Shock Wave Mach Number Research Articles

A reconsideration of the transition criterion from mach to regular reflection over cylindrical concave surfaces

Following the idea forwarded by Ben-Dor and Takayama (1985) a new propagation path was assumed for the corner generated signals. In addition to the new chosen propagation path, one of the simplified assumptions used by Ben-Dor and Takayama, namely thatu+a remains constant behind the incident shock wave, was further simplified, i.e., bothu anda were assumed to be constant behind the incident shock wave. This new path and the further simplified assumption led to a new transition criterion from Mach to regular reflection over a cylindrical wedge which unlike the two criteria developed in Ref. by Ben-Dor and Takayama (1985) has the ability to predict the transition wedge angle quite accurately in the entire range of the incident shock wave Mach numbers which were investigated in Ref. by Ben-Dor and Takayama(1985) and in this study.

Experimental investigation of the aerodynamic breakup of liquid drops

An experimental investigation was made of the deformation and the mechanism of stripping-type breakup of liquid drops. Experiments were conducted in a 60 X 150 mm cross-sectional shock tube equipped with pulsed laser holographic interferometry. Water drops having diameters of 1030 and 4300 fim were examined for shock wave Mach numbers from 1.3 to 1.5 in atmospheric air. The Weber and Reynolds numbers under these conditions were in the range of 600 to 7600 and 1.38 to 10.4 x 10, respectively. In previous works, the experimental data of the stripping-type breakup were obtained by spark shadowgraphs and streak schlieren methods and high-speed movies. However, in previous works, due to the effect of light scattering through the micromist, the structure of disintegrating drops could not be visualized. The purpose of the present work is, by using holographic interferometry, to re-examine the classical problem of the stripping-type breakup of liquid drops. As a result, a four-stage mechanism of the stripping-type breakup of liquid drops was established.

Similarity solution for low Mach number spherical shocks

A nonlinear progressive wave equation (NPE) describes the evolution of a low Mach number shock wave. The NPE is the nonlinear time domain counterpart of the frequency domain linear parabolic wave equation (PE) for small angle propagation. The NPE in spherical symmetry admits a similarity solution that specifies both the shape of the pulse and the shock strength as a function of range. For finite amplitude spherical waves whether self-similar or not, the theory predicts constancy of an impulse integral corresponding to that of linear theory. For the self-similar waves, theory and available data are in qualitative agreement in the following areas: (1) The shock strength decreases with range as an approximate power law; (2) the temporal behavior of the solution at fixed range is a shock discontinuity followed by a roughly exponential decay; and (3) the effective relaxation time behind the shock is in reasonable agreement with data for slightly more than the first decade in range.

Further analytical considerations of weak planar shock wave reflections over a concave wedge

A simplified analytical model of the Mach reflection of a planar shock wave over a concave cylindrical wedge has been used to predict the triple point trajectory and the triple point trajectory angle at glancing incidence.Comparison with experimental results suggests that the simplified approach for predicting the triple point trajectory is good for an incident shock wave Mach number, Mi, less than 1.1. However, the prediction of the triple point trajectory angle at glancing incidence is good for any value of Mi as long as the assumption of a perfect gas is valid.

Some aspects of shock-wave research

incident shock wave in shock tube kink in CMR reflected wave R first and second Mach stems in DMR self-similar Mach number (M) , = (u + v)/a Mach reflection (SMR, CMR, DMR, TDMR) incident shock wave (/) Mach number static pressure reflection point; point where RR =wave angle X,x' = first and second triple-point trajectory angles

Pseudostationary oblique shock-wave reflections in sulphur hexafluoride (SF 6 ): interferometric and numerical results

Pseudostationary oblique shock-wave reflections in SF 6 were investigated experimentally and numerically. Experiments were concluded in the UTIAS 10 x 18 cm Hypervelocity Shock Tube in the range of incident shock wave Mach number 1.25 < M s < 8.0 and wedge angle 4° < θ w < 47° with initial pressure 4 < P 0 < 267 Torr (0.53-35.60 kPa) at temperatures T 0 near 300 K. The four major types of shock-wave reflection, i. e. regular reflection (RR), single-Mach (SMR), complex-Mach (CMR) and double-Mach reflections (DMR), were observed. These were studied by using infinite-fringe interferograms from a Mach-Zehnder interferometer with a 23 cm diameter field of view. The isopycnics and the density distributions along the wedge surface are presented for the various types of reflection. The analytical transition boundaries between the four types of shock-wave reflection were established up to M s = 10.0 for frozen and equilibrium vibrational SF 6 . An examination of the relaxation length under the present experimental conditions indicated that a vibrational-equilibrium analysis was required. Comparisons of experiment with analysis for transition-boundary maps, reflection angle δ and the first triple-point trajectory angle X verify that the reflections were in vibrational equilibrium. The excellent agreement between the present interferometric results and the numerical results obtained by H. M. Glaz et al . ( Proc. int. colloq. on dynamics of explosives and reactive systems [ Berkeley ] (1985)) with real-gas effects also supports the vibrational equilibrium hypothesis for shocked SF 6 . The behaviour of the angle between the two triple-point trajectories ( X ' — X ) is discussed and the unique pattern of DMR with X ' = 0 was verified experimentally. A numerical analysis for the second triple-point system is obtained for the first time. It is shown that, for a given incident shock Mach number, the highest wedge-surface pressure is achieved through a DMR instead of an RR at high M s .

