Showing papers on "Oblique shock published in 1998"

Fermi Acceleration at the Fast Shock in a Solar Flare and the Impulsive Loop-Top Hard X-Ray Source

Abstract: Because of its high injection energy, Fermi acceleration has not been considered to be viable to explain nonthermal electrons (20-100 keV) produced in solar flares. Here we propose that nonthermal electrons are efficiently accelerated by the first-order Fermi process at the fast shock, as a natural consequence of the new magnetohydrodynamic picture of the flaring region revealed with Yohkoh. An oblique fast shock is naturally formed below the reconnection site and boosts the acceleration to significantly decrease the injection energy. The slow shocks attached to the reconnection X-point heat the plasma up to 10-20 MK, exceeding the injection energy. The combination of the oblique shock configuration and the preheating by the slow shock allows bulk electron acceleration from the thermal pool. The accelerated electrons are trapped between the two slow shocks due to the magnetic mirror downstream of the fast shock, thus explaining the impulsive loop-top hard X-ray source discovered with Yohkoh. The acceleration timescale is ~0.3-0.6 s, which is consistent with the timescale of impulsive bursts. When these electrons stream away from the region enclosed by the fast shock and the slow shocks, they are released toward the footpoints and may form the simultaneous double-source hard X-ray structure at the footpoints of the reconnected field lines.

Fermi acceleration at fast shock in a solar flare and impulsive loop-top hard X-ray source

Abstract: We propose that non-thermal electrons are efficiently accelerated by first-order Fermi process at the fast shock, as a natural consequence of the new magnetohydrodynamic picture of the flaring region revealed with Yohkoh. An oblique fast shock is naturally formed below the reconnection site, and boosts the acceleration to significantly decrease the injection energy. The slow shocks attached to the reconnection X-point heat the plasma up to 10--20 MK, exceeding the injection energy. The combination of the oblique shock configuration and the pre-heating by the slow shock allows bulk electron acceleration from the thermal pool. The accelerated electrons are trapped between the two slow shocks due to the magnetic mirror downstream of the fast shock, thus explaining the impulsive loop-top hard X-ray source discovered with Yohkoh. Acceleration time scale is ~ 0.3--0.6 s, which is consistent with the time scale of impulsive bursts. When these electrons stream away from the region enclosed by the fast shock and the slow shocks, they are released toward the footpoints and may form the simultaneous double-source hard X-ray structure at the footpoints of the reconnected field lines.

Shock oscillation in underexpanded screeching jets

Abstract: The periodic oscillation of the shock waves in screeching, underexpanded, supersonic jets, issuing from a choked, axisymmetric, nozzle at fully expanded Mach numbers (Mj) of 1.19 and 1.42, is studied experimentally and analytically. The experimental part uses schlieren photography and a new shock detection technique which depends on a recently observed phenomenon of laser light scattering by shock waves. A narrow laser beam is traversed from point to point in the flow field and the appearance of the scattered light is sensed by a photomultiplier tube (PMT). The time-averaged and phase-averaged statistics of the PMT data provide significant insight into the shock motion. It is found that the shocks move the most in the jet core and the least in the shear layer. This is opposite to the intuitive expectation of a larger-amplitude shock motion in the shear layer where organized vortices interact with the shock. The mode of shock motion is the same as that of the emitted screech tone. The instantaneous profiles of the first four shocks over an oscillation cycle were constructed through a detailed phase averaged measurement. Such data show a splitting of each shock (except for the first one) into two weaker ones through a ‘moving staircase-like’ motion. During a cycle of motion the downstream shock progressively fades away while a new shock appears upstream. Spark schlieren photographs demonstrate that a periodic convection of large organized vortices over the shock train results in the above described behaviour. An analytical formulation is constructed to determine the self-excitation of the jet column by the screech sound. The screech waves, while propagating over the jet column, add a periodic pressure fluctuation to the ambient level, which in turn perturbs the pressure distribution inside the jet. The oscillation amplitude of the first shock predicted from this linear analysis shows reasonable agreement with the measured data. Additional reasons for shock oscillation, such as a periodic perturbation of the shock formation mechanism owing to the passage of the organized structures, are also discussed.

