Ree Ection Research Articles

Incidental Displays of Cultural Knowledge in the Nonnative-English-Speaking Teacher's Classroom

This article examines incidental cultural knowledge displays by two nonnative-English-speaking teachers (NNESTs) in their intensive English program classrooms. Surprisingly, until recently, NNESTs have received little empirical attention in the literature, even though questions continue to arise about their language competence, pedagogical knowledge, and cultural orientation. This study goes beyond existing work on the impressions, ree ections, and beliefs of NNESTs to ask, What is the nature of the discourse produced in ESL classes taught by NNESTs? And, more to the point, does an analysis of this discourse suggest real problems with language, teaching, or culture? An analysis of videotaped classroom data of two teachers indicated that a wide and unpredictable range of cultural topics arose. Although with one exception they were able to deal with the cultural topics by displaying knowledge in a competent manner, the analysis suggested missed opportunities in that the teachers did not admit “ I don’ t know,” thus providing the chance for their students to become cultural informants in the classroom.

Propagation properties of inertia–gravity waves through a barotropic shear layer and application to the Antarctic polar vortex

AbstractThe propagation of inertia–gravity waves (IGWs) through a dynamical transport barrier, such as the Antarctic polar vortex edge is investigated using a linear wave model. The model is based on the linearized, inviscid hydrostatic equations on an f‐plane. Typical values for the parameters that are appropriate to the Antarctic polar vortex are given. The background flow U is assumed to be barotropic and its horizontal shear is represented by a hyperbolic tangent background wind profile. The wave equation that describes the latitudinal structure of a monochromatic disturbance contains two singularities. The first corresponds to the occurrence of a critical level where the intrinsic wave frequency Ω=ω−kU becomes zero. ω is the absolute wave frequency and k its longitudinal wave number in the direction of U. The second is an apparent singularity and does not give rise to singular wave behaviour. It becomes zero whenever the square of the intrinsic wave frequency Ω2=f(f−Uy), f being the Coriolis frequency and Uy the horizontal shear of the flow. The wave equation is solved numerically for different values of the angles of incidence of the wave upon the background flow, of the wave frequency, of the horizontal wave number and of the Rossby number. Reflection (|R|) and transmission (|T|) coefficients are determined as a function of these parameters. The results depend on whether the flow is inertially stable or not. They also depend on the presence and location of the turning levels, where the wave becomes evanescent, with respect to the location of the Ω‐critical levels. For inertially stable flows, the wave totally reflects at the turning level and never reaches the critical level. If the background flow is inertially unstable, turning levels can disappear and the wave can now reach the critical level. Then over‐reflection, over‐transmission and absorption can occur. Copyright © 2003 Royal Meteorological Society

Two-Dimensional Model for Simulating Shock-Wave Interaction with Rigid Porous Materials

A two-dimensional numerical model for predicting the characteristics of the e owe eld during an unsteady interaction between a shock wave moving through air and a rigid saturated porous matrix is developed. The numerical modelincludesa Lagrangianschemecombined with aremeshand an interfacetrackingtechnique.Thepredictions of the numerical simulations are compared with the predictions of a two-dimensional analytical model for regular ree ection from a rigid porous surface in pseudosteady e ow. The numerical and the physical model are also validated by comparing the one- and two-dimensional predictions of the model with experimental results. In general the predictions of the numerical simulations reveal very good to excellent agreement with both the experimental results and the predictions of the analytical model. Nomenclature C® = specie c heat capacities ratio for the ® phase c® = shape factor for the ® phase

Optimal Control Law for Interplanetary Trajectories with Nonideal Solar Sail

An indirect method is applied to minimize the trip time of interplanetary and escape missions using solar sails. A e at sail and specular ree ection of the incident radiation is assumed. The theory of optimal control provides the equation that gives the optimal orientation of the sail at each point of the trajectory. An analytical solution can be obtained in the ideal case (unit ree ectivity ) when all of the incident radiation is specularly ree ected. The equation is numerically solved when the incident radiation is partially absorbed (nonunit ree ectivity ). Once the ree ectivity has been assumed, the optimal trajectory only depends on the sail characteristic acceleration. The optimization procedure has been used to estimate the performance of different missions; the ine uence of the sail ree ectivity is in particular pointed out. Transfers to Mars and Mercury and escape from the solar system are presented.

