Showing papers on "Rarefaction published in 2008"

Compressibility and Rarefaction Effects on Drag of a Spherical Particle

Abstract: A review of compressibility and rarefaction effects on spherical particle drag was conducted based on existing experimental data, theoretical limits, and direct simulation Monte Carlo method results. The data indicated a nexus point with respect to effects of Mach number and Knudsen number. In particular, it was found that a single drag coefficient (of about 1.63) is obtained for all particle conditions when the particle Reynolds number is about 45, and is independent of compressibility or rarefaction effects. At lower Reynolds numbers, the drag is dominated by rarefaction, and at higher Reynolds numbers, it is dominated by compressibility. The nexus, therefore, allows construction of two separate models for these two regimes. The compression-dominated regime is obtained using a modification of the Clift-Gauvin model to specifically incorporate Mach number effects. The resulting model was based on a wide range of experimental data and showed superior prediction robustness compared with previous models. For the rarefaction-dominated regime, the present model was constructed to directly integrate the theoretical creeping flow limits, including the incompressible continuum flow limit (Stokes drag), the incompressible weak rarefaction limit (Basset-Knudsen correction), and the incompressible free-molecular flow limit (Epstein theory). Empirical correlations are used to extend this model to finite particle Reynolds numbers within the rarefaction-dominated regime.

Discovering and rediscovering the sample-based rarefaction formula in the ecological literature

Abstract: Rarefaction has long represented a powerful tool for detecting species richness and its variation across spatial scales. Some authors recently reintroduced the mathematical expression for calculating sample-based rarefaction curves. While some of them did not claim any advances, others presented this formula as a new analytical solution. We provide evidence about formulations of the sample-based rarefaction formula older than those recently proposed in ecological literature.

Gas flow through an elliptical tube over the whole range of the gas rarefaction

Abstract: A rarefied gas flow trough a long tube with an elliptical cross section is studied on the basis of the BGK kinetic model equation in the whole range of the Knudsen number varying from the free molecular regime to the hydrodynamic one. A wide range of the aspect ratio is considered. The mass flow rate is calculated as a function of the pressures on the tube ends.

Third-order nonlinear dispersive equations: Shocks, rarefaction, and blowup waves

Abstract: Shock waves and blowup arising in third-order nonlinear dispersive equations are studied. The underlying model is the equation in (0.1) $$ u_t = (uu_x )_{xx} in\mathbb{R} \times \mathbb{R}_ + . $$ It is shown that two basic Riemann problems for Eq. (0.1) with the initial data $$ S_ \mp (x) = \mp \operatorname{sgn} x $$ exhibit a shock wave (u(x, t) ≡ S −(x)) and a smooth rarefaction wave (for S +), respectively. Various blowing-up and global similarity solutions to Eq. (0.1) are constructed that demonstrate the fine structure of shock and rarefaction waves. A technique based on eigenfunctions and the nonlinear capacity is developed to prove the blowup of solutions. The analysis of Eq. (0.1) resembles the entropy theory of scalar conservation laws of the form u t + uu x = 0, which was developed by O.A. Oleinik and S.N. Kruzhkov (for equations in x ∊ ℝ N ) in the 1950s–1960s.

Lattice Boltzmann simulation of surface roughness effect on gaseous flow in a microchannel

Abstract: At the microscale level, it is impossible to obtain a completely smooth wall surface, and the effect of surface roughness may be a main factor responsible for some different characteristics between fluid flow in the microchannels and that in conventional size channels. In the present work, the lattice Boltzmann method is applied to investigate the gaseous flow in a microchannel with surface roughness which is modeled by an array of rectangular modules. The effects of relative surface roughness, roughness distribution, and rarefaction on gaseous flow are studied, but the compressibility effect is neglected since the Mach number is less than 0.2. It was shown that the surface roughness had an important influence on friction factor and mass flow rate. In particular, this effect becomes more significant with the decrease of the Knudsen number. This is because the rarefaction reduces the interaction between the gas molecules and the channel walls, which results in reduction of the surface roughness effect.