Ion leakage from quasiparallel collisionless shocks: Implications for injection and shock dissipation

Abstract: A simplified model of particle transport at a quasiparallel one-dimensional collisionless shock is suggested. In this model the MHD-turbulence behind the shock is dominated by a circularly polarized, large amplitude Alfv\'en wave originated upstream from the turbulence excited by particles leaking from the downstream medium. It is argued that such a wave having significantly increased its magnetic field during the transmission through the shock interface can effectively trap thermal ions, regulating their leakage upstream. Together with a background turbulence this wave also plays a fundamental role in thermalization of the incoming ion flow. The spectrum of leaking particles and the amplitude of the wave excited by these particles are selfconsistently calculated. The injection rate into the first order Fermi acceleration based on this leakage mechanism is obtained and compared with computer simulations. The related problem of shock energy distribution between thermal and nonthermal components of the shocked plasma is discussed. The chemical composition of the leaking particles is studied.

Nature, properties, and origin of low‐frequency waves from an oblique shock to the inner magnetosheath

Abstract: We analyze the high time-resolution profiles of the electron density and of the magnetic field and the plasma parameters recorded by ISEE 1 and 2 during a crossing of the Earth's magnetosheath at 1430 LT. Compressive and Alfven ion cyclotron modes (AIC modes) are identified by comparing the measured magnetic polarization and electron parallel compressibility with the results of calculations in an unstable kinetic linear model. A criterion to discuss the accuracy of the wave vector direction of mirror modes is established; an efficient method to disentangle mirror and AIC modes is presented and applied. From the bow shock to the inner sheath we identify successively (1) compressive modes and AIC modes in the oblique shock, (2) a pure AIC mode region of circularly and elliptically polarized waves in a layer 0.3 RE thick adjacent to the undershoot, (3) a mixed region 2 RE thick where both mirror modes and AIC modes are observed, (4) a pure mirror mode region. The nature of the dominant mode appears to be controlled by the depth in the magnetosheath, more than by the local values of βp and the proton temperature anisotropy Tp⊥/Tp‖. In the outer sheath the unusual identification of a pure Alfvenic region for a large average proton beta βp = 13 and a moderate proton temperature anisotropy could be explained by a relatively low density of α particles. The mirror modes are three-dimensional structures with their major axis along the magnetic field and with their minor axis nearly perpendicular to the magnetopause surface. We estimate the dimensions of ordered structures observed in the middle of the magnetosheath for a βp around 7 ± 1 and Tp⊥/Tp‖ around 1.5; the minor axis of regular mirror modes is typically between 1300 and 1900 km long; the intermediate dimension is larger than either 2200 or 2700 km, while the major axis is larger than either 2700 or 3400 km. For the first time the measured parallel compressibility of the pure mirror modes is shown to be in relatively good agreement with the linear model predictions for 4 < βp < 11. The absence of AIC modes in the inner sheath suggests that these modes cannot grow or propagate in regions where mirror modes are well developed and that AIC wave energy is not transferred across a large-amplitude mirror mode region.

Complex magnetohydrodynamic bow shock topology in field-aligned low-β flow around a perfectly conducting cylinder

Abstract: Two-dimensional ideal magnetohydrodynamic (MHD) simulations are presented that demonstrate several novel phenomena in MHD shock formation. The stationary symmetrical flow of a uniform, planar, field-aligned, low-β and superfast magnetized plasma around a perfectly conducting cylinder is calculated. The velocity of the incoming flow is chosen such that the formation of fast switch-on shocks is possible. Using a time marching procedure, a stationary bow shock is obtained, composed of two consecutive interacting shock fronts. The leading shock front has a dimpled shape and is composed of fast, intermediate and hydrodynamic shock parts. A second shock front follows the leading front. Additional intermediate shocks and tangential discontinuities are present in the downstream part of the flow. The intermediate shocks are of the 1–3, 1–4, 2–4 and 1=2–3=4 types. This is a confirmation in two dimensions of recent results on the admissibility of these types of shocks. Recently it has also been shown that the 1=2–3=...

