ABC Flow Research Articles

An augmented subspace based adaptive proper orthogonal decomposition method for time dependent partial differential equations

In this paper, we propose an augmented subspace based adaptive proper orthogonal decomposition (POD) method for solving the time dependent partial differential equations. We use the difference between the approximation obtained in the augmented subspace and that obtained in the original POD subspace to construct an error indicator, by which we obtain a general framework for augmented subspace based adaptive POD method. We then provide two strategies to construct the augmented subspaces, the residual type augmented subspace and the coarse-grid approximation type augmented subspace. We apply our new methods to two typical 3D advection-diffusion equations with the advection being the Kolmogorov flow and the ABC flow. Numerical results show that both the residual type augmented subspace based adaptive POD method and the coarse-grid approximation type augmented subspace based adaptive POD method are more efficient than the existing adaptive POD methods, especially for the advection dominated models.

Variational construction of tubular and toroidal streamsurfaces for flow visualization

Approximate streamsurfaces of a three-dimensional velocity field have recently been constructed as isosurfaces of the closest first integral of the velocity field. Such approximate streamsurfaces enable effective and efficient visualization of vortical regions in three-dimensional flows. Here we propose a variational construction of these approximate streamsurfaces to remove the limitation of Fourier series representation of the first integral in earlier work. Specifically, we use finite-element methods to solve a partial differential equation that describes the best approximate first integral for a given velocity field. We use several examples to demonstrate the power of our approach for three-dimensional flows in domains with arbitrary geometries and boundary conditions. These include generalized axisymmetric flows in the domains of a sphere (spherical vortex), a cylinder (cylindrical vortex) and a hollow cylinder (Taylor–Couette flow) as benchmark studies for various computational domains, non-integrable periodic flows (ABC and Euler flows) and Rayleigh–Bénard convection flows. We also illustrate the use of the variational construction in extracting momentum barriers in Rayleigh–Bénard convection.

Anisotropic Mean Flow Enhancement and Anomalous Transport of Finite-Size Spherical Particles in Turbulent Flows.

We investigate the influence of dispersed solid spherical particles on the largest scales of the turbulent Arnold-Beltrami-Childress (ABC) flow. The ABC flow is an ideal instance of a complex flow: it does not have solid boundaries, but possesses an inhomogeneous and three-dimensional mean shear. By tuning the parameters of the suspension, we show that particles modulate the largest scales of the flow toward an anisotropic, quasi-two-dimensional and more energetic state. In this regime, particles move along quasistraight trajectories and exhibit anomalous transport.

Bifurcation Analysis Software and Chaotic Dynamics for Some Problems in Fluid Dynamics Laminar–Turbulent Transition

The analysis of bifurcations and chaotic dynamics for nonlinear systems of a large size is a difficult problem. Analytical and numerical approaches must be used to deal with this problem. Numerical methods include solving some of the hardest problems in computational mathematics, which include system spectral and algebraic problems, specific nonlinear numerical methods, and computational implementation on parallel architectures. The software structure that is required to perform numerical bifurcation analysis for large-scale systems was considered in the paper. The software structure, specific features that are used for successful bifurcation analysis, globalization strategies, stabilization, and high-precision implementations are discussed. We considered the bifurcation analysis in the initial boundary value problem for a system of partial differential equations that describes the dynamics of incompressible ABC flow (3D Navier–Stokes equations). The initial stationary solution is characterized by the stability and connectivity to the main solutions branches. Periodic solutions were considered in view of instability transition problems. Finally, some questions of higher dimensional attractors and chaotic regimes are discussed.

A kind of Lagrangian chaotic property of the Arnold–Beltrami–Childress flow

Three-dimensional steady-state Arnold–Beltrami–Childress (ABC) flow has a chaotic Lagrangian structure, and also satisfies the Navier–Stokes (NS) equations with an external force per unit mass. It is well known that, although trajectories of a chaotic system have sensitive dependence on initial conditions, i.e. the famous ‘butterfly effect’, their statistical properties are often insensitive to small disturbances. This kind of chaos (such as governed by the Lorenz equations) is called normal-chaos. However, a new concept, i.e. ultra-chaos, has been reported recently, whose statistics are unstable to tiny disturbances. Thus, ultra-chaos represents higher disorder than normal chaos. In this paper, we illustrate that ultra-chaos widely exists in Lagrangian trajectories of fluid particles in steady-state ABC flow. Moreover, solving the NS equation when $Re=50$ with the ABC flow plus a very small disturbance as the initial condition, it is found that trajectories of nearly all fluid particles become ultra-chaotic when the transition from laminar to turbulence occurs. These numerical experiments and facts highly suggest that ultra-chaos should have a relationship with turbulence. This paper identifies differences between ultra-chaos and sensitivity of statistics to parameters. Possible relationships between ultra-chaos and the Poincaré section, ultra-chaos and ergodicity/non-ergodicity, etc., are discussed. The concept of ultra-chaos opens a new perspective of chaos, the Poincaré section, ergodicity/non-ergodicity, turbulence and their inter-relationships.

