ABC Flow Research Articles

Inertial particles distribute in turbulence as Poissonian points with random intensity inducing clustering and supervoiding

This work considers the distribution of inertial particles in turbulence using the point-particle approximation. We demonstrate that the random point process formed by the positions of particles in space is a Poisson point process with log-normal random intensity ("log Gaussian Cox process" or LGCP). The probability of having a finite number of particles in a small volume is given in terms of the characteristic function of a log-normal distribution. Corrections due to discreteness of the number of particles to the previously derived statistics of particle concentration in the continuum limit are provided. These are relevant for dealing with experimental or numerical data. The probability of having regions without particles, i.e. voids, is larger for inertial particles than for tracer particles where voids are distributed according to Poisson processes. Further, the probability of having large voids decays only log-normally with size. This shows that particles cluster, leaving voids behind. At scales where there is no clustering there can still be an increase of the void probability so that turbulent voiding is stronger than clustering. The demonstrated double stochasticity of the distribution originates in the two-step formation of fluctuations. First, turbulence brings the particles randomly close together which happens with Poisson-type probability. Then, turbulence compresses the particles' volume in the observation volume. We confirm the theory of the statistics of the number of particles in small volumes by numerical observations of inertial particle motion in a chaotic ABC flow. The improved understanding of clustering processes can be applied to predict the long-time survival probability of reacting particles. Our work implies that the particle distribution in weakly compressible flow with finite time correlations is a LGCP, independently of the details of the flow statistics.

Clustering of long flexible fibers in two-dimensional flow fields for different Stokes numbers

The present work studies numerically the concentration distributions of long flexible fibers in two analytical flow fields. Fibers are modeled as a set of rigid cylinders, resulting in a continuous flexible fiber. Forces are applied to the center of mass of each cylinder, thus determining cylinder motion and fiber deformation using a fourth-order Runge-Kutta method to solve the system of ordinary differential equations. The numerical code is validated against experimental data available in the literature. The analytical velocity fields are steady and two-dimensional and correspond to the Arnold-Beltrami-Childress (ABC) flows. Simulations are performed for a wide range of Stokes numbers (St), which relate the characteristic time of the fibers to the characteristic time of the flow. Previous studies with spherical particles from other authors show that there is a preferential concentration of particles in regions of low vorticity, with a maximum preferential concentration for St values around 1. In the present study, it is shown that preferential concentration of fibers for the studied ABC flows has a maximum for St=0.3. Fibers tend to concentrate also in regions of low vorticity in both cases studied, and the frequency of caustics formation correlates strongly with the increase of preferential concentration.

A Theoretical Framework for Lagrangian Descriptors

This paper provides a theoretical background for Lagrangian Descriptors (LDs). The goal of achieving rigorous proofs that justify the ability of LDs to detect invariant manifolds is simplified by introducing an alternative definition for LDs. The definition is stated for [Formula: see text]-dimensional systems with general time dependence, however we rigorously prove that this method reveals the stable and unstable manifolds of hyperbolic points in four particular 2D cases: a hyperbolic saddle point for linear autonomous systems, a hyperbolic saddle point for nonlinear autonomous systems, a hyperbolic saddle point for linear nonautonomous systems and a hyperbolic saddle point for nonlinear nonautonomous systems. We also discuss further rigorous results which show the ability of LDs to highlight additional invariants sets, such as [Formula: see text]-tori. These results are just a simple extension of the ergodic partition theory which we illustrate by applying this methodology to well-known examples, such as the planar field of the harmonic oscillator and the 3D ABC flow. Finally, we provide a thorough discussion on the requirement of the objectivity (frame-invariance) property for tools designed to reveal phase space structures and their implications for Lagrangian descriptors.

