Generalized Euler Equations Research Articles

Molecular Rotation and Generalized Euler Equations

Molecular Rotation and Generalized Euler Equations

On Liouville type theorems for the self-similar solutions to the generalized Euler equations

We study the backward discretely self-similar solutions to the generalized Euler equations, where usual Biot-Savart kernel representing the velocity in terms of the vorticity is replaced by various power of the laplacian. Under milder sufficient conditions than the previous results on the Euler equations we show that Liouville type theorems hold for the time periodic solutions to the profile equations, which means that there exists no backward discretely self-similar solutions to the generalized Euler system having such profile.

A Simple Method for Constructing a Family of Hamiltonian Conservative Chaotic Systems

Conservative chaotic systems (CCSs) have unique advantages over dissipative chaotic systems (DCSs) in the fields of secure communication and pseudo-random number generators (PRNGs), etc. However, there are relatively fewer reports on CCSs than DCSs. To this end, this paper proposes an effective method for constructing a family of Hamiltonian conservative chaotic systems (HCCSs) by letting any three of the four sub-bodies denoted by 4D generalized Euler equations share a rotation axis. From theoretical analysis to experimental verification, one of the proposed HCCSs is studied thoroughly to demonstrate the effectiveness of this method. The example system has an infinite number of equilibrium points, which belong to either centers or saddles, resulting in hidden chaos. Besides, through the bifurcation diagram, parametric chaotic set, and Lyapunov exponent, richly dynamic behaviors related to parameters are displayed. Moreover, the 3D phase portraits verify that the Hamiltonian energy is conservative. Numerous energy-related coexisting orbits are discovered in this system, such as the coexistence of quasi-periodic orbits, chaotic orbits, and chaotic quasi-periodic orbits. Furthermore, the breadboard-based circuit is implemented to illustrate the HCCS’s physical feasibility. Finally, the PRNG based on the HCCS has excellent randomness in terms of NIST and TESTU01 test results.

Attitude dynamics and control of a large flexible space structure by means of a minimum complexity model

A technique for deriving a low-order model of a large, deformable space vehicle, with a configuration resembling that of the International Space Station, is proposed. The modeling approach is based on a hybrid Newtonian–Lagrangian approach, where a generalized Euler equation is written for rotational degrees of freedom, whereas the dynamics of states associated to deformation by means of a Galërkin approach is derived by means of a Lagrangian formulation. The assumed modes method is adopted, where modal characteristics of the main deformable structure are estimated on the basis of its real characteristics, estimated by means a finite element model of the actual truss configuration. A cluster of control moment gyroscopes is considered as the actuator for attitude control. Open- and closed-loop maneuvers are considered, in order to highlight coupling between rotational and deformation degrees of freedom. The modeling tools allows for a quick search of gains which minimizes structural excitations during large angle slews.

The atypical startup characteristics of a pump as turbine

AbstractA generalized Euler equation of pump as turbine is derived, and the transient hydraulic performance of a centrifugal pump as turbine during an atypical startup period is measured. Meanwhile, the dimensionless analysis and the additional theoretical head calculation are also conducted. It is found that the transient characteristics of pump as turbine during atypical startup period mainly come from the rotational acceleration effect, rather than the flow inertia effect. The final stable values of dimensionless flowrate coefficient and dimensionless head coefficient are both closely related to the steady rotating speed and flowrate after startup process, but the dimensionless power coefficient is not remarkably related to them. In an atypical startup, the impact phenomena occur in both flowrate curve and outlet static pressure curve. At the same steady rotating speed, the larger the steady flowrate is, the more serious the flowrate impact is. Under the same steady flowrate, the lower the steady rotating speed is, the more remarkable the static pressure impact at the outlet is.

Limit Theorems and Fluctuations for Point Vortices of Generalized Euler Equations

We prove a mean field limit, a law of large numbers and a central limit theorem for a system of point vortices on the 2D torus at equilibrium with positive temperature. The point vortices are formal solutions of a class of equations generalising the Euler equations, and are also known in the literature as generalised inviscid SQG. The mean-field limit is a steady solution of the equations, the CLT limit is a stationary distribution of the equations.

