Propagation of shock waves in a medium with exponentially varying density

The results of Raizer [1], Hays [2], and Chernous'ko [3] are generalized to-the case of self-similar propagation of shock waves in a gas with exponentially varying density and constant pressure. A solution is found by the method of successive approximations. The zero-order approximation coincides with the Whitham method [4]. The first-order approximation is in good agreement with numerical calculations in [2]. The non-selfsimilar motion of a weak shock wave is investigated in the framework of linear theory.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 48–54, November–December, 1970.     

Area change effects on shock wave propagation

Experimental testing was conducted for a planar shock wave of incident Mach number \(M_\mathrm{s} = 1.33\) propagating through one of three compound parabolic profiles of 130, 195 or 260 mm in length, all of which exhibit an 80 % reduction in area. Both high-resolution single shot and low-resolution video were used in a schlieren arrangement. Results showed three main types of flow scenarios for propagation through a gradual area reduction, and an optimal net increase of 12.7 % in shock Mach number was determined for the longest profile, which is within 5 % of theoretical predictions using Milton’s modified Chester–Chisnell–Whitham relation.     

Analysis of unsteady wave processes in a rotating channel

The impact of passage rotation on the gasdynamic wave processes is analyzed through a numerical simulation of ideal shock-tube flow in a closed rotating-channel containing a gas in an initial state of homentropic solid-body rotation. Relevant parameters of the problem such as wheel Mach number, hub-to-tip radius ratio, length-to-tip radius ratio, diaphragm temperature ratio, and diaphragm pressure ratio are varied. It is shown that for a fixed geometry and initial conditions, the contact interface acquires a distorted three-dimensional time-dependent orientation at non-zero wheel Mach numbers. At a fixed wheel Mach number, the level of distortion depends primarily on the density ratio across the interface and also the hub-to-tip radius ratio. The nature of the rarefaction and shock wave propagation is one-dimensional, although the acoustic waves are diffracted due to the radially varying propagation speed. Under conditions of initially homentropic solid-body rotation, a degree of similarity exists between rotating and stationary shock-tube flows. This similarity is exploited to arrive at an approximate analytical solution to the Riemann problem in a rotating shock-tube.     

STUDY ON THE FUEL AIR MIXING INDUCED BY A SHOCK WAVE PROPAGATING INTO A H_2-AIR INTERFACE

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HLLE++格式在高马赫数空腔流动模拟中的应用

空腔流动存在剪切层运动、涡脱落与破裂,以及激波与激波、激波与剪切层、激波与膨胀波和激波/涡/剪切层相互干扰等现象,流动非常复杂,特别是高马赫数(M>2)时,剪切层和激波更强,激波与激波干扰更严重,对数值格式的要求更高,既需要格式耗散小,对分离涡等有很高的模拟精度,又需要格式在激波附近具有较大的耗散,可以很好地捕捉激波,防止非物理解的出现。Roe和HLLC等近似Riemann解格式在高马赫数强激波处可能会出现红玉现象,而HLLE++格式大大改善了这种缺陷,在捕捉高超声速激波时避免了红玉现象的发生,同时还保持在光滑区域的低数值耗散特性。本文在结构网格下HLLE++格式的基础上,通过改进激波探测的求解,建立了基于非结构混合网格的HLLE++计算方法,通过无粘斜坡算例,验证了HLLE++格式模拟高马赫数流动的能力,并应用于高马赫数空腔流动的数值模拟,开展了网格和湍流模型影响研究,验证了方法模拟高马赫数空腔流动的可靠性和有效性。     

The formation of a secondary shock wave behind a shock wave diffracting at a convex corner

This paper deals with the formation of a secondary shock wave behind the shock wave diffracting at a two-dimensional convex corner for incident shock Mach numbers ranging from 1.03 to 1.74 in air. Experiments were carried out using a 60 mm 150 mm shock tube equipped with holographic interferometry. The threshold incident shock wave Mach number () at which a secondary shock wave appeared was found to be = 1.32 at an 81° corner and = 1.33 at a 120° corner. These secondary shock waves are formed due to the existence of a locally supersonic flow behind the diffracting shock wave. Behind the diffracting shock wave, the subsonic flow is accelerated and eventually becomes locally supersonic. A simple unsteady flow analysis revealed that for gases with specific heats ratio the threshold shock wave Mach number was = 1.346. When the value of is less than this, the vortex is formed at the corner without any discontinuous waves accompanying above the slip line. The viscosity was found to be less effective on the threshold of the secondary shock wave, although it attenuated the pressure jump at the secondary shock wave. This is well understood by the consideration of the effect of the wall friction in one-dimensional duct flows. In order to interpret the experimental results a numerical simulation using a shock adaptive unstructured grid Eulerian solver was also carried out. Received 1 May 1996 / Accepted 12 September 1996     