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.

ABC-like flows on the 3-torus.

We introduce a 3-parameter family of vector fields on the 3-torus as a linear combination of unit eigenfields of the curl operator for the eigenvalue 2. For this family reminiscent of the classical ABC flow, we study the existence of stationary points, we give numerical evidence for the existence of chaotic regions, and we present an integrable case. Our main result is that the non-vanishing members of the family are associated with tight contact structures.

A Semi-Lagrangian Computation of Front Speeds of G-equation in ABC and Kolmogorov Flows with Estimation via Ballistic Orbits

The Arnold-Beltrami-Childress (ABC) flow and the Kolmogorov flow are three dimensional periodic divergence free velocity fields that exhibit chaotic streamlines. We are interested in front speed enhancement in G-equation of turbulent combustion by large intensity ABC and Kolmogorov flows. We give a quantitative construction of the ballistic orbits of ABC and Kolmogorov flows, namely those with maximal large time asymptotic speeds in a coordinate direction. Thanks to the optimal control theory of G-equation (a convex but non-coercive Hamilton-Jacobi equation), the ballistic orbits serve as admissible trajectories for front speed estimates. To study the tightness of the estimates, we compute the front speeds of G-equation based on a semi-Lagrangian (SL) scheme with Strang splitting and weighted essentially non-oscillatory (WENO) interpolation. Time step size is chosen so that the Courant number grows sublinearly with the flow intensity. Numerical results show that the front speed growth rate in terms of the flow intensity may approach the analytical bounds from the ballistic orbits.

Network and geometric characterization of three-dimensional fluid transport between two layers.

We consider transport in a fluid flow of arbitrary complexity but with a dominant flow direction. One of the situations in which this occurs is when describing by an effective flow the dynamics of sufficiently small particles immersed in a turbulent fluid and vertically sinking because of their weight. We develop a formalism characterizing the dynamics of particles released from one layer of fluid and arriving in a second one after traveling along the dominant direction. The main ingredient in our study is the definition of a two-layer map that describes the Lagrangian transport between both layers. We combine geometric approaches and probabilistic network descriptions to analyze the two-layer map. From the geometric point of view, we express the properties of lines, surfaces, and densities transported by the flow in terms of singular values related to Lyapunov exponents, and define a specific quantifier, the finite depth Lyapunov exponent. Within the network approach, degrees and an entropy are considered to characterize transport. We also provide relationships between both methodologies. The formalism is illustrated with numerical results for a modification of the ABC flow, a model commonly studied to characterize three-dimensional chaotic advection.

Gereja dan Pandemi Covid-19

Nearly all people of the church is exposed to pandemic covid-1, instead they are not exposed to sars virus covid-19, they are exposed to predisposition, frailty, and any other social impacts. Indicators of ABC, Attendance (crowded by visitors), Building and Cash flow currently deemed as measurement of successful church service abruptly changed amid covid-19. In fact, church visitors drastically decreased, church building would be vacant and cash flow particularly income definitely declines. If in history the world was suffered from pandemic for several times, it suggested that a pandemic may happens in such a prolonged time, then church is unlikely to dwell on this condition, church is unlikely to be underdeveloped with the same pattern. This study offered the possibility to church to switch to solidarity services, the mission transfers to the deepest part of the church. There is a challenge for the church to transform into more humanity, pursue solidarity, try to be more gracious, be concerned about other. Identification of solidarity and brotherhood becomes indicators of missional Church. The mission shifted into any different form of loving service, particularly for those who are the most vulnerable.

