Energy dynamics in a turbulent channel flow using the Karhunen‐Loéve approach

The dynamical equations for the energy in a turbulent channel flow have been developed by using the Karhunen‐Loéve modes to represent the velocity field. The energy balance equations show that all the energy in the flow originates from the applied pressure gradient acting on the mean flow. Energy redistribution occurs through triad interactions, which is basic to understanding the dynamics. Each triad interaction determines the rate of energy transport between source and sink modes via a catalyst mode. The importance of the proposed method stems from the fact that it can be used to determine both the rate of energy transport between modes as well as the direction of energy flow. The effectiveness of the method in determining the mechanisms by which the turbulence sustains itself is demonstrated by performing a detailed analysis of triad interactions occurring during a turbulent burst in a minimal channel flow. The impact on flow modification is discussed. Copyright © 2002 John Wiley & Sons, Ltd.     

On the Stefan-problem in a finite and moveable system

The one-dimensional Stefan-problem in a finite and moveable system with different densities of the two phases is investigated. The resulting convection leads to important dynamical effects. The corresponding interdependence of temperature, pressure, and movement is treated numerically using a finite-difference-method and is completely discussed with the aid of three non-dimensional numbers, the Fourier-number, the Stefan-number, and a newly introduced dynamical parameter. As technical applications of the theory thermal ink jets and other thermal-based injection elements are given. The theoretical results agree well with experimental observations and the technical possibilities of thermal-based injection elements are discussed. Received on 3 May 1999     

Dynamic behavior of the interface between two solid phases in an elastic bar

Using the continuum mechanical model of solid-solid phase transitions of Abeyaratne and Knowles, this paper examines the large time dynamical behavior of a phase boundary. The problem studied concerns a semi-infinite elastic bar initially in an equilibrium state that involves two material phases separated by a phase boundary at a given location. Interaction between the phase boundary and the elastic waves generated by an impact at the end of the bar and subsequent reflections is studied in detail, and an exact solution of the dynamical problem, which is governed by a nonlinear resursive formula, is obtained. It is shown that the phase boundary reaches a new equilibrium state for large time. Numerical calculations based on the recursive formula are carried out to illustrate analytical results.Address after August 15, 1995: Department of Engineering Science and Mechanics, Virginia Polytechnic Institute and State University, Blacksburg, VA24061, USA.     

一类生物流体力学连续系统的分岔研究

连续动力系统的非线性动力学研究,由于其应用的广泛性与问题的复杂性,近年来越来越受到重视。本文对一类生物流体力学中的连续系统-动脉局部狭窄时血液流动的分岔特性进行了研究,采用有限差分方法,将由偏微分方程组描述的边境动力系统约化为由常微分方程组描述的高维离散动力系统。求得了离散动力系统的平衡解并分析其稳定性,同时讨论了流场中变量空间分布的变化情况。求得了离散动力系统的前三个Lyapunov指数,以此作为系统是否发生混沌的判别条件。     

Macroscale Properties of Porous Media from a Network Model of Biofilm Processes

We demonstrate how a network model can predict porosity and permeability changes in a porous medium as a result of biofilm buildup in the pore spaces. A biofilm consists of bacteria and extracellular polymeric substances (EPS) bonded together and attached to a surface. In this case, the surface consists of the walls of the porous medium, which we model as a random network of pipes.Our model contains five species. Four of these are bacteria and EPS in both fluid and adsorbed phases. The fifth species is nutrient, which we assume to reside in the fluid phase only. Bacteria and EPS transfer between the adsorbed and fluid phases through adsorption and erosion or sloughing. The adsorbed species influence the effective radii of the pipes in the network, which affect the porosity and permeability.We develop a technique for integrating the coupled system of ordinary and partial differential equations that govern transport of these species in the network. We examine ensemble averages of simulations using different arrays of pipe radii having identical statistics. These averages show how different rate parameters in the biofilm transport processes affect the concentration and permeability profiles.     

