Shape of a shock wave front diffracting on a perforated wall

 The shape of a shock wave front diffracting on a perforated wall is determined by comparing numerical data and experimental findings. Experiments were conducted in a 60 mm×150 mm cross sectional area shock tube equipped with a double-exposure holographic interferometer. The numerical simulation was conducted using a TVD upwind finite difference scheme. First, a discharge coefficient for the mass flow through the perforations was determined by comparing the numerical results with those obtained using a simplified quasi-one-dimensional analysis. This value agreed well with the experimentally obtained value. Finally, the shape of a backward inclined incident shock wave over a perforated wall was successfully determined by employing this discharge coefficient and the numerical result. Received: 17 March 1995/Accepted: 13 August 1997     

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     

Unsteady flow with separation behind a shock wave diffracting over curved walls

The unsteady separation of the compressible flow field behind a diffracting shock wave was investigated along convex curved walls, using shock tube experimentation at large length and time scales, complemented by numerical computation. Tests were conducted at incident shock Mach numbers of $M_{\hbox {s}} =$ 1.5 and 1.6 over a 100 mm radius wall over a dimensionless time range up to $\tau \le $ 6.45. The development of the near wall flow at $M_{\hbox {s}} =$ 1.5 has been described in detail and is very similar to that observed for slightly lower $\tau $ ’s at $M_{\hbox {s}} =$ 1.6. Computations were performed at wall radii of 100 and 200 mm and for incident shock Mach numbers from 1.5 up to and including Mach 2.0. Comparing dimensionless times for different size walls shows that for a given value of $\tau $ the flow field is very similar for the various wall radii published to date and tested in this study. Previously published results that were examined alongside the results from this study had typical values of $1.6 < \tau < 3.2$ . At the later times presented here, flow features were observed that previously had only been observed at higher Mach numbers. The larger length scales allowed for a degree of Reynolds number independence in the results published here. The effect of turbulence on the numerical and experimental results could not be adequately examined due to limitations of the flow imaging system used and a number of questions remain unanswered.     

Passage of shock waves through a curved boundary separating two media

We consider the interaction of shock waves with a mildly curved surface of separation, and also the motion of this surface due to the passage of shock waves.Moscow. Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 111–121, January–February, 1972.     

Unsteady interaction of shock wave diffracting around a circular cylinder in air

The reflection and diffraction of a planar shock wave around a circular cylinder are a typical problem of the complex nonlinear shock wave phenomena in literature. It has long been studied experimentally, analytically as well as numerically. Takayama in 1987 obtained clear experimental pictures of isopycnics in shock tube under the condition that the impinging shock wave propagates as far as 3 diameters away from the cylinder. To know more completely the whole unsteady process, it is desirable to get experimental results in a region which is more than 10 diameters away from the cylinder. This is what has been done in this paper by using the pulsed laser holographic interferometry for several shock Mach numbers of the impinging shock. Results for several moments are shown, giving more knowledge about the whole unsteady flow field. This is useful for a reliable and complete understanding of the changing force acting on the cylinder, and provides interesting data to check the performance of many recently developed high resolution numerical methods for unsteady shock wave calculation. The project suported partially by National Natural Science Foundation of China     

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.      

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.     

Imploding conical shock waves

The behaviour of conical shock waves imploding axisymmetrically was first studied numerically by Hornung (J Fluid Mech 409:1–12, 2000) and this prompted a limited experimental investigation into these complex flow patterns by Skews et al. (Shock Waves 11:323–326, 2002). Modification of the simulation boundary conditions, resulting in the loss of self-similarity, was necessary to image the flow experimentally. The current tests examine the temporal evolution of these flows utilising a converging conical gap of fixed width fed by a shock wave impinging at its entrance, supported by CFD simulations. The effects of gap thickness, angle and incident shock strength were investigated. The wave initially diffracts around the outer lip of the gap shedding a vortex which, for strong incident shock cases, can contain embedded shocks. The converging shock at exit reflects on the axis of symmetry with the reflected wave propagating outwards resulting in a triple point developing on the incident wave together with the associated shear layer. This axisymmetric shear layer rolls up into a mushroom-shaped toroidal vortex ring and forward-facing jet. For strong shocks, this deforms the Mach disk to the extent of forming a second triple point with the primary shock exhibiting a double bulge. Separate features resembling the Richtmeyer–Meshkov and Kelvin–Helmholtz instabilities were noted in some tests. Aside from the incident wave curvature, the reflection patterns demonstrated correspond well with the V- and DV-types identified by Hornung although type S was not clearly seen, possibly due to the occlusion of the reflection region by the outer diffraction vortex at these early times. Some additional computational work explicitly exploring the limits of the parameter space for such systems has demonstrated the existence of a possible further reflection type, called vN-type, which is similar to the von Neumann reflection for plane waves. It is recommended that the parameter space be more thoroughly explored experimentally.     

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.     

Submerged circular cylindrical shell subjected to two consecutive shock waves: Resonance-like phenomena

A submerged evacuated circular cylindrical shell subjected to a sequence of two external shock waves generated at the same source is considered. A semi-analytical model combining the classical methods of mathematical physics with the finite-difference methodology is developed and employed to simulate the interaction. Both the hydrodynamic and structural aspects of the problem are considered, and it is demonstrated that varying the delay between the first and second wavefronts has a very significant effect on the stress–strain state of the structure. In particular, it is shown that for certain values of the delay, the constructive superposition of the elastic waves travelling around the shell results in a ‘resonance-like’ increase of the structural stress in certain regions. The respective stress can be so high that it sometimes exceeds the overall maximum stress observed in the same structure but subjected to a single-front shock wave with the same parameters, in some cases by as much as 50%. A detailed parametric analysis of the observed phenomenon is carried out, and an easy-to-use diagram summarizing the finding is proposed to aim the pre-design analysis of engineering structures.     

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.     

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