Stability of converging cylindrical shock waves

An experimental and numerical study was made of converging cylindrical shock waves. The goal of the present study was to clarify the movement and instability of the converging cylindrical shock waves. Experiments were conducted in an annular shock tube of 230 mm o.d. and 210 mm i.d. connected to a cylindrical test section of 210 mm diameter. Double exposure holographic interferometry was used to visualize the converging cylindrical shock waves. Incident shock Mach numbers ranged between 1.1 and 2.0 in air. A numerical simulation was conducted using the TVD finite difference scheme. It was found in the experiments that although the initial shock wave configuration looked cylindrical, it was gradually deformed with propagation towards the center and finally showed mode-four instability. This is attributable to the existence of initial disturbances which were introduced by the struts which supported the inner tube of the annular shock tube. This trend was significant for stronger shock waves indicating that at the last stage of shock wave convergence the initial perturbations of the converging cylindrical shock wave were amplified to form the triple point of Mach reflection. The numerical results correctly predicted the experimental trend.This article was processed using Springer-Verlag T_EX Shock Waves macro package 1990.     

The effects that changes in the diaphragm aperture have on the resulting shock tube flow

In a conventional shock tube, the driver and the driven sections have similar (if not identical) cross-sectional area and the diaphragm opened area, upon rupturing, is practically equal to the tube cross-sectional area. Such geometry results in generating a well-formed shock wave in the tube’s driven section. The present experimental work checks the effects that changes in the diaphragm ruptured area have on the generated shock and rarefaction waves. Experiments were conducted in an 80?mm by 80?mm cross section shock tube generating incident shock waves having Mach numbers within the range from 1.06 to 1.25. In each run, pressure histories were recorded along the driven and the driver sections of the shock tube. The recorded pressures reveal that progressive reduction in the diaphragm open space resulted in a weaker shock and both longer time and distance until the compression waves generated close to the diaphragm coalesces into a shock wave. In addition, reducing the open space of the diaphragm resulted in a significant slow down in the high pressure reduction prevailing in the driver section.     

Study on the Fuel Air Mixing Induced by a Shock Wave Propagating into a H2-Air Interface

2nd-order upwind TVD scheme was used to solve the laminar, fully Navier-Stokes equations. The numerical simulations were done on the propagation of a shock wave with Ma _S = 2 and 4 into a hydrogen and air mixture in a duct and a duct with a rearward step. The results indicate that a swirling vortex may be generated in the lopsided interface behind the moving shock. Meanwhile, the complex shock system is also formed in this shear flow region. A large swirling vortex is produced and the fuel mixing can be enhanced by a shock wave at low Mach number. But in a duct with a rearward step, the shock almost disappears in hydrogen for Ma _S = 2. The shock in hydrogen will become strong if Ma _S is large. Similar to the condition of a shock moving in a duct full of hydrogen and air, a large vortex can be formed in the shear flow region. The large swirling vortex even gets through the reflected shock and impacts on the lower wall. Then, the distribution of hydrogen behind the rearward step is divided into two regions. The transition from regular reflection to Mach reflection was observed as well in case Ma _S = 4.     

Diffuse holographic interferometric observation of shock wave reflection from a skewed wedge

The pattern of shock wave reflection over a wedge is, in general, either a regular reflection or a Mach reflection, depending on wedge angles, shock wave Mach numbers, and specific heat ratios of gases. However, regular and Mach reflections can coexist, in particular, over a three-dimensional wedge surface, whose inclination angles locally vary normal to the direction of shock propagation. This paper reports a result of diffuse double exposure holographic interferometric observations of shock wave reflections over a skewed wedge surface placed in a 100 × 180 mm shock tube. The wedge consists of a straight generating line whose local inclination angle varies continuously from 30° to 60°. Painting its surface with fluorescent spray paint and irradiating its surface with a collimated object beam at a time interval of a few microseconds, we succeeded in visualizing three-dimensional shock reflection over the skewed wedge surface. Experiments were performed at shock Mach numbers, 1.55, 2.02, and 2.53 in air. From reconstructed holographic images, we estimated critical transition angles at these shock wave Mach numbers and found that these were very close to those over straight wedges. This is attributable to the flow three-dimensionality.      