Dust grains and the structure of steady C-type magnetohydrodynamic shock waves in molecular clouds

Abstract: I examine the role of dust grains in determining the structure of steady, cold, oblique C-type shocks in dense molecular gas. Gas pressure, the inertia of the charged components and changes in ionization are neglected. The grain charge and rate coefficients for electron–neutral and grain–neutral elastic scattering are assumed constant at values appropriate to the shock interior. An MRN size distribution is accounted for by estimating an effective grain abundance and Hall parameter for single-size grains. A one-parameter family of intermediate shocks exists for each shock speed vs between the intermediate signal speed vA cos θ and √2vA cot θ, where vA is the pre-shock Alfven speed and θ is the angle between the pre-shock magnetic field and the normal to the shock front. In addition, there is a unique fast shock for each vs > vA. If the pre-shock density nH ≳ 105 cm−3 and the pre-shock magnetic field satisfies B(mG)/nH(105 cm−3 ≲ 1, grains are partially decoupled from the magnetic field and the field and velocity components within fast shocks do not lie in the plane containing the pre-shock field and the shock normal. The resulting shock structure is significantly thinner than in models that do not take this into account. Existing models systematically underestimate the grain–neutral drift speed and the heating rate within the shock front. At densities in excess of 108 cm −3 these effects may be reduced by the nearly equal abundances of positive and negative grains.

The location of a shock in rimming flow

Abstract: In rimming flow, a thin film of viscous liquid is entrained on the inside of a horizontally rotating cylinder. We give an explicit criterion for determining whether or not shock solutions occur and show that the location and height of these shocks can be determined using a simple lubrication theory.

Flowfield Measurements in a Slot-Bled Oblique Shock Wave and Turbulent Boundary-Layer Interaction

Abstract: An experimental investigation was conducted to determine the flowfield inside a bleed slot used to control an oblique shock-wave and turbulent boundary-layer interaction. The slot was oriented normal to the primary flow direction and had a width of 1.0 cm (primary flow direction), a length of 2.54 cm, and spanned 16.5 cm. The approach boundary layer upstream of the interaction was nominally 3.0 cm thick. Two operating conditions were studied: M = 1.98 with a shock generator deflection angle of 6 deg and M= 2.46 with a shock generator deflection angle of 8 deg. Measurements include surface and flowfield static pressure, Pitot pressure, and total mass-flow through the slot. The results show that despite an initially two-dimensional interaction for the zero bleed-flow case, the slot does not remove mass uniformly in the spanwise direction. Inside the slot, the flow is characterized by two separation regions which significantly reduce the effective flow area. The upper separation region acts as an aerodynamic throat resulting in supersonic flow through much of the slot.

Numerical study on blast flowfields induced by supersonic projectiles discharged from shock tubes

Abstract: In this paper we report on a numerical study of the blast flowfield generated by a supersonic projectile released from the open-end of a shock tube into ambient air. The Euler equations, assuming axisymmetric flows, were solved using a dispersion-controlled scheme implemented with moving boundary conditions. Two initial test cases were calculated. One of them is for validation of the numerical method and the other for verification of the moving boundary conditions. After good agreement was achieved, four further cases were calculated for examining effects of various projectile speeds and different release times of the projectile after the precursor shock wave was discharged. The present numerical study confirms that complicated transient phenomena exist in the initial stages shortly after projectile release, and that the blast flowfield is much more complex than that which can be inferred from muzzle blast studies where combustion products obscure the flow.