Towards Effective Transnational Labor Solidarity between NAFTA North and NAFTA South

U.S. political scientist Dale Hathaway’s article originated in his decision in the mid1990s to tackle the internationalist tasks integral to radical politics in the post-NAFTA (North American Free Trade Agreement) era. Like his book Allies across the Border (2000), it breathes a spirit of militancy and heart-felt commitment to the struggles of working and middle class peoples in today’s world. As bee ts the new Labor History’s forum format, the piece eschews cautiously balanced judgments anchored in an impregnable structure of academic argument. Instead, Hathaway offers bold if schematic political ree ections about a pressing issue of general importance—the unionization of “runaway” shops in Mexico—and a straightforward partisan advocacy for one voice, the Frente Autentico del Trabajo (FAT), within Mexico’s massive and heterogeneous labor movement.

Numerical Study of Blast-Wave Propagation in a Double-Bent Duct

Ahigh-resolution Eulersolverhas been used to investigatethee owe eld induced by a blast wavein a double-bent duct. A blast-wave/vortex interaction and shock/shock interactions are also studied. A pair of vortices with an opposite rotating direction is found. The trajectories of the induced main vortices are explored for different initial pressure ratios. It was found that thevortex could deform theshock front of the blast wave becauseof compression and expansion involved in their interaction. Moreover, after blast-wave diffraction around the e rst convex corner a locally maximum wall pressure could occur at the second concave corner, and a globally maximum wall pressure could occur at the e rst concave corner, resulting in a large loading at the compression corners. NGINEERS and scientists are quite interested in the problem of blast-wave propagation in ducts because the phenomena of blast-wave propagation often take place,for example,in the exhaust pipes of internal combustion engines or, caused by explosions, in- side a building. A fundamental study of blast-wave propagation in a duct is helpful in the understanding of the blast-wave dynamics. In this study we investigated the unsteady e ow of planar blast-wave propagation in a double-bent duct with two 90-deg turns. The prob- lem involved a complicated unsteady e owe eld of blast-wave/vortex interactions, shock/shock interactions, and blast-wave ree ections. Inthepasttherewereveryfewpapersthatreportedthedetailofthe e owe eldinducedbyablastwaveinadouble-bentduct.Somerelated papers are referred to here. Howard and Mathews 1 experimentally investigated the vortex generation caused by shock-wave diffrac- tion around a corner. Schardin 2 studied a ree ected shock/vortex interaction by using the shadowgraph method. His experimental results provided basic understanding of shock/vortex interaction. Mandella et al. 3 conducted a series of experiments for studying the vortex induced by a shock wave near a corner. They reported that the vortex generation was not caused by the e uid viscosity. Sivier et al. 4 numerically simulated the problem studied by Schardin and obtained good agreement between the numerical and experimental results. Igra et al. 5 used both experimental and numerical methods to study the intensity of a diffracted wave in a branched duct and concluded that the e ow behind the diffracted wave could not be assumed to be quasi-one-dimensional. Ellzey and Henneke 6 inves- tigated the sound generation caused by the shock/vortex interaction and its mechanism. They found that the shock-wave deformation and a compression effect involved in the interaction are the two main factors for sound generation. Grasso and Pirozzoli 7 numeri- callystudiedashock/vortexinteractionandclassie edtheinteraction intothree types.They also characterized theshockandvortexdefor- mationsandthemechanismofsoundgeneration intheinteractionof a cylindrical shock wave with a vortex. The numerical scheme they used is a weighted essentially nonoscillatory (WENO) scheme of

