Acceleration Terms Research Articles

Reduction of the linear stability problem for general zonal flows with any Rossby and Richardson numbers to a PDE

The stability to linear disturbances of a general laterally and/or vertically sheared zonal flow in thermal wind balance is considered for any Rossby number and any Richardson number. The problem for normal mode perturbations is reduced to a second‐order partial differential equation (PDE) for a single dependent variable, the pressure perturbation, in which the complex phase speed is an eigenvalue. It is shown that the boundary conditions of no normal flow are expressible in terms of the pressure and its first‐order derivatives, and that the PDE can be derived from the linearized equation for conservation of the Ertel potential vorticity. The derivations are performed first for an incompressible Boussinesq fluid in Cartesian coordinates allowing the perturbations to be either hydrostatic or non‐hydrostatic. They are shown to generalize to a perfect gas and to motions on a sphere. A PDE for general (non‐normal mode) linear perturbations can also be derived. The Ertel potential vorticity of the zonal flow is shown to arise in the component of the momentum equation parallel to the zonal flow in a generalized Coriolis acceleration term. An ordinary differential equation (ODE) is also derived for the vertical velocity perturbation to laterally uniform zonal flows which is consistent with previous results. The differences between the singularities of the equations for pressure and vertical velocity perturbations are discussed. Integral constraints on the perturbations are used to prove the standard necessary condition for symmetric instabilities and a generalization of the Charney–Stern–Pedlosky integral constraints.

Mechanisms of jet instability: role of deceleration

This paper shows that decrease of velocity magnitude downstream enhances the shear-layer instability in jetlike single- and two-fluid flows. This feature has not been recognized in stability studies, performed in the parallel and quasi-parallel flow approximations, because the deceleration effect is strongly non-parallel. It was first revealed in conical similarity flows. The similarity allows for reducing the base-flow and stability problems to ordinary differential equations exactly conserving the acceleration terms. This helps reveal the deceleration effect. The development of numerical technique for stability studies of two-dimensional flows helped understand the instability nature of swirling flows in sealed containers. It was found for an elongated cylinder with one rotating end disk and stationary other walls that the instability develops in the counterflow near the rotating disk, where the backflow decelerates and the shear-layer and deceleration effects cooperate. An interesting stabilizing effect of acceleration is observed in a model heat exchanger. The thermal convection develops in a rotating cylindrical container whose sidewall is adiabatic and the end disks have different temperatures. The centrifugal acceleration suppresses, while the temperature difference stimulates, the shear-layer instability of this flow. Next, the review discusses two-fluid flows in a sealed vertical container, driven by either bottom (whirlpool model) or lid (waterspout model) rotation. As the rotation speeds up, the shear-layer instability develops in a jet-like boundary layer in the air–water flows. The jet forms near a rotating disk, goes to the interface near the sidewall, and converges to the axis near the interface. New circulatory cells emerge in both fluids and the interface significantly bends. The instability develops after such flow pattern becomes well formed. The instability focuses near the interface where the jet-like motion approaches a new cell, decelerates, and diverges—hence the deceleration and shear-layer effects work together. In oil–water flows, the instability develops in the lower fluid, either in its depth or near the interface depending on the water volume fraction. Thus, the review shows that deceleration enhanced the shear-layer instability in a wide group of flows.

Development of a moving reference frame-based gas-kinetic BGK scheme for viscous flows around arbitrarily moving bodies

In this paper, a moving reference frame-based gas-kinetic BGK (MRF–BGK) scheme is developed for simulation of viscous flows around arbitrarily moving bodies. Different from most of existing gas-kinetic schemes, the BGK model is now constructed in a general moving reference frame. Due to the non-inertial effects, there will be an additional acceleration term in the Boltzmann equation, which should be carefully defined in order to correctly recover the concerned Navier–Stokes (N–S) equations. This recovery actually serves as a basis for developing the present method. Within the finite volume framework, the BGK model with the inclusion of the acceleration term is locally solved and the numerical fluxes at the cell interface can be evaluated. Through this way, the source (acceleration) term effects can be naturally included into the gas evolution process and the resulting fluxes. Considering that explicit BGK schemes may show low efficiency for certain unsteady applications where the physical time scales are significantly larger than the spatial scales, the dual time-stepping approach together with an implicit BGK scheme is employed. Numerical tests show that the developed MRF–BGK scheme can be well applied for several incompressible and compressible viscous flows. The accuracy is shown to be comparable to the moving grid BGK method, however the mesh movement is avoided in the present method. Also as compared with the immersed boundary BGK method, it is more efficient to deal with high-Reynolds number flows.

