A Global Foliation of Einstein–Euler Spacetimes with Gowdy-Symmetry on <Emphasis Type="Italic">T</Emphasis><Superscript>3</Superscript>

We investigate the initial value problem for the Einstein–Euler equations of general relativity under the assumption of Gowdy symmetry on T ³, and we construct matter spacetimes with low regularity. These spacetimes admit both impulsive gravitational waves in the metric (for instance, Dirac mass curvature singularities propagating at light speed) and shock waves in the fluid (that is, discontinuities propagating at about the sound speed). Given an initial data set, we establish the existence of a future development, and we provide a global foliation in terms of a globally and geometrically defined time-function, closely related to the area of the orbits of the symmetry group. The main difficulty lies in the low regularity assumed on the initial data set which requires a distributional formulation of the Einstein–Euler equations.     

On the development of a kinetic model for a gas with allowance for nonequilibrium processes

The problems of developing a kinetic model of a medium (gas and plasma) are considered from the viewpoint of choice of the most important physicochemical processes. For the problem of a direct shock wave propagating in the atmosphere, kinetic models are selected with allowance for the error in specifying reaction-rate constants. The investigation was performed using an automated system that incorporates structured bases of physicochemical data, a generator of kinetic equations, a complex of programs for direct calculation, and program modules for determining, from a set of admissible solutions, the one satisfying specified criteria. Institute of Mechanics, Moscow State University, Moscow 119899. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 40, No. 4, pp. 36–43, July–August, 1999.     

Shear Instability of the Structure of Media Possessing Viscous Fluidity

Instability of the structure of viscous fluids in the Couette flow regime (spontaneous formation of bands, which are tangential discontinuities in terms of viscosity) is experimentally found. This process is demonstrated to be analogous to formation of shear bands in polymethylmethacrylate (Plexiglas) under plastic shear strains. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 47, No. 2, pp. 70–76, March–April, 2006.     

Turbulent Flow and Dispersion of Inertial Particles in a Confined Jet Issued by a Long Cylindrical Pipe

In this work we examine first the flow field of a confined jet produced by a turbulent flow in a long cylindrical pipe issuing in an abrupt angle diffuser. Second, we examine the dispersion of inertial micro-particles entrained by the turbulent flow. Specifically, we examine how the particle dispersion field evolves in the multiscale flow generated by the interactions between the large-scale structures, which are geometry dependent, with the smaller turbulent scales issued by the pipe which are advected downstream. We use Large-Eddy-Simulation (LES) for the flow field and Lagrangian tracking for particle dispersion. The complex shape of the domain is modelled using the immersed-boundaries method. Fully developed turbulence inlet conditions are derived from an independent LES of a spatially periodic cylindrical pipe flow. The flow field is analyzed in terms of local velocity signals to determine spatial coherence and decay rate of the coherent K–H vortices and to make quantitative comparisons with experimental data on free jets. Particle dispersion is analyzed in terms of statistical quantities and also with reference to the dynamics of the coherent structures. Results show that the particle dynamics is initially dominated by the Kelvin–Helmholtz (K–H) rolls which form at the expansion and only eventually by the advected smaller turbulence scales.     

Modeling of hard disk drives for shock and vibration analysis – consideration of nonlinearities and discontinuities

This paper reviews recent advances (mostly after year 2000) in shock and vibration analysis of hard disk drives (HDD) considering the presence of nonlinearities and discontinuities. Components and dynamic phenomena in HDD where effects of mechanical nonlinearity and discontinuities are significant are discussed, e.g., head actuator suspension, dimple and slider, head–disk interface, fluid dynamic bearing, spinning disks, and load/unload dynamics. Ways to model these nonlinearities and discontinuities are reviewed in detail. Our research on modeling an entire HDD in operating mode subject to shock and vibration using a flexible multibody dynamics formulation is also presented. The numerical simulation of the shock response of a 1-in. form factor HDD is presented. A half-sinusoidal acceleration shock is applied at the base of the HDD. Response of the flying height for different shock amplitudes and duration times is simulated.     

