An experiment on imploding conical shock waves

Over the past decade and a half there have been a number of numerical studies of the reflection of oblique axisymmetric shock waves from the axis of symmetry. Many of these have shown a complex Mach reflection pattern together with a strong toroidal vortex which significantly distorts the Mach disk. This geometry has never been captured experimentally. This note describes a special rig for the generation of strong imploding conical shocks, and presents photographs of the predicted reflection pattern. Received 4 October 2001 / Accepted 13 November 2001     

Lifting bodies using the flow behind axisymmetric conical shock waves

One of the methods of designing aircraft with supersonic flight speeds involves solving an inverse problem by means of the well-known flow schemes and the substitution of rigid surfaces for the flow surfaces. Lifting bodies using the flows behind axisymmetric shock waves belong to these configurations. All lifting bodies using the flow behind a conical shock wave can be divided into two types [1]. Bodies whose leading edge passes through the apex of the conical shock wave pertain to the first type and those whose leading edge lies below the apex of the conical shock wave, to the second. For small apex angles of the basic cone at hypersonic flow velocities an approximate solution of the variation problem was obtained, which showed that the lift-drag ratio of lifting bodies of the second type is higher than that of the first [2]. The present paper gives a numerical solution of the problem for flow past lifting bodies of the second type using the flow behind axisymmetric conical shock waves with half-angles of the basic cone _S=9.5 and 18° The upper surfaces of the bodies are formed by intersecting planes parallel to the velocity vector of the oncoming flow.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 135–138, March–April, 1986.     

On one-dimensional shock waves

In this paper uniform asymptotic expansions for the solutions of a system of differential equations are obtained in the domain containing a shock wave. It is shown, in particular, that the function θ(t,x)/ε contained in the expansions and describing the behavior of the solution in the neighborhood of the wave front has, generally speaking, a discontinuity of derivatives at the front. The results are applicable to one-dimensional problems in gas dynamics with low viscosity and heat-conductivity.     

Enhancement of shock waves

The flow field developed behind a shock wave propagating inside a constant cross-section conduit is solved numerically for two different cases. First, when the density of the ambient gas into which the shock propagates has a logarithmic change with distance. In the second, and the more practical case, the ambient gas is composed of pairs of air–helium layers having a continually decreasing width. It is shown that in both cases meaningful pressure amplification can be reached behind the transmitted shock wave. It is especially so in the second case. By proper choice of the number of air–helium layers and their width reduction ratio, pressure amplification as high as 7.5 can be obtained.      

Unsteady oblique shock waves

A spatially and temporally local analysis is provided for unsteady, oblique shock waves, in which the flow is assumed to be two-dimensional or axisymmetric. Three unsteady parameters, in a laboratory frame, are viewed as the known independent variables. These are the upstream Mach number, the shock Mach number, and the angle of the shock relative to the instantaneous upstream velocity. Other steady and unsteady parameters, such as the velocity turn angles and downstream Mach numbers, are evaluated in closed form, in terms of these three quantities. Trends are assessed, and a sensitivity analysis is provided. It is suggested that the theory may find application in converting a shock capturing algorithm, at an early time during the computational process, into a shock fitting algorithm. Received 30 April 1999 / Accepted 29 November 1999     

Propagation of weak shock waves

An asymptotic method is developed in order to treat the evolution of weak shock waves. One obtains a geometrical theory according to which weak shock waves propagate along rays and satisfy a transport law.     

Revisiting shock waves in metals

We cast Wallace's theory of thermoplastic flow in conservative form. We point out the difference between our formulation, which accounts for contact with an external energy reservoir, and previous formulations of thermoplastic flow. The theory is exploited to show that the experiments of Johnson and Barker on 6062-T6 Al can be interpreted as a weak shock wave that splits into an infinite sequence of “infinitesimal”, shocks, caused by increasing plasticity, leading to the observed smooth temporal velocity profile (a dispersed wave). We predict that overdriven shock waves in metals will split as well. We also re-examine the need for invoking a heat dissipation mechanism for overdriven shocks. It is briefly pointed out that our approach of casting the theory of thermoplastic flow in divergence form can be generalized easily to account for heat release in energetic solids. Received 25 March 1996 / Accepted 20 August 1996     

Decay of shock waves in a dusty-gas shock tube

A numerical study is performed for the unsteady nonequilibrium flow of a gas-particle mixture in a shock tube, where a semi-empirical formula for a single particle is assumed to calculate the drag and heat transfer rate of the particle cloud. To simulate actual flows of the mixture in which the size of the particles is distributed over a finite range, the motion of the particles is analyzed by dividing them into several groups according to their different diameters. It is shown that the particles of diameter larger than the average value cause a significant delay in the relaxation of the gas-particle flow. Good agreement is obtained between the numerical and the experimental results of the decrease in the shock propagation velocity, except for strong shock waves transmitted into dusty gas with a high loading ratio.     

Distortion of the shock front in a shock tube with a divergent conical transition section

The profile of the leading shock front in a gas has been experimentally investigated in shock tubes of variable cross section. It is shown that the presence of a conical transition section between the high-pressure and low-pressure chambers leads to the retention of inhomogeneities on the surface of the wave front (slopes, twists, and bends) over a length of 20–30 diameters.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 141–147, July–August, 1991.     

Propagation characteristics of shock waves driven by gaseous detonation waves

We experimentally investigated propagation characteristics of the shock wave driven by a gaseous detonation wave emerging from the open end of a cylindrical detonation tube. In the present study, we visualized the shock wave and exhaust flowfields using a shadowgraph optical system and we obtained peak overpressure in the tube axial direction and the continuous shape transformation of shock waves around the tube open end. We also obtained overpressure histories of the shock wave using piezo-pressure transducers within 201 m from the open end of the tube. We normalized and classified these results by four regions using non-dimensional pressure and distance which are independent of variety of mixture and tube diameter. In the vicinity of the open end of the tube, the shock wave is nearly planar and does not significantly attenuate, and the peak overpressure maintains approximately C–J pressure. Subsequently, the shock wave attenuates rapidly, transforming from quasi-spherical to spherical. Farther from the tube open end, the shock wave propagates with approximately sound characteristic so that the peak overpressure decreases proportional to 1/r. Eventually, the shock wave begins to attenuate more rapidly than ideal sound attenuation, which may be due to the viscous effect.     

Multigrid simulations of detached shock waves

The use of multigrid convergence acceleration techniques is investigated for supersonic and hypersonic flows over blunt bodies. In these cases detached shock waves occur that usually complicate convergence for multigrid methods. It is shown that a simple damping of the restricted residual error enables convergence without the application of expensive upwind restriction or prolongation operators. The achieved reduction in CPU time increases with increasing free stream Mach number. A similar technique may be used for reactive flows where the restricted residual error is damped in regions of high chemical activity. A spherical nose projectile moving at Mach 6.46 in a stochiometric hydrogen–air mixture serves as a test case to demonstrate the efficiency of this approach in case of combustion. Copyright© 2004 John Wiley & Sons, Ltd.