Shock Shape Research Articles

Models of the circumstellar medium of evolving, massive runaway stars moving through the Galactic plane

At least 5 per cent of the massive stars are moving supersonically through the interstellar medium (ISM) and are expected to produce a stellar wind bow shock. We explore how the mass loss and space velocity of massive runaway stars affect the morphology of their bow shocks. We run two-dimensional axisymmetric hydrodynamical simulations following the evolution of the circumstellar medium of these stars in the Galactic plane from the main sequence to the red supergiant phase. We find that thermal conduction is an important process governing the shape, size and structure of the bow shocks around hot stars, and that they have an optical luminosity mainly produced by forbidden lines, e.g. [OIII]. The Ha emission of the bow shocks around hot stars originates from near their contact discontinuity. The H$\alpha$ emission of bow shocks around cool stars originates from their forward shock, and is too faint to be observed for the bow shocks that we simulate. The emission of optically-thin radiation mainly comes from the shocked ISM material. All bow shock models are brighter in the infrared, i.e. the infrared is the most appropriate waveband to search for bow shocks. Our study suggests that the infrared emission comes from near the contact discontinuity for bow shocks of hot stars and from the inner region of shocked wind for bow shocks around cool stars. We predict that, in the Galactic plane, the brightest, i.e. the most easily detectable bow shocks are produced by high-mass stars moving with small space velocities.

Shock loss model and blade profile optimization design of a supersonic cascade

With the improvement of single-stage compressor load, the inflow velocity and relative Mach number at blade tip of compressor are further increased. Using supersonic blade profile to design highly loaded compressor is an effective method to satisfy the requirements of the compressor aerodynamic design. In the internal flow of a blade row, detached shock is caused at the leading edge of supersonic cascade, and complex passage shock structure is produced by shock and boundary layer interaction at blade surface. Establishing shock loss calculation method for detached shock and passage shock and optimizing the blade profile according to the shock loss mechanisms are effective means to design high-efficiency compressors. The earlier studies of shock loss calculation model in supersonic compressor cascade have shown that shock loss of the detached shock at leading edge is related to leading edge radius and inlet Mach number. Therefore, several shock loss models for detached shock are established and proved to be effective. In previous studies on passage shock loss, the changes in geometry of passage shock, which is caused by shock and boundary layer interaction, are not fully considered. So, it is necessary to do further study on passage shock loss model. Based on the internal flow characteristics in supersonic cascade, passage shock loss model is proposed. Physics-based passage shock geometry, which is determined by static pressure rise, Mach number before wave fronts and blade profile curvature, is applied for this passage shock loss model. Near the blade surface, passage shock geometry is transformed into lambda-type shock that is composed of two branch oblique shocks, and oblique shock relationships are used to calculate their total pressure losses. Away from the blade surface, passage shock geometry is very close to the normal shock shape, so that normal shock relationships are used for loss calculation.

Numerical integration techniques for discontinuous manufactured solutions

When applying the method of manufactured solutions (MMS) on computational fluid dynamic software, determining the exact solutions and source terms for finite volume codes where the stored value is an integrated average over the control volume is non-trivial and not frequently discussed. MMS with discontinuities further complicates the problem of determining these values. In an effort to adapt the standard MMS procedure to solutions that contain discontinuities we show that Newton–Cotes and Gauss quadrature numerical integration methods exhibit high error, first order limitations. We propose a new method for determining the exact solutions and source terms on a uniform structured grid containing shock discontinuities by performing linearly and quadratically exact transformations on split cells. Transformations are performed on triangular and quadrilateral elements of a systematically divided discontinuous cell. Using a quadratic transformation in conjunction with a nine point Gauss quadrature method, a minimum of 4th order accuracy is achieved for fully general solutions and shock shapes. A linear approximation of curved shocks is also experimentally shown to be 2nd order accurate. The numerical integration method is then applied to a CFD code using simple discontinuous manufactured solutions which return consistent 1st order convergence values. The result is an important step towards being able to use MMS to verify solutions with discontinuities. This work also highlights the use of higher order numerical integration techniques for continuous and discontinuous solutions that are required for MMS on higher order finite volume codes.

