Two-dimensional Periodic Domain Research Articles

Spontaneous emergence of two-dimensional quasibreathers in a nonlinear Schrödinger equationwith nonlocal derivatives.

We consider the nonlinear Schrödinger equationwith nonlocal derivatives in a two-dimensional periodic domain. For certain orders of derivatives, we find a type of quasi-breather solution dominating the field evolution at low nonlinearity levels. With the increase of nonlinearity, the structures break down, giving way to Rayleigh-Jeans (or wave turbulence) spectra. Phase-space trajectories associated with the quasibreather solutions are found to be close to that of the linear system and almost periodic. We employ two methods to search for nearby periodic solutions (e.g., exact breathers), yet none are found. Given these distinguishing behaviors, we interpret this structure in the context of Kolmogorov-Arnold-Moser (KAM) theory.

Exponential mixing for random dynamical systems and an example of Pierrehumbert

We consider the question of exponential mixing for random dynamical systems on arbitrary compact manifolds without boundary. We put forward a robust, dynamics-based framework that allows us to construct space-time smooth, uniformly bounded in time, universal exponential mixers. The framework is then applied to the problem of proving exponential mixing in a classical example proposed by Pierrehumbert in 1994, consisting of alternating periodic shear flows with randomized phases. This settles a longstanding open problem on proving the existence of a space-time smooth (universal) exponentially mixing incompressible velocity field on a two-dimensional periodic domain while also providing a toolbox for constructing such smooth universal mixers in all dimensions.

Profile of a two-dimensional vortex condensate beyond the universal limit.

It is well known that an inverse turbulent cascade in a finite (2π×2π) two-dimensional periodic domain leads to the emergence of a system-sized coherent vortex dipole. We report a numerical hyperviscous study of the spatial vorticity profile inside one of the vortices. The exciting force was shortly correlated in time, random in space, and had a correlation length l_{f}=2π/k_{f} with k_{f} ranging from 100 to 12.5. Previously, it was found that in the asymptotic limit of small-scale forcing, the vorticity exhibits the power-law behavior Ω(r)=(3ε/α)^{1/2}r^{-1}, where r is the distance to the vortex center, α is the bottom friction coefficient, and ε is the inverse energy flux. Now we show that for a spatially homogeneous forcing with finite k_{f} the vorticity profile becomes steeper, with the difference increasing with the pumping scale but decreasing with the Reynolds number at the forcing scale. Qualitatively, this behavior is related to a decrease in the effective pumping of the coherent vortex with distance from its center. To support this statement, we perform an additional simulation with spatially localized forcing, in which the effective pumping of the coherent vortex, on the contrary, increases with r, and show that in this case the vorticity profile can be flatter than the asymptotic limit.

A Simple Thermal Convection System Showing Subcritical Transition and Localized Turbulences in Two-Dimensional Periodic Domains

A Simple Thermal Convection System Showing Subcritical Transition and Localized Turbulences in Two-Dimensional Periodic Domains

Struwe-like solutions for an evolutionary model of magnetoviscoelastic fluids

In this work we investigate the existence and uniqueness of Struwe-like solutions for a system of partial differential equations modeling the dynamics of magnetoviscoelastic fluids. The considered system couples a Navier-Stokes type equation with a dissipative equation for the deformation tensor and a Landau–Lifshitz–Gilbert type equation for the magnetization field.The main purpose is to establish a well-posedness theory in a two-dimensional periodic domain under standard assumption of critical regularity for the (possibly large) initial data. We prove that the considered weak solutions are everywhere smooth, except for a discrete set of time values.The proof of the uniqueness is based on suitable energy estimates for the solutions within a functional framework which is less regular than the one of the Struwe energy level. These estimates rely on several techniques of harmonic analysis and paradifferential calculus.

Moving efficiently through a crowd: A nature-inspired traffic rule.

In this article, we propose a traffic rule inspired from nature that instructs how a crowd made up of inert agents should respond to an elite agent to facilitate its motion through the crowd. When an object swims in a fluid medium or an intruder is forced through granular matter, characteristic flow fields are created around them. We show that if inert agents made small movements based on a traffic rule derived from these characteristic flow fields, then they efficiently reorganize and transport enough space for the elite to pass through. The traffic rule used in the article is a dipole field which satisfactorily captures the features of the flow fields around a moving intruder. We study the effectiveness of this dipole traffic rule using numerical simulations in a two-dimensional periodic domain, where one self-propelled elite agent tries to move through a crowd of inert agents that prefer to stay in a state of rest. Simulations are carried out for a wide range of strengths of the traffic rule and packing densities of the crowd. We characterize and analyze four regions in the parameter space-free-flow, motion due to cooperation and frozen and collective drift regions-and discuss the consequence of the traffic rule in light of the collective behavior observed. We believe that the proposed method can be of use in a swarm of robots working in constrained environments.

