Shock Reflection Research Articles

Nonexistence of the von Neumann Reflection Configuration for the Triple Point Paradox

In an attempt to resolve the von Neumann triple point paradox, a reflection, called von Neumann reflection (vNR), was reported by Colella and Henderson [J. Fluid Mech., 213 (1990), pp. 71–94] in investigating numerically the weak shock reflection. They reported that in this type of reflection configuration the incident shock and the Mach stem are seen to be smoothly merged and the reflected disturbance is not a shock but a curved band of self-similar compression wave. One should like to know whether or not the vNR configuration is a mathematically possible flow pattern. In this paper we prove that the reflection configuration in which the Mach shock is smoothly merged into the incident shock at a point and the wave behind this point is smooth is a mathematically impossible flow pattern for the two-dimensional (2D) self-similar potential flow equation and the 2D self-similar Euler equations.

Numeric study of flow separation and shock reflection hysteresis in planar nozzles

A numerical study of the flow in overexpanded planar nozzles shows the existence of Mach reflection hysteresis inside an ideal nozzle while in tapered (constant angle) nozzles it does not appear. When including the geometry of the nozzle in the simulation it becomes evident that flow separation will occur before the transition from regular to Mach reflection for all relevant Mach numbers. The simulation reveals complex changes in the flow structure as the pressure ratio between the ambient and the stagnation is increased and decreased. Detailed examination of the pressure in the region of flow separation reveals an interaction between the shock cell structure and the location of the separation point. The pressure along the nozzle wall downstream of the separation point was found to be less than the ambient pressure with the affect being more pronounced in the case of the ideal nozzle. The formation of closed circulation bubbles may generate reverse flow separation.

High order weighted essentially non-oscillatory WENO-Z schemes for hyperbolic conservation laws

In [10], the authors have designed a new fifth order WENO finite-difference scheme by adding a higher order smoothness indicator which is obtained as a simple and inexpensive linear combination of the already existing low order smoothness indicators. Moreover, this new scheme, dubbed as WENO-Z, has a CPU cost which is equivalent to the one of the classical WENO-JS [2], and smaller than that of the mapped WENO-M, [5], since it involves no mapping of the nonlinear weights. In this article, we take a closer look at Taylor expansions of the Lagrangian polynomials of the WENO substencils and the related inherited symmetries of the classical lower order smoothness indicators to obtain a general formula for the higher order smoothness indicators that allows the extension of the WENO-Z scheme to all (odd) orders of accuracy. We further investigate the improved accuracy of the WENO-Z schemes at critical points of smooth solutions as well as their distinct numerical features as a result of the new sets of nonlinear weights and we show that regarding the numerical dissipation WENO-Z occupies an intermediary position between WENO-JS and WENO-M. Some standard numerical experiments such as the one dimensional Riemann initial values problems for the Euler equations and the Mach 3 shock density-wave interaction and the two dimensional double-Mach shock reflection problems are presented.

Specular refection at a non-stationary shock: A simple model

Analytic treatments of a particle encountering a collisionless shock have commonly been based on the assumption that the shock surface is quasi-planar with length scales larger than the particle gyroradius. Within this framework, the particle distribution function width is supposed to be conserved in any shock reflection process. It is well known, however, that the thermal energy associated with backstreaming ions upstream of Earth's bow shock is significantly larger than the incident solar wind thermal energy. In a previous study, we found that non-thermal features of ions reflected quasi-adiabatically can be accounted for by considering the effect of small, normally distributed fluctuations of the shock normal over short temporal or spatial scales. The strong dependence of the particle acceleration on shock geometry leads to an increase in the temperature and to a non-thermal tail. Here, we conduct a similar analysis to investigate the effects of small, normally distributed fluctuations in the shock normal direction for specularly reflected ions. This later mechanism is considered of first importance in the dissipation process occurring at quasi-perpendicular shocks. We have derived the probability distribution functions f ( v ∥ ) and f ( v ⊥ ) of ions issued from a specular reflection of incident solar wind in the presence of normal direction fluctuations. These distributions deviate weakly from a Maxwellian, in agreement with the observations. In particular, a qualitative agreement with the ion thermal energy is obtained for fluctuations of the normal orientation in the 5–8° range about the nominal direction. Also, we have found that the shock θ B n has a weak effect on the shape of the distribution. While, not a strong determinant of the reflected distribution characteristics, the dynamical shock structure at ion scales cannot be ignored when accounting for the shock-accelerated particle thermal energy.

