Beltrami Flows Research Articles

Energy Conservation for 3D Euler and Navier–Stokes Equations in a Bounded Domain: Applications to Beltrami Flows

Energy Conservation for 3D Euler and Navier–Stokes Equations in a Bounded Domain: Applications to Beltrami Flows

Exact solutions of the Navier–Stokes equations generated by the eigenfunctions of the Stokes operator

We derive a class of exact solutions to the incompressible unsteady Navier–Stokes equations. These solutions are generated by the eigenfunctions of the Stokes operator and can be applied to studying boundary value problems for the Navier–Stokes equations related to pipe and channel flows. We explore the relation between the spectrum of the Stokes operator, generalized Beltrami flows, and exact solutions of the Navier–Stokes equations. Relevant properties of the Stokes eigenfunctions are also discussed in some detail.

Spectral Total-variation Processing of Shapes—Theory and Applications

We present a comprehensive analysis of total variation (TV) on non-Euclidean domains and its eigenfunctions. We specifically address parameterized surfaces, a natural representation of the shapes used in 3D graphics. Our work sheds new light on the celebrated Beltrami and Anisotropic TV flows and explains experimental findings from recent years on shape spectral TV [Fumero et al. 2020 ] and adaptive anisotropic spectral TV [Biton and Gilboa 2022 ]. A new notion of convexity on surfaces is derived by characterizing structures that are stable throughout the TV flow, performed on surfaces. We establish and numerically demonstrate quantitative relationships between TV, area, eigenvalue, and eigenfunctions of the TV operator on surfaces. Moreover, we expand the shape spectral TV toolkit to include zero-homogeneous flows, leading to efficient and versatile shape processing methods. These methods are exemplified through applications in smoothing, enhancement, and exaggeration filters. We introduce a novel method that, for the first time, addresses the shape deformation task using TV. This deformation technique is characterized by the concentration of deformation along geometrical bottlenecks, shown to coincide with the discontinuities of eigenfunctions. Overall, our findings elucidate recent experimental observations in spectral TV, provide a diverse framework for shape filtering, and present the first TV-based approach to shape deformation.

Dynamical property and triadic interaction of Beltrami-type rotating waves

In this work, on the one hand, we continue with the study of the dynamic property of Beltrami flows [R. González, “Dynamics of non-axisymmetric Beltrami flows,” Phys. Fluids 26, 114104 (2014)], extending its scopes to non-stationary flows in the rotating system, which allowed us to classify the rotating waves on the basis of their phase velocities. On the other hand, and in accordance with this classification, we study the resonant triadic interaction of these waves. For this purpose, we use the expansion in a Chandrasekhar–Kendall basis for an infinite tube, in an analogous procedure to the one carried out by Waleffe [“The nature of triad interactions in homogeneous turbulence,” Phys. Fluids A 4, 350–363 (1992)], to study the interaction of plane Beltrami waves. Taking an equilibrium of the resulting non-linear equations for the amplitudes of the waves, we consider their linear and non-linear stability. Regarding their linear stability, we see that unlike plane Beltrami waves, their stability depends not only on the relative helicities but also on an interaction factor that depends on the properties and the co-rotating or counter-rotating character of the interacting waves. On the other hand, for non-linear stability dependent on the same parameters as those of linear stability, we find, for one case of analysis, that there is non-linear instability only for some interactions of two co-rotating waves with a counter-rotating one and we exhibit criteria that are sufficient conditions of non-linear stability or non-linear instability.

Chaos from Symmetry: Navier Stokes Equations, Beltrami Fields and the Universal Classifying Crystallographic Group

The core of this paper is the group-theoretical approach, initiated in 2015 by one of the present authors in collaboration with Alexander Sorin that brings into the classical field of mathematical fluid-mechanics a brand new vision, allowing for a more systematic classification and algorithmic construction of Beltrami flows on torii R3/Λ where Λ is a crystallographic lattice. Here this new hydro-theory based on the focal idea of a Universal Classifying Group UGΛ is revised, reorganized, improved and extended. In particular we construct the so far missing UGΛHex for the hexagonal lattice and we advocate that, mastering the cubic and hexagonal instances of this group, we can cover all cases. The relation between the classification of Beltrami Flows with that of contact structures is enlightened. The recent developments about the framework of b-manifolds are considered and it is shown that the choice of the allowed critical surfaces for the b-deformation of a Beltrami field seems to be strongly related to the group-theoretical structure of the latter. This opens new directions of investigations about a group theoretical classification of critical surfaces. Apart from that the most promising research direction opened by the present work streams from the fact that the Fourier series expansion of a generic Navier-Stokes solution can be regrouped into an infinite sum of contributions Wr, each associated with a spherical layer of quantized radius r in the momentum lattice. Each Wr is the superposition of a Beltrami field Wr+ plus an anti-Beltrami field Wr−. These latter have a priori exactly the same decomposition into irreps of the group UGΛ that are variously repeated on higher layers. This crucial property enables the construction of generic Fourier series with prescribed hidden symmetries as candidate solutions of the NS equations. Alternatively the Fourier series representation of known solutions can be analyzed from the point of view of such symmetries. As a further result of this research program a complete and versatile system of MATHEMATICA Codes named AlmafluidaNSPsystem has been constructed and is now available through the site of the Wolfram Community. The exact solutions presented in this paper have to be considered as an illustration of the new conceptions and ideas that have emerged and of what can be further done utilizing the computer codes as an instrument. The main message streaming from our constructions is that the more symmetric is the Beltrami Flow the highest is the probability of the on-set of chaotic trajectories.