Detection of Multiple Damages by Prestack Reverse-Time Migration

An approach to detect and image multiple damages in a platelike structure is presented. A structural health monitoring system (SHMS) with a linear array of actuators/sensors is proposed. The integrated actuators/sensors are used to activate/receive lowest-order antisymmetric A0 mode Lamb waves. A migration technique used in geophysical exploration and seismic prospecting is adopted to interpret the backscattering wave e eld and to image thee awsinthestructure.Theproposed approachismodeledbya two-dimensional explicite nitedifferencemethod both in simulating the ree ection waves and in implementing the prestack migration. An analytical solution based on Mindlin plate theory (Mindlin, R. D., “ Ine uence of Rotary Inertia and Shear on Flexural Motions of Isotropic, Elastic Plates,” Journal of Applied Mechanics , Vol. 18, No. 1, 1951, pp. 31 ‐38) is derived to verify the accuracy of the numerical algorithm. An excitation-time imaging condition specie cally for the migration of waves in a plate is introduced based on a ray tracing concept, and prestack reverse-time migration is proceeded to propagate the ree ection energy back to the damages. The plate is imaged in terms of the velocity of transverse deformation after migration,thus, thelocations, dimensions, and seriousness of thee awscan bevisually displayed. Numerical results show that multiple damages can be successfully detected and the image of the damages correlate well with the target damages. The conclusion is that prestack migration can be a prospective technique in SHMS applications.

Numerical Study of Cylindrical Blast-Wave Propagation and Reflection

The problem of an unsteady cylindrical blast-wave interaction with a e at plate is numerically investigated. The numerical simulation aims at the understanding of blast-wave propagation, ree ection, and its transition phenomenonaswellasthee owfeaturesinducedbytheblastwave.Ae fth-orderweightedessentiallynonoscillatory scheme for spatial discretization associated with a fourth-order Runge ‐Kutta method for time integration is employed for solving the two-dimensional Euler/Navier ‐Stokes equations in a e nite volume fashion. To verify the accuracy of the numerical solver developed, several problems were tested. The computed results for the test problems are shown to be accurate in comparison with experimental data. To study the e owe eld of a blast wave, theproblemofblast-wavepropagation in a freee eld isexplorede rst.Twotypesofinitial conditionsareconsidered. A contact-surface instability that developed around a contact surface was found in the problem with the e rst kind of initial condition. Next, the problem of the blast-wave interaction with the e at plate is investigated. The e ow structure of a shock-shock interaction induced by the unsteady blast wave and its evolution are studied in detail. It was found that the lower the height of burst or the higher the shock Mach number, the earlier the occurrence of the transition from a regular ree ection to a Mach ree ection and the higher the triple points.

Making sense of economists: from falsification to rhetoric and beyond

To make sense of what economists do. Is there any other reason for studying economicsasa science?It wasthemain reason for me, and it stillis. Ofcourse, other reasons have presented themselves, like being able to distinguish myself from fellow economists around me, becoming a cone dant for the disgruntled among them, having the sheer pleasure of reading philosophy and an excuse for not doing econometrics, being in the company of interesting and likeminded people, getting published, being paid, slaying the dragon ofscientism, and indulgingmyree exiveinclinations.And it seemed that I had somethingto contribute to the conversation among economists. At least so I thought at e rst. Whatever, making sense of the confusing scene that economists create with their sciencehasalways seemed the best reason to justify methodological discussions and my ongoing involvement in them. When I try to make sense of developments and events in the e eld of methodology, the objective of this essay, I can’ t help but take my own involvement as vantage point, and apply the notions and insights that I have gained over the course of time. I ask the reader to forgive me this indulgence. How truthful this reconstruction is, is hard to tell. Being impressionistic and based on (deliberate and researched) ree ection the narrative can hardly do justice to everything that has happened in the e eld. Yet I hope that in these personal ruminations a more general account emanates that illustrates the usefulness of the rhetorical approach and shows the future that I see for ree exive practices in the context of economics.

Adaptive QUICK-Based Scheme to Approximate Convective Transport

A new adaptive high-order-accurate convection scheme that satise es the unsteady total variation diminishing principle has been developed. The second-order accuracy of the proposed scheme was verie ed by numerical experiments on the unsteady linear and nonlinear scalar equations. The new scheme was further applied to onedimensional unsteady compressible shock tube and steady nozzle e ows and a two-dimensional shock ree ection problem. Predicted results indicate that the present method returns a high order of accuracy and oscillation free results.