Large eddy simulations of high-magnetic Reynolds number magnetohydrodynamic turbulence for non-helical and helical initial conditions: A study of two sub-grid scale models

Pseudo-spectral large eddy simulation (LES) calculations of high-magnetic Reynolds number (Rem) incompressible magnetohydrodynamic (MHD) turbulence are carried out for two initial conditions, namely, the non-helical Orszag–Tang vortex and the strongly helical Arnold–Beltrami–Childress (ABC) flows using two eddy-viscosity-based sub-grid scale (SGS) approaches: the cross-helicity (CH) and dynamic Smagorinsky (DS) models. Validation is conducted through comparisons of 1923 LES calculations with in-house 5123 direct numerical simulations (DNS) at Reynolds number, Re=Rem=800. The results show that the CH model performs better than the DS model. The performance of the SGS models at higher Re is further evaluated by carrying out 3843 LES calculations at Re=Rem=7500. Various quantities including turbulent kinetic energy, turbulent magnetic energy, cross-helicity, helicity, vorticity structures, cosine of angle between velocity and magnetic field, cosine of angle between velocity and vorticity field, kinetic and magnetic energy spectra, and energy fluxes are analyzed to understand the capability of the two LES models in predicting the evolution of MHD turbulence. The higher Reynolds number flow shows a delay in the maximum dissipation with increased transfer of energy toward small scales, resulting in a −5/3 Kolmogorov inertial sub-range scaling. In addition, the effect of Reynolds number on the alignment between velocity and magnetic field, and the energy transfer between kinetic and magnetic energy, is studied. With the ABC flow having strong helicity and zero cross-helicity at low and high Reynolds numbers, a strong dynamo effect is also observed using the LES models, which is consistent with previous DNS.

Two-Grid Based Adaptive Proper Orthogonal Decomposition Method for Time Dependent Partial Differential Equations

In this article, we propose a two-grid based adaptive proper orthogonal decomposition (POD) method to solve the time dependent partial differential equations. Based on the error obtained in the coarse grid, we propose an error indicator for the numerical solution obtained in the fine grid. Our new method is cheap and easy to be implement. We apply our new method to the solution of time-dependent advection–diffusion equations with the Kolmogorov flow and the ABC flow. The numerical results show that our method is more efficient than the existing POD methods.

Collisions and caustics frequencies of long flexible fibers in two-dimensional flow fields

The present work analyzes numerical simulations of long flexible fibers in two-dimensional flows. The focus is on the frequency at which caustics occur and the characterization of the collisions among fibers. The continuous flexible fibers are modeled as a set of connected rigid cylinders, where forces are applied to the center of mass of each cylinder determining their motion and fiber deformation. A domestic numerical code, which implements a fourth-order Runge–Kutta method to solve the system of ordinary differential equations, was validated against experimental data in a previous study. The two-dimensional steady analytical velocity fields correspond to the Arnold–Beltrami–Childress flows. Simulations are performed for a wide range of Stokes numbers ranging from 0.05 to 10.87. In a previous study, the frequency of caustics formation was shown to correlate strongly with fiber clustering. Now, a dimensionless caustics frequency is related to the Stokes number. The number of collisions correlates negatively with the Stokes number and positively with the clustering and the frequency of caustics formation. At low Stokes numbers, collisions are preferentially perpendicular, decreasing the angle of collision as the Stokes number increases. Most of the collisions occur at low relative velocities between the two colliding fibers. The angle of collision is strongly determined by vorticity and inertia, while the relative velocities of the fibers colliding seem to be determined by the characteristic velocity field.

On approximate solutions of the equations of incompressible magnetohydrodynamics

Inspired by an approach proposed previously for the incompressible Navier–Stokes (NS) equations, we present a general framework for the a posteriori analysis of the equations of incompressible magnetohydrodynamics (MHD) on a torus of arbitrary dimension d; this setting involves a Sobolev space of infinite order, made of C∞ vector fields (with vanishing divergence and mean) on the torus. Given any approximate solution of the MHD Cauchy problem, its a posteriori analysis with the method of the present work allows one to infer a lower bound on the time of existence of the exact solution, and to bound from above the Sobolev distance of any order between the exact and the approximate solution. In certain cases the above mentioned lower bound on the time of existence is found to be infinite, so one infers the global existence of the exact MHD solution. We present some applications of this general scheme; the most sophisticated one lives in dimension d=3, with the ABC flow (perturbed magnetically) as an initial datum, and uses for the Cauchy problem a Galerkin approximate solution in 124 Fourier modes. We illustrate the conclusions arising in this case from the a posteriori analysis of the Galerkin approximant; these include the derivation of global existence of the exact MHD solution with the ABC datum, when the dimensionless viscosity and resistivity are equal and stay above a critical value.

A Numerical Third-Order Method for Solving the Navier–Stokes Equations with Respect to Time

A linearly implicit (Rosenbrock-type) numerical method for the integration of three-dimensional Navier–Stokes equations for compressible fluid with respect to time is proposed. The method has four stages and third order of accuracy with respect to time. As the benchmark, the Cauchy problem on a 3D torus is solved. The computed distributions are compared with the solution specified by the ABC flow.