Large- to small-scale dynamo in domains of large aspect ratio: kinematic regime

The Sun's magnetic field exhibits coherence in space and time on much larger scales than the turbulent convection that ultimately powers the dynamo. In this work, we look for numerical evidence of a large-scale magnetic field as the magnetic Reynolds number, Rm, is increased. The investigation is based on the simulations of the induction equation in elongated periodic boxes. The imposed flows considered are the standard ABC flow (named after Arnold, Beltrami & Childress) with wavenumber ku = 1 (small-scale) and a modulated ABC flow with wavenumbers ku = m, 1, 1 ± m, where m is the wavenumber corresponding to the long-wavelength perturbation on the scale of the box. The critical magnetic Reynolds number |$R_{\rm m}^{{\rm crit}}$| decreases as the permitted scale separation in the system increases, such that |$R_{\rm m}^{{\rm crit}} \propto [L_x/L_z]^{-1/2}$|⁠. The results show that the α-effect derived from the mean-field theory ansatz is valid for a small range of Rm after which small scale dynamo instability occurs and the mean-field approximation is no longer valid. The transition from large- to small-scale dynamo is smooth and takes place in two stages: a fast transition into a predominantly small-scale magnetic energy state and a slower transition into even smaller scales. In the range of Rm considered, the most energetic Fourier component corresponding to the structure in the long x-direction has twice the length-scale of the forcing scale. The long-wavelength perturbation imposed on the ABC flow in the modulated case is not preserved in the eigenmodes of the magnetic field.

High-order visualization of three-dimensional lagrangian coherent structures with DG-FTLE

Abstract The two-dimensional DG-FTLE algorithm for computing Finite-Time Lyapunov Exponent (FTLE) fields developed in [Nelson and Jacobs, J. Comp. Phys. 295 (2015)] is extended to and tested in three-dimensions and for complex geometries with curved boundaries. In three-dimensions, flow deformation maps, whose eigenvalues determine the FTLE, develop steep gradients over shorter finite times than in two-dimensions. Higher-order DG approximations of the steep gradients yield Gibbs oscillations and poor conditioning of interpolation matrices. To reduce Gibbs oscillations, an exponential filter is used that smoothens FTLE fields. h -adaptivity is shown to improve conditioning. To improve computational efficiency of large scale three-dimensional computations, the DG-FTLE algorithm is adapted to parallel architectures. Three test cases assess the performance of DG-FTLE on curved geometries and in three-dimensions including the two- and three-dimensional viscous flow over a NACA 65-(1)412 airfoil and the ABC flow. The test cases show the efficacy of the exponential filter and h -adaptivity in removing Gibbs oscillations and improving conditioning. The cases also show the ability of DG-FTLE to visualize FTLE fields in three-dimensional, unsteady flows over complex geometries.

Optimal Length Scale for a Turbulent Dynamo.

We demonstrate that there is an optimal forcing length scale for low Prandtl number dynamo flows that can significantly reduce the required energy injection rate. The investigation is based on simulations of the induction equation in a periodic box of size 2πL. The flows considered are the laminar and turbulent ABC flows forced at different forcing wave numbers k_{f}, where the turbulent case is simulated using a subgrid turbulence model. At the smallest allowed forcing wave number k_{f}=k_{min}=1/L the laminar critical magnetic Reynolds number Rm_{c}^{lam} is more than an order of magnitude smaller than the turbulent critical magnetic Reynolds number Rm_{c}^{turb} due to the hindering effect of turbulent fluctuations. We show that this hindering effect is almost suppressed when the forcing wave number k_{f} is increased above an optimum wave number k_{f}L≃4 for which Rm_{c}^{turb} is minimum. At this optimal wave number, Rm_{c}^{turb} is smaller by more than a factor of 10 than the case forced in k_{f}=1. This leads to a reduction of the energy injection rate by 3 orders of magnitude when compared to the case where the system is forced at the largest scales and thus provides a new strategy for the design of a fully turbulent experimental dynamo.

Ballistic Orbits and Front Speed Enhancement for ABC Flows

We study the two main types of trajectories of the ABC flow in the near-integrable regime: spiral orbits and edge orbits. The former are helical orbits which are perturbations of similar orbits that exist in the integrable regime, while the latter exist only in the nonintegrable regime. We prove existence of ballistic (i.e., linearly growing) spiral orbits by using the contraction mapping principle in the Hamiltonian formulation, and we also find and analyze ballistic edge orbits. We discuss the relationship of existence of these orbits with questions concerning front propagation in the presence of flows, in particular the question of linear (i.e., maximal possible) front speed enhancement rate for ABC flows.