Mean shape of interplanetary shocks deduced from in situ observations and its relation with interplanetary CMEs

Context. Shocks are frequently detected by spacecraft in the interplanetary space. However, the in situ data of a shock do not provide direct information on its overall properties even when a following interplanetary coronal mass ejection (ICME) is detected. Aims. The main aim of this study is to constrain the general shape of ICME shocks with a statistical study of shock orientations. Methods. We first associated a set of shocks detected near Earth over 10 years with a sample of ICMEs over the same period. We then analyzed the correlations between shock and ICME parameters and studied the statistical distributions of the local shock normal orientation. Supposing that shocks are uniformly detected all over their surface projected on the 1 AU sphere, we compared the shock normal distribution with synthetic distributions derived from an analytical shock shape model. Inversely, we derived a direct method to compute the typical general shape of ICME shocks by integrating observed distributions of the shock normal. Results. We found very similar properties between shocks with and without an in situ detected ICME, so that most of the shocks detected at 1 AU are ICME-driven even when no ICME is detected. The statistical orientation of shock normals is compatible with a mean shape having a rotation symmetry around the Sun-apex line. The analytically modeled shape captures the main characteristics of the observed shock normal distribution. Next, by directly integrating the observed distribution, we derived the mean shock shape, which is found to be comparable for shocks with and without a detected ICME and weakly a ected by the limited statistics of the observed distribution. We finally found a close correspondence between this statistical result and the leading edge of the ICME sheath that is observed with STEREO imagers. Conclusions. We have derived a mean shock shape that only depends on one free parameter. This mean shape can be used in various contexts, such as studies for high-energy particles or space weather forecasts.

Area change effects on shock wave propagation

When a planar shock wave propagating down a channel encounters a decrease in cross-sectional area, not only is the shock strengthened but the shock shape and post-shock flow are disturbed. The current research investigates how various area reduction profiles affect the shock strength and shape, as well as the uniformity of the post-shock flow for planar shocks. Bird [1] and Russell [2] investigated experimentally the effects of wall shaping and strengthening by convergence respectively using strong incident shock waves; however the current study looks at relatively low Mach numbers of 1 < M < 2. By optimising the wall shape an area reduction can be used to increase the strength of a shock significantly without compromising on the quality of the post-shock flow, which is of particular importance in the design of experimental shock tube testing. More importantly, by analysing different profiles using numerical studies, the technique could then potentially be generalized to examine to what extent a shock shape may be purposefully manipulated to a required profile by suitable wall shaping. In order to provide a comprehensive study of the topic, numerical, analytical and experimental analyses are conducted. A comparison of computational fluid dynamic (CFD) simulations and Milton’s corrected ray-shock theory [3] is examined in detail.

On the shape and motion of the Earth's bow shock

Multipoint-measurements by the magnetic field Cluster-FGM (Flux Gate Magnetometer) are used to determine the local shock normal, and in turn allow the study of shock location shape and the velocity of the Earth's bow shock. The shock crossings cover orbits in which the spacecraft separation is of the order of ~600km or less. A data selection of 133 bow shock crossings, ranging from quasi-steady perpendicular to moderately noisy oblique geometries, have been analyzed using a standard timing analysis. Prior to applying the timing technique, the magnetic field fluctuations, when present, are suppressed using low band-pass filtering. The present study contributes to similar studies conducted in the past and available in the literature through the inclusion of a larger data set. The shock standoff distance is determined conjointly with a paraboloid model and the results from a timing analysis. A statistical study reveals a standoff distance well in agreement with the standard gas dynamics model prediction for high Mach number MA. We have also found that for about half the crossings, the timing shock normals agree, within 11°, with a conic-based shock model. Our results strongly indicate that the motion of the shock is predominantly along the Sun–Earth direction; a departure from this direction is not related to the shock-crossing location. Shock velocities below ~80km/s satisfactorily follow a nearly Gaussian distribution with zero mean and a standard deviation of ~42km/s. Finally, we show that high speed motions are correlated with sharp increases in the solar wind upstream ram pressure, and are consistent with gas dynamics model predictions.