Numerical simulations of shock-vortex interactions in supersonic jet screech

Abstract: Supersonic jet screech is a form of jet noise which adversely impacts both the environment and the life of aircraft structures. A basic understanding of the mechanisms which generate the screech tone and determine its amplitude is therefore important. In the present study we perform direct numerical simulations (DNS) of the interaction of an oblique shock with instability waves of a finite thickness supersonic shear layer. We thereby retain the basic elements of an isolated jet screech source. The simulations are carried out in two dimensions using a high order accurate spatial scheme with nonreflecting boundary conditions. The unsteady shock motion is resolved with the essentially non-oscillatory (ENO) discontinuity capturing scheme. The shear layer (M = 1.2, Re = 1000 based on initial vorticity thickness) is forced at the most unstable frequency such that the instability waves develop into fully formed vortices upstream of the shock. The interaction of the vortices with the shock causes streamwise oscillations in the shock near its tip. On the subsonic side of the shear layer, the acoustic wave is released as the shock tip deflects upstream through the braid region between vortices. This acoustic field is approximately cylindrical and its directivity is nearly uniform. The acoustic waveform is comprised of a sharp compression followed by a longer expansion. On the supersonic side of the shear layer, a complex wave field is observed. Cases in which the incident shock is replaced by a nearly isentropic compression wave produce qualitatively similar behavior. Acoustic pressure amplitude is found to Copyright ©1998 Ted A. Manning and Sanjiva K. Lele. Published by the American Institute of Aeronautics and Astronautics with permission. talso affiliated with the Department of Mechanical Engineering, Stanford University. scale with compression wave strength. Acoustic wave form is insensitive to compression wave amplitude and profile width for cases examined. This research is directed toward modeling the screech generation process.

Attenuation of weak shock waves along pseudo-perforated walls

Abstract: In order to attenuate weak shock waves in ducts, effects of pseudo-perforated walls were investigated. Pseudo-perforated walls are defined as wall perforations having a closed cavity behind it. Shock wave diffraction and reflection created by these perforations were visualized in a shock tube by using holographic interferometer, and also by numerical simulation. Along the pseudo-perforated wall, an incident shock wave attenuates and eventually turns into a sound wave. Due to complex interactions of the incident shock wave with the perforations, the overpressure behind it becomes non-uniform and its peak value can locally exceed that behind the undisturbed incident shock wave. However, its pressure gradient monotonically decreases with the shock wave propagation. Effects of these pseudo-perforated walls on the attenuation of weak shock waves generated in high speed train tunnels were studied in a 1/250-scaled train tunnel simulator. It is concluded that in order to achieve a practically effective suppression of the tunnel sonic boom the length of the pseudo-perforation section should be sufficiently long.

The acceleration of pickup ions at shock waves: Test particle-mesh simulations

Abstract: A mechanism for the acceleration of pickup ions by repeated reflection from the electrostatic cross-shock potential of a quasi-perpendicular shock was proposed independently by Zank et al. [1996b] and Lee et al. [1996]. The acceleration mechanism, known variously as Multiply Reflected Ion (MRI) acceleration or shock surfing, was studied by these authors in the limit of an idealized shock, which possessed neither fine-scale structure (distinct foot, ramp, or overshoot) nor pickup ion scattering turbulence. Here the acceleration of pickup ions at cometary and interplanetary shocks and at the termination shock is studied in the test particle limit on the basis of a particle-mesh simulation. All simulations assume a shell distribution for the pickup ions (either pickup protons or helium), and the dynamics of pickup ions propagating in a fixed electromagnetic field profile are investigated. The effect of a shock foot, ramp and overshoot on the acceleration of pickup ions at perpendicular and oblique shocks is described. The acceleration of pickup ions in the presence of strong turbulence inside the foreshock region is also addressed. For quasi-perpendicular shocks with structure, we obtained the following results. First, the accelerated H + and He + spectrum is a very hard power law E -k , k = 0.92 -1.2, which is much harder than that predicted by diffusive shock acceleration. Also, as θ bn , the angle between the upstream magnetic field and shock normal, decreases from 90°, the accelerated pickup ion spectrum flattens until MRI acceleration ceases. Second, the fine structure of the shock is found to reduce slightly the maximum energy gain for an accelerated pickup ion compared to that gained at an unstructured shock. Third, a flat turbulence spectrum is found to lead to an increase in the maximum energy for transmitted pickup ions, whereas a power law turbulence spectrum leads to a modest reduction in the maximum pickup ion energy gain. However, the basic MRI acceleration mechanism continues to operate in the presence of turbulent magnetic fluctuations and the characteristic hard spectra are preserved. Fourth, MRI acceleration is found to work only for those shocks for which θ bn > 60° - 70°. The results from the test particle simulations described here are also used to interpret particle acceleration in self-consistent hybrid simulations.

