Two-time scales inner solutions and motion of a geostrophic vortex

Based on the authors' previous work, in this paper the systematical analyses on the motion and the inner solutions of a geostrophic vortex have been presented by means of thematched asymptotic expansion method with multiple time scales (S/gh001/2 and α S/gh001/2) and space scales. It has been shown that the leading inner solutions to the core structure in two-time scales analyses are identified with the results in normal one-time scale analyses. The time averages of the first-order solutions on short time variable τ are the same as the first-order solutions obtained in one normal time scale analyses. The geostrophic vortex induces an oscillatory motion in addition to moving with the background flow. The period, amplitude andthe deviation from the mean trajectory depend on the core structure and the initial conditions. The velocity of the motion of vortex center varies periodically and the time average of the velocity on short time variable τ is equal to the value of the local mean velocity.

“Deborah Numbers”, Coupling Multiple Space and Time Scales and Governing Damage Evolution to Failure

Two different spatial levels are involved concerning damage accumulation to eventual failure. nucleation and growth rates of microdamage nN* and V*. It is found that the trans-scale length ratio c*/L does not directly affect the process. Instead, two independent dimensionless numbers: the trans-scale one * * ( V*)including the * **5 * N c V including mesoscopic parameters only, play the key role in the process of damage accumulation to failure. The above implies that there are three time scales involved in the process: the macroscopic imposed time scale tim = /a and two meso-scopic time scales, nucleation and growth of damage, (* *4) N N t =1 n c and tV=c*/V*. Clearly, the dimensionless number De*=tV/tim refers to the ratio of microdamage growth time scale over the macroscopically imposed time scale. So, analogous to the definition of Deborah number as the ratio of relaxation time over external one in rheology. Let De be the imposed Deborah number while De represents the competition and coupling between the microdamage growth and the macroscopically imposed wave loading. In stress-wave induced tensile failure (spallation) De* < 1, this means that microdamage has enough time to grow during the macroscopic wave loading. Thus, the microdamage growth appears to be the predominate mechanism governing the failure. Moreover, the dimensionless number D* = tV/tN characterizes the ratio of two intrinsic mesoscopic time scales: growth over nucleation. Similarly let D be the “intrinsic Deborah number”. Both time scales are relevant to intrinsic relaxation rather than imposed one. Furthermore, the intrinsic Deborah number D* implies a certain characteristic damage. In particular, it is derived that D* is a proper indicator of macroscopic critical damage to damage localization, like D* ∼ (10–3~10–2) in spallation. More importantly, we found that this small intrinsic Deborah number D* indicates the energy partition of microdamage dissipation over bulk plastic work. This explains why spallation can not be formulated by macroscopic energy criterion and must be treated by multi-scale analysis.

Time Correlations Of Pressure In Isotropic Turbulence

The time correlations of pressure modes in stationary isotropic turbulence are investigated under the Kraichnan and Tennekes "random sweeping" hypothesis. A simple model is obtained which predicts a universal form for the time correlations. It implies that the decorrelation process of pressure fluctuations in time is mainly dominated by the sweeping velocity, and the pressure correlations have the same decorrelation time scales as the velocity correlations. These results are verified using direct numerical simulations of isotropic turbulence at two moderate Reynolds numbers; the mode correlations collapse to the universal form when the time separations are scaled by wavenumber times the sweeping velocity, and the ratios of the correlation coefficients of pressure modes to those of velocity modes are approximately unity for the entire range of time separation. (c) 2008 American Institute of Physics.

On the computation of space-time correlations by large-eddy simulation

The effect of subgrid-scale (SGS) modeling on velocity (space-) time correlations is investigated in decaying isotropic turbulence. The performance of several SGS models is evaluated, which shows superiority of the dynamic Smagorinsky model used in conjunction with the multiscale large-eddy simulation (LES) procedure. Compared to the results of direct numerical simulation, LES is shown to underpredict the (un-normalized) correlation magnitude and slightly overpredict the decorrelation time scales. This can lead to inaccurate solutions in applications such as aeroacoustics. The underprediction of correlation functions is particularly severe for higher wavenumber modes which are swept by the most energetic modes. The classic sweeping hypothesis for stationary turbulence is generalized for decaying turbulence and used to analyze the observed discrepancies. Based on this analysis, the time correlations are determined by the wavenumber energy spectra and the sweeping velocity, which is the square root of the total energy. Hence, an accurate prediction of the instantaneous energy spectra is most critical to the accurate computation of time correlations. (C) 2004 American Institute of Physics.

The motion and the core structure of a geostrophic vortex - one-time scale inner solution and the motion of the vortex

By means of the matched asymptotic expansion method with one-time scale analysis we have shown that the inviscid geostrophic vortex solution represents our leading solution away from the vortex. Near the vortex there is a viscous core structure, with the length scale O(a). In the core the viscous stresses (or turbulent stresses) are important, the variations of the velocity and the equivalent height are finite and dependent of time. It also has been shown that the leading inner solutions of the core structure are the same for two different time scales of S/(ghoo)1/2 and S/a (ghoo)1/2. Within the accuracy of O(a) the velocity of a geostrophic vortex center is equal to the velocity of the local background flow, where the vortex is located, in the absence of the vortex. Some numerical examples demonstrate the contributions of these results.