A multidimensional positive definite remapping for Lagrangian solutions of the Noh problem

A remapping based on the multidimensional positive definite advection transport algorithm (MPDATA), implemented for ALE methods, is used to model the Noh problem. Typical solutions in the Lagrangian reference frame contain heating errors which arise during the simulation of a shock reflection originating at a wall. The paper shows that the inherent properties of MPDATA can be exploited in the remapping to reduce wall heating errors. The resulting increase in accuracy and symmetry of solutions is demonstrated.

Existence and regularity of the solution of a mixed boundary value problem for the Keldysh equation with a nonlinear absorption term

The Keldysh equation is a more general form of the classic Tricomi equation from fluid dynamics. Its well-posedness and the regularity of its solution are interesting and important. The Keldysh equation is elliptic in y > 0 and is degenerate at the line y = 0 in R 2 . Adding a special nonlinear absorption term, we study a nonlinear degenerate elliptic equation with mixed boundary conditions in a piecewise smooth domain—similar to the potential fluid shock reflection problem. By means of an elliptic regularization technique, a delicate a priori estimate and compact argument, we show that the solution of a mixed boundary value problem of the Keldysh equation is smooth in the interior and Lipschitz continuous up to the degenerate boundary under some conditions. We believe that this kind of regularity result for the solution will be rather useful.

Experimental and Numerical Modeling of the Bow Shock Interaction with Pulse-Heated Air Bubbles

The influence of various parameters characterizing the single- and double-pulse energy deposition (ED) upstream of a blunt body (distance between the deposition point and the body, amount of the energy and time delay between the pulses) on the topology of the supersonic flow at Mach 2 is studied experimentally and numerically. The obtained pressure-time diagram at the upstream stagnation point of the body as well as some significant topological properties of the bow-shock / heated-bubble interaction, like shock deformation and reflection, as well as the evolution of the heated bubble into the vortex ring downstream, are analyzed as resulting from the shock-decelerated spherical light-gas inhomogeneity. The evolution and topology of the interfering heated bubbles by double-pulse ED show the significance of vorticity generation initiated by blast-waves coming from the neighboring bubbles.

Velocimetry Measurements of Unstart of an Inlet-Isolator Model in Mach 5 Flow

The dynamics of unstart in a floor-mounted inlet-isolator model in a Mach 5 flow are investigated experimentally using particle image velocimetry and fast-response wall pressure measurements. The inlet compression is obtained with a 6-deg ramp and the isolator is a rectangular straight duct that is 25.4 mm high by 50.8 mm wide by 242.3 mm long. Unstart is initiated from the scramjet mode (fully supersonic in the isolator) by deflecting a motorized flap at the downstream end of the isolator. With the flap fully down, the particle image velocimetry data of the started flow capture the characteristics of the isolator boundary layers and the initial inlet reflected shock system. During unstart, the unstart shock system propagates upstream through the inlet-isolator. The particle image velocimetry data reveal a complex, three-dimensional flow structure that is strongly dependent on viscous mechanisms. Particularly, the unstart shock system propagates upstream and induces significant boundary-layer separation. Side-view particle image velocimetry data show that the locations of strongest separation during unstart correlate with the impingement locations of the initial inlet shock as it reflects down the isolator. For example, in the middle of unstart, the unstart shock system is associated with massive separation of the ceiling boundary layer that begins where the first inlet shock reflection impinges on the ceiling. The observation that separation increases at the inlet shock reflection impingement locations is likely due to the fact that the boundary layers in these locations are subject to larger adverse pressure gradients, thus making them more susceptible to separation. During the unstart process, large regions of separated flow form near the floor and ceiling with reverse flow velocities up to about 0.4U ∞ . These regions of separated, subsonic flow appear to extend to the isolator exit, creating a path by which the isolator exit boundary condition can be communicated upstream. Plan-view particle image velocimetry data show the unstart process begins with separation of the isolator sidewall boundary layers. Overall, the unstart flow structure is highly three-dimensional.

