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Discovering exact, gauge-invariant, local energy-momentum conservation laws for the electromagnetic gyrokinetic system by high-order field theory on heterogeneous manifolds
Gyrokinetic theory is arguably the most important tool for numerical studies of transport physics in magnetized plasmas. However, exact local energy-momentum conservation laws for the electromagnetic gyrokinetic system have not been found despite the continuous effort. Without such local conservation laws, energy and momentum can be instantaneously transported across spacetime, which is unphysical and casts doubt on the validity of numerical simulations based on the gyrokinetic theory. The standard Noether procedure for deriving conservation laws from corresponding symmetries does not apply to gyrokinetic systems because the gyrocenters and electromagnetic fields reside on different manifolds. To overcome this difficulty, we develop a high-order field theory on heterogeneous manifolds for classical particle-field systems and apply it to derive exact, local conservation laws, in particular the energy-momentum conservation laws, for the electromagnetic gyrokinetic system. A weak Euler-Lagrange equation is established to replace the standard Euler-Lagrange equation for the particles. It is discovered that an induced weak Euler-Lagrange current enters the local conservation laws. And it is the new physics captured by the high-order field theory on heterogeneous manifolds. A recently developed gauge-symmetrization method for high-order electromagnetic field theories using the electromagnetic displacement-potential tensor is applied to render the derived energy-momentum conservation laws electromagnetic gauge-invariant.
Formation of the skyrmionic polaron by Rashba and Dresselhaus spin-orbit coupling
Skyrmions in reduced dimensions such as thin layers and interfaces are of both fundamental and technological importance. In these systems, itinerant electrons are often present together with the Rashba and Dresselhaus spin-orbit coupling (SOC). Here, in this study, we show that an itinerant electron in the presence of these interactions can nucleate the skyrmion state, even when the standard Dzyaloshinskii-Moriya interaction (DMI) is absent, and the electron can become self-trapped in the skyrmion core, forming the “skyrmionic polaron” (SkP). The formation of the SkP is investigated from a continuum model of the electron, exchange coupled to the lattice spins, by solving the appropriate Euler-Lagrange equations. The skyrmion (antiskyrmion) texture is favored by the Rashba (Dresselhaus) SOC, with the binding energy increasing quadratically with the strength of the interaction. In contrast, if the skyrmion is already formed due to a nonzero DMI, the electron is delocalized and avoids the skyrmion core until the strength of the Rashba or Dresselhaus SOC exceeds a critical value. Below this critical value, the electron is not bound to the skyrmion core, the polaron does not form, and the electron has little effect on the skyrmion state. Our work envisions the possibility of manipulating the skyrmion state in device applications by altering the strength of the Rashba or Dresselhaus interactions, e.g., by an external electric field.
Efficacy of vibro-impact energy harvesting absorbers on controlling dynamical systems under vortex-induced vibrations and base excitation
Here this study investigates improving the efficacy of an energy harvesting absorber's ability to control a structure under vortex-induced vibrations, base excitation, and a combination of the two by including mechanical amplitude stoppers. The nonlinear reduced-order model is developed through modifying trilinear spring models to represent the impact forces, a modified van der Pol oscillator to represent the forcing due to the vortex-induced vibrations and using the Euler-Lagrange principle to express the equations of motion. It is seen that a soft stopper stiffness and a 5mm gap performs the most effectively of increasing the power generated from the absorber while still greatly reducing the primary structure's amplitude. By changing the stopper's location towards the middle of the energy harvesting absorber, the large effects of the impact forces are reduced and improves the efficacy of medium and hard stopper stiffnesses to generate near the amount of power the soft stopper does, while greatly improving the control of the primary structure. When the system is under combined loadings, the large oscillations of the synchronization region cause the effective configuration to be that of a 27.5 mm gap with soft stiffnesses. The results shows that medium stiffness stoppers with small gaps generate large aperiodic regions due to the high impact force. When the oscillations are close to the stoppers, the beating phenomenon is observed and is not overpowered by the vibro-impact force.