The Contributions of the Economics of Information to Twentieth Century Economics

In the e eld of economics, perhaps the most important break with the past— one that leaves open huge areas for future work— lies in the economics of information. It is now recognized that information is imperfect, obtaining information can be costly, there are important asymmetries of information, and the extent of information asymmetries is affected by actions of e rms and individuals. This recognition deeply affects the understanding of wisdom inherited from the past, such as the fundamental welfare theorem and some of the basic characterization of a market economy, and provides explanations of economic and social phenomena that otherwise would be hard to understand. I. INTRODUCTION The century coming to a close has seen vast changes in economics in both ideas and methodology. Upon ree ection, it is remarkable, however, how many of the seeds of advances in this century were sowed in the previous. I would argue that perhaps the most important break with the past— one that leaves open huge areas for future work— lies in the economics of information. The recognition that information is imperfect, that obtaining information can be costly, that there are important asymmetries of information, and that the extent of information asymmetries is affected by actions of e rms and individuals, has had profound implications for the wisdom inherited from the past, and has provided explanations of economic and social phenomena that otherwise would be hard to understand. In this essay I wish to argue that information economics has had— directly and indirectly— a profound effect on how we think about economics today. Eighteenth and Nineteenth-Century Antecedents To be sure, Marshall and other nineteenth century economists talked about problems of imperfect information. But with one exception, discussions of information were obiter dicta, caveats at the end of the analysis; they were never at the center.

Application of Bethe-Zel'dovich-Thompson Fluids in Organic Rankine Cycle Engines

Thenovel concept ofimproving organicRankinecycleengines (ORCEs)by exploiting the unusual e uid dynamic behaviorofaspecialclassofworkinge uidsknownasBethe ‐Zel’dovich‐Thompson(BZT)e uidsisexplored.ORCEs are currently manufactured and used for numerous remote terrestrial applications requiring reliable, unattended power.OneofthemajorlossmechanismsinORCEsisshockwavesgenerated intheturbinestage.Operatingunder theproperthermodynamicconditions, a BZT working e uid can potentially weaken oreliminate shock waves. This would reduce losses due to both the wave drag from shocks as well as losses from boundary-layer separation due to shock ree ections on the turbine blades. Two-dimensional e owe elds through realistic symmetric impulse turbinecascades arecompared using a e ux-limited, e nite differencenumerical schemeto assess differences in e ow efe ciencies. Resultsshowsignie cantimprovementsin turbineefe cienciesforBZTworkinge uidsoverconventional ORCE working e uids.

Excitation of Thermoacoustic Instabilities by Interaction of Acoustics and Unstable Swirling Flow

Unstablethermoacousticmodeswereinvestigatedandcontrolledinanexperimentallow-emissionswirlstabilized combustor, in which theacousticboundary conditions weremodie ed to obtain combustion instability. Theacoustic boundary conditions of the exhaust system could be adjusted from almost anechoic (ree ection coefe cient jj rjj < 0:2) to open-end ree ection. Several axisymmetric and helical unstable modes were identie ed for fully premixed and diffusion-typecombustion. Theseunstablemodeswereassociated with e ow instabilities related to therecirculation wake-like region on the combustor axis and shear-layer instabilities at the sudden expansion (dump plane). The combustion structure associated with the different unstable modes was visualized by phase-locked images of OH chemiluminescence. The axisymmetric mode showed large variation of the heat release during one cycle, whereas the helical modes showed variations in the radial location of maximal heat release. The axisymmetric mode was the dominant one during unstable combustion. It was obtained by forcing a longitudinal low-frequency acoustic resonance. Helical modes could only be obtained when the axisymmetric mode was suppressed by using a nonree ectingboundarycondition.Aclosed-loopactivecontrolsystemwasemployedtosuppressthethermoacoustic pressure oscillations and to reduce NO x and CO emissions. Microphones were used to monitor the pressure oscillations during thecombustion process and provide input to thecontrol system. An acousticactuation was used to modulate the aire ow and thus affected the mixing process and the combustion. Upstream excitation modie ed the shear-layer structure and was shown to be superior to downstream excitation, which combined less effective shear-layer excitation with noise cancellation. Suppression levels of up to 5 dB in the pressure oscillations and a concomitant 24% reduction of NO x emissions were obtained in premixed combustion using an acoustic power of less than 0.002% of the combustion power. The control of the diffusion e ame was less effective, and NO x emissions increased at the phase that was most effective in suppressing the pressure oscillations. The differences between the behavior of the control system in the two combustion modes was caused by different levels of interaction between the combustion process and the shear layer.

