Evolution Of Small Perturbations Research Articles

A bi-directional visual angle car-following model considering collision sensitivity

From the perspective of driver’s visual characteristics, the potential collision conflicts in car-following and lane-changing driving behaviors are analyzed. On this basis, a bi-directional visual angle car-following model considering the collision sensitivity is proposed. The effect of collision sensitivity coefficient, the proportion of backward-looking visual angle, and many types of vehicles on the dynamics and safety of the newly proposed model is explored. A linear local stability analysis was carried out by perturbation analysis, and the critical stability condition of the improved model was deduced. The evolution of small perturbation during the car-following and lane-changing process is numerically simulated. The same conclusion as the linear stability analysis is obtained: The driver observes different types of vehicles in the vehicle queue through the bi-directional visual angle as feedback stimuli, so as to optimize the velocity and perceive potential collision conflicts, which has significant practical significance for improving the stability of the traffic system and alleviating traffic congestion. Besides, the safety of the new model is evaluated by the improved visual time-to-collision (TTC) index. The results show that the newly introduced factors can effectively reduce the potential collision risk in multiple scenarios.

Perturbation Growth Rate in Turbulent Couette Flow

The evolution of small perturbations in turbulent Couette flow is investigated numerically at the Reynolds numbers Reτ from 35 to 125. Steady-state turbulent flows calculated on the basis of solving the Navier-Stokes equations are used as the basic flow to study the process of development of the perturbations against their background. The values of the senior Lyapunov exponent λ1 which characterizes the maximum growth rate of small perturbations in the stochastic systems are determined. It is found that the exponent, being normalized by the near-wall scales, is equal to λ1 ~ 0.02. This is in agreement with the results of the previous investigations of turbulent flows in the circular pipe and plane channel. It is shown that the λ1, whose value is smaller by three times and which was obtained earlier in calculating the Lyapunov spectrum in Couette flow at Reτ = 35 in the so-called “minimum channel”, can be explained by insufficient dimension of the computational domain but not by smallness of the Reynolds number.

An extended macro model accounting for acceleration changes with memory and numerical tests

Considering effect of acceleration changes with memory, an improved continuum model of traffic flow is proposed in this paper. By applying the linear stability theory, we derived the new model’s linear stability condition. Through nonlinear analysis, the KdV–Burgers equation is derived to describe the propagating behavior of traffic density wave near the neutral stability line. Numerical simulation is carried out to study the extended traffic flow model, which explores how acceleration changes with memory affected each car’s velocity, density and fuel consumption and exhaust emissions. Numerical results demonstrate that acceleration changes with memory have significant negative effect on dynamic characteristic of traffic flow. Furthermore, research results verify that the effect of acceleration changes with memory will deteriorate the stability of traffic flow and increase cars’ total fuel consumptions and emissions during the whole evolution of small perturbation.

KdV-Burgers equation in the modified continuum model considering the “backward looking” effect

Considering the backward looking effect, a modified continuum model is put forward. The stability criterion of this continuum model is deduced by performing linear stability analysis. The theoretical result demonstrates that the traffic flow is stabilized considering backward looking effect. The KdV-Burgers equation is obtained through perturbed nonlinear analysis, which could show the propagating process of density wave. Numerical simulation is carried out to explores how backward looking affected each car’s velocity, density and energy consumption. Numerical results demonstrate that backward looking effect has significant impact on traffic dynamic characteristic. In addition, the energy consumptions of this modified continuum model are also studied. Numerical simulation results indicate that the effect of backward looking will suppress the traffic jam and decrease cars’ energy consumptions during the whole evolution of small perturbation.

Estimates of the evolution of small perturbations in the radial spread (drain) of a viscous ring

This paper studies the evolution of small perturbations in the kinematic and dynamic characteristics of the radial flow of a flat ring filled with a homogeneous Newtonian fluid or an ideal incompressible fluid. When the flow rate is specified as a function of time, the main motion is completely determined by the incompressibility condition regardless of the properties of the medium. A biparabolic equation for the stream function with four homogeneous boundary conditions which simulate adhesion to the expanding (contracting) walls of the ring is derived. Upper bounds for the perturbation are obtained using the method of integral relations for quadratic functionals. The case of an exponential decay of initial perturbations is considered in a finite or infinite time interval. The admissibility of the inviscid limit in this problem is proved, and upper and lower bounds for this limit are estimated.