Mesoscopic structure of mixed type domain walls in multiaxial ferroelectrics
The structure of a 180° uncharged rotational domain wall in a multiaxial ferroelectric film is studied in the framework of an analytical Landau-Ginzburg-Devonshire (LGD) approach. Finite element modeling (FEM) is used to solve numerically the system of the coupled nonlinear Euler-Lagrange (EL) second-order differential equations for two components of polarization. We show that the structure of the domain wall and corresponding metastable or stable phase of the film are controlled by a single parameter—the dimensionless ferroelectric anisotropy factor μ. We fitted the static profile of a solitary domain wall, calculated by FEM, with kinklike functions for polarization components, and extracted the five μ-dependent parameters from the fitting to FEM curves. The surprisingly high accuracy of the fitting results for two polarization components in the entire μ range allows us to conclude that the analytical functions, which are trial functions in the direct variational method, can be treated as a high-accuracy variational solution of the static EL equations. We further derive exact two-component analytical solutions of the static EL equations for a polydomain 180° domain structure in a multiaxial ferroelectric film. Using these, we derive analytical expressions for the system free energy and analyze its dependence on the film thickness and boundary conditions at the film surfaces. The single-domain state is ground for zero polarization derivative at the surfaces, while the polydomain states minimize the system energy for zero polarization at the surfaces. Counterintuitively, the energy of the polydomain states split into two levels, “0” and “1,” for zero polarization at the surfaces, and each of the levels contains a large number of close-energy sublevels, whose structure is characterized by a different number and type of domain walls. The analytical solutions can become a useful tool for Bayesian analysis of high-resolution scanning transmission electron microscopy images in ferroelectric films.
The discrete Green's function paradigm for two-way coupled Euler-Lagrange simulation
We outline a methodology for the simulation of particle-laden flows whereby the dispersed and fluid phases are two-way coupled. The drag force which couples fluid and particle momentum depends on the undisturbed fluid velocity at the particle location, and this latter quantity requires modelling. We demonstrate that the undisturbed fluid velocity, in the low particle Reynolds number limit, can be related exactly to the discrete Green's function of the discrete Stokes equations. The method is general in that it can be extended to other partial differential equations which may be associated with particle-laden flows, such as the thermal energy equation or Maxwell's equations. In this work, we demonstrate the method of discrete Green's functions by obtaining these functions for the Navier-Stokes equations at low particle Reynolds number in a two-plane channel geometry. We perform verification at low and finite particle Reynolds number for the case of a point-particle settling under gravity parallel to a plane wall, for different wall normal separations. In comparing to other point-particle schemes the discrete Green's function approach is the most robust at low particle Reynolds number, accurate at all wall-normal separations and is the most accurate in the near wall region at finite Reynolds number. We discuss how the accuracy away from the wall at finite Reynolds number could be improved by appealing to Oseen-like discrete Green's functions. Finally we demonstrate that the discrete Green's function approach can have important implications on statistics of particle-laden turbulent channel flow.
Settling of two-way momentum and energy coupled particles subject to Boussinesq and non-Boussinesq heating
This work establishes a procedure to accurately compute heat transfer between an Eulerian fluid and Lagrangian point-particles. Recent work has focused on accurately computing momentum transfer between fluid and particles. The coupling term for momentum involves the undisturbed fluid velocity at the particle location which is not directly accessible in the simulation. Analogously, in the context of thermal coupling, the undisturbed fluid temperature at the particle location is not directly accessible in simulations and must be estimated. In this paper, we develop a scheme to accurately estimate the undisturbed fluid temperature of a point-particle exchanging thermal energy with a surrounding fluid. Furthermore, the temperature disturbance is correlated with the enhanced temperature curvature in the vicinity of the particle and is formally valid in the low heating, low convection limit. We conduct extensive verification of the correction procedure for a settling particle subject to radiation. This setup allows the simultaneous testing of thermal and momentum corrections. By considering equations of drag and Nusselt number extended to finite Péclet and Boussinesq numbers, we establish a large range over which the correction procedure can be applied.
