Shock Mach Number Ms Research Articles

Characteristics of shock tube generated compressible vortex rings at very high shock Mach numbers

Compressible vortex rings are usually formed at the open end of a shock tube. They show exciting flow phenomena during their formation, evolution, and propagation depending on the shock Mach number (Ms) and exit flow conditions. This study considers high shock tube pressure ratio (PR) cases showing hitherto unknown, spectacular flow structures. With hydrogen as a driver section gas at high PR, a supersonic compressible vortex ring having vortex ring Mach number (Mv) >1 is obtained for the first time. The formation of multiple triple points and the corresponding slipstream shear layers and, thus, multiple counter-rotating vortex rings (CRVRs) behind the primary vortex ring at different radial locations, in addition to the usual CRVRs, appears to be a unique characteristic for high Mach number vortex rings. During the formation stage, a vortex layer of reverse circulation than that of the primary vortex ring gets generated from the outer wall of the shock tube. The instability of such a vortex layer creates another series of opposite circulation vortices, which later interfere with the primary vortex core considerably. Also, a near stationary slipstream vortex and multiple fast-moving tiny vortices of opposite circulation to the slipstream vortex are observed near the central zone. Mechanisms for the formation of these complex vortical structures are identified. The implications of these phenomena on the vortex ring's geometric and kinematic characteristics, such as ring diameter, core diameter, circulation, and translational velocity, are discussed in detail, illustrating their differences with low vortex ring Mach number cases considering 0.31 < Mv < 1.08.

On the structures of compressible vortex rings generated by the compressible starting jet from converging and diverging nozzles

Compressible vortex rings (CVRs) are one fundamental supersonic flow structure, which is usually generated by a compressible starting jet from a nozzle. Changes in the nozzle geometry provide efficient passive flow control to modulate the structures of CVRs. In this paper, a shock-tube apparatus with three open-ended nozzle geometries (straight, diverging, and converging nozzles) was designed to generate CVRs. The evolution of CVRs was recorded by using the schlieren technique. Three shock Mach numbers (Ms), including Ms=1.28, 1.48, and 1.59, were designed to generate the three typical CVRs. For Ms=1.28, shock-free CVRs are generated by the three nozzles. The CVR generated by the diverging nozzle has the largest size and was followed by a trailing jet, while the CVR generated by the converging nozzle has the smallest size with a strong turbulent trailing jet. For Ms=1.48, a CVR with embedded shock is generated by the straight nozzle, and the CVR is followed by a counter-rotating vortex ring (CRVR) for Ms=1.59. When the diverging nozzle is applied, multiple shock-cells (including barrel shock, Mach disk, oblique shock) appear in the trailing jet following the CVR, but quickly disappear as the CVR evolves. In contrast, as for the converging nozzle, strong shock-cells form in the trailing jet and interact with primary CVR. In addition, the strongest CRVRs are generated owing to the interaction between trailing shock-cells/vortices and CVRs for Ms=1.59. By identifying the CVR locations, it is found that the CVRs generated by converging nozzle have the largest average propagating velocity. However, as Ms increases, the change in the averaged propagation velocity brought by converging or diverging nozzles becomes slighter.

Similarity in Mach stem evolution and termination in unsteady shock-wave reflection

Abstract

Turbulent structures of shock-wave diffraction over 90° convex corner

The turbulent structures and long-time flow dynamics of shock diffraction over 90° convex corner associated with an incident shock Mach number Ms = 1.5 are investigated by large eddy simulation (LES). The average evolution of the core of the primary vortex is in agreement with the previous two dimensional studies. The Type-N wall shock structure is found to be in excellent agreement with the previous experimental data. The turbulent structures are well resolved and resemble those observed in the experimental findings. Subgrid scale dissipation and subgrid scale activity parameter are quantified to demonstrate the effectiveness of the LES. An analysis based on turbulent-nonturbulent interface reveals that locally incompressible regions exhibit the universal teardrop shape of the joint probability density function of the second and third invariants of the velocity gradient tensor. Stable focus stretching (SFS) structures dominate throughout the evolution in these regions. Stable node/saddle/saddle structures are found to be predominant at the early stage in locally compressed regions, and the flow structures evolve to more SFS structures at later stages. On the other hand, the locally expanded regions show a mostly unstable nature. From the turbulent kinetic energy, we found that the pressure dilatation remains important at the early stage, while turbulent diffusion becomes important at the later stage. Furthermore, the analysis of the resolved vorticity transport equation reveals that the stretching of vorticity due to compressibility and stretching of vorticity due to velocity gradients plays an important role compared to diffusion of vorticity due to viscosity as well as the baroclinic term.