Gradients at a curved shock in reacting flow

The inviscid equations of motion for the flow at the downstream side of a curved shock are solved for the shock–normal derivatives. Combining them with the shock–parallel derivatives yields gradients and substantial derivatives. In general these consist of two terms, one proportional to the rate of removal of specific enthalpy by the reaction, and one proportional to the shock curvature. Results about the streamline curvature show that, for sufficiently fast exothermic reaction, no Crocco point exists. This leads to a stability argument for sinusoidally perturbed normal shocks that relates to the formation of the structure of a detonation wave. Application to the deflection–pressure map of a streamline emerging from a triple shock point leads to the conclusion that, for non–reacting flow, the curvature of the Mach stem and reflected shock must be zero at the triple point, if the incident shock is straight. The direction and magnitude of the gradient at the shock of any flow quantity may be written down using the results. The sonic line slope in reacting flow serves as an example. Extension of the results – derived in the first place for plane flow – to three dimensions is straightforward. Received 12 February 1997 / Accepted 10 June 1997     

NND schemes and numerical simulation of axial symmetric free jet flows

Through a study on one-dimensional Navier-Stokes equations, it was found that the spurious oscillations occuring near shock waves with finite difference equations are related to the dispersion term in the corresponding modified differential equations. If the sign of dispersion coefficient is properly adjusted so that the sign changes across shock waves, the undesirable oscillations can be totally suppressed. Based on this finding, the non-oscillatory, containing no free parameters and dissipative shheme (NND scheme) is developed. This scheme is one of “TVD”. The axisymmetric free jet flows are simulated numerically using this scheme. The results obtained by the present scheme are compared with the experimental picture. It is shown that the agreement is very good, and that this scheme has advantages of high resolution for capturing shocks and contact discontinuities. Project supported by National Science Foundation of China     

The quasi-stationary structure of radiating shock waves. I. The one-temperature fluid

We calculate the quasi-stationary structure of a radiating shock wave propagating through a spherically symmetric shell of cold gas by solving the time-dependent equations of radiation hydrodynamics on an implicit adaptive grid. We show that this code successfully resolves the shock wave in both the subcritical and supercritical cases and, for the first time, we have reproduced all the expected features – including the optically thin temperature spike at a supercritical shock front – without invoking analytic jump conditions at the discontinuity. We solve the full moment equations for the radiation flux and energy density, but the shock wave structure can also be reproduced if the radiation flux is assumed to be proportional to the gradient of the energy density (the diffusion approximation), as long as the radiation energy density is determined by the appropriate radiative transfer moment equation. We find that Zel'dovich and Raizer's (1967) analytic solution for the shock wave structure accurately describes a subcritical shock but it underestimates the gas temperature, pressure, and the radiation flux in the gas ahead of a supercritical shock. We argue that this discrepancy is a consequence of neglecting terms which are second order in the minimum inverse shock compression ratio [, where is the adiabatic index] and the inaccurate treatment of radiative transfer near the discontinuity. In addition, we verify that the maximum temperature of the gas immediately behind the shock is given by , where is the gas temperature far behind the shock. Received 21 September 1998/ Accepted 2 February 1999     

Effects of Chemical Reaction and Double Dispersion on Non-Darcy Free Convection Heat and Mass Transfer

In this article, the effects of chemical reaction and double dispersion on non-Darcy free convection heat and mass transfer from semi-infinite, impermeable vertical wall in a fluid saturated porous medium are investigated. The Forchheimer extension (non-Darcy term) is considered in the flow equations, while the chemical reaction power–law term is considered in the concentration equation. The first order chemical reaction (n = 1) was used as an example of calculations. The Darcy and non-Darcy flow, temperature and concentration fields in this study are observed to be governed by complex interactions among dispersion and natural convection mechanisms. The governing set of partial differential equations were non-dimensionalized and reduced to a set of ordinary differential equations for which Runge–Kutta-based numerical technique were implemented. Numerical results for the detail of the velocity, temperature, and concentration profiles as well as heat transfer rates (Nusselt number) and mass transfer rates (Sherwood number) are presented in graphs.     

Transitive flows on orientable surfaces

We consider smooth vector fields on closed orientable surfaces with a fixed collection of singularities and a finite number of separatrices none of which connects the equilibrium states. We prove that, on an orientable surface of arbitrary genus g ≥ 2, there exists a vector field with an admissible set of singularities (degenerate saddles) whose trajectory is everywhere dense on the surface. __________ Translated from Neliniini Kolyvannya, Vol. 9, No. 2, pp. 178–186, April–June, 2006.     

Modelling gas dynamics in 1D ducts with abrupt area change

Most gas dynamic computations in industrial ducts are done in one dimension with cross-section-averaged Euler equations. This poses a fundamental difficulty as soon as geometrical discontinuities are present. The momentum equation contains a non-conservative term involving a surface pressure integral, responsible for momentum loss. Definition of this integral is very difficult from a mathematical standpoint as the flow may contain other discontinuities (shocks, contact discontinuities). From a physical standpoint, geometrical discontinuities induce multidimensional vortices that modify the surface pressure integral. In the present paper, an improved 1D flow model is proposed. An extra energy (or entropy) equation is added to the Euler equations expressing the energy and turbulent pressure stored in the vortices generated by the abrupt area variation. The turbulent energy created by the flow–area change interaction is determined by a specific estimate of the surface pressure integral. Model’s predictions are compared with 2D-averaged results from numerical solution of the Euler equations. Comparison with shock tube experiments is also presented. The new 1D-averaged model improves the conventional cross-section-averaged Euler equations and is able to reproduce the main flow features.     