The stability of inviscid Beltrami flow between parallel free-slip impermeable boundaries

We present a family of exact inviscid three-dimensional Beltrami flows in a horizontally periodic domain lying between two parallel free-slip boundaries. Significantly, these flows are not stress free: the horizontal vorticity varies on each boundary. Using direct numerical simulations (employing horizontal hyperdiffusion only for numerical stability), we find that the largest-scale member of the family is unstable and breaks down into anisotropic turbulence, with relatively large horizontal vorticity at and near each boundary, and associated surface frontal features. We conjecture that all members of the family are similarly unstable. The free-slip boundaries play an important role in the late stages of the instability by constraining the deformation of vortex lines near the boundaries. This study appears to be the first to consider the role of boundary horizontal vorticity in an inviscid context.

Knots and links of invariant tori in Beltrami flows with cubic symmetry

Beltrami flows with crystallographic cubic symmetry were demonstrated numerically to have knots or links of invariant tori, which are not only stream tubes but also vortex tubes. The cubic symmetry is associated with the ABC flow with its three parameters equalised. The knots formed by invariant tori are the trefoil knot, the granny knot, true-lovers' knot, etc. The linked invariant tori are shown to form a variety of chainlike structures.

CLASIFICACIÓN DINÁMICA DE FLUJOS DE BELTRAMI

We study four configurations of Beltrami flows (BFs) defined as ∇ × v = ±γ ±v, where γ > 0 is an eigenvalue and which have a progressive rotating wave dynamics (PRWs) that satisfies the dynamic property (DP) [1], which allows us to classify them on the basis of the eigenvalues that result in each configuration. The first configuration corresponds to an infinite volume domain without contours. The classifier eigenvalue is γ ±ph = 2/ vph± where k is the modulus of the wave vector that forms an angle θ with the rotation axis. The result is a finite-amplitude, transverse, dispersive, circularly polarised, planar PRWs with a continuous spectrum. The second configuration has the same domain as configuration one. The classifying eigenvalue is γ ±ph = 2/ vph± with vph being the phase velocity, with vph+ < 0 and vph− > 0. They are axi-symmetric or non-axi-symmetric along the axis of rotation, of finite amplitude, non-dispersive and with motion between concentric cylinders at which the radial velocity equals zero. In the third configuration the fluid is confined in an infinite cylinder. The classifying eigenvalue is again γ ±ph but it results discretized by the boundary conditions on the cylinder wall. Classification is exemplified for vph+ = −0.1 and three rotating modes with m = 0, m = 1 y m = 2. These are finite amplitude dispersive PRWs. The fourth configuration consists of a rotational-translational flow, characterized by the Rossby number R0 (=U/a Ω) which is an intake flow to a semi-infinite cylinder. The classifying eigenvalue is γ ±ph with vph± = ∓R0. These are PRWs, of the same type as in the infinite cylinder, but dependent on R0. It is shown that these waves exist only in the interval R0 ∈ (0,0.642]. Where for R0 = 0.642 one has only the mode with m = 1 and as R0 decreases the modes m = 0 and m ≥ 2 arise successively. It is observed that, for the same R0, waves of the same sign of frequency do not exchange energy. For each configuration the possibilities and conditions of resonant triadic interactions are analyzed.

Chiral anomaly in Euler fluid and Beltrami flow

We show that barotropic flows of a perfect, charged, classical fluid exhibit an anomaly analogous to the chiral anomaly known in quantum field theories with Dirac fermions. A prominent effect of the chiral anomaly is the transport electric current in the fluid at equilibrium with the chiral reservoir. We find that it is also a property of celebrated Beltrami flows — stationary solutions of the Euler equation with an extensive helicity.