Impacts of road conditions on the energy consumption of electric vehicular flow

In this paper, we use the electricity consumption model for electric vehicular flow [H. Xiao, H. J. Huang and T. Q. Tang, Mod. Phys. Lett. B 30 (2016) 1650325] to study the effects of road conditions on the electricity consumption of electric vehicular flow during the evolutions of shock, rarefaction wave and small perturbation. The numerical results indicate that road conditions have negative influences on the electricity consumption during the evolutions of shock and rarefaction wave (i.e. the electricity consumption increases when road conditions become better) and positive impacts on the electricity consumption during the evolution of small perturbation when the traffic flow is unstable (i.e. the electricity consumption produces oscillation, but its amplitude decreases when road conditions become better).

Influence of orientation on the evolution of small perturbations in compressible shear layers with inflection points.

We investigate the influence of orientation on the evolution of small perturbations in compressible shear layers with inflection points. By using linear analysis, we demonstrate that perturbations along the shear plane are most affected by compressibility. The influence of compressibility gradually diminishes with increasing obliqueness of the perturbations with respect to the shear plane. It is demonstrated that the effective gradient Mach number is an appropriate compressibility parameter. We establish that spanwise perturbations, orthogonal to the shear plane, are impervious to compressibility effects. Direct numerical simulations of compressible mixing layers subject to the perturbations at various obliqueness angles verify the analytical findings.

On the Origin of Rotation in the Universe

The evolution of small perturbations produced by gravitational instability in ideal gases is examined in a Newtonian approach. As opposed to the classical theory, here effects associated with fluctuationinduced fluxes of matter are taken into account. It is shown that for sufficiently small wave numbers, both longitudinal and vortex perturbations can grow and this explains the origin of galactic rotation in an initially uniform, isotropic, irrotational universe.

An extended macro traffic flow model accounting for the driver’s bounded rationality and numerical tests

Abstract In this paper, we propose a macro traffic flow model to explore the effects of the driver’s bounded rationality on the evolutions of traffic waves (which include shock and rarefaction waves) and small perturbation, and on the fuel consumption and emissions (that include CO, HC and NO X ) during the evolution process. The numerical results illustrate that considering the driver’s bounded rationality can prominently smooth the wavefront of the traffic waves and improve the stability of traffic flow, which shows that the driver’s bounded rationality has positive impacts on traffic flow; but considering the driver’s bounded rationality reduces the fuel consumption and emissions only at the upstream of the rarefaction wave while enhances the fuel consumption and emissions under other situations, which shows that the driver’s bounded rationality has positive impacts on the fuel consumption and emissions only at the upstream of the rarefaction wave, while negative effects on the fuel consumption and emissions under other situations. In addition, the numerical results show that the driver’s bounded rationality has little prominent impact on the total fuel consumption, and emissions during the whole evolution of small perturbation.

Flow-driven instabilities during pattern formation of Dictyostelium discoideum

The slime mold Dictyostelium discoideum is a well known model system for the study of biological pattern formation. In the natural environment, aggregating populations of starving Dictyostelium discoideum cells may experience fluid flows that can profoundly change the underlying wave generation process. Here we study the effect of advection on the pattern formation in a colony of homogeneously distributed Dictyostelium discoideum cells described by the standard Martiel–Goldbeter model. The external flow advects the signaling molecule cyclic adenosine monophosphate (cAMP) downstream, while the chemotactic cells attached to the solid substrate are not transported with the flow. The evolution of small perturbations in cAMP concentrations is studied analytically in the linear regime and by corresponding numerical simulations. We show that flow can significantly influence the dynamics of the system and lead to a flow-driven instability that initiate downstream traveling cAMP waves. We also show that boundary conditions have a significant effect on the observed patterns and can lead to a new kind of instability.