Ambipolar acceleration of ions in a magnetic nozzle

Abstract: This paper describes a magnetic nozzle with a magnetic mirror configuration that transforms a collisionless subsonic plasma flow into a supersonic jet expanding into the vacuum. The nozzle converts electron thermal energy into the ion kinetic energy via an ambipolar electric field. The ambipolar potential in the expanding plume involves a time-dependent rarefaction wave. Travelling through the rarefaction wave, electrons lose some kinetic energy and can become trapped downstream from the mirror throat. This work presents a rigorous adiabatic description of the trapped electron population. It examines the impact of the adiabatic cooling of the trapped electrons on the ambipolar potential and the ensuing ion acceleration. The problem is formulated for an arbitrary incoming electron distribution and then a “water-bag” electron distribution is used to obtain a closed-form analytical solution.

Transonic shock formation in a rarefaction riemann problem for the 2D compressible euler equations

Abstract: It is perhaps surprising for a shock wave to exist in the solution of a rarefaction Riemann problem for the compressible Euler equations in two space dimensions. We present numerical evidence and generalized characteristic analysis to establish the existence of a shock wave in such a 2D Riemann problem, defined by the interaction of four rarefaction waves. We consider both the customary configuration of waves at the right angle and also an oblique configuration for the rarefaction waves. Two distinct mechanisms for the formation of a shock wave are discovered as the angle between the waves is varied.

Lattice Boltzmann modelling Knudsen layer effect in non-equilibrium flows

Abstract: Due to its intrinsically kinetic nature, lattice Boltzmann (LB) approach to simulating non-equilibrium gas flows has recently attracted significant research interest. Compared with other kinetic methods, it can offer a significantly smaller computational cost. To capture the nonlinear high-order rarefaction phenomena in gas flows, a geometry-dependent gas local mean free path has been proposed to be implemented in our "high-order" LB model. A series of tests on rarefaction effects and the Knudsen layer interference have been carried out and the simulation results demonstrate our LB model's capability for highly non-equilibrium flows.

Nonlinear stability of rarefaction waves for the compressible Navier-Stokes equations with large initial perturbation

Abstract: The expansion waves for the compressible Navier-Stokes equations have recently been shown to be nonlinear stable. The nonlinear stability results are called local stability or global stability depending on whether the H1 -norm of the initial perturbation is small or not. Up to now, local stability results have been well established. However, for global stability, only partial results have been obtained. The main purpose of this paper is to study the global stability of rarefaction waves for the compressible Navier-Stokes equations. For this purpose, we introduce a positive parameter to in the construction of smooth approximations of the rarefaction wave solutions for the compressible Euler equations so that the quantity £ = ty (6 denotes the strength of the rarefaction waves) is sufficiently large to control the growth induced by the nonlinearity of the system and the interaction of waves from different families. Then by using the energy method together with the continuation argument, we obtain some nonlinear stability results provided that the initial perturbation satisfies certain growth conditions as £ ► +oo. Notice that the assumption that the quantity £ can be chosen to be sufficiently large implies that either the strength of the rarefaction waves is small or the rarefaction waves of different families are separated far enough initially.

Admissibility region for rarefaction shock waves in dense gases

Abstract: In the vapour phase and close to the liquid–vapour saturation curve, fluids made of complex molecules are expected to exhibit a thermodynamic region in which the fundamental derivative of gasdynamic ? is negative. In this region, non-classical gasdynamic phenomena such as rarefaction shock waves are physically admissible, namely they obey the second law of thermodynamics and fulfil the speed-orienting condition for mechanical stability. Previous studies have demonstrated that the thermodynamic states for which rarefaction shock waves are admissible are however not limited to the ? <0 region. In this paper, the conditions for admissibility of rarefaction shocks are investigated. This results in the definition of a new thermodynamic region – the rarefaction shocks region – which embeds the ? <0 region. The rarefaction shocks region is bounded by the saturation curve and by the locus of the states connecting double-sonic rarefaction shocks, i.e. shock waves in which both the pre-shock and post-shock states are sonic. Only one double-sonic shock is shown to be admissible along a given isentrope, therefore the double-sonic states can be connected by a single curve in the volume–pressure plane. This curve is named the double sonic locus. The influence of molecular complexity on the shape and size of the rarefaction shocks region is also illustrated by using the van der Waals model; these results are confirmed by very accurate multi-parameter thermodynamic models applied to siloxane fluids and are therefore of practical importance in experiments aimed at proving the existence of rarefaction shock waves in the single-phase vapour region as well as in future industrial applications operating in the non-classical regime.