Shock Waves and Energy Dissipation in Magnetohydrodynamic Turbulence

Abstract Shock waves play an important role in turbulent astrophysical media by compressing the gas and dissipating the turbulent energy into the thermal energy. Here, we study shocks in magnetohydrodynamic turbulence using high-resolution simulations. Turbulent Mach numbers of and initial magnetic fields of plasma beta β 0 = 0.1–10 are considered, targeting turbulences in interstellar and intracluster media. Specifically, we present the statistics of fast and slow shocks, such as the distribution of shock Mach numbers (M s) and the energy dissipation at shocks, based on refined methodologies for their quantifications. While most shocks form with low M s, strong shocks follow exponentially decreasing distributions of M s. More shocks appear for larger and larger β 0. Fast shock populations dominate over slow shocks if β 0 ≫ 1, but substantial populations of slow shocks develop in the cases of β ≲ 1, i.e., strong background fields. The shock dissipation of turbulent energy occurs preferentially at fast shocks with M s ≲ of a few to several, and the dissipation at strong shocks shows exponentially decreasing functions of M s. The energy dissipation at shocks, normalized to the energy injection, ϵ shock/ϵ inj, is estimated to be in the range of ∼0.1–0.5, except for the case of and β 0 = 0.1, where the shock dissipation is negligible. The fraction decreases with it is close to ∼0.4–0.6 for , while it is ∼0.1–0.25 for . The rest of the turbulent energy is expected to dissipate through the turbulent cascade. Our work will add insights into the interpretations of physical processes in turbulent interstellar and intracluster media.

Kinetic energy of the rotational flow behind an isolated rippled shock wave

The kinetic energy of the perturbation velocity field generated by a corrugated isolated shock is analyzed as a function of the shock Mach number (Ms) and the fluid compressibility (characterized here with the ideal gas ratio of specific heats γ) within the framework of a linear theory. The shock front dynamics is analyzed, evaluating the zeros and critical points of the temporal evolution of the pressure perturbations behind the shock front. These characteristic times are essential in determining the spatial distribution of the velocity fluctuations in the inhomogeneously compressed fluid. The velocity perturbations are followed in time and space behind the corrugated shock front until an asymptotic stage emerges. Graf’s addition theorem of the Bessel functions is seen to be an adequate mathematical tool with which to evaluate the detailed temporal evolution. The kinetic energy density is analyzed in time and space well into the asymptotic stage. Exact criteria are given, based on the properties of the asymptotic velocity field, in order to determine the critical points of the kinetic energy density. The space integral of the kinetic energy density is studied, and an explicit analytical formula can be written in finite form and in terms of known functions. The integrated kinetic energy (KE) is analyzed at the limits of very weak and very strong shocks. For very weak shocks the kinetic energy scales as , where ui is the asymptotic normal velocity perturbation at the surface x = 0. For very strong fronts, the scaling changes to . The logarithmic factor is due to contribution of the bulk vorticity profile, which becomes important at high compressions.

Amplification and attenuation of shock wave strength caused by homogeneous isotropic turbulence

We study the pressure increase across a planar shock wave with shock Mach numbers Ms of 1.1, 1.3, and 1.5 propagating through homogeneous isotropic turbulence at a low turbulent Mach number (Mt ∼ 10−4) based on direct numerical simulations (DNSs). Fluctuation in the pressure increase, Δp′, on a given shock ray is induced by turbulence around the ray. A local amplification of the shock wave strength, measured with the pressure increase, is caused by the velocity fluctuation opposed to the shock wave propagating direction with a time delay, while the velocity in the opposite direction attenuates the shock wave strength. The turbulence effects on the shock wave are explained based on shock wave deformation due to turbulent shearing motions. The spatial distribution of Δp′ on the shock wave has a characteristic length of the order of the integral scale of turbulence. The influence of turbulent velocity fluctuation at a given location on Δp′ becomes most significant after the shock wave propagates from the loc...