Periodic Orbits of the ABC Flow with A=B=C=1

In this paper, we prove that the ODE system $$ \begin{align*} \dot x &=\sin z+\cos y\\ \dot y &= \sin x+\cos z\\ \dot z &=\sin y + \cos x, \end{align*} $$ whose right-hand side is the Arnold-Beltrami-Childress (ABC) flow with parameters $A=B=C=1$, has periodic orbits on $(2\pi\mathbb T)^3$ with rotation vectors parallel to $(1,0,0)$, $(0,1,0)$, and $(0,0,1)$. An application of this result is that the well-known G-equation model for turbulent combustion with this ABC flow on $\mathbb R^3$ has a linear (i.e., maximal possible) flame speed enhancement rate as the amplitude of the flow grows.

An objective perspective for classic flow classification criteria

Four classic criteria used to the classification of complex flows are discussed here. These criteria are useful to identify regions of the flow related to shear, elongation or rigid-body motion. These usual criteria, namely $Q$, $\Delta$, $\lambda_{2}$ and $\lambda_{cr}/\lambda_{ci}$, use the fluid's rate-of-rotation tensor, which is known to vary with respect to a reference frame. The advantages of using objective (invariant with respect to a general transformation on the reference frame) criteria are discussed in the present work. In this connection, we construct versions of classic criteria replacing standard vorticity, a non-objective quantity, by effective vorticity, a rate of rotation tensor with respect to the angular velocity of the eigenvectors of the strain rate tensor. The classic criteria and their corresponding objective versions are applied to classify two complex flows: the transient ABC flow and the flow through the abrupt 4:1 contraction. It is shown that the objective versions of the criteria provide richer information on the kinematics of the flow

Classification of Arnold-Beltrami flows and their hidden symmetries

In the context of mathematical hydrodynamics, we consider the group theory structure which underlies the so named ABC flows introduced by Beltrami, Arnold and Childress. Main reference points are Arnold’s theorem stating that, for flows taking place on compact three manifolds ℳ3, the only velocity fields able to produce chaotic streamlines are those satisfying Beltrami equation and the modern topological conception of contact structures, each of which admits a representative contact one-form also satisfying Beltrami equation. We advocate that Beltrami equation is nothing else but the eigenstate equation for the first order Laplace-Beltrami operator ★ g d, which can be solved by using time-honored harmonic analysis. Taking for ℳ3, a torus T 3 constructed as ℝ3/Λ, where Λ is a crystallographic lattice, we present a general algorithm to construct solutions of the Beltrami equation which utilizes as main ingredient the orbits under the action of the point group B A of three-vectors in the momentum lattice *Λ. Inspired by the crystallographic construction of space groups, we introduce the new notion of a Universal Classifying Group $$\mathfrak{G}\mathfrak{A}_\Lambda$$ which contains all space groups as proper subgroups. We show that the ★ g d eigenfunctions are naturally arranged into irreducible representations of $$\mathfrak{G}\mathfrak{A}_\Lambda$$ and by means of a systematic use of the branching rules with respect to various possible subgroups $$H_i \subset \mathfrak{G}\mathfrak{A}_\Lambda$$ we search and find Beltrami fields with non trivial hidden symmetries. In the case of the cubic lattice the point group is the proper octahedral group O24 and the Universal Classifying Group $$\mathfrak{G}\mathfrak{A}_{cubic}$$ is a finite group G1536 of order |G1536| = 1536 which we study in full detail deriving all of its 37 irreducible representations and the associated character table. We show that the O24 orbits in the cubic lattice are arranged into 48 equivalence classes, the parameters of the corresponding Beltrami vector fields filling all the 37 irreducible representations of G1536. In this way we obtain an exhaustive classification of all generalized ABC-flows and of their hidden symmetries. We make several conceptual comments about the need of a field-theory yielding Beltrami equation as a field equation and/or an instanton equation and on the possible relation of Arnold-Beltrami flows with (supersymmetric) Chern-Simons gauge theories. We also suggest linear generalizations of Beltrami equation to higher odd-dimensions that are different from the non-linear one proposed by Arnold and possibly make contact with M-theory and the geometry of flux-compactifications.