Shock Graph and Shape matching

The aim of this paper to develop a theory for generic representation of 2-D shape, where structural descriptions are derived from the shocks (singularities) of a curve evolution process, acting on bounding contours. The shocks are organized into a directed, acyclic shock graph and complexity is managed by attending to the most significant (central) shape components first. The space of all such graphs is highly structured and can be characterized by the rules of a shock graph grammar which permits a reduction of a shock graph to a unique rooted shock tree. A novel tree matching algorithm which finds the best set of corresponding nodes between two shock trees in polynomial time.

激光冲击软模大面积微弯曲成形方法

To implement the large area micro bending forming of metal foils,this paper presents a large area micro bending method by laser shock wave based on advantages of laser shock bending forming and soft punch forming.With this method,the metal sheets would be deformed under the soft punch.In the experiments,apulse Nd-YAG laser(Innolas Gmbit Spitlight 2000THG)with Gaussian distribution beam was employed,and the polyurethane rubber with a thickness of 250μm were used as the soft punch.The U-shaped grooves with a depth of 120μm were machined on a printed circuit board with an engraving machine made in Germany(LPKF-ProtoMat-C60).After one impact,three U-shaped grooves were replicated from the mold on cooper foils with a thickness of30μm.Observed by the digital measurement system(KEYENCE VHX-1000C),the deformed micro grooves in the work piece own good contour shape.Besides,the numerical simulation was used tounderstand the micro-bending process by the ANSYS/LS-DYNA software.The experiment and simulation results show that the work pieces with soft punch have more uniform multi-groove contour shapes and good surface quality.The depth of the deformed micro grooves in the work piece reaches as deep as 110μm,higher than that from the laser direct shock shape(88μm).Therefore,it implies that this kind of process improves the work piece forming ability and quality.

Hypervelocity shock standoff on spheres in air

To provide data for the validation of computational fluid dynamics models, measurements of the shock standoff distance on spheres in hypervelocity flows have been made. Test flows of air at 8.7 and 9.7 km/s were generated in the X2 expansion tunnel fitted with a Mach 10 nozzle. High-speed video images were analysed with a least-squares shape-fitting algorithm. Assuming a spherical shock shape near the nose enabled increased resolution measurements beyond the native pixel size. Normalised shock standoff distances, \(\Delta \)/\(D\), in the range 0.03–0.04 were measured, with sphere diameters, \(D\), of 40, 60 and 80 mm.

PARAMETRIC SYSTEM IDENTIFICATION APPLIED TO THE BOW SHOCK WAVE SHAPE

In this paper it is addressed the prediction of form and location of detached shock waves ahead of two-dimensional and axially symmetric bodies at an zero angle of attack. As shown in Figs. 5 and 6, results show a very good agreement with experimental data. In this context an approximate method, based on a simplified form of the continuity relation, is developed to predict the location of detached shock waves ahead of two-dimensional and axially symmetric bodies. In order to reduce the problem to an equivalent onedimensional form, it is assumed that: (1) The form of the shock between its foremost point and its sonic point is adequately represented by a hyperbola asymptotic to tile free-stream Mach lines; and (2) the sonic line between the shock and the body is straight and inclined at a constant angle. Although the new methodology has some points of contact with earlier methodologies, the novelty here is that it is used Missile Datcom code as an aid to find out sonic point on body and also it is adopted Parametric System Identification (PSI) in the determination of bow shock shape which uses the Matlab® optimizer fmincon function and an active set strategy to minimize an error in a rms sense subject to simple constraint placed on the parameters by the user. The optimizer function calls a user written function which calculates the shape of the shock wave using the current parameters supplied by optimizer. Also for the shock distance L, the methodology presented here allows to select the value of the mentioned constant angle consistently based either in aerodynamics literature or through physical considerations. As the L value is previously known from measurements or aerodynamics literature, it was used an optimizer to minimize the error between predicted and known result varying a parameter which absorbs all inconsistencies that arise when it is used the basic Moeckel’s model considered here. Once the principal characteristics of the shock wave are calculated, an error value is returned to optimizer function based on the differences between predicted and known results.