Modeling of Three-Dimensional Viscous Compressible Turbomachinery Flows Using Unstructured Hybrid Grids

An advanced numerical model for the simulation of steady and unsteady viscous compressible e ows for turbomachinery applications is described. The compressible Favre-averaged Navier ‐Stokes equations are used together with a one-equation turbulence model. The e ow domain is discretized using unstructured hybrid grids that can contain a mixture of hexahedral, pentahedral, tetrahedral, and triangular prismatic cells. The e ow equations are discretized using a node-centered e nite volume scheme that relies on representing the mesh using an edge-based data structure. A dual time stepping technique is applied to a point implicit formulation so that time accuracy can be maintained with large Courant ‐Friedrichs‐Lewy numbers. Nonree ecting boundary conditions are applied at the ine ow and oute ow boundaries to prevent any spurious ree ections of the outgoing waves. The model was validated against measured data for two cases. Radial proe les of pressure and temperature rise were determined from the steady e ow analysis of a rig fan blade, and these were found to be in very good agreement with the measured quantities. A rotor/stator interaction was studied next. Detailed comparisons were carried out against measured steady and unsteady e ow data and good agreement was obtained in all cases.

Experiment to Support the Formulation and Validation of Compressor-Face Boundary Conditions

The reliability of unsteady inlet e ow computations may be seriously degraded by the lack of experimentally validated compressor-face (outeow) boundary conditions. The commonly imposed oute ow conditions require a e ow variableto beconstant (pressure, velocity, Mach number,etc. )at theoute owboundary, butthereis littleif any documented evidenceto support theseassumptions, norarethey likely to actually occurduring a rapid transient of arealinlet/enginesystem.Measurementsarepresentedofacousticree ectioncoefe cientsforanoperatingmultistage compressor, a quantity appropriate for the characterization of the compressor face for computational purposes. The experiment used an impulse method, in which short-duration, large-amplitude acoustic pulses (1 ms, with a peak value of nearly 4% of the mean static pressure, respectively ) were generated in a constant-area, annular inlet. The pulse and its ree ection from the compressor face were tracked by fast-response pressure transducers. Frequency-domain analysis of the data yields transfer functions that may be thought of as frequency-resolved ree ection coefe cients. None of the currently availablecompressor-face boundary conditions accurately predict the data obtained in this study, indicating that current practices concerning oute ow boundary conditions are in need of revision.

Restricted Shock Separation in Rocket Nozzles

In overexpanded rocket nozzles the e ow separates from the nozzle wall at a certain pressure ratio of wall pressure to ambient pressure. Flow separation and its theoretical prediction have been the subject of several experimental and theoretical studies in the past decades. Two distinctive e ow separation phenomena, the freeshock and restricted-shock separation, were observed in experiments with nozzles. Both phenomena are discussed in detail, and the system of recompression shocks and expansion waves is described. For the free-shock case three different shock structures in theplume can occur, namely the regular shock ree ection, the Mach disk, or a cap-like shock pattern. Theappearanceofthesedifferentplumepatternsis discussed. Theseshock structuresareconserved for the full-e owing, but overexpanded, nozzle. Numerical results obtained for existing rocket nozzles, e.g., Space ShuttleMain EngineorVulcain, show a qualitativegood agreement with experimental photographs.Furthermore, an explanation for the appearance of restricted shock separation, which has been widely unknown up to now, is given, analyzing why and under what conditions it occurs. The type of nozzle contour strongly ine uences this form of e ow separation, and restricted shock separation also occursin full-scale, thrust-optimized rocket nozzles. Based on the results established for e ow separation, an outlook on the generation of side loads is given.