Design of the Dense Gas Flexible Asymmetric Shock Tube

Abstract: This paper presents the conceptual design of the flexible asymmetric shock tube (FAST) setup for the experimental verification of the existence of nonclassical rarefaction shock waves in molecularly complex dense vapors. The FAST setup is a Ludwieg tube facility composed of a charge tube that is separated from the discharge vessel by a fast-opening valve. A nozzle is interposed between the valve and the charge tube to prevent disturbances from the discharge vessel to propagate into the tube. The speed of the rarefaction wave generated in the tube as the valve opens is measured by means of high-resolution pressure transducers. The provisional working fluid is siloxane D6 (dodecamethylcyclohexasiloxane, C12H36O6Si6). Numerical simulations of the FAST experiment are presented using nonideal thermodynamic models to support the preliminary design. The uncertainties related to the thermodynamic model of the fluid are assessed using a state-of-the-art thermodynamic model of fluid D6. The preliminary design is confirmed to be feasible and construction requirements are found to be well within technological limits.

Cluster observations of fast shocks in the magnetosheath launched as a tangential discontinuity with a pressure increase crossed the bow shock

Abstract: [1] The interaction of a tangential discontinuity (TD) and accompanying dynamic pressure increase with the Earth's bow shock launches a fast shock that travels ahead of the TD in the magnetosheath and carries a significant portion of the pressure change. In this event study, we use observations from the Cluster spacecraft and magnetohydrodynamic simulations to identify the fast shock and its properties and to track the TD in the magnetosheath. Velocities of the fast shock and the TD were determined by triangulation using the four distant Cluster spacecraft. The fast shock is a planar structure, traveling nearly perpendicular to B at the magnetosonic speed in the plasma rest frame. Changes in density and ∣B∣ are correlated, with about a 20% increase in each. A current was observed tangential to the plane of the fast shock, and the positive E •J there provided an electromagnetic energy source for the observed heating of the ions. The fast shock is generated by the pressure change and determines the timing of the initial response of the magnetopause to that change. The TD was moving nearly in the −XGSE direction and was being compressed as it moved inward. The passage of the TD ushered in large-scale compressive structure in the magnetosheath magnetic field, which satisfied the mirror mode instability criterion. Velocities of a fast rarefaction wave, reflected from the magnetopause, and an additional slow-mode structure, which was not a product of the initial interaction with the bow shock, were determined by triangulation.

Numerical Analyses of Exhaust and Refill Processes of a Laser Pulse Jet

Abstract: The exhaust-refill processes in a laser pulse jet with a conical nozzle are simulated using computational fluid dynamics to clarify the analytically unpredictable decreasing tendency of a momentum-coupling coefficient with an increasing nozzle apex angle. Because the exhaust-refill processes result from an adiabatically expanding blast wave after laser heating, a valid explosion source based on a measurement is used to drive the blast wave instead of a laser-absorption process. Results of computations show that processes that occur until the shock front of a blast wave reaches the nozzle exit are similar, irrespective of the nozzle apex angle. However, after the blast wave leaves the nozzle edge, the behavior of the rarefaction wave induced behind the shock wave depends greatly on the nozzle apex angle. In the case of small apex angles, successive refilling mechanisms with a vortex are activated by the prominently evolved rarefaction wave. In contrast, in the case of large apex angles, this mechanism disappears, due to the moderate evolution of the rarefaction wave. This difference creates a tendency of the momentum-coupling coefficient to decrease with the apex angle.

The spatial distribution of upstream ion events from the Earth's bow shock measured by ACE, Wind, and STEREO

Abstract: [1] Using simultaneous measurements of >40 keV upstream ions observed at ACE, Wind, and STEREO-A during the solar minimum period of 2007, day 1 through 2007, day 181, we investigate their spatial distributions by calculating their occurrence probabilities as a function of lateral and radial separation between L1 and STEREO-A. Our main results are given as follows: (1) STEREO-A observed upstream events even when it was separated from Earth by ∼1750 RE and ∼3800 RE in the radial and lateral directions, respectively. (2) The occurrence probability (∼20–30%) for measuring simultaneous upstream events at L1 and STEREO-A was far greater than that expected from accidental coincidences. (3) The occurrence rate of simultaneous upstream events at L1 and STEREO-A is significantly higher inside rarefaction regions of high-speed solar wind flows (>500 km s−1) that follow corotating compression regions and when there exist antisunward propagating Alfven waves. These new results confirm the global nature of the source region and place limits on the spatial size of the interplanetary structures that could either accelerate the ions in the first place or at the very least provide them with easier access by facilitating their scatter-free transport from the Earth's foreshock into the far upstream regions traversed by STEREO-A. We suggest that the existence of large amplitude Alfven waves with spatial scales of the order of 0.03 AU that are embedded in and get convected past the Earth by high-speed solar wind streams plays a critical and necessary role in the occurrence of upstream ion events near the Earth.