Asymptotic behaviors of global solutions to the two-dimensional non-resistive MHD equations with large initial perturbations

This paper is concerned with the asymptotic behaviors of global strong solutions to the incompressible non-resistive viscous magnetohydrodynamic (MHD) equations with large initial perturbations in two-dimensional periodic domains in Lagrangian coordinates. First, motivated by the odevity conditions imposed in Pan et al. (2018) [24], we prove the existence and uniqueness of strong solutions under some class of large initial perturbations, where the strength of impressive magnetic fields depends increasingly on the H2-norm of the initial perturbation value of both the velocity and magnetic field. Then, we establish time-decay rates of strong solutions. Moreover, we find that H2-norm of the velocity decays faster than the perturbed magnetic field. Finally, by developing some new analysis techniques, we show that the strong solution converges in a rate of the field strength to a solution of the corresponding linearized problem as the strength of the impressive magnetic field goes to infinity. In addition, an extension of similar results to the corresponding inviscid case with damping is presented.

Tear-film breakup: The role of membrane-associated mucin polymers.

Mucin polymers in the tear film protect the corneal surface from pathogens and modulate the tear-film flow characteristics. Recent studies have suggested a relationship between the loss of membrane-associated mucins and premature rupture of the tear film in various eye diseases. This work aims to elucidate the hydrodynamic mechanisms by which loss of membrane-associated mucins causes premature tear-film rupture. We model the bulk of the tear film as a Newtonian fluid in a two-dimensional periodic domain, and the lipid layer at the air-tear interface as insoluble surfactants. Gradual loss of membrane-associated mucins produces growing areas of exposed cornea in direct contact with the tear fluid. We represent the hydrodynamic consequences of this morphological change through two mechanisms: an increased van der Waals attraction due to loss of wettability on the exposed area, and a change of boundary condition from an effective negative slip on the mucin-covered areas to the no-slip condition on exposed cornea. Finite-element computations, with an arbitrary Lagrangian-Eulerian scheme to handle the moving interface, demonstrate a strong effect of the elevated van der Waals attraction on precipitating tear-film breakup. The change in boundary condition on the cornea has a relatively minor role. Using realistic parameters, our heterogeneous mucin model is able to predict quantitatively the shortening of tear-film breakup time observed in diseased eyes.

Universal profile of the vortex condensate in two-dimensional turbulence.

An inverse turbulent cascade in a restricted two-dimensional periodic domain creates a condensate-a pair of coherent system-size vortices. We perform extensive numerical simulations of this system and carry out theoretical analysis based on momentum and energy exchanges between the turbulence and the vortices. We show that the vortices have a universal internal structure independent of the type of small-scale dissipation, small-scale forcing, and boundary conditions. The theory predicts not only the vortex inner region profile, but also the amplitude, which both perfectly agree with the numerical data.

Two-dimensional photonic and phononic crystals based on lithium niobate

Specific features of the preparation and application of photonic and phononic crystals containing two-dimensional periodic domain structures are considered. Particular attention is given to the formation of two-dimensional periodic domain structures in oxide ferroelectrics using strongly focused laser beams. Specific features of the nonlinear properties of such two-dimensional structures during the propagation of optical and acoustic beams through them are described.

Filtered models for scalar transport in gas–particle flows

We employ a kinetic-theory based two-fluid model to develop a filtered two-fluid model for scalar transport in gas–particle flows. The filtering procedure gives rise to terms describing the filtered interphase heat/mass transfer and filtered scalar diffusion, which need to be constituted in order to close the filtered transport equations. In this work, the closure for these terms is accomplished by performing fine-grid simulations of the two-fluid model in a two-dimensional periodic domain. Filtered scalar diffusion is investigated by imposing a lateral mean gradient in the scalar for each phase. Interphase energy/mass transport is investigated by prescribing a heat/species source (sink) in the solids (gas) phase, such that the energy/species content of the mixture is preserved. The variation of the filtered transport coefficients with respect to filtered particle volume fraction, and scaling with respect to filter size, filtered scalar shear rate, and filtered slip velocity is discussed. We find the filtered interphase heat transfer coefficient to be as much as two orders of magnitude smaller than the microscopic interphase heat transfer coefficient. The model for filtered scalar diffusion is found to have a form very similar to that calculated for single phase turbulent flows. We also calculate the filtered Prandtl number for each phase.

An immersed boundary energy-based method for incompressible viscoelasticity

Incompressible viscoelastic materials are prevalent in biological applications. In this paper we present a method for incompressible viscoelasticity in which the elasticity of the material is described in Lagrangian form (i.e. in material coordinates), and Eulerian (spatial) coordinates are used for the equations of motion and to enforce the incompressibility condition. The elastic forces are computed directly from an energy functional without the use of stress tensors, and the immersed boundary method is used to communicate between Lagrangian and Eulerian variables. The method is first applied to a warm-up problem, in which a viscoelastic incompressible material fills a two-dimensional periodic domain. For this problem, we study convergence of the velocity field, the deformation map, and the Eulerian force density. The numerical results indicate that the velocity field and deformation map converge strongly at second order and the Eulerian force density converges weakly at second order. Incompressibility is well maintained, as indicated by area conservation in this 2D problem. Finally, the method is applied to a three-dimensional fluid–structure interaction problem with two different materials: an isotropic neo-Hookean model and an anisotropic fiber-reinforced model.