Parameters Affecting the Postshock Wave Relaxation Zone in an Oxygen Carbon Particle Suspension

A parametric study was conducted regarding the effects of the initial carbon concentration, the initial carbon particle diameter, the shock wave Mach number and the preshock suspension pressure on the suspension behavior in the relaxation zone. The suspension was composed of oxygen gas seeded with small carbon particles. It was found that changing either the initial carbon concentration or the shock wave Mach number has a marked effect on the suspension behavior in the relaxation zone and on the eventually reached postshock equilibrium state. Changes in the initial carbon particle diameter (at a constant shock wave Mach number and carbon concentration) has no effect on the postshock equilibrium state, but it does affect the way in which the suspension reaches this equilibrium state. Changes in the preshock suspension pressure has only minor effects on the suspension behavior in the relaxation zone.

Evaluation of assumptions and criteria in pseudostationary oblique shock-wave reflections

Many experiments in various gases have now been performed on regular and Mach reflection of oblique shock waves in pseudostationary flow. Experimental agreement with the analytical boundaries for such reflec­tions with two- and three-shock theories is reasonable but not precise enough over the entire range of incident shock-wave Mach numbers ( M s ) and compression wedge angle ( θ W ) used in the experiments. In order to improve the agreement, the assumptions and criteria employed in the analysis were critically examined by the use of the experimental data for nitrogen (N 2 ), argon (Ar), carbon-dioxide (CO 2 ), air and sulphurhexa-fluoride (SF 6 ). The assumptions regarding the excitation of the internal degrees of freedom were evaluated based on a relation between the relaxation lengths and a characteristic length of the flow. The ranges in which the frozen-gas and vibrational-equilibrium-gas assumptions can be applied were verified by comparing the experimental and numerical values of δ, the angle between the incident and the reflected shock waves. The deviations of the experimental orientation of the Mach stem at the triple point from a line perpendicular to the wedge surface were considered. A new criterion for the transition from single-Mach to complex-Mach reflection improved the agreement with experiments in the ( M S , θ W )-transition-boundary map. The effects of the shock-induced boundary layer on the wedge surface on the reflected-wave angle and the persistence of regular reflection into the Mach reflection region (‘von Neumann paradox’) were evaluated.

Particle acceleration in solar flares

A model for second-phase acceleration in solar flares is presented. It is argued that particles are accelerated by the flare shock wave as it passes through the corona, and that the combined effects of first-order Fermi acceleration at shock waves and second-order Fermi acceleration by resonant scattering off hydromagnetic waves successfully explain the observed interplanetary proton and electron spectra. The model reproduces quantitatively the correlations of nonrelativistic with relativistic electron spectral indices and the correlation of nonrelativistic electrons with proton indices as a function of one parameter, the Alfvenic shock wave Mach number MA. The range in MA needed to fit the observations is 1.1-3.6, indicating shock velocities of 1000-2000 km/s. It is shown that the model allows the acceleration of electrons to tens, and of protons to hundreds, of MeV in times on the order of seconds. 39 references.

Resonant ion acceleration by oblique magnetosonic shock wave in a collisionless plasma

A magnetosonic shock wave propagating obliquely to a magnetic field is studied by theory and simulation, with particular attention to the resonant ion acceleration (the vp×B acceleration) by the shock. Theoretical analysis based on a two-fluid model shows that, in the laminar shock, the electric field strength in the direction normal to the wave is about (mi/me)1/2 times larger for the quasiperpendicular shock than that for the quasiparallel shock, which is a reflection of the fact that the width of the quasiperpendicular shock is much smaller than that of the quasiparallel shock. Time evolution of a totally self-consistent magnetosonic shock wave is studied by using a 2 1/2 -dimensional fully relativistic, fully electromagnetic particle simulation with full ion and electron dynamics. Even the low Mach number shock wave can significantly accelerate some ions by the vp×B acceleration. The resonant ion acceleration occurs more strongly in the quasiperpendicular shock, because the magnitude of this acceleration is proportional to the electric field strength.