Rod-packing arrangements of invariant tori in solenoidal vector fields with cubic symmetries

The arrangements of invariant tori that resemble rod packings with cubic symmetries are considered in three-dimensional solenoidal vector fields. To find them systematically, vector fields whose components are represented in the form of multiple Fourier series with finite terms are classified using magnetic groups. The maximal magnetic group compatible with each arrangement is specified on the assumption that the cores of the nested invariant tori are straight and located on the lines corresponding to the central axes of the rods packed. Desired rod-packing arrangements are demonstrated by selecting vector fields whose magnetic groups are the maximal ones and by drawing their integral curves that twine around invariant tori. In the demonstration of chiral arrangements, Beltrami flows (or force-free fields in plasma physics), which have the strongest chirality of all solenoidal vector fields satisfying the same vector Helmholtz equation, are used. As by-products, several chain-like arrangements of closed invariant tori were found. One of the chains consists of knotted invariant tori. In all vector fields (chiral or achiral) selected for the demonstration, the volume percentages of ordered regions formed by invariant tori in a unit cell were roughly measured with the aid of a supervised machine learning technique.

On the role of continuous symmetries in the solution of the three-dimensional Euler fluid equations and related models.

We review and apply the continuous symmetry approach to find the solution of the three-dimensional Euler fluid equations in several instances of interest, via the construction of constants of motion and infinitesimal symmetries, without recourse to Noether’s theorem. We show that the vorticity field is a symmetry of the flow, so if the flow admits another symmetry then a Lie algebra of new symmetries can be constructed. For steady Euler flows this leads directly to the distinction of (non-)Beltrami flows: an example is given where the topology of the spatial manifold determines whether extra symmetries can be constructed. Next, we study the stagnation-point-type exact solution of the three-dimensional Euler fluid equations introduced by Gibbon et al. (Gibbon et al. 1999 Physica D 132, 497–510. (doi:10.1016/S0167-2789(99)00067-6)) along with a one-parameter generalization of it introduced by Mulungye et al. (Mulungye et al. 2015 J. Fluid Mech. 771, 468–502. (doi:10.1017/jfm.2015.194)). Applying the symmetry approach to these models allows for the explicit integration of the fields along pathlines, revealing a fine structure of blowup for the vorticity, its stretching rate and the back-to-labels map, depending on the value of the free parameter and on the initial conditions. Finally, we produce explicit blowup exponents and prefactors for a generic type of initial conditions.This article is part of the theme issue ‘Mathematical problems in physical fluid dynamics (part 2)’.

An improved Liouville type theorem for Beltrami flows

In this note, we improved the Liouville type theorem for the Beltrami flows. Two different methods are used to prove it, one is the monotonicity method, and another is proof by contradiction. The conditions that we proposed on Beltrami flows are significantly weaker than previously known conditions.

Dynamos driven by modified Beltrami flows, and a search for related planar flow dynamos

Bachtiar, Ivers and James [Planar velocity dynamos in a sphere. Proc. R. Soc. Lond. 2006, A462, 2439–2456]showed that a planar fluid velocity v can support dynamo action in a conducting sphere, however their modelp1Y22DM12 exhibited slow convergence. Seeking more planar flow dynamos, they also considered convertingsome historical flows to planar flows by a process they termed ‘planarising‘. In particular they studied oneof many flows considered by Pekeris, Accad and Shkoller [Kinematic dynamos and the earth's magnetic field. Phil. Trans. R. Soc. Lond. 1973, A275, 425–461]. This PAS flow was chosen because it produced adynamo at low truncation levels and low critical magnetic Reynolds number R c, properties enhanced bythe flow being a helical Beltrami flow. The original aim of the present project was to find a planar flowdynamo with faster convergence than p1Y22DM12. Whilst the PAS flow can only be partly planarised, wehave constructed two modifications, labelled biPAS and quasiPAS, which can be fully planarised. We havestudied 128 models using the PAS, biPAS, quasiPAS flows, and their part and full planarisations. We haveverified 12 known dynamos, found 84 new dynamos, but found no dynamos with fully planar flows. Thepartly-planarised PAS, and the biPAS- and quasiPAS-based flows are not Beltrami. But 20 (normalisedusing rms( v ), or 22 using max | v |) of the associated dynamos have lower R c than the original Beltrami PASdynamos from which they were derived, showing that the Beltrami property is not essential for low R c. Acomparison is also made herein with optimal enstrophy-normalised dynamos.