Interaction of a gas-droplet turbulent jet with a cocurrent high-velocity high-temperature gas flow

A mathematical model and a method for calculating a gas-droplet turbulent jet with allowance for velocity nonequilibrium and virtual mass of the condensed phase during turbulent fluctuations and also heat and mass transfer within the three-temperature scheme are developed. Methodical calculations are performed. The results of these calculations are in reasonable agreement with available experimental data. The structure of the gas-droplet jet in a cocurrent high-velocity high-temperature gas flow is studied by numerical methods. The ratio of intensities of heat and mass transfer between the phases and turbulent diffusion transfers of substances is found to be different at the initial, transitional, and basic segments of the jet. This difference is responsible for the nonmonotonic axial distribution of vapor density and the lines of the halved mass flow of the condensed phase. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 49, No. 3, pp. 85–94, May–June, 2008.     

A method for calculating the lyapunov exponent spectrum of a periodically excited non-autonomous dynamical system

The relation between the Lyapunov exponent spectrum of a periodically excited non-autonomous dynamical system and the Lyapunov exponent spectrum of the corresponding autonomous system is given and the validity of the relation is verified theoretically and computationally. A direct method for calculating the Lyapunov exponent spectrum of non-autonomous dynamical systems is suggested in this paper, which makes it more convenient to calculate the Lyapunov exponent spectrum of the dynamical system periodically excited. Following the definition of the Lyapunov dimensionD _L ^(A) of the autonomous system, the definition of the Lyapunov dimensionD _L of the non-autonomous dynamical system is also given, and the difference between them is the integer 1, namely,D _L ^(A) −D_L=1. For a quasi-periodically excited dynamical system, similar conclusions are formed. Project supported by the National Natural Science Foundation of China (No. 19772027), the Science Foundation of Shanghai Municipal Commission of Education (99A01) and the Science Foundation of Shanghai Municipal Commission of Science and Technology (No. 98JC14032).     

Dynamics analysis and hybrid function projective synchronization of a new chaotic system

This paper introduces a novel three-dimensional autonomous chaotic system by adding a quadratic cross-product term to the first equation and modifying the state variable in the third equation of a chaotic system proposed by Cai et al. (Acta Phys. Sin. 56:6230, 2007). By means of theoretical analysis and computer simulations, some basic dynamical properties, such as Lyapunov exponent spectrum, bifurcations, equilibria, and chaotic dynamical behaviors of the new chaotic system are investigated. Furthermore, hybrid function projective synchronization (HFPS) of the new chaotic system is studied by employing three different synchronization methods, i.e., adaptive control, system coupling and active control. The proposed approaches are applied to achieve HFPS between two identical new chaotic systems with fully uncertain parameters, HFPS in coupled new chaotic systems, and HFPS between the integer-order new chaotic system and the fractional-order Lü chaotic system, respectively. Corresponding numerical simulations are provided to validate and illustrate the analytical results.     

Helical shell models for MHD

A shell model for magnetohydrodynamics (MHD) is derived directly from the dynamical system driving the evolution of three helical modes interacting in a triad. The use of helical modes implies that two shell variables are required for the velocity as well as for the magnetic field. The advantage of the method is the automatic conservation of all the ideal quadratic MHD invariants. The number of coupling constants is however larger than in traditional shell models. This difficulty is worked around by introducing an averaging procedure that allows to derive the shell model coupling constants directly from the MHD equations. The resulting shell model is used to explore the influence of a helical forcing on the global properties of MHD turbulence close to the onset of the dynamo regime.     

On nonlinear oscillations of a thin bar

Summary Based on one of the simplest mathematical model of a solid, nonlinear interactions between waves in a rectilinear bar are investigated, in order to reveal and display a number of dynamic properties inherent not only to the bar, but also to most weakly nonlinear mechanical systems with internal resonances. The presence of internal resonances in the bar is twofold. Firstly, there exists a slow periodic energy exchange between the longitudinal and the two quasi-harmonic bending waves involved in the resonant triad due to the phase matching, secondly, triple-frequency envelope solitons can be created from the resonant triad with the same modal state. The paper investigates the evolution of waves in the bar with the aim to classify the elementary type of wave triplet resonant interactions and define their existence and coesistence areas.The research described here has been made possible in part by Grant N R9B000 from the International Science Foundation. The authors would like to thank Professor G.A. Maugin for having sent copies of his papers, in particular [23], as well as for his permanent interest in our work.     