Toward an asymptotic behaviour of the ABC dynamo

The ABC flow was originally introduced by Arnol'd to investigate Lagrangian chaos. It soon became the prototype example to illustrate magnetic-field amplification via fast dynamo action, i.e. dynamo action exhibiting magnetic-field amplification on a typical timescale independent of the electrical resistivity of the medium. Even though this flow is the most classical example for this important class of dynamos (with application to large-scale astrophysical objects), it was recently pointed out (Bouya Ismaël and Dormy Emmanuel, Phys. Fluids, 25 (2013) 037103) that the fast dynamo nature of this flow was unclear, as the growth rate still depended on the magnetic Reynolds number at the largest values available so far . Using state-of-the-art high-performance computing, we present high-resolution simulations (up to 40963) and extend the value of up to . Interestingly, even at these huge values, the growth rate of the leading eigenmode still depends on the controlling parameter and an asymptotic regime is not reached yet. We show that the maximum growth rate is a decreasing function of for the largest values of we could achieve (as anticipated in the above-mentioned paper). Slowly damped oscillations might indicate either a new mode crossing or that the system is approaching the limit of an essential spectrum.

Anomalous diffusion of field lines and charged particles in Arnold-Beltrami-Childress force-free magnetic fields

The cosmic magnetic fields in regions of low plasma pressure and large currents, such as in interstellar space and gaseous nebulae, are force-free in the sense that the Lorentz force vanishes. The three-dimensional Arnold-Beltrami-Childress (ABC) field is an example of a force-free, helical magnetic field. In fluid dynamics, ABC flows are steady state solutions of the Euler equation. The ABC magnetic field lines exhibit a complex and varied structure that is a mix of regular and chaotic trajectories in phase space. The characteristic features of field line trajectories are illustrated through the phase space distribution of finite-distance and asymptotic-distance Lyapunov exponents. In regions of chaotic trajectories, an ensemble-averaged variance of the distance between field lines reveals anomalous diffusion—in fact, superdiffusion—of the field lines. The motion of charged particles in the force-free ABC magnetic fields is different from the flow of passive scalars in ABC flows. The particles do not necessarily follow the field lines and display a variety of dynamical behavior depending on their energy, and their initial pitch-angle. There is an overlap, in space, of the regions in which the field lines and the particle orbits are chaotic. The time evolution of an ensemble of particles, in such regions, can be divided into three categories. For short times, the motion of the particles is essentially ballistic; the ensemble-averaged, mean square displacement is approximately proportional to t2, where t is the time of evolution. The intermediate time region is defined by a decay of the velocity autocorrelation function—this being a measure of the time after which the collective dynamics is independent of the initial conditions. For longer times, the particles undergo superdiffusion—the mean square displacement is proportional to tα, where α > 1, and is weakly dependent on the energy of the particles. These super-diffusive characteristics, both of magnetic field lines and of particles moving in these fields, strongly suggest that theories of transport in three-dimensional chaotic magnetic fields need a shift from the usual paradigm of quasilinear diffusion.

Hyperbolic and elliptic transport barriers in three-dimensional unsteady flows

We develop a general theory of transport barriers for three-dimensional unsteady flows with arbitrary time-dependence. The barriers are obtained as two-dimensional Lagrangian Coherent Structures (LCSs) that create locally maximal deformation. Along hyperbolic LCSs, this deformation is induced by locally maximal normal repulsion or attraction. Along shear LCSs, the deformation is created by locally maximal tangential shear. Hyperbolic LCSs, therefore, play the role of generalized stable and unstable manifolds, while closed shear LCSs (elliptic LCSs) act as generalized KAM tori or KAM-type cylinders. All these barriers can be computed from our theory as explicitly parametrized surfaces. We illustrate our results by visualizing two-dimensional hyperbolic and elliptic barriers in steady and unsteady versions of the ABC flow.

Detecting invariant manifolds as stationary Lagrangian coherent structures in autonomous dynamical systems

Normally hyperbolic invariant manifolds (NHIMs) are well-known organizing centers of the dynamics in the phase space of a nonlinear system. Locating such manifolds in systems far from symmetric or integrable, however, has been an outstanding challenge. Here, we develop an automated detection method for codimension-one NHIMs in autonomous dynamical systems. Our method utilizes Stationary Lagrangian Coherent Structures (SLCSs), which are hypersurfaces satisfying one of the necessary conditions of a hyperbolic LCS, and are also quasi-invariant in a well-defined sense. Computing SLCSs provides a quick way to uncover NHIMs with high accuracy. As an illustration, we use SLCSs to locate two-dimensional stable and unstable manifolds of hyperbolic periodic orbits in the classic ABC flow, a three-dimensional solution of the steady Euler equations.