Three-dimensional structure of stabilization of oblique detonation wave in hypersonic flow

Abstract: Detailed oblique detonation wave structure produced by a wedge at hypersonic speed (Mach number M=8.3) is studied experimentally using the oblique shock tube facility of the Laboratoire de Combustion et de Detonique (LCD) Laboratory. The flowfield induced by the wedge is visualized using multiframe schlieren and planar laser-induced fluorescence (PLIF) imaging. In addition, the story of the establishment and propagating phase of the structure is followed by smoked foil technique. Experiments performed in H2+2.38O2 mixture show that onset of oblique detonation is triggered by oblique shock wave, after a delay corresponding to the chemical induction time. The basic configuration appears as a three waves structure issued from a triple point, that is, the oblique shock wave, the oblique detonation wave, and a transverse detonation wave. The transverse detonation wave is found especially overdriven. For relatively short induction time corresponding to an initial pressure of po=0.5 bar, the triple point is fairly stable. Reducing the initial pressure (p0≤0.4 bar) causes the apparition of periodic instabilities: the triple point oscillates around an average position. These periodic instabilities seem to be linked to the chemical induction time. These instabilities correspond to an explosion behind the oblique shock wave and ahead of the triple point structure, giving birth to a new triple point, replacing the old one. At that time, computational fluid dynamics (CFD) models failed to simulate this structure and linked mechanism of instabilities. New improvements in modeling may provide better results.

Prediction of Supersonic Jet Noise

Abstract: Noise radiated from a supersonic jet is computed using the Parabolized Stability Equations (PSE) method. The evolution of the instability waves inside the jet is computed using the PSE method and the noise radiated to the far field from these waves is calculated by solving the wave equation using the Fourier transform method. We performed the computations for a cold supersonic jet of Mach number 2.1 which is excited by disturbances with Strouhal numbers St=.2 and .4 and the azimuthal wavenumber m=l. Good agreement in the sound pressure level are observed between the computed and the measured (Troutt and McLaughlin 1980) results. In this work we computed the noise radiated from supersonic turbulent jets using the PSE (Parabolized Stability Equations) method. Jet noise can be divided into three categories: (1) shock-induced screech tone noise; (2) shock-induced broad band noise; and (3) turbulent mixing noise. The screech tone appears in an imperfectly expanded jet as discrete band in the front part of the jet. The screech tone phenomena is very complex and it is believed that the toroidal and helical vortices which shed at the lip of the nozzle interact with the shocks when they propagate downstream and makes the shock to oscillate and this radiates sound in the upstream direction at discrete frequency. In an imperfectly expanded jet, the shockcells formed by the oblique shocks or the expansion fans generated at nozzle lip interact with the large scale turbulence and generate broad band noise. The dominant part of the broad band shock-associated noise is comprised of a spectral peak with a relatively narrow half-width. Turbulent mixing noise is the noise component which is contributed from the largescale and small scale turbulence in the jet. For a perfectly expanded supersonic jet, the noise is completely generated by the turbulence in the jet and the predominant part of the noise is radiated in the downstream direction in the range between 25^5°. It is observed that at low Reynolds numbers the noise is radiated at a discrete frequency which is closer to the most unstable instability wave for that jet. At moderate and high Reynolds numbers there is discernible peak but they become broad band. These similarities between the noise generated from the high and low Reynolds number jets imply that the noise generation mechanisms in supersonic jets are same at different Reynolds numbers. Several experiments were performed to identify these mechanisms ( McLaughlin et al. 1975, 1977, Morrision and McLaughlin 1979, 1980, Troutt and McLaughlin 1982, Seiner et al. 1993) and it is concluded that the dominant part of the turbulent mixing noise of high Reynolds number supersonic jets is generated by the large-scale coherent structures. It is also concluded that these