Tangential discontinuities of parameters of a polar fluid under shear strain

Possible formation of tangential discontinuities of parameters of a deformable polar fluid is examined by the example of glycerin. It is experimentally established that glycerin under weak shear loads possesses the properties of a non-Newtonian elastoviscoplastic fluid, and formation of tangential discontinuities in viscosity is possible. In the discontinuity region, glycerin has the properties of a low-viscosity fluid, and the structure of the medium is reconstructed after unloading. A rheological equation of the examined fluid is derived, which allows one to analyze the behavior of the medium in different modes of its deformation, including the formation of a local region with reduced viscosity and a tensile stress field. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 46, No. 3, pp. 41–49, May–June, 2005.     

Flow in a Porous Matrix with Anisotropic Structure

The flow of a viscous fluid through a porous matrix undergoing only infinitesimal deformation is described in terms of intrinsic variables, namely, the density, velocity and stress occurring in coherent elements of each material. This formulation arises naturally when macroscopic interfaces are conceptually partitioned into area fractions of fluid–fluid, fluid–solid, and solid–solid contact. Such theory has been shown to yield consistent jump conditions of mass, momentum and energy across discontinuities, either internal or an external boundary, unlike the standard mixture theory jump conditions. In the previous formulation, the matrix structure has been considered isotropic; that is, the area fractions are independent of the interface orientation. Here, that is not assumed, so in particular, the cross-section area of a continuous fluid tube depends on its orientation, which influences the directional fluxes, and in turn the directional permeability, anisotropy of the structure. The simplifications for slow viscous flow are examined, and particularly for an isotropic linearly elastic matrix in which area partitioning induces anisotropic elastic response of the mixture. A final specialization to an incompressible fluid and stationary matrix leads to potential flow, and a simple plane flow solution is presented to illustrate the effects of anisotropic permeability.     

Level set method for numerical simulation of a cavitation bubble,its growth,collapse and rebound near a rigid wall

A level set method of non-uniform grids is used to simulate the whole evolution of a cavitation bubble, including its growth, collapse and rebound near a rigid wall. Single-phase Navier–Stokes equation in the liquid region is solved by MAC projection algorithm combined with second-order ENO scheme for the advection terms. The moving interface is captured by the level set function, and the interface velocity is resolved by “one-side” velocity extension from the liquid region to the bubble region, complementing the second-order weighted least squares method across the interface and projection inside bubble. The use of non-uniform grid overcomes the difficulty caused by the large computational domain and very small bubble size. The computation is very stable without suffering from large flow-field gradients, and the results are in good agreements with other studies. The bubble interface kinematics, dynamics and its effect on the wall are highlighted, which shows that the code can effectively capture the “shock wave”-like pressure and velocity at jet impact, toroidal bubble, and complicated pressure structure with peak, plateau and valley in the later stage of bubble oscillating. The project supported by the National Natural Science Foundation of China (10272032 and 10672043). The English text was polished by Keren Wang.     

A third-order compact nonlinear scheme for compressible flow simulations

Reynolds-averaged Navier-Stokes simulations based on second-order numerical methods are widely used by commercial codes and work as dominating tools for most industrial applications. They, however, suffer from limitations in accurate and reliable predictions of skin-friction drag and aerodynamic heating, as well as in simulations of complex flows such as large-scale separation and transition. A remedy for this is the development of high-order schemes, by which numerically induced dissipation and dispersion errors of low-order schemes can be effectively reduced. Weighted compact nonlinear schemes (WCNSs) are a family of high-resolution nonlinear shock-capturing methods. A stencil-selection procedure is introduced in the proposed work with an aim to improve the nonlinear weight of the third-order WCNS. By using the approximate dispersion relation analysis, it is demonstrated that the new scheme has reduced dissipation and dispersion errors, compared with WCNSs using two typical nonlinear weights. Improvements are also achieved by the new scheme in numerical tests such as the double Mach reflection problem and the Rayleigh-Taylor instability simulation, which are characterized by strong shock discontinuities and rich small scales, respectively. The new scheme is therefore highly favored in the simulation of flow problems involving strong discontinuities and multiscales phenomena.     