A correction procedure for self-induced velocity of a finite-sized particle in two-way coupled Euler–Lagrange simulations
The importance of incorporating a correction to undo the self-induced perturbation velocity of a particle, when its size becomes comparable to the Eulerian grid, in a two-way coupled Euler–Lagrange (EL) simulation is now well appreciated. The present work improves upon the prior correction procedures in a few important ways. First, the past correction procedures have been scalar-based with the assumption that the quasi-steady force is the source of self-induced velocity perturbation. Here we generalize to a vector correction procedure and thereby the directions of feedback force and relative velocity can be different. This allows the correction procedure to be used even in the presence of added-mass, history, and lift forces. Second, the effect of a nearby wall has been systematically included in the correction procedure. The correction procedure depends on fundamental Oseen solutions of streamwise and transverse regularized feedback forces. We present a Fourier transform-based analytical approach to obtaining these regularized Oseen solutions. We also present a step-by-step numerical procedure for obtaining the Oseen solutions in any EL code. With the analytical or numerical Oseen functions, the correction procedure can be easily implemented in any EL code. Iterations are required in solving the implicit correction equations and it is demonstrated that the correction procedure converges rapidly within three or four iterations. In conclusion, a simple empirical approach is also presented to account for unsteady effects in the correction procedure.
Two-way coupled particle-turbulence interaction: Effect of numerics and resolution on fluid and particle statistics
Euler-Lagrange point-particle simulation has emerged as a premier methodology for studying dispersed particle-laden flows. This method's popularity stems from its ability to resolve fine-scale fluid structures while also tracking individual particles at reduced cost using an appropriate particle acceleration model. However, the point-particle model has known convergence issues in that refinement of the fluid grid can lead to changes in the predicted statistics. The reasons for non-convergence are twofold: the point-particle two-way coupling force in the Navier-Stokes equations requires a numerical regularization and, without careful implementation, yields a singular force on the fluid with grid refinement. The second factor that yields grid-dependent statistics is that the point-particle force model in general depends on the undisturbed fluid velocity. When the undisturbed fluid velocity is not robustly modeled in a grid-insensitive way, the calculated force for both particles and fluid will be grid-dependent, contaminating their respective statistics. While the first issue regarding regularizing the point-particle source term has received attention in the literature, the consequences of robustly modeling the undisturbed velocity in the context of grid refinement of turbulence has received little attention. Here, we consider decaying homogeneous isotropic turbulence laden with particles at different Stokes numbers. For a given Stokes number, we systematically refine the grid and demonstrate that explicitly modeling the undisturbed fluid velocity yields relative grid insensitivity for the energy of the particle and fluid phases, as well as acceleration of the particles. We also demonstrate that an appropriately defined dissipation rate is also grid-insensitive when an undisturbed fluid velocity correction is used. In contrast, when the undisturbed fluid velocity is modeled using the conventional approach of interpolating the local fluid velocity to the particle location, we show this procedure yields divergent statistics with grid refinement. In particular, we show that higher-order interpolation of the fluid velocity in two-way coupled problems is worse than lower-order interpolation, in the absence of a correction procedure to estimate the undisturbed fluid velocity. We also examine velocity derivative statistics of the fluid phase and demonstrate that these statistics are not in general convergent even when the undisturbed fluid velocity is explicitly modeled. Collectively, the observations in this paper are used to present a philosophy on the types of questions which are answerable with point-particle methods.
Closed-Form Solutions for the Equations of Motion of the Heavy Symmetrical Top with One Point Fixed
The Equations of Motion (EOM) for the Heavy Symmetrical Top with One Point Fixed are highly non-linear. The literature describes the numerical methods that are used to resolve this Classical system including modern tools i.e. the Runge-Kutta Fourth Order method. It is more difficult to derivate closed-form solutions for the EOM and as mentioned in the literature it is not always possible to find the close-form solution for all the EOM. Fortunately, there are a few examples available that will serve as a guide to move further on in this topic. It is the purpose of this paper to find a methodology that will produce the solutions for a given subset of EOM that fulfill certain requisites. The report is organized as follows: it starts with a very short summary of the literature available on this topic and quickly follows into the derivation of the EOM using the Euler-Lagrange method. The Routhian will be used to reduce the size of the expression. It continues with the formulation of the Classical cubic function (f(u)) through a novel process. The roots of f(u) are of the outmost importance to be able to find the EOM closed-form solution, and once the final roots are selected the general method that will produce the closed-form solutions is presented. Two sets of examples are included to show the validity of the process and comparisons of the results from the closed-form solutions vs. the numerical results for these examples are shown.