Computation of weak steady shock reflections by means of an unstructured shock-fitting solver

Weak steady Mach reflections are numerically simulated using unstructured grids by means of either “shock-fitting” or “shock-capturing” techniques. It is shown that shock-fitting allows using coarser meshes than those required by shock-capturing, since the latter needs mesh refinement in the direction normal to the discontinuities, which is not needed using the former approach. The shock-fitted solution is also free from the numerical disturbances that arise along the captured discontinuities and pollute the captured solution in the smooth flowfield region. Finally, the shock-fitting solutions show the presence of a small region next to the Mach stem and the reflected shock downstream of the triple point, characterized by very high gradients.

Effects of Wavy Surface Roughness on the Performance of Micronozzles

Recent trends in small-scale (~1 dm 3 ) satellites motivate the further development of microscale propulsion subsystems. In the present paper, we focus on flow dynamics simulations of conical convergent-divergent micronozzles and on the increased importance of wall effects due to the decrease in the characteristic length of such small systems. The inefficiency associated with viscous losses due to the developing boundary layer and the effect of sinusoidal surface roughness due to the employed microelectromechanical-system fabrication techniques are studied through computational fluid dynamics simulations for nonturbulent, nonrarefied flow conditions. Depending on the specific nature of the surface roughness, the formation and reflection of several weak shocks and, as a consequence, a decreased performance are observed. © 2010 by the American Institute of Aeronautics and Astronautics, Inc.

Shock-induced ignition with single step Arrhenius kinetics

Shock-initiated ignition is studied numerically for single step Arrhenius kinetics. This is relevant to hydrogen safety, because hydrogen detonates easily, and detonation in shocked mixture may occur in deflagration to detonation transition scenarios, due to shock reflections on obstacles subsequent to flame acceleration. Simulation of ignition behind a shock moving into combustible mixture is difficult because of the singular nature of the initial conditions. The solution method includes two components. First, space as an independent variable is replaced by the ratio space over time, and second, initial conditions at a small non-zero time are used, obtained in closed form from short time asymptotics. This way, the initial singularity is effectively removed and the early process is well resolved. This method was used to study how the leading shock strength, and the heat release, affects the shock-initiated ignition process. The Essentially Non-Oscillatory algorithm used captures ignition, hot spot growth, birth of a secondary shock and transition into a detonation. Results show that for weaker shock cases as well as for higher heat release the evolution is more rapid, that a secondary shock forms closer to the contact surface and quickly becomes a detonation wave.

RESISTIVE MAGNETOHYDRODYNAMIC SIMULATIONS OF RELATIVISTIC MAGNETIC RECONNECTION

Resistive relativistic magnetohydrodynamic (RRMHD) simulations are applied to investigate the system evolution of relativistic magnetic reconnection. A time-split Harten--Lan--van Leer method is employed. Under a localized resistivity, the system exhibits a fast reconnection jet with an Alfv\'{e}nic Lorentz factor inside a narrow Petschek-type exhaust. Various shock structures are resolved in and around the plasmoid such as the post-plasmoid vertical shocks and the "diamond-chain" structure due to multiple shock reflections. Under a uniform resistivity, Sweet--Parker-type reconnection slowly evolves. Under a current-dependent resistivity, plasmoids are repeatedly formed in an elongated current sheet. It is concluded that the resistivity model is of critical importance for RRMHD modeling of relativistic magnetic reconnection.

A CONTINUOUS, TWO-WAY FREE BOUNDARY IN THE UNSTEADY TRANSONIC SMALL DISTURBANCE EQUATIONS

We formulate a problem for the unsteady transonic small disturbance equations which describes a situation analogous to the reflection of a weak shock off a wedge, with the incident shock replaced by an incident rarefaction. We linearize this problem and solve it exactly, and we compute a numerical solution of the full nonlinear problem. The solution of this problem has several features in common with the solution of the weak shock reflection problem, known as Guderley Mach reflection. In both cases, a rarefaction wave reflects off a sonic line and forms a transonic shock. There is transonic coupling between the supersonic and subsonic regions across the sonic line and shock. In both situations, this sonic line/shock can be considered a free boundary in the formulation of a new type of free boundary problem which has not previously been formulated or analyzed. The free boundary problem that arises in the context of the problem considered here is, however, simpler than the free boundary problem that arises in the weak shock reflection problem.