THOR 2.0: Major Improvements to the Open-source General Circulation Model

Abstract THOR is the first open-source general circulation model (GCM) developed from scratch to study the atmospheres and climates of exoplanets, free from Earth- or solar-system-centric tunings. It solves the general nonhydrostatic Euler equations (instead of the primitive equations) on a sphere using the icosahedral grid. In the current study, we report major upgrades to THOR, building on the work of Mendonça et al. First, while the horizontally explicit and vertically implicit integration scheme is the same as that described in Mendonça et al., we provide a clearer description of the scheme and improve its implementation in the code. The differences in implementation between the hydrostatic shallow, quasi-hydrostatic deep, and nonhydrostatic deep treatments are fully detailed. Second, standard physics modules are added: two-stream, double-gray radiative transfer and dry convective adjustment. Third, THOR is tested on additional benchmarks: tidally locked Earth, deep hot Jupiter, acoustic wave, and gravity wave. Fourth, we report that differences between the hydrostatic and nonhydrostatic simulations are negligible in the Earth case but pronounced in the hot Jupiter case. Finally, the effects of the so-called “sponge layer,” a form of drag implemented in most GCMs to provide numerical stability, are examined. Overall, these upgrades have improved the flexibility, user-friendliness, and stability of THOR.

Solution property preserving reconstruction BVD+MOOD scheme for compressible euler equations with source terms and detonations

In “Solution Property Reconstruction for Finite Volume scheme: a BVD+MOOD framework”, Int. J. Numer. Methods Fluids, 2020, we have designed a novel solution property preserving reconstruction, so-called multi-stage BVD-MOOD scheme. The scheme is able to maintain a high accuracy in smooth profile, eliminate the oscillations in the vicinity of discontinuity, capture sharply discontinuity and preserve some physical properties like the positivity of density and pressure for the Euler equations of compressible gas dynamics. In this paper, we present an extension of this approach for the compressible Euler equations supplemented with source terms (e.g., gravity, chemical reaction). One of the main challenges when simulating these models is the occurrence of negative density or pressure during the time evolution, which leads to a blow-up of the computation. General compressible Euler equations with different type of source terms are considered as models for physical situations such as detonation waves. Then, we illustrate the performance of the proposed scheme via a numerical test suite including genuinely demanding numerical tests. We observe that the present scheme is able to preserve the physical properties of the numerical solution still ensuring robustness and accuracy when and where appropriate.

Limiting behavior of scaled general Euler equations of compressible fluid flow

The aim of this article is to study the limiting behavior of the solutions for the scaled generalized Euler equations of compressible fluid flow. When the initial data is of Riemann type, we showed the existence of solution which consists of shock waves and rarefaction waves and that the distributional limit of the solutions for this system converges to the solution of a non-strictly hyperbolic system, called one dimensional model for large scale structure formation of universe as the scaling parameter vanishes. An explicit entropy and entropy flux pair are also constructed for the particular flux function (Brio system) and it is shown that the solution constructed is entropy admissible. This is a continuation of our work[23].

Generalized Euler Equation Errors for Discrete Time Dynamic Portfolio Choice Models

The solution to dynamic portfolio choice models can be formulated in terms of a value function by the Bellman principle of optimality, which reduces the multi-period optimal policy choice problem to a sequence of one-period maximization problems. For two adjacent periods, economists compute the error of numerically obtained policies by measuring how much these policies violate the intertemporal first order conditions for the optimal policy choice problem---so-called Euler equation errors. In this paper, we derive generalized Euler equation errors for the solution to a broad class of discrete time dynamic portfolio choice models where the policies are continuous choice variables. Our key precondition is that the gradient of the value function with respect to the state variables can be approximated. This is, for example, the case when a global polynomial basis or B-spline basis functions are used for the approximation of the value function within the discrete time dynamic programming approach. We apply our theoretical results to exemplary, representative dynamic portfolio choice models.