Dark energy – a critical value

Experimental evidence has converged towards a flat or Euclidean geometry for the universe with an Ω parameter or ratio of universe density to critical density of precisely 1. A considerable portion of that density is provided by the so-called dark energy which has added an acceleration term to the Hubble redshift. Measurements by the Planck satellite give values of order 68 % for the dark energy contribution. This value is intriguingly close to 2 / 3, a critical value which would make the redshift velocity and acceleration related by the Hubble constant. The same value would also be obtained by applying Mach’s principle to explain inertia and using the equivalence principle to relate the inertial and gravitational values of the mass of an object at the centre of a Hubble universe.

A Unified Gas Kinetic Scheme for Transport and Collision Effects in Plasma

In this study, the Boltzmann equation with electric acceleration term is discretized and solved by the unified gas-kinetic scheme (UGKS). The charged particle transport driven by electric field is included in the electric acceleration term. To capture non-equilibrium distribution function, the probability distribution functions of gas is discretized in a discrete velocity space. After discretization, the numerical flux for distribution function is computed to update the microscopic and macroscopic states. The flux is decided by an integral solution of Boltzmann equation based on characteristic problem. An electron-ion collision model is introduced in the Boltzmann Bhatnagar-Gross-Krook (BGK) equation. This finite volume method for the UGKS couples the free transport and long-range interaction between particles. For simplicity, the electric field induced by charged particles is controlled by the Poisson’s equation, which is solved using the Green’s function for two dimensional plasma system subjected to the symmetry or periodic boundary conditions. Two numerical cases, linear Landau damping and Gaussian beam, are carried out to validate the proposed method. The linear electron plasma wave damping is simulated based on electron-ion collision operator. Comparison results show good accuracy and higher efficiency than particle based methods. Difference between Poisson’s equation and complete electromagnetic Maxwell equation is presented by numerical results based on the two models. Highly non-equilibrium and rarefied plasma flows, such as electron flows driven by electromagnetic field, can be simulated easily. The UGKS-Poisson model is proved to be promising in plasma flow simulation.

Parametric resonance analysis of rectangular plates subjected to moving inertial loads via IHB method

This paper deals with the induced instability due to parametric resonance of rectangular plates traversed by inertial loads and lying on elastic foundations. The extended Hamilton's principle is employed to derive the partial differential equation associated with the transverse motion of the plate. Subsequently, this equation is transformed into a set of ordinary differential equations by the Galerkin method. Including vertical, centripetal and Coriolis acceleration terms related to the moving mass transition in the analysis leads to governing equations with time-varying mass, damping and stiffness coefficients. Particularly, the intermittent passage of masses along rectilinear paths, or the motion of an individual mass along an orbiting path, permits to subcategorize the problem as a parametrically excited system with periodic coefficients. By applying the incremental harmonic balance (IHB) method, the stability of the induced plate vibrations is investigated, revealing an emersion of instability tongues in the parameters plane. Semi-analytical results are provided for various boundary conditions of the plate which got verified through direct numerical simulations and other results reported in the literature.

A novel structure-adaptive intelligent grey forecasting model with full-order time power terms and its application

Abstract To solve the problem that traditional grey models cannot simulate accurately any given non-homogeneous exponential sequence with velocity and acceleration terms, a novel grey forecasting model with full-order time power terms (abbreviated as FOTP-GM(1,1)) is proposed. Firstly, two forms of sequence functions of the restored values are brought forward based respectively on whitenization method and connotation method. Then, Four forecasting properties are put forward to demonstrate that FOTP-GM(1,1) is a more general model with higher accuracy and adaptability than traditional models. Then a visual comparison method is introduced to facilitate selection of a more reasonable structure from all possible structures of the FOTP-GM(1,1) model. To verify its feasibility and efficiency, performance comparisons and suitability analyses are given by 2 examples. The first example shows that the simulative accuracy given by connotation method is higher than that by whitenization method, and it confirms that NDGM(1,1) and SAIGM(1,1) models are all special cases of the FOTP-GM(1,1) model. Then by quantitative analysis and visual comparison, the second example shows that FOTP-GM(1,1) model has better adaptability and broader universality. In the last, FOTP-GM(1,1) model is employed to forecast the potential total production volume of hydropower, nuclear power and wind power from 2017 to 2021 in China. Thus practicality of the proposed model is tested.