Sound Production by Vorticity Fluctuations in a Stagnation Point Flow

A linear mechanism for the emergence of acoustic waves from aperiodic vorticity disturbances is presented for a simple stagnation point flow. The non-modal approach, which made possible the discovery of a similar mechanism in unbounded shear flows, is employed. The phenomenon is shown to be closely related to the non-normal nature of the linearized operator governing the evolution of small perturbations about the mean flow. The vorticity disturbance acts as an acoustic source by interacting with the mean velocity gradient and drives the acoustic waves in a non-resonant manner. The physical mechanism by which the acoustic and vortical components of the disturbance exchange energy with the mean flow is presented. The proposed mechanism is relevant for numerous applications involving mass addition or removal, such as aeroacoustic instabilities in a solid rocket motor. Similar phenomena are expected to occur in accelerated flows, like the flow in a nozzle. Numerical simulations using a high order, dispersion relation preserving scheme is used to demonstrate how a localized vorticity disturbance of small amplitude generates sound waves when strained by the gradients in the base flow. The simulations reveal that the vorticity disturbance acts as a quadrupole source of sound, whose origin is explained using physical arguments. Linear simulations using a spectral code verify the results of the nonlinear simulations performed at small amplitude.

Rayleigh-Taylor-Kelvin-Helmholtz combined instability at the magnetopause

The magnetopause separates the geomagnetic field from the interplanetary plasma and performs finite motions under the action of the solar wind pressure variable in time. Accelerations originating in this case result in that the necessary condition for the development of the Rayleigh-Taylor instability is formed at quite a definite motion phase. We can anticipate that the instability will develop during compression of the magnetosphere. It should be taken into account that the magnetopause is a potential tangential discontinuity. On the one hand, a plasma flow along the magnetopause results in a decrease in the Rayleigh-Taylor instability threshold. On the other hand, the Kelvin-Helmholtz instability threshold, typical of the tangential discontinuity, also decreases during the magnetosphere compression phase. Thus, if we speak about the magnetopause, it is natural and necessary to jointly consider both types of instability. Main information on the combined Rayleigh-Taylor-Kelvin-Helmholtz instability is presented, the dispersion equation determining the evolution of small perturbations is considered, and the possible geophysical applications to the theory (e.g., penetration of the solar plasma into the magnetosphere, excitation of global Pc5 oscillations) are indicated.

Propagation of large-amplitude waves on dielectric liquid sheets in a tangential electric field: Exact solutions in three-dimensional geometry

Nonlinear waves on sheets of dielectric liquid in the presence of an external tangential electric field are studied theoretically. It is shown that waves of arbitrary shape in three-dimensional geometry can propagate along (or against) the electric field direction without distortion, i.e., the equations of motion admit a wide class of exact traveling wave solutions. This unusual situation occurs for nonconducting ideal liquids with high dielectric constants in the case of a sufficiently strong field strength. Governing equations for evolution of plane symmetric waves on fluid sheets are derived using conformal variables. A dispersion relation for the evolution of small perturbations of the traveling wave solutions is obtained. It follows from this relation that, regardless of the wave shape, the amplitudes of small-scale perturbations do not increase with time and, hence, the traveling waves are stable. We also study the interaction of counterpropagating symmetric waves with small but finite amplitudes. The corresponding solution of the equations of motion describes the nonlinear superposition of the oppositely directed waves. The results obtained are applicable for the description of long waves on fluid sheets in a horizontal magnetic field.

Stability of the adiabatic compression of an ideal gas by a thin shell

A new solution has been obtained for the problems of the isentropic compression of an ideal gas in the centered wave formulated in the Lagrangian variables. The developed method makes it possible in a single, uniform way to find the solution to the gas-dynamic problem in the plane, cylindrical, and spherical cases for an initially uniform motionless gas. Equations were derived describing the evolution of the small perturbations of the shell in the case where the acceleration is time-dependent. The shell is assumed to be thin and structureless, and its mass is supposed to exceed significantly the mass of the gas surrounding the shell. The cases of the planar, cylindrical, and spherical geometries of the system are considered. The stability of the shell’s motion producing isentropic compression of the gas is considered with respect to the evolution of small perturbations of plane-wave and angularharmonic types. It is shown that under the isentropic compression of a gas, the increase in the shell’s perturbations of the plane-wave type is limited both in the planar and cylindrical geometries. The limiting growth in the amplitude of such perturbations is calculated. The increase in the perturbations of the angular-harmonic type is demonstrated to be unlimited both in cylindrical and in spherical geometries. The increment in the growth of such perturbations has been calculated.