Gas flow through a slit into a vacuum in a wide range of rarefaction

Abstract: Gas flow through a two-dimensional slit into a vacuum is investigated by a direct simulation Monte Carlo method Results for the mass flow rate are obtained as a function of the rarefaction parameter, which is inversely proportional to the Knudsen number The distributions of density, temperature and mass velocity, and streamlines are presented In the free molecular flow regime and in the hydrodynamic limit, our results agree with theoretical asymptotes, and in the transition regime, they compare well with numerical simulations by other authors

Reshocks, rarefactions, and the generalized Layzer model for hydrodynamic instabilities

Abstract: We report numerical simulations and analytic modeling of shock tube experiments on Rayleigh–Taylor and Richtmyer–Meshkov instabilities. We examine single interfaces of the type A/B where the incident shock is initiated in A and the transmitted shock proceeds into B. Examples are He/air and air/He. In addition, we study finite-thickness or double-interface A/B/A configurations such as air/SF6/air gas-curtain experiments. We first consider conventional shock tubes that have a “fixed” boundary: A solid endwall which reflects the transmitted shock and reshocks the interface(s). Then we focus on new experiments with a “free” boundary—a membrane disrupted mechanically or by the transmitted shock, sending back a rarefaction toward the interface(s). Complex acceleration histories are achieved, relevant for inertial confinement fusion implosions. We compare our simulation results with a generalized Layzer model for two fluids with time-dependent densities and derive a new freeze-out condition whereby accelerating ...

Effects of Rarefaction and Compressibility on Fluid Flow at Slip Flow Regime by Direct Simulation of Roughness

Abstract: A two dimensional numerical simulation has been performed for incompressible and compressible fluid flow through microchannels in slip flow regime. The Navier-Stokes equations have been solved in conjunction with Maxwell slip conditions for modeling flow field associated with slip flow regime. The wall roughness is simulated with triangular microelements distributed on wall surfaces to study the effects of roughness on fluid flow. Various Mach and Knudsen numbers are used to investigate the effects of rarefaction as well as compressibility. It is found that rarefaction has more significant effect on flow field in microchannels with higher relative roughness. It is also found that compressibility has more significant effects on Poiseuille number when relative roughness increases. In addition, similar to incompressible models the increase in average fRe is more significant at low Knudsen number flows but the increase of Poiseuille number duo to relative roughness is sharper for compressible models. The numerical results have also validated with some available theoretical and experimental relations and good agreements have been seen. Keywords—Relative roughness, slip flow, Poiseuille number.

Oblique shock waves incident on an interface between two materials for general equations of state

Abstract: Numerical schemes have been devised to solve the problem of an oblique shock scattering off the interface between two solid materials. The problem of a steady three-shock structure is first solved analytically using equations of state (EOS) where the shock speed and particle speed are related through a linear function. The formulation is then generalized to accept general EOS, which requires the incorporation of a more general algorithm to explore possible shock configurations based on the mechanical and thermodynamic parameters in the EOS. The correct configuration is found from the equality of pressure behind the reflected and transmitted waves as well as the equality of boundary deflection due to flow in the upper and lower halves of the three-shock structure. For the case of a reflected rarefaction in material described by general EOS, numerical integration over the release is used for accuracy where the rarefaction has a finite thickness as opposed to a discontinuous “expansion” shock. The results of...