Effects of Fine Water Mist on a Confined Blast

A computational, multi-phase, model has been developed to study the interactions between water droplets and radial expansion of a gas cloud in a spherical chamber. Initial conditions for the gas cloud are specified based on chemical equilibrium calculations for the detonation of a high explosive (RDX). Mono-dispersed water droplets are injected at uniform concentration into the chamber prior to the expansion. A Lagrangian model is used to track the breakup of the parent drops near the shock front to form child drops, which are extremely small. The Navier–Stokes solutions show that the child droplets accumulate near the shock front and evaporate at 100 times higher rate than the parent droplets. Latent heat absorption is the dominant mechanism followed by the sensible heat absorption by the water vapor (and droplets), and momentum absorption from the high velocity gases by the child droplets. The simulations also show that the water vapor formed by the evaporation increases the gas density at the shock front. The increased density and reduced gas temperature (cooling) have opposite effects on the pressure at the shock front. This leads to only a modest suppression in the pressure. At realistic droplet concentrations (0.08 kg droplets/m3 of air), the water mist is shown to evaporate completely in a short time (2.42 ms) prior to shock reflection at the chamber wall mainly due to the breakup at the shock front. High concentrations of mist may be desirable, but are difficult to achieve in practice at the total flooding conditions.

Simulation of shock-initiated ignition

The scenario of detonative ignition in shocked mixture is significant because it is a contributor to deflagration to detonation transition, for example following shock reflections. However, even in one dimension, simulation of ignition between a contact surface or a flame and a shock moving into a combustible mixture is difficult because of the singular nature of the initial conditions. Initially, as the shock starts moving into reactive mixture, the region filled with reactive mixture has zero thickness. On a fixed grid, the number of grid points between the shock and the contact surface increases as the shock moves away from the latter. Due to initial lack of resolution in the region of interest, staircasing may occur, whereby the resulting plots consist of jumps between few values a few grid points, and these numerical artifacts are amplified by the chemistry which is very sensitive to temperature, leading to unreliable results. The formulation is transformed, replacing time and space by time and space over time as the independent variables. This frame of reference corresponds to the self-similar formulation in which the non-reactive problem remains stationary, and the initial conditions are well-resolved. Additionally, a solution obtained from short time perturbation is used as initial condition, at a time still short enough for the perturbation to be very accurate, but long enough so that there is sufficient resolution. The numerical solution to the transformed problem is obtained using an essentially non-oscillatory algorithm, which is adequate not only for the early part of the process, but also for the latter part, when chemistry leads to appearance of a shock and eventually a detonation wave is formed. A validation study was performed and the results were compared with the literature, for single step Arrhenius chemistry. The method and its implementation were found to be effective. Results are presented for values of activation energy ranging from mild to stiff.

Shock wave interaction with convex circular cylindrical surfaces

The reflection of shock waves off cylindrical surfaces of different radii is examined with the help of time-resolved flow visualization. The primary diagnostic is a newly developed technique based on the tracking of deliberately introduced small perturbations in the flow. The main focus of the investigation is to determine at which position the shock wave receives information about the shape of the wall that it reflects off. In the pseudo-steady shock reflection off a plane wall, it is commonly accepted that the reflection changes from regular to irregular as soon as sonic signals generated behind the reflection point catch up with the reflection point, and the common interpretation is that this corresponds to the transition from regular to Mach reflection (the so-called sonic criterion). The results obtained here for convex circular surfaces show that this ‘catch-up’ condition occurs at wall angles considerably higher than in the plane wall case, while a visible Mach stem occurs only further along the surface at wall angles distinctively lower than those for plane walls. Tests are also conducted on a surface where a cylindrical portion is followed by a fixed angle plane section. The Mach numbers are chosen to be on either side of the plane wall transition condition so as to examine the adjustment from reflection on the cylindrical portion to that on the plane wall. These tests confirm that the wall angle at which sonic ‘catch-up’ to the reflection point occurs on the cylindrical surface is much higher than the corresponding wall angle predicted by the sonic criterion for a plane wall. While the transition from regular to irregular reflection is not the main concern of this contribution, the present results show that the transition criteria developed for steady and pseudo-steady flows are only of limited use in the fully unsteady flows such as the one investigated here.

Langmuir waves upstream of interplanetary shocks: Dependence on shock and plasma parameters

We have examined 178 interplanetary shocks observed by the Wind spacecraft to establish which shock and plasma parameters are favorable for the production of upstream Langmuir waves and therefore to determine which shocks are likely to generate interplanetary Type II radio bursts. Of the 178 shocks included in this study, 43 produced upstream Langmuir waves, as evinced by enhancements in wave power near the plasma frequency. The large number of observed shocks permits the use of statistical tests to determine which parameters control the upstream activity. The best predictor of activity is the de Hoffmann‐Teller speed, a result consistent with the fast Fermi model of electron acceleration. Several other parameters, including the magnetic field strength and the level of solar activity (but not the Mach number), are also correlated with upstream activity. These additional parameters may be associated with an increased level of shock front curvature or upstream structure, leading to the formation of upstream foreshock regions, or with the generation of an upstream electron population favorable for shock reflection.