Numerical investigations of transient nozzle flow separation

Large-eddy simulations (LES) are carried out to investigate the shock induced transient flow through a planar nozzle mimicking a shock tube experimental setup at shock Mach number Ms=1.86. A fifth-order Weighted Essentially Non-oscillatory (WENO) scheme based 3D numerical flow solver equipped with an immersed boundary method and Wall-Adapting Local Eddy-viscosity (WALE) model is used for this purpose. A comparative study is presented to show the effect of different flow initializations namely, i) without flow fluctuation, ii) with white random noise and iii) with homogeneous isotropic turbulence superimposed on the flow-field. Results are in good agreement with the experimentally measured speeds of the primary shock wave and the following secondary shock wave. It is found that an improper initialization of the flow-field may lead to erroneous predictions of the flow characteristics, particularly the location of the separation point and the unsteady shock/boundary layer interaction. Substantial improvement in the prediction of the early-stage Mach reflection, complex shock/boundary layer interaction is observed with superimposed turbulent fluctuations as an initial flow-field. This is when a homogeneous incompressible isotropic turbulence superimposed on the shocked section is assigned as initial fluctuating field. An additional test case with the latter method, having an approximately four times higher mesh resolution, is used for detailed investigation of the unsteady flow fields, turbulent statistics and boundary layer separation. Results show considerable improvements in the prediction of the secondary shock and the flow separation location compared to the previous findings which dealt with lower mesh resolution. The turbulent flow structures are depicted using the mean flow-field which is spatially averaged over the span-wise homogeneous direction. Time-averaging on the fly turns out to be inadequate, since it leads to a spatial shift of the separation bubble. The reason for this is the use of past flow information only, due to the lack of future information. This in-turn invokes the need of phase-averaging to extract suitable physically statistics of the turbulent flow-field for further deeper analysis.

Dynamics of cylindrical converging shock waves interacting with aerodynamic obstacle arrays

Cylindrical converging shock waves interacting with an array of aerodynamic obstacles are investigated numerically for diverse shock strengths and for different obstacle configurations in air in standard conditions. The considered number of obstacles N is 4, 6, 8, 16, and 24. Obstacles are lenticular airfoils with thickness-to-chord ratios tc of 0.07, 0.14, and 0.21. The distances of the airfoil leading edge from the shock focus point rLErLEref are 1, 2, and 2.5, where rLEref=7 is the dimensionless reference distance from the origin. Considered impinging shock Mach numbers Ms are 2.2, 2.7, and 3.2 at the reference distance from the origin. The reference experimental configuration (N=8,tc=0.14,rLE=7,Ms=2.7) was proposed by Kjellander et al. [“Thermal radiation from a converging shock implosion,” Phys. Fluids 22, 046102 (2010)]. Numerical results compare fairly well to available one-dimensional models for shock propagation and to available experimental results in the reference configuration. Local reflection types are in good agreement with the classical criteria for planar shock waves. The main shock reshaping patterns are identified and their dependence on the shock strength and obstacle configuration is exposed. In particular, different shock patterns are observed after the leading edge reflection, which results in polygonal shock wave with N, 2N, 3N, and 4N sides. The largest temperature peak at the origin is obtained for the 8- and the 16-obstacle configurations and for the smallest thickness to length ratio, 0.07, located at distance from the origin of 2rLEref. In terms of compression efficiency at the origin, the 16-obstacle configuration is found to perform slightly better than the reference 8-obstacle configuration—with an efficiency increase of about 2%-3%, which is well within the model accuracy—thus confirming the goodness of the obstacle arrangement proposed by Kjellander and collaborators.

Numerical study of shock waves propagating through right angled multiple elbows

The details of the shock propagated through the right angled multiple elbows (mainly double elbow and fourfold elbow) are investigated by the numerical study. Computations were carried out by solving the two-dimensional compressible Navier-Stokes equations by using the total variation diminishing (TVD) scheme. Computations were performed for six types of elbow (two types of double elbow and four types of fourfold elbow) and two incident shock Mach numbers (Ms=1.5, 2.3). The effects of the direction and number of the bends in the stabilizing, merging and attenuating processes of the shock are numerically explored by various diagrams such as pressure contours, pressure distributions on the walls and time histories of the pressure contours, and the strength of the shock front at the location of about 20 times the length for each type of elbows was displayed.

Numerical simulation of Richtmyer-Meshkov instability

The compressible Navier-Stokes equations discretized with a fourth order accurate compact finite difference scheme with group velocity control are used to simulate the Richtmyer-Meshkov (R-M) instability problem produced by cylindrical shock-cylindrical material interface with shock Mach number Ms=1.2 and density ratio 1:20 (interior density/outer density). Effect of shock refraction, reflection, interaction of the reflected shock with the material interface, and effect of initial perturbation modes on R-M instability are investigated numerically. It is noted that the shock refraction is a main physical mechanism of the initial phase changing of the material surface. The multiple interactions of the reflected shock from the origin with the interface and the R-M instability near the material interface are the reason for formation of the spike-bubble structures. Different viscosities lead to different spike-bubble structure characteristics. The vortex pairing phenomenon is found in the initial double mode simulation. The mode interaction is the main factor of small structures production near the interface.

A Study on Expansion Wave Formed by Reflection of Shock Wave at the Open-End of Tube.

