{1/4}$ and $E^{1/3}$ control the shape of the shear layers that are associated with these modes. These scales point out the key role of the small parameter $E^{1/12}$ in these oscillatory flows. With a simplified model of the viscous Poincaré equation, we can give an approximate analytical formula that reproduces the velocity field in such shear layers. Finally, we also present an analysis of the quasi-regular modes whose frequencies are close to $\sin(π/4)$ and explain why a fluid inside a spherical shell cannot respond to any periodic forcing at this frequency when viscosity vanishes.

Explore related subjects

physics.flu-dynastro-ph.EPastro-ph.SR

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Michel Rieutord, Lorenzo Valdettaro. 2018-02-21. Axisymmetric inertial modes in a spherical shell at low Ekman numbers. https://doi.org/10.1017/jfm.2018.201

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

KEEP EXPLORING

Related papers

arXiv2026

Lacunarity in a Two-Dimensional Atomizing Liquid Jet

Atomization stretches and folds the liquid--gas interface before fragmenting it into ligaments and droplets, making fractal measures a natural descriptor of the lacunarity of the final breakup state. We examine this idea in two-dimensional (2D) VOF-DNS of a liquid jet with adaptive mesh refinement in Basilisk. Box-counting of the full resolved interface does not yield a single scale-independent exponent. Instead, two scaling ranges appear, separated by a crossover near the box-counting level \(L_{\mathrm{box}}\simeq7\): coarser boxes measure the folded connected jet envelope, whereas finer boxes increasingly sample ligaments, droplets, and nearly smooth local interface segments. Decomposing the interface into detached droplets, ligaments, and the connected main body shows that the relevant effective dimension is structure dependent. Droplets are most lacunar with dimension below one, ligaments occupy an intermediate level, and the main body carries the largest coarse-scale dimension. This hierarchy persists over $Re_l=100$--$10^4$ at fixed $\We_g=200$. The lacunarity of droplets is reminiscent of that of the object positions in the night sky hypothesized by de Vaucouleurs.

physics.flu-dyn↗
arXiv2026

Generalizable turbulence closures across bluff-body shapes by PINN-based solver-agnostic training

Data-driven turbulence closures are commonly calibrated by embedding a CFD solver within an inverse procedure that requires a converged flow solution at every optimization iteration, which is fragile and costly. We instead formulate closure identification as a PDE-constrained, data-driven inverse problem within a physics-informed neural network (PINN), and subsequently stabilize the inferred closure for deployment in a numerical solver. This separation eliminates the training cost of repeated CFD solves and associated adjoint or ensemble evaluations, providing substantial flexibility to investigate alternative closure forms. Using this flexibility, we develop four closures: three model the Reynolds stress on a realizable tensor basis---a local map, a non-local model transporting the turbulent kinetic energy and recovering the out-of-plane normal stress, and the same with a learned length scale $\ell$---and a fourth models the Reynolds force $\mathbf{F}=-\nabla\cdot\boldsymbolτ$ directly, free of the realizability constraint. All four are trained across six two-dimensional bluff-body wakes at $Re=10^4$ and deployed frozen in a standard finite-element solver, stabilized primarily by input-gradient smoothing. Under a strict leave-one-shape-out (LOSO) protocol, all four reduce the field errors of a steady SST $k$-$ω$ baseline, most strongly on the mean velocity. The learned-length-scale closure is most accurate on the stress fields, while the force model generalizes best on the mean velocity and drag (LOSO drag error $\sim 7\%$). The closures also train efficiently on Particle Image Velocimetry data, enabling geometries intractable for DNS.

physics.flu-dyn↗
arXiv2026

Effect of leading edge geometry on aerodynamic performance of a flat plate

Flow separation around flat plate is influenced by the thickness and leading edge geometry, which creates the shear stresses and pressure distribution manifesting into unsteady aerodynamic forces. Leading edge geometries have been widely reported but the influence of leading edge curvature at low Reynolds numbers (Re) have found less attention. The present study uses two dimensional numerical simulation to investigate the effect of a hemispherical curvature (protuberance) at the leading edge and reports how vortex shedding, shedding frequencies and aerodynamic forces are affected over a range of angles of attack ($0^\circ \leq α\leq 25^\circ$) at Re = 1000. Thicker vortical regions are formed near the plate with curvature at the leading edge. Onset of shedding is noted at angle of attack of $5^\circ$ for the plate with curvature at the leading edge and it gets delayed for simple flat plate. The temporal force coefficients on the plates show multiple frequencies at higher angles of attack. The plate with curvature at the leading edge has a glide ratio of 1 at $0^\circ$ angle of attack and has equivalent glide ratio to that of simple flat plate at higher angles of attack. The simple flat plate has better glide ratios at moderate angles of attack like $5^\circ, 10^\circ$ and $15^\circ$. The results will help in making engineering informed design decisions for smaller unmanned aerial/aquatic vehicles cruising at low Reynolds numbers.

physics.flu-dyn↗