Effect of Partial Closure of the Channel on the Pressure Pulse in a Shock Wave Emerging from a Duct

The temporal-spatial development of the flow pattern behind a shock wave emerging from open-ended and half-closed ducts at M₀=1.15 to 3.0 Mach number is investigated both experimentally and numerically and the action of the diffracted wave on a barrier placed at different distances from the duct exit is studied. Flow toeplerograms are obtained and the pressure on the barrier is measured. The distinctive features of the interaction between the shock wave and a barrier mounted perpendicular to the duct axis are established. It is found that partial closing of the duct results in a decrease in the barrier pressure in the case of a strong shock (M₀ > 2.2) and leads to a pressure decrease when a weak shock is diffracted (M₀=1.1 to 1.7). A dependence characterizing the dynamic action of the shock wave on the barrier and specifying the threshold value of a combination of the shock Mach number and the distance from the barrier determining whether the pressure pulse on the barrier increases or decreases, is obtained.     

Experimental and numerical investigation of inclined air/SF6 interface instability under shock wave

The shock tube experiments of inclined air/SF6 interface instability under the shock wave with the Mach numbers 1.23 and 1.41 are conducted. The numerical simulation is done with the parallel algorithm and the multi-viscous-fluid and turbulence (MVFT) code of the large-eddy simulation (LES). The developing process of the interface accelerated by the shock wave is reproduced by the simulations. The complex wave structures, e.g., the propagation, refraction, and reflection of the shock wave, are clearly revealed in the flows. The simulated evolving images of the interface are consistent with the experimental ones. The simulated width of the turbulent mixing zone (TMZ) and the displacements of the bubble and the spike also agree well with the experimental data. Also, the reliability and effectiveness of the MVFT in simulating the problem of interface instability are validated. The more energies are injected into the TMZ when the shock wave has a larger Mach number. Therefore, the perturbed interface develops faster.     

Mach number effects on shock-bubble interaction

An investigation of Mach number effects on the interaction of a shock wave with a cylindrical bubble, is presented. We have conducted simulations with the Euler equations for various incident shock Mach numbers () in the range of , using high-resolution Godunov-type methods and an implicit solver. Our results are found in a very good agreement with previous investigations and further reveal additional gasdynamic features with increasing the Mach number. At higher Mach numbers larger deformations of the bubble occur and a secondary-reflected shock wave arises upstream of the bubble. Negative vorticity forms at all Mach numbers, but the “c-shaped” vortical structure appeared at gives its place to a circular-shaped structure at higher Mach numbers. The computations reveal that the (instantaneous) displacements of the upstream, downstream and jet interfaces are not significantly affected by the incident Mach number for values (approximately) greater than . With increasing the incident Mach number, the speed of the jet (arising from the centre of the bubble during the interaction) also increases. Received 21 December 2000 / Accepted 23 April 2001     

The non-uniform piston-driven magnetohydrodynamic shock wave

Summary The effects of small variations in cross-sectional area and piston velocity on the propagation of a shock wave into a duct have been considered. The motion of the perfectly conducting fluid is subjected to a transverse magnetic field. In particular an expression for the pressure perturbation behind the shock has been derived.     

Effect of thermal conductivity on structure and critical parameters of shock waves in plasma

Results are presented for a numerical solution of the problem of shock-wave propagation in a cold, low-density plasma across a magnetic field with finite conductivity and electron thermal conductivity present; a comparison is made with results obtained from a solution without consideration of thermal conductivity. It is shown that the effect of thermal conductivity can be neglected for small Mach numbers (M<2.5). An isomagnetic density discontinuity is obtained for Mach numbers 2.8 M 3.3. Increase in the magnetic field amplitude at the boundary of the plasma leads to a breakdown of the isomagnetic discontinuity. The critical Mach numbers which characterize the shock wave in this case are M_* > 3.4.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 2, pp. 8–14, March–April, 1972.In conclusion, the authors thank R. Z. Sagdeev for interest in this work.     