Numerical Investigation of Twin-Nozzle Rocket Plume Phenomenology

The generalized implicit e ow solver (GIFS)computerprogram has been modie ed and applied fortheanalysis of three-dimensionalreactingtwo-phasee owsimulationproblems.TheintentoftheoriginalGIFSdevelopmenteffort was to provide the joint Army, Navy, NASA, Air Force community with a standard computational methodology to simulate the complete e owe eld of propulsion systems including multiple nozzle/plume e owe eld phenomena and other three-dimensional effects. The Van Leer e ux splitting option has been successfully implemented into the existing GIFS model and provides a more robust solution scheme, making application of the model more reasonable for engineering applications. Signie cant results of several twin-nozzle/plume simulations using the GIFS code are reported. Eight simulations of Titan II plume e owe elds have been completed to assess the effects of three dimensionality, turbulent viscosity, afterburning, near-e eld shock structure, e nite rate kinetic chemistry, internozzlegeometricspacing,nozzleexitplaneproe le,andmissilebodyonthesubsequentplumeexhauste owe eld. The results of these calculations indicate that the viscous stress model; kinetic chemistry, particularly at lower altitudes; and nozzle exit proe le may be important parameters that should be considered in the analyses and the interpretation of radar signature calculations. Three dimensionality is also an important ine uence, which can substantially ine uence the interpretation of the results. If three-dimensional effects are oversimplie ed in the model, analyses of the spatial results can be misinterpreted and misapplied. In addition, the missile body effect and internozzle geometric spacing ine uence the expansion shock ree ection location that can signie cantly affect the plume/plume impingement shock location, inviscid shock structure, and shear-layer growth.

