Shock Shape Research Articles

A simple physical theory of weak Mach reflection over plane surfaces

It is shown that simple physical principles coupled with the inviscid shock jump relations can be applied to the problem of weak Mach reflection to the extent that the triple point path can be predicted from the incident shock Mach number \(M_{\rm i}\), gas specific heat ratio \(\gamma\) and the inclination angle \(\theta_{\rm w}\) of the reflecting surface to the shock normal. Comparison with the Euler code data and with experiments show close agreement for conditions both far and close to transition and that the general shape of the reflected and Mach stem shocks follow simple curves except in the neighbourhood of the triple point. The conflict at the triple point in matching the flow deflection angles and pressures across the contact discontinuity remains. It is shown however that the simple model presented here gives a close match to the cfd and experimental overall shock and contact surface shapes although it cannot predict these or the flow properties in any detail.

Self‐consistent model of the solar wind interaction with two‐component circumsolar interstellar cloud: Mutual influence of thermal plasma and galactic cosmic rays

The interaction of the solar wind with the two‐component interstellar medium consisting of thermal plasma and galactic cosmic rays is studied numerically using a soft fitting technique based on the high‐resolution Godunov scheme and the physical process splitting method. Mutual influence of plasma and cosmic ray components on the solar wind termination is considered. It is shown that cosmic rays can considerably modify the shape and structure of the solar wind termination shock and the bow shock, and change the positions of the heliopause and bow shock if no H atoms are taken into account.

The Dynamics of the S&P 500 Implied Volatility Surface

This empirical study is motivated by the literature on “smile-consistent” arbitrage pricing with stochastic volatility. We investigate the number and shape of shocks that move implied volatility smiles and surfaces by applying Principal Components Analysis. Two components are identified under a variety of criteria. Subsequently, we develop a “Procrustes” type rotation in order to interpret the retained components. The results have implications for both option pricing and hedging and for the economics of option pricing.

Control of Pseudo-Shock by the Slot-Injection.

It is well known that the boundary layer separation, which is induced by the adverse pressure gradient of pseudo-shock, affects the dynamical feature of the pseudo-shock such as the shape of the first shock, the pressure recovery rate and the self-induced oscillation. So, it is considered that the suppression of the separation is effective on the shock wave/boundary layer interaction. In this research, the boundary layer control with the slot injection is studied experimentally and numerically. The expriments were carried out in a blow-down supersonic wind tunnel at a free stream Mach number of 2.0. Numerical simulations were made with the 2-D compressible Navier-Stokes code. In sloving the equations, 2nd-order accurate Harten-Yee's upwind TVD scheme and κ-ε turbulence model modifed for compressibility effects were used. The result shows that the separtion is appreciably suppressed and the first shock changes in its shape, so that the wall pressure fluctuation is greatly suppressed. In addition, the small slot high results in the lower total pressure loss.

Sonoluminescence: How Bubbles Turn Sound into Light

Sonoluminescence, the transduction of sound into light, is a phenomenon that pushes fluid mechanics beyond its limit. An initial state with long wavelength and low Mach number, such as is realized for a gas bubble driven by an audible sound field, spontaneously focuses the energy density so as to generate supersonic motion and a different phase of matter, from which are then emitted picosecond flashes of broad-band UV light. Although the most rational picture of sonoluminescence involves the creation of a “cold” dense plasma by an imploding shock wave, neither the imploding shock nor the plasma has been directly observed. Attempts to attack sonoluminescence from the perspective of continuum mechanics have led to interesting issues related to bubble shape oscillations, shock shape instabilities, and shock propagation through nonideal media, and chemical hydrodynamics. The limits of energy focusing that can be achieved from collapsing bubbles in the far-off equilibrium motion of fluids have yet to be determined either experimentally or theoretically.

3D multi‐fluid MHD studies of the solar wind interaction with Mars

The interaction of the solar wind with planets with no or only very weak intrinsic magnetic fields, such as Mars and Venus, involves their ionospheres. Single fluid MHD models, which incorporate some of the important mass‐loading processes, have been useful in reproducing numerous observed features at these planets, such as the location of the bow shock, but they do have obvious limitations. Our present 3D MHD model is a two‐fluid one, which considers protons and the dominant heavy ions in the ionosphere, separately. We have used this model to study the interaction processes at Mars. The model results are in general agreement with the average observed bow shock shape and position and predict reasonable locations for the ionopause. The calculated oxygen escape flux down the tail was estimated to be 2.7×1025s−1, which is consistent with Phobos‐2 estimates.