When the propagating shock wave reaches at the exit portal of a tube, the impulsive wave discharges to surrounding area and the expansion wave occurs from an open-end of a tube in the condition of subsonic flow behind the shock wave. The problem related to the reflection of propagating shock wave at the open-end has become more important subject associated with pulse combus-tors, pressure exchangers and high-speed railway train/tunnel systems. For the case of shock Mach number Ms less than 1.34, the reflected expansion wave generated at the open-end propagates toward upstream of a tube against the direction of the incident shock wave. For the cases of reflection at an open-end with no-baffled tube, Rudinger suggested the empirical equation concerning the imaginary center of reflected expansion wave. In this paper, the improved empirical equation considering the influence of the diameter of a baffle plate at the open-end was proposed. The experiment study using the simple shock tube and the numerical calculatdon using the TVD method were carried out.

On the shock-vortex interaction in Schardin's problem

In this study we revisit Schardin's problem by investigating experimentally shock waves diffract- ing over a finite wedge and interacting with the tip vortices in a complicated manner. Holographic inter- ferometry and shadowgraphy were used in a shock tube for a shock Mach number Ms =1 .34. Numerical simulations were carried out to obtain complementary flow data. The experimental results show that di- verging acoustic waves are generated due to the interaction between shock waves and vortexlets along the slip layer. By means of the computational results obtained for short time intervals, and the corresponding optical images, analysis of the shock-vortex interactions became possible for extended time periods.

Numerical Study of the Shock Wave Propagating through the Plenum Chamber with Borda's Mouthpiece.

In this paper, the shock wave propagating through the plenum chamber with Borda's mouthpiece was studied by numerical simulation. Computations were carried out by solving the two-dimensional compressible Navier-Stokes equations by using the total variation diminishing (TVD) scheme. Computations were performed for five types of the plenum chamber with Borda's mouthpiece [coaxial mouthpiece, off axis mouthpiece, oblique edge mouthpiece, no mouthpiece (coaxis, off axis)] and two incident shock Mach numbers (Ms=1.5, 2.0). The flow fields were numerically visualized by the pressure and vorticity contours, velocity vectors and sonic lines. The pressure distributions and time histories of the pressure contours in the wall and the axis of the duct were also studied, and the attenuating process of the shock front and the diminishing process of the nonuniformity of the shock front were explored.

Numerical Study of Shock Waves Propagating in Bmanched Ducts. The Generating and Diminishing Process of the Nonuniformity of the Shock Front.

In this paper, the generating process and the diminishing process of the nonuniformity regarding the strength and the shape of the shock wave front propagating through branched ducts were explored by numerical simulation. Computations were carried out by solving the two-dimensional compressible Navier-Stokes equations by using the total variation diminishing (TVD) scheme. Computations were performed for six types of branch (90° branch, 1/2 size 90° branch, T-shape (1/1, 1/2, 1/3 size) branch, 45° branch) and two incident shock Mach numbers (Ms=1.3, 2.0). The flow field were numerically visualized by the pressure and vorticity contours, velocity vectors, pressure distributions on the walls and time histories of the pressure contours on the walls. The mechanism of the generating process and the diminishing process of the nonuniformity of propagating shock front was clarified.

Deformation Problems of Gas Interfaces of Different Densities Generated by Shock Waves. Light/Heavy Interface.

This paper reports numerical and analytical studies of the interaction problem between a gas interface of different densities and shock waves. The interface is formed between air and SF6 (light/heavy interface, i. e., the incident shock wave propagates from air to SF6), and there is a sinusoidal perturbation on the initial interface. The interface deformation process and the growth rate of a sinusoidal perturbation on the interface are investigated after shock wave interaction. Numerical simulation is carried out using the second-order TVD finite difference scheme, and the Euler equations are solved. Theoretical impulsive growth rate obtained using the stable velocities on the interface after shock interaction is compared with results of the numerical simulation. In the case of incident shock Mach numbers Ms < 2 and amplitudes of perturbed interface η0k < 1, good agreement is obtained.

Blast wave reflection trajectories from a height of burst

Consideration is given to an explosive charge (TNT) detonated at various heights of burst above a perfect reflecting planar surface in air. Variations of the incident shock Mach number Ms of the spherical blast wave front as it decays and the corresponding wedge angle Ow are plotted on a two-dimensional shock wave reflection transition map in the (Ms, Ow) plane. It is shown that all four types of shock wave reflection (regular, single Mach, complex Mach, and double Mach) can occur in a free-air explosion. However, if the height of burst is increased past a certain limit, only two types of shock wave reflection can occur (regular and single Mach).