Numerical solution for spherical laser-driven shock waves

Abstract: The history of the flow behind a laser-driven shock is investigated in the context of variable energy blast waves. Thereby the total laser energy absorbed by the blast is assumed to vary proportionally to some power of time. Due to the high temperatures and pressures occurring in the initial phase of the flow a real gas model has been employed. It accounts for vibration, dissociation, electronic excitation, ionization and intermolecular forces. Radiative and conductive heat transfer are considered as well. The numerical computations were carried out using the method of characteristics. A self-similar strong shock solution serves as initial condition. It turns out that the exponent which determines the time-dependent addition of energy at the shock front is limited for physical reasons. The computed far-field solutions expand the temporal scope of the self-similar solution domain, which has been the main subject of the classical literature, into the non-self-similar domain at late time. The differences between the solutions obtained for real gas and perfect gas are less significant than in the case of the classical point explosion.

Riemann problems for the two-dimensional unsteady transonic small disturbance equation

Abstract: We study a two-parameter family of Riemann problems for the unsteady transonic small disturbance (UTSD) equation, also called the two-dimensional Burgers equation, which is used to model the transition from regular to Mach reflection for weak shock waves The related initial-value problem consists of oblique shock data in the upper half-plane, with two parameters a and b corresponding to the slopes of the initial shock waves The study of quasi-steady solutions leads to a problem that changes type when written in self-similar coordinates The problem is hyperbolic in the region where the flow is supersonic, and elliptic where the flow is subsonicIn this paper we give a complete description of the flow in the hyperbolic region by resolving the hyperbolic wave interactions in the form of quasi-one-dimensional Riemann problems In the region of physical space where the flow is subsonic, we pose the related free-boundary problems and discuss the behavior of the subsonic solution using results from our previo

Propagation and reflection of shock waves

Abstract: Structure and basic properties of shock waves in gases shock wave propagation through a gas interaction of a plane shock wave with disturbances and stability of shock waves reflection of a shock wave from a smooth body of an arbitrary shape reflection of a shock wave from a concave body and shock focusing propagation of shock waves through a weakly ionized plasma.

Gradients at a curved shock in reacting flow

Abstract: 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.

Nonclassical Dense Gas Flows for Simple Geometries

Abstract: Two-dimensional shock wave reflection and refraction phenomena for dense gases in the high pressure and density region near the thermodynamic critical point are compared and contrasted to shock phenomena over well-known configurations for dilute, perfect gases. Both transient and steady wave fields are simulated using a time-accurate, predictor-corrector total variation diminishingscheme that solves the Euler equations incorporating the van der Waals thermodynamic model. Detailed displays of wave field structure for both perfect gas and dense gas flowfields are shown. Nonclassical wave phenomena, such as disintegrating shocks, expansion shocks, and shock-fan composite waves for wedges, ramps, circular arcs, and other geometries, demonstrate significant differences in dense gas flowfields from those of perfect gases

Shock wave propagation in a branched duct

Abstract: The propagation of a planar shock wave in a 90° branched duct is studied experimentally and numerically. It is shown that the interaction of the transmitted shock wave with the branching segment results in a complex, two-dimensional unsteady flow. Multiple shock wave reflections from the duct's walls cause weakening of transmitted waves and, at late times, an approach to an equilibrium, one-dimensional flow. While at most places along the branched duct walls calculated pressures are lower than that existing behind the original incident shock wave, at the branching segment's right corner, where a head on-collision between the transmitted wave and the corner is experienced, pressures that are significantly higher than those existing behind the original incident shock wave are encountered. The numerically evaluated pressures can be accepted with confidence, due to the very good agreement found between experimental and numerical results with respect to the geometry of the complex wave pattern observed inside the branched duct.

