Stability of small perturbations for a modified two-dimensional quasi-gasdynamic model of traffic flows

2010 ◽  
Vol 2 (6) ◽  
pp. 776-781
Author(s):  
A. A. Zlotnik
Keyword(s):  
Two Dimensional ◽  
Traffic Flows ◽  
Small Perturbations ◽  
Gasdynamic Model ◽  
Stability Of Small Perturbations

On the stability of plane shock waves

1957 ◽  
Vol 2 (4) ◽  
pp. 397-411 ◽  
Author(s):  
N. C. Freeman
Keyword(s):  
Shock Wave ◽  
Mach Number ◽  
Shock Waves ◽  
Plane Shock ◽  
Two Dimensional ◽  
Small Perturbations ◽  
Diverging Channel ◽  
Maximum Stability ◽  
The Stability ◽  
Fourier Type

The decay of small perturbations on a plane shock wave propagating along a two-dimensional channel into a fluid at rest is investigated mathematically. The perturbations arise from small departures of the walls from uniform parallel shape or, physically, by placing small obstacles on the otherwise plane parallel walls. An expression for the pressure on a shock wave entering a uniformly, but slowly, diverging channel already exists (given by Chester 1953) as a deduction from the Lighthill (1949) linearized small disturbance theory of flow behind nearly plane shock waves. Using this result, an expression for the pressure distribution produced by the obstacles upon the shock wave is built up as an integral of Fourier type. From this, the shock shape, ξ, is deduced and the decay of the perturbations obtained from an expansion (valid after the disturbances have been reflected many times between the walls) for ξ in descending power of the distance, ζ, travelled by the shock wave. It is shown that the stability properties of the shock wave are qualitatively similar to those discussed in a previous paper (Freeman 1955); the perturbations dying out in an oscillatory manner like ζ−3/2. As before, a Mach number of maximum stability (1·15) exists, the disturbances to the shock wave decaying most rapidly at this Mach number. A modified, but more complicated, expansion for the perturbations, for use when the shock wave Mach number is large, is given in §4.In particular, the results are derived for the case of symmetrical ‘roof top’ obstacles. These predictions are compared with data obtained from experiments with similar obstacles on the walls of a shock tube.

scholarly journals Impact of Asymmetric Dynamical Processes on the Structure and Intensity Change of Two-Dimensional Hurricane-Like Annular Vortices

2013 ◽  
Vol 70 (2) ◽  
pp. 559-582 ◽  
Author(s):  
Konstantinos Menelaou ◽  
M. K. Yau ◽  
Yosvany Martinez
Keyword(s):  
Inner Core ◽  
Critical Radius ◽  
Normal Modes ◽  
Intensity Change ◽  
Two Dimensional ◽  
Wind Maximum ◽  
Small Perturbations ◽  
Sensitivity Tests ◽  
Eigenmode Analysis ◽  
External Impulse

Abstract In this study, a simple two-dimensional (2D) unforced barotropic model is used to study the asymmetric dynamics of the hurricane inner-core region and to assess their impact on the structure and intensity change. Two sets of experiments are conducted, starting with stable and unstable annular vortices, to mimic intense mature hurricane-like vortices. The theory of empirical normal modes (ENM) and the Eliassen–Palm flux theorem are then applied to extract the dominant wave modes from the dataset and diagnose their kinematics, structure, and impact on the primary vortex. From the first experiment, it is found that the evolution and the lifetime of an elliptical eyewall, described by a stable annular vortex perturbed by an external wavenumber-2 impulse, may be controlled by the inviscid damping of sheared vortex Rossby waves (VRWs) or the decay of an excited quasimode. The critical radius and structure of the quasimode obtained by the ENM analysis are shown to be consistent with the predictions of a linear eigenmode analysis of small perturbations. From the second experiment, it is found that the outward-propagating VRWs that arise due to barotropic instability and the inward mixing of high vorticity in the unstable annular vortex affect the primary circulation and create a secondary ring of enhanced vorticity that contains a secondary wind maximum. Sensitivity tests performed on the spatial extent of the initial external impulse verifies the robustness of the results. That the secondary eyewall occurs close to the critical radius of some of the dominant modes emphasizes the important role played by the VRWs.