Local representation and construction of Beltrami fields

A Beltrami field is an eigenvector of the curl operator. Beltrami fields describe steady flows in fluid dynamics and force free magnetic fields in plasma turbulence. By application of the Lie–Darboux theorem of differential geometry, we prove a local representation theorem for Beltrami fields. We find that, locally, a Beltrami field has a standard form amenable to an Arnold–Beltrami–Childress flow with two of the parameters set to zero. Furthermore, a Beltrami flow admits two local invariants, a coordinate representing the physical plane of the flow, and an angular momentum-like quantity in the direction across the plane. As a consequence of the theorem, we derive a method to construct Beltrami fields with given proportionality factor. This method, based on the solution of the eikonal equation, guarantees the existence of Beltrami fields for any orthogonal coordinate system such that at least two scale factors are equal. We construct several solenoidal and non-solenoidal Beltrami fields with both homogeneous and inhomogeneous proportionality factors.

Recurrence in three dimensional magnetohydrodynamic plasma

We report on a numerical observation of recurrence phenomenon in a three dimensional magneto-hydrodynamic (MHD) plasma for certain classes of initial flow profiles. Our simulations discover such a behaviour for an initial Taylor-Green (TG) type of flow whereas under identical conditions an initial Arnold-Beltrami-Childress (ABC) flow fails to show recurrent behaviour. This difference in the dynamical behaviours is traced to the nature of the Rayleigh quotient in each case. The Rayleigh quotient which is an approximate measure of the number of active degrees of freedom in a dynamical system is found to remain small and bounded in time for the TG flow but grows indefinitely in time for the ABC flow. An estimate of the effective degrees of freedom for the TG flow case is provided and its practical implications for the nonlinear behaviour of the continuum nonlinear MHD plasma system is discussed.

Beltrami flows equipped with the same crystallographic symmetries as blue phases of cholesteric liquid crystals

Abstract The Arnold–Beltrami–Childress (ABC) flow is the best-known example of a Beltrami flow (or a force-free field in plasma physics) periodic in three orthogonal directions. When its three parameters are equalised, its set of streamlines has the same crystallographic symmetry as a model of a blue phase (BP) of cholesteric liquid crystals. This commonality appears in the similarity between the arrangements of invariant tori in the ABC flow and of double-twist cylinders in the BP model. In this paper, three Beltrami flows whose sets of streamlines have the same crystallographic symmetries as three other BP models are derived by using cubic space groups. The first and the third Beltrami flows have invariant tori arranged similarly to double-twist cylinders in two of the BP models. The second Beltrami flow has invariant tori whose arrangement has been neither theoretically nor experimentally known in liquid crystals. Heteroclinic orbits in the three flows are also studied.

Gyrotactic suppression and emergence of chaotic trajectories of swimming particles in three-dimensional flows

We study the effects of imposed {three-dimensional flows} on the trajectories and mixing of gyrotactic swimming micro-organisms, and identify new phenomena not seen in flows restricted to two dimensions. Through numerical simulation of Taylor--Green and ABC flows, we explore the role that the flow and the cell shape play in determining the long-term configuration of the cells' trajectories, which often take the form of multiple sinuous and helical `plume-like' structures, even in the chaotic ABC flow. This gyrotactic suppression of Lagrangian chaos persists even in the presence of random noise. Analytical solutions for a number of cases reveal the how plumes form and the nature of the competition between torques acting on individual cells. \note{Furthermore, studies of Lyapunov exponents reveal that as the ratio of cell swimming speed relative to the flow speed increases from zero, the initial chaotic trajectories are first suppressed and then give way to a second unexpected window of chaotic trajectories at speeds greater than unity, before suppression of chaos at high relative swimming speeds.

Polynomial interpolation and a priori bootstrap for computer-assisted proofs in nonlinear ODEs

In this work, we introduce a method based on piecewise polynomial interpolation to enclose rigorously solutions of nonlinear ODEs. Using a technique which we call a priori bootstrap, we transform the problem of solving the ODE into one of looking for a fixed point of a high order smoothing Picard-like operator. We then develop a rigorous computational method based on a Newton-Kantorovich type argument (the radii polynomial approach) to prove existence of a fixed point of the Picard-like operator. We present all necessary estimates in full generality and for any nonlinearities. With our approach, we study two systems of nonlinear equations: the Lorenz system and the ABC flow. For the Lorenz system, we solve Cauchy problems and prove existence of periodic and connecting orbits at the classical parameters, and for ABC flows, we prove existence of ballistic spiral orbits.