Initial and boundary value problems of hyperbolic heat conduction

This is a study on the initial and boundary value problem of a symmetric hyperbolic system which is related to the conduction of heat in solids at low temperatures. The nonlinear system consists of a conservation equation for the energy density e and a balance equation for the heat flux , where e and are the four basic fields of the theory. The initial and boundary value problem that uses exclusively prescribed boundary data for the energy density e is solved by a new kinetic approach that was introduced and evaluated by Dreyer and Kunik in [1], [2] and Pertame [3]. This method includes the formation of shock fronts and the broadening of heat pulses. These effects cannot be observed in the linearized theory, as it is described in [4]. The kinetic representations of the initial and boundary value problem reveal a peculiar phenomenon. To the solution there contribute integrals containing the initial fields as well as integrals that need knowledge on energy and heat flux at a boundary. However, only one of these quantities can be controlled in an experiment. When this ambiguity is removed by continuity conditions, it turns out that after some very short time the energy density and heat flux are related to the initial data according to the Rankine Hugoniot relation. Received October 6, 1998     

Towards a new type of energy trap: Classical analog of quantum Landau-Zener tunneling

We present a novel type of an energy trap providing targeted energy transfer in a system of weakly coupled pendulums. Our approach is based on the analogy we revealed between the behavior of two weakly coupled classical parametric pendulums (in linear approximation) and the nonadiabatic Landau-Zener tunneling in a two-state quantum system. This analogy leads us to the prediction of an efficient irreversible transfer of vibration energy from one subsystem to another, when the eigenfrequency of at least one of them changes in time, so that the coupled subsystems pass through an internal resonance. The existence of such a phenomenon is not restricted to coupled pendulums, but is inherent to a wide class of both linear and nonlinear parametric oscillatory systems. This opens up the possibility of designing new types of energy traps and absorbers for the dynamic protection of various mechanical systems.     

Suspended load and bed-load transport of particle-laden gravity currents: the role of particle–bed interaction

The development of particle-enriched regions (bed-load) at the base of particle-laden gravity currents has been widely observed, yet the controls and relative partitioning of material into the bed-load is poorly understood. We examine particle-laden gravity currents whose initial mixture (particle and fluid) density is greater than the ambient fluid, but whose interstitial fluid density is less than the ambient fluid (such as occurs in pyroclastic flows produced during volcanic eruptions or when sediment-enriched river discharge enters the ocean, generating hyperpycnal turbidity currents). A multifluid numerical approach is employed to assess suspended load and bed-load transport in particle-laden gravity currents under varying boundary conditions. Particle-laden flows that traverse denser fluid (such as pyroclastic flows crossing water) have leaky boundaries that provide the conceptual framework to study suspended load in isolation from bed-load transport. We develop leaky and saltation boundary conditions to study the influence of flow substrate on the development of bed-load. Flows with saltating boundaries develop particle–enriched basal layers (bed-load) where momentum transfer is primarily a result of particle–particle collisions. The grain size distribution is more homogeneous in the bed-load and the saltation boundaries increase the run-out distance and residence time of particles in the flow by as much as 25% over leaky boundary conditions. Transport over a leaky substrate removes particles that reach the bottom boundary and only the suspended load remains. Particle transport to the boundary is proportional to the settling velocity of particles, and flow dilution results in shear and buoyancy instabilities at the upper interface of these flows. These instabilities entrain ambient fluid, and the continued dilution ultimately results in these currents becoming less dense than the ambient fluid. A unifying concept is energy dissipation due to particle–boundary interaction: leaky boundaries dissipate energy more efficiently at the boundary than their saltating counterparts and have smaller run-out distance.

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Unique Global Solvability for¶Initial-Boundary Value Problems¶in One-Dimensional Nonlinear Thermoviscoelasticity

The balance laws of mass, momentum and energy are considered for a broad class of one-dimensional nonlinear thermoviscoelastic materials. For the initial-boundary value problem corresponding to pinned endpoints held at constant temperature, we establish existence and uniqueness of temporally global classical solutions for initial data of unrestricted size. Our approach also applies to all boundary conditions involving pinned or stress-free endpoints which are either held at constant temperature or insulated. An additional and novel feature of the theory is that solid-like and gaseous materials are treated in a unified way. Accepted: June 24, 1999     

Detonability of fuel-oxygen and fuel-air mixtures

An analysis of theoretical models and experimental investigations of the detonability of unconfined detonation in uniform gaseous mixtures shows a disparity in results. The present study is limited to propane, acetylene and methane diluted with oxygen or air in variable proportions and initial pressures at ambient temperature conditions. Because of the disparity in results, a simple and general formulation of critical initiation energy for gaseous detonations has been investigated. The problem has been formulated using the conservation equation of total energy enclosed by the shock. From this, a simple form for the critical energy has been deduced. This approach leads to a good simulation in uniform mixtures, regardless of initiation conditions. Some applications are presented in this paper. A new experimental study on the detonability of methane/oxygen mixtures diluted with propane and/or nitrogen is reported. The gaseous mixtures are confined in a cylindrical vessel. The initial conditions are various equivalence ratio and pressure under room temperature. In the case of methane/oxygen mixtures, the predetonation radius varies directly with the cell width. The constant ratio is in the order of 18, slightly different from the classical relation R _c= 20λ. For propane the slope variation of the critical energy versus initial pressure depends on the dilution. We have compared the critical energy obtained by several authors with the theoretical values. Fuel ratio and initial pressure are the chosen parameters. These comparisons show that the formulation allows for the prediction of the critical energy of detonation of uniform mixtures with a good estimation range. The correlation between the different geometries has been deduced and a test has been conducted as well in the case of stoichiometric methane/oxygen and acetylene/oxygen mixtures versus initial pressure for a cylindrical detonation. Received 9 January 1996 / Accepted 24 January 1997     