Mercury's magnetopause and bow shock from MESSENGER Magnetometer observations

AbstractWe have established the average shape and location of Mercury's magnetopause and bow shock from orbital observations by the MESSENGER Magnetometer. We fit empirical models to midpoints of boundary crossings and probability density maps of the magnetopause and bow shock positions. The magnetopause was fit by a surface for which the position R from the planetary dipole varies as [1 + cos(θ)]−α, where θ is the angle between R and the dipole‐Sun line, the subsolar standoff distance Rss is 1.45 RM (where RM is Mercury's radius), and the flaring parameter α = 0.5. The average magnetopause shape and location were determined under a mean solar wind ram pressure PRam of 14.3 nPa. The best fit bow shock shape established under an average Alfvén Mach number (MA) of 6.6 is described by a hyperboloid having Rss = 1.96 RM and an eccentricity of 1.02. These boundaries move as PRam and MA vary, but their shapes remain unchanged. The magnetopause Rss varies from 1.55 to 1.35 RM for PRam in the range of 8.8–21.6 nPa. The bow shock Rss varies from 2.29 to 1.89 RM for MA in the range of 4.12–11.8. The boundaries are well approximated by figures of revolution. Additional quantifiable effects of the interplanetary magnetic field are masked by the large dynamic variability of these boundaries. The magnetotail surface is nearly cylindrical, with a radius of ~2.7 RM at a distance of 3 RM downstream of Mercury. By comparison, Earth's magnetotail flaring continues until a downstream distance of ~10 Rss.

High energy concentration by symmetric shock focusing

High-energy concentrations in gas are achieved experimentally in a specially constructed shock tube facility at KTH Mechanics. The high-energy concentration is manifested by a formation of a hot, light-emitting gas core. Experimental, numerical and theoretical investigations show that the shape of the imploding shock is of pivotal importance for the final energy concentration. Cylindrical shocks are unstable. Symmetric polygonal shocks are shown to be dynamically stable and are produced by various methods, e.g. thin wing profiles placed radially in the test section. Such symmetric polygonal shocks are able to produce extremely high energy levels at the focal point. Spectral data from 60 nanosecond short intervals of 8 microsecond light pulse give temperatures in the range of 6,000 K.

A stationary bow shock model for plasmas: The spherical blunt obstacle problem

We present an analytic model of a stationary bow shock which describes the interaction between a supermagnetosonic ambient wind and an obstacle with spherical-like frontal shape. We develop expressions for the bow shock’s geometry and the physical properties of the plasma sheath as functions of the upstream conditions. The solution is limited to magnetic fields parallel to the upstream velocity. The model allows to use any value of the upstream alfvenic and sonic Mach numbers and the polytropic index (γ), pointing out the influence of γ for the magnetosheath compression and the bow shock shape. When both Mach numbers are small, the upstream magnetic field intensity affects also the bow shock shape. We compare our results with other models finding important consistencies. We also compare our results with in-situ data, we fund a reasonable qualitative agreement; however, it seems that our model underestimates the magnetosheath size.