The fractional dimension identification method of critical bifurcated parameters of bearing-rotor system

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Rich dynamics in a China Telecom–cable television model

The policy and schedule of the integration of the three networks, i.e., Internet, telecom, and cable television (catv), were formally released by the Chinese government. However, under this circumstance, the dynamical behaviors between the telecom and catv industry have not been studied. As a result, a discrete-time China Telecom–cable television model by qualitative analysis and numerical simulation is investigated. It is verified that there are phenomena of the transcritical bifurcation, flip bifurcation types, and chaos. The results obtained show that such a model can have rich dynamical behavior and the market behavior cannot be well predicted due to the existence of chaos. It may suggest that the telecom and the catv industry should operate their traditional business more efficiently and strengthen the development and propagation of novel businesses in their own industry.     

Numerical Simulation of the High-Velocity Impact of a Cosmic Dust Particle on a Barrier

The process of high-velocity collision of a cosmic dust particle against a semi-infinite barrier is numerically simulated. The interaction velocities considered are such that the entire projectile and a considerable portion of the solid target material pass into the gaseous phase. A finite volume method based on the Godunov breakdown scheme with improved resolution is used for calculating the axisymmetric problem. The basic flow singularities are fitted; these are the main shocks, the interfaces between different substances and phases of the same substance, and the boundaries with a vacuum. The entire computational domain is divided into a number of subdomains, correlated with the flow pattern and confined by movable boundaries, and movable difference grids are introduced. An algorithm for solving the Riemann problem on the common sides of adjacent cells is developed for a complicated equation of state. Marked corpuscles, or simply markers, are taken into consideration; tracking their evolution makes it possible to analyze in more detail the numerical results and to check their accuracy. The calculated results for the impact of a porous SiO₂ dust particle on a silver target are discussed for the case in which the particle moves at a velocity of 80 km/s normal to the target plane.     

Structure of tropical variability from a vertical mode perspective

A composite mesoscale precipitation event and a convectively coupled Kelvin wave produced by a diabatically accelerated cloud resolving model are compared. Special emphasis is placed on the vertical structure of density and moisture perturbations and the interaction of these perturbations with the composited dynamical fields. Both composites share the same general features, a gradual deepening and strengthening of convection followed by deep convection and a stratiform region, quite similar in character to observations and some recent idealized models. Composited frozen moist static energy (FMSE) perturbations are several times larger than virtual temperature perturbations, suggesting moisture is a dominant regulator of convection. An empirically derived two vertical mode decomposition of the dynamical and moisture fields is found to reproduce both composites quite well. The leading vertical modes of FMSE and virtual temperature variability are strongly correlated with the modes of vertical velocity variability; these correlations are strongest at near-zero time lags. Deep convection is associated with moistening in the lower and middle troposphere, while shallow convection is associated with a moist lower troposphere and dry middle and upper troposphere. To the extent that our numerical model is realistic, the empirical modal decomposition provides support for the use of two-mode idealized models for convective interaction with large-scale circulations and guidance for formulating feedbacks between convection and the thermodynamic profile in such models. The FMSE budget leads to an interpretation of the convective life-cycle as a recharge–discharge mechanism in column-integrated FMSE. The budget analysis places diabatic forcing, surface and radiative fluxes into the moist energetic framework. In particular, these fluxes are seen to prolong active convection, but play a passive role in its initiation. The modally decomposed FMSE budget highlights the dynamical importance of the second baroclinic mode in moistening the lower and middle troposphere before convective onset (recharging), and then discharging stored FMSE in the stratiform region.     