Transnational American Studies in the Digital Age

Transnational American Studies in the Digital Age YANOULA K. ATHANASSAKIS and ERIC LARS MARTINSEN Technology is moving at lightning speed, and as editors of an online journal we find ourselves trying to keep up with the pace and scope of both life and work, all of which hold so much more than we had imagined. Near the end of a talk on robotics and warfare at the University of California at Santa Barbara in January 2010, international relations scholar Peter W. Singer motioned to the stream of images behind him that displayed machines and robots resembling those of the Star Wars movies (old and new). 1 Cognizant of the audience’s awestruck reaction, he commented that, while we might assume he was speaking of the future, he was actually referencing the past. We, too, seek to use the past, and the ever‐changing present, to illuminate the future. The second issue of the Journal of Transnational American Studies, or JTAS, comes in the wake of worldwide outbreaks of natural and manmade devastation and economic depression. These events are global in scope, yet hit close to home, thereby crystallizing what the editors of JTAS are working hard to illuminate: the multivalent entangled connections between and among events and cultural phenomena across the world—or what Shelley Fisher Fishkin points to as the “transnational turn” in American Studies. 2 We write this at a critical time in American Studies, when warring ideologies seem to be drowning out reasoned public policy, and purveyors of divisive rhetoric from around the world are outshouting voices of reason and compromise. We write this at a time when it has become increasingly obvious that American bodies, American borders, and American policy are all being submerged in acts of violence both perpetrated against the US and stemming from it. 3 This turbulent epoch calls ever more for an examination and reflection of the seismic shocks that remain and have manifested themselves in the stories of our times. Theorist Judith Butler suggests in her 2004 book Precarious Life: The Powers of Mourning and Violence that most Americans “have probably experienced something

Global solutions of shock reflection by large-angle wedges for potential flow

When a plane shock hits a wedge head on, it experiences a reflectiondiffraction process and then a self-similar reflected shock moves outward as the original shock moves forward in time. Experimental, computational, and asymptotic analysis has shown that various patterns of shock reflection may occur, including regular and Mach reflection. However, most of the fundamental issues for shock reflection have not been understood, including the global structure, stability, and transition of the different patterns of shock reflection. Therefore, it is essential to establish the global existence and structural stability of solutions of shock reflection in order to understand fully the phenomena of shock reflection. On the other hand, there has been no rigorous mathematical result on the global existence and structural stability of shock reflection, including the case of potential flow which is widely used in aerodynamics. Such problems involve several challenging difficulties in the analysis of nonlinear partial differential equations such as mixed equations of elliptic-hyperbolic type, free boundary problems, and corner singularity where an elliptic degenerate curve meets a free boundary. In this paper we develop a rigorous mathematical approach to overcome these difficulties i nvolved and establish a global theory of existence and stability for shock reflection by large-angle wedges for potential flow. The techniques and ideas developed here will be useful for other nonlinear problems involving similar difficulties.

Hydrodynamic fields induced by the shock response of a fluid-filled submerged cylindrical shell containing a rigid co-axial core

The hydrodynamic fields induced by the response to an external shock wave of a system consisting of a submerged fluid-filled cylindrical shell and a rigid cylindrical co-axial core are considered. The primary focus of the study is on the complexity brought into the interaction by the presence of the core, and on the analysis of the multitude of the respective shock wave propagation and reflection phenomena. It is shown that when the core is small, its overall impact on the interaction is insignificant, although the hydrodynamic patterns observed exhibit some interesting features even in that case. As the radius of the core increases, its effect on the interaction becomes more and more pronounced, with the internal wave pattern eventually becoming dramatically different from what is observed in the no-core case. As a part of this investigation, the exact reasons for the significant reduction of the tensile stress reported for a larger core in earlier studies are identified. Finally, when the core becomes sufficiently large to dominate the internal volume, the fluid between the shell and the core starts to behave as a fluid layer, and exhibits some properties similar to those of a waveguide.