A computational study of shock speeds in high-performance shock tubes

Abstract: This paper describes U2DE, a finite-volume code that numerically solves the Euler equations. The code was used to perform multi-dimensional simulations of the gradual opening of a primary diaphragm in a shock tube. From the simulations, the speed of the developing shock wave was recorded and compared with other estimates. The ability of U2DE to compute shock speed was confirmed by comparing numerical results with the analytic solution for an ideal shock tube. For high initial pressure ratios across the diaphragm, previous experiments have shown that the measured shock speed can exceed the shock speed predicted by one-dimensional models. The shock speeds computed with the present multi-dimensional simulation were higher than those estimated by previous one-dimensional models and, thus, were closer to the experimental measurements. This indicates that multi-dimensional flow effects were partly responsible for the relatively high shock speeds measured in the experiments.

Development of a modified Runge-Kutta scheme with TVD limiters for the ideal two-dimensional MHD equations

Abstract: A fourth-order modified Runge-Kutta scheme augmented with TVD models have been developed to solve the ideal two-dimensional MHD equations. The numerical scheme is applied to supersonic flow within a channel including compression and expansion corners in the current investigation. The flowfield includes oblique shocks, expansion waves, shock reflections, shock/expansion interactions, and secondary wave patterns due to magnetic field. The solutions are compared to analytical solutions when available. The numerical scheme is shown to be accurate with ability to capture physical phenomena occurring in the flowfield without any or with minimum oscillations in the solution.

Mach reflection wave configuration in two-dimensional supersonic jets of overexpanded nozzles

Abstract: The Mach reflection wave configurations and the flowfields associated with two-dimensional supersonic free jets of overexpanded nozzles are analysed. Based on the two- and the three-shock theories along with the classical gasdynamics theory, an analytical model for calculating the height of Mach stem and the cell size of the jet are proposed

Method and system for treatment with acoustic shock waves

Abstract: A method and a system for the treatment of a target area within the body of a human being or animal with acoustic shock waves, consisting of the step of generating shock waves outside the body, passing the shock waves through at least some body tissue, focusing the shock waves on the target area, putting at least some of the body tissue through which the shock waves pass under a positive pressure which is greater than the surrounding air pressure and a system.

Shock induced turbulence in composite materials at moderate Reynolds numbers

Abstract: Numerical simulation is used to study the turbulence generated by the passage of strong shocks (typical Mach number 7.3) through an inhomogeneous fluid at moderate Reynolds numbers. Before passage of the shock, the material consists of mass-density inhomogeneities embedded in a background fluid. The entire system is initially at uniform temperature, pressure, and number density, with the nonuniform mass density resulting from differing mass species in different regions. In the present application, the substances are treated as ideal gases, though in the motivating physical problems they are more complex materials. The shock retains its identity and a sharp front, but leaves behind it a turbulent state whose locally averaged properties only slowly become spatially uniform. The shock acquires a turbulent “thickness” (the linear dimension of the nonuniform region behind the shock front) that seems ultimately damped by viscous and thermally conducting properties that are dependent on transport coefficients and (highly uncertain) Reynolds numbers. Typically, the turbulence is highly compressible, with comparable mean divergences and curls in the velocity field, and fractional rms density fluctuations of the order of 0.25 in the parameter ranges studied. The rms vorticity generated can be estimated reasonably well from dimensional considerations. The effect of the high density inhomogeneities is primarily to create a wide region of compressible turbulence behind the shock. The inhomogeneities create both a succession of reflected shocks and considerable vorticity.