Exact Beltrami flows in a spherical shell

Abstract Exact flows of an incompressible fluid satisfying the Beltrami equation inside a spherical shell are constructed in the Cartesian coordinates in terms of elementary functions. Two scale-invariant equations defining two infinite series of eigenvalues λ n and λ ̃ m ${\tilde {\lambda }}_{m}$ of the operator curl in the shell with the nonpenetration boundary conditions on the boundary spheres are derived. The corresponding eigenfields are presented in explicit form and their symmetries are investigated. Asymptotics of the eigenvalues λ n and λ ̃ m ${\tilde {\lambda }}_{m}$ at n, m → ∞ are obtained.

Existence and structure of symmetric Beltrami flows on compact 3-manifolds

We show that for almost every given symmetry transformation of a Riemannian manifold there exists an eigenvector field of the curl operator, corresponding to a non-zero eigenvalue, which obeys the symmetry. More precisely, given a smooth, compact, oriented Riemannian 3-manifold (M¯,g) with (possibly empty) boundary and a smooth flow of isometries ϕt:M¯→M¯ we show that, if M¯ has non-empty boundary or if the infinitesimal generator is not purely harmonic, there is a smooth vector field X, tangent to the boundary, which is an eigenfield of curl and satisfies (ϕt)⁎X=X, i.e. is invariant under the pushforward of the symmetry transformation. We then proceed to show that if the quantities involved are real analytic and (M¯,g) has non-empty boundary, then Arnold's structure theorem applies to all eigenfields of curl, which obey a symmetry and appropriate boundary conditions. More generally we show that the structure theorem applies to all real analytic vector fields of non-vanishing helicity which obey some nontrivial symmetry. A byproduct of our proof is a characterisation of the flows of real analytic Killing fields on compact, connected, orientable 3-manifolds with and without boundary.

Explicit superposed and forced plane wave generalized Beltrami flows

AbstractWe revisit and present new linear spaces of explicit solutions to incompressible Euler and Navier–Stokes equations on ${{{\mathbb{R}}}}^n$, as well as the rotating Boussinesq equations on ${{{\mathbb{R}}}}^3$. We cast these solutions are superpositions of certain linear plane waves of arbitrary amplitudes that also solve the nonlinear equations by constraints on wave vectors and flow directions. For $n\leqslant 3$, these are explicit examples for generalized Beltrami flows. We show that forcing terms of corresponding plane wave type yield explicit solutions by linear variation of constants. We work in Eulerian coordinates and distinguish the two situations of vanishing and of gradient nonlinear terms, where the nonlinear terms modify the pressure. The methods that we introduce to find explicit solutions in nonlinear fluid models can also be used in other equations with material derivative. Our approach offers another view on known explicit solutions of different fluid models from a plane wave perspective and provides transparent nonlinear interactions between different flow components.

Typical field lines of Beltrami flows and boundary field line behaviour of Beltrami flows on simply connected, compact, smooth manifolds with boundary

We characterise the boundary field line behaviour of Beltrami flows on compact, connected manifolds with vanishing first de Rham cohomology group. Namely we show that except for an at most nowhere dense subset of the boundary, on which the Beltrami field may vanish, all other field lines at the boundary are smoothly embedded 1-manifolds diffeomorphic to {mathbb {R}}, which approach the zero set as time goes to pm , infty. We then drop the assumptions of compactness and vanishing de Rham cohomology and prove that for almost every point on the given manifold, the field line passing through the point is either a non-constant, periodic orbit or a non-periodic orbit which comes arbitrarily close to the starting point as time goes to pm infty. During the course of the proof, we in particular show that the set of points at which a Beltrami field vanishes in the interior of the manifold is countably 1-rectifiable in the sense of Federer and hence in particular has a Hausdorff dimension of at most 1. As a consequence, we conclude that for every eigenfield of the curl operator, corresponding to a non-zero eigenvalue, there always exists exactly one nodal domain.

Global solutions of 3D tropical climate model with finite energy

In this paper, we construct a family of finite energy classical solutions to the 3D tropical climate model equations. The approach is by studying the steady-state Beltrami flows and using the standard cut-off technique to establish the global regularity of the systems.

On the global stability of smooth solutions of the Navier–Stokes equations

In Morosi and Pizzocchero (2015) and previous papers by the same authors, a general smooth setting was proposed for the incompressible Navier–Stokes (NS) Cauchy problem on a torus of any dimension d⩾2, and the a posteriori analysis of its approximate solutions. In this note, using the same setting I propose an elementary proof of the following statement: global existence and time decay of the NS solutions are stable properties with respect to perturbations of the initial datum. Fully explicit estimates are derived, using Sobolev norms of arbitrarily high order. An application is proposed, in which the initial data are generalized Beltrami flows. A comparison with the related literature is performed.

Objective barriers to the transport of dynamically active vector fields

Abstract