Front shock behavior of stable curved detonation waves in rectangular-cross-section curved channels

The propagation of curved detonation waves of gaseous explosives stabilized in rectangular-cross-section curved channels is investigated. Three types of stoichiometric test gases, C2H4+3O2, 2H2+O2, and 2C2H2+5O2+7Ar, are evaluated. The ratio of the inner radius of the curved channel (ri) to the normal detonation cell width (λ) is an important factor in stabilizing curved detonation waves. The lower boundary of stabilization is around ri/λ=23, regardless of the test gas. The stabilized curved detonation waves eventually attain a specific curved shape as they propagate through the curved channels. The specific curved shapes of stabilized curved detonation waves are approximately formulated, and the normal detonation velocity (Dn)−curvature (κ) relations are evaluated. The Dn nondimensionalized by the Chapman–Jouguet (CJ) detonation velocity (DCJ) is a function of the κ nondimensionalized by λ. The Dn/DCJ−λκ relation does not depend on the type of test gas. The propagation behavior of the stabilized curved detonation waves is controlled by the Dn/DCJ−λκ relation. Due to this propagation characteristic, the fully-developed, stabilized curved detonation waves propagate through the curved channels while maintaining a specific curved shape with a constant angular velocity. Self-similarity is seen in the front shock shapes of the stabilized curved detonation waves with the same ri/λ, regardless of the curved channel and test gas.

Revised model coefficients for vibrational relaxation in a nitrogen–oxygen gas mixture

A numerical investigation of thermal non-equilibrium flows requires species specific relaxation rates, which are often calculated using the Landau–Teller model. This model requires the determination of collision specific relaxation times, which can be computed using Millikan and White’s empirical formula. The coefficients used in this formula for each specific collision pair form a set of coefficients, which are assessed here. The focus of the investigation lies on their performance in hypersonic low-temperature (300–2,500 K) flows that occur at shock-tunnel nozzle exits or in supersonic combustion ramjets (scramjets) before combustion. Two experimental validation cases are chosen; a shock-tunnel nozzle and a sharp cone in hypersonic cross-flow experiment. A comparison of the experimentally measured vibrational temperatures at the nozzle exit against numerical data shows large discrepancies for two commonly used coefficient sets. A revised set of coefficients is proposed that greatly improves the agreement between the numerical and experimental results. Furthermore, the numerically generated shock shape over the sharp cone using the revised set of coefficients correlates well with the experimental measurements.

A CORONAL HOLE'S EFFECTS ON CORONAL MASS EJECTION SHOCK MORPHOLOGY IN THE INNER HELIOSPHERE

We use STEREO imagery to study the morphology of a shock driven by a fast coronal mass ejection (CME) launched from the Sun on 2011 March 7. The source region of the CME is located just to the east of a coronal hole. The CME ejecta is deflected away from the hole, in contrast with the shock, which readily expands into the fast outflow from the coronal hole. The result is a CME with ejecta not well centered within the shock surrounding it. The shock shape inferred from the imaging is compared with in situ data at 1 AU, where the shock is observed near Earth by the Wind spacecraft, and at STEREO-A. Shock normals computed from the in situ data are consistent with the shock morphology inferred from imaging.

An independent distance estimate to CW Leonis

CW Leo has been observed six times between October 2009 and June 2012 with the SPIRE instrument on board the Herschel satellite. Variability has been detected in the flux emitted by the central star with a period of 639 \pm 4 days, in good agreement with determinations in the literature. Variability is also detected in the bow shock around CW Leo that had previously been detected in the ultraviolet and Herschel PACS/SPIRE data. Although difficult to prove directly, our working hypothesis is that this variability is directly related to that of the central star. In this case, fitting a sine curve with the period fixed to 639 days results in a time-lag in the variability between bow shock and the central star of 402 \pm 37 days. The orientation of the bow shock relative to the plane of the sky is unknown (but see below). For an inclination angle of zero degrees, the observed time-lag translates into a distance to CW Leo of 130 \pm 13 pc, and for non-zero inclination angles the distance is smaller. Fitting the shape of the bow shock with an analytical model (Wilkin 1996), the effect of the inclination angle on the distance may be estimated. Making the additional assumption that the relative peculiar velocity between the interstellar medium (ISM) and CW Leo is determined entirely by the star space velocity with respect to the local standard of rest (i.e. a stationary ISM), the inclination angle is found to be (-33.3 \pm 0.8) degrees based on the observed proper motion and radial velocity. Using the Wilkin model, our current best estimate of the distance to CW Leo is 123 \pm 14 pc. For a distance of 123 pc, we derive a mean luminosity of 7790 \pm 150 Lsol (internal error).