The 1986 hypervelocity impact symposium: October 21–24, 1986 municipal auditorium host hotel: St. Anthony inter-continental hotel San Antonio, Texas

A critical appraisal is made of the design, research, development, and operation of the novel UTIAS implosion-driven hypervelocity launchers and shock tubes. Explosively driven (PbN6-leadazide, PETN-pentaerythritetetranitrate) implosions in detonating stoichiometric hydrogen-oxygen mixtures have been successfully developed as drivers for hypervelocity launchers and shock tubes in a safe and reusable facility. Intense loadings at very high calculated pressures (almost megabar range), densities (g/cm3) and temperatures (thousands °K), at the implosion centre, cause severe problems with projectile integrity. Misalignment of the focal point can occur and add to the difficulty in using small calibre (6–8 mm dia.) projectiles. In addition, the extreme driving conditions cause barrel expansion, erosion, and possible gas leakage from the base to the head of the projectile which cut the predicted muzzle velocities to half or a third of the lossless calculated values. However, in the case of a shock-tube operation these difficulties are minimized or eliminated and the possibilities of approaching Jovian re-entry velocities are encouraging. In a recent run using about 100 g of explosive PETN and 400 psi 2H2 + O2 a shock Mach number Ms ≈ 60 was obtained in air at an initial pressure of 1 torr in a 1.0-in.-dia. shock-tube channel. In addition, the use of focused, explosive-driven implosion waves may have many other physical and technological applications.

An update on non-stationary oblique shock-wave reflections: actual isopycnics and numerical experiments

Nonstationary oblique shock-wave reflections over compressive wedges in air and argon were investigated using infinite-fringe interferometric techniques. These allowed direct, continuous and accurate observations of the isopycnics (lines of constant density) of the flow field. The initial pressures for these experiments were made as high as possible (15 to 250 torr) in order to increase the number of isopycnics and to enhance their details and distribution along the wedge surface over a shock-Mach-number range 2.0 &lt; Ms [les ] 8.7. Included in the study were two cases of regular reflection (RR) and one of each single Mach reflection (SMR), complex Mach reflection (CMR) and double-Mach reflection (DMR) for air, and one RR, SMR, CMR and DMR for argon. These particular cases, which we investigated previously in N2 and Ar using a finite-fringe technique, have been used by computational fluid dynamicists to check their finite-difference results against our experimental data. It will be shown that the isopycnic structure previously reported by us differs in detail, in most cases, from that of the present study. The major difference arises from the fact that it was only possible previously to obtain discrete points on isopycnics and along the wedge surface. Consequently, the results obtained before were not as accurate. Comparisons were made of actual wall-density distributions with numerical simulations of the density contours of the various flows obtained by a number of authors. Each numerical method displays its advantages and disadvantages in describing the details of the flow fields. The present experimental results for air are new. They are of great interest from a practical viewpoint. The experiments in argon were redone to provide better data for a gas free from real-gas effects in the range of initial conditions considered, in order to simplify the computations in the numerical simulations. Although the recent numerical simulations are better than those reported previously, additional efforts are required to improve the predictions of the shape, location and values of the isopycnics and other flow isolines in the various regions and along the wall, and to render the predictions free of computer ‘noise’. It is worth noting that real-gas effects did not play any significant role in determining the various wave systems in RR, SMR, CMR and DMR; a different claim was made in our previous work. Relaxation of nitrogen in air can be seen however, at the highest shock Mach numbers (Ms = 7.19 and 8.70), with relaxation lengths in good agreement with accepted predictions.

Coupled interactions of shock-wave structure with laminar boundary layers in ionizing-argon flows

Analyses are made of the mutual interactions between shock structure and the sidewall laminar boundary layer and their effects on the quasi-steady flat-plate laminar boundary layer in ionizing-argon shock-tube flows. The mutual interactions are studied using effective quasi-one-dimensional equations derived from an area-averaged-flow concept in a finite-area shock tube. The effects of mass, momentum and energy nonuniformities and the wall dissipations in the ionization and relaxation regions on the argon shock structure are discussed. The new results obtained for shock structure, shock-tube laminar side-wall and quasi-steady flat-plate boundary-layer flows are compared with dual-wavelength interferometric data obtained from the UTIAS 10 × 18 cm Hypervelocity Shock Tube. It is shown that the difference between the results obtained from the present method and those obtained by Enomoto based on Mirels’ perfect-gas boundary-layer solutions are significant for lower shock Mach numbers (Ms ∼ 13) where the relaxation lengths are large (∼ 10cm). In general, the present results agree better with our experimental data than our previous results for uncoupled ionizing flows.