Light propagation in 2PN approximation in the field of one moving monopole I. Initial value problem

In this investigation the light propagation in the gravitational field of one arbitrarily moving body with monopole structure is considered in the second post-Newtonian approximation. It is found that the light trajectory depends on the acceleration of the body. Some of these acceleration terms are important in order to get well-defined logarithmic functions with dimensionless arguments, while all the other acceleration terms are negligible on the pico-second level of accuracy in time-delay measurements. The expressions of the observables total light deflection and time delay are determined.

Evaluation of Offshore Wind Turbine Tower Dynamics with Numerical Analysis

A dynamic behaviour of a cylindirical wind tower with variable cross section is investigated under environmental and earthquake forces. The ground acceleration term is represented by a simple cosine function to investigate both normal and parallel components of the earthquake motions located near ground surface. The function of earthquake force is simplified to apply Rayleigh’s energy method. Wind forces acting on above the water level and wave forces acting on below this level are utilized in computations considering earthquake effect for entire structure. The wind force is divided into two groups: the force acting on the tower and the forces acting on the rotor nacelle assembly (RNA). The drag and the inertial wave forces are calculated with water particle velocities and accelerations due to linear wave theory. The resulting hydrodynamic wave force on the tower in an unsteady viscous flow is determined using the Morison equation. The displacement function of the physical system in which dynamic analysis is performed by Rayleigh’s energy method is obtained by the single degree of freedom (SDOF) model. The equation of motion is solved by the fourth‐order Runge–Kutta method. The two‐way FSI (fluid‐structure interaction) technique was used to determine the accuracy of the numerical analysis. The results of computational fluid dynamics and structural mechanics are coupled in FSI analysis by using ANSYS software. Time‐varying lateral displacements and the first natural frequency values which are obtained from Rayleigh’s energy method and FSI technique are compared. The results are presented by graphs. It is observed from these graphs that the Rayleigh model can be an alternative way at the prelimanary stage of the structural analysis with acceptable accuracy.

Acoustic nonreciprocity in Coriolis mean flow systems.

One way to break acoustic reciprocity is to have a moving wave propagation medium. If the acoustic wave vector and the moving fluid velocity are collinear, the wave vector shift caused by the fluid flow can be used to break. In this paper, an alternative approach is investigated in which the fluid velocity enters the differential equation of the system as a cross product term with the wave vector. A circular field where the fluid velocity increases radially has a Coriolis acceleration term. In such a system, the acoustic wave enters from the central wall and exits from the perimeter wall. In this paper, the differential equation is solved numerically and the effect of fluid velocity on the nonreciprocity factor is examined.

Prediction of the turning and zig-zag maneuvering performance of a surface combatant with URANS

The main objective of this study is to investigate the turning and zig-zag maneuvering performance of the well-known naval surface combatant DTMB (David Taylor Model Basin) 5415 hull with URANS (Unsteady Reynolds-averaged Navier-Stokes) method. Numerical simulations of static drift tests have been performed by a commercial RANS solver based on a finite volume method (FVM) in an unsteady manner. The fluid flow is considered as 3-D, incompressible and fully turbulent. Hydrodynamic analyses have been carried out for a fixed Froude number 0.28. During the analyses, the free surface effects have been taken into account using VOF (Volume of Fluid) method and the hull is considered as fixed. First, the code has been validated with the available experimental data in literature. After validation, static drift, static rudder and drift and rudder tests have been simulated. The forces and moments acting on the hull have been computed with URANS approach. Numerical results have been applied to determine the hydrodynamic maneuvering coefficients, such as, velocity terms and rudder terms. The acceleration, angular velocity and cross-coupled terms have been taken from the available experimental data. A computer program has been developed to apply a fast maneuvering simulation technique. Abkowitz\'s non-linear mathematical model has been used to calculate the forces and moment acting on the hull during the maneuvering motion. Euler method on the other hand has been applied to solve the simultaneous differential equations. Turning and zig-zag maneuvering simulations have been carried out and the maneuvering characteristics have been determined and the numerical simulation results have been compared with the available data in literature. In addition, viscous effects have been investigated using Eulerian approach for several static drift cases.