Space-time correlations of fluctuating velocities in turbulent shear flows

Space-time correlations or Eulerian two-point two-time correlations of fluctuating velocities are analytically and numerically investigated in turbulent shear flows. An elliptic model for the space-time correlations in the inertial range is developed from the similarity assumptions on the isocorrelation contours: they share a uniform preference direction and a constant aspect ratio. The similarity assumptions are justified using the Kolmogorov similarity hypotheses and verified using the direct numerical simulation DNS of turbulent channel flows. The model relates the space-time correlations to the space correlations via the convection and sweeping characteristic velocities. The analytical expressions for the convection and sweeping velocities are derived from the Navier-Stokes equations for homogeneous turbulent shear flows, where the convection velocity is represented by the mean velocity and the sweeping velocity is the sum of the random sweeping velocity and the shearinduced velocity. This suggests that unlike Taylor’s model where the convection velocity is dominating and Kraichnan and Tennekes’ model where the random sweeping velocity is dominating, the decorrelation time scales of the space-time correlations in turbulent shear flows are determined by the convection velocity, the random sweeping velocity, and the shear-induced velocity. This model predicts a universal form of the spacetime correlations with the two characteristic velocities. The DNS of turbulent channel flows supports the prediction: the correlation functions exhibit a fair good collapse, when plotted against the normalized space and time separations defined by the elliptic model.

Characteristic Dimensionless Numbers in Multi-Scale and Rate-Dependent Processes

For this sake, the macroscopic equations of mechanics and the kinetic equations of the microstructural transformations should form a unified set that be solved simultaneously. As a case study of coupling length and time scales, the trans-scale formulation

Recovery of the solitons using a lattice Boltzmann model

We formulate a lattice Boltzmann model which simulates Korteweg-de Vries equation by using a method of higher moments of lattice Boltzmann equation. Using a series of lattice Boltzmann equations in different time scales and the conservation law in time scale to, we obtain equilibrium distribution function. The numerical examples show that the method can be used to simulate soliton.

Trans-Scale Mechanics: Looking For The Missing Links Between Continuum And Micro/Nanoscopic Reality

Problems involving coupled multiple space and time scales offer a real challenge for conventional frameworks of either particle or continuum mechanics. In this paper, four cases studies (shear band formation in bulk metallic glasses, spallation resulting from stress wave, interaction between a probe tip and sample, the simulation of nanoindentation with molecular statistical thermodynamics) are provided to illustrate the three levels of trans-scale problems (problems due to various physical mechanisms at macro-level, problems due to micro-structural evolution at macro/micro-level, problems due to the coupling of atoms/molecules and a finite size body at micro/nano-level) and their formulations. Accordingly, non-equilibrium statistical mechanics, coupled trans-scale equations and simultaneous solutions, and trans-scale algorithms based on atomic/molecular interaction are suggested as the three possible modes of trans-scale mechanics.

Molecular/Cluster Statistical Thermodynamics Methods To Simulate Quasi-Static Deformations At Finite Temperature

The rapid evolution of nanotechnology appeals for the understanding of global response of nanoscale systems based on atomic interactions, hence necessitates novel, sophisticated, and physically based approaches to bridge the gaps between various length and time scales. In this paper, we propose a group of statistical thermodynamics methods for the simulations of nanoscale systems under quasi-static loading at finite temperature, that is, molecular statistical thermodynamics (MST) method, cluster statistical thermodynamics (CST) method, and the hybrid molecular/cluster statistical thermodynamics (HMCST) method. These methods, by treating atoms as oscillators and particles simultaneously, as well as clusters, comprise different spatial and temporal scales in a unified framework. One appealing feature of these methods is their "seamlessness" or consistency in the same underlying atomistic model in all regions consisting of atoms and clusters, and hence can avoid the ghost force in the simulation. On the other hand, compared with conventional MD simulations, their high computational efficiency appears very attractive, as manifested by the simulations of uniaxial compression and nanoindenation. (C) 2008 Elsevier Ltd. All rights reserved.

Instability of the vertically forced surface wave in a circular cylindrical container

The nonlinear amplitude equation, which was derived by Jian Yongjun employing expansion of two-time scales in inviscid fluids in a vertically oscillating circular cylindrical vessel, is modified by introducing a damping term due to the viscous dissipation of this system. Instability of the surface wave is analysed and properties of the solutions of the modified equation are determined together with phase-plane trajectories. A necessary condition of forming a stable surface wave is obtained and unstable regions are illustrated. Research results show that the stable pattern of surface wave will not lose its stability to an infinitesimal disturbance.