THE SPECTRUM OF SCALAR FLUCTUATIONS OF A BOUNCING UNIVERSE

In the last years, the idea of the existence of a collapsing phase previous to the actual expanding one has attracted attention in many different contexts (being very active!). There are many reasons for this, which concerns the standard model and its difficulties in dealing with a singularity which — in the words of the creator of general relativity — means the failure of the equations of general relativity to represent the gravitational field in those regions of extraordinary high curvature. However, we would like to point out just one: the possibility of deciding the existence of such primordial collapsing phase by observational tests due to the inprint it could be left in the inhomogeneous structure that constitutes the actual distribution of galaxies and cluster of galaxies. In this vein, the purpose of the present work is to analyze a particular bouncing universe and the evolution of small perturbations. To realize such analysis when the geometry has a bouncing (that is, the associated Hubble parameter — that measures the rate of the velocity of the expansion through [Formula: see text] — has a zero) the standard Lifshitz–Bardeen–Mukhanov variables/method is not the best one. Instead, we use the most well-behaved standard quasi-Maxwellian equations of perturbation introduced by Hawking and developed by Ellis et al. and Novello et al.

Evolution of distortions of the spherical shape of a cavitation bubble in acoustic supercompression

The evolution of small perturbations of the spherical shape of a vapor bubble in the process of its single strong expansion and compression in deuterated acetone is studied. In the mathematical model used the motion of vapor and liquid is broken down into the spherical component and its small nonspherical perturbation. The spherical component is described by the fluid dynamics equations with account for time-dependent heat conduction and evaporation and condensation on the liquid-vapor interface using equations of state constructed from experimental data. In describing the nonspherical component the liquid viscosity and the surface tension are taken into account, while the effect of the bubble content is disregarded. Certain simple analytical formulas are presented which describe the bubble radius at the moment of maximum expansion, its variation in the compression stage, and the evolution of the bubble sphericity distortion in compression.

Self-similar shapes of the free boundary of a nonlinear-viscous band under uniaxial tension

An equation of evolution of small perturbations of the free boundary of a nonlinear-viscous band under quasi-static uniaxial tension is derived for studying the necking problem in metals under superplasticity conditions. It is shown that the group of symmetry of this linear parabolic equation is equivalent to the group of symmetry of the linear equation of heat conduction with an arbitrary material parameter of the model. Self-similar solutions are obtained in the form of simple and complicated steady localized structures transferred together with the material of the stretched band.

Aggregation of chemotactic organisms in a differential flow

We study the effect of advection on the aggregation and pattern formation in chemotactic systems described by Keller-Segel-type models. The evolution of small perturbations is studied analytically in the linear regime complemented by numerical simulations. We show that a uniform differential flow can significantly alter the spatial structure and dynamics of the chemotactic system. The flow leads to the formation of anisotropic aggregates that move following the direction of the flow, even when the chemotactic organisms are not directly advected by the flow. Sufficiently strong advection can stop the aggregation and coarsening process that is then restricted to the direction perpendicular to the flow.

Effect of a strong longitudinal magnetic field on capillary disintegration of a magnetic liquid film covering the inner surface of a narrow cylindrical tube

The effect of magnetic forces on the viscous regime of disintegration of a thin cylindrical ferroliquid film magnetized to saturation is studied on the basis of an asymptotic model describing the evolution of small perturbations during capillary disintegration of this film. A formula describing the dependence of the wavelength of the most rapidly increasing mode on the physical parameters characterizing this phenomenon is derived analytically.

On the rate of spatial predictability in near-wall turbulence

Spatial evolution of small perturbations introduced into an inlet cross-section of fully developed turbulent flow in a long straight circular pipe is investigated via direct numerical simulation (DNS). The turbulent inflow field is extracted from an auxiliary streamwise-periodic simulation running in parallel with the main spatial simulation. It is shown that mean perturbation amplitude ϵ increases exponentially with distance downstream. The growth rate is found to be constant when normalized by viscous length, ϵ ~ exp(0.0021x+) over the considered Reynolds-number range 140 ≤ Reτ ≤ 320. The universal character of perturbation growth is confirmed by channel-flow simulations.