A Ffowcs Williams and Hawkings formulation for hydroacoustic analysis of propeller sheet cavitation

A novel hydroacoustic formulation for the prediction of tonal noise emitted by marine propellers in presence of unsteady sheet cavitation, is presented. The approach is based on the standard Ffowcs Williams and Hawkings equation and the use of transpiration (velocity and acceleration) terms, accounting for the time evolution of the vapour cavity attached on the blade surface. Drawbacks and potentialities of the method are tested on a marine propeller operating in a nonhomogeneous onset flow, by exploiting the hydrodynamic data from a potential-based panel method equipped with a sheet cavitation model and comparing the noise predictions with those carried out by an alternative numerical approach, documented in literature. It is shown that the proposed formulation yields a one-to-one correlation between emitted noise and sheet cavitation dynamics, carrying out accurate predictions in terms of noise magnitude and directivity.

Dynamical Effects of the Scale Invariance of the Empty Space: The Fall of Dark Matter?

Abstract The hypothesis of the scale invariance of the macroscopic empty space, which intervenes through the cosmological constant, has led to new cosmological models. They show an accelerated cosmic expansion after the initial stages and satisfy several major cosmological tests. No unknown particles are needed. Developing the weak-field approximation, we find that the here-derived equation of motion corresponding to Newton’s equation also contains a small outward acceleration term. Its order of magnitude is about Newton’s gravity (ϱ being the mean density of the system and the usual critical density). The new term is thus particularly significant for very low density systems. A modified virial theorem is derived and applied to clusters of galaxies. For the Coma Cluster and Abell 2029, the dynamical masses are about a factor of 5–10 smaller than in the standard case. This tends to leave no room for dark matter in these clusters. Then, the two-body problem is studied and an equation corresponding to the Binet equation is obtained. It implies some secular variations of the orbital parameters. The results are applied to the rotation curve of the outer layers of the Milky Way. Starting backward from the present rotation curve, we calculate the past evolution of the Galactic rotation and find that, in the early stages, it was steep and Keplerian. Thus, the flat rotation curves of galaxies appear as an age effect, a result consistent with recent observations of distant galaxies by Genzel et al. and Lang et al. Finally, in an appendix we also study the long-standing problem of the increase with age of the vertical velocity dispersion in the Galaxy. The observed increase appears to result from the new small acceleration term in the equation of the harmonic oscillator describing stellar motions around the Galactic plane. Thus, we tend to conclude that neither dark energy nor dark matter seems to be needed in the proposed theoretical context.

Toward Position-Only Time-Delayed Control for Uncertain Euler–Lagrange Systems: Experiments on Wheeled Mobile Robots

This letter addresses the various practical design issues of a continuous-time time-delayed control (TDC) and proposes a new controller to make the TDC more suitable and applicable for real-life systems. While TDC has been renowned for its robust performance and simplicity in form, it requires state-derivatives feedback which may not be available explicitly in practice; and multiple numerical differentiations of the noisy state data deteriorate performance by invoking measurement error. In this letter, position-only time-delayed control (POTDC) has been proposed for a class of Euler–Lagrange systems which encompasses a large class of practical systems such as robotic manipulators and unmanned mobile robots. Contrary to the conventional TDC, the proposed POTDC eliminates explicit requirement of velocity and acceleration feedbacks and uses only position information of present and past instances to estimate the velocity and acceleration terms. It thus alleviates the measurement error arising from a numerical computation of state-derivatives. Moreover, based on the Razumikhin theorem, continuous-time stability is rigorously analyzed with consideration of the time-delay element in POTDC, which indeed establishes a selection criterion for the sampling interval and provides the designer a range of sampling intervals for same choice of controller gains. This allows POTDC to be suitable for systems which specifically operate with high sampling intervals due to application requirement. Accordingly, experimental validations of POTDC are provided in comparison with TDC under various sampling intervals, using a wheeled mobile robot.

Experimental and numerical investigation on steady and transient flow in the flat-walled micro-diffuser/nozzle

In this paper, experiments and numerical simulations have been conducted to investigate the flow characteristics in the flat-walled micro-diffuser/nozzle on conditions of angle (0°–60°), Reynolds number (100–1500) and excitation frequency (0–5000 Hz). In steady flow, the diffuser efficiency increases with an increase in Reynolds number as θ ≤ 10°, but the optimal diffuser efficiency is found as 20° ≤ θ ≤ 60°, and the internal flow characteristics at the optimal diffuser efficiency are recorded. Based on statistics, it is concluded that the interaction of adverse pressure gradient and diverging flow results in the optimal angle, and the optimal angle decreases with an increase in Reynolds number. While in transient flow, the excitation frequency has great impacts on the optimal angle, the higher frequency results in the larger optimal angle. As the Reynolds numbers are 100 and 1500, the variations of diffuser efficiency with excitation frequency show the opposite trends. At low Reynolds number, the acceleration term caused by the excitation frequency increases the flow loss and reduces the diffuser efficiency. However, at high Reynolds number, the energy of diverging flow is stored in the vortexes and used to prevent the converging flow, this contributes to the great increase in the diffuser efficiency. Therefore, the influence of excitation frequency on the diffuser efficiency is determined by the location and intensity of the vortexes.