Shock-fitted multidomain solution of supersonic flows

Shock fitting is presented as a way to compute shocks with spectral methods. With fitting, spectral approximation is performed on smooth portions of the flow and a precise description of a discontinuous shock is possible. Coupled with spectral domain decomposition, one has the ability to resolve efficiently and without stretching transformations, boundary layers and other regions in the flow where increased local resolution is needed. A supersonic blunt body problem is solved to provide an example of the use of the approach. Comparisons between computations and experimental measurements are made for body surface quantities, shock shape, and body normal direction profiles.

Computational Vision at Yale

We present a brief introduction to the five articles that make up this special issue: Shock Graphs and Shape Matching, The Bas-Relief Ambiguity, Incremental Focus of Attention for Robust Vision Based Tracking, What Tasks can be Performed with an Uncalibrated Stereo Vision System, and Volumetric Deformation Analysis Using Mechanics-Based Data Fusion: Application in Cardiac Motion Recovery. Tjis introduction and accompanying articles provide a by no means exhaustive, but hopefully representative sampling of the computational vision at Yale University.

次世代スペースシャトルの試験機Ｘ‐３３周りの流れの放電発光法による可視化

Visualizations of flowfield around the X-33 test vehicle have been performed utilizing the electric discharge method. The electric discharge method has been invented and developed by the authors and others for visualizing three dimensional hypersonic flowfield phenomena. In this experiment the objectives of visualization are three dimensional shock shapes, temperature distributions, and flow patterns near the model surface. It is considered that these visualized results are useful for verifying the theoretical and numerical results of the flowfield around the model. These visualizations are carried out using the hypersonic gun tunnel whose main characteristics are that the Mach number is 10 and the duration is 10ms.

Heating characteristics of blunt swept fin-induced shock wave turbulent boundary layer interaction

An experimental study was conducted on shock wave turbulent boundary layer interactions caused by a blunt swept fin-plate configuration at Mach numbers of 5.0, 7.8, 9.9 for a Reynolds number range of (1.0.similar to 4.7) x 10(7)/m. Detailed heat transfer and pressure distributions were measured at fin deflection angles of up to 30 degrees for a sweepback angle of 67.6 degrees. Surface oil flow patterns and liquid crystal thermograms as well as schlieren pictures of fin shock shape were taken. The study shows that the flow was separated at deflection of 10 degrees and secondary separation were detected at deflection of theta greater than or equal to 20 degrees. The heat transfer and pressure distributions on flat plate showed an extensive plateau region followed by a distinct dip and local peak close to the fin foot. Measurements of the plateau pressure and heat transfer were in good agreement with existing prediction methods, but pressure and heating peak measurements at M greater than or equal to 6 were significantly lower than predicted by the simple prediction techniques at lower Mach numbers.

A Blunt-Nosed Thin Body in Hypersonic Flow

An inverse problem is posed to simulate the influence of the nose bluntness on the hypersonic inviscid steady flow field around a slender two-dimensional section. In this formulation a bow shock is given in advance whereas the body contour comes at the end of analysis being identified with a particular streamline. An equation for the shock shape involves two terms in the form of different powers of the distance measured along the direction of oncoming stream. The leading term derives from a classical similar solution for the strong viscous/inviscid interaction regime; a correction to it models bluntness effects at large distances downstream. Matched asymptotic expansions are used to solve the problem within the hypersonic small- disturbance theory. In the outer region bounded by the shock, two sets of ordinary differential equations control the pressure, density, and velocity distributions. The second-order approximation admits of an explicit integral obtainable from a consideration of the finite drag exerted on the blunted nose of a section. The use of the momentum conservation law allows us to predict the power of the exponent of the correction term entering the shock equation. The asymptotic behavior of both first- and second-order approximations is established and employed for providing conditions for a solution in the inner region occupied by a high-entropy layer. Governing equations here are solved, explicitly determining a dependence of the body shape on the correction in the shock representation. A thorough analysis of the Newtonian approach reveals certain limitations inherent in this sim- plified treatment of steady hypersonic flows.