Statistical Measures and Selective Decay Principle for Generalized Euler Dynamics: Formulation and Application to the Formation of Strong Fronts

In this work we investigate the statistical mechanics of a family of two dimensional (2D) fluid flows, described by the generalized Euler equations, or $\alpha$-models. These models describe both nonlocal and local dynamics, with one example of the latter given by the Surface Quasi Geostrophy (SQG) model for which the existence of singularities is still under discussion. Furthermore, SQG is relevant both for atmosphere and ocean dynamics, and in particular, it is proposed to understand the oceanic submesoscale structures, front and filaments, associated with horizontal gradients of buoyancy. We discuss under which conditions the statistical theory suggests a principle of selective decay for the whole family of turbulent models, and then we explore the selective decay principle numerically, the transition to equilibrium and the formation of singularities, starting from initial conditions (i.c.s) corresponding to a hyperbolic saddle. We study the topological transitions in the flow configuration induced by filaments breaking. Furthermore we compare the theoretical equilibrium states, and the functional relation between the generalized vorticity $q$ and its correspondent stream function $\psi$, with the results of the simulations. For the particular i.c.s investigated, and domain used for the simulations, we have not noticed transition from tanh-like to sinh-like $\psi-q$ functional relation, which would be expected by the emergence of coherent structures that are, however, filtered by the time averages used to compute the mean fields.

Modeling of a Hamiltonian conservative chaotic system and its mechanism routes from periodic to quasiperiodic, chaos and strong chaos

In this paper, the important role of 3D Euler equation playing in forced-dissipative chaotic systems is reviewed. In mathematics, rigid-body dynamics, the structure of symplectic manifold, and fluid dynamics, building a four-dimensional (4D) Euler equation is essential. A 4D Euler equation is proposed by combining two generalized Euler equations of 3D rigid bodies with two common axes. In chaos-based secure communications, generating a Hamiltonian conservative chaotic system is significant for its advantage over the dissipative chaotic system in terms of ergodicity, distribution of probability, and fractional dimensions. Based on the proposed 4D Euler equation, a 4D Hamiltonian chaotic system is proposed. Through proof, only center and saddle equilibrium lines exist, hence it is not possible to produce asymptotical attractor generated from the proposed conservative system. An analytic form of Casimir power demonstrates that the breaking of Casimir energy conservation is the key factor that the system produces the aperiodic orbits: quasiperiodic orbit and chaos. The system has strong pseudo-randomness with a large positive Lyapunov exponent (more than 10 K), and a large state amplitude and energy. The bandwidth for the power spectral density of the system is 500 times that of both existing dissipative and conservative systems. The mechanism routes from quasiperiodic orbits to chaos is studied using the Hamiltonian energy bifurcation and Poincaré map. A circuit is implemented to verify the existence of the conservative chaos.

On the initial-boundary value problem for the Euler equations in presence of a rarefaction wave

We study the initial-boundary value problem for the general non-isentropic 3D Euler equations with data which are incompatible in the classical sense, but are “rarefaction-compatible”. We show that such data are also rarefaction-compatible of infinite order and the initial-boundary value problem has a piece-wise smooth solution containing a rarefaction wave.

Compressible and Choked Flows in Rotating Passages

The present study revisits the rotating duct problem examined by Polihronov and Straatman (J. Polihronov and A. G. Straatman, Phys. Rev. Lett. v. 109, p. 054504 (2012)). Starting from the general compressible Euler equations in a non-stationary reference frame closed form expressions for velocity, temperature, density and pressure along the duct are determined. The present results are more general than those obtained by Polihronov and Straatman, as the change of in-frame kinetic energy has been retained. The improvement of the present results over Polihronov and Straatman’s is demonstrated by comparison with the results of a computational fluid dynamics study. The new results have been further generalized to the case of a rotating duct with varying cross-sectional area, and again for a general curved passage in three-dimensional space. The work required or derived from the rotating duct has also been computed. The choked flow condition within the passage of varying cross-sectional area has been identified, along with the constraints which must be placed on the Mach, Rossby, and tip Mach numbers to avoid choked flow. Finally, a straightforward technique to identify any locations where an ideal rotating flow in a constrained passage will become sonic has been presented.

Collapse of generalized Euler and surface quasigeostrophic point vortices.