Shock wave interaction with porous plates

Previous detailed studies of the interaction of a shock wave with a perforated sheet considered the impact of a shock wave on a plate with regularly spaced slits giving area blockages of 60 and 67%, at various angles of incidence, and resulting in both regular and Mach reflection. The current work extends this study to a much wider variety of plate geometries. Blockage ratios of 20, 25, 33, 50, and 67 and inclinations of 45, 60, 75, and 90° to the shock wave were tested. Four different thicknesses of plate were tested at the same frontal blockage in order to assess the effects of gap guidance. Tests were conducted at two shock Mach numbers of 1.36 and 1.51 (inverse pressure ratios of 0.4 and 0.5). It is found that secondary reflected and transmitted waves appear due to the complex interactions within the grid gaps, and that the vortex pattern which is generated under the plate is also complex due to these interactions. The angle of the reflected shock, measured relative to the plate, decreases with plate blockage and the angle of inflow to the plate reduces with increasing blockage. By analysing the flow on the underside of the plate the pseudo-steady flow assumption is found to be a reasonable approximation. Both the pressure difference and the stagnation pressure loss across the plate are evaluated. It is found that over the range tested the plate thickness has a minimal effect.     

Three-dimensional propagation of the transmitted shock wave in a square cross-sectional chamber

An investigation into the three-dimensional propagation of the transmitted shock wave in a square cross-section chamber was described in this paper, and the work was carried out numerically by solving the Euler equations with a dispersion-controlled scheme. Computational images were constructed from the density distribution of the transmitted shock wave discharging from the open end of the square shock tube and compared directly with holographic interferograms available for CFD validation. Two cases of the transmitted shock wave propagating at different Mach numbers in the same geometry were simulated. A special shock reflection system near the corner of the square cross-section chamber was observed, consisting of four shock waves: the transmitted shock wave, two reflection shock waves and a Mach stem. A contact surface may appear in the four-shock system when the transmitted shock wave becomes stronger. Both the secondary shock wave and the primary vortex loop are three-dimensional in the present case due to the non-uniform flow expansion behind the transmitted shock.PACS: 43.40.Nm     

Incidence of plane supersonic jet on a plane at an arbitrary angle

A solution method is described and results are presented of numerical calculations for the problem of determining the subsonic part of the flow with incidence of a plane uniform supersonic jet on a plane at an arbitrary angle, which corresponds to the flow regime with a a detached shock wave. For the problem solution we use the method of integral relations in the first approximation in a polar coordinate system. The calculation results (pressure distribution on the plate, shock wave shape, and velocity gradient magnitude at the stagnation point) are presented for Mach numbers of 5 and 20, in a range of incidence angles from 0° to 35°, and also for M=3 for an incidence angle 0° (angles measured from normal to the plate).In conclusion the authors wish to thank G. I. Taganov for guidance in this work.     

The interaction between shock waves and solid spheres arrays in a shock tube

When a shock wave interacts with a group of solid spheres, non-linear aerodynamic behaviors come into effect. The complicated wave reflections such as the Mach reflection occur in the wave propagation process. The wave interactions with vortices behind each sphere‘s wake cause fluctuation in the pressure profiles of shock waves. This paper reports an experimental study for the aerodynamic processes involved in the interaction between shock waves and solid spheres. A schlieren photography was applied to visualize the various shock waves passing through solid spheres. Pressure measurements were performed along different downstream positions. The experiments were conducted in both rectangular and circular shock tubes. The data with respect to the effect of the sphere array, size, interval distance, incident Mach number, etc., on the shock wave attenuation were obtained.     

Effects of a Single-pulse Energy Deposition on Steady Shock Wave Reflection

The effects of energy deposition in the free stream on steady regular and Mach shock wave reflections are studied numerically. A short-duration laser pulse is focused upstream of the incident shock waves. It causes formation of the expanding blast wave and the residual hot-spot interacting in a complex way with the steady shock wave reflection. It was found that the laser energy addition in the free stream may force the transition from regular to Mach reflection in the dual solution domain. In contrast to previously reported numerical results, the transition from Mach to regular reflection has not been reproduced in our refined computations since the Mach reflection is restored after the flow perturbation.