An experimental study on shock waves propagating through a dusty gas. The flow in a horizontal dusty-gas shock tube.

An experimental study was conducted on shock waves propagating through a dusty gas in a horizontal channel using a horizontal dusty-gas shock tube. As dusty gas, air containing fly ash (mean diameters of 3 μm and 4μm) was used. Pressures and particle concentrations behind the shock waves were simultaniously measured by means of pressure transducers and a laser beam attenuation method. Variations of particle concentrations behind the frozen shock waves and the effect of particle loading ratios on the frozen shock wave Mach numbers are shown, and schlieren photographs of shock waves in a dusty gas taken with an Imacon 790 camera are shown.

The initiation process of heterogeneous detonation waves (Kerosene spray and film)

Experiments were performed on the initiation of a heterogeneous detonation wave propagating through mixtures of liquid fuel and a gaseous oxydizer. The liquid fuel tested was kerosene in the forms of a spray and a film attached on tube walls. The ignition was performed by oxyhydrogen detonation. The effects of the content of oxygen in the oxydizer were investigated. It was found that for pure oxygen as an oxydizer, detonation waves were established by the combustion of kerosene, but for an oxydizer with a small amount of oxygen, only low Mach number shock waves with combustion waves following after were formed. For the former case, the ignition was initiated by the strong shock wave front, but for the latter case, the liquid phase seemed to be ignited by the combustion products of the initiator gas.

A numerical study of oblique shock-wave reflections with experimental comparisons

A direct comparison is made for several occurrences of oblique shock-wave reflections between interferometric results obtained at the University of Toronto Institute for Aerospace Studies (UTIAS) 10 cm x 18 cm hyper­-velocity shock tube and numerical results obtained by using a current computational method for solving the Euler equations. Very good qualitative agreement is obtained for equilibrium and frozen flow fields except in small regions where the experiments were dominated by viscous flow. The quantitative agreement is very close in some cases but can be out by 10–15% in cases with non-equilibrium flow or viscous structures or both. Additional parametrized sequences of calculations are presented to assess the utility of the present numerical method in constructing the various reflection–transition lines for perfect inviscid flows in the shock-wave Mach number, wedge-angle (Ms,θw)-plane, and the validity of the ‘boundary-layer defect’ theory.

炭酸ガス中の弱い衝撃波構造の観測

Structures of weak shock waves reflected from various models in carbon dioxide are observed by using a Mach-Zehnder Interferometer, whose light source is a pulsed nitrogen laser (wave length 337.1 nm). Shock waves of the equilibrium shock Mach number of 1.015-1.097 are generated for initial pressure of about 2 atm in a shock tube of 40×40 mm 2 cross section. The observed structures of fully and partly dispersed waves due to vibrational relaxation of carbon dioxide, reflected from a porous wall, a hemicylinder or an inclined plate, exhibit the 'nonstationary effect' and the 'N-wave effect', which are common phenomena for weak waves in dissipative and dispersive media.

The relaxation zone behind normal shock waves in a dusty reacting gas. Part 2. Diatomic gases

The propagation of a strong normal shock wave into a quiescent mixture of nitrogen gas seeded with small, spherical inert dust particles is studied. While crossing the shock front, the gaseous phase of the suspension experiences a sudden change in temperature, pressure, density and velocity. (These changes can easily be evaluated using the Rankine-Hugoniot relations.) The solid phase of the suspension (dust) is initially unaffected by the shock wave. As a result, immediately behind the shock front, one phase of the suspension (the nitrogen gas) is in a state of relatively high temperature and low velocity while the other (the dust) is in a state of relatively low temperature and high velocity. Owing to these differences in temperature and velocity, intense heat transfer and viscous interactions between the two phases take place leading eventually to a new state of equilibrium that is reached farther downstream of the shock front. The flow field where these interactions take place, the relaxation zone, is solved numerically. It is shown that the spatial extent of this zone is strongly affected by the mass concentration of the dust in the suspenson and its physical properties (size, density and specific-heat capacity). These parameters also affect the post-shock equilibrium suspension properties. It was found that increasing the dust concentration results in a shorter kinematic relaxation zone, higher post-shock suspension pressure, density and temperature, and lower velocity, as compared to a similar pure-gas case. Increasing the dust particle density or its diameter results in a longer relaxation zone and a higher post-shock equilibrium suspension pressure, density and temperature. Changes in the dust specific-heat capacity affect the extent at the thermal relaxation length and the suspension temperature and density; they do not affect the extent of the kinematic relaxation length or the post-shock suspension pressure and velocity. For the range of dust concentration, size, density, specific-heat capacity and shock-wave Mach number investigated, the kinematic relaxation zone is always longer than the thermal relaxation zone.