Two-dimensional macroscopic model of traffic flows

2009 ◽  
Vol 1 (6) ◽  
pp. 669-676 ◽  
Author(s):  
A. B. Sukhinova ◽  
M. A. Trapeznikova ◽  
B. N. Chetverushkin ◽  
N. G. Churbanova
Keyword(s):  
Macroscopic Model ◽  
Two Dimensional ◽  
Traffic Flows

The growth, saturation, and scaling behaviour of one- and two-dimensional disturbances in fluidized beds

1998 ◽  
Vol 362 ◽  
pp. 83-119 ◽  
Author(s):  
M. F. GÖZ ◽  
S. SUNDARESAN
Keyword(s):  
Fluidized Bed ◽  
Growth Rates ◽  
Fluidized Beds ◽  
Secondary Growth ◽  
Two Dimensional ◽  
Small Perturbations ◽  
One Dimensional ◽  
Gas Fluidized Beds ◽  
The Waves ◽  
Primary Instability

It is well-known that fluidized beds are usually unstable to small perturbations and that this leads to the primary bifurcation of vertically travelling plane wavetrains. These one-dimensional periodic waves have been shown recently to be unstable to two-dimensional perturbations of large transverse wavelength in gas-fluidized beds. Here, this result is generalized to include liquid-fluidized beds and to compare typical beds fluidized with either air or water. It is shown that the instability mechanism remains the same but there are big differences in the ratio of the primary and secondary growth rates in the two cases. The tendency is that the secondary growth rates, scaled with the amplitude of a fully developed plane wave, are of similar magnitude for both gas- and liquid-fluidized beds, while the primary growth rate is much larger in the gas-fluidized bed. This means that the secondary instability is accordingly stronger than the primary instability in the liquid-fluidized bed, and consequently sets in at a much smaller amplitude of the primary wave. However, since the waves in the liquid-fluidized bed develop on a larger time and length scale, the primary perturbations need longer time and thereby travel farther until they reach the critical amplitude. Which patterns are more amenable to being visually recognized depends on the magnitude of the initially imposed disturbance and the dimensions of the apparatus. This difference in scale plays a key role in bringing about the differences between gas- and liquid-fluidized beds; it is produced mainly by the different values of the Froude number.

Support stability of maximizing measures for shifts of finite type

2019 ◽  
pp. 1-12
Author(s):  
JULIANO S. GONSCHOROWSKI ◽  
ANTHONY QUAS ◽  
JASON SIEFKEN
Keyword(s):  
Finite Type ◽  
Fundamental Difference ◽  
Probability Measures ◽  
Two Dimensional ◽  
Small Perturbations ◽  
Subshift Of Finite Type ◽  
Local Configuration ◽  
Ergodic Optimization ◽  
Full Shift ◽  
Subshifts Of Finite Type

This paper establishes a fundamental difference between $\mathbb{Z}$ subshifts of finite type and $\mathbb{Z}^{2}$ subshifts of finite type in the context of ergodic optimization. Specifically, we consider a subshift of finite type $X$ as a subset of a full shift  $F$ . We then introduce a natural penalty function  $f$ , defined on  $F$ , which is 0 if the local configuration near the origin is legal and $-1$ otherwise. We show that in the case of $\mathbb{Z}$ subshifts, for all sufficiently small perturbations, $g$ , of  $f$ , the $g$ -maximizing invariant probability measures are supported on $X$ (that is, the set $X$ is stably maximized by  $f$ ). However, in the two-dimensional case, we show that the well-known Robinson tiling fails to have this property: there exist arbitrarily small perturbations, $g$ , of  $f$ for which the $g$ -maximizing invariant probability measures are supported on $F\setminus X$ .

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