Full well-posedness of point vortex dynamics corresponding to stochastic 2D Euler equations

The motion of a finite number of point vortices on a two-dimensional periodic domain is considered. In the deterministic case it is known to be well posed only for almost every initial configuration. Coalescence of vortices may occur for certain initial conditions. We prove that when a generic stochastic perturbation compatible with the Eulerian description is introduced, the point vortex motion becomes well posed for every initial configuration, in particular coalescence disappears.

Random attractors for a stochastic hydrodynamical equation in Heisenberg paramagnet

This article studies the asymptotic behaviors of the solution for a stochastic hydrodynamical equation in Heisenberg paramagnet in a two-dimensional periodic domain. We obtain the existence of random attractors in ▪ 1.

Surface Hexagonally Poled Lithium Niobate Waveguides

We report on surface hexagonally poled lithium niobate for two-dimensional non-linear interactions in optical waveguide structures. A method for surface inversion of ferroelectric domains has been applied for the fabrication of a hexagonal two-dimensional periodic domain structure with a period of 7.6 μm on an annealed proton exchanged lithium niobate waveguide. This periodic pattern is suitable for quasi-phase-matched second harmonic generation at the fundamental wavelength of 1.06 μm by means of the first order (G10) reciprocal lattice vector.

Coupled mean flow-amplitude equations for nearly inviscid parametrically driven surface waves.

Nearly inviscid parametrically excited surface gravity-capillary waves in two-dimensional periodic domains of finite depth and both small and large aspect ratio are considered. Coupled equations describing the evolution of the amplitudes of resonant left- and right-traveling waves and their interaction with a mean flow in the bulk are derived, and the conditions for their validity established. In general the mean flow consists of an inviscid part together with a viscous streaming flow driven by a tangential stress due to an oscillating viscous boundary layer near the free surface and a tangential velocity due to a bottom boundary layer. These forcing mechanisms are important even in the limit of vanishing viscosity, and provide boundary conditions for the Navier-Stokes equation satisfied by the mean flow in the bulk. The streaming flow is responsible for several instabilities leading to pattern drift.

Nonlinear stability of Euler flows in two-dimensional periodic domains

The non-quadratic conservation laws of the two-dimensional Euler equations are used to show that the gravest modes in a doubly-periodic domain with aspect ratio L = 1 are stable up to translations (or structurally stable) for finite-amplitude disturbances. This extends a previous result based on conservation of energy and enstrophy alone. When L 1, a saturation bound is established for the mode with wavenumber |k| = L −1 (the next-gravest mode), which is linearly unstable. The method is applied to prove nonlinear structural stability of planetary wave two on a rotating sphere.

Fully Conservative Higher Order Finite Difference Schemes for Incompressible Flow

Conservation properties of the mass, momentum, and kinetic energy equations for incompressible flow are specified as analytical requirements for a proper set of discrete equations. Existing finite difference schemes in regular and staggered grid systems are checked for violations of the conservation requirements and a few important discrepancies are pointed out. In particular, it is found that none of the existing higher order schemes for a staggered mesh system simultaneously conserve mass, momentum, and kinetic energy. This deficiency is corrected through the derivation of a general family of fully conservative higher order accurate finite difference schemes for staggered grid systems. Finite difference schemes in a collocated grid system are also analyzed, and a violation of kinetic energy conservation is revealed. The predicted conservation properties are demonstrated numerically in simulations of inviscid white noise, performed in a two-dimensional periodic domain. The proposed fourth order schemes in a staggered grid system are generalized for the case of a non-uniform mesh, and the resulting scheme is used to perform large eddy simulations of turbulent channel flow.

An inverse energy cascade in two-dimensional low Reynolds number bubbly flows

Two direct numerical simulations of several buoyant bubbles in a two-dimensional periodic domain are presented. The average rise Reynolds number of the bubbles is close to 2, and surface tension is high, resulting in small bubble deformation. The void fraction is relatively high, and the bubbles interact strongly. Simulations of the motion of both 144 and 324 bubbles show the formation of flow structures much larger than the bubble size, and a continuous increase in the energy of the low-wavenumber velocity modes. Plots of the energy spectrum show a range of wavenumbers with an approximately -5/3 slope. This suggests that a part of the work done by the buoyant bubbles is not dissipated, but instead increases the energy of flow structures much larger than the bubbles. This phenomenon, which is also seen in numerical simulation of forced two-dimensional turbulence, prevents the appearance of a statistically steady-state motion that is independent of the size of the computational domain.

Conservative Properties of Finite Difference Schemes for Incompressible Flow. 3rd Report. Numerical Tests.

In this report, numerical tests of the conservation properties of finite difference schemes are performed. Inviscid flow simulations in a two-dimensional periodic domain are performed using several finite difference schemes that are discussed in the first and second reports. The momentum and kinetic energy are conserved when the proper schemes in regular and staggered grid systems are used. The defect of the finite difference schemes in a collocated grid system is emphasized in the simulation. In addition, plane channel flows at Re=180 and 2000 are simulated using the proper fourth-order accurate finite difference scheme in a staggered grid system and the results are better than those of the second-order accurate algorithm and an existing fourth-order accurate scheme.