ON A DUSTY GAS SHOCK TUBE*

AbstractAnalytical and numerical methods were used to investigate the flow induced by a shock wave in a shock tube channel containing air laden with suspended small solid particles. Exact results are given for the frozen and equilibrium shock wave properties as a function of the diaphragm‐pressure ratio and shock wave Mach numbers. The driver contained air at high pressure. A modified random choice method, together with an operator‐splitting technique, shows clearly the decay both of a discontinuous frozen shock wave and of a contact discontinuity and the formation both of a stationary shock structure and of an effective contact front of finite thickness. The effects of particle diameter, particle‐number density, and diaphragm‐pressure ratio on the transitional behavior of the flow are investigated in detail. The alteration of the flow properties owing to the presence of particles is discussed thoroughly and compared with classical shock tube flows.

On a dusty-gas shock tube

Analytical and numerical methods were used to investigate the flow in­duced by a shock wave in a shock-tube channel containing air laden with suspended small solid particles. Exact results are given for the frozen and equilibrium shock-wave properties as a function of diaphragm-pressure ratio and shock-wave Mach numbers. The driver contained air at high pressure. A modified random-choice method together with an operator-splitting technique show clearly both the decay of a discontinuous frozen shock wave and a contact discontinuity, and the formation of a stationary shock structure and an effective contact front of finite thickness. The effects of particle diameter, particle-number density and diaphragm- pressure ratio on the transitional behaviour of the flow are investigated in detail. The alteration of the flow properties owing to the presence of particles is discussed thoroughly and compared with classical shock-tube flows.

Transonic dip mechanism of flutter of a sweptback wing. II

Conclusions An experimental investigation of the RR—-MR and MR-*RR transitions over concave and convex, smooth and rough wedges revealed that 1) as the surface roughness increases, the wedge angle at which transition takes place (for a given Mach number) decreases and 2) for a mesh 40 sand paper type surface roughness, it seems that the transition angle becomes independent of the incident shock wave Mach number, i.e., a) MR—>RR transition occurs at 6W =^54 deg and b) RR-MR transition takes place at Ow-21 deg. These results should be of great importance to those dealing with reflections of blast waves, since this type of problem involves nonstationary flows and high degrees of surface (ground) roughness.

Solution of the Boltzmann equation at the upstream and downstream singular points in a shock wave

A solution of the Boltzmann equation is obtained at the upstream and downstream singular points in a shock wave, for the case of Maxwell molecules. The fluid velocityu, rather than the spatial co-ordinatex, is used as the independent variable, and an equation for ∂f/∂uat a singular point is obtained from the Boltzmann equation by taking the appropriate limit. This equation is solved by using the methods of Grad and of Wang Chang & Uhlenbeck; and it is observed that the two methods are the same, since they involve not only an equivalent system of moment equations but also the same closure relations. Because many quantities are zero at a singular point, the problem becomes sufficiently simple to allow the solution to be carried out to any desired order. At the supersonic singular point, the solution converges very slowly for strong shock waves; but a simple modification to Grad's method provides a rapidly convergent solution. The solution shows that the Navier-Stokes relations, or the first-order Chapman-Enskog results, do not apply unless the shock-wave Mach number is unity, and that they are grossly in error for strong shock waves. The solution confirms the existence of temperature overshoot in a strong shock wave; shows that the critical Mach number in Grad's solution increases monotonically with the order of the solution; provides a simple explanation as to why Grad's closure relations fail and shows how they can be improved; and provides exact boundary values that can be used to guide future numerical solutions of the Boltzmann equation for shock-wave structure.

Pioneer 10 observations of the solar wind interaction with Jupiter

Pioneer 10 Plasma Analyzer experiment flight data during the Jupiter flyby are presented. The observations show that the interaction of Jupiter's magnetic field with the solar wind is similar in many ways to that at earth, but the scale size is over 100 times larger. Jupiter is found to have a detached standing bow shock wave of high Alfven Mach number. Jupiter has a prominent magnetopause which deflects the magnetosheath plasma and excludes its direct entry into the Jovian magnetosphere. The sunward hemisphere of Jupiter's outer magnetosphere is found to be highly inflated with thermal plasma and a high beta region which is highly responsive to changes in solar wind dynamic pressure. Observational arguments are presented which tend to discount a thin disklike magnetosphere but, rather, favor a Jovian magnetosphere, albeit probabily considerably flattened as compared to the earth's magnetosphere, yet still with reasonable thickness. Results concerning the shock jump conditions, the magnetosheath flow field and inferred internal magnetospheric plasma are presented.