A Multiscale-Domain Decomposition Method for High Reynolds Number Flows

The high Reynolds number flow contains a wide range of length and time scales, and the flow domain can be divided into several sub-domains with different characteristic scales. In some sub-domains, the viscosity dissipation scale can only be considered in a certain direction; in some sub-domains, the viscosity dissipation scales need to be considered in all directions; in some sub-domains, the viscosity dissipation scales are unnecessary to be considered at all. For laminar boundary layer region, the characteristic length scales in the streamwise and normal directions are L and L Re-1/ 2 , respectively. The characteristic length scale and the velocity scale in the outer region of the boundary layer are L and U, respectively. In the neighborhood region of the separated point, the length scale l<scales. Then the different dominant equation of all sub-domains is defined according to the diffusion parabolized (DP) theory of viscous flow. Finally these different equations are solved simultaneously in whole computational region. For numerical tests of high Reynolds numerical flows, two-dimensional supersonic flows over rearward and frontward steps as well as an interaction flow between shock wave and boundary layer were solved numerically. The pressure distributions and local coefficients of skin friction on the wall are given. The numerical results obtained by the multiscale-domain decomposition algorithm are well agreement with those by NS equations. Comparing with the usual method of solving the Navier-Stokes equations in the whole flow, under the same numerical accuracy, the present multiscale domain decomposition method decreases CPU consuming about 20% and reflects the physical mechanism of practical flow more accurately.

On the origin of shear banding instability in metallic glasses

To uncover the physical origin of shear-banding instability in metallic glass (MG), a theoretical description of thermo-mechanical deformation of MG undergoing one-dimensional simple shearing is presented. The coupled thermo-mechanical model takes into account the momentum balance, the energy balance and the dynamics of free volume. The interplay between free-volume production and temperature increase being two potential causes for shear-banding instability is examined on the basis of the homogeneous solution. It is found that the free-volume production facilitates the sudden increase in the temperature before instability and vice versa. A rigorous linear perturbation analysis is used to examine the inhomogeneous deformation, during which the onset criteria and the internal length and time scales for three types of instabilities, namely free-volume softening, thermal softening and coupling softening, are clearly revealed. The shear-banding instability originating from sole free-volume softening takes place easier and faster than that due to sole thermal softening, and dominates in the coupling softening. Furthermore, the coupled thermo-mechanical shear-band analysis does show that an initial slight distribution of local free volume can incur significant strain localization, producing a shear band. During such a localization process, the local free-volume creation occurs indeed prior to the increase in local temperature, indicating that the former is the cause of shear localization, whereas the latter is its consequence. Finally, extension of the above model to include the shear-induced dilatation shows that such dilatation facilitates the shear instability in metallic glasses.

Effects of solar UV radiation on morphology and photosynthesis of filamentous cyanobacterium Arthrospira platensis

To study the impact of solar UV radiation (UVR) (280 to 400 nm) on the filamentous cyanobacterium Arthrospira (Spirulina) platensis, we examined the morphological changes and photosynthetic performance using an indoor-grown strain (which had not been exposed to sunlight for decades) and an outdoor-grown strain (which had been grown under sunlight for decades) while they were cultured with three solar radiation treatments: PAB (photosynthetically active radiation [PAR] plus UVR; 280 to 700 nm), PA (PAR plus UV-A; 320 to 700 nm), and P (PAR only; 400 to 700 nm). Solar UVR broke the spiral filaments of A. platensis exposed to full solar radiation in short-term low-cell-density cultures. This breakage was observed after 2 h for the indoor strain but after 4 to 6 h for the outdoor strain. Filament breakage also occurred in the cultures exposed to PAR alone; however, the extent of breakage was less than that observed for filaments exposed to full solar radiation. The spiral filaments broke and compressed when high-cell-density cultures were exposed to full solar radiation during long-term experiments. When UV-B was screened off, the filaments initially broke, but they elongated and became loosely arranged later (i.e., there were fewer spirals per unit of filament length). When UVR was filtered out, the spiral structure hardly broke or became looser. Photosynthetic 0, evolution in the presence of UVR was significantly suppressed in the indoor strain compared to the outdoor strain. UVR-induced inhibition increased with exposure time, and it was significantly lower in the outdoor strain. The concentration of UV-absorbing compounds was low in both strains, and there was no significant change in the amount regardless of the radiation treatment, suggesting that these compounds were not effectively used as protection against solar UVR. Self-shading, on the other hand, produced by compression of the spirals over adaptive time scales, seems to play an important role in protecting this species against deleterious UVR. Our findings suggest that the increase in UV-B irradiance due to ozone depletion not only might affect photosynthesis but also might alter the morphological development of filamentous cyanobacteria during acclimation or over adaptive time scales.

Photoinduced spin alignment of the magnetic ions in (Ga,Mn)As

Time-resolved Kerr rotation measurement in the (Ga,Mn)As diluted magnetic semiconductor allows direct observation of the dynamical properties of the spin system of the magnetic ions and the spin-polarized holes. Experimental results show that the magnetic ions can be aligned by the polarized holes, and the time scales of spin alignment and relaxation take place in tens and hundreds of picoseconds, respectively. The Larmor frequency and effective g factor obtained in the Voigt geometry show an unusual temperature dependence in the vicinity of the Curie temperature due to the exchange coupling between the photoexcited holes and magnetic ions. Such a spin coherent precession can be amplified or destructed by two sequential excitation pulses with circularly copolarized or oppositely polarized helicity, respectively. (c) 2006 American Institute of Physics.