Point-vortex models are presented for the generalized Euler equations, which are characterized by a fractional Laplacian relation between the active scalar and the stream function. Special focus is given to the case of the surface quasigeostrophic (SQG) equations, for which the existence of finite-time singularities is still a matter of debate. Point-vortex trajectories are expressed using Nambu dynamics. The formulation is based on a noncanonical bracket and allows for a geometrical interpretation of trajectories as intersections of level sets of the Hamiltonian and Casimir. Within this setting, we focus on the collapse of solutions for the three-point-vortex model. In particular, we show that for SQG the collapse can be either self-similar or non-self-similar. Self-similarity occurs only when the Hamiltonian is zero, while non-self-similarity appears for nonzero values of the same. For both cases, collapse is allowed for any choice of circulations within a permitted interval. These results differ strikingly from the classical point-vortex model, where collapse is self-similar for any value of the Hamiltonian, but the vortex circulations must satisfy a strict relationship. Results may also shed a light on the formation of singularities in the SQG partial differential equations, where the singularity is thought to be reached only in a self-similar way.

On uniqueness of time-consistent Markov policies for quasi-hyperbolic consumers under uncertainty

We give a set of sufficient conditions for uniqueness of a time-consistent stationary Markov consumption policy for a quasi-hyperbolic household under uncertainty. To the best of our knowledge, this uniqueness result is the first presented in the literature for general settings, i.e. under standard assumptions on preferences, as well as some new condition on a transition probability. This paper advocates a “generalized Bellman equation” method to overcome some predicaments of the known methods and also extends our recent existence result. Our method also works for returns unbounded from above. We provide a few natural extensions of optimal policy uniqueness: convergent and accurate computational algorithm, monotone comparative statics result and generalized Euler equation.

Operator Newton polynomials and well-solvable problems for generalized Euler equation

The study of well-solvable operator equations in a Banach space, which was initiated by the authors in [4, 5], is continued. Namely, it is proved by means of Maslov’s operator method that a polynomial equation with abstract Newton polynomials is well solvable in the sense of Hadamard. The obtained results are applied to prove that a large class of problems for differential equations with variable coefficient having a singularity (such equations are called generalized Euler equations in the paper) are well solvable.

On Uniqueness of Time-Consistent Markov Policies for Quasi-Hyperbolic Consumers Under Uncertainty

We give a set of sufficient conditions for uniqueness of a time-consistent Markov stationary consumption policy for a quasi-hyperbolic household under uncertainty. To the best of our knowledge, this uniqueness result is the first presented in the literature for general settings, i.e. under standard assumptions on preferences, as well as some new condition on a transition probability. This paper advocates a Bellman equation method to overcome some predicaments of the known methods and also extends our recent existence result. Our method also works for returns unbounded from above. We provide few natural followers of optimal policy uniqueness: convergent and accurate computational algorithm, monotone comparative statics results and generalized Euler equation.

Ruling Out Multiplicity of Smooth Equilibria in Dynamic Games: A Hyperbolic Discounting Example

The literature that conducts numerical analysis of equilibrium in models with hyperbolic (quasi-geometric) discounting reports difficulties in achieving convergence. Surprisingly, numerical methods fail to converge even in a simple, deterministic optimal growth problem that has a well-behaved, smooth closed-form solution. We argue that the reason for nonconvergence is that the generalized Euler equation has a continuum of smooth solutions, each of which is characterized by a different integration constant. We propose two types of restrictions that can rule out the multiplicity: boundary conditions and shape restrictions on equilibrium policy functions. With these additional restrictions, the studied numerical methods deliver a unique smooth solution for both the deterministic and stochastic problems in a wide range of the model’s parameters.

Axion as a cold dark matter candidate: analysis to third order perturbation for classical axion

We investigate aspects of axion as a coherently oscillating massive classical scalar field by analyzing third order perturbations in Einstein's gravity in the axion-comoving gauge. The axion fluid has its characteristic pressure term leading to an axion Jeans scale which is cosmologically negligible for a canonical axion mass. Our classically derived axion pressure term in Einstein's gravity is identical to the one derived in the non-relativistic quantum mechanical context in the literature. We present the general relativistic continuity and Euler equations for an axion fluid valid up to third order perturbation. Equations for axion are exactly the same as that of a zero-pressure fluid in Einstein's gravity except for an axion pressure term in the Euler equation. Our analysis includes the cosmological constant.