Multidimensional, Magnetohydrodynamic Simulation of Solar-Generated Disturbances: Space Weather Forecasting of Geomagnetic Storms

A short review is given, first, of an extension of classical, continuum gasdynamics to solar system magnetohy-drodynamics (MHD) in both the steady and time-dependent contexts. A slightly longer review is given of two-and three-dimenslonal, numerical, MHD simulations of transient, solar-generated disturbances in the inner solar system. Validation of this approach is provided by making use of several examples wherein shock shapes and energetic particle fluxes (generated by shocks) are empirically confirmed. Although these studies are based on an ideal, one-fluid (proton) plasma, we suggest that a three-dimensional MHD model be employed to make use of presently available solar observations as approximate physical parameters for initialization conditions. Such a model may be useful, within the space weather national implementation plan, for geomagnetic storm forecasts.

Magnetic field near Venus: A comparison between Pioneer Venus Orbiter magnetic field observations and an MHD simulation

Pioneer Venus Orbiter (PVO) measurements revealed the shape and the changing location of the Venus bow shock with solar cycle and provided a detailed picture of the magnetic field pileup in the dayside magnetosheath. Nevertheless, the reason for the increase of the terminator shock position to the observed distances has evaded our understanding, and the “magnetic barrier” region has been studied primarily by comparisons with gasdynamic models due to the difficulty of using more sophisticated treatments. In this study we investigate the extent to which a three‐dimensional magnetohydrodynamic (MHD) model of the Venus‐solar wind interaction, with and without “mass loading” by photoionization of the atomic oxygen upper atmosphere, can reproduce some of the basic features of the dayside magnetic field observed on PVO. The ideal MHD model uses a conducting sphere to represent the basic Venus ionospheric obstacle to the solar wind flow. We adopt the viewpoint that during solar maximum, a conducting obstacle with oxygen mass loading is appropriate, while a no‐mass loading case is a good first approximation to the solar minimum situation. The MHD simulations are found to give a realistic picture of both the shape of the bow shock and its observed elliptical cross section at the terminator. The introduction of the oxygen mass loading moves the shock position to that observed at solar maximum. The magnetic field strength on the dayside has a dependence on solar zenith angle similar to that found in statistical analyses of the PVO data, although the field is stronger than that measured. The mass loading creates a layer near the planet where the magnetic pressure is replaced with the thermal pressure much like observed. Our studies also raise the question of the role of nightside flow vortices in the formation of the effective obstacle boundary. Overall, our results illustrate that many features seen in the MHD model are consistent with the previously reported observations of the Pioneer Venus Orbiter.

Shape of the heliospheric termination shock: Effects of latitude variations of solar wind dynamic pressure

We investigate the consequences of internal solar wind latitude variations on the heliospheric termination shock and the flow of the gas beyond the shock. We have developed a simple gasdynamic model, assuming the solar wind to be a steady, axially symmetric radial outflow of gas that passes through a termination shock and flows incompressibly beyond the shock. We ignore any latitude variations external to the heliosphere (i.e., due to the local interstellar medium) by requiring that the stagnation pressure infinitely far away must be spherically symmetric. Analysis of the model leads to three broad conclusions: (1) The shape of the heliospheric shock is qualitatively similar to what one would predict using the “naive” assumption that the heliocentric distance of the shock is proportional to the square root of the scaled solar wind dynamic pressure ρvr2. (2) However, the existence of an internal latitude dependence of the scaled dynamic pressure requires that the shock must be oblique at some latitudes, and this obliquity produces an outward “bulge” in the shape of the termination shock; this conclusion is completely general, and in particular is true for the case of an oblate shock. (3) For a prolate termination shock the far‐down‐stream flow is deflected toward the equator, and solar wind originating in the poleward half of (say) the northern hemisphere would fill more than half of the volume of the same hemisphere beyond the termination shock; the deflection would be in the opposite sense (poleward) for an oblate termination shock.

Multidimensional, magnetohydrodynamic simulation of solar-generated disturbances - Space weather forecasting of geomagnetic storms

A short review is given, first, of an extension of classical, continuum gasdynamics to solar system magnetohy-drodynamics (MHD) in both the steady and time-dependent contexts. A slightly longer review is given of two-and three-dimenslonal, numerical, MHD simulations of transient, solar-generated disturbances in the inner solar system. Validation of this approach is provided by making use of several examples wherein shock shapes and energetic particle fluxes (generated by shocks) are empirically confirmed. Although these studies are based on an ideal, one-fluid (proton) plasma, we suggest that a three-dimensional MHD model be employed to make use of presently available solar observations as approximate physical parameters for initialization conditions. Such a model may be useful, within the space weather national implementation plan, for geomagnetic storm forecasts.