Generalized contact and improved frictional heating in the material point method

The material point method (MPM) has proved to be an effective particle method for computational mechanics modeling of problems involving contact, but all prior applications have been limited to Coulomb friction. This paper generalizes the MPM approach for contact to handle any friction law with examples given for friction with adhesion or with a velocity-dependent coefficient of friction. Accounting for adhesion requires an extra calculation to evaluate contact area. Implementation of velocity-dependent laws usually needs numerical methods to find contacting forces. The friction process involves work which can be converted into heat. This paper provides a new method for calculating frictional heating that accounts for interfacial acceleration during the time step. The acceleration terms is small for many problems, but temporal convergence of heating effects for problems involving vibrations and high contact forces is improved by the new method. Fortunately, the new method needs few extra calculations and therefore is recommended for all simulations.

Dynamic contact of a thermoelastic Mindlin–Timoshenko beam with a rigid obstacle

We concentrate on the dynamics of a thermoelastic Mindlin–Timoshenko beam striking a rigid obstacle. We state classical formulations involving complementarity conditions. Weak formulations are in the form of systems consisting of a hyperbolic variational inequality for a deflection, a hyperbolic and a parabolic equation for an angle of rotation and a thermal strain, respectively. The penalization method is applied to solve the unilateral problem. The time derivative of the function representing the deflection of the beam’s middle line is not continuous due to the hitting the obstacle. The acceleration term has the form of a vector measure.

Revisiting the velocity-dependent one-scale model for monopoles

We revisit the physical properties of global and local monopoles and discuss their implications in the dynamics of monopole networks. In particular, we review the Velocity-dependent One-Scale (VOS) model for global and local monopoles and propose physically motivated changes to its equations. We suggest a new form for the acceleration term of the evolution equation of the root-mean-squared velocity and show that, with this change, the VOS model is able to describe the results of radiation and matter era numerical simulations of global monopole networks with a single value of the acceleration parameter $k$, thus resolving the tension previously found in the literature. We also show that the fact that the energy of global monopoles is not localized within their cores affects their dynamics and, thus, the Hubble damping terms in the VOS equations. We study the ultra-relativistic linear scaling regime predicted by the VOS equations and demonstrate that it cannot be attained either on radiation or matter eras and, thus, cannot arise from the cosmological evolution of a global monopole network. We also briefly discuss the implications of our findings for the VOS model for local monopoles.

Time-dependent interaction between a two-level atom and a su(1,1) Lie algebra quantum system

The problem of the interaction between a two-level atom and a two-mode field in the parametric amplifier-type is considered. A similar problem appears in an ion trapped in a two-dimensional trap. The problem is transformed into an interaction governed by su(1,1) Lie algebraic operators with phase and coupling parameter depending on time. Under an integrability condition, that relates phase and coupling, a solution to the wavefunction is obtained using the Schrödinger equation. The effects of the functional dependence of the coupling and the initial state of the two-level atom on atomic inversion, the degree of entanglement, the fidelity and the Glauber second-order correlation function are investigated. It is shown that the acceleration term plays an important role in controlling the function behavior of the considered quantities.

Decentralized Multi-Robot Formation Control with Communication Delay and Asynchronous Clock

This paper investigates the leader–follower formation control problem for a group of networked nonholonomic mobile robots that are subject to bounded time-varying communication delays and an asynchronous clock. First we convert the formation control problem into a trajectory tracking problem, and then a fully distributed unified control framework based on the receding horizon control is implemented to converge the tracking errors. By adding an auxiliary acceleration term into the receding horizon controller, the framework is able to solve the impractical velocity jump problem. Considering the time-varying delays, the timing and order features of the messages are utilized to guarantee their logical correctness. To compensate for the delay effect, an improved control framework that exploits the predictability of the receding horizon controller is proposed. The asynchronous clock problem, which makes the communication delay unmeasurable, is studied. We give a definition of the syn point that is inspired from investigation of the property that messages are received out of order in a bounded time-varying delayed network. A novel method that detects the occurrence of syn points is integrated into the control framework to solve the asynchronous clock problem. Finally the effectiveness of the proposed approaches is demonstrated in the Player/Stage simulation environment.