Dynamo action in the ABC flows using symmetries

This paper concerns kinematic dynamo action by the 1:1:1 ABC flow, in the highly conducting limit of large magnetic Reynolds number . The flow possesses 24 symmetries, with a symmetry group isomorphic to the group of orientation-preserving transformations of a cube. This can be exploited to break up the linear eigenvalue problem into five distinct symmetry classes or irreducible representations, which we label I–V. The paper discusses how to reduce the scale of the numerical problem to a subset of Fourier modes for a magnetic field in each representation, which then may be solved independently to obtain distinct branches of eigenvalues and magnetic field eigenfunctions. Two numerical methods are employed: the first is to time step a magnetic field in a given symmetry class and obtain the growth rate and frequency by measuring the magnetic energy as a function of time. The second method involves a more direct determination of the eigenvalue using the eigenvalue solver ARPACK for sparse matrix systems, which employs an implicitly restarted Arnoldi method. The two methods are checked against each other and compared for efficiency and reliability. Eigenvalue branches for each symmetry class are obtained for magnetic Reynolds numbers up to together with spectra and magnetic field visualisations. A sequence of branches emerges as increases and the magnetic field structures in the different branches are discussed and compared. In a parallel development, results are presented for the corresponding fluid stability problem as a function of the Reynolds number .

Inverse cascades in turbulence and the case of rotating flows

We first summarize briefly several properties concerning the dynamics of two-dimensional (2D) turbulence, with an emphasis on the inverse cascade of energy to the largest accessible scale of the system. In order to study a similar phenomenon in 3D turbulence undergoing strong solid-body rotation, we test a previously developed large eddy simulation (LES) model against a high-resolution direct numerical simulation of rotating turbulence on a grid of 30723 points. We then describe new numerical results on the inverse energy cascade in rotating flows using this LES model and contrast the case of 2D versus 3D forcing, as well as non-helical forcing (i.e. with weak overall alignment between velocity and vorticity) versus the fully helical Beltrami case, for both deterministic and random forcing. The different scaling laws for the inverse energy cascade can be attributed to the dimensionality of the forcing, with either a k−3⊥ or a k−5/3⊥ energy spectrum of slow modes at large scales, k⊥ referring to a direction perpendicular to that of rotation. We finally invoke the role of shear in the case of a strongly anisotropic deterministic forcing, using the so-called ABC flow; in that case, a k−5/3⊥ is again observed for the slow modes, together with a k−1 spectrum for the total energy associated with enhanced shear at a large scale [].

Helicity dynamics in stratified turbulence in the absence of forcing

A numerical study of decaying stably stratified flows is performed. Relatively high stratification (Froude number ≈10(-2)-10(-1)) and moderate Reynolds (Re) numbers (Re≈ 3-6×10(3)) are considered and a particular emphasis is placed on the role of helicity (velocity-vorticity correlations), which is not an invariant of the nondissipative equations. The problem is tackled by integrating the Boussinesq equations in a periodic cubical domain using different initial conditions: a nonhelical Taylor-Green (TG) flow, a fully helical Beltrami [Arnold-Beltrami-Childress (ABC)] flow, and random flows with a tunable helicity. We show that for stratified ABC flows helicity undergoes a substantially slower decay than for unstratified ABC flows. This fact is likely associated to the combined effect of stratification and large-scale coherent structures. Indeed, when the latter are missing, as in random flows, helicity is rapidly destroyed by the onset of gravitational waves. A type of large-scale dissipative "cyclostrophic" balance can be invoked to explain this behavior. No production of helicity is observed, contrary to the case of rotating and stratified flows. When helicity survives in the system, it strongly affects the temporal energy decay and the energy distribution among Fourier modes. We discover in fact that the decay rate of energy for stratified helical flows is much slower than for stratified nonhelical flows and can be considered with a phenomenological model in a way similar to what is done for unstratified rotating flows. We also show that helicity, when strong, has a measurable effect on the Fourier spectra, in particular at scales larger than the buoyancy scale, for which it displays a rather flat scaling associated with vertical shear, as observed in the planetary boundary layer.