A model of the Earth's distant bow shock

We present a new global model of the Earth's bow shock that is parametrized by the solar wind conditions. We begin with a conic section base model taken to be correct for average solar wind conditions. Then we apply modifications to the base model, based on physical arguments, to account for the changes in the size and shape of the bow shock caused by changes in the prevailing solar wind dynamic pressure, Alfvén and sonic Mach numbers, and interplanetary magnetic field orientation. We show that our model matches the location and timing of shock crossings observed, at large downtail distances, by the Galileo spacecraft during December 1990 and December 1992 and by the Pioneer 7 spacecraft during September 1966. Magnetic field and plasma moments in a shock normal coordinate system change across the model shock surface as required by conservation laws.

The Effects of Vibrational Relaxation on Hypersonic Flow Past Blunt Bodies

SummaryThe analysis of Freeman is extended to the hypersonic flow of an inviscid, vibrationally relaxing gas past a bluff body. Expressions for the shock shape, streamline shapes and stand-off distance are derived; these expressions have been evaluated for a sphere for various values of an appropriate non-equilibrium parameter Λ.

Analytical Solution of the Type IV Shock Interaction

An analytical model for the type IV shock interaction in high Mach number, calorically perfect air is developed. The model solves the e owe eld around the transmitted shock, using the oblique shock relations, and the e ow inside the jet, using Prandtl ‐ Meyer waves. The length of the transmitted shock and jet geometry are derived explicitly by calculating the shape of the bow shocks using a continuity methodology. Unlike previous analytical methods, no empirical or experimental data on the interaction are required. The model for the jet impingement gives a prediction for the location and value of the peak pressure on the cylinder. Comparison with experimental results shows good agreement for the impingement location, peak pressure, and shock shapes, and the calculation is performed very quickly with minimal computing resources. A parametric study is conducted, demonstrating the variation of key interaction results such as peak pressure, impingement location and lengths of the transmitted shock, and the terminating shock of the jet, as a result of impinging shock location and strength, freestream Mach number, or cylinder radius. Many trends between the dependent and independent variables in this study were apparent, providing useful and insightful results for design purposes.

Observation and analysis of selected IMP 8 and WIND bow shock crossings in late 1994

Nine bow shock crossings from late 1994 are examined using data from the IMP 8 spacecraft as input to the Rankine‐Hugoniot (RH) conservation equations. In addition, two crossings from the same time period are examined using WIND spacecraft data. A nonlinear chi‐square minimization method is used to obtain the best fit bow shock (BS) normal and speed for each crossing. The shock normals and speeds are also calculated neglecting temperature, as well as using only the magnetic coplanarity and velocity coplanarity techniques. All of the methods generally give mutually consistent results. The orientations of the shock normals are consistent with the nominal bow shock shape. The shocks were expected to move according to a simple “breathing model in which the bow shock moves inward and outward from Earth according to changes in external solar wind conditions. However, the results show outward motion of the bow shock for all but one of all the crossings considered. Moreover, the direction of motion of the BS depends on the exact choice of the shock normal so that a slightly different shock normal can give a different sense of motion of the BS. We conclude that the RH equations that we use may be missing some terms which would provide an accurate description of the balance on both sides of the BS. In particular, the BS acceleration at its flanks is not taken into account by these equations.

Reaction-zone structure of a steady-state detonation wave in a cylindrical charge

This paper reports numerical simulations of the structure of the reaction zone and adjoining flow of a steady-state two-dimensional detonation wave in an unconfined cylindrical charge in order to study the shock shape, sonic locus, streamlines, and characteristic lines. The reactive fluid dynamic equations are solved by the flux-corrected transport technique and making no a priori assumptions about the flow structure. The polytropic reactive fluid with the simple heat-release law is used as a model. The results of the simulations, when compared to the theoretical analysis, elucidate some fundamental features of the reaction-zone structure. In particular, the calculations show that two different forms of the sonic loci are possible, depending on the charge diameter. Near the charge edge, the sonic locus is always tangential to the upstream limiting characteristic of the rarefaction fan. In the vicinity of the sonic surface, the streamlines have inflection points, and unresolved singularities at the characteristic lines are expected. The sonic surface limits the part of the reaction zone which has any influence on the detonation front.