Predictive model of magnetosheath plasma flow and its validation against Cluster and THEMIS data

Abstract. An analytical model of magnetosheath plasma flow is described and compared with a large dataset of magnetosheath ion flow velocity measurements from Cluster and THEMIS spacecraft. The model is based on previous works by Kobel and Flückiger (1994) and Génot et al. (2011) and has been modified to overcome the restrictions of these models on the shape of model magnetopause and bow shock. Our model is compatible with any parabolic bow shock model and arbitrary magnetopause model. The model is relatively simple to implement and computationally inexpensive, and its only inputs are upstream solar wind parameters. Comparison with observed data yields a good correspondence: median error in the direction of flow velocity is comparable with the instrumental error, and flow magnitude is predicted with a reasonable accuracy (relative error in flow speed was less than 25% for 86.5% of observations).

A streamline approach to two-dimensional steady non-ideal detonation: the straight streamline approximation

Detonations in non-ideal explosives tend to propagate significantly below the nominal one-dimensional detonation speed. In these cases, multi-dimensional effects within the reaction zone are important. A streamline-based approach to steady-state non-ideal detonation theory is developed. It is shown in this study that, given the streamline shapes, the two-dimensional problem reduces to an ordinary differential equation eigenvalue problem along each streamline, the solution of which determines the local shock shape that, in turn, leads to the solution of the detonation speed as a function of charge diameter. A simple approximation of straight but diverging streamlines is considered. The results of the approximate theory are compared with those of high-resolution direct numerical simulations of the problem. It is shown that the straight streamline approximation is remarkably predictive of highly non-ideal explosive diameter effects. It is even predictive of failure diameters. Given this predictive capability, one potential use of the method is in the determination of rate law parameters by fitting to data from unconfined rate stick experiments. This is illustrated by using data for ammonium nitrate fuel oil explosives.

Stable detonation wave propagation in rectangular-cross-section curved channels

The detonation propagation phenomena in curved channels were experimentally studied in order to determine the stable propagation condition. A stoichiometric ethylene–oxygen mixture gas and five types of rectangular-cross-section curved channels with different inner radii of curvature were employed. The detonation waves propagating through the curved channels were visualized using a high-speed video camera. Multi-frame short-time open-shutter photography (MSOP) was developed in the present study to simultaneously observe the front shock shape of the detonation wave and the trajectories of triple points on the detonation wave. The detonation wave became more stable under the conditions of a higher filling pressure of the mixture gas and/or a larger inner radius of curvature of the curved channel. The critical condition under which the propagation mode of the detonation wave transitioned from unstable to stable was having an inner radius of curvature of the curved channel ( r i) equivalent to 21–32 times the normal detonation cell width ( λ). In the stable propagation mode, the normal detonation velocity ( D n) increased with the distance from the inner wall of the curved channel and approached the velocity of the planar detonation propagating through the straight section of the curved channel ( D str). The smallest D n was observed on the inner wall and decreased with decreasing r i/ λ. The distribution of D n on the detonation wave in the stable mode was approximately formulated. The approximated D n given by the formula agreed well with the experimental results. The front shock shape of the detonation wave could be reconstructed accurately using the formula. The local curvature of the detonation wave ( κ) nondimensionalized by λ decreased with increasing distance from the inner wall. The largest λκ was observed on the inner wall and increased with increasing r i/ λ. D n/ D str decreased with increasing λκ. This nondimensionalized D n– κ relation was nearly independent of r i/ λ.