Leaving flatland: Diagnostics for Lagrangian coherent structures in three-dimensional flows

Finite-time Lyapunov exponents (FTLE) are often used to identify Lagrangian Coherent Structures (LCS). Most applications are confined to flows on two-dimensional (2D) surfaces where the LCS are characterized as curves. The extension to three-dimensional (3D) flows, whose LCS are 2D structures embedded in a 3D volume, is theoretically straightforward. However, in geophysical flows at regional scales, full prognostic computation of the evolving 3D velocity field is not computationally feasible. The vertical or diabatic velocity, then, is either ignored or estimated as a diagnostic quantity with questionable accuracy. Even in cases with reliable 3D velocities, it may prove advantageous to minimize the computational burden by calculating trajectories from velocities on carefully chosen surfaces only. When reliable 3D velocity information is unavailable or one velocity component is explicitly ignored, a reduced FTLE form to approximate 2D LCS surfaces in a 3D volume is necessary. The accuracy of two reduced FTLE formulations is assessed here using the ABC flow and a 3D quadrupole flow as test models. One is the standard approach of knitting together FTLE patterns obtained on adjacent surfaces. The other is a new approximation accounting for the dispersion due to vertical (u,v) shear. The results are compared with those obtained from the full 3D velocity field. We introduce two diagnostic quantities to identify situations when a fully 3D computation is required for an accurate determination of the 2D LCS. For the ABC flow, we found the full 3D calculation to be necessary unless the vertical (u,v) shear is sufficiently small. However, both methods compare favorably with the 3D calculation for the quadrupole model scaled to typical open ocean conditions.

Kinetic helicity needed to drive large-scale dynamos

Magnetic field generation on scales that are large compared with the scale of the turbulent eddies is known to be possible via the so-called α effect when the turbulence is helical and if the domain is large enough for the α effect to dominate over turbulent diffusion. Using three-dimensional turbulence simulations, we show that the energy of the resulting mean magnetic field of the saturated state increases linearly with the product of normalized helicity and the ratio of domain scale to eddy scale, provided this product exceeds a critical value of around unity. This implies that large-scale dynamo action commences when the normalized helicity is larger than the inverse scale ratio. Our results show that the emergence of small-scale dynamo action does not have any noticeable effect on the large-scale dynamo. Recent findings by Pietarila Graham et al. [Phys. Rev. E 85, 066406 (2012)] of a smaller minimal helicity may be an artifact due to the onset of small-scale dynamo action at large magnetic Reynolds numbers. However, the onset of large-scale dynamo action is difficult to establish when the kinetic helicity is small. Instead of random forcing, they used an ABC flow with time-dependent phases. We show that such dynamos saturate prematurely in a way that is reminiscent of inhomogeneous dynamos with internal magnetic helicity fluxes. Furthermore, even for very low fractional helicities, such dynamos display large-scale fields that change direction, which is uncharacteristic of turbulent dynamos.

Finite Element Computation of KPP Front Speeds in 3D Cellular and ABC Flows

We carried out a computational study of propagation speeds of reaction-diffusion-advection fronts in three dimensional (3D) cellular and Arnold-Beltrami-Childress (ABC) flows with Kolmogorov-Petrovsky-Piskunov(KPP) nonlinearity. The variational principle of front speeds reduces the problem to a principal eigenvalue calculation. An adaptive streamline diffusion finite element method is used in the advection dominated regime. Numerical results showed that the front speeds are enhanced in cellular flows according to sublinear power law O (δ p ), p ≈ 0.13, δ the flow intensity. In ABC flows however, the enhancement is O (δ ) which can be attributed to the presence of principal vortex tubes in the streamlines. Poincare sections are used to visualize and quantify the chaotic fractions of ABC flows in the phase space. The effect of chaotic streamlines of ABC flows on front speeds is studied by varying the three parameters (a,b,c ) of the ABC flows. Speed enhancement along x direction is reduced as b (the parameter controling the flow variation along x ) increases at fixed (a,c ) > 0, more rapidly as the corresponding ABC streamlines become more chaotic.