Asymptotic decay toward rarefaction wave for a hyperbolic-elliptic coupled system on half space

Abstract: We consider the asymptotic behavior of solutions to a model of hyperbolicelliptic coupled system on the half-line R+ = (0,∞), ut + uux + qx = 0, −qxx + q + ux = 0, with the Dirichlet boundary condition u(0, t) = 0. S. Kawashima and Y. Tanaka [Kyushu J. Math., 58(2004), 211-250] have shown that the solution to the corresponding Cauchy problem behaviors like rarefaction waves and obtained its convergence rate when u − < u+. Our main concern in this paper is the boundary effect. In the case of null-Dirichlet boundary condition on u, asymptotic behavior of the solution (u, q) is proved to be rarefaction wave as t tends to infinity. Its convergence rate is also obtained by the standard L-energy method and L-estimate. It decays much lower than that of the corresponding Cauchy problem.

LED chip package structure with different LED spacings and a method for making the same

Abstract: An LED chip package structure with different LED spacing includes a substrate unit, a light-emitting unit, and a package colloid unit. The light-emitting unit has a plurality of LED chips electrically arranged on the substrate unit, and the LEDs are separated from each other by totally different spacing or partially different spacing. For example, the spacings between each two LED chips are from rarefaction to condensation, from condensation to rarefaction, from center rarefaction to outer condensation, from center condensation to outer rarefaction, alternate rarefaction and condensation, or alternate condensation and rarefaction. The package colloid unit covers the LED chips.

Asymptotic stability of the rarefaction wave for the generalized KdV–Burgers–Kuramoto equation

Abstract: In this paper, we will study the Cauchy problem for the generalized KdV–Burgers–Kuramoto equation, which represents a dissipative, stroboscopic and unstable system in physics. When the initial data is a small disturbance of a rarefaction wave of the inviscid Burgers equation, we prove the global existence of the solution to the corresponding Cauchy problem and asymptotic stability of the rarefaction wave. The analysis is based on a priori estimates and the L 2 -energy method.

Effect of Rarefaction, Dissipation, and Accommodation Coefficients on Heat Transfer in Microcylindrical Couette Flow

Abstract: This paper examines the effects of rarefaction, dissipation, curvature, and accommodation coefficients on flow and heat transfer characteristics in rotating microdevices. The problem is modeled as a cylindrical Couette flow with a rotating shaft and stationary housing. The housing is maintained at uniform temperature while the rotating shaft is insulated. Thus, heat transfer is due to viscous dissipation only. An analytic solution is obtained for the temperature distribution in the gas filled concentric clearance between the rotating shaft and its stationary housing. The solution is valid in the slip flow and temperature jump domain defined by the Knudsen number range of 0.001 < Kn < 0.1. The important effect of the momentum accommodation coefficient on velocity reversal and its impact on heat transfer is determined. The Nusselt number was found to depend on four parameters: the momentum accommodation coefficient of the stationary surface σ uo , Knudsen number Kn, ratio of housing to shaft radius r o /r i , and the dimensionless group [γ/(γ+1)](2σ to -1)/(σ to Pr). Results indicate that curvature, Knudsen number, and the accommodation coefficients have significant effects on temperature distribution, heat transfer, and Nusselt number.

Magnetohydrodynamic Effects in Propagating Relativistic Jets: Reverse Shock and Magnetic Acceleration

Abstract: We solve the Riemann problem for the deceleration of an arbitrarily magnetized relativistic flow injected into a static unmagnetized medium in one dimension. We find that for the same initial Lorentz factor, the reverse shock becomes progressively weaker with increasing magnetization \sigma (the Poynting-to kinetic energy flux ratio), and the shock becomes a rarefaction wave when \sigma exceeds a critical value, \sigma_c, defined by the balance between the magnetic pressure in the flow and the thermal pressure in the forward shock. In the rarefaction wave regime, we find that the rarefied region is accelerated to a Lorentz factor that is significantly larger than the initial value. This acceleration mechanism is due to the strong magnetic pressure in the flow. We discuss the implications of these results for models of gamma-ray bursts and active galactic nuclei.

Experimental investigations of convection heat transfer in microporous media

Abstract: The convection heat transfer between the solid particles and the fluid in microporous media were studied experimentallyThe experimentally measured convection heat transfer Nusselt numbers for air in the microporous media with average diameters of 200μm to 40μm agree well with the published resultsThe experimental values for the Nusselt numbers for air in the microporous media with 10μm and 20μm average diameters are much less than the publish resultsThe results show that rarefaction effects occur in convection heat transfer for air in the microporous media with particle diameters less than 20μmA new correlation for Nusslet number was proposed with consideration of the influence of Kn