Nonlinear effect of external low-frequency acoustics on eigen-oscillations in a supersonic boundary layer

A method of simulation and results of numerical calculation of the evolution of hydrodynamic disturbances in a supersonic boundary layer on a flat plate under the influence of external acoustic waves at Reynolds numbersRe=220–640 and Mach numberM=2 are described. The solution is constructed by the method of expansion with respect to the small parameter; the contribution of linear and quadratic terms to the solution is taken into account. The method developed allows one to estimate the admissible level of the acoustic field, which does not affect the development of eigen-oscillations in the boundary layer. Institute of Theoretical and Applied Mechanics, Siberian Division, Russian Academy of Sciences, Novosibirsk 630090. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 40, No. 5, pp. 99–105, September–October, 1999.     

Coarsening Fronts

We characterize the spatial spreading of the coarsening process in the Allen–Cahn equation in terms of the propagation of a nonlinear modulated front. Unstable periodic patterns of the Allen–Cahn equation are invaded by a front, propagating in an oscillatory fashion, and leaving behind the homogeneous, stable equilibrium. During one cycle of the oscillatory propagation, two layers of the periodic pattern are annihilated. Galerkin approximations and the Conley index for ill-posed spatial dynamics are used to show existence of modulated fronts for all parameter values. In the limit of small amplitude patterns or large wave speeds, we establish uniqueness and asymptotic stability of the modulated fronts. We show that the minimal speed of propagation can be characterized by a dichotomy which depends on the existence of pulled fronts. The main tools here are an Evans function type construction for the infinite-dimensional ill-posed dynamics and an analysis of the complex dispersion relation based on Sturm–Liouville theory.     

Study of the shock-induced acceleration of hexane droplets

An experimental study of the interaction of a shock wave with a hexane droplet is presented. The main goal of the experiments was to record images of the process and measure basic parameters describing movement, dispersion and evaporation of the droplets engulfed by a shock wave propagating in air. A shock tube with a visualization section was used for this research. Photography of the process allowed one to measure the positions, velocities and sizes of mist clouds created by the interaction processes. Analysis of the pictures shows that there is no qualitative difference between cases for different size droplets, but shock Mach number had a significant effect on the process. Quantitative analysis shows that under certain conditions, a catastrophic breakup mechanism of dispersion occurred. The droplets are shattered into a mist cloud before they achieve mechanical equilibrium with the surrounding gas. The approximate time for the complete dispersion and acceleration of the fuel droplet varies from 300 to 500 μs, and depends both on the droplet diameter and shock velocity. The dispersion time is controlled principally by the droplet diameter, and to a lesser extent, the shock Mach number. This paper is based on work that was presented at the 20th International Colloquium on the Dynamics of Explosions and Reactive Systems, Montreal, Canada, July 31–August 5, 2005.     

Transition to detonation in dynamic phase changes

Subsonically propagating phase boundaries (kinks) can be modelled by material discontinuities which satisfy integral conservation laws plus an additional jump condition governing the phase-change kinetics. The necessity of an additional jump condition distinguishes kinks from the conventional shocks which satisfy the Lax criterion. We study stability of kinks with respect to the breakup (splitting) into a sequence of waves. We assume that all conventional shocks are admissible and that admissible kinks are selected by a prescribed kinetic relation. As we show, regardless of a particular choice of the kinetic relation, sufficiently fast-phase boundaries are unstable. The mode of instability includes an emission of a centered Riemann wave followed by a sonic shock (Chapman-Jouguet type phase boundary).     

Wave Packets, Signaling and Resonances in a Homogeneous Waveguide

We solve the initial-boundary-value linear stability problem for small localised disturbances in a homogeneous elastic waveguide formally by applying a combined Laplace – Fourier transform. An asymptotic evaluation of the solution, expressed as an inverse Laplace – Fourier integral, is carried out by means of the mathematical formalism of absolute and convective instabilities. Wave packets, triggered by perturbations localised in space and finite in time, as well as responses to sources localised in space, with the time dependence satisfying e^−iωt + O(e^−ɛt ), for t → ∞, where Im ω₀ = 0 and ω > 0 , that is, the signaling problem, are treated. For this purpose, we analyse the dispersion relation of the problem analytically, and by solving numerically the eigenvalue stability problem. It is shown that due to double roots in a wavenumber k of the dispersion relation function D(k, ω), for real frequencies ω, that satisfy a collision criterion, wave packets with an algebraic temporal decay and signaling with an algebraic temporal growth, that is, temporal resonances, are present in a neutrally stable homogeneous waveguide. Moreover, for any admissible combination of the physical parameters, a homogeneous waveguide possesses a countable set of temporally resonant frequencies. Consequences of these results for modelling in seismology are discussed. This revised version was published online in August 2006 with corrections to the Cover Date.