Absorption of Sound near Abrupt Area Expansions

Sound incident onto an abrupt area expansion in a channel is investigated both numerically and analytically. In the presence of a mean e ow, the incident sound leads to unsteady vortex shedding from the lip of the expansion, thereby converting acoustic into vortical energy. We use an acoustic analogy and Green' s functions to determine the sound ree ected and transmitted across the area change. We compare predictions obtained from threedifferent Green' sfunctionswithsourcetermsderivedeitherusinga simpleanalyticalmodelorfrom anumericalcalculation. Thecompact Green' s function, with zero normalderivative on theductwalls, givesthe bestresultsfora low-Mach- number e ow. This Green' s function contains a singularity at the lip of the expansion (and hence acoustic sources neartheliphavethegreatesteffect ).Thismeansthatourestimateoftheoverallvorticitye eldcanberelativelycrude, when using the compact Green' s function, provided it is accurate near the lip. Therefore, although predictions for the radiated sound e eld made using all three Green' s functions are formally correct, the solution made using the compact Green' s function is less susceptible to errors in the source terms and gives more accurate results. In addition, we e nd that thereisa Strouhal number at which sound absorption is maximized and that this absorption can be enhanced by multiple ree ections from the duct ends. Our predictions are compared with an experiment. In this paper we investigate this energy exchange for a simple two-dimensional e ow past a backward-facing step with e ow using anacousticanalogyfromHowe 14 torelatetheacousticsourcestothe shed vorticity. Initially, we consider two semi-ine nite channels of heighth and H(h < H) joinedat y1 = 0.There is a mean e ow along the channelthatseparatesatthe rearward-facing step.Wedetermine the transmission and ree ection of an incident sound wave atthe step including the effects of the mean e ow. This problem is investigated both numerically and analytically. Vortex-sound interaction is of fundamental interest in aeroacous- tics, and this simple geometry enables us to investigate solution techniques and compare theoretical results with experiment. To re- late the sources from the acoustic analogy to the sound that radiates away from the junction, we must introduce a suitable Green' s func- tion for the pipework system. The choice of Green' s function is crucial to the solution, and three different Green' s functions are cal- culated.Two ofthesearederivedfromexpansionsintermsofmodes of straight-walled semi-ine nite pipes, whereas the remaining one, the compact Green' s function, satise es boundary conditions of zero normal velocity on all rigid surfaces. The compact Green' s function has a singularity at the lip of the expansion, and so sources near the lip have a greater effect on the sound that radiates to the far e eld than those farther away. This is important because it implies that, when using the compact Green' s function, our prediction for the vorticity e eld, whether from an analytical model or from a numer- ical calculation, can be relatively crude provided that the vorticity near the lip is accurately predicted. As a result, we postulate that the most accurate results will be obtained using the compact Green' s function. The calculation of these Green' s functions is discussed in Sec. III. The vorticity e eld is obtained in two ways: using an analytical model (Sec. IV.A) and using computational e uid dynamics (CFD; Sec. IV.B) with comparison between the two. The analytical model that we use for the vorticity is that of Howe 17 and assumes that the strength of the shed vorticity is unchanged as it convects down- stream and it forms an ine nitely thin vortex sheet whose strength is determinedbytheapplicationoftheKuttacondition.Incontrast,our numerical calculation uses a crude zero-equation turbulence model and the strength of the vorticity diffuses and decays as it convects downstream as a result of numerical diffusion. Despite the apparent differences between the two models, we show in Sec. V that the radiated sound e elds, which they predict, are very similar because of the importance of the region near the lip, provided the compact Green' s function is used in the calculation of the far-e eld sound. Our initial calculations are for two semi-ine nite pipes, but pre- dictions for e nite length pipes are obtained in Sec. VI by the in- troduction of a ree ection coefe cient at the pipe exit. Solutions for e nite length channels are easily derived from the ine nite channel solutions, which isimportant because it implies that, fora particular expansion ratio and Mach number, the ine nite channel solution is all that is required; the solution for any length channel can then be deduced from this. We justify our results by comparison with an experiment and discuss the implication for the absorption of incident sound waves. In particular, we e nd, for ine nite channels, there is an optimum Strouhal number at which maximum absorption is achieved. When wave ree ection from the ends of e nite length pipes is included, channel resonances can amplify this absorption.

Numerical Study of Shock-Reflection Hysteresis in an Underexpanded Jet

Shock-ree ection hysteresis and plume structure in a low-density, axisymmetric highly underexpanded air jet is examined using a Navier ‐Stokes e ow solver. This type of jet is found in a number of applications, e.g., rocket exhausts and fuel injectors. The plume structure is complex, involving the interaction of several e ow features, making this a demanding problem. Two types of shock ree ection appear to occur in the plume, regular and Mach, depending on the jet pressure ratio. The existence of a dual solution domain where either type may occur has been predicted, in agreement with experiment where the same phenomenon has been observed for a nitrogen jet. There is a hysteresis in the shock-ree ection type; the ree ection type observed in the dual-solution domain depends on the time history of the plume development. A quasi-steady approach is employed to calculate the entire hysteresis loop. An implicit, multiblock structured, e nite volume e ow solver is used. The results of the computational study are used to examine the structure of the plume and are compared with experimental data where possible. Some e ow features not initially recognized from experiment have been identie ed, notably curvature of the Mach disk, recirculation behind the Mach disk, and the regular ree ection having Mach-ree ection characteristics.

Impedance and Its Time-Domain Extensions

The classical concept of impedance as a means to characterize the ree ectivity and absorption of a surface to the incident waves has always been a complex-valued quantity akin to plane harmonic waves and the analysis of their ree ections. A new look at this classical quantity irrespective of the type of incident waves is presented. It points out that a direct operational inversion of impedance into time-domain boundary operators will, in general, lead to an unstable system, whereas the inversion of the corresponding ree ection coefe cient will result in stable, easily implementable boundary operators for time-domain prediction of wave ree ection. It reveals the equivalent, pulselike, causal process of time convoluting the incident wave into the ree ected wave. Excellent agreement with analytical solutions derived here and from other sources supports the validity of the approach and demonstrates the accuracy of the implementations.