Overall electromechanical properties of a binary composite with 622 symmetry constituents.: Antiplane shear piezoelectric state

A binary composite is studied here, where the electroelastic properties of the constituent materials belong to the crystal class 622. A square arrangement of long continuous circular cylinders, the fiber phase, embedded in a homogeneous medium is consider here. The composite is in a state of antiplane shear piezoelectricity, that is, a coupled state of out-of-plane mechanical displacement and in-plane electric field, which is characterized by three electroelastic parameters: longitudinal shear modulus, shear stress piezoelectric coefficient and transverse dielectric constant. Our interest here lies in the determination of its effective properties. They are derived by means of the method of two spatial scales. Closed-form expressions are obtained for them. Only one of the four local (or canonical) problems that arise is needed. Two properties are thus found. The Milgrom–Shtrikman compatibility relation is used to fix the remaining one. The local problem is solved using potential methods of a complex variable. The solution involves doubly periodic Weierstrass elliptic and related functions. The final formulae for the overall properties show explicitly the dependence on (i) the properties of the phases, (ii) the radius of the cylindrical fiber and (iii) the lattice sums associated with the square array. The shear modulus is shown to depend explicitly not only on the rigidity of the phases but also on their piezoelectric and dielectric coefficients. Some natural organic substances have the symmetry 622 like collagen. Recently Silva et al. measured its electroelastic properties. Their data is used to show some numerical results of the derived formulae as a function of the fiber volumetric fraction.     

Bifurcation, response scenarios and dynamic integrity in a single-mode model of noncontact atomic force microscopy

The nonlinear dynamical behavior of a single-mode model of noncontact AFM is analyzed in terms of attractors robustness and basins integrity. The model considered for the analyses, proposed in (Hornstein and Gottlieb in Nonlinear Dyn. 54:93–122, 2008), consistently includes the nonlinear atomic interaction and is studied under scan excitation (which appears as parametric excitation) and vertical excitation (which is prevalently external). Local bifurcation analyses are carried out to identify the overall stability boundary in the excitation parameter space as the envelope of system local escapes, to be compared with the one obtained via numerical simulations. The dynamical integrity of periodic bounded solutions is studied, and basin erosion is evaluated by means of two different integrity measures. The obtained erosion profiles allow us to dwell on the possible lack of homogeneous safety of the stability boundary in terms of robustness of the attractors, and to identify practical escape thresholds ensuring an a priori design safety target.     

Infiltration of liquid droplets into porous media: Effects of dynamic contact angle and contact angle hysteresis

We present a detailed theory for infiltration, which accounts for a general model for the dynamic contact angle between the droplet and the porous medium as well as contact angle hysteresis, and analyze the resulting equations of motion. The theory shows that infiltration of droplets into dry porous media involves three phases due to contact angle hysteresis: (1) An increasing drawing area (IDA) phase during which the interface between the droplet and the porous medium increases, (2) a constant drawing area (CDA) phase during which the contact line of the droplet remains pinned, and (3) a decreasing drawing area (DDA) phase. The theory is based on the following assumptions: (1) The droplet has the shape of a spherical cap, (2) the porous medium consists of a bundle of vertical tubes of same size, and (3) the pressure within the droplet is uniform. We find that infiltration always consists of a cascade process formed by the IDA, CDA, and DDA phases, where the entire process may begin or end in any of the three phases. The entire process is formulated with four nondimensional parameters: Three contact angles (initial, advancing, and receding) and a porous permeability parameter. A comparison of our theory to experimental data suggests that one should use different parameterizations for the dynamic contact angle models of the IDA and DDA phases. In general, the IDA and DDA phases are described by integro-differential equations. A numerical-solution approach is presented for solving the dynamic equations for infiltration. The total time of infiltration and the time dependence of drawing area are critically affected by the occurrence of the IDA, CDA, and DDA phases as well as by the permeability. With ordinary differential equations (ODEs), we are able to approximate the IDA phase and to describe exactly infiltration processes that start out with the CDA or DDA phase.