The self-preservation of homogeneous shear flow turbulence

An analysis of the equations governing homogeneous shear flow shows the possibility of solutions which are self-preserving at all scales of motion, and that these solutions are dependent on the initial conditions. The appropriate velocity scale is the one obtained from the turbulence kinetic energy, q ²/2, while the length scale is the Taylor microscale, . Two cases of self-preserving flow are identified: one corresponding to constant mean shear, the other to a mean shear which is inversely proportional to time. For the first case (the only one considered in detail) the principal results of the postulated similarity are that is constant, while q ² varies exponentially with time. The ratio of the turbulence energy production rate to its dissipation rate remains constant. It is also shown that the energy spectra scale over all wavenumbers with q ² and , and that they have shapes determined by the initial conditions. The experimental evidence is generally consistent with the theory.     

Low-thrust Lambert transfer based on two-stage constant-vector thrust control method

The low-thrust Lambert transfer refers to that the spacecraft achieves the orbital transfer whose boundary conditions are represented by two sets of orbital elements at initial and final time by the low-thrust propulsion system. The modulus and direction of the low-thrust solutions in previous methods change with time, which leads to high control requirements for the engine. In this paper, to reduce the requirements of the engine, a practical two-stage constant-vector thrust control method is proposed, in which the magnitude and direction of the thrust are deemed as segmental constant value in TNH frame, where three components of the thrust are f_t, f_n, and f_h. First, the mathematical model of the two-stage constant-vector thrust is formulated, and a rapid algorithm is presented to obtain the solution based on the linearized sensitivity matrix, which describes the relationship between the constant-vector thrust and the change of the orbital elements approximately. Furthermore, two low-thrust Lambert strategies based on the two-stage constant-vector thrust are presented for cases of short-time transfer and long-time transfer. A sequence of numerical simulations demonstrated the efficiency of the proposed approaches. The proposed control strategies are solved rapidly, and they are also suitable for different types of orbits with J₂ perturbation, which are practical options for engineering applications.

     

On energy transfer between vibrating systems under linear and nonlinear interactions

The present study deals with energy transfer in a dissipative mechanical system. Numerical results are given by considering two different potentials and periodical excitation. Specifically, we show energy transfer from linear oscillator to another one, depending on initial conditions. Also, energy transfer from linear to nonlinear (energy pumping), as well as from nonlinear to linear, oscillator is analyzed, under linear and nonlinear interactions.     

Dynamic coefficient evaluation for an algebraic subgrid stress model

Static model coefficients for an algebraic subgrid stress (SGS) model are determined using a dynamic approach, based on results from simulations of isotropic decaying turbulence. The study was motivated by the discrepancies in energy transfer predictions using the previously documented coefficients (Bhushan and Warsi, Int. J. Numer. Meth. Fluids 2005; 49 : 489–519). The discrepancies are identified to be due to inconsistent filter functions used in the analytic estimates and the simulations. The study emphasizes that SGS model development should use filter functions compatible with those inherent in CFD application solvers. The dynamic approach predicts consistent model and transfer coefficients for different grid resolutions and is judged to be a reliable basis for model coefficient adjustments. The predicted Leonard's stress coefficient and associated energy transfer coefficients agree very well with the analytic estimates using a Gaussian/cutoff combination filter. This suggests that the modeling of Leonard's stress term using a truncated Taylor series expansion is robust and may not benefit significantly from dynamic modeling. Validation simulations were performed for turbulent channel flow at Re_τ = 180 and 590. The dynamic approach was found to be reliable only for the lower log‐layer of the Re_τ = 590 case, where the scale invariance condition was satisfied. Nonetheless, in this narrow range, the model and transfer coefficients compare well with the isotropic case. The static coefficient algebraic model with new adjusted coefficients shows improved predictions compared with the previous coefficients, for both isotropic decaying turbulence and channel flow cases. Copyright © 2013 John Wiley & Sons, Ltd.     

Transient in a two-DOF nonlinear system

Dynamics of a system containing a linear oscillator, linearly coupled to an essentially nonlinear attachment, is considered. A damping is taken into account. It is assumed that some initial excitation implies vibrations of the linear oscillator. Envelops of the subsystem’s kinetic energies are selected to use the numerical investigation of transient in the system. The parametrical optimization approach is used to obtain regions of effective energy transfer in the system parameter space. It is demonstrated that this efficient energy transfer may be obtained for a rather small value of the attachment mass.     

Constitutive Relations for Monatomic Gases¶Based on a¶Generalized Rational Approximation¶to the Sum of the Chapman-Enskog Expansion

. We study the asymptotic behavior as time goes to infinity of solutions to the initial‐boundary‐value problem on the half space for a one‐dimensional model system for the isentropic flow of a compressible viscous gas, the so‐called p‐system with viscosity. As boundary conditions, we prescribe the constant state at infinity and require that the velocity be zero at the boundary . When the velocity at infinity is negative and satisfies a condition on the magnitude, we prove that if the initial data are suitably close to those for the corresponding outgoing viscous shock profile, which is suitably far from the boundary, then a unique solution exists globally in time and tends toward the properly shifted viscous shock profile as the time goes to infinity. The proof is given by an elementary energy method. (Accepted March 2, 1998)