Direct Acceleration of Pickup Ions at the Solar Wind Termination Shock: the Production of Anomalous Cosmic Rays

Abstract: We have modeled the injection and acceleration of pickup ions at the solar wind termination shock and investigated the parameters needed to produce the observed Anomalous Cosmic Ray (ACR) fluxes. A non-linear Monte Carlo technique was employed, which in effect solves the Boltzmann equation and is not restricted to near-isotropic particle distribution functions. This technique models the injection of thermal and pickup ions, the acceleration of these ions, and the determination of the shock structure under the influence of the accelerated ions. The essential effects of injection are treated in a mostly self-consistent manner, including effects from shock obliquity, cross-field diffusion, and pitch-angle scattering. Using recent determinations of pickup ion densities, we are able to match the absolute flux of hydrogen in the ACRs by assuming that pickup ion scattering mean free paths, at the termination shock, are much less than an AU and that modestly strong cross-field diffusion occurs. Simultaneously, we match the flux ratios He$^{+}$/H$^{+}$ or O$^{+}$/H$^{+}$ to within a factor $\sim 5$. If the conditions of strong scattering apply, no pre-termination-shock injection phase is required and the injection and acceleration of pickup ions at the termination shock is totally analogous to the injection and acceleration of ions at highly oblique interplanetary shocks recently observed by the Ulysses spacecraft. The fact that ACR fluxes can be modeled with standard shock assumptions suggests that the much-discussed "injection problem" for highly oblique shocks stems from incomplete (either mathematical or computer) modeling of these shocks rather than from any actual difficulty shocks may have in injecting and accelerating thermal or quasi-thermal particles.

Accurate computations of hypersonic flows using ausmpw+ scheme and shock-aligned grid technique

Abstract: There are two problems in computation of hypersonic flow. One is the inaccuracy of the physical modeling and the other is an error due to inaccurate numerical dissipation. Until now the distribution of species can not calculate accurately in the region of the strong interaction between vibration and chemical reaction. The computation in large expansion region such as nozzle is also inaccurate due to the inaccurate reaction rate coefficient. From the view of numerical difficulty, a stiff pressure discontinuity may cause large numerical oscillations and degenerate robustness and convergence. Numerical dissipation is especially sensitive in boundary layer since it can easily contaminate physical viscosity, leading to the inaccurate resolution of skin friction or heat transfer coefficients. In this paper we focus on the issue of numerical discretization. A newly improved scheme of AUSMPW, so called AUSMPW+, and ShockAligned Grid Technique (SAGT) are proposed for an accurate computation of hypersonic flows. Compared to AUSMPW, AUSMPW+ scheme is more efficient to implement while it maintains the same level of robustness and accuracy in capturing the stiff pressure discontinuity or boundary layer. SAGT combined with AUSMPW+ can capture the oblique shock as well as the normal shock with little numerical dissipation.

Nonlinear Strong Shock Interactions: A Shock-Fitted Approach

Abstract: This paper addresses nonlinear effects which result from the interaction of shock waves with vortices. A series of experiments are carried out, which involve the interaction of a strong shock wave with a single plane vorticity wave and a randomly distributed wave system. These experiments are first conducted in the linear regime to obtain a mutual verification of theory and computation. They are subsequently extended into the nonlinear regime. A systematic study of the interaction of a plane shock wave and a single vortex is then conducted. Specifically, we investigate the conditions under which nonlinear effects become important, both as a function of shock Mach number, M1, and incident vortex strength (characterized by its circulation Γ). The shock Mach number is varied from 2 to 8, while the circulation of the vortex is varied from infinitesimally small values (linear theory) to unity. Budgets of vorticity, dilatation, and pressure are obtained. They indicate that nonlinear effects become more significant as both the shock Mach number and the circulation increase. For Mach numbers equal to 5 and above, the dilatation in the vortex core grows quadratically with circulation. An acoustic wave propagates radially outward from the vortex center. As circulation increases, its upstream-facing front steepens at low Mach numbers, and its downstream-facing front steepens at high Mach numbers. A high Mach number asymptotic expansion of the Rankine--Hugoniot conditions reveals that nonlinear effects dominate both the shock motion and the downstream flow for ΓM1 > 1.